On implementing passivity in discrete linear systems

Recent publications have shown that under some conditions continuous linear time-invariant systems become strictly positive real with constant feedback. This paper expands the applicability of this result to discrete linear systems. The paper shows the sufficient conditions that allow a discrete system to become stable and strictly passive via static (constant or nonstationary) output feedback. However, as the passivity conditions require a direct input-output connection that ends in an algebraic loop that includes the adaptive or nonlinear controller, they have been considered to be impossible to implement in realistic discrete-time systems. Therefore, this paper also finally solves the apparently inherent algebraic loop, thus allowing satisfaction of the passivity condition and implementation of adaptive and nonlinear control techniques in discrete-time positive real systems.

[1]  Alexander L. Fradkov.  Adaptive stabilization of minimal-phase vector-input objects without output derivative measurements , 1994 .

[2]  Alexander L. Fradkov.  Passification of Non-square Linear Systems and Feedback Yakubovich - Kalman - Popov Lemma , 2003, Eur. J. Control.

[3]  Ezra Zeheb,et al.  A sufficient condition of output feedback stabilization of uncertain systems , 1986 .

[4]  Petros A. Ioannou,et al.  Frequency domain conditions for strictly positive real functions , 1987 .

[5]  I. Barkana,et al.  Output feedback stability and passivity in nonstationary and nonlinear systems , 2005, 2005 International Conference on Control and Automation.

[6]  J. Wen.  Time domain and frequency domain conditions for strict positive realness , 1988 .

[7]  Itzhak Barkana.  ON OUTPUT FEEDBACK STABILITY AND PASSIVITY IN DISCRETE LINEAR SYSTEMS , 2005 .

[8]  H. Kaufman,et al.  Global stability and performance of a simplified adaptive algorithm , 1985 .

[9]  Itzhak Barkana,et al.  On Gain Conditions and Convergence of Simple Adaptive Control , 2003 .

[10]  David H. Owens,et al.  Positive-Real Structure and High-Gain Adaptive Stabilization , 1987 .

[11]  Zenta Iwai,et al.  Robust and simple adaptive control systems , 1992 .

[12]  Itzhak Barkana,et al.  Simple adaptive controlߞA stable direct model reference adaptive control methodology - brief survey , 2007, ALCOSP.

[13]  Liu Hsu,et al.  Mitigation of symmetry condition in positive realness for adaptive control , 2006, Autom..

[14]  Liu Hsu,et al.  MIMO direct adaptive control with reduced prior knowledge of the high frequency gain , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[15]  M. Corless,et al.  Output feedback stabilization of uncertain dynamical systems , 1984 .

[16]  Murat Arcak,et al.  Constructive nonlinear control: a historical perspective , 2001, Autom..

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

[18]  Itzhak Barkana,et al.  Classical and Simple Adaptive Control for Nonminimum Phase Autopilot Design , 2005 .

[19]  Dimitri Peaucelle,et al.  ROBUST PASSIFICATION VIA STATIC OUTPUT FEEDBACK – LMI RESULTS , 2005 .

[20]  Ezra Zeheb,et al.  Modified output error identification-elimination of the SPR condition , 1995, IEEE Trans. Autom. Control..

[21]  I. Bar-Kana.  Parallel feedforward and simplified adaptive control , 1987 .

[22]  Howard Kaufman,et al.  Direct Adaptive Control Algorithms , 1998 .

[23]  Petros A. Ioannou,et al.  Design of strictly positive real systems using constant output feedback , 1999, IEEE Trans. Autom. Control..

[24]  G. Gu.  Stabilizability conditions of multivariable uncertain systems via output feedback control , 1990 .

[25]  Itzhak Barkana.  Comments on "Design of strictly positive real systems using constant output feedback" , 2004, IEEE Trans. Autom. Control..

[26]  Izhak Bar-Kana,et al.  Absolute Stability and Robust Discrete Adaptive Control of Multivariable Systems , 1989 .

[27]  Jason L. Speyer,et al.  System characterization of positive real conditions , 1994, IEEE Trans. Autom. Control..

[28]  A. L. Fradkov.  Quadratic Lyapunov functions in the adaptive stability problem of a linear dynamic target , 1976 .

[29]  David J. Hill,et al.  Exponential Feedback Passivity and Stabilizability of Nonlinear Systems , 1998, Autom..

[30]  H. Kaufman,et al.  Implicit Adaptive Control for a Class of MIMO Systems , 1982, IEEE Transactions on Aerospace and Electronic Systems.