PART 6 Stability , Stabilization

The following notational conventions and terminology will be in force. Inequalities for vectors are understood component-wise. Given two matrices M and N with the same number of columns, the notation col(M, N) denotes the matrix obtained by stacking M over N. Let M be a matrix. The sub-matrix M JK of M is the matrix whose entries lie in the rows of M indexed by the set J and the columns indexed by the set K. For square matrices M , M JJ is called a principal submatrix of M. A symmetric matrix M is said to be non-negative (nonpositive) definite if x T M x ≥ 0 (x T M x ≤ 0) for all x. It is said to be positive (negative) definite if the equalities hold only for x = 0. Sometimes, we write M > 0 (M ≥ 0) to indicate that M is positive definite (non-negative definite), respectively. We say that a square matrix M is Hurwitz if its eigenvalues have negative real parts. A pair of matrices (A, C) is observable if the corresponding system ˙ x = Ax, y = Cx is observable, equivalently if col(C, CA, · · · , CA n−1) is of rank n where A is of order n. Lyapunov stability theory is one of the ever green topics in systems and control. For (finite dimensional) linear systems, the following theorem is very well-known. Theorem 1:[3, Theorem 1.2]: The following conditions are equivalent. 1. The system ˙ x = Ax is asymptotically stable. 2. The Lyapunov equation A T P + P A = Q has a positive definite symmetric solution P for any negative definite symmetric matrix Q. As a refinement, we can replace the last statement by 2. The Lyapunov equation A T P +P A = Q has a positive definite symmetric solution P for any nonpositive definite symmetric matrix Q such that the pair (A, Q) is observable. An interesting application is to the stability of the so-called switched systems. Consider the system ˙ x = A σ x (1) where the switching signal σ : [0, ∞) → {1, 2} is a piecewise constant function. We assume that it has a finite number of discontinuities over finite time intervals in order to rule out infinitely fast switching. A strong notion of stability for the system (1) is the requirement of stability for arbitrary switching …