Popov-Belevitch-Hautus type controllability tests for linear complementarity systems

It is well-known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems. This paper deals with the controllability problem of a class of piecewise linear systems, known as linear complementarity systems. By exploiting the underlying structure and employing the results on the controllability of the so-called conewise linear systems, we present a set of inequalitytype conditions as necessary and sufficient conditions for controllability of linear complementarity systems. The presented conditions are of Popov–Belevitch–Hautus type in nature. © 2006 Elsevier B.V. All rights reserved.

[1]  Bernard Brogliato,et al.  Some results on the controllability of planar variational inequalities , 2005, Syst. Control. Lett..

[2]  G. Smirnov.  Introduction to the Theory of Differential Inclusions , 2002 .

[3]  Alberto Bemporad,et al.  Observability and controllability of piecewise affine and hybrid systems , 2000, IEEE Trans. Autom. Control..

[4]  Kyun K. Lee,et al.  On the controllability of piecewise-linear hypersurface systems , 1987, 1987 American Control Conference.

[5]  M. Kanat Camlibel,et al.  Algebraic Necessary and Sufficient Conditions for the Controllability of Conewise Linear Systems , 2008, IEEE Transactions on Automatic Control.

[6]  Eduardo Sontag.  Controllability is harder to decide than accessibility , 1988 .

[7]  Dragan Nesic.  Controllability for a class of simple Wiener–Hammerstein systems , 1999 .

[8]  John N. Tsitsiklis,et al.  Complexity of stability and controllability of elementary hybrid systems , 1999, Autom..

[9]  R. Brammer.  Controllability in Linear Autonomous Systems with Positive Controllers , 1972 .

[10]  Aristotle Arapostathis,et al.  On the controllability of piecewise-linear hypersurface systems , 1987 .

[11]  Richard W. Cottle,et al.  Linear Complementarity Problem , 2009, Encyclopedia of Optimization.

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

[13]  J. Schumacher,et al.  A nine-fold canonical decomposition for linear systems , 1984 .

[14]  M.L.J. Hautus,et al.  Controllability and observability conditions of linear autonomous systems , 1969 .

[15]  Wpmh Maurice Heemels,et al.  Hybrid Systems: Computation and Control , 2003 .

[16]  M. Kanat Camlibel,et al.  On the Controllability of Bimodal Piecewise Linear Systems , 2004, HSCC.

[17]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[18]  Michel C. Delfour,et al.  CONTROLLABILITY AND OBSERVABILITY FOR INFINITE-DIMENSIONAL SYSTEMS* , 1972 .

[19]  W. P. M. H. Heemels,et al.  The Complementarity Class of Hybrid Dynamical Systems , 2003, Eur. J. Control.

[20]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[21]  H. Hermes,et al.  Foundations of optimal control theory , 1968 .

[22]  W.P.M.H. Heemels,et al.  Stability and controllability of planar bimodal linear complementarity systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[23]  J. M. Schumacher,et al.  Complementarity systems in optimization , 2004, Math. Program..

[24]  F. Vasca,et al.  Modelling switching power converters as complementarity systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[25]  R. Kálmán.  On the general theory of control systems , 1959 .

[26]  Wilson J. Rugh,et al.  Mathematical Description of Linear Systems , 1976 .

[27]  J. Yorke,et al.  Controllability of linear oscillatory systems using positive controls , 1971 .