Optimal Tracking Design and Performance Analysis for LTI Systems with Quantization Effects

This paper studies the tracking problem for linear time-invariant multi-input single-output (MISO) discrete-time systems with quantization effects. Logarithmic quantization laws are adopted in the systems. The tracking performance is measured by the energy of the error response between the output of the plant and the reference signal. Our goals are to design an optimal controller for the tracking problem and to find an explicit formula of the minimum tracking cost. It turns out that the optimal state feedback law can be obtained by solving a modified discrete-time Riccati equation associated with the state space model of the plant and the features of the quantization law. Furthermore, from the unique positive solution of the modified Riccati equation, we obtain an analytic expression for the minimum tracking cost in terms of the nonminimum phase zeros and the bound of quantization error. When the quantization error approaches zero, the minimum tracking cost degrades to the minimum tracking cost of the system without quantization effects, which is presented some existing works. The results obtained in this work explicitly show how is the optimal tracking performance limited by the quantization error.