High-order Moment State Estimation For Markov Jump System

The practical industrial process is influenced by the state evolution and the random jump driven by discrete events.For this kind of stochastic Markov jump system affected by time and events jointly,the state estimation is more complicated and challenge with the consideration of the stochastic mode jumping of the system.Mostly,the state estimation is utilized for the first and second-order moment in the existing result,which is commonly known as the mean and mean square state estimation.However,sometimes the existing results are unavailable to satisfy the requirement of estimation,even result in inaccuracy only by the first and second moment in the practical applications.To meet the requirements,this paper transforms the multi-mode stochastic system into a single-mode system by the derandomization method,and with the help of the cumulant generating function from the statistic theorem,the high-order moment information of the state is proposed.Compare to the existing first and second-moment state estimation approaches,the high-order moment state estimation introduces the high-order moment information,which improves the estimation performances and compensates for the drawbacks of the existing methods.The main contribution of this paper is as follow:1.In the Markov jump system,the derandomization method and cumulant generating func-tion are utilized,which transformed the multi-mode stochastic jumping system into a single-mode deterministic system.The transformed state is recursively estimated according to the high-order moment Kalman filter.Compared with the Markov jump system's transformation techniques in the existing results,the derandomization method introduces the transition proba-bility into the transformed deterministic system with the help of the state's expectation.Thus,the norm of the transformed state and the original state are the same.To determine the state's high-order moment,the cumulant generating function in statistics is introduced,which obtains the high-order moment component by Taylor expansion.In the end,the high-order moment Kalman filter is used to estimate the state of the transformed system in high-order moment component form.Compare with the mean and mean square state estimation method in the current results,the high-order moment recursive state estimation methods compensate for the drawback of first and second-order moment and obtain a more accurate result.2.With the consideration of nonlinearities,the Markov jump system is more complicated than before.To deal with such a problem,the cumulant generating function is improved so that the more complicated nonlinearities in the stochastic system can be transformed into a nonlinear single-mode system.Then,with the help of Bayesian theory,the high-order moment component form of the nonlinear Markov jump system is estimated.With the existence of the nonlinearity,the system parameters and the states are difficult to separate.In the improved cumulant gen-erating function,the state vector is separated into several state elements so that the aim of the high-order moment of the state vector is simplified into several elements.Then the high-order moment of the system state with nonlinearities is proposed.Because of the existence of the nonlinearities,the traditional Kalman is unavailable to deal with the state estimation problem here.Consequently,the recursive estimation algorithms based on the Bayesian framework are given,which solves the complication caused by the nonlinearities and improves the accuracy of the estimation algorithm with the high-order moment state estimation.3.To solve the estimation problem for Markov jump system with time-correlated measure-ment noise,the high-order moment H_?state estimation problem in finite time is considered.Depending on the measurement differencing method,the time-correlated measurement noise se-quence is converted to a new time-independent measurement noise sequence by a linear transfor-mation,which means these two sequences are equivalent.For the finite time performance index of the transformed deterministic system,the finite time theory is utilized,and the requirements are transformed into linear matrix inequalities,which can be solved by the toolbox in Matlab.Compare with the former studies for the infinity time performance,the transient response is considered,which makes estimation performance more accurate and satisfies performance index in certain time domain.4.With the consideration of the fault detection state estimation problem of the Markov jump system,a finite frequency fault detection state estimator is designed.After the stochastic Markov jump system transforming to a deterministic system with high-order moment component form,the fault detection state estimation problem is solved by general KYP lemma.The fault and noise usually occur in a certain frequency domain in practical applications.The correspond-ing performance indices are given.With the equivalent transformation of the GKYP lemma and the inequalities of the projection lemma,the performance indices with frequency constraints are transformed into linear matrix inequalities.For the reason that the high-order moment infor-mation is considered in the certain frequency range,the fault detection state estimator is more flexible,which largely improves the estimation performances.5.This section studies the Markov jump system high-order moment fusion state estima-tion problem in multi-sensor case.After the high-order moment information of state,which is estimated from different sensors,the fusion step is presented based on the maximum entropy theory.Since the high-order moment state estimation is proposed in the estimation step,more information of the state is provided,and the estimation accuracy is improved,then the fusion estimator can be more accurate with details known base on the information entropy theory.In this way,the uncertainty of the state information is decreased.Therefore,a more accurate high-order moment fusion state estimation result is proposed.