Output stabilization of nonlinear disturbed system

1. INTRODUCTION

One of the classical problems of control theory is the problem of stabilization of nonlinear systems. The absolute stability problem, formulated by Lurie and coworkers in the 40’s, has been a well-studied and fruitful area of research, as presented in [1] – [11]. Since works of Lurie [6], [7], interest of researches of control systems has been attracted by structures including linear block and nonlinear feedback static block. It is possible to mark big enough number of works devoted to solution of problems of nonlinear systems stabilization  for a case when output of the nonlinear block is given as control input of the linear block (see the review [12]). It is also possible to mark a block of works [13] – [23] in which nonlinear part and static nonlinearity adjust with an input on control. However, approaches [13] – [23] are focused on stabilization of systems with nonlinearity, resulted to an input of system, and do not allow one to solve more general problems.

Today problems of control of nonlinear systems in which nonlinearity is not coordinated with control are of interest. Among works devoted to these subjects one can allocate papers [24] – [33] in which similar problem is considered. In papers [24] – [26], a nonlinear system consisting of a linear block described by known constant matrixes and a nonlinear negative feedback block with known static characteristic is considered and lays in a region: the first and the third sectors. Authors of papers [24] – [26] offer to choose a proportional controller on measurements of the state vector with addition of a feedback with known nonlinearity. For a case of measurability of the state vector and certain parameters of the linear part, in papers [24] – [26] conditions of the solution of stabilization problem for the specified conditions on plant are proposed. In comparison with results obtained by Kokotovic group [24] – [26], the approach in [28] – [33] ensures an output stabilization of nonlinear systems without measurement of state variables. But in comparison with results obtained in [28] – [33], the approach in [24] – [26] ensures stabilization of nonlinear systems for general case (in papers [28] – [33]) authors consider the nonlinear system in triangular canonical form).

In this paper a robust version of the results on high-gain stabilization of nonlinear systems published in [16] is extended to a class of nonlinear systems without matching conditions. In the given work, developing approaches obtained in [13] – [33], solution of the problem of stabilization of nonlinear disturbed system consisting of structures including a linear block and nonlinear feedback static block is offered. Assuming, parameters of the linear part to be unknown, the output is measured