Mathematical Problems in Engineering

Volume 2015, Article ID 181389, 9 pages

http://dx.doi.org/10.1155/2015/181389

## Quasilinear Extreme Learning Machine Model Based Internal Model Control for Nonlinear Process

Institute of Automation, Beijing University of Chemical Technology, No. 15 East Road of the North 3rd Ring-Road, Chao Yang District, Beijing 100029, China

Received 17 August 2014; Revised 21 October 2014; Accepted 22 October 2014

Academic Editor: Jiuwen Cao

Copyright © 2015 Dazi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new strategy for internal model control (IMC) is proposed using a regression algorithm of quasilinear model with extreme learning machine (QL-ELM). Aimed at the chemical process with nonlinearity, the learning process of the internal model and inverse model is derived. The proposed QL-ELM is constructed as a linear ARX model with a complicated nonlinear coefficient. It shows some good approximation ability and fast convergence. The complicated coefficients are separated into two parts. The linear part is determined by recursive least square (RLS), while the nonlinear part is identified through extreme learning machine. The parameters of linear part and the output weights of ELM are estimated iteratively. The proposed internal model control is applied to CSTR process. The effectiveness and accuracy of the proposed method are extensively verified through numerical results.

#### 1. Introduction

Internal model control is to design control strategy based on a kind of mathematical model of the process. Because of its obvious dynamic and static performance, as well as simple structure and strong robustness, internal model control plays an increasingly significant effect in control area [1, 2]. Two crucial problems in the inverse system approach are identification of plant model and determination of controller settings. For a complex nonlinear system, it is difficult to obtain an accurate internal model and its inverse model. In recent years, much effort has been devoted to nonlinear system modeling based on artificial neural networks (NNs) and support vector machine (SVM) [3–5]. It is widely applied to use the solution of trained inverse model as a nonlinear controller [6]. However, the disadvantages of the dynamic gradient method lie in its long training time, minor update of weights, and high probability of training failure. SVM method based on standard optimization often suffers from parameter adjustment difficulties. Moreover, the update information based on errors in internal model and inverse model also leads to decrease of the control performance [7].

To deal with the above problems, extreme learning machine (ELM) proposed by Huang et al. [8–10] shows great advantages. Its simplified neural network structure makes the learning speed fast. A smaller training error can be obtained via a canonical equation. The advantage of ELM is its low computational effort and high generalization ability. Therefore, ELM has been successfully applied in many areas, such as classification of EEG signals and protein sequence [11], building regression model [12, 13], and fault diagnosis [14, 15].

For the nonlinear modeling, the key point is to find a suitable model structure. Volterra model is a kind of crucial nonlinear system model [16]. It provides an elaborate mathematical description for a great many of nonlinear systems. Recently, some researchers proposed the proof of inverse theory for IMC based on Volterra model [17]. However, the obvious shortcoming that limits its application is its high complexity in the identification of kernel function.

In recent years, some block-oriented models have been proposed and applied widely, such as Wiener model and Hammerstein model [18–21] which consist of a static nonlinear function and a linear dynamic subsystem. Both of them are of simple structures and can be used to identify some highly nonlinear process, such as pH neutralization process [22] and fermentation process [23]. However, sometimes it is difficult to separate the system concerned into a linear dynamic block and a memoryless nonlinear one. Another class of methods based on local linearization of the structure, combining the nonlinear nonparametric models with some conventional statistical models, has achieved some great results. McLoone et al. [24] proposed an off-line hybrid training algorithm for feed-forward neural networks. Peng et al. [25, 26] proposed hybrid pseudolinear RBF-AR, RBF-ARX models. A cascaded structure of the ARX-NN model is proposed by Hu et al. [27]. The idea of these two different classes of methods is to separate the linear and nonlinear identification, so as to facilitate the inverse computation.

However, these models show high nonlinear characteristics, which are difficult to analysis in theory, without exploiting some good linearity properties. It is well known that simple structure, such as ARX model, has a lot of advantages in modeling. Firstly, its linear properties will significantly simplify the parameter estimation. Secondly, it is convenient to deduce regression predictor. Furthermore, linearity structure is also convenient for control design as well as the control law derivation. A good representation cannot only approximate the nonlinear function accurately, but also simplify the identification process. Hu et al. [28, 29] proposed a quasilinear model constructed by a linear structure using a quasilinear ARX model for nonlinear process mapping. From a macrostandpoint, the model can be seen as a linear structure which is a redundant for the regression ability. Its complex coefficients reflect the nonlinearity of the system. The model has a great flexibility to deal with the system nonlinearity.

Inspired by this kind of quasilinear ARX model as well as the thought of separate identification, the motivation of this paper is intended to propose a class of quasilinear ELM model, which can be separated into a linear part and a nonlinear kernel part. It cannot only identify ordinary nonlinear system, but also simplify the identification process via separating the model complexity. In this paper, a novel internal model control based on quasilinear-ELM (QL-ELM) structure is proposed for CSTR system. Taking advantage of separate identification, the quasilinear model consists of a linear part and a nonlinear kernel part. The parameters of nonlinear part are estimated by ELM, which increases the flexibility of the model. The linear parameters are estimated by using the RLS method. A recursive algorithm is conducted to estimate the parameters in both parts. Moreover, QL-ELM is used to set up the internal and inverse model of nonlinear CSTR systems. Through the establishment of the inverse model, the control action is obtained to achieve fixed-point control and tracking control of concentration. Taking the advantage of its characteristics of high modeling accuracy and less human interference, the closed-loop system control is more stable and has less steady-state deviation. Simulation results demonstrate the dynamic performance and tracking ability of the proposed QL-ELM based IMC strategy.

This paper is organized in six sections. Following the introduction, the traditional extreme learning machine is illustrated in Section 2. In Section 3 the algorithm of QL-ELM is presented. IMC with QL-ELM is described in Section 4. To show the applicability of the proposed method, simulations results for CSTR are presented in Section 5. Finally, the conclusion is presented in Section 6.

#### 2. Extreme Learning Machine: Basic Principles

For the input nodes and output nodes the single-hidden layer feed-forward neural networks (SLFNs) with hidden nodes and activation function can be expressed aswhere is the vector of weights between hidden layer and the output nodes. is the vector of weights between input vectors and hidden layer. In addition, is the bias of the th hidden node. ELM with wide types of activation functions can get high regression accuracy. Unlike other traditional implementations, the input weights and biases are randomly chosen in extreme learning machine. The output of the hidden layer is written as a matrix , and (1) can be rewritten aswhere

With the theorems proposed in [8, 9], the input weights and the hidden layer biases are randomly generated without further tuning. It is the main idea of the ELM that training problem is simplified to find a least square solution. According to the Moore-Penrose generalized inverse theory, the output can be calculated by using the following equation:

It must be the smallest norm solution among all the solutions. The one step algorithm can produce best generalization performance and learn much faster than traditional learning algorithms. It can also avoid local optimum.

#### 3. The Quasilinear ELM Model Treatment

A quasilinear ELM model can be seen as a SLFN embedded in the coefficients of a linear model. The feature of the quasilinear ELM model is that it has both good approximation abilities and easy-to-use properties. For a nonlinear SISO system described bywhere . is the regression vector, , are the order of the system. , . is a stochastic noise with zero-mean.

Assume that is continuous and differentiable at a small region around . By using Taylor equation [30], can be further expanded as

Set ; . Picking up common factor from (6), then the following can be obtained:where . It can be seen as the coefficient of nonlinear function .

The quasilinear model has a linear structure ARX model with a functional coefficient . It can be separated into a nonlinear part and a linear part described aswhere .

For case of near linear system, nonlinear part is the supplement for nonlinear feature, so good regression results can be achieved. For case of the nonlinear system, nonlinear network as an interpolated coefficient can be used to expend the regression space. Equation (10) can be seen as the linear form with a nonlinear coefficient , which is actually a problem of function approximation from a multidimensional input space into a one-dimensional scalar space . Using ELM to estimate nonlinear part parameters will be more convenient and concise. Replacing by ELM, the model in (7) can be rewritten aswhere ; then the quasilinear ELM model can be further expressed aswhere the activation function is chosen as . The whole identification process based on QL-ELM is described in Figure 1, where , and are orders of the input and output, and are weight matrices of the input and output layer, is bias vector of hidden nodes, and is the parameter of linear part and also can be seen as the bias vector of output nodes. The parameters of two submodels are updated during each iterative process until the ultimate goal to make the error between the output of actual model and the QL-ELM model minimized. The deviation between and is used to update the nonlinear part through ELM learning. The deviation between and is used to update the linear part through recursive least squares.