An approach to the analysis of observability and controllability in nonlinear systems via linear methods

Abstract The paper is devoted to the problem of observability and controllability analysis in nonlinear dynamic systems. Both continuous- and discrete-time systems described by nonlinear differential or difference equations, respectively, are considered. A new approach is developed to solve this problem whose features include (i) consideration of systems with non-differentiable nonlinearities and (ii) the use of relatively simple linear methods which may be supported by existing programming systems, e.g.,Matlab. Sufficient conditions are given for nonlinear unobservability/uncontrollability analysis. To apply these conditions, one isolates the linear part of the system which is checked to be unobservable/uncontrollable and, if the answer is positive, it is examined whether or not existing nonlinear terms violate the unobservability/uncontrollability property.

[1]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[2]  Eduardo Sontag,et al.  Linear systems with sign-observations , 1993 .

[3]  H. Nijmeijer.  Observability of autonomous discrete time non-linear systems: a geometric approach , 1982 .

[4]  Domenico D'Alessandro,et al.  Observability and Forward–Backward Observability of Discrete-Time Nonlinear Systems , 2002, Math. Control. Signals Syst..

[5]  A. Zhirabok.  Analysis of observability and controllability of nonlinear dynamic systems by linear methods , 2010 .

[6]  Philip E. Sarachik.  Controllability and observability of linear systems , 1997 .

[7]  A. Schaft.  Observability and Controllability for Smooth Nonlinear Systems , 1982 .

[8]  Kurt Schlacher,et al.  Possible Non-Integrability of Observable Space for Discrete-Time Nonlinear Control Systems , 2008 .

[9]  L. Minchenko,et al.  Controllability of Nonsmooth Discrete Systems with Delay , 2002 .

[10]  W. M. Wonham,et al.  Linear Multivariable Control , 1979 .

[11]  Eduardo Sontag,et al.  Controllability of Nonlinear Discrete-Time Systems: A Lie-Algebraic Approach , 1990, SIAM Journal on Control and Optimization.

[12]  W. Wonham,et al.  Topics in mathematical system theory , 1972, IEEE Transactions on Automatic Control.

[13]  Liang Xu,et al.  A quantitative measure of observability and controllability , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[14]  A. Krener,et al.  Nonlinear controllability and observability , 1977 .

[15]  Alexey N. Zhirabok,et al.  FAULT DIAGNOSIS IN NONLINEAR DYNAMIC SYSTEMS VIA LINEAR METHODS , 2002 .

[16]  Wei Kang,et al.  Analyzing Control Systems By Using Dynamic Optimization , 2010 .

[17]  Joel W. Burdick,et al.  Nonsmooth controllability theory and an example , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[18]  Alexey Zhirabok,et al.  Linear Methods in Observability and Controllability of Nonlinear Systems , 2010 .

[19]  Eduardo Sontag.  On the Observability of Polynomial Systems, I: Finite-Time Problems , 1979 .

[20]  Maamar Bettayeb,et al.  Controllability and Observability of Linear Discrete-Time Fractional-Order Systems , 2008, Int. J. Appl. Math. Comput. Sci..

[21]  J. Klamka.  Uncontrollability and unobservability of composite systems , 1973 .

[22]  Héctor J. Sussmann,et al.  Single-input observability of continuous-time systems , 1978, Mathematical systems theory.

[23]  J. Klamka.  On the global controllability of perturbed nonlinear systems , 1975 .

[24]  H. Nijmeijer.  Observability of autonomous discrete-time nonlinear systems: a geometric approach : Preprint , 1982 .

[25]  Jerzy Klamka,et al.  Controllability of nonlinear discrete systems , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[26]  Gerhard Jank.  CONTROLLABILITY, OBSERVABILITY AND OPTIMAL CONTROL OF CONTINUOUS-TIME 2-D SYSTEMS , 2002 .

[27]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[28]  Brian Roffel,et al.  Linear Multivariable Control , 2004 .

[29]  A. Isidori.  Nonlinear Control Systems , 1985 .

[30]  Yu Kawano,et al.  Global Observability of Discrete-Time Polynomial Systems , 2010 .

[31]  Arthur J. Krener,et al.  Measures of unobservability , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.