Set invariance under output feedback: a set-dynamics approach

The article offers a framework to define and analyse the notion of invariance with respect to output feedback under non-parametric disturbances. The motivation is that the straightforward generalisation of the definition of invariance under state feedback to the output feedback framework, namely relying only on feedback from the output, does not yield a useful notion. Our model follows standard feedback invariance considerations with, however, a crucial modification that is needed when only an observation of the state, rather than the state itself, is available. The model incorporates information gathered by the controller during the process; this is in similarity with the observer-based dynamics model; however our framework represents the information within a set dynamics. The evolution of the resulting information sets determines invariant sets and attractors of the state dynamics. The framework in this article is discrete-time control systems. We offer an analysis of the notion with results on existence of, and convergence to, output feedback invariant sets; illustrative examples related to potentially practical feedback rules are exhibited.

[1]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[2]  David Q. Mayne,et al.  Robust output feedback model predictive control of constrained linear systems , 2006, Autom..

[3]  Panos J. Antsaklis,et al.  Set-valued observer design for a class of uncertain linear systems with persistent disturbance and measurement noise , 2003 .

[4]  H. Witsenhausen.  A minimax control problem for sampled linear systems , 1968 .

[5]  A. B. Kurzhanskii,et al.  The problem of measurement feedback control , 2004 .

[6]  S. Gutman.  Synthesis of min-max strategies , 1985 .

[7]  Tamer Başar,et al.  H1-Optimal Control and Related Minimax Design Problems , 1995 .

[8]  Jeff S. Shamma,et al.  Set-valued observers and optimal disturbance rejection , 1999, IEEE Trans. Autom. Control..

[9]  T. Basar,et al.  H∞-0ptimal Control and Related Minimax Design Problems: A Dynamic Game Approach , 1996, IEEE Trans. Autom. Control..

[10]  D. Mayne,et al.  Set Robust Control Invariance for Linear Discrete Time Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[11]  Christopher I. Byrnes,et al.  Directions in Mathematical Systems : Theory and Optimization , 2003 .

[12]  Alexander B. Kurzhanski,et al.  The Principle of Optimality in Measurement Feedback Control for Linear Systems , 2003 .

[13]  H. Witsenhausen.  Sets of possible states of linear systems given perturbed observations , 1968 .

[14]  D. Bertsekas,et al.  On the minimax reachability of target sets and target tubes , 1971 .

[15]  D. Bertsekas,et al.  Recursive state estimation for a set-membership description of uncertainty , 1971 .

[16]  D. Bertsekas,et al.  Sufficiently informative functions and the minimax feedback control of uncertain dynamic systems , 1973 .

[17]  Vladimir M. Veliov,et al.  Optimal control of discrete-time uncertain systems with imperfect measurement , 2002, IEEE Trans. Autom. Control..

[18]  B. Bank,et al.  Non-Linear Parametric Optimization , 1983 .

[19]  Zvi Artstein,et al.  Feedback and invariance under uncertainty via set-iterates , 2008, Autom..

[20]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[21]  David Q. Mayne,et al.  Invariant approximations of the minimal robust positively Invariant set , 2005, IEEE Transactions on Automatic Control.

[22]  F. Schweppe,et al.  Control of linear dynamic systems with set constrained disturbances , 1971 .

[23]  Elmer G. Gilbert,et al.  The minimal disturbance invariant set: Outer approximations via its partial sums , 2006, Autom..

[24]  Mato Baotic,et al.  Multi-Parametric Toolbox (MPT) , 2004, HSCC.

[25]  W. Schmitendorf,et al.  Minmax control of systems with uncertainty in the initial state and in the state equations , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[26]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[27]  F. Schweppe.  Recursive state estimation: Unknown but bounded errors and system inputs , 1967 .

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

[29]  M. Corless,et al.  Adaptive control of systems containing uncertain functions and unknown functions with uncertain bounds , 1983 .

[30]  Sasa V. Rakovic,et al.  Minkowski algebra and Banach Contraction Principle in set invariance for linear discrete time systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[31]  E. Gilbert,et al.  Theory and computation of disturbance invariant sets for discrete-time linear systems , 1998 .