Rational ℒ1 suboptimal compensators for continuous-time systems

The persistent disturbance rejection problem (L¹ optimal control) for continuous time-systems leads to nonrational compensators, even for single input/single output systems. As noted in Dahleh and Pearson (1987), the difficulty of physically implementing these controllers suggests that the most significant application of the continuous time L¹ theory is to furnish achievable performance bounds for rational controllers. In this paper the authors 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 with a guaranteed cost. The main results of the paper show that (i) the L ¹ norm of a continuous-time system is bounded above by the l ¹ norm of an auxiliary discrete-time system obtained by using the transformation z=1+rs and (ii) the proposed rational compensators yield L¹ cost arbitrarily close to the optimum, even in cases where the design procedure proposed in the above mentioned paper fails due to the existence of plant zeros on the stability boundary