Joint identification of system parameter and noise parameters in quantized systems

This paper investigates the joint identification problem of unknown system parameter and noise parameters in quantized systems when the noises involved are Gaussian with unknown variance and mean value. Under such noises, previous investigations show that the unknown system parameter and noise parameters are not jointly identifiable in the single-threshold quantizer case. The joint identifiability in the multi-threshold quantizer case still remains an open problem. This paper proves that the unknown system parameter, the noise variance and the mean value are jointly identifiable if and only if there are at least two thresholds. Then, a decomposition-recombination identification algorithm is proposed to jointly identify the unknown system parameter and noise parameters. Firstly, a technique is designed to convert the identification problem with unknown noise parameters into an extended parameter identification problem with standard Gaussian noises. Secondly, the extended parameter is identified by a stochastic approximation method for quantized systems. For the effectiveness, this paper obtains the strong consistency and the $L^p$ convergence for the algorithm under non-persistently exciting inputs and without any a priori knowledge on the range of the unknown system parameter. The almost sure convergence rate is also obtained. Furthermore, when the mean value is known, the unknown system parameter and noise variance can be jointly identified under weaker conditions on the inputs and the quantizer. Finally, the effectiveness of the proposed algorithm is demonstrated by simulation.