Observability and observers for nonlinear systems with time delays

In modern control theory, it has been a common practice to use ordinary differential equations to model dynamic systems. However, most of real systems have delays associated -normally introduced by natural response time (of sensors, actuators, etc, . . .) , and transport (of information, mass, etc, . . .) phenomenaand a more accurate model would require the use of functional-differential equations. A lot of examples have been reported in the literature in a wide variety of applications like obtaining models in biological sciences (ecology, epidemiology, immunology, physiology, . . . [see, e.g. [3] and references therein]), engineering (cold rolling mills, artificial neural networks, optimal control of flow [traffic, water resources systems], shunted transmission lines, information transmission over the internet, . . . [see, e. g. [9, 11, 17] and references therein]) chemistry [2, 10, 16], just to name some. These systems which are referred to as time-delay systems have been attracting an increasing number of researchers in recent years. In fact, some general results are already available specially in the linear case. Some results are also available when dealing with nonlinear time-delay systems in particular for control problems like feedback linearization, disturbance decoupling, and noninteracting control see contributions in [6, 18, 21]. However, the proposed solutions assume that the full state variables are available for measurement and can be used in the control design. Such a request is seldom fulfilled in practice and one issue then consists in the design of a nonlinear observer. For linear time-delay systems, the observability has been widely investigated [5, 15, 22]. However, in the nonlinear case, and despite a couple of contributions [1, 7], this important problem remains open.