FP2 zyxwvu - 1750 A zyxwvutsrqponmlk NOTE ON COMPUTING THE MlNIMUM DISTANCE BETWEEN LYAPUNOV FUNCTIONS Albert0 Cavallo and Ciuseppe De Maria Dipartimento di Informatica e Sistemistica Universita degli Studi di Napoli Federico zyxwvu I1 via Claudio 21, 80125 Naples, Italy INTRODUCTION. Many system stability and robustness problems can be reduced to find solutions of particular structure of Lyapunov equations. Problems such as absolute stability and passivity are of this type. Indeed, there exists a positive definite quadratic Lyapunov function which establishes absolute stability if and only if a particular set of matrices is simultaneously Lyapunov stable. Simultaneous Lyapunov stability problem can zyxwvutsrq be formulated as follows: given a set of matrices {A ,...Ak}, such a set is said simultaneously Lyapunov stable if and only if there exists a single Lyapunov function V(x)=xTPx establishing stability of the matrices A, ... A, [I], 121. In 111 methods which allows us to establish the above property are detailed. Design methods which rely on controller scheduling can be characterized in terms of the simultaneous Lyapunov stability problem. Indeed, such an approach foresees the design of several controllers for a fixed number of operating points of the plant, and then a switch from one controller to another during the maneuver occurs. Such a method is traditionally used in aircraft control. If the maneuver is rapid, stability is not guaranteed unless any couple of contiguous controlled systems is simultaneously Lyapunov stable. Such a requirement may lead to the use of a massive controller scheduling during the maneuver and a simple digital implementation of the switching controller is not allowed 131. Taking into account this problem, by supposing that between two operating points the controlled system can be described by means of the convex combination of the systems in the given points, we can look for an upper bound on the maneuver velocity which guarantees stability. Since this upper bound depends on the distance between the Lyapunov functions in the operating points, in this note, following an approach similar to that in 111, we present an algorithm which allows us to minimize such a distance in a finite number of steps. PRELIMINARIES. Consider a n-th order time-varying autonomous system t=t, t*tl+T We say that during the interval tlatStl+T the system exhibits a zyxwvutsrqpo commutation. Suppose that the matrices A, and A, are stable, then two positive Lyapunov functions V,(t)=xT(t)P,x(t) and V2(t)=xT(t)P2x(t) exist such that V,(t) and t,(t) are negative definite. To check the stability of the system (1) we will choose as a Lyapunov function during the commutation CH3229-2/92/0000-3462$1 .OO zyxwvutsrqpon 0 1992 IEEE t-tl P(t) = [ 1 - - t;] PI + - P,. (3) Obviously P(t) id p.d. Then t ( t l during the commutation is given by Being in order to make V(tl n.d. we must minimize the last term. It can be bounded as follows zyxwvu (6) Then the problem of finding an upper bound on the switching time T is reduced to that of minimizing &P,-P,), where PI and P, are symmetric p.d. matrices solutions of the Lyapunov equations ATP, + PIA: = -Q1 AZP, + P,AZ = -Qz zyxw (7) where Q1 and Q2 are symmetric p.d. matrices. FORMULATION OF THE PROBLEM. The minimization problem min G(P~-P,) subject to is a convex optimization problem, being both the constraint and the objective function convex. Moreover i t is possible to show that the upper bound on T does not depend on E in (91, then we can impose &=l. To solve problem (8)-(9), let B,, ..., Br, r=n(n+l)/Z, be a basis of the subspace of symmetric nxn matrices; then Q, and Q2 can be expressed as r r Q1 = ~a,B,, Qz = lal+,BI, a=(+ ..., a2,) E IR* (10) I=, i=1 Moreover PI can be computed via the following expression (A: c A:) vec(P,) = -vec(Q,) (11) By using eq. (11) and suitable selection matrices, it is possible to compute a matrix K, such that 3462