Finite bisimulations for switched linear systems

Abstract

Abstract-In this paper, we consider the problem of constructing a finite bisimulation quotient for a discrete-time switched linear system in a bounded subset of its state space. Given a set of observations over polytopic subsets of the state space and a switched linear system with stable subsystems, the proposed algorithm generates the bisimulation quotient in a finite number of steps with the aid of sublevel sets of a polyhedral Lyapunov function. Starting from a sublevel set that includes the origin in its interior, the proposed algorithm iteratively constructs the bisimulation quotient for the region bounded by any larger sublevel set. We show how this bisimulation quotient can be used for synthesis of switching laws and verification with respect to specifications given as syntactically co-safe Linear Temporal Logic formulae over the observed polytopic subsets.

Finite Uniform Bisimulations for Linear Systems With Finite Input Alphabets

“…• Our definition of finite uniform bisimulation is in accordance with the definitions of finite bisimulation introduced in [6,9]. However, the sufficient conditions for existence of finite bisimulations derived in [9] concern linear vector fields, and as such correspond to special cases of (6) where B is the zero matrix, whereas the present contribution addresses the more general case where B is nonzero.…”

“…Given system (6), under what conditions on A, B, U does there exist a finite uniform bisimulation ∼ on some invariant set S of system (6)?…”

“…Multiple complementary approaches have been developed and used [8,10,16,23,24] including qualitative models and l-complete approximations [11][12][13], approximating automata [2][3][4], exact or approximate bisimulation and simulation abstractions [6,18], and ρ/µ approximations [20][21][22].…”

“…In [19], the authors show that certain controllable linear systems admit finite bisimulations. In [6] the authors propose an algorithm, based on polyhedral Lyapunov functions, to generate finite bisimulations for switched linear systems with stable subsystems.…”

Symbolic Design of Networked Control Systems with State Prediction

“…The stability of the controlled system with the approximate simulation is investigated with a Lyapunov-like function [5]. The approximate (bi)simulation-based abstraction has been studied for nonlinear systems [6], [7], switched linear systems [8], and time-delay systems [9].…”