A Characterization of Convex Problems in Decentralized Control$^ast$

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.

[1]  E. C. Titchmarsh Introduction to the Theory of Fourier Integrals , 1938 .

[2]  J. Partington,et al.  Introduction to Functional Analysis , 1981, The Mathematical Gazette.

[3]  R. Radner,et al.  Team Decision Problems , 1962 .

[4]  H. Witsenhausen A Counterexample in Stochastic Optimum Control , 1968 .

[5]  H. Witsenhausen Separation of estimation and control for discrete time systems , 1971 .

[6]  R. Saeks,et al.  The analysis of feedback systems , 1972 .

[7]  Y. Ho,et al.  Team decision theory and information structures in optimal control problems--Part II , 1972 .

[8]  D.L. Elliott,et al.  Feedback systems: Input-output properties , 1976, Proceedings of the IEEE.

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

[10]  C. Desoer,et al.  On the stabilization of nonlinear systems , 1982, 1982 21st IEEE Conference on Decision and Control.

[11]  Christos Papadimitriou,et al.  Intractable problems in control theory , 1985, 1985 24th IEEE Conference on Decision and Control.

[12]  B. Francis,et al.  A Course in H Control Theory , 1987 .

[13]  Stephen P. Boyd,et al.  Linear controller design: limits of performance , 1991 .

[14]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[15]  J. Speyer,et al.  Centralized and decentralized solutions of the linear-exponential-Gaussian problem , 1994, IEEE Trans. Autom. Control..

[16]  Mohammad Aldeen,et al.  Stabilization of decentralized control systems , 1997 .

[17]  Petros G. Voulgaris Control under structural constraints: An input-output approach , 1998, Robustness in Identification and Control.

[18]  S. Mitter,et al.  Information and control: Witsenhausen revisited , 1999 .

[19]  John N. Tsitsiklis,et al.  A survey of computational complexity results in systems and control , 2000, Autom..

[20]  Srdjan S. Stankovic,et al.  Decentralized overlapping control of a platoon of vehicles , 2000, IEEE Trans. Control. Syst. Technol..

[21]  T. Başar Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses , 2001 .

[22]  P. Voulgaris A convex characterization of classes of problems in control with specific interaction and communication structures , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[23]  Petros G. Voulgaris,et al.  OPTIMAL DISTRIBUTED CONTROL WITH DISTRIBUTED DELAYED MEASUREMENTS , 2002 .

[24]  S. Lall,et al.  Decentralized control information structures preserved under feedback , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[25]  Guang-Hong Yang,et al.  Optimal symmetric H2 controllers for systems with collocated sensors and actuators , 2002, IEEE Trans. Autom. Control..

[26]  S. Lall,et al.  Decentralized control of unstable systems and quadratically invariant information constraints , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[27]  S. Lall,et al.  Decentralized control subject to communication and propagation delays , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[28]  Murti V. Salapaka,et al.  Structured optimal and robust control with multiple criteria: a convex solution , 2004, IEEE Transactions on Automatic Control.

[29]  S. Lall,et al.  On computation of optimal controllers subject to quadratically invariant sparsity constraints , 2004, Proceedings of the 2004 American Control Conference.

[30]  M. Rotkowitz Tractable problems in optimal decentralized control , 2005 .