Variational Bayesian identification for bilinear state space models with Markov‐switching time delays

Abstract

This article studies the parameter identification problem for bilinear state space models with time-varying time delays. Considering the correlation of time delays, the Markov chain switching mechanism is adopted to model the delay sequence. Based on the observer canonical form, the bilinear state space model is transformed into a regressive form. A bilinear state observer is designed to estimate the state variables. Under the variational Bayesian scheme, the system parameters, discrete delays, and the Markov transition probabilities are identified by using the measurement data. A numerical example and a continuous stirred tank reactor simulation are employed to validate the effectiveness of the proposed algorithm.

Augmented flexible least squares algorithm for time‐varying parameter systems

“…Remark 6. When 𝜇 > 0 and L ⩾ 1, the P × P matrix R(L, 𝜇) is positive definite, thus the parameter estimates 𝜣(L) can always be computed by Equation (15).…”

“…Time-varying parameter system identification is more challenging when comparing with the constant system identification, because the parameters evolve unanticipatedly. [14][15][16] Recently, most of the identification methods for time-varying parameter systems usually have assumed that the parameters keep unchanged in a fixed interval and evolve quickly to another mode, and the number of the collected input-output data in the fixed interval is larger than that of the unknown parameters. 17,18 For example, Yang and Yin 19 proposed an EM algorithm for linear parameter varying dual-rate systems, the identity of the sub-model is estimated in the E-step, and the parameters of each sub-model are updated in the M-step based on the estimated identities.…”

Expectation‐maximization algorithm for bilinear state‐space models with time‐varying delays under non‐Gaussian noise

“…[4][5][6] Furthermore, the bilinear system can theoretically approximate a class of input-affine dynamic system and is more accurate than traditional linear approximation. [7][8][9] Thus, the research on the identification algorithm of bilinear SSM has high academic value.…”

“…As a particular kind of nonlinear system, the bilinear SSM contains the coupled term of the state vector and the control vector, and has an obvious changeable structure, which can describe the nuclear reaction process, population evolution process, cell division, and others 4‐6 . Furthermore, the bilinear system can theoretically approximate a class of input‐affine dynamic system and is more accurate than traditional linear approximation 7‐9 . Thus, the research on the identification algorithm of bilinear SSM has high academic value.…”