Modern Koopman Theory for Dynamical Systems

Abstract

The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated in terms of measurements, making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.

Full text

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS∗
STEVEN L. BRUNTON†, MARKO BUDIˇ
SI´
C‡, EURIKA KAISER§,AND J. NATHAN KUTZ¶
Abstract. The field of dynamical systems is being transformed by the mathematical tools
and algorithms emerging from modern computing and data science. First-principles derivations
and asymptotic reductions are giving way to data-driven approaches that formulate models in op-
erator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant
perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-
dimensional linear operator acting on the space of all possible measurement functions of the system.
This linear representation of nonlinear dynamics has tremendous potential to enable the prediction,
estimation, and control of nonlinear systems with standard textbook methods developed for linear
systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the
dynamics appear approximately linear remains a central open challenge. The success of Koopman
analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical
geometric approaches for dynamical systems, 2) the approach is formulated in terms of measurements,
making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful
numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and
extended to reduce Koopman theory to practice in real-world applications. In this review, we pro-
vide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic
developments and highlighting these methods with a diverse range of applications. We also discuss
key advances and challenges in the rapidly growing field of machine learning that are likely to drive
future developments and significantly transform the theoretical landscape of dynamical systems.
Key words. Dynamical systems, Koopman operator, Data-driven discovery, Control theory,
Spectral theory, Operator theory, Dynamic mode decomposition, Embeddings.
AMS subject classifications. 34A34, 37A30, 37C10, 37M10, 37M99, 37N35, 47A35, 47B33
1. Introduction. Nonlinearity is a central challenge in dynamical systems, re-
sulting in diverse phenomena, from bifurcations to chaos, that manifest across a range
of disciplines. However, there is currently no overarching mathematical framework
for the explicit and general characterization of nonlinear systems. In contrast, lin-
ear systems are completely characterized by their spectral decomposition (i.e., ei-
genvalues and eigenvectors), leading to generic and computationally efficient algo-
rithms for prediction, estimation, and control. Importantly, linear superposition
fails for nonlinear dynamical systems, leading to a variety of interesting phenom-
enon including frequency shifts and the generation of harmonics. The Koopman
operator theory of dynamical systems provides a promising alternative perspective,
where it may be possible to enable linear superposition even for strongly nonlinear
dynamics via the infinite-dimensional, but linear, Koopman operator. The Koop-
man operator is linear, advancing measurement functions of the system, and its
spectral decomposition completely characterizes the behavior of the nonlinear sys-
tem. Finding tractable finite-dimensional representations of the Koopman operator
is closely related to finding effective coordinate transformations in which the nonlin-
ear dynamics appear linear. Koopman analysis has recently gained renewed inter-
∗Uploaded to arXiv February 23, 2021.
Funding: SLB acknowledges support from the Army Research Office (W911NF-17-1-0306
and W911NF-19-1-0045). SLB and JNK acknowledge support from the Defense Advanced Research
Projects Agency (DARPA contract HR011-16-C-0016) and the UW Engineering Data Science Insti-
tute, NSF HDR award #1934292.
†Dept. of Mechanical Engineering, University of Washington, Seattle, WA (sbrunton@uw.edu).
‡Dept. of Mathematics, Clarkson University, Potsdam, NY (mbudisic@clarkson.edu).
§Dept. of Mechanical Engineering, University of Washington, Seattle, WA (eurika@uw.edu).
¶Dept. of Applied Mathematics, University of Washington, Seattle, WA (kutz.edu).
1
arXiv:2102.12086v1 [math.DS] 24 Feb 2021

2S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
est [234,231,284,57,55,232,180], in large part because of its strong connections to
data-driven modeling. In this review, we provide an overview of modern Koopman
theory for dynamical systems, including an in-depth analysis of leading computational
algorithms, such as the dynamic mode decomposition (DMD).
Koopman introduced his operator theoretic perspective of dynamical systems
in 1931 to describe the evolution of measurements of Hamiltonian systems [170],
and this theory was generalized by Koopman and von Neumann to systems with a
continuous eigenvalue spectrum in 1932 [171]. Koopman’s 1931 paper was central to
the celebrated proofs of the ergodic theorem by von Neumann [248] and Birkhoff [28,
29]. The history of these developments is fraught with intrigue, as discussed by
Moore [239]. In his original paper [170], Koopman drew connections between the
Koopman eigenvalue spectrum and conserved quantities, integrability, and ergodicity.
For Hamiltonian flows, the Koopman operator is unitary. It is no surprise then that
DMD, the leading numerical algorithm for approximating the Koopman operator, is
built on the discrete Fourier transform (DFT) and the singular value decomposition
(SVD), which both provide unitary coordinate transformations [49].
The operator theoretic framework discussed here complements the traditional
geometric and probabilistic perspectives on dynamical systems. For example, level
sets of Koopman eigenfunctions form invariant partitions of the state-space of a dy-
namical system [55]; in particular, eigenfunctions of the Koopman operator may be
used to analyze the ergodic partition [235,54]. Koopman analysis has also been
shown to generalize the Hartman–Grobman theorem to the entire basin of attrac-
tion of a stable or unstable equilibrium point or periodic orbit [185]. The Koopman
operator is also known as the composition operator, which is formally the pull-back
operator on the space of scalar observable functions [5], and it is the dual, or left-
adjoint, of the Perron–Frobenius (PF) operator, or transfer operator, which is the
push-forward operator on the space of probability density functions. When a poly-
nomial basis is chosen to represent the Koopman operator, then it is closely related
to Carleman linearization [63,64,65], which has been used extensively in nonlinear
control [311,177,22,328].
1.1. An overview of Koopman theory. In this review, we will consider dy-
namical systems of the form
d
dtx(t) = f(x(t)),(1.1)
where x∈ X ⊆ Rnis the state of the system and fis a vector field describing the
dynamics. In general, the dynamics may also depend on time t, parameters β, and
external actuation or control u(t). Although we omit these here for simplicity, they
will be considered in later sections.
A major goal of modern dynamical systems is to find a new vector of coordinates
zsuch that either
(1.2) x=ϕ(z) or z=ϕ(x)
where the dynamics are simplified, or ideally, linearized:
d
dtz=Lz.(1.3)
While in geometric dynamics, one asks for homeomorphic (continuously invertible)
or even diffeomorphic coordinate maps, which trivialize the choice between the two

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 3
options in (1.2), there is little hope for global coordinate maps of this sort. Rather,
we contend with embeddings ϕthat lift the dynamics into a higher-dimensional space
of zvariables, allowing for “unfolding” of nonlinearities.
In practice, we typically have access to measurement data of our system, discretely
sampled in time. This data is governed by the discrete-time dynamical system
xk+1 =F(xk),(1.4)
where xk=x(tk) = x(k∆t). Also known as a flow map, the discrete-time dynamics
are more general than the continuous-time formulation in (1.1), encompassing discon-
tinuous and hybrid systems as well. In this case, the goal is still to find a linearizing
coordinate transform so that yk+1 =Kyk. These coordinates are given by eigen-
functions of the discrete-time Koopman operator, K, which advances a measurement
function g(x) of the state forward in time through the dynamics:
Kg(xk):=g(F(xk)) = g(xk+1).(1.5)
For an eigenfunction ϕof K, corresponding to an eigenvalue λ, this becomes
Kϕ(xk) = λϕ(xk) = ϕ(xk+1).(1.6)
Thus, a tremendous amount of effort has gone into characterizing the Koopman op-
erator and approximating its spectral decomposition from measurement data.
The coordinates ϕand the linear operator Lare closely related to the continuous-
time generator Lof the Koopman operator K. In particular, eigenfunctions ϕjof
Lprovide such a linearizing coordinate system, and the operator Lis obtained by
restricting the operator Lto the span of these functions. Spectral theory provides
a complete description of the dynamics in terms of the eigenstructure of the linear
operator L. Thus, transforming the system into coordinates where the dynamics are
linear dramatically simplifies all downstream analysis and control efforts.
1.2. An illustrative example: The Duffing oscillator. Although the objec-
tive of Koopman theory is easily expressed mathematically, it is helpful to explore
its application on a simple dynamical system. Consider the nonlinear Duffing system
¨x=x−x3with state space representation
˙x1=x2
(1.7a)
˙x2=x1−x3
1,(1.7b)
and the corresponding phase portrait in Figure 1.1. This example has three fixed
points, a saddle at the origin with Jacobian eigenvalues λ±=±1 and two centers
at (x1, x2) = (±1,0) with eigenvalues ±√2i. These results, shown in Figure 1.1(a),
can be obtained by a local phase-plane analysis [36], and these local linearizations are
valid in a small neighborhood of each fixed point, illustrated by the shaded regions.
There is no homeomorphic coordinate transformation that captures the global
dynamics of this system with a linear operator, since any such linear operator Lhas
either one fixed point at the origin, or a subspace of infinitely many fixed points [48],
but never three isolated fixed points. Instead, the Koopman operator can provide a
system of coordinate transformations that extend the local neighborhoods where a
linear model is valid to the full basin around them [185], as shown in Figure 1.1(b).
This interpretation may lead to seemingly unintuitive results, as even the most
obvious observables, such as g(x) = x, may require different expansions in different

4S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
(c) Global Koopman eigenfunction (energy)
H(x,·
x= 0)
H
(a) Linearization domains (b) Koopman linearization domains
(d) Coordinate transformation to linearize
φ1
φ2
x
·
x
x
·
x
x
·
x
φ1
φ2
t
𝓓1
𝓓2
𝓓3
𝓓1
𝓓2
𝓓3
Fig. 1.1: Different Koopman perspectives for the Duffing oscillator, ¨x=x−x3, the
equation for a particle in a double potential well. (a) Traditional linearization near
the fixed points gives small regions where the system is approximately linear. (b)
Koopman theory may extend the Hartman–Grobman theorem to enlarge the domain
of linearity until the next fixed point [185]. (c) There are also global Koopman eigen-
functions, like the Hamiltonian energy, although these lose information about which
basin the solution is in. (d) Yet a third perspective seeks a coordinate transformation
to rescale space and time until dynamics live on a hypertoroid.
areas of the state space, as recently explored by Page and Kerswell [259]. Indeed,
systems that possess multiple simple invariant solutions cannot be represented by a
Koopman expansion that is globally uniformly convergent, implying that any choice
of a representation of the Koopman operator Kin a system of coordinates may not
hold everywhere, for any given observable of interest.
Even if there are no globally convergent linear Koopman representations, the
Koopman eigenfunctions maybe be globally well defined, and even regular in the entire
phase space. For example, the Hamiltonian energy function H=x2
2−x2
1/2 + x4
1/4 is
a Koopman eigenfunction for this conservative system, with eigenvalue λ= 0, as
shown in Figure 1.1(c). In a sense, this global eigenfunction exploits symmetry in
the dynamics to represent a global dynamic quantity, valid in all regions of phase
space. This example establishes a connection between the Koopman operator and
Noether’s theorem [251], as a symmetry in the dynamics gives rise to a new Koopman

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 5
eigenfunction with eigenvalue λ= 0. In addition, the constant function ϕ≡1 is a
trivial eigenfunction corresponding to λ= 0 for every dynamical system.
A final interpretation of Koopman theory is shown in Figure 1.1(d), related to
Figure 1.1(b). In this case, we see that the Koopman operator establishes a change
of coordinates, in space and time, in which the original nonlinear trajectories become
linear. When obtaining these approximate coordinate systems from data, only ap-
proximately periodic dynamics will persist for long times. Rescaling space and time
to make dynamics approximately periodic is the approach taken in [202,186].
1.3. Dynamics in the big data era. The recent interest in Koopman op-
erator theory is inherently linked to a growing wealth of data. Measurement tech-
nologies across every discipline of the engineering, biological, and physical sciences
have revolutionized our ability to interrogate and extract quantities of interest from
complex systems in real time with rich multi-modal and multi-fidelity time-series
data. From broadband sensors that measure at exceptionally high sampling rates to
high-resolution imaging, the front lines of modern science are being transformed by
unprecedented quality and quantities of data. These advances are based on three
foundational technologies: (i) improved sensors capable of measurement with higher
quality and quantity, (ii) improved hardware for high-performance computing, data
storage, and transfer, and (iii) improved algorithms for processing the data. Taken to-
gether, these advances form the underpinnings of the big data era and are driving the
fourth paradigm of scientific discovery [135], whereby the data itself drives discovery.
Data science is now a targeted growth area across almost all academic disciplines.
However, it has been almost six decades since John Tukey, co-developer of the fast
Fourier transform with James Cooley, first advocated for data science as its own
discipline [342,89]. Not surprisingly, the 1960s also coincided with pioneering devel-
opments of Gene Golub and co-workers on numerical algorithms for computing the
singular value decomposition of a matrix [312], enabling one of the earliest data sci-
ence exploration tools: principal component analysis (PCA). Thus the mathematical
foundations for the big data era have been long in development. Indeed, its matu-
rity is reflected in the emergence of the two cultures [37] of statistical learning and
machine learning. In the former, the primary focus is on the development of inter-
pretable models of data, while in the latter, accuracy is of paramount importance.
Although accuracy and interpretability are not mutually exclusive, often the refine-
ment of one comes at the expense of the other. For example, modern machine learning
and artificial intelligence algorithms are revolutionizing computer vision and speech
processing through deep neural network (DNN) architectures. DNNs have produced
performance metrics in these fields far beyond any previous algorithms. Although
individual components of DNNs may be interpretable, the integration across many
layers with nonlinear activation functions typically render them opaque and uninter-
pretable. In contrast, sparsity promoting algorithms, such as the LASSO [336], are
examples of statistical learning where interpretable variable selection is achieved. As
will be highlighted throughout this review, Koopman theory is amenable to many of
the diverse algorithms developed in both statistical learning and machine learning.
Dynamical systems theory has a long history of leveraging data for improving
modeling insights, promoting parsimonious and interpretable models, and generating
forecasting capabilities. In the 1960s, Kalman introduced a rigorous mathematical
architecture [159,160] whereby data and models could be combined through data
assimilation techniques [102,188], which is especially useful for forecasting and con-
trol. Thus the integration of streaming data and dynamical models has a nearly seven

6S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
decade history. In the modern era, it is increasingly common to build the dynamical
models from the data directly. This is especially important in complex systems where
first principles models are not available, or where it is not even known what the cor-
rect state-space variable should be. Biological systems, such as those that arise in
neuronal recordings in the brain, are well suited for such data-driven model discovery
techniques, as whole-brain imaging provides insight into how the microscale dynamics
of individual neurons produces large scale patterns of spatio-temporal brain activity.
Such systems are ideally suited for leveraging the modern data-driven modeling tools
of machine learning to produce dynamical models characterizing the observed data.
1.4. Koopman objectives and applications. The ultimate promise of Koop-
man spectral theory is the ability analyze, predict, and control nonlinear systems with
the wealth of powerful techniques from linear systems theory. This overarching goal
may be broken into several specific goals:
Diagnostics. Spectral properties of the Koopman operator may be used effectively
to characterize complex, high-dimensional dynamical systems. For example, in fluid
mechanics, DMD is used to approximate the Koopman mode decomposition, resulting
in an expansion of the flow as a linear combination of dominant coherent structures.
Thus, Koopman mode decomposition can be used for dimensionality reduction and
model reduction, generalizing the space-time separation of variables that is classically
obtained via either Fourier transform or singular value decomposition [49].
Prediction. One of the major benefits of dynamical systems over other statistical
models is that they provide a mechanistic understanding of how the future evolves,
based on the current state of a system. Prediction is thus one of the central goals
of any dynamical system framework. The prediction of a nonlinear system presents
numerous challenges, such as sensitive dependence on initial conditions, parameter
uncertainty, and bifurcations. Koopman operator theory provides insights into which
observables are easier or harder to predict, and suggests robust numerical algorithms
for such time-series prediction.
Estimation and control. In many systems, we not only seek to understand the
dynamics, but also to modify, or control, their behavior for some engineering goal. In
modern applications, it is rare to have complete measurements of a high-dimensional
system, necessitating the estimation of these quantities from limited sensor measure-
ments. By viewing the system through the linearizing lens of Koopman eigenfunctions,
it is possible to leverage decades of results from linear estimation and control theory
for the analysis of nonlinear systems. Indeed, linear systems are by far the most well
studied and completely characterized class of dynamical systems for control.
Uncertainty quantification and management. Uncertainty is a hallmark feature
of the real world, and dynamical systems provide us with a framework to forecast,
quantify, and manage uncertainty. However, modeling uncertainty is challenging for
strongly nonlinear and chaotic systems. Integrating uncertainty quantification into
the Koopman operator framework is an important avenue of ongoing research.
Understanding. Beyond the practical goals value of prediction and control, dy-
namical systems provide a compelling framework with which to understand and model
complex systems. Normal forms, for example, distill essential structural information
about the dynamics, while remaining as simple as possible. For Koopman theory to
be embraced, it will need to facilitate similar intuition and understanding.
Across a wide range of applications, researchers are developing and advancing
Koopman operator theory to address these challenges. Some of these applications
include fluid dynamics [294,284,232], epidemiology [277], neuroscience [45,7], plasma

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 7
(a) Fluids (b) Epidemiology (c) Plasmas (d) Video
(e) Neuroscience (f) Chemistry (g) Power Grid (h) Robotics
Fig. 1.2: Overview of applications of data-driven Koopman analysis via DMD.
Figures reproduced with permission from: (a) top [294], bottom [284]; (b) [277]; (c)
[162]; (d) [98]; (e) [45]; (f) from Emw https://commons.wikimedia.org/wiki/File:
Protein PCMT1 PDB 1i1n.png; (g) from Henk Monster https://commons.wikimedia.
org/wiki/File:Power grid masts besides the new Waalbridge Nijmegen - panoramio.
jpg; (h) from Daderot https://commons.wikimedia.org/wiki/File:Minsky’
s robot arm, late 1960s, view 2 - MIT Museum - DSC03759.JPG.
physics [335,162], finance [213], robotics [27,4,44,43], and the power grid [325,320];
a number of these are shown in Figure 1.2 and will be explored more in section 3.
1.5. Organization and goals of review. A primary goal of this review is
to make modern Koopman operator theory accessible to researchers in many diverse
fields. We seek to provide the reader with a big-picture overview of the state of the art,
expediting the process of getting up to speed in this rapidly developing field. Further,
we explore the dual theoretical and applied perspectives for understanding Koopman
operator theory. To this end, each section will be approached from the perspective
of providing a unified overview of major theoretical, methodological, numerical, and
applied developments. Furthermore, we have compiled a number of striking success
stories along with several outstanding challenges to help guide readers wishing to enter
this field. Finally, it is our goal to highlight connections with existing and emerging
methods from other fields, including geometric and probabilistic dynamical systems,
computational science and engineering, and machine learning.
This review begins with relatively standard material and builds up to advanced
concepts. Section 2 provides a practical introduction Koopman operator theory, in-
cluding simple examples, the Koopman mode decomposition, and spectral theory.
Similarly, section 3 provides an overview of dynamic mode decomposition, which is
the most simple and widely used numerical algorithm to approximate the Koopman
operator. Section 4 then introduces many of the most important concepts in modern
Koopman theory. Advanced numerical and data-driven algorithms for representing
the Koopman operator are presented in section 5. Thus, section 4 and section 5
provide advanced material extending section 2 and section 3, respectively. Section 6
investigates how Koopman theory is currently extended for advanced estimation and
control. Section 7 provides a discussion and concluding remarks. This section also
describes several ongoing challenges in the community, along with recent extensions
and motivating applications. The goal here is to provide researchers with a quick
sketch of the state of the art, so that they may quickly get to the frontier of research.

8S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
2. A practical introduction to the Koopman operator framework.
2.1. Definitions and vocabulary. The Koopman operator advances measure-
ment functions of the state of a dynamical system with the flow of the dynamics.
To explain the basic properties of the Koopman operator, we begin with an autono-
mous ordinary differential equation (1.1) on a finite-dimensional state space X ⊆ Rn.
The flow map operator, or time-tmap, Ft:X → X advances initial conditions x(0)
forward along the trajectory by a time t, so that trajectories evolve according to
(2.1) x(t) = Ft(x(0)).
The family of Koopman operators Kt:G(X)→ G(X), parameterized by t, are given
by
(2.2) Ktg(x) = g(Ft(x)),
where G(X) is a set of measurement functions g:X → C. Another name for g,
derived from this framework’s origin in quantum mechanics, is an observable function,
although this should not be confused with the unrelated observability from control
theory. We can interpret (2.2) as defining a family of functions
(2.3) gt:=Ktg, g0:=g
that corresponds to the trajectory t7→ gtin the set G(X) of measurement functions.
In most applications, the set of functions G(X) is not defined a priori, but is
loosely specified by a set of properties it should satisfy, e.g., that it is a vector space,
that it possesses an inner product, that it is complete, or that it contains certain
functions of interest, such as continuous functions on X. Hilbert spaces, such as
L2(X, dµ) or reproducing kernel Hilbert spaces (RKHS), are a common choice in
modern applied mathematics, although historically other Banach spaces, such as in-
tegrable functions L1or continuous functions C(X) have also been used. The choice
of the space, whether explicit or implicit, can have consequences on the properties of
the operator and its approximations. In all cases, however, G(X) is of significantly
higher dimension than X, i.e., countably or uncountably infinite.
The most significant property of the Koopman operator is that it is linear when
G(X) is a linear (vector) space of functions:
(2.4) Kt(α1g1(x) + α2g2(x)) = α1g1Ft(x)+α2g2Ft(x)
=α1Ktg1(x) + α2Ktg2(x).
This property holds regardless of whether Ft:X → X is linear itself, as it is simply a
consequence of definition (2.2), since the argument function gis on the “outside” of the
composition, allowing linearity to carry over from the vector space of observables. In
this sense, the Koopman framework obtains linearity of Ktdespite the nonlinearity of
Ftby trading the finite-dimensional state space Xfor an infinite-dimensional function
space G(X).
When time tis discrete, t∈N, and the dynamics are autonomous, then Ftis a
repeated t-fold composition of F≡F1given by Ft(x) = F(F(·· ·(F(x)))), so that
Ktgis likewise generated by repeated application of K ≡ K1. The generator Kof
the composition semigroup is then called the Koopman operator, which results in a
dynamical system
(2.5) gk+1 =Kgk,

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 9
analogous to xk+1 =F(xk), except that (2.5) is linear and infinite-dimensional.
When time tis continuous, the flow map family satisfies the semigroup property
(2.6) Ft+s(x) = Ft(Fs(x)),∀x, t, s ≥0,
which can be strengthened to a group property t, s ∈Rif the flow map is invertible.
The Koopman operator family Ktinherits these properties as well. Given a continuous
and sufficiently smooth dynamics, it is also possible to define the continuous-time
infinitesimal generator of the Koopman operator family as
(2.7) Lg:= lim
t→0Ktg−g
t= lim
t→0
g◦Ft−g
t.
As our goal here is to provide an introduction, we omit the mathematical scaffolding
that accompanies a careful definition of an operator derivative; all details in the
context of Koopman operator theory are available in [187,§7.5].
The generator Lhas been called the Lie operator [170], as it is the Lie derivative
of galong the vector field f(x) when the dynamics is given by (1.1) [5,70]. This
follows from applying the chain rule to the time derivative of g(x):
d
dtg(x(t)) = ∇g·˙
x(t) = ∇g·f(x(t))(2.8)
and equating with
d
dtg(x(t)) = lim
τ→0
g(x(t+τ)) −g(x(t))
τ=L(g(x(t))),(2.9)
resulting in
Lg=∇g·f.(2.10)
The adjoint of the Lie operator is called the Liouville operator, especially in
Hamiltonian dynamics [115,116], while the adjoint of the Koopman operator is the
Perron–Frobenius operator [87,84,106,107]. In many ways, the operator-theoretic
framework for applied dynamical systems has two dual perspectives, corresponding
either to the Koopman operator or the Perron–Frobenius operator. In subsection 4.3
we discuss these parallels in more detail.
Similar to (2.5), Linduces a linear dynamical system in continuous-time:
d
dtg=Lg.(2.11)
The linear dynamical systems in (2.5) and (2.11) are analogous to the dynamical
systems in (1.1) and (1.4), respectively.
An important special case of an observable function is the projection onto a
component of the state g(x) = xior, with a slight abuse of notation, g(x) = x. In
this case, the left-hand side of (2.11) is plainly ˙
x, but the right hand side Lxmay
not be simple to represent in a chosen basis for the space G(X). It is typical in real
problems that representing Lxwill involve an infinite number of terms. For certain
special structures, this may not be the case, as will be demonstrated in subsection 2.4.
In summary, the Koopman operator is linear, which is appealing, but is infinite
dimensional, posing issues for representation and computation. Instead of capturing
the evolution of all measurement functions in a function space, applied Koopman

10 S. L. BRUNTON, M. BUDIˇ
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analysis attempts to identify key measurement functions that evolve linearly with the
flow of the dynamics. Eigenfunctions of the Koopman operator provide just such a set
of special measurements that behave linearly in time. In fact, a primary motivation
to adopt the Koopman framework is the ability to simplify the dynamics through the
eigendecomposition of the operator.
2.2. Eigenfunctions and the spectrum of eigenvalues. A Koopman eigen-
function ϕ(x) corresponding to an eigenvalue λsatisfies
(2.12) ϕ(xk+1) = Kϕ(xk) = λϕ(xk).
In continuous-time, a Lie operator eigenfunction ϕ(x) satisfies
(2.13) d
dtϕ(x) = Lϕ(x) = µϕ(x),
where µis a continuous-time eigenvalue. In general, eigenvalues and eigenvectors are
complex-valued scalars and functions, respectively, even when the state space Xand
dynamics F(x) are real-valued.
It is simple to show that Koopman eigenfunctions ϕ(x) that satisfy (2.12) for
λ6= 0 are also eigenfunctions of the Lie operator, although with a different eigenvalue.
Applying the Lie operator (2.7) to such a ϕleads to
(2.14) Ktϕ=λtϕ=⇒ Lϕ= lim
t→0Ktϕ−ϕ
t= lim
t→0
λt−1
tϕ= log(λ)ϕ.
Conversely, the induced dynamics (2.11) applied to an eigenfunction of Lleads to
(2.15) Lϕ=µϕ =⇒d
dtϕ=Lϕ=µϕ.
An eigenfunction ϕof Lwith eigenvalue µis then an eigenfunction of Ktwith eigen-
value λt= exp(µt). Thus, we will typically not make a distinction between Lie and
Koopman eigenfunctions in the context of autonomous dynamical systems.
Eigenfunctions of K,Lthat are induced by already-linear dynamics further illus-
trate the connection between linear discrete dynamics xn+1 =Axnwith analogous
concepts for gn+1 =Kgn. Given a left-eigenvector ξ>A=λξ>, we form a corre-
sponding Koopman eigenfunction as
ϕ(x):=ξ>x(2.16)
since
Kϕ(x) = ϕ(Ax) = ξ>Ax =λξ>x=λϕ(x).(2.17)
In other words, while right-eigenvectors of Agive rise to time-invariant directions in
the state space X, the left-eigenvectors giver rise to Koopman eigenfunctions, which
are similarly time-invariant directions in the space of observables G(X).
In general systems, a set of Koopman eigenfunctions may be used to generate
more eigenfunctions. In discrete time, we find that the product of two eigenfunctions
ϕ1(x) and ϕ2(x) is also an eigenfunction:
(2.18) K(ϕ1(x)ϕ2(x)) = ϕ1(F(x))ϕ2(F(x))
=λ1λ2ϕ1(x)ϕ2(x)

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 11
with a new eigenvalue λ1λ2given by the product of the two eigenvalues of ϕ1(x) and
ϕ2(x). The corresponding relationship for Lcan be found by applying (2.14),
(2.19) λ1λ2=eµ1eµ2=eµ1+µ2,
resulting in
(2.20) L(ϕ1(x)ϕ2(x)) = (µ1+µ2)ϕ1(x)ϕ2(x).
A simple consequence is that a complex conjugate pair of eigenfunctions of L, (µ, ϕ),
(¯µand ¯ϕ), additionally implies the existence of a real-valued eigenfunction |ϕ|=√ϕ¯ϕ
with an associated eigenvalue (µ+ ¯µ)/2 = Re µ, thus leading to a non-oscillatory
growth/decay of the eigenvalue.
Algebraically, the set of Koopman eigenfunctions establishes a commutative mon-
oid1under point-wise multiplication. Thus, depending on the dynamical system, there
may be a finite set of generator eigenfunction elements that may be used to construct
all other eigenfunctions. The corresponding Koopman eigenvalues form a multiplica-
tive lattice, or an additive lattice for Lie eigenvalues due to (2.14).
Observables that can be formed as linear combinations of eigenfunctions, i.e.,
g∈span{ϕk}K
k=1 have a particularly simple evolution under the Koopman operator
(2.21) g(x) = X
k
vkϕk=⇒ Ktg(x) = X
k
vkλt
kϕk.
This implies that the subspace span{ϕk}K
k=1 is invariant under the action of K.
These simple relationships enable the analysis of the phase portrait of a dynamical
system in terms of level sets, and they also enable the analysis of the evolution of
general observables through the lens of their decomposition into eigenfunctions.
2.2.1. Level sets of eigenfunctions. Level sets of eigenfunctions can clarify
the relationships between sets in the domain X. In particular, (sub-)level sets asso-
ciated with eigenfunctions map into each other. To see this, we write the eigenvalue
and the eigenfunction in polar form
ϕ(x) = R(x)eiΘ(x)
(2.22)
λ=reiθ
(2.23)
and define the associated sublevel sets to be
Mϕ(C):={x∈ X :R(x)≤C},(2.24)
Aϕ(α):={x∈ X : Θ(x)≤α}.(2.25)
Given x∈ X, we can evaluate the eigenfunction on its successor x+=F(x), to find
ϕ(x+) = Kϕ(x) = λϕ(x) = (rR(x))ei(θ+Θ(x)) ,(2.26)
so that if x∈Mϕ(C) then x+∈Mϕ(rC) and if x∈Aϕ(α) then x+∈Aϕ(α+θ), or
more succinctly
(2.27) F(Mϕ(C)) = Mϕ(rC),F(Aϕ(α)) = Aϕ(α+θ).
1Monoids are groups without guaranteed inverses, or semigroups with an identity element.

12 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Fig. 2.1: Koopman eigenfunctions constructed as (2.16) for a linear matrix ODE
˙
x=Ax with a saddle-type fixed point at the origin. Velocity field and sample orbits
are overlaid in white.
For the particular case of λ= 1 these relationships simplify to
(2.28) F(Mϕ(C)) = Mϕ(C),F(Aϕ(α)) = Aϕ(α),
implying that level sets of invariant eigenfunctions are invariant sets.
When the eigenvalue is on the unit circle, with a phase θ/2π=p/q ∈Q, then the
level sets of phase satisfy
(2.29) F(Aϕ(α)) = Aϕα+2πp
qFq(Aϕ(α)) = Aϕ(α+ 2πp) = Aϕ(α),
and, therefore, identify chains of q-periodic sets where the value of the phase indicates
the order in which they are visited by a trajectory xk.
To illustrate these concepts, Figure 2.1 shows two eigenfunctions constructed from
two left eigenvectors of a planar linear system ˙
x=Ax with saddle-type dynamics,
as in (2.16). In contrast to right eigenvectors of A, which act as invariant manifolds
attracting/repelling the trajectories, Koopman eigenfunctions foliate the space into
leaves (level sets), providing a time-ordering on the domain associated with each
eigenvalue.
When system matrix Ahas complex conjugate eigenvalues λ, ¯
λ, the associated
eigenfunctions ξ>xand ¯
ξ>xare themselves complex conjugates. Visualizing the as-
sociated modulus R(x) and phase Θ(x) (2.22) for planar dynamics with a focus equi-
librium, as in Figure 2.2, demonstrates that the modulus acts as a Lyapunov function
for the stable focus, as sub-level sets provide a time-ordering on the plane implying
that trajectories converge to the origin. Indeed, |ξ>x|is itself an eigenfunction of
the Koopman operator Ktassociated with eigenvalues et<λ, therefore capturing the
exponential envelope of oscillating trajectories. Level sets of |ξ>x|are called isosta-
bles [225]. An analogous definition of an isostable for real-valued eigenvalues gives
points that converge to the same trajectory in the stable/unstable manifold.
Level sets of the argument (angle) provide a cyclic foliation of the domain, acting
as isochrons [225], i.e., collections of initial conditions that converge to the origin
with a common phase. We will revisit the concepts of isochrons and isostables as
tools for nonlinear reduction of order, in particular as a foundation for understanding
the synchronization and control of oscillators, in subsection 4.1.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 13
Fig. 2.2: Modulus R(x) and argument Θ(x) of the Koopman eigenfunction construc-
ted from a complex eigenvector of a linear matrix ODE ˙
x=Ax with a focus-type
fixed point at the origin.
2.2.2. Computing eigenfunctions. There are several approaches to approxi-
mate Koopman eigenfunctions computationally. Given the evolution of an observable
g(x(t)) = g(Ft(x)) it is possible to compute an eigenfunction associated with an
eigenvalue eiω by forming the following long-term harmonic or Fourier average
(2.30) ˜gω(x):= lim
T→∞
1
TZT
0
gFt(x)e−iωtdt.
If ω= 0, harmonic averages reduce to the trajectory (ergodic) average. When the
space of observables G(X) is taken as the L1space of integrable functions with re-
spect to a dynamically-preserved measure µ, then pointwise convergence µ-almost
everywhere is guaranteed by the Birkhoff ergodic theorem [28].
More generally, Yosida’s theorem [57] implies that in Banach spaces the limit
not only exists, but that g7→ ˜gωis a projection onto the eigenspace associated with
the eigenvalue eiω . Said another way, eigenfunctions associated with eigenvalues on
the unit circle |λ|= 1 can be computationally approximated through (2.30), as em-
ployed in [234,231,55,193]. Since such eigenvalues correspond to neutrally-stable
steady-state behavior, they are of practical importance to analyze dynamics. This
approach can, in principle, be extended to compute eigenfunctions associated with
other eigenvalues as well, with conditions and restrictions given in [238]. Recently,
[82] also provided an extension and reframing of such results to RKHS, including both
theoretical and computational results.
Instead of the iteration (2.30) that requires simulating the evolution gt, in certain
cases it is possible to solve for eigenfunctions directly. The eigenfunction relation
Kϕ(x) = ϕ(F(x)) = λϕ(x) is a compositional algebraic equation that is challeng-
ing to solve directly. The relation (2.10) implies that the analoguous relation for
eigenfunctions of Lis the PDE
(2.31) ∇ϕ(x)·f(x) = λϕ(x).
With this PDE, it is possible to approximate eigenfunctions, either by solving for the
Laurent series as in subsection 2.5 or with data via regression as in section 5. This
formulation assumes that the dynamics are both continuous and differentiable.

14 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
2.3. Koopman mode decomposition and finite representations. Until
now, we have considered scalar measurements of a system, and we explored special
eigen-measurements (i.e., Koopman eigenfunctions) that evolve linearly in time. How-
ever, we often take multiple measurements of a system, which we will arrange in a
vector g:
g(x) = 




g1(x)
g2(x)
.
.
.
gp(x)




.(2.32)
Each of the individual measurements may be expanded in terms of a basis of eigen-
functions ϕj(x):
gi(x) = ∞
X
j=1
vij ϕj(x).(2.33)
Thus, the vector of observables, g, may be similarly expanded:
g(x) = 




g1(x)
g2(x)
.
.
.
gp(x)




=∞
X
j=1
ϕj(x)vj,(2.34)
where vjis the j-th Koopman mode associated with the eigenfunction ϕj.
For conservative dynamical systems, such as those governed by Hamiltonian dy-
namics, the Koopman operator is unitary on the space G(X) of square-integrable
functions with respect to the conserved measure. Thus, the Koopman eigenfunctions
form an orthonormal basis for conservative systems, and it is possible to compute the
Koopman modes vjdirectly by projection:
vj=




hϕj, g1i
hϕj, g2i
.
.
.
hϕj, gpi




,(2.35)
where h·,·i is the standard inner product of functions in G(X). These modes have
a physical interpretation in the case of direct spatial measurements of a system,
g(x) = x, in which case the modes are coherent spatial modes that behave linearly
with the same temporal dynamics (i.e., oscillations, possibly with linear growth or
decay).
Given the decomposition in (2.34), it is possible to represent the dynamics of the
measurements gas follows:
g(x(t)) = Ktg(x0) = Kt∞
X
j=1
ϕj(x0)vj
(2.36a)
=∞
X
j=1 Ktϕj(x0)vj
(2.36b)
=∞
X
j=1
λt
jϕj(x0)vj.(2.36c)

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 15
This sequence of triples {(λj, ϕj,vj)}∞
j=1 is the Koopman mode decomposition and was
introduced by Mezi´c in 2005 [231]. The Koopman mode decomposition was later con-
nected to data-driven regression via the dynamic mode decomposition (DMD) [284],
which is explored in section 3, where the modes vare called dynamic modes.
2.3.1. Invariant eigenspaces and finite-dimensional models. Instead of
capturing the evolution of all measurement functions in a Hilbert space, applied Koop-
man analysis approximates the evolution on an invariant subspace spanned by a finite
set of measurement functions. A Koopman-invariant subspace is defined as the span
of a set of functions {g1, g2,·· · , gp}if all functions gin this subspace
g=α1g1+α2g2+· ·· +αpgp
(2.37)
remain in this subspace after being acted on by the Koopman operator K:
Kg=β1g1+β2g2+· ·· +βpgp.(2.38)
It is possible to obtain a finite-dimensional matrix representation of the Koopman
operator by restricting it to an invariant subspace spanned by a finite number of
functions {gj}p
j=1. The matrix representation Kacts on a vector space Rp, with the
coordinates given by the values of gj(x). This induces a finite-dimensional linear
system.
Any finite set of eigenfunctions of the Koopman operator will span an invariant
subspace. Discovering these eigenfunction coordinates is, therefore, a central chal-
lenge, as they provide intrinsic coordinates along which the dynamics behave linearly.
In practice, it is more likely that we will identify an approximately invariant subspace,
given by a set of functions {gj}p
j=1, where each of the functions gjis well approximated
by a finite sum of eigenfunctions: gj≈Pp
k=1 vjk ϕk.
2.4. Example of a simple Koopman embedding. Here, we consider an ex-
ample system with a single fixed point from Tu et al. [341] that is explored in more
detail in Brunton et al. [48], given by:
˙x1=µx1
(2.39a)
˙x2=λ(x2−x2
1).(2.39b)
For λ < µ < 0, the system exhibits a slow attracting manifold given by x2=x2
1. It is
possible to augment the state xwith the nonlinear measurement g=x2
1, to define a
three-dimensional Koopman invariant subspace. In these coordinates, the dynamics
become linear:
d
dt 
y1
y2
y3
=
µ0 0
0λ−λ
0 0 2µ

y1
y2
y3
for 
y1
y2
y3
=
x1
x2
x2
1
.(2.40a)
The full three-dimensional Koopman observable vector space is visualized in Fig-
ure 2.3. Trajectories that start on the invariant manifold y3=y2
1, visualized by the
blue surface, are constrained to stay on this manifold. There is a slow subspace,
spanned by the eigenvectors corresponding to the slow eigenvalues µand 2µ; this sub-
space is visualized by the green surface. Finally, there is the original asymptotically
attracting manifold of the original system, y2=y2
1, which is visualized as the red
surface. The blue and red parabolic surfaces always intersect in a parabola that is

16 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Fig. 2.3: Visualization of three-dimensional linear Koopman system from (2.40a) along
with projection of dynamics onto the x1-x2plane. The attracting slow manifold is
shown in red, the constraint y3=y2
1is shown in blue, and the slow unstable subspace
of (2.40a) is shown in green. Black trajectories of the linear Koopman system in y
project onto trajectories of the full nonlinear system in xin the y1-y2plane. Here,
µ=−0.05 and λ= 1. Reproduced from Brunton et al. [48].
inclined at a 45◦angle in the y2-y3direction. The green surface approaches this 45◦
inclination as the ratio of fast to slow dynamics become increasingly large. In the full
three-dimensional Koopman observable space, the dynamics produce a single stable
node, with trajectories rapidly attracting onto the green subspace and then slowly
approaching the fixed point.
The left eigenvectors of the Koopman operator yield Koopman eigenfunctions.
The Koopman eigenfunctions of (2.40a) corresponding to eigenvalues µand λare:
ϕµ=x1,and ϕλ=x2−bx2
1with b=λ
λ−2µ.(2.41)
The constant bin ϕλcaptures the fact that for a finite ratio λ/µ, the dynamics
only shadow the asymptotically attracting slow manifold x2=x2
1, but in fact follow
neighboring parabolic trajectories. This is illustrated more clearly by the various
surfaces in Figure 2.3 for different ratios λ/µ.
In this way, a set of intrinsic coordinates may be determined from the observable
functions defined by the left eigenvectors of the Koopman operator on an invariant
subspace. Explicitly,
ϕα(x) = ξ>
αy(x),where ξ>
αK=αξ>
α.(2.42)
These eigen-observables define observable subspaces that remain invariant under the
Koopman operator, even after coordinate transformations. As such, they may be
regarded as intrinsic coordinates [352] on the Koopman-invariant subspace.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 17
2.5. Analytic series expansions for eigenfunctions. Given the dynamics in
(1.1), it is possible to solve the PDE in (2.31) using standard techniques, such as
recursively solving for the terms in a Taylor or Laurent series. A number of simple
examples are explored below.
2.5.1. Linear dynamics. Consider the simple linear dynamics
(2.43) d
dtx=x.
Assuming a Taylor series expansion for ϕ(x):
ϕ(x) = c0+c1x+c2x2+c3x3+·· ·
then the gradient and directional derivatives are given by:
∇ϕ=c1+ 2c2x+ 3c3x2+ 4c4x3+·· ·
∇ϕ·f=c1x+ 2c2x2+ 3c3x3+ 4c4x4+·· ·
Solving for terms in the Koopman eigenfunction PDE (2.31), we see that c0= 0 must
hold. For any positive integer λin (2.31), only one of the coefficients may be nonzero.
Specifically, for λ=k∈Z+, then ϕ(x) = cxkis an eigenfunction for any constant c.
For instance, if λ= 1, then ϕ(x) = x.
2.5.2. Quadratic nonlinear dynamics. Consider a nonlinear dynamical sys-
tem
d
dt =x2.(2.44)
There is no Taylor series that satisfies (2.31), except the trivial solution ϕ= 0 for
λ= 0. Instead, we assume a Laurent series:
ϕ(x) = · ·· +c−3x−3+c−2x−2+c−1x−1+c0+c1x+c2x2+c3x3+··· .
The gradient and directional derivatives are given by:
∇ϕ=· ·· − 3c−3x−4−2c−2x−3−c−1x−2+c1+ 2c2x+ 3c3x2+ 4c4x3+···
∇ϕ·f=· ·· − 3c−3x−2−2c−2x−1−c−1+c1x2+ 2c2x3+ 3c3x4+ 4c4x5+··· .
Solving for the coefficients of the Laurent series that satisfy (2.31), we find that all
coefficients with positive index are zero, i.e. ck= 0 for all k≥1. However, the
nonpositive index coefficients are given by the recursion λck+1 =kck, for negative
k≤ −1. Thus, the Laurent series is
ϕ(x) = c01−λx−1+λ2
2x−2−λ3
3! x−3+·· ·=c0e−λ/x .
This holds for all values of λ∈C. There are also other Koopman eigenfunctions that
can be identified from the Laurent series.
2.5.3. Polynomial nonlinear dynamics. For a more general nonlinear dy-
namical system
d
dt =axn,(2.45)
ϕ(x) = eλ
(1−n)ax1−nis an eigenfunction for all λ∈C.
As mentioned in subsection 2.2, it is also possible to generate new eigenfunctions
by taking powers of these primitive eigenfunctions; the resulting eigenvalues generate
a lattice in the complex plane.

18 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
3. Dynamic mode decomposition. Dynamic mode decomposition, originally
introduced by Schmid [295,294] in the fluid dynamics community, has rapidly be-
come the standard algorithm to approximate the Koopman operator from data [284,
341,180]. Rowley et al. [284] established the first connection between DMD and
the Koopman operator. The DMD algorithm was originally developed to identify
spatio-temporal coherent structures from high-dimensional time-series data, as are
commonly found in fluid dynamics. DMD is based on the computationally efficient
singular value decomposition (SVD), also known as proper orthogonal decomposition
(POD) in fluid dynamics, so that it provides scalable dimensionality reduction for
high-dimensional data. The SVD orders modes hierarchically based on how much
of the variance of the original data is captured by each mode; these modes remain
invariant even when the order of the data is shuffled in time. In contrast, the DMD
modes are linear combinations of the SVD modes that are chosen specifically to ex-
tract spatially correlated structures that have the same coherent linear behavior in
time, given by oscillations at a fixed frequency with growth or decay. Thus, DMD
provides dimensionality reduction in terms of a low-dimensional set of spatial modes
along with a linear model for how the amplitudes of these modes evolve in time. In
this way, DMD may be thought of as a combination of SVD/POD in space with the
Fourier transform in time, combining the strengths of each approach [69,180].
There are a number of factors that have led to the widespread adoption of
DMD as a workhorse algorithm for processing high-dimensional spatiotemporal data.
The DMD algorithm approximates the best-fit linear operator that advances high-
dimensional measurements of a system forward in time [341]. Thus, DMD approxi-
mates the Koopman operator restricted to the measurement subspace given by direct
measurements of the state of a system. DMD is valid for both experimental and
simulated data, as it is based entirely on measurement data and does not require
knowledge of the governing equations. In addition, DMD is highly extensible because
of its simple formulation in terms of linear algebra, resulting in innovations related
to control, compressed sensing, and multi-resolution, among others. Because of these
strengths, DMD has been applied to a wide range of diverse applications beyond
fluid mechanics, including neuroscience, disease modeling, robotics, video processing,
power grids, financial markets, and plasma physics. Many of these extensions and
applications will be discussed more here and in section 5.
3.1. The DMD algorithm. The DMD algorithm seeks a best fit linear oper-
ator Athat approximately advances the state of a system, x∈Rn, forward in time
according to the linear dynamical system
xk+1 =Axk,(3.1)
where xk=x(k∆t), and ∆tdenotes a fixed time step that is small enough to resolve
the highest frequencies in the dynamics. Thus, the operator Ais an approximation
of the Koopman operator Krestricted to a measurement subspace spanned by direct
measurements of the state x.
DMD is fundamentally a data-driven algorithm, and the operator Ais approx-
imated from a collection of snapshot pairs of the system, {(x(tk),x(t0
k)}m
k=1, where
t0
k=tk+∆t. A snapshot is typically a measurement of the full state of the system, such
as a fluid velocity field sampled at a large number of spatially discretized locations,
that is reshaped into a column vector of high dimension. The original formulation of
Schmid [294] required data from a single trajectory with uniform sampling in time,
so that tk=k∆t. Here we present the exact DMD algorithm of Tu et al. [341], which

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 19
works for irregularly spaced data and data concatenated from multiple different time
series. Thus, in exact DMD, the times tkneed not be sequential or evenly spaced,
but for each snapshot x(tk) there is a corresponding snapshot x(t0
k) one time step ∆t
in the future. These snapshots are arranged into two data matrices, Xand X0:
X=
x(t1)x(t2)·· · x(tm)

(3.2a)
X0=
x(t0
1)x(t0
2)·· · x(t0
m)
.(3.2b)
Equation (3.1) may be written in terms of these data matrices as
X0≈AX.(3.3)
The best fit operator Aestablishes a linear dynamical system that approximately
advances snapshot measurements forward in time, which may be formulated as an
optimization problem
A= argmin
AkX0−AXkF=X0X†
(3.4)
where k·kFis the Frobenius norm and †denotes the pseudo-inverse. The pseudo-
inverse may be computed using the SVD of X=UΣV∗as X†=VΣ−1U∗. The
matrices U∈Cn×nand Vm×mare unitary, so that U∗U=Iand V∗V=I, where ∗
denotes complex-conjugate transpose. The columns of Uare known as POD modes.
The matrix Ahas n2elements, so for high-dimensional data it may be intractable
to represent this operator. Instead, the DMD algorithm seeks the leading spectral
decomposition (i.e., eigenvalues and eigenvectors) of Awithout ever explicitly con-
structing it. The data matrices Xand X0typically have far more rows than columns,
i.e. mn, so that Awill have at most mnonzero eigenvalues and non-trivial eigen-
vectors. In practice, the effective rank of the data matrices, and hence the operator
A, may be even lower, given by r < m. Instead of computing Ain (3.4), we may
project Aonto the first rPOD modes in Urand approximate the pseudo-inverse
using the rank-rSVD approximation X≈UrΣrV∗
r:
˜
A=U∗
rAUr
(3.5a)
=U∗
rX0X†Ur
(3.5b)
=U∗
rX0VrΣ−1
rU∗
rUr
(3.5c)
=U∗
rX0VrΣ−1
r.(3.5d)
The leading spectral decomposition of Amay be approximated from the spectral
decomposition of the much smaller ˜
A:
˜
AW =WΛ.(3.6)
The diagonal matrix Λcontains the DMD eigenvalues, which correspond to eigenval-
ues of the high-dimensional matrix A. The columns of Ware eigenvectors of ˜
A, and
provide a coordinate transformation that diagonalizes the matrix. These columns may
be thought of as linear combinations of POD mode amplitudes that behave linearly
with a single temporal pattern given by the corresponding eigenvalue λ.

20 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
−1 0 1 2 3 4 5 6 7 8
2
1
0
−1
−2
−1 0 1 2 3 4 5 6 7 8
2
1
0
−1
−2
−1 0 1 2 3 4 5 6 7 8
2
1
0
−1
−2
2
1
0
1
2
2
1
0
1
2
2
1
0
1
2
2
1
0
1
2
2
1
0
1
2
X0=2
4x2x3···xm
3
5
X=2
4x1x2···xm1
3
5
Experiment Collect Data DMD
Data Regression
A=X0X†
...
a) Diagnostics
b) Future state prediction
Dynamic modes
Time dynamics
past future
xk+1=Axk
t
x1
x2
xm
t
1m
x3
Fig. 3.1: Overview of DMD illustrated on the fluid flow past a circular cylinder at
Reynolds number 100. Reproduced from Kutz et al. [180].
The eigenvectors of Aare the DMD modes Φ, and they are reconstructed using
the eigenvectors Wof the reduced system and the time-shifted data matrix X0:
Φ=X0˜
V˜
Σ−1W.(3.7)
Tu et al. [341] proved that these DMD modes are eigenvectors of the full Amatrix
under certain conditions. This approach is illustrated for a fluid flow in Figure 3.1.
There are also several open-source DMD implementations [180,88].
3.1.1. Spectral decomposition and the DMD expansion. Once the DMD
modes and eigenvalues are computed, it is possible to represent the system state in
terms of the DMD expansion:
xk=
r
X
j=1
φjλk−1
jbj=ΦΛk−1b,(3.8)
where φjare eigenvectors of A(DMD modes), λjare eigenvalues of A(DMD eigen-
values), and bjare the mode amplitudes. The amplitudes in bare given by
b=Φ†x1.(3.9)
Alternative approaches to compute b[69,150,16] will be discussed in subsection 3.1.2.
The spectral expansion in (3.8) may be converted to continuous time by intro-
ducing the continuous eigenvalues ω= log(λ)/∆t:
x(t) =
r
X
j=1
φjeωjtbj=Φexp(Ωt)b,(3.10)
where Ωis a diagonal matrix containing the continuous-time eigenvalues ωj. Thus,
the data matrix Xmay be represented as
(3.11)
X≈
| |
φ1·· · φr
| | 



b1
...
br





eω1t1·· · eω1tm
.
.
.....
.
.
eωrt1·· · eωrtm


=Φdiag(b)T(ω).

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 21
3.1.2. Alternative optimizations for DMD. The DMD algorithm is purely
data-driven, and is thus equally applicable to experimental and numerical data. When
characterizing experimental data with DMD, the effects of sensor noise and stochastic
disturbances must be accounted for. Bagheri [19] showed that DMD is particularly
sensitive to the effects of noisy data, and it has been shown that significant and sys-
tematic biases are introduced to the eigenvalue distribution [93,18,83,131]. Although
increased sampling decreases the variance of the eigenvalue distribution, it does not
remove the bias [131]. This noise sensitivity has motivated several alternative opti-
mization algorithms for DMD to improve the quality and performance of DMD over
the standard optimization in (3.4), which is a least-square fitting procedure involv-
ing the Frobenius norm. These algorithms include the total least-squares DMD [131],
forward-backward DMD [83], variable projection [16], and robust principal component
analysis [292].
One of the simplest ways to remove the systematic bias of the DMD algorithm
is by computing it both forward and backward in time and averaging the equivalent
operator, as proposed by Dawson et al. [83]. Thus the two following approximations
are considered
X0≈A1Xand X≈A2X0
(3.12)
where A−1
2≈A1for noise-free data. Thus the operator A2is the inverse, or backward
time-step, mapping the snapshots from tk+1 to tk. The forward and backward time
operators are then averaged, removing the systematic bias from the measurement
noise:
(3.13) A=1
2A1+A−1
2
where the optimization (3.4) can be used to compute both the forward and backward
mapping A1and A2. This optimization can be formulated as
(3.14) A= argmin
A
1
2kX0−AXkF+kX−A−1X0kF,
which is highly nonlinear and non-convex due to the inverse A−1. An improved
optimization framework was developed by Azencot et al. [17] which proposes
(3.15) A= argmin
A1,A2
1
2(kX0−A1XkF+kX−A2X0kF) s.t. A1A2=I,A2A1=I,
to circumvent some of the difficulties of the optimization in (3.14).
Hemati et al. [131] formulate another DMD algorithm, replacing the original least-
squares regression with a total least-squares regression to account for the possibility
of noisy measurements and disturbances to the state. This work also provides an
excellent discussion on the sources of noise and a comparison of various denoising
algorithms. The subspace DMD algorithm of Takeishi et al. [331] compensates for
measurement noise by computing an orthogonal projection of future snapshots onto
the space of previous snapshots and then constructing a linear model. Extensions
that combine DMD with Bayesian approaches have also been developed [329].
Good approximations for the mode amplitudes bin (3.11) have also proven to be
difficult to achieve, with and without noise. Jovanovi´c et al. [150] developed the first
algorithm to improve the estimate of the modal amplitudes by promoting sparsity.

22 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
In this case, the underlying optimization algorithm is framed around improving the
approximation (3.11) using the formulation
(3.16) argmin
b
(kX−Φdiag(b)T(ω)kF+γkbk1)
where k·k1denotes the `1-norm penalization which promotes sparsity of the vector
b. More recently, Askham and Kutz [16] introduced the optimized DMD algorithm,
which uses a variable projection method for nonlinear least squares to compute the
DMD for unevenly timed samples, significantly mitigating the bias due to noise. The
optimized DMD algorithm solves the exponential fitting problem directly:
(3.17) argmin
ω,ΦbkX−ΦbT(ω)kF.
This has been shown to provide a superior decomposition due to its ability to optimally
suppress bias and handle snapshots collected at arbitrary times. The disadvantage of
optimized DMD is that one must solve a nonlinear optimization problem.
3.1.3. Krylov subspace perspective. In the original formulation [294,284],
the matrices Xand X0were formed from sequential snapshots, evenly spaced in time:
X=
x1x2·· · xm

(3.18a)
X0=
x2x3·· · xm+1
.(3.18b)
The columns of Xbelong to a Krylov subspace generated by Aand x1:
X≈
x1Ax1·· · Am−1x1
.(3.19)
Thus, DMD is related to Arnoldi iteration to find the dominant eigenvalues and
eigenvectors of a matrix A.
The matrices Xand X0are also related through the shift operator Sby
X0=XS.(3.20)
Thus, Sacts on columns of X, as opposed to A, which acts on rows of X. The shift
operator S, also known as a companion matrix, is given by
S=






000·· · 0a1
100·· · 0a2
010·· · 0a3
.
.
..
.
..
.
.....
.
..
.
.
000·· · 1am







.(3.21)
In other words, the first m−1 columns of X0are obtained by shifting the last m−1
columns of X, and the last column is obtained as a best-fit combination of the m
columns of Xthat minimizes the residual. The shift operator may be viewed as a
matrix representation of the Koopman operator, as it advances snapshots forward in
time. The m×mmatrix Shas the same non-zero eigenvalues as A, so that Smay
be used to obtain dynamic modes and eigenvalues. However, computations based on
Sare not as numerically stable as the DMD algorithm presented above.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 23
3.2. Methodological extensions. The DMD algorithm has been successful in
large part because of its simple formulation in terms of linear regression and because it
does not require knowledge of governing equations. For these reasons, DMD has been
extended to include several methodological innovations [180] presented here. Several
of these extensions, including to nonlinear systems, delay measurements, and control,
will be explored in much more detail in later sections. Algorithms to handle noisy
data were already discussed in subsection 3.1.2.
3.2.1. Including inputs and control. Often, the goal of obtaining reduced-
order models is to eventually design more effective controllers to manipulate the be-
havior of the system. Similarly, in many systems, such as the climate, there are
external forcing variables that make it difficult to identify the underlying unforced
dynamics. Proctor et al. [275] introduced the DMD with control (DMDc) algorithm
to disambiguate the natural unforced dynamics and the effect of forcing or actuation
given by the variable u. This algorithm is based on
xk+1 ≈Axk+Buk,(3.22)
which results in another linear regression problem. This algorithm was motivated
by epidemiological systems (e.g., malaria or polio), where it is not possible to stop
intervention efforts, such as vaccinations and bed nets, in order to characterize the
unforced dynamics [274]. DMDc will be explored extensively in subsection 6.2.1.
3.2.2. Compression and randomized linear algebra. The DMD algorithm
is fundamentally based on the assumption that there are dominant low-dimensional
patterns even in high-dimensional data, such as fluid flow fields. Randomized al-
gorithms [124] are designed to exploit these patterns to accelerate numerical linear
algebra. In the randomized DMD algorithm [30,100] data is randomly project into a
lower-dimensional subspace where computations may be performed more efficiently.
The existence of patterns also facilitate more efficient measurement strategies based
on principles of sparsity to reduce the number of measurements required in time [340]
and space [52,123,98]. This has the broad potential to enable high-resolution char-
acterization of systems from under-resolved measurements. In 2014, Jovanovic et
al. [150] used sparsity promoting optimization to identify the fewest DMD modes re-
quired to describe a data set; the alternative, testing and comparing all subsets of
DMD modes, is computationally intractable. Finally, libraries of DMD modes have
also been used to identify dynamical regimes [178], based on the sparse representa-
tion for classification [358], which was used earlier to identify dynamical regimes using
libraries of POD modes [38,53].
3.2.3. Nonlinear measurements and latent variables. The connection be-
tween DMD and the Koopman operator [284,341,180] has motivated several exten-
sions for strongly nonlinear systems. The standard DMD algorithm is able to accu-
rately characterize periodic and quasi-periodic behavior, even in nonlinear systems.
However, DMD models based on linear measurements of the system are generally not
sufficient to characterize truly nonlinear phenomena, such as transients, intermittent
phenomena, or broadband frequency cross-talk. In Williams et al. [352,353], DMD
measurements were augmented to include nonlinear measurements of the system, en-
riching the basis used to represent the Koopman operator. There are also important
extensions of DMD to systems with latent variables. Although DMD was developed
for high-dimensional data, it is often desirable to characterize systems with incom-
plete measurements. As an extreme example, consider a single measurement that

24 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
oscillates as a sinusoid, x(t) = sin(ωt). Although this would appear to be a perfect
candidate for DMD, the algorithm incorrectly identifies a real eigenvalue because the
data does not have sufficient rank to extract a complex conjugate pair of eigenvalues
±iω. This paradox was first explored by Tu et al. [341], where it was discovered that a
solution is to stack delayed measurements into a larger matrix to augment the rank of
the data and extract phase information. Delay coordinates have also been used effec-
tively to extract coherent patterns in neural recordings [45]. The connections between
delay DMD and Koopman were subsequently investigated [47,14,81,161]. Nonlinear
measurements and latent variable will both be explored extensively in section 5.
3.2.4. Multiresolution. DMD is often applied to complex, high-dimensional
dynamical systems, such as fluid turbulence or epidemiological systems, that exhibit
multiscale dynamics in both space and time. Many multiscale systems exhibit tran-
sient or intermittent phenomena, such as the El Ni˜no observed in global climate
data. These transient dynamics are not captured accurately by DMD, which seeks
spatio-temporal modes that are globally coherent across the entire time series of data.
To address this challenge, the multiresolution DMD (mrDMD) algorithm was intro-
duced [181], which effectively decomposes the dynamics into different timescales, iso-
lating transient and intermittent patterns. Multiresolution DMD modes were recently
shown to be advantageous for sparse sensor placement by Manohar et al. [215].
3.2.5. Streaming and parallelized codes. Because of the computational bur-
den of computing the DMD on high-resolution data, several advances have been made
to accelerate DMD in streaming applications and with parallelized algorithms. DMD
is often used in a streaming setting, where a moving window of snapshots are pro-
cessed continuously, resulting in savings by eliminating redundant computations when
new data becomes available. Several algorithms exist for streaming DMD, based on
the incremental SVD [132], a streaming method of snapshots SVD [267], and rank-one
updates to the DMD matrix [365]. The DMD algorithm is also readily parallelized, as
it is based on the SVD. Several parallelized codes are available, based on the QR [290]
and SVD [100,101,99].
3.3. Domain applications. DMD has been widely applied to a diverse range
of applications. We will explore key applications in fluid dynamics, epidemiology,
neuroscience, and video processing. In addition, DMD has been used for robotics [27,
4,44,111], finance [213], power grids [303,324,321,326,319,322,176], and plasma
physics [335,162].
3.3.1. Fluid dynamics. DMD originated in the fluid dynamics community [294],
and has since been applied to a wide range of flow geometries (jets, cavity flow, wakes,
channel flow, boundary layers, etc.), to study mixing, acoustics, and combustion,
among other phenomena. In Schmid [295,294], both a cavity flow and a jet were
considered; the cavity flow example is shown in Figure 3.2. Rowley et al. [284] in-
vestigated a jet in cross-flow; modes are shown in Figure 3.3 with the corresponding
eigenvalue spectrum. Thus, it is no surprise that DMD has subsequently been used
broadly in both cavity flows [294,203,297,24,23] and jets [25,298,296,293].
DMD has also been applied to wake flows, including to investigate frequency
lock-on [339], the wake past a gurney flap [260], the cylinder wake [18], and dynamic
stall [94]. Boundary layers have also been extensively studied with DMD [255,291,
236]. In acoustics, DMD has been used to capture the near-field and far-field acoustics
that result from instabilities observed in shear flows [308]. In combustion, DMD has
been used to understand the coherent heat release in turbulent swirl flames [237] and

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 25
16 P. J. Schmid
Outflow
0 5 10 15 20 25 30
10−7
10−6
10−5
10−4
10−3
Residual norm r
Number
(
N
)
of snapshots
(a)(b)
Figur e 4. (a) Sketch of cavity geometry with subdomain indicated by the blue dashed line.
(b) Residual history of the dynamic mode decomposition for cavity flow.
−15 −10 −5 0 5 10 15
−3
−2
−1
0
1
2
λi
λr
Figur e 5. DMD spectrum for cavity flow at a Reynolds number Re = 4500.The colour and
symbol size of the eigenvalues indicate the coherence of the associated dynamic modes.
of the number of included snapshots. Rapid convergence is observed; only about
20 snapshots are needed to determine the growth rates and phase velocities with
sufficient accuracy. The extracted spectrum (i.e. the spectrum of ˜
S) is displayed in
figure 5. The spectrum appears symmetric with respect to the imaginary axis λi=0,
which is a consequence of processing real-valued data. If general complex-valued data
are processed (e.g. after a Fourier transformation along a homogeneous or periodic
direction), the spectrum will generally appear asymmetric with respect to λi=0.
For the chosen parameters (Re = 4500 based on the uniform inflow velocity and the
cavity length), a pair of unstable modes has been identified. In addition, we observe a
typical spectrum for a cavity, consisting of a parabolic branch containing the unstable
eigenvalues (the unstable branches) and a parabolic branch describing the dynamics
inside the cavity (the stable branch). This type of spectrum should be compared
with spectra from a global stability analysis for similar flow configurations (see e.g.
˚
Akervik et al. 2007 for a shallow cavity, and Sipp & Lebedev 2007 and Barbagallo
et al. 2009 for a square cavity). The symbol size and colouring of the eigenvalues
in figure 5 indicates a coherence measure of the associated modes and is intended
to separate large-scale energetic structures (in red) from smaller-scale less-energetic
Dynamic mode decomposition of numerical and experimental data 17
(a)(b)
(c)(d)
(e)(f)
Figu re 6. Representative dynamic modes, visualized by the streamwise velocity component,
for flow over a cavity at Re = 4500.(a)Mostunstabledynamicmode,(b–d)dynamicmode
from the unstable branch, (e,f) dynamic modes from the stable branch. Because data from
linearized Navier–Stokes simulations have been processes, the dynamic modes are equivalent
to global modes.
structures. The criterion is given by a projection of a specific dynamic modes Φi
onto the POD basis U,computed from the data sequence VN−1
1; the modulus of the
coefficients of this projection measures the presence of various POD modes and thus
gives a measure of coherence. It is important to realize, however, that modes with
a moderate to small projection onto a POD basis (blue symbols) can still play a
significant dynamic role within the snapshot sequence.
Representative dynamic modes are displayed in figure 6 using the streamwise
velocity component; their respective eigenvalues are given in table 1 (second and third
columns). The unstable mode (figure 6a) is clearly located in the shear layer of the
flow and shows the characteristic streamwise wavelength of the observed instability.
Other modes from the unstable branch (figure 6b–d) have significant components in
the shear layer, but also show features inside the cavity. These features are related
to the instability of the shear layer detaching from the right edge of the cavity.
Dynamic modes from the stable branch (figure 6e,f)containsimilarcharacteristics:
vortical structures coincidental with the mean shear layer on top of the cavity and
features linked to the vortex inside the cavity. Modes from the stable branch show
increasingly more small-scale features inside the cavity, as the frequency λiincreases,
which is consistent with observations of Barbagallo et al. (2009).
Fig. 3.2: (top) DMD eigenvalue spectrum for the cavity flow at Reynolds number
Re = 4500. (bottom) Corresponding DMD modes, visualized by the streamwise
velocity. (a) Most unstable dynamic mode, (b–d) DMD modes from the unstable
branch, (e,f) DMD modes from the stable branch. Modified with permission, from
Schmid 2010 Journal of Fluid Mechanics [294].
to analyze a rocket combustor [143]. DMD has also been used to analyze non-normal
growth mechanisms in thermoacoustic interactions in a Rijke tube. DMD has been
compared with POD for reacting flows [285]. DMD has also been used to analyze
more exotic flows, including a simulated model of a high-speed train [242]. Shock
turbulent boundary layer interaction (STBLI) has also been investigated with DMD to
identify a pulsating separation bubble that is accompanied by shockwave motion [121].
DMD has also been used to study self-excited fluctuations in detonation waves [219].
Other problems include identifying hairpin vortices [333], decomposing the flow past
a surface mounted cube [243], modeling shallow water equations [31], studying nano
fluids past a square cylinder [289], and measuring the growth rate of instabilities in
annular liquid sheets [92].

26 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Fig. 3.3: DMD eigenvalues for jet-in-crossflow plotted on the unit circle (a) and as a
power spectrum versus Strouhal number (b). Two DMD modes (m1) and (m2) are
pictured below. Modified with permission, from Rowley et al. 2009 Journal of Fluid
Mechanics [284].
It is interesting to note that the noise effects that were carefully analyzed by
Bagheri [19] explain the eigenvalue spectrum observed earlier by Schmid et al. [293]
for a turbulent jet. This comparison is shown in Figure 3.4.
3.3.2. Epidemiology. Epidemiological data often consists of high-dimensional
spatiotemporal time series measurements, such as the number of infections in a given
neighborhood or city. Thus, DMD provides particularly interpretable decomposi-
tions for these systems, as explored by Proctor and Eckhoff [277] and illustrated in
Figure 3.5. Modal frequencies often correspond to yearly or seasonal fluctuations.
Moreover, the phase of DMD modes gives insight into how disease fronts propagate
spatially, potentially informing future intervention efforts. The application of DMD
to disease systems also motivated the DMD with control algorithm [275], since it is
infeasible to stop vaccinations in order to identify the unforced dynamics.
3.3.3. Neuroscience. Complex signals from neural recordings are increasingly
high-fidelity and high dimensional, with advances in hardware pushing the frontiers
of data collection. DMD has the potential to transform the analysis of such neural
recordings, as evidenced in a recent study by B. Brunton et al. [45] that identified
dynamically relevant features in ECOG data of sleeping patients, shown in Figure 3.6.
Since then, several works have applied DMD to neural recordings or suggested possible
implementation in hardware [6,39,337].
3.3.4. Video processing. DMD has also been used for segmentation in video
processing, as a video may be thought of as a time-series of high-dimensional snapshots
(images) evolving in time. Separating foreground and background objects in video is

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 27
094104-2 Shervin Bagheri Phys. Fluids 26,094104(2014)
period—on the degree of non-normality4of the linearized and deterministic Floquet system. As we
show herein and first derived in Gaspard,5
Q≈T2
π"
1
|S|,(2)
where Sis the “sensitivity” to white Gaussian noise of amplitude ""1. It can be shown5that the
expression for Sis given by
S=−fT
1˜
u(T)
fT
1e1
,(3)
where ˜
u(T)isobtainedbyintegratingthelinearizedequations(presentedinSec.III C). The vectors
f1and e1are the first left (i.e., adjoint) and right eigenvectors of the linearized Floquet system.
In particular, if the system is highly non-normal then fT
1e1"1, giving S$1, e.g., a very large
sensitivity of the limit cycle period to noise. Thus, although phase diffusion is a stochastic effect
observed in the nonlinear system, Qis directly dependent on the local linear stability properties
of the deterministic limit cycle. This suggests that some oscillators may in fact be very sensitive
to external fluctuations, despite the common viewpoint that oscillators are “insensitive” to small
amplitude external forcing (in contrast to noise amplifiers).6
The second objective of this paper is to use the presented theory to show the effects of weak
noise on the spectrum of the Koopman operator.7,8This linear operator describes the evolution of
observables and provides a full description of nonlinear dynamics. In Figure 1,thespectrumofthe
operator for an oscillator is shown in the absence (square symbols/red ) and presence (circles/black
symbols) of noise. In the former case, the eigenvalues are integer multiples of the fundamental
frequency, i.e.,
λm=imω,
where m=0, ±1, ±2, . . . . As shown in Bagheri,9these Koopman eigenvalues10 can be computed
using the dynamic mode decomposition (DMD) algorithm.11 The eigenvalues correspond to global
modes, characterizing the different harmonics of the oscillating flow. In the presence of noise
however, the eigenvalues can be approximated by
λm=imω−"|S|
2Tω2m2+O("2),(4)
where Swas defined in (3). For a noisy dynamical system containing a limit cycle, the spectrum
(at the leading order) given by (4) becomes damped, i.e., the eigenvalues have a nonzero real part,
which is proportional to the sensitivity Sand increases quadratically with higher harmonics of ω.
Since any experiment—numerical or laboratory—is subject to some level of random perturbations,
we may—in practice—expect damped Koopman eigenvalues more often than eigenvalues on the
imaginary axis (or on the unit circle). Thus, in the presence of noise one should observe parabolic
shapes in the spectrum, which happens to be the form of many DMD spectra in literature.12,13 The
reason for damped eigenvalues is that the Koopman operator—which in the absence of noise is linear
advection operator—becomes a linear advection-diffusion operator in the presence of noise.
Noisy limit cycle
Deterministic limit cycle
Spectrum of evolution operator
0
0
1 2-1-2
1/τ
Re(λ )
x iω
Τ
1
m
FIG. 1. Koopman eigenvalues λmof an oscillator with period T=2π/ω1in the presence and absence of noise.
094104-8Shervin Bagheri Phys. Fluids 26,094104(2014)
FIG. 6. The top frame shows four snapshots of a helium jet; it uses Schlieren techniques to visualize the dynamics of the
fluid structure; thus, only a scalar field—proportional to the density gradient and quantified by its location on a grey-scale
colormap—is being processed. The first snapshot (most left) also contains the size and location (box marked with an arrow)
of the interrogation window for the subsequent DMD-analysis. Note from this snapshots that the flow is turbulent away from
the nozzle, but initially shows very well structured coherent motion, that is also oscillatory in time. The bottom frame in the
figure shows the spectrum extracted using the DMD algorithm. Figure taken from Schmid et al.12 Reprinted with permission
from P. Schmid et al. Copyright 2009 Springer Science and Business Media.
the eigenvalues associated with the deterministic limit cycle obtained from the DMD in Bagheri9
where all located on the imaginary axis. However, in practice and often for experimental data,
the DMD spectrum shows damped eigenvalues. We show in Figure 6atypicalDMDspectrum
found in literature.12 The spectrum shows the trait of an advection-diffusion operator, i.e., parabolic
branches are clearly observed. These parabolic shapes are indeed very similar to the spectrum of
anoisylimitcycleasoutlinedinSec.II B. In the context of the theory presented herein, one may
hypothesize that the parabolic branches observed in DMD spectra are due to the presence of noise.
In that case, one would further expect that the decay rate of the first eigenvalue in the spectrum
provides an estimate of the quality factor. However, whereas the parabolic shapes in the spectrum of
Adefined by (10) are related to expected values of an observable, classical DMD is often based on
sampling directly the observable (not its statistics). As shown in this paper, for a noisy and nonlinear
oscillator, each realization of a trajectory (even with the same initial condition) is different, and
hence the DMD spectrum will vary for different realizations of the same system. One way to make
the algorithm robust and the spectrum unique is to use statistical observables, such as expectation
values and correlation functions. Using such a probabilistic approach, one is effectively investigating
the evolution of a set of trajectories. One effort to generalize DMD to datasets based on more than
one trajectory is the recent DMD theory presented in Tu et al.,16 which provides Ritz vectors and
values of multiple trajectories.
III. WEAK-NOISE THEORY
This section provides the necessary theory to derive Eqs. (3) and (4).Inplaceswherelengthy
but rather straightforward derivations are required, we refer to other papers. This is because the
key points of this paper are the consequences of weak-noise theory linking quality factor, Floquet
analysis and Koopman spectrum. The main theory in a more general and rigorous form can be found
in Gaspard.5Specifically, the structure of this section is as follows. The first part shows that Qin (1)
may be approximated by the first eigenvalue λ1of an evolution operator. Moreover, Qdepends on
Fig. 3.4: (top) Koopman eigenvalue spectrum for oscillator with noise. (middle, bot-
tom) DMD spectrum for jet flow, exhibiting similar noise pattern. Reproduced with
permission, from Bagheri 2014 Physics of Fluids [19]; (bottom, middle) panels origi-
nally from Schmid et al., 2011 Theoretical and Computational Fluid Dynamics [293].
Figure 2. The panels describe the data and output of the dynamic mode decomposition (DMD) on three examples: Google Flu, pre- vaccination measles
in the UK, and type 1 paralytic polio cases. In each panel, two plots are included to visualize the data: the top left plot shows four time-histories from
different locations; the bottom left is a visualization of all the locations in time. The time histories in the bottom left are normalized, described in the text.
The three plots illustrate the output of DMD: how to select the mode based on a power calculation, the eigenvalue spectrum of
˜
A, a dynamic mode
φplotted as a map.
International Health
143
Fig. 3.5: Results of DMD analysis for Polio cases in Nigeria. Reproduced with permis-
sion, from Proctor and Eckhoff 2014 International health [277].

28 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Fig. 3.6: Illustration of DMD applied to human brain data from electroencephalogram
(ECOG) measurements. Reproduced with permission, from B. Brunton et al. 2016
Journal of Neuroscience Methods [45].
a common task in surveillance applications. Real-time separation is a challenge that
is only exacerbated by increasing video resolutions. DMD provides a flexible approach
for video separation, as the background may be approximated by a DMD mode with
zero eigenvalue [122,98,267], as illustrated in Figure 3.7.
3.4. Connections to other methods. Linear regression models are common
throughout the sciences, and so it is not surprising to find many methods that are
closely related to DMD. These related methods have been developed in a diverse set of
scientific and mathematical disciplines, using a range of mathematical formulations.
3.4.1. System identification. DMD may be viewed as a form of system iden-
tification, which is a mature subfield of control theory. Tu et al. [341] established a
connection between DMD and the eigensystem realization algorithm (ERA) [152].
Subsequently, DMD with control [275] was also linked to ERA and the observer
Kalman identification methods (OKID) [272,271]. ERA and OKID construct input-
output models using impulse response data and continuous disturbance data, respec-
tively [152,153,272,271,151]. Both of these methods were developed to construct
input-output state-space models for aerospace applications where the systems tend
to have higher rank than the number of sensor measurements [137,152]. In contrast,
DMD and DMDc were developed for systems with a large number of measurements
and low-rank dynamics. ERA and OKID have also been categorized as subspace
identification methods [278], which include the numerical algorithms for subspace
system identification (N4SID), multivariable output error state space (MOESP), and
canonical variate analysis (CVA) [347,348,163,278]. Algorithmically, these meth-
ods involve regression, model reduction, and parameter estimation steps, similar to
DMDc. There are important contrasts regarding the projection scaling between all of
these methods [275], but the overall viewpoint is similar among these diverse methods.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 29
Fig. 3.7: DMD may be used to process videos, extracting dominant background modes
with zero frequency. Reproduced with permission, from Erichson et al. 2016 Journal
of Real Time Processing [98].
3.4.2. ARIMA: Auto-regressive integrating moving average. Autoregres-
sive moving average (ARMA) models and the generalized autoregressive integrated
moving average (ARIMA) models [35] are commonly used in statistics and economet-
rics. These models leverage time-series data to build models to forecast predictions
into the future. ARIMA models are often applied to data that are non-stationary. Like
DMD, ARMA and ARIMA models are characterized by a number of key parameters,
one of them being the number of past time points used for forecasting a future point.
However, DMD correlates each time snapshot directly to the previous time snapshot.
Autoregressive models have a number of useful variants, including a generalization to
a vector framework, i.e. the VARIMA (vector ARIMA) model, and a generalization
which includes seasonal effects, i.e. the SARIMA (seasonal ARIMA) model. In the
DMD architecture, seasonal variations are automatically included. Moreover, if the
mean of the data matrix Xis subtracted, then the companion matrix DMD formula-
tion from subsection 3.1.3 has been shown to be equivalent to a Fourier decomposition
of the vector field in time [69,136]. DMD can be thought of as taking advantage of
both the vector nature of the data and any oscillatory (seasonal) variations. Further,
the real part of the DMD spectra allows one to automatically capture exponential
trends in the data.
3.4.3. LIM: Linear inverse modeling. Linear inverse models (LIMs) have
been developed in the climate science community. LIMs are essentially identical to
the DMD architecture under certain assumptions [341]. By construction, LIMS rely on
low-rank modeling, like DMD, using low-rank truncation which are known as empiri-
cal orthogonal functions (EOFs) [201]. Using EOFs, Penland [268] derived a method
to compute a linear dynamical system that approximates data from a stochastic linear

30 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
Markov system, which later came to be known as LIM [269]. Under certain circum-
stances, DMD and LIM may be considered equivalent algorithms. The equivalence
of projected DMD and LIM gives us yet another way to interpret DMD analysis. If
the mean of the data is removed, then the low-rank map that generates the DMD
eigenvalues and eigenvectors is simply the same map that yields the statistically most
likely state in the future. This is the case for both the exact and projected DMD
algorithms, as both are built on the same low-order linear map. LIM is typically
performed for data where the mean has been subtracted, whereas DMD is valid with
or without mean subtraction. Regardless, there is a strong enough similarity between
the two methods that the communities may find their particular innovations valuable
to one another.
3.4.4. PCR: Principal component regression. In the statistical sciences,
principal component regression (PCR) [147] is a regression analysis technique that is
once again based on the SVD (specifically PCA). PCR regresses the outcome (also
known as the response or, the dependent variable) on the principal components of
a set of covariates (also known as predictors or, explanatory variables or, indepen-
dent variables) based on a standard linear regression model. Instead of regressing the
dependent variable on the explanatory variables directly, PCR uses the principal com-
ponents of the explanatory variables as regressors. One typically uses only a subset
of all the principal components for regression, thus making PCR a regularized proce-
dure. Often the principal components with larger variances are selected as regressors;
these principal components correspond to eigenvectors with larger eigenvalues of the
sample variance-covariance matrix of the explanatory variables. However, for the pur-
pose of predicting the outcome, the principal components with low variances may also
be important, in some cases even more important [147,146]. Unlike the DMD/LIM
literature, which has been traditionally steeped in dynamics, the statistics literature
is often concerned with regression of static data, mapping generic input data to tar-
get data. Thus PCR is typically not specifically applied to time-series data, but is
instead a general regression procedure that may or may not be applied to data from
a dynamical system. In some sense, the first two steps of the DMD algorithm may
be viewed as performing PCR on snapshot data from a high-dimensional dynamical
system. However, PCR does not include the additional steps of eigendecomposition
of the matrix ˜
Aor the reconstruction of high-dimensional coherent modes. This last
step is what relates DMD to the Koopman operator, connecting the data analysis to
nonlinear dynamics.
3.4.5. Resolvent analysis. DMD and Koopman operator theory have also
been connected to the resolvent analysis from fluid mechanics [299,134]. Resol-
vent analysis seeks to find the most receptive states of a dynamical system that will
be most amplified by forcing, along with the corresponding most responsive forc-
ings [338,149,228,148]. Sharma, Mezi´c, and McKeon [299] established several im-
portant connections between DMD, Koopman theory, and the resolvent operator, in-
cluding a generalization of DMD to enforce symmetries and traveling wave structures.
They also showed that the resolvent modes provide an optimal basis for the Koop-
man mode decomposition. Typically, resolvent analysis is performed by linearizing
the governing equations about a base state, often a turbulent mean flow. However,
this approach is invasive, requiring a working Navier-Stokes solver. Herrmann et
al. [134] have recently developed a purely data-driven resolvent algorithm, based on
DMD, that bypasses knowledge of the governing equations.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 31
4. Koopman operator and modern nonlinear dynamics.
4.1. Eigenfunctions as nonlinear coordinate changes. Koopman eigen-
functions provide an explicit coordinate transformation between the nonlinear dy-
namical system ˙
x=f(x) (1.1) and a factor of the Koopman (Lie) dynamical system
˙g=Lg(2.11), where functions gare restricted to the span of a subset of Koopman
eigenfunctions. In this section, we provide an overview of results that exploit linearity
of Koopman dynamics to arrive at conclusions about the original nonlinear dynamical
system.
Consider a more general setting between two dynamical systems ˙
x1=f1(x1) for
x1∈ X1and ˙
x2=f2(x2) for x2∈ X2, respectively inducing flows Fk:Xk→ Xk
given by xk(t) = Ft
k(xk(0)), k = 1,2. If there exists a homeomorphism (continuous
function with a continuous inverse) h:X1→ X2such that its composition with the
flows
(4.1) h◦Ft
1≡Ft
2◦h
holds everywhere, the two dynamical systems are topologically conjugate.2The orbit
structure of two topologically conjugate dynamical systems is qualitatively the same.
Therefore, if one identifies a pair of such systems where one of them is easier to
analyze analytically, numerically, or is already familiar, it makes it possible to port
results from the “simpler” to the more complicated system. One seminal result of
this sort is the Hartman–Grobman theorem [350,§19.12A], which establishes that
a nonlinear system ˙
x=f(x) in an open neighborhood of a hyperbolic fixed point
x0is locally topologically conjugate to its linearization ˙
z=Az, where A=Df(x0),
formally justifying the use of the Jacobian in inferring stability of hyperbolic nonlinear
equilibria.
Lan and Mezi´c [185] recognized that the conjugacy relationship resulting from
the Hartman–Grobman theorem
(4.2) h(Ft(x)) = eAth(x),
where eAtis the flow of the linearization, implies the existence of a (vector-valued)
Koopman eigenfunction h, as it matches the definition (2.12). By diagonalization
of A, the components hkof hare precisely scalar-valued Koopman eigenfunctions.
Although H–G establishes that (4.2) holds only in some neighborhood of the ori-
gin, [185] show that trajectories emanating from the boundary of that neighborhood
(backward-in-time) can be used to extend the definition of eigenfunctions hkup to
the edge of the basin of attraction or up to the end of the interval of existence of
trajectories, therefore proving an extension of the Hartman–Grobman theorem to the
entire basin of attraction. By proving analogous results for discrete-time dynamics,
linearization of periodic and periodically-forced dynamical systems follows.
In [33], this idea is taken further, to construct the conjugacy h(·) between two
nonlinear systems that are known to be topologically conjugate. Explicit construction
of conjugacies is generally a difficult task that is often eschewed when existence of
conjugacy can be inferred from other properties of the system. It is demonstrated that
while for one-dimensional systems the construction is straightforward, conjugacies on
2Restricting the domain to a strict subset of X1,2leads to local conjugacy. Requiring higher
regularity (differentiability, smoothness, analyticity) of hand h−1, or extending hto also convert
between the time domains of two systems leads to related concepts in the theory of differential
equations and dynamical systems covered by standard textbooks, such as [350,270,229].

32 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
28 A. Mauroy et al. / Physica D 261 (2013) 19–30
Fig. 8. For a fixed point with a complex eigenvalue 1, the isostables (black curves)
and the isochrons (red curves) of the fixed point are the level sets of |s1(x)|and
6s1(x), respectively. In the vicinity of the fixed point, the isostables are ellipses and
the isochrons are straight lines. (The numerical computations are performed for the
FitzHugh–Nagumo model, with the parameters considered in Section 4.1.2; the blue
dot represents the fixed point.) (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
determined by their topological properties: they define the unique
periodic partition of the state space (of period T1). In contrast, more
care was needed to define the isostables as the level sets of the
unique smooth Koopman eigenfunction s1.
Isostables and isochrons appear to be two different but
complementary notions. On one hand, the isostables are related
to the stability property of the system and provide information
on how fast the trajectories converge toward the attractor. On the
other hand, the isochrons are related to a notion of phase and
provide information on the asymptotic behavior of the trajectories
on the attractor. Given (11), the isostables are related to the
property
d
dt |s1(t(x))|=1|s1(t(x))|(22)
while the isochrons are characterized by
d
dt
6s1(t(x)) =!1.(23)
In the case of fixed points, it is clear that the isochrons are
not relevant to characterize the synchronous convergence of the
trajectories, a fact that stresses the importance of considering the
isostables instead.
5.2. Action–angle coordinates and global linearization
For a two-dimensional dynamical system which admits a
spiral sink (i.e. with two complex eigenvalues), the two families
of isostables and isochrons provide an action–angle coordinates
representation of the dynamics. More precisely, (22) and (23)
imply that, with the variables (r,✓)=(|s1(x)|,6s1(x)), the system
is characterized by the (action–angle) dynamics
˙
r=1r
˙
✓=!1
in the basin of attraction of the fixed point. For systems of higher
dimension, the action–angle dynamics are obtained with several
Koopman eigenfunctions, i.e. (rj,✓j)=(|sj(x)|,6sj(x)) leads to
˙
rj=jrj,˙
✓j=!j. Note that this was also shown in Section 2.1.2 in
the case of linear systems with 162 R.
When expressed in the action–angle coordinates, the dynamics
become linear. This is in agreement with the recent work [20]
showing that a coordinate system which linearizes the dynamics is
naturally provided by the eigenfunctions of the Koopman operator
Fig. 9. The coordinates z1(black curves) and z2(red curves) correspond to Cartesian
coordinates in the vicinity of the fixed point but are deformed when far from the
fixed point. (The numerical computations are performed for the FitzHugh–Nagumo
model, with the parameters considered in Section 4.1.2; the blue dot represents the
fixed point.) (For interpretation of the references to color in this figure legend, the
reader is referred to the web version of this article.)
(see also the Appendix). Namely, in the new variables yj=sj(x),
the system dynamics are given by
d
dt 0
B
@
y1
.
.
.
yn
1
C
A=0
B
@
10
...
02
1
C
A0
B
@
y1
.
.
.
yn
1
C
A.
Moreover, the linear change of coordinates
0
B
@
z1
.
.
.
zn
1
C
A=V0
B
@
y1
.
.
.
yn
1
C
A,(24)
where the columns of Vare the eigenvectors vjof the Jacobian
matrix Jat the fixed point, leads to the linear dynamics
d
dt 0
B
@
z1
.
.
.
zn
1
C
A=J0
B
@
z1
.
.
.
z2
1
C
A.
For the two-dimensional FitzHugh–Nagumo model, the coordi-
nates (z1,z2)are represented in Fig. 9 and are equivalent to the ac-
tion–angle coordinates (r,✓)(Fig. 8). They correspond to Cartesian
coordinates in the vicinity of the fixed point, where the linearized
dynamics are a good approximation of the nonlinear dynamics (see
also (A.3) in the Appendix). But owing to the nonlinearity, the co-
ordinates are deformed as their distance from the fixed point in-
creases. The comparison between these coordinates and regular
Cartesian coordinates therefore appears as a measure of the system
nonlinearity.
In the case of two-dimensional systems with a stable spiral sink,
the derivation of action–angle coordinates and the global lineariza-
tion are obtained through the isostables and the isochrons, that
is, with only the first Koopman eigenfunction s1(x). For higher-
dimensional systems (or two-dimensional systems with a sink
node), global linearization involves several Koopman eigenfunc-
tions sj(x)(see [20] for a detailed study), which can be obtained
through the generalized Laplace averages (see Remark 3). In the
context of model reduction, or when the dynamics are significantly
slow in one particular direction, the first eigenfunction—related to
the isostable—is however sufficient to retain the main information
on the system behavior.
Fig. 4.1: Isostables and isochrons for the FitzHugh–Nagumo model acting as a de-
formed rectifiable coordinate system in vicinity of a focus (left) and node (right).
Reproduced with permission, from Mauroy et al. 2013 Physica D [225].
state spaces in Rnrequire a more delicate approach, especially when dealing with
eigenvalues with nontrivial geometric multiplicity.
In the setting of model/order reduction, relations (4.1) and (4.2) are required to
hold for non-invertible maps or on closed (or even singular) subsets of the domain.
In this case the full orbit structure of systems is not typically qualitatively the same;
however, much can be gained by studying a simpler, often significantly-lower dimen-
sional, system, and transporting its properties onto the original dynamics.
In this context, Mauroy et al. [222,225] demonstrate that the coordinate trans-
formations hkcan be numerically computed by forward integration of a trajectory
and a Laplace average of an observable
(4.3) ˜gλ(x):= lim
T→∞
1
TZT
0
gt(x)e¯
λtdt,
which is an extension of the harmonic Fourier average (2.30). The computed isostables
(level sets of the absolute value of eigenfunctions) and isochrons (level sets of the
arguments of complex eigenfunctions) act either as rectifiable Cartesian coordinate
systems for node equilibria, or as rectifiable polar (action-angle) coordinate systems
for focus-type equilibria, as shown in Figure 4.1, with clear generalizations to higher
dimensional systems.
Further theoretical developments have led to extensions to nonlinear stability
analysis and optimal control of dynamical systems [310,221,224,309,223]. Papers
[354,357] apply this concept to synchronization of oscillators by extending the phase-
response curves using isostables and isochrons computed as Koopman eigenfunctions.
Notably, these developments demonstrate that Koopman eigenfunctions are a viable
and practical path both in analytic [356] and in data-driven approaches [355] to syn-
chronization of oscillators.
4.2. Phase portrait and symmetries. Discussions of Hartman–Grobman the-
ory and conjugacies are typically concerned with the behavior of two dynamical sys-
tems in the vicinity of an object of interest, such as a basin of attraction/repulsion
of a fixed point or periodic orbit. Here we describe how Koopman eigenfunctions can
reflect the structure of the entire phase portrait of the dynamical system.
The phase portrait of a dynamical system on state space Xis a collection of all
orbits o(x) = {Ft(x)}t∈Remanating from each initial condition xby the flow map Ft.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 33
(a) (b)
Fig. 4.2: (a) Approximate ergodic partition of the Chirikov Standard Map. Repro-
duced with permission, from Levnaji´c and Mezi´c 2010 Chaos [192]. (b) Sketch of
ordering of orbits by ergodic quotient trajectories. Reproduced with permission, from
Budiˇsi´c and I. Mezi´c 2012 Physica D [55].
It is immediately clear that two initial conditions y,zon the same orbit o(x) generate
the same orbit, o(x) = o(y) = o(z). However, computationally pointwise equality
between orbits may be challenging to verify when orbits are merely approximated
by numerical integration, and when orbits contain thousands of points. Instead, it
may be practical to verify that orbital averages of a subset of functions G(X) are
sufficiently close, resp. distant, to declare that two collections of points belong, resp.
do not belong, to the same orbit.
As mentioned in subsection 2.2, the orbital average, or ergodic average, of any
observable g∈ G(X) projects the observable on the invariant eigenspace of the Koop-
man operator. Therefore the heuristic technique for comparing orbits just described
can be interpreted by comparing whether points yand zare mapped to the same
point by a vector valued function P:X → Rp,p≤ ∞ whose components are all
invariant Koopman eigenfunctions:
(4.4) Q(x) = q1(x)q2(x). . . qp(x),where Kqj≡qj.
Mezi´c [235] showed that when the number of averaged functions grows to infinity, this
procedure manages to separate ergodic sets that are measure-theoretic counterparts to
orbits. This approach was applied to visualize the phase portrait [235,192,55], where
it is possible to assign pseudocolors to trajectories, generating visually appealing and
detailed figures as in Figure 4.2a.
Using the embedding function (4.4) it is further possible to treat the ergodic
quotient as a geometric object and compute its local coordinates. Such coordinates
parametrize the space of invariant (ergodic) sets; when continuous, they act as a
way of ordering level sets of conserved quantities, even when no explicit (or global)
formulas for conserved quantities is known. The crucial step is to treat the elements
qi(x) in (4.4) as Fourier coefficients and define a metric using a Sobolev norm on
the associated space. The resulting geometric space can then be coordinatized using
manifold learning algorithms.
The ergodic partition is a process of classifying global information, assuming ac-
cess to many trajectories, into how they relate locally. The opposite direction, where
local information is stitched into a global picture is equally as relevant. Consider two
disjoint invariant sets A, B ⊂ X of the dynamics Ft:X → X. A typical application

34 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
of DMD starts from a single trajectory and constructs a finite-dimensional approxi-
mation of the Koopman operator. Depending on whether the initial condition is in
Aor in B, the resulting operator is indistinguishable from the restricted Koopman
operator Ft|A:A→Aor Ft|B:B→B. The associated Koopman operators KA
and KBseemingly have nothing in common. Indeed, it was typical of early papers to
either assume that the system has a quasiperiodic attractor, so that all trajectories
quickly converge to this set, or to assume that the trajectory used for DMD is ergodic,
so that it visits close to any other point in the state space, ruling out the existence of
disjoint invariant sets A, B of positive measure.
If DMD is indeed computed twice, based on trajectories contained in disjoint
ergodic sets, it is possible to “stitch” the two operators KA,KBtogether by placing
them into a block-diagonal stitched Koopman operator. Assuming that the respective
spaces of observables are L2(A, µA), L2(B, µB), the joint space over X=A∪Bcan
be taken as
(4.5) G(X) = L2(A∪B, µA+µB) = L2(A, µA)⊕L2(B , µB),
since any function f∈ G(X) can be decomposed into disjoint components owing to
A, B being disjoint. Reference [245] gives further theoretical backing to this process, as
well an incremental and a non-incremental version of the data-driven DMD procedure
built in this fashion. For practical purposes, the assumption that Aand Bare truly
dynamically separate is not needed; rather, it is simply sufficient to choose pairs of
DMD trajectory samples in a trajectory from two disjoint sets in order to construct
such restricted approximations that can be stitched [304].
A particularly important application of stitching across invariant sets concerns
phase spaces of systems with symmetries. When the orbit structure is symmetric with
respect to a transformation, the analysis of the entire phase space can be dramatically
simplified. Symmetry of the differential equations (1.1) with respect to a symmetry
group Γ, or Γ-equivariance, is defined as a conjugacy
(4.6) f(γx) = γf(x),∀γ∈Γ,
where γrepresent action by a group element on the state space. An analogous re-
lationship holds for discrete dynamics (1.4). In both cases, the implication is that
given any orbit {x(t)}t∈R, there exist symmetry-related counterparts {γx(t)}t∈Rgen-
erated by applying any γ∈Γ to the orbit. Stability and asymptotic properties of the
symmetry-related orbits are the same, which allows us to study just a portion of the
state space in detail, and then export those results to other parts of the state space
by symmetry.
Although symmetry-based arguments have long been used to simplify dynami-
cal systems, especially in classification of bifurcations, explicit connections with the
Koopman operator framework are fairly recent [155,287,230,304]. In all cases, the
central role is played by connections between the definition of the Koopman opera-
tor and the conjugacy in the definition of equivariance. The following two theorems
appear as Thm. III.1 and its corrolary in [287], and Thm. 1 and Prop. 2 in [304].
Theorem 4.1. For a Γ-equivariant dynamical system, Kcommutes with the ac-
tion of all γ∈Γfor any observable g∈ G(X)
(4.7) [γ◦(Kg)](x)=[K(γ◦g)](x).
Proposition 4.2. Any eigenspace of Kfor a Γ-equivariant dynamical system is
Γ-invariant.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 35
Fig. 4.3: Invariant sets for a dynamical system with Z2×Z2symmetry whose Hamil-
tonian is H(p, q) = p4/4−9p2/2−q4/4+9q2/2. Reproduced with permission, from
Sinha et al. [304].
Proofs of both statements follow by manipulation of the definitions of an eigenfunction
(2.12) and equivariance (4.6), and we omit them here. For cyclic groups, i.e., finite
groups Zn={1, γ, γ 2,··· , γn−1}, a more detailed argument is given in [230], in
particular, stating that Koopman modes associated with a particular eigenvalue are
similarly symmetric.
The symmetry arguments fit well with the idea of “stitching” the global Koopman
operator from “local” Koopman operators, because symmetry removes the need for
simulating additional trajectories in order to explore various invariant sets. As [304]
show, if two disjoint invariant sets A, B ⊂ X are related by a linear group action τγ,
i.e., A=τγB, the corresponding Koopman matrices KA,KBare conjugate (similar)
(4.8) KA=T−1
γKBTγ
where the conjugating matrix Tis the group action on the space of observables.
Assuming it is possible to construct such a matrix, KAcan be computed without the
additional simulation of trajectories, once the DMD matrix KBhas been computed.
Even in the case that the full Koopman operator approximation may be directly
computed, converting it to a block-diagonal form simplifies the task of computing
its spectral properties as eigendecomposition may be performed on individual blocks
instead of the entire operator. Commutativity of two linear operators, the Koop-
man operator and its matrix representation, implies that they preserve each others
eigenspaces, further implying that even when the space of observables G(X) is not
constructed as a direct product (4.5), it is possible to perform a change of basis based
on the symmetry group such that G(X) decomposes into so-called isotypic subspaces,
invariant with respect to K. Consequently, this block-diagonalizes a finite-dimensional
representation of the Koopman operator. In [287], this idea is taken as a starting point
for a practical block-diagonalization of matrices appearing in DMD.
An example of isotypic decomposition of functions g∈ G(X) associated with the

36 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Fig. 4.4: Sparsity structure of DMD matrices before (left) and after (right) block-
diagonalization procedure applied to a coupled Duffing oscillator system with Z2×D3
symmetry. Reproduced with permission, from Salova et al. 2019 Chaos [287].
group Z2, where γ(x) = −x, is a decomposition into odd and even components
(4.9) g(x) = g(x) + g(−x)
2
| {z }
=:ge(x)
+g(x)−g(−x)
2
| {z }
=:go(x)
Generalization of this procedure to arbitrary finite groups is used as a precon-
ditioning step in a modified eDMD (see subsection 5.1) procedure in [287], resulting
in a block-diagonalized form of the associated DMD matrix that approximates the
Koopman operator. Consequently, this reduces the computational effort needed to
compute eigenvalues of the Koopman operator.
4.3. Adjoint: The Perron–Frobenius operator. The Perron–Frobenius (PF)
operator, also known as the Ruelle transfer operator, is a linear representation of non-
linear dynamics that traces its roots to the mathematics underpinning statistical and
quantum physics, paralleling the development of the Koopman operator. Instead of
evolving measurement functions (observables) taking values from the domain of the
dynamics, as the Koopman operator does, the PF operator evolves measures (distribu-
tions) supported on the domain of the dynamics. As the PF and Koopman operators
can be shown to be formally adjoint in appropriately defined function spaces, we sum-
marize the basic concepts related to the PF operator here. The fundamentals are well
documented in textbooks and monographs [187,34,115,78] and we point to them for
a more precise and general introduction of these topics.
Let the domain of dynamics Xbe given the structure of a Borel measurable space.
Given a probability measure µ, and any measurable set A, define the Perron–Frobenius
operator as
(4.10) Ptµ(A):=µ(F−t(A)),
where F−t(A):={x∈ X :Ft(x)∈A}is the pre-image of Athrough the dynamics.
Similar to the Koopman operator, the family Ptforms a monoid. An alternative
formulation by Lasota and Mackey [187] replaces the action on the space of probability
measures with an action on a function space. This assumes that the flow map Ft
is nonsingular with respect to some ground measure m, e.g., a Lebesgue measure,
meaning
(4.11) m(A)6= 0 =⇒m(F−t(A)) 6= 0.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 37
Interpreting g∈L1(X, dm) as densities of measures, i.e., dµ =gdm, it is possible to
define the PF operator Pt:L1(X, dm)→L1(X, dm) as
(4.12) ZAPtg(x)dm =ZF−t(A)
g(x)dm,
for any Borel set A. If the flow map is additionally smooth, this definition is equivalent
to
(4.13) Ptg(x) = ZF−t(x)
g(s)
|∇Ft(s)|dm(s),
where |∇Ft|indicates the determinant of the Jacobian matrix of derivatives of the
flow map.
The two formulations (4.10) and (4.12) are connected by interpreting g∈L1as
densities defining probability measures absolutely continuous with respect to dm, i.e.,
dµ =gdm.
Assuming that the dynamics can be restricted to a space of finite measure, it
holds that L∞(X, dm)⊂L2(X, dm)⊂L1(X, dm). In this setting, the Koopman and
PF operators can be defined, respectively, on L∞(X, dm) and L1(X, dm), or both on
L2(X, dm), and one can show that they are adjoint to each other:
(4.14) Ptf, g=f, Ktg,where hf , gi:=ZX
¯
f(x)g(x)dm.
Since the proof [187,§3.3] proceeds by the standard argument of approximation by
simple functions, i.e., linear combinations of indicator functions, this relationship ex-
tends to a wider range of spaces. Adjoint operators have the same spectrum, although
their eigenfunctions do differ, as is the case in general for eigenvectors of matrices and
their adjoints (transposes). This connection is partially responsible for the parallel
development of Koopman and PF techniques in various contexts of applied mathe-
matics.
The Perron–Frobenius operator lends itself to a particularly straightforward ap-
proximation by a stochastic matrix, using a technique termed Ulam’s method after
a conjecture by S. Ulam [343,§IV.4], ultimately proved to be correct by Y. Li [195].
First, assume that the flow Ftpreserves a finite probability measure mon a bounded
set X, partition Xinto subsets {Si}, each with the associated charateristic function
χi. Choosing a fixed time T, we then form the stochastic Ulam matrix
(4.15) U= (uij ), uij :=m(F−T(Si)∩Sj).
Entries uij can be approximately computed by seeding a random (large) collection of
initial values in each set Sjand evolving by FT. The proportion of resulting endpoints
of trajectories that land in a set Siis entered in uij .
The described process amounts to a Monte Carlo integration of the integral
m(F−T(Si)∩Sj) = RSjχ(F−T(Si))dm(x) [86,85], or more generally, a procedure
to compute a Galerkin approximation of the PF operator using piecewise constant
functions [87]. A computationally efficient implementation as a code GAIO [84] was
shown to be able to approximate the eigenvalue spectrum of the PF operator and its
eigenfunctions [87] both in the L2-space and in fractional Sobolev spaces [107].
Eigenfunctions of the PF operator correspond to complex-valued distributions of
points in state space that evolve according to the associated eigenvalue
(4.16) Ptρ=λtρ.

38 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
When λ= 1, ρare invariant densities, which can be used to estimate the sets contain-
ing dynamical attractors, Sinai–Ruelle–Bowen measures, and partition the dynamics
into invariant sets. Even away from the limit of the Galerkin approximation, the
Ulam matrix Ucan be interpreted as a Markov chain transition matrix on a directed
graph, which allows for a definition of almost-invariant sets in the state space [106]
and a variational formulation of the problem of locating invariant sets in the state
space. Eigenfunctions for eigenvalues with |λ| 6= 1 are associated with the mixing
properties and escape rates in the state space of dynamical systems [109].
The infinitesimal generator for PF is the Liouville operator A, defined analogously
to (2.7), which satisfies
(4.17) Pt=etA.
Similar to the Lie generator of the Koopman operator (2.10), when dynamics are
specified using a velocity field ˙
x=f(x) the Liouville operator can be shown to satisfy
(4.18) Aρ=−div(ρf),
further leading to a partial differential equations that eigenfunctions of the Liouville
operator and PF operator must satisfy
(4.19) µρ + div(ρf)=0,
whenever (4.16) is satisfied with λ=eµ∈C/{0}.
Approximating Ainstead of Pleads to so-called simulation-free numerical tech-
niques that can be interpreted either as finite-volume methods for the advection of
PDEs, or as spectral collocation methods [108,32].
Invariant eigenfunctions of both the Koopman and Perron–Frobenius operators
have been used to extract invariant sets in the state space of dynamical systems.
In function spaces where these operators are dual, eigenfunctions of both operators
theoretically contain the same information. However, in reality, the choice may be
made based on practical constraints. For example, approximation of invariant sets via
Koopman eigenfunctions in [55,192,193] relies on long-duration trajectories, while
Ulam’s approximation of PF typically requires short bursts of trajectories but seeded
densely in the domain.
4.4. Spectrum beyond eigenvalues. Spectral characterization of infinite di-
mensional operators requires a separate treatment of two concepts: of the spectrum,
which generalizes the set of eigenvalues of finite dimensional operators, and of in-
tegration against spectral measures, which takes the role of the eigenvalue-weighted
summation appearing in the spectral decomposition theorem for normal matrices.
Standard textbooks on functional analysis commonly provide an introductory treat-
ment of these concepts; however, among them we highlight [282,283,189], which
include examples relating to the Koopman operator.
For operators T:G → G acting on a finite-dimensional space G, the spectrum
σ(T) is synonymous with the set of eigenvalues λ, that is scalars λsuch that
(4.20) Tξ=λξ,or (T−λI)ξ= 0,
for some vector ξ∈ G, termed the eigenvector. To extend the concept of a spectrum
to operators on Banach spaces, we interpret (4.20) as a statement that eigenvalues
λare those scalars for which the operator T−λIdoes not have a bounded inverse.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 39
The spectrum σ(T) can be further classified into subsets based on the reason for
why (T−λI)−1(the resolvent ) fails to exist as a bounded operator. For λ∈σp(T)
(the point spectrum)T−λIis non-injective; this coincides with eigenvalues and is
equivalent to a finite-dimensional spectrum. For λ∈σc(T) (the continuous spectrum)
the range of T−λIis not the whole codomain (non-surjective), however it is dense in
the codomain. For λ∈σr(T) (the residual spectrum) the range of T−λIis not the
whole codomain (non-surjective), and it is not even dense in the codomain. There are
standard examples of Koopman operators that have continuous and residual spectra,
e.g., those equivalent to multiplication operators and shift operators.
The spectral decomposition theorem for normal matrices, which have orthogonal
eigenvectors, states that the action of the matrix can be represented by the decom-
position
(4.21) Tng=X
λ∈σ(T)
λnξλhξλ,gi.
The operators ξλhξλ,·i are orthogonal spectral projections onto eigenvectors ξλ. Gen-
eralizing (4.21) to (infinite dimensional) Koopman operators requires that the opera-
tors are normal, which holds for certain dynamical systems. For example, when the
flow Fis invertible and preserves a finite measure µ, e.g., when the associated velocity
field is divergence-free on a bounded domain, working with the Hilbert space Hof
square-integrable observables g∈ H =L2(X, µ) results in a unitary, and therefore
normal, Koopman operator [249]. Then, the classical spectral resolution theorem (due
to Hilbert, Stone, and Hellinger) applies to K[234,231,189] as
(4.22) Kng=Zπ
−π
einωd[E(ω)g] = X
k
einωkPkg
| {z }
atomic
+Zπ
−π
einωd[Ec(ω)g]
| {z }
continuous
,
where the operator-valued spectral measure Eforms a partition of unity and can be
separated into atomic projections Pkand the continuous part Ec. This setting covers
a wide-range of steady-state dynamics [96,187,76].
The atomic part of the spectral measure E(ω) is supported on frequencies ωkthat
yield eigenvalues eiωkof Kand are associated with regular dynamics. As for matrices,
when the eigenvalues are simple, the eigenspace projections Pkcan be written using
eigenfunctions ϕkof Kas
(4.23) Pkg=hg, ϕkiϕk.
The atomic part of the spectral decomposition therefore aligns with the framework
described in section 2.
There is no counterpart to the continuous spectral measure Ec(ω) in finite-dimensional
settings; therefore, to interpret Ec(ω) and its connections to the dynamics requires
routes that do not involve eigenvalues. In the remainder of this section we summa-
rize (a) how existence/non-existence of Ec(ω) is connected to asymptotic statistical
properties of the dynamics, (b) how local structure of Ec(ω) connects to evolution of
Koopman on subspaces of observables, and (c) how to modify the space of observables
in order to convert the continuous spectrum into a continuum of eigenvalues.
4.4.1. Global properties. An operator-valued measure Ec(ω) can be converted
to a “plain” scalar measure by choosing a function (observable) g∈L2(X, µ) and

40 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
studying its autocorrelation sequence hKng, gi. Applying (4.22) here yields
(4.24) hKng, gi=Zπ
−π
einωdhE(ω)g , gi
| {z }
=:σg(ω)
,
where σg(ω) is the Fourier or spectral measure for the evolution of g.
If the space L2(X, µ) is defined with respect to an ergodic measure µ, the auto-
correlation function of the observable can be computed along trajectory initialized at
any x
(4.25) Cg(n):= lim
K→∞
1
K
K−1
X
k=0
g(xk+n)g(xk) = lim
K→∞
1
K
K−1
X
k=0
[Kng](xk)g(xk)
and is directly related to the Fourier coefficients of σg
Zπ
−π
einωdσg(ω) = hKng, gi= lim
K→∞
1
K
K−1
X
k=0
[Kng](xk)g(xk) = Cg(n).(4.26)
In other words, the Fourier measure σg(ω) is the Fourier transform of the autocorrela-
tion function Cg(n), which allows for characterization of irregular dynamics in terms
of the measure σg[171].
In general E, and therefore some σg, contain all three components that are mu-
tually singular (e.g., the Lebesgue decomposition of a measure [314]):
•atomic component, supported on frequencies of eigenvalues;
•absolutely continuous component, having a spectral density and correspond-
ing to mixing (stochastic-like) behavior;
•singularly continuous component, having a fractal structure.
If σgis absolutely continuous, it has a spectral density (the Radon–Nikodym deriva-
tive) and then Cg(n)→0, so the time-separated samples g(xk) will be asymptotically
equivalent to independent random variables. If the same holds for all non-constant
observables g∈1⊥, the dynamics is mixing. If σghas neither atomic components
nor a spectral density, it is singularly continuous. In this case, samples in the time
trace are correlated infinitely often no matter their separation in time, but the corre-
lation occurs rarely enough that on average they appear uncorrelated. In this case,
the measure can be thought of as having a fractal structure as its support is nei-
ther a full interval, nor a collection of discrete points. If this holds for all g∈1⊥
then the dynamics is weakly mixing; this is the signature behavior of the so-called
weakly-anomalous transport [364,363].
4.4.2. Local properties. To interpret how the local variation of σg(ω) is associ-
ated with the dynamics, we investigate integrals over spectral intervals [a, b]⊂[−π, π]
Rb
aeinωdσg(ω). It can be shown that such restrictions on the spectral domain are
equivalent to restrictions of Kto certain subspaces of observables [280, Prop. 2.7].
More precisely, there exists an orthogonal pro jection P[a,b]such that the following
equality holds
(4.27) Zb
a
einωdσg(ω) = Zπ
−π
einω1[a,b]dσg(ω) = hKnP[a,b]g, P[a,b]gi.
The range of the projection P[a,b]is the invariant subspace
(4.28) H[a,b]={g∈L2(X, dµ): σg(T/[a, b]) = 0}.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 41
M. Korda et al. / Appl. Comput. Harmon. Anal. 48 (2020) 599–629 625
Fig. 11. Lorenz system – N= 100. Left: density approximation with the CD kernel. Middle: distribution function approximation.
Right: Singularity indicator ∆Ndefined in (37 ).
Fig. 12. Lorenz system – Observable f(x) =x3. Left: approximation of the atomic part ζN/(N+1). Middle: approximation of the
density ζN. Right: singularity indicator ∆N.
with an almost-periodic motion of the x3component during the time that the state resides in either of
the two lobes, with switches between the lobes occurring in a chaotic manner. In Fig. 13 we depict the
approximation of the spectral projection P[a,b)f(see Section 5.2) with [a, b) =[0.24 , 0.28] and f(x) =x3,
i.e., we are projecting on a small interval around the peak in the spectrum of x3. This function will evolve
almost linearly with frequency of the peak, i.e., (P[a,b)f)(x(t +τ)) ≈eiωτ (P[a,b)f)(x(t)) with ω≈8.17 rad/s.
626 M. Korda et al. / Appl. Comput. Harmon. Anal. 48 (2020) 599–629
Fig. 13. Lorenz system – Approximation of the spectral projection P[a,b)fwith f(x) =x3and [a, b] =[0.24 , 0.28] and N= 100,
M=10
5.
7.3. Cavity flow
In this example we study the 2-D model of a lid-driven cavity flow; see [2]for a detailed description of the
example and the data generating process. As in [2], the goal is to document the changes in the spectrum of the
Koopman operator with increasing Reynolds number which are manifestations of the underlying bifurcations,
going from periodic through quasi-periodic to fully chaotic behavior. For each Reynolds number, the data
available to us is in the form of the so called stream function of the flow evaluated on a uniform grid
of points in the 2-D domain with equidistant temporal sampling. This leaves us with a very large choice
of observables since the value of the stream function at any of the grid points (as well as any nonlinear
function of the values of the stream function) is a candidate observable. In general, one wishes to choose
the observable fsuch that its spectral content is as rich as possible, preferably such that fis ∗-cyclic (see
Eq. (9)), which is, however difficult to test numerically. For example, for Re = 13 ·10 3, exhibit periodic
behavior with a single (or very dominant) harmonic component and hence might not contain the full spectral
content of the operator (i.e., fis not ∗-cyclic). Therefore, for each value of the Reynolds number we chose
as the observable the stream function at a grid point where the time evolution is complex and hence the
spectral content of this observable is likely to be rich. A more careful numerical study, such as the one
carried out in [2], should analyze a whole range of observables (perhaps the values of the stream function
at all grid points). However, here, already one suitably chosen observable allows us to draw interesting
conclusions on the behavior of the spectrum of the operator as a function of the Reynolds number. The
point spectrum approximation results ζN/(N+1)are depicted in Fig. 14 . Since the observable fis real,
the spectrum is symmetric around the point θ=0.5and hence we depict it only for θ∈[0 , 0.5]; in addition,
we change coordinates from θto ω=2πθ/Ts, where Ts=0.5 sis the sampling period. Finally, in order to
better discern very small atoms, we also show the point spectrum approximation on a logarithmic scale.
Based on Theorem 1, whether or not there is an atom at a given frequency ω, can be assessed based on
the proximity of the values of ζN/(N+1) for two different N: When there is an atom, we expect the two
values to be closed to each other; otherwise we expect the value of ζN/(N+ 1) to be significantly smaller
since in that case ζN/(N+ 1) →0. Fig. 14 suggests that there is a very strong atomic component of the
spectrum for Re = 13 ·10 3and Re =16 ·10 3and even for Re =19 ·10 3as the atomic part accounts for
at least 80 % of the energy of the given observable (i.e., 80 % of the mass of µf). This is confirmed by the
approximations of the distribution function which are piecewise constant for these values of the Reynolds
number. For Re =30 ·10 3, on the other hand, the spectrum appears to be purely continuous. In order to
assess whether the spectrum is purely absolutely continuous or has a singular continuous part, we also plot
Fig. 4.5: Approximation to the density of spectral measure for the Lorenz’63 model
(left) and the real part of the projection of the evolution onto the cyclic vector of
the subspace associated with spectral interval [0.24,0.28] (right). Reproduced with
permission, from Korda et al. 2020 Appl. Comput. Harmon. Anal. [175].
In other words, localizing the spectral integral results in a compression of Ktto some
dynamically-invariant subspace of the space of observables. This holds more generally
not only for intervals, but for arbitrary measurable sets in the spectral domain.
In applications, the selection of observables often comes before the analysis of dy-
namics. If the chosen observable happens to already be in some subspace H[a,b], the re-
stricted integral would be equivalent to its full evolution hKng, gi=hKnP[a,b]g, P[a,b]gi.
In other words, if one would try to infer the “full” evolution of the Kfrom a single ob-
servable, an unintentional choice of the observable gfrom an invariant subspace H[a,b]
may result that instead of the full operator, K, only its compression P>
[a,b]KP[a,b]may
be reconstructed. On the other hand, there exists a subspace of observables for which
associated Fourier measures are “maximal”, in the sense that any zero set for a max-
imal measure, i.e. spectral “bandwidth” exhibiting no dynamics, is a zero set for
any other Fourier measure. This implies that a judicious choice of observable can
make it possible to fully characterize statistical properties of the system from the
autocorrelation timeseries of a single observable.
If the content for many spectral intervals [a, b]⊂Tis of interest, we may want to
approximate the weak derivative dσg/dθ and visualize it. In [175] the Fourier coeffi-
cients (or trigonometric moments), computed by ergodic averages (4.26), are used to
formulate the moment problem for the density dσg/dθ, which is solved using a varia-
tional approach based on the Christoffel–Darboux kernel. Based on the approximated
density, for any given interval [a, b] in the spectral domain one can construct a type
of an ergodic average that computes the associated projection P[a,b]of the evolution,
resulting in the analog of eigenvectors for the continuous spectral measure. Construc-
tions involved relate to the HAVOK algorithm [47,14] (see subsection 5.2) due to the
correlation function that connects time-delayed samples of the observable. Figure 4.5
demonstrates this approach on an example of the Lorenz’63 dynamical system that
is known to be mixing on the attractor [204] by computing its spectral density and a
spectral projection for a spectral interval containing significant dynamical content.
4.4.3. Removing the continuous spectrum by “rigging”. To illustrate
that existence of the continuous spectrum (as defined by non-surjectivity of K−λI) is
not necessarily connected with irregular behavior, [233] studies in detail the pendulum
(4.29) ˙α=v, ˙v=−sin θ

42 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
which can be converted to action-angle variables with I≥0 representing the conserved
quantity, and a periodic variable θ∈S1the momentum,
(4.30) ˙
I= 0,˙
θ=I.
The exact solution is
(4.31) I(t) = I0, θ(t) = I0t+θ0(mod 2π).
Any observable g:R+×S1→Rwithout dependence on the angle coordinate g(I, θ) = g(I)
is clearly an eigenfunction of Ktwith λ= 1, since
(4.32) Ktg(I0) = g(I(t)) = g(I0),
implying that the eigenvalue λ= 1 has an infinite multiplicity. This rules out ergod-
icity, and therefore mixing, with respect to the L2space of observables defined over
any annulus [I1, I2]×S1of positive area. In other words, the atomic spectrum detects
regular behavior of dynamics.
At the same time, no function with a variation in the angular direction is an
eigenfunction, as the only translation-invariant function on S1is a constant, despite
all trajectories being clearly periodic. However, if observables are taken from a space
that includes generalized functions (distributions), then a Dirac-δsupported on a
single level set of I
(4.33) gc(I, θ) = δ(I−c)eiθ
would indeed be a (generalized) eigenfunction as
(4.34) Ktgc(I, θ) = δ(I−c)eiIt+θ=eict δ(I−c)eiθ =eictgc(I , θ).
The example above illustrates how a rigged3, or equipped Hilbert space can con-
vert a continuous spectrum (containing no eigenvalues) to a continuum of eigenval-
ues [12,315,334,210]. Instead of a single L2space of observables, the approach
employs a so-called Gelfand triple Γ ⊂L2⊂Γ†, where “the rigging” consists of Γ, a
subset of judiciously-chosen test functions, and Γ†its dual. In the example above, Γ†
contains generalized functions (distributions). By enlarging the domain of the Koop-
man operator, surjectivity of (K − λI) can be mostly restored, resulting in shrinking
of the continuous spectrum to a set of discrete values [306].
As a result, the continuous projection measure in the Hellinger–Stone spectral
theorem (4.22) can be replaced by an integral against eigenvector-based projections,
analogous to (4.21)
(4.35) hρ, Kgi=X
λ
λhρ, ψλihϕλ, gi,
for an observable g∈Γ and a density ρ∈Γ†, while ϕλ, ψλare elements of a biorthonor-
mal basis in L2. The extended spectrum λnow contains both the L2-eigenvalues of
Kand Ruelle–Pollicott resonances [286,273] associated with infinitesimal stochastic
perturbations of the Koopman operator [67,68]. For example, while the map x7→ 2x
on S1has only a constant eigenfunction and only the eigenvalue at 1, a representation
3This metaphor is intended to evoke a utilitarian rigging of a ship, rather than a nefarious rigging
of a match.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 43
in the rigged Hilbert space where Γ are analytic functions yields functions ϕk, ψkthat
are related to Bernoulli polynomials, and values λk= 2−k[11].
While the rigged Hilbert space framework has existed since the 1950s and Gelfand,
the approach has been used to analyze the Koopman and Perron–Frobenius operators
only since the 1990s, and for quantum theory a decade later [209,210]. Only in
the past two years have the modern numerical approaches such as DMD, started to
connect to this theory [233,306], so we expect the growth of interest in this area in
the coming years.
4.5. Koopman operators for nonautonomous and stochastic dynamics.
The theory developed so far was based on the time-independent or autonomous dy-
namics (1.1) and (1.4). This clearly does not cover most models used in practice.
Common sources of time variability include changes in system parameters, the pres-
ence of input forcing, stochastic noise, and control or actuation. Time variability
induced by feedback control is highly structured, and thus Koopman theory can be
developed in more detail depending on the structure, as described in section 6. Even
though the original justification for DMD-style algorithms was based on the autono-
mous Koopman framework, the algorithms were applied to data generated by nonau-
tonomous dynamics, either by tacitly assuming that the time variation is negligible,
or by employing various sliding window techniques.
4.5.1. Sliding and multiresolution analysis. Consider the nonautonomous
dynamics
(4.36) ˙
x=F(x, t),x∈ X,
and assume that over a time window t∈[τ , τ +T] the function Fremains approxi-
mately constant
(4.37) F(·, t)≈F(·, τ ).
Furthermore, assume that this holds over a continuous range of starting points τ,
while maintaining a constant window size T.
Asliding window implies that the snapshots of observations generated by data
collected over each time window [τi, τi+T], for some i= 1,2, . . ., are separately
processed by a DMD algorithm to produce eigenvalues λk(τ, T ) and DMD modes
φk(τ, T ) that depend on the parameters of the time window. This approach is neither
new nor unique to Koopman analysis. In signal processing, computing the discrete
Fourier transform over a sliding window goes under several names, including the
Short Time Fourier Transform,spectrogram,Sliding Discrete Fourier Transform, or
Time-Dependent Fourier Transform, and is a standard topic in many signal processing
textbooks [254,313]. If, in addition to the starting point τ, the length of the window T
is systematically varied as well, this is known as multiresolution or multiscale analysis.
The first systematic treatment of multiresolution analysis in the context of DMD
presented the basic sliding strategy of computing DMD in several passes over the same
set of snapshots [180,182]. Within each pass, the window size was kept constant, and
the starting point of the window was moved in non-overlapping fashion. Several
additional strategies for sampling the data and moving the window were discussed,
including connections to classical signal processing techniques, such as the Gabor
transform [182,46]. The overlap between consecutive windows can be exploited to
compute a more accurate global reconstruction [95], and to reduce the computational
effort required to compute the DMD [365]. Sliding window strategies have been found

44 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
useful even in autonomous systems [69]; for example, analytic solutions of autonomous
dynamics that follow heteroclinic connections between an unstable and a stable fixed
point were analyzed, where a well-tuned sliding window is able to discern the difference
in DMD spectra near each of the fixed points [259].
4.5.2. Process formulation for the nonautonomous Koopman operator.
The extension of definitions associated with the Koopman operator follow two paths
for extending the general theory of dynamical systems: the so-called process formu-
lation, or the skew-product formulation.
For an autonomous system, such as ˙
x(t) = Ax(t),x(t0) = x0, the time depen-
dence of the corresponding flow map x07→ x0eA(t−t0)can be written in terms of the
duration t−t0of the time interval over which the dynamics is evolving. The semigroup
property (2.6) captures the consistency between advancing dynamics t→t+ ∆tin
one step, or in two steps t→t+ ∆t/2 and then t+ ∆t/2→t+ ∆t.
To illustrate the process formulation of nonautonomous systems [166,62,207],
consider the simple nonautonomous ODE
˙x(t) = cos(t)x, x(t0) = x0
(4.38)
solved by
x(t) = Ft
t0(x0) = x0esin(t0)−sin(t).(4.39)
The time dependence of the flow map cannot be expressed simply in terms of the
duration t−t0. The consistency of advancing dynamics across adjoining time intervals
is now captured by the cocycle property
(4.40) Ft
t0(x) = Ft
τ(Fτ
t0(x)),∀x,0≤t0≤τ≤t.
Koopman operator can now naturally be formed as a composition operator with
this two-parameter flow map [208]
(4.41) Kt
t0g=g◦Ft
τ.
Its Lie generator, given by
(4.42) Lt0g:= lim
t→t0
Kt
t−t0g−g
t,
itself depends on time t0, in contrast to the autonomous case, and in turn eigen-
values and eigenvectors also depend on time. Such a construction of the Koopman
operator appears to match the sliding-window approaches to DMD discussed earlier.
However, [208] demonstrate that a sliding-window approach may make large mistakes,
especially when the window overlaps the region of rapid change in system parameters;
the same source describes an algorithm that is able to detect the local error of a DMD
approximation and adjust accordingly, which is particularly effective for so-called hy-
brid systems, in which the time dependency of the equations is discontinuous.
A word of caution is needed here; in nonautonomous systems (finite- or infinite-
dimensional) eigenvalues of the system matrix do not always correctly predict the
stability of trajectories. For example, it is possible to formulate a finite-dimensional
dynamical system ˙
x=A(t)xsuch that eigenvalues of A(t) (the generator of the flow)
are time-independent and have negative real value, while the system admits a subspace
of unstable solutions [218,229]. Furthermore, [208] find that methods based on Krylov
subspaces, e.g., snapshot-based DMD algorithms, result in substantial errors in the
real parts of eigenvalues when the time-dependence of eigenvalues is pronounced, and
suggest that the problem can be mitigated by a guided selection of observables.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 45
4.5.3. Skew-product formulation for the nonautonomous Koopman op-
erator. We turn now to the skew-product formulation, which can incorporate both
deterministic and stochastic nonautonomous systems. Consider again the ODE (4.38)
but now introduce an additional periodic variable y∈S1
(4.43) ˙x(t) = cos(y)x,
˙y(t) = 1.
The added variable plays the role of time, re-establishing the semigroup property of
the associated flow map which now acts on the extended space Ft:X ×S1→ X ×S1.
The dynamics of y(t) is itself autonomous and is sometimes referred to as the driver
for the driven dynamics of x(t). The skew-product construction appeared in classical
ergodic theory, for example [76].
General analysis of skew-products does not require that y(t) is as simple as
in (4.43); rather, an extension of (1.4) to the more general skew form
(4.44) xk+1 =F(xk,yk)
yk+1 =G(yk)
can be treated in an analogous fashion. Assuming that the driving dynamics evolves
on a compact state space, or otherwise is measurable with a well-defined invariant mea-
sure, is sufficient to justify computation of eigenfunctions using ergodic averages (2.30)
for time-periodic systems and for systems driven by quasiperiodic dynamics [56,323].
There are two possible formulations of the Koopman operator associated with
(4.44). The first formulation treats the skew-flow as an autonomous system on the
extended state space and acts by composing an observable g∈ G(X × Y) with the
flow
(4.45) [Kg](x,y):=g(F(x,y),G(y)).
Since any particular observable on the original state space h:X 7→ Ccan be trivially
extended to h(x,y) = h(x), this formulation is sufficient for studying dynamics of
a finite collection of observables. However, since the space G(X × Y) is larger than
G(X), representing Kin a particular basis of observables requires working with a
practically larger set, e.g., instead of monomials xk, k = 1,2, . . ., one has to work with
xkyj, k = 1,2, . . . , j = 1,2, . . .. The problem is, of course, more acute the higher the
dimension of Y.
4.5.4. Stochastic Koopman operator. An alternative formulation of the Koop-
man operator for (4.44) acts on the observables in the original state variable only,
G(X), but retains a parametric dependence of the Koopman operator on the ini-
tial condition of the driving system. The skew-flow map used in this case is the
interpretation of the second argument in F(x,y) as a parameter for the flow map
Fy(x):=F(x,y). The replacement for the semigroup property of autonomous sys-
tems is then the skew-flow property
(4.46) Ft+s
y=Ft
Gs(y)◦Fs
y.
The Koopman operator is then defined as the composition operator with respect to
the flow
(4.47) [Kt
yg](x):=g(Ft
y(x)).

46 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
In contrast to the “cocycle Koopman operator” (4.41), in which time-dependence is
represented by an additional time-like parameter, in this “skew Koopman operator”
the time-dependence is an added state-like parameter.
Depending on the properties of the driving system, this framework can encom-
pass not only systems such as (4.43), but also so-called random dynamical systems
(RDS) [15,62] for which the driving system on Yis simply assumed to be a measure-
preserving dynamical system [206,77].
While several special cases have been discussed in recent literature [77], we focus
here on the case of Markovian RDS. This is the case for dynamics generated by a
nonlinear stochastic differential equation
(4.48) dx(t) = f(x)dt +σ(x)dw(t).
It is then possible to define a stochastic Koopman operator by computing the expec-
tation of the skew-Koopman operator (4.47) (with continuous time domain) over the
invariant probability of the driving flow [15,234]:
(4.49) [Kt
Sg](x):=Ey{g(Ft
y(x))}.
When the driving system is a stochastic system, this establishes the Koopman operator
as the action of the dynamics, averaged over the distribution of the stochastic input.
Furthermore, assuming continuous and bounded functions on the right-hand side
of the SDE (4.48), it can be shown that (4.49) is a strongly-continuous semigroup,
which is a consequence of the Chapman–Kolmogorov equation, with a well-defined
generator LSacting on a space of twice-differentiable observables. In this case, the
analogue of the PDE formulation of the generator (2.10) is given by
(4.50) LSg=∇gf+1
2Tr(σ∇2gσ>).
Several DMD algorithms have been adapted to the RDS setting for the Koopman
operator [77,331], with convergence assurances. Additionally, [302] gives an explicit
optimization-based approximation of the Koopman operator in this context.
In summary, the theory behind DMD-style algorithms for nonautonomous and
stochastic dynamical systems have seen a rapid development in recent years, bringing
about both justification for applying such algorithms at first glance “off-the-label”,
and providing additional guidance for reduction of bias and errors in computation of
eigenvalues and eigenmodes.
4.6. Partial differential equations. There already exists a clear connection
of Koopman theory to PDEs [183]. Just as with ordinary differential equations, the
goal is to discover a linearizing transformation of the governing nonlinear PDE to a
new PDE model which evolves linearly in the new coordinate system. Historically,
this has been done in a number of contexts, specifically the Cole–Hopf transforma-
tion for Burgers’ equation with diffusive regularization and the inverse scattering
transform (IST) for completely integrable PDEs. The connection of these two ana-
lytic transformations is considered below. Such linearizing transformations have been
difficult to achieve in practice. However, data-driven methods have opened new path-
ways for constructing these transformations. Neural networks, diffusion maps, and
time-delay embeddings all allow for the data-driven construction of mappings capable
of transforming nonlinear PDEs into linear PDEs whose Koopman operator can be
constructed. In this section, we consider the connection of some of the historically

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 47
developed methods to Koopman theory. In section 5, we show how such linearizing
embeddings are constructed with modern data-driven methods.
The dynamics of nonlinear PDEs evolve on manifolds which are often difficult
to characterize and are rarely known analytically. However, an appropriate choice of
coordinates can in some cases, linearize the dynamics. For instance, the nonlinear
evolution governed by Burgers’ PDE equation can be linearized by the Cole–Hopf
transformation [140,74], thus providing a linear model that can trivially describe the
evolution dynamics. Such exact solutions to nonlinear PDEs are extremely rare and do
not often exist in practice, with the inverse scattering transform for Korteweg–deVries,
nonlinear Schr¨odinger, and other integrable PDEs being the notable exceptions [2].
As will be shown in subsection 5.4, neural networks provide a data-driven method to
learn coordinate embeddings such as those analytically available from the Cole–Hopf
transform and IST.
To demonstrate the construction of a specific and exact Koopman operator, we
consider Burgers’ equation with diffusive regularization and its associated Koopman
embedding [183,259,258,21]. The evolution is governed by diffusion with a nonlinear
advection [58]:
(4.51) ut+uux−uxx = 0  > 0, x ∈[−∞,∞].
When = 0, the evolution can lead to shock formation in finite time. The presence of
the diffusion term regularizes the PDE, ensuring continuous solutions for all time. In
independent, seminal contributions, Hopf [140] and Cole [74] derived a transformation
that linearizes the PDE. The Cole–Hopf transformation is defined as follows
(4.52) u=−2vx/v .
The transformation to the new variable v(x, t) replaces the nonlinear PDE (4.51) with
the linear, diffusion equation
(4.53) vt=vxx
where it is noted that  > 0 in (4.51) in order to produce a well-posed PDE. Denoting
ˆv= ˆv(k, t) as the Fourier transform of v(x, t) with wavenumber kgives the analytic
solution
(4.54) ˆv= ˆv0exp(−k2t)
where ˆv0= ˆv(k, 0) is the Fourier transform of the initial condition v(x, 0). Thus to
construct the Koopman operator, we can then combine the transform to the variable
v(x, t) from (4.52) with the Fourier transform to define the observable g(u) = ˆv. This
gives the Koopman operator
(4.55) K= exp(−k2t).
This is one of the rare instances where an explicit expression for the Koopman oper-
ator and the observables can be constructed analytically. As such, Burgers’ equation
allows one to build explicit representations of Koopman operators that characterize
its nonlinear evolution [183,258].
The inverse scattering transform [2] for other canonical and integrable PDEs,
such as the Korteweg–deVries and nonlinear Schr¨odinger equations, also can lead to
an explicit expression for the Koopman operator, but the scattering transform and

48 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
its inversion are much more difficult to construct in practice. Peter Lax developed
a general mathematical framework that preceded IST theory and which provided a
general principle for associating nonlinear evolutions with linear operators so that
the eigenvalues of the linear operator are integrals of the nonlinear equation [190].
The scattering theory and its association with nonlinear evolution equations was then
placed on more rigorous foundations by the seminal contribution of Ablowitz, Kaup,
Newell and Segur known as the AKNS scheme [1].
In brief, the method developed by Lax for constructing analytic solutions for
nonlinear evolution equations involved constructing suitable linear operators whose
compatibility condition was the evolution equation itself. For a given nonlinear evo-
lution equation
(4.56) ut=N(u, ux, uxx,···),
the goal is to posit a Lax pair of operators
Lφ =λφ(4.57a)
φt=Mφ(4.57b)
where Land Mare linear operators. Specifically, the operator Lis a spatial operator
that is self-adjoint and does not depend explicitly on t, while the operator Mis a
time-evolution linear operator. Importantly, L,Mand the evolution equation for
u(x, t) must be all self-consistent, or compatible, in order for the Lax theory to hold.
Self-consistency is achieved by taking the time-derivative of (4.57a) with respect to
time and enforcing solutions that have an iso-spectral evolution with respect to these
operators so that λt= 0. This then gives
(4.58) Lt+ [L, M ]=0
where [L, M ] = LM −ML represents the commutator of the operators. Importantly,
within this framework, the operators Land Mare linear. Thus once found, the
evolution dynamics in the transformed coordinate system is linear, much like what
occurs in the Burgers’ example. Of course, such a general mathematical framework
only holds for integrable PDEs [1]. However, it does show that the Koopman operator
framework is directly associated with the Lax pairs, and in particular with the linear
time evolution operator M=K, connecting IST and Koopman theory explicitly.
Parker and Page [262] recently developed a detailed analysis of fronts and solitons in
a variety of systems and explicitly connected their findings to the IST. Moreover, they
showed how two Koopman decompositions, upstream and downstream of the localized
structure, can be used to derive a full Koopman decomposition that leverages the IST
mathematical machinery. As will be shown in subsection 5.4, neural networks provide
an ideal, data-driven mathematical construct to learn coordinate embeddings such as
those analytically available from the Cole–Hopf transform and IST. Indeed, this has
been done even for the Kuramoto-Sivashinsky equation [118]. Additional connections
between the Cole–Hopf transform and Koopman eigenfunctions in the context of
ODEs are discussed in [33].

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 49
5. Data-driven observable selection and embeddings. Linearizing a non-
linear dynamical system near fixed points or periodic orbits provides a locally linear
representation of the dynamics [139], enabling the limited use of linear techniques
for prediction, estimation, and control [49]. The Koopman operator seeks globally
linear representations that are valid far from fixed points and periodic orbits. In the
data-driven era, this amounts to finding a coordinate system, or embedding, defined
by nonlinear observable functions that span a Koopman-invariant subspace. Finding,
approximating, and representing these observables and embeddings is still a central
challenge in modern Koopman theory. Dynamic mode decomposition [294,284,180]
from section 3 approximates the Koopman operator restricted to a space of linear mea-
surements with a best-fit linear model advancing these measurements from one time
to the next. However, linear DMD alone is unable to capture many essential features
of nonlinear systems, such as multiple fixed points and transients. It is thus common
to augment DMD with nonlinear functions of the measurements [352], although there
is no guarantee that these functions will form a closed subspace under the Koopman
operator [48]. In this section, we cover several leading approaches to identify and
represent Koopman embeddings from data, including the extended dynamic mode
decomposition [352] and methods to directly identify eigenfunctions [154], the use of
time-delay coordinates [47], as well as machine learning approaches such as diffusion
maps [117] and deep neural networks [349,217,330,359,256,202].
5.1. Extended DMD. Although DMD [294] has become a standard numerical
approach to approximate the Koopman operator [284,341,180], it is based on linear
measurements of the system and is unable to identify nonlinear changes of coordinates
necessary to approximate the Koopman operator for strongly nonlinear systems. The
extended dynamic mode decomposition (eDMD) [352,353,167] was introduced by
Williams et al. to address this issue, so that the best-fit linear DMD regression is
performed on an augmented vector containing nonlinear measurements of the state.
The eDMD approach was recently shown to be equivalent to the earlier variational ap-
proach of conformation dynamics (VAC) [250,252,253] from the molecular dynamics
literature, as explored in the excellent review by Klus et al. [168].
In eDMD, an augmented state zis constructed from nonlinear measurements of
the state xgiven by the functions gk:
(5.1) z=g(x) = 




g1(x)
g2(x)
.
.
.
gp(x)




.
The vector zmay contain the original state xas well as nonlinear measurements, so
often pn. Next, two data matrices are constructed, as in DMD:
Z=
z1z2·· · zm
,Z0=
z2z3·· · zm+1
.(5.2a)
Here zk=g(xk) = g(x(k∆t)), where we assume data is sampled at regular intervals
in time, for simplicity. As in DMD, a best-fit linear operator AZis constructed that
maps Zinto Z0:
AZ= argmin
AZkZ0−AZZkF=Z0Z†.(5.3)

50 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Because the augmented vector zmay be significantly larger than the state x, it is
typically necessary to employ kernel methods to compute this regression [353]. In
principle, the functions {gk}p
k=1 form an enriched basis in which to approximate the
Koopman operator. In the limit of infinite data, the extended DMD operator con-
verges to the Koopman operator projected onto the subspace spanned by these func-
tions [172]. However, if these functions do not span a Koopman invariant subspace,
then the projected operator will have spurious eigenvalues and eigenvectors that differ
from the true Koopman operator [48].
For example, consider the trivial example of a diagonalized linear system with
eigenvalues λ∈ {1,2,5}, eigenvectors ξjin the coordinate directions xj, and a naive
measurement that mixes the first two eigenvectors:
d
dt 
x1
x2
x3
=
100
020
005

x1
x2
x3
with y=110
x1
x2
x3
.(5.4)
In this case, DMD will predict a spurious eigenvalue of 3, which is the sum of the
first two eigenvalues λ= 1 and λ= 2, since the measurement is a sum of the first two
eigenvectors. Therefore, it is essential to use validation and cross-validation techniques
to ensure that eDMD models are not overfit, as discussed below.
Eigenfunctions of the Koopman operator form a Koopman invariant subspace
and provide an ideal basis, or coordinate system, in which to represent the dynamics.
However, Koopman eigenfunctions may not admit a finite representation in any stan-
dard basis. Thus, these eigenfunctions may only be approximately represented in a
given finite basis. It is possible to approximate an eigenfunction ϕ(x) as an expansion
in terms of the set of candidate functions {gk(x)}p
k=1 from (5.1) as:
(5.5) ϕ(x)≈
p
X
k=1
ξkgk(x) = ξTg(x).
In discrete-time, a Koopman eigenfunction ϕ(x) evaluated on a trajectory of snapshots
{x1,·· · ,xm}will satisfy:
λϕ(x1)ϕ(x2)·· · ϕ(xm)=ϕ(x2)ϕ(x3)·· · ϕ(xm+1).(5.6)
Expanding the eigenfunction ϕusing (5.5) this equality becomes
λξTz1ξTz2·· · ξTzm=ξTz2ξTz3·· · ξTzm+1
(5.7)
which is possible to write as a matrix system of equations in terms of the data matrices
Zand Z0:
(5.8) λξTZ−ξTZ0=0.
If we seek a least-squares fit to (5.8), this reduces to extended DMD [353,352]:
(5.9) λξT=ξTZ0Z†.
The eigenfunctions ϕ(x) are formed from left eDMD eigenvectors ξTof Z0Z†as
ϕ≈ξTzin the basis {gk(x)}p
k=1. The right eigenvectors are the eDMD modes.
It is essential to confirm that predicted eigenfunctions actually behave linearly
on trajectories, by comparing them with the predicted dynamics ϕ(xk+1) = λϕ(xk),

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 51
as the regression above will result in spurious eigenvalues and eigenvectors unless the
basis elements gkspan a Koopman invariant subspace [48]. It is common to include
the original state xin the augmented eDMD vector z. However, it was shown that
including the state xin eDMD results in closure issues for systems with multiple
fixed points, periodic orbits, or other attractors, because these systems cannot be
topologically conjugate to a finite-dimensional linear eDMD system with a single fixed
point [48]. For example, the Duffing oscillator in Figure 1.1 has three fixed points, so
no finite linear system can accurately evolve the state xnear all three of these fixed
points. However, eigenfunctions like the Hamiltonian, may be accurately expanded
in a basis. Thus, it is critical to sort out the accurate and spurious eigenfunctions
in eDMD; often eigenfunctions corresponding to lightly damped eigenvalues can be
better approximated, as they have a significant signature in the data.
One approach to prevent overfitting is to promote sparsity, as in the sparse iden-
tification of nonlinear dynamics (SINDy) [51]. This principle of parsimony may also
be used to identify Koopman eigenfunctions by selecting only the few most important
terms in the basis {gk(x)}p
k=1 needed to approximate ϕ[154].
As with standard DMD, the data in Zdoes not need to be generated from a single
trajectory, but can instead be sampled more efficiently, such as with latin hypercube
sampling or sampling from a distribution over the phase space. In this case, the data
in Z0must be obtained by advancing the data in Zone time step forward. Moreover,
reproducing kernel Hilbert spaces (RKHS) can be employed to describe ϕ(x)locally
in patches of state space.
For continuous-time dynamics, the eigenfunction dynamics
d
dtϕ(x) = λϕ(x)(5.10)
may be written in terms of the approximation ϕ(x)≈ξTg(x):
d
dtξTg(x) = λξTg(x).(5.11)
Applying the chain rule results in
ξTΓ(x,˙
x) = λξTg(x)(5.12)
where Γis given by:
Γ(x,˙
x) = 




∇g1(x)·˙
x
∇g2(x)·˙
x
.
.
.
∇gp(x)·˙
x




.(5.13)
Each term is a directional derivative, representing the possible terms in ∇ϕ(x)·f(x)
from (2.31). It is then possible to construct a data matrix Γevaluated on the trajec-
tories from Xand ˙
X:
Γ=




∇g1(x1)·˙
x1∇g1(x2)·˙
x2·· · ∇g1(xm)·˙
xm
∇g2(x1)·˙
x1∇g2(x2)·˙
x2·· · ∇g2(xm)·˙
xm
.
.
..
.
.....
.
.
∇gp(x1)·˙
x1∇gp(x2)·˙
x2·· · ∇gp(xm)·˙
xm




.

52 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
The Koopman eigenfunction equation then becomes:
(5.14) λξTZ−ξTΓ=0.
Note that here we use notation where Γis the transpose of the notation in Kaiser et
al. [154] to be consistent with the eDMD notation above.
5.2. Time delay coordinates. The DMD and eDMD algorithms are based on
the availability of full-state measurements, which are typically quite high-dimensional.
However, it is often the case that only partial observations of the system are available,
so that there are hidden, or latent, variables. In this case, it is possible to use time-
delayed measurements of the system to build an augmented state vector, resulting in
an intrinsic coordinate system that forms a Koopman-invariant subspace [47]. The
use of time-delay coordinates as a Koopman coordinate system relies on the conditions
of the Takens embedding theorem [332] being satisfied, so that the delay-embedded
attractor is diffeomorphic to the attractor in the original full-state coordinates.
The time-delay measurement scheme is illustrated schematically in Figure 5.1 on
the Lorenz’63 system. In this example, we have a single scalar measurement signal
x(t) from the original three-state Lorenz system. It is possible to construct a Hankel
matrix Hfrom a time-series of this scalar measurement:
H=




x(t1)x(t2)·· · x(tp)
x(t2)x(t3)·· · x(tp+1)
.
.
..
.
.....
.
.
x(tq)x(tq+1)··· x(tm)




.(5.15)
Each column of Hmay be obtained by advancing the previous column forward in time
by one time step. Thus, we may re-write (5.15) in terms of the Koopman operator K:
H=




x(t1)Kx(t1)·· · Kp−1x(t1)
Kx(t1)K2x(t1)·· · Kpx(t1)
.
.
..
.
.....
.
.
Kq−1x(t1)Kqx(t1)·· · Km−1x(t1)




.(5.16)
For a sufficient volume of data, the system will converge to an attractor, so that the
columns of Hbecome approximately linearly dependent. In this case, it is possible
to obtain a Koopman-invariant subspace by computing the SVD of H:
H=UΣV∗.(5.17)
The columns of Uand Vfrom the SVD are arranged hierarchically by their ability
to model the columns and rows of H, respectively. The low-rank approximation in
(5.17) provides a data-driven measurement system that is approximately invariant
to the Koopman operator for states on the attractor. By definition, the dynamics
map the attractor into itself, making it invariant to the flow. Often, Hwill admit
a low-rank approximation by the first rcolumns of Uand V, so that these columns
approximate a Koopman-invariant subspace. Thus, the columns of (5.15) are well-
approximated by the first rcolumns of U. The first rcolumns of Vprovide a time
series of the magnitude of each of the columns of UΣ in the data. By plotting the
first three columns of V, we obtain an embedded attractor for the Lorenz system, as
in Figure 5.1.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 53
x
Measure
Delay
Coordinates
2
6
6
6
6
6
4
v1
v2
v3
.
.
.
vr
3
7
7
7
7
7
5
2
6
6
6
6
6
4
v1
v2
v3
.
.
.
vr
3
7
7
7
7
7
5
2
6
6
6
6
6
4
v1
v2
v3
.
.
.
vr
3
7
7
7
7
7
5
25 30 35 40 45 50 55 60 65
-5
0
5×10-3
25 30 35 40 45 50 55 60 65
-0.01
-0.005
0
0.005
0.01
25 30 35 40 45 50 55 60 65
0
0.5
1×10-4
Forcing Active
Forcing Inactive
25 30 35 40 45 50 55 60 65
-5
0
5×10-3
25 30 35 40 45 50 55 60 65
-0.01
-0.005
0
0.005
0.01
25 30 35 40 45 50 55 60 65
0
0.5
1×10-4
Forcing Active
Forcing Inactive
0 5 10 15
2
4
6
8
10
12
14
-60
-30
0
30
60
-5 0 5 10
2
4
6
8
10
12
14
d
dt
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
v1
v2
v3
v4
v5
v6
v7
v8
v9
v10
v11
v12
v13
v14
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
v1
v2
v3
v4
v5
v6
v7
v8
v9
v10
v11
v12
v13
v14
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
v15
+
0 5 10 15
2
4
6
8
10
12
14
-60
-30
0
30
60
60
-60
-30
30
0
d
dt
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
v1
v2
v3
v4
v5
v6
v7
v8
v9
v10
v11
v12
v13
v14
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
=
d
dt
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
v1
v2
v3
v4
v5
v6
v7
v8
v9
v10
v11
v12
v13
v14
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
=
Regression model
Prediction
t
x(t)
v1
|v15|2
lobe switching
Fig. 5.1: Decomposition of chaos into a linear system with forcing. A time series x(t)
is stacked into a Hankel matrix H. The SVD of Hyields a hierarchy of eigen time
series that produce a delay-embedded attractor. A best-fit linear regression model
is obtained on the delay coordinates v; the linear fit for the first r−1 variables is
excellent, but the last coordinate vris not well-modeled as linear. Instead, vris an
input that forces the first r−1 variables. Rare forcing events correspond to lobe
switching in the chaotic dynamics. This architecture is called the Hankel alternative
view of Koopman (HAVOK) analysis, from [47]. Figure modified from Brunton et
al. [47].
Because the columns of H, and hence U, form a Koopman-invariant subspace, it
is possible to perform DMD on the two matrices formed from the first p−1 and last
p−1 columns of H. In practice, we recommend performing DMD on a similar set
of two matrices formed from the first p−1 and last p−1 columns of V, since the
columns of Vare the coordinates of the system in the UΣ frame. This results in a
linear regression model on the variables in V
d
dtv(t) = Av(t).(5.18)
Champion et al. [66] showed that this linear system captures the dynamics of weakly
nonlinear systems. For chaotic systems, however, even with an approximately Koopman-
invariant measurement system, there remain challenges to identifying a closed linear
model. A linear model, however detailed, cannot capture multiple fixed points or the
unpredictable behavior characteristic of chaos with a positive Lyapunov exponent [48].
Instead of constructing a closed linear model for the first rvariables in V, we build a
linear model on the first r−1 variables and impose the last variable, vr, as a forcing
term [47]:
d
dtv(t) = Av(t) + Bvr(t),(5.19)
where v=v1v2·· · vr−1Tis a vector of the first r−1 eigen-time-delay coordi-
nates. In all of the chaotic examples explored [47], the linear model on the first r−1

54 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
terms is accurate, while no linear model represents vr. Instead, vris an input forcing
to the linear dynamics in (5.19), which approximates the nonlinear dynamics. The
statistics of vr(t) are non-Gaussian, with long tails correspond to rare-event forcing
that drives lobe switching in the Lorenz system; this is related to rare-event forcing
distributions observed and modeled by others [211,288,212].
The Hankel matrix has been used for decades in system identification, for exam-
ple in the eigensystem realization algorithm (ERA) [152] and the singular spectrum
analysis (SSA) [42]. Computing DMD on a Hankel matrix was first introduced by Tu
et al. [341] and was used by B. Brunton et al. [45] in the field of neuroscience. The
connection between the Hankel matrix and the Koopman operator, along with the lin-
ear regression models in (5.19), was established by Brunton et al. [47] in the Hankel
alternative view of Koopman (HAVOK) framework. Several subsequent works have
provided additional theoretical foundations for this approach [14,81,161,66,136].
Hirsh et al. [136] established connections between HAVOK and the Frenet-Serret
frame from differential geometry, motivating a more accurate computational model-
ing approach. The HAVOK approach is also often referred to as delay-DMD [341] or
Hankel-DMD [14]. A connection between delay embeddings and the Koopman opera-
tor was established as early as 2004 by Mezi´c and Banaszuk [234], where a stochastic
Koopman operator is defined and a statistical Takens theorem is proven. Other work
has investigated the splitting of dynamics into deterministic linear, and chaotic sto-
chastic dynamics [231]. The use of delay coordinates may be especially important
for systems with long term memory effects and where the Koopman approach has
recently been shown to provide a successful analysis tool [327].
5.3. Diffusion maps for Koopman embeddings. Diffusion maps are a re-
cently developed nonlinear dimensionality-reduction technique for embedding high-
dimensional data on nonlinear manifolds [73,72,71,244]. Diffusion maps leverage
the underlying geometry, and in particular its local similarity structure, to create an
organization of data that is heavily influenced by local structures. The distance be-
tween data is measured by a kernel, for example the Gaussian kernel, which takes the
form of the fundamental Green’s function solution of the heat equation and is pro-
portional to the connectivity between two data points. The diffusion kernel is given
by
(5.20) k(xj,xk) = exp −kxj−xkk
α,
where xjand xkare two data points and αdetermines the range of influence of
the kernel. Thus, points that are not sufficiently close have an approximately zero
connectivity since the kernel decays as a Gaussian between data points.
The diffusion kernel plays the role of a normalized likelihood function. It also
has important properties when performing spectral analysis of the distance matrix
constructed from the embedding. These include symmetry, k(x,y) = k(y,x), and
positivity preserving, k(x,y)≥0. The basic diffusion mapping algorithm computes a
kernel matrix Kwhose elements are given by Kj,k =k(xj,xk). After normalization
of the rows of the kernel matrix, the eigenvalues and eigenvectors of Kare computed
and the data is projected onto the dominant r-modes. This r-dimensional subspace is
the low-dimensional embedding of the diffusion map. Coifman and Lafon [72] demon-
strated that this mapping gives a low dimensional parametrization of the geometry
and density of the data. In the field of data analysis, this construction is known as
the normalized graph Laplacian.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 55
(a) (b)
xk
ykyk+1 xk+1 xk
ykyk+1
xk+1
ϕKψ ϕ Kψ
Fig. 5.2: Competing neural network architectures to approximate the Koopman op-
erator. (a) Key Koopman eigenfunctions are extracted with a deep auto-encoder
network. (b) Alternatively, the system is lifted to a higher dimension where a linear
model is identified. In these architectures ϕis the encoder and ψis the decoder.
Diffusion maps thus provide a dimensionality reduction method that exploits the
geometry and density of the data. The diffusion map can be directly used to construct
a Koopman model by using a DMD regression on the time evolution in the diffusion
coordinates. The methodology can also be used for forecasting [59], for example
leveraging time-delay embeddings to provide a nonparametric forecasting method for
data generated by ergodic dynamical systems [117]. Such a representation is based
upon the Koopman and Perron-Frobenius groups of unitary operators in a smooth
orthonormal basis which is acquired from time-ordered data through the diffusion
maps algorithm. Giannakis [117] establishes in such a representation a correspondence
between Koopman operators and Laplace-Beltrami operators constructed from data in
Takens delay-coordinate space, using this correspondence to provide an interpretation
of diffusion-mapped delay coordinates for ergodic systems.
5.4. Neural networks for Koopman embeddings. Despite the promise of
Koopman embeddings, obtaining tractable representations has remained a central
challenge. Even for relatively simple dynamical systems, the eigenfunctions of the
Koopman operator may be arbitrarily complex and will only be approximately repre-
sented in a finite basis. Deep learning is well-suited for representing such arbitrarily
complex functions, and has recently shown tremendous promise for discovering and
representing Koopman embeddings [349,217,330,359,256,194,202].
There are two leading deep neural network architectures that have been proposed
for Koopman embeddings, shown in Figure 5.2. In the first architecture, the deep
auto-encoder architecture extracts a few key latent variables y=ϕ(x) to parameterize
the dynamics. In the second architecture, the high-dimensional input data is lifted
to an even higher dimension, where the evolution is approximately linear. In either
Koopman neural network, an additional constraint is enforced so that the dynamics
must be linear on these latent variables, given by the linear operator K. The constraint
of linear dynamics is enforced by the loss function kϕ(xk+1)−Kϕ(xk)k, where Kis
a matrix. In general, linearity is enforced over multiple time steps, so that additional
terms kϕ(xk+p)−Kpϕ(xk)kare added to the loss function.
Autoencoder networks have the advantage of a low-dimensional latent space,
which may promote interpretable solutions. Autoencoders are already widely used
to model complex systems, for example in fluid mechanics [50], and they may be
viewed as nonlinear extensions of the singular value decomposition, which is central
to the DMD algorithm. In this way, a deep Koopman network based on an autoen-
coder may be viewed as a nonlinear generalization of DMD. Similarly, if the matrix
Kis diagonalized, then the embedding functions ϕcorrespond to Koopman eigen-

56 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
ℂ
iω
−iω
X
X
ℂ
iω1
−iω1
iω2
−iω2
X
X
X
X
t
t
(a) Discrete spectrum (b) Continuous spectrum
Fig. 5.3: Comparison of discrete vs. continuous spectrum dynamics. Right panel
reproduced from Lusch et al. [202].
functions [202,118]. Variational autoencoders are also used for stochastic dynamical
systems, such as molecular dynamics, where the map back to physical configuration
space from the latent variables is probabilistic [349,217]. In contrast, the second
paradigm, where measurements are lifted to a higher-dimensional space, is related
to many results in machine learning, such as Cover’s theorem [186], where nonlinear
problems tend to become more linear in higher-dimensional embeddings.
For simple systems with a discrete eigenvalue spectrum, a compact representation
may be obtained in terms of a few autoencoder variables. However, dynamics with
continuous eigenvalue spectra defy standard low-dimensional Koopman representa-
tions, including the autoencoder network above. Continuous spectrum dynamics are
ubiquitous, ranging from the simple pendulum to nonlinear optics and broadband
turbulence. For example, the classical pendulum, given by
¨x=−sin(ωx)(5.21)
exhibits a continuous range of frequencies, from ωto 0, as the amplitude of the
pendulum oscillation is increased, as illustrated in Figure 5.3. Thus, the continuous
spectrum confounds a simple description in terms of a few Koopman eigenfunctions.
Indeed, away from the linear regime, an infinite Fourier sum is required to approximate
the continuous shift in frequency, which may explain why the high-dimensional lifting
approach has been widely used in Koopman neural networks.
In a recent work by Lusch et al. [202], an auxiliary network is used to parameterize
the continuously varying eigenvalue, enabling a network structure that is both parsi-
monious and interpretable. In contrast to other network structures, which require a
large autoencoder layer to encode the continuous frequency shift with an asymptotic
expansion in terms of harmonics of the natural frequency, the parameterized network
is able to identify a single complex conjugate pair of eigenfunctions with a varying
imaginary eigenvalue pair. If this explicit frequency dependence is unaccounted for,
then a high-dimensional network is necessary to account for the shifting frequency
and eigenvalues. Recently, this framework has been generalized to identify linearizing
coordinate transformations for PDE systems [118], such as the Cole-Hopf transform of
the nonlinear Burgers’ equation into the linear heat equation. Related work has been
developed to identify the analogues of Green’s functions for nonlinear systems [119].

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 57
6. Koopman theory for control. The Koopman operator framework is espe-
cially relevant for engineering applications in control [226,257], for which it offers
new opportunities for the control of nonlinear systems by circumventing theoreti-
cal and computational limitations due to nonlinearity. Nonlinear control methods,
such as feedback linearization and sliding mode control, overcome some of these lim-
itations. However, these approaches often do not generalize beyond a narrow class
of systems, and deriving stability and robustness conditions, for instance, can be-
come a demanding exercise. Koopman-based methods provide a linear framework
that can exploit mature theoretical and computational methods, with successes al-
ready demonstrated in a wide range of challenging applications, including fluid dy-
namics [13,265], robotics [3,4,44,111], power grid [176,247], traffic [197], biol-
ogy [128], logistics [138], and chemical processes [246]. Koopman analysis achieves
this by representing the nonlinear dynamics in a globally linear framework, without
linearization. Thus Koopman analysis is able to generalize the Hartman-Grobman
theorem to the entire basin of attraction of a stable or unstable equilibrium or peri-
odic point [185]. Further, as the Koopman operator acts on observables, it is amenable
to data-driven (model free) approaches which have been extensively developed in re-
cent years [275,351,173,276,316,154,155,264,4]. The resulting models have
been shown to reveal insights into global stability properties [309,224], observabil-
ity/controllability [345,120,360], and sensor/actuator placement [305,301] for the
underlying nonlinear system.
Koopman theory is closely related to Carleman linearization [63], which also em-
beds finite-dimensional dynamics into infinite-dimensional linear systems. Carleman
linearization has been used for decades to obtain truncated linear (and bilinear)
state estimators [179,40,9] and to examine stability, observability, and controlla-
bility of the underlying nonlinear system [200,22,79,80]. However, the applica-
bility is restricted to polynomial (or analytical) systems. In contrast, the Koop-
man operator framework does not rely on the analyticity of the vector field, but
applies to general nonlinear systems, including systems with discontinuities. Extend-
ing Koopman operator theory for actuated systems was first noted in [234], which
interpreted the stochastic forcing in random dynamical systems as actuation. The
first Koopman-based control schemes were published more than a decade later, pow-
ered by the algorithmic development of DMD [275]. More recently, Koopman models
have been increasingly used in combination with LQR [48,111,112], state-dependent
LQR [154], and model predictive control (MPC) [173,156]. Other directions include
optimal control for switching control problems [264,265], Lyapunov-based stabiliza-
tion [141,142], eigenstructure assignment [130], and, more recently, active learning [4].
MPC [114,191,227,241,279,113,60,8,97] stands out as a main facilitator for
the success of Koopman-based control, with applications including power grids [176],
high-dimensional fluid flows [13,145], and electrical drives [126].
In this section, we review the mathematical formulation for a Koopman operator
control framework, beginning with model-based control and moving to data-driven
methods. We will describe several approaches for identifying control-oriented models
including dynamic mode decomposition with control (DMDc), extended DMD with
control (eDMDc), extensions based on SINDy, and the use of delay coordinates. Fur-
ther, we compare these approaches on numerical examples and discuss the use of the
Koopman operator for analyzing important properties, such as stability, observability,
and controllability, of the underlying system.

58 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Plant
Output
˙
x=f(x,u)
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Input
u
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y
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Model predictive control
Plant model
ˆ
xk+1 =Aˆ
xk+Bˆ
uk
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Future input
k
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...
<latexit sha1_base64="pLXFuThevrw27Kh2nIwSE813uM0=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRSUG9FLx4r2A9oQ9lsNu3azW7YnQil9D948aCIV/+PN/+N2zYHbX0w8Hhvhpl5YSq4Qc/7dgpr6xubW8Xt0s7u3v5B+fCoZVSmKWtSJZTuhMQwwSVrIkfBOqlmJAkFa4ej25nffmLacCUfcJyyICEDyWNOCVqp1RORQtMvV7yqN4e7SvycVCBHo1/+6kWKZgmTSAUxput7KQYTopFTwaalXmZYSuiIDFjXUkkSZoLJ/Nqpe2aVyI2VtiXRnau/JyYkMWachLYzITg0y95M/M/rZhhfBRMu0wyZpItFcSZcVO7sdTfimlEUY0sI1dze6tIh0YSiDahkQ/CXX14lrYuqX6te39cq9Zs8jiKcwCmcgw+XUIc7aEATKDzCM7zCm6OcF+fd+Vi0Fpx85hj+wPn8Ab//j0Q=</latexit>
ˆ
u
<latexit sha1_base64="oNi8y+6tp+GVDv79Nl6TyKfU200=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4r2A9IStlsN+3SzSbsToQS+jO8eFDEq7/Gm//GbZuDtj4YeLw3w8y8MJXCoOt+O6W19Y3NrfJ2ZWd3b/+genjUNkmmGW+xRCa6G1LDpVC8hQIl76aa0ziUvBOO72Z+54lrIxL1iJOU92I6VCISjKKV/GBEMQ/CiGTTfrXm1t05yCrxClKDAs1+9SsYJCyLuUImqTG+56bYy6lGwSSfVoLM8JSyMR1y31JFY256+fzkKTmzyoBEibalkMzV3xM5jY2ZxKHtjCmOzLI3E//z/Ayj614uVJohV2yxKMokwYTM/icDoTlDObGEMi3srYSNqKYMbUoVG4K3/PIqaV/Uvcv6zcNlrXFbxFGGEziFc/DgChpwD01oAYMEnuEV3hx0Xpx352PRWnKKmWP4A+fzBzuskT4=</latexit>
Future output
k
<latexit sha1_base64="ESqfkpoOzQTKxIV+zY20hF3GUbw=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoN6KXjy2YGuhDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjto5TxbDFYhGrTkA1Ci6xZbgR2EkU0igQ+BCMb2f+wxMqzWN5byYJ+hEdSh5yRo2VmuN+ueJW3TnIKvFyUoEcjX75qzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NApObPKgISxsiUNmau/JzIaaT2JAtsZUTPSy95M/M/rpia88jMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZkQ/CWX14l7YuqV6teN2uV+k0eRxFO4BTOwYNLqMMdNKAFDBCe4RXenEfnxXl3PhatBSefOYY/cD5/ANZljPs=</latexit>
0
<latexit sha1_base64="2Fw0tYkoHQj8k1S+Z8AszaOALHo=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoN6KXjy2YGuhDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjto5TxbDFYhGrTkA1Ci6xZbgR2EkU0igQ+BCMb2f+wxMqzWN5byYJ+hEdSh5yRo2Vmm6/XHGr7hxklXg5qUCORr/81RvELI1QGiao1l3PTYyfUWU4Ezgt9VKNCWVjOsSupZJGqP1sfuiUnFllQMJY2ZKGzNXfExmNtJ5Ege2MqBnpZW8m/ud1UxNe+RmXSWpQssWiMBXExGT2NRlwhcyIiSWUKW5vJWxEFWXGZlOyIXjLL6+S9kXVq1Wvm7VK/SaPowgncArn4MEl1OEOGtACBgjP8ApvzqPz4rw7H4vWgpPPHMMfOJ8/fPmMwA==</latexit>
1
<latexit sha1_base64="wQ+2foIZ8ymAbto+K5qO6s2mJBk=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoN6KXjy2YGuhDWWznbRrN5uwuxFK6C/w4kERr/4kb/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrW9sbhW3Szu7e/sH5cOjto5TxbDFYhGrTkA1Ci6xZbgR2EkU0igQ+BCMb2f+wxMqzWN5byYJ+hEdSh5yRo2Vml6/XHGr7hxklXg5qUCORr/81RvELI1QGiao1l3PTYyfUWU4Ezgt9VKNCWVjOsSupZJGqP1sfuiUnFllQMJY2ZKGzNXfExmNtJ5Ege2MqBnpZW8m/ud1UxNe+RmXSWpQssWiMBXExGT2NRlwhcyIiSWUKW5vJWxEFWXGZlOyIXjLL6+S9kXVq1Wvm7VK/SaPowgncArn4MEl1OEOGtACBgjP8ApvzqPz4rw7H4vWgpPPHMMfOJ8/fn2MwQ==</latexit>
N1
<latexit sha1_base64="hYyQUlc3USMgnKgDct56tTjc6mY=">AAACDHicbVDLSsNAFL3xWeur6tJNsAhuWhItqLuiG1dSwT6gDWUyvWmHTiZhZiKU0A9w46+4caGIWz/AnX/jtM1CWw8MHM45lzv3+DFnSjvOt7W0vLK6tp7byG9ube/sFvb2GypKJMU6jXgkWz5RyJnAumaaYyuWSEKfY9MfXk/85gNKxSJxr0cxeiHpCxYwSrSRuoXibWegYkIxLTll54yJcWlecE3K0CnsReJmpAgZat3CV6cX0SREoSknSrVdJ9ZeSqRmlOM430kUmhVD0se2oYKEqLx0eszYPjZKzw4iaZ7Q9lT9PZGSUKlR6JtkSPRAzXsT8T+vnejgwkuZiBONgs4WBQm3dWRPmrF7TCLVfGQIoZKZv9p0QCSh2vSXNyW48ycvksZp2a2UL+8qxepVVkcODuEITsCFc6jCDdSgDhQe4Rle4c16sl6sd+tjFl2yspkD+APr8wdAOpnb</latexit>
...
<latexit sha1_base64="pLXFuThevrw27Kh2nIwSE813uM0=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRSUG9FLx4r2A9oQ9lsNu3azW7YnQil9D948aCIV/+PN/+N2zYHbX0w8Hhvhpl5YSq4Qc/7dgpr6xubW8Xt0s7u3v5B+fCoZVSmKWtSJZTuhMQwwSVrIkfBOqlmJAkFa4ej25nffmLacCUfcJyyICEDyWNOCVqp1RORQtMvV7yqN4e7SvycVCBHo1/+6kWKZgmTSAUxput7KQYTopFTwaalXmZYSuiIDFjXUkkSZoLJ/Nqpe2aVyI2VtiXRnau/JyYkMWachLYzITg0y95M/M/rZhhfBRMu0wyZpItFcSZcVO7sdTfimlEUY0sI1dze6tIh0YSiDahkQ/CXX14lrYuqX6te39cq9Zs8jiKcwCmcgw+XUIc7aEATKDzCM7zCm6OcF+fd+Vi0Fpx85hj+wPn8Ab//j0Q=</latexit>
...
<latexit sha1_base64="pLXFuThevrw27Kh2nIwSE813uM0=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSRSUG9FLx4r2A9oQ9lsNu3azW7YnQil9D948aCIV/+PN/+N2zYHbX0w8Hhvhpl5YSq4Qc/7dgpr6xubW8Xt0s7u3v5B+fCoZVSmKWtSJZTuhMQwwSVrIkfBOqlmJAkFa4ej25nffmLacCUfcJyyICEDyWNOCVqp1RORQtMvV7yqN4e7SvycVCBHo1/+6kWKZgmTSAUxput7KQYTopFTwaalXmZYSuiIDFjXUkkSZoLJ/Nqpe2aVyI2VtiXRnau/JyYkMWachLYzITg0y95M/M/rZhhfBRMu0wyZpItFcSZcVO7sdTfimlEUY0sI1dze6tIh0YSiDahkQ/CXX14lrYuqX6te39cq9Zs8jiKcwCmcgw+XUIc7aEATKDzCM7zCm6OcF+fd+Vi0Fpx85hj+wPn8Ab//j0Q=</latexit>
ˆ
y
<latexit sha1_base64="CfFjhV5JZKxLrkBPPqizqZGdYjI=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0lEUG9FLx4r2A9IQtlsN+3SzW7YnQgh9Gd48aCIV3+NN/+N2zYHbX0w8Hhvhpl5USq4Adf9dipr6xubW9Xt2s7u3v5B/fCoa1SmKetQJZTuR8QwwSXrAAfB+qlmJIkE60WTu5nfe2LacCUfIU9ZmJCR5DGnBKzkB2MCRRDFOJ8O6g236c6BV4lXkgYq0R7Uv4KholnCJFBBjPE9N4WwIBo4FWxaCzLDUkInZMR8SyVJmAmL+clTfGaVIY6VtiUBz9XfEwVJjMmTyHYmBMZm2ZuJ/3l+BvF1WHCZZsAkXSyKM4FB4dn/eMg1oyBySwjV3N6K6ZhoQsGmVLMheMsvr5LuRdO7bN48XDZat2UcVXSCTtE58tAVaqF71EYdRJFCz+gVvTngvDjvzseiteKUM8foD5zPH0HAkUI=</latexit>
ˆ
y0:= y(t)
<latexit sha1_base64="7ssmqYPu8ToIwKqQaulnxXR1Uak=">AAACAnicbZDLSsNAFIYn9VbrLepK3AwWoW5KIgUvIBTduKxgL9CEMJlO2qGTCzMnQgnFja/ixoUibn0Kd76N0zYLbf1h4OM/53Dm/H4iuALL+jYKS8srq2vF9dLG5tb2jrm711JxKilr0ljEsuMTxQSPWBM4CNZJJCOhL1jbH95M6u0HJhWPo3sYJcwNST/iAacEtOWZB86AQOb4AR6NPevyKscKnHhm2apaU+FFsHMoo1wNz/xyejFNQxYBFUSprm0l4GZEAqeCjUtOqlhC6JD0WVdjREKm3Gx6whgfa6eHg1jqFwGeur8nMhIqNQp93RkSGKj52sT8r9ZNITh3Mx4lKbCIzhYFqcAQ40keuMcloyBGGgiVXP8V0wGRhIJOraRDsOdPXoTWadWuVS/uauX6dR5HER2iI1RBNjpDdXSLGqiJKHpEz+gVvRlPxovxbnzMWgtGPrOP/sj4/AHss5Z/</latexit>
u(t):=ˆ
uopt
0
<latexit sha1_base64="3lWcZylYbsFi9p5+XbjNsBy+c9U=">AAACEXicbVDLSsNAFJ3UV62vqks3g0Wom5JIwQcIRTcuK9gHNDVMppN26OTBzI1QQn7Bjb/ixoUibt2582+ctFlo64GBM+fcy733uJHgCkzz2ygsLa+srhXXSxubW9s75d29tgpjSVmLhiKUXZcoJnjAWsBBsG4kGfFdwTru+DrzOw9MKh4GdzCJWN8nw4B7nBLQklOuJrbr4TitwvHFpT0ikP8d8z6xfQIj6SdhBGnqlCtmzZwCLxIrJxWUo+mUv+xBSGOfBUAFUapnmRH0EyKBU8HSkh0rFhE6JkPW0zQgPlP9ZHpRio+0MsBeKPULAE/V3x0J8ZWa+K6uzJZU814m/uf1YvDO+gkPohhYQGeDvFhgCHEWDx5wySiIiSaESq53xXREJKGgQyzpEKz5kxdJ+6Rm1Wvnt/VK4yqPo4gO0CGqIgudoga6QU3UQhQ9omf0it6MJ+PFeDc+ZqUFI+/ZR39gfP4AnbKdkw==</latexit>
Optimizer
min J
<latexit sha1_base64="VuE7UqgsQf9nh/61K6E5zcWw1N0=">AAAB+HicbVDLSsNAFL3xWeujUZdugkVwISWRgroruhFXFewDmlAm00k7dGYSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPnDBhVGnX/bZWVtfWNzZLW+Xtnd29ir1/0FZxKjFp4ZjFshsiRRgVpKWpZqSbSIJ4yEgnHN/kfueRSEVj8aAnCQk4GgoaUYy0kfp2xedIjyTPOBVT/+yub1fdmjuDs0y8glShQLNvf/mDGKecCI0ZUqrnuYkOMiQ1xYxMy36qSILwGA1Jz1CBOFFBNgs+dU6MMnCiWJontDNTf29kiCs14aGZzGOqRS8X//N6qY4ug4yKJNVE4PmhKGWOjp28BWdAJcGaTQxBWFKT1cEjJBHWpquyKcFb/PIyaZ/XvHrt6r5ebVwXdZTgCI7hFDy4gAbcQhNagCGFZ3iFN+vJerHerY/56IpV7BzCH1ifP+Grk0A=</latexit>
constraints
+
State
estimator
ˆ
x0
<latexit sha1_base64="I0Kz/NnYo/mxrH1D7YSoEv567fc=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgKexKQL0FvXiMYB6QXcLsZDYZMvtwpjcYlnyHFw+KePVjvPk3TpI9aGJBQ1HVTXeXn0ih0ba/rcLa+sbmVnG7tLO7t39QPjxq6ThVjDdZLGPV8anmUkS8iQIl7ySK09CXvO2Pbmd+e8yVFnH0gJOEeyEdRCIQjKKRPHdIMXP9gDxNe3avXLGr9hxklTg5qUCORq/85fZjloY8Qiap1l3HTtDLqELBJJ+W3FTzhLIRHfCuoRENufay+dFTcmaUPgliZSpCMld/T2Q01HoS+qYzpDjUy95M/M/rphhceZmIkhR5xBaLglQSjMksAdIXijOUE0MoU8LcStiQKsrQ5FQyITjLL6+S1kXVqVWv72uV+k0eRxFO4BTOwYFLqMMdNKAJDB7hGV7hzRpbL9a79bFoLVj5zDH8gfX5A2vxkeQ=</latexit>
ˆ
yk=Cˆ
xk
<latexit sha1_base64="lRbWhnhbkIfbEaOH+6usbf703ys=">AAACPnicbVDPS8MwGE39OeevqkcvwSF4cbQ6UA/CcBePE+w2WMdIs3QLS9OSpGIp/cu8+Dd48+jFgyJePZp1RXTzQeDxvvf48j0vYlQqy3o2FhaXlldWS2vl9Y3NrW1zZ7clw1hg4uCQhaLjIUkY5cRRVDHSiQRBgcdI2xs3JvP2HRGShvxWJRHpBWjIqU8xUlrqm447Qip1PR8mWX/sjmSEMEmPrap1SnkGL+GslHsb2U/sXsfmTH2zolkOOE/sglRAgWbffHIHIY4DwhVmSMqubUWqlyKhKGYkK7uxJHrDGA1JV1OOAiJ7aX5+Bg+1MoB+KPTjCubq70SKAimTwNPOAKmRnJ1NxP9m3Vj5572U8ihWhOPpIj9mUIVw0iUcUEGwYokmCAuq/wrxCAmElW68rEuwZ0+eJ62Tql2rXtzUKvWroo4S2AcH4AjY4AzUwTVoAgdg8ABewBt4Nx6NV+PD+JxaF4wiswf+wPj6BkkOrfs=</latexit>
y=c(x,u)
<latexit sha1_base64="x7PR1bLt51smNQfcoSlc8qpy0yk=">AAACCXicbVDLSsNAFL2pr1pfUZduBotQQUoiBXUhFN24rGAf0JYymU7aoZMHMxMxhGzd+CtuXCji1j9w5984TbPQ1gvDOZxzL3fucULOpLKsb6OwtLyyulZcL21sbm3vmLt7LRlEgtAmCXggOg6WlDOfNhVTnHZCQbHncNp2JtdTv31PhWSBf6fikPY9PPKZywhWWhqYKOk5LorTywxJWsnwIT3JMEqPB2bZqlpZoUVi56QMeTUG5ldvGJDIo74iHEvZta1Q9RMsFCOcpqVeJGmIyQSPaFdTH3tU9pPskhQdaWWI3EDo5yuUqb8nEuxJGXuO7vSwGst5byr+53Uj5Z73E+aHkaI+mS1yI45UgKaxoCETlCgea4KJYPqviIyxwETp8Eo6BHv+5EXSOq3aterFba1cv8rjKMIBHEIFbDiDOtxAA5pA4BGe4RXejCfjxXg3PmatBSOf2Yc/ZXz+ADoimWo=</latexit>
Fig. 6.1: Schematic of the receding horizon model predictive control framework.
6.1. Model-Based Control. Beginning with model-based control theory, con-
sider the non-affine nonlinear control system
˙
x=f(x,u),x(0) = x0
(6.1a)
y=c(x,u),(6.1b)
where x∈ X ⊆ Rnis the state vector, u∈ U ⊆ Rqis the vector of control inputs, and
y∈ Y ⊆ Rpis the output. Unless noted otherwise, we assume full-state measurements
y=x. Equation (6.1) represents a non-autonomous dynamical system, where the
input umay be interpreted as a perturbed parameter or control actuation. In the
context of control, we seek typically to determine a feedback control law d:Y → U,
(6.2) u=d(y),
that maps measurements yto control inputs uto modify the behavior of the closed-
loop system.
6.1.1. Model predictive control. MPC is one of the most successful model-
based control schemes [114,191,227,241,279,113,60,8,97]. Over the last two
decades, MPC has gained increasing popularity due to its success in a wide range of
applications, its ability to incorporate customized cost functions and constraints, and
extensions to nonlinear systems. In particular, it has become the de-facto standard
advanced control method in process industries [279] and has gained considerable trac-
tion in the aerospace industry due to its versatility [97]. Many of the successes of
Koopman-based controls leverage the MPC framework, with its adaptive nature al-
lowing one to compensate for modeling discrepancies and to account for disturbances.
MPC is an adaptive control procedure solving an open-loop optimization problem
over a receding horizon (see schematic in Figure 6.1). The optimization problem aims
to solve for a sequence of control inputs {ˆ
u0,ˆ
u2,...,ˆ
uN−1}over the time horizon

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 59
T=N∆tthat minimizes a pre-defined objective function J. Typically, only the
first control input ˆ
uopt
0is applied and then a new measurement is collected. At
each time instant a new measurement is collected, the optimization problem is re-
initialized and, thus, adaptively determines optimal control actions adjusting to model
inaccuracies and changing conditions in the environment. The most critical part
of MPC is the identification of a dynamical model that accurately and efficiently
represents the system behavior in the presence of actuation. If the model is linear,
minimization of a quadratic cost functional subject to linear constraints results in a
tractable convex problem. There exist several variants of MPC, including nonlinear
MPC, robust MPC, and explicit MPC, which are, however, more computationally
expensive and thus limit their real-time applicability. Combining Koopman-based
models with linear MPC has the potential to significantly extend the reach of linear
MPC for nonlinear systems.
The receding-horizon optimization problem can be stated as follows. Linear MPC
aims to minimize the following quadratic objective function
min
ˆ
u(·|y)∈UJ= min
ˆ
u(·|y)∈U
N−1
X
k=0 ||ˆ
yk−rk||2
Q+||ˆ
uk||2
R+||∆ˆ
uk||2
R∆
(6.3)
subject to discrete-time, linear system dynamics
ˆ
xk+1 =Aˆ
xk+Bˆ
uk,(6.4a)
ˆ
yk=Cˆ
xk,(6.4b)
and state and input constraints
ymin ≤ˆ
yk≤ymax,(6.5a)
umin ≤ˆ
uk≤umax,(6.5b)
(6.5c)
where ∆ˆ
uk:=ˆ
uk−ˆ
uk−1is the control input rate. Each term in the cost function (6.3)
is computed as the weighted norm of a vector, i.e. ||y||2
Q:=yTQy. In the model (6.4),
A:X → X is the state transition matrix, B:U → X is the control matrix, and
C:X → Y the measurement matrix. The weight matrices R∈Rq×q,R∆∈Rq×q,
and Q∈Rn×nare positive semi-definite and penalize the inputs, input rates, and
deviations of the predicted output ˆ
yalong a trajectory r, respectively, and set their
relative importance. We define the control sequence to be solved over the receding
horizon as ˆ
u(0, . . . , N −1|y):={ˆ
u0,ˆ
u2,...,ˆ
uN−1}given the measurement y. The
measurement yis the current output of the plant, whose dynamics are governed by a
generally nonlinear system (6.1), and is used to estimate the initial condition ˆ
x0for
the optimization problem. The general feedback control law (6.2) is then:
(6.6) d(y) = ˆ
uopt(0|y) = ˆ
uopt
0
given a specific yand selecting the first entry of the optimized control sequence.
Two primary research thrusts, which can be roughly categorized into approaches
for either discrete or continuous inputs, have integrated Koopman theory and MPC.
For the latter, the Koopman-MPC framework is schematically depicted in Figure 6.2.
Output measurements are lifted into a higher-dimensional space using a nonlinear
transformation. Dynamics are modeled in the lifted space, typically by solving a lin-
ear least-squares regression problem, and the resulting model is employed in the MPC

60 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Plant
Output
˙
x=f(x,u)
<latexit sha1_base64="6Kbor9sG75vO9W8BXLYIU3WmrUk=">AAACD3icbVDLSgMxFL1TX7W+Rl26CRalgpQZKagLoejGZQX7gHYomTTThmYeJBmxDPMHbvwVNy4UcevWnX9jOp2FVg+EHM659yb3uBFnUlnWl1FYWFxaXimultbWNza3zO2dlgxjQWiThDwUHRdLyllAm4opTjuRoNh3OW2746up376jQrIwuFWTiDo+HgbMYwQrLfXNw94gVEnP9dB9ii5Qxry0kivH2R2nR32zbFWtDOgvsXNShhyNvvmpB5PYp4EiHEvZta1IOQkWihFO01IvljTCZIyHtKtpgH0qnSTbJ0UHWhkgLxT6BApl6s+OBPtSTnxXV/pYjeS8NxX/87qx8s6chAVRrGhAZg95MUcqRNNw0IAJShSfaIKJYPqviIywwETpCEs6BHt+5b+kdVK1a9Xzm1q5fpnHUYQ92IcK2HAKdbiGBjSBwAM8wQu8Go/Gs/FmvM9KC0beswu/YHx8Az8Im4s=</latexit>
Input
u
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y
<latexit sha1_base64="EFmHDPXvPUqfUdG67YSl3rnjoLo=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4r2A9oQ9lsN+3SzSbsToQQ+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O6W19Y3NrfJ2ZWd3b/+genjUNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7mZ+54lrI2L1iFnC/YiOlAgFo2ilTt4PQpJNB9WaW3fnIKvEK0gNCjQH1a/+MGZpxBUySY3peW6Cfk41Cib5tNJPDU8om9AR71mqaMSNn8/PnZIzqwxJGGtbCslc/T2R08iYLApsZ0RxbJa9mfif10sxvPZzoZIUuWKLRWEqCcZk9jsZCs0ZyswSyrSwtxI2ppoytAlVbAje8surpH1R9y7rNw+XtcZtEUcZTuAUzsGDK2jAPTShBQwm8Ayv8OYkzovz7nwsWktOMXMMf+B8/gA0b4+B</latexit>
Model predictive control
Plant model
ˆ
zk+1 =Aˆ
zk+Bˆ
uk
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u(t):=ˆ
uopt
0
<latexit sha1_base64="3lWcZylYbsFi9p5+XbjNsBy+c9U=">AAACEXicbVDLSsNAFJ3UV62vqks3g0Wom5JIwQcIRTcuK9gHNDVMppN26OTBzI1QQn7Bjb/ixoUibt2582+ctFlo64GBM+fcy733uJHgCkzz2ygsLa+srhXXSxubW9s75d29tgpjSVmLhiKUXZcoJnjAWsBBsG4kGfFdwTru+DrzOw9MKh4GdzCJWN8nw4B7nBLQklOuJrbr4TitwvHFpT0ikP8d8z6xfQIj6SdhBGnqlCtmzZwCLxIrJxWUo+mUv+xBSGOfBUAFUapnmRH0EyKBU8HSkh0rFhE6JkPW0zQgPlP9ZHpRio+0MsBeKPULAE/V3x0J8ZWa+K6uzJZU814m/uf1YvDO+gkPohhYQGeDvFhgCHEWDx5wySiIiSaESq53xXREJKGgQyzpEKz5kxdJ+6Rm1Wvnt/VK4yqPo4gO0CGqIgudoga6QU3UQhQ9omf0it6MJ+PFeDc+ZqUFI+/ZR39gfP4AnbKdkw==</latexit>
Optimizer
min J
<latexit sha1_base64="VuE7UqgsQf9nh/61K6E5zcWw1N0=">AAAB+HicbVDLSsNAFL3xWeujUZdugkVwISWRgroruhFXFewDmlAm00k7dGYSZiZCDf0SNy4UceunuPNvnLRZaOuBgcM593LPnDBhVGnX/bZWVtfWNzZLW+Xtnd29ir1/0FZxKjFp4ZjFshsiRRgVpKWpZqSbSIJ4yEgnHN/kfueRSEVj8aAnCQk4GgoaUYy0kfp2xedIjyTPOBVT/+yub1fdmjuDs0y8glShQLNvf/mDGKecCI0ZUqrnuYkOMiQ1xYxMy36qSILwGA1Jz1CBOFFBNgs+dU6MMnCiWJontDNTf29kiCs14aGZzGOqRS8X//N6qY4ug4yKJNVE4PmhKGWOjp28BWdAJcGaTQxBWFKT1cEjJBHWpquyKcFb/PIyaZ/XvHrt6r5ebVwXdZTgCI7hFDy4gAbcQhNagCGFZ3iFN+vJerHerY/56IpV7BzCH1ifP+Grk0A=</latexit>
constraints
+
ˆ
z0:= z(t)
<latexit sha1_base64="PIrpEHiHG8LZK9v5k6uQ3mDwCgA=">AAACAnicbZDLSsNAFIYnXmu9RV2Jm8Ei1E1JpOAFhKIblxXsBZoQJtNJO3RyYeZEqKG48VXcuFDErU/hzrdx2mahrT8MfPznHM6c308EV2BZ38bC4tLyymphrbi+sbm1be7sNlWcSsoaNBaxbPtEMcEj1gAOgrUTyUjoC9byB9fjeuueScXj6A6GCXND0ot4wCkBbXnmvtMnkDl+gB9GnnVxmWMZjj2zZFWsifA82DmUUK66Z3453ZimIYuACqJUx7YScDMigVPBRkUnVSwhdEB6rKMxIiFTbjY5YYSPtNPFQSz1iwBP3N8TGQmVGoa+7gwJ9NVsbWz+V+ukEJy5GY+SFFhEp4uCVGCI8TgP3OWSURBDDYRKrv+KaZ9IQkGnVtQh2LMnz0PzpGJXK+e31VLtKo+jgA7QISojG52iGrpBddRAFD2iZ/SK3own48V4Nz6mrQtGPrOH/sj4/AHvzpaB</latexit>
Nonlinear
transform.
z=⇥T(y)
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y=c(x,u)
<latexit sha1_base64="x7PR1bLt51smNQfcoSlc8qpy0yk=">AAACCXicbVDLSsNAFL2pr1pfUZduBotQQUoiBXUhFN24rGAf0JYymU7aoZMHMxMxhGzd+CtuXCji1j9w5984TbPQ1gvDOZxzL3fucULOpLKsb6OwtLyyulZcL21sbm3vmLt7LRlEgtAmCXggOg6WlDOfNhVTnHZCQbHncNp2JtdTv31PhWSBf6fikPY9PPKZywhWWhqYKOk5LorTywxJWsnwIT3JMEqPB2bZqlpZoUVi56QMeTUG5ldvGJDIo74iHEvZta1Q9RMsFCOcpqVeJGmIyQSPaFdTH3tU9pPskhQdaWWI3EDo5yuUqb8nEuxJGXuO7vSwGst5byr+53Uj5Z73E+aHkaI+mS1yI45UgKaxoCETlCgea4KJYPqviIyxwETp8Eo6BHv+5EXSOq3aterFba1cv8rjKMIBHEIFbDiDOtxAA5pA4BGe4RXejCfjxXg3PmatBSOf2Yc/ZXz+ADoimWo=</latexit>
ˆ
yk=Cˆ
zk
<latexit sha1_base64="TBmag/WlDvmWMAlghT3GmnS/td0=">AAACPnicbVDPS8MwGE39OeevqUcvwSF4cbQ6UA/CcBePE+w2WEtJs3QLS9OSpEIt/cu8+Dd48+jFgyJePZptRXTzQeDxvvf48j0/ZlQq03w2FhaXlldWS2vl9Y3Nre3Kzm5bRonAxMYRi0TXR5IwyomtqGKkGwuCQp+Rjj9qjuedOyIkjfitSmPihmjAaUAxUlryKrYzRCpz/ACmuTdyhjJGmGTHZs08pTyHl3BWmnib+U/sXsfmTF6lqtkEcJ5YBamCAi2v8uT0I5yEhCvMkJQ9y4yVmyGhKGYkLzuJJHrDCA1IT1OOQiLdbHJ+Dg+10odBJPTjCk7U34kMhVKmoa+dIVJDOTsbi//NeokKzt2M8jhRhOPpoiBhUEVw3CXsU0GwYqkmCAuq/wrxEAmElW68rEuwZk+eJ+2TmlWvXdzUq42roo4S2AcH4AhY4Aw0wDVoARtg8ABewBt4Nx6NV+PD+JxaF4wiswf+wPj6Bkw+rf0=</latexit>
Fig. 6.2: Schematic of the model predictive control framework incorporating a model
based on the Koopman operator.
optimization procedure. Besides the goal of achieving increased predictive power via
a Koopman-based model, this approach further provides the possibility to readily in-
corporate nonlinear cost functions and constraints in a linear fashion by incorporating
these directly in the set of observables [351].
6.1.2. Koopman operator theory for control systems. Koopman theory
for control requires disambiguating the unforced dynamics from the effect of actuation.
The first Koopman-based approaches were developed for discrete-time systems, which
are more general and form a superset containing those induced by continuous-time
dynamics. Such discrete-time dynamics are often more consistent with experimental
measurements and actuation, and may be preferred for numerical analysis. However,
there has been increasing effort in developing formulations building on the Lie op-
erator, i.e. the infinitesimal generator of the semigroup of Koopman operators, for
system identification and control.
Consider the non-affine, continuous-time control system given by (6.1a) that is
fully observable, i.e. y=xin (6.1b). For every initial condition x∈ X and control
function u∈ U, there exists a unique solution Ft(x,u) at time twith initial condition
F0(x,u) = x. System (6.1) represents a family of differential equations parameterized
by the control functions u. In order to analyze this from a Koopman perspective, it
is convenient to introduce the control flow, which defines (6.1) as a single dynamical
system (for more details about the dynamical systems perspective of control theory
we refer to Colonius et al. [75]). The control flow ˜
Ft(x,u) : R× X × U → X × U
associated with (6.1) is given by the map:
(6.7) ˜
Ft(x,u) = (Ft(x,u),Θt(u)),
where Θt(u) is the shift on U, so that Θt(u)(s) = u(s+t), s∈R. Skew-product flows,
such as ˜
Ft(x,u), arise in topological dynamics to study non-autonomous systems, e.g.
with explicit time dependency or parameter dependency. Actuation renders the dy-
namical system and its associated Koopman family (or its generator) non-autonomous.
By defining the Koopman operator on the extended state ˜
x:= [xT,uT]T, the Koop-
man operator becomes autonomous and is equivalent to the Koopman operator as-

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 61
sociated with the unforced dynamics. Under further special considerations, standard
numerical schemes for autonomous systems become readily applicable for system iden-
tification. Beyond considering the extended state ˜
x, we also make the dependency on
xand uexplicit to disambiguate the state and inputs.
Let g(x,u) : X × U → Cbe a scalar observable function of the extended state
space. Each observable is an element of an infinite-dimensional Hilbert space and the
semigroup of Koopman operators Kt:G(X,U)→ G(X,U) acts on these observables
according to:
(6.8) g(x(t),u(t)) = Ktg(x0,u0) = g(˜
Ft(x0,u0)).
Here, it is assumed that the Koopman operator acts on the extended state space in
the same manner as the Koopman operator associated with the unforced, autonomous
dynamical system. A Koopman eigenfunction ϕ(x,u) corresponding to eigenvalue λ
then satisfies
(6.9) ϕ(x(t),u(t)) = Ktϕ(x0,u0) = λtϕ(x0,u0).
Further, a vector-valued observable
(6.10) g(x,u):=


g1(x,u)
.
.
.
gp(x,u)



can be written in terms of the infinite Koopman expansion as
(6.11) g(x(t),u(t)) = Ktg(x(0),u(0)) = ∞
X
j=1
λt
jϕj(x0,u0)vj,
where vj= [hϕj, g1i,...,hϕj, gpi]. This representation encompasses dynamics on u
itself, which may appear due to external perturbations when uis interpreted as a per-
turbed parameter to the system. While the actuation dynamics are typically known
or set for both open-loop and closed-loop control, it provides a convenient starting
point for system identification. Indeed, it is a useful representation for data-driven
approaches that identify the underlying system dynamics and control simultaneously.
Depending on the choice of observable functions, further simplifications are possible
to identify a model for the state dynamics by incorporating the effect of control, which
is discussed in subsection 6.2.
We consider special cases in the discrete-time and continuous-time settings which
are modeled within the Koopman with inputs and control (KIC) framework [276]. The
time-varying actuation input may evolve dynamically according to ˙
u=h(u) or it may
be governed by a state feedback control law (6.2), u=d(x), as in closed-loop control
applications.
Discrete-time formulation. For a discrete-time control system
xk+1 =F(xk,uk),(6.12a)
with initial condition x0, the Koopman operator advances measurement functions
according to
(6.13) Kg(xk,uk) = g(F(xk,uk),uk+1) = g(xk+1 ,uk+1).

62 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Koopman eigenpairs (ϕ, λ) associated with (6.13) satisfy:
(6.14) Kϕ(xk,uk) = ϕ(F(xk,uk),uk+1) = ϕ(xk+1 ,uk+1) = λϕ(xk,uk).
By defining the control flow ˜
Fon the extended state space ˜
x:= [xT,uT]T, the dy-
namics become autonomous and can be written as
(6.15) Kg(˜
xk) = g(˜
F(˜
xk)) = g(˜
xk+1).
If uk+1 =H(uk), then g(F(xk),H(uk)) = g(xk+1,uk+1 ), thus allowing for a suitable
choice of observable functions that can simultaneously model and identify dynamics
for xand u. In many situations, we are not interested in the dynamics of uitself
but only of x, e.g. in control where uis usually a design variable. For instance,
if the dynamics of uis prescribed by a specific state-feedback law uk=D(xk),
then Kg(xk,uk) = g(F(xk,uk),D(xk)) = g(xk+1,D(xk+1 )). By defining FD(x):=
F(x,D(x)) and restricting the observable to be a function solely of the state, the
Koopman operator is associated with the autonomous dynamics FDfor a given control
law D:Kg(xk) = g(FD(xk)) = g(xk+1). If instead, we consider a constant exogenous
forcing or discrete control action ¯
u∈ U, where Uis the set of discrete inputs. The
Koopman operator may be defined for each discrete input separately: K¯
ug(xk,¯
u) =
g(F(xk,¯
u),¯
u) = g(xk+1,¯
u). In general, the Koopman operator and its associated
eigenfunctions are parameterized by the discrete control input ¯
uand the dynamics
are autonomous for each ¯
u. The flow map F¯
u(x):=F(x,¯
u) is then defined for each
¯
u. Considering only the reduced dynamics on observable functions of the state, we
then obtain
(6.16) K¯
ug(xk) = g(F¯
u(xk)) = g(xk+1).
By switching from continuous to discrete inputs, a single model is replaced with a
family of models, so that the specific control dependency of the state does not have
to be captured. Instead of optimizing the input itself, one may then optimize the
switching times between inputs, as in Peitz et al. [264].
Lie operator formulation. There has been increasing interest in the control for-
mulation for the infinitesimal generator of the Koopman operator family. It can be
shown [187] that if observables gare continuously differentiable with compact support,
then they satisfy the first-order partial differential equation (2.31). The Lie operator
Lg=∇˜
xg·˜
fassociated with the dynamics of the control system (6.1) induces the
dynamics
(6.17) d
dtg(x,u) = Lg(x,u).
Similarly, smooth eigenfunctions corresponding to the eigenvalue µsatisfy
(6.18) d
dtϕ(x,u) = Lϕ(x,u) = µϕ(x,u).
Note that the smooth eigenfunction of the generator is also a Koopman eigenfunc-
tion (6.9) and their eigenvalues are connected via µ= log(λ) (see also section 2).
We may rewrite the Lie operator in (6.17) explicitly using the chain rule
(6.19) Lg(x,u) = ∇xg(x,u)·˙
x+∇ug(x,u)·˙
u.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 63
The derivatives ˙
x=fand ˙
u=hare both velocity vectors (state or action change
per time unit) and generally depend on xand u. These describe local changes in
the observable function due to local changes in xand external forcing via u. For
a dynamically evolving uit may be possible to approximate the Koopman opera-
tor on the extended state using the system identification framework based on the
generator formulation [220]. It is also possible to consider ˙
uas the new input to
the system [240], while applying more traditional methods. Equation (6.19) repre-
sents the adjoint equation of the controlled Liouville equation (see e.g. [41]), which
describes how a density function evolves in state space under the effect of exter-
nal forcing. Using the analogy to the scattering of particles, these transport op-
erators have been used to model and derive optimal control for agent-based sys-
tems [184]. For the state-feedback law u=d(x), then the right-hand side of (6.17)
becomes Lg(x,u)=[∇xg(x,d(x)) + ∇dg(x,d(x)) · ∇xd(x)] ·f(x,d(x)). Since this
is solely a function of the state, the (autonomous) Koopman operator associated
with a particular state-feedback law d(x) can be defined on the reduced state x:
Lg(x) = ∇xg(x)·f(x,d(x)).
Constant exogenous forcing or discrete control actions ¯
u∈ U render the system
autonomous for each ¯
uso that L¯
ug(x,¯
u) = ∇xg(x,¯
u)·f(x,¯
u) with ˙
u=0. In order
to apply most control methods a model for the reduced state xis required. Thus, by
restricting the space of observables defined on the reduced state x, we obtain
(6.20) Lg(x) = ∇xg(x)·f(x,¯
u).
This representation is the starting point for data-driven formulations, e.g., to estimate
the associated Koopman operator by assuming the zero-order hold for the input across
consecutive snapshots, or Koopman operators parameterized by the discrete control
input so that a gain-scheduled or interpolated controller may be enacted.
Bilinearization. Many applications are modeled by control-affine systems
(6.21) ˙
x=f0(x) +
q
X
j=1
fj(x)uj.
Using Carleman linearization, it is possible to transform this system into a (generally
infinite-dimensional) bilinear model using multivariable monomials of the state as ob-
servables, which is then truncated at a suitable order. However, this may nevertheless
lead to systems with an undesirably high dimensionality for the required accuracy. A
general nonlinear system ˙
x=f(x,u) with analytic fcan also be transformed into an
infinite-dimensional bilinear system of the form ˙
z=Az +˙
uBz, where components
of zconsist of multivariable monomials xk
iul
j. Note the similarity to the Koopman
generator PDE (6.19): considering a smooth vector-valued observable g(x,u), inter-
preting ˙
uas new input, and assuming that ∇ug(x,u) lies in the span of g(x,u), this
too leads to a bilinear equation and is equivalent to Carleman linearization.
In general, any vector-valued smooth observable with compact support that is
solely a function of the state x, which evolves according to (6.21), satisfies
(6.22) d
dtg(x) = Lug(x) = ∇xg(x)·f0(x) + ∇xg(x)·

q
X
j=1
fj(x)uj
,
where Ludenotes the non-autonomous Lie operator due to the external control input
u. If ∇xg(x)·fj(x) lies in the span of g(x) for all j= 1, . . . , q, then there exists a set

64 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
of constant matrices Bjso that
(6.23) d
dtg(x) = Ag(x) +
q
X
j=1
ujBjg(x).
leading to a bilinear equation for the observable g. Here, Aand Bjdecompose the Lie
operator into an unforced term Aand forcing terms Bj(compare also Figure 6.3).
The following theorem appears as Thm. 2 in [120] and states the bilinearizability
condition for Eqns. (6.21) and (6.23):
Theorem 6.1. Suppose there is a finite set of Koopman eigenfunctions associated
with the unforced vector field f0,ϕj(x)for j= 1, . . . , n, and these form an invariant
subspace of the Lie derivatives Lfj,j= 1, . . . , q. Then there exists a finite-dimensional
bilinear representation for (6.21), and in turn also for (6.23).
A special class of observables are Koopman eigenfunctions (or eigenfunctions of the
Lie operator), which behave linearly in time by definition. Specifically, smooth eigen-
functions of the Koopman operator associated with the uncontrolled system satisfy
(6.24) d
dtϕ(x) = µϕ(x) + ∇xϕ(x)·

q
X
j=1
fj(x)uj
,
with unforced dynamics determined by its associated eigenvalue µand where the
second terms reflects explicitly how the control input affects these eigenfunctions.
Koopman eigenfunctions are associated with the global behavior of the system and
represent a Koopman-invariant subspace. Equation (6.24) may also represent a bilin-
ear equation (for further details see subsection 6.2.4). Importantly, for systems with
low-rank structure it is possible to describe the dynamics in terms of a small number
of eigenfunctions which sufficiently well describe the global behavior and avoid the
explosion of observables of the Carleman linearization. Even though the system may
not be completely bilinearizable, approximate bilinearizability may be sufficient for
accurate prediction [120]. Furthermore, there exists a vast literature of diagnostic and
control techniques tailored to bilinear systems that can be utilized.
6.2. Data-Driven Control. In contrast to model-based control, there are
many emerging advances in equation-free, data-driven system identification that lever-
age Koopman theory. The relationships between some of the more prominent methods
and their connection to the Koopman operator are shown in Figure 6.3.
6.2.1. Dynamic mode decomposition with control. A major strength of
DMD is the ability to describe complex and high-dimensional dynamical systems in
terms of a small number of dominant modes, which represent spatio-temporal coher-
ent structures. Reducing the dimensionality of the system from n(often millions or
billions) to r(tens or hundreds) enables faster and lower-latency prediction and esti-
mation, which generally translates directly into controllers with higher performance
and robustness. Thus, compact and efficient representations of complex systems such
as fluid flows have been long-sought, resulting in the field of reduced-order modeling.
However, the original DMD algorithm was designed to characterize naturally evolving
systems, without accounting for the effect of actuation and control.
The dynamic mode decomposition with control (DMDc) algorithm by Proctor et
al. [275] extends DMD to disambiguate between the natural unforced dynamics and

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 65
2
6
6
6
6
6
6
6
6
6
6
4
L(x0)
N(x0)
N(x0,u0)
L(u0)
N(u0)
3
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
4
11 12 ⇤14 15
21 22 ⇤24 25
⇤⇤⇤⇤⇤
⇤⇤⇤44 45
⇤⇤⇤⇤⇤
3
7
7
7
7
7
7
7
7
7
7
5
| {z }
K
2
6
6
6
6
6
6
6
6
6
6
4
L(x)
N(x)
N(x,u)
L(u)
N(u)
3
7
7
7
7
7
7
7
7
7
7
5
<latexit sha1_base64="AHxtxodCjCdp20TXMN4iJPo+T6A=">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</latexit>
A B
H
EDMDc
DMDc
KIC
SINDYc
Fig. 6.3: Data-driven finite-dimensional approximation of the Koopman operator de-
fined on the extended state space (x,u) in practice. As with eDMDc it is also possible
to identify evolution dynamics u0=H(x,u) for the control input within the KIC
framework. A state-feedback law u=D(x) may be hidden within Aand can be
identified with knowledge of u. Vector-valued observables are splitted into a linear L
and nonlinear Npart, e.g. g(x):= [LT(x)NT(x)]T.
the effect of actuation. This generalizes the DMD regression to the control form
xk+1 ≈Axk+Buk.(6.25)
For DMDc, snapshot pairs {xj,xj+1 ,uj}m
j=1 of the state variable xand actuation
command uare collected and organized into the following data matrices
(6.26) X=
| |
x1·· · xm−1
| | 
,X0=
| |
x2·· · xm
| | 
,Υ=
| |
u1·· · um−1
| | 
,
where X0is a time-shifted version of X. Then the system matrices Aand Bare
determined jointly by solving the following least-squares optimization problem
min
hA Bi

X0−A BX
Υ


2
2,(6.27)
where the solution is given as [A B] = X0[X
Υ]†.
Originally motivated by intervention efforts in epidemiology [274], DMDc based
on linear and nonlinear measurements of the system has since been used with MPC
for enhanced control of nonlinear systems by Korda and Mezi´c [173] and by Kaiser
et al. [156], with the DMDc method performing surprisingly well, even for strongly
nonlinear systems. DMDc allows for a flexible regression framework, which can accom-
modate undersampled measurements of an actuated system and identify an accurate
and efficient low-order model using compressive system identification [20], which is
closely related to the eigensystem realization algorithm (ERA) [152].
6.2.2. Extended dynamic mode decomposition with control. The DMDc
framework provides a simple and efficient numerical approach for system identifica-
tion, in which the Koopman operator is approximated with a best-fit linear model
advancing linear observables and linear actuation variables. As discussed in subsec-
tion 5.1, eDMD is an equivalent approach to DMD that builds on nonlinear observ-
ables. An extension of eDMD for controlled systems was first introduced in Williams

66 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
et al. [351] to approximate the Koopman operator associated with the unforced system
and to correct for inputs affecting the system dynamics and data. Inputs are han-
dled as time-varying system parameters and the Koopman operator is modeled as a
parameter-varying operator drawing inspiration from linear parameter varying (LPV)
models. This approach has been generalized in Korda et al. [173] to identify the system
matrices Aand Bin the higher-dimensional observable space which disambiguates
the unforced dynamics and control on observables. In particular, the Koopman op-
erator is defined as an autonomous operator on the extended state ˜
x:= [xT,uT]Tas
in (6.15) and observables can be nonlinear functions of the state and input, i.e. g(x,u).
However, in practice, simplifications are employed to allow for convex formulations of
the control problem.
Assuming that the observables are nonlinear functions of the state and linear
functions of the control input, i.e. g(x,u):= [θ1(x), . . . , θp(x), u1, . . . , uq]T∈Rp+q,
and by restricting the dynamics of interest to the state observables θ(x) themselves,
the linear evolution equation to be determined is
zk+1 ≈Azk+Buk,(6.28)
where z∈Rpis the vector-valued observable as in subsection 5.1 defined as
z:=ΘT(x) = 




θ1(x)
θ2(x)
.
.
.
θp(x)




.(6.29)
Analogous to DMDc (6.26), the (time-shifted) data matrices in the lifted space, Z=
ΘT(X) and Z0=ΘT(X0), are evaluated given data X0,X,Υ. The system matrices
A,Bare determined from the least-squares regression problem
min
hA Bi

Z0−A BZ
Υ


2
2
(6.30)
where the solution is given as [A B] = Z0[Z
Υ]†=ΘT(X0)[ ΘT(X)
Υ]†. The state xis
often included in the basis or space of observables, e.g. z= [xT,Θ(x)]T, and can
then be estimated by selecting the appropriate elements of the observable vector z,
so that x=Cz with the measurement matrix C= [In×n0]. If the state vector is not
included as observable, one may approximate the measurement matrix by solving an
equivalent least-squares problem:
min
C

X−CZ


2
2
(6.31)
Alternatively, it has been shown that the state can be estimated using, e.g., multi-
dimensional scaling [164]. If full-state information is not available, but only input-
output data is available, the observable vector zoften needs to be augmented with past
input and output values, as is typically done in system identification [199], following
attractor reconstruction methods, such as the Takens embedding theorem [332].
As eDMD is known to suffer from overfitting, it is important to regularize the
problem based on, for instance, the L1,2-norm for group sparsity [351] or the L1-
norm [154]. eDMDc yields a linear model for the controlled dynamics in the space of
observables through a nonlinear transformation of the state. As with DMDc, it can

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 67
be combined with any model-based control approach. In particular, when combined
with MPC for a quadratic cost function, the resulting MPC optimization problem can
be shown [173] to be independent of the number of observables and to outperform
MPC with a model based on a local linearization or Carleman linearization.
6.2.3. Generalizations of DMDc and eDMDc. The simple formulation of
DMDc and eDMDc as a linear regression problem allows for a number of general-
izations that can be easily exploited for integration into existing linear model-based
control methods. A number of these generalizations are highlighted here.
Mixed observables. eDMDc allows for generalization by using a suitable choice
of observable functions. It has also been proposed [173,276] to include nonlinear
observables of the input, e.g g(u) = u2, or mixed observables of the state and input,
e.g g(x, u) = xu. The advantage of incorporating these measurements has yet to be
demonstrated and a subsequent control optimization is not straightforward. However,
for systems forced by external inputs or parameters this may provide new opportu-
nities to identify their underlying dynamics, e.g. ˙
u=h(x,u) or a state-feedback law
u=d(x) (see also Figure 6.3).
Input and output spaces. Within the KIC framework [276], different domain and
output spaces of the Koopman operator approximation have been examined. This
develops more formally what has been implicitly carried out in eDMDc-like methods
to obtain models for only the state itself. If only observables that are functions of the
state and not of the input are considered, the output space of the Koopman operator
can be restricted to a subspace of the Hilbert space spanned by observables which are
solely functions of the state and not of the extended state space:
(6.32) zx,k+1 =Kxzk=Kx
g(xk)
g(xk,uk)
g(uk)

with vector-valued observables zx:=g(x) and z:= [gT(x),gT(x,u),gT(u)]T. This
is reasonable as the prediction of the future actuation input uis not of interest for the
purpose of control. Analogously, Koopman eigenfunction expansions of observables
must be distinguished based on the domain and output spaces.
Time delay coordinates. Time delay coordinates provide an important and uni-
versal class of measurement functions for systems which display long-term memory
effects. They are especially important in real-world applications where limited access
to full state measurements are available (see subsection 5.2), and they have demon-
strated superior performance for control, compared with models using monomials
as observables [156]. For simplicity, we will consider a sequence of scalar input-
output measurements u(t) and x(t). From these, we may construct a delay vec-
tor of inputs uk:= [uk, uk−1, uk−2, . . . , uk−mu]Tand outputs zk:=g(xk) =
[xk, xk−1, xk−2, . . . , xk−mx]T, respectively. Here, it is assumed m=mx=mu
for simplicity. The dynamics may then be represented as
zk+1 =Azk+Buk,(6.33a)
yk=1 0 . . . 0zk=xk,(6.33b)
where the current state xis recovered from the first component of zk. Both the system
matrix Aand the control matrix Bmust satisfy an upper triangular structure to not
violate causality; otherwise, current states will depend on future states and inputs.
Here, ukis constructed from past inputs, while effectively the system has only a

68 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
single input. Thus, it is recommended to augment zwith past inputs, i.e. zk:=
g(xk,uk−1)=[xk,(x, u)k−1,(x, u)k−2,...,(x, u)k−m], so that the current actuation
value ukappears as a single input to the system [173,157]:
zk+1 =ˆ
Azk+ˆ
buk,(6.34a)
yk=1 0 . . . 0zk=xk.(6.34b)
This delay observable zk=g(xk,uk−1) has also been used in eDMDc if only input-
output data is available [173]. Without the delay information, eDMDc may fail despite
the lifting into a higher-dimensional observable space (as for the examples in subsec-
tion 6.3). By combining (6.33a) and (6.33b) into a single equation, we obtain an
equivalent description to (6.32) with different domain and output spaces:
(6.35) z1,k+1 =yk+1 =CA CBzk
uk.
This formulation is analogous to autoregressive-exogenous (ARX) models in linear
system identification [199] , where the current output is represented as a linear super-
position of past measurements and inputs.
Parametrized models. The control input can also be restricted to a finite set of
discrete values. The dynamics are then modeled as a linear parameter-varying model,
so that each discrete input is associated with an autonomous Koopman operator (6.16)
which can be approximated using DMD or eDMD [264]. The control input is thus
not modeled explicitly, but implicitly, by switching the model whenever the actuation
command switches. The advantage of this formulation is that each model associated
with a discrete control input remains autonomous. Recently, this approach has been
extended to continuous control input by interpolating between the finite set of control
actions, so that the system dynamics are predicted using interpolated models based on
the infinitesimal generator of the Koopman operator, leading to bilinear models [266]
similar to (6.23). This also alleviates performance issues associated with the time
discretization of the models when optimizing their switching times [266,169].
Deep Koopman models for control. Finding a suitable basis or feature space that
facilitates a Koopman-invariant subspace remains challenging. Due to their flexibil-
ity and rich expressivity [281], there is increased interest in using neural networks to
learn complex Koopman representations (see also subsection 5.4). Extending these
architectures to incorporate the effect of control requires careful handling of the con-
trol variable to ensure tractability of the subsequent control optimization, which a
nonlinear transformation of the control variable generally impedes. Building on their
work on deep DMD [359] and Koopman Gramians [360], Liu et al. [198] propose to
decompose the space of observables as three independent neural networks, associated
with parameters θx,θxu, and θu, respectively, to which the input is either the state
variable alone, the control variable alone, or both:
(6.36) zx,k+1 =Kzk=KxKxu Ku
zx,k
zxu,k
zu,k 
with
zx:=g(x,θx),
zxu :=g(x,u,θxu),
zu:=g(u,θu).
The unknown Koopman matrices K•and neural network parameters θ•, where •=x,
xu, or u, can then be optimized jointly [198] or separately by switching alternately
between model update and parameter update, as in [125,196]. While the model
is linear in the latent space, it is not tractable for prediction and control of the

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 69
original system. In order to map back to the original state for predictions and to
solve the control optimization problem, the states and control inputs are also directly
included as observables, so that zx:= [x,g(x,θx)]Tand zu:= [u,g(u,θu)]T. While
zxu :=gT(θxu) is kept the same, it is also possible to neglect Kxu under certain
conditions. In subsequent work [361,128], slightly modified approaches are studied
while maintaining the tractability of the control synthesis problem.
Analogous to eDMDc, a multi-layer perceptron neural network can be used solely
to lift the state space, while the effect of the control input in the lifted space re-
mains linear [125], i.e. zx,k+1 =Azx,k +Bu, where Kx=A,Ku=B,Kxu = 0,
zx:=g(x,θx), and zu:=u. The transformation from the latent space to the origi-
nal state is determined via a least-squares optimization problem analogous to (6.31)
where Zthen represents the output of the neural network zx. Autoencoders provide
a natural framework for learning Koopman representations [202] (see also subsec-
tion 5.4) combining the state lifting operation and inverse transformation. Recently,
this approach has been extended for control by incorporating the control term [125],
so that only the state is encoded into zx,k and the shifted feature state zx,k+1 is de-
coded. These formulations facilitate immediate application of standard linear control
methods. Graph neural networks (GNN) [196] have also been exploited to encode and
decode the compositional structure of the system, modeling similar physical interac-
tions with the same parameters, resulting in a block-structured, scalable Koopman
matrix. The GNNs yield a nonlinear transformation of the state, while the effect of
the control input is assumed to be linear in the transformed latent space as above,
resulting in a quadratic program for control synthesis.
6.2.4. Control in intrinsic Koopman eigenfunction coordinates. Eigen-
functions of the Koopman operator represent a natural set of observables, as their
temporal behavior is linear by design. Furthermore, these intrinsic observables are
associated with global properties of the underlying system. Koopman eigenfunc-
tions have been used for observer design within the Koopman canonical transform
(KCT) [316,317] and within the Koopman reduced-order nonlinear identification and
control (KRONIC) framework [154], which both typically yield a global bilinear rep-
resentation of the underlying system. The Koopman eigenfunctions that are used to
construct a reduced description are those associated with the point spectrum of the
Koopman operator and are restricted to the unactuated, autonomous dynamics. In
particular, we consider control-affine systems of the form
˙
x=f(x) +
l
X
j=1
hj(x)uj,(6.37)
where hj(x) are the control vector fields.
The observable vector is defined as
z:=T(x) = 




ϕ1(x)
ϕ2(x)
.
.
.
ϕp(x)




,(6.38)
where T:Rn→Crepresents a nonlinear transformation of the state xinto eigenfunc-
tion coordinates ϕ(x), which are associated with the unforced dynamics ˙
x=f(x). If
the ϕ’s are differentiable at x, their evolution equation can be written in terms of the

70 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Lie operator (2.15), i.e. ˙ϕ(x) = Lfϕ(x) = f(x)· ∇ϕ(x) = µϕ(x). Hence, we obtain
(6.39) LfT(x) = ΛT(x),
where Λ= diag(λ1, . . . , λp) is a matrix, with diagonal elements consisting of the eigen-
values λiassociated with the eigenfunctions ϕj. Then, the dynamics for observables
zfor the controlled system satisfy (see also (6.24))
˙
z=
f(x) +
l
X
j=1
hj(x)uj
· ∇T (x) = LfT(x) +
l
X
j=1 LhjT(x)uj
(6.40a)
=Λz +
l
X
j=1
bj(z)uj=Λz +B(z)u,(6.40b)
where bj(z):=LhjT(x)|x=Cz and x=Cz estimates the state xfrom observables z.
The system (6.40) is bilinearizable under certain conditions, e.g. if LhjT(x) lies in the
span of the set of eigenfunctions for all j[141]. This admits a representation of the Lie
operators of the control vector fields in terms of the Koopman eigenfunctions [316].
For u=0the dynamics (6.40b) are represented in a Koopman-invariant subspace
associated with the unforced system. For u6=0the control term B(z) describes how
these eigenfunctions are affected by u. The advantage of this formulation is that
the dimension of the system scales with the number of eigenfunctions, and often a
few dominant eigenfunctions may be sufficient to capture the principal behavior. In
general, the eigenfunctions can be identified using eDMD, kernel-DMD or other vari-
ants, and the model (6.40b) may be well represented as long as their span contains
f(x), hj(x), and x[318]. Alternatively, the KRONIC framework seeks to learn eigen-
functions directly, which are sparsely represented in a dictionary of basis functions.
The control term B(z) can either be determined from the gradient of the identified
eigenfunctions for known hj(x) or identified directly [155] for unknown hj(x) by sim-
ilarly expanding it in terms of a dictionary. Multiple works have since been published
focusing on the direct identification of Koopman eigenfunctions [174,127,261]. In
any case, validating the discovered eigenfunctions, i.e. ensuring that these behavior
linearly as predicted by their associated eigenvalue, is critical for prediction tasks.
6.3. Koopman-based control strategies. In contrast to linear systems, the
control of nonlinear systems remains an engineering grand challenge. Many real-world
systems represent a significant hurdle to nonlinear control design, leading to linear
approximations which produce suboptimal solutions. Linear Koopman-based models
aim to capture the dynamics of a nonlinear system, and thus leverage linear control
methods. The predictive power of linear models is increased within the Koopman
operator framework by replacing (or augmenting) the state space with nonlinear ob-
servables. Nonlinear observables are not completely new in the context of control
theory, but appear also as the (control) Lyapunov function for stability, the value
function in optimal control, or as measurement functions in input-output lineariza-
tion. Normally the state itself is included in the set of observables. While this is
problematic from the perspective of finding a linear, closed form approximation to
the Koopman operator associated with the underlying nonlinear system [48], it allows
one to easily measure the state given the model output and the cost function associ-
ated with the original control problem. If the state is not included, a measurement
matrix or function mapping from the nonlinear observables to the state vector needs
to estimated along with the system matrices, typically introducing additional errors.

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 71
Fig. 6.4: Tail-actuated robotic fish (top left corner) developed by the Smart Microsys-
tems Lab at Michigan State University for trajectory tracking using Koopman LQR
(top row) and backstepping (bottom row). Reference trajectory is displayed in blue,
controlled trajectory in red. Reproduced with permission, from Mamakoukas et al.
2019 Robotics: science and systems [111].
Koopman-based frameworks are amenable to the application of standard linear
estimation and control theory, and these have been increasingly used in combination
with optimal control [48,111,4,154,173,155,13,264,3,264] using DMDc and eD-
MDc models (see subsection 6.2.1 and 6.2.2). In the simplest case, a linear quadratic
regulator may be used, which has been successfully demonstrated in experimental
robotics [111,112], as in Figure 6.4. In addition to modifying the eigenvalues of the
closed-loop system as in pole placement, the shape of Koopman eigenfunctions may
be modified directly using eigenstructure assignment [130]. Several other approaches
build on a global bilinearization [316,154,141] of the underlying system in terms
of Koopman eigenfunctions (see subsection 6.2.4). Under certain conditions, this
may allow one to stabilize the underlying dynamical system with feedback lineariza-
tion [120]. Feedback stabilization for nonlinear systems can also be achieved via a
control Lyapunov function [316] expressed in Koopman eigenfunctions, for which the
search can be formulated as a convex optimization [141,142]. Stabilizing solutions
have also been determined by constructing a Lyapunov function from stable Koop-
man models obtained by imposing stability constraints on the Koopman matrix [110].
Sequential action control [10] is an alternative to MPC in experimental robotics [3,4],
which optimizes both the control input and the control duration. This is a convex
approach applicable to control-affine systems, and as such, may also be combined with
Koopman bilinearizations.
Although the predictive power of Koopman models can be sensitive to the particu-
lar choice of observables and the training data, in particular receding-horizon methods
such as MPC (see subsection 6.1.1) provide a robust control framework that system-
atically compensate for model uncertainties by continually updating the optimization
problem and taking into account new measurements. Incorporating Koopman-based
models into MPC was first introduced in Korda & Mezi´c [173] using eDMDc. The
eDMDc results in a convex quadratic programming problem whose computational
cost remains comparable to that of linear MPC and is independent of the number of
observables if the optimization problem [173]. MPC has also been used for control
in Koopman eigenfunction coordinates [154,155], for optimal switching control using

72 S. L. BRUNTON, M. BUDIˇ
SI´
C, E. KAISER, J. N. KUTZ
Fig. 6.5: Model predictive control of compositional systems by combining Koopman
operators with graph neural networks for an object-centric embedding. Reproduced
with permission from Li et al. 2019 ICLR [196].
linear parameter-varying eDMD models [264] associated with discrete control inputs,
and interpolated Koopman models [263,266] extending to continuous control inputs.
The latter approach combines the data efficiency of estimating autonomous Koop-
man models associated with discrete control inputs with continuous control, which
is achieved by relaxation of the switching control approach. Convergence can be
proven, albeit assuming an infinite-data limit and an infinite number of basis func-
tions. In general, guarantees on optimality, stability, and robustness of the controlled
dynamical system remain limited. Of note are Koopman Lyapunov-based MPC that
guarantees closed-loop stability and controller feasability [246,307]. More recently,
deep learning architectures have been increasingly used to represent the nonlinear
mapping into the observable space. These have also been combined with MPC with
promising results [196] (see also Figure 6.5).
The success of Koopman-based MPC belies the practical difficulties of approxi-
mating the Koopman operator. On the one hand, there exist only a few systems with
a known Koopman-invariant subspace and eigenfunctions, on which models could
be analyzed and evaluated. On the other hand, even the linearity property of the
Koopman eigenfunctions associated with the model is rarely validated and MPC ex-
hibits an impressive robustness when handling models with very limited predictive
power. Thus, there have been impressive practical successes, but there are open
gaps in understanding how well the Koopman operator is actually approximated
and what possible error bounds can be given. In addition to theoretical advances,
Koopman-based control has been increasingly applied in real-world problems such
as power grids [176,247], high-dimensional fluid flows [205,13,265,266], biologi-
cal systems [128], chemical processes [246,307], human-machine systems [39], and
experimental robotic systems [4,44,111,43,104].
Multiple variants of Koopman-based control have been demonstrated in numer-
ical and experimental settings. Here, we illustrate model predictive control (MPC,
see subsection 6.1.1 for details) combined with different Koopman-based models for
several nonlinear systems including a bilinear DC motor (see also [173]), the forced
Duffing oscillator, and the van der Pol oscillator. An overview of the parameters,

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 73
models and controls are presented in Tab. 6.1 with results summarized in Figure 6.6
and Figure 6.7. Consider the single-input single-output (SISO) system
Examples
Bilinear Forced Duffing Van der Pol
DC motor oscillator oscillator
La= 0.314, a= 1, b=−1, µ= 2
Ra= 12.345, d=−0.3,
f0= 0.5,
km= 0.253, ω= 1.2
J= 0.00441,
B= 0.00732,
τl= 1.47, ua= 60
Training
Timestep ∆t= 0.01 ∆t= 0.1 ∆t= 0.1
Domain x∈[−1,1]2x∈[−1.5,−1]2x∈[−2,2]2
Input u∈[−1,1] u∈[−0.5,0.5] u∈[−5,5]
Model
DMDc z=x z =x z =x
p= 3 p= 3 p= 3
DDMDc z=g1(y, u)z=g1(y, u)z=g1(y , u)
p= 3 p= 3 p= 3
z=g16(y, u) — z=g8(y, u)
p= 33 p= 17
eDMDc z= [g1,ξ]z= [g1,ξ]z= [g1,ξ]
p= 103 p= 103 p= 103
Control
Reference r(t) = 1
2cos(2
3πt)r(t) = −sin(t)r(t) = −sin(t)
Horizon T= 0.1T= 1 T= 3
Weights Q= 1, R= 0.1Q= 1, R= 0.1Q= 1, R= 0.01
Constraints −0.4≤x2≤0.4−0.8≤x2≤0.8−0.8≤x2≤0.8
−1≤u≤1−0.5≤u≤0.5−5≤u≤5
Table 6.1: Parameters for numerical control examples.
˙
x=f(x, u, t)(6.41a)
y=Cx(6.41b)
to be controlled using MPC to track a time-varying reference signal r(t). The cost
function is defined as
(6.42) J=||yN−rN||2
Q+
N−1
X
k=0 ||yk−rk||2
Q+R||uk||2,
where the first term represents the terminal cost and the subscript kcorresponds
to the kth timestep tk:=k∆t. The control sequence {u0, u1, . . . ,uN−1}over the
N−step horizon is determined by minimizing the cost function subject to the model
dynamics and state and control constraints, ymin ≤y≤ymax and umin ≤u≤umax,
respectively. These results are compared with a local linearization of the system
dynamics (referred to as LMPC in the following).

74 S. L. BRUNTON, M. BUDIˇ
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(b) Forced Duffing Osc.(a) Bilinear DC motor
PredictionControl
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˙x1=Ra
La
x1km
La
x2u+ua
La
˙x2=B
Jx2+km
Jx1u⌧l
J
y=x2
(c) Van der Pol Oscillator
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˙x1=x2
˙x2=ax1+bx3
1+dx2+f0cos(!t)
+ (3 + cos(x1))u
y=x2
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˙x1=x2
˙x2=x1µ(x2
11)x2+u
y=x2
/
Fig. 6.6: Reference tracking using MPC combined with different Koopman operator
models illustrated for three dynamical systems.
While DMDc relies on full-state measurements, only the output y=Cx is evalu-
ated in the cost function (6.3). For the remaining models, time delay coordinates are
employed and we define the time-delayed observable vector as
(6.43) gm(y, u)=[yk,(y, u)k−1,...,(y , u)k−m]T
with (y, u)k:= [yk, uk] and where mdenotes the number of delays. The DDMDc

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 75
(b) Forced Duffing Osc.(a) Bilinear DC motor (c) Van der Pol Oscillator
L-MPCDMDcEDMDcDDMDc
(m=1)
Fig. 6.7: N-step controllable set for the reference tracking problems in Figure 6.6.
The trajectory of the unforced system is displayed in gray in the top row. The same
initial condition (gray circle) is used in Figure 6.6.
model, which is short for DMDc applied to delay coordinates, is based on time delay
coordinates discussed in subsection 6.2.3. However, past control values are also in-
cluded in the observable vector gm(y, u). The observable vector associated with the
eDMDc models is defined as z= [gm(y, u),ξ(y , u)]T(analogous to [173]), so that the
output yitself is a coordinate of the observable. The variable ξ(y, u) represents a
particular choice of lifted nonlinear observables. Here, we employ thin plate spline ra-
dial basis functions, ξj=r2ln ||gm(y, u)−cj|| with r= 1 and 100 randomly sampled
centers within the considered domain. If the output is not included as an observable,
an estimate of the mapping from ξ(y, u) to the output yis required in order to evalu-
ate the cost function. By incorporating the output in the model this can be avoided;
however, this also restricts the model [48].
Training data for all examples is collected from 102initial conditions uniformly
distributed within the examined domain stated in Tab. 6.1. A different random,
binary actuation signal within the range of the input constraints is applied to sample

76 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
each trajectory in the training set. Each trajectory consists of 103samples, so that in
total the training set consists of 105samples. Data efficiency in system identification
is an important topic, as only limited measurements may be available to train models
and each model may have different requirements. For instance, as little as 10 samples
may be sufficient to train a DMDc model that successfully stabilizes an unstable fixed
point using MPC (see, e.g., Fig.7 in [156]).
The prediction and control performance is summarized in Figure 6.6. The first
part illustrates the prediction accuracy for a single trajectory and shows the mean
relative prediction error e=1
NPN
k=1 (yk−ˆyk)/ykwith N=T/δt, where yand ˆy
are the true and predicted values, respectively. The bar displays the median value
of efor each model evaluated over 300 random initial conditions, and the black lines
indicate extremal values. The second part shows the control performance for the same
initial condition. The cost is evaluated by (6.3). We note that the model based on a
local linearization (LMPC) can be used to successfully control the bilinear DC motor
without constraint violations or infeasibility issues, in contrast to [173], by using a
smaller prediction horizon for MPC. A significantly larger horizon would exceed the
predictability of the model and result in infeasibility and performance issues. While
the prediction accuracy may be comparable with other models, DMDc-MPC fails
for the DC motor and is inferior in the other examples for tracking a time-varying
reference signal. However, it may be possible to stabilize an unstable fixed point using
DMDc in certain cases, despite having limited predictive power [156]. DDMDc−m
models, where mdenotes the number of delays, achieves comparable predictive and
control capabilities as eDMDc; intriguingly, even with only one delay m= 1 for the
Duffing and van der Pol systems. MPC is surprisingly robust with respect to model
errors. Even in cases with limited predictive power (compare, e.g., the early divergence
of LMPC for the van der Pol oscillator), the system is successfully controlled with
MPC. The robustness of MPC w.r.t. modeling inaccuracies may hide the fact that
the model may actually be a poor representation of the underlying dynamics or its
Koopman operator.
In Figure 6.7, the control performance is assessed over the domain of the training
data, as the prediction performance can vary significantly depending on the initial
condition. Each initial condition is color-coded with the cost J(6.3), which is eval-
uated for a duration of 1, 4, and 3 time units for the three examples (left to right),
respectively. DDMDc−1 and eDMDc, both using m= 1 delays, are nearly indis-
tinguishable, while the model dimension is p= 3 for the first and p= 103 for the
latter. This raises the important, and so far relatively unanswered question, of which
set of basis functions should be employed to train a Koopman model. Time delay
coordinates perform extremely well for prediction and control in a variety of systems
and have the benefit of well-developed theoretical foundations. More sophisticated
basis functions may perform similarly well, but it is not clear that they warrant the
increased design effort given the comparable performance.
6.4. Stability. Stability is central in the analysis and control of real-world sys-
tems. The Lyapunov function plays an essential role in global stability analysis and
control design of nonlinear systems. Finding or constructing a Lyapunov function
poses a significant challenge for general nonlinear systems. Importantly, spectral prop-
erties of the Koopman operator have been associated with geometric properties such
as Lyapunov functions and measures, the contracting metric, and isostables which
are used to study global stability of the underlying nonlinear system. For instance,
building on [344], it has been shown [225,223] that Lyapunov’s second method im-

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 77
plicitly uses an operator-theoretic approach and Lyapunov functions represent special
observables advanced through the Koopman operator.
Consider a dynamical system ˙
x=f(x) with fixed point x∗. A Lyapunov function
V:Rn→Rmust satisfy
(6.44) ˙
V(x) = ∇V·f(x)≤0 for all x6=x∗.
If ˙
V(x)≤0 the system is asymptotically stable in the sense of Lyapunov. There is
also an equivalent definition of the Lyapunov function for discrete-time systems. The
dynamics of the Lyapunov equation (6.44) can be formulated directly in terms of the
generator of the Koopman operator family acting on a nonnegative observable (6.19)
(here without input u=0). Thus, the Lyapunov function decays everywhere under
action of the Lie operator and is related to its spectral properties and those of the
Koopman operator semigroup. The global stability of a fixed point can be estab-
lished through the existence of a set of C1eigenfunctions of the Koopman operator
associated with the eigenvalues of the Jacobian of the vector field with the Koopman
eigenfunctions being used to define a Lyapunov function and contracting metric [223].
A corresponding numerical scheme based on a Taylor expansion may then be used
to compute stability properties, including the domain of attraction. These methods
have been extended to characterize the global stability properties of hyperbolic fixed
points, limit cycles, and non-analytical eigenfunctions [224].
6.5. Observability and controllability. Observability and controllability play
a crucial role in the design of sensor-based estimation and control. However, they have
limited validity when applied to linearized nonlinear systems. There exist analogous
(local) controllability and observability criteria for nonlinear systems using Lie deriv-
atives [133,165]. However, these criteria are typically restricted to a specific class
of systems and their evaluation remains challenging for even low-dimensional sys-
tems, becoming computationally intractable for high-dimensional systems. Operator-
theoretic approaches provide new opportunities to assess observability and control-
lability of high-dimensional nonlinear systems using linear techniques and obtaining
corresponding estimates from given data. Generally, two states are indistinguishable
if their future output sequences are identical. Thus, a nonlinear system is considered
nonlinearly observable if any pair of states is distinguishable. Common criteria may
be divided into two classes: rank conditions and Gramians [49]. Rank conditions yield
a binary decision, while Gramians are used to examine the degree of observability and
controllability. The latter is also used for balanced truncation to obtain reduced-order
models that balance a joint observability and controllability measure. In contrast, cor-
responding metrics for nonlinear systems rely on the derivation of its linear equivalent
using Lie derivatives, which can be challenging. Reformulating a nonlinear system into
a linear system allows one to apply linear criteria in a straightforward manner. Thus,
Koopman operator-theoretic techniques have been increasingly studied in the context
of observability and controllability of nonlinear systems.
An immediate extension of linear rank conditions to nonlinear systems is achieved
by applying the linear criteria to the representation in the lifted state space. In [317],
nonlinear observability is evaluated in a Koopman-based observer, which provides
a linear representation of the underlying system in Koopman eigenfunction coordi-
nates (see also subsection 6.2.4). The underlying nonlinear system is then considered
nonlinearly observable if the pair (A,CH) is observable, which can be determined
via the rank condition of the corresponding observability matrix. These ideas have
been applied to study pedestrian crowd flow [26], extended further to input-output

78 S. L. BRUNTON, M. BUDIˇ
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nonlinear systems resulting in bilinear or Lipschitz formulations [316], and used to
compute controllability and reachability [120]. The observability and controllability
Gramians can also be computed in the lifted observable space [317]. In this case,
the observability/controllability of the underlying system is related to the observabil-
ity/controllability of the observables. The underlying assumption is that the state,
input, and output are representable in terms of a few Koopman eigenfunctions. More
recently, a deep learning DMD framework [359] has been extended to incorporate
Koopman Gramians to maximize internal subsystem observability and disturbance
rejection from unwanted noise from other subsystems [362].
As shown in the following example, if a system can be represented in a Koopman-
invariant subspace, linear criteria applied to the Koopman representation can be
equivalent to the corresponding nonlinear criteria applied to analyze the underly-
ing the nonlinear system. However, a Koopman-invariant subspace is rarely obtained
and open questions remain, e.g., whether an approximate Koopman-invariant sub-
space may be sufficient and how observability and controllability estimates depend on
the specific choice of observable functions.
Example: Nonlinear system with single fixed point and a slow manifold. We ex-
amine the relationship between controllability of the original nonlinear system and
the Koopman system using the corresponding controllability criteria. We consider
the system examined in subsection 2.4, with the addition of control:
(6.45) ˙
x=f0(x) + f1(x)u=µx1
λ(x2−x2
1)+0
1u.
The additional control input uactuates only the second state x2. The system becomes
unstable for either λ > 0 or µ > 0. Choosing observables y= (x1, x2, x2
1)T, the system
admits a fully linear, closed description in a three-dimensional Koopman invariant
subspace:
(6.46) d
dt 
y1
y2
y3
=A
y1
y2
y3
+Bu=
µ0 0
0λ−λ
0 0 2µ

y1
y2
y3
+
0
1
0
u
with constant control vector B.
The controllability for a control-affine system ˙
x=f0(x) + Pq
j=1 fj(x)ujcan be
assessed through the following theorem. We define the Lie derivative of gwith re-
spect to fas Lfg:=∇g·f, the Lie bracket as [f,g]:=∇g·f− ∇f·g, and the
recursive Lie bracket as adfg= [f,g] = ∇g·f− ∇f·g. Hunt’s theorem (1982) [144]
states that a nonlinear system is (locally) controllable if there exists an index ksuch
that C= [f0,f1,...,fq,adf0f1,...,adf0fq,...,adfk
0f1,...,adfk
0fq] has nlinearly inde-
pendent columns. The controllability matrix for the nonlinear system (6.45) is given
by
(6.47) C=f0adf0f1. . . adfk
0f1=0
1 0
−λ 0
λ2. . . 0
(−1)kλk,
which has rank 1 for any k. The system is uncontrollable in x1direction.
Analogously, we can construct the linear controllability matrix for the Koopman
system (6.46):
(6.48) C=B AB A2B=
0 0 0
1λ λ2
0 0 0 
,

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 79
which is also of rank 1. Another very useful test is the Popov-Belevitch-Hautus
(PBH) test connecting controllability to a relationship between the columns of Band
the eigenspace of A. The PBH test states that the pair (A,B) is controllable iff the
column rank of [(A−αI)B] is equal to nfor all eigenvalues α(A)∈C. This test
confirms that the rank is n= 3 for the eigenvalue λand 1 for both eigenvalues µ
and 2µassociated with (6.46). Note that (6.45) can still be stabilized as long as µis
stable.
6.6. Sensor and actuator placement. Optimizing sensor and actuator loca-
tions for data collection and decision-making is a crucial and challenging task in any
real-world application. Optimal placement of sensors and actuators amounts to an in-
tractable brute force search, as these represent combinatorial problems suffering from
the curse of dimensionality. Rigorous optimization remains an open challenge for even
linear problems; thus, approaches generally rely on heuristics. In this realm, sparse
sampling and greedy selection techniques, such as gappy POD [103] and DEIM [91],
and more recently sparsity-promoting algorithms, such as compressed sensing [61,90],
have played an increasingly important role in the context of sensor/actuator selec-
tion [214]. These methods rely on exploiting the ubiquitous low-rank structure of
data typically found in high-dimensional systems. Operator-theoretic methods fit well
into this perspective, as they are able to provide a tailored feature basis capturing
global, dynamically persistent structures from data. This can be combined with exist-
ing heuristic selection/sampling methods which have been demonstrated on a range
of practical applications, such as environmental controls in buildings [346,105,300],
fluid dynamics [158,215], and biology [129].
Importantly, operator-theoretic methods generalize to nonlinear systems, e.g.
for estimating nonlinear observability and controllability, providing practical means
to systematically exploit nonlinear system properties for sensor/actuator placement
within a linear framework. As already noted, controllability and observability Grami-
ans can be generalized for nonlinear systems based on the Koopman and Perron-
Frobenius operators (or Liouville and adjoint Liouville operators as their continuous-
time counterpart) and subsequently used for sensor and actuator placement by max-
imizing the support (or the L2 norm) of the finite-time Gramians. In [346,305],
set-oriented methods have been utilized to approximate the (adjoint) Lie opera-
tors, i.e. the domain is discretized into small cells and few cells are selected as sen-
sors/actuators, and the location can be optimized by solving a convex optimization
problem. A greedy heuristic approach based on these ideas is proposed in [105], which
further investigates different criteria such as maximizing the sensing volume (sensor
coverage), response time and accuracy (relative measure transported to the sensor
in finite time) and incorporating spatial constraints. The framework has been fur-
ther extended to incorporate uncertainty [300]. More recently, observability Gramians
based on the Koopman operator have been used to inform optimal sensor placement
in the presence of noisy and temporally sparse data [128]. Sensor placement is here
facilitated by maximizing the finite-time output energy in the lifted space. More
broadly, balanced truncation and model reduction has recently been used for efficient
placement of sensors and actuators to simultaneously maximize the controllability
and observability Gramians [216], which may be promising direction to extend to
nonlinear systems by using Gramians in the lifted observable space.

80 S. L. BRUNTON, M. BUDIˇ
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C, E. KAISER, J. N. KUTZ
7. Discussion and outlook. In this review, we have explored the use of Koop-
man operator theory to characterize and approximate nonlinear dynamical systems in
a linear framework. Finding linear representations of nonlinear systems has a broad
appeal, because of the potential to enable advanced prediction, estimation, and con-
trol using powerful and standardized linear techniques. However, there appears to
be a conservation of difficulty with Koopman theory, where nonlinear dynamics are
traded for linear but infinite dimensional dynamics, with their own associated chal-
lenges. Thus, a central focus of modern Koopman analysis is to find a finite set of
nonlinear measurement functions, or coordinate transformations, in which the dynam-
ics appear linear, and the span of which may be used to approximate other quantities
of interest. In this way, Koopman theory follows in the tradition of centuries of work
in mathematical physics to find effective coordinate systems to simplify the dynam-
ics. Now, these approaches are augmented with data-driven modeling, leveraging
the wealth of measurement data available for modern systems of interest, along with
high-performance computing and machine learning architectures to process this data.
This review has also described several leading data-driven algorithms to approxi-
mate the Koopman operator, based on the dynamic mode decomposition (DMD) and
variants. DMD has several advantages that have made it widely adopted in a variety
of disciplines. First, DMD is purely data-driven, as it does not require governing
equations, making it equally applicable to fluid dynamics and neuroscience. In ad-
dition, the DMD algorithm is formulated in terms of simple linear algebra, so that
it is highly extensible, for example to include control. For these reasons, DMD has
been applied broadly in several fields. Although it is possible to apply DMD to most
time-series data, researchers are still developing diagnostics to assess when the DMD
approximation is valid and to what extent it is useful for prediction and control.
Despite the widespread use of DMD, there are still considerable challenges as-
sociated with applying DMD to strongly nonlinear systems, as linear measurements
are often insufficient to span a Koopman-invariant subspace. Although significant
progress has been made connecting DMD to nonlinear systems [353], choosing non-
linear measurements to augment the DMD regression is still not an exact science.
Identifying measurement subspaces that remain closed under the Koopman opera-
tor is an ongoing challenge [48]. In the past decade, several approaches have been
proposed to extend DMD, including with nonlinear measurements, time-delayed ob-
servations, and deep neural networks to learn nonlinear coordinate transformations.
These approaches have had varying degrees of success for systems characterized by
transients and intermittent phenomena, as well as systems with broadband frequency
content associated with a continuous eigenvalue spectrum. It is expected that neural
network representations of dynamical systems, and Koopman embeddings in partic-
ular, will remain a growing area of interest in data-driven dynamics. Combining the
representational power of deep learning with the elegance and simplicity of Koop-
man embeddings has the potential to transform the analysis and control of complex
systems.
One of the most exciting areas of development in modern Koopman theory is
around its use for the control of nonlinear systems [257]. Koopman-based control is
an area of intense focus, as even a small amount of nonlinearity often makes con-
trol quite challenging, and an alternative linear representation may enable dramatic
improvements to robust control performance with relatively standard linear control
techniques. Model predictive control using DMD and Koopman approximations is
particularly interesting, and has been applied to several challenging systems in recent
years. However, there are still open questions about how much of this impressive

MODERN KOOPMAN THEORY FOR DYNAMICAL SYSTEMS 81
performance is due to the incredible robustness of MPC as opposed to the improved
predictive capabilities of approximate Koopman models. The goal of more effective
nonlinear control will likely continue to drive applied Koopman research.
Despite the incredible promise of Koopman operator theory, there are still several
significant challenges that face the field. In a sense, these challenges guarantee that
there will be interesting and important work for decades to come. There are still
open questions about how the choice of observables impacts what can be observed
in the spectrum. Similarly, there is no return path from a Koopman representation
back to the governing nonlinear equations. More generally, there is still little known
about how properties about the nonlinear system ˙
x=f(x) carry over to the Koop-
man operator, and vice versa. For example, relationships between symmetries in the
nonlinear dynamics and how they manifest in the Koopman spectrum is at the cusp
of current knowledge. A better understanding about how to factor out symmetries
and generalize the notion of conserved quantities and symmetries to near-invariances
may provide insights into considerably more complex systems. The community is still
just now beginning to rectify the local and global perspectives on Koopman. In addi-
tion, most theory has been formulated for ODEs, and connections to spatiotemporal
systems are still being developed. Finally, there are open questions around whether
or not there is a Heisenberg uncertainty principle for the Koopman operator.
In addition to these theoretical open questions, there are several areas of applied
research that are under active development. Applied Koopman theory, driven by the
dynamic mode decomposition, has largely been applied within the fluid dynamics
community. Koopman’s original theory was a critical piece in resolving Boltzmann’s
ergodic hypothesis for gas dynamics. It is likely that Koopman theory will provide
similar enabling technology for the characterization of fully turbulent fluid dynamics,
which have defied macroscopic coarse-graining for over a century. Unifying algorithmic
innovations is also an ongoing challenge, as it is not always obvious which algorithm
should be used for a particular problem. Several open source software libraries are
being developed to ease this burden, including
•PyDMD (https://github.com/mathLab/PyDMD); and
•PyKoopman (https://github.com/dynamicslab/pykoopman).
As a parting thought, it is important to note that nonlinearity is one of the most fas-
cinating features of dynamical systems, providing a wealth of rich phenomena. In a
sense, nonlinearity provides an amazing way to parameterize dynamics in an extremely
compact and efficient manner. For example, with a small amount of cubic nonlinear-
ity in the Duffing equation, it is possible to parameterize continuous frequency shifts
and harmonics, whereas this may require a comparatively complex Koopman param-
eterization. Realistically, the future of dynamical systems will utilize the strengths
of both traditional nonlinear, geometric representations along with emerging linear,
operator theoretic representations.
Acknowledgements. We would like to thank Igor Mezi´c for early discussions
and encouragement about this review. We also thank Shervin Bagheri, Bing Brunton,
Bethany Lusch, Frank Noe, Josh Proctor, Clancy Rowley, and Peter Schmid for many
fruitful discussions on DMD, Koopman theory, and control.

82 S. L. BRUNTON, M. BUDIˇ
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