Strongly Regular Differential Variational Systems

A differential variational system is defined by an ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution of a finite-dimensional variational inequality containing the state variable of the system. This paper addresses two system-theoretic topics for such a nontraditional nonsmooth dynamical system; namely, (non-)Zenoness and local observability of a given state satisfying a blanket strong regularity condition. For the former topic, which is of contemporary interest in the study of hybrid systems, we extend the results in our previous paper, where we have studied Zeno states and switching times in a linear complementarity system (LCS). As a special case of the differential variational inequality (DVI), the LCS consists of a linear, time-invariant ODE and a linear complementarity problem. The extension to a nonlinear complementarity system (NCS) with analytic inputs turns out to be non-trivial as we need to use the Lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state. Further extension to a differential variational inequality is obtained via its equivalent Karush-Kuhn-Tucker formulation. For the second topic, which is classical in system theory, we use the non-Zenoness result and the recent results in a previous paper pertaining to the B-differentiability of the solution operator of a nonsmooth ODE to obtain a sufficient condition for the short-time local observability of a given strongly regular state of an NCS. Refined sufficient conditions and necessary conditions for local observability of the LCS satisfying the P-property are obtained

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

[2]  Jong-Shi Pang,et al.  Linear Complementarity Systems: Zeno States , 2005, SIAM J. Control. Optim..

[3]  M. Kanat Camlibel,et al.  Conewise Linear Systems: Non-Zenoness and Observability , 2006, SIAM J. Control. Optim..

[4]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

[5]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[6]  W. Heemels Linear complementarity systems : a study in hybrid dynamics , 1999 .

[7]  M. Egerstedt,et al.  On the regularization of Zeno hybrid automata , 1999 .

[8]  Johannes Schumacher,et al.  Rational complementarity problem , 1998 .

[9]  Arjan van der Schaft,et al.  The complementary-slackness class of hybrid systems , 1996, Math. Control. Signals Syst..

[10]  Jong-Shi Pang,et al.  Solution dependence on initial conditions in differential variational inequalities , 2008, Math. Program..

[11]  Héctor J. Sussmann Bounds on the number of switchings for trajectories of piecewise analytic vector fields , 1982 .

[12]  M. Çamlibel Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems , 2001 .

[13]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[14]  S. Sastry,et al.  Zeno hybrid systems , 2001 .

[15]  M.K. Camlibel,et al.  On the Zeno behavior of linear complementarity systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[16]  Karl Henrik Johansson,et al.  Towards a Geometric Theory of Hybrid Systems , 2000, HSCC.

[17]  W. P. M. H. Heemels,et al.  Linear Complementarity Systems , 2000, SIAM J. Appl. Math..

[18]  M. Kanat Camlibel,et al.  On Linear Passive Complementarity Systems , 2002, Eur. J. Control.

[19]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[20]  A. Schaft,et al.  Switched networks and complementarity , 2003 .

[21]  Jong-Shi Pang,et al.  Newton's Method for B-Differentiable Equations , 1990, Math. Oper. Res..

[22]  Pavol Brunovský,et al.  Regular synthesis for the linear-quadratic optimal control problem with linear control constraints , 1980 .

[23]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[24]  E. Hille,et al.  Lectures on ordinary differential equations , 1968 .

[25]  P. Hartman Ordinary Differential Equations , 1965 .

[26]  A. J. van der Schaft,et al.  Complementarity modeling of hybrid systems , 1998, IEEE Trans. Autom. Control..

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

[28]  Bernard Brogliato,et al.  Some perspectives on the analysis and control of complementarity systems , 2003, IEEE Trans. Autom. Control..

[29]  Jong-Shi Pang,et al.  Differential variational inequalities , 2008, Math. Program..

[30]  Johannes Schumacher,et al.  Well-posedness of a class of linear networks with ideal diodes , 2000 .

[31]  A. Krener,et al.  Nonlinear controllability and observability , 1977 .

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

[33]  O. Mangasarian On Concepts of Directional Differentiability , 2004 .