Relationship between coupling and the controllability Grammian in co-design problems

Design of smart products requires optimization of both the physical device, or artifact, and its controller. While some components of coupling can be computed a priori, the existence and strength of coupling between these problems over the entire Pareto frontier currently cannot be computed until they are solved. If coupling is expected to be present, then the problem is often solved as a simultaneous, or all-in-one, optimization. This solution process is more difficult, computationally intensive, and operationally inconvenient than a sequential solution method. Consequently, knowing in advance whether coupling is weak or nonexistent is useful.In this paper, a general formulation for the control problem is given, which includes components for the speed of response, accuracy of the response, and control effort. While this general formulation does not permit a priori determination of coupling, the existence and strength of coupling can be determined for several special cases, which represent common and important control problems. Controllability is a property of the uncontrolled system and does not depend on the controller design. These relationships between coupling and controllability can be utilized in problem formulation, choice of solution method, and development of Control Proxy Functions (CPFs) for each of these problem types. The concepts developed are demonstrated via design of a positioning gantry and a MEMS actuator.