Recursive Parsimonious Subspace Identification for Closed-Loop Hammerstein Nonlinear Systems

Abstract

In this paper, a recursive closed-loop subspace identification method for Hammerstein nonlinear systems is proposed. To reduce the number of unknown parameters to be identified, the original hybrid system is decomposed as two parsimonious subsystems, with each subsystem being related directly to either the linear dynamics or the static nonlinearity. To avoid redundant computations, a recursive least-squares (RLS) algorithm is established for identifying the common terms in the two parsimonious subsystems, while another two RLS algorithms are established to estimate the coefficients of the nonlinear subsystem and the predictor Markov parameters of the linear subsystem, respectively. Subsequently, the system matrices of the linear subsystem are retrieved from the identified predictor Markov parameters. The convergence of the proposed method is analyzed. Two illustrative examples are shown to demonstrate the effectiveness and merit of the proposed method.

Identification of Time-Varying Hammerstein-Wiener Systems

“…Corollary 1. When applying algorithm (23)- (25) to the time-invariant noisy H-W systems parameterized by (4) on Assumptions 1-5, if (29)- (31) and (41) are fulfilled, t   is exponentially bounded in mean square and bounded with probability one. Proof: If the H-W system is time-invariant, we have…”

“…The more complex types, namely Hammerstein-Wiener (H-W) and Wiener-Hammerstein (W-H), consist of their combinations. Many works have been published to identify these kinds of systems, such as twostage method [5]- [7], maximum likelihood estimation method [8], [9], iterative method [10]- [15], recursive method [16]- [25], etc. All of the contributions mentioned above studied the identification of time-invariant block oriented systems.…”

Approximate GP Inference for Nonlinear Dynamical System Identification Using Data-Driven Basis Set

“…, k(y 2 , z)], ψ j (y n ) is j-th eigenvector of N × N covariance function matrix evaluated at N data points, corresponding to eigenvalue λ j , andψ j (y n ) = √ N λ j ψ j (y n ). We can interpret j-th eigenfunction in (20), as linear combination of kernel values evaluated at N data points, where the weights of linear combinations are provided bỹ ψ j (y n ). There are in general as many eigenfunctions as data points N i.e.…”

“…Once we have generated these points, we form N × N kernel covariance matrix for each state, input and output variable. Performing eigenvalue decomposition of these n x + n u + n y kernel matrices produces N (n x + n u + n y ) eigenfunctions according to (20). We label N eigenfunctions corresponding to each state variable x i as ψ…”

“…A large number of research efforts have been made for nonlinear system identification especially by proposing specific system classes such as based on Volterra models [14], nonlinear ARMAX models [13], neural networks [15], fuzzy models [16], [17], linear parameter varying (LPV) models [18], nonlinear error in variable (EIV) models [19], Hammerstein-Wiener block structured models [20] and state-space models (SSM) [21]. In these classes, parametric form of identification relies strongly on model parameter estimation for a given fixed model structure candidate [22].…”