ZHANG Yan (張燕), LIANG Xiuxia (梁秀霞), YANG Peng (楊鵬), CHEN Zengqiang (陳增強(qiáng)) and YUAN Zhuzhi (袁著祉)

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Modeling and Control of Nonlinear Discrete-time Systems Based on Compound Neural Networks*

ZHANG Yan (張燕)1,**, LIANG Xiuxia (梁秀霞)1, YANG Peng (楊鵬)1, CHEN Zengqiang (陳增強(qiáng))2and YUAN Zhuzhi (袁著祉)2

1Department of Automation, Hebei University of Technology, Tianjin 300130, China2Department of Automation, Nankai University, Tianjin 300071, China

An adaptive inverse controller for nonliear discrete-time system is proposed in this paper. A compound neural network is constructed to identify the nonlinear system, which includes a linear part to approximate the nonlinear system and a recurrent neural network to minimize the difference between the linear model and the real nonlinear system. Because the current control input is not included in the input vector of recurrent neural network (RNN), the inverse control law can be calculated directly. This scheme can be used in real-time nonlinear single-input single-output (SISO) and multi-input multi-output (MIMO) system control with less computation work. Simulation studies have shown that this scheme is simple and affects good control accuracy and robustness.

adaptive inverse control, compound neural network, process control, reaction engineering, multi-input multi-output nonlinear system

1 INTRODUCTION

The other approach is by using a linear state observer and an error compensator to approximate the error between the linear model and the process based on NN. The nonlinear dynamics are inverted directly [14]. The extension of output feedback employing a high-gain observer is given [15]. An error observer approach with-modification and a filter with nonlinearly parameterized NN and modification are introduced to decrease oscillation. A linear dynamic compensator is designed to stabilize the linearized system [16]. A temperature controller with Takagi- Sugeno-Kang-type recurrent fuzzy network (TRFN) designed by direct inverse modeling approach is proposed [17]. The recurrent property of TRFN enables it to be directly applied to a dynamic plant control without a priori knowledge of the plant order. An application of neural networks based additive nonlinear autoregressive exogenous (NNANARX) structure for modeling of nonlinear MIMO systems is presented [18]. Then, the problem of the inverse function calculation in the control algorithm is solved and applied to control nonlinear MIMO systems. Most of these approaches are limited to system in strict-feedback form, and accuracy of the plant model is critical when an inverse is used.

Motivated by this, a novel NN based adaptive inverse control approach is proposed for both SISO and MIMO nonlinear nonaffine discrete systems in this paper. A compound neural network (CNN) is constructed to model the system and an adaptive inverse control strategy is presented. Simulation results can demonstrate the effectiveness and advantages.

2 SYSTEM DESCRIPTION AND PRELIMINARIES

2.1 System representation and identification

The following SISO nonaffine nonlinear discrete- time system is considered [16]:

For convenience of analysis, the future output is determined by a number of past observations of the inputs and outputs. An equivalent input-output representation can be written as the nonlinear auto regressive moving average with exogenous inputs (NARMAX) system:

Classically this controlled system can be realized by the NN-based model at moment:

2.2 Compound neural network

To identify the system shown in Eq. (2), many kinds of NNs can be used. Here, in order to get an inverse controller directly and avoid using another control neural network, a compound neural network (CNN) is proposed. Its structure is shown in Fig. 1. It is composed of two parts: a two-layer linear feedforward neural network (LFNN) and a recurrent neural network (RNN). The output of the identifier can be expressed as:

Figure 1 The structure of compound neural network

The identifier output is rewritten as follows:

In the approximation process, suppose() is the weight matrix of the whole CNN. It is trained by minimizing the following index function.

2.3 Error correction

3 Adaptive inverse controller

Unlike the general NARMAX model, this compound structure can be easily controlled by the dynamic feedback theory. If the model is given by Eq. (2), the control signal is calculated by the following equation:

From Eq. (11), the desired output feedback is given by

Figure 2 Control system architecture

Step 1 Initialize the structure of the compound neural network. Select the relative coefficient: the learning rate, the initial weight vector of CNN, and the error proportional coefficient.

Step 3 Put the trained CNN into the closed loop control, and then, the control signal at timecan be calculate by Eq. (13).

Step 4 Update the CNN weight vector by Eq. (9).

4 MIMO system inverse controller

The controlled nonlinear MIMO system withinputs andoutputs is considered. The controlled system can be represented by the NARMAX model as follows:

Similarly as SISO nonlinear system, a compound neural network can be used to approximate the MIMO NARMAX model. The input vector of the LFNN and RNN are defined as:

The output of CNN can be written as follows:

In the approximation process,() is supposed to be the weight matrix of the whole CNN. The gradient decent method is used to train().

In the control process, as SISO system, the error at timeis introduced to predict the output of the system. The predictive output can be written as:

Because the current control signal() is one part of the input vector of LFNN in the CNN, the control law for the MIMO case can be determined straightforwardly under the inverse control thought:

5 SIMULATION RESULTS

Example 1 A liquid level system is described by the following equations [16]:

The NARMAX model of the process should be written as:

Figure 3 CNN training error

From Fig. 3, it can be seen that the CNN has the accurate identification capability. The nonlinear model of the plant was placed into the inverse control loop. The initial conditions of the plant are set to random variables. Here, the parameters of the CNN used in the inverse control are the same as in the CNN training. The error proportional coefficientis set as 0.7. Fig. 4 presents the control results and the control signal in the process obtained by the proposed inverse control method for tracking square signal. It shows a good tracking result.

Example 2 Isothermal reactor [20]

The following reaction occurs in an ideal stirred tank reactor:

where A is in excess. The reaction rate equation is given by the following equation:

The cross-sectional area of the tank is 1.0 m2and the sampling time is 1.0 min. After simplification, the model becomes:

In the process of the model for the MIMO nonlinear system, 12 nodes are chosen in the hidden layer of RNN for the CNN identifier. First, the following input signals are used to approximate the MIMO nonlinear discrete systems:

In this case, the learning rateis selected as 0.2. From Fig. 5, it can be seen that the CNN model can describethe MIMO nonlinear system with high accuracy.

Figure 5 Identification results of Example 2

Second, the trained CNN is put into the closed-loop inverse control. The control results are illustrated in Fig. 6. It shows that the system can track the reference signals with acceptable approximation errors. The proposed CNN inverse control method shows good trackingperformance for the MIMO nonlinear chemical system.

6 CONCLUSIONS

An adaptive inverse control scheme has been proposed in this paper. To use inverse theory directly, a new compound neural network is proposed. The linear feed-forward neural network is used to approximate the nonlinear controlled process. The recurrent neural network is used to minimize the error between the LFNN and the real nonlinear process. Based on this kind of neural network to approximate the controlled process, an adaptive inverse control scheme can be directly implemented. During this process, an error correction method is proposed to reduce the predictive error. The less computation work is needed since no further training task is required for the neural inverse controller in the I/O domain. This scheme can be used to control both nonlinear dynamic discrete-time SISO and MIMO systems in real time. Simulation results exploit that the proposed scheme is effective and practical.

Figure 6 Tracking performance of Example 2

NOMENCLATURE

the error proportional coefficient

Bconcentration of B in the reactor example

error between the system output and the identifier

a smooth nonlinear function vector

a smooth nonlinear function

Ea smooth linear function vector

Na smooth nonlinear function vector

La smooth linear function

Na smooth nonlinear function

a sigmodal activation function

Lthe input vector of the LFNN in CNN

Nthe input vector of the RNN in CNN

the number of layers of RNN

the degree of system input

the degree of system output

a smooth nonlinear function

the system input

weight matrix of CNN

the system output

Lthe LFNN’s output

Nthe RNN’s output

the learning rate for the weight vector

Subscripts

B the tank B

L the LFNN in the CNN

N the RNN in the CNN

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2008-06-24,

2009-03-20.

the National Natural Science Foundation of China (60575009, 60574036).

** To whom correspondence should be addressed. E-mail: yzhangzz@yahoo.com.cn