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

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.

MSC:

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