A tutorial account of variable structure control with sliding mode is presented. The purpose is to introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control system design approach. This approach is particularly attractive for the control of nonlinear systems. Prominent characteristics such as invariance, robustness, order reduction, and control chattering are discussed in detail. Methods for coping with chattering are presented. Both linear and nonlinear systems are considered. Future research areas are suggested and an extensive list of references is included. >

Variable structure control: a survey

Survey paper: Set invariance in control

The design of variable-structure control (VSC) systems for a class of multivariable, nonlinear, time-varying systems is presented. Using the Utkin-Drazenovic method of equivalent control and generalized Lyapunov stability concepts, the VSC design is described in a unified manner. Complications that arise due to multiple inputs are examined, and several approaches useful in overcoming them are developed. Recent developments are investigated, as is the kinship of VSC and the deterministic approach to the control of uncertain systems. All points are illustrated by numerical examples. The recent literature on VSC applications is surveyed. >

Variable structure control of nonlinear multivariable systems: a tutorial

Robust control of a class of uncertain nonlinear systems

We introduce a concept of input-to-output practical stability (IOpS) which is a natural generalization of input-to-state stability proposed by Sontag. It allows us to establish two important results. The first one states that the general interconnection of two IOpS systems is again an IOpS system if an appropriate composition of the gain functions is smaller than the identity function. The second one shows an example of gain function assignment by feedback. As an illustration of the interest of these results, we address the problem of global asymptotic stabilization via partial-state feedback for linear systems with nonlinear, stable dynamic perturbations and for systems which have a particular disturbed recurrent structure.

Small-gain theorem for ISS systems and applications

This paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems. The vector of uncertain parameters $q( \cdot )$ is time-varying, and its values $q(t)$ lie within a prespecified bounding set Q in $R^p $. Furthermore, no statistical description of $q( \cdot )$ is assumed, and the controller is shown to render the closed loop system “practically stable” in a so-called guaranteed sense; that is, the desired stability properties are assured no matter what admissible uncertainty $q( \cdot )$ is realized. Within the perspective of previous research in this area, this paper contains one salient feature: the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time-invariant. In contrast, in much of the previous literature (see, for example, [1], [2], [7], and [9]), a linear system is stabilized via nonlinear control. Another feature of this paper is the fact that the...

A New Class of Stabilizing Controllers for Uncertain Dynamical Systems

Given a strictly Hurwitz polynomial f(\lambda) = \lambda^{n} + a_{n-1} \lambda^{n-1} + a_{n-2}\lambda^{n-2}+...+ a_{1}\lambda + a_{0} , it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict Hurwitz property. For systems with n \leq 4 , maximal intervals of the a i are given in a recent paper by Guiver and Bose [1]. In this note, a theorem of Kharitonov is exploited to obtain a general result for polynomials of any degree.

Invariance of the strict Hurwitz property for polynomials with perturbed coefficients

This paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems. The vector of uncertain parameters q(?) is time-varying, and its values q(t) lie within a prespecified bounding set Q in Rp. Furthermore, no statistical description of q(?) is assumed, and the controller is shown to render the closed loop system "practically stable" in a so-called guaranteed sense; that is, the desired stability properties are assured no matter what admissible uncertainty q(?) is realized. Within the perspective of previous research in this area, this paper contains one salient feature: the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time-invariant. In contrast, in much of the previous literature (see, for example, [1],[2],[7] and [9]), a linear system is stabilized via nonlinear control. Another feature of this paper is the fact that the methods of analysis and design do not rely on transforming the system into a more convenient canonical form; e.g., see [3]. It is also interesting to note that a linear stabilizing controller can sometimes be constructed even when the system dynamics are nonlinear. This is illustrated with an example.

A new class of stabilizing controllers for uncertain dynamical systems

Given a strictly Hurwitz polynomial f(?) = ?n+ an-1?n-1+an-2?n-2+...+a1?+a0, it is of interest to know how much the coefficients ai can be perturbed while simultaneously preserving the strict Hurwitz property. For systems with n ? 4, maximal intervals of the ai are given in a recent paper by Guiver and Bose [1]. In this note, a theorem of Kharitonov is exploited to obtain a general result for polynomials of any degree.

B. Barmish

Papers

A New Class of Stabilizing Controllers for Uncertain Dynamical Systems

Invariance of the strict Hurwitz property for polynomials with perturbed coefficients

A new class of stabilizing controllers for uncertain dynamical systems

Invariance of the strict Hurwitz property for polynomials with perturbed coefficients

A Note on Avoidance Control