The problem of observer design for nonlinear,
deterministic, finite-dimensional systems is discussed. We initially
provide a survey of existing results for linear and nonlinear systems,
pointing out open problems and directions of research. We then focus on
the problem of global observer design for nonlinear systems,
highlighting the role of normal forms, of input-state-output
properties, of invariant manifolds and of an extended notion of
homogeneity. The classical results in [5], [6] and Luenberger's ideas
[9], [10], with their nonlinear enhancement proposed in [8] and further
developed in [12], [1], are the point of departure for the construction
of global observers.
We discuss in detail three global design methodologies. The design in
[4], applicable to uniformly observable systems, exploits the notion of
output-to-state-stability to construct a state norm estimator which is
used to tune on-line a high-gain observer. This design guarantees
global convergence of the estimation error within the domain of
definition of the trajectories. The design in [7], applicable to
general nonlinear systems exploits Luenberger's ideas (and their
nonlinear counterpart) and the notion of invariant manifold to
construct globally convergent (adaptive) state and parameter
estimators. The design in [2], applicable to systems in triangular
forms, relies upon the newly developed notion of homogeneity in the
bi-limit, and uses observer backstepping/ and dynamic scaling to
recursively design globally convergent observers. The applicability of
these global observer designs to the problem of global output feedback
stabilization is also discussed [11], [2], [3].
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