Conditional maximum likelihood identification for state space system

In this paper, we investigate the use of conditional maximum likelihood identification in the context of identifying one general state space system, being parametrized by one unknown parameter vector. The process of modifying the common state space system into our general form is presented, and the traditional negative log-likelihood function for identifying unknown parameter vector is constructed with only observed output variables. To combine state variables and output variables simultaneously, the conditional maximum likelihood estimate based on the conditional probability density and the total probability theorem is proposed here. Further, when the prior distribution of that parameter vector is flat, we continue to obtain the joint maximum a posteriori estimate. To maximize a negative log-likelihood function, the classical Robbins- Monro algorithm from stochastic approximation theory is applied to avoid the computation of the second-order derivative of conditional likelihood function.