Online identification of nonlinear spatiotemporal systems using kernel learning approach

Online Identification of Nonlinear Stochastic Spatiotemporal System With Multiplicative Noise by Robust Optimal Control-Based Kernel Learning Method

In this paper, we propose a novel kernel method for the online identification of stochastic nonlinear spatiotemporal dynamical systems using the robust control approach. By the difference method, the stochastic spatiotemporal (SST) systems driven by multiplicative noise are first transformed into a class of multi-input-multi-output-partially linear kernel models (PLKMs) with heterogeneous random terms. With the help of techniques established for reproducing kernel Hilbert space, the online learning problem is reasonably considered as an output feedback control problem for a group of time varying linear dynamical systems. We develop an effective algorithm to address the learning problem of PLKM and SST systems by employing the model predictive control theory. Compared with the existing learning methods, the new one can achieve adaptive, robust, and fast convergent online modeling performance for the spatiotemporal dynamics with multiplicative noise, which greatly facilitates the characterization of physical characteristics of the system. Moreover, this investigation for the first time addresses the learning problems for SST systems with novel robust control techniques, which can provide some novel insights into the design of kernel machine learning methods from the perspective of optimal control theory. Numerical studies for benchmark systems are presented to illustrate the effectiveness and efficiency of our new method.

An Alternating Identification Algorithm for a Class of Nonlinear Dynamical Systems

While modeling nonlinear systems by combining a linear model with a nonlinear compensation term, namely, virtual unmodeled dynamics (VUD), the parameter estimation of the linear model and the learning-based VUD estimate influences and interacts with each other simultaneously. This paper aims to develop an alternating identification scheme for resolving such a challenging problem, where a projection algorithm is employed to identify the linear model and a feedforward neural network is used to model the VUD of a class of nonlinear dynamical systems. An open-loop estimation algorithm on the VUD is first presented under the known linear model, followed by an alternating identification algorithm for completely unknown nonlinear systems. Algorithm description is given and some simulation studies on multiple input and multiple output nonlinear systems are carried out to illustrate the effectiveness of our proposed modeling techniques.

Identification of Nonlinear Spatiotemporal Dynamical Systems With Nonuniform Observations Using Reproducing-Kernel-Based Integral Least Square Regulation

The identification of nonlinear spatiotemporal dynamical systems given by partial differential equations has attracted a lot of attention in the past decades. Several methods, such as searching principle-based algorithms, partially linear kernel methods, and coupled lattice methods, have been developed to address the identification problems. However, most existing methods have some restrictions on sampling processes in that the sampling intervals should usually be very small and uniformly distributed in spatiotemporal domains. These are actually not applicable for some practical applications. In this paper, to tackle this issue, a novel kernel-based learning algorithm named integral least square regularization regression (ILSRR) is proposed, which can be used to effectively achieve accurate derivative estimation for nonlinear functions in the time domain. With this technique, a discretization method named inverse meshless collocation is then developed to realize the dimensional reduction of the system to be identified. Thereafter, with this novel inverse meshless collocation model, the ILSRR, and a multiple-kernel-based learning algorithm, a multistep identification method is systematically proposed to address the identification problem of spatiotemporal systems with pointwise nonuniform observations. Numerical studies for benchmark systems with necessary discussions are presented to illustrate the effectiveness and the advantages of the proposed method.

Numerical analysis of modeling based on improved Elman neural network

A modeling based on the improved Elman neural network (IENN) is proposed to analyze the nonlinear circuits with the memory effect. The hidden layer neurons are activated by a group of Chebyshev orthogonal basis functions instead of sigmoid functions in this model. The error curves of the sum of squared error (SSE) varying with the number of hidden neurons and the iteration step are studied to determine the number of the hidden layer neurons. Simulation results of the half-bridge class-D power amplifier (CDPA) with two-tone signal and broadband signals as input have shown that the proposed behavioral modeling can reconstruct the system of CDPAs accurately and depict the memory effect of CDPAs well. Compared with Volterra-Laguerre (VL) model, Chebyshev neural network (CNN) model, and basic Elman neural network (BENN) model, the proposed model has better performance.

Model-based online learning with kernels

New optimization models and algorithms for online learning with Kernels (OLK) in classification, regression, and novelty detection are proposed in a reproducing Kernel Hilbert space. Unlike the stochastic gradient descent algorithm, called the naive online Reg minimization algorithm (NORMA), OLK algorithms are obtained by solving a constrained optimization problem based on the proposed models. By exploiting the techniques of the Lagrange dual problem like Vapnik's support vector machine (SVM), the solution of the optimization problem can be obtained iteratively and the iteration process is similar to that of the NORMA. This further strengthens the foundation of OLK and enriches the research area of SVM. We also apply the obtained OLK algorithms to problems in classification, regression, and novelty detection, including real time background substraction, to show their effectiveness. It is illustrated that, based on the experimental results of both classification and regression, the accuracy of OLK algorithms is comparable with traditional SVM-based algorithms, such as SVM and least square SVM (LS-SVM), and with the state-of-the-art algorithms, such as Kernel recursive least square (KRLS) method and projectron method, while it is slightly higher than that of NORMA. On the other hand, the computational cost of the OLK algorithm is comparable with or slightly lower than existing online methods, such as above mentioned NORMA, KRLS, and projectron methods, but much lower than that of SVM-based algorithms. In addition, different from SVM and LS-SVM, it is possible for OLK algorithms to be applied to non-stationary problems. Also, the applicability of OLK in novelty detection is illustrated by simulation results.

Online kernel-based learning for task-space tracking robot control

Task-space control of redundant robot systems based on analytical models is known to be susceptive to modeling errors. Data-driven model learning methods may present an interesting alternative approach. However, learning models for task-space tracking control from sampled data is an ill-posed problem. In particular, the same input data point can yield many different output values, which can form a nonconvex solution space. Because the problem is ill-posed, models cannot be learned from such data using common regression methods. While learning of task-space control mappings is globally ill-posed, it has been shown in recent work that it is locally a well-defined problem. In this paper, we use this insight to formulate a local kernel-based learning approach for online model learning for task-space tracking control. We propose a parametrization for the local model, which makes an application in task-space tracking control of redundant robots possible. The model parametrization further allows us to apply the kernel-trick and, therefore, enables a formulation within the kernel learning framework. In our evaluations, we show the ability of the method for online model learning for task-space tracking control of redundant robots.

Approximate solutions to ordinary differential equations using least squares support vector machines

In this paper, a new approach based on least squares support vector machines (LS-SVMs) is proposed for solving linear and nonlinear ordinary differential equations (ODEs). The approximate solution is presented in closed form by means of LS-SVMs, whose parameters are adjusted to minimize an appropriate error function. For the linear and nonlinear cases, these parameters are obtained by solving a system of linear and nonlinear equations, respectively. The method is well suited to solving mildly stiff, nonstiff, and singular ODEs with initial and boundary conditions. Numerical results demonstrate the efficiency of the proposed method over existing methods.