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.