PWL identification of dynamical systems: some examples

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

A method for identifying nonlinear dynamic systems with noise is proposed by using probabilistic universal learning networks (PrULNs). PrULNs are extensions of universal learning networks (ULNs). ULNs form a superset of neural networks and were proposed to provide a universal framework for modeling and control of nonlinear large-scale complex systems. But the ULN does not provide any stochastic characteristics of the signals propagating through it. The PrULNs are equipped with machinery to calculate stochastic properties of signals and to train network parameters so that the signals behave with the pre-specified stochastic properties. On the other hand it is generally recognized that there exists an overfitting problem when identification of nonlinear dynamic systems with noise is done by neural networks. In this paper, it is shown from simulation results of identification of a nonlinear robot dynamics that PrULNs are useful for avoiding the overfitting.

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

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.

Time–frequency characterization of nonlinear normal modes and challenges in nonlinearity identification of dynamical systems

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.

A recursive algorithm for identification of nonlinear dynamic systems with backlash is proposed in this paper. In this method, the backlash, which is a non‐smooth function, is decomposed into a combination of a group of piecewise linearized models so that all the parameters of the backlash can be estimated separately. Moreover, the model of the backlash is embedded into a Hammerstein‐type model. Thus, a pseudo‐Hammerstein model with backlash is constructed. The estimation of the parameters for such a non‐smooth nonlinear system can be implemented through a so‐called recursive general identification algorithm (RGIA). Then, the corresponding convergence analysis of the RGIA for the model with backlash is also investigated. After that, two examples are presented to show the performance of the proposed method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

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.

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.

The goal of this paper is to introduce a new neural network architecture called Sigmoid Diagonal Recurrent Neural Network (SDRNN) to be used in the adaptive control of nonlinear dynamical systems. This is done by adding a sigmoid weight victor in the hidden layer neurons to adapt of the shape of the sigmoid function making their outputs not restricted to the sigmoid function output. Also, we introduce a dynamic back propagation learning algorithm to train the new proposed network parameters. The simulation results showed that the (SDRNN) is more efficient and accurate than the DRNN in both the identification and adaptive control of nonlinear dynamical systems.

In recent years, multilayer neural networks and recurrent networks have emerged as important components for representing nonlinear transformations and have proved particularly successful in pattern recognition and optimization problems. The authors explore methods for incorporating such networks in adaptive systems for the identification and control of complex nonlinear dynamical systems. >