Path-Complete Graphs and Common Lyapunov Functions

A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time arbitrary switching systems. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. the converse however is not true even for discrete-time linear systems: we present such a system where a max-of-2 quadratics Lyapunov function exists while no corresponding Path-Complete Lyapunov function with 2 quadratic pieces exists. In light of this, we investigate when it is possible to decide if a Path- Complete Lyapunov function is less conservative than another. By analyzing the combinatorial and algebraic structure of the graph and the pieces respectively, we provide simple tools to decide when the existence of such a Lyapunov function implies that of another.

[1]  Tingshu Hu,et al.  Dual Matrix Inequalities in Stability and Performance Analysis of Linear Differential/Difference Inclusions , 2006 .

[2]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[3]  Victor Kozyakin,et al.  The Berger-Wang formula for the Markovian joint spectral radius , 2014, 1401.2711.

[4]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[5]  Pierre-Alexandre Bliman,et al.  Stability Analysis of Discrete-Time Switched Systems Through Lyapunov Functions with Nonminimal State , 2003, ADHS.

[6]  Jamal Daafouz,et al.  Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach , 2002, IEEE Trans. Autom. Control..

[7]  Amir Ali Ahmadi,et al.  A Characterization of Lyapunov Inequalities for Stability of Switched Systems , 2016, IEEE Transactions on Automatic Control.

[8]  J. Tsitsiklis,et al.  The boundedness of all products of a pair of matrices is undecidable , 2000 .

[9]  W. P. M. H. Heemels,et al.  Observability and Controllability Analysis of Linear Systems Subject to Data Losses , 2018, IEEE Transactions on Automatic Control.

[10]  Amir Ali Ahmadi,et al.  Joint Spectral Radius and Path-Complete Graph Lyapunov Functions , 2011, SIAM J. Control. Optim..

[11]  Anders Rantzer,et al.  Computation of piecewise quadratic Lyapunov functions for hybrid systems , 1997, 1997 European Control Conference (ECC).

[12]  M. Lazar,et al.  Alternative stability conditions for switched discrete time linear systems , 2014 .

[13]  Raphaël M. Jungers,et al.  Stability of discrete-time switching systems with constrained switching sequences , 2015, Autom..

[14]  A. Jadbabaie,et al.  Approximation of the joint spectral radius using sum of squares , 2007, 0712.2887.

[15]  Robert Shorten,et al.  Stability Criteria for Switched and Hybrid Systems , 2007, SIAM Rev..

[16]  Christos G. Cassandras,et al.  Introduction to Discrete Event Systems , 1999, The Kluwer International Series on Discrete Event Dynamic Systems.

[17]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[18]  Ji-Woong Lee,et al.  Uniform stabilization of discrete-time switched and Markovian jump linear systems , 2006, Autom..

[19]  Geir E. Dullerud,et al.  Control of Linear Switched Systems With Receding Horizon Modal Information , 2014, IEEE Transactions on Automatic Control.

[20]  Rapha L. Jungers.  The Joint Spectral Radius , 2009 .