Control Proxy Functions for Sequential Design and Control Optimization

Abstract

Optimal system design of "smart" products requires optimization of both the artifact and its controller. When the artifact and the controller designs are independent, the system solution is straightforward through sequential optimization. When the designs are coupled, combined simultaneous optimization can produce system-optimal results, but presents significant computational and organizational complexity. This paper presents a method that produces results comparable with those found with a simultaneous solution strategy, but with the simplicity of the sequential strategy. The artifact objective function is augmented by a control proxy function (CPF), representing the artifact's ease of control. The key to successful use of this method is the selection of an appropriate CPF. Four theorems that govern the choice and evaluation of a CPF are given. Each theorem is illustrated using a simple mathematical example. Specific CPFs are then presented for particular problem formulations, and the method is applied to the optimal design and control of a micro-electrical mechanical system actuator.

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

“…It is known, in this case, that a sequential solution method will not guarantee optimality, and a method that accounts for coupling should be used. Such methods include simultaneous problem formulations, methods involving decomposition and coordination of sub-systems, and the use of a Control Proxy Function (CPF) [25][26][27]. The a priori knowledge of coupling is particularly useful in the CPF method, since it provides a basis for choosing an appropriate CPF.…”

“…A modified sequential approach, such as that proposed in [25][26][27][28], may provide system-optimal solutions; however, this method requires the specification of an appropriate Control Proxy Function (CPF). An appropriate CPF needs to capture the fundamental control limitations, and has been shown to be related to the coupling vector [25,27]. Thus, in addition to indicating which solution methods might be most appropriate for a given problem, knowledge of coupling can be used to determine what CPF may be used to implement this particular method.…”

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.…”

“…Bode (1945); Freudenberg and Looze (1985); Serön et al (1997). Indeed, co-design problems can find a great number of applications, such as the optimal design and control of aerospace crafts (Hale et al, 1985;Messac, 1998), smart buildings (Lu and Skelton, 2000;Skelton and Kim, 1992), electromechanical devices (da Silva et al, 2009;Peters et al, 2011;Reyer and Papalambros, 2002), and robotics (Ravichandran et al, 2006).…”