NARX model based nonlinear dynamic system identification using low complexity neural networks and robust H∞ filter

This paper proposes NARX (nonlinear autoregressive model with exogenous input) model structures with functional expansion of input patterns by using low complexity ANN (artificial neural network) for nonlinear system identification. Chebyshev polynomials, Legendre polynomials, trigonometric expansions using sine and cosine functions as well as wavelet basis functions are used for the functional expansion of input patterns. The past input and output samples are modeled as a nonlinear NARX process and robust H_∞ filter is proposed as the learning algorithm for the neural network to identify the unknown plants. H_∞ filtering approach is based on the state space modeling of model parameters and evaluation of Jacobian matrices. This approach is the robustification of Kalman filter which exhibits robust characteristics and fast convergence properties. Comparison results for different nonlinear dynamic plants with forgetting factor recursive least square (FFRLS) and extended Kalman filter (EKF) algorithms demonstrate the effectiveness of the proposed approach.     

Identification of multiscale spatio-temporal dynamical systems using a wavelet multiresolution analysis

In this article, a new algorithm for the multiscale identification of spatio-temporal dynamical systems is derived. It is shown that the input and output observations can be represented in a multiscale manner based on a wavelet multiresolution analysis. The system dynamics at some specific scale of interest can then be identified using an orthogonal forward least-squares algorithm. This model can then be converted between different scales to produce predictions of the system outputs at different scales. The method can be applied to both multiscale and conventional spatio-temporal dynamical systems. For multiscale systems, the method can generate a parsimonious and effective model at a coarser scale while considering the effects from finer scales. Additionally, the proposed method can be used to improve the performance of the identification when the measurements are noisy. Numerical examples are provided to demonstrate the application of the proposed new approach.     

基于小波网络的实时稳定器研究

提出一种应用小波神经网络进行动态实时调整侍统电力系统稳定器参数的设计方法。由于小波神经网络所具有的非线性逼近能力及良好的时频分析能力，系统能精确地辨识动态特性，映射更复杂的控制策略。仿真结果表明小波网络电力系统稳定器比传统电力系统稳定器更有效。     

Identification of partial differential equation models for a class of multiscale spatio-temporal dynamical systems

In this article, the identification of a class of multiscale spatio-temporal dynamical systems, which incorporate multiple spatial scales, from observations is studied. The proposed approach is a combination of Adams integration and an orthogonal least squares algorithm, in which the multiscale operators are expanded, using polynomials as basis functions, and the spatial derivatives are estimated by finite difference methods. The coefficients of the polynomials can vary with respect to the space domain to represent the feature of multiple scales involved in the system dynamics and are approximated using a B-spline wavelet multi-resolution analysis. The resulting identified models of the spatio-temporal evolution form a system of partial differential equations with different spatial scales. Examples are provided to demonstrate the efficiency of the proposed method.     

Estimation of spatial derivatives and identification of continuous spatio-temporal dynamical systems

A new approach for the estimation of spatial derivatives and the identification of a class of continuous spatio-temporal dynamical systems from experimental data is presented in this study. The proposed identification approach is a combination of implicit Adams integration and an orthogonal forward regression algorithm (OFR), in which the operators are expanded using polynomials as basis functions. The noisy experimental data are de-noised by using biorthogonal spline wavelet filters and the spatial derivatives are estimated using a multiresolution analysis method. Finally, a bootstrap method is applied to refine the identified parameters from the OFR algorithm. The resulting identified models of the spatio-temporal evolution form a system of partial differential equations. Examples are provided to demonstrate the efficiency of the proposed method.     

Modeling and complexity in neural networks

In this paper, we study nonlinear spatio-temporal dynamics in synchronous and asynchronous chaotic neural networks from the viewpoint of the modeling and complexity of the dynamic brain. First, the possible roles and functions of spatio-temporal neurochaos are considered with a model of synchronous chaotic neural networks composed of a neuron model with a chaotic map. Second, deterministic point-process dynamics with spikes of action potentials is demonstrated with a biologically more plausible model of asynchronous chaotic neural networks. Last, the possibilities of inventing a new brain-type of computing system are discussed on the basis of these models of chaotic neural networks. This work was presented, in part, at the Third International Symposium on Artificial Life and Robotics, Oita, Japan, January 19–21, 1998.     

A multi-channel spatio-temporal Hammerstein modeling approach for nonlinear distributed parameter processes

Modeling of distributed parameter processes is a challenging problem because of their complex spatio-temporal nature, nonlinearities and uncertainties. In this study, a spatio-temporal Hammerstein modeling approach is proposed for nonlinear distributed parameter processes. Firstly, the static nonlinear and the distributed dynamical linear parts of the Hammerstein model are expanded onto a set of spatial and temporal basis functions. In order to reduce the parametric complexity, the Karhunen–Loève decomposition is used to find the dominant spatial bases with Laguerre polynomials selected as the temporal bases. Then, using the Galerkin method, the spatio-temporal modeling will be reduced to a traditional temporal modeling problem. Finally, the unknown parameters can be easily estimated using the least squares estimation and the singular value decomposition. In the presence of unmodeled dynamics, a multi-channel modeling framework is proposed to further improve the modeling performance. The convergence of the modeling can be guaranteed under certain conditions. The simulations are presented to show the effectiveness of this modeling method and its potential to a wide range of distributed processes.     

Diffusion-Inspired Shrinkage Functions and Stability Results for Wavelet Denoising

We study the connections between discrete one-dimensional schemes for nonlinear diffusion and shift-invariant Haar wavelet shrinkage. We show that one step of a (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusion-inspired shrinkage functions are among the best for translation-invariant multiscale wavelet denoising. Moreover, by transferring stability notions from diffusion filtering to wavelet shrinkage, we derive conditions on the shrinkage function that ensure that shift invariant single-level Haar wavelet shrinkage is maximum–minimum stable, monotonicity preserving, and variation diminishing.First online version published in June, 2005     

Wavelet reduced order observer based adaptive tracking control for a class of uncertain nonlinear systems using reinforcement learning

This Paper investigates the mean to design the reduced order observer and observer based controllers for a class of uncertain nonlinear system using reinforcement learning. A new design approach of wavelet based adaptive reduced order observer is proposed. The proposed wavelet adaptive reduced order observer performs the task of identification of unknown system dynamics in addition to the reconstruction of states of the system. Reinforcement learning is used via two wavelet neural networks (WNN), critic WNN and action WNN, which are combined to form an adaptive WNN controller. The “strategic” utility function is approximated by the critic WNN and is minimized by the action WNN. Owing to their superior learning capabilities, wavelet networks are employed in this work for the purpose of identification of unknown system dynamics. Using the feedback control, based on reconstructed states, the behavior of closed loop system is investigated. By Lyapunov approach, the uniformly ultimate boundedness of the closed-loop tracking error is verified. A numerical example is provided to verify the effectiveness of theoretical development.     

Wavelet networks for nonlinear system modeling

This study presents a nonlinear systems and function learning by using wavelet network. Wavelet networks are as neural network for training and structural approach. But, training algorithms of wavelet networks is required a smaller number of iterations when the compared with neural networks. Gaussian-based mother wavelet function is used as an activation function. Wavelet networks have three main parameters; dilation, translation, and connection parameters (weights). Initial values of these parameters are randomly selected. They are optimized during training (learning) phase. Because of random selection of all initial values, it may not be suitable for process modeling. Because wavelet functions are rapidly vanishing functions. For this reason heuristic procedure has been used. In this study serial-parallel identification model has been applied to system modeling. This structure does not utilize feedback. Real system outputs have been exercised for prediction of the future system outputs. So that stability and approximation of the network is guaranteed. Gradient methods have been applied for parameters updating with momentum term. Quadratic cost function is used for error minimization. Three example problems have been examined in the simulation. They are static nonlinear functions and discrete dynamic nonlinear system.     

基于小波网络的动态系统辨识方法及应用*

本文介绍了一种多输入非线性动态系统辨识算法，基于该算法的非线性辨识系统成功用于局部地区短时暴雨的预报。在这个系统中我们采用一种小波网络来追踪非线性系统的动态性，用一种基于小波逼近的非参数估计方法用于系统的状态空间模型的辨识中。从实验结果可看出，与传统的神经网络方法相比，该系统在速度、可靠性以及精确度上都有了很大的提高。     

Nonsingular Terminal Sliding Mode Control of Robot Manipulators Using Fuzzy Wavelet Networks

This paper presents an adaptive nonsingular terminal sliding mode (NTSM) tracking control design for robotic systems using fuzzy wavelet networks. Compared with linear hyperplane-based sliding control, terminal sliding mode controller can provide faster convergence and higher precision control. Therefore, a terminal sliding controller combined with the fuzzy wavelet network, which can accurately approximate unknown dynamics of robotic systems by using an adaptive learning algorithm, is an attractive control approach for robots. In addition, the proposed learning algorithm can on-line tune parameters of dilation and translation of fuzzy wavelet basis functions and hidden-to-output weights. Therefore, a robust control law is used to eliminate uncertainties including the inevitable approximation errors resulted from the finite number of fuzzy wavelet basis functions. The proposed controller requires no prior knowledge about the dynamics of the robot and no off-line learning phase. Moreover, both tracking performance and stability of the closed-loop robotic system can be guaranteed by Lyapunov theory. Finally, the effectiveness of the fuzzy wavelet network-based control approach is illustrated through comparative simulations on a six-link robot manipulator     

Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform global asymptotic, and input-to-state stability of fast time-varying dynamics. We also construct strict Lyapunov functions for our systems using a strictification approach. We illustrate our results using several examples.     

Local recurrence based performance prediction and prognostics in the nonlinear and nonstationary systems

This paper presents a local recurrence modeling approach for state and performance predictions in complex nonlinear and nonstationary systems. Nonstationarity is treated as the switching force between different stationary systems, which is shown as a series of finite time detours of system dynamics from the vicinity of a nonlinear attractor. Recurrence patterns are used to partition the system trajectory into multiple near-stationary segments. Consequently, piecewise eigen analysis of ensembles in each near-stationary segment can capture both nonlinear stochastic dynamics and nonstationarity. The experimental studies using simulated and real-world datasets demonstrate significant prediction performance improvements in comparison with other alternative methods.     

Error-transformation-based consensus algorithms of multi-agent systems: connectivity-preserving approach

This paper investigates distributed connectivity-preserving consensus problems of networked multi-agent systems with limited communication ranges. Compared with existing literature, a main contribution of this paper is to present a new nonlinear transformation approach of consensus errors for preserving the initial interaction patterns of multi-agent systems. Both the consensus and the connectivity preservation can be achieved by using one transformed error function. Based on the proposed nonlinear error transformation, we derive distributed connectivity-preserving consensus algorithms for single-integrator dynamics, double-integrator dynamics and second-order nonlinear systems. The asymptotic stability of consensus errors and the connectivity preservation among agents are established through Lyapunov stability analysis.     

基于能量密度的小波神经网络

本文提出了基于能量密度构造单隐层前向小波网络用以逼近复杂非线性函数。在时频定位分析的基础上，引入了能量密度的概念，用其作为选择小波元的标准。在本文中给出了网络构造算法及相应的学习算法，并与其它小波网及ＢＰ网进行了比较。实验结果证明了该方法是可行的，且具有小波元数目相对较少、学习收敛速度快等特点，并就其在实际应用中应注意的问题提出了我们的观点。     

Adaptive fuzzy backstepping output feedback control for strict feedback nonlinear systems with unknown sign of high-frequency gain

In this paper, an adaptive fuzzy robust output feedback control approach is proposed for a class of SISO nonlinear strict-feedback systems with unknown sign of high-frequency gain and the unmeasured states. The nonlinear systems addressed in this paper are assumed to possess the unmodeled dynamics, dynamical disturbances and unknown nonlinear functions, where the unknown nonlinear functions are not linearly parameterized, and no prior knowledge of their bounds is available. In the recursive designing, fuzzy logic systems are used to approximate the unknown nonlinear functions, K-filters are designed to estimate the unmeasured states, and a dynamical signal and Nussbaum gain functions are introduced to handle the unmodeled dynamics and the unknown sign of the high-frequency gain, respectively. Based on Lyapunov function method, a stable adaptive fuzzy output feedback control scheme is developed. It is mathematically proved that the proposed adaptive fuzzy control approach can guarantee that all the signals of the closed-loop system are uniformly ultimately bounded, the output converges to a small neighborhood of the origin. The effectiveness of the proposed approach is illustrated by the simulation examples.     

A new class of wavelet networks for nonlinear system identification

A new class of wavelet networks (WNs) is proposed for nonlinear system identification. In the new networks, the model structure for a high-dimensional system is chosen to be a superimposition of a number of functions with fewer variables. By expanding each function using truncated wavelet decompositions, the multivariate nonlinear networks can be converted into linear-in-the-parameter regressions, which can be solved using least-squares type methods. An efficient model term selection approach based upon a forward orthogonal least squares (OLS) algorithm and the error reduction ratio (ERR) is applied to solve the linear-in-the-parameters problem in the present study. The main advantage of the new WN is that it exploits the attractive features of multiscale wavelet decompositions and the capability of traditional neural networks. By adopting the analysis of variance (ANOVA) expansion, WNs can now handle nonlinear identification problems in high dimensions.     

Perceptual hash function for scalable video

Perceptual hash functions are important for video authentication based on digital signature verifying the originality and integrity of videos. They derive hashes from the perceptual contents of the videos and are robust against the common content-preserving operations on the videos. The advancements in the field of scalable video coding call for efficient hash functions that are also robust against the temporal, spatial and bit rate scalability features of the these coding schemes. This paper presents a new algorithm to extract hashes of scalably coded videos using the 3D discrete wavelet transform. A hash of a video is computed at the group-of-frames level from the spatio-temporal low-pass bands of the wavelet-transformed groups-of-frames. For each group-of-frames, the spatio-temporal low-pass band is divided into perceptual blocks and a hash is derived from the cumulative averages of their averages. Experimental results demonstrate the robustness of the hash function against the scalability features and the common content-preserving operations as well as the sensitivity to the various types of content differences. Two critical properties of the hash function, diffusion and confusion, are also examined.     

A novel approach for high dimension 3D object representation using Multi-Mother Wavelet Network

In this paper, we present a novel approach for 3D objects representation. Our idea is to prove that wavelet networks are capable for reconstruction and representing irregular 3D objects used in computer graphics. The major contribution consist to transform an input surface vertices into signals and to provide instantaneously an estimation of the output values for input values. To prove this, we will use a new structure of wavelet network founded on several mother wavelet families. This structure uses several mother wavelet, in order to maximize best wavelet selection probability. An algorithm to construct this structure is presented. First, Data is taken from 3D object. The vertices and their corresponding normal values of a 3D object are used to create a training set. To this stage, the training set can be expressed according to three functions, which interpolates all their vertices. Second we approximate each function using wavelet network. To achieve a better approximation, the network is trained several iterations to optimize wavelet selection for every mother. To guarantee a small error criterion, we adjust wavelet network parameters (weight, translation and dilation) by using an improved Orthogonal Least Squares method version. We consider our proposed approach on some 3D examples to prove that the new approach is able to approximate 3D objects with a good approximation ability.