INTRODUCTION
Nowadays, a wide variety of advanced control methods is used in industries
in order to control important process variables. Many engineers have applied
advanced control methods (ElKouatly and Salman, 2008;
Farivar et al., 2009; Otero,
2004) and prefer these methods to conventional methods (Astrom
and Hagglund, 1995; Astrom and Wittenmark, 1989).
Different people may have different understandings of advanced control methods
according to their uses and applications.
Generally, selection of a control method depends on where it is to be used. Nevertheless, the objective is always reaching to the desired conditions. Up to late 90’s, monitoring in industrial systems was done manually. Advances in technology, make these systems more and more automated.
Nowadays, PID controllers are used in more than 80% of feedback control systems
in different industries (ChengChing, 1999;
Bao et al., 1999; Shi and Chen, 2001; Shu
and Pi, 2005). Astrom and Hagglund (1995) has shown
how to adjust the parameters according to frequency responses and experimental
rules. In case of process parameters change, Astrom and Wittenmark in their
novel work (Astrom and Wittenmark, 1989) show adaptive
methods for adjusting the parameters of a controller. The adaptive methods for
controlling such systems are growing rapidly (Barzamini
et al., 2009). It can be proved that a PI controller can be used
for controlling a first order system but for controlling a second order system
a PID controller is needed. In practice, the characteristics of the real systems
are nonlinear and timevarying, so that linear model of the system is almost
useless. In order to adjust the parameters of the controller in such systems,
some solutions are proposed. One of these methods is to store many parameters
in the system and to select a number of them according to the circumstances.
In other words, at the time when it is detected that the system is moving from
a region to another, system parameters are switched to new ones; but in general,
this method cannot be applied to continuous and modelfree systems.
On the other hand in adaptive methods, parameters of the model are determined
frequently according to the process situation. These parameters contain the
characteristics of the system at each moment. These parameters are used for
calculating and adjusting the parameters of the controller and the controller
adjustments change frequently as the process model change.
MATERIALS AND METHODS
In this study, which is the result of extensive research in the field of neural network and its application as the controller on industrial plants during 2005 to 2008 in Power and Water University of Technology, we aimed to propose a new neuralnetwork based adaptive controller. The proposed controller is capable of solving the problems and shortages of the other similar methods.
Fundamentals of neural networksbased adaptive controller design: Due to the nonlinearity involved in the system under investigation, it is preferred to propose a method that dose not need the linear model of the system. Therefore, the system is considered as a black box, where only the inputs and outputs are assumed to be available at all time instants. Having considered the above assumption, the introduced method is supposed to be applicable in many industrial systems.
The goal of this study is to substitute the conventional PID controller with
a neuralnetworkbased controller that is capable of dealing with nonlinear
plants. A neural networkbased controller dose not needs a priori information
about the dynamic of the system and only inputoutput data would be enough for
running the control scheme (Chen and Cheng, 1996; Chen
et al., 1995).
Introduction to general PID neural network: Several static neural network
based identifiers and controllers have been introduced in the literature for
identification and control of dynamic systems (Chen
et al., 1990; Shu and Guo, 2004). Nowadays,
the use of neuron with dynamic structure is being rapidly increased. By using
dynamic elements embedded in a static neuron structure, like adaptive delays
(Yazdizadeh and Khorasani, 2000, 2002),
embedded adaptive filter at the out put of each neuron (Yazdizadeh
and Khorasani, 1998; Cong and Li, 2005), the desired
dynamics can be created.
Structure of the dynamic neuron in order to create dynamics of a PID controller
is changed and proposed. Each neuron includes some inputs and outputs (Fig.
1). The governing equation of the neuron that relates the inputs to the
outputs are given. Proportional, Integral and Differential type neurons which
are given are basically different in their governing equations.

Fig. 1: 
Schematic of a neuron 
Structure of dynamic neurons: The general structure of three different types of the neurons is shown in Fig. 1. The governing equation of each neuron is presented as follows.
Proportional type neuron: This simple neuron is characterized by its activation function as o_{j}(t) = net_{j}(t) for continuous time or as o_{j}(k) = net_{j}(k) for discrete time where its input is given by
This neuron has a linear activation function and acts as an adaptive weighted adder. In fact, the PType neuron is an adaptive gain unit.
Integral type neuron: The IType neuron acts as an integrator and the
output of the neuron is, in fact, the weighted integral of the input. The relationship
between the input and the output in continuous and discrete form are presented
by Eq. 1 and 2, respectively.
Differential type neuron: The Dtype neuron acts as a derivative operator.
The relationship between the input and output of this neuron is presented by
Eq. 3 and 4, respectively:
Proportional integral differential type neural network controller: The
general structure of this controller is shown in Fig. 2. The
network has two separate layers where the hidden layer consists of all three
types of neurons (P, I and D). The output layer consists of a Ptype neuron
and computes the weighted sum of the outputs of the hidden layer. In order to
show the inputoutput representation of the network, the weights are initially
set as in the following equations Eq. 5, 6:
Generalization of the PID neural network controller: Here, a generalized
PID neural network controller consisting of several previously introduced PID
neural networks which can be used as a PID controller for MultiInput MultiOutput
systems is introduced (Mehrafrooz and Yazdizadeh, 2007).
Figure 3 shows the structure of the controller and the multivariable
system as a feedback loop.
Regardless of the type of the multivariable system and due to the importance of the proposed controller, we take a closer look at the neurocontroller. Figure 4 shows the structure of the MIMO type of the proposed controller. In Fig. 4, r and y represent the desired values and the output respectively and the second layer of the neural network generates the control command. It can be easily concluded that:
• 
The network can operate for systems with n inputs and n outputs
(the systems in which the number of inputs and the number of outputs are
equal) 
• 
The number of neurons in the output layer represents the number
of inputs of the system and can be changed depending on the application 

Fig. 4: 
Structure of the PID neural controller for MIMO systems with
p inputs and q outputs 
• 
Therefore, it may be inferred that the network is capable
of controlling systems with p inputs and q outputs and in fact in this method
there is no obligation on the equivalency of the number of inputs and outputs 
Dynamic neural network controller with local feedback: The structure introduced here, like one is discussed earlier, has the structure of a PID controller. In this section a neural network with only 3 neurons in the hidden layer is proposed. Also, in this controller, which we call it as Dynamic Neural Network Controller with Local Feedbacks, internal feedbacks are considered as output and activation feedbacks (Fig. 5).
By using the above structure, the number of neurons is decreased that in turn
leads to a faster adjusting algorithm. The hidden layer of the network can act
as a P, PI, PD and PID controller. The proposed controller like generalized
neural network PID controller has the capability of automatic adjustment of
weights by using adaptive algorithms and can be used for controlling nonlinear
systems. Having combined the simple structure of the PID controllers with the
automatic learning capability of the neural networks in the proposed method,
a very efficient structure is achieved.
Structure of a dynamic neural network controller with local feedbacks:
As shown in Fig. 5, the structure consists of a hidden layer
and an output layer. In the hidden layer there are 3 neurons called as a_{1},
a_{2} and a_{3}. The output layer generates the output signal
by using a single neuron. The activation functions in input and output layers
are linear. The outputs of the neurons of the first layer are sent to the hidden
layer neurons. There are 3q weights: w_{i1}, w_{i2} and w_{i3},
1≤i≤q. Neuron a_{1} in the hidden layer is an integral neuron
whose feedback causes a delay. Neuron a_{3} in the same layer is a derivative
neuron that sends the weighted values to a_{3} with a delay feedback.
Neuron a_{2} in the hidden layer with a linear activation function and
without any feedback acts as the proportional part in the PID controller. Having
considered the above explanation and according to Fig. 5,
the outputs in n_{1}, n_{2} and n_{3} are as follows:
It can be seen that in contrary with the former neural network, discussed in
the beginning of the study, the activation functions are linear and originpassing
and the hidden layer neurons may be distinguished based on their feedbacks.
The output layer neuron generates the control command as:
where,
Hence the equations may be rewritten as:
where, 1z^{1} as the denumerator shows the integration behaviour;
that’s the reason for calling the neuron as integral neuron. Also we have:
where, 1z^{1} shows the derivative behavior. That’s the reason for calling the neuron as derivative neuron. By using a_{1}, a_{2} and a_{3} neurons in the hidden layer the inputoutput representation of a PID controller in its multiinput multioutput case is achieved.
Weight adjustment algorithm for the dynamic neural network controller with
local feedbacks: Like all industrial controllers, the goal of using a Dynamic
Neural Network Controller with Local Feedbacks is to minimize the difference
between the outputs and the desired trajectory. The closed loop structure of
this controller is depicted in Fig. 6. The error for each
one of the outputs of the system is:
Assuming r_{j} as the desired output, the cost function for each output
is defined as follows:
And the total cost function which presents the error in each sampling is:

Fig. 6: 
The closed loop structure of dynamic neural network controller
with local feedbacks 
The gradient of the cost function with respect to the weights is calculated
by using the following equation:
Since, there is no explicit relationship between the input and output of the
system, in the above gradient the following approximation is used:
Moreover, it is prefered to use the sign of the above expression instead of
the value itself in weight adjustemnt algorithm, since the effects can be compensated
for in the weight adjustment rule. This prevents the algorithm from getting
stuck in the steps in which the above value is small and accelerates and eases
the calculations of the learning of the weights.
The same procedure may be used for adjusting the parameters of the network
(Mehrafrooz and Yazdizadeh, 2007).
Stability of the closed loop system: Among many research works in the
field of application of neural network in control systems, only few consider
the stability issue. Hourfar and Salahshoor (2009) have
introduced a novel technique for controlling a continuous stirred reactor based
on the very well known feedback linearization technique. In their proposed method
the plant is modeled by an adaptive neural network. Mehrafrooz
and Yazdizadeh (2007) in their research work show the stability of such
a system for the first time. The use of Lypunov theorem for analysis and proof
of the stability of nonlinear systems is a popular and well established method
(Yazdizadeh et al., 2009). In order to prove
the stability of the proposed method, the following Lyapunov function is defined:
and therefore for a sampling time k we have:
and according to:
we have:
On the other hand, we have:
By defining:
we have:
It is now clear that since we have
therefore, V(k)<0 and the system is stable provided that:
where:
in which λ_{j} can be negative or positive. Having assumed λ_{j}>0,
the stability condition is:
and when λ_{j}<0 the condition will be:
On the other hand, we have:
By defining:
and according to the equation:
and Eq. 37, we have:
and the learning rate η, should be in the following interval:
RESULTS AND DISCUSSION
Conventional PID controllers are widely used in industrial plants. It is very
hard to convince engineers in industry to substitute the conventional controllers
by modern and more advanced techniques like NeuroPID controllers instead. Toward
this, we first selected one of the challenging issues in the steam power plant.
In order to show the performance of the proposed method in a real life physical
system, we extracted almost the exact model of the system under investigation.
In order to compare the results of the proposed method with the conventional
method, we applied a conventional PID controller to the system as well. As it
is shown here, due to the nonlinear behaviour of the proposed neural network
based method and also due to adaptive characteristic of the proposed method,
the time domain response of the system is improved in all important aspects.
Application of the proposed controllers to neka power plant model: NEKA
Power Plant is a large steam power plant in Northern Province Mazandaran in
Iran. There is a number of processes and control loop in large scale system
like a steam power plant. Among them, one may refer to polishing plant that
is responsible for chemical process for water and steam closed circle in the
power plant. Although, some other important variables are controlled in this
section, tank level control is one of the challenging issues (ElKouatly
and Salman, 2008; AlGallaf, 2002).
As shown in Fig. 7, the system consists of a tank and a pump
which pumps the fluid into the tank from above. This method of filling the tank
is called as nongravitational filling. The pump has a variable speed rate and
its speed rate depends on the voltage it receives. The term bV represents the
input flow to the tank where b is a constant that shows characteristic of the
pump and V is the voltage applied to the pump.

Fig. 7: 
Tank and the Pump in water level control system 
The rate at which the fluid exits the tank, due to the gravity force, is proportional
to the square root of the height of the fluid and is represented by a in which a is a constant coefficient and depends on the characteristics of the
output pipe. The change of the tank fluid level depends on the difference between
the input and output flow, therefore:
where, vol represent the volume of the fluid in the tank. On the other hand:
where A is the area of the base of the tank. Therefore:
It can be seen in Eq. 37 that the term
makes the system nonlinear:
Implementation of the proposed controllers for NEKA power plant water tanks:
The tanks have a circular base with a radius of 11 m and are 11 m high. Each
tank is filled by using a control valve and the water exits each tank at the
rate of 55 m^{3} h^{1} by a pump (Mehrafrooz
and Yazdizadeh, 2007). To apply the controllers, it is assumed that the
desired level of the fluid in the tank is 5 m. The pump starts to work at t_{0}.
The equation of the nonlinear state for the system is given by:
where, b depends on the input flow and c is the control signal applied to the
control valve with a magnitude of 4 to 20 mA and A, h, a and F are the area
of the base of the tank, height of the tank, characteristics coefficient of
the output pipe and the output flow that is pumped by the pump, respectively.
It is clear that at t_{0} the pump and the gravitational force causes
the fluid level to drop. These parameters are defined as follows:
At first, the system is controlled by a conventional PID controller. Proportional coefficients, derivation and the integration for this controller are 2, 3 and 0.2, respectively based on experiments.
Figure 8 shows the water level of the tank. As depicted in Fig. 8, when the pump starts, the water level drops about 8 cm that represents a significant amount of water according to the large volume of the tank.
To compare the results of the Generalized Neural Network Controller with the
results of the conventional PID controller, the weight adjustment algorithm
is run 1000 iterations to achieve the following values:
The above weights are proportional, integral and derivative coefficients of the PID controller, respectively. It should be mentioned that the initial values are the same initial values used in the conventional controller.
The out put signal, namely, the water level of the tank is shown in Fig. 9.
Compared to the traditional method, the overshoot and undershoot are less. In other words the overshoot decreases from 1bout 40 to 30 cm and the undershoot decreases from 80 to 40 cm. In addition, rest time of the fluid is reduced from 150 to 100 sec.
In industrial plants the control signal is very important. It is shown in Fig.
10. From industrial point of view, this control command is acceptable to
be applied to the actuator.

Fig. 9: 
The output signal curve of the generalized PID controller 

Fig. 10: 
The control valve signal 
Figure 11 shows the error which is the difference between the output signal (real fluid level) and the desired output (5 m). The magnitude of the error decreases as the number of iterations in weight adjustment algorithm increases. As shown in Fig. 4, the error reduction rate slows down as the number of iteration reaches to 500. Therefore, in case that we need more speed for the system we may decrease the number of iteration to 500.

Fig. 11: 
Errors for different number of iterations 
In order to show the performance of the proposed algorithm, even with only 500 iteration, the system is simulated under this condition. The curve for the control signal applied to the control valve is depicted in Fig. 12. It can be seen that the control valve is completely opened (20 mA) about 10 sec after the pump starts working and finally reaches to more steady state condition at 12 mA(10+4). From industrial point of view, this control command is a very usual type of command which is applied to the actuator.
CONCLUSION
In this study, a neural network based PID controller is proposed. The inputoutput representation of the network matches the PID governing equation. The proposed method is generalized to multi inputmulti output case which is not applicable in conventional PID controller. The stability of the proposed method is shown by using Lyapunov technique. To show the effectiveness of the proposed method, it is applied to an industrial plant, namely water level control in a tank in NEKA steam power plant. The simulation results show very good performance of the controller in the sense that it is more accurate and due to the system performance enhancement control valve life time is increased and energy consumption is reduced.