Journal of Robotics, Networking and Artificial Life
Vol. 7(1); June (2020), pp. 68–72
DOI: https://doi.org/10.2991/jrnal.k.200512.014; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Design of a Data-Driven Multi PID Controllers using
Ensemble Learning and VRFT
Takuya Kinoshita*, Yuma Morota, Toru Yamamoto
Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-hiroshima city, Hiroshima, Japan

ARTICLE INFOABSTRACTArticle History

Data-driven control has been proposed for directly calculating control parameters using experimental data. Specifically, theVirtual Reference Feedback Tuning (VRFT) has been proposed for linear time-invariant systems. In the field of machinelearning, the ensemble learning was proposed to improve the accuracy of prediction by using multiple learners. In this study,a design scheme of data-driven controllers using the ensemble learning and VRFT is newly proposed for linear time-varying
systems. The ensemble learning can divide the linear time-varying system into some sections that can be regarded locally as
linear time-invariant systems.

Received 06 November 2019Accepted 17 March 2020
Keywords
Data-driven control
PID control
ensemble learning© 2020 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
Data-driven control schemes have been proposed for directly designing a controller using a set of experimental data. Virtual Reference
Feedback Tuning (VRFT) [1] and Fictitious Reference Iterative Tuning(FRIT) [2] have been proposed as data-driven control schemes.
According to these schemes, control parameters can be directly calculated using a set of experimental data. However, VRFT and FRIT work
well for only linear time-invariant systems, and it is difficult to obtain
good control performance for linear time-varying systems.
Meanwhile, the effectiveness of the deep and machine learning havebeen demonstrated in the field of image recognition. The ensemble
learning such as random forest [3] and adaptive boosting [4] is amachine learning scheme, and it has been proposed to improve theaccuracy of prediction by using multiple learners.
This paper proposes a design scheme of data-driven controllers
using ensemble learning and VRFT for linear time-varying systems. Specifically, a linear time-invariant system is first divided
into some linear systems by applying ensemble learning based on
decision tree learning. Second, VRFT is applied to each divided
linear system for obtaining multiple linear controllers.

2. DESIGN OF DATA-DRIVEN CONTROLLER
USING ENSEMBLE LEARNING
Figure 1 shows the schematic figure of the proposed data-driven control system where multiple controllers (controller-1, controller-2,…, controller-n) are designed using ensemble learning using theCorresponding author. Email: kinoshita-takuya@hiroshima-u.ac.jp*

initial closed-loop data. First, the closed-loop data is divided intosome linear time-invariant systems based on time t (t1, t2, …, tn−1).
Next, control parameters are calculated using VRFT for eachdivided data (Divided data-1, Divided data-2, …, Divided data-n).

2.1. Virtual Reference Feedback Tuning
Figure 2 shows the block diagram of VRFT. P(z−1) is a system plantand C(z−1, q ) shows the controller. q is a control parameters. The
optimization problem is adjusting q so that closed-loop transfer
function W(z−1) closed to the following desired reference modelGm(z−1) [5].
z -1T (1)		 Gm (z -1 ) =(1)T (z -1 )		 T (z -1 ) = 1 + t1z -1 + t 2 z -2 , (2)whereì
æ r ö æ 4m - 1 ö
ït1 = -2 exp ç r÷÷ cos ç÷ïè 2 m ø çè 2 møïæöït = exp - rç÷. (3)	    í 2
è møï
Tsï
ïr = sï
î m = 0.25(1 - d ) + 0.51d
Ts is the sampling time, σ and δ are the user-specified parameters
related to the rise characteristic and attenuation characteristic ofcontrol system.

T. Kinoshita et al. / Journal of Robotics, Networking and Artificial Life 7(1) 68–72		

69u (t ) = - K P y0 (t ) + K I x(t )(10)= qj (t )Here, the following equations denote q and j(t).

q = [K P K I ]T (11)		 		 j (t ) = [- y0 (t ) x(t )] (12)The control parameter q  * which minimizes the objective function
J is obtained as Equation (13).		 q * = (FT F)-1 FTU (13)T		 F = [j (1)j (2)¼j (N )] (14)Figure 1 | Schematic figure of the proposed control system.

T		 U = [u0 (1)u0 (2) ¼ u0 (N )] (15)2.2. Design of Learner to Divide
Closed-loop Data
In this paper, a learner is designed to divide closed-loop data into nsystems in Figure 3 is designed. Decision tree learning is a schemeof assigning a data set allocated to a parent node to a child nodeaccording to a split function h* and creating a decision tree such asFigure 4. The split function h* is obtained as follows:
h* = arg max I (16)		  Figure 2 | Block diagram of VRFT.

In order to match W(z−1) to Gm(z−1), one-shot experimental input
u0(t) and output y0(t) is obtained. Next, a pseudo reference inputr(t ) is calculated as follows:1		 r(t ) =y0 (t ). (4)Gm (z -1 )I is an evaluation function showing the degree of variation of theclass in the child node which divided the parent node. Based on
the division scheme of the decision tree, the optimum time to split
closed-loop data t * (t1* , t 2* ,¼, tn* -1 ) is obtained using the evaluation
function I as follows:t * = arg max I (t ) (17)		  The output u (t ) is denoted as Equation (5).
C(z -1 ,q )u (t ) =e(t ) 		 D(5)		 e(t ) = r(t ) - y0 (t ) (6)The following equation denotes the objective function J.
NFigure 3 | Learning instrument.

1J={e (t ,q )}2 (7)		 2

å
t =1e (t ,q ) = u0 (t ) - u (t ) (8)		 In this paper, I − P controller is used and the control parameterθ which minimizes the evaluation function J. From Figure 2, u (t )becomes given byu (t ) = - K P y0 (t ) + K I		 e(t )(9)D

By putting e(t ) / D = x(t ), Equation (9) is rewritten as follows:

Figure 4 | Decision tree.

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T. Kinoshita et al. / Journal of Robotics, Networking and Artificial Life 7(1) 68–72

Here, the evaluation function I is defined as follows:n			 I (t ) = -åJ (18)*kk =1J k* is the minimum value in Equation (7) of the kth system obtainedby dividing the closed-loop data at time t.

Ensemble learning is a scheme of integrating multiple learners togenerate one learner. If the accuracy of each learner is higher than50%, it is known that higher accuracy is achieved than one learnersince the learner that estimates erroneously becomes a minority.
In this paper, ‘fminsearch.m’ in MATLAB & Simulink Ver.
9.4.0813654 (R2018a), Optimization Toolbox is used to find t*.

2.3. Controller Selection Method based onthe Controller’s Plane
Equation (10) is as follows when the closed-loop transfer function
at the top of Figure 2 and the transfer function Gm(z−1) are equal.		 -K P y0 (t ) + K I

ì100 (0 £ t < 200)ï		 r (t ) = í150 (200 £ t < 400) (23)ï50 (400 £ t £ 600)
îWhite Gaussian noise with zero mean and a variance of (1/3)2 isadded to the controlled object. The reference model Gm(z−1) was set
as Equation (24).

2.2.1. Ensemble learning		 u0 (t ) = - K P y0 (t ) + K I

The reference signal r(t) is given as follows:

e(t )
(19)
De(t )- u0 (t ) = 0 (20)D-1		 Gm (z ) =0.0392z -1

1 - 1.6057 z -1 + 0.6449z -2

(24)3.2. Simulation Result
The control result with fixed PI gains which are calculated by VRFT
is shown in Figure 5. PI gains are calculated as follows:		 K P = -0.45, K I = 0.10 (25)
KP is negative gain because the system has three characteristics ofEquation (22). In Figure 5, the overshoot has occurred on the output
with fixed PI gains and good control results are not obtained. The
sum of squared errors is 708.7 in Table 1.
Figure 6 shows the control result applying the proposed scheme.
Here, the number of weak learners is 10. Figures 7 and 8 respectively show the estimation result of the system change step inthe weak learners and the transition of PI gains in the proposedscheme. Figure 6 shows that good control result can be obtained

The following equation defined the equation of the plane passing
the origin with x, y, z as axes.			 ax + by + cz = 0 (21)From Equations (20) and (21) can be regarded as an equation of aplane passing the origin with y, e/Δ, and u as axes. From Equation(20), a linear time-invariant system can be represented by a plane
and the coefficients consist of PI gains. From this property, input
and output data are plotted in a space with y, e/Δ, and u as axes, anda controller corresponding to a plane closest to the plotted points issequentially selected.

3. NUMERICAL EXAMPLE
3.1. Control Object and Setting Parameters
In this numerical example, the following system is discussed.
ì 5
ï 1 + 20sïï 3		 G(s) = íï 1 + 50s
ï 1
ï 1 + 10sî

(0 £ t < 150)

Figure 5 | Control results using the fixed PI controller.

Table 1 | Sum of squared errors(150 £ t < 350) (22)(350 £ t £ 600)

¬ e(t)2
Conventional scheme (Figure 5)
Proposed scheme (Figure 6)

708.7
653.9

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using the proposed scheme. In addition, Figures 7 and 8 show thatthe system change step is estimated accurately and PI gains arechanged at the proper timing. Finally, the sum of squared errors
is 653.9 in Table 1, and the proposed scheme is better than theconventional scheme.

4. CONCLUSION
In this paper, a new control scheme has been proposed. It is ascheme of dividing a linear time-varying system into multiple linear
time-invariant systems and applying VRFT to calculate multiple

Figure 8 | Trajectories of PI gains.

linear controllers. In the numerical example, the effectiveness ofthe proposed scheme has been shown.
In the future works, it is necessary to optimize the number of divisions for an unknown system. In addition, the least squares method
is susceptible to noises other than normal distribution. Therefore,
further consideration is needed such as incorporating bagging [6]to suppress the effects of noise.

CONFLICTS OF INTEREST
Figure 6 | Control results using the proposed scheme.

The authors declare they have no conflicts of interest.

REFERENCES
[1]
[2]

[3]
[4]
[5]
Figure 7 | Estimation result of ensemble learning.

[6]

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.
S. Soma, O. Kaneko, T. Fujii, A new approach to parameter tuningof controllers by using one-shot experimental data - a proposalof fictitious reference iterative tuning, Trans. Inst. Syst. Control
Inform. Eng. 17 (2004), 528–536 (in Japanese).
L. Breiman, Random forests, Mach. Learn. 45 (2001), 5–32.
Y. Freund, R.E. Schapire, A decision-theoretic generalization of
on-line learning and an application to boosting, J. Comput. Syst.
Sci. 55 (1997), 119–139.
T. Yamamoto, S.L. Shah, Design and experimental evaluation ofa multivariable self-tuning PID controller, IEEE Proc. Control
Theory Appl. 151 (2004), 645–652.
L. Breiman, Bagging predictors, Mach. Learn. 24 (1996), 123–140.

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AUTHORS INTRODUCTION
Dr. Takuya Kinoshita

Prof. Toru Yamamoto

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 Societyfor the Promotion of Science) in 2017. Heis currently an Assistant Professor withthe Department of System Cybernetics,Graduate School of Engineering, Hiroshima
University, Japan. His research interests are performance drivencontrol.

He received the B.Eng. and M.Eng.
degrees from the University of Tokushima,
Tokushima, Japan, in 1984 and 1987, respectively, and the D.Eng. degree from Osaka
University, Osaka, Japan, in 1994. He iscurrently a Professor with the Departmentof System Cybernetics, Graduate Schoolof Engineering, Hiroshima University,
Japan. He was a Visiting Researcher with the Department ofMathematical Engineering and Information Physics, Universityof Tokyo, Tokyo, Japan, in 1991. He was an Overseas Research
Fellow of the Japan Society for Promotion of Science with theUniversity of Alberta for 6 months in 2006. His current researchinterests are in the area of data-driven control, and process control. He was the recipient of the Commendation for Science andTechnology by the Minister of Education, Culture, Sports andTechnology in 2009.

Mr. Yuma Morota
He received his B.Eng. From Hiroshima
University in Japan in 2019. He is currently
a Master course student in Hiroshima
University, Japan.