# Christopher I. Byrnes's research while affiliated with KTH Royal Institute of Technology and other places

## Publications (161)

Article
Periodic phenomena play a pervasive role in natural and in man-made systems. They are exhibited, for example, in simple mathematical models of the solar system and in the observed circadian
Article
This work is concerned with a problem of tracking regulation for a one-dimensional Kuramoto–Sivashinsky equation. The objective is to design dynamic and static controllers to drive the state of the plant at the ends of the spatial domain to desired reference signals which may be time dependent. The particular case of constant reference signals is r...
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Over the past three decades there has been interest in using Padé approximants K with n = deg(K) < deg(G) = N as “reduced-order models” for the transfer function G of a linear system. The attractive feature of this approach is that by matching the moments of G we can reproduce the steady-state behavior of G by the steady-state behavior of K, for ce...
Conference Paper
Full-text available
The moment problem matured from its various special forms in the late 19th and early 20th Centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have h...
Conference Paper
We recently introduced a new approach, zero dynamics inverse (ZDI) design, for designing a feedback compensation scheme achieving asymptotic regulation, i.e., asymptotic tracking and/or disturbance rejection, for a linear or nonlinear distributed parameter system (DPS) in the case when only the value of the signal w(t) to be tracked or rejected are...
Chapter
The existence and nature of nonlinear oscillations for periodically forced nonlinear differential equations has historically attracted quite a bit of attention in both the pure and the applied mathematics literature. In control theory, it encompasses the study of the steady-state response of control systems to periodic inputs, generalizing the freq...
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The moment problem as formulated by Krein and Nudel’man is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment problem and the moment problem arising in Nevanlinna-Pick interpolation. Motivated by classical applications and examples, in both finite and infinite dime...
Article
Full-text available
In this paper we consider the Jacobian conjecture for a map f of complex affine spaces of dimension n. It is well known that if f is proper, then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if f is proper onto its image.
Article
In [2], Roger Brockett derived a necessary condition for the existence of a feedback control law asymptotically stabilizing an equilibrium for a given nonlinear control system. The intuitive appeal and the ease with which it can be applied have made this criterion one of the standard tools in the study of the feedback stabilizability of nonlinear c...
Article
Full-text available
The moment problem matured from its various special forms in the late 19th and early 20th Centuries to a general class of problems that continues to exert profound inuence on the nite-dimensional subspace P of the Banach space C(a;b) and a \positive" sequences c, but with a new wrinkle inspired by the applications to systems and control. We seek to...
Chapter
This paper is concerned with the development of basic concepts and constructs for a nonequilibrium theory of nonlinear control. Motivated by an example of nonstabilizability of rigid spacecraft about an equilibrium (reference attitude) but stabilizability about a revolute motion, we review recent work on the structure of those compact attractors wh...
Article
In this book, an enhancement of the first author's 2004 Ph.D. dissertation, the authors treat the problem of output regulation for a nonlinear control system. There are a number of approaches to, and alternatives in the formulation of, such problems. One of the things that set this book apart is that the authors' willingness to discuss and define t...
Chapter
One of the modern geometric views of dynamical systems is as vector fields on a manifold, with or without boundary. The starting point of this paper is the observation that, since one-forms are the natural expression of linear functionals on the space of vector fields, the interaction between the two makes some aspects of the study of equilibria an...
Article
In a recent series of papers, Drager and Martin have shown global observability for a variety of particular classes of systems (including both discrete and continuous time systems). The underlying philosophy behind their methods is apparently that if a flow is ergodic and the system observation has a sufficiently special value, the system is comple...
Article
Variational problems and the solvability of certain nonlinear equations have a long and rich history beginning with calculus and extending through the calculus of variations. In this paper, we are interested in "well-connected" pairs of such problems which are not necessarily related by critical point considerations. We also study constrained probl...
Chapter
The paper deals with the problem of synthesizing a feedback loop matching the input-output behavior of a prescribed linear model and being internally stable, in the sense that the response to any bounded excitation is also bounded. The required condition on the plant is that of having a stable (in the same sense) inverse.
Article
In this work the authors introduce a notion of zero dynamics for distributed parameter systems governed by linear parabolic equations on bounded domains with controls implemented through first order linear boundary conditions. The idea of zero dynamics presented here is motivated by classical root-locus constructs from finite dimensional linear sys...
Article
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In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a refor-mulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering appli-cations typically demand a solution that is the ratio o...
Chapter
The problems of determining the minimal order of a stabilizing compensator for a fixed linear, multivariable system and for the generic p m system of fixed degree are considered. An elementary geometric argument gives sufficient conditions for the generic stabilizability by a compensator of order q. A more delicate geometric argument, involving pol...
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Without Abstract
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In this paper, I outline a program, initiated in independent and in joint efforts by A.J. Krener, A. Isidori, and myself, whose goal is to extend the linear (A,B)-invariant subspace theory to nonlinear problems beyond the existing local theory, including particularly those problems which are either globally formulated or which require an analysis o...
Chapter
In this paper we use a recently proven general position lemma for transfer functions to derive several important qualitative properties, some new, of the root-locus map for multivariable systems. Among the immediate applications which we derive is that it is not in general possible to develop a formula, involving rational operations and the extract...
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Algebraic geometry plays an important role in the theory of linear systems for (at least) three reasons. First, the Laplace transform turns expressions about linear differential systems into expressions involving rational functions. In addition, many of the transformations studied in linear systems theory, like changes of coordinates or feedback, t...
Article
In nonlinear control theory, the equilibrium of a system is semiglobally practically stabilizable if, given two balls centred at the equilibrium, one of arbitrarily large radius and one of arbitrarily small radius, it is possible to design a feedback so that the resulting closed-loop system has the following property: all the trajectories originati...
Article
Full-text available
In a seminal paper, Sarason generalized some classical interpolation problems for H<sup>∞</sup> functions on the unit disc to problems concerning lifting onto H<sup>2</sup> of an operator T that is defined on $\cal{K} = H^{2}\ominus \phi H^2$ (φ is an inner function) and commutes with the (compressed) shift S. In particular, he showed that interpol...
Article
In this paper we show how to use nonlinear internal models in the design of output regulators for nonlinear systems. This result provides a significant enhancement of the non-equilibrium theory for output regulation.
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This paper is dedicated to Arthur Krener – a great researcher, a great teacher and a great friend – on the occasion of his 60th birthday. In this work we study the generalized moment problem with complexity constraints in the case where the actual values of the moments are uncertain. For example, in spectral estimation the moments correspond to est...
Article
In this paper, we lay the foundations for a nonequilibrium theory of nonlinear output regulation, giving a more general (nonequilibrium) definition of the problem, deriving necessary conditions, and, using these necessary conditions, we present a set of sufficient conditions and a design methodology for the solution of the problem in question. Our...
Book
This volume provides a compilation of recent contributions on feedback and robust control, modeling, estimation and filtering. They were presented on the occasion of the sixtieth birthday of Anders Lindquist, who has delivered fundamental contributions to the fields of systems, signals and control for more than three decades. His contributions incl...
Chapter
Our interest in suppression of harmonic disturbances arose in the development of feedback control strategies for next generation aircraft. The control objective is to track a prescribed trajectory while suppressing the disturbance produced by a harmonic exogenous system. This is a slight modification of the standard problem of output regulation, in...
Article
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Traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates. Here, we present a new approach to spectral estimation, which is based on the use of filter banks as a means of obtaining spectral interpolation data. Such data replaces standard covariance estimates. A computational procedure for obtaining suitab...
Article
Full-text available
In this paper we present a universal solution to the generalized moment problem, with a nonclassical complexity constraint. We show that this solution can be obtained by minimizing a strictly convex nonlinear functional. This optimization problem is derived in two different ways. We first derive this in-trinsically, in a geometric way, by path inte...
Article
In this short paper we present an example of the geometric theory of output regulation applied to solve a tracking problem for a plant consisting of a boundary controlled distributed parameter system (heat equation on a rectangle) with unbounded input and output maps and signal to be tracked generated by an infinite dimensional exosystem. The exosy...
Article
Full-text available
In this paper, we study the well-posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients, covariance lags, or Markov param- eters. For example, we determine whether there exists a shaping filter with a prescribed window of Markov parameters anda prescribedwindow of covariance lags. We sh...
Article
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Thetrigonom5KO1 mrig tproblem is a classicalmass tproblem with numMO8S applications inm9M86VC5KS9 physics, and engineering. The rational covariance extensionproblem is a constrained version of thisproblem with the constraints arisingfrom the physical realizability of the corresponding solutions. Although themeCM um entropym ethod gives one well-kno...
Conference Paper
In this paper we apply the methods of geometric output regulation to regulate a linear control system with a delay. For this example our design objective is to track a harmonic reference signal and while rejecting a constant disturbance. The signal to be tracked and disturbance to be rejected are generated by a finite dimensional exogenous system....
Article
Full-text available
The trigonometric moment problem is a classical moment problem with numerous applica-tions in mathematics, physics, and engineering. The rational covariance extension problem is a constrained version of this problem, with the constraints arising from the physical re-alizability of the corresponding solutions. Although the maximum entropy method giv...
Article
Full-text available
We present a generalized entropy criterion for solving the rational Nevanlinna-Pick problem for n+1 interpolating conditions and the degree of interpolants bounded by n. The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to...
Article
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In this paper we describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this CovarianceEiancebfi EEnceb is based on a recent complete parameterization of all strictly positive real solutions to the rational covariance extension problem, a...
Article
In this paper, we derive a necessary condition for local asymptotic stability of equilibria of nonlinear systems with parameters. As a corollary of our general result, we deduce Brockett's necessary condition for local asymptotic stability of equilibria of nonlinear autonomous systems. The proof we give, however, is quite different from the existin...
Article
In this report we summarize our recent research on the development of a systematic methodology for the design of feedback laws achieving stabilization and regulation of nonlinear control systems. We consdier the stabilization and control of both lumped nonlinear systems and nonlinear distributed parameter systems. The principal control objective is...
Article
Full-text available
Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, as well as in circuit synthesis, spectral analysis, and speech processing. For this reason, results about positive real transfer functions and their realizations typically have many applications and manifestations. In this paper, we stud...
Article
A method and apparatus for analyzing and synthesizing speech includes a programmable lattice-ladder notch filter which may be programmed to exhibit both filter poles and filter zeros and thereby exhibit a power spectral density with a better fit to that of a speech frame such that, when energized by a selected signal sample, a more accurate regener...
Article
In this work we show that the now standard lumped non-linear enhancement of root-locus design still persists for a non-linear distributed parameter boundary control system governed by a scalar viscous Burgers' equation. Namely, we construct a proportional error boundary feedback control law and show that closed-loop trajectories tend to trajectorie...
Article
Full-text available
.<F3.802e+05> In this paper we present a convex optimization problem for solving the rational covariance extension problem. Given a partial covariance sequence and the desired zeros of the modeling filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solv...
Conference Paper
We present a constructive theory for solving the Nevanlinna-Pick problem with a degree constraint. The theory relies on convex optimization, and the method generates all solutions with degree less than the number of interpolation conditions. The degree of the interpolant relates to the dimension of the feedback control system or of a modeling filte...
Article
Full-text available
We review a new approach to spectral estimation, based on the use of filter banks as a means of obtaining spectral interpolation data. This data replaces the standard covariance estimates used in traditional maximum entropy spectral estimation. The new method is based on our recent theory of analytic interpolation with degree constraint and produce...
Conference Paper
In this chapter, we present a number of topics in linear systems theory from a global, geometric perspective. Among the topics studied are the geometry of spaces of scalar and multivariable systems, the scalar and matrix valued Hermite-Hurwitz Theorem, and the geometry of the deterministic partial realization problem. Inverse eigenvalue problems ar...
Article
The recent solution of the output regulation problem for nonlinear control systems gives necessary and sufficient conditions for the local existence of a feedback/feedforward law in terms of the solvability of an "off-line" system of partial differential equations, the "regulator equations." The regulator equations are the nonlinear analogue of the...
Article
Full-text available
In this paper we consider a boundary control problem for a forced Burgers' equation on a finite interval. The controls enter as gain parameters in the boundary conditions as in [7, 6] and the forcing term is allowed to be time dependent and square integrable in the spatial variable for all time. The uncontrolled problem is obtained by equating the...
Conference Paper
For certain single input single output (SISO) distributed parameter systems we provide a useful solvability criteria for the regulator problem in terms of the eigenvalues of the exo-system and the transmission zeros. Namely, we show that the regulator problem is solvable provided that the eigenvalues of the exo-system do not coincide with the syste...
Conference Paper
It is well known that the solvability of the regulator problem, for finite dimensional systems, is related to the system zeros, or in other words, to the system zero dynamics. We demonstrate the existence of the zero dynamics for a class of SISO distributed parameter systems and provide a result relating the eigenvalues of the zero dynamics and the...
Article
Full-text available
We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a c...
Article
The lack of a systematic methodology for the design of feedback laws capable of controlling complex dynamical systems has been a limiting factor in several current and emerging DoD missions. The research carried out by the principal investigators in this three year research effort has focused on analyzing and computing the steady-state behavior of...
Article
Full-text available
In this paper, we give a new proof of the solution of the rational covariance extension problem, an interpolation problem with historical roots in potential theory, and with recent application in speech synthesis, spectral estimation, stochastic systems theory, and systems identification. The heart of this problem is to parameterize, in useful syst...
Conference Paper
It makes little sense to discuss future directions in applied mathematics without emphasizing similar opportunities and challenges in mathematics as a whole. A few decades ago, someone who should have known better wrote an article entitled “Applied mathematics is bad mathematics.” Nowadays, clearer thinking sometimes prevails. On one hand, it is cl...
Chapter
Full-text available
Abstra ct. Positive real rational functions play a central role in both deterministic and stochastic linear systems theory, arising in circuit synthesis, filtering, interpolation, spectral analysis, speech process-ing, stability theory, stochastic realization theory and systems iden-tification – to name just a few. For this reason, results about po...
Book
The mathematical theory of networks and systems has a long, and rich history, with antecedents in circuit synthesis and the analysis, design and synthesis of actuators, sensors and active elements in both electrical and mechanical systems. Fundamental paradigms such as the state-space real­ ization of an input/output system, or the use of feedback...
Chapter
The purpose of this Chapter is to show that the existence of solutions for the pair of equations $$\begin{array}{*{20}{c}} {\frac{{\partial \pi }}{{\partial w}}s\left( w \right) = f\left( {\pi \left( w \right),c\left( w \right),w} \right)} \\ {0 = h\left( {\pi \left( w \right),w} \right),} \end{array}$$ (3.1) which, as we have seen in the previou...
Book
Introduction: the basic ingredients of asymptotic output regulation the computation of the steady-state response highlights of output regulation for linear systems. Output regulation of nonlinear systems: the regulator equations the internal model necessary and sufficient conditions for local output regulation the special case of harmonic exogenus...
Chapter
The purpose of this Chapter is to study problems of output regulation in the presence of parameter uncertainties. In order to facilitate the exposition of the material, we proceed by addressing problems of increasing complexity, beginning with the solution of a problem of local output regulation in the presence of small parameter variations, then c...
Article
In this paper we present an integral-invariance principle generalizing LaSalle's invariance principle for nonlinear systems. The principal new ingredients are the use of observation functions and certain integrability conditions, which are particularly well suited for dynamical systems involving control and observations. The integral-invariance pri...
Article
This paper presents necessary and sufficient conditions under which a discrete-time autonomous system with outputs is locally state equivalent to an observable linear system or a system in the nonlinear observer form (Krener and Isidori, 1983). In particular, an open problem raised in Lee and Nam (1991), namely the observer linearization problem, i...
Article
This paper deals with the problem of H∞ control via measurement feedback with internal stability for discrete-time nonlinear systems. Using the theory of discrete-time dissipative systems and differential game, we first present a parameterization of a family of static state feedback H∞ nonlinear controllers. Then, we provide sufficient conditions f...
Article
By using the theory of discrete-time passive systems and the concept of feedback equivalence, we present an approach toward deriving sufficient conditions for the global stabilization of discrete-time nonlinear systems in the form x(k + 1) = tf(x(k)) + g(x(k))u(k).The central feature of the approach is that, a unifying set of stability criteria can...
Article
This paper studies discrete-time invariant autonomous systems ∑:xk + 1 = tf(xk)yk = h(xk)We investigate the problem of when asymptotic stability of ∑ can be characterized by means of zero-state observability of ∑ and square summability of the output function h(xk). The relationship among observability, square summable series and stability of discre...
Article
Using the center manifold theory for maps in a systematic way, the authors present in this note a necessary condition and sufficient conditions for discrete-time nonlinear systems to be stabilizable via smooth state feedback. Then, the authors derive a necessary and sufficient condition characterizing the existence of exponential observers for Lyap...
Article
In this paper, a fairly complete parallel of the finite-dimensional root locus theory is presented for quite general, nonconstant coefficient, even order ordinary differential operators on a finite interval with control and output boundary conditions representative of a choice of collocated point actuators and sensors. Root-locus design methods for...
Article
Full-text available
The main purpose of this paper is to address a fundamental open problem in linear filtering and estimation, namely, what is the steady-state or asymptotic behavior of the Kalman filter, or the Kalman gain, when the observed stationary stochastic process is not generated by a finite-dimensional stochastic system, or when it is generated by a stochas...
Article
We consider a boundary control problem for Burgers’ equation on a finite interval. The controls enter as gain parameters in the boundary conditions. The uncontrolled problem is obtained by equating the control parameters to zero while the zero dynamics system is obtained by equating the control parameters to infinity, or (intuitively) as the “high...
Article
We describe a complete parametrization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation, which is analyzed by using topological methods. Our analysis of this covariance extension equation is based on a recent complete parametrization of all strictly positive real solutions to the ratio...
Article
For a class of nonlinear discrete-time systems of the form Σ: x(k+1)=ƒ(x(k))+g(x(k))u(k), we investigate conditions under which a nonlinear system can be rendered globally asymptotically stable via smooth state feedback. Our main result is that any nonlinear system with Lyapunov-stable unforced dynamics can always be globally stabilized by smooth s...
Article
Motivated by several examples arising in linear systems theory, the problem considered here is the inverse eigenvalue problem for an arbitrary square matrix and for arbitrary additive perturbations belonging to a matrix Lie algebra. For an algebraically closed field with characteristic zero, the main theorem gives necessary and sufficient condition...
Article
This paper is concerned with the use of a boundary control to stabilize a viscous Burgers’ equation on a finite interval. The main result of the paper parallels a result found in [1], The main differences are that our original uncontrolled system is not asymptotically stable. Indeed, using a center manifold argument, it can be shown that the trajec...
Article
This paper has its origin in the confluence of two themes which are central in the use of optimization methods to generate feedback laws meeting certain desired design criteria.
Article
In this paper we explicitly describe, by generators and relations, the cohomology ring of the manifold n,m (F) of controllable linear systems having m inputs and state-space dimension n. It is shown that the cohomology ring of n,m (F) is isomorphic to the invariant cohomology ring of a product of projective spaces. Estimates for the cup length of...
Article
This final report surveys the research accomplishments of a three year effort on the adaptive stabilization and control of distributed parameter systems and on the development of a systematic feedback design methodology for nonlinear control systems. In both areas, significant advances have been made by the use of concepts and techniques from dynam...
Article
Full-text available
The derivation and analysis of the Riccati partial differential equation for optimal control has been recently developed in a series of papers and announcements. In addition to finite escape time, which already is an issue for linear quadratic problems, the existence of shock waves for solutions of these Riccati partial differential equations is th...
Article
In this paper, we settle in the negative a longstanding problem concerning the existence of a smooth (static or dynamic) state variable feedback law locally asymptotically stabilizing a rigid spacecraft with two controls about a desired reference attitude. Modelling a spacecraft actuated by three thruster jets, one of which has failed, this well st...
Book
In its broadest sense, nonlinear synthesis involves in fact the synthesis of sometimes so­ phisticated or complex control strategies with the aim of prescribing, or at least influencing, the evolution of complex nonlinear systems. Nonlinear synthesis requires the development of methodologies for modeling complex systems, for the analysis of nonline...
Article
In the paper [4] we initiated an analysis of the discrete-time Kalman filter as a nonlinear dynamical system, motivated by a desire to understand the asymptotic dependence of the Kalman filter on the parameters determining it. Since in the Kalman filtering of systems in statistical steady state the filtering equations often rely on estimates of eit...
Article
Shaping the response of a control system has long been a central problem in the analysis and design of feedback systems. The widespread use of both frequency domain techniques and state-space methods is at least in part due to the relative ease and intuitive content of these methods in addressing problems such as asymptotic tracking and disturbance...
Article
In finite dimensional linear systems theory, stability and Transient performance of a closed-loop control system are directly related to the location of the closed-loop roots of the characteristic equation in the complex plane. Frequently, it is necessary to adjust one or more system parameters in order to obtain desirable root locations. Therefore...
Article
Recently, a combination of methods drawn from geometric nonlinear control theory and from nonlinear dynamics was developed to give a local soLution to the nonlinear regulator problem, yielding necessary and sufficient conditions for nonlinear regulation for the class of detectable and stabilizable nonlinear systems[1][2]In section 2, we review the...
Article
The problem of controlling a fixed nonlinear plant in order to have its output track (or reject) a family of reference (or disturbance) signal produced by some external generator is discussed. It is shown that, under standard assumptions, this problem is solvable if and only if a certain nonlinear partial differential equation is solvable. Once a s...
Article
Full-text available
We discuss questions concerning the geometry of the Kimura-Georgiou para-metrizalion of the set ( + (n) of degree n positive real transfer functions with the first n coefficients in the Laurent expansion about infinity prescribed. For example, one interesting question which has been raised is whether this set is star-shaped about the maximum entrop...
Article
Proceeding from the problem posed by the need to stabilize the motion of two helicopters maneuvering a single load, a methodology is developed for the stabilization of classes of decentralized systems based on a more algebraic approach, which involves the external symmetries of decentralized systems. Stabilizing local-feedback laws are derived for...
Article
Using the general methodology of nonlinear zero dynamics we derive globally stabilizing state feedback laws for broad classes of nonlinear systems. While it is tempting to reinterpret this design philosophy in terms of high gain feedback, we present an example of a globally stabilizable nonlinear feedback system which cannot be stabilized, even loc...
Article
The development of a comprehensive feedback design methodology for nonlinear control systems, similar both in scope and in intuitive appeal to classical automatic control, is a long sought after goal in modern systems and control theory. The classical approaches to the control of finite dimensional linear systems made heavy use, in one or another f...
Article
In this note we briefly review the notion of a nonlinear minimum-phase system, i.e. a system which when constrained in such a way as to produce zero output, evolves with an asymptotically stable dynamics. The main purpose of the paper is to show that any minimum-phase nonlinear system can always be locally stabilized by smooth state-feedback
Conference Paper
In this paper, we present a solution to the problem of achieving both decoupling of an output of a nonlinear system from (nonlinear) disturbances and uniform BIBO stability of the closed-loop state equations driven by the disturbances. While this involves asymptotic stabilization of the nonlinear system as a special case, BIBO stability entails muc...
Conference Paper
For finite dimensional nonlinear systems, an enhancement of root-locus methods has recently been developed for the analysis and design of nonlinear feedback systems. Central to this approach was a dynamical systems interpretation of zeroes, suggested by geometric control theory. For flexible systems, an analogous physical interpretation of system z...

## Citations

... Therefore, differential flatness theory is applied to non-linear systems to reduce and simplify the order of the model. Consequently, the alternative model allows the dynamics of the trajectories to be characterized based on [42]. Therefore, the non-linear system can be considered flat by applying this theory. ...