Interval Observer Design for Discrete-Time Nonlinear Dynamic Systems

Abstract

The paper considers the problem of interval observer design for nonlinear dynamic systems described by discrete-time models under external disturbances, measurement noise, and parametric uncertainties. The problem is to design the observer with fewer dimensions than that of the original system; such an observer must generate upper and lower bounds of admissible values of the prescribed nonlinear function of the original system state vector. To solve the problem, special mathematical tool is used. The advantage of this tool is that it allows studying the systems described by models with non-smoo th nonlinearities. To construct interval observer, the reduced-order model of the original system insensitive or having minimal sensitivity to the disturbances is designed. The designing procedure is based on two algorithms: the first one is intended to design the model of minimal sensitivity; the second one is used to reduce the dimension of the model. The rules are formulated to ensure stability based on the prescribed set of the desirable eigenvalues and feedback. The interval observer consists of two subsystems: the first one generates the lower bound, the second one the upper bound. The relations describing both subsystems are given. To construct such an observer in the nonlinear case, the terms of positive and negative influence of variables describing the model are introduced. These terms allow finding out how the upper and lower bounds of these variables will appear in the interval observer. The conditions under which the observer can be designed are given. The theoretical results are illustrated by an example of three tank system. Simulation results based on the package Matlab show the effectiveness of the developed theory.