Maarten Schoukens

Parametric Identification of Parallel Hammerstein Systems

This paper proposes a parametric identification method for parallel Hammerstein systems. The linear dynamic parts of the system are modeled by a parametric rational function in the z - or s-domain, while the static nonlinearities are represented by a linear combination of nonlinear basis functions. The identification method uses a three-step procedure to obtain initial estimates. In the first step, the frequency response function of the best linear approximation is estimated for different input excitation levels. In the second step, the power-dependent dynamics are decomposed over a number of parallel orthogonal branches. In the last step, the static nonlinearities are estimated using a linear least squares estimation. Furthermore, an iterative identification scheme is introduced to refine the estimates. This iterative scheme alternately estimates updated parameters for the linear dynamic systems and for the static nonlinearities. The method is illustrated on a simulation and a validation measurement example.

Identification of Wiener-Hammerstein systems by a nonparametric separation of the best linear approximation

Wiener-Hammerstein models are flexible, well known and often studied. The main challenge in identifying a Wiener-Hammerstein model is to distinguish the linear time invariant (LTI) blocks at the front and the back. This paper presents a nonparametric approach to separate the front and back dynamics starting from the best linear approximation (BLA). Next, the nonparametric estimates of the LTI blocks in the model can be parametrized, taking into account a phase shift degeneration. Once the dynamics are known, the estimation of the static nonlinearity boils down to a simple linear least squares problem. The consistency of the proposed approach is discussed and the method is validated on the Wiener-Hammerstein benchmark that was presented at the IFAC SYSID conference in 2009.

Parametric Identification of Parallel Wiener Systems

This paper proposes a parametric identification method for parallel Wiener systems. The linear dynamic parts of the Wiener system are modeled by a parametric rational function in the Laplace or z-domain. The static nonlinearity is represented by a linear combination of multiple-input single-output nonlinear basis functions. The identification method uses a three-step procedure to obtain initial estimates. In the first step, the frequency response function of the best linear approximation is estimated for different input excitation levels. In the second step, the power-dependent dynamics are decomposed over a number of parallel orthogonal branches. In the last step, the static nonlinearities are estimated using a linear least squares estimation. Furthermore, a nonlinear optimization method is implemented to refine the estimates. The method is illustrated on a simulation and a validation measurement example.

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.

Parametric identification of parallel Wiener-Hammerstein systems

Block-oriented nonlinear models are popular in nonlinear modeling because of their advantages to be quite simple to understand and easy to use. To increase the flexibility of single branch block-oriented models, such as Hammerstein, Wiener, and Wiener-Hammerstein models, parallel block-oriented models can be considered. This paper presents a method to identify parallel Wiener-Hammerstein systems starting from input-output data only. In the first step, the best linear approximation is estimated for different input excitation levels. In the second step, the dynamics are decomposed over a number of parallel orthogonal branches. Next, the dynamics of each branch are partitioned into a linear time invariant subsystem at the input and a linear time invariant subsystem at the output. This is repeated for each branch of the model. The static nonlinear part of the model is also estimated during this step. The consistency of the proposed initialization procedure is proven. The method is validated on real-world measurements using a custom built parallel Wiener-Hammerstein test system.

Cross-term Elimination in Parallel Wiener Systems Using a Linear Input Transformation

Multivariate polynomials are often used to model nonlinear behavior, e.g., in parallel Wiener models. These multivariate polynomials are mostly hard to interpret due to the presence of cross terms. These polynomials also have a high amount of coefficients, and the calculation of an inverse of a multivariate polynomial with cross terms is cumbersome. This paper proposes a method to eliminate the cross terms of a multivariate polynomial using a linear input transformation. It is shown how every homogeneous polynomial described using tensors can be transformed to a canonical form using multilinear algebraic decomposition methods. Such tensor decomposition methods have already been used in nonlinear system modeling to reduce the complexity of Volterra models. Since every polynomial can be written as a sum of homogeneous polynomials, this method results in a decoupled description of any multivariate polynomial, allowing a model description that is easier to interpret, easier to use in a design, and easier to invert. This paper first describes a method to represent and decouple multivariate polynomials using tensor representation and tensor decomposition techniques. This method is applied to a parallel Wiener model structure, where a multiple-input-single-output polynomial is used to describe the static nonlinearity of the system. A numerical example shows the performance of the proposed method.

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.

Study of the effective number of parameters in nonlinear identification benchmarks

This paper discusses the importance of the notion of effective number of parameters as a measure of model complexity. Exploiting this concept allows a fair comparison of models obtained from different model classes. Several illustrative examples of linear and nonlinear models are presented to provide more insight in the problem. As one possible way of showing that model complexity can be reduced without having to pull any parameters to zero, an approach for rank reduced estimation based on the truncated SVD is also discussed. These ideas are then applied to two nonlinear real world problems: the Wiener-Hammerstein and the Silverbox benchmarks.

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.