Hardware Implementation of the Neural Network Predictive Controller for Coupled Tank System

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2.1. System Modelling

The twin-connected tanks system is a nonlinear dynamical system and the governing dynamical equations can be written as

(1)

(2)

where is the level in the first tank;

is the level in the second tank;

is the flow rate from the first tank to the second tank;

is the flow rate from the second tank to a reservoir;

is the inlet flow rate;

g is the gravitational constant;

A is the cross-section area of first tank and second tank;

a₂is the cross-section area of the outlet orifice;

a₁₂is the cross-area of the coupled orifice.

2.2. System Constraints

The pump can only pump the water into the first thank, therefore, the inlet flow rate q is always positive. The constraint on the inlet flow rate is given by

(3)

In the steady state, for constant water level set points , the respective derivatives must be zero separately i.e.,

(4)

At the steady state, Equation. 1 and Equation 2 cab be rewritten as

(5)

(6)

where Q is the steady state inflow rate. From Equation 5 and Equation 6, to satisfy the constrain in Equation 3, the following constrain should be satisfy as

(7)

3. Neural Network Predictive Control

To design the neural network predictive control, two steps should be carried out:

• Determine the neural network plant model for the given nonlinear system (system identification).

• The neural network plant model is used by the controller to predict future performance.

The following subsection described the steps design of the NNPC.

3.1. Nonlinear System Modeling Using Neural Networks

System identification is the process of developing a mathematical model of a dynamic system based on the input and output data from the actual process. An important stage in control system design is the development of a mathematical model of the system to be controlled. Previous studies in system identification have demonstrated that neural networks are successful in modelling many non-linear systems ^[13]. Before neural networks are investigated for identification, linear techniques such as auto regressive with exogenous input (ARX) and auto regressive moving average with exogenous input (ARMAX) will be applied to nonlinear system.

The model of any nonlinear system in the form of NARMA is given as

(8)

where f(.) represents an unknown nonlinear system, d represented the time delay of the system, n_y and n_u are the orders of the system. For an unknown model plant, many forms of NN model can be applied. In this work, the multilayer perceptron neural network is used to model the nonlinear system as shown in Figure 2. The neural network design to identify the plant is expressed as

(9)

Figure 2. Nonlinear system identification

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Where is the output of the neural network system, represents the l^th input of the neural network; , w, b₁and b₂ are the weights and biases of the NN (the adjustable parameters of the NN); F(.) and g(.) are nonlinear functions ; q and m are the number of the nodes in the hidden and input layer, respectively.

The adjustable parameters of the NN are determined from a set of examples through the process called training. The examples, or the training data as they are usually called, are a set of inputs, u(t), and corresponding desired outputs, y(t).

Specify the training set by

(10)

where N is the number of the samples in training set.

The objective of training is then to determine a mapping from the set of training data to the set of possible weights. Therefore, the neural network will produce predictions , which in some sense are “close” to the true outputs y(t)

Because the training of neural network is often operated off-line, any training algorithms in batch version can be developed to minimize the following criterion

(11)

where is the trainable parameters vector.

The weights are then found as. By some kind of iterative minimization scheme:

(12)

Where specified the current iterate; is the search direction; and the step size.

A large number of training algorithms exist, each of which is characterized by the way in which search direction and step size are selected. The MATLAB Neural toolbox provides a lot of training algorithms such as: Backpropagation algorithm, Variable Learning Rate algorithm, Resilient Backpropagation algorithm, Conjugate Gradient algorithm, Quasi-Newton Algorithm, Levenberg-Marquardt algorithm, and Reduced Memory Levenberg-Marquardt algorithm.

3.2. Predictive Controller

The objective of the predictive control strategy using neural predictors is twofold: (i) to estimate the future output of the plant and (ii) to minimize a cost function based on the error between the predicted output of the processes and the reference trajectory. The cost function, which may be different from case to case, is minimized in order to obtain the optimum control input that is applied to the non-linear plant. In most of the predictive control algorithms a quadratic form is utilized for the cost function:

(13)

with following requirements:

(14)

where N_u is the control horizon; N₁ is the minimum prediction horizon; N₂is the prediction horizon; i is the order of the predictor; r is the reference trajectory; is weight factor; and ∆ is the differentiation operator.

The command u may be subject to amplitude constraints

(15)

The cost function is often used with the weight factor . A very important parameter in the predictive control strategy is the control horizon , which specifies the instant time, since when the output of the controller should be kept at a constant value.

The output sequence of the optimal controller is obtained over the prediction horizon by minimizing the cost function with respect to the vector U. This can be achieved by setting

(16)

The analytical approach to the optimization problem needs for the differentiation of the cost function and, finally, leads to a nonlinear algebraic equation; unfortunately this equation cannot be solved by any analytic procedure. This is why a computational method is preferred for the minimization of the cost function, also complying with the typical requirements of the real-time implementations (guaranteed convergence, at least to a sub-optimal solution, within a given time interval). Figure 3 depicts the structure of the NNPC. The controller consists of the neural network plant model and the optimization block. The optimization block determines the values of u' that minimize , and then the optimal u is inputted to the plant (nonlinear system).

Figure 3. Structure of neural network predictive control

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4. Hardware Implementation of the Neural Predictive Control

During the last years, consumer digital devices have been built using either application specific hardware modules (ASICs) or general purpose software programmed microprocessors, or a combination of them. Hardware implementations offer high speed and efficiency but they are tailored for a specific set of computations. If an alternative implementation is needed, a new and expensive design process has to be performed. On the contrary, software implementations can be modified freely during the life-cycle of a device, through patches and updates. However, they are much more inefficient in terms of speed and area. Reconfigurable computing is intended to fill the gap between hardware and software, achieving potentially much higher performance than software, while maintaining a higher level of flexibility than hardware. Reconfigurable devices, including Field-Programmable Gate Arrays (FPGAs), contain an array of computational elements whose functionality is determined through multiple programmable configuration bits. These elements, usually called logic blocks, are connected using a set of routing resources that are also programmable.

Neural networks can be implemented using analogy or digital systems. The digital implementation is more popular as it has the advantage of higher accuracy, better repeatability, lower noise sensitivity, better testability, higher flexibility, and compatibility with other types of pre-processors. Two implementation methods can be used to implement the NNPC: (i) FPGA implementation; and (ii) DSP implementation. FPGA is a suitable hardware for neural network implementation as it preserves the parallel architecture of the neurons in a layer and offers flexibility in reconfiguration.

The VHDL coding developed is tested using Xilinx 10.1 software tool to design the NNPC. This software is synthesised the VHDL codes and generated a configuration file that used to program the FPGA board. The ModelSim 6.4b software tool is used to simulate the output response of the FPGA boards. The same NNPC is designed and simulated using MATLAB software tool. This Software-Based Controller (SBC) will be used to make a comparison with the proposed design. This comparison is important because it tells us to which extent our FPGA-Based Controller (FBC) is close to similar controller designed as a computer program.

5. Simulation and Results

In this work, the Simulink Toolbox is used to simulate the NNPC and coupled tank system. Figure 4 depicts the MATLAB Simulink model of coupled tank system. While, Figure 5 shows the connection between NNPC and coupled tank system.

Figure 4. MATLAB Simulink model of coupled tank system

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Figure 5. MATLAB Simulink model of NNPC with coupled tank system

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As it is mentioned before, to design the neural network predictive control, two steps should be carried out:

• Determine the neural network plant model for the given nonlinear system (system identification).

• The neural network plant model is used by the controller to predict future performance.

To carry out the first step, the neural network is used to design the identification system for the coupled tank system. The training data should be generated by applying a random step inputs to the coupled tank Simulink model. Figure 6 shows the input/output data set that is used to design the neural network identifier.

The Levenberg-Marquardt Algorithm (LMT) is an iterative technique that locates the minimum of a multivariate function expressed as the sum of squares of nonlinear real-valued functions. It has become a standard technique for nonlinear least-squares problems ^[15], widely adopted in a broad spectrum of disciplines.

To design the NN identifier, we assume that it has four inputs (y(t-1),y(t-2),u(t-1)) and u(t-2) and one output . The size on hidden layer is assumed seven.

In this work, the Levenberg-Marquardt algorithm is used as training algorithm to adjust the trainable parameters of the NN. After 166 optimization iteration steps, the objective function has been minimized to very small value as shown in Figure 7.

Figure 6. Input/output data training set

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Figure 7. Objective Function during optimization steps

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The optimization problem was addressed in accordance with the computational scenario built in Subsection 3.1. With respect to the notations introduced in Subsection 3.1, the following concrete values were chosen for the tuning parameters of the predictive control algorithm: N1=1, N2=10, Nu=5, and =0.05. The NNPC is generated the suitable control signal to force the liquid level of the second tank to track the reference input signal. In this work, the random step input is used as the reference input signal. In order to investigate the effectiveness of the neural network based predictive controller, the output response of the controlled system with NNPC is compared with output response of the controlled system with PID controller as shown in Figure 8. The Internal model control theory is used to design the PID controller. Figure 8 shows that the overshoot and settle time of the output response are minimized.

Figure 8. the output response of the coupled tank system with different control system

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After the design of neural network predictive controller has been completed, the FPGA board can be used to implement the proposed control system. First, VHDL codes, which describe the operation of NNP controller, are written (the flowchart of VHDL program is shown in Figure 9). This program is downloaded into the FPGA chip (XILINX Spartan XC3S700AN) by using USB cable. Then, the Hirose 100-pin FX2 Edge connector (one port of the FPGA board) can be used to connect the FPGA board with the tank system. The error between the reference input and the system output can be applied as input to the A/D converter. The digital output of the A/D converter has been applied as input data to the FPGA boards. The FPGA board generates digital inputs to the NNP controller (error and error rate). The NNP controller generates a suitable digital control signal based on the program that is stored in the FPGA chip. The digital control signal is sent to the D/A convertor to generate an analogy control signal which is applied as input to the pump of tank system. Figure 10 shows the control signal obtained from ModelSim program, while, Figure 11 shows the output signal of the NNP controller obtained from MATLAB simulation. By comparing Figure 10 and Figure 11, it can be seen that, the two output signals are identical.

Figure 9. the flowchart of describe the Neural network using VHDL

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Figure 10. Control signal obtained from FPGA Board

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Figure 11. Control signal obtained from MALAB Simulation

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