A qualitative analysis of the behavior of dynamic nonlinear networks: Steady-state solutions of nonautonomous networks

Several theorems are presented which predict in a qualitative manner the transient and steady-state behavior of a large class of nonautonomous dynamic nonlinear networks containing coupled and multiterminal resistors, inductors, capacitors, and time-varying sources. The input waveforms of the sources may be aperiodic, periodic, or almost periodic. 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, for all bounded inputs. In particular, conditions are given such that a "small-signal" input will give rise to a "small-signal" output, thereby justifying the validity of the small-signal analysis approach for this class of networks. The properties of reciprocity and monotonicity (local passivity) are invoked to guarantee that the network has a unique steady-state response for all bounded excitations. In particular, conditions are given such that a periodic (respectively, asymptotically almost periodic) input gives rise to a unique periodic (respectively, asymptotically almost periodic) output. Moreover, conditions are given such that the frequency spectrum of the output waveforms is a subset of all linear combinations of the frequency components of the input waveforms. A further imposition of a growth condition guarantees that all solution waveforms will converge to each other asymptotically. In particular, the magnitude of the differences between any two solutions is shown to be bounded between two exponential waveforms for all time t > 0 . The main features of the majority of the theeorems 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 graph-theoretic 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" condition. Several counter examples are given to demonstrate that most of the theorems presented in this paper are the best possible for the class of networks under consideration in the sense that a slight weakening of any one hypothesis would invalidate the theorems.