Minimum rate coding for LTI systems over noiseless channels

This paper studies rate requirements for state estimation in linear time-invariant (LTI) systems where the controller and the plant are connected via a noiseless channel with limited capacity. Using information theoretic arguments, we obtain first for scalar systems, and subsequently for multidimensional systems, lower bounds on the data rates required for state estimation under three different stability criteria, namely monotonic boundedness of entropy, asymptotic stability of distortion, and support size stability. Further, the minimum data rate achievable by any source-encoder is computed under each of these criteria, and the best rate achievable with quantization is shown to be in agreement with the information-theoretic bounds in some specific cases (such as if the system coefficient is an integer or if the criterion is an asymptotic one). Existence of optimal variable-length and fixed-length quantizers are studied and optimal quantizers are constructed under each of these criteria. One observation is that, the uniform quantizer is, in addition to being simple, efficient in linear control systems.