Closed-Loop Convex Formulation of Classical and Singular Value Loop Shaping

We show that control system design via classical loop shaping and singular value loop shaping can be formulated as a closed-loop convex problem [4, 5, 22, 15]. Consequently, loop shaping problems can be solved by e cient numerical methods. In particular, these numerical methods can always determine whether or not there exists a compensator that satis es a given set of loop shaping speci cations. Problems such as maximizing bandwidth subject to given margin and cuto speci cations can be directly solved. Moreover, any other closed-loop convex speci cations, such as limits on step-response overshoot, tracking errors, and disturbance rejection, can be simultaneously considered. These observations have two practical rami cations. First, closed-loop convex design methods can be used to synthesize compensators in a framework that is familiar to many control engineers. Second, closed-loop convex design methods can be used to aid the designer using classical loop shaping by computing absolute performance limits against which a classical design can be compared. To appear as a chapter in Advances in Control Systems, edited by C. T. Leondes, 1993. Research supported in part by NSF under ECS-85-52465 and AFOSR under 89-0228.