A Unified Approach for the Identification of Wiener, Hammerstein, and Wiener–Hammerstein Models by Using WH-EA and Multistep Signals

Wiener, Hammerstein, and Wiener–Hammerstein structures are useful for modelling dynamic systems that exhibit a static type nonlinearity. Many methods to identify these systems can be found in the literature; however, choosing a method requires prior knowledge about the location of the static nonlinearity. In addition, existing methods are rigid and exclusive for a single structure. This paper presents a unified approach for the identification of Wiener, Hammerstein, and Wiener–Hammerstein models. This approach is based on the use of multistep excitation signals and WH-EA (an evolutionary algorithm for Wiener–Hammerstein system identification). The use of multistep signals will take advantage of certain properties of the algorithm, allowing it to be used as it is to identify the three types of structures without the need for the user to know a priori the process structure. In addition, since not all processes can be excited with Gaussian signals, the best linear approximation (BLA) will not be required. Performance of the proposed method is analysed using three numerical simulation examples and a real thermal process. Results show that the proposed approach is useful for identifying Wiener, Hammerstein, and Wiener–Hammerstein models, without requiring prior information on the type of structure to be identified.

Identification of Nonlinear Systems Structured by Wiener-Hammerstein Model

Wiener-Hammerstein systems consist of a series connection including a nonlinear static element sandwiched with two linear subsystems. The problem of identifying Wiener-Hammerstein models is addressed in the presence of hard nonlinearity and two linear subsystems of structure entirely unknown (asymptotically stable). Furthermore, the static nonlinearity is not required to be invertible. Given the system nonparametric nature, the identification problem is presently dealt with by developing a two-stage frequency identification method, involving simple inputs.

Identification of Nonlinear Systems Structured by Wiener-Hammerstein Model

Wiener-Hammerstein systems consist of a series connection including a nonlinear static element sandwiched with two linear subsystems. The problem of identifying Wiener-Hammerstein models is addressed in the presence of hard nonlinearity and two linear subsystems of structure entirely unknown (asymptotically stable). Furthermore, the static nonlinearity is not required to be invertible. Given the system nonparametric nature, the identification problem is presently dealt with by developing a two-stage frequency identification method, involving simple inputs.

Principal Dynamic Mode Analysis of the Hodgkin–Huxley Equations

We develop an autoregressive model framework based on the concept of Principal Dynamic Modes (PDMs) for the process of action potential (AP) generation in the excitable neuronal membrane described by the Hodgkin–Huxley (H–H) equations. The model's exogenous input is injected current, and whenever the membrane potential output exceeds a specified threshold, it is fed back as a second input. The PDMs are estimated from the previously developed Nonlinear Autoregressive Volterra (NARV) model, and represent an efficient functional basis for Volterra kernel expansion. The PDM-based model admits a modular representation, consisting of the forward and feedback PDM bases as linear filterbanks for the exogenous and autoregressive inputs, respectively, whose outputs are then fed to a static nonlinearity composed of polynomials operating on the PDM outputs and cross-terms of pair-products of PDM outputs. A two-step procedure for model reduction is performed: first, influential subsets of the forward and feedback PDM bases are identified and selected as the reduced PDM bases. Second, the terms of the static nonlinearity are pruned. The first step reduces model complexity from a total of 65 coefficients to 27, while the second further reduces the model coefficients to only eight. It is demonstrated that the performance cost of model reduction in terms of out-of-sample prediction accuracy is minimal. Unlike the full model, the eight coefficient pruned model can be easily visualized to reveal the essential system components, and thus the data-derived PDM model can yield insight into the underlying system structure and function.