A qualitative analysis of the behavior of dynamic nonlinear networks: Stability of autonomous networks

Several theorems are presented which predict in a qualitative manner the behavior of a large class of dynamic nonlinear networks containing coupled and multiterminal resistors, inductors, and capacitors. A very general and rather surprising result is presented which guarantees that most autonomous and nonautonomous dynamic nonlinear active networks of practical interest have no finite "forward" escape time solutions. In the case of autonomous networks, sufficient conditions are given which guarantee that the solution waveforms possess various forms of stability properties. The concepts of eventual passivity and eventual strict passivity are invoked to guarantee that all solution waveforms are bounded and eventually uniformly bounded, respectively. The properties of reciprocity and monotonicity (local passivity) are invoked to guarantee that all solutions are completely stable. The further imposition of a growth condition guarantees that all solutions will converge to a globally asymptotically stable equilibrium point. In this case, the magnitude of all solutions is shown to be bounded between two exponential waveforms for all time
t > 0
. An algorithm is presented which computes for the maximum "transient decay" time constant associated with the upper bounding exponential. The main features of the majority of the theorems presented in this paper are that their hypotheses are simple and easily verifiable-often by inspection. The hypotheses are of two types: first, very general conditions on the network state equations and second, conditions on the individual element characteristics and their interconnections. The hypotheses and proofs of the latter type of theorems depend heavily upon the graphtheoretic results of an earlier paper [14] and involve solely the examination of the global nature of each element's constitutive relation and the verification of a topological "loop-cutset" conditions.