Stabilization of linear parameter-dependent systems using eigenvalue expansion with application to two time-scale systems

An eigenvalue expansion method is applied to the stability analysis and stabilization of linear parameter-dependent systems in the neighborhood of a specified system parameter. The system, at the parameter of interest, is assumed to possess distinct eigenvalues on the imaginary axis. The asymptotic stability conditions, which rely on Taylor series expansions of the continuous extensions of the critical eigenvalues with respect to system parameters, are algebraic and are given in terms of the system dynamics and the critical eigenvectors. Asymptotically stabilizing control laws are designed for the case in which the system at the specified parameter has uncontrollable eigenvalues on the imaginary axis. Application of these results yields computational algorithms for the stability analysis and stabilization design of two-time scale linear systems