Optimal tracking design for a linear system with a quantized control input

This paper studies the optimal tracking performance of a unstable linear system with a quantized control signal. The plant under consideration is discrete-time linear time-invariant (LTI) unstable and the reference signal in the tracking problem is a step signal. The tracking performance is measured by the energy of the error between the output of the plant and the reference. Two degree of freedom(2DOF) is adopted. To achieve asymptotical tracking, the quantization scheme includes two parts: one is quantized steady-state control signal transmitted to the plant at initial time and the other is a logarithmic quantizer which quantizes the error between the control signal and its steady-state value. The logarithmic quantization error is assumed to be a product of the original signal and a white noise with a uniform distribution over a given range. By using dynamic programming approach, discrete-time algebraic Riccati equation (ARE) is obtained. The best attainable tracking performance is obtained, in terms of the space equation of given system and the unique solution of the discrete-time ARE. At last, the minimum quantization density which guarantees the closed-loop system in tracking problem via a two degree of freedom controller is quadratic stable and has optimal tracking performance is obtained.