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Overcoming the coupling among variables is greatly necessary to obtain accurate, rapid and independent control of the real nonlinear systems. In this paper, the main methodology, on which the method is based, is dynamic neural networks (DNN) and adaptive control with the Lyapunov methodology for the time-varying, coupling, uncertain, and nonlinear system. Under the framework, the DNN is developed to accommodate the identification, and the weights of DNN are iteratively and adaptively updated through the identification errors. Based on the neural network identifier, the adaptive controller of complex system is designed in the latter. To guarantee the precision and generality of decoupling tracking performance, Lyapunov stability theory is applied to prove the error between the reference inputs and the outputs of unknown nonlinear system which is uniformly ultimately bounded (UUB). The simulation results verify that the proposed identification and control strategy can achieve favorable control performance.

Coupling is a widespread phenomenon existing in nonlinear systems. Due to the existence of the coupling, the variables among systems often suffer impact from each other’s fluctuations. Besides, time-varying and time delay is frequently encountered in many real control systems, and these may be the root of instability in the performance of closed-loop system. If the problems which have attracted many researchers cannot be solved effectively, they would not only delay achieving the steady states, but also realize the goal of independent control at all. Thereby, in order to achieve accurate, rapid, and independent control, it is essential to decouple among these variables and take the related methods. However, how to select the proper methodology according to the characteristics of control object is a thorny question.

In the open pieces of literature, the traditional decoupling ways to a multi-input multioutput (MIMO) system are primarily represented by frequency domain methods such as state variable method, diagonal dominance matrix, characteristic curve method, inverse Nyquist array, and relative gain analysis method [

With the development of decoupling control, many other decoupling approaches, such as adaptive decoupling [

In practice, allowing for a complicated MIMO nonlinear system with uncertainty and strong coupling, the common PID controller with fixed parameters can hardly achieve the desired steady sate at all. At the same time, the physical system is often difficult to obtain accurate and faithful mathematical model so that the conventional control schemes based on precise mathematical model can hardly achieve good performance in the real control process. Motivated by the seminal paper [

In this paper, we focus on developing an indirect adaptive NN controller for complex nonlinear systems including strong coupling, unknown or uncertain models, and disturbances simultaneously. The proposed method is the combination of NN-based identifier and adaptive controller, and the controller is designed based on the identified NN model. The main merits of this paper can be summarized as follows.

A novel and generalized decoupling control strategy based on indirect adaptive control is presented for nonlinear systems. Firstly, we construct a dynamic neural network (DNN) identifier without coupling to replace the real coupled systems. Then, we design the adaptive controller to deal with the nonlinear systems based on DNN identifier models.

According to the Lyapunov methodology, the online weights updating laws of DNN are developed to accommodate the identification and to guarantee that the error between the DNN identifier and the real unknown systems is UUB.

According to the Lyapunov methodology, the adaptive control laws are designed to deal with modeling uncertainties, system nonlinearities, and external disturbances and to guarantee stable tracking performance of the real outputs related to the reference inputs.

This paper is structured in the following way. In Section

The equation of MIMO continuous-time-varying nonlinear coupling system can be generally described as

In this study, the following assumptions are imposed.

The architecture of the MIMO nonlinear system.

In order to analyze the dynamic characteristic of nonlinear system more conveniently, we use the state-space equation to describe system (

By qualitatively analyzing the above model (

To identify the coupling, uncertain, and nonlinear dynamic system (

The architecture of system identification based on DNN.

We consider a single-layer, fully interconnected DNN as follows [

The architecture of DNN-based identifier.

The real system is a coupling, time-varying, and uncertain nonlinear system. Since neural network is a universal approximator which is capable of approximating any nonlinear function to any desired degree of accuracy, we can use DNN model without coupling to replace the real coupled system. This idea motivates a novel and generalized decoupling control strategy based on indirect adaptive control as described in what follows. Using this basic idea, the specific design for DNN-based identifier can be developed in the next section.

The nonlinear system (

In the process of approximating the time-varying, coupling, nonlinear system, DNN model (

The identification errors are defined as

From (

In model (

If

If

Considering the identification model (

Consider a Lyapunov function candidate as

The time derivative of

Now applying the error dynamics equation (

According to the properties (

In the view of Lemma

Then, substituting (

Namely,

As

The time derivative of

Using the updating laws (

At the same time, it is clear that

Substituting (

In the view of Lemma

Using the characteristics of the positive define matrices (

We choose the following Lyapunov function

Make

Therefore we arrive at

We assume that

As

According to the Boundedness Theorem [

In this section, the aim of controller design is to drive outputs of system properly following a prespecified trajectory. In addition, model errors of DNN-based identifier and external disturbances should be considered. The architecture of indirect adaptive control for the time-varying, coupling, and nonlinear system is shown in Figure

The architecture of indirect adaptive decoupling control.

In Figure

In Section

The desired reference inputs

The states errors

So we can obtain the errors dynamics equation as follows:

Then, we design the control action

If model errors and external disturbances are zero or negligible,

The states errors (

Substituting (

Consider the Lyapunov function candidate of controller design as follows:

We can obtain the time derivative of the Lyapunov function candidate (

Then, substituting (

Thus, we have

The online computational algorithm for the system identification and controller design will be described in next section.

If considering the identification and control as a whole process, we can prove the system stability by defining the final Lyapunov function candidate as

In this section, a step by step procedure is listed to implement the identification and control strategy.

Assign the initial values of gain matrices

Based on the initial states

Calculate the new parameter values of

Choose the suitable control gain

Go to Step

This is the algorithm of online identification and control scheme for the MIMO system with time varying, coupling, and nonlinearity.

In this section, in order to verify the effectiveness of indirect adaptive controller based on DNN, we choose a coupled two-input two-output, time-varying, nonlinear system as the simulation model

It is assumed that the structure of two-input two-output system is a black box system. In this experiment, this test was to validate the feasibility of the proposed DNN-based identifier to approximate the unknown, coupled, and nonlinear system.

In this study,

Identification result for system state

Identification result for system state

The real system (

Although the DNN has the ability to approximate any nonlinear systems, however, the DNN model errors and environment disturbances are inevitable in the real systems. In the control point of view, we should consider these factors which include neural network model errors and uncertain disturbance forces acting on the real nonlinear systems.

The control objective is to make the real outputs of the unknown, coupled, and MIMO nonlinear systems tracking the perspecified inputs; namely, the system states

In the view of this paper, the real systems can be formulated as

In this study, first we simply define

In fact, the desired inputs

By using the DNN-based identifier to approximate the real systems, the adaptive controller is designed as follows:

Assigned the initials

Trajectory tracking result for system state

Trajectory tracking result for system state

In Figures

For the purpose of verifying the generality and effectiveness of the proposed method for more complex nonlinear signal, we choose another reference input signal as follows:

The lump model errors and disturbances

And the other conditions are the same with the above. The response curves of trajectory tracking are shown in Figures

Trajectory tracking result for system state

Trajectory tracking result for system state

From Figures

In this paper, we present the DNN identification and adaptive control strategy for the real systems which is of nonlinearity, coupling, and uncertain environment disturbances. According to the Lyapunov methodology, the weights updating laws of DNN-based identifier and the control laws of indirect adaptive controller have been derived to ensure the stability of decoupling control and to achieve favorable tracking performance for the real system. The simulation results have indicated that the success of decoupling and the proper dynamic response of the plant states to follow the desired input trajectories.

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 (no. CDJZR10170007) and the National Key Basic Research and Development Program of China (973 Program) (no. 2012CB215202).