Page 145 - IJOCTA-15-3

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.3, pp.517-534 (2025 )
https://doi.org/10.36922/IJOCTA025080034

RESEARCH ARTICLE

Proportional integral derivative plus control for nonlinear
discrete-time state-dependent parameter: Industrial applications

E. M. Shaban 1,2*

1
Department of Mechanical Engineering Technology, College of Applied Industrial Technology, Jazan
University, Jazan, Saudi Arabia
2
On leave from Faculty of Engineering (Mataria), Helwan University, Cairo, Egypt
eeshaban@jazanu.edu.sa

ARTICLE INFO ABSTRACT
Article History:
Received: February 21, 2025 In modern industrial automation, control of nonlinear systems with complex
1st revised: March 2, 2025 dynamics poses significant challenges, especially when dealing with discrete-
2nd revised: March 20 , 2025 time models that incorporate state-dependent parameters. Addressing this
Accepted: April 11, 2025 need, this paper explores the Proportional-integral-derivative-plus (PID+) con-
Published Online: July 16, 2025 trol approach applied to nonlinear systems characterized by state-dependent
Keywords: parameter (SDP) discrete-time models. Two industrial applications are
Proportional-integral-derivative control demonstrated as follows: a bitumen tank system and a reeling/packing machine
used in a bitumen membrane sheet production line. Both systems are modeled
State-dependent parameter models
using discrete-time transfer functions with SDP structures. The present work
Non-minimal state space
extends the novel SDP-PID+ approach by formulating its control algorithms
State variable feedback
and integrating additional proportional and input compensators. This en-
AMS Classification:
hancement enables effective and intuitive handling of processes characterized
37H05, 37M05, 37M10, 37M15, 37N35, by discrete-time transfer functions with any order and sampling time delay.
37N40, 65P40, 70Q05, 70K70 The approach enables a straightforward implementation of the SDP-PID+ al-
gorithm across two distinct industrial applications, considering their varying
response times. The approach reduces the time required to design the SDP-
PID+ method for the selected applications while also demonstrating enhanced
robustness and performance. It effectively mitigates disturbances and accom-
modates nonlinearities, higher-order dynamics, and delays.

9
1. Introduction nonlinearities. The proportional integral deriv-
ative plus (PID+) control approach was devel-
Proportional integral derivative (PID) controllers oped to address these challenges as an extension
have been widely used in industrial processes since of conventional PID methods. It integrates addi-
their development in the mid-1950s. 1,2 Today, tional dynamic feedback and input compensators
they remain the most prevalent method for con- to handle processes with higher-order dynamics
trolling process variables due to their simplicity or pure time delays exceeding unity. 10 Unlike tra-
3
and reliability. Research has continually refined ditional PID control, PID+ employs state vari-
PID tuning methods, introducing approaches able feedback (SVF) techniques, replacing man-
such as auto-tuning, 4,5 genetic tuning, 5,6 robust ual tuning with pole assignment or linear qua-
7
8
tuning, and optimal tuning. However, signifi- dratic (LQ) optimization strategies. 11 These en-
cant challenges arise when addressing processes hancements improve control performance in sys-
with high-order transfer functions (TFs), n ≥ 3, tems that conventional PID tuning often struggles
time delays greater than unity, δ ≥ 2, or to manage effectively. 10,12
*Corresponding Author
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