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

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.

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. $\square$