Further results on rational approximations of ℒ1 optimal controllers

The continuous-time persistent disturbance rejection problem (L¹ optimal control) leads to nonrational compensators, even for SISO systems. The difficulty of physically implementing these controllers suggests that the most significant applications of the continuous time L¹ theory is to furnish bounds for the achievable performance of the plant. Previously, two different rational approximations of the optimal L¹ controller were developed by Ohta et al. (1992) and by Blanchini and Sznaier (1994). In this paper the authors explore the connections between these two approximations. The main result of the paper shows that both approximations belong to the same subset Ω_T of the set of rational approximations, and that the method proposed by Blanchini and Sznaier gives the best approximation, in the sense of providing the tightest upper bound of the approximation error, among the elements of this subset. Additionally, the authors exploit the structure of the dual to the L¹ optimal control problem to obtain rational approximations with approximation error smaller than a prespecified bound