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In this research, a comparative study of two recurrent neural networks, nonlinear autoregressive with exogenous input (NARX) neural network and nonlinear autoregressive moving average (NARMA-L2), and a feedforward neural network (FFNN) is performed for their ability to provide adaptive control of nonlinear systems. Three dynamical nonlinear systems of different complexity are considered. The aim of this work is to make the output of the plant follow the desired reference trajectory. The problem becomes more challenging when the dynamics of the plants are assumed to be unknown, and to tackle this problem, a multilayer neural network-based approximate model is set up which will work in parallel to the plant and the control scheme. The network parameters are updated using the dynamic backpropagation (BP) algorithm.

Linear control methods are based on the existence of an analytical model of the system. However, most physical systems have nonlinearity, and their mathematical model is unknown or partially known and variable in time. So, conventional methods suffer some limitation in terms of stabilization and performance [

In the field of control engineering for a class of nonlinear discrete-time systems, a neural network has recently emerged as adjustable approximators capable of reproducing the complex behavior of nonlinear systems. Artificial neural networks have been effectively used as tracking controllers for unknown linear and nonlinear dynamic plants [

There are different types of ANNs proposed in the literature: feedforward neural networks (FFNNs), Kohonen self-organizing network, radial basis function (RBF), and recurrent neural network [

Various structures of neural networks have been proposed in the literature. In [

The main contribution of this paper is to propose a strong nonlinear adaptive controller of unknown nonlinear dynamical systems based on the approximate models. Another objective is to present a prescriptive method for the dynamic adjustment of the parameters based on backpropagation. The big advantage of the proposed control system is that it does not require previous knowledge of the model. Our ultimate goal is to determine the control input using only the values of the input and output.

In our research study, the strong learning capability of the dynamic neural network in identification and control is combined with the functionality of the approximate model controller structure to propose a novel online NN-based controller for nonlinear single-input single-output (SISO) dynamical systems. The main issues in the field of ANN-based control are the choice of neural network structure to be used as a controller. In our paper, an attempt has been made to compare the three types of neural networks: two recurrent neural networks, nonlinear autoregressive with exogenous input (NARX) neural network and nonlinear autoregressive and moving average (NARMA-L2), and a feedforward neural network (FFNN). The control system configuration employed in our paper consists of an ANN-based controller in cascade with the plant, and the training is performed online.

This study differentiates from the studies in the literature in terms of mathematically obtaining nonlinear SISO controller parameters. Thus, the main novelty of this paper is that the parameters of the nonlinear SISO controller can be identified as neural network expressions for nonlinear SISO systems. All neural network simulations are performed in a Matlab environment. The results indicate that the online NARMA-L2, NARX, and FFNN controllers attain good modeling and control performances.

The remainder of the paper is organized as follows: Section

Generally, there are many different neural networks (NN) of nonlinear models. In this research study, the recurrent and feedforward neural networks are presented. Since a lot of literature exists on NARX, NARMA-L2, and FFNN, in this section, a brief introduction regarding their mathematical formulation is given. Figures

A feedforward neural network structure.

Nonlinear autoregressive with exogenous input (NARX) model.

Nonlinear autoregressive and moving average.

In a feedforward neural network, the information moves only in one direction, from the input layer to the output layer, but not vice versa as presented in Figure

The NARX model represents a generic recurrent neural network having one step ahead output,

Note that in Figure

The nonlinear autoregressive moving average (NARMA) (Figure

The NARMA-L2 is obtained by

Equations (

The two subfunctions,

The control configuration based on the NN approximate model is shown in Figure

Adaptive control using neural network approximate models.

The process of training consists in modifying the weights in an organized way using an appropriate algorithm. During the training process, a specified number of inputs and their desired output are introduced in the network. Then, the weights are tuned so that the neural network produces an output close to the target values. The fundamental training algorithm for multilayer networks is the backpropagation (BP) algorithm [

Therefore,

If we define

Thus, each element in

The error

Simulation studies have been carried out using the Matlab software environment to verify the performance of the proposed methods. Three different nonlinear dynamical systems are presented to evaluate the performance of the NN controllers. Two hidden layers are used with

Consider the nonlinear discrete time which is described by the third-order difference equation [

The system can be reformulated as

A reference model is described by the third-order difference equation:

If

The simulation reported in Figures

Example 1: (a) response for no control action (with

Example 1: (a) response of the reference model, (b) outputs of the reference model and the plant with control action, (c) NARX controller output, and (d) control error.

Example 1: (a) response of the reference model, (b) outputs of the reference model and the plant with control action, (c) NARMA-L2 controller output, and (d) control error.

The performance index of the different controllers with various neural network models is given in Table

Comparison of the performance index of the proposed controller and other controllers (example 1).

Controller | MSE | Nbr. of iterations | Learning rate |
---|---|---|---|

RBFN [ | 0.012 | 2000 | 0.005 |

DRNN [ | 0.0253 | 2000 | 0.0154 |

FFNN | 100 | 0.1 | |

NARX | 100 | 0.1 | |

NARMA-L2 | 0.25 | 100 | 0.1 |

Consider the nonlinear discrete-time system [

The order of the plant is

The control results for different epochs and learning rates along with control error are shown in Figure

Plant output and reference model response for

The model of discrete-time plants introduced here can be also described by the following nonlinear difference equations:

The same proposed scheme in Figure

The control action is required to get

In this example, performance comparisons of the BP-NNs are carried out with the three approximate models, FFNN, NARX, and NARMA-L2. The results are shown in Figures

Example 2: (a) response of an ANN-NARX identification model, (b) response of the plant under the NARX controller, (c) NARX controller output, and (d) control error with the NARX model (using

Example 2: (a) response of the reference model, (b) response of the plant under the NARMA-L2 controller, (c) NARMA-L2 controller output, and (d) control error with the NARMA-L2 model (using

Example 2: (a) response of an ANN-FFNN identification model, (b) response of the plant under the FFNN controller, (c) FFNN controller output, and (d) control error with the FFNN model (using

The numerical parameters used in the second system are presented in Table

Comparison of the performance index of the proposed controller (example 2).

Controller | MSE | Nbr. of iterations | Learning rate |
---|---|---|---|

FFRN | 0.0698 | 100 | 0.01 |

NARX | 300 | 0.01 | |

NARMA-L2 | 100 | 0.01 |

The second-order differential equation which describes the dynamics of a single-link robotic manipulator is given as

The angular position of the robotic manipulator arm is represented by

The difference equation of the one-link robotic manipulator with a sampling period is given by

The desired reference model dynamics is given by the following difference equation:

Before control action, the sampling period

Reference model and open-loop response of the plant without a controller (example 3).

In the simulation studies,

From equation (

The mean square error (MSE) obtained with different controllers for example 3 is presented in Table

Comparison of the performance index of the proposed controller and other controllers (example 3).

Controller | MSE | Nbr. of iterations | Learning rate |
---|---|---|---|

RBFN [ | 0.1427 | 2000 | 0.027 |

DRNN [ | 0.0288 | 800 | 0.0354 |

FFNN | 1000 | 0.05 | |

NARX | 500 | 0.1 | |

NARMA-L2 | 100 | 0.01 |

In this study, all the three methods provide simultaneous structure and parameter learning within the approximate models. The learning rate,

Example 3: (a) response of the reference model, (b) response of the plant under the adaptive controller, (c) FFNN controller output, and (d) control error (using

Example 3: (a) response of the reference model, (b) response of the plant under the adaptive controller, (c) NARX controller output, and (d) control error (using

Example 3: (a) response of the reference model, (b) response of the plant under the adaptive NARMA-L2 controller, (c) controller output, (d) control error (using

This paper describes the application of NN models as a controller. Both feedforward and two recurrent NARX and NARMA-L2 predictors are proposed, tested, and compared. The method for the adjustment of parameters in generalized neural networks is treated, and the concept of dynamic back propagation is introduced. The training of all the three neural networks is done online. Simulation and comparative studies demonstrate the superior performance of the proposed approaches. It can be concluded that the ANNs can be successfully utilized for controlling different nonlinear dynamic systems. In addition, future works will explore the adaptive neural network control for nonlinear MIMO systems under disturbances using the hybrid learning method.

No data were used to support this study.

The authors declare that they have no conflicts of interest.