Strictly Positive Realness-Based Feedback Gain Design Under Imperfect Input-Output Feedback Linearization in Prioritized Control Problem

The prioritized control problem is a process to find a control strategy for a dynamical system with prioritized multiple outputs, so that it can operate outside its nonsingular domain. Singularity typically leads to imperfect inversion in the prioritized control problem, which in turn results in imperfect input-output feedback linearization. In this paper, we propose a method based on the Kalman-Yakubovich-Popov lemma that compensates nonlinear feedback terms caused by the imperfect inversion of the prioritized control problem. In order to realize this idea, we prove existence of a feedback gain matrix that gives a strictly positive real transfer function whose output matrix is identical to the feedback gain matrix. Our proof is constructive so that a set of such matrices can be found. Also, we provide a numerical approach that gives a larger set of feedback gain matrices and validate the result with numerical examples.