Abstract
In this chapter, funnel control will be introduced for systems of class \(\mathcal {S}_1\) and of class \(\mathcal {S}_2\) and their respective extensions to systems with input saturation and exponentially bounded perturbation (see Definitions 9.2 and 9.13 of system classes \(\mathcal {S}_1^{{{\mathrm{sat}}}}\) and \(\mathcal {S}_2^{{{\mathrm{sat}}}}\), respectively).
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- 1.
Note that the overshoot can be reduced by accelerating the gain adaption, i.e. \(\dot{k}(t)=q_1 d_{\lambda }(|e(t)|)^{q_2}\) with \(q_1 \gg 1\) and/or \(q_2 \gg 2\). Nevertheless, a priori, no upper bound on the output (or the states) can be specified.
- 2.
Note that ‘\( - \psi _1(t) + \tfrac{\delta }{2} \ge 0\) for all \(t \in [t_0,\, t_1]\)’ is possible for e.g. \(0<\psi _1(t)=\lambda _1 < \tfrac{\delta }{2}\) and \(\dot{\psi }_0(t) = \delta \). Clearly, \(\dot{\psi }_1(t) \ge -\dot{\psi }_0(t) + \delta \) also holds true.
- 3.
Albeit not shown, very similar results were obtained for the exponential boundary (9.80) with \(\varLambda _0=1.5\), \(\lambda _0 = 0.1\), \(T_\mathrm{exp}= 0.379\, {\text {s}}\) and \(\lambda _1 = 3 \, \frac{1}{\text {s}}\).
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Hackl, C.M. (2017). Funnel Control. In: Non-identifier Based Adaptive Control in Mechatronics. Lecture Notes in Control and Information Sciences, vol 466. Springer, Cham. https://doi.org/10.1007/978-3-319-55036-7_9
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DOI: https://doi.org/10.1007/978-3-319-55036-7_9
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