State dimension reduction and analysis of quantized estimation systems

• We design jointly the state dimension reduction matrix and quantizer for state estimation.
• We analyze the state estimability based on quantized innovarions.
• The quantized estimability Gramian is derived and its convergence is proved.
• It is shown that even a coarse 1-bit quantizer can preserve the estimability of the original unquantized system.

The problem of state dimension reduction and quantizer design under communication constraints is discussed for state estimation in quantized linear systems. Subject to the limited signal power, number and bandwidth of the parallel channels, a differential pulse code modulation (DPCM)-like structure is adopted to generate the quantized innovations as the transmitted signals, and the multi-level quantized Kalman filter (MLQ-KF) is used to serve as the pre- and post-filters. The dimension reduction matrix and quantizer are designed jointly under the MMSE criterion of estimation at the channel receiver. To demonstrate the validity of state estimation under the adopted framework, the state estimability based on quantized innovations is analyzed by using information theoretic method. This leads to a sufficient and necessary condition of a certain estimability Gramian matrix having full rank. The quantized Gramian is proved to converge to that of the original unquantized system when the quantization intervals turn to zero. Our work also provides an auxiliary analytic support for the estimation under 1-bit quantization. Simulations show that under communication constraints, the estimation performance is satisfactory when the designed dimension reduction method and quantizer are applied. The analytic conclusion of estimability is also verified by illustrative simulations.