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Optimal Robust Control for Unstable Delay System

Rihem Farkh1,2,*, Khaled A. Aljaloud1, Moufida Ksouri2, Faouzi Bouani2
1 King Saud University, Riyadh, 11451, Saudi Arabia
2 Laboratory for Analysis, Conception and Control of Systems, LR-11-ES20, Department of Electrical Engineering, National Engineering School of Tunis, Tunis El Manar University, Tunis, 1002, Tunisia
* Corresponding Author: Rihem Farkh. Email:

Computer Systems Science and Engineering 2021, 36(2), 307-321.

Received 14 September 2020; Accepted 08 November 2020; Issue published 05 January 2021


Proportional-Integral-Derivative control system has been widely used in industrial applications. For uncertain and unstable systems, tuning controller parameters to satisfy the process requirements is very challenging. In general, the whole system’s performance strongly depends on the controller’s efficiency and hence the tuning process plays a key role in the system’s response. This paper presents a robust optimal Proportional-Integral-Derivative controller design methodology for the control of unstable delay system with parametric uncertainty using a combination of Kharitonov theorem and genetic algorithm optimization based approaches. In this study, the Generalized Kharitonov Theorem (GKT) for quasi-polynomials is employed for the purpose of designing a robust controller that can simultaneously stabilize a given unstable second-order interval plant family with time delay. Using a constructive procedure based on the Hermite-Biehler theorem, we obtain all the Proportional-Integral-Derivative gains that stabilize the uncertain and unstable second-order delay system. Genetic Algorithms (GAs) are utilized to optimize the three parameters of the PID controllers and the three parameters of the system which provide the best control that makes the system robust stable under uncertainties. Specifically, the method uses genetic algorithms to determine the optimum parameters by minimizing the integral of time-weighted absolute error ITAE, the Integral-Square-Error ISE, the integral of absolute error IAE and the integral of time-weighted Square-Error ITSE. The validity and relatively effortless application of presented theoretical concepts are demonstrated through a computation and simulation example.


Unstable time-delay system; interval plants; generalized Kharitonov theorem; PID controller; Hermite-Biehler theorem; stability region; genetic algorithms; optimum PID controller; optimum system parameters

Cite This Article

R. Farkh, K. A. Aljaloud, M. Ksouri and F. Bouani, "Optimal robust control for unstable delay system," Computer Systems Science and Engineering, vol. 36, no.2, pp. 307–321, 2021.


This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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