The Frobenius theorem of differential geometry provides necessary and sufficient conditions for a distribution to be completely integrable. This has important implications on the design of controllers and observers for nonlinear systems. In this presentation we go through all necessary definitions in order to state the Frobenius theorem. These are the notions of a Lie bracket, distribution, involutive and completely integrable distribution, codistribution and annihilator of a distribution. We then state the Frobenius theorem and explain its logical depth.
The local inner triangular decomposition of a nonlinear system is the basis for our analysis. We introduce the notion of a state observer as a separate dynamical system whose role is to reconstruct the state of a given system using its output. This way we arrive at the Observer Linearization problem. Its solvability is tackled using the Frobenius theorem. Then, the observer canonical form of the nonlinear system is used to design a state observer, that is to solve the observer linearization problem.
Finally we introduce the design approach known as Extended Linearization - somewhat dual to State feedback linearization of a control system.
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Download the PDF file of the presentation from http://users.ntua.gr/chvng/ac2/obs.pdf.
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