Abstract
In this chapter, we briefly discuss the basic notions of input–output stability for nonlinear systems described by input–output maps. Also the stability of input–output systems in standard feedback closed-loop configuration is treated.
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- 1.
Throughout we will identify functions which are equal except for a set of Lebesgue measure zero. Thus conditions imposed on functions are always to be understood in the sense of being valid for all \(t\in \mathbbm {R}^{+}\) except for a set of measure zero.
- 2.
A function \(f:\mathbbm {R}^{+}\rightarrow \mathbbm {R}\) is measurable if it is the pointwise limit (except for a set of measure zero) of a sequence of piecewise constant functions on \(\mathbbm {R}^{+}\).
- 3.
Note the abuse of notation, with, e.g., \(u \in U\) denoting the value of the input and on the other hand \(u \in L_{qe}(U)\) denoting a time function \(u: \mathbbm {R}^+ \rightarrow U\).
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van der Schaft, A. (2017). Nonlinear Input–Output Stability. In: L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-49992-5_1
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DOI: https://doi.org/10.1007/978-3-319-49992-5_1
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49991-8
Online ISBN: 978-3-319-49992-5
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