| Among the many nonlinear systems,bilinear state space systems have received widespread attention because they contain the product of states and inputs,can describe a class of nonlinear dynamic properties,and have a simple structure that most closely resembles the structure of a linear model.The bilinear system can be widely used in industrial process,medicine,biology,architecture and other fields.In the actual production process,the sampling data often has non-ideal conditions such as unpredictable states,random missing,unknown delays,outliers.If these conditions are ignored directly,the identification accuracy will be reduced and the realization of the controller will be affected.Therefore,the study of parameter identification algorithms for bilinear systems with non-ideal data has important theoretical significance and practical value.In this paper,the expectation-maximization(EM)algorithm is used to study the identification problem of bilinear state space system under non-ideal data,the mainly research works and innovations are as follows:(1)For the bilinear state space system with unmeasurable state variables,an EM iterative identification algorithm based on Kalman smoothing algorithm is proposed.The difficulty of identification is that the model contains unknown parameters,internal states and the coupling term of input and state.In view of the above problem,the discussed model is converted into a linear parameter varying model using the particularity of the bilinear system,and the timevarying gain Kalman smoothing algorithm is derived to estimate the state.Combined with the Kalman smoothing algorithm,under the framework of EM algorithm,the unknown state is regarded as a hidden variable,thus realizing the joint estimation of parameters and states.The effectiveness of the algorithm is verified by a numerical simulation and a continuous stirred tank reactor process.(2)The parameter identification of bilinear state space systems is investigated in the presence of random missing output data.Missing data not only affects the stability of the control system,but also makes traditional identification algorithms unable to be directly used.Therefore,based on the interactive estimation theory and EM algorithm,the interactive estimation of missing data,internal states and unknown parameters is realized.In the E-step,the estimates of the state and missing data are obtained by the Kalman smoothing algorithm based on parameter estimation,and the M-step updates the system parameters by maximizing the Q-function.A numerical simulation and a three-tank system experiment show that the algorithm is effective in reducing the impact of missing data on identification and provides a good predictive model.(3)A bilinear state observer based EM algorithm with t-distribution is presented for the bilinear system in the presence of outliers and time-varying delays.In order to weaken the influence of outliers and time-varying delays on the accuracy of parameter identification,the noise is modeled with a heavy-tailed t-distribution rather than the traditional Gaussian distribution,the first-order Markov chain is used to control the switching law of the time delays.Based on the auxiliary model identification idea,a state observer is designed to estimate the unknown states by using the measurable data set.In the framework of EM algorithm,combined with the state observer based on parameter estimation,the joint estimation of system parameters,noise variance,internal states,time-varying delays and probability transfer matrix is implemented.A numerical simulation and a continuous stirred tank reactor process demonstrate that the proposed algorithm has good immunity against outliers and time-varying delays,and offers good estimation accuracy for the bilinear system. |
