Controller design for optimal tracking response in discrete-time systems

The problem of designing a controller, which results in a closed-loop system response with optimal time-domain characteristics, is considered. In the approach presented in this paper, the controller order is fixed (higher than pole-placement order) and we seek a controller that results in closed-loop poles at certain desired and pre-specified locations; while at the same time the output tracks the reference input in an optimal way. The optimality is measured by requiring certain norms on the error sequence – between the reference and output signals – to be minimum. Several norms are used. First, l2-norm is used and the optimal solution is computed in one step of calculations. Second, l∞-norm (i.e. minimal overshot) is considered and the solution is obtained by solving a constrained affine minimax optimization problem. Third, the l1-norm (which corresponds to the integral absolute error-(IAE)-criterion) is used and linear programming techniques are utilized to solve the problem. The important case of finite settling time (i.e. deadbeat response) is studied as a special case. Examples that illustrate the different design algorithms and demonstrate their feasibility are presented.