A Convex Approach Solving Simultaneous Mechanical Structure and Control System Design Problems With Multiple Closed-loop Performance Specifications

In this paper, a new integrated design method, referred to as the extended multiple simultaneous specification (EMSS) method, is proposed to solve simultaneous mechanical structure and control system design problems in which a set of $n$ multiple closed-loop performance specifications must be simultaneously satisfied. To utilize this approach, all closed-loop performance specifications considered must have the property that they are convex with respect to the closed-loop system transfer matrix. With the proposed approach, a simply implemented two-stage design approach is used to determine a set of open-loop mechanical system design parameters and a closed-loop controller which simultaneously satisfies a set of $n$ closed-loop performance specifications. In the first stage, for each closed-loop performance specification, one “sample system,” i.e., the closed-loop system with one set of mechanical design parameters with a closed-loop controller chosen from the set of all linear controllers, is determined by trial and error, such that the specification is satisfied. In the second stage, the transfer matrix of the final system, which satisfies all $n$ performance specifications, is determined through the convex combination of the transfer matrices of $n$ sample systems. A linear programming problem is solved to give the combination vector for this convex combination. With the closed-loop transfer matrix given, the mechanical design parameters, the closed-loop controller structure and its gains, are solved algebraically. In this paper, we establish conditions for the existence of a solution to this integrated design problem as well as prove that the EMSS approach retains the stability properties of the sample systems. Experimental results of the EMSS method, carried out on a linear positioning system are given, verifying the effectiveness of the proposed method. We note that the proposed EMSS method works well when the number of design parameters to be determined is small. Further, the proposed EMSS method also has some utility as a controller design method, to determine a closed-loop controller that satisfies a set of $n$ multiple closed-loop performance specifications, given a fixed mechanical system structure.