On the stabilization of nonlinear systems using state detection

Abstract

In this paper, we study the problem of stabilizing a nonlinear control system by means of a feedback control law, in cases where the entire state of the system is not available for measurement. The proposed method of stabilization consists of three parts: 1) determine a stabilizing control law based on state feedback, assuming the state vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> can be measured; 2) construct a state detection mechanism, which generates a vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z(t)</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z(t)-x(t)\rightarrow 0</tex> as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t\rightarrow \infty</tex> and 3) apply the previously determined control law to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z(t)</tex> . This scheme is well established for linear time-invariant systems, and its global convergence has previously been studied in the case of nonlinear systems. Hence, the contribution of this paper is in showing that such a scheme works in the absence of any linearity assumptions, and in studying both local asymptotic stability and global asymptotic stability.