Necessary and Sufficient Conditions for Stabilizability subject to Quadratic Invariance

In this paper we deal with the problem of stabilizing linear, time-invariant plants using feedback control configurations that are subject to sparsity constraints. Recent results show that given a strongly stabilizable plant, the class of all stabilizing controllers that satisfy certain given sparsity constraints admits a convex representation via Zames's Q-parametrization. More precisely, if the pre-specified sparsity constraints imposed on the controller are quadratically invariant with respect to the plant, then such a convex representation is guaranteed to exist. The most useful feature of the aforementioned results is that the sparsity constraints on the controller can be recast as convex constraints on the Q-parameter, which makes this approach suitable for optimal controller design (in the ℋ2 sense) using numerical tools readily available from the classical, centralized optimal ℋ2 synthesis. All these procedures rely crucially on the fact that some stabilizing controller that verifies the imposed sparsity constraints is a priori known, while design procedures for such a controller to initialize the aforementioned optimization schemes are not yet available. This paper provides necessary and sufficient conditions for such a plant to be stabilizable with a controller having the given sparsity pattern. These conditions are formulated in terms of the existence of a doubly coprime factorization of the plant with additional sparsity constraints on certain factors. We show that the computation of such a factorization is equivalent to solving an exact model-matching problem. We also give the parametrization of the set of all decentralized stabilizing controllers by imposing additional constraints on the Youla parameter. These constraints are for the Youla parameter to lie in the set of all stable transfer function matrices belonging to a certain linear subspace.

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

[2]  P. Khargonekar.  Control System Synthesis: A Factorization Approach (M. Vidyasagar) , 1987 .

[3]  Petros G. Voulgaris.  Control under a hierarchical decision making structure , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[4]  C. Jacobson,et al.  A connection between state-space and doubly coprime fractional representations , 1984 .

[5]  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).

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

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

[8]  Paul Van Dooren,et al.  A novel numerical method for exact model matching problem with stability , 2006, Autom..

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

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

[11]  Murti V. Salapaka,et al.  Architectures for Distributed Controller With Sub-Controller Communication Uncertainty , 2010, IEEE Transactions on Automatic Control.

[12]  Nuno C. Martins,et al.  On the closest quadratically invariant constraint , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[13]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

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

[15]  Petros G. Voulgaris,et al.  Control of nested systems , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[16]  Petros G. Voulgaris,et al.  A convex characterization of distributed control problems in spatially invariant systems with communication constraints , 2005, Syst. Control. Lett..

[17]  Jr. G. Forney,et al.  Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems , 1975 .

[18]  Cristian Oară,et al.  Minimal indices cancellation and rank revealing factorizations for rational matrix functions , 2009 .

[19]  Sanjay Lall,et al.  A Characterization of Convex Problems in Decentralized Control$^ast$ , 2005, IEEE Transactions on Automatic Control.

[20]  Panos J. Antsaklis,et al.  On stable solutions of the one- and two-sided model matching problems , 1989 .

[21]  Alan J. Laub,et al.  Matrix analysis - for scientists and engineers , 2004 .