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Introduction, Motivation, and Background

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Book cover Tautological Control Systems

Part of the book series: SpringerBriefs in Electrical and Computer Engineering ((BRIEFSCONTROL))

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Abstract

One can study nonlinear control theory from the point of view of applications, or from a more fundamental point of view, where system structure is a key element. From the practical point of view, questions that arise are often of the form, “How can we...”, for example, “How can we steer a system from point \(A\) to point \(B\)?” or, “How can we stabilise this unstable equilibrium point?” or, “How can we manoeuvre this vehicle in the most efficient manner?” From a fundamental point of view, the problems are often of a more existential nature, with, “How can we” replaced with, “Can we”. These existential questions are often very difficult to answer in any sort of generality.

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Notes

  1. 1.

    We understand that there are many ways of formulating system equivalence. But here we are content to be, not only vague, but far from comprehensive.

  2. 2.

    Sussmann actually has a more sophisticated notion of degree, but for this example it boils down to the one we give.

  3. 3.

    The terminology “tautological” arises from two different attributes of our framework. First of all, when one makes the natural connection from our systems to standard control systems, we encounter the identity map (Example 5.2–2). Second, in our framework we prove that the only pure feedback transformation is the identity transformation (cf. Proposition 5.39).

  4. 4.

    A bornology on a set \(\mathcal {S}\) is a family \(\fancyscript{B}\) of subsets of \(\mathcal {S}\), called bounded sets, and satisfying the axioms:

    1. 1.

      \(\mathcal {S}\) is covered by bounded sets, i.e., \(\mathcal {S}=\cup _{B\in \fancyscript{B}}B\);

    2. 2.

      subsets of bounded sets are bounded, i.e., if \(B\in \fancyscript{B}\) and if \(A\subseteq B\), then \(A\in \fancyscript{B}\);

    3. 3.

      finite unions of bounded sets are bounded, i.e., if \(B_1,\dots ,B_k\in \fancyscript{B}\), then \(\cup _{j=1}^kB_j\in \fancyscript{B}\).

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  1. Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

    Andrew D. Lewis

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Lewis, A.D. (2014). Introduction, Motivation, and Background. In: Tautological Control Systems. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-08638-5_1

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