Research Of The Identification Methods For Markov Jump Systems

Due to the intrinsic complexity of the controlled process and the interaction of the internal mechanism and external environment,the processes in engineering practice of control generally exhibit nonlinearity.Markov jump systems can be used for the modeling of a class of nonlinear processes with multiple-model switching property.On the other hand,the unavailability of information would be frequently encountered in practical industrial processes because of the variation of the operation condition,the failure of the sensors,the transmission of the network,and other disturbances.Meanwhile,some information of the processes is difficult to be obtained.When a part of the information of a process is missing,we need to infer the unobserved information based on the observed data to estimate the parameters of a process or identify its model.This paper investigates the parameter identification problems of the Markov jump systems when part of the information is missed or unavailable.The paper is mainly focused on the following topics:(1)The identification problems of the Markov jump autoregressive exogenous(MJARX)systems with unknown time-delay are discussed.Markov jump systems are a special kind of hybrid systems,in which the continuous dynamics and the discrete modes coexist.The discrete modes evolve according to a Markov chain.The unknown time-delay,which often appears in practice,will influence the modeling of the systems and degenerate the performance of the identification methods.During the identification of a system,the unknown time-delay should be carefully considered.In this paper,the stochastic modes of the Markov jump system and the unknown time-delay are treated as the unobserved information and dealt with under the framework of the expectation-maximization(EM)algorithm and the variational Bayesian(VB)inference algorithm.The continuous dynamics of the local modes are approximated with the autoregressive exogenous(ARX)models in this paper.Finally,the point estimations and the posterior distributions of the parameters of Markov jump systems are respectively derived with the above methods.The effectiveness of the algorithms is verified with a numerical example and a simulated continuous fermentation reactor process.(2)The online identification problem of the time-delay Markov jump autoregressive exogenous systems is discussed using the recursive EM algorithm.The conventional EM algorithm is an iterative algorithm,in which the data with missing information are coped with in a batch manner.Although the conventional EM algorithm can solve the problem of information missing adequately,it can hardly be adapted to the online identification problem.In contrast,the recursive EM algorithm,in which the parameters are online updated with the recursive sufficient statistics,overcomes the deficiency of the batch EM algorithm in the online estimation.In this paper,the recursive Q-function of the time-delay Markov jump autoregressive exogenous system is derived,based on which the sufficient statistics are recursively updated.Furthermore,the online updating equations of the parameters and the transition probabilities of the modes are obtained.The effectiveness of the proposed algorithms is verified with a numerical example and a continuous fermentation reactor example.(3)The identification problems of the Markov jump ARX systems contaminated with abnormal measurements are considered.Abnormal measurements are important issues that frequently encountered in engineering,which should be paid attention to.In this paper,two kinds of abnormal measurements,which are respectively referred to as the missing measurements and the outliers,are dealt with the variational Bayesian inference.For the missing measurements,the posterior expectations are employed during the parameters' updating.The posterior distributions of the parameters are obtained.As for the outliers,the Student's-t distribution is implemented to describe the characteristic of the measurement noise,which results in a robust variational Bayesian inference algorithm for the Markov jump ARX system.A numerical example and a simulated continuous fermentation reactor process are given to illustrate the effectiveness of the proposed algorithms.In summary,this paper considered the identification problems of the Markov jump autoregressive exogenous systems with some practical issues,such as time-delay,missing measurements,and outliers,using the EM algorithm,the variational Bayesian inference approach,and recursive EM algorithm.The above problems will be investigated in detail in the following context.A prospect of the future work will be provided.