Abstract
This chapter introduces adaptive \(\lambda \)-tracking control for minimum-phase systems with relative degree one (see Definition 8.1 of class \(\mathcal {S}_1\)) and with relative degree two (see Definition 8.8 of class \(\mathcal {S}_2\)). The considered systems have a known sign of the high-frequency gain and are subject to nonlinear but sector-bounded functional perturbations.
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Notes
- 1.
A system of form (8.54) is said to have relative degree r at a point \(\varvec{x}^{\star } \in \mathbb {R}^n\) if the following two conditions hold: (i) \(L_{\varvec{g}}L_{\varvec{f}}^{i}h(\varvec{x}) = 0\) for all \(\varvec{x}\) in a neighborhood of \(\varvec{x}^{\star }\) and all \(i \in \{1,\ldots r-1\}\) and (ii) \(L_{\varvec{g}}L_{\varvec{f}}^{r-1}h(\varvec{x}) \ne 0\) (see [185, p. 137]) where \(L_{\varvec{f}}^{i}h(\varvec{x}):= (\partial L_{\varvec{f}}^{i-1}h(\varvec{x})/\partial \varvec{x})\varvec{f}(\varvec{x})\) represents the i-th Lie derivative of \(h(\cdot )\) along \(\varvec{f}(\varvec{x})\) (see [203, pp. 509, 510]). If both conditions hold for any \(\varvec{x}^{\star } \in \mathbb {R}^n\), the relative degree r is globally defined.
- 2.
Let \(n\in \mathbb {N}\). A function \(\varvec{\varPhi }:\, \mathbb {R}^n \rightarrow \mathbb {R}^n\) is called a global diffeomorphism if (i) \(\varvec{\varPhi }\) is invertible, i.e. \(\varvec{\varPhi }^{-1}(\varvec{\varPhi }(\varvec{x})) = \varvec{x}\) for all \(\varvec{x} \in \mathbb {R}^n\) and (ii) \(\varvec{\varPhi }(\cdot )\), \(\varvec{\varPhi }(\cdot )^{-1} \in \mathcal {C}^{\infty }(\mathbb {R}^n;\mathbb {R}^n)\) (see [185, p. 11]).
- 3.
ITAE is used in favor of integral squared error (ISE) or integral absolute error (IAE) (see [217, p. 218]) due to time-weighting: Non-zero errors at future times have greater influence on the performance measure than those at earlier times.
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Hackl, C.M. (2017). Adaptive \(\lambda \)-Tracking 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_8
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DOI: https://doi.org/10.1007/978-3-319-55036-7_8
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