Perfect regulation with cheap control for uncertain linear systems: a Riccati equation approach

A robust linear quadratic regulation problem with cheap control is studied for a class of uncertain systems with norm-bounded uncertainty or integral quadratic constraint uncertainty. A Riccati equation approach is employed as a tool to investigate the limiting case in which a scalar weighting coefficient on the control input in the quadratic cost functional approaches zero. Some results concerning the monotonicity properties and the limiting behaviour of the minimal positive-definite (stabilising) solution to the Riccati equation are given by using a well-known comparison theorem for Riccati equations. Using the limiting behaviour of the minimal positive-definite stabilising solution to the Riccati equation, it is found that perfect regulation with cheap control can be achieved if the uncertain system has a particular structure.