Lyapunov approximation of reachable sets for uncertain linear systems

Abstract

A method is presented for calculating an approximation to the reachable set from the origin of a class of linear control systems. The method involves the construction of a Lyapunov-like function, and ultimately requires the numerical solution of a simple optimization problem. Advantages of the method are that it is applicable to n-dimensional systems and does not require calculation of orbits or a choice of boundary conditions. Further, the system need not be completely controllable for the method to be applicable. However, while the method ensures that the reachable set lies within a specified set, a closeness-of-fit criterion is not available.

Under-estimates of reachable sets for linear control systems

“…Then, the largest set S of the form (5) which is contained in ~K is defined Proof Define the scalar function V(x) = x'Px for system (3b) where P is the solution of (6b). Then,…”

“…For linear continuous-time systems, over-estimates have been obtained by using Liapunov methods [4][5][6][7][8] as well as parallelepipeds [9] to approximate reachable sets. A method for obtaining both under and over-estimates of the projection of a reachable set onto a one or two dimensional subspace in the case of a scalar control has been proposed in [3] and [10].…”

A survey of techniques for approximating reachable and controllable sets

“…Such procedures were described first by Goh (1976) and Grantham (1980Grantham ( ,1981 in the context of nonlinear control systems. Summers (1985) refined the process for linear systems with A stable and scalar control Iu I I 1. In that context, the idea is as follows: Let …”