Robustness in the gap metric and coprime factor perturbations for LTV systems

In this paper, we study the problem of robust stabilization for linear time-varying (LTV) systems subject to time-varying normalized coprime factor uncertainty. Operator theoretic results which generalize similar results known to hold for linear time-invariant (infinite-dimensional) systems are developed. In particular, we compute a tight upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a time-varying Hankel structure. We point to a necessary and sufficient condition which guarantees compactness of the TV Hankel operator, and in which case singular values and vectors can be used to compute the time-varying stability margin and TV controller.

[1]  Mathukumalli Vidyasagar,et al.  Robust controllers for uncertain linear multivariable systems , 1984, Autom..

[2]  Glenn Vinnicombe,et al.  Linear feedback systems and the graph topology , 2002, IEEE Trans. Autom. Control..

[3]  J. Doyle,et al.  Linear control theory with an H ∞ 0E optimality criterion , 1987 .

[4]  J. Doyle,et al.  Essentials of Robust Control , 1997 .

[5]  Cornelis Praagman,et al.  Sufficient conditions for robust BIBO stabilization: given by the gap metric , 1988 .

[6]  T. Georgiou,et al.  Optimal robustness in the gap metric , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[7]  Stephen C. Power,et al.  Commutators with the Triangular Projection and Hankel Forms on Nest Algebras , 1985 .

[8]  K. Glover,et al.  Robust stabilization of normalized coprime factor plant descriptions with H/sub infinity /-bounded uncertainty , 1989 .

[9]  Tryphon T. Georgiou,et al.  Geometric techniques for robust stabilization of linear time-varying systems , 1990, 29th IEEE Conference on Decision and Control.

[10]  Wilbur N. Dale,et al.  Stabilizability and existence of system representations for discrete-time time-varying systems , 1993 .

[11]  R. Douglas.  Banach Algebra Techniques in Operator Theory , 1972 .

[12]  Tryphon T. Georgiou,et al.  On the computation of the gap metric , 1988 .

[13]  Yanyan Li,et al.  On Robust Stabilization in the Gap Metric for LTV Systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[14]  R. B. Vinter.  SYSTEM THEORY A Hilbert Space Approach , 1984 .

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

[16]  Tryphon T. Georgiou,et al.  Robust stability of feedback systems: a geometric approach using the gap metric , 1993 .

[17]  Avraham Feintuch.  Robustness for time-varying systems , 1993, Math. Control. Signals Syst..

[18]  A. Feintuch.  Robust control theory in Hilbert space , 1997 .

[19]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[20]  Seddik M. Djouadi,et al.  On optimal performance for linear time-varying systems , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[21]  R. Schatten,et al.  Norm Ideals of Completely Continuous Operators , 1970 .

[22]  Tryphon T. Georgiou,et al.  Robust stabilization in the gap metric: controller design for distributed plants , 1992 .

[23]  W. Arveson.  Interpolation problems in nest algebras , 1975 .

[24]  A. Feintuch,et al.  The gap metric for time-varying systems , 1991 .