# ISS Lyapunov Functions for Cascade Switched Systems and Sampled-Data Control

## TL;DR

This paper develops ISS Lyapunov functions for cascade switched systems with state resets and applies these tools to analyze stability and design dynamic sampling algorithms for observer-based feedback control in nonlinear systems.

## Contribution

It introduces a novel method for constructing ISS Lyapunov functions for cascade switched systems with resets and applies it to event-based sampling control.

## Key findings

- Constructed ISS Lyapunov functions for cascade switched systems with resets.
- Derived lower bounds on average dwell-time for stability.
- Designed dynamic sampling algorithms ensuring stability in nonlinear systems.

## Abstract

Input-to-state stability (ISS) of switched systems is studied where the individual subsystems are connected in a serial cascade configuration, and the states are allowed to reset at switching times. An ISS Lyapunov function is associated to each of the two blocks connected in cascade, and these functions are used as building blocks for constructing ISS Lyapunov function for the interconnected system. The derivative of individual Lyapunov functions may be bounded by nonlinear decay functions, and the growth in the value of Lyapunov function at switching times may also be a nonlinear function of the value of other Lyapunov functions. The stability of overall hybrid system is analyzed by constructing a newly constructed ISS-Lyapunov function and deriving lower bounds on the average dwell-time. The particular case of linear subsystems and quadratic Lyapunov functions is also studied. The tools are also used for studying the observer-based feedback stabilization of a nonlinear switched system with event-based sampling of the output and control inputs. We design dynamic sampling algorithms based on the proposed Lyapunov functions and analyze the stability of the resulting closed-loop system.