Approximation of the asymptotic stability boundary of discrete-time control systems using an inverse transformation approach

Abstract

The objective of this paper is to describe the use of the direct method of Lyapunov together with an inverse transformation process as a basis for determining the stability domain and for estimating the stability boundary of the null solution of a set of autonomous difference equations which describe the dynamics of a class of discrete-time control systems. Two theorems are developed which give sufficient conditions for the existence of an asymptotic stability boundary, which is a hypersurface enclosing a simply-connected domain, provided that a Lyapunov function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\upsilon(x)</tex> exists in a neighborhood of the origin. The proofs of these theorems indicate that an original subdomain of asymptotic stability, obtained by using a Lyapunov function, can be enlarged by applying an inverse transformation to the boundary of the subdomain. The inverse transformation is defined from the given set of difference equations as giving the state at the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> th instant in terms of the state at the ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k+1</tex> )th instant. The exact boundary of asymptotic stability is theoretically obtained in the limit by carrying out the inverse transformation an infinite number of times. Analogous to the work on stability, two theorems give sufficient conditions for the existence of an instability boundary which is a hypersurface surrounding a simply connected domain that includes the origin. It is shown that a known domain of instability found by using a Lyapunov function can be increased in size by applying the inverse transformation to its boundary. An important property of this approach is that each time the inverse transformation is applied, an enlarged region of known asymptotic stability or instability is found. In addition to giving stability information, the successive plots of the boundaries can also be used to give the settling time of the system.