Control With Communications Constraints: Measuring the Instability in Parametric Linear Systems

This paper investigates the instability measure of linear systems defined as the sum of the unstable eigenvalues in the continuous-time case and the product of the unstable eigenvalues in the discrete-time case. The problem consists of determining the largest instability measure in systems depending polynomially on parameters constrained in a semialgebraic set. It is shown that upper bounds of the sought measure can be established via linear matrix inequality feasibility tests. Moreover, a priori and a posteriori conditions for establishing nonconservatism are proposed. Finally, two special cases of the proposed methodology are investigated—the first one concerns systems with a single parameter, and the second one concerns the determination of the largest spectral abscissa and radius. Three applications in control with communications constraints are discussed.

[1]  John Baillieul,et al.  Robust and efficient quantization and coding for control of multidimensional linear systems under data rate constraints , 2006, CDC 2006.

[2]  Graziano Chesi,et al.  Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach , 2014, Autom..

[3]  K. Mahler.  An application of Jensen's formula to polynomials , 1960 .

[4]  Wei Chen,et al.  Stabilization of networked control systems with multirate sampling , 2013, Autom..

[5]  Robin J. Evans,et al.  Stabilizability of Stochastic Linear Systems with Finite Feedback Data Rates , 2004, SIAM J. Control. Optim..

[6]  Lihua Xie,et al.  The sector bound approach to quantized feedback control , 2005, IEEE Transactions on Automatic Control.

[7]  Carsten W. Scherer,et al.  Computing optimal fixed order H∞-synthesis values by matrix sum of squares relaxations , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[8]  L. Qiu,et al.  Stabilization of networked control systems with finite data rate , 2009, 2009 7th Asian Control Conference.

[9]  G. Stengle.  A nullstellensatz and a positivstellensatz in semialgebraic geometry , 1974 .

[10]  Andrey V. Savkin,et al.  Multirate Stabilization of Linear Multiple Sensor Systems via Limited Capacity Communication Channels , 2005, SIAM J. Control. Optim..

[11]  Nicola Elia,et al.  Stabilization of linear systems with limited information , 2001, IEEE Trans. Autom. Control..

[12]  Pablo A. Parrilo,et al.  Introducing SOSTOOLS: a general purpose sum of squares programming solver , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[13]  Graziano Chesi,et al.  On the Gap Between Positive Polynomials and SOS of Polynomials , 2007, IEEE Transactions on Automatic Control.

[14]  Graziano Chesi,et al.  Worst-Case Mahler Measure in Polytopic Uncertain Systems , 2012, IEEE Transactions on Automatic Control.

[15]  Li Qiu,et al.  Quantify the Unstable , 2010 .

[16]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

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

[18]  Olga Taussky-Todd.  SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM , 1996 .

[19]  Graziano Chesi.  Instability Analysis of Uncertain Systems via Determinants and LMIs , 2015, IEEE Transactions on Automatic Control.

[20]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[21]  A. Papachristodoulou,et al.  Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach , 2004, 2004 5th Asian Control Conference (IEEE Cat. No.04EX904).

[22]  Richard H. Middleton,et al.  Feedback stabilization over signal-to-noise ratio constrained channels , 2007, Proceedings of the 2004 American Control Conference.

[23]  A. Vicino,et al.  On convexification of some minimum distance problems , 1999, 1999 European Control Conference (ECC).

[24]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[25]  G. Chesi,et al.  LMI Techniques for Optimization Over Polynomials in Control: A Survey , 2010, IEEE Transactions on Automatic Control.

[26]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..