ABSTRACT
A class of motion control problems can be formulated as a linear quadratic optimization problem, but with a series of structural constraints imposed on the composite gain matrix. Since the standard formulae for the classical LQR problem can only handle the case when there is no structural constraint on the gain matrix, there is no standard closed-form solution for the constrained linear quadratic optimization problem. In this chapter, theoretical results are given to calculate the gradient of the objective function with respect to the gain matrix for the standard LQR problem in both finite and infinite horizons. To cater to the constraints in the composite gain matrix, the projection approach is introduced, and the determination of the projection gradient matrix onto the constrained hyperplane is summarized. A gradient-based optimization algorithm is presented based on the direct computation of the projection gradient matrix and line search of the optimal step size. As a case study, the proposed optimization algorithm is applied to integrally design the flexure joints and the controllers in a DHG and the effectiveness of the proposed optimization algorithm is successfully validated by comparative experiments.
