Zusammenfassung
In vielen Anwendungen der Regelungstechnik werden Beobachter eingesetzt, um interne Zustandsgrößen oder Parameter zu schätzen oder Fehler zu detektieren. Solche Beobachter lassen sich auch für nichtlineare Systeme systematisch auf Basis der Beobachter- oder der Beobachtbarkeitsnormalform entwerfen. Letztere existiert für eine größere Systemklasse. Allerdings ist das Vektorfeld in der Beobachtbarkeitsnormalform nicht unbedingt an allen Punkten definiert oder Lipschitz-stetig, selbst wenn diese Eigenschaften auf die ursprüngliche Systemdarstellung zutreffen. Durch die Einbettung in höherdimensionale Räume ist es möglich, die Normalform einerseits zu konstruieren und gegebenenfalls gewisse singuläre Punkte zu vermeiden. In diesem Beitrag wird gezeigt, wie dies systematisch für polynomiale Systeme mit mehreren Ein- oder Ausgängen bewerkstelligt werden kann.
Abstract
Observers are used in a variety of control applications. This includes estimating a systems state, system parameters, or fault detection. Systematic observer design is applicable on basis of the observer- or observability normal form. While the former normal form is preferable because of the easier observer design, it exists for a smaller subset of dynamical systems than the latter one. For nonlinear systems the vector field in the observability normal form may possess singularities or may fail a Lipschitz condition. This can sometimes be avoided by embedding the system in a higher-dimensional state space. In this contribution this embedding and its implications are discussed for polynomial multiple input or output systems.
Über die Autoren
Daniel Gerbet ist wissenschaftlicher Mitarbeiter am Institut für Regelungs- und Steuerungstheorie an der Fakultät Elektrotechnik und Informationstechnik der Technischen Universität Dresden. Er beschäftigt sich mit dem Reglerentwurf durch Methoden aus der algebraischen Geometrie und der Quantorenelimination.
Prof. Klaus Röbenack ist Direktor des Instituts für Regelungs- und Steuerungstheorie an der Fakultät Elektrotechnik und Informationstechnik der Technischen Universität Dresden. Seine Arbeitsgebiete umfassen den Entwurf nichtlinearer Regler und Beobachter sowie das wissenschaftliche Rechnen.
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Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) — 417698841.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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