Structural stability of linear dynamically varying (LDV) controllers

LDV systems are linear systems with parameters varying according to a nonlinear dynamical system. This paper examines the robust stability of such systems in the face of perturbations of the nonlinear system. Three classes of perturbations are examined: differentiable functions, Lipschitz continuous functions and continuous functions. It is found that in the first two cases the system remains stable. Whereas, if the perturbations are among continuous functions, the closed-loop may not be asymptotically stable, but, instead, is asymptotically bounded with the diameter of the residual set bounded by a function that is continuous in the size of the perturbation. It is also shown that in the case of differential perturbations, the resulting optimal LDV controller is continuous in the size of the perturbation. An example is presented that illustrates the continuity of the variation of the controller in the case of a nonstructurally stable dynamical system. (C) 2001 Elsevier Science B.V. All rights reserved.