Adaptive fixed-time trajectory tracking control for Mars entry vehicle

Abstract

This paper develops a fixed-time trajectory tracking control scheme for Mars entry vehicle under uncertainty. First, a novel fixed-time nonsingular terminal sliding mode (FTNTSM) surface with bounded convergence time independent on the initial conditions is developed, which not only averts the singularity problem but also assures fast convergence. Second, based on the FTNTSM surface and adaptive technique, a continuous adaptive fixed-time nonsingular terminal sliding mode control (AFTNTSMC) method is proposed. Under the control scheme, the fixed-time convergence of tracking error is assured, and the chattering phenomenon is alleviated. Furthermore, by estimating the square of upper bound of the uncertainty, the designed AFTNTSMC method averts the use of boundary layer technique as imposed in most existing literature on adaptive fixed-time control. The effectiveness of the developed control approach is confirmed by numerical simulations.

Appendix A

Proof of Lemma 6

Since $0< \tau =p_1/q_1 < 1$, $p_1$ and $q_1$ are positive odds, it follows that $\left| {x - {\tilde{x}}} \right| \left| {{x^{\frac{p_1}{q_1}}} - {{{\tilde{x}}}^{\frac{p_1}{q_1}}}} \right| = \left( {x - {\tilde{x}}} \right) \left( {{x^{\frac{p_1}{q_1}}} - {{{\tilde{x}}}^{\frac{p_1}{q_1}}}} \right)$. Based on Lemma 3, it can be obtained that

$$\begin{aligned} {\tilde{x}}{{\hat{x}}^{\frac{p_1}{q_1}}}&\le \frac{1}{{1 + p_1/q_1}}\left[ {{x^{1 + p_1/q_1}} - {{(x - {\tilde{x}})}^{1 + p_1/q_1}}} \right] \nonumber \\&\le \frac{q_1}{{p_1 + q_1}}\Big ({x^{\frac{{p_1 + q_1}}{q_1}}} - {2^{\frac{p_1}{q_1} - 1}} ( {x - {\tilde{x}}} ) \Big ( {{x^{\frac{p_1}{q_1}}} - {{{\tilde{x}}}^{\frac{p_1}{q_1}}}} \Big ) \Big ) \nonumber \\&= \frac{q_1}{{p_1 + q_1}}\Big ({x^{\frac{{p_1 + q_1}}{q_1}}} - {2^{\frac{p_1}{q_1} - 1}}{x^{\frac{{p_1 + q_1}}{q_1}}} + {2^{\frac{p_1}{q_1} - 1}}x{{{\tilde{x}}}^{\frac{p_1}{q_1}}}\nonumber \\&\quad + {2^{\frac{p_1}{q_1} - 1}}{\tilde{x}}{x^{\frac{p_1}{q_1}}} - {2^{\frac{p_1}{q_1} - 1}}{{{\tilde{x}}}^{\frac{{p_1 + q_1}}{q_1}}}\Big ). \end{aligned}$$

(35)

In view of Lemma 5, inequality (35) is transformed into

$$\begin{aligned}&{\tilde{x}}{{\hat{x}}^{\frac{p_1}{q_1}}} \le \frac{q_1}{{p_1 + q_1}} \Big ({x^{\frac{{p_1 + q_1}}{q_1}}} - {2^{\frac{p_1}{q_1} - 1}}{x^{\frac{{p_1 + q_1}}{q_1}}} - {2^{\frac{p_1}{q_1} - 1}}{{{\tilde{x}}}^{\frac{{p_1 + q_1}}{q_1}}} \nonumber \\&\quad + \,\frac{q_1}{{p_1 + q_1}}{\left( {2^{\frac{p_1}{q_1} - 1}}{\tilde{x}}\right) ^{\frac{{p_1 + q_1}}{q_1}}} + \frac{q_1}{{p_1 + q_1}}{\left( {2^{ - \frac{{{{(p_1 - q_1)}^2}}}{{{q_1^2}}}}}x\right) ^{\frac{{p_1 + q_1}}{q_1}}}\nonumber \\&\quad +\, \frac{p_1}{{p_1 + q_1}}{x^{\frac{{p_1 + q_1}}{q_1}}} + \frac{p_1}{{p_1 + q_1}}{\left( {2^{\frac{p_1}{q_1} - 1}}{\tilde{x}}\right) ^{\frac{{p_1 + q_1}}{q_1}}} \Big ) \nonumber \\&= \frac{q_1}{{p_1 + q_1}}\left[ {{{\left( {2^{\frac{p_1}{q_1} - 1}}\right) }^{\frac{{p_1 + q_1}}{q_1}}} - {2^{\frac{p_1}{q_1} - 1}}} \right] {{{\tilde{x}}}^{\frac{p_1}{q_1} + 1}} + \frac{{\bar{l}}_2}{2}{x^{\frac{p_1}{q_1} + 1}} \nonumber \\&= - \frac{{\bar{l}}_1}{2}{{{\tilde{x}}}^{\tau + 1}} + \frac{{\bar{l}}_2}{2}{x^{\tau + 1}}. \end{aligned}$$

(36)

where ${\bar{l}}_2=\frac{2}{{\tau + 1}}\left[ {1 - {2^{\tau - 1}} + \frac{\tau }{{\tau + 1}} + \frac{1}{{\tau + 1}}{2^{ - {{(\tau - 1)}^2}(\tau + 1)}}} \right]$ and ${\bar{l}}_1= \frac{2}{{\tau + 1}}\left[ {{2^{\tau - 1}}- {2^{(\tau - 1)(\tau + 1)}}} \right]$. Moreover, it is noting that $(\tau - 1) - (\tau - 1)(\tau + 1) > 0$ and $1-2^{\tau -1}>0$, thus both ${\bar{l}}_1$ and ${\bar{l}}_2$ are positive constants. $\square$