Journal of Robotics, Networking and Artificial Life
Vol. 8(3); December (2021), pp. 229–233
DOI: https://doi.org/10.2991/jrnal.k.210922.015; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Design of a Nonlinear Generalized Predictive Controller
based on Database-Driven Approach
Zhe Guan1,*, Toru Yamamoto2
KOBELCO Construction Machinery Dream-Driven Co-Creation Research Center, Hiroshima University, Higashi-Hiroshima 739-8527, Japan
Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

1
2

ARTICLE INFOABSTRACT

Article History

This paper presents a Generalized Predictive Control (GPC) design scheme based on database-driven approach for nonlinear
systems. In industrial systems, lots of controlled plants with unknown time-delay and strong nonlinearity, are difficult to behandled in terms of control performance. The databased-driven approach has been attracted attention to tackle non-linearity
issue without system information. On the other hand, GPC is considered in cases where the time-delay is unknown. The
proposed scheme is validated through a numerical simulation.

Received 14 November 2020
Accepted 22 August 2021
Keywords
GPC
database-driven approach
nonlinear systems© 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
The tuning of controller for controlled processes is of increased
attentions for industrial process. A variety of controllers are
required to deal with broad range of applications to maintainthe process performance. These controllers are either classicalthe Proportional-Integral-Derivative (PID) controller, or moreadvanced adaptive predictive controller. For simple processes, theconventional PID controller can achieve satisfied performance,
however, it may degrade the performance in the case of somechanges or uncertainties for some complex processes. Maintaining
system performance becomes an essential issue in industrial processes. Unfortunately, it is typically complicated and expensive.
Predictive control is considered as an advanced control strategy
since it uses a long-range prediction idea. As a result, the excellent performance can be achieved. One of popular in the familyof long-range predictive controllers is the Generalized Predictive
Control (GPC) algorithm [1] which adapts to process time-delay
and model order based on the long-range prediction. Therefore,
it is shown to be particularly effective for the self-tuning control
of industrial processes [2], in which are widely applied. However,regarding to the existing GPC, the excellent control behavior
can be achieved, provided that the number of parameters in themodel are required. It is important to include a reasonable representation of disturbances and other unknown dynamics, such thatthe correct controller can be designed. In other words, the GPC
is model-based technique, so good performance needs a highquality process model. Unfortunately, as it is well known, modeling for an industrial process using first principles or identification

Corresponding author. Email: guanzhe@hiroshima-u.ac.jp*

is a difficult task. Further, even the mathematical model of theindustrial process is established, it only approximates the process
but not involves entire process dynamics. Moreover, it is time- andcost-consuming to build the relative accurate complicate model ofindustrial process. Therefore, some challenges have been addressedin the model-based technique.
The industrial processes have undergone significant changes alongwith the rapid development of information technology. It is possible to store and process huge amounts of process data with the helpof computers. In recent years, it is a trend to design the controller
by directly using online or offline Input/Output (I/O) data obtained
from the controlled system or knowledge, instead of the explicit
mathematical model of the controlled process [3]. Several methods[4,5] have been proposed by considering the above idea. Moreover,the data-oriented GPC controller [6] has been proposed, where theinitial polynomials parameters are estimated from the I/O data,
and then the polynomials are calculated based on the iterative calculation. However, the controller is operated in an offline manner,which the controller parameters are not modified during the operation. Consequently, it leads to an issue that the control performance
may be degraded for some systems with strong non-linearity.
The database-driven approach [7,8] is one of control strategies
applied in systems with strong non-linearity since it can tune thecontrol parameters rapidly at each equilibrium point. Furthermore,
the database-driven approach performs the tuning in on-line
manner to cope with the operation condition along the time, suchthat the performance deterioration can be avoided. Successful
implementations of the data-driven approach have been presented
in the technical literatures [9–11].
Based on the above observation, this paper newly presents a nonlinear generalized predictive controller based on databased-driven

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Z. Guan and T. Yamamoto / Journal of Robotics, Networking and Artificial Life 8(3) 229–233

approach. The main contributions lie in twofold. One is that the
proposed controller can deal with time-delay involved in nonlinear systems. The other one is that the proposed controller is operated in on-line manner by exploiting the measured I/O data, which
results in improving the performance.
The remainder of this paper is as follows. The Section 2 states the
problem. The Section 3 gives the details of the proposed scheme.
In Section 4, the numerical simulation and comparative study are
conducted to verify the effectiveness, and Section 5 concludes thepaper.

the controller generates u(t) to control the system Equation (1)based on database-driven approach aiming at tracking the reference signal w(t).

3. PROPOSED SCHEME DESIGN
3.1. Generalized Predictive Control
The GPC law is derived based on the minimization of the criterion
in the following equation:
éN2
J = E êå { y(t + d + j ) - w(t + d + j )}
ë j =1Nù+ å l ( j){Du(t + j - 1)}2 ú ,j =1
û2. PROBLEM STATEMENT
Consider the following nonlinear system:		 y(t ) = f (f (t − 1)) + x (t ) (1)where y(t), f(∙) and x (t) denote the system output, the nonlinear
function and the white Gaussian noise with zeros mean and variance s 2, respectively. Additionally, f (t − 1) is called information
vector, which is defined as follows:f(t - 1) := [ y(t - 1), ¼, y(t - ny ),u(t - d - 1), ¼, u(t - d - nu )], 

When considering regulation about a particular operation point, it
is reasonable to assume that the nonlinear system can be described
as a locally-linearized model by the following equation:A(z -1 ) y(t ) = z - (d +1) B(z -1 )u(t ) +x (t ),D(3)

where z−1 represents the backward shift operator which implies
z−1y(t) = y(t − 1), and Δ denotes the differential operator definedby ∆ := 1 − z−1. A(z−1) and B(z−1) are the polynomials which areformulated as follows.
-1-1…
	   A(z ) = 1 + a1z + + any z

- ny

,

(4)

− nu
−1
−1
…
	   B(z ) = b0 + b1z + + anu z . (5)Based on the above statement, the problem to be addressed inthis paper is as follows. The block diagram of proposed scheme isdepicted in Figure 1. Use the batch of data, collected the historical data, to generate the initial database. Then the nonlinear predictive controller can be updated by exploiting the I/O data, and(6)where E(·) is the expectation value and [1, N] denotes the range ofcontrol horizon and prediction horizon. w(t) is the reference value
and l (j) is a user-specified control weighting sequence.
To derive the j-step ahead predictive output y(t + j) based on system(3), the following Diophantine equations are introduced.

(2)

where u(t) and d represent the control input and system delay time.
Also, ny and nu are the order of the system output and control input,
respectively.



1 = DA(z -1 )E j (z -1 ) + z - j Fj (z -1 ),E j (z -1 )B(z -1 ) = R j (z -1 ) + Z - j S j (z -1 ), (7)

E j (z -1 ) = 1 + e1z -1 + … + e j -1z - ( j -1)-n
Fj (z -1 ) = f 0j + f1j z -1 + … + fnjy z y	 R j (z -1 ) = r0 + r1z -1 + … + rj -1z - ( j -1)
S j (z -1 ) = s0j + s1j z -1 + … + snju -1z - (nu -1) .

(8)

where

Multiplying the system (3) by Ej(z−1) and (7), and then obtain thefollowing equation:y(t + j) = Fj (z −1 ) y(t ) + E j (z −1 )B(z −1 )∆u(t + j − 1)  
+ E (z −1 )x (t + j).
j(9)Consider the definition Ej(z−1) B(z−1) = Gj(z−1). And the future noise
term is independent with current step. The above equation can be
written as:-1-1  y(t + j) = Fj (z ) y(t ) + G j (z )Du(t + j - 1). (10)It can also be re-written through (7) by:y(t + j) = Fj (z -1 ) y(t ) + R j (z -1 )Du(t + j - 1)

+ S j (z -1 )Du(t - 1).

(11)

Then the above equations can be expressed in a vector form as
follows.
		 y = Ru + Fy(t ) + SDu(t - 1), 

(12)

where

Figure 1 | Block diagram of the proposed scheme.

y = [ y(t + 1), y(t + 2), …, y(t + N )]T ,u = [ ∆u(t ), …, ∆u(t + N − 1)]
(13)F = [F1 , F2 , … FN ]T ,S = [S1 , S2, … SN ]T ,

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231

with the matrix R[STEP 1] Generate the initial database

ér0
êr
1
		 R := êêêërN -1

The historical data can be obtained by employing some conventional methods. The initial data set then is used to identify thepolynomial parameters when j = 1, namely F1(z−1) and Gj(z−1).
Then the input and output data, reference signal and F1(z−1) andGj(z−1) are stored in the database.

0
0 ù
ú
r0
ú
(14)

 ú
rN - 2 r0 úû
The minimization of J by the partial derivative method with respectto ũ and obtain the control sequence.
u = [R R + Λ] R {w − Fy(t ) − S∆u(t − 1)} . 
	 
T

−1

T

(15)

It should be noted that the first element is the ∆u(t), then the current control u(t):    u(t ) = u(t - 1) + [1, 0, , 0]u. (16)

3.2. Derivation of Output by
Control Parameters(17)

which reveals the relation between the one-step ahead predictive
output and polynomials F1(z−1) and Gj(z−1). In the existing work[6], the identified polynomials parameters remain the same, which
results that the performance may not be satisfactory when thesystem property changes or is of strong nonlinearity.
Therefore, the database-driven approach is introduced to address
this issue. The database-driven approach works in an on-line
manner, which means the polynomials F1(z−1, t) and Gj(z−1, t) will
be updated based on the database.
The following subsection will review the database-driven approachby involving the parameters F1(z−1, t) and Gj(z−1, t). The above
Equation (17) is re-written in the form of information vector as:y(t + 1) = f (t )q (t ) 
		 
T

(18)

with F1(z−1, t) and Gj(z−1, t) are the components in θ(t), andf(t - 1) := [ y(t - 1), ¼, y(t - n y ),
D(t - 1), ¼, Du(t - nu )].

(21)[STEP 2] Calculate the distance and select the neighbors
The distance between the query and the information vector f(t ) is
calculated by the following ℒ1-norm.
f l (t ) - f l (q)n y + nu +1,

å max f (m) - min f (m) 
l =1

m

l

j = 1, 2, , N t

m

l

(22)

where Nt is the dimension of the database, f l (t ) is the l th elementof the query at current step t, and f l (q) is the l th element of theqth information vector. Moreover, max f l (m) is the maximum
element among all l th element of all information vector, and thesimilar explanation with min f l (m). Therefore, the database isarranged by an ascended order based on the calculated distance,
and then the Ne-pieces neighbors can be selected.
[STEP 3] Re-calculate the initial polynomial parameters
In this step, the selected Ne-pieces neighbors’ data are used to
re-calculate the parameters F1(z−1, t) and Gj(z−1, t) in an on-line
manner by linearly weighted average [12] as follows:Neq old (t ) = å wiq (i),
	 
i =1

Ne

åw = 1 
i =1

i

(23)

where
		

exp(- ¡ i )wi = N e.
å exp(- ¡i )

(24)

i =1

(19)

3.3. Database-driven Approach
The detail of this approach will be reviewed in this subsection. The
query f(t ), that is the information vector in the current step, isintroduced firstly.
f(t ) := [w(t + 1), y(t ),, y(t - ny + 1),Du(t - 1),, Du(t - nu )],q = 1, 2, , N 0 

where N0 is the number of data set in the initial database. Note that,in the initial stage, each set of parameters F1(z−1) and Gj(z−1) remain
the same.

   ¡(f (t ),f (q)) =Recall Equation (10) and for j = 1 to give the one-step ahead predictive output as:y(t + 1) = F1 (z -1 ) y(t ) + G1 (z -1 )Du(t ), F(q) := [f (q),q (q)],(20)[STEP 4] Parameters’ adjustment
The parameters obtained in the last STEP θold(t) are updated in an
online manner to achieve satisfactory performance by the following steepest descent method.
	   q

new

(t ) = q old (t ) - h¶J (t + 1)
¶q (t )

(25)

with η learning rate, and J(t + 1) denotes the prediction error andis defined by:where w(t + 1) is the reference signal at (t + 1).

1
2
		 J (t + 1) := 2 e (t + 1) , (26)

The procedure of the database-driven approach is summarized in
the following.

		 e (t + 1) := y(t + 1) - y(t + 1). (27)

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4. NUMERICAL SIMULATIONIn this section, we have conducted the simulation and illustrate theeffectiveness of the proposed scheme. The Hammerstein model[13] is considered and is given in the following form:y(t ) = 0.1 y(t - 1) - 0.2 y(t - 2)
+ 1.2 x(t - d - 1) - 0.1x(t - d - 2) + x (t ) x(t ) = u(t ) - u2 (t ) + 0.5u3 (t )(28)

where x(t) is the Gaussian noise with zero mean and variance 0.012,
and delay time d is set as 3 in this study. The reference signals are
given by:
1.0(0 ≤ t < 100)0.5(100 ≤ t < 200)		 w(t ) = 2(200 ≤ t < 300)
1.5(300 ≤ t < 400)

(29)

Figure 3 | Control results using the proposed scheme.
Table 2 | The user-specified parameters

The comparative study is firstly tested in the case where the polynomials F1(z−1) and Gj(z−1) at step j = 1 are the fixed value. The
user-specified parameters are shown in Table 1.
The simulation results are shown in Figure 2, where shows theoscillation when the reference signal w(t) = 2. The tracking performance is suppressed in this area due to the strong nonlinearity.

The orders of information vector
Time-delay
Weighting factor
Predictive horizon
Initial database
Learning rate for F1(z−1)
Learning rate for Gj(z−1)

ny= 2
nu= 5
d=3
λ=4
10
150
0.0001
0.1

The proposed scheme then is used, and the control results areshown in Figure 3. The user-specified are given in Table 2.
The control result is shown in Figure 3 where shows the stable
tracking property, particularly when the reference signal w(t) = 2.
The trajectories of polynomial F1(z−1, t) and Gj(z−1, t) are illustrated
in Figures 4 and 5, respectively.
The control results from the proposed scheme outperform the onesin comparative study, which demonstrates the effectiveness of the
proposed scheme.

Table 1 | The user-specified parameters
The orders of information vector
Time-delay
Weighting factor
Predictive horizon

ny= 2
nu= 5
d=3
λ=4
10

Figure 2 | Control results using the data-oriented GPC.

Figure 4 | Trajectory of F1(z−1, t).

Figure 5 | Trajectory of G1(z−1, t).

Z. Guan and T. Yamamoto / Journal of Robotics, Networking and Artificial Life 8(3) 229–233

5. CONCLUSION
In this paper, the design of a nonlinear GPC controller based on
database-driven approach has been presented. First, according
to the proposed scheme, the model information is not requiredin designing the controller. Then, the database-driven approachis considered to deal with systems with strong non-linearity andprovides a way for the online utilization of the proposed scheme.
Furthermore, the proposed scheme inherits the advantage of GPC
algorithm such that good performance can be achieved regardless of time-delay. Finally, the effectiveness has been verified ona numerical simulation and comparative study. The experimental
evaluation will be in our future work.

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

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AUTHORS INTRODUCTION
Dr. Zhe Guan

Prof. Toru Yamamoto

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, Government 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).