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A Comparative Study of Various Intelligent Controllers’ Performance for
Systems Based on Bat Optimization Algorithm
Article · June 2020
DOI: 10.30684/etj.v38i6A.622
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Engineering and Technology Journal Vol. 38, Part A (2020), No. 06, Pages 938-950
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Publishing rights belongs to University of Technology’s Press, Baghdad, Iraq.
| A Comparative Study of Various Intelligent Controllers’ Performance for Systems Based on Bat Optimization Algorithm |
||
| Luay T. Rasheed a | ||
| a University of Technology, Control and Systems Engineering Department-Iraq. [email protected] |
||
| Submitted: 26/09/2019 | Accepted: 16/01/2020 | Published: 25/06/2020 |
| K E Y W O R D S | A B S T R A C T | |
| PID controller, PIDA controller, Bat Optimization Algorithm, Higher-order systems. |
The aim of this paper is to demonstrate the performance of two intelligent controllers; the proportional-integral-derivative (PID) controller, and the proportional-integral-derivative-acceleration (PIDA) controller, based on optimization algorithm for higher order systems. In this work, bat control algorithm has been utilized to find and tune the optimal weight parameters of the controllers as simple and fast tuning technique to find the best unsaturated state and smooth control action for the systems based on the intelligent controllers. The simulation results using (Matlab Package) show that both controllers with the bat control algorithm can give excellent performance but the performance of the PIDA controller is better than that of the PID controller in terms of reducing the rising time (Tr), peak time (Tp), settling time (Ts), maximum overshoot (Mp), and steady state error (Ess). Furthermore, the fitness evaluation value is reduced. |
|
| How to cite this article: L. T. Rasheed, “A comparative study of various intelligent controllers’ performance for systems based on bat optimization algorithm,” Engineering and Technology Journal, Vol. 38, No. 06, pp. 938-950, 2020. DOI: https://doi.org/10.30684/etj.v38i6A.622 |
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1. Introduction
In the last few decades, the PID controller has been widely used both in industry and academia in
spite of several advanced control methods that have been proposed. Presently, more than 90% control
systems are still PID controllers in several advanced parts of the industry, such as power systems,
process control, robotics, and motion control. The majority of the controllers are still simple PID
control systems because of their ease of understanding, simplicity of usage, and its effective
performance [1].
The performance and stability of a PID-based control system may change drastically by the controller
parameters (proportional, integral, and derivative). Today, the PID controller based on tuning control
algorithms provides much convenience in engineering and therefore, several types of tuning control
algorithms have been proposed for PID controllers such as Genetic Algorithm (GA) [2], Particle
Swarm Optimization (PSO) [3], Ant Colony Optimization (ACO) [4], and Fruit Fly Optimization
(FOA) [5].
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The PIDA controller Proposed by Jung and Dorf in 1996 an extension to the conventional PID
controller which has four control parameters (proportional, integral, derivative and Acceleration
control parameters) and with this new term, a closed-loop system can respond faster with less
overshoot [6, 7]. Similar to the PID controllers several different tuning algorithms have been
proposed for the PIDA Controllers such as Flower Pollination Algorithm (FPA) [8], Genetic
Algorithm (GA) [6], Bat Optimization Algorithm (BOA) [9], and Spider Monkey Optimization
(SMO) algorithm [10].
The main contribution of this work is to enhance and develop the performance of the PID controller
because, in some particular situations, the PID controllers are not suitable especially for higher- order
control systems. Therefore, the PIDA controller was introduced in 1996 and utilized to deliver faster
and smoother response. In addition, it is more suitable for the higher-order plants compared to the
PID controller. This is done by adding an additional zero to the standard PID structure to derive the
PIDA structure of the controller. The addition of an extra zero to the PID controller will change the
root locus of the third-order plant in order to make dominant roots more dominant by eliminating the
effects of non-dominant roots [6, 7].
The remainder of this paper is organized with five sections: Section two describes the design of the
intelligent PID and PIDA controllers. Section three explains in details the Bat optimization
algorithm. In section four, the performance of the intelligent PID and PIDA controllers are presented
through the simulation results and discussion. Finally, the conclusions are explained in section five.
2. Intelligent Controllers Design
A PID controller is commonly employed in several industrial applications because of its ability to
enhance the dynamic behavior as well as to diminish or eliminate the steady-state error. This is
because the derivative part adds a finite zero to the open loop transfer function and enhances the
transient response and the integral part adds a pole at the origin, thus increasing system type by one
and diminishing the steady state error due to a step function to zero. Besides, it is easy to use, reliable
with strong adaptation performance, and its gains can be separately and simply adjusted [11].
The block diagram of this intelligent controller which composed of a PID controller and BAT
algorithm is shown in Figure 1. The BAT algorithm has a strong adaptation performance, high
dynamic characteristics, and good robustness performance because of its ability to find and tune the
PID control parameters.
Figure 1: The block diagram of the intelligent PID controller for the higher-order system.
The output of the PID controller in the time domain and its transfer function are illustrated as in Eqs.
(1 and 2) [12, 13]:
| ( ) ( ) ∫ ( ) | ( ) | (1) |
( ) ( )
( ) (2)
Where, represents the proportional gain, represents the integration gain, represents the
differentiation gain, ( ) is the error signal and ( ) is the control action.
Engineering and Technology Journal Vol. 38, Part A, (2020), No. 06, Pages 938-950
940
The equation of the PID controller in the discrete-time can be obtained by taking the first derivative
of Eq. (1) as follows:
̇ ( ) ̇( ) ( ) ̈( ) (3)
Applying the Backward difference formula to ̇ ( ) , ̇( ) and ̈( ) as follows:
| ̇ ( ) ( ) ( ) | (4) |
̇( ) ( ) ( ) (5)
̈( ) ̇( ) ̇( ) (6)
Where, is the sampling time.
Substitute Eqs. (4, 5, and 6) in Eq. (3) gives Eq. (7) as follows:
( ) ( ) ( ) ( )
( ) ̇( ) ̇( ) (7)
Applying the Backward difference formula on ̇( ) and ̇( ) in Eq. (7) gives Eq. (8) as follows:
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
(8)
Solving for ( ) finally the equation of the PID controller in the discrete-time is written as
follows:
( ) ( ) ( ( ) ( )) ( ) ( ( ) ( ) ( )) (9)
Where, , , and
The PID controller structure based on Eq. (9) for controlling the higher order systems is shown in
Figure 2.
Figure 2: The structure of the PID controller
The block diagram of the PIDA intelligent controller which composed of a PIDA controller and bat
algorithm is depicted in Figure 3.
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Figure 3: The block diagram of the intelligent PIDA controller for the higher-order system.
The output of the PIDA controller in the time domain and its Laplace Transform (transfer function) are
symbolized in Eqs. (10, 11, and 12) [8]:
| ( ) ( ) ∫ ( ) | ( ) ( ) | (10) |
( ) ( )
( ) ( ) ( )( ) (11)
( ) ( )
( )
( )( )( )
( )( ) (12)
According to equation (12), the PIDA controller has three zeros and three poles. Once (a,b,c) <<
(d,e), the two poles at and are neglected in design but this additional zero changes
the root locus of the third-order system by making dominant roots more dominant. Due to this, the
PIDA transfer function in Eqs. (11 and 12) can be rewritten as follows:
| ( ) ( ) ( ) |
(13) |
( ) ( )
( ) (14)
The equation of the PIDA controller in the discrete-time can be obtained by taking the first derivative
of Eq. (10) as in the following:
̇ ( ) ̇( ) ( ) ̈( ) ⃛( ) (15)
Applying the Backward difference formula to ̇ ( ), ̇( ) , ̈( ) , and ⃛( ) as in Eqs. (4, 5, 6, and 16)
the Eq. (15) is rewritten as in Eq. (17):
| ⃛( ) ̈( ) ̈( ) | (16) |
( ) ( ) ( ) ( )
( ) ̇( ) ̇( ) ̈( ) ̈( ) (17)
Applying the Backward difference formula on ̇( ) , ̇( ) , ̈( ) and ̈( ) in Eq. (17) gives
Eq. (18) as follows:
Engineering and Technology Journal Vol. 38, Part A, (2020), No. 06, Pages 938-950
942
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ̇ ( ) ̇ ( ) ̇ ( ) ̇ ( )
(18)
Applying the Backward difference formula once again on ̇( ) , ̇( ) in Eq. (18) with some
rearrangement and Solving for ( ) finally gives the discrete-time PIDA controller as in Eq. (19):
( ) ( ) ( ( ) ( )) ( ) ( ( ) ( ) ( ))
( ( ) ( ) ( ) ( )) (19)
Where, , , , and
The PIDA controller structure based on Eq. (19) for controlling the higher-order systems is described
in Figure 4.
Figure 4: The structure of the PIDA controller
3.BAT Optimization Algorithm
Bat algorithm is a relatively new meta-heuristic swarm algorithm for global optimization and it is
introduced and developed in 2010 by Xin-She Yang. This algorithm is inspired by the echolocation
behavior of the microbats, with varying pulse rates of emission and loudness [14].
The bat optimization algorithm can be explained in three main rules: The first rule is estimating the
optimal distance to the food using the echolocation phenomena. In the Second rule, the bats fly
randomly in the search space with a certain velocity at a certain position with a fixed frequency.
However, the wavelength and bat loudness can vary according to their distance between food and the
entire bat current position. Finally, the third rule followed by bat algorithm is linearly decreasing the
behavior of bat loudness factor [18]. The steps of bat optimization algorithm are given as follows [15,
16]:
Step 1: Initialize the algorithm parameters such as dimension of the problem ( ), population size
( ), maximum number of iterations ( ), minimum frequency ( ), maximum frequency ( ),
loudness of bats ( ), the initial Pulse emission rate ( ), pulse emission rate of bats ( ), and the
initial velocity of bats ( ).
Step 2: In BA, each bat position represents a feasible solution and the initial population for each bat
can be generated randomly as in Eq. (20):
| () | ( (20) | ) |
| Where; , , | is the lower boundary of the dimension and equals to 0, and |
is the upper boundary of dimension and equals to 10.
Step 3: Evaluate the cost function ( ) as in Eq. (21) for each bat and store the results in ( )
vector.
( ) ∑ ( ) (21)
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Where; is the desired signal, is the output signal, is the population size.
Step 4: Find the best bat position which is the bat that has the smallest ( ) value among all
bats in ( ) vector and consider its ( ) value as ( ).
| Step 5: Update the frequency, velocity, and position for the respectively. |
bat as shown in Eqs. (22, 23, and 24) |
| ( ) () ( ) Where; . Step 6: A local search is carried out if the rate of pulse emission by the |
(22) (23) (24) |
| bat ( ) is less than a |
| randomly generated number, a new solution is generated for the improve the variability of the possible solutions as in Eq. (25) |
bat via a random walk to |
| (25) | |
| Where; is a scaling factor generated randomly of interval [-1, 1], and is the average | |
| loudness of all bats at iteration. Step 7: Evaluate the cost function value ( ) for the ) variable. |
bat as in equation (21) and store the result |
| Step 8: For the | bat, if its new cost function value ( ) is smaller than its previous cost |
in ( function value and its loudness ( ) is bigger than a randomly generated number then the new values
of its cost function, loudness and Pulse rate are calculated as in Eqs. (26, 27, and 28) as follows:
(26)
| (27) | |
| ( ( )) Where; and are constants between [0, 1]. |
(28) |
| Step 9: For the | bat, if its ( ) is smaller than then the best bat position ( ) and |
| minimum frequency ( ) can be updated according to the Eqs. (29 and 30) as follows: | |
| (29) |
(30)
Step 10: If the maximum number of bats ( ) is reached go to Step 11. Otherwise, go to Step 5.
Step 11: Stop if the maximum number of iterations ( ) is reached. Otherwise, Step 5 to Step 11 is
repeated.
The typical flow chart of bat algorithm for the PID and PIDA controllers is shown in Figure 5.
Engineering and Technology Journal Vol. 38, Part A, (2020), No. 06, Pages 938-950
944
Figure 5: The flowchart of the bat optimization algorithm
4. Simulation Results and Discussions
The simulation results were obtained using Matlab package to show the performance of the two
intelligent controllers. The performance of the PIDA controller based on the bat control algorithm is
compared with that of the PID controller based on the same control algorithm. The parameters of the
bat algorithm are: Population size is 25, Maximum iteration number is 30, Dimension of the problem
is 3 for PID controller and 4 for PIDA controller, and are 0.5, Initial Pulse emission rate is
0.001, Minimum frequency is 0, Maximum frequency is 0.01, and the loudness is selected as a
random number. The comparison has been done through the dynamic behavior of the two higherorder systems as follows:
Higher-Order System 1
The first higher-order system has been taken from [11] and can be symbolized as follows:
| ( ) | ( ) |
(31)
The Matlab simulation is carried out for the PID and PIDA controllers based on the bat control
algorithm as shown in Figures 1 and 3, respectively. The bat tuning control algorithm is used off-line
in the simulation in order to follow a desired signal for the higher-order system. By applying the
Shannon theorem, the sampling time ( ) equals to based on the time constant of the
Engineering and Technology Journal Vol. 38, Part A, (2020), No. 06, Pages 938-950
945
system which equals to depending on natural frequency and the
damping ratio of the system.
Figure 6 shows the unit step change open loop response for the system which has unstable response.
Figure 6: The open loop response of the higher-order system 1.
The optimal tuning parameters and the dynamic higher-order system behavior such as rising time,
peak time, settling time, overshoots, and steady state error after applying both intelligent controllers
are depicted in Figure 7 and Table 1.
Figure 7: The output responses of the intelligent controllers for the higher-order system 1.
Table 1: Sets of the intelligent controllers’ parameters with the output response for system 1.
| Parameter | PID | PIDA |
| 0.4483 | 0.7793 | |
| 0.0013 | 0.0016 | |
| 1.5960 | 0.1558 | |
| – | 0.5821 | |
| Tr | 0.7644 Sec |
0.6445 Sec |
| Tp | 0 Sec |
0 Sec |
| Ts 2% error |
1.3800 Sec |
1.100 Sec |
| Mp % | 0 | 0 |
| Ess | 0 | 0 |
Engineering and Technology Journal Vol. 38, Part A, (2020), No. 06, Pages 938-950
946
In Figure 8, the error signals of both closed-loop controllers system were a small value in the
transient response and it has become zero at the steady-state response.
Figure 8: The error signals of the intelligent controllers for the higher-order system 1
The action responses of both intelligent controllers were smooth without oscillation response and no
spikes behavior occurs as shown in Figure 9.
Figure 9: The control actions of the intelligent controllers for the higher-order system 1.
Figure 10 clearly shows the improved performance indices of the intelligent controllers based on the
Mean Square Error.
Figure 10: The performance indices (MSE) of the intelligent controllers for the higher-order system 1.
Engineering and Technology Journal Vol. 38, Part A, (2020), No. 06, Pages 938-950
947
Higher-Order System 2
The second higher-order system has been taken from [8] and can be expressed as follows:
| ( ) |
(32)
(33)
| ( ) |
(34)
The Matlab simulation is carried out for the PID and PIDA controllers based on the bat control
algorithm as shown in Figures 1 and 3, respectively. The bat tuning control algorithm is used off-line
in the simulation in order to follow a desired signal for the higher-order system. By applying the
Shannon theorem, the sampling time ( ) equals to based on the time constant of the
system which equals to depending on natural frequency and the
damping ratio of the system.
The unit step change open loop response for the system is shown in Figure 11 which reveals that the
system is stable.
Figure 11: The open loop response of the higher-order system 2.
The optimal tuning parameters and the dynamic higher-order system behavior such as rising time,
peak time, settling time, overshoots, and steady state error after applying both intelligent controllers
are depicted in Figure 12 and Table 2.
Figure 12: The output responses of the intelligent controllers for the higher-order system 2.
Engineering and Technology Journal Vol. 38, Part A, (2020), No. 06, Pages 938-950
948
Table 2: Sets of the intelligent controllers’ parameters with the output response for system 2
| Parameter | PID | PIDA |
| 1.0633 | 1.4915 | |
| 0.0786 | 0.0780 | |
| 3.8904 | 2.4550 | |
| – | 1.3517 | |
| Tr | 6.7840 Sec |
6.0325 Sec |
| Tp | 9.8895 Sec |
7.4415 Sec |
| Ts 2% error |
16.3084 Sec |
8.4662 Sec |
| Mp % | 8.8425 | 2.3670 |
| Ess | 0 | 0 |
The error signals of both closed-loop controllers system were a small value in the transient response
and it has become zero at the steady-state response as shown in Figure 13.
Figure 13: The error signals of the intelligent controllers for the higher-order system 2.
Figure 14 shows the action responses of both intelligent controllers were smooth without oscillation
response and no spikes behavior occurs.
Figure 14: The control actions of the intelligent controllers for the higher-order system 2.
Figure 15 clearly shows the improved performance indices of the intelligent controllers based on the
Mean Square Error.
Engineering and Technology Journal Vol. 38, Part A, (2020), No. 06, Pages 938-950
949
Figure 15: The performance indices (MSE) of the intelligent controllers for the higher-order system 2
5. Conclusion
The numerical simulation results of the PID and PIDA controllers based on bat optimization
algorithm have been presented in this paper for the higher-order systems. The bat tuning control
algorithm shows the following capabilities:
1. The off-line bat control algorithm has the ability of fast finding and tuning the optimal parameters
of the controller with the minimum fitness evaluation number.
2. A proper control action was obtained as a smooth without oscillation response and no spikes
behavior occurs.
3. The numerical simulation results for both controllers based on bat tuning control algorithm show
that both controllers can give excellent performance but the performance of the PIDA controller for
higher order systems is better than that of the PID controller in terms of reducing the rising time,
peak time, settling time, overshoot and steady-state error. Moreover, the fitness evaluation number is
reduced.
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