The constrained Lyapunov problem and its application to robust output feedback stabilization

Given a dynamical system whose description includes time-varying uncertain parameters, it is often desirable to design an output feedback controller leading to asymptotic stability of a given equilibrium point. When designing such a controller, one may consider static (i.e., memoryless) or dynamic compensation. In this paper, we show that solvability of various output feedback design problems is implied by existence of a solution to a certain constrained Lyapunov problem (CLP). The CLP can be stated in purely algebraic terms. Once the CLP is described, we provide necessary and sufficient conditions for its solution to exist. Subsequently, we consider application of the CLP to a number of robust stabilization problems involving static output feedback and observer-based feedback.

[1]  Uri Shaked,et al.  On the stability robustness of the continuous-time LQG optimal control , 1985 .

[2]  M. Corless,et al.  Output feedback stabilization of uncertain dynamical systems , 1984, The 23rd IEEE Conference on Decision and Control.

[3]  A. Jameson,et al.  Conditions for nonnegativeness of partitioned matrices , 1972 .

[4]  Theodore Djaferis Robust observers and regulation for systems with parameters , 1984, The 23rd IEEE Conference on Decision and Control.

[5]  S. Bhattacharyya Parameter invariant observers , 1980 .

[6]  G. Leitmann,et al.  Stabilizing feedback control for dynamical systems with bounded uncertainty , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[7]  B. Ross Barmish,et al.  Robustness of Luenberger Observers: Linear Systems Stabilized via Nonlinear Control , 1984 .

[8]  D. Luenberger An introduction to observers , 1971 .

[9]  S. Bhattacharyya The structure of robust observers , 1976 .

[10]  Raymond T. Stefani Reducing the sensitivity to parameter variations of a minimum-order reduced-order observer , 1982 .

[11]  G. Leitmann Guaranteed Asymptotic Stability for Some Linear Systems With Bounded Uncertainties , 1979 .

[12]  B. Barmish Necessary and sufficient conditions for quadratic stabilizability of an uncertain system , 1985 .

[13]  A. Andry,et al.  Modalized observers , 1984 .

[14]  George Leitmann,et al.  On ultimate boundedness control of uncertain systems in the absence of matching assumptions , 1982 .

[15]  B. Barmish,et al.  Dynamic compensation in robust stabilization using observers , 1984, The 23rd IEEE Conference on Decision and Control.

[16]  B. Barmish,et al.  On guaranteed stability of uncertain linear systems via linear control , 1981 .

[17]  B. Barmish Stabilization of uncertain systems via linear control , 1983 .

[18]  Peter Dorato,et al.  Observer feedback for uncertain systems , 1974, CDC 1974.

[19]  B. R. Barmish,et al.  Robust asymptotic tracking for linear systems with unknown parameters , 1986, Autom..