Identification of nonlinear dynamic systems by using probabilistic universal learning networks

Teaching Aids for Modeling and Control of Hybrid Systems (CAMCHS)

This article presents a theoretical framework for the identification of nonlinear dynamic MISO systems with orthonormal base function models on fundamental basics of the Volterra theory. In the past the Volterra theory was used for the identification of nonlinear dynamic SISO systems (e.g. Hammerstein and Wiener models). In principle it is possible to extend these approaches to systems with more than one input (i.e. MISO systems). In former times this failed due to a lack of computational performance. In this paper an approach is presented which allows the identification of arbitrary coupled Hammerstein and Wiener models with multiple inputs. Some fundamental considerations for the identification of MISO systems based on arbitrary coupled Hammerstein models are made and extended to MISO systems based on arbitrary coupled Wiener models. This extended Volterra theory results in equations where the unknown parameters can be separated of the input values in a linear manner. In order to approximate the truncated impulse responses of the linear dynamic systems and to reduce the number of unknown parameters orthonormal base functions (OBFs) are introduced. As an example the proposed identification method is applied to a MISO system based on forward and backward coupled Hammerstein and Wiener models.

Nonlinear dynamic systems are characterized with uncertainties in terms of structure and parameters. These uncertainties cannot be described by deterministic models. The modelling and identification of nonlinear dynamic systems through the measured experimental data is a problem in engineering and technical processes. Therefore, field of system identification have become an important area of research. Fuzzy technology is an effective tool for dealing with complex nonlinear processes that are characterized with uncertain factors. In this paper, a novel approach based on Local Linear method learning in dynamical filter weights neurons for the identification of non-linear dynamic systems is presented. The fuzzy wavelet neural network combines wavelet theory with fuzzy logic and neural networks. Learning fuzzy rules and parameter update in fuzzy wavelet neural network is based on gradient decent method. The proposed approach is said to be Fuzzy Local Linear Wavelet Neural Network based model. It has been explained through examples. The structure is tested for the identification with both wavelet neural network and Fuzzy Local Linear Wavelet Neural Network that shows the comparative performance.KeywordsSystem IdentificationNon-linear SystemWavelet Neural NetworkFuzzy Wavelet Neural NetworkLocal Linear Wavelet Neural NetworkFuzzy Local Linear Wavelet Neural Network

Presents a modeling scheme for nonlinear black-box systems based on universal learning networks (ULN). The ULN, a superset of all kinds of neural networks, consists of two kinds of elements: nodes and branches corresponding to equations and their relations in a traditional description of dynamic systems. Following the idea of ULN, a nonlinear black-box system is first represented by a set of related unknown equations, and then treated as the ULN with nodes and branches. Each unknown node function in the ULN is re-parameterized by using an adaptive fuzzy model. One of distinctive features of the black-box model constructed in this way is that it can incorporate prior knowledge obtained from input-output data into its modeling and thus its parameters to be trained have explicit meanings useful for estimation and application.

A procedure based on the use of artificial neural networks for the identification of nonlinear dynamic systems is developed and applied to the damped Duffing oscillator under deterministic excitation. The “generalization” ability of neural networks is invoked to predict the response of the same nonlinear oscillator under stochastic excitations of differing magnitude. The analogy between the neural network approach and a qualitatively similar nonparametric identification technique previously developed by the authors is illustrated. Some of the computational aspects of identification by neural networks, as well as their fault-tolerant nature, are discussed. It is shown that neural networks provide high-fidelity mathematical models of structure-unknown nonlinear systems encountered in the applied mechanics field.

Discovering nonlinear PDEs with sparse identification of nonlinear dynamical systems (SINDy) is hindered by high dimensionality, noise, and expensive data acquisition. We propose the greedy sampling neural network for sparse identification of nonlinear PDEs (GN-SINDy), a three–stage framework that integrates strategic sampling, differentiable surrogate modelling, and sparse equation discovery. First, a two–way Q-DEIM–based greedy strategy selects maximally informative space–time samples from snapshot data, drastically reducing data requirements. Second, a deep neural network (DNN) is trained as a differentiable surrogate of the solution field, enabling noise–robust analytic derivatives via automatic differentiation. Third, sparse regression with sparsity–promoting estimators [Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016a). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932–3937; Rudy, S. H., Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2017). Data-driven discovery of partial differential equations. Science Advances, 3(4), e1602614.] is applied to recover the governing PDE. Building on the DeepMoD paradigm, GN-SINDy embeds greedy sampling into data acquisition and stabilises coefficient estimation through neural–enhanced differentiation. We analyze noise robustness, structural stability, and QR–based sampling strategies to guide sampler and hyperparameter selection. Experiments on Burgers', Allen–Cahn, and Korteweg–de Vries equations show that GN-SINDy reliably recovers governing PDEs using under 1 % of the data, outperforming DeepMoD in efficiency, support recovery, and robustness to noise.

In this study, nonlinear dynamic systems are identified by using artificial bee colony (ABC) algorithm and adaptive neuro fuzzy inference system (ANFIS). ABC algorithm is used in training and updating of ANFIS. The most appropriate model is formed by optimizing the antecedent and conclusion parameters that are found in the structure of ANFIS. The dynamic systems that consist of one input and one output (SISO) are used for the identification of nonlinear dynamic systems. The obtained results are compared with fuzzy neural network, neural network and ANFIS-based methods such as RSONFIN, DFNN, RSEFNN-LF, WRFNN and RFNN. The simulation results show that the proposed method is successful in the identification of considered nonlinear dynamic systems.

The paper demonstrates that radial basis function network (RBFN) with adaptive centers and width can be used effectively for identification of nonlinear dynamic system. The proposed RBFN is trained by hybrid learning algorithm, which uses conjugate gradient optimization algorithm to obtain the center and width of each radial basis function and the least squares method to obtain the weights. To avoid capturing a local optimum, regularization error energy function is used and the centers of basis functions are initialized using a fuzzy C-means clustering method. Simulation results reveal that the identification schemes based on RBFN gives considerably better performance and show faster learning in comparison to previous methods

An identification method of nonlinear dynamic system is studied in this paper. First, suppose that the original nonlinear dynamic system is described by Hammerstein model. Then, an intermediate model is generated. Next, Parameters of the intermediate model are obtained using a Bacterial Chemotaxis Optimization (BCO) approach. Finally, through the relationships of the parameters of intermediate model and those of Hammerstein model, we derive the parameters of the system. Consequently, the original nonlinear dynamic system is identified. The feasibility and efficiency of the presented algorithm are demonstrated using numerical simulations.

This paper compares radial basis function networks for identification of nonlinear dynamic systems with classical methods derived from the Volterra series. The performance of these different approaches, such as Hammerstein, Wiener and NDE models, is analysed. Since the centres and variances of the Gaussian radial basis functions will be fixed before learning and only the weights are learned, a linear optimization problem arises. Therefore training the network and parameter estimation becomes comparable in computational effort. It is shown that the classical methods can compete or even perform better than the neural network, if the assumptions for the structure are valid. However, in practical applications when the structure is not known the radial basis function network performs much better than the classical methods.

An approach to fuzzy identification of discrete-time nonlinear dynamical systems with a suitable formulation, based on the Takagi-Sugeno (TS) model is proposed. To form the fuzzy model from samples of a nonlinear dynamical system where the consequent parameters are modified by an adaptive weighted instrumental variable (WIV) algorithm based on the numerically robust orthogonal Householder transformation, offline and online schemes are developed. To show the consistency, high speed of convergence, tracking of the output that vary with time and the high accuracy of the output estimate, important in adaptive control design applications, simulations are performed.

This paper presents developed engineering software tools used for nonparametric identification of nonlinear dynamical systems based on Volterra models. The polyharmonic test sequences with different amplitudes are used for test procedures in frequency domain. The wavelets application allowed enlarging computational stability of the identification method for measurement noises filtering of received responses and characteristics of the system being identified. The methodology proposed in pervious works and developed software tools applied for construction communication channel models of different orders.

The paper proposes a general framework for modelling non-linear dynamic systems based on a support vector machine (SVM): it first provides a short introduction to regression SVMs, then uses a standard SVM to model a non-linear auto-regressive and moving average (NARMAX) model, and contains a theoretical discussion about its robustness under low and high noise by its properties. The simulation results indicate that the SVM method can reduce the effect of samples and noise for modelling, and its performance is better than that of the neural network modelling method.

Structure Identification of Continuous-Time Block-Oriented Nonlinear Systems in the Frequency Domain

In this paper, advanced concepts for the identification of complex nonlinear systems are discussed. Three major problems are addressed: The nonlinearity of the system, noise in the data upon which the model has to be built, and the potential to incorporate qualitative and quantitative prior knowledge about the system. As an integrated solution approach, local model networks (LMNs) with appropriate parameter estimation schemes are proposed. LMNs generally offer a versatile structure for the identification of nonlinear dynamic systems. In order to account for a realistic situation when noise is present both in input and output data, an equality constrained generalised total least squares algorithm for the local model parameter estimation of the LMN is presented; the incorporation of equality constraints allows to mathematically enforce desired system properties. As an application and benchmark problem, the vertical dynamics of a vehicle is considered. After training the LMN on a rough road, excellent predictions of the behaviour of the vehicle at crossing a single obstacle are obtained, thus proving the effectiveness of the proposed algorithm. It is illustrated how both the application of a proper parameter estimation scheme and the integration of system constraints systematically improve the performance of the model.