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

Abstract

a b s t r a c tDesign 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.

An iterative approach to the optimal co-design of linear control systems

“…Papalambros and Wilde (2000) study the existence of an optimal solution of a system design problem using the Weierstrass Theorem, which requires the compactness of the feasible set. Some researchers have been endeavoring in developing necessary conditions for local optimal solutions (Alyaqout et al, 2007;Fathy et al, 2001;Patil et al, 2012;Peters et al, 2010Peters et al, , 2011. In terms of how to compute an optimal solution, one of the earliest studies of Problem 1 can be found in (Salama et al, 1988), where a gradient method was developed to numerically search for the optimal solution.…”

“…An earlier approach is to numerically and simultaneously optimize the parameters of both the plant and the control policy using nonlinear programming (NLP) strategies (Onoda and Haftka, 1987;Salama et al, 1988). Necessary optimality conditions and the coupling of co-design problems have been studied (Alyaqout et al, 2007;Fathy et al, 2001;Patil et al, 2010Patil et al, , 2012Peters et al, 2010Peters et al, , 2011. In the aforementioned prior art, the non-convex co-design problem is tackled by direct transcript to NLP solvers, which suffer some well-known weaknesses, for instance sensitivity to initial guesses, no convergence guarantee etc.…”

A new formulation for co-design of linear systems with system matrices having affine design variables

“…However, the algorithm guaranteed only local optimality of the computed solution. In a recent work [9], a relationship between coupling and controllability grammian of the system was developed. Coupling is an interdependence between the plant and its controller.…”

“…It may be a vector based on optimality conditions or a matrix based on global sensitivity equations. The work [9] demonstrated the methodology wherein certain co-design problems could be decoupled into separate design and control sub-problems based on coupling. A sequential approach to solve certain LQR based co-design problems was proposed [10].…”

“…The past work mainly considers co-design problems which are either convexified [7], [8] or are contingent on specific system properties or the initial state of the system [9], [10]. The resultant optimization process also involves dynamic constraints which enhances the complexity of the co-design problem [5].…”