Conditions on the Stability of a Class of Second-Order Switched Systems
IEEE Transactions on Automatic Control
For a special class of systems, it is shown that the existence of a common quadratic Lyapunov function (CQLF) is necessary and sufficient for the stability of an associated switched system under arbitrary switching. Furthermore, it is shown that the existence of a CQLF for ( 2) subsystems is equivalent to the existence of a CQLF for every pair of subsystems. An algorithm is proposed to compute a CQLF for the subsystems, when it exists, using the left and right eigenvectors of a critical matrix
... a critical matrix obtained from a matrix pencil. Index Terms-Common quadratic Lyapunov function (CQLF), M-matrix, stability, switched systems. I. PROBLEM STATEMENT Consider the switched system where x(t) 2 < 2 is the state, and A i 2 < 222 , i = 1; 2; . . . ; N are the system matrices for the subsystems 6 i : _ x(t) = A i x(t); i= 1; 2; . . . ; N: (2) Throughout this note, the negative of each matrix A i (i.e., 0A i ) is assumed be an M-matrix. A matrix W is called an M-matrix if it has nonpositive off-diagonal elements and eigenvalues in the open right half of the complex plane  1 . M-matrices arise in many areas of application and have a close link with the positive matrices -. In particular, an equivalent representation of an M-matrix W is W = I 0 P , > (P ) where (P ) denotes the spectral radius of a nonnegative matrix P . For many other interesting properties of M-matrices, the reader is referred to -. The objective of this note is to derive necessary and sufficient conditions for the stability of the switched system (1) under arbitrary switching between the system matrices Ai, i = 1; 2; . . . ; N. It is necessary that each subsystem (2) is stable; otherwise, the switched system utilizing the unstable subsystem only would also be unstable. In this note, this necessary condition is satisfied by assuming that the negative of each system matrix is an M-matrix. A detailed discussion on the stability of switched systems under arbitrary switching is in  . Clearly, if a common quadratic Lyapunov function (CQLF) exists for the subsystems 6 i , i = 1; 2; . . . ; N, then the switched system (1) is stable under arbitrary switching. The converse of this statement is not true, in general ,  . However, in this note, we prove that the converse is indeed true for a specific class of systems.