Co-Design of Time-Invariant Dynamical Systems

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. Co-design involves simultaneous optimization of the design and the control variables with respect to certain structural property as constraint. The structural property may be in the form of stability, observability or controllability leading to different types of co-design problems. Co-design optimization problems are non-convex optimization problems involving bilinear matrix inequality (BMI) constraints and are NP-hard in general. In this dissertation, four interrelated research tasks in the area of co-design are undertaken. In the first research task, a theoretical and computational framework is developed to co-design a class of linear time invariant (LTI) dynamical systems. A novel solution procedure based on an iterative combination of generalized Benders decomposition and gradient projection method is developed guaranteeing convergence to a solution in a finite number of iterations which is within a tolerance bound from the nearest local/global minimum. In the second research task, the sparse and structured static feedback design problem is modeled as a co-design problem. A formulation based on the alternating direction method of multipliers is used to solve the sparse feedback design problem which has given robustness as a constraint. In the third research task, the optimal actuator placement problem is formulated as a co-design problem. The actuator positions are modeled as $0/1-$binary design variables and result in a mixed integer nonlinear programming (MINLP) problem. In the fourth research task, a heuristic procedure to place sensors and design observer is developed for a class of Lipschitz nonlinear systems. The procedure is based on the relation between Lipschitz constant, sensor locations and observer gain. The vast and diverse application potential of co-design across all engineering branches is the primary motivation and relevance of the research work carried out in this dissertation.