A continuous feedback approach to global strong stabilization of nonlinear systems. (English) Zbl 1012.93053

Qian, Chunjiang; Lin, Wei

IEEE Trans. Autom. Control 46, No. 7, 1061-1079 (2001).

The paper addresses the problem of the global stabilization by means of continuous feedback of a class of nonlinear (possibly non-affine and nonsmooth) systems which in general cannot be stabilized by smooth (i.e., at least \(C^1\)) feedback. The basic result concerns systems which can be represented as a chain of power integrators perturbed by a vector field in triangular form \[ \begin{aligned} \dot x_1 &= d_1(t) x^{p_1}_2+ f_1(t, x_1,x_2)\\ &\vdots\\ \dot x_i &= d_i(t) x^{p_i}_{i+1}+ f_i(t, x_1,\dots, x_{i+1})\\ &\vdots\\ \dot x_n &= d_n(t) u^{p_n}+ f_n(t, x_1,\dots, x_n,u)\end{aligned} \] where the \(p_i\)’s are odd, the \(d_i(t)\) are unknown but constrained to a bounded interval, and \[ f_i(t, x_1,\dots, x_{i+1})= \sum^{p_i-1}_{j=0} x^j_{i+1} a_{ij}(t, x_1,\dots, x_i) \] (\(x_{n+1}\) stands for \(u\)). The functions \(a_{ij}\) are subject to some other technical assumptions. The proof is based on an iterative procedure and exploits the theory of homogeneous systems. It uses the method of adding a power integrator in order to explicitly construct a continuous feedback and generate a \(C^1\) Lyapunov function. The paper contains also some extensions of the basic result and a rich variety of interesting examples.

Reviewer: A.Bacciotti (Torino)

MSC:

93D15	Stabilization of systems by feedback
93D30	Lyapunov and storage functions
93C10	Nonlinear systems in control theory

Keywords:

triangular form perturbation; nonlinear systems; global stabilization; continuous feedback; chain of power integrators; homogeneous systems; \(C^1\) Lyapunov function