Variable structure systems in control and optimization

Michael Peter Glazos, Purdue University

Abstract

We examine two meaningful applications of variable structure systems. The first application addresses the problem of practically stabilizing nonlinear and/or uncertain continuous-time dynamical systems when the norm of the control input is subject to a given fixed bound. We consider a class of systems whose nominal part is linear and whose nonlinear/uncertain part does not satisfy the matching condition. In this treatment no statistical information regarding the uncertainties is needed. Only a norm-bound on each nonlinear/uncertain quantity is assumed. We propose a bounded state-feedback controller whose design incorporates elements from both variable structure control theory and the deterministic control methodology. Using the second method of Lyapunov we obtain sliding domains, regions of uniform ultimate boundedness, and estimates of the domain of attraction for systems employing this controller. The closed-loop stability analysis as well as the design of the controller is facilitated by introducing a special coordinate transformation. The approach is illustrated by a numerical example. The second application concerns the use of analog neural networks to solve certain constrained optimization problems. We examine a class of analog optimizers that can be classified as variable structure systems. We analyze the dynamic behavior of these networks when applied to a broad class of convex programming problems. In carrying out the analysis we utilize concepts from the theory of differential equations with discontinuous right-hand sides and Lyapunov stability theory. We show that irrespective of the initial state of the network the state converges to a solution of the convex programming problem. The dynamic behavior of the networks is illustrated by two numerical examples.

Degree

Ph.D.

Advisors

Zak, Purdue University.

Subject Area

Electrical engineering

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