Comments on "A Markovian reset problem"

Abstract

(in the sense of Lyapunov) should be considered in the physical state space, not in the state space of another system which is a lower dimensional realization of some input-output relation. The dimension of the stat.e space depends on t,he stability property one is interested in. It. is, hence, not correct to state that the dimension of the st.ate space is (an - 1) for a nonuniformly damped power system, and only (2n - 2) for a uniformly damped or undamped power system; eveh this is a priori' expect,ed for the power system model, since t.he physical variables are t.he speeds and the load angle differences, but not the load angles themselves. However, I usel Anderson’s t.heorem only to derive a Lyapunov funct.ion for the power syst,em stability problem and not, to conclude its sign dehiteness; the validity and the usefulness of the Lyapunov function are checked a posteriori. Because of t.he reasons pointed out by Sastry and Murthy, it would indeed be incorrect to conclude that for the power syst,em case a positive-definite mat.rix P is obtained from Anderson’s technique. My purpose is to develop a systematic procedure to derive a useful Lyapunov function. Actually, once the Lyapunov function is obtained, one can forget about Anderson’s theorem and investigate the sign definiteness of the tentative Lyapunov function. This function should be considered in the (2n - l>dimensional state space with