On the stability of multiloop feedback systems

Some extensions on recent work in passive stability theory by Desoer and the author are presented. The stability of feedback systems is considered where the forward-loop and return-loop subsystems each have been subdivided into several systems operating in parallel. Results are obtained in the areas of Lyapunov stability, L
2
input L
2
output stability, and bounded-input bounded-state stability. Before applying these results to a specific multiloop feedback system, a pseudoenergy analysis must be done on each sub sub-system. Two specific types of systems are so analyzed. The first is a general linear first-order time-varying system. The second is a linear time-varying infinite-dimensional system; the analysis of this system takes the form of a Kalman-Yacubovich-type lemma. By using these two new systems, along with several others that have been analyzed previously, stability theorems for many specific multiloop feedback systems can be proven. One such example is given.