2.1 Notations

Throughout this paper, for a scalar, $|{\cdot}|$ means the absolute value; for any vectors, $\textbf{\textit{x}} = [ {x_1},\break \ldots, {x_k}]^T$ and $\boldsymbol\alpha = {\left[ {{a_1}, \ldots ,{a_k}} \right]^T}$ , $si{g^a}(\textbf{\textit{x}})$ is expressed as $si{g^a}(\textbf{\textit{x}}) = {\left[ {si{g^{{a_1}}}({x_1}),\break \ldots ,si{g^{{a_k}}}({x_k})} \right]^T}$ , where $si{g^{{a_i}}}({x_i}) = sgn({x_i}){\left| {{x_i}} \right|^{{a_i}}}(i\,=\,1, \ldots ,k)$ , with $sgn( \cdot )$ being the signum function. Define $\left\| \cdot \right\|$ as the Euclidean norm of a vector or 2-norm of a matrix.

2.2 RBF NNs approximation

For any unknown smooth nonlinear function $f(\textbf{\textit{x}}):{\mathbb{R}^m} \to \mathbb{R}$ , the following RBF NNs can approximate it over a compact set $\Omega \subset {\mathbb{R}^m}$ :

(1) \begin{equation}f(\textbf{\textit{x}}) = {\boldsymbol\theta ^{*T}}\boldsymbol\phi (\textbf{\textit{x}}) + \epsilon(\textbf{\textit{x}})\end{equation}

where $\epsilon (\textbf{\textit{x}})$ is a bounded function denoting the approximation error. By supplying sufficient hidden neurons, an acceptable small value of the approximation error $\boldsymbol\epsilon(\textbf{\textit{x}})$ can be obtained.^{(Reference Ge, Hang, Lee and Zhang7)} $\textbf{\textit{x}} \in \boldsymbol\Omega $ indicates the input vector and the radial basis function vector is represented by $\boldsymbol\phi (\textbf{\textit{x}}) = {\left[ {{\phi _1}(\textbf{\textit{x}}), \ldots ,{\phi _k}(\textbf{\textit{x}})} \right]^T} \in {\mathbb{R}^k}$ , with $k\,>\,1$ indicating the number of nodes and ${\phi _i}(\textbf{\textit{x}})$ chosen as the Gaussian function

(2) \begin{equation}{\phi _i} = exp\left( { - {\frac{{{\left\| {\textbf{\textit{x}} - {\textbf{\textit{p}}_i}} \right\|}^2}}{\textbf{\textit{q}}_i^2}}} \right)\end{equation}

where ${\textbf{\textit{p}}_i} \in {\mathbb{R}^m}$ is the centre and q _i denotes the spread. The ideal weight vector ${\boldsymbol\theta ^*} \in {\mathbb{R}^k}$ is typically computed as

(3) \begin{equation}{\boldsymbol\theta^*} = arg\mathop {min}\limits_{\hat{\boldsymbol\theta} \in {\mathbb{R}^k}} \left\{ {\mathop {sup}\limits_{x \in \Omega } \left| {f(\textbf{\textit{x}}) - {{\hat{\boldsymbol\theta}}^T}\boldsymbol\phi (\textbf{\textit{x}})} \right|} \right\}\end{equation}

where $\hat{\boldsymbol\theta} $ is the estimation of ${\boldsymbol\theta ^*}$ .

2.3 Definitions and lemmas

Lemma 1 Lemma 1 Consider the finite-time convergent differentiator ^{(Reference Yi, Wang and Li29)}described as

(4) \begin{equation}\left\{ \begin{matrix} {{\dot{x}}_1} = {x_2} - {\zeta _1}{{\left| {{x_1} - {x^*}} \right|}^{1 - 1/p}}sgn({x_1} - {x}^*) \\[3pt] {{\dot{x}}_2} = - {\zeta _2}{{\left| {{x_1} - {x^*}} \right|}^{1 - 2/p}}sgn({x_1} - {x}^*) \end{matrix} \right.\end{equation}

where ${\zeta _1}\,>\,0$ , ${\zeta _2}\,>\,0$ , and $p\,>\,2$ are design parameters and x ₁and x ₂are the estimations of x ^*and ${\dot{x}^*}$ respectively. If ${\ddot x^*}$ is bounded, namely, $0 \le \left| {{{\ddot x}^*}} \right| < L$ , where L is an unknown positive constant, then the estimation errors ${e_1} = {x^*} - {x_1}$ and ${e_2} = {\dot{x}^*} - {x_2}$ are uniformly ultimately bounded and can approach zero within a finite time.

Lemma 2 Lemma 2 For any $x \in \mathbb{R}$ and $\Theta \in {\mathbb{R}^+ }$ , we have ^{(Reference Jin9)}

(5) \begin{equation}0 \le \left| x \right| \le \Theta + {\frac{{x^2}}{\sqrt {{x^2} + \Theta } }}\end{equation}

To achieve a guaranteed transient and steady-state tracking performance, the following speed function is employed:

(6) \begin{equation}\beta (t) = \left\{ \begin{matrix} {\dfrac{{T^4}{e^t}}{(1 - {c_f}){{(T - t)}^4} + {c_f}{T^4}{e^t}}} & {,0 \le t \le T} \\ \\[-6pt] {\frac{1}{{c_f}}} & {,t \ge T} \\ \end{matrix} \right.\end{equation}

where $T\,>\,0,0 < {c_f} \ll 1$ represent the preselected finite settling time and pre-chosen error transformation coefficient, respectively. Referring to the work of Zhao et al. ^{(Reference Zhao, Song, Ma and He34)}, some essential properties relative to the speed function $\beta (t)$ are directly offered here without proof.

3.1 Stratospheric airship model

The main structure of the stratospheric airship investigated herein contains a teardrop-shaped helium balloon, aerodynamic control surfaces, propellers, and an equipment tank, as exhibited in Fig. 1. The equipment tank is suspended from the body, whereas the rudders are attached to the empennage surfaces and provide steering torques. The propellers supply the primary propulsive forces for flight and are fixed on both sides of the equipment tank. Due to the existence of fans and valves, the airship can maintain a stable shape during operation. Thus, we consider the airship as a rigid body and neglect the aeroelastic effects. In this paper, we concentrate on the horizontal motions of the airship and assume that the airship is flying in a buoyancy-weight balance condition.

Figure 1. Sketch of the stratospheric airship.

To derive the kinematic and dynamic models of an airship, it is essential to define the coordinate systems as follows: An arbitrary point on the earth is chosen suitably as the origin of the earth reference frame (ERF), with x _g pointing north and y _g pointing east. The airship’s centre of volume (CV) is coincident with the origin of the body reference frame (BRF), with x _b pointing forward and y _b perpendicular to x _b, pointing right. According to Zheng et al.^{(Reference Zheng, Guan, Ma and Zhu37)}, the horizontal motion of an airship can be modelled as

(7) \begin{equation}\left\{ \begin{array}{l} {\dot{\boldsymbol\eta} = \textbf{\textit{R}}(\psi )\boldsymbol\nu } \\[3pt] {\dot{\boldsymbol\nu} = \textbf{\textit{F}}(\boldsymbol\nu ) + {\textbf{\textit{M}}^{ - 1}}\boldsymbol\tau + \textbf{\textit{d}}} \end{array} \right.\end{equation}

(8) \begin{equation}\textbf{\textit{R}}(\psi ) = \left[ \begin{array}{c@{\quad}c@{\quad}c} {cos\psi } & { - sin\psi } & 0 \\[2pt] {sin\psi } & {cos\psi } & 0 \\[2pt] 0 & 0 & 1 \end{array} \right]\!,\quad \textbf{\textit{M}} = \left[ \begin{array}{c@{\quad}c@{\quad}c} {{m_u}} & 0 & 0 \\[2pt] 0 & {{m_v}} & 0 \\[2pt] 0 & 0 & {m_r} \\[2pt] \end{array} \right]\end{equation}

In Equations 7 and 8, $\boldsymbol\eta = {[x,y,\psi ]^T}$ denotes the airship’s relative position and attitude under the ERF and $v = {[u,v,r]^T}$ represents linear velocities and yaw velocity under the BRF. M stands for the known inertia matrix, and $\textbf{\textit{F}}(\textbf{\textit{v}}) = {[{{\kern1.3pt}f_u}(\textbf{\textit{v}}),{f_v}(\textbf{\textit{v}}),{f_r}(\textbf{\textit{v}})]^T}$ is an unknown nonlinear function vector, the components of which are forces and moments acting on the airship’s body caused by gravity, buoyancy, aerodynamic forces, Coriolis forces, and Coriolis torques. It should be noted that the airship is actuated in only two degrees of freedom (DOF). Thus, the airship is referred to as an underactuated system. $\textbf{\textit{d}} = {[{d_u},{d_v},{d_r}]^T}$ indicates unknown environmental disturbances satisfying the following assumption.

3.2 Control objective

To explicitly clarify the control objective of this article, as illustrated in Fig. 2, the position tracking error E and attitude tracking error ${\psi _e}$ are defined as

(9) \begin{align} {x_e} &= {x_d} - x,\ {y_e} = {y_d} - y \nonumber\\ \psi _e &= {\psi _r} - \psi,\ E = \sqrt{x_e^2 + y_e^2} \end{align}

where ${\boldsymbol{\eta}_d} = {[{x_d},{y_d},{\psi _d}]^T}$ denotes a desired time-varying trajectory and ${\psi _r} \in ( - \pi ,\pi ]$ is the desired azimuth angle of the airship, calculated by

(10) \begin{equation}{\psi _r} = arctan2[{{\kern1.3pt}y_e},{x_e})\end{equation}

In this paper, our goal is to develop a proper control scheme to steer the airship forward and track the horizontal reference trajectory with prescribed performance. Therefore, we give the following logical assumptions, which are quite helpful in the design of the control framework.

Assumption 2 Assumption 2 ${\boldsymbol\eta _d}$ and its derivatives up to ${\ddot{\boldsymbol\eta}_d}$ are all known and bounded.

Figure 2. Geometrical relationship for trajectory tracking.

4.1 Position control

This subsection focuses on computing the virtual control signal $a_u^d$ to stabilise the position tracking error and calculating ${\tau _u}$ generated by the propellers for linear velocity tracking. To strictly guarantee the prescribed performance of position tracking, two error transformations are first defined as

(11) \begin{align} {\xi _E} &= \beta E \nonumber\\[3pt] {Z_E} &= {L_E}{\xi _E} \end{align}

where

(12) \begin{equation}{L_E} = \frac{1}{({k_{bE}} + {\xi _E})({k_{aE}} - {\xi _E})}\end{equation}

Note that the parameters ${k_{aE}}\,>\,0$ and ${k_{bE}}\,>\,0$ are given by the operator, representing the asymmetric error constraints on E or ${\xi _E}$ . It is obvious that Z _E is valid in the compact set ${\Omega _{{\xi _E}}}: = \{ {\xi _E}:{-k_{bE}} < {\xi _E} < {k_{aE}}\} $ and for any

(13) \begin{equation} -\!{k_{bE}} < {\xi _E}(0) = E(0) < {k_{aE}}\end{equation}

if Z _E is bounded for all $t \ge 0$ and since $\beta $ is a monotone increasing function with respect to time, then the prescribed constraints are guaranteed, i.e., $ - {\beta ^{ - 1}}{k_{bE}} < E < {\beta ^{ - 1}}{k_{aE}}$ for all $t \ge 0$ and $ - {k_{bE}} < {\xi _E} < {k_{aE}}$ for all $t \ge 0$ . Then, calculating the derivative of E along the trajectories of Equation 7 yields

(14) \begin{equation}\dot{E} = {\dot{x}_d}cos{\psi _d} + {\dot{y}_d}sin{\psi _d} - ucos{\psi _e} - vsin{\psi _e}\end{equation}

Thus, we have the derivative of the transformed position tracking error Z _E as

(15) \begin{equation}{\dot{Z}_E} = {\mu _E}\beta ({\beta ^{ - 1}}\dot{\beta} E + {\dot{x}_d}cos{\psi _d} + {\dot{y}_d}sin{\psi _d} - ucos{\psi _e} - vsin{\psi _e})\end{equation}

where

(16) \begin{equation}{\mu _E} = {\frac{{k_{aE}}{k_{bE}} + \xi _E^2}{{{({k_{bE}} + {\xi _E})}^2}{{({k_{aE}} - {\xi _E})}^2}}}\end{equation}

Since the velocity information of the airship is not measurable in real time and, specifically, at the current moment, to facilitate the design of the controller, an observer is constructed here to obtain exact velocity information. With the results in Ref. (Reference Yi, Wang and Li29), the nonlinear state observer is constructed as

(17) \begin{equation}\left\{ \begin{array}{l} {{{\dot{\boldsymbol\chi}}_1} = {\boldsymbol\chi_2} - {\boldsymbol{\lambda}_1}si{g^{{\boldsymbol\alpha_1}}}({\boldsymbol\chi _1} - \boldsymbol\eta )} \nonumber\\[3pt] {{{\dot{\boldsymbol\chi} }_2} = - {\boldsymbol\lambda_2}si{g^{{\boldsymbol\alpha_2}}}({\boldsymbol\chi _1} - {\boldsymbol\eta} )} \\ \end{array} \right.\end{equation}

where ${\boldsymbol\lambda _1} = diag({\lambda _{1u}},{\lambda _{1v}},{\lambda _{1r}})$ and ${\boldsymbol\lambda_2} = diag({\lambda _{2u}},{\lambda _{2v}},{\lambda _{2r}})$ are positive definite coefficient matrices and ${\boldsymbol\alpha_1}$ and ${\boldsymbol\alpha_2}$ are design parameter vectors, described by

(18) \begin{align} & {\boldsymbol\alpha _1} = {[1 - 1/{p_u},1 - 1/{p_v},1 - 1/{p_r}]^T} \nonumber\\[3pt] & {\boldsymbol\alpha _2} = {[1 - 2/{p_u},1 - 2/{p_v},1 - 2/{p_r}]^T} \end{align}

With ${p_i}\,>\,2$ , $i = u,v,r$ , and ${\boldsymbol\chi_1}$ and ${\boldsymbol\chi_2}$ are state vectors of the observer, denoting the estimations for $\boldsymbol\eta $ and $\dot{\boldsymbol\eta}$ , respectively. The observation errors are defined as

(19) \begin{align} & \tilde{\boldsymbol\eta} = \boldsymbol\eta - \hat{\boldsymbol\eta} \nonumber\\[3pt] & \tilde{\dot{\boldsymbol\eta}} = \dot{\boldsymbol\eta} - \hat{\dot{\boldsymbol\eta}} \end{align}

Clearly, by applying Lemma 1 and Assumption 3, an unknown positive constant $\vartheta $ exists, such that

(20) \begin{equation}\left\| {\tilde{\dot{\boldsymbol\eta}}} \right\| \le \vartheta \end{equation}

Let us gain more insight into Equation 7. Utilising the fact that $\left\| {\textbf{\textit{R}}(\psi )} \right\|\,=\,1$ , we can conclude that the estimation error $\tilde{v}$ is governed by $\vartheta $ , namely,

(21) \begin{equation}\left| {\tilde{v}} \right| \le {\vartheta _u} \le \left\| {\tilde{\textbf{\textit{v}}}} \right\| = \left\| {{\textbf{\textit{R}}^T}(\psi )(\dot{\boldsymbol{\eta}} - \hat{\dot{\boldsymbol{\eta}}} )} \right\| \le \vartheta \end{equation}

where $\tilde{\textbf{\textit{v}}} = \textbf{\textit{v}} - \hat{\textbf{\textit{v}}}$ , $\tilde{v} = v - \hat{v}$ and ${\vartheta _u}$ denotes an unknown positive constant. From Equations 15 and 21, virtual control $a_u^d$ can be designed as

(22) \begin{align}a_u^d &= {(cos{\psi _e})^{ - 1}}({\beta ^{ - 1}}\dot{\beta} E + {\dot{x}_d}cos{\psi _d} + {\dot{y}_d}sin{\psi _d} - \hat{v}sin{\psi _e}) \nonumber\\[3pt] &\quad + {(cos{\psi _e})^{ - 1}}\left( {{k_E}\mu _E^{ - 1}{L_E}E + {\frac{{Z_E}{\mu _E}\beta }{2{\gamma _u}}} + {\frac{{Z_E}{\mu _E}\beta {{\hat{\vartheta} }_u}}{\sqrt {Z_E^2\mu _E^2{\beta ^2} + \Theta } }}} \right)\end{align}

where ${k_E}\,>\,0$ , ${\gamma _u}\,>\,0$ , and $\Theta \,>\,0$ are design parameters and the update law for the estimate of ${\vartheta _u}$ is chosen as

(23) \vspace*{-3pt}\begin{equation}{\dot{\hat{\vartheta}}_u} = - {k_{\vartheta u}}{\hat{\vartheta} _u} + {\frac{Z_E^2\mu _E^2{\beta ^2}}{\sqrt {Z_E^2\mu _E^2{\beta ^2} + \Theta } }}\end{equation}

In this paper, we utilise the dynamic surface control method^{(Reference Swaroop, Hedrick, Yip and Gerdes23)} to avoid complicated analytic differentiation of the virtual control $a_u^d$ . The first-order filter can be expressed as

(24) \vspace*{-3pt}\begin{equation}{T_u}\dot{a}_u^c + a_u^c = a_u^d\end{equation}

where ${T_u}\,>\,0$ denotes the time constant. To carry out recursive design, the surge speed tracking error and the filter error are defined as

(25) \vspace*{-3pt}\begin{align} {u_e} &= u - a_u^c \nonumber\\ {y_u} &= a_u^c - a_u^d \end{align}

Then, the derivative of y _u is given by

(26) \vspace*{-3pt}\begin{align} {{{\dot{y}}_u}} & = \dot{a}_{.u}^c - \dot{a}_u^d \nonumber\\[3pt] & = - {\frac{{y_u}}{{T_u}}} + {B_u}({\psi _r},{{\dot{\psi} }_r},{{\dot{x}}_d},{{\ddot x}_d},{{\dot{y}}_d},{{\ddot y}_d},E,\dot{E},\beta ,\dot{\beta} ,\ddot \beta ,{\mu _E},{{\dot{\mu} }_E},{L_E},{{\dot{L}}_E},{Z_E},{{\dot{Z}}_E},{{\dot{\psi} }_e},{{\ddot \psi }_e},{{\hat{\vartheta} }_u},\hat{v},\dot{\hat{v}}) \end{align}

where ${B_u}( \cdot )$ is a continuously bounded function on a compact set on a compact set^{(Reference Wang and Huang24)} ${\Omega _{{y_u}}} \in {\mathbb{R}^{22}}$ , i.e., $\left| {{B_u}( \cdot )} \right| < {M_u}$ with M _u being an unknown positive constant. To facilitate a discussion of the design process, now consider the simple quadratic Lyapunov function candidate

(27) \vspace*{-3pt}\begin{equation}{V_E} = \frac{1}{2}Z_E^2 + \frac{1}{2}\tilde{\vartheta} _u^2 + \frac{1}{2}y_u^2\end{equation}

where ${\tilde{\vartheta} _u} = {\vartheta _u} - {\hat{\vartheta} _u}$ . Taking the derivative of V _E and using Equation 15, we obtain

(28) \begin{align}{\dot{V}_E} &= {Z_E}{\mu _E}\beta ({\beta ^{ - 1}}\dot{\beta} E + {\dot{x}_d}cos{\psi _d} + {\dot{y}_d}sin{\psi _d} - {u_e}cos{\psi _e} - {y_u}cos{\psi _e} - a_u^d - vsin{\psi _e}) + {y_u}{\dot{y}_u}\nonumber\\ &\quad - {\tilde{\vartheta} _u}{\dot{\hat{\vartheta}} _u}\end{align}

Substituting Equations 22, 23, and 26 into 28, and noting that the following inequalities hold:

(29) \begin{gather} - {Z_E}{\mu _E}\beta {y_u}cos{\psi _e} \le {{Z_E^2\mu _E^2{\beta ^2}} \over {2{\gamma _u}}} + {{{\gamma _u}y_u^2} \over 2} \nonumber\\[2pt] - {Z_E}{\mu _E}\beta \tilde{v}\textit{sin}{\psi _e} \le \left| {{Z_E}{\mu _E}\beta } \right|{\vartheta _u} \le \Theta {\vartheta _u} + {{Z_E^2\mu _E^2{\beta ^2}{\vartheta _u}} \over {\sqrt {Z_E^2\mu _E^2{\beta ^2} + \Theta } }} \nonumber\\[2pt] {{\tilde{\vartheta} }_u}{{\hat{\vartheta} }_u} \le - {{\tilde{\vartheta} _u^2} \over 2} + {{\vartheta _u^2} \over 2} \nonumber\\[2pt] {y_u}{B_u} \le {{y_u^2M_u^2} \over {4{b_u}}} + {b_u} \end{gather}

the derivative of V _E can be given by

(30) \begin{equation}{\dot{V}_E} \le - {k_E}Z_E^2 - {{{k_{{\vartheta _u}}}\tilde{\vartheta} _u^2} \over 2} - \left( {{1 \over {{T_u}}} - {{M_u^2} \over {4{b_u}}} - {{{\gamma _u}} \over 2}} \right)y_u^2 - {Z_E}{\mu _E}\beta {u_e}\textit{cos} {\psi _e} + \Theta {\vartheta _u} + {{{k_{{\vartheta _u}}}\vartheta _u^2} \over 2} + {b_u}\end{equation}

Consider the surge speed dynamics in Equation 7; evaluating the derivative of u _e gives

(31) \begin{equation}{\dot{u}_e} = {f_u}(\boldsymbol\nu ) + m_u^{ - 1}{\tau _u} + {d_u} - \dot{a}_u^c\end{equation}

It should be noted that the first equation in 7 can be rewritten as $\boldsymbol\nu = {\textbf{\textit{R}}^T}(\psi )\dot{\boldsymbol\eta} $ ; thus, the function ${f_u}(\boldsymbol\nu )$ can be restated as a continuous function ${f_u}(\boldsymbol\zeta )$ , where $\boldsymbol\zeta = {[{\boldsymbol\eta ^T},{\dot{\boldsymbol\eta} ^T}]^T}$ . Since the nonlinear ${f_u}(\boldsymbol\zeta )$ is indefinite, we use the RBF NNs to model the nonlinear functions ${f_u}(\boldsymbol\zeta )$ as^{(Reference Li, Tong, Liu and Feng13)}

(32) \begin{align} {{\hat{f}}_u}(\hat{\boldsymbol\zeta} ) & = \hat{\textbf{\textit{W}}}_u^T{\boldsymbol{\phi}_u}(\hat{\boldsymbol\zeta} ) \nonumber\\[2pt] & = \left[ {{\hat{w}}_{u1}},\ {{\hat{w}}_{u2}},\ \ldots ,\ {{\hat{w}}_{uk}} \right]\left[ \begin{array}{c} {\phi _{u1}}(\hat{\boldsymbol\zeta} ) \\[3pt] {\phi _{u2}}(\hat{\boldsymbol\zeta} ) \\[3pt] \vdots \\[3pt] {\phi _{uk}}(\hat{\boldsymbol\zeta} ) \end{array}\right] \end{align}

Where $\hat{\boldsymbol\zeta} = {\left[ {{{\hat{\boldsymbol\eta} }^T},{{\hat{\dot{\boldsymbol\eta}}}^T}} \right]^T}$ and ${\boldsymbol\phi _u}(\hat{\boldsymbol\zeta} )$ denotes the radial basis function vector with k being the number of nodes. ${\hat{\textbf{\textit{W}}}_u}$ is the estimation of the ideal weight vector $\textbf{\textit{W}}_u^*$ , which is computed by

(33) \begin{equation}\textbf{\textit{W}}_u^* = arg\,\mathop {min}\limits_{{{\hat{w}}_u} \in {\Omega _u}} \left\{ {\mathop {sup }\limits_{(\hat{\zeta} ,\zeta ) \in {\Xi _u}} \left| {{f_u}(\boldsymbol\zeta ) - \hat{\textbf{\textit{W}}}_u^T{\boldsymbol\phi _u}(\hat{{\boldsymbol\zeta}})} \right|} \right\}\end{equation}

Thus, the function ${f_u}(\boldsymbol\zeta )$ can be expressed as

(34) \begin{equation}{f_u}(\boldsymbol\zeta ) = \textbf{\textit{W}}_u^{*T}{\boldsymbol\phi _u}(\hat{\boldsymbol\zeta} ) + {\epsilon _u}\end{equation}

Where ${\epsilon _u}$ denotes the minimum approximation error satisfying $\left| {{\epsilon _u}} \right| < \epsilon _u^*$ , with $\epsilon _u^*$ being an unknown positive constant.

Therefore, the derivative of ${u_e}$ can be rewritten as

(35) \begin{equation}{\dot{u}_e} = \textbf{\textit{W}}_u^{*T}{\boldsymbol\phi _u}(\hat{\boldsymbol\zeta} ) + m_u^{ - 1}{\tau _u} + {\delta _u} - \dot{a}_u^c\end{equation}

Note that the following inequality holds:

(36) \begin{equation}\left| {\textbf{\textit{W}}_u^{*T}{\boldsymbol\phi _u}(\hat{\boldsymbol\zeta} ) + {\delta _u}} \right| \le \left\| {\textbf{\textit{W}}_u^{*T}} \right\|\left\| {{\boldsymbol\phi _u}(\hat{\boldsymbol\zeta} )} \right\| + {\Delta _u} = {k_u}{\iota _u}\end{equation}

where ${\kappa _u} = max\left\{ {\left\| {\textbf{\textit{W}}_u^{*T}} \right\|\!,{\Delta _u}} \right\}$ and ${\iota _u}\,=\,1 + \left\| {{\phi _u}} \right\|$ . Choosing the composite Lyapunov function for the position tracking control loop as

(37) \begin{equation}{V_P} = {V_E} + \frac{1}{2}u_e^2 + {1 \over {2{{\varpi}_u}}}\tilde{\Gamma} _u^2\end{equation}

where ${\Gamma _u} = \kappa _u^2$ and ${\tilde{\Gamma} _u} = {\Gamma _u} - {\hat{\Gamma} _u}$ , we obtain

(38) \begin{equation}{\dot{V}_P} = {\dot{V}_E} + {u_e}(\textbf{\textit{W}}_u^{*T}{\boldsymbol\phi _u} + m_u^{ - 1}{\tau _u} + {\delta _u} - \dot{a}_u^c) - {1 \over {{{\varpi}_u}}}{\tilde{\Gamma} _u}{\dot{\hat{\Gamma}} _u}\end{equation}

Then, the control input command for surge velocity tracking and the adaptive law can be designed as

(39) \begin{equation}{\tau _u} = {m_u}\left( {\dot{a}_u^c - {k_u}{{\hat{u}}_e} + {Z_E}{\mu _E}\beta cos{\psi _e} - 2{{\hat{\Gamma} }_u}{\Phi _u}{{\hat{u}}_e}} \right)\end{equation}

(40) \begin{equation}\hspace*{-60pt}{\dot{\hat{\Gamma}} _u} = {\varpi _u}\left[ {{\Phi _u}\hat{u}_e^2 - {\sigma _u}({{\hat{\Gamma} }_u} - {{\hat{\Gamma} }_u}(0))} \right]\end{equation}

where ${k_u}$ , ${\varpi _u}$ , ${\sigma _u}$ and ${\hat{\Gamma} _u}(0)$ are user-defined positive parameters, ${\hat{u}_e} = \hat{u} - a_u^c$ and ${\Phi _u} = \iota _u^2/2{c_u}$ , with ${c_u}\,>\,0$ . Based on the properties of RBF NNs, it should be clear that Φ_u has an unknown positive upper bound ${\Phi _{Mu}}$ , namely, $\left| {{\Phi _u}} \right| < {\Phi _{Mu}}$ . Using Young’s inequality, the following inequalities hold:

(41) \begin{align} {{{\tilde{\Gamma} }_u}{\Phi _u}u_e^2 - {{\tilde{\Gamma} }_u}{\Phi _u}\hat{u}_e^2} & {\,=\,2{{\tilde{\Gamma} }_u}{\Phi _u}{u_e}\tilde{u} - {{\tilde{\Gamma} }_u}{\Phi _u}{{\tilde{u}}^2}} \nonumber\\[2pt] & { \le 2\tilde{\Gamma} _u^2 + \Phi _{Mu}^2{\vartheta ^2}u_e^2 + {1 \over 4}\Phi _{Mu}^2{\vartheta ^4}} \nonumber\\[2pt] {{{\tilde{\Gamma} }_u}{\Phi _u}{{\tilde{u}}^2}} & { = {\Gamma _u}{\Phi _u}{{\tilde{u}}^2} - {{\tilde{\Gamma} }_u}{\Phi _u}{{\tilde{u}}^2}} \nonumber\\[2pt] {} & { \le ({\Gamma _u} + 1){\Phi _{Mu}}{\vartheta ^2} + {{{\Phi _{Mu}}{\vartheta ^2}} \over 4}\tilde{\Gamma} _u^2} \nonumber\\[2pt] {{\sigma _u}{{\tilde{\Gamma} }_u}({\Gamma _u} - {{\hat{\Gamma} }_u}(0))} & { \le {{{\sigma _u}} \over 2}\tilde{\Gamma} _u^2 + {{{\sigma _u}} \over 2}{{({\Gamma _u} - {{\hat{\Gamma} }_u}(0))}^2}} \end{align}

Using Equations 38, 39, and 40 and inequality 41 results in

(42) \begin{align} {{{\dot{V}}_P}} & = {{\dot{V}}_E} - {k_u}{{\hat{u}}_e}{u_e} + {u_e}(\textbf{\textit{W}}_u^{*T}{\boldsymbol\phi _u} + {\delta _u}) - 2{{\tilde{\Gamma} }_u}{\Phi _u}\hat{u}_{e} u_{e} - {{\tilde{\Gamma} }_u}{\Phi _u}\hat{u}_e^2\nonumber\\[2pt] &\quad + {\sigma _u}{{\tilde{\Gamma} }_u}({{\tilde{\Gamma} }_u} - {{\hat{\Gamma} }_u}(0)) + {Z_E}{\mu _E}\beta cos{\psi _e}{u_e} \nonumber\\[2pt] & \le {{\dot{V}}_E} - {k_u}u_e^2 + {k_u}{u_e}\tilde{u} + {\Gamma _u}{\Phi _u}u_e^2 - 2{{\hat{\Gamma} }_u}{\Phi _u}u_e^2 + 2{{\hat{\Gamma} }_u}{\Phi _u}{u_e}\tilde{u} \nonumber\\[3pt] & \quad - {{\tilde{\Gamma} }_u}{\Phi _u}\hat{u}_e^2 + {\sigma _u}{{\tilde{\Gamma} }_u}({{\hat{\Gamma} }_u} - {{\hat{\Gamma} }_u}(0)) + {Z_E}{\mu _E}\beta cos{\psi _e}{u_e} + {{{c_u}} \over 2} \nonumber\\[2pt] & \le {{\dot{V}}_E} - {{{k_u}} \over 2}u_e^2 + {{\tilde{\Gamma} }_u}{\Phi _u}u_e^2 + {{\hat{\Gamma} }_u}{\Phi _u}{{\tilde{u}}^2} - {{\tilde{\Gamma} }_u}{\Phi _u}\hat{u}_e^2 - {\sigma _u}\tilde{\Gamma} _u^2 \nonumber\\[2pt] &\quad + {\sigma _u}{{\tilde{\Gamma} }_u}({\Gamma _u} - {{\hat{\Gamma} }_u}(0)) + {Z_E}{\mu _E}\beta cos{\psi _e}{u_e} + {{{k_u}} \over 2}{\vartheta ^2} + {{{c_u}} \over 2} \nonumber\\[2pt] & \le {{\dot{V}}_E} - \left( {{{{k_u}} \over 2} - \Phi _{Mu}^2{\vartheta ^2}} \right)u_e^2 - \left( {{{{\sigma _u}} \over 2} - 2 - {1 \over 4}{\Phi _{Mu}}{\vartheta ^2}} \right)\tilde{\Gamma} _u^2 + {Z_E}{\mu _E}\beta cos{\psi _e}{u_e} \nonumber\\[2pt] &\quad + {\Phi _{Mu}}{\vartheta ^2}\left( {{\Gamma _u} + 1 + {1 \over 4}{\Phi _{Mu}}{\vartheta ^2}} \right) + {{{\sigma _u}} \over 2}{{({\Gamma _u} - {{\hat{\Gamma} }_u}(0))}^2} + {{{k_u}} \over 2}{\vartheta ^2} + {{{c_u}} \over 2} \end{align}

Substituting inequality 30 into 42, we obtain

(43) \begin{align} & {\dot{V}_P} \le - {k_E}Z_E^2 - {{{k_{\vartheta u}}\tilde{\vartheta} _u^2} \over 2} - \left( {{1 \over {{T_u}}} - {{M_u^2} \over {4{b_u}}} - {{{\gamma _u}} \over 2}} \right)y_u^2 - \left( {{{{k_u}} \over 2} - \Phi _{Mu}^2{\vartheta ^2}} \right)u_e^2 \nonumber\\[3pt] &\quad - \left( {{{{\sigma _u}} \over 2} - 2 - {1 \over 4}{\Phi _{Mu}}{\vartheta ^2}} \right){\tilde{\Gamma} ^2} + {\Game_u}\end{align}

where

(44) \begin{equation}{\Game_u} = \Theta {\vartheta _u} + {\Phi _{Mu}}{\vartheta ^2}\left( {{\Gamma _u} + 1 + {1 \over 4}{\Phi _{Mu}}{\vartheta ^2}} \right) + {{{\sigma _u}} \over 2}{\left( {{\Gamma _u} - {{\hat{\Gamma} }_u}(0)} \right)^2} + {{{k_u}} \over 2}{\vartheta ^2} + {{{k_{\vartheta u}}} \over 2}\vartheta _u^2 + {{{c_u}} \over 2} + {b_u}\end{equation}

The relevant conclusions in Equation 43 are needed for the stability analysis in subsection 4.3.

4.2 Attitude control

This subsection details the calculations of virtual control $a_r^d$ to stabilise the attitude tracking error and ${\tau _r}$ generated by aerodynamic control surfaces for yaw velocity tracking. Similarly, the following error transformations are defined as

(45) \begin{align} {\xi _{{\psi _e}}} &= \beta {\psi _e} \nonumber\\ {Z_{{\psi _e}}} &= {L_{{\psi _e}}}{\xi _{{\psi _e}}} \end{align}

where

(46) \begin{equation}{L_{{\psi _e}}} = {1 \over {\left( {{k_{b{\psi _e}}} + {\xi _{{\psi _e}}}} \right)\left( {{k_{a{\psi _e}}} - {\xi _{{\psi _e}}}} \right)}}\end{equation}

In Equation 46, ${k_{a{\psi _e}}}\,>\,0$ and ${k_{b{\psi _e}}}\,>\,0$ are user-defined parameters, standing for the asymmetric error constraints on ${\psi _e}$ or ${\xi _{{\psi _e}}}$ . Since the condition $\left| {{\psi _e}} \right|\,<\,0.5\pi $ should never be violated, the design parameters ${k_{a{\psi _e}}}$ and ${k_{b{\psi _e}}}$ should be selected appropriately such that

(47) \begin{equation} - 0.5\pi \le {k_{b{\psi _e}}} < {\xi _{{\psi _e}}}(0) = {\psi _e}(0) < {k_{a{\psi _e}}} \le 0.5\pi \end{equation}

The relevant analyses of the error transformations in Equation 45 are similar to the process applied to E and thus are omitted here. Following Equation 7, the derivative of ${\psi _e}$ satisfies

(48) \begin{equation}{\dot{\psi} _e} = {\dot{\psi} _r} - r\end{equation}

Then, computing the derivative of ${Z_{{\psi _e}}}$ results in

(49) \begin{equation}{\dot{Z}_{{\psi _e}}} = {\mu _{{\psi _e}}}\beta \left( {{\beta ^{ - 1}}\dot{\beta} {\psi _e} + {{\dot{\psi} }_r} - r} \right)\end{equation}

where

(50) \begin{align} {\mu _{{\psi _e}}} & = {{{k_{a{\psi _e}}}{k_{b{\psi _e}}} + \xi _{{\psi _e}}^2} \over {{{\left( {{k_{b{\psi _e}}} + {\xi _{{\psi _e}}}} \right)}^2}{{\left( {{k_{a{\psi _e}}} - {\xi _{{\psi _e}}}} \right)}^2}}} \nonumber\\[3pt] {{\dot{\psi} }_r} &= - {{\dot{x}}_d}{\jmath_y} + {{\dot{y}}_d}{\jmath_x} + \dot{x}{\jmath_y} - \dot{y}{\jmath_x} \end{align}

In Equation 50, ${j_x} = {x_e}/\left( {x_e^2 + y_e^2} \right)$ and ${j_y} = {y_e}/\left( {x_e^2 + y_e^2} \right)$ . Since the estimation error $\tilde{\boldsymbol\nu} $ is uniformly bounded, we obtain

(51) \begin{equation}\left| {\tilde{\dot{x}}} \right| + \left| {\tilde{\dot{y}}} \right| \le {\vartheta _r}\end{equation}

Where ${\vartheta _r}$ is an unknown positive constant. From Equations 49, 50, and 51, the virtual control $a_r^d$ and the update law are defined as

(52) \begin{equation}a_r^d = {\beta ^{ - 1}}\dot{\beta} {\psi _e} - {{\dot{x}}_d}{j_y} + {{\dot{y}}_d}{j_x} + \hat{\dot{x}}{j_y} - \dot{\hat{y}}{j_x} + {k_{{\psi _e}}}\mu _{{\psi _e}}^{ - 1}{L_{{\psi _e}}}{\psi _e} + {{{Z_{{\psi _e}}}{\mu _{{\psi _e}}}\beta } \over {2{\gamma _r}}} + {{{Z_{{\psi _e}}}{\mu _{{\psi _e}}}\beta {{\hat{\vartheta} }_r}} \over {\sqrt {Z_{{\psi _e}}^2\mu _{{\psi _e}}^2{\beta ^2} + \Theta } }}\end{equation}

(53) \begin{equation}{{\dot{\hat{\vartheta}} }_r} = - {k_{\vartheta r}}{{\hat{\vartheta} }_r} + {{Z_{{\psi _e}}^2\mu _{{\psi _e}}^2{\beta ^2}} \over {\sqrt {Z_{{\psi _e}}^2\mu _{{\psi _e}}^2{\beta ^2} + \Theta } }}\end{equation}

where ${k_{{\psi _e}}}\,{>}\,0$ , ${\gamma _r}\,{>}\,0$ , $\Theta \,{>}\,0$ and ${k_{\vartheta r}}\,{>}\,0$ are design parameters. The following first-order filter is introduced to circumvent the problem of ‘explosion of complexity’ and generate $a_r^c$ and $\dot{a}_r^c$ :

(54) \begin{equation}{T_r}\dot{a}_r^c + a_r^c = a_r^d\end{equation}

where ${T_r}\,>\,0$ represents the time constant, and the following three error variables are defined as:

(55) \begin{align} {r_e} &= r - a_r^c \nonumber\\ {{\hat{r}}_e} &= \hat{r} - a_r^c \nonumber\\ {y_r} &= a_r^c - a_r^d \end{align}

Taking the derivative of ${y_r}$ gives

(56) \begin{align} {{\dot{y}}_r} &= \dot{a}_r^c - \dot{a}_r^d \nonumber\\ & = - {{{y_r}} \over {{T_r}}} + {B_r}\big( {\psi _e},{{\dot{\psi} }_e},\beta ,\dot{\beta} ,\ddot \beta ,{\mu _{{\psi _e}}},{{\dot{\mu} }_{{\psi _e}}},{L_{{\psi _e}}},{{\dot{L}}_{{\psi _e}}},{Z_{{\psi _e}}},{{\dot{Z}}_{{\psi _e}}},{x_e},{{\dot{x}}_e},{{\dot{x}}_d},{{\ddot x}_d},{y_e},{{\dot{y}}_e},{{\dot{y}}_d},{{\ddot y}_d},\hat{\dot{x}}, \nonumber\\[3pt] & \qquad \dot{\hat{\dot{x}}},\dot{\hat{y}},\dot{\dot{\hat{y}}} \big) \end{align}

where ${B_r}( \cdot )$ is a continuous and bounded function on a compact set ${\Omega _{{y_r}}} \in \mathbb{R}{^{23}}$ , namely, ${\left| {{B_r}} \right| \le {M_r}}$ . According to the state-space Equation 7, the derivative of ${r_e}$ is given by

(57) \begin{equation}{\dot{r}_e} = {f_r}(\boldsymbol\nu ) + m_r^{ - 1}{\tau _r} + {d_r} - \dot{a}_r^c\end{equation}

Similarly, we still utilise RBF NNs to approximate the function ${f_r}(\boldsymbol\nu )$ and employ the back-stepping techniques to achieve the control law. To avoid procedure design process descriptions, we directly give the results of the control input command ${\tau _r}$ and the update law ${\dot{\hat{\Gamma}} _r}$ , where all the variables have the same meaning as in the position tracking control loop. Then, the control input command ${\tau _r}$ and the adaptive law ${\dot{\hat{\Gamma}} _r}$ are developed in the following forms:

(58) \begin{equation}{\tau _r} = {m_r}\left( {\dot{a}_r^c - {k_r}{{\hat{r}}_e} + {Z_{{\psi _e}}}{\mu _{{\psi _e}}}\beta - 2{{\hat{\Gamma} }_r}{\Phi _r}{{\hat{r}}_e}} \right)\end{equation}

(59) \begin{equation}{\dot{\hat{\Gamma}} _r} = {\varpi _r}\left( {{\Phi _r}\hat{r}_e^2 - {\sigma _r}\left[ {{{\hat{\Gamma} }_r} - {{\hat{\Gamma} }_r}(0)} \right]} \right)\end{equation}

where ${k_r}$ , ${\varpi _r}$ , ${\sigma _r}$ and ${\hat{\Gamma} _r}(0)$ are positive design constants and $\left| {{\Phi _r}} \right| \le {\Phi _{Mr}}$ with ${\Phi _{Mr}}$ being an unknown positive constant. For the attitude tracking control loop, if we choose the Lyapunov candidate function to be

(60) \begin{equation}{V_A} = \frac{1}{2}Z_{{\psi _2}}^2 + \frac{1}{2}\tilde{\vartheta} _r^2 + \frac{1}{2}y_r^2 + \frac{1}{2}r_e^2 + \frac{1}{2{\varpi _r}}\tilde{\Gamma} _r^2\end{equation}

then computing the first-order derivative of V _A and using Equations 49 and 55 gives

(61) \begin{equation}{\dot{V}_A} = {Z_{{\psi _e}}}{\mu _{{\psi _e}}}\beta \left( {{\beta ^{ - 1}}\dot{\beta} {\psi _e} + {{\dot{\psi} }_r} - {r_e} - {y_r} - a_r^d} \right) - {\tilde{\vartheta} _r}{\dot{\hat{\vartheta}} _r} + {y_r}{\dot{y}_r} + {r_e}{\dot{r}_e} - \frac{1}{{\varpi _r}}{\tilde{\Gamma} _r}{\dot{\hat{\Gamma}} _r}\end{equation}

Substituting Equations 52, 53, 56, 57, 58 and 59 into 61 and considering the following inequalities

(62) \begin{align} {Z_{{\psi _e}}}{\mu _{{\psi _e}}}\beta \left( {\tilde{\dot{x}}.{\jmath_y} - \tilde{\dot{y}}.{\jmath_x}} \right) & \le |{Z_{{\psi _e}}}{\mu _{{\psi _e}}}\beta |{\vartheta _r} \le \Theta {\vartheta _r} + {\frac{Z_{{\psi _e}}^2\mu _{{\psi _e}}^2{\beta ^2}{\vartheta _r}}{\sqrt {Z_{{\psi _e}}^2\mu _{{\psi _e}}^2{\beta ^2} + \Theta } }} \nonumber\\ - {Z_{{\psi _e}}}{\mu _{{\psi _e}}}\beta {y_r} & \le {\frac{Z_{{\psi _e}}^2\mu _{{\psi _e}}^2{\beta ^2}}{2{\gamma _r}}} + \frac{{\gamma _r}\gamma _r^2}{2} \nonumber\\ {{\tilde{\vartheta} }_r}{{\hat{\vartheta} }_r} & \le \frac{\tilde{\vartheta} _r^2}{2} + \frac{\vartheta _r^2}{2} \nonumber\\ {y_r}{B_r} & \le {\frac{y_r^2M_r^2}{4{b_r}}} + {b_r} \nonumber\displaybreak\\ {{\tilde{\Gamma} }_r}{\Phi _r}r_e^2 - {{\tilde{\Gamma} }_r}{\Phi _r}\hat{r}_e^2 & = 2{{\tilde{\Gamma} }_r}{\Phi _r}{r_e}\tilde{r} - {{\tilde{\Gamma} }_r}{\Phi _r}{{\tilde{r}}^2} \le 2\tilde{\Gamma} _r^2 + \Phi _{Mr}^2{\vartheta ^2}r_e^2 + \frac{1}{4}\Phi _{Mr}^2{\vartheta ^4} \nonumber\\ {{\tilde{\Gamma} }_r}{\Phi _r}{{\tilde{r}}^2} & = {\Gamma _r}{\Phi _r}{{\tilde{r}}^2} - {{\tilde{\Gamma} }_r}{\Phi _r}{{\tilde{r}}^2} \le ({\Gamma _r} + 1){\Phi _{Mr}}{\vartheta ^2} + \frac{{\Phi _{Mr}}{\vartheta ^2}}{4}\tilde{\Gamma} _r^2 \nonumber\\ {\sigma _r}{{\tilde{\Gamma} }_r}\left( {{\Gamma _r} - {{\hat{\Gamma} }_r}(0)} \right) & \le {{{\sigma _r}} \over 2}\tilde{\Gamma} _r^2 + {{{\sigma _r}} \over 2}{{\left( {{\Gamma _r} - {{\hat{\Gamma} }_r}(0)} \right)}^2}\end{align}

we thus arrive at

(63) \begin{align} {{\dot{V}}_{{\psi _e}}} &\le - {k_{{\psi _e}}}Z_{{\psi _e}}^2 - {{{k_{\vartheta r}}\tilde{\vartheta} _r^2} \over 2} - \left( {{1 \over {{T_r}}} - {{M_r^2} \over {4{b_r}}} - {{{\gamma _r}} \over 2}} \right)y_r^2 - {k_r}{{\hat{r}}_e}{r_e} + {r_e}\left( {W_r^{*T}{\phi _r} + {\delta _r}} \right) \nonumber\\ &\quad - 2{{\hat{\Gamma} }_r}{\Phi _r}{{\hat{r}}_e}{r_e} - {{\tilde{\Gamma} }_r}{\Phi _r}\hat{r}_e^2 + {\sigma _r}{{\tilde{\Gamma} }_r}\left[ {{{\hat{\Gamma} }_r} - {{\hat{\Gamma} }_r}(0)} \right] + \Theta {\vartheta _r} + {{{k_{\vartheta r}}\vartheta _r^2} \over 2} + {b_u} \nonumber\\ &\le - {k_{{\psi _e}}}Z_{{\psi _e}}^2 - {{{k_{\vartheta r}}\tilde{\vartheta} _r^2} \over 2} - \left( {{1 \over {{T_r}}} - {{M_r^2} \over {4{b_r}}} - {{{\gamma _r}} \over 2}} \right)y_r^2 - {k_r}r_e^2 + {k_r}{r_e}\tilde{r} + {\Gamma _r}{\Phi _r}r_e^2 - {{\tilde{\Gamma} }_r}{\Phi _r}\hat{r}_e^2 \nonumber\\ &\quad - 2{{\hat{\Gamma} }_r}{\Phi _r}r_e^2 + 2{{\hat{\Gamma} }_r}{\Phi _r}{r_e}\tilde{r} + {\sigma _r}{{\tilde{\Gamma} }_r}\left[ {{{\hat{\Gamma} }_r} - {{\hat{\Gamma} }_r}(0)} \right] + \Theta {\vartheta _r} + {{{k_{\vartheta r}}\vartheta _r^2} \over 2} + {b_u} + {{{c_r}} \over 2} \nonumber\\ & \le - {k_{{\psi _e}}}Z_{{\psi _e}}^2 - {{{k_{\vartheta r}}\tilde{\vartheta} _r^2} \over 2} - \left( {{1 \over {{T_r}}} - {{M_r^2} \over {4{b_r}}} - {{{\gamma _r}} \over 2}} \right)y_r^2 - \left( {{{{k_r}} \over 2} - \Phi _{Mr}^2{\vartheta ^2}} \right)r_e^2 \nonumber\\ &\quad - \left( {{{{\sigma _r}} \over 2} - 2 - {1 \over 4}{\Phi _{Mr}}{\vartheta ^2}} \right)\tilde{\Gamma} _r^2 + {\Game_r} \end{align}

where

(64) \begin{equation}{\Game_r} = {\Phi _{Mr}}{\vartheta ^2}\left( {{\Gamma _r} + 1 + {1 \over 4}{\Phi _{Mr}}{\vartheta ^2}} \right) + {{{\sigma _r}} \over 2}{\left( {{\Gamma _r} - {{\hat{\Gamma} }_r}(0)} \right)^2} + {{{k_r}} \over 2}{\vartheta ^2} + {{{k_{\vartheta r}}} \over 2}\vartheta _r^2 + {{{c_r}} \over 2} + \Theta {\vartheta _r} + {b_r}\end{equation}

The conclusions in Equation 63 are also needed for stability analysis of the controller in subsection 4.3.

4.3 Stability analysis

The properties of the proposed trajectory tracking control algorithm are summarised in the next theorem.

Proof. Assign the complete Lyapunov function candidate as

(65) \begin{equation}V = {V_P} + {V_A}\end{equation}

Given Equations 43 and 63, the time derivative of V satisfies

(66) \begin{align} \dot{V} &\le - {k_E}Z_E^2 - {k_{{\psi _e}}}Z_{{\psi _e}}^2 - {{{k_{\vartheta u}}\tilde{\vartheta} _u^2} \over 2} - {{{k_{\vartheta r}}\tilde{\vartheta} _r^2} \over 2} - \left( {{{{k_u}} \over 2} - \Phi _{Mu}^2{\vartheta ^2}} \right)u_e^2 \nonumber\\ &\quad - \left( {{{{k_r}} \over 2} - \Phi _{Mr}^2{\vartheta ^2}} \right)r_e^2 - \left( {{1 \over {{T_u}}} - {{M_u^2} \over {4{b_u}}} - {{{\gamma _E}} \over 2}} \right)y_u^2 - \left( {{1 \over {{T_r}}} - {{M_r^2} \over {4{b_r}}} - {{{\gamma _\psi }} \over 2}} \right)y_r^2 \nonumber\\ &\quad - \left( {{{{\sigma _u}} \over 2} - 2 - {1 \over 4}{\Phi _{Mu}}{\vartheta ^2}} \right)\tilde{\Gamma} _u^2 - \left( {{{{\sigma _r}} \over 2} - 2 - {1 \over 4}{\Phi _{Mr}}{\vartheta ^2}} \right)\tilde{\Gamma} _r^2 + \Game \end{align}

where $ \Game= {\Game_u} + {\Game_r}$ . Choosing appropriate design parameters such that

(67) \begin{align} &{{{k_i}} \over 2} - \Phi _{Mi}^2{\vartheta ^2} \ge 0 \nonumber\\[2pt] &{{{\sigma _i}} \over 2} - 2 - {1 \over 4}{\Phi _{Mi}}{\vartheta ^2} \ge 0 \nonumber\\[2pt] &{1 \over {{T_i}}} - {{M_i^2} \over {4{b_i}}} - {{{\gamma _i}} \over 2} \ge 0 \end{align}

where $i = u,r$ , inequality 66 can be rewritten as

(68) \begin{equation}\dot{V} \le - \ell V + \Game\end{equation}

Where $\ell = \textsf{min} \left\{ {2{k_j},{k_{\vartheta i}},{k_i} - 2\Phi _{Mi}^2{\vartheta ^2},\frac{2}{{{T_i}}} - \frac{{M_i^2}}{{2{b_i}}} - {\gamma _i},,{\sigma _i}{\gamma _i} - 4{\gamma _i} - \frac{{{\gamma _i}}}{2}{\Phi _{Mi}}{\vartheta ^2}} \right\}$ with $i = u,r$ , $j = E,{\psi _e}$ .

Integrating both sides of inequality 68 results in

(69) \begin{equation}0 \le V\left( t \right) \le {\Game \over \ell } + \left( {V\!\left( 0 \right) - {\Game \over \ell }} \right){e^{ - \ell t}}\end{equation}

Consequently, it is evident that the signals ${Z_E}$ , ${Z_{{\psi _e}}}$ , ${\tilde{\vartheta} _u}$ , ${\tilde{\vartheta} _r}$ , ${u_e}$ , ${r_e}$ , ${y_u}$ , ${y_r}$ , ${\tilde{\Gamma} _u}$ and ${\tilde{\Gamma} _r}$ are all semi-globally uniformly ultimately bounded. Therefore, all the signals in the loop system are uniformly bounded.

It should be highlighted that tracking error ${Z_i}\left( {i = E,{\psi _e}} \right)$ being bounded for all $t \ge 0$ implies that ${\xi _i}\left( {i = E,{\psi _e}} \right)$ always remains in the compact set $ - {k_{b{\xi _i}}} < {\xi _i} < - {k_{a{\xi _i}}}\left( {i = E,{\psi _e}} \right)$ as long as the initial conditions stated in Equations 13 and 47 are satisfied, which further indicates that the tracking-error-constrained requirements are not violated, i.e., $ - {k_{bE}} < E < {k_{aE}}$ and $ - {k_{b{\psi _e}}} < {\psi _e} < {k_{a{\psi _e}}}$ , $\forall t \ge 0$ .

Table 1 Airship simulation parameters

According to ${\xi _E} = \beta E$ and noting that ${\xi _E}$ is confined to the set $ - {k_{b{\xi _E}}} < {\xi _E} < - {k_{a{\xi _E}}}$ all the time, the following inequalities hold:

(70) \begin{align}\left\{ \begin{array}{ll} - \left( {1 - {c_f}} \right){{\left( {\frac{{T - t}}{T}} \right)}^4}{e^{ - t}}{k_{bE}} - {c_f}{k_{bE}}&\\[2pt] \qquad\quad < E < & ,0 \le t < T \\[2pt] \left( {1 - {c_f}} \right){{\left( {\frac{{T - t}}{T}} \right)}^4}{e^{ - t}}{k_{aE}} + {c_f}{k_{aE}}& \\[2pt] - {c_f}{k_{bE}} < E < {c_f}{k_{aE}}&,t \ge T \end{array}\right. \end{align}

where $\left( {1 - {c_f}} \right){\left( {\frac{{T - t}}{T}} \right)^4}{e^{ - t}}$ represents a preassigned convergence rate. From Equations 69 and 70, we understand that the convergence rate of position error E is not only ruled by T and c _f but also influenced by parameter $\ell$ . By adjusting the design parameters, we can make the error convergence rate greater than $\left( {1 - {c_f}} \right){\left( {\frac{{T - t}}{T}} \right)^4}{e^{ - t}}$ . However, regardless of the final value of parameter $\ell$ , the error convergence rate will not be less than $\left( {1 - {c_f}} \right){\left( {\frac{{T - t}}{T}} \right)^4}{e^{ - t}}$ , which signifies the main merit of this controller. Moreover, Equation 70 reveals that the proposed control scheme ensures that the tracking error E converges to the prescribed compact set ${\Omega _E}: = \left\{ {E: - {c_f}{k_{bE}} < E < {c_f}{k_{bE}}} \right\}$ within a finite time T. Adopting a similar reasoning process, we can also reach the same conclusion for ${\psi _e}$ . Thus, Theorem 1 is proved.

Figure 3. Trajectory of the airship.

Figure 4. Trajectory tracking errors.