Control of discrete chaotic systems based on echo state network modeling with an adaptive noise canceler

Abstract

In this paper, we present a new method based on echo state network (ESN) to control discrete chaotic systems. ESN could achieve very high precision in chaotic time series prediction and overcome most issues encountered in using traditional artificial neural networks, especially local minima and overfitting. In order to achieve good control effect when there is noise in chaotic systems, an adaptive noise canceler is introduced to eliminate the effect of the noise and perturbation. The support vector machine (SVM) is adopted to identify inverse model of the controlled plant as the adaptive noise canceler. Simulation results show that the proposed method could achieve very good control effect, possess a good stability and completely reduce the adverse effect.

Introduction

Chaos is a very interesting nonlinear phenomenon and has already been studied extensively during the past two decades. It is associated with deterministic dynamics, however, it looks like random behavior usually seen in stochastic systems. The fundamental characteristic of chaotic systems is their sensitive dependence on initial conditions; in other words, a small shift in the initial states may lead to extraordinary perturbation in the system states. Therefore it could easily lead systems to instability, performance degradation, or even catastrophic situations. In such cases, chaos is considered as an unexpectable phenomenon and should be entirely avoided or completely eliminated. In recent years, many engineering applications of chaotic systems have been studied including secure communication, digital communication, power electronic devices, power quality, biological systems, chemical reaction analysis and information processing. The control and synchronization of chaotic systems have attracted increasing research attention, and many methods have been applied to the identification [1], prediction [2], [3] and control [4], [5], [6] of a variety of chaotic systems. Due to the universal approximation property of artificial neural networks (ANN) [7], [8], [9], [10], it has been widely applied in the chaotic system. However, the free parameters of ANN are learnt from the given training samples by the gradient descent algorithms, which are relatively slow and have many issues related to its local minimum and overfitting. Owing to these shortages, it limits the development of conventional artificial neural network.

Recently, a novel and surprisingly easy to use network structure of recurrent neural networks (RNNs) was proposed independently by Jaeger in 2004, which is called echo state network (ESN) [2]. The main idea of ESN is to separate architectures of RNNs into two crucial parts: a recurrent structure which is hidden layer and called the dynamical reservoir, and a liner output layer that is called readout neuron. The dynamical reservoir owns the echo state property, which is guaranteed by randomly generating a large network of the hidden layer neurons with a special degree of sparsity. By employing a sparse and fixed hidden layer, we could analytically calculate the output weights. ESN training procedure is a simple adjustment of output weights to fit training data. ESN has been successfully applied in a variety of domains [11], [12], [13], [14], [15], especially in the prediction of chaotic systems [16], for example, ESN could achieve the Normalized Root Mean Square Error (NRMSE) 10^−4.2 in the classical benchmark of Mackey–Glass 84-step prediction, however, the prediction accuracy of conventional neural network is only 10^−1.2–10^−1.7 [2]. At the same time, the prediction precision by ESN is also very high in other chaotic time series prediction [17], [18].

However, currently ESN in chaos prediction research is limited to the systems which contain no noise or high signal-to-noise ratio of time sequence. There is always noise and perturbation in most practical applications, so we introduced an adaptive noise canceler which is the inverse model of the controlled plant in order to reduce the influence of noise and perturbation.

Support vector machine (SVM) is a new and valid machine learning algorithm [19], [20]. Due to overcoming most shortcomings of conventional neural networks, it could be used in regression estimation as well as in classification [21], [22], [23], which possesses better generalization than the conventional neural network. Nowadays, it has become one of the classic nonlinear system identification tools in machine learning [24], [25], [26], [27], [28], [29]. However the training speed of the SVM is very slow and it is not suitable for real-time control application.

In view of the advantages and disadvantages of the two new learning machines, we combined the two techniques to control discrete chaotic system and eliminate noise.

The paper is organized as follows: the echo state machine and support vector machine are reviewed in Section 2. Section 3 describes the process of designing controller and canceling noise. Section 4 shows the simulation and its results. Finally, Section 5 makes a summary of this paper.