Optimal Tracking Performance of a Linear System with a Quantized Control Input

This paper studies the optimal tracking performance of a linear system with a quantized control signal. The plant under consideration is linear time-invariant (LTI) stable 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. To achieve asymptotical tracking, a new quantization scheme is proposed. This 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. The best attainable tracking performance is obtained, in terms of characteristics of the given plant and the quantizer. This result gives a clear answer to a fundamental question in networked control system: How large quantization error is allowed for a step tracking problem with a given tracking error energy level in an averaged sense?