Koen Tiels

Structure discrimination in block-oriented models using linear approximations

In this paper we show that it is possible to retrieve structural information about complex block-oriented nonlinear systems, starting from linear approximations of the nonlinear system around different setpoints. The key idea is to monitor the movements of the poles and zeros of the linearized models and to reduce the number of candidate models on the basis of these observations. Besides the well known open loop single branch Wiener-, Hammerstein-, and Wiener-Hammerstein systems, we also cover a number of more general structures like parallel (multi branch) Wiener-Hammerstein models, and closed loop block oriented models, including linear fractional representation (LFR) models.

Wiener system identification with generalized orthonormal basis functions

Many nonlinear systems can be described by a Wiener-Schetzen model. In this model, the linear dynamics are formulated in terms of orthonormal basis functions (OBFs). The nonlinearity is modeled by a multivariate polynomial. In general, an infinite number of OBFs are needed for an exact representation of the system. This paper considers the approximation of a Wiener system with finite-order infinite impulse response dynamics and a polynomial nonlinearity. We propose to use a limited number of generalized OBFs (GOBFs). The pole locations, needed to construct the GOBFs, are estimated via the best linear approximation of the system. The coefficients of the multivariate polynomial are determined with a linear regression. This paper provides a convergence analysis for the proposed identification scheme. It is shown that the estimated output converges in probability to the exact output. Fast convergence rates, in the order O p ( N F - n r e p / 2 ) , can be achieved, with N F the number of excited frequencies and n r e p the number of repetitions of the GOBFs.

Identification of parallel Wiener-Hammerstein systems with a decoupled static nonlinearity

Abstract Block-oriented models are often used to model a nonlinear system. This paper presents an identification method for parallel Wiener-Hammerstein systems, where the obtained model has a decoupled static nonlinear block. This decoupled nature makes the interpretation of the obtained model more easy. First a coupled parallel Wiener-Hammerstein model is estimated. Next, the static nonlinearity is decoupled using a tensor decomposition approach. Finally, the method is validated on real-world measurements using a custom built parallel Wiener-Hammerstein test system.

Identifying a Wiener system using a variant of the Wiener G-Functionals

This paper concerns the identification of nonlinear systems using a variant of the Wiener G-Functionals. The system is modeled by a cascade of a single input multiple output (SIMO) linear dynamic system, followed by a multiple input single output (MISO) static nonlinear system. The dynamic system is described using orthonormal basis functions. The original ideas date back to the Wiener G-functionals of Lee and Schetzen. Whereas the Wiener G-Functionals use Laguerre orthonormal basis functions, in this work Takenaka-Malmquist orthonormal basis functions are used. The poles that these basis functions contain, are estimated using the best linear approximation of the system. The approach is illustrated on the identification of a Wiener system.

Decoupling static nonlinearities in a parallel Wiener-Hammerstein system: A first-order approach

We present a method to decompose a static MIMO (multiple-input-multiple-output) nonlinearity into a set of SISO (single-input-single-output) polynomials acting on internal variables that are related to the inputs and outputs of the MIMO nonlinearity by linear transformations. The method is inspired on the small-signal analysis of nonlinear circuits and proceeds by collecting first-order information of the MIMO function into a set of Jacobian matrices. A simultaneous diagonalization of the set of Jacobian matrices is computed using a tensor decomposition, providing the required linear transformations, after which also the coefficients of the internal SISO polynomials can be computed. The method is validated on measurements of a parallel two-branch Wiener-Hammerstein identification setup.

Initial estimates for Wiener-Hammerstein models using phase-coupled multisines

Block-oriented models are often used to model nonlinear systems. These models consist of linear dynamic (L) and nonlinear static (N) sub-blocks. This paper addresses the generation of initial estimates for a Wiener-Hammerstein model (LNL cascade). While it is easy to measure the product of the two linear blocks using a Gaussian excitation and linear identification methods, it is difficult to split the global dynamics over the individual blocks. This paper first proposes a well-designed multisine excitation with pairwise coupled random phases. Next, a modified best linear approximation is estimated on a shifted frequency grid. It is shown that this procedure creates a shift of the input dynamics with a known frequency offset, while the output dynamics do not shift. The resulting transfer function, which has complex coefficients due to the frequency shift, is estimated with a modified frequency domain estimation method. The identified poles and zeros can be assigned to either the input or output dynamics. Once this is done, it is shown in the literature that the remaining initialization problem can be solved much easier than the original one. The method is illustrated on experimental data obtained from the Wiener-Hammerstein benchmark system.

Identification of block-oriented nonlinear systems starting from linear approximations: A survey

Abstract Block-oriented nonlinear models are popular in nonlinear system identification because of their advantages of being simple to understand and easy to use. Many different identification approaches were developed over the years to estimate the parameters of a wide range of block-oriented nonlinear models. One class of these approaches uses linear approximations to initialize the identification algorithm. The best linear approximation framework and the ϵ -approximation framework, or equivalent frameworks, allow the user to extract important information about the system, guide the user in selecting good candidate model structures and orders, and prove to be a good starting point for nonlinear system identification algorithms. This paper gives an overview of the different block-oriented nonlinear models that can be identified using linear approximations, and of the identification algorithms that have been developed in the past. A non-exhaustive overview of the most important other block-oriented nonlinear system identification approaches is also provided throughout this paper.

Incorporating Best Linear Approximation within LS-SVM-based Hammerstein System Identification

Hammerstein systems represent the coupling of a static nonlinearity and a linear time invariant (LTI) system. The identification problem of such systems has been a focus of research during a long time as it is not a trivial task. In this paper a methodology for identifying Hammerstein systems is proposed. To achieve this, a combination of two powerful techniques is used, namely, we combine Least Squares Support Vector Machines (LS-SVM) and the Best Linear Approximation (BLA). First, an approximation to the LTI block is obtained through the BLA method. Then, the estimated coefficients of the transfer function from the LTI block are included in a LS-SVM formulation for modeling the system. The results indicate that a good estimation of the underlying nonlinear system can be obtained up to a scaling factor.

Comparison of several data-driven non-linear system identification methods on a simplified glucoregulatory system example

In this paper, several advanced data-driven nonlinear identification techniques are compared on a specific problem: a simplified glucoregulatory system modeling example. This problem represents a challenge in the development of an artificial pancreas for T1DM treatment, since for this application good nonlinear models are needed to design accurate closed-loop controllers to regulate the glucose level in the blood. Block-oriented as well as state-space models are used to describe both the dynamics and the nonlinear behavior of the insulin-glucose system, and the advantages and drawbacks of each method are pointed out. The obtained nonlinear models are accurate in simulating the patients behavior, and some of them are also sufficiently simple to be considered in the implementation of a model-based controller to develop the artificial pancreas.

Generating initial estimates for Wiener-Hammerstein systems using phase coupled multisines

Abstract Block oriented nonlinear models capture the dynamics of a nonlinear system with linear dynamic sub-systems (L), the nonlinear behavior is modelled using static nonlinear sub-blocks (N). In this paper we study the generation of initial estimates for the linear dynamic blocks of a Wiener-Hammerstein system that has a cascaded LNL structure. While it is very easy to identify the product of the transfer functions of the first and last dynamic block using linear system identification methods, it turns out to be very difficult to split the global dynamics over these individual blocks. In this paper a method is proposed that allows the poles of the best linear approximation to be assigned to the first or second linear block. Once this split is made, it is shown in the literature that the remaining initialization problem can be solved much easier than the original one. The first step of the method is the design of a special random phase multisine excitation, using pair-wise coupled random phases. Next, a modfied best linear approximation will be estimated on a shifted frequency grid. It will be shown that this procedure shifts the poles and zeros of the first linear sub-block with a known frequency offset, while those of the second sub-block are not changed. The shifted poles and zeros result in a transfer function with complex coefficients that can be identified using a modified frequency domain estimation method. This results in a simple initialization method, based on a linear system identification step.