Journal of Robotics, Networking and Artificial Life
Vol. 8(3); December (2021), pp. 180–185
DOI: https://doi.org/10.2991/jrnal.k.210922.006; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Design of an Optimized GMV Controller Basedon Data-Driven Approach
Liying Shi1,*, Zhe Guan2, Toru Yamamoto3
Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, HigashiHiroshima 739-8527, JapanKOBELCO Construction Machinery Dream-Driven Co-Creation Research Center, Hiroshima University, 1-4-1 Kagamiyama,
HigashiHiroshima 739-8527, Japan
3
Graduate School of Advanced Science and Engineering, Hiroshima, 1-4-1 Kagamiyama, HigashiHiroshima 739-8527, Japan
1
2

ARTICLE INFOABSTRACT

Article History

This paper presents a data-driven scheme that can obtain the optimized Generalized Minimum Variance (GMV) control
parameters by applying the Nelder–Mead (NM) method based on Proportional-Integral Derivative (PID) controller for linear
system. An adjustable l is included in the GMV-PID controller. In the existing GMV-PID controller, the PID parameters
are calculated by simply changing l manually. Therefore, it is hard to get desirable control performance. The NM method is
introduced to improve the control performance. The application of NM method can optimize the calculation of l. However, theobjective function in NM method needs the output calculation. In other words, the model information is inevitably to obtainoutput. To achieve model-free design scheme, the estimation of closed-loop response method is introduced in database-driven
approach. Using obtained closed-loop data can predict the output and then substitute it into the objective function without any
model information of the process. The effectiveness is verified by experiment.

Received 14 November 2020
Accepted 21 August 2021
Keywords
PID controller
GMV
Nelder–Mead methoddata-driven approach© 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
Despite the new results in control theory that have been achievedby researchers year-by-year all over the world, the ProportionalIntegral Derivative (PID) controller [1] in all control design techniques is the most popular controller used in the process industries.
It has a long history in the automatic control field and can assure
satisfactory performances with simple algorithm for a wide rangeof processes. The structure is very simple, and it is very easy to
grasp the physical meaning of PID parameters. Therefore, over thelast half-century, a great deal of academic and industrial effort has
focused on defining PID parameters.
In recent years, some methods for calculating PID parameters havebeen proposed, but these methods are model-based and require
system identification to calculate the PID parameters. However, it ishard to tune these parameters because most real systems generally
have high-order lag factors and uncertainness. However, some databased methods have gradually been proposed. It does not require
system identification to calculate PID parameters only throughclosed-loop data, which greatly reduces the calculation cost.
Among the existing control strategies, using the Minimum
Variance (MV) as the benchmark can claim to be successful. The
basic idea behind the MV index is to only consider the controller error variance. However, few of the available techniques in use
Corresponding author. Email: m192383@hiroshima-u.ac.jp*

take the control effort or the manipulative variable activity into
account. Then, the Generalized Minimum Variance (GMV) [2]is proposed, and it takes into account controller error variance
as well as the manipulating variable variance, which can achievemodel-free. In the existing GMV-PID controller [3], an adjustableparameter l (a weighting factor penalty on the manipulating variable) is included, and the PID parameters are calculated by simply
changing l manually. Therefore, it is hard to get desirable control
performance.
To improve the control performance, the Nelder–Mead (NM)method [4] is introduced. The NM method is different from thesimplex algorithm in linear programming. It does not use any derivative operation and the algorithm is relatively simple. It can optimize the calculation of the most suitable parameter l. However, theobjective function in NM method needs the output calculation. In
other words, the model information is inevitably to obtain output.
To achieve model-free design scheme, the estimation of closedloop response method is introduced in database-driven approach.
Using obtained closed-loop data can predict the output and thensubstitute it into the objective function without any model information of the process. The effectiveness of the proposed scheme isverified by the experiment of tank system.
The rest of this paper is organized as follows: the problem is formulated in Section 2; the proposed scheme is the main topic in
Sections 3 and 4; Section 5 provides the experiment and analysis,and Section 6 concludes the paper.

L. Shi et al. / Journal of Robotics, Networking and Artificial Life 8(3) 180–185

2. OVERVIEW OF THE DATA-DRIVEN
GMV CONTROLLER
This paper gives the details of the proposed method which
describes a design scheme for data-driven GMV-PID controller
which can obtain GMV control parameters by applying the NM
method and the estimation of closed-loop response based on PID
controller. In Figure 1, an adjustable parameter l is included in the
GMV-PID controller, therefore PID parameters are calculated bysimply changing l. Using the method of estimation of closed-loop
response to predict new output, then the application of NM method
can optimize the calculation of the most suitable l according to thepredict output and get optimal PID parameters without any modelof the process.

3.1. System Description
Consider the following system:x (t )A(z ) y(t ) = z B(z )u(t ) +
	 

D
-1

-1

-1

where w (t) denotes the reference signal, k is the minimum time-lag
which is estimated by an operator (k is set as 0 when the time-lag is
unknown). In addition, l is the weight coefficient for the varianceof the control input and it is set by a user arbitrarily. Next, the following Diophantine equation is introduced.
P (z −1 ) = ∆E(z −1 )A(z −1 ) + z − (1+ k )F (z −1 ) 	 (6)

E(z -1 ) = 1 + e1 z -1 + … + ek z - k 
	 

(7)

	 
F (z -1 ) = f 0 + f1z -1 + f 2 z -2 

(8)Moreover, P(z−1) is the design polynomial and it is designed basedon the following equation. Where, the order of E(z−1) and F(z−1) are
set to decide these coefficients uniquely from ΔA(z−1) and P(z−1).
-1-1-2		 P (z ) = 1 + p1z + p2 z 

3. GENERALIZED MINIMUMVARIANCE CONTROL(1)

where A(z−1) and B(z−1) are given by the following polynomials:A(z -1 ) = 1 + a1z -1 + a2 z -2 
		 

(2)

B(z -1 ) = b0 + ¼ + bm z - m 
		 

(3)

181

(9)

 r 
4m − 1p1 = − 2exp −
cos
r
	 

 2m 
2m

(10)

 r
−
		 p2 = exp  m  (11)T
r := s 
		 
s

(12)

	   m := 0.25(1 − c ) + 0.51 c (13)where u(t) and y(t) are respectively the input and the corresponding output; x(t) is the Gaussian white noise which has zero mean;z−1, the backshift operator which implies z−1 y(t) = y(t − 1); Δ, thedifferencing operator (Δ = 1 − z−1); and m, the order of B(z−1).

s and c are parameters which express the rise time and the damping property which are set by an operator respectively. Moreover,s denotes the rise time that the system output attains about 60%of the value of the step reference signal. From (1), (4) and (5), thepredictive value of the generalized output after k + 1 at t can beobtained as following equation:

3.2. Control Designf (t + k + 1) = F (z −1 ) y(t ) + {E(z −1 )B(z −1 ) + l }  ∆u(t ) − P (1)w (t ) + E(z −1 )x (t + k + 1).

The GMV control law for the system (1) can be derived by minimizing the following cost function:
2

		 J = E [f (t + k + 1)].   f (t + k + 1) = P (z -1 ) y(t + k + 1) + lDu(t ) - P (1)w (t ) Here, the optimum predictive value at t is defined byf(t + k + 1|t ) := F (z -1 ) y(t ) - P (1)w (t ) +(G(z -1 ) + l )Du(t ).

(4)Here, f(t + k + 1) is the generalized output given by following
equation:
(5)

(14)

(15)

where,
-1
-1
-1
-1
-n
   G(z ) = E(z )B(z ) = g 0 + g 1z + … g n z . (16)The control law is described as the following equation:
Du(t ) =P (1)F (z -1 )w (t ) y(t ). 
-1
G( z ) + l
G(z -1 ) + l(17)

To obtain an optimum predictive value f(t + k + 1 | t ), it is necessary to compute E(z−1) and F(z−1). However, it needs system parameters to calculate these coefficients, therefore it has to do a system
identification. The error between output and the optimum predictive value is defined as
Figure 1 | The block diagram of data-driven GMV controller.

∆e (t + k + 1) = f (t + k + 1) − f(t + k + 1|t ) (18)

L. Shi et al. / Journal of Robotics, Networking and Artificial Life 8(3) 180–185

182

The least-square method is applied to the obtained closed-loop
data to minimize the square sum of these residual errors, thenthe parameters of G(z−1) and F(z−1) are identified without systemparameters. The velocity type of PID control is given by the following equation:
Du(t ) = K I e(t ) - K P Dy(t ) - K D D 2 y(t )		 e(t ) = w (t ) - y(t ) 

The proposed control system can be designed by calculate themost suitable l by the steps based on the mentioned procedure.
The proposed control system can be designed by calculating themost suitable l by the steps based on the mentioned procedure in
Figures 2–6. The above algorithm can be summarized as follows:

(19)
(20)

KP, KI, and KD denotes the proportional gain, the integral gain
and the derivative gain. By replacing the polynomial G(z−1) by thesteady-state term G(1), the following equation can be obtained:∆u(t ) =P (1)F (z −1 )w (t ) −y(t ). G(1) + l
G(1) + l(21)

Figure 2 | Reflection of the Nelder–Mead method.

Therefore, PID parameters can be calculated as follows:f + 2 f2
KP = 1		 
G(1) + l(22)f + f + f0
KI = 1 2
		 
G(1) + l

(23)

f2
KD =
		 

G(1) + l(24)

Figure 3 | Expansion of the Nelder–Mead method.

4. EVOLUTIONARY COMPUTATIONUSING NELDER-MEAD METHOD
4.1. Nelder–Mead Method
The Nelder–Mead method is briefly explained to calculate theuser-specified parameter l. Each top of the triangle is determined
as Si = li. In this paper, the objective function is determined by
		 H (Si ) = ∑ t =1 { ym (t ) − yi (t )} M

2Figure 4 | The contraction of the Nelder–Mead method.

(25)

where ym(t) denotes the reference model output and M denotes
the evaluation horizon. yi(t) is the output corresponding to Si. The
objective function can be determined. The top with the largest
target function value H(Si), the second large top and the smallest
top are respectively assumed to be SH, SM, and SL. Furthermore, thecenter between SM and SL is determined as SG. Four operations arebriefly explained as follows:

Figure 5 | Reduction of the Nelder–Mead method.

•• Reflection
SR is determined as SR = (1 + b )SG − b SH, where b is set as b > 0,
and it corresponds to the ratio of distances SRSG and SHSG.
•• Expansion
SE is determined as SE = (1 − γ)SG + γSH, where γ is set as γ > 1,
and it corresponds to the ratio of distances SESG and SRSG.
•• Contraction
SC is determined as SC = (1 − η)SG + ηSH, where η is set as 1 > η> 0, and it corresponds to the ratio of distances SCSG and SHSG.
•• Reduction
SH and SM are moved in the direction of SL.

Figure 6 | Procedure of the Nelder–Mead method.

L. Shi et al. / Journal of Robotics, Networking and Artificial Life 8(3) 180–185

1.		 Construct the initial working simplex S.
2.		 Repeat the following steps until the termination test is satisfied. Calculate the termination test information.

y 0 = [ y0 (0) y0 (1) ¼ y0 (N - 1)]T (32)Ty1 = éë0 y0 (0) y0 (1) ¼ ¼ y0 (N - 2)ùû 

3.		 If the termination test is not satisfied, then transform theworking simplex.
4.		 Return the best vertex of the current simplex S and the associated function value.

183

⁝
T

y N -1 = éë0 0 ¼ ¼ ¼ 0 y0 ùû (34)

We also define the vectors ud and yd.
T

4.2. Estimation of Closed-Loop Response
To achieve model-free design scheme, the estimation of closed-loop
response method [5] is introduced in data-driven approach. Using
obtained closed-loop data can predict the output and then substitute it into the objective function without any model informationof the process. The estimation of closed-loop response method isused to predict the output using only the initial data without any
model of the process.
The superposition principle is valid for input–output relations in
Linear Time Invariant (LTI) systems. In this method, the response
is predicted by describing the predicted input and output using
the superposition of the input–output relations already obtained.
First, we describe the prediction of the response of an arbitrary LTI
system, and then we describe the prediction of the response of aclosed-loop system composed of LTI systems. Response prediction
for linear systems suppose that the input–output data pairs u0, y0(t = 0, 1, …, N − 1) of an arbitrary discrete-time LTI system G are
obtained. The relationship between u0, y0 is given by:
			 y0 = Gu0 (26)

We consider predicting the response yd from u0, y0 when an arbitrary input ud given by the designer is applied to G. Where ud, yd(t = 0, 1, …, N − 1) satisfies the following equation.
yd (t ) = Gud (t ) 
		  

(27)

Since G is an LTI system, if u0, y0 and the different input-output
relations u1, y1 are known, the following relations hold for any constants k and i.
		 ky0 + iy1 = G(ku0 + iu1 ) (28)

The relationship between both equation is commonly referred toas the superposition principle. When the data length N of ud andyd is large enough, it is difficult to determine k and i such that ud
and yd are obtained from only two sets of input and output data. To
adjust each time series, it is desirable to prepare N sets of input andoutput data. Therefore, it is desirable to prepare N pairs of input
and output data to adjust each time series, so that u0 and y0 can be
adjusted by k(k = 0, 1, 2, …, N − 1) of u0 and y0, respectively. They
are then introduced in vector form as in equations.
Tu0 = éëu0 (0) u0 (1) ¼ u0 (N - 1)ùû 
	 

(29)
T

	 
u1 = éë0 u0 (0) u0 (1) ¼ ¼ u0 (N - 2)ùû (30)⁝
	   uN -1 = [0 0 ... ... ... 0 u0 ] 
T

(31)

(33)

	 
ud = éëud (0) ud (1) ¼ ud (N - 1)ùû T

y d = éë yd (0) yd (1) ¼ yd (N - 1)ùû 
	 

(35)
(36)

If u0(0) is non-zero, then the vectors uk and yk are linearly independent for each k. Therefore, ud can be expressed using the coefficient
vectors a and uk as in equation.
ud = a0u0 + a1u1 + ¼ + aN -1uN -1 (37)		 a = [a0 a1 ¼ a(N -1) ] 

(38)

T

(39)

T

		 U = [u0 u1 ¼ u(N -1) ] 

Here, U is regular if u0 is non-zero. Solving from the relationship,
ud(0) = a0u0(0), ud(1) = a0u0(1) + a1u0(0), a, which corresponds to k
and i, can be obtained sequentially and uniquely determined by thedesigner by giving ud. Using the coefficient spectrum k obtainedin this way, yd can be obtained from the superposition principle asshown in next equation.
	   yd = a0 y 0 + a1 y1 + ¼ + aN -1 y N -1 T		 Y = é y 0 y1 ¼ y( N -1) ù 
ë
û

(40)
(41)

U and Y are described by the previously obtained input and output
data u0, y0. Using the known data, the output yd of system G for any
input ud can be predicted.
y d = YU -1ud 
		 

(42)

In this paper, the pseudo reference signal is described as the following equation:	   r (t ) =C(z -1 )-1
u0 (t ) + y0 (t ) D(43)

where, u0 denotes the initial input and y0 denotes the initial output.
The predicted output is determined as the following equation:
 -1v 		 yd (t ) = YR

(44)

 denotes [r r … r ], Y denotes [y y … y ].
where R
0 1
N−1
0 1
N−1

5. EXPERIMENTAL EVALUATION
The proposed method is applied to a tank system which is shownin Figure 7. The schematic diagram of the process system is shownin Figure 8. In this system, there are three pipes. When the wateris flowing along with the time, the temperatures at three points

184

L. Shi et al. / Journal of Robotics, Networking and Artificial Life 8(3) 180–185

Figure 7 | Photo of the temperature control system.
Figure 9 | Control result by the Ziegler-Nichols method.

Figure 8 | Schematic diagram of the temperature control system.

Figure 10 | Control result by using manually adjusted l (l = 4.5).

mentioned above are measured and displayed in Temp.1, Temp.2
and Temp.3. The temperature sensor in the tank system transmits
the sensed water temperature signal to the controller. The controller compares the measured water temperature signal with theset signal to obtain the deviation. Then according to the nature of
the deviation, the hot water, cold water, and electric valve sends
out “open” and “close” commands to ensure that the tank systemreaches the set temperature. It should be noted that the cold water
value is set as 10%.
TS is the sampling time and is set as TS = 5 s in this experiment.
Then, the Ziegler-Nichols method [6] is applied considering thestability of the system when the parameters are identified. As aresult of the step response, the system parameters are estimatedand at the same time the PID parameters can be obtained by applying the Ziegler-Nichols method. The control result by the ZieglerNichols method is shown in Figure 9, and the corresponding PID
gains are shown as follows:
K P = 4.3963, K I = 0.0925, K D = 8.2747
Firstly, using manually adjusted l to calculate the PID parameters.
When l = 4.5, the control result is shown in Figure 10. PID gains
are shown as follows:
K P = 0.6275, K I = 0.0796, K D = 4.5127
calculated as 3.772 by the proposed scheme, and then the PID gains
can be obtained as follows:

Figure 11 | Control result by the proposed method (l = 3.772).

K P = 2.7873, K I = 0.1896, K D = 10.2434
Figure 11 shows the control result when l = 3.772. In comparing
the control results obtained by manually adjusting l and calculating l. It is clear that the PID parameters by using the calculated
l can get desirable control performance, and it takes a short timeto achieve the control effect. However, the control result by using
manually adjusting l takes a relative long time to reach the target
value.

L. Shi et al. / Journal of Robotics, Networking and Artificial Life 8(3) 180–185

6. CONCLUSIONCONFLICTS OF INTEREST

In this paper, a design scheme of data-driven GMV-PID controller has been proposed, which can obtain control parameters byapplying the Nelder–Mead method based on PID controller. The
features of the newly proposed control scheme are summarized
as follows:

The authors declare they have no conflicts of interest.

•• The user-specified parameter included in GMV-PID
control system, is calculated by the NM method for a liner
system.
•• Only using one-short data can calculate l. The estimation of closed-loop response is introduced in data-driven
approach.

185

REFERENCES
[1]
[2]
[3]
[4]

•• The effectiveness of the proposed scheme has been evaluated on
the simulation example and experiment.

[5]

Furthermore, this system can only calculate linear systems, andfuture research will be devoted to the development of nonlinear
systems.

[6]

A.J. Michael, H.M. Mohammad, PID Control: New Identification
and Design Methods, Springer-Verlag, London, 2005.
D.W. Clarke, P.J. Gawthrop, Self-tuning control, IEE Proc. 126(1979), 633–640.
T. Yamamoto, S.L. Shah, Design and experimental evaluationof a multivariable self-tuning PID controller, IEE Proc. Control
Theory Appl. 151 (2004), 645–652.
J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 7 (1965), 308–313.
T. Sakatoku, K. Yubai, D. Yashiro, S. Komada, Data-driven controller tuning with closed-loop response estimation, IEEJ Trans.
Electr. Electron. Eng. 16 (2021), 1397–1406.
J.G. Ziegler, N.B. Nichols, Optimum settings for automatic controllers, Trans. ASME 64 (1942), 759–768.

AUTHORS INTRODUCTION
Ms. Liying Shi

Prof. Toru Yamamoto
She received her Master’s degree from the
Graduate School of Engineering, Hiroshima
University, Japan in 2019. Her main research is
GMV-PID controller design.

Dr. Zhe Guan
He received the M.S. degree and D. Eng.
degree in Control System Engineering fromthe Hiroshima University, Japan, in 2015
and 2018, respectively. He is currently working in KOBELCO Construction Machinery
Dream-Driven Co-Creation Research Center,
Hiroshima University, Japan. He was a visiting postdoc with the Department of Electrical and Computer
Engineering, University of Alberta from April to June in 2019.
His current research interests are in area of adaptive control,
data-driven control, Reinforcement Learning and their applications. He is a member of Society of Instrument and Control
Engineers in Japan (SICE).

He received the B. Eng. and M. Eng. degrees
from the Tokushima University, Japan, in 1984
and 1987, respectively, and the D. Eng. degreefrom Osaka University, Japan, in 1994. He iscurrently a Professor with the Department
of Smart Innovation, Graduate School of
Advanced Science and Engineering, Hiroshima
University, Japan, and a leader of the National Project on
Regional Industry Innovation with support from the Cabinet
Office, Goverment of Japan. He was a Visiting Researcher with
the Department of Mathematical Engineering and Information
Physics, University of Tokyo, Japan, in 1991, and an Overseas
Research Fellow of the Japan Society for Promotion of Science(JSPS) with the Department of Chemical and Materials
Engineering, University of Alberta for 6 months in 2006. His
current research interests are in area of self-tuning & learning control, data-driven control, and their implementation forindustrial systems. He was a National Organizing Committee
Chair of 5th International Conference on Advanced Control ofIndustrial Processes (ADCONIP 2014) and General Chair of
SICE (Society of Instrument and Control Engineers in Japan)
Annual Conference 2019, both held in Hiroshima. He is a
Fellow of SICE, Institute of Electrical Engineers of Japan (IEEJ)
and Japan Society of Mechanical Engineers (JSME).