2.1 Preliminaries

The notation $\mathbb R_{>0}$ is referred to the set of positive real numbers, and the notation $I_{n}\in \mathbb R^{n\times n}$ represents the unit matrix. $|{\cdot}|$ is the absolute value of a scalar, while $\left\|\cdot\right\|$ is the Euclidean 2-norm of a vector or the induced 2-norm of a matrix. Given $\imath>0$ and $\ell=\left[ \ell_{1},\ell_{2},\ldots,\ell_{n}\right]^{T}\in\mathbb R^{n}$ , $|\ell|$ , $|\ell|^{\imath}$ , and $\lfloor \ell \rceil$ refer to $|\ell|=\left[ |\ell_{1}|,|\ell_{2}|,\ldots,|\ell_{n}|\right]^{T}$ , $|\ell|^{\imath}=\left[ |\ell_{1}|^{\imath},|\ell_{2}|^{\imath},\ldots,|\ell_{n}|^{\imath}\right]^{T}$ , and $\lfloor \ell \rceil=diag\left\lbrace \ell_{1},\ell_{2},\ldots,\ell_{n}\right\rbrace $ , respectively. Denote $sig^{\imath}\!\left(\ell\right)=\left[ sig^{\imath}\!\left(\ell_{1}\right),sig^{\imath}\!\left(\ell_{2}\right),\ldots,\right.$ $\left.sig^{\imath}\!\left(\ell_{n}\right)\right]^{T}$ , where $sig^{\imath}\!\left(\ell_{i}\right)=|\ell_{i}|^{\imath}sgn\!\left( \ell_{i}\right)\left(i=1,\ldots,n\right)$ , and $sgn\!\left( \cdot\right)$ is the standard signum function given by

(1) \begin{equation} sgn\!\left(x\right)=\left\lbrace \begin{aligned} -1, &\quad if\!\!\quad x<0\\[3pt] 0, &\quad if \!\!\quad x=0\\[3pt] 1, &\quad if \!\!\quad x>0 \end{aligned}. \right.\end{equation}

At this stage, we provide some useful definitions and lemmas used later.

Consider the dynamical system

(2) \begin{equation}\dot{x}=f\!\left(x\!\left(t\right) \right), x\!\left(0\right)=x_{0},\ \ f\!\left(0\right)=0,x\in \mathbb U_{0} \subset \mathbb R^{n},\end{equation}

where x is a state vector, the time variable t varies from 0 to $\infty$ , $\mathbb U_{0}$ is a finite open set containing the origin $x=0$ , and $f({\cdot})\,:\,\mathbb R^{n}\to\mathbb R^{n}$ , well-defined on $\mathbb U_{0}$ , is a continuous differentiable nonlinear vector function.

Lemma 1 (see the work of Sun et al. [Reference Sun, Liu, Qiu and Feng16]). Suppose there exists a Lyapunov function $V\!\left(x\right) $ defined in domain $\mathbb U_{0}$ , and the time derivative of $V\!\left(x\right) $ along the trajectory of system (2) satisfies

(3) \begin{equation} \dot{V}\!\left(x\right)\le-qV^{g}\!\left(x\right)+p, \end{equation}

where $\left\lbrace q, p\right\rbrace\in\mathbb R_{>0}$ , and $0<g<1$ , then the system (2) is finite-time stable.

Lemma 2 (see the work of Sun et al. [Reference Sun, Liu, Qiu and Feng16]). For any $\left\lbrace x,y \right\rbrace\in\mathbb R $ , the following inequality holds:

(4) \begin{equation} \begin{aligned} |x|^{m}|y|^{n}\le&\frac{m}{m+n}s|x|^{m+n}+\frac{n}{m+n}s^{-\frac{m}{n}}|y|^{m+n}, \end{aligned} \end{equation}

where $\left\lbrace m,n,s \right\rbrace\in \mathbb R_{>0}$ ,

Lemma 3 (see the work of Zheng et al. [Reference Zheng, Feroskhan and Sun34]). Let $\xi_{i}\in \mathbb R$ , $i=1,2,\ldots,n$ . Then

(5) \begin{equation} \begin{aligned} &\left(\sum_{i=1}^{n}|\xi_{i}|\right)^\mu \le \sum_{i=1}^{n}|\xi_{i}|^{\mu}\le n^{1-\mu}\!\left(\sum_{i=1}^{n}|\xi_{i}|\right)^\mu &, \mu\in\left(0,1\right], \\ &\sum_{i=1}^{n}|\xi_{i}|^{\mu} \le \left(\sum_{i=1}^{n}|\xi_{i}|\right)^\mu\le n^{\mu-1}\sum_{i=1}^{n}|\xi_{i}|^{\mu} &, \mu\in\left(1,\infty\right). \end{aligned} \end{equation}

Lemma 4 (see the work of Du et al. [Reference Du, Zhang, Wu, Zhu, Li and Chu30]). If $\vartheta_{1}>0$ and $0<\vartheta_{2}\le1$ , then

(6) \begin{equation} |sig^{\vartheta_{1}\vartheta_{2}}\!\left(x\right)-sig^{\vartheta_{1}\vartheta_{2}}\!\left(y\right)|\le2^{1-\vartheta_{2}}|sig^{\vartheta_{1}}\!\left(x\right)-sig^{\vartheta_{1}}\!\left(y\right) |^{\vartheta_{2}}, \forall\left\lbrace x,y\right\rbrace\in \mathbb R. \end{equation}

Lemma 5 (see the work of Du et al. [Reference Du, Zhang, Wu, Zhu, Li and Chu30]). For any $\varsigma\in \mathbb R_{+}$ and $z\in \mathbb R$ , we have

(7) \begin{equation} \frac{d}{dt}|z|^{\varsigma+1}=\left(\varsigma+1\right)sig^{\varsigma}\!\left(z\right)\dot{z}, \frac{d}{dt}sig^{\varsigma+1}\!\left(z\right)=\left(\varsigma+1\right)|z|^{\varsigma}\dot{z}. \end{equation}

2.2 Airship model

Figure 1 displays the stratospheric airship with a typical streamline ballonet. The helium-filled ballonet generates an upward lift for the airship. The cargo bay fixed below the ballonet aims to house the on-board systems. The propulsive units mounted on both sides of the gondola furnish thrust for flight. The control surfaces (elevators and rudders) installed on the tail offer yawing and pitching moments.

Figure 1. Depiction of the stratospheric airship.

To investigate the motion control of the airship, it is reasonable to establish the earth and body-fixed coordinate systems; see Fig. 1. The earth reference frame (ERF) has its origin $o_{g}$ at a fixed point on the earth, the $o_{g}x_{g}$ -axis points north, the $o_{g}y_{g}$ -axis points east, and the $o_{g}z_{g}$ -axis points to the earth’s centre perpendicular to the plane $o_{g}x_{g}y_{g}$ . The body-fixed reference frame (BRF) moving with the airship sets its origin o at the centre of volume (CV), the $ox_{b}$ -axis points to the nose of the airship, the $oy_{b}$ -axis points to the starboard side of the airship, and the $oz_{b}$ -axis lying on the longitudinal axisymmetric plane of the airship normal to the plane $ox_{b}y_{b}$ .

Neglect the aeroelastic influences and regard the airship as a rigid body. Taken from the airship modeling technique [Reference Khoury2, Reference Han, Wang, Chen and Duan9, Reference Wu, Wang, Duan, Xie and Wei10, Reference Xiao, Han, Wang, Zhou and Duan14, Reference Yang15, Reference Zheng and Xie17, Reference Yang, Wu and Zheng22, Reference Azinheira, Moutinho and De Paiva23, Reference Zheng, Feroskhan and Sun34], the airship model built around the horizontal motion can be directly given here, which is formulated by [Reference Zheng, Guan, Ma and Zhu6, Reference Yan, Weidong, Yugeng and Zili18, Reference Yang, Wu and Zheng36–Reference Yang, Wu and Zheng38]

(8) \begin{equation} \begin{aligned} \begin{bmatrix} \dot{x}\\[3pt] \dot{y}\\[3pt] \dot{\psi} \end{bmatrix}&=\begin{bmatrix} \cos\!\left(\psi\right) & \quad -\sin\!\left(\psi\right)&\quad 0\\[3pt] \sin\!\left(\psi\right) &\quad \cos\!\left(\psi\right)& \quad 0\\[3pt] 0&\quad 0& \quad 1 \end{bmatrix} \begin{bmatrix} u\\[3pt] v\\[3pt] r \end{bmatrix}=J\!\left(\psi\right)\nu,\\[3pt] \begin{bmatrix} m_{u}\dot{u}\\[3pt] m_{v}\dot{v}\\[3pt] m_{r}\dot{r} \end{bmatrix}&=\begin{bmatrix} m_{v}vr\\[3pt] -m_{u}ur\\[3pt] m_{uv}uv \end{bmatrix}-\begin{bmatrix} d_{u}u\\[3pt] d_{v}v\\[3pt] d_{r}r \end{bmatrix}+\begin{bmatrix} \tau_{u}\\[3pt] 0\\[3pt] \tau_{r} \end{bmatrix}+\begin{bmatrix} \delta_{u}\\[3pt] \delta_{v}\\[3pt] \delta_{r} \end{bmatrix}.\end{aligned}\end{equation}

In the above equations, x, y are the CV’s positions in the ERF, and $\psi$ is the yaw attitude in the BRF; u, v, r are the surge velocity, lateral velocity and yaw angular velocity with respect to the BRF, respectively; $m_{i}$ and $d_{i}$ ( $i=u,v,r$ ) denote the mass and damping term [Reference Khoury2, Reference Yamada, Adachi and Funahashi19, Reference Funk, Lutz and Wagner39], respectively; $m_{uv}=m_{u}-m_{v}$ ; $\tau=[\tau_{u},\tau_{r}]^{T}$ is the actuating signal. $\delta=[\delta_{u},\delta_{v},\delta_{r}]^{T}$ characterises all the modeling imperfections and disturbances. To facilitate subsequent discussions, some of most fundamental assumptions are delineated below.

Assumption 2 (see other works [Reference Zheng, Guan, Ma and Zhu6, Reference Yamada, Adachi and Funahashi19, Reference Funk, Lutz and Wagner39]). In this work, we suppose that $\delta_{i}$ (i = u,v,r) takes the form

(9) \begin{equation}\left\lbrace\begin{aligned} \delta_{u}=\delta^{\prime}_{u}+m^{\prime}_{v}vr-d^{\prime}_{u}u-m^{\prime}_{u}\dot{u}+\delta_{dis_{u}}\\[4pt] \delta_{v}=\delta^{\prime}_{v}-m^{\prime}_{u}ur-d^{\prime}_{v} v-m^{\prime}_{v}\dot{v}+\delta_{dis_{v}}\\[4pt] \delta_{r}=\delta^{\prime}_{r} +m^{\prime}_{uv} uv-d^{\prime}_{r} r-m^{\prime}_{r}\dot{r}+\delta_{dis_{r}} \end{aligned}\right., \end{equation}

where $\delta^{\prime}_{i}$ is the cross-coupling term, $m^{\prime}_{i}$ and $d^{\prime}_{i}$ are the uncertain part of $m_{i}$ and $d_{i}$ , respectively, and $\delta_{dis_{i}}$ characterises the slow time-varying wind resistance.

2.3 Control objective

In this article, our control objective is to generate a control law for $\tau$ to be employed by the stratospheric airship in underactuated mode, such that, despite the adverse influences of unmeasured velocities and uncertainties, the position of the airship $\eta=[x,y]^{T}$ tracks with the scheduled route $\eta_{d}=[x_{d},y_{d}]^{T}$ with a priori designate performance, i.e. such that the tracking error $e=\eta-\eta_{d}=[x_{e},y_{e}]^{T}$ fulfills

(10) \begin{equation} \max\left\lbrace|x_{e}|, |y_{e}| \right\rbrace \le\epsilon,\forall t\ge T_{f},\end{equation}

where $\epsilon$ represents the preassigned tracking accuracy, and $0<T_{f}<\infty$ denotes the finite settling time. Meanwhile, all the closed-loop signals maintain bounded $\forall t\ge0$ .

4.1 FTO

In this paper, the FTO is formulated by

(21) \begin{equation} \left\lbrace\begin{aligned} \dot{\hat{\varkappa}}_{1}&=\hat{\varkappa}_{2}+p_{1}sig^{r_{1}}\!\left(\varkappa_{1}-\hat{\varkappa}_{1} \right)+q_{1}sig^{z_{1}}\!\left(\varkappa_{1}-\hat{\varkappa}_{1} \right)\\[4pt] \dot{\hat{\varkappa}}_{2}&=R\!\left(\psi\right)M_{\epsilon}\tau+\hat{\delta}_{lu}+p_{2}sig^{r_{2}}\!\left(\varkappa_{1}-\hat{\varkappa}_{1} \right) +q_{2}sig^{z_{2}}\!\left(\varkappa_{1}-\hat{\varkappa}_{1} \right)\\[4pt] \dot{\hat{\delta}}_{lu}&=p_{3}sig^{r_{3}}\!\left(\varkappa_{1}-\hat{\varkappa}_{1} \right) +q_{3}sig^{z_{3}}\!\left(\varkappa_{1}-\hat{\varkappa}_{1} \right)+\Upsilon sign\!\left( \varkappa_{1}-\hat{\varkappa}_{1}\right) \\ \end{aligned}\right.,\end{equation}

where $\hat{\varkappa}_{1}$ , $\hat{\varkappa}_{2}$ , and $\hat{\delta}_{lu}$ are the estimates of $\varkappa_{1}$ , $\varkappa_{2}$ , and $\delta_{lu}$ , respectively. The parameters in Equation (21) satisfy $0<r_{i}<1$ , $z_{i}>1$ , $r_{i}=ir_{0}-\left( i-1\right) $ , $z_{i}=iz_{0}-\left( i-1\right) $ , $i=1,2,3$ , $0<r_{0}<1-\sigma_{1}$ , $0<z_{0}<1+\sigma_{2}$ , $\sigma_{1}\in\mathbb R_{>0}$ and $\sigma_{2}\in\mathbb R_{>0}$ are sufficiently small constants, and $\Upsilon\ge B_{\delta_{lu}}$ . The observer gains are selected to ensure the matrices

(22) \begin{equation} P=\begin{bmatrix} -p_{1}& \quad 1&\quad 0\\[4pt] -p_{2}&\quad 0&\quad 1\\[4pt] -p_{3}&\quad 0&\quad 0 \end{bmatrix} \text{and}\ \ Q=\begin{bmatrix} -q_{3}&\quad 1&\quad 0\\[4pt] -q_{3}&\quad 0&\quad 1\\[4pt] -q_{3}&\quad 0&\quad 0 \end{bmatrix}\end{equation}

are Hurwitz. Based on the above contents, we obtained the main results of the FTO.

Proof of Theorem 1. Define $e_{o_{1}}=\varkappa_{1}-\hat{\varkappa}_{1}$ , $e_{o_{2}}=\varkappa_{2}-\hat{\varkappa}_{2}$ , and $e_{o_{3}}=\delta_{lu}-\hat{\delta}_{lu}$ as observer errors; therefore, together with Equation (18), the observer error dynamics can be computed as

(23) \begin{equation}\left\lbrace \begin{aligned} \dot{e}_{o_{1}}&=e_{o_{2}}-p_{1}sig^{r_{1}}\left(e_{o_{1}} \right)-q_{1}sig^{z_{1}}\!\left(e_{o_{1}} \right)\\[3pt] \dot{e}_{o_{2}}&=e_{o_{3}}-p_{2}sig^{r_{2}}\!\left(e_{o_{1}} \right)-q_{2}sig^{z_{2}}\!\left(e_{o_{1}} \right)\\[3pt] \dot{e}_{o_{3}}&=\dot{\delta}_{lu}-p_{3}sig^{r_{3}}\!\left(e_{o_{1}} \right)-q_{3}sig^{z_{3}}\!\left(e_{o_{1}} \right)-\Upsilon sign\!\left( e_{o_{1}}\right)\\ \end{aligned}\right.. \end{equation}

The reminder of this proof is quite similar to that of Theorem 1 given by Basin et al. [Reference Basin, Yu and Shtessel43], and therefore, it is omitted here for space.

4.2 Control algorithm design

The entire design procedure is elaborated as follows.

Step 1. Design a stabilising function for $e_{t_{1}}$ . To begin with, let

(24) \begin{equation} e_{t_{2}}=\varkappa_{2}-\dot{\eta}_{d}, {e}^{\dagger}_{t_{2}}=\hat{\varkappa}_{2}-\dot{\eta}_{d}.\end{equation}

Consider the simple quadratic Lyapunov function candidate $V_{1}=\frac{1}{2}e^{T}_{t_{1}}e_{t_{1}}$ . Evaluating the time derivative of $V_{1}$ by using Equations (18) and (24) results in

(25) \begin{equation} \dot{V}_{1}=e^{T}_{t_{1}}e_{t_{2}}=e^{T}_{t_{1}}\!\left( \varkappa_{2}-\dot{\eta}_{d}\right)=e^{T}_{t_{1}}\!\left(\hat{\varkappa}_{2}+e_{o_{2}}-\dot{\eta}_{d} \right) =e^{T}_{t_{1}} e^{\dagger}_{t_{2}}+e^{T}_{t_{1}}e_{o_{2}} .\end{equation}

Adopting the virtual control law

(26) \begin{equation} e^{*}_{t_{2}}=-\lfloor \kappa_{1} \rceil sig^{\alpha}\!\left( e_{t_{1}} \right) \end{equation}

for $e_{t_{1}}$ produces

(27) \begin{equation}\begin{aligned} \dot{V}_{1}&=e^{T}_{t_{1}}\!\left(e^{\dagger}_{t_{2}}- e^{*}_{t_{2}}\right)-e^{T}_{t_{1}}\lfloor \kappa_{1}\rceil sig^{\alpha}\!\left(e_{t_{1}}\right)+e^{T}_{t_{1}}e_{o_{2}}=e^{T}_{t_{1}}\!\left(e^{\dagger}_{t_{2}}- e^{*}_{t_{2}}\right)-\sum_{i=1}^{2}\kappa_{1_{i}}\left|e_{t_{1_{i}}}\right|^{1+\alpha}+e^{T}_{t_{1}}e_{o_{2}},\end{aligned}\end{equation}

where $\kappa_{1}\in \mathbb R^{2}$ and $\alpha\in\!\left(0,1\right) $ . Define the intermediate variable $\varpi$ as

(28) \begin{equation} \varpi=sig^{\frac{1}{\alpha}}\!\left(e^{\dagger}_{t_{2}} \right)-sig^{\frac{1}{\alpha}}\!\left(e^{*}_{t_{2}} \right),\end{equation}

and together with Lemma 4, we get

(29) \begin{equation}\begin{aligned} e^{T}_{t_{1}}\!\left(e^{\dagger}_{t_{2}}-e^{*}_{t_{2}}\right)&=\sum_{i=1}^{2}e_{t_{1_{i}}}\!\left(e^{\dagger}_{t_{2_{i}}}-e^{*}_{t_{2_{i}}}\right)\le\sum_{i=1}^{2}\left|e_{t_{1_{i}}}\right| \Big|e^{\dagger}_{t_{2_{i}}}-e^{*}_{t_{2_{i}}}\Big|\\ &=\sum_{i=1}^{2}\left|e_{t_{1_{i}}}\right| \Big|sig^{\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{\dagger}_{t_{2_{i}}}\right) \right) -sig^{\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)\Big| \\ &\le\sum_{i=1}^{2}2^{1-\alpha}\left|e_{t_{1_{i}}}\right| \Big|sig^{\frac{1}{\alpha}}\!\left( e^{\dagger}_{t_{2_{i}}}\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right)\Big|^{\alpha}\le\sum_{i=1}^{2}2^{1-\alpha}\left|e_{t_{1_{i}}}\right| |\varpi_{i}|^{\alpha}.\end{aligned}\end{equation}

In view of Lemma 2, Equation (29) can be rewritten as

(30) \begin{equation} e^{T}_{t_{1}}\!\left(\bar{e}_{t_{2}}-\bar{e}^{*}_{t_{2}}\right)\le\sum_{i=1}^{2}\!\left( \frac{2^{1-\alpha}}{1+\alpha}\left|e_{t_{1_{i}}}\right|^{1+\alpha}+\frac{2^{1-\alpha}\alpha}{1+\alpha}|\varpi_{i}|^{1+\alpha}\right).\end{equation}

Substituting Equation (30) for Equation (27), the derivative $\dot{V}_{1}$ becomes

(31) \begin{equation}\begin{aligned} \dot{V}_{1}\le&-\sum_{i=1}^{2}\!\left(\kappa_{1_{i}}-\frac{2^{1-\alpha}}{1+\alpha} \right)\left|e_{t_{1_{i}}}\right|^{1+\alpha}+\sum_{i=1}^{2}\frac{2^{1-\alpha}\alpha}{1+\alpha}|\varpi_{i}|^{1+\alpha}+e^{T}_{t_{1}}e_{o_{2}}.\end{aligned}\end{equation}

Step 2. Design a fixed-time control law for $\tau$ . To this end, select the complete Lyapunov function candidate as

(32) \begin{equation} V=V_{1}+\sum_{i=1}^{2}V_{2_{i}},\end{equation}

where $V_{2_{i}}$ , $i=1,2$ , takes the form

(33) \begin{equation} V_{2_{i}}=\int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds.\end{equation}

In the sequel, we demonstrate that $V_{2_{i}}$ is a scalar positive function through analysing the following cases.

• Case 1: $e^{\dagger}_{t_{2_{i}}}> e^{*}_{t_{2_{i}}}\ge0$ . From $sig^{\imath}({\cdot})=|{\cdot}|^{\imath}sign({\cdot}) $ and $s\in\left[ e^{\dagger}_{t_{2_{i}}},e^{*}_{t_{2_{i}}} \right] $ we thus get

  (34) \begin{equation} s\ge e^{*}_{t_{2_{i}}}\ge 0,\ \ sig^{\frac{1}{\alpha}}\!\left( s\right)=s^{\frac{1}{\alpha}},\ \ \text{and}\ \ sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) = e^{*\frac{1}{\alpha}}_{t_{2_{i}}}. \end{equation}
  This implies that
  (35) \begin{equation} V_{2_{i}}=\int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}\Big|s^{\frac{1}{\alpha}}-e^{*\frac{1}{\alpha}}_{t_{2_{i}}}\Big|^{2-\alpha}sign\!\left(s^{\frac{1}{\alpha}}-e^{*\frac{1}{\alpha}}_{t_{2_{i}}} \right)ds= \int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}\!\left( s^{\frac{1}{\alpha}}-e^{*\frac{1}{\alpha}}_{t_{2_{i}}}\right) ^{2-\alpha}ds, \end{equation}
  where we used the fact that the power function $f\!\left(x \right)=x^{\frac{1}{\alpha}} $ is strictly increasing when $x>0$ . This, together with the well-known mean value theorem, gives
  (36) \begin{equation} V_{2_{i}}=\left( s^{*\frac{1}{\alpha}}_{1}-e^{*\frac{1}{\alpha}}_{t_{2_{i}}}\right) ^{2-\alpha}\!\left(e^{\dagger}_{t_{2_{i}}}-e^{*}_{t_{2_{i}}} \right)>0 \end{equation}
  with $s^{*}_{1}\in\left( e^{*}_{t_{2_{i}}},e^{\dagger}_{t_{2_{i}}}\right) $ .

• Case 2: $e^{\dagger}_{t_{2_{i}}}\ge0> e^{*}_{t_{2_{i}}}$ . Rewrite Equation (33) as

  (37) \begin{equation} V_{2_{i}}=\int_{0}^{e^{\dagger}_{t_{2_{i}}}}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds+\int_{e^{*}_{t_{2_{i}}}}^{0}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds. \end{equation}
  Regarding the first term on the right side of Equation (37), if $e^{\dagger}_{t_{2_{i}}}=0$ , then $\int_{0}^{e^{\dagger}_{t_{2_{i}}}}sig^{2-\alpha} \!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-\right.$ $\left.sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds=0$ . Otherwise, we can easily verify that there exists a strictly positive constant $s^{*}_{2}\in\left( 0,e^{\dagger}_{t_{2_{i}}}\right) $ such that
  (38) \begin{equation} \begin{aligned} &\int_{0}^{e^{\dagger}_{t_{2_{i}}}}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds=\int_{0}^{e^{\dagger}_{t_{2_{i}}}}\big|s^{\frac{1}{\alpha}}+\big| e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}\big|^{2-\alpha}sign\!\left(s^{\frac{1}{\alpha}}+\big| e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}} \right)ds=\\ &\int_{0}^{e^{\dagger}_{t_{2_{i}}}}\!\left(s^{\frac{1}{\alpha}}+\big| e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}} \right)^{2-\alpha}ds=\left(s^{*\frac{1}{\alpha}}_{2}+\big| e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}} \right)^{2-\alpha}e^{\dagger}_{t_{2_{i}}}>0, \end{aligned} \end{equation}
  where $e^{*}_{t_{2_{i}}}<0\Rightarrow sign\!\left(e^{*}_{t_{2_{i}}}\right)=-1$ , $\forall s\in\left[ 0, e^{\dagger}_{t_{2_{i}}}\right] \Rightarrow sign(s)=1 $ , and the mean value theorem have been used.

  As for the second term, let us now consider the integration by substitution technique. By making the substitution $s=-g$ , we have

  (39) \begin{equation} \begin{aligned} &\int_{e^{*}_{t_{2_{i}}}}^{0}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds=\int_{0}^{\big|e^{*}_{t_{2_{i}}}\big|}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( -g\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)dg=\\ &\int_{0}^{\big|e^{*}_{t_{2_{i}}}\big|}sig^{2-\alpha}\!\left(|g|^{\frac{1}{\alpha}}sign\!\left({-}g \right)-\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}sign\!\left(e^{*}_{t_{2_{i}}} \right) \right)dg. \end{aligned} \end{equation}
  Further consider that $sign\!\left(e^{*}_{t_{2_{i}}}\right)=-1$ and $\forall g\in\left[ 0, \big|e^{*}_{t_{2_{i}}}\big|\right] $ means $ sign({-}g)=-1$ and $\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}>|g|^{\frac{1}{\alpha}}$ . Consequently, Equation (39) simply becomes
  (40) \begin{equation} \begin{aligned} &\int_{0}^{\big|e^{*}_{t_{2_{i}}}\big|}sig^{2-\alpha}\!\left(|g|^{\frac{1}{\alpha}}sign\!\left({-}g \right)-\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}sign\!\left(e^{*}_{t_{2_{i}}} \right) \right)dg=\int_{0}^{\big|e^{*}_{t_{2_{i}}}\big|}sig^{2-\alpha}\!\left(\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}-|g|^{\frac{1}{\alpha}}\right) dg=\\[4pt] &\int_{0}^{\big|e^{*}_{t_{2_{i}}}\big|}\!\left(\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}-|g|^{\frac{1}{\alpha}}\right)^{2-\alpha}sign\!\left(\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}-|g|^{\frac{1}{\alpha}} \right)dg= \int_{0}^{\big|e^{*}_{t_{2_{i}}}\big|}\!\left(\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}-|g|^{\frac{1}{\alpha}}\right)^{2-\alpha}dg. \end{aligned} \end{equation}
  Proceeding similarly to get Equation (36) leads to
  (41) \begin{equation} \int_{e^{*}_{t_{2_{i}}}}^{0}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds=\left( \big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}-g_{1}^{*\frac{1}{\alpha}}\right)^{2-\alpha}\big|e^{*}_{t_{2_{i}}}\big|>0 \end{equation}
  with $g^{*}_{1}\in\left(0,\big|e^{*}_{t_{2_{i}}}\big|\right) $ . Summarising the results in Equations (38) and (41) gives $V_{2_{i}}>0$ in the case of $e^{\dagger}_{t_{2_{i}}}\ge0> e^{*}_{t_{2_{i}}}$ .

• Case 3: $0\ge e^{\dagger}_{t_{2_{i}}}> e^{*}_{t_{2_{i}}}$ . Clearly, in this case, $0\le\big|e^{\dagger}_{t_{2_{i}}}\big|<\big|e^{*}_{t_{2_{i}}}\big|$ holds. Let us again use the integration by substitution technique. Setting $s=-g$ results in

  (42) \begin{equation} V_{2_{i}}=\int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left(s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds=\int_{\big|e^{\dagger}_{t_{2_{i}}}\big|}^{\big|e^{*}_{t_{2_{i}}}\big|}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( -g\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)dg. \end{equation}
  Following the same lines to obtain Equations (40) and (41), it can also be shown that
  (43) \begin{equation} V_{2_{i}}=\int_{\big|e^{\dagger}_{t_{2_{i}}}\big|}^{\big|e^{*}_{t_{2_{i}}}\big|}\!\left(\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}-|g|^{\frac{1}{\alpha}}\right)^{2-\alpha}dg=\left(\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}}-g_{2}^{*\frac{1}{\alpha}}\right)^{2-\alpha}\big(\big|e^{*}_{t_{2_{i}}}\big|-\big|e^{\dagger}_{t_{2_{i}}}\big|\big)>0 \end{equation}
  with $g^{*}_{2}\in\left(\big|e^{\dagger}_{t_{2_{i}}}\big|,\big|e^{*}_{t_{2_{i}}}\big|\right) $ .

• Case 4: $e^{*}_{t_{2_{i}}}\ge0>e^{\dagger}_{t_{2_{i}}}$ . We now switch the upper and lower bounds of integral (33) and thus obtain

  (44) \begin{equation} \begin{aligned} V_{2_{i}}&=-\int_{e^{\dagger}_{t_{2_{i}}}}^{e^{*}_{t_{2_{i}}}}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds\\ &=-\int_{0}^{e^{*}_{t_{2_{i}}}}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds-\int_{e^{\dagger}_{t_{2_{i}}}}^{0}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds. \end{aligned} \end{equation}
  The first term on the right side of Equation (44) is identically equal to zero if $e^{*}_{t_{2_{i}}}=0$ ; if not, proceeding as before, it can be simplified as
  (45) \begin{equation} \begin{aligned} &-\int_{0}^{e^{*}_{t_{2_{i}}}}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds=-\int_{0}^{e^{*}_{t_{2_{i}}}}\Big|s^{\frac{1}{\alpha}}-e^{*\frac{1}{\alpha}}_{t_{2_{i}}}\Big|^{2-\alpha}sign\!\left( s^{\frac{1}{\alpha}}-e^{*\frac{1}{\alpha}}_{t_{2_{i}}}\right)ds=\\ &\left( e^{*\frac{1}{\alpha}}_{t_{2_{i}}}-s_{3}^{*\frac{1}{\alpha}}\right) ^{2-\alpha}e^{*}_{t_{2_{i}}}>0, \end{aligned} \end{equation}
  where $s^{*}_{3}\in\left(0,e^{\dagger}_{t_{2_{i}}}\right) $ and we have used the fact $\forall s\in\left[ 0,e^{*}_{t_{2_{i}}}\right] \Rightarrow s^{\frac{1}{\alpha}}\le e^{*\frac{1}{\alpha}}_{t_{2_{i}}}$ .

  And the second term, given the substitution $s=-g$ , satisfies

  (46) \begin{align} &-\int_{e^{\dagger}_{t_{2_{i}}}}^{0}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds=\int_{\big|e^{\dagger}_{t_{2_{i}}}\big|}^{0}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( -g\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)dg=\nonumber\\ &-\int_{0}^{\big|e^{\dagger}_{t_{2_{i}}}\big|}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( -g\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)dg=\int_{0}^{\big|e^{\dagger}_{t_{2_{i}}}\big|}\left(g^{\frac{1}{\alpha}}+\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}} \right)^{2-\alpha}dg=\nonumber\\ & \left(g^{*\frac{1}{\alpha}}_{3}+\big|e^{*}_{t_{2_{i}}}\big|^{\frac{1}{\alpha}} \right)^{2-\alpha}\big|e^{\dagger}_{t_{2_{i}}}\big|>0, \end{align}

  where $g^{*}_{3}\in\left(0,\big|e^{*}_{t_{2_{i}}}\big|\right) $ and we have used the facts $sign\!\left({-}g \right)=-1 $ and $sign\!\left(e^{*}_{t_{2_{i}}} \right)=-1 $ . Taking Equations (44) and (45) into account, we know that $V_{2_{i}}>0$ when $e^{*}_{t_{2_{i}}}\ge0>e^{\dagger}_{t_{2_{i}}}$ .

Evidently, the above discussion guarantees that $V_{2_{i}}$ is positively defined. Differentiating $V_{2_{i}}$ with respect to time and applying Lemma 5 lead to

(47) \begin{equation} \dot{V}_{2_{i}} = sig^{2-\alpha}\!\left(\varpi_{i}\right)\dot{e}^{\dagger}_{t_{2_{i}}}-\left(2-\alpha\right)\frac{\mathrm{d} sig^{\frac{1}{\alpha}}\!\left(e^{*}_{t_{2_{i}}}\right) }{\mathrm{d} t}\int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}\left|sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right)\right|^{1-\alpha}ds. \end{equation}

Noting that

(48) \begin{equation} sig^{\frac{1}{\alpha}}\!\left(\kappa_{1_{i}}sig^{\alpha}\!\left(e_{t_{1_{i}}}\right) \right)=\kappa_{1_{i}}^{\frac{1}{\alpha}}|\left|e_{t_{1_{i}}}\right|^{\alpha}sign\!\left(e_{t_{1_{i}}} \right) | ^{\frac{1}{\alpha}}sign\!\left(\left|e_{t_{1_{i}}}\right|^{\alpha}sign\!\left(e_{t_{1_{i}}} \right) \right)=\kappa_{1_{i}}^{\frac{1}{\alpha}}e_{t_{1_{i}}}\end{equation}

and

(49) \begin{equation} \frac{\mathrm{d} sig^{\frac{1}{\alpha}}\!\left(e^{*}_{t_{2_{i}}}\right) }{\mathrm{d} t}=-\frac{\mathrm{d} sig^{\frac{1}{\alpha}}\!\left(\kappa_{1_{i}}sig^{\alpha}\!\left(e_{t_{1_{i}}}\right)\right) }{\mathrm{d} e_{t_{1_{i}}}}e_{t_{2_{i}}}=-\kappa_{1_{i}}^{\frac{1}{\alpha}}e_{t_{2_{i}}},\end{equation}

we have

(50) \begin{equation} \dot{V}_{2_{i}} = sig^{2-\alpha}\!\left(\varpi_{i}\right)\dot{e}^{\dagger}_{t_{2_{i}}}+\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)e_{t_{2_{i}}}\int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}\left|sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right)\right|^{1-\alpha}ds.\end{equation}

The second term in the right side of Equation (50) satisfies

(51) \begin{equation}\begin{aligned} & \left|\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)e_{t_{2_{i}}}\int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}\right|sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right)|^{1-\alpha}ds|\\[4pt] \le& \kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)\left|e_{t_{2_{i}}}\right| \left|e^{\dagger}_{t_{2_{i}}}-e^{*}_{t_{2_{i}}}\right| \left|sig^{\frac{1}{\alpha}}\!\left( e^{\dagger}_{t_{2_{i}}}\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right)\right|^{1-\alpha}\\[4pt] \le& \kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)\left|e_{t_{2_{i}}}\right| \left|sig^{\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{\dagger}_{t_{2_{i}}}\right) \right) -sig^{\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)\right|\times\left|sig^{\frac{1}{\alpha}}\!\left( e^{\dagger}_{t_{2_{i}}}\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right)\right|^{1-\alpha}\\[4pt] \le& \kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha}\left|e_{t_{2_{i}}}\right| |\varpi_{i}|\end{aligned}\end{equation}

with

(52) \begin{equation}\begin{aligned} \left|e_{t_{2_{i}}}\right|&=\left|e^{\dagger}_{t_{2_{i}}}+e_{o_{2_{i}}} \right|\le\big|e^{\dagger}_{t_{2_{i}}}\big|+\left|e_{o_{2_{i}}}\right|\le|e^{\dagger}_{t_{2_{i}}}-e^{*}_{t_{2_{i}}}|+|\kappa_{1_{i}}sig^{\alpha}\!\left( e_{t_{1_{i}}}\right) |+\left|e_{o_{2_{i}}}\right|\\ &\le|sig^{\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{\dagger}_{t_{2_{i}}}\right) \right) -sig^{\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)|+\kappa_{1_{i}}\left|e_{t_{1_{i}}}\right|^{\alpha}+\left|e_{o_{2_{i}}}\right|\\ &\le2^{1-\alpha}|\varpi_{i}|^{\alpha}+\kappa_{1_{i}}\left|e_{t_{1_{i}}}\right|^{\alpha}+\left|e_{o_{2_{i}}}\right|,\end{aligned}\end{equation}

where Equations (24), and (26), (28), and Lemma 4 have been used. Coupling Equations (51) and (52), we have

(53) \begin{equation}\begin{aligned} &\left|\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)e_{t_{2_{i}}}\int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}\right|sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right)|^{1-\alpha}ds|\\ &\le\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha} |\varpi_{i}|\left( 2^{1-\alpha}|\varpi_{i}|^{\alpha}+\kappa_{1_{i}}\left|e_{t_{1_{i}}}\right|^{\alpha}+\left|e_{o_{2_{i}}}\right|\right) \\ &=\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{2-2\alpha}|\varpi_{i}|^{1+\alpha}+\kappa_{1_{i}}^{1+\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha}|\varpi_{i}| \left|e_{t_{1_{i}}}\right|^{\alpha}+\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha}|\varpi_{i}| \left|e_{o_{2_{i}}}\right|\\ &\le\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{2-2\alpha}|\varpi_{i}|^{1+\alpha}+\frac{\left(2-\alpha\right)2^{1-\alpha}\alpha}{1+\alpha}\left|e_{t_{1_{i}}}\right|^{1+\alpha}+\frac{\left(2-\alpha\right)2^{1-\alpha}}{1+\alpha}\kappa_{1_{i}} ^{1+\alpha}|\varpi_{i}|^{1+\alpha}\\ &+\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha}|\varpi_{i}| \left|e_{o_{2_{i}}}\right|,\end{aligned}\end{equation}

where Lemma 2 has been used. Then, the differentiation of $V_{2_{i}}$ and the utilisation of Equation (53) yield

(54) \begin{equation}\begin{aligned} \dot{V}_{2i}\le& \ sig^{2-\alpha}\!\left(\varpi_{i}\right)\dot{e}^{\dagger}_{t_{2_{i}}}+\frac{\left(2-\alpha\right)2^{1-\alpha}\alpha}{1+\alpha}\left|e_{t_{1_{i}}}\right|^{1+\alpha}\\ &+\left( \kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{2-2\alpha}+\frac{\left(2-\alpha\right)2^{1-\alpha}}{1+\alpha}\kappa_{1_{i}} ^{1+\alpha} \right) |\varpi_{i}|^{1+\alpha} +\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha}|\varpi_{i}| \left|e_{o_{2_{i}}}\right|.\end{aligned}\end{equation}

Together with Equations (18), (23), (24), (27), and (54), the definition of V, and the dynamics of $e^{\dagger}_{t_{2}}$

(55) \begin{equation} \dot{e}^{\dagger}_{t_{2}} = R\!\left(\psi\right)M_{\epsilon}\tau+\delta_{lu} -\ddot{\eta}_{d}+p_{2}sig^{r_{2}}\!\left(e_{o_{1}} \right)+q_{2}sig^{z_{2}}\!\left(e_{o_{1}} \right)-e_{o_{3}},\end{equation}

we get

(56) \begin{equation}\begin{split} \dot{V}\le&-\sum_{i=1}^{2}\!\left(\kappa_{1_{i}}-\gamma_{1_{i}} \right)\left|e_{t_{1_{i}}}\right|^{1+\alpha}+\sum_{i=1}^{2}\frac{\alpha}{1+\alpha}\left|e_{o_{2_{i}}}\right|^{1+\frac{1}{\alpha}}+\sum_{i=1}^{2}\gamma_{2_{i}} |\varpi_{i}|^{1+\alpha}+\sum_{i=1}^{2}\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha}|\varpi_{i}| \left|e_{o_{2_{i}}}\right|\\ &+\left( sig^{2-\alpha}\!\left( \varpi\right) \right) ^{T} \!\left( R\!\left(\psi\right)M_{\epsilon}\tau+\delta_{lu} -\ddot{\eta}_{d}+p_{2}sig^{r_{2}}\!\left(e_{o_{1}} \right)+q_{2}sig^{z_{2}}\!\left(e_{o_{1}} \right)-e_{o_{3}} \right)\end{split}\end{equation}

with $\gamma_{1_{i}}=\frac{2^{1-\alpha}(1+\left(2-\alpha\right)\alpha)+1}{1+\alpha}$ and $\gamma_{2_{i}}=\frac{2^{1-\alpha}\alpha}{1+\alpha}+\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{2-2\alpha}+\frac{\left(2-\alpha\right)2^{1-\alpha}}{1+\alpha}\kappa_{1_{i}} ^{1+\alpha}$ , $i=1, 2$ . According to the structure of $\dot{V}$ , the trajectory tracking control law for $\tau$ is designed as

(57) \begin{equation} \begin{split} \tau=-M_{\epsilon}^{-1}R^{-1}\!\left(\psi\right)\left(\hat{\delta}_{lu} -\ddot{\eta}_{d} +\left( \lfloor\kappa_{2}\rceil+\lfloor\gamma_{2}\rceil\right) sig^{2\alpha-1}\!\left(\varpi\right)+\varrho\!\left(t\right)\left( sig^{\alpha}\!\left(\varpi\right)+ sig^{2-\alpha}\!\left(\varpi\right)\right) \right) ,\end{split}\end{equation}

where $\rho\!\left(t\right)=\left\lbrace \begin{array}{cl}((T_{\rho}-t)/T_{\rho})^\frac{1}{\alpha_{\rho}}(\rho_{0}-\rho_{\infty})+\rho_{\infty}&,0\le t\le T_{\rho}\\ \rho_{\infty}&,t>T_{\rho}\end{array}\right. $ with $\left\lbrace\rho_{0},\rho_{\infty},\alpha_{\rho},T_{\rho} \right\rbrace\in\mathbb R_{>0} $ , and $ \kappa_{2} \in \mathbb R^{2}_{>0}$ . The main results of this brief are included in the following theorem.

Theorem 2. Consider the stratospheric airship (8) subject to underactuated dynamics, unmeasured velocities, modeling imperfections, and exogenous disturbances, with the coordinate transformation (11), the FTO (21), the virtual control law (26), and the control action (57). Suppose that Assumptions 1–4 hold and the control parameters are selected such that

(58) \begin{equation}\begin{aligned} \kappa_{1_{i}}>\gamma_{1_{i}}+\kappa^{\prime}_{1_{i}}, \end{aligned}\end{equation}

where $ \kappa^{\prime}_{1_{i}} \in R_{>0}$ , $i=1,2$ . Then, the position error $e=\eta-\eta_{d}$ converges to a preassigned small vicinity of the origin within a finite time $T_{f}$ , while all the closed-loop states maintain bounded for $\forall t\ge0$ .

Proof of Theorem 2. Let us prove first that these signals do not go to infinity when $t\in\left[0,T_{o}\right] $ . Substituting the control action (57) with design parameters satisfying the condition (58), we have

(59) \begin{equation}\begin{aligned} \dot{V}\le&-\sum_{i=1}^{2}\kappa^{\prime}_{1_{i}}|e_{t_{2_{i}}}|^{1+\alpha}-\sum_{i=1}^{2}\kappa_{2i}|\varpi_{i}|^{1+\alpha}+\sum_{i=1}^{2}\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha}|\varpi_{i}| \left|e_{o_{2_{i}}}\right|+\sum_{i=1}^{2}\frac{\alpha}{1+\alpha}\left|e_{o_{2_{i}}}\right|^{1+\frac{1}{\alpha}}\\ &+\left( sig^{2-\alpha}\!\left(\varpi\right) \right) ^{T} \left( p_{2}sig^{r_{2}}\!\left(e_{o_{1}} \right)+q_{2}sig^{z_{2}}\!\left(e_{o_{1}} \right) \right)-\varrho\!\left(t\right)\left( sig^{2-\alpha}\!\left(\varpi\right) \right) ^{T}\left( sig^{\alpha}\!\left(\varpi\right)+ sig^{2-\alpha}\!\left(\varpi\right)\right) ,\end{aligned}\end{equation}

where the fact

(60) \begin{equation} sig^{2\alpha-1}\!\left( \varpi_{i}\right)sig^{2-\alpha}\!\left( \varpi_{i}\right)=|\varpi_{i}|^{1+\alpha}\end{equation}

has been used. Noticing that the signals $e_{o_{1}}$ and $e_{o_{2}}$ decay to zero after a fixed-time time $T_{0}$ , hence $\left\lbrace e_{o_{1_{i}}},e_{o_{2_{i}}}\right\rbrace\in\mathbb{L}^{\infty}$ . Accordingly, there exists a constant $B_{e_{o}}\in\mathbb{R}_{>0}$ such that $\sup_{t\in\left[0,\infty \right) }\big\lbrace |e_{o_{1_{i}}}|,\left|e_{o_{2_{i}}}\right|,$ $\left.|e_{o_{1_{i}}}|^{r_{2}},\left|e_{o_{2_{i}}}\right|^{z_{2}}\right\rbrace \le B_{e_{o}}$ , $i=1,2$ . This, combined with Young’s inequality, gives

(61) \begin{equation} \sum_{i=1}^{2}\kappa_{1_{i}}^{\frac{1}{\alpha}}\!\left(2-\alpha\right)2^{1-\alpha}|\varpi_{i}| |e_{o_{1_{i}}}|\le\sum_{i=1}^{2}\!\left(\varrho\!\left(t\right)|\varpi_{i}|^{2}+\frac{\kappa_{1_{i}}^{\frac{2}{\alpha}}\!\left(2-\alpha\right)^{2}2^{2-2\alpha}}{4\varrho\!\left(t\right)}B_{e_{o}}^{2} \right),\end{equation}

(62) \begin{equation}\begin{aligned} \left( sig^{2-\alpha}\!\left(\varpi\right) \right) ^{T} \!\left( p_{2}sig^{r_{2}}\!\left(e_{o_{1}} \right)+q_{2}sig^{z_{2}}\!\left(e_{o_{1}} \right) \right) \le\sum_{i=1}^{2}\!\left( \varrho\!\left(t \right) |\varpi_{i}|^{4-2\alpha}+\frac{p_{2}^{2}+z_{2}^{2}}{4\varrho\!\left(t \right)}|B_{e_{o}}|^{2}\right).\end{aligned} \end{equation}

In view of Equations (61) and (62), Equation (59) can be restated as

(63) \begin{equation}\begin{aligned} \dot{V}\le&-\sum_{i=1}^{2}\kappa^{\prime}_{1_{i}}\left|e_{t_{1_{i}}}\right|^{1+\alpha}-\sum_{i=1}^{2}\kappa_{2i}|\varpi_{i}|^{1+\alpha}+\sum_{i=1}^{2}\frac{\alpha}{1+\alpha}B_{e_{o}}^{1+\frac{1}{\alpha}}+\sum_{i=1}^{2}\frac{\kappa_{1_{i}}^{\frac{2}{\alpha}}\!\left(2-\alpha\right)^{2}2^{2-2\alpha}}{4\varrho\!\left(t\right)}B_{e_{o}}^{2}\\&+\sum_{i=1}^{2}\frac{p_{2}^{2}+z_{2}^{2}}{4\varrho\!\left(t \right)}|B_{e_{o_{n}}}|^{2}.\end{aligned}\end{equation}

Apparently, a constant $B_{s}^{[0,T_{o}]}\in\mathbb{R}_{>0}$ exists, which is an upper bound on the sum of the last three terms in Equation (63) when $t\in[0,T_{o}]$ . Besides, Recalling the definition of function $V_{2i}$ , we can easily see

(64) \begin{equation}\begin{aligned} V_{2i}&=\int_{e^{*}_{t_{2_{i}}}}^{e^{\dagger}_{t_{2_{i}}}}sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( s\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)ds\le\left|e^{\dagger}_{t_{2_{i}}}-e^{*}_{t_{2_{i}}}\right| \left|sig^{2-\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{\dagger}_{t_{2_{i}}}\right)-sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right)\right|\\ &\le|sig^{\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{*}_{t_{2_{i}}}\right) \right) -sig^{\alpha}\!\left(sig^{\frac{1}{\alpha}}\!\left( e^{\dagger}_{t_{2_{i}}}\right) \right)| |\varpi_{i}|^{2-\alpha}\le 2^{1-\alpha}|\varpi_{i}|^{2}\le2|\varpi_{i}|^{2},\end{aligned}\end{equation}

and therefore the complete Lyapunov function candidate V satisfies $V\le2e^{T}_{t_{1}}e_{t_{1}}+2\varpi^{T}\varpi$ . Since $\alpha\in\left( 0,1\right)\Rightarrow \frac{1+\alpha}{2}\in\left(\frac{1}{2},1\right) $ , this further leads to

(65) \begin{equation}\begin{aligned} V^{\frac{1+\alpha}{2}}&\le\left(2\sum_{i=1}^{2}e_{t_{1_{i}}}^{2}+2\sum_{i=1}^{2}\varpi_{i}^{2}\right)^{\frac{1+\alpha}{2}}\le2^{\frac{1+\alpha}{2}}\sum_{i=1}^{2}\left|e_{t_{1_{i}}}\right|^{1+\alpha}+2^{\frac{1+\alpha}{2}}\sum_{i=1}^{2}|\varpi_{i}|^{1+\alpha}\end{aligned}\end{equation}

When $V>1$ , summarising the results in Equations (59), (63), (64) and (65) gives

(66) \begin{equation} \dot{V}\le-\lambda_{0}V^{\frac{1+\alpha}{2}}+B_{s}^{[0,T_{o}]}\le-\lambda_{0} V+B_{s}^{[0,T_{o}]},\end{equation}

where Lemma 3 has been used, $\lambda_{0}=\min\!\left(\frac{1}{2}\right)^{\frac{1+\alpha}{2}}\times\left\lbrace \kappa^{\prime}_{1_{i}},\kappa_{2i}\right\rbrace $ , $i=1,2$ . Multiplying Equation (66) by $e^{ \lambda_{0}t}$ and integrating it over $\left[0,t\right] $ , we have

(67) \begin{equation} 0\le V\!\left(t\right) \le \left(V\!\left(0\right)-\frac{B_{s}^{[0,T_{o}]}}{\lambda_{0}} \right)e^{- \lambda_{0} t}+\frac{B_{s}^{[0,T_{o}]}}{ \lambda_{0}}.\end{equation}

Besides, in the case of $V\le1$ , the system states $e_{t_{i}}$ , $i=1,2$ , and $\varpi$ are obviously bounded. As a result, we can conclude that these signals will not drift to infinity as $t\in\left[0,T_{o}\right] $ .

Now, we proceed to prove Theorem 2. Theorem 1 reveals that $\hat{\varkappa}_{2}=\varkappa_{2}$ and $\hat{\delta}_{lu}=\delta_{lu}$ for $\forall t\ge T_{0}$ . Along with the fact that $\varrho\!\left(t\right)\in\mathbb{R}_{>0} $ , Equation (59) becomes

(68) \begin{equation} \begin{aligned} \dot{V}\le&-\sum_{i=1}^{2}\kappa^{\prime}_{1_{i}}\left|e_{t_{1_{i}}}\right|^{1+\alpha}-\sum_{i=1}^{2}\kappa_{2i}|\varpi_{i}|^{1+\alpha}-\sum_{i=1}^{2}\varrho\!\left(t \right) \left(\left( |\varpi_{i}|^2+ |\varpi_{i}|^{4-2\alpha}\right)\right)\le-\lambda_{0}V^{\frac{1+\alpha}{2}}, \end{aligned} \end{equation}

Consequently, all of the system states remain bounded for $t\ge T_{o}$ . Besides, it is noteworthy that:

1. 1. the finite-time tracking task is fulfilled. Given Lemma 1, we can conclude that both $e_{t_{1}}$ and $\varpi$ decay to zero within a finite time $T_{f}$ estimated by

   (69) \begin{equation} T_{f}\le\frac{2V^{\frac{1-\alpha}{2} }\!\left(0\right)}{\lambda_{0}(1-\alpha)}+T_{o}; \end{equation}

2. 2. a priori assignment of tracking accuracy is assured. Recalling Equation (20), we get

   (70) \begin{equation} \|e_{t_{1}}\|=0, \forall t\ge T_{f}\Rightarrow\max\left\lbrace|x_{e}|,|y_{e}| \right\rbrace\le\epsilon, \forall t\ge T_{f}, \end{equation}
   indicating that we can specify the accuracy bound with the parameter $\epsilon$ .

Table 1. Airship model parameters

5.1 Simulation condition

In this section, numerical simulations are performed to intuitively evaluate the effectiveness of the preceding theoretical results with the running time $T_{final}=100$ s. The physical parameters of the stratospheric airship are borrowed from Zheng et al. [Reference Zheng, Guan, Ma and Zhu6] and Zhang et al. [Reference Yan, Weidong, Yugeng and Zili18], and all of them are listed in Table 1.

In this simulations, the airship starts from $\left[x\!\left(0\right),y\!\left(0\right)\right]^{T}= \left[340{\rm m},390{\rm m}\right]^{T}$ with initial heading $\psi\!\left(0\right)=0.1$ rad, and initial velocity $\nu\!\left(0\right)= \left[u\!\left(0\right),v\!\left(0\right),r\!\left(0\right)\right]^{T}$ $=\left[0{\rm m/s},0{\rm m/s},0{\rm rad/s}\right]^{T}$ , and is required to track the reference trajectory

(71) \begin{equation}\eta_{d}=\begin{bmatrix}500\sin\!\left(0.0052t+\dfrac{\pi}{4}\right)\\[9pt] 500\cos\!\left(0.0052t+\dfrac{\pi}{4}\right)\end{bmatrix},\end{equation}

with the preassigned accuracy $\epsilon=0.2$ . To test the robustness, the disturbance $\delta_{dis}$ induced by environmental forces is mathematically assumed as [Reference Zheng, Guan, Ma and Zhu6, Reference Wu, Wang, Duan, Xie and Wei10]:

(72) \begin{equation}\delta_{dis}=\begin{bmatrix}\delta_{dis_{u}}\\[4pt] \delta_{dis_{v}}\\[4pt] \delta_{dis_{r}}\end{bmatrix}=\begin{bmatrix} \left[1.4+2.1\sin\!\left(0.1t\right)+1.1\cos\!\left(0.06t\right) \right]\times40\\[4pt] \left[-0.8+1.5\sin\!\left(0.1t\right)+0.3\cos\!\left(0.06t\right) \right]\times40\\[4pt] -\left[2.3\sin\!\left(0.1t\right)+1.9\cos\!\left(0.06t\right) \right]\times110\end{bmatrix}.\end{equation}

To fulfill this mission, the control parameters are selected as $p_{1}=q_{1}=24$ , $p_{2}=q_{2}=216$ , $p_{3}=q_{3}=864$ , $r_{1}=0.8$ , $r_{2}=0.6$ , $r_{3}=0.4$ , $z_{1}=1.2$ , $z_{2}=1.4$ , $z_{3}=1.6$ , $\Upsilon=5$ , $\epsilon=0.2$ , $\alpha=0.7$ , $\kappa_{1}=\left[2,2\right]^{T}$ , $\kappa_{2}=\left[0.1,0.1\right]^{T}$ , $\rho_{0}=1$ , $\rho_{\infty}=0.01$ , $\alpha_{\rho}=2$ , and $T_{\rho}=1$ . The initial conditions of the FTO (21) are set to zero.

Figure 3. Trajectories for horizontal tracking.

Figure 4. Actual and reference positions.

Figure 5. Position errors $x_{e}$ and $y_{e}$ .

Figure 6. Absolute values of the position errors $x_{e}$ and $y_{e}$ .

Figure 7. Velocities in surge u, sway v, and yaw r.

5.2 Simulation result

Applying the FTO (21) and the control action (57) to the airship model (8), we reach some simulation outcomes, illuminated by Figs 3-13. From Figs 3-6, we see that the airship can move to the desired trajectory swiftly and smoothly, and the position errors $x_{e}=x-x_{d}$ and $y_{e}=y-y_{d}$ decay toward a close vicinity of zero within a finite time. Then, the airship flies along the reference trajectory precisely, irrespective of underactuated dynamics, modeling imprecisions, and exceptional disturbances. The simulation results for velocities are shown in Fig. 7, with the three curves corresponding to velocities in the surge, sway and yaw, respectively, all of which are bounded for $\forall t\ge0$ but do not enter a steady state. The reason is twofold. First, a circular path is allocated to track, yielding the desired velocities time-varying. Second, the persistent perturbations (9) consisting in the airship model (8) affect the system dynamics directly. Figures 8-9 plot the unmeasured velocity $\dot{\eta}_{t}$ (i.e., $\varkappa_{2}$ ), the lumped disturbances $\delta_{lu}$ , and their observed values, which show that the FTO (21) can supply the exact observations of $\dot{\eta}_{t}$ and $\delta_{lu}$ in a finite time. Figure 10 presents the necessary control action, from which we see that the surge force and yaw torque are continuous, and no control chattering exists. It should be noted that, compared with the actuating signal $\tau_{r}$ in the steady-state phase, it appears more aggressive in the initial stage. The reason is that the mass $m_{r}$ is huge (12,176kg $\cdot$ m2), and the velocity is initialised to zero. As a result, a large control torque is required in the initial stage to accelerate the airship to the reference route. Nevertheless, it decreases quickly, as shown in Fig. 10. The trajectory tracking responses for different initial positions are depicted in Figs 11-13. It can be observed that the finite-time convergence is assured, and the requirement for tracking accuracy, featured in Equation (70), is also met. The above conclusions successfully affirm that a good tracking performance is achieved under our method.

Figure 8. Unmeasured state $\dot{\eta}_{t}$ (i.e., $\varkappa_{2}$ ) and its finite-time observation.

Figure 9. Lumped disturbance $\delta_{lu_{i}}$ $(i=x,y)$ and its finite-time observation.

Figure 10. Control signal $\tau$ .

Figure 11. Trajectories with different initial positions: [340m,390m] (position 1); [300m,310m] (position 2); [280m,460m] (position 3).

Figure 12. Position errors $x_{e}$ and $y_{e}$ with different initial positions: [340m,390m] (position 1); [300m,310m] (position 2); [280m,460m] (position 3).

Figure 13. Absolute values of the position errors $x_{e}$ and $y_{e}$ with different initial positions: [340m,390m] (position 1); [300m,310m] (position 2); [280m,460m] (position 3).

Aimed at comparison, a standard command-filter backstepping (CFB) controller

(73) \begin{equation}\left\lbrace \begin{aligned}&z_{1}=\varkappa_{1}-\eta_{d},z_{2}=\varkappa_{2}-x_{2r}\\&\bar{z}_{1}=z_{1}-c_{1},\bar{z}_{2}=z_{2}-c_{2}\\&\dot{c}_1=-k_{1}c_{1}+x_{2r}-x_{2r}^{0},\dot{c}_{2}=-k_{2}c_{2}+R(\psi)M_{\epsilon}\left(\tau-\tau^{0}\right)\\ &x_{2r}^{0}=a_{1}-c_{2},\tau^{0}=a_{2}\\ &a_{1}=-k_{1}z_{1}+\dot{\eta}_{d},\tau^{0}=a_{2}=(R(\psi)M_{\epsilon})^{-1}\left({-}k_{2}z_{2}-\bar{z}_{1}-\digamma+\dot{x}_{2r}\right)\\\end{aligned}\right.\end{equation}

formulated by Han et al. [Reference Han, Wang, Chen and Duan9] is introduced in this paper, where the control parameters $k_{1}=k_{2}=eye(3)$ are identical to those used by Han et al. [Reference Han, Wang, Chen and Duan9]. The comparison results are illustrated in Figs 14-17, where FI control shorts for our method. Moreover, to further display the comparative simulations, we summarise the quantisation indexes in Table 2, where the integrated absolute error, IAE (defined as IAE $=\int_{0}^{T_{final}}|j_{e}\!\left(t\right) |dt,j=x,y$ ), the integrated time absolute error, ITAE (defined as ITAE $=\int_{0}^{T_{final}}t|j_{e}\!\left(t\right) |dt,j=x,y$ ), and the mean integrated absolute control, MIAC (defined as MIAC $=\frac{1}{T_{final}}\int_{0}^{T_{final}}|\tau_{j}\!\left(t\right) |dt,j=u,i$ ), devote to assess steady-state performance, transient performance, and control effort, respectively. As seen in Figs 14-17 and Table 2, it is clear that the tracking performance is not satisfactory under the CFB controller and large position errors emerge in the steady state, although the CFB controller ensures the boundedness of position errors. Particularly, the robustness of the CFB controller cannot be warranted in default of a compensation mechanism versus modeling imprecisions and disturbances, as demonstrated in Figs 15-16. In addition, compared to our method, the CFB controller has the biggest IAE, ITAE and MIAC values. This indicates that our method can offer a better tracking quality with less control energy consumption.

Figure 14. Trajectories based on FT control and CFB control.

Figure 15. Position errors $x_{e}$ and $y_{e}$ based on FT control and CFB control.

Figure 16. Absolute values of the position errors $x_{e}$ and $y_{e}$ based on FT control and CFB control.

Figure 17. Control signal $\tau$ based on CFB control.

Table 2. Performance indices comparisons

5.3 Discussion

This paper studies the horizontal trajectory tracking control problem of airships. The system model is built upon Assumption 1, which is widely made in the related literature [Reference Yang, Yang and Deng3, Reference Zheng, Guan, Ma and Zhu6, Reference Wu, Wang, Duan, Xie and Wei10, Reference Zheng and Xie17, Reference Yan, Weidong, Yugeng and Zili18, Reference Yang, Wu and Zheng22, Reference Azinheira, Moutinho and De Paiva23]. Assumption 2 is introduced to give some characterisation of unmodeled dynamics and external disturbances, as the exact model of airships is not always attainable in reality, and the persistent wind field directly affects the motion of airships and thus cannot be neglected in the design of a tracker. From Equation (8), we easily find that the airship, flying in the horizontal plane, is a typical underactuated system. To this end, we start our research with a coordinate transformation (11). After a series of reasoning shown in Equations (12)–(17), a fully actuated EL model (18) is obtained. From Figs 3-6, we see that the coordinate transformation (11) does solve the underactuated problem. We then present an FTO (21) that requires Assumption 4 to realise the velocity-free control and the dynamical compensation. The main property of the FTO (21) is given in Theorem 1, and the reasonability of Assumption 4 is given in Remark 5. Figures 8-9 demonstrate the effectiveness of the FTO (21). Note that in the current literature on API technique [Reference Du, Zhang, Wu, Zhu, Li and Chu30–Reference Fu, Wang and Wang35], certain control parameters are restricted to ratios of positive odd integers. Based on Lemma 4, we relax this restriction. The selection for $\alpha$ shows this advantage. Finally, in the light of Lemmas 1–5, we propose an API-based finite trajectory tracking control algorithm for underactuated airships without command filters or dynamic surfaces. Figures 11–12 illustrate the finite-time convergence of position errors to a preassigned residual set, validating the theoretical predictions in Remark 4 and Theorem 2.