Controllability, Reachability, and Stabilizability of Finite Automata: A Controllability Matrix Method

This paper investigates the controllability, reachability, and stabilizability of finite automata by using the semitensor product of matrices. Firstly, by expressing the states, inputs, and outputs as vector forms, an algebraic form is obtained for finite automata. Secondly, based on the algebraic form, a controllability matrix is constructed for finite automata. Thirdly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of finite automata by using the controllability matrix. Finally, an illustrative example is given to support the obtained new results

Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We provide approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs, establishing as a byproduct a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has gone through two major rounds of revision. In particular, a section on the performance of our algorithm on application-motivated problems has been added and a more comprehensive literature review is presente

Reachability of Consensus and Synchronizing Automata

We consider the problem of determining the existence of a sequence of matrices driving a discrete-time consensus system to consensus. We transform this problem into one of the existence of a product of the transition (stochastic) matrices that has a positive column. We then generalize some results from automata theory to sets of stochastic matrices. We obtain as a main result a polynomial-time algorithm to decide the existence of a sequence of matrices achieving consensus.Comment: Update after revie

Global exponential stabilization of language constrained switched linear discrete-time system based on the s-procedure approach

This paper considers global exponential stabilization (GES) of switched linear discrete-time system under language constraint which is generated by non-deterministic finite state automata. A technique in linear matrix inequalities called S-procedure is employed to provide sufficient conditions of GES which are less conservative than the existing Lyapunov-Metzler condition. Moreover, by revising the construction of Lyapunov matrices and the corresponding switching control policy, a more flexible result is obtained such that stabilization path at each moment might be multiple. Finally, a numerical example is given to illustrate the effectiveness of the proposed results