Co-design of linear systems using Generalized Benders Decomposition

Abstract

Design of a physical system and its controller has significant ramifications on the overall system performance. The traditional approach of first optimizing the physical design and then the controller may lead to sub-optimal solutions. This is due to the interdependence between the physical design and control parameters through the dynamic equations. Recognition of this fact paved the way for investigation into the “Co-Design” research theme wherein the overall system’s physical design and control are simultaneously optimized. In this paper, a novel approach to address the co-design problem for a class of Linear Time Invariant (LTI) dynamic systems controlled by a Linear Quadratic Regulator (LQR) feedback is presented. The considered co-design problem is formulated as a non-convex optimization problem with Algebraic Riccati Equation (ARE) constraint and convex design objective function. Using Semi-Definite Programming (SDP) duality, the ARE constraint is reduced into equivalent Bilinear Matrix Inequality (BMI) constraints. This reformulated co-design problem is solved using an iterative algorithm based on the Generalized Benders Decomposition (GBD) and Gradient Projection Method. The proposed algorithm converges to a solution which is within a specified tolerance from the nearest local minimum (in special cases global minimum) in a finite number of iterations. Necessary and sufficient conditions are developed to test minimality. Three examples are presented to show efficacy of the proposed algorithm.