LYAPUNOV-LIKE EQUATIONS AND REACHABILITY OBSERVABILITY GRAMIANS FOR DESCRIPTOR SYSTEMS

Reachability and observability Gramians are defined for the descriptor systems Ex//k// plus //1 equals Ax//k plus Bu//k, y//k equals Cx//k and Ex(t) equals Ax(t) plus Bu(t), y(t) equals Cx(t). The pencils (zE - A) and (sE - A) are assumed to be regular. If E is singular, these Gramians do not, in general, satisfy the expected Lyapunov-like (Sylvester) equations (e. g. , EPA**T plus APE**T equals - BB**T if P is the continuous-time reachability Gramian). This is shown by expressing the solution of the descriptor system in terms of the parameters of the Laurent expansion of (zE - A)-**1 or (sE - A)-**1. The Lyapunov-like equations which these Gramians satisfy are derived.