Uniqueness of a Basic Nonlinear Structure

I N NONLINEAR systems theory two types of operators are especially important: linear time-inuariant (LTI) operators and memoryless or static nonlinear operators. Many important and well-known results pertain to systems which are interconnections of these operators, for example, the Popov criterion for the Lur’e structure. Indeed if multi-input multi-output (MIMO) operators are considered, all dynamical systems are included. In this paper we consider what is perhaps the simplest interconnection of these operators, shown in Fig. 1, and ask the question: in what sense are such systems unique, that is, under what conditions could two such systems have the same input-output (I/O) map? Some conditions are easy to think of, for example, we can rescale the operators or distribute any delay in A and C arbitrarily between them (A = (Y exp (sT)A, C = y exp (sT)C, B(x) = k’B(y-lx)). W e s h ow that these are the only ways these systems fail to be unique. Rugh and others [l]-[5] have shown that certain systems containing lumped LTI operators and memoryless power nonlinearities or multipliers are unique in a certain sense, and this paper is inspired by their work. Our emphasis, however, is slightly different: we consider memoryless nonlinearities as opposed to multipliers and pure power nonlinearities, and general as opposed to lumped LTI operators.