Finite-time velocity-free trajectory tracking control for a stratospheric airship with preassigned accuracy

Abstract

This paper concentrates on the trajectory tracking problem for a stratospheric airship subject to underactuated dynamics, unmeasured velocities, modeling inaccuracies and environmental disturbances. First, a coordinate transformation is performed to solve the underactuated issue, which simultaneously permits a priori assignment of the tracking accuracy. Second, a finite-time observer is integrated into the control structure to offer the exact information of unmeasured velocities and uncertainties in an integral manner. Then, by combining the backstepping technique with the method of adding a power integrator, a new output-feedback control strategy is derived with several salient contributions: (1) the airship’s position errors fall into a predetermined residual region near zero within a finite settling time and stay there, while all the closed-loop signals maintain bounded during operation; and (2) no artificial neural networks and filters are adopted, resulting in a low-complexity control property. Furthermore, the presented method can be extended readily to a broad range of second-order mechanical systems as its design builds upon a transformed system model. Rigorous mathematical analysis and simulations demonstrate the above theoretical findings.

Nomenclature

NN

neural network

LOS

line of sight

API

adding a power integrator

FTO

finite-time observer

ERF

earth reference frame

BRF

body-fixed reference frame

CV

centre of volume

AUV

autonomous underwater vehicle

EL

Euler–Lagrange

CFB

command-filter backstepping

IAE

integrated absolute error

ITAE

integrated time absolute error

MIAC

mean integrated absolute control

$ \mathbb{R}_{>0} $

set of positive real numbers

$ I_{n} $

identity matrix of size $n$

$ \mathbb{R}^{n} $

$n$ -dimensional Euclidean space

$ o_{g}x_{g}y_{g}z_{g} $

earth reference frame (ERF)

$ ox_{b}y_{b}z_{b} $

body-fixed reference frame (BRF)

$ {[x,\,y]} $

positions of the CV in ERF

$ \psi $

yaw attitude in BRF

$ {[u,\,v,\,r]} $

velocities in BRF

$ m_{i} $

mass, $ i=u,v,r $

$ d_{i} $

damping term, $ i=u,v,r $

$ \tau_{i} $

control input, $ i=u,r $

$ \delta_{i} $

system uncertainties and external disturbances, $ i=u,v,r $

$ \delta^{'}_{i} $

cross-coupling term, $ i=u,v,r $

$ m^{'}_{i} $

uncertain part of $ m_{i}, i=u,v,r $

$ d^{'}_{i} $

uncertain part of $ d_{i}, i=u,v,r $

$ \delta_{dis_{i}} $

wind resistance, $ i=u,v,r $

$ {[x_{d},\,y_{d}]}$

reference trajectory

1.0 Introduction

Recently, the stratosphere has received considerable attention from the modern aviation and aerospace industry that endeavors to exploit its stable atmospheric conditions [1–3]. This dramatically accelerates the development of the long-dwell stratospheric airship [4], which is a typical lighter-than-air aircraft that plays various roles from telecommunication to space-like observation outpost, similar to satellites [5–8]. To accomplish diverse mission objectives, driving the airship to reach and follow a time parameterised reference route, also termed as trajectory tracking control, is the most fundamental flight control task [9–11]. However, high nonlinearities, strong couplings, modeling inaccuracies, and unpredictable disturbances render the trajectory tracking control design quite intractable.

To date, many powerful control methodologies have been applied to solve this problem, such as adaptive control [12], backstepping method [9, 13], and sliding mode control [14]. Taking several kinds of uncertainty into consideration, Xiao et al. [14] proposed an adaptive integral sliding mode controller for an airship. However, no effective modification technique was provided to eliminate control chattering, thus yielding its implementation impossible. In other works [6, 10, 15–17], the unmodeled dynamics and external disturbances were identified and compensated by neural networks (NNs) or fuzzy logic systems (FLSs). However, the employment of NNs and FLSs will inevitably make the controller computationally expensive due to their inherent attributes. Furthermore, it should be emphasised that these controllers only apply to fully actuated airships, which, in general, cannot guarantee the tracking behaviour of underactuated ones.

In reality, most of the potential application scenarios of stratospheric airships are always associated with horizontal motion, and the stratospheric airship can automatically maintain the cruise altitude via an independent lift adjustment system alone [3, 15, 17, 18]. Given this fact, this work focuses on the horizontal trajectory tracking design for one kind of airship. At present, several challenging issues concerning this subject are still open, three of which are discussed in this paper. The first one is the underactuated problem. The conventional teardrop-shaped airship, operating at a proper flight altitude, is a typically underactuated system [6, 17], primarily due to the non-existence of an independent actuator producing the lateral force to command the sway dynamics. This poses new challenges as the lateral underactuation imposes a non-integrable restriction on the acceleration of the airship, and therefore it has become an important topic of research [3, 19–22]. Toward underactuated airships and other types of underactuated vehicles, several controllers have been proposed, both of which have great reference values for us, including the waypoint navigation method [23], the transverse function control [20], and the line of sight (LOS) approach [6, 10, 17, 21]. The controller grounded on the waypoint navigation method [23] demands pre-planning an optimal course that aligns with the direction of the wind, which may be suitable for hovering control rather than trajectory tracking control. The transverse function control also needs dynamic extension [20] to accommodate the lateral underactuation, which admittedly complicates the plant model. Although the LOS approach is deemed an efficient guidance law for underactuated vehicles, LOS-based trajectory tracking controllers require confining the tracking error of yaw angle $\psi_{e}$ to the interval $\left( -\frac{\pi}{2},\frac{\pi}{2}\right) $ for $\forall t\ge0$ . Generally, the error-constrained problem is unavoidably linked to rather complex nonlinear mappings. Some prime examples can be found in the works of Jia et al. [21] and Wu et al. [10], both of which introduced barrier functions and error-dependent transformations to meet such a restrictive condition.

The second issue is about the performance specification. Note that some previous designs [6, 9, 14, 15, 18, 23] can only ensure the equilibrium point of error dynamics is of asymptotic stability, namely, that the settling time, a critical indicator in control design for airships, is infinity. To this end, the finite-time stability and stabilisation theory was established [12, 24–26], permitting a bounded convergence time. Until now, the finite-time control has been an active research area because of its good control qualities, such as the faster decay rate, higher tracking accuracy, and better disturbance rejection property, and has been rapidly applied in various fields, including strict-feedback or nonstrict-feedback nonlinear systems [16, 27, 28], uncertain manipulators [29], and rigid spacecrafts [30, 31]. Nevertheless, own to the coupled nonlinearities in the kinematic equation of airships, extending these finite-time controllers to the trajectory tracking control design for airships is nontrivial, especially for underactuated ones. For fully actuated airships, several finite-time trajectory tracking or path following control algorithms were constructed [10, 17], where the hard computation of time derivatives of virtual control laws was obviated through filter tools. Though the filter can avoid repeated differentiation, it still structurally increases the complexity of control systems. Attributed to the adding a power integrator (API) method, a range of finite-time control strategies were formulated for airships and other mechanical systems without filters [30–35]. It should be noted that the power terms used in these API-based methods are strictly constrained to be an even integer or a ratio of two odd integers. Furthermore, in the above finite-time controllers, the prior designation of the size of the residual set is infeasible in that the steady-state accuracy counts on some unknowable model parameters and uncertainty bounds.

The third issue is related to the output feedback. Careful reviews of the above results reveal that most entail full state measurement. However, such a demand is hardly guaranteed in some practical applications. For example, the velocity information of airships cannot always be available at each instant coming from considerations of sensor faults. Inspired by this observation, some notable works used high-gain observers [21], sliding model observers [10], and fuzzy observers [27] to achieve output-feedback control. Although the maturity of the current output-feedback control, the finite-time trajectory tracking control design for airships subject to underactuated dynamics, unmeasured velocities, modeling imprecisions, and external disturbances is still a challenging control problem that needs more in-depth research.

Motivated by the above discussion, this paper proposes a novel approach by a combined application of the backstepping method, the API technique, and the idea of coordinate transformation. Our contributions are as follows:

1. 1. Compared to the asymptotic control algorithms [6, 9, 14, 15, 18, 23], our approach allows tracking behaviour to be preassigned by the operator, i.e. it drives the position errors of the airship into a preset range near zero within a finite settling time.

2. 2. By employing some useful lemmas, we relax the strong constraints placed on the power terms [30–35], broadening the set of possible design parameters. Moreover, unlike the sliding mode controllers [11, 14, 15], our control signal is continuous and chattering-free.

3. 3. A coordinate transformation is performed herein. Consequently, this work forsakes the extra dynamics needed in the transverse function control [20], and lifts the restriction of LOS-based controllers [10, 21]. Furthermore, the presented controller can be extended easily to a wide range of second-order mechanical systems as its design counts on a transformed equivalent model.

4. 4. Our method is structurally less demanding; no tools for filtering [6, 9, 14, 15, 18, 23] are involved, and no arduous computation of analytic differentiation required in the backstepping technique [20] is performed. Furthermore, this work realises velocity-free control and is robust enough in that it establishes a finite-time observer (FTO) to reconstruct unmeasured velocities and unpredictable uncertainties in a integral manner. In contrast to NN or FLS approximation [6, 10, 15–17, 27], the FTO can sharply lighten the calculational burden, making it particularly appealing for control applications.

Section 2 presents the preliminaries and control objective. Section 3 elucidates the coordinate conversion to cope with lateral underactuation. The major control design procedure and Lyapunov analysis are given in Section 4. Section 5 delineates the simulation results. Section 6 concludes this brief.

2.0 Preliminaries and problem formulation

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.

Definition 1 (see the work of Sun et al. [16]). If the equilibrium point $x=0$ of system (2) is referred to as a (locally) asymptotic stable node and for any initial state $x_{0}\in \mathbb U_{0} $ , there exist $\varepsilon\in\mathbb R_{>0}$ and a settling time function $T\!\left(\varepsilon,x_{0}\right)< \infty$ such that $\|x(t)\|\le\varepsilon, \forall t>T\!\left(\varepsilon,x_{0}\right)$ , then it is true that system (2) has a (locally) finite-time stable equilibrium point at $x=0$ . Furthermore, if $\mathbb U_{0}=\mathbb R^{n}$ , then $x=0$ is globally finite-time stable.

Lemma 1 (see the work of Sun et al. [16]). 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. [16]). 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. [34]). 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. [30]). 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. [30]). 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 [2, 9, 10, 14, 15, 17, 22, 23, 34], the airship model built around the horizontal motion can be directly given here, which is formulated by [6, 18, 36–38]

(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 [2, 19, 39], 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 1 (see other works [3, 10, 22]). The airship is flying at a proper cruising altitude with stable meteorological conditions, while maintaining buoyancy-weight balance. The pitch and roll angle are very small such that the associated dynamics can be neglected.

Assumption 2 (see other works [6, 19, 39]). 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.

Remark 1. From Equation (8), the airship, operating in three degrees of freedom, only has two independent actuating signals ( $\tau_{u},\tau_{r}$ ) in surge and yaw, which poses a prominent obstacle to steering the airship alone through a scheduled trajectory with stringent time and performance requirements. Indeed, it is physically apparent that most surface vehicles (SVs) and autonomous underwater vehicles (AUVs) propelling themselves on a horizontal plane are underactuated, and the motion equations of SVs and AUVs are pretty similar to that of the underactuated airship [20, 21]. When designing the motion control algorithms for SVs and AUVs, the performance specifications, such as the convergent time and the residual set, always exist. Therefore, this work is constructive for motion control design for SVs and AUVs to some extent.

Remark 2. Some effective altitude control techniques [3, 10, 22], such as inflating and deflating valves, make Assumption 1 mild and realistic. Assumption 2 is frequently made in the works on horizontal motion control of airships [6, 19, 39]. It is also important to point out that the term $\delta^{\prime}_{i}$ ( $i=u,v,r$ ) in Equation (9) is used to depict the coupling effects of pitch and roll.

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 $\forall t\ge0$ .

Assumption 3. The reference trajectory $\eta_{d}$ and its derivatives up to $\ddot{\eta}_{d}$ are bounded, continuous, and available for $\forall t\ge0$ .

3.0 Coordinate transformation

To fix the underactuated issue, a coordinate transformation is first conducted, specified as

(11) \begin{equation}\begin{aligned} x_{t}=x+\epsilon\cos\!\left(\psi\right), y_{t}=y+\epsilon\sin\!\left(\psi\right),\end{aligned}\end{equation}

where $\eta_{t}=\left[x_{t},y_{t}\right]^{T}$ denotes a new position, and except for having the definition in Equation (10), $\epsilon$ also is the distance between $\eta$ and $\eta_{d}$ (see Fig. 2). Calculating the second-order time-derivative of $\eta_{t}$ and taking Equation (8) into consideration yield

Figure 2. Coordinate transformation.

(12) \begin{equation}\begin{aligned} \ddot{x}_{t}=\dot{u}\cos\!\left(\psi\right)-\left(\dot{v}+\epsilon\dot{r}\right)\sin\!\left(\psi\right)-ur\sin\!\left(\psi\right)-\left(vr+\epsilon r^{2}\right)\cos\!\left(\psi\right),\\ \ddot{y}_{t}=\dot{u}\sin\!\left(\psi\right)+\left(\dot{v}+\epsilon\dot{r}\right)\cos\!\left(\psi\right)+ur\cos\!\left(\psi\right)-\left(vr+\epsilon r^{2}\right)\sin\!\left(\psi\right).\end{aligned}\end{equation}

Then, substituting Equation (8) for Equation (12), we obtain

(13) \begin{equation}\begin{aligned} \ddot{x}_{t}&=\frac{\cos\!\left(\psi\right)}{m_{u}}\tau_{u}-\frac{\epsilon\sin\!\left(\psi\right)}{m_{r}}\tau_{r}+\digamma_{x}+\delta_{tf_{x}},\\[5pt] \ddot{y}_{t}&=\frac{\sin\!\left(\psi\right)}{m_{u}}\tau_{u}+\frac{\epsilon\cos\!\left(\psi\right)}{m_{r}}\tau_{r}+\digamma_{y}+\delta_{tf_{y}},\end{aligned}\end{equation}

where

(14) \begin{equation}\begin{aligned} \digamma_{x}=&\frac{m_{v}vr-d_{u}u }{m_{u}}\cos\!\left(\psi\right)+\frac{m_{u}ur+d_{v}v }{m_{v}}\sin\!\left(\psi\right)-\frac{m_{uv}uv-d_{r}r}{m_{r}} \epsilon\sin\!\left(\psi\right) \\&-ur\sin\!\left(\psi\right)-\left(vr+\epsilon r^{2}\right)\cos\!\left(\psi\right),\end{aligned}\end{equation}

(15) \begin{equation}\begin{aligned} \digamma_{y}=&\frac{m_{v}vr-d_{u}u }{m_{u}}\sin\!\left(\psi\right)-\frac{m_{u}ur+d_{v}v }{m_{v}}\cos\!\left(\psi\right)+\frac{m_{uv}uv-d_{r}r}{m_{r}} \epsilon\cos\!\left(\psi\right) \\&+ur\cos\!\left(\psi\right)-\left(vr+\epsilon r^{2}\right)\sin\!\left(\psi\right),\end{aligned}\end{equation}

(16) \begin{equation}\delta_{tf_{x}}=\frac{\delta_{u}}{m_{u}}\cos\!\left(\psi\right)-\left(\frac{\delta_{v}}{m_{v}}+\frac{\epsilon \delta_{r}}{m_{r}} \right)\sin\!\left(\psi\right),\end{equation}

(17) \begin{equation}\delta_{tf_{y}}=\frac{\delta_{u}}{m_{u}}\sin\!\left(\psi\right)+\left(\frac{\delta_{v}}{m_{v}}+\frac{\epsilon \delta_{r}}{m_{r}} \right)\cos\!\left(\psi\right).\end{equation}

Denoting $\varkappa_{1}=\eta_{t}$ , $\varkappa_{2}=\dot{\eta}_{t}$ , $\digamma=\left[\digamma_{x},\digamma_{y}\right]^{T}$ , $\delta_{tf}=\left[\delta_{tf_{x}},\delta_{tf_{y}}\right]^{T}$ , and $\tau=\left[\tau_{u},\tau_{r} \right]^{T} $ , Equation (13) then can be rewritten in the synthetic form

(18) \begin{equation}\begin{aligned} \dot{\varkappa}_{1}&=\varkappa_{2},\\ \dot{\varkappa}_{2}&=R\!\left(\psi\right)M_{\epsilon}\tau+\digamma+\delta_{tf},\end{aligned}\end{equation}

where

(19) \begin{equation}R\!\left(\psi\right)=\begin{bmatrix} \cos\!\left(\psi\right)& \quad -\sin\!\left(\psi\right)\\[4pt] \sin\!\left(\psi\right)&\quad \cos\!\left(\psi\right)\end{bmatrix}, \ \ M_{\epsilon}=\begin{bmatrix} \dfrac{1}{m_{u}}&\quad 0\\[10pt] 0&\quad \dfrac{\epsilon}{m_{r}}\end{bmatrix}.\end{equation}

Obviously, $R\!\left(\psi\right)R^{T}\!\left(\psi\right)=I_{2}$ , and $M_{\epsilon}$ is a positive-definite diagonal matrix if, and only if, $\epsilon\in\mathbb R_{>0}$ . Note that in this paper, we suppose that the velocity cannot be measured, and therefore, the term $\digamma=\left[\digamma_{x},\digamma_{y}\right]^{T}$ is actually unavailable for control design. To this end, we consider it as a part of uncertainties, and define the lumped disturbances $\delta_{lu}$ as $\delta_{lu}=\digamma+\delta_{tf}$ , which will be estimated by an observation mechanism designed later. For the convenience of observer design, the following assumption is provided.

Assumption 4 (see other works [40–42]). A bounded positive constant $B_{\delta_{lu}}$ exists such that the lumped disturbances $\delta_{lu}$ satisfy $\|\dot{\delta}_{lu}\|\le B_{\delta_{lu}}$ .

Remark 3. From Equation (18), the original motion model for the underactuated airship has been transformed into a fully actuated uncertain Euler–Lagrange (EL) model, formally defined as $M\!\left(q\right)\ddot{q}+C\!\left(q,\dot{q} \right)\dot{q}+G\!\left(q\right)=\tau$ , where $q\in \mathbb R^{n}$ is the generalised coordinate, $M\!\left(q\right)\in \mathbb R^{n\times n}$ is the known matrix, $C\!\left(q,\dot{q}\right)\in \mathbb R^{n}$ is the known nonlinear dynamic, $G\!\left(q\right)\in \mathbb R^{n}$ accounts for uncertain dynamics and disturbances, and $\tau\in \mathbb R^{n}$ is the control variable; essentially, the EL model can describe various physical systems [33], such as robot manipulators [12, 24, 29] and spacecrafts [30, 31]. Thus, the presented method can be readily extended to a series of mechanical systems in second-order form.

Remark 4. It is emphasised that the coordinate transformation does not weaken the control quality; conversely, it is a potent approach to resolve the underactuated problem and realise the preassigned accuracy simultaneously. This is especially clear if we now construct a control algorithm that succeeds in forcing the signal $e_{t_{1}}=\varkappa_{1}-\eta_{d}$ to shrink to zero within a finite time $T_{f}$ and maintain it there for $t\ge T_{f}$ , i.e., $\|e_{t_{1}}\|=0, \forall t\ge T_{f}$ , then the actual position error e will satisfy

(20) \begin{align} \|e\| = & \|\eta-\eta_{d}\|=\Big\|\varkappa_{1}-\epsilon\begin{bmatrix} \cos\!\left(\psi\right)\\[4pt] \sin\!\left(\psi\right) \end{bmatrix}-\eta_{d}\Big\|=\Big\|e_{t_{1}}-\epsilon\begin{bmatrix} \cos\!\left(\psi\right)\\[4pt] \sin\!\left(\psi\right) \end{bmatrix}\Big\|\le\|e_{t_{1}}\|\nonumber\\[5pt] & +\Big\|\epsilon\begin{bmatrix} \cos\!\left(\psi\right)\\[4pt] \sin\!\left(\psi\right) \end{bmatrix}\Big\|\le\epsilon, \forall t\ge T_{f}. \end{align}

Evidently, decreasing $\epsilon$ yields a higher tracking precision.

Remark 5. It is necessary to remark that this paper views the airship as a rigid body and therefore the linear velocities u and v and the angular velocity r cannot change suddenly. In addition, the rate of change of the uncertainties $\delta$ is restrained as the stratospheric climate is stable and has limited energy. As a result, Assumption 4 is reasonable in reality and is widely employed in current literature [40–42] to facilitate the output feedback control realisation.

4.0 Main result

In this section, we first build an FTO to offer the exact information of the unmeasured velocity $\varkappa_{2}$ and the lumped disturbances $\delta_{lu}$ in an integrated manner. Afterward, in conjunction with the transformed model, the designed FTO, the backstepping technique, and the API method, a novel trajectory tracking control algorithm for stratospheric airships is proposed. Finally, a Lyapunov analysis is carried out to prove the closed-loop system stability.

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.

Theorem 1. Using the FTO (21) under Assumption 4, the velocity $\varkappa_{2}$ and the lumped disturbances $\delta_{lu}$ can be estimated accurately; more specifically, the estimation errors $e_{o_{1}}$ , $e_{o_{2}}$ , and $e_{o_{3}}$ can be driven to zero with a finite reaching time $T_{o}$ .

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. [43], and therefore, it is omitted here for space.

Remark 6. The FTO (21) essentially is a uniform robust exact differentiator. Historically, the concept of uniform exact convergence was proposed by Cruz-Zavala et al. [44] for the first time. Note that chattering, measurement noise, sampling step and small delay are out of the scope of this paper.

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 $\epsilon$ .

Remark 7. In the related works on API method [30–35], the power is circumscribed to be an even integer or a ratio of two odd integers. However, with the benefit of Lemma 4, such curtailments are removed in our work.

Remark 8. In contrast to the sliding mode control designs [11, 14, 15, 42], the control action (57) is chattering-free. In this note, although the signum operator $sign({\cdot})$ is employed, the highly undesirable control activity is avoided as the fractional power term $sig^{\varsigma}\!\left(z\right)=|z|^\varsigma sign\!\left( z\right) $ is a non-smooth but continuous function of z, where $\varsigma>0$ . This is vital for the long-time flight of the airship as control chattering may shorten the lifespan of the devices. Moreover, from the definition of $\rho\!\left( t\right)$ , we can find that $\rho\!\left( t\right)$ is a strictly decreasing positive function and it takes a very small value $\rho_{\infty}$ when $t>T_{\rho}$ . In this sense, we can avoid unnecessary control effort to some extent by incorporating it into the control action (57).

Table 1. Airship model parameters

5.0 Simulation

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. [6] and Zhang et al. [18], 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 [6, 10]:

(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. [9] is introduced in this paper, where the control parameters $k_{1}=k_{2}=eye(3)$ are identical to those used by Han et al. [9]. 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 [3, 6, 10, 17, 18, 22, 23]. 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 [30–35], 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.

6.0 Conclusion

This brief presented a novel finite-time velocity-free trajectory tracking control algorithm for an underactuated airship under the condition of modeling imperfections and environmental disturbances. First, a coordinate transformation was conducted to address the underactuated problem, which make the presented approach can be extended easily to a wide range of second-order mechanical systems. Second, an FTO was built to form a output-feedback control structure with disturbance estimation and attenuation ability. Finally, we blended the backstepping technique and API method into a Lyapunov design, which successfully guaranteed the finite-time convergence of the position errors $x_{e}$ and $y_{e}$ into a preassigned residual set around zero. The control design did not cover any analytically formidable calculation, filters, or self-tuning mechanisms (e.g. FLSs or NNs), leading to a structurally simple control attribute. Our future work will focus on extending this approach to a stratospheric airship with actuator faults.