Robust control by two Lyapunov functions

A new method of designing a robust control law is proposed for a general class of non-linear or linear systems with bounded uncertainties. The method uses the property that the Lyapunov function is not unique for a stable or stabilizable system. It is shown that the proposed control law normally guarantees the stability of the system if there are two Lyapunov functions whose null sets have a trivial intersection. The null set of a Lyapunov function is defined to be the set in state space in which the product of the transpose of the system input matrix and the gradient of the Lyapunov function is equal to zero. The robust control results impose no restriction on the structure and size of the input-unrelated uncertainties. Moreover, it is shown that asymptotic stabilization of the nominal system is not necessarily required in this method.

[1]  P. Hartman.  Ordinary Differential Equations , 1965 .

[2]  S. Gutman.  Uncertain dynamical systems--A Lyapunov min-max approach , 1979 .

[3]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[4]  M. Corless,et al.  Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems , 1981 .

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

[6]  M. Corless,et al.  A new class of stabilizing controllers for uncertain dynamical systems , 1982, 1982 21st IEEE Conference on Decision and Control.

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

[8]  Y.H. Chen.  Reducing the Measure of Mismatch for Uncertain Dynamical Systems , 1986, 1986 American Control Conference.

[9]  Ian R. Petersen,et al.  A riccati equation approach to the stabilization of uncertain linear systems , 1986, Autom..

[10]  Harold Stalford,et al.  Robust control of uncertain systems in the absence of matching conditions: Scalar input , 1987, 26th IEEE Conference on Decision and Control.

[11]  C. Hollot,et al.  lth-Step Lyapunov Min-Max Controllers: Stabilizing Discrete-Time Systems under Real Parameter Variations , 1987, 1987 American Control Conference.

[12]  G. Leitmann,et al.  Robustness of uncertain systems in the absence of matching assumptions , 1987 .

[13]  W. Schmitendorf.  Designing stabilizing controllers for uncertain systems using the Riccati equation approach , 1988 .

[14]  I. Petersen.  Stabilization of an uncertain linear system in which uncertain parameters enter into the input matrix , 1988 .