Identification Of Discrete-time Nonlinear Systems Research Articles

Kernel-Based Identification of Incrementally Input-to-State Stable Nonlinear Systems

Methods based on Reproducing Kernel Hilbert Spaces (RKHS) have proven to be a valuable tool for the identification of linear time-invariant systems in both discrete- and continuous-time. In particular, unlike most other techniques, they enable to systematically confer a priori desirable properties, such as stability, on the estimated models. However, existing RKHS methods mainly target impulse responses and, hence, do not extend well to the context of nonlinear systems. In this work, we propose a novel RKHS-based methodology for the identification of discrete-time nonlinear systems guaranteeing that the identified system is incrementally input-to-state stable (δISS). We model the identified system using a predictor function that, given past input and output samples, yields the output prediction at the next time instant. The predictor is selected from an RKHS by solving a constrained optimization problem that guarantees its δISS properties. The proposed approach is validated via numerical simulations.

A Sparse Bayesian Approach to the Identification of Nonlinear State-Space Systems

This technical note considers the identification of nonlinear discrete-time systems with additive process noise but without measurement noise. In particular, we propose a method and its associated algorithm to identify the system nonlinear functional forms and their associated parameters from a limited number of time-series data points. For this, we cast this identification problem as a sparse linear regression problem and take a Bayesian viewpoint to solve it. As such, this approach typically leads to nonconvex optimisations. We propose a convexification procedure relying on an efficient iterative re-weighted $\ell_1$-minimisation algorithm that uses general sparsity inducing priors on the parameters of the system and marginal likelihood maximisation. Using this approach, we also show how convex constraints on the parameters can be easily added to our proposed iterative re-weighted $\ell_1$-minimisation algorithm. In the supplementary material \cite{appendix}, we illustrate the effectiveness of the proposed identification method on two classical systems in biology and physics, namely, a genetic repressilator network and a large scale network of interconnected Kuramoto oscillators.

An Improved Global Harmony Search Algorithm for the Identification of Nonlinear Discrete-Time Systems Based on Volterra Filter Modeling

This paper describes an improved global harmony search (IGHS) algorithm for identifying the nonlinear discrete-time systems based on second-order Volterra model. The IGHS is an improved version of the novel global harmony search (NGHS) algorithm, and it makes two significant improvements on the NGHS. First, the genetic mutation operation is modified by combining normal distribution and Cauchy distribution, which enables the IGHS to fully explore and exploit the solution space. Second, an opposition-based learning (OBL) is introduced and modified to improve the quality of harmony vectors. The IGHS algorithm is implemented on two numerical examples, and they are nonlinear discrete-time rational system and the real heat exchanger, respectively. The results of the IGHS are compared with those of the other three methods, and it has been verified to be more effective than the other three methods on solving the above two problems with different input signals and system memory sizes.

Identification of continuous-time nonlinear systems: The nonlinear difference equation with moving average noise (NDEMA) framework

Although a vast number of techniques for the identification of nonlinear discrete-time systems have been introduced, the identification of continuous-time nonlinear systems is still extremely difficult. In this paper, the Nonlinear Difference Equation with Moving Average noise (NDEMA) model which is a general representation of nonlinear systems and contains, as special cases, both continuous-time and discrete-time models, is first proposed. Then based on this new representation, a systematic framework for the identification of nonlinear continuous-time models is developed. The new approach can not only detect the model structure and estimate the model parameters, but also work for noisy nonlinear systems. Both simulation and experimental examples are provided to illustrate how the new approach can be applied in practice.

On-line parameter estimator derived from modified matrix pseudoinverse

A method based on the matrix pseudoinverse is considered for on-line parameter identification of discrete-time nonlinear systems. A recursive scheme was previously reported but assuming some restrictions on the input–output data. By resorting to the formulae developed for modified matrix pseudoinverse, an iterative algorithm is derived in this work. This algorithm applies to the general class of proposed systems while keeping well known desired properties for parameter identification.

Identification of nonlinear discrete-time systems using raised-cosine radial basis function networks

An effective technique for identifying nonlinear discrete-time systems using raised-cosine radial basis function (RBF) networks is presented. Raised-cosine RBF networks are bounded-input bounded-output stable systems, and the network output is a continuously differentiable function of the past input and the past output. The evaluation speed of an n-dimensional raised-cosine RBF network is high because, at each discrete time, at most 2n RBF terms are nonzero and contribute to the output. As a consequence, raised-cosine RBF networks can be used to identify relatively high-order nonlinear discrete-time systems. Unlike the most commonly used RBFs, the raised-cosine RBF satisfies a constant interpolation property. This makes raised-cosine RBF highly suitable for identifying nonlinear systems that undergo saturation effects. In addition, for the important special case of a linear discrete-time system, a first-order raised-cosine RBF network is exact on the domain over which it is defined, and it is minimal in terms of the number of distinct parameters that must be stored. Several examples, including both physical systems and benchmark systems, are used to illustrate that raised-cosine RBF networks are highly effective in identifying nonlinear discrete-time systems.

Random and pseudorandom inputs for Volterra filter identification

This paper studies input signals for the identification of nonlinear discrete-time systems modeled via a truncated Volterra series representation. A Kronecker product representation of the truncated Volterra series is used to study the persistence of excitation (PE) conditions for this model. It is shown that i.i.d. sequences and deterministic pseudorandom multilevel sequences (PRMS's) are PE for a truncated Volterra series with nonlinearities of polynomial degree N if and only if the sequences take on N+1 or more distinct levels. It is well known that polynomial regression models, such as the Volterra series, suffer from severe ill-conditioning if the degree of the polynomial is large. The condition number of the data matrix corresponding to the truncated Volterra series, for both PRMS and i.i.d. inputs, is characterized in terms of the system memory length and order of nonlinearity. Hence, the trade-off between model complexity and ill-conditioning is described mathematically. A computationally efficient least squares identification algorithm based on PRMS or i.i.d. inputs is developed that avoids directly computing the inverse of the correlation-matrix. In many applications, short data records are used in which case it is demonstrated that Volterra filter identification is much more accurate using PRMS inputs rather than Gaussian white noise inputs. >

A joint frequency-position domain structure identification of nonlinear discrete-time systems by neural networks

A new technique is proposed to identify the structure and the parameters of nonlinear discrete-time system models. The structure is represented in a frequency-position domain of Gabor basis functions (GBFs). A simplification to the GBF is also presented, where the spatial Gaussian envelope of GBF is replaced with a triangular one. A modification to the GBF has also been introduced in order to suppress the effects of noise on the procedure. A three-layered neural network, augmented with nonuniform sampling, is described for solving the system identification problem. >

Non-linear systems identification using radial basis functions

This paper investigates the identification of discrete-time non-linear systems using radial basis functions. A forward regression algorithm based on an orthogonal decomposition of the regression matrix is employed to select a suitable set of radial basis function centers from a large number of possible candidates and this provides, for the first time, fully automatic selection procedure for identifying parsimonious radial basis function models of structure-unknown non-linear systems. The relationship between neural networks and radial basis functions is discussed and the application of the algorithms to real data is included to demonstrate the effectiveness of this approach.

Non-linear system identification using neural networks

Multi-layered neural networks offer an exciting alternative for modelling complex non-linear systems. This paper investigates the identification of discrete-time nonlinear systems using neural networks with a single hidden layer. New parameter estimation algorithms are derived for the neural network model based on a prediction error formulation and the application to both simulated and real data is included to demonstrate the effectiveness of the neural network approach.

On the Volterra-series functional identification of non-linear discrete-time systems

The problem of determining the kernels in the discrete Volterra-series representation of a time-invariant non-linear discrete-time system is considered. The identification problem is transformed to an optimal control problem, which is then solved using dynamic-programming formulation. Examples illustrating the use of the techniques are presented.