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As a kind of novel feedforward neural network with single hidden layer, ELM (extreme learning machine) neural networks are studied for the identification and control of nonlinear dynamic systems. The property of simple structure and fast convergence of ELM can be shown clearly. In this paper, we are interested in adaptive control of nonlinear dynamic plants by using OS-ELM (online sequential extreme learning machine) neural networks. Based on data scope division, the problem that training process of ELM neural network is sensitive to the initial training data is also solved. According to the output range of the controlled plant, the data corresponding to this range will be used to initialize ELM. Furthermore, due to the drawback of conventional adaptive control, when the OS-ELM neural network is used for adaptive control of the system with jumping parameters, the topological structure of the neural network can be adjusted dynamically by using multiple model switching strategy, and an MMAC (multiple model adaptive control) will be used to improve the control performance. Simulation results are included to complement the theoretical results.

Modeling and controlling of nonlinear systems are always the main research field in control theory and engineering [

The training of batch processing ELM algorithms can be realized only after all the training samples are ready. But in actual application, the training data may come one by one or chunk by Huang et al. [

To the best of our knowledge, a large number of researches about ELM are focused on the regression and classification [

The paper is organized as follows. The structure of ELM neural network and OS-ELM algorithm are introduced in Section

Let us consider standard single-hidden-layer feedforward neural networks (SLFNs) with

Single-hidden-layer feedforward networks.

To approximate these

The hidden node parameters

As mentioned above in Section

Assign random parameters of hidden nodes

Calculate the initial hidden-layer output matrix

Calculate the initial output weight

Set

(b) Calculate the partial hidden-layer output matrix

Calculate the output weight

Set

It can be seen from the above OS-ELM algorithm that OS-ELM becomes the batch ELM when

Consider the following SISO nonlinear discrete-time system:

Let

Due to the existence of unmodelling dynamics, the nonlinear system (

Given the system described by (

(i)

(ii)

(a)

(b)

For the detailed proof of Theorem

In this section we discuss the adaptive control problem using OS-ELM neural networks. For the system described as

Assign random parameters of hidden nodes

Calculate the initial hidden-layer output matrix

where

Calculate the initial output weight as follows:

where

Set

(b) Calculate the partial hidden-layer output matrix

(c) Calculate the output weight

(d) Set

Define the error between the neural network identification state and the reference model state as

To sum up, we find that OS-ELM neural networks have the capability of identification of and controling nonlinear systems. However, as we know, the conventional adaptive control systems are usually based on a fixed or slowly adaptive model. It cannot react quickly to abrupt changes and will result in large transient errors before convergence. In this case, MMAC algorithm is presented as a useful tool. We can construct multiple OS-ELM neural networks to cover the uncertainty of the parameters of the plant by initializing OS-ELM neural network, respectively, in different position. Meanwhile, an effective index function can be used to select the optimal identification model and the optimal controller. MMAC based on OS-ELM algorithm can improve the control property of the system greatly and avoid the influence of jumping parameter on the plant.

Multiple model adaptive control can be regarded as an extension of conventional indirect adaptive control. The control system contains

Structure of multiple models adaptive control.

Referring to paper [

In this section, we present results of simulations of adaptive control nonlinear discrete-time systems by using OS-ELM neural networks. The nonlinear systems will be considered as below:

For OS-ELM control algorithm, in Step 1, we choose

Following this, in Step 2, both identification and control are implemented. The response of the controlled system with a reference input

Nonlinear systems adaptive control by using OS-ELM.

Control error.

In simulation process, we find that OS-ELM is sensitive to the initial training data. Initial training data determine the initial value of adaptive control directly. When we select different initial training samples, the performance of the control results varies greatly. Figure

OS-ELM control results with different initial training samples.

In Figure

Data set. (a) Output results with different input data. (b) Different ranges of input values and the corresponding output values concentrate in a single area.

On one hand, in the actual production process, we have a large number of field data which is distributed over a large range. On the other hand, we generally know the objective output range of the controlled plant. Then, according to the output range of the controlled plant, we can choose a specific data area to train the neural network. In this way, the accuracy of control can be improved and the computation time can be saved.

From the above simulation result, we find that OS-ELM neural networks show perfect capability of identification. But from the OS-ELM adaptive control algorithm we can see that it contains two steps (initialization phase and adaptive control phase). This algorithm just shows the perfect capability of identification with fixed or slowly time-varying parameters. For the system with jumping parameters, once the change of parameters happens, the OS-ELM adaptive control algorithm needs to implement Step 1 again or the control performance will be very poor. To solve this kind of problem, MMAC (multiple models adaptive control) will be presented as a very useful tool.

Consider the controlled plant as follows:

Multiple OS-ELM neural network models adaptive control.

In Figure

The application of OS-ELM in the identification and control of nonlinear dynamic system is studied carefully. OS-ELM neural network can improve the accuracy of identification and the control quality dramatically. Since the training process of OS-ELM neural network is sensitive to the initial training data, different scope of training data can be used to initialize OS-ELM neural network based on the output range of the system. The topological structure of OS-ELM neural network can be adjusted dynamically by using multiple model switching strategy, when it is used to control systems with jumping parameters. Simulation results show that MMAC based on OS-ELM which is presented in this paper can improve the control performance dramatically.

If

Defining

Summing both sides of (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is partially supported by the Fund of National Natural Science Foundation of China (Grant no. 61104013), Program for New Century Excellent Talents in Universities (NCET-11-0578), the Fundamental Research Funds for the Central Universities (FRF-TP-12-005B), and Specialized Research Fund for the Doctoral Program of Higher Education (20130006110008).