Multi-Objective LQG Design with Primal-Dual Method

The goal of this paper is to study a multi-objective linear quadratic Gaussian (LQG) control problem. In particular, we consider an optimal control problem minimizing a quadratic cost over a finite time horizon for linear stochastic systems subject to control energy constraints. To solve the problem, we suggest an efficient bisection line search algorithm which is computationally efficient compared to other approaches such as the semidefinite programming. The main idea is to use the Lagrangian function and Karush-Kuhn-Tucker (KKT) optimality conditions to solve the constrained optimization problem. The Lagrange multiplier is searched using the bisection line search. Numerical examples are given to demonstrate the effectiveness of the proposed methods.