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Robust Adaptive Finite-Time Stabilization of Nonlinearly Parameterized Nonholonomic Systems

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Abstract

This paper investigates the problem of adaptive finite-time stabilization of nonlinearly parameterized nonholonomic systems. By skilly using the parameter separation, input-state-scaling, and adding a power integrator techniques, an adaptive state feedback controller is obtained. Based on switching strategy to eliminate the phenomenon of uncontrollability, the proposed controller can guarantee that the system states globally finite-time converge to the origin, while other signals remain bounded. Simulation examples demonstrate the effectiveness and the robust features of the proposed approach.

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Acknowledgements

This work has been supported in part by National Nature Science Foundation of China under Grant 61073065 and Nature Science Foundation of Henan Province under Grant 092300410145.

The author would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper.

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Correspondence to Fangzheng Gao.

Appendix

Appendix

Proof of Proposition 3

Recall that \(\xi_{i}=z_{i}^{1/q_{i}}-z_{i}^{*1/q_{i}}\) and \(z_{i}^{*}=\xi_{i-1}^{q_{i}}\beta_{i-1}\). By Lemma 3, for i=2,…,k,

$$ | z_{i}| \leq \bigl| \xi_{i}+z_{i}^{*1/q_{i}}\bigr|^{q_{i}} \leq| \xi_{i}|^{q_{i}}+|\xi_{i-1}|^{q_{i}}| \beta_{i-1}| $$
(66)

Using (34), Lemma 4 and 0<q k <⋯<q 2<q 1=1, we have

(67)

where \(\bar{\gamma}_{k}\) is a nonnegative smooth function. □

Proof of Proposition 4

From Proposition 1, we have

(68)

Moreover, it follows from (34) that

(69)

where D k,1 is a nonnegative smooth function.

The equation \(z_{k}^{*1/q_{k}}=\xi_{k-1}\beta_{k-1}^{1/q_{k}}\) implies that \(z_{k}^{*1/q_{k}}\) is a smooth function and

$$ \everymath{\displaystyle} \begin{array}{l@{\quad}l} \frac{\partial z_{k}^{*1/q_{k}}}{\partial x_{0}} (x_{0},0,\ldots,0, \hat{\varTheta}_{0},\hat{\varTheta})=0, & \frac{\partial^{2} z_{k}^{*1/q_{k}}}{\partial x_{0}^{2}}(x_{0},0,\ldots,0,\hat{\varTheta}_{0},\hat{ \varTheta})=0; \\\noalign{\vspace{6pt}} \frac{\partial z_{k}^{*1/q_{k}}}{\partial\hat{ \varTheta}_{0}}(x_{0},0,\ldots,0,\hat{\varTheta}_{0}, \hat{\varTheta})=0 & \frac{\partial^{2}z_{k}^{*1/q_{k}}}{\partial\hat{ \varTheta}_{0}^{2}}(x_{0},0,\ldots,0, \hat{\varTheta}_{0},\hat{\varTheta})=0 \end{array} $$
(70)

Base on the Mean Value Theorem and Lemma 4, we arrive at

$$ \biggl| \frac{\partial W_{k}}{\partial x_{0}}\dot{x}_{0}+\frac{\partial W_{k}}{\partial\hat{ \varTheta}_{0}} \dot {\hat{ \varTheta}}_{0}\biggr| \leq\frac{1}{8} \sum_{i=1}^{k-1}\xi_{i}^{d}+ \xi_{k}^{d}D_{k,2}\varTheta $$
(71)

where D k,2 is a smooth nonnegative function.

Then, combining (69) and (71) yields

$$ \Biggl| \sum_{i=1}^{k-1} \frac{\partial W_{k}}{\partial z_{i}}\dot{z}_{i}+\frac{\partial W_{k}}{\partial x_{0}}\dot{x}_{0}+ \frac{\partial W_{k}}{\partial\hat{ \varTheta}_{0}}\dot {\hat{ \varTheta}}_{0} \Biggr| \leq\frac{1}{4}\sum _{i=1}^{k-1}\xi_{i}^{d}+ \xi_{k}^{d}\bar{\rho}_{k}\varTheta $$
(72)

where \(\bar{\rho}_{k}\geq D_{k,1}+D_{k,2}\) is a smooth nonnegative function. □

Proof of Proposition 5

Since

(73)

by (36) and (68), we have

$$ \biggl| \frac{\partial W_{k}}{\partial\hat{\varTheta}}\biggr| | \varPsi_{k}| \leq\frac{1}{4}\sum_{i=1}^{k-1} \xi_{i}^{d}+ \xi_{k}^{d}D_{k,3} $$
(74)

Combining (73) and (74) yields

where \(\tilde{\phi}_{k}\geq D_{k,3}+(\bar{\gamma}_{k}+\bar{\rho}_{k})\eta_{k-1} \) is a nonnegative smooth function. □

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Gao, F., Shang, Y. & Yuan, F. Robust Adaptive Finite-Time Stabilization of Nonlinearly Parameterized Nonholonomic Systems. Acta Appl Math 123, 157–173 (2013). https://doi.org/10.1007/s10440-012-9759-2

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