Neural Network based Direct MRAC Technique for Improving Tracking Performance for Nonlinear Pendulum System

This paper investigates the application of a neural network-based model reference adaptive intelligent controller for controlling the nonlinear systems. The idea is to control the plant by minimizing the tracking error between the desired reference model and the nonlinear system using conventional model reference adaptive controller by estimating the adaptation law using a multilayer backpropagation neural network. In the conventional model reference adaptive controller block, the controller is designed to realize the plant output converges to reference model output based on the plant, which is linear. This controller is effective for controlling the linear plant with unknown parameters. However, controlling of a nonlinear system using MRAC in real-time is difficult. The Neural Network is used to compensate the nonlinearity and disturbance of the nonlinear pendulum that is not taken into consideration in the conventional MRAC therefore, the proposed paper can significantly improve the system behaviour and force the system to behave the reference model and reduce the error between the model and the plant output. Adaptive law using Lyapunov stability criteria for updating the controller parameters online has been formulated. The behaviour of the proposed control scheme is verified by developing the simulation results for a simple pendulum. It is shown that the proposed neural network-based Direct MRAC has small rising time, steady-state error and settling time for a different disturbance than Conventional Direct MRAC adaptive control.


Introduction
In the adaptive control, controlling of the nonlinear system with present-day sophistication and complexities has often been an important research area due to the difficulty in modelling, nonlinearities, and uncertainties. Model reference adaptive control is the best scheme used in the adaptive control technique. Recently MRAC has received considerable attention and many new approaches have been applied to the practical process [2]. In the MRAC scheme, the controller is designed to realize the plant output converges to reference model output based on assumption that plant can be linearized [3], [4], and [5]. Therefore, direct MRAC is best controller for controlling linear plants with unknown parameters. However, it may not guarantee for controlling nonlinear plants (Pendulum) with unknown structure. In recent years, an artificial neural network (ANN) has become very popular in many control applications due to their higher computation rate and ability to handle nonlinear systems [6].
The adaptive controller is designed to realize a plant output tracks to reference model output based on assumption that the plant can be linearized. [8], [9], [3] However, as most industrial processes are highly nonlinear, non-minimum, and with various type of uncertainties and load disturbances the performance of the linear MRAC may deteriorate, and suitable nonlinear control may have to be used.
In [11], the output of neural networks then adaptively adjusts the gain of the sliding mode controller so that the effects of system uncertainties eliminated and the output tracking error between the plant output and the desired reference signal can be asymptotically converging to zero. However, the sliding mode control action can lead to high frequency oscillations called chattering which may excite un-modeled dynamics, energy loss, and system instability and sometimes it may lead to plant damage.

Mathematical Modeling
Consider the simple pendulum shown in Figure 1, where l denotes the length of the rod and m denotes the mass of the bob.
Assume the rod is rigid and has zero mass. Let θ denote the angle subtended by the rod and the vertical axis through the pivot point. The pendulum is free to swing in the vertical plane. The bob of the pendulum moves in a circle of radius l. To write the equation of motion of the pendulum, let us identify the forces acting on the bob. There is a downward gravitational force equal to mg, where g is acceleration due to gravity. There is also a frictional force resisting the motion (by air and any other frictions), which we assume to be proportional to the speed of the bob with a coefficient of friction b.
The mathematical model for a simple pendulum is that The state space form of the pendulum is given by, On the other hand the state space form is given by , Λ control input uncertainty

Adaptive control
Adaptive control is the best control method used by a controller, which must adapt to a controlled system with parameters, which vary or are initially uncertain [12], [16]. For example, as an aircraft flies, its mass will slowly decrease because of fuel consumption; a control law is needed that adapts itself to such changing conditions.

1 Model Reference Adaptive Control
MRAC is one of adaptive control classification technique as shown in figure 2 below. When designing a controller for a system, a control designer typically would like to know how the system behaves physically. This knowledge is usually captured in the form of a mathematical model.
There are generally two classes of adaptive control: direct adaptive control and indirect adaptive controller [1]. Direct adaptive controller methods adjust the control gains directly and indirect adaptive controller methods estimate unknown system parameters for use in the update of the control gains. Asymptotic tracking is the fundamental property of model-reference adaptive control, which guarantees that the tracking error tends to approximately zero in the limit.

Multilayer Neural Networks (Back propagation Algorithm)
The input signals are propagated in a forward direction on a layer-by-layer basis. Learning in a multilayer network proceeds the same way as for a perception. In a back-propagation neural network, the learning algorithm has two phases. First, a training input pattern is presented to the network input layer [19]. The network propagates the input pattern from layer to layer until the output layer generates the output pattern as shown in figure 3 below.
Second, if this pattern is different from the desired output, an error is calculated and then propagated backward through the network from the output layer to the input layer. The weights are modified as the error is propagated.

The Back propagation training algorithm
Back propagation is a common method for training a neural network, the goal of back propagation is to optimize the weights so that the neural network can learn how to correctly map arbitrary inputs to outputs. [19] Back propagation method contains the following steps: Journal of Informatics Electrical and Electronics Engineering (JIEEE) A2Z Journals Step 1: Initialization; set all the weights and threshold levels of the network to random numbers uniformly distributed inside a range: Where is the total number of inputs of neuron in the network.
Calculate the actual outputs of the neurons in the hidden layer: Where n is the number of inputs of neuron j in the hidden layer.
Calculate the actual outputs of the neurons in the output layer: Where m is the number of inputs of neuron k in the output layer Calculate the weight corrections: Update the weights at the hidden neurons: Step 4: Iteration:-Increase iteration p by one, go back to Step 2 and repeat the process until the selected error criterion is satisfied.

Adaptation law design for simple pendulum
From state space equation form of the simple pendulum in the equation shown below. ̇1 = 0 1 + 2 (4.1) We can express as a general expression of ̇= because the parameters of and are unknown or vary with time. Therefore, we can assign as the estimates of the parameter 1 ( ℎ 1), 2 ( ℎ 2) respectively and the known basis function 1 and 2 to 1 ( ), 2 ( ) respectively. From the above Let the adaptive controller  The general block diagram for conventional direct model reference control in figure 4 below.

Conventional Direct Model Reference Adaptive control for nonlinear pendulum result
The simulation result for conventional model reference adaptive control as shown below in figure 5 and figure 6.

Neural network based direct MRAC for nonlinear pendulum result
In this paper, Artificial NN is used to improve the performance     Finally, we can observe from the above graphs an ANN controller is a powerful controller to stabilize a nonlinear simple pendulum system when compared to Conventional MRAC. The conventional MRAC and an artificial neural network controller are designed for the stabilization of simple pendulum systems.

Conclusion
In this paper, modeling and designing of neural network based direct MRAC of nonlinear systems using MATLAB software to overcome the tracking performance of conventional direct model reference adaptive control to an equilibrium point for different disturbances have been investigated. Adaptive law using Lyapunov stability criteria for updating the controller parameters online has been formulated. Both the transient and steady state performances of the nonlinear system are improved by updating the parameters of neural networks (weight and biases) controller.