Volterra Expansions for Time-Varying Nonlinear Systems

Volterra Expansions for Time-Varying Nonlinear Systems By I. W. SANDBERG (Manuscript received August 13, 1981) Recent results show for the first time the existence of a locally convergent Volterra-series representation for the input-output relation of a certain important large class of time-invariant nonlinear systems containing an arbitrary finite number of nonlinear elements. (Systems of the type considered arise, for example, in the area of communication channel modeling.) Here corresponding results are given for time-varying systems, which arise frequently. A key hypothesis of our main theorem, which asserts that a convergent Volterra expansion exists under certain specified conditions, has the useful property that it is met if a certain "linearized subgraph" of the system is bounded-input bounded-output stable. I. INTRODUCTION This paper is a continuation of the study initiated in Ref. 1 concerning operator-type models of dynamic nonlinear physical systems, such as communication channels and control systems. Reference 1 addresses the problem of determining conditions under which there exists a power-series-like expansion, or a polynomial-type approximation, for a system's outputs in terms of its inputs. Related problems concerning properties of the expansions are also considered, and nonlocal as well as local results are presented. In particular, the results in Ref. 1 show for the first time the existence of a locally convergent Volterra-series representation for the input-output relation of a certain important large class of time-invariant systems containing an arbitrary finite number of nonlinear elements.