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Barrier Lyapunov functions-based fixed-time stabilization of nonholonomic systems with unmatched uncertainties and time-varying output constraints

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Abstract

The fixed-time stabilization problem is addressed in this paper for a kind of nonholonomic systems in chained form with unmatched uncertainties and time-varying output constraints. A novel tan-type barrier Lyapunov function is introduced to deal with time-varying output constraints. Under the unified framework of the considered system with and without output constraints, a state feedback controller is designed with the aid of adding a power integrator technique and switching control strategy. It is shown that the suggested controller ensures the states of closed-loop system to zero in a given fixed time without disobeying the constraints. Finally, simulation results are given to confirm the efficacy of the presented control scheme.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China under Grants 61873120, 61673243 and 61703232, the young teacher training plan of colleges and universities of Henan Province under Grant 2019GGJS192, the Natural Science Foundation of Jiangsu Higher Education Institutions of China under Grant 18KJB510016, the High-level Talent Initial Funding of Nanjing Institute of Technology under Grant YKJ201824 and the Qing Lan project of Jiangsu Province.

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Correspondence to Fangzheng Gao.

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Appendix

Appendix

Proof of (33)

Since \(0<r_{i}<1 \), by Lemma 2.4, one has

$$\begin{aligned} \displaystyle |x_{i}-x_{i}^{*}| \le 2^{1-r_{i}}\Big | [x_{i}]^{\frac{1}{r_{i}}}-[x_{i}^{*}]^{\frac{1}{r_{i}}}\Big |^{r_{i}}=2^{1-r_{i}}|\xi _{i}|^{r_{i}}.\nonumber \\ \end{aligned}$$
(63)

It can be directly deduced from (63) and Lemma 2.3 that

$$\begin{aligned} \displaystyle \lceil \xi _{i-1}\rceil ^{2-\tau -r_{i-1}}c_{0}^{*}(x_{i}-x_{i}^{*}) \le \frac{1}{3}|\xi _{i-1}|^{2}+|\xi _{i}|^{2}h_{i1}, \end{aligned}$$
(64)

where \(h_{i1} \ge 0\) is a smooth function. \(\square \)

Proof of (34)

From (6), (29) and Lemma 2.2, one has

$$\begin{aligned}&|{\varDelta }_{i}|\le \displaystyle {\bar{\varphi }}_{i}\sum \limits _{j=1}^{i}\left( |\xi _{j}|+\alpha _{j-1}^{\frac{1}{r_{j}}}|\xi _{j-1}|\right) ^{r_{i}+\tau } \nonumber \\&\quad \le \displaystyle {\tilde{\varphi }}_{i}\sum \limits _{j=1}^{i}|\xi _{j}|^{r_{i}+\tau }, \end{aligned}$$
(65)

where \({\bar{\varphi }}_{i}\) and \({\hat{\varphi }}_{i}\) are the nonnegative smooth functions.

Based on (65) and Lemma 2.3, one gets

$$\begin{aligned} \displaystyle [\xi _{i}]^{2-r_{i+1}}|{\varDelta }_{i} \displaystyle \le \frac{1}{3}\sum \limits _{j=1}^{i-1}|\xi _{j}|^{2}+|\xi _{i}|^{2}h_{i2}, \end{aligned}$$
(66)

where \(h_{i2}\ge 0\) is a smooth function. \(\square \)

Proof of (35)

Noting that

$$\begin{aligned} \displaystyle x_{2}^{*}=- \alpha _{1}\lceil \xi _{1}\rceil ^{r_{2}}, \end{aligned}$$
(67)

using the inductive argument, it is easily obtained that

$$\begin{aligned} \displaystyle \lceil x_{i}^{*}\rceil ^{\frac{1}{r_{i}}}=-\sum \limits _{l=1}^{i-1}B_{il}\lceil \xi _{l}\rceil ^{\frac{1}{r_{l}}}, \end{aligned}$$
(68)

for some nonnegative smooth functions \(B_{il}, l=1,\ldots ,i-1\).

Therefore, by (29), (65), (68) and Lemma 2.2, one has

$$\begin{aligned}&\displaystyle \sum \limits _{j=1}^{i-1}\frac{\partial W_{i}}{\partial x_{j}}(c_{0}^{*}x_{j+1}+{\varDelta }_{j}) \nonumber \\&\quad \displaystyle \le (2-r_{i+1})|\xi _{i}|^{1-r_{i+1}}|x_{i}-x_{i}^{*}| \nonumber \\&\qquad \displaystyle \times \sum \limits _{j=1}^{i-1}\left| \frac{\partial \left( \lceil x_{i}^{*}\rceil ^{\frac{1}{r_{i}}}\right) }{\partial x_{j}}\right| |c_{0}^{*}x_{j+1}+{\varDelta }_{j}| \nonumber \\&\quad \displaystyle \le \frac{1}{3}\sum \limits _{j=1}^{i-1}|\xi _{j}|^{2}+|\xi _{i}|^{2}h_{i3}, \end{aligned}$$
(69)

where \(h_{i3} \ge 0\) is a smooth function. \(\square \)

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Yao, H., Gao, F., Huang, J. et al. Barrier Lyapunov functions-based fixed-time stabilization of nonholonomic systems with unmatched uncertainties and time-varying output constraints. Nonlinear Dyn (2020) doi:10.1007/s11071-019-05450-3

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Keywords

  • Nonholonomic systems
  • Time-varying output constraints
  • Barrier Lyapunov function (BLF)
  • Adding a power integrator (API)
  • Fixed-time stabilization