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Finite-time containment control without velocity and acceleration measurements

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Abstract

This paper studies the distributed finite-time containment control for a group of mobile agents modeled by double-integrator dynamics under multiple dynamic leaders with bounded unknown acceleration inputs. A class of distributed finite-time containment protocols is proposed without relying velocity and acceleration measurements. This kind of protocols can drive the states of the followers to track the convex hull spanned by those of the leaders in finite time under the constraint that the leaders’ acceleration inputs are unknown but bounded for all the followers. Further, by computing the value of the Lyapunov function at the initial point, the finite settling time can also be theoretically estimated for the second-order finite-time containment control problems. Finally, the effectiveness of the results is illustrated by numerical simulation.

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Acknowledgments

The authors would like to thank the editor and the anonymous reviewers for their valuable comments and constructive suggestions. This work was supported by the National Science Foundation of China under Grants 61225013, 11332001.

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Authors and Affiliations

  1. School of Automation, Northwestern Polytechnical University, Xi’an, 710072, China

    Yu Zhao

  2. State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871, China

    Zhisheng Duan

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  1. Yu Zhao
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Correspondence to Yu Zhao.

Appendix

Appendix

The proof of Lemma 4

By choosing a symmetric matrix \(P={\left( \begin{array}{c@{\quad }c} 2y &{} -2 \\ -2 &{} 1 \\ \end{array} \right) }\), \(P\) is positive definite if \(y>2\). Then, substituting \(P\) into (3), one has

$$\begin{aligned} Q=\left( \begin{array}{c@{\quad }c} 2l_1y-4l_2-4-\varepsilon ^2 &{} \;\;\;\; -l_1+l_2-y+2 \\ -l_1+l_2-y+2&{} 1 \\ \end{array} \right) . \end{aligned}$$

Thus, \(Q>0\) if and only if there exists \(y>2\) such that \(2l_1y-4l_2-4-\varepsilon ^2-(-l_1+l_2-y+2)^2>0\), which are equivalent to

$$\begin{aligned}&y>2, \end{aligned}$$
(33)
$$\begin{aligned}&y^2-2(l_2+2)y+(l_1-l_2-2)^2+4l_2+4+\varepsilon ^2<0. \end{aligned}$$
(34)

The conditions (33) and (34) are satisfied if and only if

$$\begin{aligned}&l_2+2+\sqrt{-l_1^2+2(l_2+2)l_1-4l_2-4-\varepsilon ^2}>2, \end{aligned}$$
(35)
$$\begin{aligned}&\Delta _y\triangleq -l_1^2+2(l_2+2)l_1-4l_2-4-\varepsilon ^2>0. \end{aligned}$$
(36)

It is easy to see that (35) holds if and only if (36) and \(l_2>0\). Note that (36) is satisfied if and only if

$$\begin{aligned}&\Delta _{l_1}\triangleq l_2^2-\varepsilon ^2>0, \end{aligned}$$
(37)
$$\begin{aligned}&l_2+2-\sqrt{l_2^2-\varepsilon ^2}<l_1<l_2+2+ \sqrt{l_2^2-\varepsilon ^2}, \end{aligned}$$
(38)

which are equivalent to conditions (5) and (6). Thus, there exists a positive definite matrix \(P\) such that (3) is feasible if (5) and (6) hold.\(\square \)

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Zhao, Y., Duan, Z. Finite-time containment control without velocity and acceleration measurements. Nonlinear Dyn 82, 259–268 (2015). https://doi.org/10.1007/s11071-015-2154-6

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  • DOI: https://doi.org/10.1007/s11071-015-2154-6

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