Journal of Robotics, Networking and Artificial Life
Vol. 8(3); December (2021), pp. 170–174
DOI: https://doi.org/10.2991/jrnal.k.210922.004; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Design of a Data-Driven Control System based on Reference
Model using Predicted Input/Output Responses
Yuki Nakatani, Takuya Kinoshita*, Toru Yamamoto
Graduate School of Advanced Science and Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Higashi-Hiroshima,
Hiroshima 739-8527, Japan

ARTICLE INFOABSTRACT

Article History

In recent years, data-driven control schemes that do not require system modeling have been actively studied. These schemes use
only a set of experimental data to design a controller that realizes the desired reference output offline. However, it is necessary toconsider the output response and the input response since there is a limit of the actuator performance in actual machines. This
paper proposes a new data-driven control scheme that can predict the input/output responses of an unknown system in offline
using a set of operating data. The effectiveness of the proposed scheme is numerically verified by a simulation example.

Received 15 November 2020
Accepted 18 June 2021
Keywords
Data-driven control
extended output
predicted data
offline tuning© 2021 The Authors. Published by Atlantis Press International B.V.
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
In industrial systems, data-driven control schemes [1–4] have been
actively studied to achieve the desired control performance for thecontrolled system with unknown structure and parameters. These
schemes use only a set of experimental data to design a controllerthat satisfies the desired control performance specified by the reference model offline. Generally, the reference model is designed
to focus on only the output response. However, both input/output
responses should be considered in designing the control systembecause there is a limit of the actuator performance in the actualmachine. If the input response could be estimated in advance, itwould be very useful in designing the reference model.
The Estimated Response Iterative Tuning (ERIT) scheme [5] hasbeen proposed to predict the input/output response before applying the adjusted control parameters. However, this scheme can only
be applied to Two-Degree-of-Freedom (2DOF) control systems.
On the other hand, it is important to design One-DOF (1DOF)
controller because there are also many 1DOF control systems inindustries.
In this paper, the new data-driven control scheme is proposed
to design a 1DOF controller considering input response forunknown structure systems. The features of the proposed scheme
are as follows:(i) The 1DOF control system design that predicts the input
response in advance, even for unknown system.
(ii) Adjusting the desired input/output responses with one
parameter l.
Corresponding author. Email: kinoshita-takuya@hiroshima-u.ac.jp*

Finally, the effectiveness of the proposed scheme is verified in
Section 4.

2. OVERVIEW OF THE PROPOSEDSCHEME
Figure 1 shows an overview of the proposed data-driven control
scheme. In the proposed scheme, the controller is designed by thefollowing procedure:**(I) Obtaining the desired predicted data y (t ) and u (t ).
*(II) Constructing the reference model Gm(z−1) based on y (t ) and*u (t ) obtained in (I).

(III) Designing a controller using the reference model Gm(z−1)
obtained in (II).
The detailed procedure is explained in the next section.

3. DESIGN OF THE CONTROLLER
3.1. Generating Predicted the Data
using the ERIT Scheme
In this section, the ERIT scheme [5] is described to generate thepredicted data. This scheme is theoretically only applicable tothe 2DOF control system. In the proposed scheme, the ERIT
scheme is utilized for only predicting the input/output data in
Figure 2. Note that the controller is not designed by the ERIT
scheme.

Y. Nakatani et al. / Journal of Robotics, Networking and Artificial Life 8(3) 170–174

171

*
*
Specifically, y (t ) and u (t ) are calculated by solving the following
minimization problem of the evaluation norm based on the integral of time squared absolute error scheme [6] for tuning CFF(z−1):
N

N

t =0

t =0

J des = åt r (t ) - y(t ) + l åt D u(t ) , 
	 (5)where r(t), ∆(:= 1 − z−1), and N denote the reference signal, thedifference operator, and the number of data.
Firstly, r(t) − ŷ(t) in Equation (5) shows the error between thereference signal and the predicted output response. The desiredpredicted response is obtained by reducing it. Secondly, ∆û(t) =û(t) − û(t − 1) in Equation (5) represents the difference of input
response. Therefore, a smaller value of l increases the responsiveness and a larger value decreases the responsiveness dependingon the adjustable parameter l. Thus, the input/output responses
can be easily adjusted with l. Additionally, it is easy for the user to**select the optimal u (t ) by changing l because predicted
y (t ) and 
data of unknown systems can be calculated in the ERIT scheme.

Figure 1 | Overview of the data-driven control system by the proposed
scheme.

Figure 2 | 2DOF control system.

First, the initial output y0(t) is obtained in a 1DOF control system.
In the case of the feedforward controller CFF(z−1) = 0 in Figure 2,
y0(t) is given as follows:	   y0 (t ) =G(z -1 )CFB (z -1 )r (t ). 
1 + G(z -1 )CFB (z -1 )(1)Then, extending to a 2DOF control system for predicting the data,
the predicted output ŷ(t) is defined as:G(z -1 )CFF (z -1 ) + G(z -1 )CFB (z -1 )y(t ) =r (t ). 
1 + G(z -1 )CFB (z -1 )(2)Here, system identification is required to obtain an exactly predicted output ŷ(t) because Equation (2) contains a controlled system
G(z−1). Therefore, substituting Equation (1) into Equation (2)yields the following equation:-1

CFF (z )y (t ) + y0 (t ). 	   y(t ) =CFB (z -1 ) 0(3)Consequently, the predicted data ŷ(t) can be derived offline without G(z−1) by using the initial data y0(t) by tuning CFF(z−1). û(t) can
also be derived in the same procedure using u0(t) as follows:u(t ) =
	   

CFF (z -1 )u (t ) + u0 (t ). 
CFB (z -1 ) 0(4)From Equations (3) and (4), it is possible to obtain the predicteddata ŷ(t) and û(t) offline corresponding to the various CFF(z−1) byusing the initial data y0(t) and u0(t). Note that the 2DOF control
system was only implemented virtually offline to obtain the predicted data ŷ(t) and û(t).

The adjustment of l is currently a trial and error process. However,l can be easily adjusted offline. First, l is set to 0 in Equation (5).
As a result, the predicted input response û(t) is large, because theinput term in Equation (5) is not considered. Then, l is gradually
increased to satisfy the desired predicted input/output response
ŷ(t) and û(t).

3.3. Design of Gm(z−1) based on the Desired
Predicted Data y (t )*In this section, Gm(z−1) is designed based on the desired predicted*data y (t ). The evaluation norm for designing Gm(z−1) is defined asfollows:
1 N *
2
	   J ref = å( y (t ) - ym (t )) (6)
N t =0		 ym (t ) = Gm (z -1 )r (t ). 
Gm(z−1) is designed by minimizing Jref.

3.4. Design of a PID Controller basedon the Extended Output ϕ(t)In this section, a 1DOF controller C(z−1) is designed as an I–PD
controller [7] by using the reference model Gm(z−1) that considers
the input response obtained in the previous section. It is possible
to design the controller by using any data-driven approach. In thispaper, the Proportional-Integral-Differential (PID) control scheme
based on extended output [8] is used.
I-PD controller is given as follows:Du(t ) = K I e(t ) - K P Dy(t ) - K D D 2 y(t ), 
	 3.2. Obtaining the Desired Predicted
Input/Output Data*y (t ) andIn this section, obtaining the desired predicted data *u (t ) is described by using the predicted data ŷ(t) and û(t).

(7)

(8)

where KP, KI, and KD are the proportional gain, integral gain, andderivative gain, respectively. In this scheme, C(z−1) is designed byminimizing the following evaluation norm:
1 N
J = å (t )2 		  
N t =0(9)

Y. Nakatani et al. / Journal of Robotics, Networking and Artificial Life 8(3) 170–174172−1		  (t ) = Gm (z )f (t ) − y(t ), 

(10)where ϵ(t) denotes the difference between the output y(t) of theclosed-loop transfer function and the output Gm(z−1)ϕ(t) of the reference. C(z−1) can be designed by minimizing J.
For reasons of space, the details are omitted and refer to the PID
control scheme based on extended output ϕ(t) [8].

4. NUMERICAL EXAMPLE
4.1. Controlled System
The effectiveness of the proposed scheme is verified using numerical simulation. The controlled system is the experimental thermal
equipment for simulating bag-and-bound welding owned by ourlaboratory. The model of the system is given as follows:
T
ü
- s ï
ïì
R í1 - e RH ý z -1ïþ		 y(t ) = ïîu(t ) Ts-

1 + e RH z

(11)

q 1 + q 2 z −1−1	   Gm (z ) = 1 + q z −1 + q z −2 , 
3
4

(12)

where R, H, d(t), and TS are the thermal resistance, the heat capacity, the room temperature, and the sampling time, respectively. And
ξ(t) is Gaussian white noise with mean 0 and variance 0.1. The coefficients R, H were derived by system identification. Table 1 shows
the value of thermal resistance R and heat capacity H.

where the value of θi (i = 1, 2, 3, 4) is determined by using the ’fminsearch.m’ to minimize Jref in Equation (6).
Finally, the tuned proportional gain K Ptuned and integral gain K Ituned
were designed by minimizing J in Equation (9).
In this numerical example, the control results are compared bychanging l in Equation (5), and Table 2 shows the feedforward*
controller CFF(z −1 ), the reference model Gm(z−1), the tuned protuned
portional gain K Ptuned and the tuned integral gain K I respectively
in the cases of l = 0.5 and 5.

4.2. Control Result
Figures 3 and 4 show simulation results corresponding to thecases of l = 0.5 and 5, where y0(t) and u0(t) denote initial I/O data,**u (t ) denote the desired predicted I/O data, and y (t)y (t ) and 
and u1(t) denote I/O data given by the proposed scheme. The predicted response can be adjusted with one parameter according to l
because the responsiveness is high because l = 0.5 in Figure 3, andthe responsiveness is low because l = 5 in Figure 4.
Table 2 | Obtained parameters in the simulationl = 0.5

The proposed scheme is applied to the system. The reference signal
is r(t) = 100(°C), the sampling time is TS = 0.04(s), and the room
temperature is d(t) = 20(°C). In this numerical example, I–P controller was used, whose initial proportional and integral gains wereset as follows:

a1
a2
a3
θ1
θ2
θ3
θ4

-3
ini
ini
		 K P = 1.5, K I = 2.0 ´ 10 . 

K Ptuned

ini
P

(15)

1

-1

		 y(t ) = y(t ) + d(t ) + x (t ), 

Furthermore, Gm(z−1) was determined as follows:

(13)

ini
I

Note that the initial PI gains of K and K are set to obtain thestable input/output data y0(t), u0(t). Additionally, the initial datay0(t) and u0(t) were filtered by using 1 / {1 + (4/3)s}.

K Itunedl=5−22.55 × 10−1.30
3.25 × 10−1
−4.08 × 10−4
4.76 × 10−4
−1.98
9.80 × 10−1
6.18

3.64 × 10−3
−1.01
1.77 × 10−2
−2.24 × 10−4
2.39 × 10−4
−1.99
9.93 × 10−1
2.12

2.08 × 10−2

4.77 × 10−3*
(z -1 ) was calculated by the
Next, the feedforward controller CFF
proposed scheme as follows:a1-1*	   CFF (z ) = 1 + a z -1 + a z -2 . 
(14)
2
3

The order of the denominator was set to second-order because
the low order term significantly affects the output response of thecontrolled system. Furthermore, the value of ai (i = 1, 2, 3) was
determined by minimizing Jdes in Equation (5), for example, using
the Matlab/Simulink Ver.9.8.0.1396136(R2020a), Optimization
Toolbox ’fminsearch.m’.
Table 1 | Derived parameters by system identification
H(J/°C)
R(°C/W)

12.18
0.1612 { y(t )} + 2.448

Figure 3 | Simulation results by applying the proposed scheme wherel = 0.5.

Y. Nakatani et al. / Journal of Robotics, Networking and Artificial Life 8(3) 170–174

173

the actuator by predicting the input in advance. In the future, thescheme how to choose l and the controller design considering the
input saturation will be studied.

CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.

REFERENCES[1][2]
Figure 4 | Simulation results by applying the proposed scheme wherel = 5.

Furthermore, it is possible to design a controller that considers the*y (t )
input/output responses because the desired predicted data *u (t ) , and the results of implementing a 1DOF controller y1(t)and and u1(t) are almost identical in Figures 3 and 4. The results y1(t) andu1(t) oscillate more for l = 0.5 in Figure 3 than for l = 5 in Figure 4
because l = 0.5 in Figure 3 has a larger PI gain.

5. CONCLUSION
In this paper, the new data-driven control scheme has been proposed to design a 1DOF by considering input response for unknown
structure systems. The effectiveness has been numerically verified
by the simulation example. It is possible to predict the appropriate amount of input response to correspond the performance of

[3]
[4]
[5]

[6]
[7]
[8]

T. Yamamoto, O. Kaneko, S. Wakitani, T. Kinoshita, Y. Ohnishi,
H. Kugemoto, et al., Data-oriented PID control (in Japanese),
Morikita Publish, Tokyo, 2020.
O. Kaneko, Data-driven controller tuning: FRIT approach, IFAC
Proc. Vol. 46 (2013), 326–336.
H. Hjalmarsson, M. Gevers, S. Gunnarsson, O. Lequin, Iterative
feedback tuning: theory and applications, IEEE Control Syst.
Mag. 18 (1998), 26–41.
M.C. Campi, A. Lecchini, S.M. Savaresi, Virtual reference feedback tuning: a direct method for the design of feedback controllers, Automatica 38 (2002), 1337–1346.
O. Kaneko, T. Nakamura, T. Ikezaki, A new approach to updating
feedforward controllers in two-degree-of-freedom control systems estimated response iterative tuning (ERIT) (in Japanese),
Trans. Soc. Instrum. Control Eng. 54 (2018), 857–864.
D. Graham, R.C. Lathrop, The synthesis of optimum response:
criteria and standard forms, part 2, Trans. AIEE 72 (1953),
273–288.
M. Ogawa, T. Katayama, A robust tuning method for I-PD controller incorporating a constraint on manipulated variable, Trans.
Soc. Instrum. Control Eng. E-1 (2001), 265–273.
A. Yoichiro, S. Wakitani, T. Yamamoto, Design of an augmented
output-based multiloop self-tuning PID control system, Ind. Eng.
Chem. Res. 58 (2019), 11474–11484.

AUTHORS INTRODUCTION
Mr. Yuki Nakatani
He received his B. Eng. from Hiroshima
University in Japan in 2020. He is currently a
Master course student in Hiroshima University
in Japan.

Dr. Takuya Kinoshita
He received his B. Eng., M. Eng. and D. Eng.
from Hiroshima University in Japan in 2013,2015 and 2017, respectively. He was postdoctoral fellow of JSPS (Japan Society for the
Promotion of Science) in 2017. He is currently
an Assistant Professor with the Departmentof Graduate School of Advanced Science and Engineering,
Hiroshima University, Japan. His research interests are
performance-driven control.

174

Y. Nakatani et al. / Journal of Robotics, Networking and Artificial Life 8(3) 170–174

Prof. Toru Yamamoto
He received the B. Eng. and M. Eng. degrees
from Tokushima University, Tokumshima,
Japan, in 1984 and 1987, respectively, and theD. Eng. degree from Osaka University, Osaka,
Japan, in 1994. He is currently a Professor with
the Graduate School of Advanced Science andEngineering, Hiroshima University, Japan. He was a Visiting
Researcher with the Department of Mathematical Engineering
and Information Physics, University of Tokyo, Tokyo, Japan, in
1991. He was an Overseas Research Fellow of the Japan Society
for Promotion of Science with the University of Alberta for6 months in 2006. His current research interests are in the areaof data-driven control, and process control. He was the recipient of the Commendation for Science and Technology by the
Minister of Education, Culture, Sports and Technology in 2009.