ROBUST CONTROL FOR UNCERTAIN DELAY SYSTEM ¤ RIHEM FARKH y,{ , KAOUTHER LAABIDII z and MEKKI KSOURI x Research unit on Systems Analysis and Control, National School Engineering of Tunis, BP, 37, Le Belvedere, 1002 Tunis, Tunisia y Rihem.Farkh@enit.rnu.tn z Kaouther.Laabidi@enit.rnu.tn x Mekki.Ksouri@mrstdc.gov.tn Received 6 February 2010 Accepted 30 November 2010 In this paper, the Generalized Kharitonov Theorem for quasi-polynomials is exploited for the purpose of synthesizing a robust controller. The aim here is to develop a controller to simul- taneously stabilize a given interval plant family with unknowing and bounded time delay. Using a constructive procedure based on HermiteBiehler theorem, we obtain all PI gains that stabilize an uncertain ¯rst-order delay system. An application example is presented for the temperature control of a heated air stream, process trainer PT 326. Keywords: Uncertain time delay system; interval plants; Generalized Kharitonov Theorem (GKT); PI controller; HermiteBiehler theorem; stability region. 1. Introduction Dead times are often encountered in various engineering systems and industry pro- cesses such as electrical and communication network, turbojet engine, chemical process, hydraulic system and nuclear reactor; In fact delays are caused by many phenomena like the time required to transport mass, energy or information, the time processing for sensors, the time needed for controllers to execute a complicated algorithm control and the accumulation of time lags in a number of simple plants connected in series. 9,15 Delays have a considerable in°uence on the behavior of closed-loop system and can generate oscillations and even instability. 5 PID con- trollers are of high interest thanks to their broad use in industrial circles. 4 An ana- lytical approach was developed in Refs. 1113 and allowed the characterization of the stability region of delayed systems controlled via PI and PID. Indeed, by using *This paper was recommended by Regional Editor Piero Malcovati. { Corresponding author. Journal of Circuits, Systems, and Computers Vol. 20, No. 3 (2011) 479499 # . c World Scienti¯c Publishing Company DOI: 10.1142/S0218126611007402 479