On The Structural Stability of Dynamical Control Systems (original) (raw)

Robustness in dynamical and control systems

2011 6th IEEE International Symposium on Industrial and Embedded Systems, 2011

We compile some results on robustness of dynamical and control systems. As control theory is preoccupied with stability problems, the robustness put forward in this paper is related to stability. We ask the question whether an asymptotically stable system remains asymptotically stable when perturbations are affecting it. We analyze robustness of control systems by examining vector fields in C r topology, by studying associated Lyapunov functions, and by studying corresponding input-output maps. In the first case, we conclude that there is an open set of perturbations such that the system that is affected by them stays asymptotically stable. In the second case, we estimate the size of perturbations that do not destabilize the system. In the third and last case, we provide conditions on the gains of the interconnected systems such that the closed loop system has finite gain.

On the structural properties of the bounded control set of a linear control system

Nonlinear Differential Equations and Applications NoDEA

The present paper shows that the closure of the bounded control set of a linear control system contains all the bounded orbits of the system. As a consequence, we prove that the closure of this control set is the continuous image of the cartesian product of the set of control functions by the central subgroup associated with the drift of the system.

Controllability of dynamical systems. A survey

Bulletin of the Polish Academy of Sciences: Technical Sciences, 2013

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and o...

Generic properties for two-dimensional dynamic systems

2003

This paper studies the concept of genericity for two dimensional dynamic systems. The basic model is the differential model, proposed by S. Smale, of constructing generic properties for diffeomorphisms defined on compact varieties . The generic property concept for dynamic systems defined on an open set G, is proposed. The aim of the construction is to test the conditions for which, in two-dimensional case, the structural stability of dynamic systems can be a generic property. 2000 Mathematical Subject Classification: 37C20 1 The differential model of S. Smale S. Smale [4] defined the dynamic systems by a diffeomorphism , f : M → M , with M compact differential bundle. Most of results are also available for dynamic systems defined by a first order ordinary differential equation.

A Dynamical Systems Approach to Control

Nonlinear Control Systems Design 1992, 1993

Control systems Call be viewed as dynamical systems over (infmite)• dimensional state spaces. From this point of view the long tenn behavior of control systems, such as limit sets, Mone sets. approximations on the entire time axis, ergodicity, Lyapunov exponents, stable and unstable manifolds etc. becomes accessible. This paper presents some of the underlying theory, as well as applications 1.0 the global characterization of control systems with bounded control range that are not completely controllable, to control of chaotic systems, and to exponential stability of uncertain systems.

Passivity Equivalence of Siso Systems Via Continuous Feedback

IFAC Proceedings Volumes, 1998

Passiv ity cont inuous (possibly nonsmooth) feedback equivalence is developed for a class of SISO nolinear systems, introducing the so-called generalized relative degree (GRD) systems that are Cl weakly minimum phase, in the sense of a definition that generalizes the usual one for relative degree 1 systems. It is shown that the above systems are continuous feedback equivalent to a passive system provided some conditions, stated in an intrinsically geometrical manner, are valid. These results also generalize, in the SISO case, the well known result stating that a C2 weakly minimum phase system of relative degree 1 is smooth feedback equivalent to a passive system. Copyright @1998 IFAC. Resume. Dans ce travail, on etude l'equivalence aux systemes passifs par retour d'etat statique continu (et possiblement non doux) associee a une classe de systemes non lineaires monovariables. Pour etudier ce probleme on introduit les systemes avec un grade relatif generalise (GRG) et qui ont une phase minimale C l-faible, dans le sense d ' une definition qui generalise celle utilisee pour les systemes avec un grade relatif egal a 1. On montre que, si quelques conditions enoncees d 'une fa<;on intrinsequement geometrique sont satisfees, cette classe de systemes est equivalent a une systeme passif par retour d'etat statique. Ces resultats generalisent, pour le cas monovariable, quelques resultas precedents.

Controllability properties of nonlinear behaviors

Transactions of the American Mathematical Society, 2008

This paper proposes a topological framework for the analysis of the time shift on behaviors. It is shown that controllability is not a property of the time shift, while chain controllability is. This also leads to a global decomposition.

On the Classification of Control Sets

Lecture Notes in Control and Information Sciences

The controllability behavior of nonlinear control systems is described by associating semigroups to locally maximal subsets of complete controllability, i.e., local control sets. Periodic trajectories are called equivalent if there is a 'homotopy' between them involving only trajectories. The resulting object is a semigroup, which we call the dynamic index of the local control set. It measures the different ways the system can go through the local control set.

A local theory of input/output stability of dynamical systems

Journal of the Franklin Institute, 1995

The purpose of this paper is to present a local theory of input/output stability OJ dynamical systems. More precisely, the intention is to obtain local open loop conditions for local closed loop stability of feedback interconnections. It will be shown that, by setting the problem in the context of extended binormed spaces it is possible to obtain local versions of the so-called small 9ain theorem and circle criterion for stability of nonlinear systems. The results clearly determine the extent of input~output stability, i.e. the set of input.[hnctions./br which stability is 9uaranteed.

On nonasymptotic stabilizability of controllable systems

The paper is devoted to the stabilization problem of nonlin- ear controllable systems. It has been proved, that any lo- cally controllable system is nonasymptotically stabilizable by means of discontinuous time-invariant feedback law, pro- vided that the solutions of closed-loop system are defined in the sense of A.F. Filippov. In addition, the set of disconti- nuity points of stabilizing feedback law is investigated. The idea of such investigation is based on selection of continu- ous branch from set-valued feedback control, defining stable system of differential inclusions.

On The Geometrical Description of Dynamical Stability II

Arxiv preprint math-ph/0610084, 2006

Geometrization of dynamics using (non)-affine parametization of arc length with time is inves-tigated. The two archetypes of such parametrizations, the Eisenhart and the Jacobi metrics, are applied to a system of linear harmonic oscillators. Application of the Jacobi ...

Structural stability of linear dynamically varying (LDV) controllers

Systems & Control Letters, 2001

Linear dynamically varying (LDV) controllers have been shown t o be useful in controlling nonlinear dynamical systems on compact sets, especially chaotic systems. In this paper it is shown that the ability t o stabilize a dynamical system with a n LDV controller is structurally stable in the C' topology, the Lipschitz topology and, in a restricted sense, the topology provided that these systems are near enough t o an LDV stabilizable C' dynamical system. Furthermore, the optimal LDV controller is shown t o depend continuously on the dynamical system.

Generic properties and control of linear structured systems: a survey

Automatica, 2003

In this survey paper, we consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A; B; C and D, is either a ÿxed zero or a free parameter. The location of the ÿxed zeros in these matrices constitutes the structure of the system. Indeed a lot of man-made physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly, a lot of classical properties of linear systems can be studied in terms of genericity. Moreover, these generic properties can, in general, be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system-speciÿc properties like controllability, the ÿnite and inÿnite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input-output decoupling, and so on. In this paper, we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool. We emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only. Moreover, the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well-known polynomially bounded combinatorial techniques for larger systems. ?

Systèmes non linéaires Nonlinear systems

2000

1.1 16 Pour tout renseignement s'adresser à : For further information, please contact: Topics 1. Embedded and networked systems Taking into account the network topology (multiple levels, different sizes and time scales) and constraints of communication. Development of new methodologies (modeling, control, state estimation of parameters) to ensure good performances, be robust to disturbances and reliable.

Structural Stability of Nonautonomous Systems

Differential Equations, 2003

The structural stability of systems of ordinary differential equations with arbitrary dependence on time was studied in [1-4]. However, the existence of a homeomorphism taking the solutions of the nonperturbed system to the solutions of the perturbed system was not established there. Here we prove this fact under the assumptions stated in the cited papers. Consider the system of ordinary differential equationṡ x = X(t, x), x∈ R n. (1) We assume that the vector X(t, x) and its Jacobi matrix ∂X(t, x)/∂x with respect to x are uniformly continuous on the entire space R × R n and there exists a constant M > 0 such that |X(t, x)| ≤ M, |∂X(t, x)/∂x| ≤ M for all t ∈ R and x ∈ R n. Here the symbol | • | stands for the Euclidean norm of a vector or the corresponding matrix norm. Along with system (1), we consider the perturbed systeṁ x = X(t, x) + Y (t, x), (2) where the vector Y (t, x) and its Jacobi matrix with respect to x are uniformly continuous and satisfy the relations |Y (t, x)| < δ, |∂Y (t, x)/∂x| < δ (3) for any t ∈ R and x ∈ R n. By x (t, t 0 , x 0) we denote the solution of system (1) with the initial data x (t 0 , t 0 , x 0) = x 0 , and by y (t, t 0 , y 0) we denote the solution of system (2) with the initial data y (t 0 , t 0 , y 0) = y 0. We also consider the set of linear systemṡ

Geometric Methods in Study of the Stability of Some Dynamical Systems

2009

In this paper we aim to analyse the stability of two dynamical sys- tems given by differential equations or by systems of differential equa- tions. The first model is a mechanical system which is described by a system of differential equations of the first degree. We study the sta- bility of this system using the method of the Lyapunov function. The second studied model is the model of a vibrant tool machine described by a differential equation of second degree with two delay arguments. For the study of the stability of these models, we use the stage analysis of the differential equations systems with delayed arguments.

On local linearization of control systems

Journal of Dynamical and Control Systems, 2009

We consider the problem of topological linearization of smooth (C ∞ or C ω ) control systems, i.e. of their local equivalence to a linear controllable system via point-wise transformations on the state and the control (static feedback transformations) that are topological but not necessarily differentiable. We prove that local topological linearization implies local smooth linearization, at generic points. At arbitrary points, it implies local conjugation to a linear system via a homeomorphism that induces a smooth diffeomorphism on the state variables, and, except at "strongly" singular points, this homeomorphism can be chosen to be a smooth mapping (the inverse map needs not be smooth). Deciding whether the same is true at "strongly" singular points is tantamount to solve an intriguing open question in differential topology.