Observer design for complex systems in the presence of uncertainties, nonlinearities and distributed delays D. P. GOODALL†* and Z. WANG‡ †Control Theory and Applications Centre, Coventry University, Priory Street, Coventry, UK ‡Department of Information Systems and Computing, Brunel University, Uxbridge, Middlesex, UK (Received 1 July; in final form 25 November 2005) A nonlinear full-order observer is synthesized for a class of complex nonlinear time-delay systems of the retarded type. The system under consideration is subject to delayed states, delayed output, known nonlinear disturbances and unknown perturbations. The known nonlinearities are assumed to satisfy global Lipschitz conditions and, for the unknown perturbations, bounding information, dependent upon the output and controls, is assumed. Each delay system, in the class considered, contains multiple, time-varying, discrete and distributed delays. The problem addressed is the design of full-order, state observers such that, under the dynamics of the observation error system, the zero state is globally uniformly exponentially stable. Keywords: Algebraic Riccati inequalities; Exponential stability; Nonlinear systems; Observer design; Time-delay systems 1. Introduction Most complex dynamical systems are nonlinear to some degree and often include time-delays as intrinsic components; in addition, the systems may be subject to unknown perturbations. Sometimes the states of the system cannot be measured directly and, therefore, in this case, it may be desirable to obtain estimates of such states through the design of appropriate observers. The problem of designing nonlinear observers has been the focus of many researchers in recent times and there are mainly two research areas in this field. One is the extension of the linear Luenberger observer to the nonlinear case, such as the extended Kalman filter, and the psuedo linearization technique (see Misawa and Hedrick, 1989, for a survey). The latter technique is valid in a small range around the operating point, and often also requires a great deal of real-time computation, as discussed by Raghavan and Hedrick (1994). The former technique has been applied to design exact observers for nonlinear systems using a differential geometric approach [see, for example, (Hunt and Verma 1994, Xia and Zeitz 1997)], where stringent assumptions are required. An algebraic Lyapunov-based approach was developed by Thau (1973) to investigate the observer design problem for a class of continuous-time systems with known nonlinear disturbances. For systems with unknown bounded disturbances/uncertainties, robust observer synthesis has been studied by Dawson et al. (1992) and Cheng (1997). Moreover, sliding mode techniques have been utilized for designing observers for nonlinear systems with uncertainties (see Zak and Hui 1993, Koshkouei and Zinober 2004). However, the above International Journal of General Systems ISSN 0308-1079 print/ISSN 1563-5104 online q 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/03081070600660988 *Corresponding author. Email: D.Goodall@coventry.ac.uk International Journal of General Systems, Vol. 35, No. 3, June 2006, 295–311