Synthesis of Switching Rules for Ensuring Reachability Properties of Sampled Linear Systems

Abstract

We consider here systems with piecewise linear dynamics that are periodically sampled with a given period {\tau} . At each sampling time, the mode of the system, i.e., the parameters of the linear dynamics, can be switched, according to a switching rule. Such systems can be modelled as a special form of hybrid automata, called "switched systems", that are automata with an infinite real state space. The problem is to find a switching rule that guarantees the system to still be in a given area V at the next sampling time, and so on indefinitely. In this paper, we will consider two approaches: the indirect one that abstracts the system under the form of a finite discrete event system, and the direct one that works on the continuous state space. Our methods rely on previous works, but we specialize them to a simplified context (linearity, periodic switching instants, absence of control input), which is motivated by the features of a focused case study: a DC-DC boost converter built by electronics laboratory SATIE (ENS Cachan). Our enhanced methods allow us to treat successfully this real-life example.