Journal of Robotics, Networking and Artificial Life
Vol. 5(4); March (2019), pp. 257–260
DOI: https://doi.org/10.2991/jrnal.k.190220.011; ISSN 2405-9021 print; 2352-6386 online
https://www.atlantis-press.com/journals/jrnal

Research Article

Design of a Data-Driven Controller with Evaluating
Controller Performance
Takuya Kinoshita*, Toru Yamamoto
Graduate School of Engineering, Hiroshima University, Higashihiroshima, Hiroshima, Japan

ARTICLE INFO
Article History
Received 10 November 2018
Accepted 21 November 2018
Keywords
PID controller
controller assessment
data-driven controller
least squares method

ABSTRACT
Data-driven controller systems have been proposed to achieve the desired control performance without using any system
identifications. The effectiveness of these control schemes has been shown through experimental results. For time-variant andnonlinear systems, it is important to evaluate the controller performance and redesign controller when the performance is poor.
According to the proposed scheme, the controller performance calculator and controller design are integrated using only input
and output data.
© 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
In most industries, it is very important to get desired control performance. Data-driven control schemes [1–4] have been proposedto achieve the aforementioned desired control performance without using any system identifications. For time-variant systems, performance-driven controller [5] has been proposed to improve thesteady state control performance only when the performance is poor.
However, performance-driven controller cannot evaluate the transientstate because this scheme focuses on the variance of steady state error.
In this paper, the scheme of evaluating controller performance
without using any system identifications is proposed. According to
the proposed scheme, the control performance is evaluated system
output error including transient state. Furthermore, the controller
performance and controller parameters are simultaneously calculated using only input and output data.

2. OUTLINE FIGURE OF THE
PROPOSED SCHEME
The schematic figure of the proposed control system is shown in (z -1 ) are the controller,Figure 1. C(z–1), G(z–1), Gm(z–1), and G
controlled system, reference model and estimated system model,respectively.
The purpose of the proposed control system is to achieve thedesired control performance by minimizing the following criterion
Jr [Equations (1) and (2)]:Corresponding author. Email: kinoshita-takuya@hiroshima-u.ac.jp*1J r = fr (t )22
fr (t ) = y(t ) - yr (t ),(1)(2)

where y(t) and yr(t) are the control output and reference output,
respectively.
Another purpose of the proposed scheme is to design the estimated (z -1 ) by minimizing the following Equations (3)system model Gand (4):
1J r = f(t )22f(t ) = y(t ) - y(t ),(3)(4)

 (z -1 )y (t) is the estimated output. In the proposed scheme, Gwhere can be introduced by using the controller parameters and reference
model parameters without any system identifications. Therefore, controller and estimated system model can be designed simultaneously.
In addition, users set a desired reference model expressed by thefollowing Equations (5) and (6):yr (t ) = Gm (z -1 )r (t )Gm (z -1 ) : =z- (1+ d )P (1)P (z -1 )

(5)
(6)

where r(t) is the reference signal and P(z–1) is user-specified polynomial. P(z–1) is designed based on the reference design [6] as follows [Equations (7) and (8)]:P (z −1 ) : = 1 + p1z −1 + p2 z −1

(7)

T. Kinoshita and T. Yamamoto / Journal of Robotics, Networking and Artificial Life 5(4) 257–260

258

3.1. Evaluation of the Controller Performance
The optimized controller C*(z–1) makes f(t ) equals to zero. In contrast, f(t ) becomes large when the performance of the controller
C(z–1) is poor. Hence, this paper considers the performance of thecontroller C(z–1) based on f(t ).

3.2. Tuning Scheme of the PID Gains
In this paper, the controller is utilized as following I-PD controller[Equations (12) and (13)]:

Figure 1 | Schematic figure of the proposed control system.

ìæ r ö æ 4m - 1 ör÷ï p1 = -2exp ç ÷ cos ç÷ïè 2 m ø çè 2 møïï p = exp æ - r öç÷í 2è møï
ï r := Tsïsï m := 0.25 (1 - d ) + 0.51d
î(8)3. RELATIONSHIP BETWEEN CONTROLLER
AND ESTIMATED SYSTEM MODEL
The following Equation (9) can be obtained by introducing the
optimized controller C*(z–1) which achieves ϕr (t) = 0:
(9)The controlled system G(z–1) is expressed as follows by using theaforementioned Equation (10):Gm (z -1 )-1

G( z ) ={}C * (z -1 ) 1 - Gm (z -1 )

.

(10)

 (z -1 ) is definedHere, the following estimated system model G–1* –1using a controller C(z ) instead of C (z ) because it is difficult toobtain the optimized controller C*(z–1) [Equation (11)]: (z −1 ) =G−1Gm (z −1 ){−1}C(z ) 1 − Gm (z ).

(12)e(t ) := r (t ) - y(t ),(13)

where KP, KI, and KD are the proportional gain, integral gain andderivative gain, respectively.
 (z -1 ) is expressed by using I-PD
The estimated system model G
controller is as follows [Equation (14)]:

where s is a parameter related to the rise-time and d is a parameter related to the damping oscillation. User set them arbitrarily.
s denotes the time when output reaches about 60% of the stepreference value. Moreover, d is set between 0 ≤ d ≤ 2.0 desirably.
In particular, d = 0 indicates the response of Butterworth modeland d = 1.0 indicates the response of binominal model.

G(z −1 )C * (z −1 )= Gm (z −1 ).
1 + G(z −1 )C * (z −1 )Du(t ) = K I e(t ) - K P Dy(t ) - K D D 2 y(t ) (z -1 ) =G

Therefore, the optimized controller C*(z–1) achieves ϕr (t) = 0 andf(t ) = 0 simultaneously.

KI- C(z -1 )Gm (z -1 )
D,

(14)

where C(z–1) is as follows [Equation (15)]:C(z -1 ) =K P D + K I + K D D2.
D(15)The estimated output is calculated as follows [Equation (16)]:y(t ) =Gm (z -1 )KI- C(z -1 )Gm (z -1 )Du(t ),(16)where u(t) is control input. To apply the least squares method,f(t ) = 0 is considered and the following Equation (17) is derived
by using Equations (2) and (18):ìKü
K D + K I + K D D2
Gm (z -1 )u(t ) = í I - PGm (z -1 )ý y(t )
D
îD
þ

(17)

∆Gm (z −1 )u(t ) = K I {1 − Gm (z −1 )} y(t ) − (K P ∆ + K D ∆ 2 )Gm (z −1 ) y(t ).
(18)Therefore, the PID gains are calculated by using the following leastsquares method [Equations (19) and (20)]:
p = (F TF )−1F Tm ,(19)Tìé - DGm (z -1 ) y(t ) ùïêú-1
ïy (t ) = ê 1 - Gm (z ) y(t )ú
ïê - D 2Gm (z -1 ) y(t ) úïëûíF = éy (1),y (2),,y (N )ùT
ëûï
Tïm = é DG (z -1 )u(1),, DG (z -1 )u(N )ù
m
ë m
û
ï
T
ïp = éë K P , K I , K D ùû
î

(20)

(11)

 (z -1 ) is expressed
Note that the estimated controlled system G–1using a controller C(z ) and reference model Gm(z–1) without any
system identifications.
 (z -1 ) equals to G(z–1) when the optimized controller C*(z–1) isG (z -1 ) = G(z -1 ) .
obtained. It mentions that f(t ) = 0 because of G

Gm (z -1 ){}T. Kinoshita and T. Yamamoto / Journal of Robotics, Networking and Artificial Life 5(4) 257–260

259

4. NUMERICAL EXAMPLE

4.1. Control Results

The controlled system G(s) is given as follows [Equation (21)]:

Figure 2 shows the control result by using initial PID gains ofEquation (23). The control performance is poor because control
output y(t) does not track to reference output yr(t).

1ìïG(s) = 1 + 10s (t < 100).
í2
ïG(s) =(t ³ 100)
1 + 20s
î(21)The parameters of the reference model Gm (z–1) are set as follows[Equation (22)]:s = 3.0, d = 0.

(22)

Finally, the initial PID gains are set as follows [Equation (23)]:
K P = 1.0, K I = 1.0, K D = 1.0.

(23)

In this section, the following three simulations are shown.
A) Figure 2: Control result by using initial PID gains of Equation (23).
B) Figure 3: Control result of the proposed scheme without controller retuning.
C) Figure 5: Control result of the proposed scheme with controller retuning.

Figure 2 | Control result by using the initial PID gains.

Figure 3 | Control result of the proposed scheme without controller retuning.

Figure 3 shows the control result of the proposed scheme. The
following PID gains [Equation (24)] were calculated applying
Equation (19) by using the data between t = 0 and 100 [step] inFigure 2:
K P = 6.74, K I = 2.49, K D = 0.0.

(24)

In Figure 3, the control performance is good even though thesystem G(z–1) is changed at t = 100 [step]. It mentions that it is difficult to detect the system parameters are changed. The reason forthe abovementioned, the feedback controller C(z–1) has an integrator, and it can reduce the steady state error. On the other hand,Figure 4 shows the trajectories of the estimated output y (t). It easier
to detect the system parameters are changed than Figure 3.
Finally, Figure 5 shows the control result with controller retuning at
120 [step]. Figure 6 shows the estimated output y(t), and Figure 7
shows the trajectories of the PID gains corresponding to Figure 5. InFigure 7, each PID gains are slightly adjusted, however, the estimatedoutput y(t) of Figure 6 is significantly improved. Therefore, the proposed scheme can diagnose the system parameters are changed strictly.

Figure 4 | Trajectory of the estimated output ŷ(t) corresponding to Figure 3.

Figure 5 | Control result of the proposed scheme with controller retuning
at 120 [step].

T. Kinoshita and T. Yamamoto / Journal of Robotics, Networking and Artificial Life 5(4) 257–260

260

5. CONCLUSION
This paper has proposed the design of data-driven control systemwith evaluating controller performance. The features of the proposed scheme are as follows:•• Controller and estimated system model can be designed simultaneously.
•• Controller performance is evaluated by the error between
output y(t) and estimated output y (t) without using any system
identifications.
The proposed scheme has been verified by numerical examples.

REFERENCES
Figure 6 | Trajectory of the estimated output ŷ(t) corresponding to Figure 5.

Figure 7 | Trajectories of the PID gains corresponding to Figure 5.

[1] S. Soma, O. Kaneko, T. Fujii, A new approach to controller
parameter tuning based on single experimental data, Trans. Inst.
Syst. Control Inform. Eng. 17 (2004), 528–536 (in Japanese).
[2] T. Yamamoto, K. Takao, T. Yamada, Design of a data-driven PIDcontroller, IEEE Transactions on Control Systems Technology,IEEE, USA, 2009, pp. 29–39.
[3] 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.
[4] H. Hjalmarsson, M. Gevers, S. Gunnarsson, O. Lequin, Iterative
feedback tuning: theory and applications, IEEE Control Systems
Magazine, IEEE, USA, 1998, pp. 26–41.
[5] T. Kinoshita, Y. Ohnishi, T. Yamamoto, S.L. Shah, Design of aperformance-driven control system based on the control assessment, IECON 2018 - 44th Annual Conference of the IEEE
Industrial Electronics Society, IEEE, Washington, DC, USA,2018, pp. 5283–5388.
[6] T. Yamamoto, S.L. Shah, Design and experimental evaluation ofa multivariable self-tuning PID controller, IEE Proceedings Control Theory and Applications, IET, 2004, pp. 645–652.

Authors Introduction
Dr. Takuya Kinoshita

Prof. Toru Yamamoto

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

He received the B. Eng. and M. Eng.
degrees from the University ofTokushima, Tokushima, Japan, in1984 and 1987, respectively, and theD. Eng. degree from Osaka University,Osaka, Japan, in 1994. He is currently
a Professor with the Departmentof System Cybernetics, Graduate
School of Engineering, Hiroshima
University, Japan. He was a VisitingResearcher with the Department of Mathematical Engineeringand Information Physics, University of Tokyo, Tokyo, Japan, in
1991. He was an Overseas Research Fellow of the Japan Societyfor Promotion of Science with the University of Alberta for six
months in 2006. His current research interests are in the area ofdata-driven control, and process control. Dr. Yamamoto was therecipient of the Commendation for Science and Technology by theMinister of Education, Culture, Sports and Technology in 2009.