Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems

The paper discusses stabilization of linear multimodal systems using a control law that selects a particular system mode. The systems considered are similar to jump linear systems. In a jump linear system the active mode is selected by a random process. Here, the controller uses a piecewise quadratic Lyapunov function to select the system mode to activate. The paper relates existence of a stabilizing control law to the solution of a pair of coupled Lyapunov equations. Solution of these equations is discussed and related to the location of the eigenvalues of certain matrix operators derived from the component system matrices and the coupling parameter. Relationships to earlier work by the authors (1994) is discussed