Nonlinear observers for Lipschitz continuous systems with inputs

Abstract

This paper considers the state observation problem for nonlinear dynamical systems. The proposed framework is a direct generalization of a method introduced in a recent paper for autonomous system. Its characteristic feature is that the dynamic part of the observer is linear and, as a consequence, that convergence takes place globally in the observer coordinates. The observer is completed by a static nonlinearity which maps the observer state in the original state space. An associated observation mapping is introduced and is interpreted in terms of an orthonormal expansion of the input and the output with respect to a certain basis in a suitable Hilbert space. It is shown that, by choosing the observer dimension properly, an observer with arbitrary small asymptotic observation error is obtained, provided that some compactness properties for the subset to be observed and the set of input signals hold. Under a stronger condition, the finite complexity property, an exact observer is achieved. Finally, an integral formula representation for the observer nonlinearity is given.

Luenberger Observers for Nonautonomous Nonlinear Systems

“…As far as we know, no result concerning this problem exists in the literature apart from [14], [15] which follows and extends [16]. The idea pursued in [14] belongs to the first path : the transformation is stationary and the input is seen as a disturbance which must be small enough.…”

“…As far as we know, no result concerning this problem exists in the literature apart from [14], [15] which follows and extends [16]. The idea pursued in [14] belongs to the first path : the transformation is stationary and the input is seen as a disturbance which must be small enough. Although the construction is extended in a cunning fashion to a larger class of inputs, namely those which can be considered as output of a linear generator model with small external input, this approach remains theoretic and restrictive.…”

Observer design for continuous-time dynamical systems

“…Such a problem can be recast as the problem of state-observation for dynamical systems (1) that can be put into the form ( 54), (55), in which the function ϕ n z is possibly tdependent. Following the prescriptions of Section 6.2, one can simply design a high-gain observer of the form ż = Aẑ + D K(y − Cẑ) (87) in which the copy of the term ϕ n z is ignored in the observer dynamics. The observer ( 87) is also often denoted as dirtyderivative observer, see, e.g., [225].…”

“…After first steps in [202,230] for linear time-varying systems, an extension of the Luenberger design to nonlinear controlled systems was considered in [87], following the ideas of [137]. In [87], injectivity of a time-varying transformation T is proved only under a so-called "finite-complexity" assumption, originally introduced in [137] for autonomous systems. Unfortunately, this property is very restrictive and hard to check.…”