Changing supply rates for input-output to state stable discrete-time nonlinear systems with applications
The authors consider a family of discrete-time nonlinear systems of the form \[ \begin{cases} x(k+1)= F_T(x(k), u(k))\\ y(k)= h(x(k))\end{cases}\qquad x\in\mathbb{R}^n,\;u\in\mathbb{R}^m,\;y\in \mathbb{R}^l\tag{1} \] parametrized by \(T> 0\). Systems of this form arise as discretization of continuous time systems, where \(T\) represents the sampling time. A very general notion of input-to-state stability is introduced. It includes more usual properties (such as stability, ISS, detectability, etc.) and it is related to the existence of a Lyapunov function satisfying appropriate conditions. The main contribution of the paper consists in proposing two different constructions of such a Lyapunov functions, obtained by combining two other auxiliary Lyapunov functions. This can be useful to prove input-to-state stability results for systems of the form (1), provided that the same property is enjoyed by an approximate system of the same form. Several applications are presented in the paper (an invariance principle, cascade systems, semidefinite Lyapunov functions, observer-based stabilization). Finally, the problem of a two-link manipulator is studied in this context.
scientific article; zbMATH DE number 1963702
scientific article