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Notes
- 1.
Called “all-pass” since a single-input single-output linear system is inner if and only if its transfer function has amplitude 1 for all frequencies.
- 2.
Many of the subsequent developments continue to hold for any \(C^1\) function V satisfying (9.62) (not necessarily \(\ge 0\)). However, in this case the factor system \(\Theta \) will not be inner anymore; but only cyclo-conservative with respect to the supply rate \(\frac{1}{2}\Vert \bar{y}\Vert ^2 - \frac{1}{2}\Vert y\Vert ^2\).
- 3.
If \((x_e,u_e)\) is an asymptotically stable steady state then \(||h(x_e,u_e)||\) is the steady-state value of the norm of the output y for a step input with magnitude \(u_e\).
- 4.
In this case \(\overline{\Sigma }^*\) is called an outer system: asymptotically stable with asymptotic stable zero-output constrained dynamics.
- 5.
I thank Gjerrit Meinsma for an illuminating discussion on the derivation of the Smith predictor.
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van der Schaft, A. (2017). Factorizations of Nonlinear Systems. 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_9
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DOI: https://doi.org/10.1007/978-3-319-49992-5_9
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