Almost sure stability of discrete-time linear systems subject to nested quantization under erasure channels of limited data rate

This paper investigates almost sure stability of discrete-time linear systems over packet erasure forward channel (the channel from the controller to the actuator) and packet erasure backward channel (the channel from the sensor to the controller) of limited data rate, which involves input quantization (for the controller output) and output quantization (for the plant output) with limited data rate. The interaction of output quantization and input quantization may lead to the nested quantizations, which complicates the system design immensely. Moreover, only by the plant output or the controller output received by their quantizers, the quantizers can not decide the dynamics of quantization region of the plant output and the controller output under packet erasure forward and backward channels, so some dynamical systems are constructed to represent the dynamics of quantization region for quantizing the outputs of the system and the controller. Specially, the state vector of the dynamical systems composes of not only the real outputs of the system and the controller but also their quantized values at last time. By these dynamical systems the considered system is modeled by a stochastic difference system, and the considered problem is transformed into the almost sure stability problem of the stochastic difference system. With spherical polar coordinate quantizer and the constructed systems, a quantization method is presented for solving nested quantization with limited data rate and achieving almost sure stability of the systems under the packet erasure channels. Numerical examples show the effectiveness of the results.