Back propagation (BP) neural network is used to approximate the dynamic character of nonlinear discrete-time system. Considering the unmodeling dynamics of the system, the weights of neural network are updated by using a dead-zone algorithm and a robust adaptive controller based on the BP neural network is proposed. For the situation that jumping change parameters exist, multiple neural networks with multiple weights are built to cover the uncertainty of parameters, and multiple controllers based on these models are set up. At every sample time, a performance index function based on the identification error will be used to choose the optimal model and the corresponding controller. Different kinds of combinations of fixed model and adaptive model will be used for robust multiple models adaptive control (MMAC). The proof of stability and convergence of MMAC are given, and the significant efficacy of the proposed methods is tested by simulation.

Due to the strong ability of approximation, neural network has been widely used in the identification of nonlinear system. It is also a very useful tool for prediction, pattern recognition, and control [

Adaptive control of nonlinear systems using neural network has been an active research area for over two decades [

Since MMAC was presented in 1970s, it has attracted a lot of attention of experts [

In this paper, a kind of robust MMAC is proposed for nonlinear system. Multiple BP neural networks with different weights will be used to cover the uncertainty of the parameters of the system. A performance index function based on the identification errors will be used to choose the best model and the corresponding controller. Considering the unmodeling error of neural network, a dead-zone recursive algorithm will be used, and the proof of robust property and stability of the MMAC are given. Different combinations of adaptive models and fixed models will be used for MMAC, and the effectiveness of the proposed method has been tested in simulations.

The single-input/single-output nonlinear discrete-time system can be represented as follows:

Due to the existence of noncausal problem, normally state transformation should be made first [

As Assumption 3 in [

Plant (

Consider

Let

The output of the dead-zone function is used in the following updating rule:

Suppose

the

the tracking error between the plant output and the reference command will converge to a ball of radius

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. For this kind of problem, MMAC algorithm is presented as a useful tool. The rationale for using MMAC is to ensure that there is at least one model with parameters sufficiently close to those of the unknown plant. By the switching rule, the control strategy is to determine the best model for the current environment at every instant and activate the corresponding controller. The structure of the multiple model adaptive control is shown in Figure

Structure of multiple model adaptive control.

Multiple adaptive models can be regarded as an extension of conventional indirect adaptive control. The objective is to make the control error

One has

At every instant, one of the models

Given prior knowledge of the different possible environments, the control problem is to determine suitable rules for switching and tuning these parameters to yield the best performance for the given objective while assuring stability.

The following three different combinations have been considered [

The previous method reveals that massive calculation may be produced because each adaptive model needs to adjust dynamically. Hence, if the models are fixed, the same strategy can be used in stationary and time-varying environments. However, fixed models can represent exactly only a finite number of environments. Thus,

It is commonly accepted that the convergence time of an adaptive model will be large for large initial parametric errors. Hence, in the configuration described above, a large number of fixed models may be needed to keep the transient response under control until the adaptive model has converged. If the fixed model, which is the closest to the given plant, is assumed to be known, faster convergence can be obtained by initiating a new adaptive model from the location of the former. The same objective can be achieved on-line by starting adaptation from the location of each different fixed model that is successively chosen by the switching scheme.

The reinitialized adaptive model

A natural way to decide when and to which controller one should switch is to determine performance cost indexes for each controller

Considering the unmodeling error of neural network and robustness of the adaptive controller, the specific performance index proposed has the form

The switching scheme consists of monitoring the performance indexes

Suppose

all the signals in the system are bounded,

the tracking error between

At time

The control input

So we have all the signals in the system bounded, and

Consider the following index function:

From Theorem

If

all the signals in the system are bounded,

The introduction of the reinitialized adaptive model will not affect the stability of the whole system, and the proof of the stability will be similar to the case of

PH neutralization is a very important procedure in the chemical industry. Usually, we use the logarithmic behavior to present pH characteristic; the existing nonlinearity always makes the identification and control of pH neutralization more difficult. A strong acid flows into a tank and is thoroughly mixed with a strong base whose inward rate of flow is controlled in such a way to produce a neutral outward flow from the tank. Because the acid and the base are strong, they are completely dissociated, and also the dissociation of the water can be disregarded [

We suppose that

An approximate discrete-time model can be developed, incorporating measurement and input actuator errors, as follows:

The neural networks

One adaptive mode with fixed parameter.

System output pH

Control input

After 300 sample times, the weights will converge to the following values:

As the parameters change at

One adaptive mode with variable parameter.

System output pH

Control input

Three adaptive models

The multiple models based on neural networks are chosen as in (

Three adaptive models.

System output pH

Control input

Switching scheme

In this case, three fixed models

In the process of parameter identification, this method could improve the transient response compared with the conventional adaptive control (Figures

Three fixed models and one adaptive model.

System output pH

Control input

Switching scheme

In this case, we establish three fixed models

From the simulation, we can see that this method can improve the control quality dramatically (Figures

Three fixed models, one free running adaptive model and one reinitialized adaptive model.

System output pH

Control input

Switching scheme

In this paper, multiple models are used to establish robust multiple models adaptive controller for a class of nonlinear discrete-time systems by using neural networks. Three kinds of combinations of adaptive model and fixed model are used to make the multiple model set, and a switching law is suitably defined to make the decision of the best model. The principal contribution of this paper is the proof of stability of robust MMAC by using neural networks. Multiple neural network models with different weights represent different dynamical characters of the plant when it operates in different environments, which can be described by a mount of input and output data. So the design of the model set can also be regarded as a kind of data driven problem [

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

This work was supported by the Fundamental Research Funds for the Central Universities under Grant FRF-TP-12-005B, the Program for New Century Excellent Talents in Universities under Grant NCET-11-0578, and the National Natural Science Foundation of China under Grant 61074055.