A Lower Bound of the Degree of Stability for a Matrix Polytope and Its Applications
When considering the state-space model for a linear system with uncertain parameters , we often have a polytope of matrices as possible system matrices. Stability condition of a matrix polytope, however, has not been fully developed and useful stability criteria are few . With a view to shedding light on this stability problem, this paper considers the degree of stability, a quantitative measure of stability, for a polytope of matrices. A lower bound of the degree of stability based on algebraic Lyapunov equations is proposed. It uses m positive definite symmetric matrices and aims at finding the best one that attains the maximum of the lower bound in the convex hull of the m matrices. The solution is expressed in terms of a zero-sum game, hence it can be solved easily by linear programming. The present approach makes it possible to improve the lower bound by increasing m. In this connection, condition is provided which guarantees a real improvement of the lower bound. Stability of a matrix polytope can be tested by the proposed lower bound. Moreover we can obtain a Lyapunov function common to all the members of the matrix polytope, in case the polytope is determined to be stable. This common Lyapunov function is utilized for studying the stability of a large-scale system with unknown parameters.