Stability of switched linear systems on non-uniform time domains

A recent development in Lyapunov stability theory allows for analysis of switched linear systems evolving on nonuniform, discrete time domains. The analysis makes use of an emerging mathematical framework termed dynamic equations on time scales. We will present stability conditions for a general, arbitrarily switched system and then for system with a “constrained” switching signal. The results take the form of a compute-able inequality, which imposes conditions on the time domain itself.

[1]  Naresh K. Sinha,et al.  Modern Control Systems , 1981, IEEE Transactions on Systems, Man, and Cybernetics.

[2]  John M. Davis,et al.  A Unified Approach to High-Gain Adaptive Controllers , 2009, 0901.3873.

[3]  Kejun Zhuang.  PERIODIC SOLUTIONS FOR A STAGE-STRUCTURE ECOLOGICAL MODEL ON TIME SCALES , 2007 .

[4]  J. J. Dacunha.  Stability for time varying linear dynamic systems on time scales , 2005 .

[5]  Robert J. Marks,et al.  CONTROLLABILITY, OBSERVABILITY, REALIZABILITY, AND STABILITY OF DYNAMIC LINEAR SYSTEMS , 2009, 0901.3764.

[6]  LINEAR CONTROL SYSTEMS ON TIME SCALE: UNIFICATION OF CONTINU- OUS AND DISCRETE , 2004 .

[7]  Ewa Pawluszewicz,et al.  Realizations of linear control systems on time scales , 2006 .

[8]  Ferhan Merdivenci Atici,et al.  An application of time scales to economics , 2006, Math. Comput. Model..

[9]  Ewa Pawluszewicz,et al.  Realizations of Nonlinear Control Systems on Time Scales , 2008, IEEE Transactions on Automatic Control.

[10]  Ferhan Merdivenci Atici,et al.  A production-inventory model of HMMS on time scales , 2008, Appl. Math. Lett..

[11]  R. Ruth,et al.  Stability of dynamical systems , 1988 .

[12]  John M. Davis,et al.  The Laplace transform on time scales revisited , 2007 .

[13]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

[14]  Robert J. Marks,et al.  Nonregressivity in switched linear circuits and mechanical systems , 2006, Math. Comput. Model..

[15]  A. Peterson,et al.  Dynamic Equations on Time Scales , 2001 .

[16]  C. Johnson,et al.  Unified canonical forms for matrices over a differential ring , 1991 .

[17]  William S. Levine,et al.  The Control Handbook , 2010 .

[18]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

[19]  Robert J. Marks,et al.  Bandwidth reduction for controller area networks using adaptive sampling , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[20]  Stability of non-diagonalizable switched linear systems on time scales. , 2009 .

[21]  A. Peterson,et al.  Advances in Dynamic Equations on Time Scales , 2012 .

[22]  Ravi P. Agarwal,et al.  Dynamic equations on time scales: a survey , 2002 .

[23]  Robert J. Marks,et al.  A generalized Fourier transform and convolution on time scales , 2008 .

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

[26]  Robert J. Marks,et al.  How deterministic must a real-time controller be? , 2005, 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[27]  Peter L. Ralph,et al.  Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan , 2005, math/0509270.