Partial Observers and Partial Synchronization

Traditionally, in engineering science, observer techniques most often deal with control problems. However, the potential of the theory on nonlinear observers [Njimejier & Mareels, 1997; Krener & Respondek, 1983; Xia & Gao, 1989] lies far beyond the area of control applications. Let us discuss a naive but illustrative example. Suppose a physician can monitor through a set of sensors some important characteristics of a patient, such as temperature or blood pressure. Suppose further that an important piece of information about some other characteristics (e.g. glucose concentration in the blood) is not available. Following an observer’s point of view, to design an online estimator for unknown characteristics we should design an observer for a system representing a human model. It is intuitively clear, however, that to solve this problem we do not need any information about the position of the ankle of the patient. All we need is a partial model because we do not need to estimate the whole state. A design problem of this sort constitutes what we call the partial observation problem. It is clear that the problem which we discuss is different from a reduced observer problem. The observer problem is related to the synchronization problem (see, e.g. [Nijmeijer & Mareels, 1997]). Synchronization of chaos is a topic that recently attracted much attention from various branches of the scientific community [Fujisaka & Yamada, 1983; Afraimovich et al., 1986; Pecora & Carroll, 1990]. While the effect of synchronization (of periodic systems) was known to Huijgens centuries ago [Huijgens, 1673] and since then has been intensively studied, there is still something astonishing about the fact that two chaotic and therefore unstable systems can produce coherent oscillations [Landa, 1996]. Its interest lies in many disciplines, such as statistical mechanics, because of the possible development of attraction properties that can be evaluated only on average, biology and population ecology, for the analysis of coherence in reproduction cycles of interacting populations, and engineering mechanics and electronics, because of possible applications in the design of tracking systems or secure communication devices. The interested reader can find more details on this topic in the following works [Blekhman, 1988; Pecora et al., 1997]. Great interest in the topic of chaotic synchronization started after the publication by Pecora