A notion of dissipativity for discrete event systems

Dissipativity is a useful tool for analyzing and synthesizing stable feedback systems. Briefly, the property of dissipativity is that a system only stores and dissipates energy, with respect to a given energy storage function, and does not generate its own energy. This property carries a strong reliance on a notion of time that prohibits its direct application to discrete event systems (DES). The current paper takes the concept of dissipativity and applies it to DES by replacing the notion of time with an event-based definition. Two notions of dissipativity are defined, one for finite automata and the other for a general class of DES. For each notion, properties of dissipative DES are shown that connect with existing notions of stability for DES as would be expected from the classical definition of dissipativity. Examples are provided to illustrate the methods covered in the paper.

[1]  A. Michel,et al.  Lyapunov Stability of a Class of Discrete Event Systems , 1991, 1991 American Control Conference.

[2]  P. Moylan,et al.  The stability of nonlinear dissipative systems , 1976 .

[3]  W. Wonham,et al.  Supervisory control of timed discrete-event systems , 1994, IEEE Trans. Autom. Control..

[4]  M. W. Shields.  An Introduction to Automata Theory , 1988 .

[5]  Geert Stremersch,et al.  Supervision of Petri Nets , 2001, The Springer International Series on Discrete Event Dynamic Systems.

[6]  J. Büchi.  Weak Second‐Order Arithmetic and Finite Automata , 1960 .

[7]  W. M. Wonham,et al.  The control of discrete event systems , 1989 .

[8]  P. Olver.  Nonlinear Systems , 2013 .

[9]  Panos J. Antsaklis,et al.  Lyapunov stability of a class of discrete event systems , 1994, IEEE Trans. Autom. Control..

[10]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[11]  Panos J. Antsaklis,et al.  Timed Petri Nets in Hybrid Systems: Stability and Supervisory Control , 1998, Discret. Event Dyn. Syst..

[12]  Dominique Perrin,et al.  Finite Automata , 1958, Philosophy.

[13]  Madhavan Mukund,et al.  Finite-State Automata on Infinite Inputs , 2012, Modern Applications of Automata Theory.

[14]  R. McNaughton.  Review: J. Richard Buchi, Weak Second-Order Arithmetic and Finite Automata; J. Richard Buchi, On a Decision Method in Restricted second Order Arithmetic , 1963, Journal of Symbolic Logic.

[15]  Jan C. Willems,et al.  Dissipative Dynamical Systems , 2007, Eur. J. Control.

[16]  Norman R. Howes,et al.  Modern Analysis and Topology , 1995 .

[17]  D UllmanJeffrey,et al.  Introduction to automata theory, languages, and computation, 2nd edition , 2001 .

[18]  Pao C. Chau,et al.  Process Control , 2002 .

[19]  Michael J. McCourt,et al.  Dissipativity theory for hybrid systems with applications to networked control systems , 2013 .

[20]  Thomas E Marlin,et al.  Process Control , 1995 .

[21]  J. R. Büchi.  On a Decision Method in Restricted Second Order Arithmetic , 1990 .

[22]  Panos J. Antsaklis,et al.  Supervisory Control of Discrete Event Systems Using Petri Nets , 1998, The International Series on Discrete Event Dynamic Systems.

[23]  Tadao Murata,et al.  Petri nets: Properties, analysis and applications , 1989, Proc. IEEE.

[24]  David E. Muller,et al.  Infinite sequences and finite machines , 1963, SWCT.