Topological Entropy Bounds for Switched Linear Systems with Lie Structure

In this thesis, we provide an initial investigation into bounds for topological entropy of switched linear systems. Entropy measures, roughly, the information needed to describe the behavior of a system with finite precision on finite time horizons, in the limit. After working out entropy computations in detail for the scalar switched case, we review the linear time-invariant nonscalar case, and extend to the nonscalar switched case. We assume some commutation relations among the matrices of the switched system, namely solvability, define an upper average time of activation quantity and use it to provide an upper bound on the entropy of the switched system in terms of the eigenvalues of each subsystem.

[1]  H. Haimovich,et al.  Sufficient Conditions for Generic Feedback Stabilizability of Switching Systems via Lie-Algebraic Solvability , 2013, IEEE Transactions on Automatic Control.

[2]  Christoph Kawan,et al.  A note on topological feedback entropy and invariance entropy , 2013, Syst. Control. Lett..

[3]  Fritz Colonius,et al.  Minimal Bit Rates and Entropy for Exponential Stabilization , 2012, SIAM J. Control. Optim..

[4]  Christoph Kawan,et al.  Invariance Entropy for Control Systems , 2009, SIAM J. Control. Optim..

[5]  Roy L. Adler,et al.  Topological entropy , 2008, Scholarpedia.

[6]  Daniel Liberzon,et al.  Stabilizing a switched linear system by sampled-data quantized feedback , 2011, IEEE Conference on Decision and Control and European Control Conference.

[7]  Paul R. Halmos.  Recent progress in ergodic theory , 1961 .

[8]  Daniel Liberzon,et al.  Entropy and Minimal Data Rates for State Estimation and Model Detection , 2016, HSCC.

[9]  A. Katok,et al.  Introduction to the Modern Theory of Dynamical Systems: INTRODUCTION , 1995 .

[10]  Sekhar Tatikonda,et al.  Control under communication constraints , 2004, IEEE Transactions on Automatic Control.

[11]  R. Bowen.  Entropy for group endomorphisms and homogeneous spaces , 1971 .

[12]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[13]  D. D. Siljak,et al.  Algebraic criterion for absolute stability, optimality and passivity of dynamic systems , 1970 .

[14]  Daniel Liberzon,et al.  Towards robust Lie-algebraic stability conditions for switched linear systems , 2010, 49th IEEE Conference on Decision and Control (CDC).

[15]  Loring W. Tu,et al.  An introduction to manifolds , 2007 .

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

[17]  Robert Shorten,et al.  Stability Criteria for Switched and Hybrid Systems , 2007, SIAM Rev..

[18]  C. Wall,et al.  Lie Algebras And Lie Groups , 1967, The Mathematical Gazette.

[19]  Michael Margaliot,et al.  Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions , 2006, Syst. Control. Lett..

[20]  Robin J. Evans,et al.  Topological feedback entropy and Nonlinear stabilization , 2004, IEEE Transactions on Automatic Control.

[21]  Willard Van Orman Quine,et al.  Methods of Logic , 1951 .

[22]  Yoav Sharon,et al.  Third-Order Nilpotency, Nice Reachability and Asymptotic Stability ? , 2007 .

[23]  C. Kawan,et al.  Some results on the entropy of non-autonomous dynamical systems , 2015, 1507.08590.

[24]  A. Morse,et al.  Stability of switched systems: a Lie-algebraic condition ( , 1999 .

[25]  D. Ornstein.  Ergodic theory, randomness, and dynamical systems , 1974 .

[26]  Anatole Katok,et al.  FIFTY YEARS OF ENTROPY IN DYNAMICS: 1958-2007 , 2007 .

[27]  Andrey V. Savkin,et al.  Analysis and synthesis of networked control systems: Topological entropy, observability, robustness and optimal control , 2005, Autom..

[28]  A. Matveev,et al.  Estimation and Control over Communication Networks , 2008 .

[29]  Hernan Haimovich,et al.  Feedback stabilisation of switched systems via iterative approximate eigenvector assignment , 2010, 49th IEEE Conference on Decision and Control (CDC).

[30]  Guosong Yang,et al.  Stabilizing a switched linear system with disturbance by sampled-data quantized feedback , 2015, 2015 American Control Conference (ACC).

[31]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .