Stability of discrete-time systems with quantized input and state measurements

This note focuses on linear discrete-time systems controlled using a quantized input computed from quantized measurements. Nominally stabilizing, but otherwise arbitrary, state feedback gains could result in limit cycling or nonzero equilibrium points. Although a single quantizer is a sector nonlinearity, the presence of a quantizer at each state measurement channel makes traditional absolute stability theory not applicable in a direct way. A global asymptotic stability condition is obtained by means of a result which allows us to apply discrete positive real theory to systems with a sector nonlinearity which is multiplicatively perturbed by a bounded function of the state. The stability result is readily applicable by evaluating the location of the polar plot of a system transfer function relative to a vertical line whose abcissa depends on the one-norm of the feedback gain. A graphical method is also described that can be used to determine the equilibrium points of the closed-loop system for any given feedback gain.

[1]  Wassim M. Haddad,et al.  Nonlinear control of Hammerstein systems with passive nonlinear dynamics , 2001, IEEE Trans. Autom. Control..

[2]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[3]  R. Mollin.  Fundamental number theory with applications , 1998 .

[4]  M. Gevers,et al.  Optimal finite precision implementation of a state-estimate feedback controller , 1990 .

[5]  Robert E. Skelton,et al.  Optimal Controllers for Finite Wordlength Implementation , 1990 .

[6]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[7]  Garrett Birkhoff,et al.  A survey of modern algebra , 1942 .

[8]  Jay C. Hsu,et al.  Modern Control Principles and Applications , 1968 .

[9]  Verne C. Fryklund,et al.  What systems analysis? , 1981, Nature.

[10]  B. Anderson.  A SYSTEM THEORY CRITERION FOR POSITIVE REAL MATRICES , 1967 .

[11]  K. Grigoriadis,et al.  Optimal controllers for finite wordlength implementation , 1992 .

[12]  D. Delchamps.  Stabilizing a linear system with quantized state feedback , 1990 .

[13]  James F. Whidborne,et al.  Optimal finite-precision controller realization of sampled-data systems , 2000, Int. J. Syst. Sci..

[14]  Robert S. H. Istepanian,et al.  Optimizing stability bounds of finite-precision PID controllers using adaptive simulated annealing , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[15]  Jj Dacruz,et al.  FREQUENCY-DOMAIN APPROACH TO THE ABSOLUTE STABILITY ANALYSIS OF DISCRETE-TIME LINEAR-QUADRATIC REGULATORS , 1990 .

[16]  E.A. Misawa,et al.  Stability analysis of discrete linear systems with quantized input , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[17]  S. Mitter,et al.  Quantization of linear systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[18]  Donato Trigiante,et al.  THEORY OF DIFFERENCE EQUATIONS Numerical Methods and Applications (Second Edition) , 2002 .

[19]  J. P. Lasalle,et al.  Absolute Stability of Regulator Systems , 1964 .

[20]  Michel Gevers,et al.  Comparative study of finite wordlength effects in shift and delta operator parameterizations , 1993, IEEE Trans. Autom. Control..

[21]  B. Widrow,et al.  Statistical theory of quantization , 1996 .

[22]  Daniel Liberzon,et al.  Quantized feedback stabilization of linear systems , 2000, IEEE Trans. Autom. Control..

[23]  Sheng Chen,et al.  Optimizing stability bounds of finite-precision PID controller structures , 1999, IEEE Trans. Autom. Control..

[24]  F. Fagnani,et al.  Stability analysis and synthesis for scalar linear systems with a quantized feedback , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[25]  R. M. Thrall.  Review: Garrett Birkhoff and Saunders MacLane, A Survey of Modern Algebra , 1942 .

[26]  Emmanuel G. Collins,et al.  Improved closed-loop stability for fixed-point controller realizations using the delta operator , 2001 .

[27]  Brian D. O. Anderson,et al.  Discrete positive-real fu nctions and their applications to system stability , 1969 .