On Problem of Partial Detectability for Nonlinear Discrete-Time Systems

Abstract

Discrete (finite-difference) systems are widely used in modern nonlinear control theory. One of the main problems of a qualitative study of such systems is the problem of stability of the zero equilibrium position, which has great generality. In most works, such a stability problem is analyzed with respect to all variables that determine the state of the system. However, for many cases important in applications, it becomes necessary to analyze a more general problem of partial stability: the stability of the zero equilibrium position not for all, but only with respect to some given part of the variables. Such a problem is often also considered as auxiliary problem in the study of stability with respect to all variables. In this way, the corresponding concepts and problems of detectability of the studied system arise, which play an important role in the process of analysis of nonlinear controlled systems. Then, more general problems of partial detectability were posed, within the framework of which the situation was studied when stability from a part of variables implies stability not with respect to all, but with respect to more part of the variables. This article studies a nonlinear discrete (finite-difference) system of a general form that admits a zero equilibrium position. Easily interpreted conditions are found on the structural form of the system under consideration that determine its partial detectability, for which stability over a given part of the variables of the zero equilibrium position means its stability with respect to the other, more part of the variables. In this case, the stability with respect to the remaining part of the variables is uncertain and can be investigated additionally. In the process of analyzing this problem of partial detectability, the concept of partial null-dynamics of the system under study is introduced. An application of the obtained results to the stabilization problem with respect to part of the variables of nonlinear discrete controlled systems is given.