Rational L1 Suboptimal Compensators for Continuous-Time Systems

The persistent disturbance rejection problem (L¹ Optimal Control) for continuous time-systems leads to non-rational compensators, even for SISO systems [1-3]. As noted in [2], the difficulty of physically implementing these controllers suggest that the most significant applications of the continuous time L¹ theory is to furnish bounds for the achievable performance of discrete-time controllers. However, at the present time, there are no theoretical results relating the optimal l¹ norm of the discrete time system with the actual performance obtained when the controller is used in the continuous-time system. In this paper we use the theory of positively invariant sets to provide a design procedure, based upon the use of the discrete Euler approximating system, for suboptimal rational L¹ controllers. The main results of the paper show that i) the L¹ norm of the resulting continuous-time system is bounded above by the l¹ norm of the discrete-time counterpart and ii) the proposed rational compensators yield L¹ cost arbitrarily close to the optimum, even in cases where the design procedure proposed in [2] fails due to the existence of plant zeros on the stability boundary.