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Nonlinear Observability and the Observer Design Problem

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Observer Design for Nonlinear Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 479))

Abstract

This preliminary chapter defines the problem of observer design for nonlinear systems and presents some basic notions of observability which will be needed throughout the book. It also formalizes and justifies the observer design methodology adopted in this book, which consists in looking for a reversible change of coordinates transforming the expression of the system dynamics into a target normal form, designing an observer in those coordinates, and finally deducing an estimate of the system state in the initial coordinates via inversion of the transformation.

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Notes

  1. 1.

    The need for differentiability and its order will vary locally throughout the book.

  2. 2.

    Again, this causality condition may be removed if the whole trajectory of u is explicitly known, for instance, in the case of a time-varying system where \(u(t)=t\) for all t.

  3. 3.

    We say “any solution” because \(\mathscr {F}\) being only continuous, there may be several solutions. This is not a problem as long as any such solution verifies the required convergence property.

  4. 4.

    The expression of the dynamics under the form \(F(\xi ,u,H(\xi ,u))\) can appear strange and abusive at this point because it is highly non-unique, and we should rather write \(F(\xi ,u)\). However, we will see in Part I how specific structures of dynamics \(F(\xi ,u,y)\) allow the design of an observer (1.4).

  5. 5.

    Denoting \(T^{-1}_{u,t,j}\) the jth component of \(T^{-1}_{u,t}\), take \(T^{-1}_{u,t,j}(\xi )=\min _{\tilde{\xi }\in T_u(\mathscr {X},t)} \{ T^{-1}_{u,t,j}(\tilde{\xi })+\rho (|\tilde{\xi }-\xi |)\}\) or equivalently \(T^{-1}_{u,t,j}(\xi )=\min _{x\in \mathscr {X}} \left\{ x_j+\rho (|T_u(x,t)-\xi |)\right\} \)

  6. 6.

    A function \(\gamma \) is uniformly continuous if and only if \(\lim _{n\rightarrow +\infty }|x_n-y_n|=0\) implies \(\lim _{n\rightarrow +\infty }|\gamma (x_n)-\gamma (y_n)|=0\). This property is indeed needed in the context of observer design.

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Authors and Affiliations

  1. Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi”, University of Bologna, Bologna, Italy

    Pauline Bernard

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  1. Pauline Bernard
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Correspondence to Pauline Bernard .

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Bernard, P. (2019). Nonlinear Observability and the Observer Design Problem. In: Observer Design for Nonlinear Systems. Lecture Notes in Control and Information Sciences, vol 479. Springer, Cham. https://doi.org/10.1007/978-3-030-11146-5_1

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