## Nonlinear stabilization by hybrid quantized feedback.(English)Zbl 0952.93109

Lynch, Nancy (ed.) et al., Hybrid systems: computation and control. 3rd international workshop, HSCC 2000, Pittsburgh, PA, USA, March 23-25, 2000. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1790, 243-257 (2000).
A quantizer is a piecewise constant function defined by $q_\Delta(x)= \begin{cases} k\Delta, &(k-1/2) \Delta\leq x< (k+1/2)\Delta, \quad k\in{\mathcal I},\\ M\Delta, &x\geq (M+1/2) \Delta,\\ -M\Delta, &x\leq -(M+1/2) \Delta. \end{cases}$ For the controlled system $\dot x= Ax+Bu$ the hybrid quantized feedback $$u= q_\Delta(x)$$ is considered; its stabilizing properties are analyzed via a quadratic Lyapunov function. Further the nonlinear case $\dot x= f(x,k(x+e))$ is considered, where $$k(x)$$ is the feedback law and $$e$$ is the quantization error. The input-to-state (ISS) stability is considered and applied to this system.
For the entire collection see [Zbl 0934.00029].

### MSC:

 93D15 Stabilization of systems by feedback 93C10 Nonlinear systems in control theory

### Keywords:

input-to-state stability; hybrid quantized feedback