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Input-output parametric models for non-linear systems. I: deterministic non-linear systems. (English) Zbl 0569.93011

This paper and its second part [see the article reviewed below] deal with input-output equations for discrete-time nonlinear systems. In the linear theory the concept of i/o representation via high order difference equations (as opposed to state evolution equations) is well understood. For nonlinear systems, the first work in this area appears to have been that of the reviewer [in ”Polynomial Response Maps”, Lect. Notes Control Inf. Sci. (1979; Zbl 0413.93004)]. The present work presents results in a somewhat different direction than that of the reviewer, but includes a detailed comparison (in part 2) with that approach.
The basic assumption which the authors make - in contrast to the setup in the reviewer’s work - is that the linearization of the given system at the origin have the same dimension as the system itself (the precise statement is in terms of Hankel matrices and the ”Jacobian condition” in the above reference). With this assumption, an argument based essentially on the implicit function theorem shows the existence of a nonlinear i/o equation, relating a finite number of past inputs and outputs, and valid for small enough inputs. This local validity - a consequence of the use of the implicit function theorem - is one aspect that distinguishes the obtained results from those for polynomial response maps - where tools from algebraic geometry are used in order to obtain global results. The other difference is that the results for polynomial responses imply the existence of implicit, as opposed to recursive, equations, and the latter may be of greater interest for identification applications.
Reviewer: E.Sontag

MSC:

93B15 Realizations from input-output data
93C10 Nonlinear systems in control theory
93C55 Discrete-time control/observation systems
26B10 Implicit function theorems, Jacobians, transformations with several variables
93C35 Multivariable systems, multidimensional control systems
93B10 Canonical structure
93B30 System identification
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References:

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