This paper proposes two methods for nonlinear observer design which are based on a partial nonlinear observer canonical form (POCF). Observability and integrability existence conditions for the new POCF are weaker than the well-established nonlinear observer canonical form (OCF), which achieves exact error linearization. The proposed observers provide the global asymptotic stability of error dynamics assuming that a global Lipschitz and detectability-like condition holds. Examples illustrate the advantages of the approach relative to the existing nonlinear observer design methods. The advantages of the proposed method include a relatively simple design procedure which can be broadly applied.
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The Chern-Moser (CM) normal form of a real hypersurface in $ℂ^N$ can be obtained by considering automorphisms whose derivative acts as the identity on the complex tangent space. However, the CM normal form is also invariant under a larger group (pseudo-unitary linear transformations) and it is this property that makes the CM normal form special. Without this additional restriction, various types of normal forms occur. One of them helps to give a simple proof of a (previously complicated) theorem about triviality of the scalar linear type isotropy subgroup of a nonquadratic hypersurface. An example of an analogous nontrivial subgroup for a 2-codimensional CR surface in $ℂ^4$ is constructed. We also consider the question whether the group structure that is induced on the family of normalisations to the CM normal form via the parametrisation of the isotropy automorphism group of the underlining hyperquadric coincides with the natural composition operation on the biholomorphisms.
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The paper consists of two parts. In the first part, new canonical forms are defined for singular 1D linear systems and a procedure to determine nonsingular matrices transforming matrices of singular systems to their canonical forms is derived. In the second part new canonical forms of matrices of the singular 2D Roesser model are defined and a procedure for determining realisations in canonical forms for a given 2D transfer function is presented. Necessary and sufficient conditions for the existence of a pair of nonsingular block diagonal matrices transforming the matrices of the singular 2D Roesser model to their canonical forms are established. A procedure for computing the pair of nonsingular matrices is presented.
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