Absolute stability analysis of multivariable regulators through the Popov criterion

The multivariable Popov criterion is used to derive the sectors of absolute stability for two classes of regulators in both the continuous and discrete-time cases. The first class corresponds to the well known linear quadratic regulators; in the second one a feedback control law depending on the solution of a Lyapunov equation is considered. Relatively simple reasoning shows that the absolute stability analysis can be accomplished in the frequency domain. To carry this out, necessary conditions for a given matrix transfer function to represent a specific regulator are established. It is shown that the necessary conditions play the same role in the absolute stability context as the Kalman frequency-domain equality does with respect to stability margins