Algebraic and dynamic Lyapunov equations on time scales

We revisit the canonical continuous-time and discrete-time matrix algebraic and matrix differential equations that play a central role in Lyapunov-based stability arguments. The goal is to generalize and extend these types of equations and subsequent analysis to dynamical systems on domains other than R or Z, called “time scales”, e.g. nonuniform discrete domains or domains consisting of a mixture of discrete and continuous components. In particular, we compare and contrast a generalization of the algebraic Lyapunov equation and the dynamic Lyapunov equation in this time scales setting.

[1]  Michael K. Tippett,et al.  Upper bounds for the solution of the discrete algebraic Lyapunov equation , 1999, Autom..

[2]  T. Gard,et al.  ASYMPTOTIC BEHAVIOR OF NATURAL GROWTH ON TIME SCALES , 2003 .

[3]  S. Hilger.  Analysis on Measure Chains — A Unified Approach to Continuous and Discrete Calculus , 1990 .

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

[5]  H. Abou-Kandil,et al.  Matrix Riccati Equations in Control and Systems Theory , 2003, IEEE Transactions on Automatic Control.

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

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

[9]  G. Guseinov.  Integration on time scales , 2003 .

[10]  S. Bittanti,et al.  The difference periodic Ricati equation for the periodic prediction problem , 1988 .

[11]  Douglas Bowman,et al.  q-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions , 2002 .

[12]  A. M. Lyapunov.  The general problem of the stability of motion , 1992 .

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

[14]  Christopher C. Tisdell,et al.  Stability and instability for dynamic equations on time scales , 2005 .

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

[16]  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.

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

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

[19]  G. Demidenko,et al.  On Stability of Solutions to Linear Systems with Periodic Coefficients , 2001 .

[20]  V. Lakshmikantham,et al.  Dynamic systems on measure chains , 1996 .

[21]  Alice A. Ramos.  Stability of hybrid dynamic systems: Analysis and design , 2009 .

[22]  A. Michel,et al.  Stability of Dynamical Systems — Continuous , Discontinuous , and Discrete Systems , 2008 .

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

[24]  A. Varga.  Periodic Lyapunov equations: Some applications and new algorithms , 1997 .

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

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

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

[29]  Alessandro Astolfi,et al.  Stability of Dynamical Systems - Continuous, Discontinuous, and Discrete Systems (by Michel, A.N. et al.; 2008) [Bookshelf] , 2007, IEEE Control Systems.

[30]  Roy M. Howard,et al.  Linear System Theory , 1992 .

[31]  Stability results for higher dimensional equations on time scales , 2011 .

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

[33]  Fabian R. Wirth,et al.  A spectral characterization of exponential stability for linear time-invariant systems on time scales , 2003 .

[34]  A. Sikorska-Nowak,et al.  Dynamic equations (…) on time scales , 2011 .

[35]  Michael K. Tippett,et al.  Bounds for the solution of the discrete algebraic Lyapunov equation , 1998, Autom..