Asian Journal of Control, Vol. 13, No. 3, pp. 445 448, May 2011 Published online 10 December 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.325 –Brief Paper– PRACTICAL EXPONENTIAL STABILITY OF PERTURBED TRIANGULAR SYSTEMS AND A SEPARATION PRINCIPLE Amel Benabdallah, Ines Ellouze, and Mohamed Ali Hammami ABSTRACT In this paper, we present a practical stability result for perturbed dynamic systems depending on a parameter and we study the practical exponential stability of perturbed triangular systems. These results are applied to show that a separation principle for nonlinear uncertain systems can be achieved and which considers practical global uniform exponential stability. Key Words: Perturbed systems, practical stability, separation principle. I. INTRODUCTION The separation principle involves the design of a state observer and a state feedback stabilizing controller independently. A separation principle is established if the closed loop system remains stable when the state feedback controller is implemented using state estimates. In [1, 2], authors have established a local or a global separation principle. However, global asymptotic stability by output feedback does not hold in general. Thus, some results on semi-global stability have been reported [3, 4]. In this paper, we study the separation principle for nonlinear uncertain systems with nominal linear part. For such systems, an observer has been designed in [5, 6]. A high gain observer has been used by [3] to estimate the state of the considered uncertain system. Another interesting result has been developed by [7], where they have built a practical observer; that is the error converges exponentially towards an arbitrarily small neighborhood of the origin. All these observers Manuscript received April 26, 2008; revised December 15, 2008; accepted March 29, 2009. The authors are with the Faculty of Sciences of Sfax, Depart- ment of Mathematics, BP 1171, Sfax 3000, Tunisia. Amel Benabdallah is the corresponding author (e-mail: Amel.Benabdallah@fss.rnu.tn). are parameter dependent systems. The problem of the state feedback stabilization of uncertain systems has been addressed by [8], where a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state. References [9, 10] presented controllers that guarantee exponential stability of a ball containing the origin of the state space, the radius of this ball can be made arbitrary small. The resulting closed loop systems are always parameter dependent. Getting a separation principle is closely related to the stability of cascaded parameter dependent perturbed systems. This moti- vates us to address the problem of giving sufficient conditions under which a cascaded perturbed system is practically globally uniformly exponentially stable. Dynamical systems depending on a parameter were considered for instance in [11, 12], where semi-global practical asymptotic stability was addressed. This paper is organized as follows: we start by introducing the notion of practical global uniform exponential stability of a family of systems depending on a parameter ε>0. Then, we give sufficient conditions to establish prac- tical global uniform exponential stability of perturbed systems. In Section III, under the assumption that the nominal triangular system is globally uniformly exponentially stable, we will discuss the stability of the perturbed triangular system. As an application, we will establish, in Section IV, a separation principle for two classes of uncertain systems having a nominal linear part. 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society