Control of linear systems subject to input constraints: a polynomial approach

A polynomial approach is pursued for locally stabilizing discrete-time linear systems subject to input constraints. Using the Youla-Kucera parametrization and geometric properties of polyhedra and ellipsoids, the problem of simultaneously deriving a stabilizing controller and the corresponding stability region is cast into standard convex optimization problems solved by linear, second-order cone and semidefinite programming. Key topics are touched on such as stabilization of multi-input multi-output plants or maximization of the size of the stability domain. Readily implementable algorithms are described.

[1]  H. Kwakernaak,et al.  Recent progress in polynomial methods and Polynomial Toolbox for Matlab version 2.0 , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[2]  M. Sznaier.  A set induced norm approach to the robust control of constrained systems , 1993 .

[3]  Jean-Claude Hennet,et al.  A class of invariant regulators for the discrete-time linear constrained regulation problem , 1991, Autom..

[4]  Sophie Tarbouriech,et al.  Piecewise-Linear Robust Control of Systems with Input Constraints , 1999, Eur. J. Control.

[5]  J. Hennet,et al.  On invariant polyhedra of continuous-time linear systems , 1993, IEEE Trans. Autom. Control..

[6]  V. Kučera.  Stability of Discrete Linear Feedback Systems , 1975 .

[7]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[8]  Sophie Tarbouriech,et al.  Control of Uncertain Systems with Bounded Inputs , 1997 .

[9]  M. Sznaier.  A set induced norm approach to the robust control of constrained systems , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[10]  Eugênio B. Castelan,et al.  Eigenstructure assignment for state constrained linear continuous time systems , 1992, Autom..

[11]  K. T. Tan,et al.  Linear systems with state and control constraints: the theory and application of maximal output admissible sets , 1991 .

[12]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[13]  Dante C. Youla,et al.  Modern Wiener-Hopf Design of Optimal Controllers. Part I , 1976 .

[14]  Zongli Lin,et al.  Semi-global Exponential Stabilization of Linear Systems Subject to \input Saturation" via Linear Feedbacks , 1993 .

[15]  J. Hennet.  Une extension du lemme de Farkas et son application au problème de régulation linéaire sous contrainte , 1989 .

[16]  Stephen Boyd,et al.  A new CAD method and associated architectures for linear controllers , 1988 .

[17]  Stephen P. Boyd,et al.  A New CAD Method and Associated Architectures for Linear Controllers , 1987, 1987 American Control Conference.

[18]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[19]  Komei Fukuda,et al.  Exact volume computation for polytopes: a practical study , 1996 .

[20]  Thomas Kailath,et al.  Linear Systems , 1980 .

[21]  Vladimír Kučera,et al.  Analysis and design of discrete linear control systems , 1991 .

[22]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

[23]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

[24]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[25]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[26]  Sophie Tarbouriech,et al.  Output feedback robust stabilization of uncertain linear systems with saturating controls: an LMI approach , 1999, IEEE Trans. Autom. Control..

[27]  Robert J. Vanderbei,et al.  Linear Programming: Foundations and Extensions , 1998, Kluwer international series in operations research and management service.

[28]  R. Suárez,et al.  Linear systems with bounded inputs : global stabilization with eigenvalue placement , 1997 .

[29]  D. Henrion,et al.  Stabilité des systèmes linéaires incertains à commande contrainte , 1999 .

[30]  J. Hennet,et al.  Feedback control of linear discrete-time systems under state and control constraints , 1988 .

[31]  Sophie Tarbouriech,et al.  Output feedback robust stabilization of uncertain linear systems with saturating controls , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[32]  Eugênio B. Castelan,et al.  A reduced-order framework applied to linear systems with constrained controls , 1996, IEEE Trans. Autom. Control..

[33]  Mathukumalli Vidyasagar,et al.  Control System Synthesis , 1985 .

[34]  D. Bernstein,et al.  A chronological bibliography on saturating actuators , 1995 .

[35]  G. Ziegler,et al.  Polytopes : combinatorics and computation , 2000 .

[36]  Eduardo Sontag,et al.  A general result on the stabilization of linear systems using bounded controls , 1994, IEEE Trans. Autom. Control..

[37]  C. Burgat,et al.  Regulator problem for linear discrete-time systems with non-symmetrical constrained control , 1988 .

[38]  Manfred Morari,et al.  A unified framework for the study of anti-windup designs , 1994, Autom..

[39]  Sophie Tarbouriech,et al.  Polyhedral regions of local stability for linear discrete-time systems with saturating controls , 1999, IEEE Trans. Autom. Control..

[40]  S. Tarbouriech,et al.  Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[41]  Franco Blanchini,et al.  Set invariance in control , 1999, Autom..

[42]  M. Sznaier,et al.  Suboptimal norm based robust control of constrained systems with an H/sub infinity / cost , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[43]  Vladimír Kucera,et al.  Discrete linear control: The polynomial equation approach , 1985, IEEE Transactions on Systems, Man, and Cybernetics.