A concept of local observability

Observability is a fundamental system-theoretic property which reflects the possibility of estimating internal states on the basis of input/output data. In the context of nonlinear systems, various possible ways of formalizing this property give rise to different, nonequivalent, notions. These notions differ mainly in the choice of inputs used in testing observability (see for instance a discussion in the introduction to [lo]) or in the type of local behavior desired (see [5]). For continuous-time systems the fundamental global observability results are found in [13,14], where the necessary techniques are developed in the context of the minimal realization problem. Here we shall be interested more in focal observability: intuitively, distinguishing states only from their neighbors. In that area, by far the most important work is that of Hermann and Krener in [5] (see [3] for related results), who introduced and compared various notions of local observability and obtained Lie-theoretic characterizations of some of these. The best results relate to ‘local weak observability’, which we shall review below. In what follows, we refer to this later concept simply as HK-observability. When dealing with problems of input/output stabilization [ll], where observation and control