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Further Facts About Input To State Stabilization
 IEEE Trans. Automat. Control
, 1989
"... Previous results about input to state stabilizability are shown to hold even for systems which are not linear in controls, provided that a more general type of feedback be allowed. Applications to certain stabilization problems and coprime factorizations, as well as comparisons to other results on i ..."
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Cited by 55 (17 self)
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Previous results about input to state stabilizability are shown to hold even for systems which are not linear in controls, provided that a more general type of feedback be allowed. Applications to certain stabilization problems and coprime factorizations, as well as comparisons to other results on input to state stability, are also briefly discussed. Appeared as: "Further facts about input to state stabilization", IEEE Trans. Automatic Control, 35(1990): 473476. Rutgers Center for Systems and Control December, 1988  Revised April 1 Introduction In a previous paper [3] we studied the problem of when a system on IR n , x = f(x) +G(x)u (1) with f and the entries of the n \Theta m matrix G being smooth, can be made input to state stable (ISS) in a rather strong sense to be reviewed below. Our main result there was that this system is smoothly input to state stabilizable, that is, there exists a smooth (i.e., infinitely differentiable) map K : IR n ! IR m with K(0) = 0 and su...
Stability and stabilization: discontinuities and the effect of disturbances, in “Nonlinear analysis, differential equations and control
, 1998
"... ..."
Remarks on Stabilization and InputtoState Stability
, 1989
"... This paper describes how notions of inputtostate stabilization are useful when stabilizing cascades of systems. 1 Introduction Consider a cascade as follows: (CAS) ( x = f(x; y) y = g(y; u) where f and g are smooth, x and y evolve in IR n and IR m respectively, and f(0; 0) = g(0; 0) = 0 ..."
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Cited by 37 (12 self)
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This paper describes how notions of inputtostate stabilization are useful when stabilizing cascades of systems. 1 Introduction Consider a cascade as follows: (CAS) ( x = f(x; y) y = g(y; u) where f and g are smooth, x and y evolve in IR n and IR m respectively, and f(0; 0) = g(0; 0) = 0. The input u takes values in IR k . It is natural to ask: If the system x = f(x; y) is stabilizable (with y thought of as a control) and the same is true for y = g(y; u), what can one conclude about the cascade? More particularly, what can be said if the "zeroinput" system x = f(x; 0) (1) is already known to be asymptotically stable? There are many reasons for studying these problems; see the tutorial paper [7] for motivations and references. The simplest result along these lines is local, and it states that a cascade of locally asymptotically stable systems is again asystable. More precisely, if (1) has the origin as a locally asystable point, and if in (CAS) the second equation is i...
On the InputtoState Stability Property
 Systems & Control Letters
, 1995
"... The "input to state stability" (iss) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunovtheoretic, and dissipation terms. We sketch s ..."
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Cited by 35 (5 self)
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The "input to state stability" (iss) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunovtheoretic, and dissipation terms. We sketch some applications to the stabilization of cascades of systems and of linear systems subject to control saturation. 1 Introduction There are two very conceptually different ways of formulating the notion of stability of control systems. One of them, which we may call the input/output approach, relies on operatortheoretic techniques. Among the main contributions to this area, one may cite the foundational work by Zames, Sandberg, Desoer, Safanov, Vidyasagar, and others. In this approach, a "system" is a causal operator F between spaces of signals, and "stability" is taken to mean that F maps bounded inputs into bounded outputs, or finiteenergy inputs into finiteenergy outputs. More stringe...
Feedback Stabilization Of Nonlinear Systems
 In Mathematical Theory of Networks and Systems. Birkhauser
, 1989
"... This paper surveys some wellknown facts as well as some recent developments on the topic of stabilization of nonlinear systems. 1 Introduction In this paper we consider problems of local and global stabilization of control systems x = f(x; u) ; f(0; 0) = 0 (1) whose states x(t) evolve on IR n a ..."
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Cited by 9 (0 self)
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This paper surveys some wellknown facts as well as some recent developments on the topic of stabilization of nonlinear systems. 1 Introduction In this paper we consider problems of local and global stabilization of control systems x = f(x; u) ; f(0; 0) = 0 (1) whose states x(t) evolve on IR n and with controls taking values on IR m , for some integers n and m. The interest is in finding feedback laws u = k(x) ; k(0) = 0 which make the closedloop system x = F (x) = f(x; k(x)) (2) asymptotically stable about x = 0. Associated problems, such as those dealing with the response to possible input perturbations u = k(x) + v of the feedback law, will be touched upon briefly. We assume that f is smooth (infinitely differentiable) on (x; u), though much less, for instance a Lipschitz condition, is needed for many results. The discussion will emphasize intuitive aspects, but we shall state the main results as clearly as possible. The references cited should be consulted, however, f...
Statespace and i/o stability for nonlinear systems
 Lecture Notes in Control and Information Sciences
, 1995
"... This paper surveys several alternative but equivalent definitions of “input to state stability ” (iss), a property which provides a natural framework in which to formulate notions of stability with respect to input perturbations. Relations to classical Lyapunov as well as operator theoretic approach ..."
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Cited by 6 (2 self)
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This paper surveys several alternative but equivalent definitions of “input to state stability ” (iss), a property which provides a natural framework in which to formulate notions of stability with respect to input perturbations. Relations to classical Lyapunov as well as operator theoretic approaches, connections to dissipative systems, and applications to stabilization of several cascade structures are mentioned. The particular case of linear systems subject to control saturation is singledout for stronger results. 1
External Stabilizability of Nonlinear Systems with Some Applications
 International Journal of Robust and Nonlinear Control
"... : The aim of this paper is to discuss the existence of suitable feedback laws which provide stability (in a technical sense to be specified) with respect to inputs and initial conditions of the state response of a nonlinear system. 1. Introduction Roughly speaking, a forced system is said to be exte ..."
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Cited by 2 (1 self)
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: The aim of this paper is to discuss the existence of suitable feedback laws which provide stability (in a technical sense to be specified) with respect to inputs and initial conditions of the state response of a nonlinear system. 1. Introduction Roughly speaking, a forced system is said to be externally stable if the future state evolution is bounded whenever the input is bounded. Moreover, the system is said to be externally stabilizable if the external stability property can be achieved by means of some state static feedback connection. As the classical theory of linear systems shows, external stability and stabilizability are related to the socalled internal stability properties, that is stability properties of the associated unforced system. In this work we consider nonlinear systems of the form (1) x = f(x; u) where x 2 R n , u 2 R m and f 2 C 1 (R n \Theta R m ; R n ). Admissible inputs are piecewise continuous, bounded functions u : [0; +1) ! R m . The associ...
Input/Output and StateSpace Stability
 in New Trends in Systems Theory
, 1991
"... This paper first reviews various results relating statespace (Lyapunov) stabilization to notions of input/output or "boundedinput boundedoutput" stabilization, and then provides generalizations of some of these results to the case of systems with saturating controls. 1 Various Notions of Stabilit ..."
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Cited by 1 (0 self)
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This paper first reviews various results relating statespace (Lyapunov) stabilization to notions of input/output or "boundedinput boundedoutput" stabilization, and then provides generalizations of some of these results to the case of systems with saturating controls. 1 Various Notions of Stability Problems of stabilization underlie most questions of control design. In the nonlinear control literature, a great deal of effort has been directed towards the understanding of the general problem of stabilizing systems of the type x = f(x; u) ; f(0; 0) = 0 (1) by means of feedback control laws u = k(x) ; k(0) = 0 (2) which make the closedloop system x = f(x; k(x)) (3) globally asymptotically stable about x = 0. There are many variants of this general question, which differ on the degree of smoothness required of k, as well as on the structure assumed of the original system. We call this type of problem a statespace stabilization problem. For references, see for instance the survey pap...
The Method of Liapunov Functions for Nonlinear Input Systems
"... . The purpose of this work is to survey some possible definitions of external stability for nonlinear systems and to provide the basic tools for investigating their relationship and dependence on internal stability properties. In particular, we review some early and more recent results about conv ..."
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Cited by 1 (1 self)
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. The purpose of this work is to survey some possible definitions of external stability for nonlinear systems and to provide the basic tools for investigating their relationship and dependence on internal stability properties. In particular, we review some early and more recent results about converse Liapunov theorems. To this respect, some improvements are pointed out without proof. 1. Introduction During the last few years, new directions of studies are emerged in the literature about nonlinear systems. In particular, many papers have been devoted to the investigation and characterization of certain input/output stability properties in the framework of the classical Liapunov functions method. The present work is an attempt of reviewing some important results obtained on this subject and exposing the basic material in an organized way. We are interested in physical input systems modeled by continuous time, 1 timedependent, finite dimensional ordinary differential equations (...
A Synthesis of Gradient and Hamiltonian Dynamics Applied to Learning in Neural Networks
, 1995
"... The process of model learning can be considered in two stages: model selection and parameter estimation. In this paper a technique is presented for constructing dynamical systems with desired qualitative properties. The approach is based on the fact that an ndimensional nonlinear dynamical system c ..."
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The process of model learning can be considered in two stages: model selection and parameter estimation. In this paper a technique is presented for constructing dynamical systems with desired qualitative properties. The approach is based on the fact that an ndimensional nonlinear dynamical system can be decomposed into one gradient and (n \Gamma 1) Hamiltonian systems. Thus, the model selection stage consists of choosing the gradient and Hamiltonian portions appropriately so that a certain behavior is obtainable. To estimate the parameters, a stably convergent learning rule is presented. This algorithm is proven to converge to the desired system trajectory for all initial conditions and system inputs. This technique can be used to design neural network models which are guaranteed to solve certain classes of nonlinear identification problems. Key Words Dynamical systems, System Identification Acknowledgments This research was supported by a grant from Boeing Computer Services under C...