Internationalof Computational
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Systems
Journal ofJournal
Robotics,
Networking andArtificial Life
Vol.
XX(Z);
Month(2019),(Year), pp.
pp. 118–122
xx–yy
Vol.
6(2);
September
DOI: 10.1080/XXXXXXXXXXXXXX; ISSN
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DOI: https://doi.org/10.2991/jrnal.k.190828.010;
ISSNXXXX–XXXX
2405-9021; eISSN
2352-6386
https://www.atlantis-press.com/journals/ijndc
https://www.atlantis-press.com/journals/jrnal

Research Article

Design of a PID Controller using a Fictitious Exogenous
Signal for a Fluctuation System
Masatoshi Kozui*, Takuya Kinoshita, Toru YamamotoGraduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima, JapanARTICLE INFO
Article History
Received 10 November 2018
Accepted 21 November 2018
Keywords
PID control
Fictitious Exogenous Signal
fictitious reference iterative tuning

ABSTRACTThe fictitious reference iterative tuning method is used to design Proportional–Integral–Derivative (PID) controllers to achieve
desired performance for industrial machines. However, the desired control performance cannot be obtained by this method, if
the system characteristics changed in steady state. In this paper, design method of a PID controller using the Fictitious Exogenous
Signal is proposed. The change of system characteristics is considered as an impulse-like virtual disturbance. The Fictitious
Exogenous Signal to calculate PID parameters is obtained from a set of operational data. The effectiveness of the proposed
method has been verified by numerical examples.
© 2019 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
PID controllers has been used in a lot of industrial machines.
Appropriate PID parameters are necessary for a desired performance. However, the parameters are difficult to be decided bytrial and error. Therefore, the Chien–Hrones–Reswick (CHR)
method, a conventional method, is widely used. In this method,
the proportional gain, the integral gain and the differential gain
are determined based on the parameters of the controlled system,respectively. However, this method can be applied only when thesystem parameters are known, and desired response characteristics
cannot necessarily be obtained in some cases.
Recently, the Fictitious Reference Iterative Tuning (FRIT) method
which calculates PID parameters directly using the closed-loop
data obtained by a set of operational data attracts attention [1]. The
method can be applied, even when the characteristics of the system
are unknown. The fictitious reference input is calculated fromsystem input and output in the operational data, and input to thereference model. This method designs the optimum controller byadjusting PID parameters to let the output of the reference model
be close to the output in the operational data. However, for manyindustrial machines, especially machines that operated by human,the system characteristics change when the operating environmentchanges. In such cases, it is necessary to redesign the controller
after the system fluctuation.
In this paper, the controlled system is assumed to be switched insteady state, and the controller is constituted as a PID controller.
The PID parameters before the system switching are set by theFRIT method. After the system switching, a method that setting
the PID parameters by the Fictitious Exogenous Signal is proposed.
Corresponding author. Email: kozuim@hiroshima-u.ac.jp*Specifically, the system fluctuation in steady state is regarded as animpulse-like virtual disturbance is input. At this time, the transfer
function from the disturbance to the control error is constructed.
The optimum controller is designed by adjusting the PID parameters so that the characteristics of the closed-loop system approaches
the reference model. Genetic Algorithm (GA) is used to calculate
the PID parameters. Moreover, the effectiveness of the proposed
method has been verified by numerical examples.

2. CONTROL SYSTEM
2.1. Control Objective
A switching system in which the system changing occurs in steadystate is shown in Figure 1 as the control objective. System1 is initial state, and system2 is the state after switching. Switching of thesystem is assumed to occur at a known time. Controller1 and controller2 are controllers for these systems, respectively. PID parameters are set in the controllers.

2.2. Control Law
The controllers for System1 and System2 use the following I-PD
controller [Equation (1)].
2	     Du(k) = K ie(k) - K p Dy(k) - K d D y(k) (1)u(k), e(k) and y(k) are control input, control error and system
output. Kp, Ki, and Kd are proportional gain, integral gain and differential gain, respectively. D is a differencing operator, defined by
D := 1–z−1.

M. Kozui et al. / Journal of Robotics, Networking and Artificial Life 6(2) 118–122

Furthermore, the control error e(k) is calculated as the following
Equation (2). r(k) is the reference value.		  
e(k) := r (k ) - y(k) (2)The controller1 for system1 is set as Kp1, Ki1, and Kd1 using the FRIT
method. Here, Kp1, Ki1, and Kd1 are proportional gain, integral gain
and differential gain, respectively. The controller parameters arecalculated directly by using a set of operational data, and the block
diagram is shown in Figure 2. u0(k) and y0(k) in the closed-loop
data are system input and output of a set of initial operational data.
y(k) is the output of a reference model Gm(z−1). r(k) indicates thefictitious reference input of the reference model Gm(z−1), and calculated from a set of operational data (u0(k), y0(k)) and Equation (1).
Gm(z−1) and r(k) are defined as the following Equations (3)–(5).
z -1P (1)r(k) (3)P (z -1 )P (z -1 ) = 1 + p1z -1 + p2 z -2 (4)	  r(k) = y0 (k ) +designer, arbitrarily). It is desirable that d is set between 0 £ d £ 1.
When d = 0, the Binominal model response is shown, and whend = 1, the Butterworth model response is shown.

2.2.2. Fictitious Exogenous Signal method

2.2.1. FRIT method

y(k) = Gm (z -1 )r(k) =119

Du0 (k) + K p1Dy0 (k) + K d1D 2 y0 (k)
K i1

(5)

Here, p1 and p2 are designed by the following Equation (6) [2].
 4m − 1  
 r r p1 = − 2exp  −  cos  2m  2m
 r (6)	      p2 = exp  −  m

Ts
r=
s

m = 0.225 (1 − d ) + 0.51ds relates to the rising characteristic of the control, and d is a para­
meter relating to the attenuation characteristics (it can be set by theThe controller2 for system2 is set as Kp2, Ki2 and Kd2 by a method
using the Fictitious Exogenous signal. Here, Kp2, Ki2 and Kd2 areproportional gain, integral gain and differential gain, respectively.
The controller parameters are calculated directly by using a setof operational data. The method using the Fictitious Exogenous
Signal is similar to the FRIT method, and block diagram is shownin Figure 3. The reference model Gmd(z−1) means the desired
closed-loop transfer characteristics from disturbance to control
error. d(k) is the input, and e0(k) is the output. The d(k) is a
Fictitious Exogenous Signal, and the output of the reference modelGmd(z−1) is e(k) when the input is d(k ). The Fictitious ExogenousSignal d(k ) is expressed by the following Equation (7) [3] from a
set of initial operational data (u0(k), y0(k)) measured in advanceand Equation (1). Here, the system fluctuation is regarded as animpulse-like virtual disturbance d(k ) was input. At this time,since d(k ) is a virtual value, the initial operational data that it isnecessary to calculate the d(k ) is the same as a set of operational
data (u0(k), y0(k)) measured in advance. uc(k) is calculated by thecontroller. d(k ) is calculated by the following Equation (7).
d(k) = u0 (k) - uc (k )(7)= u0 (k) - (uc (k - 1) + K i 2e0 (k ) - K p 2 Dy0 (k ) - K d 2 D 2 y0 (k))
The reference model Gmd(z−1) can be expressed by the following
Equation (8) using the Gm(z−1) in the FRIT method.
-1æK ö		 Gmd (z -1 ) = Gm (z -1 ) ç i 2 ÷ (8)
è D øTherefore, output e(k) of the reference model Gmd(z−1) is calculatedby the following Equation (9).	 
e(k) = Gmd (z -1 )d(k) = --1

z -1Pd (1) æ K i 2 ö d(k) (9)Pd (z -1 ) çè D ÷øThe setting method of Pd(z−1) is the same as the FRIT method, andparameters s and d can be set different values from system1.

Figure 1 | Switching system.

Figure 2 | Block diagram of the FRIT method.

Figure 3 | Block diagram of the proposed method using the Fictitious
Exogenous Signal.M. Kozui et al. / Journal of Robotics, Networking and Artificial Life 6(2) 118–122

120

2.3. Genetic Algorithm
The GA was used to minimize the value of the evaluation function [4,5]. The evaluation function is defined by the following
Equations (10) and (11).
 System1
		

J1 =

N

1 1
å( y0 (k) - y(k))2 (10)2 1System2
		

J2 =

N

1 2
å(e0 (k) - e(k))2 (11)2 1

N1 and N2 indicate the total number of steps of operational data inthe system1 and the system2, respectively. In the system1, a desired
response characteristic can be obtained by defining the difference
between the reference model output y(k) and y0(k) as an evaluation function and setting PID parameters to minimize the evaluation value. When the evaluation value is low, the fitness is high andPID parameters are better. The system2 is also the same.
First, initial individuals Kp, Ki and Kd as genes are randomly generated, and initial evaluation is performed. Individual with highfitness are considered elite, and carry over to the next generation.
Next, crossover, tournament selection and mutation are performed
based on the evaluation values. These procedures are repeated ineach generation, and PID parameters are obtained as the gene of
the individual with the highest fitness in the final generation. In
this paper, the GA was calculated using Matlab under the conditionof 200 individuals and 200 generations.

System2   y(k) = 0.984 y(k - 1) + 0.0165u(k - 3) + x (k ) First, a simulation result using PID parameters calculated by theCHR method was shown in Figure 4. The PID parameters areshown in Table 1.
In Figure 4, it took long time to follow the Reference r(k), and theresponsiveness of the system1 is not good. The fixed PID parameters corresponding to the system1 has particularly small proportional gain Kp. Therefore, input u(k) is moderate at the switchingtime, and the output variation after switching to the system2 is large
and recovery time is long. The control performance is not good.

3.2. Proposed Method
Figure 5 shows the control result of the proposed method. The PID
parameters for the system1 are calculated using the FRIT method,
and the PID parameters for the system2 are calculated by adjustment method using the Fictitious Exogenous Signal. u0(k) and y0(k)that obtained by the CHR method were used as a set of the operational data in order to calculate the PID parameters for the system1
and the system2. Table 2 shows the PID parameters. As shown in
Figure 5, the response of the system1 is faster than the conventional
method, and the desired output was obtained. In addition, theoutput variation after switching to the system2 is less than 1/4 ascompared with the conventional method, and good control performance is obtained. The PID parameters for system2 is larger thanthat for system1. Therefore, the response of input u(k) is faster andoutput variation can be suppressed.
It was confirmed that the responsiveness after the systemswitching was improved by considering the system fluctuation

3. NUMERICAL SIMULATION
To verify the effectiveness of the proposed method, a switching system of the first order lag systems was set as the following
Equations (12) and (13) and a simulation was performed.
 System1
		 G1 =

System2
		 G2 =

1e -2 s1 + 50s(12)1e -2 s
1 + 60s(13)

3.1. Conventional Method (CHR Method)
The following Equations (14) and (15) were calculated by discretizing the equations above at the sampling time Ts = 1 s, and areobtained, respectively. Here, the Gaussian white noise with mean
0 and variance 0.01 was given by ξ(k). Furthermore, in 250 steps,
the system was switched from the system1 to the system2. The
effectiveness of the proposed method was verified. Reference isr(k) = 10.

System1   
y(k) = 0.980 y(k - 1) + 0.0990u(k - 3) + x (k)

(15)

(14)

Figure 4 | Simulation result by the CHR method.
Table 1 | PID parameters calculated by the CHR method

Systems1 and 2KpKi

Kd

3.0

0.06

3.0

M. Kozui et al. / Journal of Robotics, Networking and Artificial Life 6(2) 118–122

121

in steady state was proposed. In the proposed method, the system
fluctuation was regarded as a disturbance input, and the transfer function from the disturbance d(k) to the control error e(k)was constructed as the reference model Gmd(z−1). The Fictitious
Exogenous Signal calculated from a set of operational data wasinput to the reference model Gmd(z−1). It was confirmed that a controller with a good disturbance responsiveness can be obtained byadjusting the PID parameters to approximate the reference model
output e(k) to the control error e0(k).
In the future, the proposed method will be applied to an actualmachine to confirm the effectiveness.

CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.
Figure 5 | Simulation result using the FRIT method and the Fictitious
Exogenous Signal.
Table 2 | PID parameters calculated by the FRIT method and Fictitious
Exogenous Signal

System1System2KpKiKd1.6018.4

0.07
0.73

3.19
20

as a disturbance input and applying the adjustment law using
the Fictitious Exogenous Signal.

4. CONCLUSION
In this paper, a method of controller adjustment law using theFictitious Exogenous Signal for a system that switched characteristics

REFERENCES
[1]

[2]
[3]
[4]
[5]

O. Kaneko, K. Yoshida, K. Matsumoto, T. Fujii, A new parametertuning for controllers based on least-squares method by using
one-shot closed loop experimental data: an extension of fictitious
reference iterative tuning, ISCIE J. 18 (2005), 400–409.
S. Hayashi, T. Yamamoto, Evolutionary design of a PID controller
using closed-loop data, IEEJ Trans. Electron. Inform. Syst. 131(2011), 794–799.
S. Masuda, K. Takeda, An FRIT method for disturbance attenuation
using input-output data generated from disturbance responses,
IEEJ Trans. Electron. Inform. Syst. 131 (2011), 788–793.
D.E. Goldberg, Genetic algorithms in search, optimization,and machine learning, Reading, MA, USA: Addison-Wesley
Publishing Company; 1989.
K. Kawada, H. Sogo, T. Yamamoto, Y. Mada, Robust PD swaycontrol of a lifted load for a crane using a genetic algorithm, IEEJ
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Authors Introduction
Mr. Masatoshi Kozui

Dr. Takuya Kinoshita
He received his B. Eng. and M. Eng.
from Hiroshima University in Japan in2009 and 2011, respectively. He joined
KOBELCO Construction MachineryCo., Ltd. in 2011. He is currently anAssistant Professor of Collaborative
Research Laboratory with the Graduate
School of Engineering, Hiroshima
University, Japan. His research interests are skill evaluation and system
control of Hydraulic Excavator.

He received his B. Eng., M. Eng. andD. Eng. from Hiroshima University
in Japan in 2013, 2015 and 2017,respectively. He was postdoctoral
fellow of Japan Society for thePromotion of Science (JSPS) in 2017.
He is currently an Assistant Professor
with the Department of System
Cybernetics, Graduate School ofEngineering, Hiroshima University,Japan. His research interests are performance driven control.

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M. Kozui et al. / Journal of Robotics, Networking and Artificial Life 6(2) 118–122

Prof. Toru Yamamoto
He received the B. Eng. and M. Eng.
degrees from the University ofTokushima, Tokushima, Japan, in
1984 and 1987, respectively, andthe D. Eng. degree from Osaka
University, Osaka, Japan, in 1994.
He is currently a Professor with theDepartment of System Cybernetics,
Graduate School of Engineering,
Hiroshima University, Japan. He
was a Visiting Researcher with theDepartment of Mathematical Engineering and Information
Physics, University of Tokyo, Tokyo, Japan, in 1991. He was an
Overseas Research Fellow of the Japan Society for Promotionof Science with the University of Alberta for 6 months in 2006.
His current research interests are in the area of data-driven control, and process control. Dr. Yamamoto was the recipient of theCommendation for Science and Technology by the Minister ofEducation, Culture, Sports and Technology in 2009.