SUGISAKAMASANORI

Journal of Robotics, Networking and Artificial Life
Vol. 10(3); December (2023), pp. 213–219
ISSN (Online): 2352-6386; ISSN (Print): 2405-9021
https://alife-robotics.org/jrnal.html

Research Article

Improvement of the Database-Driven PID Control Methodby H-Infinity Norm of the Local Model
Takumi Ueda, Kazuo Ishii

Kyushu institute of technology, 2-4 Hibikino, Wakamatsu, Kitakyushu, Fukuoka, 808-0135, Japan

ARTICLE INFO
Article History
Received 05 December 2022
Accepted 08 December 2023
Keywords
Database-driven PID
H-Infinity norm
Nonlinear system

ABSTRACTPID controllers are used in many applications and are considered an example of a fixed structure
controller. For each specific system, the controller's PID parameters must be adjusted accordingly.
Therefore, database-driven control, which automatically adjusts PID controller parameters usinga large amount of experimental data, is attracting attention. Database-driven control is one of themethods that does not require model information. Therefore, in database-driven PID (DD-PID),
the Jacobian of the system used for the steepest descent is unknown. In this paper, we propose an
optimization method using gradient descent based on a local model of the system and confirm its
effectiveness through simulation using the polystyrene reactor model as an example.
© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd.
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

model using the JIT method, and they also affect thecontrol results. Next, correct the gain obtained using the
JIT method. A nonlinear optimization method is used tomodify the gain. In this paper, we use the gradientdescent method. To be effective with the gradient descentmethod, it is important to set the appropriate coefficients.
If appropriate learning coefficients are not given, there isa problem that PID gains are calculated, resulting inunstable control results.
In this paper, we examine a database design method forthe DD-PID method. It is which identifies the local
models of the system using the Recursive Least Squares(RLS) method. And estimate to 𝐻𝐻∞ norm. It uses theestimated 𝐻𝐻∞ norm to design the database. Furthermore,
the Jacobian of the local model identified by RLS isreflected in the gradient method to optimize the PID gain
update. Finally, the proposed method is validated bynumerical simulations using the polystyrene reactor
model.
Table 1 shows a list of variables used in this paper.

1. Introduction
PID controllers are widely used in industry [1] and are
one of the typical examples of controllers with a fixed
structure. To adjust the parameters appropriately, it iseffective to identify a mathematical model of the system,
and many methods have been reported so far [2].
However, the systems in general use have various
nonlinearities, making it difficult to construct a detailed
model, and the desired control performance may not be
achieved with a conventional fixed PID controller.
Therefore, database-driven control has been actively
studied in which PID controller parameters areautomaticallyadjustedaccordingtosystemcharacteristics using a large amount of experimental data[3], [4], [5], [6].
The DD-PID method, one of the database-driven control
methods, constructs a local model based on data
accumulated in a database and updates the PID gains
using a nonlinear optimization method. This local model
is computed by extracting data from the database thatclosely resembles the query generated using the current
operational data as neighborhood data. Design of thedatabase and query is important to construct a local

Corresponding author E-mail: ueda.takumi964@mail.kyutech.jp, ishii@brain.kyutech.ac.jp213

1.1. Controller

𝐺𝐺𝑚𝑚 (𝑧𝑧) =In practical control scenes, achieving smooth response
characteristics is desirable. In this paper, we design thecontroller using an I-PD control scheme.

(1 + 𝑝𝑝1 + 𝑝𝑝2 )𝑧𝑧 −1
𝑃𝑃(1)𝑧𝑧 −1=
𝑃𝑃(𝑧𝑧)
1 + 𝑝𝑝1 𝑧𝑧 −1 + 𝑝𝑝2 𝑧𝑧 −2𝑦𝑦𝑚𝑚 (𝑡𝑡) = 𝐺𝐺𝑚𝑚 (𝑧𝑧)𝑟𝑟(𝑡𝑡)⎧ 𝑝𝑝 = −2𝑒𝑒 −2𝜇𝜇 cos ��4𝜇𝜇 − 1 𝜌𝜌�
1
2𝜇𝜇
⎪
⎪
𝜌𝜌
−
𝑝𝑝2 = 𝑒𝑒 𝜇𝜇⎨ 𝜇𝜇 = 0.25(1 − 𝛿𝛿) + 0.51𝛿𝛿
⎪𝑇𝑇⎪⎩ 𝜌𝜌 = 𝜎𝜎

(3)
(4)

𝜌𝜌

Table 1. Setting parameters for DD-PID
Specification
Symbol𝛼𝛼 - th epoch𝛼𝛼
Number of samples of epoch𝑀𝑀𝑡𝑡(𝛼𝛼)Time𝑇𝑇
Sampling time𝑦𝑦(𝑡𝑡)
Jacket temperature𝑢𝑢(𝑡𝑡)
Reaction temperature
𝑁𝑁𝑦𝑦 , 𝑁𝑁𝑢𝑢
Order of the 𝑦𝑦(𝑡𝑡) and the 𝑢𝑢(𝑡𝑡)Delay operator𝑧𝑧
DatabaseΩ(𝛼𝛼)Querry𝑞𝑞(𝑡𝑡)
Degree of similarity
𝑆𝑆�𝒒𝒒(𝑡𝑡), 𝒒𝒒(𝜏𝜏)�
𝜅𝜅
Threshold of degree of similarity
Neighborhood dataΓ(𝑡𝑡)Evaluation function
𝐿𝐿(𝛼𝛼)
Acceptable error range of 𝐿𝐿(𝛼𝛼)𝜌𝜌𝐿𝐿
PID gains𝜽𝜽 = [𝐾𝐾𝑃𝑃 , 𝐾𝐾𝐼𝐼 , 𝐾𝐾𝐷𝐷 ]Learning vector for gradient descent𝜼𝜼 = [𝜂𝜂𝑃𝑃 , 𝜂𝜂𝐼𝐼 , 𝜂𝜂𝐷𝐷 ]𝑇𝑇
Controller model𝐶𝐶(𝜽𝜽)
𝐻𝐻(𝑧𝑧)
Plant model
Model parameter of the control target
𝑎𝑎1 , 𝑎𝑎2 , 𝑏𝑏1 , 𝑏𝑏2
White noise𝜁𝜁(𝑡𝑡)
Reference model
𝐺𝐺𝑚𝑚 (𝑧𝑧)Number of the Query registered in the𝑛𝑛(⋅)
database
This paper utilizes the I-PD controller since achievingsmooth response characteristics is desirable in practicalcontrol applications [7]. Eq. (1) shows the I-PD controller.
𝑢𝑢(𝑡𝑡) = 𝑢𝑢(𝑡𝑡 − 1) + 𝐾𝐾𝐼𝐼 (𝑡𝑡)𝑒𝑒(𝑡𝑡)(1)−𝐾𝐾𝑃𝑃 (𝑡𝑡)Δ𝑦𝑦(𝑡𝑡) − 𝐾𝐾𝐷𝐷 (𝑡𝑡)Δ2 𝑦𝑦(𝑡𝑡)Here, 𝐾𝐾𝑃𝑃 (𝑡𝑡),𝐾𝐾𝐼𝐼 (𝑡𝑡), and 𝐾𝐾𝐷𝐷 (𝑡𝑡) represent the proportional,
integral, and derivative gains at each time instant. And 𝑧𝑧
is a delay operator that satisfies 𝑧𝑧 −1 𝑦𝑦(𝑡𝑡) = 𝑦𝑦(𝑡𝑡 − 1) ,where Δ = 1 − 𝑧𝑧 −1 . 𝑒𝑒(𝑡𝑡) denotes the deviation, definedby Eq. (2).
𝑒𝑒(𝑡𝑡) = 𝑟𝑟(𝑡𝑡) − 𝑦𝑦(𝑡𝑡)
(2)

(5)

1.3. Proposed method

The DD-PID method proposed in this paper updates the
PID gains by the following steps.
Step.1 Setting of the reference model and experimentusing initial PID GainsThe DD-PID method cannot update parameters when
there is no data in the database. Therefore, an initial
database is constructed using a stable controller. The
stable controller is assumed to be pre-constructed using
the CHR method or other methods. The database used inthis paper is defined by Eq. (6). In addition, ‖𝐺𝐺(𝑡𝑡)‖∞ isthe 𝐻𝐻∞ norm of the local model calculated using the RLS
method, 𝝓𝝓�𝑡𝑡(𝛼𝛼) � is the information vector, and 𝜽𝜽�𝑡𝑡(𝛼𝛼) � isthe gain vector of the controller.
Note that the 𝐻𝐻∞ norm coincides with the maximum
gain of the signal amplified by the transfer function 𝐺𝐺(𝑡𝑡).
Here, 𝑡𝑡(𝛼𝛼) represents the time of the 𝛼𝛼-th epoch.
𝑇𝑇𝛀𝛀(𝛼𝛼) = �𝝓𝝓𝑇𝑇 �𝑡𝑡(𝛼𝛼) �, �𝐺𝐺�𝑡𝑡(𝛼𝛼) ��∞ , 𝜽𝜽𝑇𝑇 �𝑡𝑡(𝛼𝛼) ��(6)The control target model to be estimated by RLS isdefined by Eq. (7).
𝑏𝑏1 (𝑡𝑡)𝑧𝑧 −1 + 𝑏𝑏2 (𝑡𝑡)𝑧𝑧 −2
𝐻𝐻(𝑧𝑧) =(7)
1 + 𝑎𝑎1 (𝑡𝑡)𝑧𝑧 −1 + 𝑎𝑎2 (𝑡𝑡)𝑧𝑧 −2
The control target assumed in this paper is assumed tohave no direct derivatives and the order of the systemdoes not vary with time. The reason for using a quadratic
delay system here is that identification by a quadratic
delay system can represent most mechanical systems.
𝐺𝐺(𝑧𝑧) is given by Eq. (8).
𝐻𝐻(𝑧𝑧)𝐶𝐶(𝜽𝜽)𝐺𝐺(𝑡𝑡) =(8)
1 + 𝐻𝐻(𝑧𝑧)𝐶𝐶(𝜽𝜽)

1.2. Reference model

The reference model for the DD-PID method is definedby Eq. (3), (4) and (5). Here 𝜇𝜇 is a parameter related toattenuation, 𝜌𝜌 is a parameter related to rising time, and 𝑇𝑇
is the sampling time [8].

Define 𝝓𝝓�𝑡𝑡(𝛼𝛼) � and 𝜽𝜽�𝑡𝑡(𝛼𝛼) � by Eq. (9) and (10).
𝝓𝝓�𝑡𝑡(𝛼𝛼) � = �𝑦𝑦�𝑡𝑡(𝛼𝛼) �, 𝑢𝑢�𝑡𝑡(𝛼𝛼) ��

214

𝑇𝑇

(9)

𝜽𝜽�𝑡𝑡(𝛼𝛼) � = �𝐾𝐾𝑃𝑃 �𝑡𝑡(𝛼𝛼) �, 𝐾𝐾𝐼𝐼 �𝑡𝑡(𝛼𝛼) �, 𝐾𝐾𝐷𝐷 �𝑡𝑡(𝛼𝛼) ��𝑇𝑇

Step. 2 Getting queries and neighborhood data

(10)

Step. 3 Local model calculation by JIT method
Using the neighborhood data selected in Step.2,
construct a local model using Eq. (16) and (17) to obtain
�(𝑡𝑡).
the PID gain 𝜽𝜽

Obtain the query 𝑞𝑞(𝑡𝑡) while operating the control target.
𝑞𝑞(𝑡𝑡) is defined by Eq. (11). Here, 𝑁𝑁𝑦𝑦 and 𝑁𝑁𝑢𝑢 are theorders of 𝑦𝑦(𝑡𝑡) and 𝑢𝑢(𝑡𝑡).Here, 𝑁𝑁𝑦𝑦 and 𝑁𝑁𝑢𝑢 are constants
that are determined by the control designer by trial anderror.
𝒒𝒒(𝑡𝑡) = �𝑦𝑦(𝑡𝑡), ⋯ , 𝑦𝑦�𝑡𝑡 − 𝑁𝑁𝑦𝑦 + 1�,(11)
𝑢𝑢(𝑡𝑡), ⋯ , 𝑢𝑢(𝑡𝑡 − 𝑁𝑁𝑢𝑢 + 1), ‖𝐺𝐺(𝑡𝑡)‖∞ ]𝑇𝑇
In this paper, control is performed using the PID gain𝜽𝜽�𝑡𝑡(𝛼𝛼 − 1)� at 𝑡𝑡(𝛼𝛼 − 1).
Using the retrieved query, the proximity distance of thedata to the information vector in the database iscalculated, and neighboring data are selected. Althoughseveral algorithms have been proposed for selecting
nearby data from a large amount of data, this paper uses
Kinoshita's [8] method for obtaining nearby data basedon similarity. Kinoshita's method uses statistical methods
and is superior to the K-NN method in that it iscomputationally less expensive. The similarity between
the information vector and the query is given by Eq. (12).
𝑆𝑆�𝒒𝒒(𝑡𝑡), 𝒒𝒒(𝜏𝜏)� =𝑁𝑁𝑦𝑦 +𝑁𝑁𝑢𝑢+1

�
𝑗𝑗=1

1

�2𝜋𝜋ℎ𝑗𝑗2

exp �−

�𝑞𝑞𝑗𝑗 (𝑡𝑡) − 𝑞𝑞𝑗𝑗 (𝜏𝜏)�
2ℎ𝑗𝑗2

2

�

𝑛𝑛(Ω)
𝑘𝑘=1

𝑆𝑆�𝒒𝒒(𝑡𝑡), 𝒒𝒒(𝜏𝜏)� ≥ 𝜅𝜅

𝑗𝑗=1

1

�2𝜋𝜋ℎ𝑗𝑗2𝑖𝑖=1

� 𝝎𝝎𝑖𝑖 = 1

𝑆𝑆�𝒒𝒒(𝑡𝑡), 𝒒𝒒(𝑖𝑖)�

𝑛𝑛�Γ(𝑡𝑡)�
∑𝑗𝑗=1 𝑆𝑆�𝒒𝒒(𝑡𝑡), 𝒒𝒒(𝑗𝑗)�

(16)

(17)

�(𝑡𝑡) calculated by the JIT method may be
The gain 𝜽𝜽�(𝑡𝑡) calculated in step
inappropriate. Therefore, the gain 𝜽𝜽
3 is corrected using the gradient method. The gain
correction by the gradient method is given by Eq. (18).
As shown in Eq. (18), the gain correction by the DD-PID
method is delayed by one step. This is because the data
used to update the gain by the gradient method is related
to 𝑢𝑢(𝑡𝑡).
𝜕𝜕𝜕𝜕(𝑡𝑡 + 1)
�(𝑡𝑡) − 𝜼𝜼𝜽𝜽(𝑡𝑡) = 𝜽𝜽
(18)
�(𝑡𝑡)
𝜕𝜕𝜽𝜽
Where 𝜼𝜼 is the learning coefficient for the gradientmethod, 𝐽𝐽(𝑡𝑡 + 1) is the evaluation function, and 𝜀𝜀(𝑡𝑡) isthe difference between the system output and thereference model output. From these relationships, thepartial derivative in Eq. (18) can be expanded using thedifferential chain rule as in Eq. (19).
𝜕𝜕𝐽𝐽(𝑡𝑡 + 1)
�(𝑡𝑡)
𝜕𝜕 𝜽𝜽
(19)
𝜕𝜕𝐽𝐽(𝑡𝑡 + 1) 𝜕𝜕𝜕𝜕(𝑡𝑡 + 1) 𝜕𝜕𝑦𝑦(𝑡𝑡 + 1) 𝜕𝜕𝑢𝑢(𝑡𝑡)=�(𝑡𝑡)𝜕𝜕𝜕𝜕(𝑡𝑡 + 1) 𝜕𝜕𝑦𝑦(𝑡𝑡 + 1) 𝜕𝜕𝑢𝑢(𝑡𝑡) 𝜕𝜕 𝜽𝜽
Here, 𝐽𝐽(𝑡𝑡) and 𝜀𝜀(𝑡𝑡)are defined by Eq. (20).
1
1
𝐽𝐽(𝑡𝑡) = 𝜀𝜀(𝑡𝑡)2 = {𝑦𝑦(𝑡𝑡) − 𝑦𝑦𝑚𝑚 (𝑡𝑡)}2(20)
2
2
The Jacobian of the system is given by Eq. (21) from Eq.
(7).
𝜕𝜕𝜕𝜕(𝑡𝑡 + 1)
(21)
= 𝑏𝑏1 (𝑡𝑡 + 1)
𝜕𝜕𝜕𝜕(𝑡𝑡)
The differential chain rule can be calculated using Eq.
(21) and (22).

(12)

(13)

1

�

𝑖𝑖=1

𝝎𝝎𝒊𝒊 =(14)ℎ𝑗𝑗 = 1.06𝜎𝜎𝑗𝑗 𝑛𝑛(Ω)−5This method of estimating the kernel density function iscalled Plug-In method.
The selection of neighboring data is based on similarity
according to Eq. (15). The designer sets 𝜅𝜅(0 ≤ 𝜅𝜅 ≤ 1).
𝑁𝑁𝑦𝑦 +𝑁𝑁𝑢𝑢 +1𝑛𝑛�Γ(𝑡𝑡)�

�(𝑡𝑡) = � 𝝎𝝎𝑖𝑖 𝜽𝜽(𝑖𝑖) ,𝜽𝜽ℎ𝑗𝑗 is the bandwidth of the probability density function,
𝑞𝑞𝑗𝑗 (𝑡𝑡) is the 𝑗𝑗-th element of the query at time 𝑡𝑡.𝜎𝜎𝑗𝑗 and ℎ𝑗𝑗
is given by Eq. (13) and (14).
12𝜎𝜎𝑗𝑗 = �� �𝑞𝑞𝑗𝑗 (𝑘𝑘) − 𝜇𝜇𝑗𝑗 �𝑛𝑛(Ω)

𝑛𝑛�Γ(𝑡𝑡)�

�𝑃𝑃 + 𝑏𝑏1 (𝑡𝑡 + 1)𝜂𝜂𝑃𝑃 𝜀𝜀(𝑡𝑡 + 1)Δ𝑦𝑦(𝑡𝑡 + 1)
𝐾𝐾𝑃𝑃 = 𝐾𝐾
�𝐼𝐼 − 𝑏𝑏1 (𝑡𝑡 + 1)𝜂𝜂𝐼𝐼 𝜀𝜀(𝑡𝑡 + 1)𝑒𝑒(𝑡𝑡 + 1)
� 𝐾𝐾𝐼𝐼 = 𝐾𝐾
�𝐷𝐷 + 𝑏𝑏1 (𝑡𝑡 + 1)𝜂𝜂𝐷𝐷 𝜀𝜀(𝑡𝑡 + 1)Δ2 𝑦𝑦(𝑡𝑡 + 1)
𝐾𝐾𝐷𝐷 = 𝐾𝐾(15)

The neighboring data selected by Eq. (15) are representedby Γ(𝑡𝑡).

215

(22)

Table 2. Setting parameters for DD-PID
Step. 4 Update the database

Sample time

To store the controlled variable, manipulated variables,and system parameters obtained from experiments in the
database.

Order of the 𝑦𝑦(𝑡𝑡) and the 𝑢𝑢(𝑡𝑡)The constant value for the RLS

Step. 5 Evaluation and Convergence Decision

Value of the forgetting coeﬃcient
Similarity threshold

Update the controller parameters by repeating Step 2
through Step 5 until the evaluation function shown in Eq.
(23) and (24) becomes sufficiently small or is repeated a
specified number of times.
𝑀𝑀

12
𝐿𝐿(𝛼𝛼) = � ��𝑦𝑦𝑚𝑚 (𝑡𝑡𝑖𝑖 ) − 𝑦𝑦�𝑡𝑡(𝛼𝛼),𝑖𝑖 ��𝑀𝑀
𝑖𝑖=1|𝐿𝐿(𝛼𝛼) − 𝐿𝐿(𝛼𝛼 − 1)| ≤ 𝜌𝜌𝐿𝐿

Gradient dissent coeﬃcient
Convergence value threshold
2.2. Simulation results(23)

𝑇𝑇 = 1[𝑠𝑠]𝑁𝑁𝑦𝑦 = 5
𝑁𝑁𝑢𝑢 = 5
𝑎𝑎 = 1000
𝑷𝑷0 = 𝛼𝛼𝑰𝑰𝜻𝜻0 = 𝑶𝑶
𝜌𝜌𝐹𝐹 = 0.99
𝜅𝜅 = 0.85
𝜂𝜂𝑃𝑃 = 13 × 10−3𝜂𝜂𝐼𝐼 = 13 × 10−3𝜂𝜂𝐷𝐷 = 8 × 10−3𝜌𝜌𝐿𝐿 = 10−4

In this paper, we setting to the parameter are 𝛼𝛼1 =0.5, 𝛼𝛼2 = 0.1, 𝑁𝑁(0) = 6, and the gradient descent gains
setting to (1) 𝜼𝜼(1) = [2 × 10−4 , 2 × 10−4 , 2 × 10−4 ]𝑇𝑇 ,(2) 𝜼𝜼(2) = [4 × 10−4 , 4 × 10−4 , 2 × 10−4 ]𝑇𝑇 , (3) 𝜼𝜼(3) =[6 × 10−4 , 6 × 10−4 , 2 × 10−4 ]𝑇𝑇 and (4) 𝜼𝜼(4) = [8 ×
10−4 , 8 × 10−4 , 2 × 10−4 ]𝑇𝑇 .
Fig. 2 shows result using the Yamamoto's DD-PID
update method for polystyrene reactor model [8], here
Fig. 2-a shows the results of the control, Fig. 2-b showstuning PID Gains, and Fig. 2-c shows the evaluation
function values. The evaluation function's value is
𝐿𝐿(34) = 1.396 using 𝜼𝜼1 , 𝐿𝐿(137) = 1.708 using 𝜼𝜼2 ,and 𝐿𝐿(46) = 1.449 using 𝜼𝜼3 .Here, when the evaluation
function value using 𝜼𝜼4 , it is diverged. The lowest values
for the Yamamoto’s method are 𝐿𝐿(33) = 1.396 using𝜼𝜼(1) . The PID Gains when 𝜼𝜼(3) is inappropriate because
the 𝐾𝐾𝐷𝐷 (𝑡𝑡) Gains are negative.
Fig. 3 shows result using the proposed method forpolystyrene reactor model [8], here Fig. 3-a shows theresults of the control, Fig. 3-b shows tuning PID Gains,
Fig. 3-c shows the evaluation function values, Fig. 3-dshows result of the 𝐻𝐻∞ norm, and Fig. 3-e shows theresult of the 𝑏𝑏1 (𝑡𝑡) .
From Fig. 3-a, its response has improved compared toFig. 2. In particular, the ability to follow the reference
model in the rising section has been improved.

(24)𝜌𝜌𝐿𝐿 is a convergence threshold, which is determined by
the user by trial and error.
2. Simulation

2.1. System parameter settings
In this paper, the effectiveness of the proposed method
is verified by numerical simulations using thepolystyrene reactor model shown in Eq. (25) [3], [6], [9].
𝑦𝑦(𝑡𝑡) = 0.804𝑦𝑦(𝑡𝑡 − 1) + 5.739 × 1015
𝐸𝐸𝑎𝑎
(25)
−
⋅ 𝑒𝑒 𝑅𝑅(𝑦𝑦(𝑡𝑡−1)+273) + 0.148𝑢𝑢(𝑡𝑡 − 1) + 𝜉𝜉(𝑡𝑡)Here,𝑢𝑢(𝑡𝑡)is the jacket temperature, 𝑦𝑦(𝑡𝑡) is the reaction
temperature, 𝐸𝐸𝑎𝑎 = 240 , 𝑅𝑅 = 0.01986 and 𝜉𝜉(𝑡𝑡) is
gaussian white noise. In this paper, temperature is the
Celsius degree. This system has ability to be strongly
non-linear at about 75 Celsius degrees or more.
Target temperature was set as in Eq. (26).
60(000 < 𝑡𝑡 ≤ 100)𝑟𝑟(𝑡𝑡) = �70(100 < 𝑡𝑡 ≤ 200)(26)
85(200 < 𝑡𝑡 ≤ 300)

The reference model is designed as in Eq. (27). Here,𝛿𝛿 = 0 and 𝜎𝜎 = 5.
(27)
𝑃𝑃(𝑧𝑧) = 1 − 1.34𝑧𝑧 −1 + 0.449𝑧𝑧 −2
The initial database is constructed using the initial PID
gain shown in Eq. (28).
𝐾𝐾𝑃𝑃 = 9.0, 𝐾𝐾𝐼𝐼 = 0.5, 𝐾𝐾𝐷𝐷 = 1.0(28)
Other parameters are shown in Table 2.
Fig. 1 shows the results of control using Eq. (28).
From Fig. 1, the ability to follow the output signal of thereference model is poor.
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(c) Results of the evaluation function 𝐿𝐿(𝛼𝛼)Fig.2 a, b, c results of the Yamamoto's method using
learning costs of 𝜂𝜂(1) , 𝜂𝜂(2) , 𝜂𝜂(3) and 𝜂𝜂(4) [3].

Fig.1 Result of 𝑦𝑦(𝑡𝑡) and 𝑢𝑢(𝑡𝑡) using the initial PID Gains(a) Result of the 𝑦𝑦(𝑡𝑡) and 𝑢𝑢(𝑡𝑡) using 𝜂𝜂(1)(a) Result of the 𝑦𝑦(𝑡𝑡) and 𝑢𝑢(𝑡𝑡)(b) Result of the PID Gains(b) Result of the PID Gains

217

changes significantly at 𝑡𝑡 = 160. This is thought to bebecause the nonlinearity suddenly becomes stronger
around 𝑡𝑡 = 160.From Fig. 3-e, the graph is positive inmost areas. The Jacobian of this system is positive asshown in Eq. (25). The negative portion is the effect of
estimation error by the RLS.
3. Conclusion
This paper proposed the gradient descent using local
model parameters for DD-PID. And the obtained 𝐻𝐻∞ ofthe system was reflected in the database. The
effectiveness of the proposed method was confirmed bynumerical simulation by adapting it to the polystyrene
reactor model discussed in Yamamoto's studies. The
Yamamoto's update method becomes 𝐿𝐿(33) = 1.396 .
The simulation results take time to converge for the
Yamamoto's update method, and 𝐿𝐿(30) = 1.02 for the
proposed method, and good results were obtained. And
the proposed method's evaluation function value is morethan the Yamamoto's method. One of the reasons for such
good results may be that the system Jacobian was given,
which enabled learning with more correct coefficients. A
side effect of giving the system Jacobian is to allow for alarger learning coefficient. This is thought to result infaster PID gain updates. These results suggest that the
proposed method is effective for better PID gain updating.
Future research is needed to introduce the proposed
algorithm into actual equipment. For this purpose, it isnecessary to consider the DD-PID method, which candirectly specify the design of stability. In addition,
database design methods need to be discussed.

(c) Result of the evaluation function 𝐿𝐿(𝛼𝛼)(d) Result of the H-Infinity norm of the system

References
1.
2.
3.

4.

(e) Result of the 𝑏𝑏1 (𝑡𝑡)Fig.3-a, b, c, d, e Results using the proposed method
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Yamamoto's method. From Fig. 3-c, the evaluation
function for the proposed method is 𝐿𝐿(30) = 1.02. The
evaluation function value and epoch number are better
than Yamamoto's method. From Fig. 3-d, the 𝐻𝐻∞

5.

6.

218

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Authors Introduction
Mr. Takumi Ueda

He received his M.E., in Engineeringfrom Kyusyu Institute of Technology,
Japan, in 2017. He is a third-year
student in the doctoral program at the
Kyushu Institute of Technology. His
research interest includes non-linear
control by PID using the database.

Dr. Kazuo Ishii

He is a Professor at the Kyushu
Institute of Technology, where he has
been since 1996. He received his Ph.D.
degree in engineering from the
University of Tokyo, Tokyo, Japan, in
1996. His research interests span bothship
marineengineeringandIntelligent Mechanics. He holds fivepatents derived from his research.

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