![Performance J [0,T ] of the AEˆxAEˆ AEˆx ae 0:T (cyan) and MHEˆxMHEˆ MHEˆx mhe 0:T (red) for different lengths N of the problem P N compared to the performance achieved by full solutionˆxsolutionˆ solutionˆx * 0:T (green).](https://www.researchgate.net/publication/388529196/figure/fig1/AS:11431281306490177@1738294064367/Performance-J-0-T-of-the-AExAE-AEx-ae-0T-cyan-and-MHExMHE-MHEx-mhe-0T-red_Q320.jpg)
Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
Performance guarantees for optimization-based
state estimation using turnpike properties
Julian D. Schiller, Lars Gr¨
une, and Matthias A. M¨
uller
Abstract—In this paper, we develop novel accuracy and
performance guarantees for optimal state estimation of
general nonlinear systems (in particular, moving horizon
estimation, MHE). Our results rely on a turnpike property
of the optimal state estimation problem, which essentially
states that the omniscient infinite-horizon solution involv-
ing all past and future data serves as turnpike for the
solutions of finite-horizon estimation problems involving a
subset of the data. This leads to the surprising observation
that MHE problems naturally exhibit a leaving arc, which
may have a strong negative impact on the estimation accu-
racy. To address this, we propose a delayed MHE scheme,
and we show that the resulting performance (both averaged
and non-averaged) is approximately optimal and achieves
bounded dynamic regret with respect to the infinite-horizon
solution, with error terms that can be made arbitrarily small
by an appropriate choice of the delay. In various simulation
examples, we observe that already a very small delay in
the MHE scheme is sufficient to significantly improve the
overall estimation error by 20–25% compared to standard
MHE (without delay). This finding is of great importance
for practical applications (especially for monitoring, fault
detection, and parameter estimation) where a small delay
in the estimation is rather irrelevant but may significantly
improve the estimation results.
Index Terms—Moving horizon estimation (MHE), estima-
tion, nonlinear systems, optimal control, turnpike property
I. INTRODUCTION
Reconstructing the internal state trajectory of a dynamical
system based on a batch of measured input-output data is
an important problem of high practical relevance. This can
be accomplished, for example, by solving an optimization
problem to find the best state and disturbance trajectories
that minimize a suitably defined cost function depending
on the measurement data. If all available data is taken into
account, this corresponds to the full information estimation
(FIE) problem. However, if the data set or the underlying
model is very large or only a limited amount of computation
time or resources is available (as is the case with, e.g., online
state estimation), the optimal solution to the FIE problem is
usually difficult (or even impossible) to compute in practice.
For this reason, it is essential to find a reasonable approx-
imation, which can be done, e.g., by means of a sequence
This work was supported by the Deutsche Forschungsgemein-
schaft (DFG, German Research Foundation), Projects 426459964 and
499435839. (Corresponding author: Julian D. Schiller)
Julian D. Schiller and Matthias A. M¨
uller are with the Leibniz University
Hannover, Institute of Automatic Control, 30167 Hannover, Germany. (e-
mail: {schiller,mueller}@irt.uni-hannover.de).
Lars Gr¨
une is with the University of Bayreuth, Chair of Applied
Mathematics, 95447 Bayreuth, Germany. (e-mail: lars.gruene@uni-
bayreuth.de).
of truncated optimal estimation problems, each of which uses
only a limited time window of the full data set. In the case
of online state estimation, this corresponds to moving horizon
estimation (MHE), where in each discrete time step an optimal
estimation problem is solved with a data set of fixed size.
Related work: Current research in the field of MHE is
primarily concerned with stability and robustness guarantees,
see, e.g., [1, Ch. 4] and [2]–[6]. These works essentially show
that under suitable detectability conditions, the estimation error
of MHE (i.e., the deviation between the estimated state and the
real unknown system state) converges to a neighborhood of the
origin, whose size depends on the true unknown disturbance.
However, results on the actual performance of MHE, and in
particular on the approximation accuracy and performance
loss (i.e., the regret) with respect to a certain challenging
benchmark estimator, are lacking.
For linear systems, regret-optimal filters for state estimation
are designed in [7], [8], that minimize the regret with respect
to a clairvoyant (acausal) benchmark filter having access to
future measurements. This approach is extended in [9], where
an exact solution to the minimal-regret observer is provided
utilizing the system level synthesis framework, compare also
[10], [11] in the context of regret-optimal control. In [12],
an MHE scheme is proposed that provides regret guarantees
with respect to an arbitrary comparative (e.g., the clairvoyant)
observer. This approach is extended to nonlinear systems
in [13], but requires a restrictive convexity condition on the
problem and disturbance- and noise-free data.
Whereas performance guarantees for state estimators are
generally rather rare and usually restricted to linear systems,
they often play an important role in nonlinear optimal con-
trol, especially when the overall goal is an economic one.
Corresponding results usually employ a turnpike property of
the underlying optimal control problem, cf. [14], [15]. This
property essentially implies that optimal trajectories stay close
to an optimal equilibrium most of the time (or in general
an optimal time-varying reference), which is regarded as the
turnpike. Turnpike-related arguments are an important tool
for assessing the closed-loop performance of model predictive
controllers with general economic costs on finite and infinite
horizons, cf. [16]–[18]. Necessary and sufficient conditions for
the presence of the turnpike phenomenon in optimal control
are discussed in, e.g., [19]–[22] and are usually based on
dissipativity, controllability, and suitable optimality conditions.
Contribution: We investigate and characterize the turnpike
phenomenon in general nonlinear optimal state estimation
problems and discuss conditions for its occurrence (Sec-
tion III). This property essentially requires that the omniscient
arXiv:2501.18385v1 [math.OC] 30 Jan 2025
(acausal) infinite-horizon solution involving all past and future
data serves as turnpike for the solutions of state estimation
problems involving only a finite subset of the data. This leads
to the surprising observation that MHE problems naturally
exhibit a leaving arc, which may have a strong negative impact
on the estimation accuracy. Based on this new insight, we
propose a delayed MHE scheme in Section IV, where the
delay effectively counteracts the leaving arc. We show that the
performance of the delayed MHE scheme is approximately
optimal and achieves bounded dynamic regret with respect
to the infinite-horizon solution, with error terms that can be
made arbitrarily small by an appropriate choice of the delay.
As a result, MHE (with delay) is able to track the accuracy
and performance of the omniscient infinite-horizon estimator
(Sections IV-A and IV-B). Moreover, we propose a novel
turnpike prior for MHE formulations with prior weighting,
which—in contrast to the standard filtering or smoothing
prior—can be shown to converge to a neighborhood around
the turnpike that can be made arbitrarily small by design
(Section IV-C). Furthermore, we consider the special case of
MHE for offline state estimation (Section V) and show that
good performance of a state estimator directly implies small
estimation errors with respect to the true unknown system
state (Section VI). We illustrate our theoretical findings with
several simulation examples from the literature (Sections VII),
including a continuously stirred tank reactor and a highly
nonlinear quadrotor model with 12 states. For each example,
we can observe that the turnpike phenomenon is present in
MHE. Moreover, we find that even a delay of very few steps
in the MHE scheme improves the overall estimation error by
20-25 % compared to standard MHE (without delay).
Compared to the preliminary conference version [23], in this
paper we focus on the case of MHE for online state estimation.
Specifically, we provide novel results on the accuracy and
performance of MHE with respect to the omniscient infinite-
horizon estimator (involving all past and future data), establish
bounded dynamic regret with respect to this benchmark, and
consider the practically relevant case of MHE formulations
with an additional prior weighting. Moreover, we link esti-
mator performance to its accuracy with respect to the true
unknown system trajectory, give further technical details, and
provide more extensive simulation examples.
Notation: We denote the set of integers by I, the set of all
(even) integers greater than or equal to a∈Iby I≥a(Ie
≥a),
and the set of integers in the interval [a, b]⊂Iby I[a,b]. The
Euclidean norm of a vector x∈Rnis denoted by |x|, and the
weighted norm with respect to a positive definite matrix Q≻0
with Q=Q⊤by |x|Q=px⊤Qx. The maximum eigenvalue
of Qis denoted by λmax(Q). We refer to a sequence {xj}b
j=a,
xj∈Rn,j∈I[a,b]with xa:b. The identity matrix of size n×n
is denoted by In. Finally, we recall that a function α:R≥0→
R≥0is of class Kif it is continuous, strictly increasing, and
satisfies α(0) = 0; if additionally α(r) = ∞for r→ ∞, it
is of class K∞. By L, we refer to the class of functions θ:
R≥0→R≥0that are continuous, non-increasing, and satisfy
lims→∞ θ(s)=0, and by KL to the class of functions β:
R≥0×R≥0→R≥0with β(·, s)∈ K and β(r, ·)∈ L for any
fixed s∈R≥0and r∈R≥0, respectively.
II. PROBL EM SE TU P
A. System description
We consider nonlinear uncertain discrete-time systems of
the following form:
xt+1 =f(xt, ut, wt),(1a)
yt=h(xt, ut) + vt(1b)
with discrete time t∈I≥0, state xt∈Rn, (known) control
input ut∈Rm, (unknown) process disturbance wt∈Rq,
(unknown) measurement noise vt∈Rp, and noisy output
measurement yt∈Rp. The functions f:Rn×Rm×Rq→Rn
and h:Rn×Rm→Rpdefine the system dynamics and output
equation, which we assume to be continuous.
In the following, we further assume that trajectories of the
system (1) satisfy
(xt, ut, wt, vt)∈ X × U × W × V, t ∈I≥0(2)
for some known sets X ⊆ Rn,U ⊆ Rm,W ⊆ Rq(where
0∈ W), V ⊆ Rp, and furthermore, that (x, u, w)∈ X ×
U × W ⇒ f(x, u, w )∈ X. Such knowledge typically arises
from the physical nature of the system (e.g., non-negativity of
certain physical quantities such as partial pressure or absolute
temperature), the incorporation of which can significantly
improve the estimation results, cf. [1, Sec. 4.4] and compare
also [4].
B. Optimal state estimation
For ease of notation, we define the input-output data (or
parameter) tuple dt:= (ut, yt)obtained from the system (1)
at time t∈I≥0. Now, consider a given batch of input-output
data d0:Nof length N+ 1 for some N∈I≥0. We aim to
compute the state and disturbance sequences ˆx0:Nand ˆw0:N−1
that are optimal in the sense that they minimize a cost function
involving the full data set d0:N. In particular, we consider the
following optimal state estimation problem
PN(d0:N) : min
ˆx0:N
ˆw0:N−1
JN(ˆx0:N,ˆw0:N−1;d0:N)(3a)
s.t. ˆxj+1 =f(ˆxj, uj,ˆwj), j ∈I[0,N −1],(3b)
ˆxj∈ X , j ∈I[0,N],(3c)
ˆwj∈ W, j ∈I[0,N−1] ,(3d)
yj−h(ˆxj, uj)∈ V, j ∈I[0,N].(3e)
For convenience, we define the combined sequence ˆz0:Nas
ˆzj:= (ˆxj,ˆwj)for j∈I[0,N −1] and ˆzN:= (ˆxN,0). The
constraints (3b)–(3e) enforce the prior knowledge about the
system model, the domain of the true trajectories, and the
disturbances/noise (note that feasibility is always guaranteed
due to our standing assumptions). In (3a), we consider the cost
function
JN(ˆx0:N,ˆw0:N−1;d0:N) =
N−1
X
j=0
l(ˆxj,ˆwj;dj) + g(ˆxN;dN)
(4)
with continuous stage cost l:X × W × U × Rp→R≥0
and terminal cost g:X × U × Rp→R≥0. Note that this
is a generalization of classical designs for state estimation,
where land gare positive definite in the disturbance input ˆw
and the fitting error y−h(ˆx, u)(cf., e.g., [1, Ch. 4]), and it
particularly includes the practically relevant case of quadratic
stage and terminal cost
l(x, w;d) = |w|2
Q+|y−h(x, u)|2
R(5)
and
g(x;d) = |y−h(x, u)|2
G,(6)
respectively, where Q, R, G are positive definite weighting
matrices. However, our results also hold for more general cost
functions land g, which allow the objective (4) to be tailored
to the specific problem at hand. Note that cost functions with
an additional prior weighting in (4) (as usual in MHE) are
considered in Section IV-C.
In the state estimation context, a cost function with terminal
cost as in (6) is usually referred to as the filtering form of
the state estimation problem. To simplify the analysis and
notation, the most recent works on FIE/MHE theory often
consider a cost function with g= 0 (i.e., without terminal
cost), which corresponds to the prediction form of the state
estimation problem, cf., e.g., [2]–[6], and see also [1, Ch. 4].
The estimation problem PNin (3) is a parametric nonlinear
program, the solution of which solely depends on the (input-
output) data provided, i.e., the sequence d0:N. We characterize
solutions to PNusing the generic solution mapping ζN:
ˆz∗
j:= ζN(j, d0:N), j ∈I[0,N ],(7)
which yields the value function VN(d0:N) = JN(ˆz∗
0:N;d0:N).
Moreover, with ζx
Nwe refer to the state variable defined by
ζNsuch that ˆx∗
j=ζx
N(j, d0:N)for all j∈I[0,N].
Remark 1 (Existence of solutions to PN): Throughout the
following, we assume that whenever we employ the solution
mapping ζN, the corresponding solution exists. Note that
under continuity of fand hin (1), this can generally be
guaranteed by choosing the stage cost land terminal cost g
such that the cost function JNis radially unbounded in
the (condensed) decision variables (which requires positive
definiteness of land gand observability1of the system with a
corresponding horizon length N, compare [1, Sec. 4.3.1]), or
under compactness of the sets Xand W, see [1, Prop. A.7].
C. Benchmark solution
We are interested in how the solution in (7) compares to a
certain (challenging) benchmark problem. For this purpose, we
interpret the measured data sequence d0:Nas a segment of an
infinite data sequence d−∞:∞that contains all past and future
data that could possibly be generated by the system (1) in the
interval I. Then, we consider the omniscient infinite-horizon
estimator, that is, the solution of the (acausal) optimal state
estimation problem
P∞(d−∞:∞) : min
ˆz−∞:∞
∞
X
j=−∞
l(ˆzj;dj)s.t. (3b)–(3e), j ∈I,
(8)
1Observability is required here because the cost function JNdoes not
contain an additional prior weighting; the case of MHE with prior weighting,
which does not require observability, is considered in Section IV-C below.
where ˆzj= (ˆxj,ˆwj),j∈I. We denote the solution to
P∞(d−∞:∞)by the infinite sequence z∞:= z∞
−∞:∞with
z∞
j= (x∞
j, w∞
j),j∈I, where we assume that z∞exists and
is unique for any possible d−∞:∞, compare Remark 1.
Note that in linear settings [8], [9], a common benchmark
for observers estimating the state xtat some time t∈I≥0
is the clairvoyant acausal observer relying on data from the
interval I[0,∞), where in particular data from I[t+1,∞)is only
fictitious (as it depends on future disturbances and noise) and
may or may not actually be measured at a future point in time
(e.g., if the experiment is terminated). We adopt this approach
and take it even further by assuming that our benchmark—the
omniscient infinite-horizon estimator—can not only predict the
future (knowing data from I[t+1,∞)), but also has a perfect
memory (knowing data from I(−∞,−1]).
In the following section, we will investigate how the
solution ˆz∗
0:N(7) of the finite-horizon estimation problem
PN(d0:N)behaves compared to the infinite-horizon bench-
mark solution z∞on the interval I[0,N].
III. TURNPIKE IN OPTIMAL STATE ESTIMATION PROBLEMS
Optimal state estimation problems (such as PNin (3)) can
be interpreted as optimal control problems using the distur-
bance ˆwas the control input (compare also [1, Sec. 4.2.3] and
[24, Sec. 4]). In particular, a cost function (4) that is positive
definite in the estimated disturbance and the fitting error (as,
e.g., in (5) and (6)) can be regarded as an economic output
tracking cost, penalizing deviations from the ideal reference
(wr
j, yr
j) = (0, yj),j∈I[0,N]. This reference, however, is
generally unreachable, as it is usually impossible to attain
zero cost VN(d0:N)=0, except for the special case where
y0:Ncorresponds to an output sequence of (1) under zero dis-
turbances w0:N−1≡0,v0:N≡0. For unreachable references,
on the other hand, it is known that the corresponding optimal
control problem exhibits the turnpike property with respect to
the best reachable reference [25], which suggests that a similar
phenomenon can also be expected in optimal state estimation
problems.
In the following Section III-A, we provide a simple example
that supports this intuition. Then, we mathematically formalize
and discuss this property in Section III-B.
A. Motivating example
Suppose that some output data y0:Tfor T= 30 is measured
from the system xt+1 =xt+wt,x0= 1,yt=xt+vt,
where wt=vt= 1 for all t∈I. We consider the finite-
horizon optimal estimation problem PN(d0:N)with quadratic
stage and terminal cost (5) and (6) using Q=R=G= 1 and
different values of N. For comparison reasons, we also con-
sider the infinite-horizon problem P∞, which we approximate
by simulating the system and computing the solution on some
extended interval I[−Te,T+Te], where we choose Tesuch that
the solution does not change on I[0,T ]if Teis further increased.
Figure 1 shows the difference between the infinite-horizon
solution x∞
jand the solution of the finite-horizon problem ˆx∗
j
for j∈I[0,N]. One can clearly see that the optimal reference
x∞
0:Tserves as turnpike for the solution ˆx∗
0:N. In particular,
0 5 10 15 20 25 30
j
0
0.2
0.4
0.6
j^x$
j!x1
jj
Fig. 1: Difference between the solution of the infinite-horizon problem P∞
and solutions of the finite-horizon problem PNfor different values of N.
ˆx∗
0:Nis constructed from three pieces: an initial transient where
ˆx∗converges to the turnpike x∞, a large phase where ˆx∗
stays near the turnpike x∞, and a transient at the end of the
horizon where ˆx∗diverges from the turnpike x∞. Figure 1
indicates that these transients2are independent of the horizon
length N. Note that a similar behavior is also observed for the
disturbance difference ˆw∗
j−w∞
j,j∈I[0,N−1] .
B. Turnpike characterization
There are in fact multiple ways to formalize the turnpike
behavior observed in the motivating example in Section III-
A. A characterization popular in the context of (receding
horizon) optimal control involves a bound (independent of the
horizon length) on the number of elements of the sequence
ˆz∗
0:Nthat lie outside an ϵ-neighborhood around the turnpike,
cf. [20], [26] and see also [23, Sec. III-B]. This definition
is particularly suitable for use in the context of economic
model predictive control (cf., e.g., [27]), and also has the
decisive advantage that the corresponding necessary condition
(dissipativity of the optimal control problem) is a global
concept. However, the resulting turnpike property is rather
weak in the sense that it is not possible to infer which
elements of the solution ˆz∗
0:Nare actually close to the turnpike,
and which are not. Such additional information (which is
crucially required in the context of state estimation, as will
be clear in Sections IV and V) is provided by exponential
(or polynomial) turnpike characterizations that involve an
explicit time-dependent bound on the difference of optimal
trajectories and the turnpike, cf. e.g., [19], [21], [28], [29]. To
cover arbitrary decay rates, we propose the following unified
turnpike property involving general KL-functions.
Definition 1 (Turnpike for optimal state estimation): The
optimal estimation problem PNexhibits the turnpike property
with respect to the infinite-horizon solution z∞if there exists
β∈ KL such that ˆz∗
j=ζN(j, d0:N)satisfies
|ˆz∗
j−z∞
j| ≤ β(|ˆz∗
0−z∞
0|, j) + β(|ˆz∗
N−z∞
N|, N −j)(9)
for all j∈I[0,N],N∈I≥0, and all possible data d−∞:∞.
2In the turnpike-related literature, the left transient is usually referred to as
approaching arc and the right transient as leaving arc.
For linear systems and quadratic cost functions as in (5)
and (6), one can establish the turnpike property from Def-
inition 1 by suitably adapting [30, Th. 5], where the KL-
function βspecializes to an exponential one. This requires
a controllability property of the system with respect to the
disturbance input and an additional regularity condition on
the underlying Hamiltonian system (the latter can also be
replaced by invertibility of the dynamics and observability);
hence, the conditions are intuitive, not particularly restrictive
in the context of state estimation, and can be easily checked.
For nonlinear systems, however, deriving sufficient condi-
tions for turnpike behavior is generally difficult and usually
requires a detailed and rather technical analysis. For example,
one may establish (9) by considering linearizations of the
extremal equations as in [28] or by exploiting the specific
graph structure of the underlying nonlinear program and use
the decaying sensitivity results from [31], [32]. For the latter,
it has been shown that exponentially decaying sensitivity is
present under standard regularity and optimality conditions
of the problem (such as a uniform second order sufficient
condition and uniform boundedness of the Lagrangian Hes-
sian, cf. [32]), which are satisfied under uniform observability
and controllability conditions, see [33] and compare also [34].
However, these approaches suffer from local validity and may
only hold in a possibly small neighborhood around the turn-
pike. In contrast, global turnpike properties of optimal control
problems could be established by combining assumptions of
global nature (such as strict dissipativity) with assumptions of
local nature that involve the linearizations at the turnpike (an
optimal equilibrium), cf. [19], [21]. Extending these results to
the more general case considered here is an interesting topic
for future research.
In the following sections, we use the turnpike property
from Definition 1 to assess the performance of MHE and
its regret with respect to the infinite-horizon benchmark z∞.
For practical applications, a reliable indicator for the presence
of turnpike behavior in non-convex optimal state estimation
problems is to simply run simulations and check whether the
turnpike property can be observed, see also the simulation
examples in Section VII.
We want to close this section with the following remark.
Remark 2 (Approaching and leaving arcs): Note that the
finite-horizon problem PN(d0:N)considers a segment of
the data set that underlies the infinite-horizon problem
P∞(d−∞:∞). Specifically, the neglected information involves
the fictitious historical data d−∞:−1and the future data
dN+1:∞, which is why, under the turnpike property from
Definition 1, finite-horizon solutions exhibit both a left ap-
proaching arc and a right leaving arc, compare Figure 1.
IV. OPTIMIZATION-BASE D STATE ES TI MATI ON
In online state estimation, one is generally interested in
obtaining, at each time instant t∈I≥0, an accurate estimate
of the current true (unknown) state xt. An intuitive solution
is to simply solve the optimal state estimation problem in (3)
based on all available historical data (by setting N=t). This
corresponds to the case of FIE, which can be formalized using
Fig. 2: Sketch of the infinite-horizon solution x∞(blue), the current FIE
solution ζt(j, d0:t)(green), and solutions of the finite-horizon estimation
problem ζN(j, dτ−N:τ)for different values of τ(red).
the solution mapping ζx
Ndefined below (7) as follows:
ˆxfie
t=ζx
t(t, d0:t), t ∈I≥0.(10)
However, repeatedly solving Pt(d0:t)for the current FIE so-
lution ˆxfie
tis generally infeasible in practice since the problem
size continuously grows with time. Instead, MHE considers
the truncated optimal estimation problem PN(dt−N:t)using
the most recent data dt−N:tonly, where the horizon length
N∈I≥0is fixed. More precisely, the MHE estimate at the
current time t∈I≥0can be formulated as
ˆxmhe
t=(ζx
N(N, dt−N:t), t ∈I≥N
ζx
t(t, d0:t), t ∈I[0:N−1].(11)
MHE constitutes a well-established method for state estima-
tion that is increasingly applied in practice. However, assuming
that the underlying optimal estimation problem exhibits the
turnpike property in the sense of Definition 1, we know that
both the FIE sequence defined by (10) and the MHE sequence
defined by (11) consist of point estimates of finite-horizon
problems that lie on the leaving arc, see Figure 2; hence, FIE
and MHE might produce estimates that are actually far away
from the turnpike.
In the following, we employ a novel performance analysis
to improve the estimation results of MHE as follows:
•Reduce the influence of the leaving arc by introducing
an artificial delay in the estimation, cf. Sections IV-A
and IV-B.
•Reduce the influence of the approaching arc by using a
turnpike-based prior weighting, cf. Section IV-C.
Our results in this section are based on the assumption that
the MHE problems exhibit the turnpike behavior.
Assumption 1: The finite-horizon optimal state estimation
problem PNin (3) exhibits the turnpike property in the sense
of Definition 1.
In the following, for the sake of simplicity we restrict
ourselves to horizons Nbeing a non-negative even number,
the set of which we denote by Ie
≥0.
A. A delay improves the estimation results
To avoid the influence of naturally appearing leaving arcs
in MHE problems (11), it seems meaningful to introduce a
delayed MHE scheme (δMHE). Specifically, for a fixed delay
δ∈I[0,N/2] with N∈Ie
≥0, we define
ˆxδmhe
t−δ=(ζx
N(N−δ, dt−N:t), t ∈I≥N
ζx
t(t−δ, d0:t), t ∈I[δ,N−1] .(12)
Note that for the special case δ= 0,δMHE reduces to standard
MHE (11). Hence, the parameter δis an additional degree
of freedom that constitutes a trade-off between delaying the
estimation results and reducing the influence of the leaving arc.
Remark 3 (δFIE): By replacing Nwith tand using only
the second case in (12), we can similarly define a delayed
FIE scheme; thus, all of the following results and implications
derived for δMHE directly carry over to δFIE.
Remark 4 (Smoothing form of MHE): Considering a fixed
delay in the estimation scheme to improve the results is
actually quite common in signal processing and filtering theory
and refers to fixed-lag smoothing algorithms [35, Ch. 5]. Early
results for linear systems can be found in, e.g., [36]; more
recent works address, e.g., robustness against model errors [37]
or extensions to certain classes of nonlinear systems [38]. For
linear systems with Gaussian noise, fixed-lag receding-horizon
smoothers are proposed in [39]–[41]. In this context, the
proposed δMHE scheme can also be interpreted as the (fixed-
lag) smoothing form of MHE; compared to the smoothing
literature, however, we consider general nonlinear systems
under arbitrary process and measurement noise and provide
performance and regret guarantees with respect to the infinite-
horizon optimal solution using turnpike arguments.
The following result shows that by a suitable choice of the
delay δ, the estimated state sequence (12) can be made arbi-
trarily close to the state sequence of the omniscient infinite-
horizon benchmark estimator z∞.
Proposition 1: Let Assumption 1 hold and Xbe compact.
Then, there exists σ∈ L such that the estimated state sequence
of δMHE in (12) satisfies
|ˆxδmhe
j−x∞
j| ≤ σ(δ), j ∈I[δ,t−δ](13)
for all t∈I≥δ,δ∈[0, N/2],N∈Ie
≥0, and all d−∞:∞.
Proof: By compactness of X, there exists C > 0such
that |x1−x2| ≤ Cfor all x1, x2∈ X . The rest of this
proof follows directly from the definition of δMHE (12) and
Assumption 1. Specifically, for all j∈I[δ,N −δ−1], we have
|ˆxδmhe
j−x∞
j|≤|ζj+δ(j, d0:j+δ)−z∞
j|
≤β(|ζx
j+δ(0, d0:j+δ)−x∞
0|, j)
+β(|ζx
j+δ(j+δ, d0:j+δ)−x∞
j+δ|, δ)
≤2β(C, δ).
For j∈I[N−δ,t−δ], on the other hand, it follows that
|ˆxδmhe
j−x∞
j| ≤ |ζN(N−δ, dj+δ−N:j+δ)−z∞
j|
≤β(|ζx
N(0, dj+δ−N:j+δ)−x∞
j+δ−N, N −δ)
+β(|ζx
N(N, dj+δ−N:j+δ)−x∞
j+δ|, δ)
≤2β(C, δ).
Combining both cases for j∈I[δ,t−δ]and defining σ(s) :=
2β(C, s),s≥0yields (13). Noting that σ∈ L establishes the
statement and hence finishes this proof.
Proposition 1 provides an estimate on the accuracy of δMHE
in the sense of how close the sequence ˆxδmhe
δ:t−δis to the
benchmark x∞
δ:t−δ. Since σ∈ L, this difference can be made
arbitrarily small by increasing the delay δ(as well as Nif
needed). This is in line with intuition, as increasing δresults
in the state estimates in (12) being closer to the turnpike.
In the following, we establish novel performance guarantees
for δMHE.
B. Performance estimates
In this section, we consider the case where the dynam-
ics (1a) are subject to additive disturbances:
f(x, u, w) = fa(x, u) + w(14)
with w∈ W =Rn. Furthermore, we impose a Lipschitz
condition on the nonlinear functions faand hon X.
Assumption 2: The functions faand hare Lipschitz in x∈
X, i.e., there exist constants Lf, Lh>0such that
|fa(x1, u)−fa(x2, u)| ≤ Lf|x1−x2|,(15)
|h(x1, u)−h(x2, u)| ≤ Lh|x1−x2|(16)
for all x1, x2∈ X uniformly for all u∈ U.
Note that Assumption 2 is not restrictive in practice under
compactness of X(as considered in Proposition 1).
The dynamics (14) ensure one-step controllability with
respect to the disturbance input w; consequently, the sequence
ˆxδmhe
δ:t−δforms a state trajectory of system (1), driven by the
disturbance input ˆwδmhe
j= ˆxδmhe
j+1 −fa(ˆxδmhe
j, uj),j∈
I[δ,t−δ−1]. For the sake of conciseness, we define ˆzδmhe
j:=
(ˆxδmhe
j,ˆwδmhe
j), j ∈I[δ,t−δ−1] and ˆzδmhe
t−δ:= (ˆxδmhe
t−δ,0).
We now specify the performance measure. To this end,
we denote with t1, t2∈I≥0the time instances defining
some interval of interest I[t1,t2]. For a given sequence ˆzt1:t2
with ˆzj= (ˆxj,ˆwj)satisfying the system dynamics ˆxj+1 =
f(ˆxj, uj,ˆwj)for j∈I[t1,t2], we consider the performance
criterion
J[t1,t2](ˆzt1:t2) :=
t2−1
X
j=t1
l(ˆzj;dj)(17)
with the stage cost lfrom (4). The following result provides a
performance estimate for δMHE, and moreover, can be used
to quantify the dynamic regret with respect to the omniscient
infinite-horizon benchmark estimator z∞.
Theorem 1 (Performance of δMHE): Consider the system
dynamics (14) and the quadratic stage cost in (5). Let As-
sumptions 1 and 2 be satisfied. Then, there exists ¯σ∈ L such
that for any choice of ϵ > 0,δMHE for some arbitrary delay
δ∈I[0,N/2] and horizon length N∈Ie
≥0satisfies the following
performance estimate:
J[t1,t2](ˆzδmhe
t1:t2)≤(1 + ϵ)J[t1,t2](z∞
t1:t2) + 1 + ϵ
ϵt∆¯σ(δ)(18)
for all t1, t2∈I[δ,t−δ], all t∈I≥δ, and all possible d−∞,∞,
where t∆=t2−t1.
Before proving Theorem 1, we want to highlight some key
properties of the performance estimate in (18).
Remark 5 (Performance of δMHE):
1) The performance of δMHE (the sequence ˆzδmhe
t1:t2) is
approximately optimal on the interval I[t1,t2]with error
terms that depend on ϵ,δ, and the interval length t∆.
2) The performance estimate in (18) directly implies a
bound on the dynamic regret (i.e., the performance loss)
of δMHE with respect to the omniscient benchmark z∞,
compare also Corollary 1 below.
3) In case of an exponential turnpike property (i.e., As-
sumption 1 holds with β(s, t) = Ksλtin Definition 1
for some K > 0and λ∈(0,1)), the L-functions σ
and ¯σin Proposition 1 and Theorem 1 decay expo-
nentially. Then, the performance of δMHE converges
exponentially to the performance of the infinite-horizon
estimator as δincreases. This behavior is also evident
in the numerical example in Section VII.
4) The performance estimate (18) grows linearly with the
size of the performance interval t∆and tends to infinity
if t∆approaches infinity. This property is to be expected
(due to the fact that the turnpike is never exactly
reached) and conceptually similar to (non-averaged)
performance results in economic model predictive con-
trol, see, e.g., [27, Sec. 5], [16]. To make meaningful
performance estimates in case t∆→ ∞, we analyze the
averaged performance in Corollary 2 below.
5) Theorem 1 is stated for the practically relevant case of
quadratic stage costs as in (5) for ease of presentation,
but can easily be extended to more general cost functions
that fulfill a weak triangle inequality.
Proof of Theorem 1: Using the definitions from (4)–(6),
the performance criterion evaluated for δMHE reads
J[t1,t2](ˆzδmhe
t1:t2) =
t2−1
X
j=t1
|ˆwδmhe
j|2
Q+|yj−h(ˆxδmhe
j, uj)|2
R.(19)
From the definition of ˆxδmhe
jin (12) and the fact that
fa(x∞
j, uj) + w∞
j−x∞
j+1 = 0 using (14), the square root
of the input cost satisfies
|ˆwδmhe
j|Q
=|ˆxδmhe
j+1 −x∞
j+1 +fa(x∞
j, uj)−fa(ˆxδmhe
j, uj) + w∞
j|Q
≤ |ˆxδmhe
j+1 −x∞
j+1|Q+Lf|x∞
j−ˆxδmhe
j|Q+|w∞
j|Q(20)
for all j∈I[t1,t2−1], where in the last step we have used the
triangle inequality and Assumption 2. Using Proposition 1 and
the fact that t1, t2∈I[δ,t−δ], it follows that
|ˆxδmhe
j−x∞
j|≤|ˆzδmhe
j−z∞
j| ≤ σ(δ), j ∈I[t1,t2],(21)
where σ∈ L. Hence, from (20) we have
|ˆwδmhe
j|Q≤ |w∞
j|Q+ (1 + Lf)λmax(Q)σ(δ), j ∈I[t1,t2−1].
Squaring both sides and using that for any ϵ > 0, it holds that
(a+b)2≤(1 + ϵ)a2+1+ϵ
ϵb2for all a, b ≥0, we obtain
|ˆwδmhe
j|2
Q≤(1 + ϵ)|w∞
j|2
Q+1 + ϵ
ϵ(1 + Lf)2λmax(Q)2σ(δ)2
(22)
for each j∈I[t1,t2−1]. A similar reasoning for the fitting error
(where we add h(x∞
j, uj)−h(x∞
j, uj)=0) yields
|yj−h(ˆxδmhe
j, uj)|R≤ |yj−h(x∞
j, uj)|R+Lhλmax(R)σ(δ)
for all j∈I[t1,t2−1]. By squaring both sides and using the
same argument that allowed us to obtain (22), we get
|yj−h(ˆxδmhe
j, uj)|2
R≤(1 + ϵ)|yj−h(x∞
j, uj)|2
R
+1 + ϵ
ϵL2
hλmax(R)2σ(δ)2(23)
for all j∈I[t1,t2−1]. The performance criterion (19) to-
gether with (22), (23), and the definition of ¯σ(s) := (1 +
Lf)2λmax(Q)2+L2
hλmax(R)2σ(s)2, s ≥0(where we note
that ¯σ∈ L) establishes the statement of this theorem.
To derive a bound on the dynamic regret and the asymp-
totic averaged performance of δMHE, we first show that
J[t1,t2](z∞
t1:t2)grows at maximum linearly in t∆.
Lemma 1: Let Wand Vbe compact. There exists A > 0
such that J[t1,t2](z∞
t1:t2)≤A(t2−t1)for any possible d−∞:∞.
Proof: Due to Wand Vbeing compact, there exist
CQ, CR>0such that |ˆw|2
Q≤CQand |y−h(ˆx, u)|2
R≤CR
for all (ˆx, ˆw, u, y)∈ X ×W×U ×Rpsuch that y−h(ˆx, u)∈ V .
Hence, the claim holds with A=CQ+CR.
The following corollary from Theorem 1 establishes
bounded regret of δMHE with respect to the benchmark z∞.
Corollary 1 (Bounded regret): Let the conditions of Theo-
rem 1 be satisfied. Assume that Wand Vare compact. Then,
the regret of δMHE can be bounded by
J[t1,t2](zδmhe
t1:t2)−J[t1,t2](z∞
t1:t2)≤t∆ϵA +1 + ϵ
ϵ¯σ(δ)
for all t1, t2∈I[δ,t−δ], all t∈I≥δ, and all possible d−∞,∞,
where ϵ > 0and ¯σ∈ L are from Theorem 1, A > 0is from
Lemma 1, and t∆=t2−t1.
Note that the regret bound provided by Corollary 1 is
linear in the interval length t∆, where the slope C(ϵ, δ) :=
ϵA +1+ϵ
ϵ¯σ(δ)can be rendered arbitrarily small by suitable
choices of ϵand δ. Again, linear dependency on t∆is to be ex-
pected as the turnpike is never exactly reached, cf. Remark 5.
The following result establishes an estimate on the averaged
performance of δMHE for the asymptotic case when t∆→ ∞.
Corollary 2 (Averaged performance): Let the conditions of
Theorem 1 be satisfied. Assume that Wand Vare compact.
Then, δMHE satisfies the averaged performance estimate
lim sup
t∆→∞
1
t∆
(J[t1,t2](zδmhe
t1:t2)−J[t1,t2](z∞
t1:t2)) ≤ϵA+1 + ϵ
ϵ¯σ(δ)
for all possible data d−∞,∞, where ϵ > 0and ¯σ∈ L are from
Theorem 1, A > 0is from Lemma 1, and t∆=t2−t1.
Corollary 2 implies that the averaged performance of the
δMHE is finite (due to the fact that J[t1,t2](z∞
t1:t2)can be
bounded as J[t1,t2](z∞
t1:t2)/t∆≤Aby Lemma 1) and approxi-
mately optimal with respect to the infinite-horizon benchmark
estimator, with error terms that can be made arbitrarily small
by suitable choices of ϵand δ. In case of an exponential
turnpike property, in the limit δ→ ∞ we can fully recover the
benchmark performance as discussed in the following remark.
Remark 6: Under an exponential turnpike property (see Re-
mark 5, Statement 3), we can easily choose the delay δ(ϵ)
such that limϵ→01+ϵ
ϵ¯σ(δ(ϵ)) = 0 (consider, e.g., δ(ϵ) =
1/ϵ). Then, in the limit ϵ→0, we are able to recover the
asymptotic averaged performance of the omniscient infinite
horizon estimator, that is, we obtain
lim sup
t∆→ ∞
ϵ→0
1
t∆J[t1,t2](zδmhe
t1:t2)−J[t1,t2](z∞
t1:t2)= 0.
Overall, Proposition 1, Theorem 1, Corollary 1, and Corol-
lary 2 imply that δMHE is able to track the solution and
the performance of the omniscient infinite-horizon estimator.
Increasing the delay δreduces the influence of the leaving arc
and improves the performance estimate; the best performance
is achieved for δ=N/2, which, on the other hand, introduces
a potentially large delay (depending on N). However, in the
practically relevant case of exponential turnpike behavior, al-
ready small values of δare expected to significantly reduce the
influence of the leaving arc and hence improve the estimation
results compared to standard MHE (without delay), which is
also evident in the simulation examples in Section VII-B.
We conclude this section by noting that while it is possible
in model predictive control to design suitable terminal ingre-
dients that yield finite non-averaged performance for t→ ∞
(cf. [42]), this does not seem to be possible here, as it would
imply that we have certain information about future data in
order to exactly reach and stay on the solution of the acausal
infinite-horizon estimation problem.
C. MHE with prior weighting
It is known that MHE schemes with a cost function as
in (4) might require relatively large estimation horizons to
achieve satisfactory estimation results, cf. [1, Sec. 4.3.2]. In
order to reduce the required horizon length and enable faster
computations, MHE formulations that leverage an additional
prior weighting are therefore usually preferred in practice.
The prior weighting can generally be seen as additional
regularization of the cost function, ensuring that the initial
state ˆx0of an estimated sequence ˆx0:Nstays in a meaningful
region. In view of our turnpike results, a well-chosen prior
weighting hence reduces the influence of the approaching arc
and ensures that solutions of truncated problems can reach the
turnpike in fewer steps.
The prior weighting is usually parameterized by a given
prior estimate ¯xand a (possibly time-varying) function
pt(x, ¯x)that is positive definite and uniformly bounded in
the difference |x−¯x|. Throughout the following, we use the
definition Nt:= min{t, N }, which is convenient here as it
avoids additional case distinctions. The (time-varying) MHE
cost function with prior weighting can then be formulated as
Jp
Nt(ˆx0:Nt,ˆw0:Nt−1;dt−Nt:t, t)
:= pt−Nt(ˆx0,¯xt−Nt) + JNt(ˆx0:Nt,ˆw0:Nt−1;dt−Nt:t),(24)
with the current decision variables ˆx0:Ntand ˆw0:Nt−1, and
where JNtis from (4) with Nreplaced by Nt. At any
time t∈I≥0, the current MHE problem to solve is given
by the optimal estimation problem (3) with Nreplaced by
Ntand the cost function Jp
Ntfrom (24), which we denote
by Pp
Nt(dt−Nt:t,¯xt−Nt, t). The corresponding solution (which
exists under mild conditions, compare Remark 1) is described
by the sequence ˜z∗
t−Nt:t, where ˜z∗
j= (˜x∗
j,˜w∗
j),j∈I[t−Nt,t−1]
and ˜z∗
t= (˜x∗
t,0). The prior ¯xt−Ntis typically chosen in terms
of a past solution of the problem Pp
Nt, which introduces a
coupling between the MHE problems. For easier reference,
it is therefore convenient to introduce double indices, where,
e.g., ˜z∗
j|t,j∈I[t−Nt,t]refers to the element ˜z∗
jof the solution
of Pp
Nt(·, t)computed at time t∈I≥0. We first consider the
standard MHE case and set the current estimate ˆxmhe,p
tto
the last state of the sequence ˜z∗
t−Nt:t|t, i.e., ˆxmhe,p
t:= ˜x∗
t|t,
t∈I≥0. In Remark 8 below, we again consider a delay to
reduce the influence of the naturally appearing leaving arc.
A popular choice for the prior weighting is
pt(x, ¯x) = |x−¯x|2
Wt,(25)
where Wtis a constant or time-varying positive definite
weighting matrix that might be updated using, e.g., covariance
update laws from nonlinear Kalman filtering, cf., [43]–[46].
Given some initial guess ¯x0∈ X , there are two common
choices for updating the prior estimate ¯xt−Nt: first, the fil-
tering prior
¯xt−Nt=(˜x∗
t−N|t−N, t ∈I≥N
¯x0, t ∈I[0,N−1] ,(26)
which corresponds to the state estimate computed Nsteps in
the past, i.e., the last element of the solution to the problem
Pp
N(·, t −N); second, the smoothing prior
¯xt−Nt=(˜x∗
t−Nt|t−1, t ∈I≥1
¯x0, t = 0,(27)
which refers to the second element of the solution to Pp
Nt(·, t−
1) computed at the previous time step t−1, cf. [1, Sec. 4.3.2].
MHE with a smoothing prior can generally recover faster from
poor initial estimates, whereas MHE with a filtering prior
essentially comprises measurements from two time horizons
and may therefore be advantageous in the long term. Note that
the filtering prior is considered in most of the recent literature
on nonlinear MHE theory, as this allows to derive a contraction
of the estimation error over the estimation horizon and thus
establish robust stability [1]–[6]. In contrast, the smoothing
prior is used in, e.g., [47], [48], and often serves in practice-
oriented works as a linearization point for computing an
improved prior estimate using Kalman filter updates, compare,
e.g., [45], [46] and see also the review article [49].
However, from our turnpike analysis, we know that both of
these two options may be unsuitable if we are interested in
approximating the infinite-horizon optimal performance; the
smoothing prior corresponds to an element of a finite-horizon
solution on the approaching arc, and the filtering prior to an
element of the solution on the leaving arc. Hence, we propose
the following turnpike prior:
¯xt−Nt=(˜x∗
t−Nt|t−N/2, t ∈I≥N/2
¯x0, t ∈I[0,N/2−1],(28)
which corresponds to the middle element of the solution of
PN(·, t −N/2) computed at time t−N/2. This avoids the
influence of the approaching and leaving arcs (for t∈I≥N). In
fact, we can show that the prior estimate ¯xt−Ntconverges to a
neighborhood of the turnpike x∞under the following modified
(exponential) turnpike property of the MHE problem Pp
Nt.
Assumption 3 (Turnpike for MHE with prior weighting):
There exist constants K > 0and λ∈(0,1) such that for
all N∈I≥0, the solutions of the finite-horizon problem
Pp
N(dτ:τ+N,¯x, τ +N)and the infinite-horizon problem
P∞(d−∞:∞)satisfy
|˜z∗
τ+j−z∞
τ+j| ≤ K|˜x∗
τ−x∞
τ|λj+|˜x∗
τ+N−x∞
τ+N|λN−j
for all j∈I[0,N],τ∈I≥0,¯x∈ X , and all possible d−∞:∞.
Assumption 3 essentially states that the infinite-horizon
solution z∞serves as turnpike for MHE problems with prior
weighting, cf. Section III-B. Note that such behavior could be
observed in all numerical examples in Section VII-B.
Remark 7 (Exponential turnpike): In contrast to Defini-
tion 1, we impose an exponential turnpike property in As-
sumption 3, which is crucially required to derive uniform
(exponential) convergence of the turnpike prior ¯xt−Ntto the
turnpike x∞in the following proposition. Note that this is
conceptually similar to recent stability results for nonlinear
MHE, which also require exponential detectability (rather than
asymptotic detectability), cf., e.g., [1], [2], [4], [5].
Proposition 2: Let Assumption 3 hold and the sets X,W,V
be compact. Suppose there exists p1, p2>0and a≥1such
that
p1|x−¯x|a≤pt(x, ¯x)≤p2|x−¯x|a(29)
for all x, ¯x∈ X uniformly for all t∈I≥0. Furthermore,
assume that there exists α1, α2, α3∈ K∞such that
l(ˆx, ˆw; (u, y)) ≤α1(|ˆw|) + α2(|y−h(ˆx, u)|),(30)
g(ˆx; (u, y)) ≤α3(|y−h(ˆx, u)|)(31)
for all ˆx∈ X ,ˆw∈ W,(u, y)∈ U ×Rpsatisfying y−h(ˆx, u)∈
V. Then, there exists σ∈ L such that for any ρ∈(0,1), there
exists ¯
N∈Ie
≥0such that the turnpike prior (28) satisfies
|¯xt−N/2−x∞
t−N/2| ≤ ρ|¯xt−N−x∞
t−N|+σ(N)(32)
for all N∈Ie
≥¯
Nand t∈I≥N.
Proof: Consider any t∈I≥N, the data sequence d−∞:∞
associated with the system (1), and an arbitrary prior ¯xt−N∈
X. Let ˜z∗
t−N:tdenote the solution of Pp
N(dt−N:t,¯xt−N, t)
and consider the infinite-horizon estimator z∞. From Assump-
tion 3, we obtain
|˜z∗
t−N+j−z∞
t−N+j| ≤ K|˜x∗
t−N−x∞
t−N|λj+Kc1λN−j(33)
for all j∈I[0,N], where c1>0satisfies |x1−x2| ≤ c1for
all x1, x2∈ X (compactness of Xensures existence of such
c1). By the triangle inequality, it follows that
|˜x∗
t−N−x∞
t−N| ≤ |˜x∗
t−N−¯xt−N|+|¯xt−N−x∞
t−N|.(34)
Using (29), the cost function (24), and optimality of ˜z∗
t−N:t,
it holds that
p1|˜x∗
t−N−¯xt−N|a≤Jp
N(˜x∗
t−N:t,˜w∗
t−N:t−1;dt−N:t, t)
≤Jp
N(x∞
t−N:t, w∞
t−N:t−1;dt−N:t, t)
=p2|x∞
t−N−¯xt−N|a+JN(x∞
t−N:t, w∞
t−N:t−1;dt−N:t).(35)
The bounds in (30) and (31), compactness of Wand V,
and similar arguments as in the proof of Lemma 1 imply
the existence of uniform constants A, B > 0such that
JN(x∞
t−N:t, w∞
t−N:t−1;dt−N:t)≤AN +B. In combination
with (34) and (35) and using the fact that the function s7→
s1/a is subadditive for s≥0and a≥1, this leads to
|˜x∗
t−N−x∞
t−N| ≤ c2|x∞
t−N−¯xt−N|+AN +B
p11/a
,
where c2:= 1 + (p2/p1)1/a . Evaluating (33) at j=N/2, we
can infer that
|˜z∗
t−N/2−z∞
t−N/2| ≤ Kc2|x∞
t−N−¯xt−N|λN/2(36)
+KAN +B
p11/a
λN/2+Kc1λN/2.
For any ρ∈(0,1), there exists ¯
N∈Ie
≥0sufficiently large
such that Kc2λN/2≤ρfor all N∈I≥¯
N. Define σ1(r) :=
K((Ar +B)/p1)1/aλr/2,r≥0. Note that σ1(r)converges
to zero for r→ ∞, since the exponential term dominates
for large enough r. Hence, there exists ¯σ1∈ L satisfying
σ1(r)≤¯σ1(r)for all r≥0. Thus, from (36), we can infer
that the updated turnpike prior ¯xt−N/2= ˜x∗
t−N/2satisfies
|¯xt−N/2−x∞
t−N/2|≤|˜z∗
t−N/2−z∞
t−N/2|
≤ρ|¯xt−N−x∞
t−N|+ ¯σ1(N) + Kc1λN/2
for all t∈I≥Nand N∈Ie
≥¯
N. Defining σ(r) := ¯σ1(r) +
Kc1λr /2,r≥0and noting that σ∈ L establishes the
statement and hence finishes this proof.
The conditions in (29)–(31) on the prior weighting, stage
cost, and terminal cost are standard (cf., e.g., [1, Ass. 4.22])
and obviously satisfied for the practically relevant case of
quadratic penalties as in (5), (6), and (25). Provided that the
horizon length Nis chosen sufficiently large, Proposition 2
implies (by a recursive application of the bound in (32)) that
the prior ¯xt−Nt(28) forms a sequence that exponentially
converges into a neighborhood of the turnpike (the infinite-
horizon benchmark solution x∞), which is also evident in
the simulation example in Section VII-B.1. Here, we want
to emphasize that the size of this neighborhood depends on
the horizon length Nand can in fact be made arbitrarily small
by choosing larger values of N(due to the fact that σ∈ L).
Overall, a properly selected prior weighting ptaccording
to Proposition 2 ensures that the initial state ˜x∗
t−Nt|tof the
solution of the finite-horizon problem Pp
Nt(·, t)is close to
the turnpike, which hence effectively reduces the approaching
arc and allows using short horizons. However, the natural
occurrence of the leaving arc can still cause the resulting
estimate ˆxmhe,p
t= ˜x∗
t|tto be again relatively far away from the
turnpike, which could again be reduced by using an artificial
delay in the MHE scheme as suggested in Section IV-A.
Remark 8 (Performance of δMHE with prior weighting):
For some fixed delay δ∈I[0,N/2], we can define δMHE (with
prior weighting) as
ˆxδmhe,p
t−δ= ˜x∗
t−δ|t, t ∈I≥δ.(37)
Under Assumption 3, it is straightforward to show that
|ˆxδmhe,p
j−x∞
j| ≤ σ(δ)for all j∈I[δ,t−δ],t∈I≥δ, with
σ∈ L from Proposition 1 (which can be easily modified to this
case). As a result, the performance estimates from Theorem 1,
Corollary 1, and Corollary 2 directly carry over to δMHE
with prior weighting. Thus, δMHE with prior weighting and a
suitably selected delay δcan recover the accuracy and perfor-
mance of the infinite-horizon estimator, with shorter horizons
compared to δMHE without prior weighting. Here, we want
to emphasize that this conclusion holds under Assumption 3,
i.e., for any choice of the prior estimate ¯xt−Ntfrom (26)–(28)
(in contrast to Proposition 2, which is an exclusive feature
of the proposed turnpike prior (28)). Our simulation results
in Section VII-B show that already (very) small values of δ
significantly improve the estimation accuracy.
V. OFFLIN E STATE ES TI MATI ON
We now want to briefly discuss the case of offline state
estimation, which can be interpreted as a special case of our
previous setup. Here, one is interested in matching an a priori
given data sequence d0:Tfor some T∈I≥0to the system
equations (1) to obtain an estimate of the true unknown state
sequence x0:T. To this end, a natural approach is to simply
solve the optimal state estimation problem in (3) with N=T.
However, if the data set (in particular, T) or the underlying
model is very large or the computations are limited in terms
of time or resources, solving the full problem PT(d0:T)for
the optimal solution is usually difficult (or even impossible)
in practice.
Instead, we can construct an approximation of the optimal
state sequence ˆx∗
0:Tbelonging to the solution of the full
problem PT(d0:T)using a sequence of smaller problems
PNof length N∈Ie
≥0and our results from Section IV.
Specifically, we define the approximate estimator
ˆxae
j=
ζx
N(j, d0:N), j ∈I[0:N/2]
ζx
N(N/2, dj−N/2:j+N/2), j ∈I[N/2+1:T−N/2−1]
ζx
N(N−T+j, dT−N:T), j ∈I[T−N/2:T],
(38)
where ζx
Nis defined below (7).
Note that for j∈I[N/2+1:T−N/2−1], the approximate
estimator in (38) corresponds to the δMHE scheme in (12)
with δ=N/2. Hence, the accuracy and performance esti-
mates established in Section IV (Proposition 1, Theorem 1,
Corollary 1, and Corollary 2) directly apply with respect to
the benchmark z∞(where the underlying data set d−∞:∞
a suitable extension of d0:Tto the interval Isuch that the
dynamics (1) and constraints (2) are satisfied for all t∈I).
Therefore, under Assumption 1, the estimates ˆxae
jare close to
the turnpike z∞for all j∈I[N/2,T −N/2]. Notice, however,
that the turnpike property imposed in Assumption 1 also
applies to the full problem PT(d0:T), which implies that the
corresponding solution is also close to z∞on the interval
I[N/2,T −N/2], compare Figure 2. Hence, Assumption 1 ensures
that the estimated sequence in (38) is approximately optimal
on I[N/2,T −N/2] with respect to the (unknown) desired solution
of the full problem PT(d0:T), compare Remark 5.
Moreover, since the individual finite-horizon estimation
problems in (38) are completely decoupled from each other,
computing the approximate estimator can be parallelized and
hence has the potential to significantly save time and resources.
In other words, the approximate estimator as defined in (38)
can be interpreted as a distributed computation of the optimal
solution PT(d0:T)with negligible error (provided that Nis
large enough), which can also be seen in the simulation
example in Section VII-A.2. This is practically relevant for,
e.g., large data assimilation problems that appear in geophysics
and environmental sciences [50], [51], but conceptually also
more general in the context of robust optimization for data-
driven decision making, cf., e.g., [52], [53].
Remark 9 (Reduced computations): We can easily gener-
alize our results by constructing the approximate estima-
tor ˆxae
0:Tby concatenating subsequences of the solutions
of the truncated problems. Specifically, from each solution
ζN(j, dτ,τ +N),j∈I[0,N ],τ∈I[1,T −N−1], instead using
only the single element at j=N/2as in (38), we take
the elements corresponding to j∈I[N/2−∆,N/2+∆] for some
∆∈I[0,N/2]. This construction allows for qualitatively similar
performance results as in Theorem 1 and Corollary 2, albeit
with slightly worse bounds depending on the length of the
subsequences (i.e., ∆). However, this approach can greatly
reduce the number of problems to be solved. In particular,
assuming that there exists k∈I≥0such that T, N , ∆satisfy
T=N+ (k+ 1)(2∆ + 1), our construction requires solving
K∆:= k+ 2 = T−N
2∆+1 + 1 truncated problems. For the
special case of ∆=0(i.e., the approximate estimator (38)),
it follows that K∆=T−N+ 1, which is quite large if
Tis large. However, increasing the value of ∆significantly
reduces the value of K∆(as K∆is proportional to 1/∆). In
fact, our simulation example in Section VII-A.2 shows that it
is sufficient to select ∆relatively close to N/2such that only
the first (resp. last) few elements on the approaching (resp.
leaving) arc of the truncated solutions are discarded, which
significantly reduces the number of problems to solve.
VI. GOOD PERFORMANCE IMPLIES ACCURATE STATE
ESTIMATES
So far, we have investigated how close the solutions of
finite-horizon state estimation problems are to the infinite-
horizon solution. In this section, we draw a direct link between
the performance of a state estimator (measured by the criterion
in (17)) and its accuracy (in terms of the estimation error).
To this end, a detectability condition is required to ensure that
the collected measurement data contains sufficient information
about the real unknown state trajectory, where we consider the
notion of incremental input/output-to-state stability (i-IOSS).
Assumption 4 (Exponential i-IOSS): The system (1) is ex-
ponentially i-IOSS, i.e., there exists a continuous function
U:X × X → R≥0together with matrices P1, P2, Q, R ≻0
and a constant η∈[0,1) such that
|x1−x2|2
P1≤U(x1, x2)≤ |x1−x2|2
P2,(39a)
U(f(x1, u, w2), f (x2, u, w2)) (39b)
≤ηU (x1, x2) + |w1−w2|2
Q+|h(x1, u)−h(x2, u)|2
R
for all (x1, u, w1),(x2, u, w2)∈ X × U × W.
Assumption (4) is a Lyapunov characterization of exponen-
tial i-IOSS, which became a standard detectability condition
in the context of MHE in recent years, see, e.g., [1]–[6].
This property implies that the difference between any two
state trajectories is bounded by the differences of their initial
states, their disturbance inputs, and their outputs. In other
words, if two trajectories of an i-IOSS system have the same
inputs and outputs, then their states must converge to each
other. Note that Assumption 4 is not restrictive in the state
estimation context; in fact, by adapting the results from [3],
[54], it is in fact necessary and sufficient for the existence
of robustly (exponentially) stable state estimators. Moreover,
Assumption 4 can be verified using LMIs, cf., e.g, [4], [55].
Given a sequence ˆx0:t,t∈I≥0produced by some state
estimator, the following result establishes a bound with respect
to the true state sequence x0:tin terms of its performance.
Proposition 3: Suppose Assumption 4 holds. Consider the
performance measure (17) for some t1, t2∈I≥0with the
quadratic stage cost lfrom (5) for some Q, R ≻0. Then,
there exist C1, C2, C3>0such that
|xτ−ˆxτ|2≤C1ητ−t1|xt1−ˆxt1|2(40)
+C2max
j∈I[t1,τ−1]
{|wj|2,|vj|2}+C3J[t1,t2](ˆzt1:t2)
for all τ∈I[t1,t2], all initial conditions x0,ˆx0∈ X , and all
input and disturbance sequences u∈ U∞,w, ˆw∈ W∞, and
v∈ V∞, where η∈(0,1) is from Assumption 4 and xj+1 =
f(xj, uj, wj),yj=h(xj, uj) + vj,ˆxj+1 =f(ˆxj, uj,ˆwj), and
ˆzj= (ˆxj,ˆwj)for all j∈I≥0.
Proof: Without loss of generality, we can assume that
the matrices Qand Rfrom the cost function (5) are the same3
matrices as in (39b). For any τ∈I[t1,t2], the sequences x0:τ
and ˆx0:τform trajectories of the i-IOSS system (1). Using the
dissipation inequality (39b) from Assumption 4, the fact that
|a−b|2
H≤2|a|2
H+ 2|b|2
Hfor any real vectors a, b and matrix
H≻0by Cauchy-Schwarz and Young’s inequality, and the
performance criterion in (17), we can infer that
U(xτ,ˆxτ)≤ητ−t1U(xt1,ˆxt1)
+ 2
τ−1
X
j=t1
ητ−j−1(|wj|2
Q+|vj|2
R)+2J[t1,t2](ˆzt1:t2).
The fact that a+b≤max{2a, 2b}for all a, b ≥0together
with the convergence property of the geometric series leads to
τ−1
X
j=t1
ητ−j−1(|wj|2
Q+|vj|2
R)≤2C
1−ηmax
j∈I[t1,τ−1]
{|wj|2,|vj|2}
with C:= max{λmax(Q), λmax (R)}. Using (39a), we ob-
tain (40) with C1=λmax(P2)
λmin(P1),C2= 4C/(λmin(P1)(1 −η)),
and C3= 2/λmin(P1), which finishes this proof.
Proposition 3 draws a link between the performance of
a state estimator (measured by the criterion in (17)) and
the corresponding estimation error. In particular, for large τ,
the error is upper bounded by the performance J[t1,t2]and
the maximum disturbance wj, vj,j∈I[t1,τ−1] that affected
the past system behavior and associated measurement data.
Hence, if the disturbances are small, we directly have that
good performance (small values of J[t1,t2]) implies accurate
estimation results (small errors |xτ−ˆxτ|). Consequently, it
is indeed advisable to design estimators that achieve good
performance in the sense of the criterion J[t1,t2].
We want to emphasize that Proposition 3 and its implica-
tions apply for any state estimator/observer design. However,
as the performance criterion appearing in the estimation error
3For any weighting matrices Q, R ≻0, the i-IOSS Lyapunov function U
can be suitably scaled such that (39b) holds with that choice of Qand R.
bound in (40) constitutes a part of the cost function used in
the infinite-horizon problem P∞, the corresponding solution
z∞provides a comparatively small bound for the estimation
accuracy for all τ∈I[t1,t2]for any t1, t2∈I≥0. Under
the turnpike condition from Assumption 1, we know that
δMHE along with a suitably selected delay δachieves nearly
the same performance as the benchmark z∞on any interval
I[t1,t2]⊆I[δ,t−δ]for all t∈I≥δ, and is hence expected to be
similarly4accurate. This highly useful feature of δMHE can
also be seen in all our simulation examples in Section VII.
VII. NUMERICAL EXAMPLES
We now illustrate our results from Sections IV–VI. The
following simulations were performed on a standard laptop
in MATLAB using CasADi [56] and the solver IPOPT [57].
In Section VII-A, we first consider the offline estimation
case by means of a nonlinear batch reactor model and a linear
system with more than 100 states. Our simulations show that
the proposed estimator (38) along with the modifications from
Remark 9 approximates the optimal solution with negligible
error. In Section VII-B, we focus on online state estimation
(in particular, MHE with prior weighting) and consider two
realistic examples from the literature: a continuous stirred-
tank reactor and a highly nonlinear 12-state quadrotor model.
In both examples, we can observe the turnpike behavior being
present in MHE problems with prior weighting. Our main
observation is that already a small delay in the MHE scheme
(one to three steps) reduces the overall estimation error with
respect to the true unknown system state by 20 –25 %.
A. Offline estimation
1) Batch reactor example:We consider the system
x+
1=x1+t∆(−2k1x2
1+ 2k2x2) + u1+w1,
x+
2=x2+t∆(k1x2
1−k2x2) + u2+w2,
y=x1+x2+v
with k1= 0.16,k2= 0.0064, and t∆= 0.1. This corresponds
to the batch-reactor example from [58, Sec. 5] under Euler
discretization with additional controls u∈R2, disturbances
w∈R2, and measurement noise v∈R. We consider a data set
d0:Twith T= 400, where the process started at x0= [3,0]⊤,
was subject to uniformly distributed disturbances and noise
satisfying w∈ {w∈R2:|wi| ≤ 0.05, i = 1,2}and v∈ {v∈
R:|v| ≤ 0.5}, and where the input ujwas used to periodically
empty and refill the reactor such that xj+1 = [3,0]⊤+wjfor
all j= 50iwith i∈I[1,7] and uj= 0 for all j= 50i. To
reconstruct the unknown state trajectory x0:T, we consider the
cost function (4)–(6) and select Q=I2and R=G= 1. In
the following, we compare the performance and accuracy of
the optimal solution ˆx∗
0:Tto the full problem PT(d0:T), the
proposed approximate estimator (AE) ˆxae
0:T(38), and standard
MHE ˆxmhe
0:T(11) for different choices of N.
From Figure 3, for small horizons (N= 40) we observe
that the AE ˆxae
0:Tachieves significantly worse performance
compared to the solution of the full problem (and MHE).
4This also applies to δMHE with prior weighting under Assumption 3.
40 70 100 130 160
N
15
20
25
J[0;T ]
full
AE
MHE
Fig. 3: Performance J[0,T ]of the AE ˆxae
0:T(cyan) and MHE ˆxmhe
0:T(red) for
different lengths Nof the problem PNcompared to the performance achieved
by full solution ˆx∗
0:T(green).
This can be attributed to the problem length Nbeing too
small, leading to the fact that the estimates contained in ˆxae
0:T
correspond to solutions of truncated problems that are far away
from the turnpike, compare also the motivating example in
Section III-A, particularly Figure 1 for small N. For increasing
values of N, the estimates are getting closer to the turnpike,
and the performance improves significantly. Specifically, we
see exponential convergence to the optimal performance. This
could be expected since the system is exponentially detectable
[4, Sec. V.A] and controllable with respect to the input w,
which suggests that the turnpike property specializes to an
exponential one and hence renders the second statement of
Remark 5 valid. Overall, a problem length of N= 130 is
sufficient to achieve nearly optimal performance. The MHE
sequence ˆxmhe
0:T, on the other hand, generally yields worse per-
formance than the AE ˆxae
0:T(for N≥70). This is completely in
line with our theory, as ˆxmhe
0:Tis a concatenation of solutions of
truncated problems that are on the right leaving arc and hence
may be far from the turnpike, see the discussion below (11).
To assess the accuracy of the estimated state sequence
with respect to the real unknown system trajectory x0:T, we
compare the sum of squared errors5(SSE) of the full solution
ˆx∗
0:T, the AE ˆxae
0:T, and MHE ˆxmhe
0:Tfor different sizes Nof the
truncated problems. The corresponding results in Table I show
qualitatively the same behavior as in the previous performance
analysis. In particular, the full solution yields the most accurate
estimates with the lowest SSE. The proposed AE yields much
higher SSE for small horizons (SSE increase of 73.5 % for
N= 40 compared to the full solution), but improves very fast
as Nincreases, and exponentially converges to the SSE of the
full solution. On the other hand, the SSE of MHE improves
much slower, and is particularly much worse than that of the
full solution and the proposed AE (for N≥70).
2) Large estimation problems:We illustrate the potential of
the proposed AE for large estimation problems where the
computation of the full solution is either time-consuming or
simply impossible. To this end, we consider the linear time-
invariant (LTI) system
x+=Ax +Bu +w, y =C x +v,
5For a given sequence ˆx0:T, we define SSE(ˆx0:T) := PT
j=0 |ˆxj−xj|2.
TABLE I: SSE for the proposed AE and MHE.
Problem length NAE MHE
40 36.598 (+73.5 %) 27.197 (+28.9 %)
70 24.265 (+15.0 %) 24.311 (+15.2 %)
100 21.708 (+2.9 %) 23.986 (+13.7 %)
130 21.320 (+1.0 %) 23.341 (+10.6 %)
160 21.176 (+0.4 %) 23.325 (+10.5 %)
Values in parentheses indicate the relative increase in the SSE compared
to the full solution x∗
0:T(which achieves SSE = 21.1).
where the matrices A, B, C correspond to a random stable
LTI system (computed using the drss command of MATLAB)
with m= 30 inputs, n∈ {30,60,120}states and p∈
{10,20,40}outputs. We consider a batch of measured input-
output data d0:Twith T= 4803, where the system was
subject to x0= 0, a known sinusoidal input sequence u0:T,
and unknown process disturbance w0:T−1and measurement
noise v0:T(the former constituting a uniformly distributed
random sequence superimposed with a deterministic sinusoidal
function that may represent unmodeled nonlinear dynamics,
and the latter a uniformly distributed random sequence). For
reconstructing the unknown state sequence x0:T, we consider
the quadratic cost function (4)–(6) and select Q=In,R=
G=Ip. In the following, we compare the full solution
of the problem PT(d0:T)with the AE (38), where we rely
on the modifications from Remark 9 and select k= 32,
N= 150,∆ = 70 (this modification reduces the number
of truncated problems to be solved from K∆= 4654 for
∆=0to merely K∆= 34, i.e., by more than 99 %). The
full and truncated optimal estimation problems can be cast as
unconstrained quadratic programs (QPs), which we solve using
the quadprog implementation of MATLAB. The truncated
QPs for the AE are additionally solved in parallel using the
Parallel Computing Toolbox. For comparison reasons, we also
consider the Kalman filter (KF) using the covariance matrices
Q−1and R−1, where the initial estimate ˆxkf
0is drawn from
an isotropic normal distribution with zero mean and identity
covariance matrix In, and the initial covariance is chosen as
ˆ
Pkf
0=In. Moreover, we consider the fixed-interval smoother
(FIS) considering the entire batch of measurements, which
provides the best possible estimates in the context of KF-
related smoothing algorithms, see [35, Ch. 5] for further details
and a description of the corresponding algorithm.
Table II shows the estimation results in terms of the perfor-
mance index J[0,T ], accuracy (SSE), and overall computation
time τfor different system dimensions. For n∈ {30,60}, we
observe that the performance J[0,T ]and the SSE of the full
solution and the AE are nearly identical. However, the AE can
be computed more than 10 times faster than the full solution
(saving more than 90 % of the computation time) due to a
(much) smaller problem size and the fact that the truncated
problems are solved in parallel. For n= 120, it was already
impossible to numerically solve the full problem PT(d0:T)
due to the problem size (in contrast to the AE). For all system
realizations, the KF is much faster, as expected (because the
QPs are replaced by simple matrix computations), however,
performs much worse than the AE (both in terms of J[0,T ]and
SSE), which is mainly due to the fact that only past data is used
TABLE II: Simulation results for the full solution (full), the proposed approx-
imate estimator (AE), the fixed-interval smoother (FIS), and the Kalman filter
(KF).
n p Estimator J[0,T ]SSE τ[s]
30 10 full 12.91 40.81 9.914
30 10 AE 12.93 40.83 0.828
30 10 FIS 15.62 53.04 0.536
30 10 KF 4513.66 69.02 0.113
60 20 full 16.30 51.91 37.648
60 20 AE 16.31 51.91 2.849
60 20 FIS 19.02 66.80 1.608
60 20 KF 4021.66 94.68 0.354
120 40 AE 21.03 84.52 11.456
120 40 FIS 29.21 148.49 4.756
120 40 KF 18403.92 202.58 0.960
to compute the corresponding estimates. In contrast, the FIS
combines the KF forward recursion with a backward recursion
so that each estimate is computed based on the entire batch of
data, which requires more computations but provides improved
estimates compared to the KF; however, the performance and
accuracy are still worse compared to the AE due to the fact
that the considered disturbance and noise distributions violate
the conditions for the FIS to be optimal.
This example shows that the modifications from Remark 9
are very effective for computing the AE in practice. Specif-
ically, to recover the performance and accuracy of the full
solution, it suffices to choose ∆close to N/2such that only
the first and last few elements of the truncated solutions (which
lie on the approaching and leaving arcs) are discarded. Overall,
it turns out that the AE approximates the full solution with
negligible error, which is particularly important in practice
when the full problem PT(d0:T)cannot be solved due to the
size or complexity of the problem and iterative solutions such
as the KF and related smoothing algorithms are not sufficiently
accurate.
B. Online estimation
1) Continuous stirred-tank reactor:We consider the continu-
ous stirred-tank reactor (CSTR) from [1, Example 1.11], where
an irreversible, first order reaction A→Boccurs in the liquid
phase and the reactor temperature is regulated with external
cooling, see also [59] for more details. The continuous-time
nonlinear state space model is given by
dc
dt =F0(c0−c)
πr2h−k0exp −E
RT c,
dT
dt =F0(T0−T)
πr2h+−∆H
ρCp
k0exp −E
RT c+2U
rρCp
(Tc−T),
dh
dt =F0−F
πr2,
where the states are c(the molar concentration of species A),
T(the reactor temperature), and h(the level of the tank),
and the control inputs are Tc(the coolant liquid temperature)
and F(the outlet flowrate). The model parameters are taken
from [1, Example 1.11]. The open-loop stable steady state
solution is xss = [0.878,323.5,0.659]⊤associated with the
input uss = [300,0.1]⊤. By using Runge-Kutta discretization
130 140 150 160 170 180
t
0
0.1
0.2
0.3
0.4
0.5
0.6
j~x$
jjt!x1
jj
Fig. 4: CSTR. Finite-horizon solutions ˜x∗
j|tcompared to x∞
j,j∈I[t−N,t]
for t∈I[130,180] using the filtering prior (blue), smoothing prior (black), and
turnpike prior (red); highlighted are particular solutions obtained at t= 149
and t= 169.
(RK4) with sampling time t∆= 0.25, we obtain the discrete-
time model in (1) with fas in (14) and h(x, u) = x2,
where we define x= [c, T, h]⊤and u= [Tc, F ]⊤, consider
additional process disturbance w∈R3, and assume that only
the temperature of the reactor Tcan be measured, subject to
the measurement noise v∈R. The real system is initialized
with x0= [0.8,295,0.7]⊤. In the following, we consider the
simulation time Ts= 200 and apply an open-loop control
sequence u0:Tswith u2=uss
2and u1as a trapezoidal input
sequence with plateaus at uss
1and uss
1−25. During the
simulations, we sample the disturbances wand vfrom uniform
distributions over {w:|w1| ≤ 5·10−3,|w2| ≤ 1,|w3| ≤
5·10−3}and {v:|v| ≤ 3}. To estimate the true unknown
state xtfrom the measurement data, we design different MHE
schemes that rely on the cost function (24) with the horizon
length N= 10 and quadratic costs (5) and (6), where we
select Q= diag(103,1,105),R=G= 1. Moreover, we use
a quadratic prior weighting (25) with time-varying matrix Wt,
t∈I≥0, initialized with W0= 10−2I3and updated using the
well-known covariance formulas of the extended Kalman Filter
(EKF). In the optimal estimation problems, we also consider
the state constraints X= [0.5,1.5] ×[200,400] ×[0.5,1.5],
but we consider the disturbance sets to be unknown and
use W=R3and V ∈ R. In the following, we compare
MHE with filtering prior (26), smoothing prior (27), and
turnpike prior (28) and additionally consider the infinite-
horizon estimator (IHE), which we approximate by solving
the clairvoyant FIE problem using all available data d0:Ts.
From Figure 4, we can observe that all MHE problems (for
all priors) exhibit the turnpike behavior with respect to the
infinite-horizon solution, with clear approaching and leaving
arcs, which is a strong indicator that Assumption 3 holds true.
We additionally compare standard MHE schemes (without
delay) with δMHE (37) using the turnpike prior for δ= 1
and δ=N/2. From Figure 5, we see that the standard MHE
schemes yield very similar estimation results in terms of the
difference to the IHE (for all priors), that δMHE with δ= 1
provides estimates that are much closer to the IHE, and that
δMHE with δ=N/2(which corresponds to the turnpike
0 50 100 150 200
t
10!3
10!2
10!1
100
j^xt!x1
tj
filtering
smoothing
turnpike
δMHE, δ=1
δMHE, δ=N/2
Fig. 5: CSTR. Distance between state estimates using different MHE schemes
and the IHE. Dots indicate values at time t, lines indicate the moving average
over a sliding window of size N+ 1.
filtering
smoothing
turnpike
δMHE, δ=1
δMHE, δ=N/2
IHE
1.5
2
2.5
3
SSE/102
Fig. 6: CSTR. Boxplot of the SSE for MHE using the filtering prior, the
smoothing prior, and the turnpike prior, δMHE using the turnpike prior for
δ= 1 and δ=N/2 = 5, and the IHE over 100 different simulations with
random disturbance/noise and randomly selected initial priors.
prior (28)) converges to a (small) neighborhood around the
IHE, which nicely illustrates Proposition 2.
We consider 100 different simulations with random distur-
bances and randomly selected initial priors ¯x0that are sampled
from a uniform distribution over the interval centered at x0
with a relative deviation of 25 % for each state. Figure 6 shows
that the standard MHE schemes are again very similar in terms
of their SSE (for all priors), significantly outperformed by
δMHE for δ= 1 (which yields a reduction in the SSE by
20 %). Moreover, we observe that the SSE of δMHE with
δ=N/2is very close to that of the IHE.
Overall, this example nicely illustrates the developed the-
ory. In particular, it shows that MHE problems with prior
weighting exhibit the turnpike behavior with respect to the
IHE (Assumption 3), with a potentially strong leaving arc,
cf. Figure 4. This motivates to incorporate an artificial delay
in the estimation scheme in order to reduce the influence of the
leaving arc. Surprisingly, already a one-step delay is sufficient
to significantly reduce the influence of the leaving arc such
that δMHE tracks the performance and accuracy of the IHE
with small error, cf. Figures 5 and 6.
2) Quadrotor:We consider a quadrotor with flexible blades
and adapt the dynamical model from [60]. In the following,
by Iwe denote the stationary inertial system with its vertical
component pointing into the Earth, where z= [z1, z2, z3]⊤
and v= [v1, v2, v3]⊤represent the position and velocity of
the quadrotor, respectively. By Bwe refer to the body-fixed
frame attached to the quadrotor, with the third component
pointing in the opposite direction of thrust generation. The
attitude of Bwith respect to Iis described by a rotation matrix
Rinvolving the roll, pitch, and yaw angle of the quadrotor,
which we denote by ξ= [ϕ, θ, ψ]⊤. The angular velocity
of the quadrotor in Bwith respect to Iis represented by
Ω = [Ω1,Ω2,Ω3]⊤. Assuming a wind-free environment, the
dynamics of the quadrotor can be described as
˙z=v , m ˙v=mge3−T R(ξ)e3−R(ξ)BΩ,
˙
ξ= Γ(ξ)Ω, J ˙
Ω = −Ω×JΩ + τ−DΩ,
where e3= [0 0 1]⊤and (·)×refers to the skew symmetric
matrix associated with the cross product such that u×v=u×v
for any u, v ∈R3. The thrust T∈Rand the torque τ∈R3
are generated by the velocities ωi,i={1,2,3,4}of the rotors
via
T
τ=
cTcTcTcT
0−lcT0lcT
lcT0−lcT0
−cQcQ−cQcQ
ω2
1
ω2
2
ω2
3
ω2
4
,
and the matrix Γis defined as
Γ(ξ) =
1 sin ϕtan θcos ϕtan θ
0 cos ϕ−sin ϕ
0 sin ϕsec θcos ϕsec θ
,
see, e.g., [60] for further details. The parameters are chosen as
m= 1.9,J= diag(5.9,5.9,10.7) ·10−3,g= 9.8,l= 0.25,
cT= 10−5,cQ= 10−6,B= 1.14·e×
3, and D= 0.0297·e3e⊤
3.
The overall model has the states x= [z⊤, ξ⊤, v⊤,Ω⊤]⊤∈
R12 and the inputs u= [ω1, ω2, ω3, ω4]⊤∈R4. Using Euler-
discretization and the sampling time t∆= 0.05, we obtain the
discrete-time model in (1) with fas in (14) and h(x, u) =
[I6,06×6]x, where we consider process disturbances w∈R12
and assume that only measurements of the position zand
orientation ξare available, subject to the noise v∈R6.
In the simulations, the disturbance wand noise vare
uniformly distributed random variables sampled from the sets
{w:|wi| ≤ 2·10−2, i ={1,2,3},|wi| ≤ 2·10−5, i =
{4,5,6},|wi| ≤ 2·10−3, i ={7,8,9},|wi| ≤ 2·10−6, i =
{10,11,12}} and {v:|vi| ≤ 2·10−1, i ={1,2,3},|vi| ≤
5·10−2, i ={4,5,6}}. We consider the simulation time Ts=
1000 and a given open-loop trajectory u0:Ts, which moves
the quadrotor spirally upwards, see Figure 7 for an exemplary
trajectory under a specific disturbance realization. To estimate
the unknown state xt, we consider the cost function (24)
with the horizon length N= 30 and quadratic costs (5) and
(6), where we select Q= blkdiag(102I3,104I3,103I3,105I3)
and R=G= blkdiag(101I3,102I3). Moreover, we use the
quadratic prior weighting (25), where Wtis initialized with
W0= 10I12 and updated using the EKF covariance formulas
for all t∈I≥0. We consider the case where no additional
Fig. 7: Quadrotor. Exemplary 3D trajectory for one specific disturbance
realization; comparison of the true trajectory (red), measurements (gray dots),
MHE with filtering prior (blue), and δMHE for δ=N/2 = 15 (cyan).
filtering
smoothing
turnpike
δMHE, δ=1
δMHE, δ=3
δMHE, δ=N/2
IHE
0.5
1
1.5
2
2.5
SSE/10
Fig. 8: Quadrotor. Boxplot of the SSE for MHE using the filtering prior, the
smoothing prior, and the turnpike prior, δMHE using the turnpike prior for
δ= 1,δ= 3, and δ=N/2 = 15, and the IHE over 100 different simulations
with random disturbance/noise and randomly selected initial priors.
information about the domains of the states and disturbances
is available and use X=R12,W=R12,V=R6in (3c)–
(3e). In the following, we examine 100 different simulations
with random disturbances, where we additionally sample the
initial prior ¯x0from a uniform distribution over the set X0=
{x:|zi| ≤ 1,|ξi| ≤ π/16, i ={1,2,3}, v = 0,Ω=0}. We
compare standard MHE with filtering prior (26), smoothing
prior (27), and turnpike prior (28), δMHE (37) with turnpike-
based prior weighting and δ= 1,δ= 3,δ=N/2, and the
IHE (which we approximate by solving the clairvoyant FIE
problem using all available data d0:Ts).
From Figure 8, we observe that MHE with filtering and
turnpike prior performs quite similarly. The fact that MHE
with smoothing prior is slightly worse can be attributed to
the fact that the movement of the quadrotor is rather slow
compared to the sampling time, while the horizon length
N= 30 is also rather small. In such setting, MHE with
filtering or turnpike prior proves beneficial, as this essentially
considers measurements from a larger estimation window,
cf. the discussion below (27). For δMHE, we can observe
a reduction of the SSE of approximately 10 % for δ= 1, of
25 % for δ= 3, and of by 50 % for δ=N/2, which is also
close the SSE of the IHE.
To conclude, this example illustrates that the developed
theory is also applicable to more complex and realistic systems
from the literature. In particular, it again shows that using the
proposed δMHE scheme with a small delay δcan already
significantly improve the estimation performance in practice.
VIII. CONCLUSION
In this paper, we developed novel accuracy and performance
guarantees for optimal state estimation of general nonlinear
systems, with a strong focus on MHE for online state es-
timation. Our results rely on a turnpike property of finite-
horizon optimal state estimation problems with respect to the
solution of the omniscient (acausal) infinite-horizon problem
involving all past and future data. This naturally causes MHE
problems to exhibit a leaving arc that can have a potentially
strong negative impact on estimation accuracy. To counteract
the leaving arc, we used an artificial delay in the MHE scheme,
and we showed that the resulting performance is approximately
optimal with respect to the infinite-horizon solution, with error
terms that can be made arbitrarily small by an appropriate
choice of the delay. We proposed a novel turnpike prior for
MHE formulations with prior weighting, proven to be a valid
alternative to the classical options (such as the filtering or
smoothing prior) with superior theoretical properties.
In our simulations, we found that MHE with the proposed
turnpike prior performs comparably well to MHE with fil-
tering or smoothing priors, while the delay resulted in a
significant improvement of the estimation results. In particular,
considering practical examples from the literature, we could
observe the turnpike phenomenon and found that a delay of
one to three steps improved the overall estimation error by 20-
25 % compared to standard MHE (without delay). For offline
estimation, the proposed delayed MHE scheme has shown to
be a useful alternative to established iterative filtering and
smoothing methods, significantly outperforming them espe-
cially in the presence of non-normally distributed noise.
An interesting topic for future work is the combination of
the delayed MHE scheme with state feedback control and in
particular the investigation of whether the delay in the closed
loop is worthwhile for significantly better estimation results.
REFERENCES
[1] J. B. Rawlings, D. Q. Mayne, and M. Diehl, Model Predictive Control:
Theory, Computation, and Design, 2nd ed. Santa Barbara, CA, USA:
Nob Hill Publish., LLC, 2020, 3rd printing.
[2] D. A. Allan and J. B. Rawlings, “Robust stability of full information
estimation,” SIAM J. Control Optim., vol. 59, no. 5, pp. 3472–3497,
2021.
[3] S. Kn¨
ufer and M. A. M¨
uller, “Nonlinear full information and moving
horizon estimation: Robust global asymptotic stability,” Automatica, vol.
150, p. 110603, 2023.
[4] J. D. Schiller, S. Muntwiler, J. K¨
ohler, M. N. Zeilinger, and M. A.
M¨
uller, “A Lyapunov function for robust stability of moving horizon
estimation,” IEEE Trans. Autom. Control, vol. 68, no. 12, pp. 7466–
7481, 2023.
[5] W. Hu, “Generic stability implication from full information estimation
to moving-horizon estimation,” IEEE Trans. Autom. Control, vol. 69,
no. 2, pp. 1164–1170, 2024.
[6] A. Alessandri, “Robust moving-horizon estimation for nonlinear sys-
tems: From perfect to imperfect optimization,” arXiv:2501.03894, 2025.
[7] G. Goel and B. Hassibi, “Regret-optimal estimation and control,” IEEE
Trans. Autom. Control, vol. 68, no. 5, pp. 3041–3053, 2023.
[8] O. Sabag and B. Hassibi, “Regret-optimal filtering for prediction and
estimation,” IEEE Trans. Signal Process., vol. 70, pp. 5012–5024, 2022.
[9] J.-S. Brouillon, F. D¨
orfler, and G. Ferrari-Trecate, “Minimal regret state
estimation of time-varying systems,” IFAC-PapersOnLine, vol. 56, no. 2,
pp. 2595–2600, 2023.
[10] A. Didier, J. Sieber, and M. N. Zeilinger, “A system level approach to
regret optimal control,” IEEE Control Syst. Lett., vol. 6, pp. 2792–2797,
2022.
[11] A. Martin, L. Furieri, F. D¨
orfler, J. Lygeros, and G. Ferrari-Trecate, “Re-
gret optimal control for uncertain stochastic systems,” Eur. J. Control,
vol. 80, Part A, p. 101051, 2024.
[12] M. Gharbi, B. Gharesifard, and C. Ebenbauer, “Anytime proximity
moving horizon estimation: Stability and regret,” IEEE Trans. Autom.
Control, vol. 68, no. 6, pp. 3393–3408, 2022.
[13] ——, “Anytime proximity moving horizon estimation: Stability and
regret for nonlinear systems,” in Proc. 60th IEEE Conf. Decision Control
(CDC), 2021, pp. 728–735.
[14] L. W. McKenzie, “Optimal economic growth, turnpike theorems and
comparative dynamics,” in Handbook of Mathematical Economics, K. J.
Arrow and M. D. Intriligator, Eds. Amsterdam, The Netherlands:
Elsevier, 1986, vol. 3, pp. 1281–1355.
[15] D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon
Optimal Control. Berlin, Heidelberg, Germany: Springer, 1991.
[16] L. Gr¨
une, “Approximation properties of receding horizon optimal con-
trol,” Jahresber. Dtsch. Math. Ver., vol. 118, no. 1, pp. 3–37, 2016.
[17] T. Faulwasser and L. Gr ¨
une, “Turnpike properties in optimal control,”
in Handbook of Numerical Analysis, E. Tr´
elat and E. Zuazua, Eds.
Amsterdam, The Netherlands: Elsevier, 2022, vol. 23, pp. 367–400.
[18] L. Gr¨
une and S. Pirkelmann, “Economic model predictive control for
time-varying system: Performance and stability results,” Optim. Control
Appl. Methods, vol. 41, no. 1, pp. 42–64, 2019.
[19] T. Damm, L. Gr¨
une, M. Stieler, and K. Worthmann, “An exponential
turnpike theorem for dissipative discrete time optimal control problems,”
SIAM J. Control Optim., vol. 52, no. 3, pp. 1935–1957, 2014.
[20] L. Gr¨
une and M. A. M¨
uller, “On the relation between strict dissipativity
and turnpike properties,” Syst. Control Lett., vol. 90, pp. 45–53, 2016.
[21] E. Tr´
elat, “Linear turnpike theorem,” Math. Control Signals Syst.,
vol. 35, no. 3, pp. 685–739, 2023.
[22] T. Faulwasser, L. Gr¨
une, J.-P. Humaloja, and M. Schaller, “The interval
turnpike property for adjoints,” Pure Appl. Funct. Anal., vol. 7, no. 4,
pp. 1187–1207, 2022.
[23] J. D. Schiller, L. Gr¨
une, and M. A. M¨
uller, “Optimal state estimation:
Turnpike analysis and performance results,” arXiv:2409.14873, 2024.
[24] D. A. Allan, “A Lyapunov-like function for analysis of model predic-
tive control and moving horizon estimation,” Ph.D. dissertation, Univ.
Wisconsin-Madison, 2020.
[25] J. K¨
ohler, M. A. M¨
uller, and F. Allg¨
ower, “Nonlinear reference tracking:
An economic model predictive control perspective,” IEEE Trans. Autom.
Control, vol. 64, no. 1, pp. 254–269, 2019.
[26] L. Gr¨
une, S. Pirkelmann, and M. Stieler, “Strict dissipativity implies
turnpike behavior for time-varying discrete time optimal control prob-
lems,” in Control Systems and Mathematical Methods in Economics,
G. Feichtinger, R. Kovacevic, and G. Tragler, Eds. Cham, Germany:
Springer, 2018, pp. 195–218.
[27] T. Faulwasser, L. Gr¨
une, and M. A. M¨
uller, “Economic nonlinear model
predictive control,” Found. Trends Syst. Control, vol. 5, no. 1, pp. 1–98,
2018.
[28] E. Tr´
elat and E. Zuazua, “The turnpike property in finite-dimensional
nonlinear optimal control,” J. Differ. Equ., vol. 258, no. 1, pp. 81–114,
2015.
[29] Z.-J. Han and E. Zuazua, “Slow decay and turnpike for infinite-horizon
hyperbolic linear quadratic problems,” SIAM J. Control Optim., vol. 60,
no. 4, pp. 2440–2468, 2022.
[30] A. Ferrante and L. Ntogramatzidis, “Employing the algebraic Riccati
equation for a parametrization of the solutions of the finite-horizon LQ
problem: the discrete-time case,” Syst. Control Lett., vol. 54, no. 7, pp.
693–703, 2005.
[31] S. Na and M. Anitescu, “Exponential decay in the sensitivity analysis
of nonlinear dynamic programming,” SIAM J. Optim., vol. 30, no. 2, pp.
1527–1554, 2020.
[32] S. Shin, M. Anitescu, and V. M. Zavala, “Exponential decay of sensi-
tivity in graph-structured nonlinear programs,” SIAM J. Optim., vol. 32,
no. 2, pp. 1156–1183, 2022.
[33] S. Shin and V. M. Zavala, “Controllability and observability im-
ply exponential decay of sensitivity in dynamic optimization,” IFAC-
PapersOnLine, vol. 54, no. 6, pp. 179–184, 2021.
[34] L. Gr¨
une, M. Schaller, and A. Schiela, “Exponential sensitivity and
turnpike analysis for linear quadratic optimal control of general evolution
equations,” J. Differ. Equ., vol. 268, no. 12, pp. 7311–7341, 2020.
[35] J. L. Crassidis and J. L. Junkins, Optimal estimation of dynamic systems,
2nd ed. Boca Raton, FL, USA: Chapman and Hall/CRC, 2011.
[36] J. B. Moore, “Discrete-time fixed-lag smoothing algorithms,” Automat-
ica, vol. 9, no. 2, pp. 163–173, 1973.
[37] S. Yi and M. Zorzi, “Robust fixed-lag smoothing under model pertur-
bations,” J. Franklin Inst., vol. 360, no. 1, pp. 458–483, 2023.
[38] O. U. Rehman and I. R. Petersen, “A robust continuous-time fixed-
lag smoother for nonlinear uncertain systems,” Int. J. Robust Nonlinear
Control, vol. 26, no. 2, pp. 345–364, 2015.
[39] D. Simon and Y. S. Shmaliy, “Unified forms for Kalman and finite
impulse response filtering and smoothing,” Automatica, vol. 49, no. 6,
pp. 1892–1899, 2013.
[40] P. S. Kim, “A computationally efficient fixed-lag smoother using recent
finite measurements,” Measurement, vol. 46, no. 1, pp. 846–850, 2013.
[41] B. Kwon and P. S. Kim, “Novel unbiased optimal receding-horizon
fixed-lag smoothers for linear discrete time-varying systems,” Appl. Sci.,
vol. 12, no. 15, p. 7832, 2022.
[42] L. Gr¨
une and A. Panin, “On non-averaged performance of economic
MPC with terminal conditions,” in Proc. IEEE Conf. Decis. Control
(CDC), 2015, pp. 4332–4337.
[43] C. V. Rao, J. B. Rawlings, and D. Q. Mayne, “Constrained state
estimation for nonlinear discrete-time systems: stability and moving
horizon approximations,” IEEE Trans. Autom. Control, vol. 48, no. 2,
pp. 246–258, 2003.
[44] C. C. Qu and J. Hahn, “Computation of arrival cost for moving horizon
estimation via unscented Kalman filtering,” J. Process Control, vol. 19,
no. 2, pp. 358–363, 2009.
[45] K. Baumg¨
artner, A. Zanelli, and M. Diehl, “A gradient condition for the
arrival cost in moving horizon estimation,” in Proc. Eur. Control Conf.
(ECC), 2020, pp. 1286–1291.
[46] P. K ¨
uhl, M. Diehl, T. Kraus, J. P. Schl¨
oder, and H. G. Bock, “A real-time
algorithm for moving horizon state and parameter estimation,” Comput.
Chem. Eng., vol. 35, no. 1, pp. 71–83, 2011.
[47] A. Alessandri, M. Baglietto, and G. Battistelli, “Moving-horizon state
estimation for nonlinear discrete-time systems: New stability results and
approximation schemes,” Automatica, vol. 44, no. 7, pp. 1753–1765,
2008.
[48] A. Alessandri and M. Gaggero, “Fast moving horizon state estimation for
discrete-time systems using single and multi iteration descent methods,”
IEEE Trans. Autom. Control, vol. 62, no. 9, pp. 4499–4511, 2017.
[49] M. Elsheikh, R. Hille, A. Tatulea-Codrean, and S. Kr¨
amer, “A com-
parative review of multi-rate moving horizon estimation schemes for
bioprocess applications,” Comput. Chem. Eng., vol. 146, p. 107219,
2021.
[50] M. Asch, M. Bocquet, and M. Nodet, Data Assimilation: Methods,
Algorithms, and Applications. Philadelphia, PA, USA: SIAM, 2016.
[51] A. Carrassi, M. Bocquet, L. Bertino, and G. Evensen, “Data assimilation
in the geosciences: An overview of methods, issues, and perspectives,”
WIREs Climate Change, vol. 9, no. 5, 2018.
[52] P. Mohajerin Esfahani, T. Sutter, D. Kuhn, and J. Lygeros, “From
infinite to finite programs: Explicit error bounds with applications to
approximate dynamic programming,” SIAM J. Optim., vol. 28, no. 3,
pp. 1968–1998, 2018.
[53] P. Mohajerin Esfahani and D. Kuhn, “Data-driven distributionally robust
optimization using the Wasserstein metric: performance guarantees and
tractable reformulations,” Math. Program., vol. 171, no. 1–2, pp. 115–
166, 2017.
[54] D. A. Allan, J. B. Rawlings, and A. R. Teel, “Nonlinear detectability
and incremental input/output-to-state stability,” SIAM J. Control Optim.,
vol. 59, no. 4, pp. 3017–3039, 2021.
[55] H. Arezki, A. Zemouche, A. Alessandri, and P. Bagnerini, “LMI design
procedure for incremental input/output-to-state stability in nonlinear
systems,” IEEE Control Syst. Lett, vol. 7, pp. 3403–3408, 2023.
[56] J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl,
“CasADi: a software framework for nonlinear optimization and optimal
control,” Math. Program. Comput., vol. 11, no. 1, pp. 1–36, 2018.
[57] A. W¨
achter and L. T. Biegler, “On the implementation of an interior-
point filter line-search algorithm for large-scale nonlinear programming,”
Math. Program., vol. 106, no. 1, pp. 25–57, 2005.
[58] M. J. Tenny and J. B. Rawlings, “Efficient moving horizon estimation
and nonlinear model predictive control,” in Proc. Am. Control Conf.,
2002, pp. 4475–4480.
[59] G. Pannocchia and J. B. Rawlings, “Disturbance models for offset-free
model-predictive control,” AIChE Journal, vol. 49, no. 2, pp. 426–437,
2003.
[60] J.-M. Kai, G. Allibert, M.-D. Hua, and T. Hamel, “Nonlinear feed-
back control of quadrotors exploiting first-order drag effects,” IFAC-
PapersOnLine, vol. 50, no. 1, pp. 8189–8195, 2017.
Julian D. Schiller received his Master degree
in Mechatronics from the Leibniz University
Hannover, Germany, in 2019. Since then, he
has been a research assistant at the Institute
of Automatic Control, Leibniz University Han-
nover, where he is currently working on his
Ph.D. under the supervision of Prof. Matthias
A. M¨
uller. His research interests are in the area
of optimization-based state estimation and the
control of nonlinear systems.
Lars Gr ¨une has been Professor for Applied
Mathematics at the University of Bayreuth, Ger-
many, since 2002. He received his Diploma and
Ph.D. in Mathematics in 1994 and 1996, respec-
tively, from the University of Augsburg and his
habilitation from the J.W. Goethe University in
Frankfurt/M in 2001. He held or holds visiting
positions at the Universities of Rome ‘Sapienza’
(Italy), Padova (Italy), Melbourne (Australia),
Paris IX - Dauphine (France), Newcastle (Aus-
tralia) and IIT Bombay (India).
Prof. Gr¨
une was General Chair of the 25th International Symposium
on Mathematical Theory on Networks and Systems (MTNS 2022), he
is Editor-in-Chief of the journal Mathematics of Control, Signals and
Systems (MCSS) and is or was Associate Editor of various other
journals, including the Journal of Optimization Theory and Applications
(JOTA), Mathematical Control and Related Fields (MCRF) and the IEEE
Control Systems Letters (CSS-L). His research interests lie in the area
of mathematical systems and control theory with a focus on numerical
and optimization-based methods for nonlinear systems.
Matthias A. M ¨uller received a Diploma degree
in engineering cybernetics from the University
of Stuttgart, Germany, an M.Sc. in electrical
and computer engineering from the University of
Illinois at Urbana-Champaign, US (both in 2009),
and a Ph.D. from the University of Stuttgart in
2014. Since 2019, he is Director of the Institute
of Automatic Control and Full Professor at the
Leibniz University Hannover, Germany.
His research interests include nonlinear con-
trol and estimation, model predictive control, and
data- and learning-based control, with application in different fields
including biomedical engineering and robotics. He has received various
awards for his work, including the 2015 European Systems & Control
PhD Thesis Award, the inaugural Brockett-Willems Outstanding Paper
Award for the best paper published in Systems & Control Letters in
the period 2014-2018, an ERC starting grant in 2020, the IEEE CSS
George S. Axelby Outstanding Paper Award 2022, and the Journal of
Process Control Paper Award 2023. He serves/d as an associate editor
for Automatica and as an editor of the International Journal of Robust
and Nonlinear Control.
