Description of switched systems by implicit representations

This paper shows that a certain class of time-dependent, autonomous switched systems can also be studied using the structural concepts and properties of the linear time-invariant implicit systems theory. We first consider the modeling aspects, and then, the control aspects are tackled. This fact contributes to the enrichment of the switched systems theory and enlarges its action field. In Section I we specify a class of time-dependent, autonomous switched systems which enables us to describe systems with a variable internal structure. In Section II we show how to translate the considered class of time-dependent, autonomous switched systems into an implicit representation and then we recall some important results, of the implicit systems theory, which enable us to control them in such a way that they behave as a specified linear time-invariant system whatever be the position of the switches. In Section IV, we consider some reachability aspects. In Section V, we conclude.

[1]  W. Wolovich.  State-space and multivariable theory , 1972 .

[2]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[3]  Robert Shorten,et al.  Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time‐invariant systems , 2002 .

[4]  Daniel Liberzon,et al.  On stability of linear switched differential algebraic equations , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[5]  George C. Verghese,et al.  Further notes on singular descriptions , 1981 .

[6]  H. Rosenbrock,et al.  State-space and multivariable theory, , 1970 .

[7]  M. Malabre,et al.  Almost rejection of internal structural variations in linear systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[8]  Diego Napp Avelli,et al.  On the stability of switched behavioral systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[9]  Michel Malabre,et al.  ON THE CONTROL OF LINEAR SYSTEMS HAVING INTERNAL VARIATIONS, PART I—REACHABILITY , 2002 .

[10]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[11]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[12]  Frank L. Lewis,et al.  A tutorial on the geometric analysis of linear time-invariant implicit systems , 1992, Autom..

[13]  Kumpati S. Narendra,et al.  Adaptive control using multiple models , 1997, IEEE Trans. Autom. Control..

[14]  J. Willems.  Input-output and state-space representations of finite-dimensional linear time-invariant systems , 1983 .

[15]  Jeffrey S. Rosenthal,et al.  Introduction to mathematical systems theory. a behavioral approach [Book Review] , 2002, IEEE Transactions on Automatic Control.

[16]  K. Narendra,et al.  A common Lyapunov function for stable LTI systems with commuting A-matrices , 1994, IEEE Trans. Autom. Control..

[17]  Michel Malabre,et al.  Structural matrix minimization algorithm for implicit descriptions , 1997, Autom..

[18]  Johannes Schumacher,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[20]  Ton Geerts.  Solvability conditions, consistency, and weak consistency for linear differential-algebraic equations and time-invariant singular systems: the general case , 1993 .

[21]  A. Morse.  Supervisory control of families of linear set-point controllers , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[22]  M. Malabre,et al.  More about non square implicit descriptions for modelling and control , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[23]  Michel Malabre,et al.  On the control of linear systems having internal variations , 2003, Autom..

[24]  Jean Jacques Loiseau,et al.  Proportional and proportional-derivative canonical forms for descriptor systems with outputs , 1994, Autom..

[25]  P. Bernhard.  On Singular Implicit Linear Dynamical Systems , 1982 .