2.1 Preliminaries

Consider the following system [Reference Yang, Su and Zhang19]:

(1) \begin{align}\dot \lambda \!\left( t \right) = f\!\left( {\lambda \!\left( t \right)} \right),\lambda \!\left( 0 \right) = {\lambda _0}\end{align}

where $\lambda = {\left[ {\begin{array}{l}{{\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}}\end{array}} \right]^{\rm{T}}}$ , $f$ ( $ \cdot )$ is a continuous nonlinear function, assuming the origin is an equilibrium point for system (1).

Lemma 1. If there exists a Lyapunov function V, such that $\dot V \le - {\mu _1}V - {\mu _2}{V^{{n_1}}}$ , ${\mu _1} \gt 0$ , ${\mu _2} \gt 0$ and $0 \lt {n_1} \lt 1$ , then the system (1) is finite-time stable at the origin, and V converges to 0 within a finite time [Reference Mu, Li, Wang, Fan, Zhao and Sun20].

(2) \begin{align}{\rm{T}} \le \frac{1}{{{\mu _1}\left( {1 - {n_1}} \right)}}{\rm{ln}}\frac{{{\mu _1}{V^{1 - {n_1}}}\!\left( 0 \right) + {\mu _2}}}{{{\mu _2}}}\end{align}

Lemma 2. If there exists a Lyapunov function V, such that $\dot V \le - {\mu _1}{V^{{n_1}}}$ , ${\mu _1} \gt 0$ and $0 \lt {n_1} \lt 1$ , then the system (1) is finite-time stable at the origin, and V converges to 0 within a finite time [Reference Wang, Deng, Xie and Pan21].

(3) \begin{align}{\rm{T}} \le \frac{{{V^{1 - {n_1}}}\left( 0 \right)}}{{{\mu _1}\left( {1 - {n_1}} \right)}}\end{align}

2.2 Problem statement

According to Ref. [Reference Ou22], the six degrees of freedom kinematic equation for an airship is given by

(4) \begin{align}\left( {\begin{array}{l}P\\[5pt] {\rm{\Omega }}\end{array}} \right) = \left( {\begin{array}{l@{\quad}l}{{R_1}} & {{0_{3 \times 3}}}\\[5pt] {{0_{3 \times 3}}}& {{R_2}}\end{array}} \right)\!\left( {\begin{array}{l}V\\[5pt] W\end{array}} \right)\end{align}

where $P = \left[ x,y,z\right]^{T},{\rm{\Omega }} = \left[\theta ,\psi ,\phi \right]^{T},V = \left[ u,v,w\right]^{T},W = \left[p,q,r\right]^{T},$

\begin{align*}{R_1} &= \left[ {\begin{array}{l@{\quad}l@{\quad}l}{{\rm{cos}}\;\psi {\rm{cos}}\;\theta } & {{\rm{cos}}\;\psi {\rm{sin}}\;\theta {\rm{sin}}\;\phi - {\rm{sin}}\;\psi {\rm{cos}}\;\phi } & {{\rm{cos}}\;\psi {\rm{sin}}\;\theta {\rm{cos}}\;\phi + {\rm{sin}}\;\psi {\rm{sin}}\;\phi }\\[5pt] {{\rm{sin}}\;\psi {\rm{cos}}\;\theta } & {{\rm{sin}}\;\psi {\rm{sin}}\;\theta {\rm{sin}}\;\phi + {\rm{cos}}\;\psi {\rm{cos}}\;\phi } & {{\rm{sin}}\;\psi {\rm{sin}}\;\theta {\rm{cos}}\;\phi - {\rm{cos}}\;\psi {\rm{sin}}\;\phi }\\[5pt] { - {\rm{sin}}\;\theta } & {{\rm{cos}}\;\theta {\rm{sin}}\;\phi } & {{\rm{cos}}\;\theta {\rm{cos}}\;\phi }\end{array}} \right],\\[5pt] &\qquad\qquad \qquad\qquad {R_2} = \left[ {\begin{array}{c@{\quad}c@{\quad}c}0 & {{\rm{cos}}\;\phi } & { - {\rm{sin}}\;\phi }\\[5pt] 0 & {{\rm{sec}}\;\theta {\rm{sin}}\;\phi } & {{\rm{sec}}\;\theta {\rm{cos}}\;\phi }\\[5pt] 1& {{\rm{tan}}\;\theta {\rm{sin}}\;\phi }& {{\rm{tan}}\;\theta {\rm{cos}}\;\phi }\end{array}} \right].\end{align*}

$x$ , $y$ , and $z$ are the airship’s displacements along the longitudinal, lateral, and vertical axes; $\theta ,\psi ,{\rm{and\;}}\phi $ are the airship’s pitch, yaw and roll angles; $u$ , $v$ and $w$ are the velocity vector components along each axis; $p$ , $q$ and $r$ are the airship’s roll, pitch, and yaw angular velocities, respectively.

Assuming the airship is a rigid body and neglecting its elastic deformation, the following ground-coordinate airship dynamic equation can be derived based on the Newton-Euler equation:

(5) \begin{align}m\frac{{d{\nu _G}}}{{dt}} = F\end{align}

(6) \begin{align}{\nu _G} = {\nu _o} + \omega \times {r_G}\end{align}

Upon substitution of Equations (6) into (5), we get

(7) \begin{align}m\frac{{d{\nu _o}}}{{dt}} + m\frac{{d\omega }}{{dt}} \times {r_G} + m\omega \times \frac{{d{r_G}}}{{dt}} = F\end{align}

where $m$ is the airship’s mass, ${\nu _G}$ is the velocity vector of the centre of gravity, ${\nu _o}$ is the velocity vector of the buoyancy centre, $\omega $ is the angular velocity vector, ${r_G}$ is the vector from the buoyancy centre to the centre of gravity, and $F$ is the force acting on the airship.

(8) \begin{align}\frac{{dH}}{{dt}} = \tau \end{align}

(9) \begin{align}H = {I_o}\omega + {r_G} \times \left( {m{v_o}} \right)\end{align}

Upon substitution of Equations (9) into (8), we obtain

(10) \begin{align}\left[ {\begin{array}{c@{\quad}c}{m{E_{3 \times 3}}}& { - mr_G^ \times }\\[5pt] {mr_G^ \times } & {{I_o}}\end{array}} \right]\left[ {\begin{array}{c}{{{\dot \nu }_o}}\\[5pt] {\dot \omega }\end{array}} \right] + \left[ {\begin{array}{c}{m{\omega ^ \times }{{\dot \nu }_o} + m{\omega ^ \times }\left( {{\omega ^ \times }{r_G}} \right)}\\[5pt] {{\omega ^ \times }\left( {{I_o}\omega } \right) + mr_G^ \times \left( {{\omega ^ \times }{\nu _o}} \right)}\end{array}} \right] = \left[ {\begin{array}{c}F\\[5pt] \tau \end{array}} \right]\end{align}

where $H$ is the moment of force relative to the buoyancy centre, ${I_o}$ is the moment of inertia matrix relative to the buoyancy centre, and $\tau $ is the moment of force acting on the airship.

By the application of the vector derivative rule, the airship dynamic equation in the body coordinate system can be derived.

Based on Equation (7), we derive

(11) \begin{align}m\frac{{d{\nu _o}}}{{dt}} + m{\omega ^ \times }{\nu _o} + m\frac{{d\omega }}{{dt}} \times {r_G} + m{\omega ^ \times }\left( {{\omega ^ \times }{r_G}} \right) = F\end{align}

where ${\omega ^ \times }$ is the cross product matrix of the angular velocity vector $\omega $ .

From Equation (10), we have

(12) \begin{align}{I_o}\frac{{d\omega }}{{dt}} + {\omega ^ \times }\left( {{I_o}\omega } \right) + m\!\left[ {r_G^ \times \frac{{d{\nu _o}}}{{dt}} + {\omega ^ \times }\left( {r_G^ \times {\nu _o}} \right)} \right] = \tau \end{align}

where $r_G^ \times $ is the cross product matrix of the angular velocity vector ${r_G}$ .

Combining Equations (11) and (12), we derive the matrix form of the dynamic equation

(13) \begin{align}\left[ {\begin{array}{c@{\quad}c}{m{E_{3 \times 3}}} & { - mr_G^ \times }\\[5pt] {mr_G^ \times } & {{I_o}}\end{array}} \right]\left[ {\begin{array}{c}{{{\dot \nu }_o}}\\[5pt] {\dot \omega }\end{array}} \right] + \left[ {\begin{array}{c}{m{\omega ^ \times }{{\dot \nu }_o} + m{\omega ^ \times }\left( {{\omega ^ \times }{r_G}} \right)}\\[5pt] {{\omega ^ \times }\left( {{I_o}\omega } \right) + mr_G^ \times \left( {{\omega ^ \times }{\nu _o}} \right)}\end{array}} \right] = \left[ {\begin{array}{c}F\\[5pt] \tau \end{array}} \right]\end{align}

If we account for the impact of additional mass and additional inertial force, the airship’s dynamic equation can be formulated as

(14) \begin{align}\left[ {\begin{array}{c@{\quad}c}{m{E_{3 \times 3}} + {M_a}} & { - mr_G^ \times }\\[5pt] {mr_G^ \times } & {{I_o} + {I_a}}\end{array}} \right]\left[ {\begin{array}{c}{{{\dot \nu }_o}}\\[5pt] {\dot \omega }\end{array}} \right] + \left[ {\begin{array}{c}{\left( {m{E_{3 \times 3}} + {M_a}} \right)\left[ {{\omega ^ \times }{{\dot \nu }_o} + {\omega ^ \times }\left( {{\omega ^ \times }{r_G}} \right)} \right]}\\[5pt] {{\omega ^ \times }\left[ {\left( {{I_o} + {I_a}} \right)\omega } \right] + mr_G^ \times \left( {{\omega ^ \times }{\nu _o}} \right)}\end{array}} \right] = \left[ {\begin{array}{c}F\\[5pt] \tau \end{array}} \right]\end{align}

where ${M_a}$ and ${I_a}$ are the additional mass matrix and additional moment of inertia matrix, respectively.

The attitude motion of the stratospheric airship in the geocentric reference frame ( ${O_b}{x_b}{y_b}{z_b}$ ) and the body reference frame ( ${O_e}{x_e}{y_e}{z_e}$ ) is shown in Fig. 1. ${C_V}$ is the buoyancy centre of the airship, while ${C_G}$ is the centre of gravity. The vector from the buoyancy centre to the centre of gravity is denoted as ${r_G} = {\left[ {\begin{array}{l}{{x_G},{y_G},{z_G}}\end{array}} \right]^{\rm{T}}}$ .

Figure 1. Attitude motion diagram.

From Equation (4) for the airship’s kinematic Equation and Equation (14) for the airship’s dynamic equation, we can obtain the mathematical model of the airship’s attitude motion

\begin{align*}\left[ {\begin{array}{c}{\dot \theta }\\[5pt] {\dot \psi }\\[5pt] {\dot \phi }\\[5pt] {\dot p}\\[5pt] {\dot q}\\[5pt] {\dot r}\end{array}} \right] = \left[ {\begin{array}{c}{ - r{\rm{sin}}\;\phi + q{\rm{cos}}\;\phi }\\[5pt] {{\rm{sec}}\;\theta \left( {r{\rm{cos}}\;\phi + q{\rm{sin}}\;\phi } \right)}\\[5pt] {p + {\rm{tan}}\;\theta \left( {r{\rm{cos}}\;\phi + q{\rm{sin}}\;\phi } \right)}\\[5pt] {\left( {{c_1}r + {c_2}p} \right)q + {c_3}\left( {L - {z_G}G{\rm{cos}}\;\theta {\rm{sin}}\;\phi } \right) + {c_4}N}\\[5pt] {{c_5}pr - {c_6}\left( {{p^2} - {r^2}} \right) + {c_7}\left( {M - {z_G}G{\rm{sin}}\;\theta } \right)}\\[5pt] {\left( {{c_8}p + {c_2}r} \right)q + {c_4}\left( {L - {z_G}G{\rm{cos}}\;\theta {\rm{sin}}\;\phi } \right) + {c_9}N}\end{array}} \right]\end{align*}

${\rm{where\;}}{c_1} = \frac{{I_{xz}^2 - {I_z}\left( {{I_y} - {I_z}} \right)}}{{I_{xz}^2 - {I_x}{I_z}}}$ ; ${c_2} = - \frac{{\left( {{I_x} - {I_y} + {I_z}} \right){I_{xz}}}}{{I_{xz}^2 - {I_x}{I_z}}}$ ; ${c_3} = - \frac{{{I_z}}}{{I_{xz}^2 - {I_x}{I_z}}}$ ; ${c_4} = - \frac{{{I_{xz}}}}{{I_{xz}^2 - {I_x}{I_z}}}$ ; ${c_5} = \frac{{{I_z} - {I_x}}}{{{I_y}}}$ ; ${c_6} = \frac{{{I_{xz}}}}{{{I_y}}}$ ; ${c_7} = \frac{1}{{{I_y}}}$ ; ${c_8} = \frac{{{I_x}\left( {{I_y} - {I_x}} \right) - I_{xz}^2}}{{I_{xz}^2 - {I_x}{I_z}}}$ ; ${c_9} = - \frac{{{I_x}}}{{I_{xz}^2 - {I_x}{I_z}}}$ ; $G$ is the gravity of the airship; $L$ , $M$ , $N$ are the roll, pitch and yaw moments applied to the airship, respectively; ${I_x}$ , ${I_y}$ , ${I_z}$ , and ${I_{xz}}$ are the generalised moments of inertia.

During actual flight, the airship may be affected by external disturbances, and there may be some deviations in the model parameters. Let ${I_{x1}} = {I_x} + {\rm{\Delta }}{I_x}$ , ${I_{y1}} = {I_y} + {\rm{\Delta }}{I_y}$ , ${I_{z1}} = {I_z} + {\rm{\Delta }}{I_z}$ , and ${I_{xz1}} = {I_{xz}} + {\rm{\Delta }}{I_{xz}}$ . Where ${\rm{\Delta }}{I_x}$ , ${\rm{\Delta }}{I_y}$ , ${\rm{\Delta }}{I_z}$ , and ${\rm{\Delta }}{I_{xz}}$ are the uncertain parts of the moments of inertia.

By selecting the state variable $x = {[\theta ,\psi ,\phi ,p,q,r]^T}$ , control variable $u = {[ - {I_z}L - {I_{xz}}N,M, - {I_{xz}}L - {I_x}N]^T}$ , output variable $h\left( x \right) = {[\theta ,\psi ,\phi ]^T}$ and external disturbance torque $d = {[{d_1},{d_2},{d_3}]^T}$ , the mathematical model of the airship attitude motion system can be described as the following nonlinear system

(15) \begin{align}\left\{ {\begin{array}{l}{\dot x = \left( {f\left( x \right) + \Delta f\left( x \right)} \right) + \left( {g\left( x \right) + \Delta g\left( x \right)} \right)u + T}\\[5pt] {y = h\left( x \right)}\end{array}} \right.\end{align}

where

\begin{align*}f\left( x \right) = \left[ {\begin{array}{c}{ - r{\rm{sin}}\;\phi + q{\rm{cos}}\;\phi }\\[5pt] {{\rm{sec}}\;\theta \left( {r{\rm{cos}}\;\phi + q{\rm{sin}}\;\phi } \right)}\\[5pt] {p + {\rm{tan}}\;\theta \left( {r{\rm{cos}}\;\phi + q{\rm{sin}}\;\phi } \right)}\\[5pt] {\left( {{c_1}r + {c_2}p} \right)q - {c_3}{z_G}G{\rm{cos}}\;\theta {\rm{sin}}\;\phi }\\[5pt] {{c_5}pr - {c_6}\left( {{p^2} - {r^2}} \right) - {c_7}{z_G}G{\rm{sin}}\;\theta }\\[5pt] {\left( {{c_8}p + {c_2}r} \right)q - {c_4}{z_G}G{\rm{cos}}\;\theta {\rm{sin}}\;\phi }\end{array}} \right],\end{align*}

\begin{align*}g\left( x \right) = \left[ {\begin{array}{c@{\quad}c@{\quad}c}0 & 0 & 0\\[5pt] 0 & 0 & 0\\[5pt] 0 &0& 0\\[5pt] {1/ - {I_x}{I_z} + I_{xz}^2}& 0 &0\\[5pt] 0& {1/{I_y}}& 0\\[5pt] 0& 0& {1/ - {I_x}{I_z} + I_{xz}^2}\end{array}} \right],\end{align*}

\begin{align*}T = \left[ {\begin{array}{c}0\\[5pt] 0\\[5pt] 0\\[5pt] {{d_1}/ - {I_x}{I_z} + I_{x{\rm{\Xi }}}^2}\\[5pt] {{d_2}/{I_y}}\\[5pt] {{d_3}/ - {I_x}{I_z} + I_{x{\rm{\Xi }}}^2}\end{array}} \right],\,{\rm{\Delta }}f\left( x \right) = \left[ {\begin{array}{c}0\\[5pt] 0\\[5pt] 0\\[5pt] {\left( {\left( {{c_{11}} - {c_1}} \right)r + \left( {{c_{22}} - {c_2}p} \right)} \right)q - \left( {{c_{33}} - {c_3}} \right){z_G}G{\rm{cos}}\;\theta {\rm{sin}}\;\phi }\\[5pt] {\left( {{c_{55}} - {c_5}} \right)pr - \left( {{c_{66}} - {c_6}} \right)\left( {{p^2} - {r^2}} \right) - \left( {{c_{77}} - {c_7}} \right){z_G}G{\rm{sin}}\;\theta }\\[5pt] {\left( {\left( {{c_{88}} - {c_8}} \right)p + \left( {{c_{22}} - {c_2}} \right)r} \right)q - \left( {{c_{44}} - {c_4}} \right){z_G}G{\rm{cos}}\;\theta {\rm{sin}}\;\phi }\end{array}} \right],\end{align*}

\begin{align*}{\rm{\Delta }}g\left( x \right) = \left[ {\begin{array}{c@{\quad}c@{\quad}c}0& 0& 0\\[5pt] 0& 0& 0\\[5pt] 0& 0& 0\\[5pt] {1/ - {I_{x1}}{I_{z1}} + I_{x1}^2 - 1/ - {I_x}{I_z} + I_{z2}^2}& 0& 0\\[5pt] 0& {1/{I_{y1}} - 1/{I_y}}& 0\\[5pt] 0& 0& {1/ - {I_{x1}}{I_{z1}} + I_{z1}^2 - 1/ - {I_x}{I_z} + I_{zz}^2}\end{array}} \right].\end{align*}

${\rm{where\;}}{c_{11}} = \frac{{I_{xz1}^2 - {I_{z1}}\left( {{I_{y1}} - {I_{z1}}} \right)}}{{I_{xz1}^2 - {I_{x1}}{I_{z1}}}}$ ; ${c_{22}} = - \frac{{\left( {{I_{x1}} - {I_{y1}} + {I_{z1}}} \right){I_{xz1}}}}{{I_{xz1}^2 - {I_{x1}}{I_{z1}}}}$ ; ${c_{33}} = - \frac{{{I_{z1}}}}{{I_{xz1}^2 - {I_{x1}}{I_{z1}}}}$ ; ${c_4} = - \frac{{{I_{xz1}}}}{{I_{xz1}^2 - {I_x}1{I_z}1}}$ ; ${c_5} = \frac{{{I_{z1}} - {I_{x1}}}}{{{I_{y1}}}}$ ; ${c_6} = \frac{{{I_{xz1}}}}{{{I_{y1}}}}$ ; ${c_7} = \frac{1}{{{I_y}1}}$ ; ${c_8} = \frac{{{I_{x1}}\left( {{I_{y1}} - {I_{x1}}} \right) - I_{xz1}^2}}{{I_{xz1}^2 - {I_{x1}}{I_{z1}}}}$ .

According to Ref. [Reference Wang, Zhou, Chen and Duan17], the nonlinear attitude system can be transformed into the following system through coordinate transformation and state feedback

(16) \begin{align}\left\{ {\begin{array}{l}{\dot \xi = A\xi + B\left( {\nu + f} \right)}\\[5pt] {y = C\xi }\end{array}} \right.\end{align}

where $v = {[{v_1},{v_2},{v_3}]^T}$ represents the pseudo-control variable; $f = {[{f_1},{f_2},{f_3}]^T}$ accounts for system parameter perturbations and external disturbances; $y$ denotes the system output; $\xi $ , $A$ , $B$ and $C$ are given by

\begin{align*}\xi &= \left( {\begin{array}{c}{{\xi _1}}\\[5pt] {{\xi _2}}\\[5pt] {{\xi _3}}\\[5pt] {{\xi _4}}\\[5pt] {{\xi _4}}\\[5pt] {{\xi _5}}\\[5pt] {{\xi _6}}\end{array}} \right) = \left( {\begin{array}{c}\theta \\[5pt] { - r{\rm{sin}}\;\phi + q{\rm{cos}}\;\phi }\\[5pt] \psi \\[5pt] {}\\[5pt] {\frac{1}{{{\rm{cos}}\;\theta }}\left( {r{\rm{cos}}\;\phi + q{\rm{sin}}\;\phi } \right)}\\[5pt] {}\\[5pt] \phi \\[5pt] {p + {\rm{tan}}\;\theta \left( {r{\rm{cos}}\;\phi + q{\rm{sin}}\;\phi } \right)}\end{array}} \right),\,A = \left( {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0& 1& 0 &0 &0& 0\\[5pt] 0& 0& 0 &0& 0 & 0\\[5pt] 0& 0& 0& 1& 0& 0\\[5pt] 0& 0& 0& 0& 0& 0\\[5pt] 0& 0& 0& 0& 0& 1\\[5pt] 0& 0& 0& 0& 0& 0\end{array}} \right),\,B = \left( {\begin{array}{c@{\quad}c@{\quad}c}0 & 0 & 0\\[5pt] 1 & 0 & 0\\[5pt] 0 & 0 & 0\\[5pt] 0 & 1 & 0\\[5pt] 0 & 0& 0\\[5pt] 0& 0& 1\end{array}} \right),\end{align*}

\begin{align*}C = \left( {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1 & 0& 0& 0& 0& 0\\[5pt] 0& 0& 1 &0& 0& 0\\[5pt] 0 &0& 0& 0& 1& 0\end{array}} \right) \qquad\qquad\qquad \end{align*}

3.1 Design of the RESO

The design of the RESO is exemplified using the pitch angle channel as an example, with similar design method applied to the other channels. Based on Equation (16), the state space representation of the pitch angle channel can be formulated as

(17) \begin{align}\left\{ {\begin{array}{l}{{{\dot \xi }_1} = {\xi _2}}\\[5pt] {{{\dot \xi }_2} = {f_1} + {v_1}}\\[5pt] {{y_1} = {\xi _1}}\end{array}} \right.\end{align}

To accurately estimate the combined effect of parameter perturbations and external disturbances [Reference Zhai, Li, Xu, Zhang and Li24], define ${\xi _7} = {f_1}$ , and extend the state variables as ${[{\xi _1},{\xi _2},{\xi _7}]^T}$ . Equation (17) can be rewritten as

(18) \begin{align}\left\{ {\begin{array}{l}{{{\dot \xi }_1} = {\xi _2}}\\[5pt] {{{\dot \xi }_2} = {\xi _7} + {\nu _1}}\\[5pt] {{{\dot \xi }_7} = {{\dot f}_1}}\\[5pt] {{y_1} = {\xi _1}}\end{array}} \right.\end{align}

In Equation (18), the formulation of the traditional nonlinear extended state observer (ESO) is as follows:

(19) \begin{align}\left\{ {\begin{array}{l}{{{\dot z}_1} = {z_2} - {b_1}fa{l_1}\left( {{e_{E1}},{a_1}} \right)}\\[5pt] {{{\dot z}_2} = {z_7} - {b_2}fa{l_2}\left( {{e_{E1}},{a_2}} \right) + {v_1}}\\[5pt] {{{\dot z}_7} = - {b_7}fa{l_7}\left( {{e_{E1}},{a_7}} \right)}\end{array}} \right.\end{align}

where ${z_1}$ represents the estimated value of the output ${y_1}$ , ${z_2}$ denotes the estimated value of the output derivative ${\dot y_1}$ , ${z_7}$ signifies the estimated value of ${f_1}$ , and ${e_{{{\rm{\;}}_{E1}}}}$ pertains to the estimation error of the pitch angle. So, ${z_1} = {\hat y_1}$ , ${z_2} = {\hat{\hat{y}}_1}$ , ${z_7} = {\hat f_1}$ , ${e_{{{\rm{\;}}_{E1}}}} = {\hat y_1} - {y_1}$ . The expression of $fa{l_i}\left( {{e_{E1}},{a_i}} \right)$ is as follows [Reference Wang, Yao and Qu25]

(20) \begin{align}fa{l_i}\left( {{e_{E1}},{a_i}} \right) = \left\{ {\begin{array}{l@{\quad}l}{{{\left| {{e_{E1}}} \right|}^{{a_i}}}{\rm{sgn}}\left( {{e_{E1}}} \right)} & {\left| {{e_{E1}}} \right| \gt \delta }\\[5pt] {\frac{{{e_{E1}}}}{{{\delta ^{\left( {1 - {a_i}} \right)}}}}} & {\left| {{e_{E1}}} \right| \lt \delta }\end{array}} \right.\end{align}

where $0 \lt {a_7} \lt {a_2} \lt {a_1} \le 1$ , $0 \lt \delta \lt 1$ .

To enhance the accuracy of estimation, the ESO typically applies a larger gain for the observer. Nonetheless, augmenting the gain amplifies the noise, consequently undermining the controller’s performance. Due to the measurability of the state variables in the attitude system, ${e_{{E_2}}} = {\dot{\hat{y}}_1} - {\dot{y}_1}$ is utilised in place of ${e_{E1}}$ to mitigate the gain and suppress the noise. The design of the RESO is as follows:

(21) \begin{align}\left\{ {\begin{array}{l}{{{\dot z}_2} = {z_7} - {{\bar b}_2}fa{l_2}\left( {{e_{E2}},{{\bar a}_2}} \right) + {v_1}}\\[5pt] {{{\dot z}_7} = - {{\bar b}_7}fa{l_7}\left( {{e_{E2}},{{\bar a}_7}} \right)}\end{array}} \right.\end{align}

To rectify the issue of poor control accuracy when the error is large, an enhancement is implemented:

(22) \begin{align}fa{l_i}\left( {{e_{E2}},{{\bar a}_i}} \right) = \left\{ {\begin{array}{l@{\quad}l}{{{\left| {{e_{E2}}} \right|}^{{{\bar a}_i}}}{\rm{tanh}}\left( {{e_{E2}}} \right)} & {\left| {{e_{E2}}} \right| \gt \delta }\\[5pt] {\frac{{{e_{E2}}}}{{{\delta ^{\left( {1 - {{\bar a}_i}} \right)}}}}} & {\left| {{e_{E2}}} \right| \lt \delta }\end{array}} \right.\end{align}

In contrast to the ESO, the RESO eliminates the need for secondary estimation of known output variable, diminishes the phase lag attributed to the observer and enhances the accuracy of disturbance estimation. Moreover, the proposed observer is characterised by a simple structure, capable of achieving commendable estimation performance with a smaller gain, and effectively reduces the engineering application cost.

3.2 Design of the CNFTSMC

Using the pitch angle channel as an illustration to design the CNFTSMC. The design method for the remaining channels is similar. Let ${y_{1d}}$ present the desired output, ${y_1}$ denote the actual output, and define the system error [Reference Jahanshahi, Yao, Khan and Moroz26] and its derivative as follows [Reference Chen, Yang and Fu27]:

(23) \begin{align}{e_1} = {y_1} - {y_{1d}}\end{align}

(24) \begin{align}{\dot e_1} = {\dot y_1} - {\dot y_{1d}}\end{align}

(25) \begin{align}{\ddot e_1} = {\ddot y_1} - {\ddot y_{1d}}\end{align}

Specify the composite sliding surfaces as follows:

(26) \begin{align}{s_{11}} = {e_1} + {\alpha _1}si{g^{\frac{{{p_1}}}{{{q_1}}}}}\left( {{e_1}} \right) + {\beta _1}{\rm{s}}i{g^{\frac{{{m_1}}}{{{n_1}}}}}\left( {{{\dot e}_1}} \right)\end{align}

(27) \begin{align}{s_{12}} = {\dot e_1} + {\alpha _2}{e_1} + {\beta _2}si{g^{{w_1}}}\left( {{e_1}} \right)\end{align}

where $si{g^{\frac{{{p_1}}}{{{q_1}}}}}\left( {{e_1}} \right) = {\rm{sgn}}\left( {{e_1}} \right)|{e_1}{|^{\frac{{{p_1}}}{{{q_1}}}}}$ , ${\alpha _1} \gt 0$ , ${\beta _1} \gt 0$ , ${\alpha _2} \gt 0$ , ${\beta _2} \gt 0$ , ${w_1} \gt 1$ , ${p_1}$ , ${q_1}$ , ${m_1}$ , and ${n_1}$ are odd numbers, and satisfy $1 \lt \frac{{{m_1}}}{{{n_1}}} \lt \frac{{{p_1}}}{{{q_1}}} \lt 2$ .

When $\left| {{e_1}} \right| \lt b$ , take the sliding surface ${s_{11}}$ ; when $\left| {{e_1}} \right| \geqslant b$ , take the sliding surface ${s_{12}}$ , where $0 \lt b \lt 1$ .

From Equation (26), it can be inferred that

(28) \begin{align}{\dot s_{11}} = {\dot e_1}\left( {1 + {\alpha _1}\frac{{{p_1}}}{{{q_1}}}si{g^{\frac{{{P_1}}}{{{q_1}}} - 1}}\left( {{e_1}} \right)} \right) + {\beta _1}\frac{{{m_1}}}{{{n_1}}}si{g^{\frac{{{m_1}}}{{{n_1}}} - 1}}\left( {{{\dot e}_1}} \right){\ddot e_1}\end{align}

By incorporating Equations (17) and (25) into Equation (28), we obtain

(29) \begin{align}{\dot s_{11}} = {\dot e_1}\left( {1 + {\alpha _1}\frac{{{p_1}}}{{{q_1}}}si{g^{\frac{{{p_1}}}{{{q_1}}} - 1}}\left( {{e_1}} \right)} \right) + {\beta _1}\frac{{{m_1}}}{{{n_1}}}si{g^{\frac{{{m_1}}}{{{n_1}}} - 1}}\left( {{{\dot e}_1}} \right)\left( {{f_1} + {\nu _1} - {{\ddot y}_{1d}}} \right)\end{align}

When the system adheres to the sliding surface, the equivalent control law can be derived as follows:

(30) \begin{align}{\nu _{1eq}} = {\ddot y_{1d}} - {f_1} - \frac{{{n_1}}}{{{\beta _1}{m_1}}}{\rm{s}}i{g^{2 - \frac{{{m_1}}}{{{n_1}}}}}\left( {{{\dot e}_1}} \right)\left( {1 + {\alpha _1}\frac{{{p_1}}}{{{q_1}}}si{g^{\frac{{{p_1}}}{{{q_1}}} - 1}}\left( {{e_1}} \right)} \right)\end{align}

Simultaneously, the subsequent switching control law is

(31) \begin{align}{\nu _{1sw}} = - {k_1}{s_{11}} - {\eta _1}sgn\left( {{s_{11}}} \right)\end{align}

where ${k_1} \gt 0$ , ${\eta _1} \gt 0$ .

Hence, when $\left| {{e_1}} \right| \lt b$ , the overall control law is

(32) \begin{align}{\nu _1} = {\ddot y_{1d}} - {f_1} - \frac{{{n_1}}}{{{\beta _1}{m_1}}}si{g^{2 - \frac{{{m_1}}}{{{n_1}}}}}\left( {{{\dot e}_1}} \right)\left( {1 + {\alpha _1}\frac{{{p_1}}}{{{q_1}}}si{g^{\frac{{{p_1}}}{{{q_1}}} - 1}}\left( {{e_1}} \right)} \right) - {k_1}{s_{11}} - {\eta _1}{\rm{sgn}}\left( {{s_{11}}} \right)\end{align}

From Equation (27), it can be inferred that

(33) \begin{align}{\dot s_{12}} = {\ddot e_1} + {\alpha _2}{\dot e_1} + {\beta _2}{w_1}si{g^{{w_1} - 1}}\left( {{e_1}} \right){\dot e_1}\end{align}

By incorporating Equations (17) and (25) into Equation (33), we obtain

(34) \begin{align}{\dot s_{12}} = \left( {{f_1} + {v_1} - {{\ddot y}_{1d}}} \right) + {\alpha _2}{\dot e_1} + {\beta _2}{w_1}{\rm{s}}i{g^{{w_1} - 1}}\left( {{e_1}} \right){\dot e_1}\end{align}

When the system adheres to the sliding surface, the equivalent control law can be derived as follows:

(35) \begin{align}{\nu _{1eq}} = {\ddot y_{1d}} - {f_1} - {\dot e_1}\left( {{\alpha _2} + {\beta _2}{w_1}si{g^{{w_1} - 1}}\left( {{e_1}} \right)} \right)\end{align}

Hence, when $\left| {{e_1}} \right| \geqslant b$ , the overall control law is

(36) \begin{align}{v_1} = {\ddot y_{1d}} - {f_1} - {\dot e_1}\left( {{\alpha _2} + {\beta _2}{w_1}si{g^{{w_1} - 1}}\left( {{e_1}} \right)} \right) - {k_1}{s_{12}} - {\eta _1}sgn\left( {{s_{12}}} \right)\end{align}

It is worth noting that the discontinuity of the sign function $sgn\left( \cdot \right)$ in the aforementioned control law can give rise to chattering, which can disrupt actuator performance and excite unmodeled high-frequency dynamic. To mitigate this phenomenon, this paper employs the hyperbolic tangent function $tanh\left( \cdot \right)$ instead of the sign function [Reference Aghababa and Akbari28]. Specifically, $sgn\left( {{s_{11}}} \right)$ is substituted with ${\rm{tanh}}\left( {{s_{11}}/\sigma } \right)$ , and $sgn\left( {{s_{12}}} \right)$ is substituted with ${\rm{tanh}}\left( {{s_{12}}/\sigma } \right)$ , where $\sigma $ is a very small positive number.

In summary, when $\left| {{e_1}} \right| \lt b$ , the composite sliding mode control law based on the RESO is

(37) \begin{align}{v_1} = {\ddot y_{1d}} - {\hat f_1} - \frac{{{n_1}}}{{{\beta _1}{m_1}}}{\rm{s}}i{g^{2 - \frac{{{m_1}}}{{{n_1}}}}}\left( {{{\dot e}_1}} \right)\left( {1 + {\alpha _1}\frac{{{p_1}}}{{{q_1}}}si{g^{\frac{{{p_1}}}{{{q_1}}} - 1}}\left( {{e_1}} \right)} \right) - {k_1}{s_{11}} - {\eta _1}{\rm{tanh}}\left( {{s_{11}}/\sigma } \right)\end{align}

When $\left| {{e_1}} \right| \geqslant b$ , the composite sliding mode control law based on the RESO is

(38) \begin{align}{v_1} = {\ddot y_{1d}} - {\hat f_1} - {\dot e_1}\left( {{\alpha _2} + {\beta _2}{w_1}si{g^{{w_1} - 1}}\left( {{e_1}} \right)} \right) - {k_1}{s_{12}} - {\eta _1}{\rm{tanh}}\left( {{s_{12}}/\sigma } \right)\end{align}

4.1 Observer stability analysis

In this section, we take the pitch channel as an example to demonstrate that the proposed improved RESO guarantees bounded estimation error, with analogous analysis applied to the remaining two channels.

Proof. Let ${\lambda _i}\left( {{e_{E2}}} \right) = \frac{{fa{l_i}\left( {{e_{{{\rm{\;}}_{E2}}}},{{\bar a}_{{{\rm{\;}}_i}}}} \right)}}{{{e_{{{\rm{\;}}_{E2}}}}}}$ , ${\dot f_1} = \varpi $ , and define $e_{E3}=\hat{f}_{1}-f_{1}$ as the estimation error of ${f_1}$ . ${e_E} = {[{e_{E2}},{e_{E3}}]^T}$ represents the state error estimation vector. By employing Equation (21), we can deduce the differential equation for ${e_E}$

(39) \begin{align}{\dot e_E} = {A_E}{e_E} - {B_E}\varpi \end{align}

where ${A_E} = \left[ {\begin{array}{l}{ - {{\bar b}_2}{\lambda _2}\left( {{e_{E2}},{{\bar a}_2}} \right)} 1\\[5pt] { - {{\bar b}_7}{\lambda _7}\left( {{e_{E2}},{{\bar a}_7}} \right)} 0\end{array}} \right]$ , ${B_E} = \left[ {\begin{array}{l}0\\[5pt] 1\end{array}} \right]$

Given that ${\bar b_2} \gt 0$ , ${\bar b_7} \gt 0$ , ${\lambda _2}\left( {{e_{{{\rm{\;}}_{E2}}}},{{\bar a}_{{{\rm{\;}}_2}}}} \right) \gt 0$ , ${\lambda _7}\left( {{e_{{{\rm{\;}}_{E2}}}},{{\bar a}_{{{\rm{\;}}_7}}}} \right) \gt 0$ , ${A_E}$ can be identified as a Hurwitz matrix. Based on the Hurwitz stability theory [Reference Yao and Deng29], it can be inferred that Equation (39) is stable, and ${e_E}$ is bounded. Consequently, the estimation error of the proposed improved RESO is bounded.

4.2 Controller stability analysis

In this section, we illustrate the convergence of the tracking error to zero within a finite time and demonstrate that the actual state can track the desired value within a finite time using the pitch channel as an example.

Based on the stability analysis of the observer, it is known that the estimation error of the proposed improved RESO is bounded. Without loss of generality, assume ${\left| {{f_1} - {{\hat f}_1}} \right|_{{\rm{max}}}} \le {L_1}$ , ${L_1} \lt {\eta _1}$ .

Proof. When $\left| {{e_1}} \right| \lt b$ , we select the Lyapunov function as

(40) \begin{align}{V_{11}} = \frac{1}{2}{s_{11}}^2\end{align}

Differentiating Equation (40) yields

(41) \begin{align}{\dot V_{11}} = {s_{11}}\left( {{{\dot e}_1}\left( {1 + {\alpha _1}\frac{{{p_1}}}{{{q_1}}}si{g^{\frac{{{p_1}}}{{{q_1}}} - 1}}\left( {{e_1}} \right)} \right) + {\beta _1}\frac{{{m_1}}}{{{n_1}}}si{g^{\frac{{{m_1}}}{{{n_1}}} - 1}}\left( {{{\dot e}_1}} \right)\left( {{f_1} + {\nu _1} - {{\ddot y}_{1d}}} \right)} \right)\end{align}

By substituting Equation (37) into Equation (41), we obtain

(42) \begin{align}{\dot V_{11}} = {\beta _1}\frac{{{m_1}}}{{{n_1}}}si{g^{\frac{{{m_1}}}{{{n_1}}} - 1}}\left( {{{\dot e}_1}} \right){s_{11}}\left( {{f_1} - {{\hat f}_1} - {k_1}{s_{11}} - {\eta _1}{\rm{tanh}}\left( {{s_{11}}/\sigma } \right)} \right)\end{align}

Since ${\left| {{f_1} - {{\hat f}_1}} \right|_{{\rm{max}}}} \le {L_1}$ , ${L_1} \lt {\eta _1}$ , according to Equation (42), we have

(43) \begin{align}{\dot V_{11}} \le - {\beta _1}\frac{{{m_1}}}{{{n_1}}}si{g^{\frac{{{m_1}}}{{{n_1}}} - 1}}\left( {{{\dot e}_1}} \right)\left( {{k_1}s_{11}^2 + \left( {{\eta _1} - {L_1}} \right)\left| {{s_{11}}} \right|} \right)\end{align}

Let ${\beta _1}\frac{{{m_1}}}{{{n_1}}}si{g^{\frac{{{m_1}}}{{{n_1}}} - 1}}\left( {{{\dot e}_1}} \right) = \varepsilon $ , since $1 \lt \frac{{{m_1}}}{{{n_1}}} \lt 2$ , then $si{g^{\frac{{{m_1}}}{{{n_1}}} - 1}}\left( {{{\dot e}_1}} \right) \gt 0$ , and thus $\varepsilon \gt 0$ .

From Equation (43), we obtain

(44) \begin{align}{\dot V_{11}} \le - 2\varepsilon {k_1}{V_{11}} - \sqrt 2 \varepsilon \left( {{\eta _1} - {L_1}} \right)V_{11}^{\frac{1}{2}}\end{align}

Let $2\varepsilon {k_1} = {\alpha _3}$ , $\sqrt 2 \varepsilon \left( {{\eta _1} - {L_1}} \right) = {\alpha _4}$ , we have

(45) \begin{align}{\dot V_{11}} \le - {\alpha _3}{V_{11}} - {\alpha _4}V_{11}^{\frac{1}{2}}\end{align}

According to Lemma 1, the sliding variable ${s_{11}}$ converges to zero within a finite time, with the convergence time satisfies

(46) \begin{align}{{\rm{T}}_1} \le \frac{2}{{{\alpha _3}}}{\rm{ln}}\frac{{{\alpha _3}V_{11}^{\frac{1}{2}}\left( 0 \right) + {\alpha _4}}}{{{\alpha _4}}}\end{align}

Given that the system state reaches the sliding surface ${s_{11}}$ at $t = {t_1}$ , where ${t_1} \le {T_1}$ , it ensures that when

(47) \begin{align}{e_1} + {\alpha _1}si{g^{\frac{{{p_1}}}{{{q_1}}}}}\left( {{e_1}} \right) + {\beta _1}si{g^{\frac{{{m_1}}}{{{n_1}}}}}\left( {{{\dot e}_1}} \right) = 0\end{align}

From Equation (47), we obtain

(48) \begin{align}{\dot e_1} = - \beta _1^{ - \frac{{{n_1}}}{{{m_1}}}}e_1^{\frac{{{n_1}}}{{{m_1}}}}{\left( {1 + {\alpha _1}si{g^{\frac{{{p_1}}}{{{q_1}}} - 1}}\left( {{e_1}} \right)} \right)^{\frac{{{n_1}}}{{{m_1}}}}}\end{align}

Choosing the Lyapunov function as

(49) \begin{align}{V_{12}} = \frac{1}{2}{e_1}^2\end{align}

Differentiating Equation (49) yields

(50) \begin{align}{\dot V_{12}} = {e_1}{\dot e_1}\end{align}

By substituting Equation (48) into Equation (50), we obtain

(51) \begin{align}{\dot V_{12}} = - \beta _1^{ - \frac{{{n_1}}}{{{m_1}}}}e_1^{\frac{{{n_1}}}{{{m_1}}} + 1}{\left( {1 + {\alpha _1}si{g^{\frac{{{p_1}}}{{{q_1}}} - 1}}\left( {{e_1}} \right)} \right)^{\frac{{{n_1}}}{{{m_1}}}}}\end{align}

From Equation 51, we obtain

(52) \begin{align}\begin{array}{l}{{{\dot V}_{12}}} { = - \beta _1^{ - \frac{{{n_1}}}{{{m_1}}}}{{\sqrt 2 }^{\frac{{{n_1}}}{{{m_1}}} + 1}}V_{12}^{\frac{{{n_1} + {m_1}}}{{2{m_1}}}}{{\left( {1 + {{\sqrt 2 }^{\frac{{{p_1}}}{{{q_1}}} - 1}}{\alpha _1}V_{12}^{\frac{{{p_1} - {q_1}}}{{2{q_1}}}}} \right)}^{\frac{{{n_1}}}{{{m_1}}}}}}\\ \\ {} \;\qquad { \le - \beta _1^{ - \frac{{{n_1}}}{{{m_1}}}}{{\sqrt 2 }^{\frac{{{n_1}}}{{{m_1}}} + 1}}V_{12}^{\frac{{{n_1} + {m_1}}}{{2{m_1}}}}}\end{array}\end{align}

Let $\beta _1^{ - \frac{{{n_1}}}{{{m_1}}}}{\sqrt 2 ^{\frac{{{n_1}}}{{{m_1}}} + 1}} = {\alpha _5}$ , we have

(53) \begin{align}{\dot V_{12}} \le - {\alpha _5}V_{12}^{\frac{{{n_1} + {m_1}}}{{2{m_1}}}}\end{align}

According to Lemma 2, the attitude tracking error ${e_1}$ converges to zero within a finite time, with the convergence time satisfies

(54) \begin{align}{{\rm{T}}_2} \le \frac{{2{m_1}}}{{{\alpha _5}\left( {{m_1} - {n_1}} \right)}}V_{12}^{\frac{{{m_1} - {n_1}}}{{2{m_1}}}}\left( 0 \right)\end{align}

In conclusion, when $\left| {{e_1}} \right| \lt b$ , the system is finite-time stable, and the overall convergence time satisfies

(55) \begin{align}{\rm{T}} \le {{\rm{T}}_1} + {{\rm{T}}_2}\end{align}

Proof. When $\left| {{e_1}} \right| \geqslant b$ , we select the Lyapunov function as

(56) \begin{align}{V_{21}} = \frac{1}{2}{s_{12}}^2\end{align}

Differentiating Equation (56) yields

(57) \begin{align}{\dot V_{21}} = {s_{12}}\left( {{f_1} + {\nu _1} - {{\ddot y}_{1d}} + {{\dot e}_1}\left( {{\alpha _2} + {\beta _2}{w_1}si{g^{{w_1} - 1}}\left( {{e_1}} \right)} \right)} \right)\end{align}

By substituting Equation (38) into Equation (57), we obtain

(58) \begin{align}{\dot V_{21}} = {s_{12}}\left( {{f_1} - {{\hat f}_1} - {k_1}{s_{12}} - {\eta _1}{\rm{tanh}}\left( {{s_{12}}/\sigma } \right)} \right)\end{align}

Since ${\left| {{f_1} - {{\hat f}_1}} \right|_{{\rm{max}}}} \le {L_1}$ , ${L_1} \lt {\eta _1}$ , according to Equation (58), we have

(59) \begin{align}{\dot V_{21}} \le - {k_1}{s_{12}}^2 - \left( {{\eta _1} - {L_1}} \right)\left| {{s_{12}}} \right|\end{align}

From Equation (59), obtain

(60) \begin{align}{\dot V_{21}} \le - 2{k_1}{V_{21}} - \sqrt 2 \left( {{\eta _1} - {L_1}} \right){V_{21}}^{\frac{1}{2}}\end{align}

Let $2{k_1} = {\alpha _6}$ , $\sqrt 2 \left( {{\eta _1} - {L_1}} \right) = {\alpha _7}$ , we have

(61) \begin{align}{\dot V_{21}} \le - {\alpha _6}{V_{21}} - {\alpha _7}{V_{21}}^{\frac{1}{2}}\end{align}

According to Lemma 1, the sliding variable ${s_{12}}$ converges to zero within a finite time, with the convergence time satisfies

(62) \begin{align}{{\rm{T}}_3} \le \frac{2}{{{\alpha _6}}}{\rm{ln}}\frac{{{\alpha _6}V_{21}^{\frac{1}{2}}\left( 0 \right) + {\alpha _7}}}{{{\alpha _7}}}\end{align}

Table 1. Model parameters

Table 2. External disturbances

Figure 3. Estimation of total disturbance under two observers.

Figure 4. Attitude angle tracking error response curves under two observers.

Figure 5. Control moment response curves under two observers.

Figure 6. Attitude angle tracking response curves.

In conclusion, when $\left| {{e_1}} \right| \geqslant b$ , the system is finite-time stable, and the overall convergence time satisfies

(63) \begin{align}{\rm{T}} \le {{\rm{T}}_1} + {{\rm{T}}_2} + {{\rm{T}}_3}\end{align}

5.1 Simulation setup

To validate the effectiveness and superiority of the proposed algorithm, this section conducts two parts of comparative experiments on the decoupled subsystems of pitch, yaw and roll. Firstly, a comparison simulation is carried out between the Composite Non-singular Fast Terminal Sliding Mode Control algorithm with the traditional Nonlinear Extended State Observer (CNFTSMC+ESO) and the Composite Non-singular Fast Terminal Sliding Mode Control algorithm with the Reduced-order Extended State Observer (CNFTSMC+RESO). Secondly, a comparison simulation is conducted among the Composite Non-singular Fast Terminal Sliding Mode Control algorithm with the Reduced-order Extended State Observer (CNFTSMC+RESO), the Non-singular Fast Terminal Sliding Mode Control algorithm with the Reduced-order Extended State Observer (NFTSMC+RESO), and the method in (30) with the Reduced-order Extended State Observer (The method in (30)+RESO). Furthermore, the method in Ref. [Reference Derrouaoui, Bouzid, Belmouhoub and Guiatni30] is identical to the method in Ref. [Reference Derrouaoui, Bouzid, Belmouhoub and Guiatni31]. The experimental airship described in Refs (Reference Wang32, Reference Wang and Shan33) serves as the research object, with the primary model parameters shown in Table 1.

The initial values of the airship attitude system are set as $\theta \left( 0 \right) = 0$ , $\psi \left( 0 \right) = 0$ , $\phi \left( 0 \right) = 0$ , $p\left( 0 \right) = 0$ , $q\left( 0 \right) = 0$ , $r\left( 0 \right) = 0$ . The desired values for pitch, yaw and roll are set as ${\theta _d} = 0.1{\rm{sin}}\left( {0.05t} \right)$ , ${\psi _d} = - 0.1{\rm{sin}}\left( {0.05t} \right)$ , ${\phi _d} = 0.1$ . The model parameter perturbations are set as ${\rm{\Delta }}{I_x} = 0.2{I_x}$ , ${\rm{\Delta }}{I_y} = 0.15{I_y}$ , ${\rm{\Delta }}{I_z} = 0.15{I_z}$ , ${\rm{\Delta }}{I_{xz}} = 0.2{I_{xz}}$ . The external disturbances applied to the system dynamics are specified in Table 2.

Taking the pitch channel as an example, the sliding surface design for the NFTSMC+RESO is given by

(64) \begin{align}{s_{12}} = {\dot e_1} + {\alpha _2}{e_1} + {\beta _2}si{g^{{w_1}}}\left( {{e_1}} \right)\end{align}

The comparative controller design is as follows:

(65) \begin{align}{v_1} = {\ddot y_{1d}} - {\hat f_1} - {\dot e_1}\left( {{\alpha _2} + {\beta _2}{w_1}si{g^{{w_1} - 1}}\left( {{e_1}} \right)} \right) - {k_1}{s_{12}} - {\eta _1}{\rm{tanh}}\left( {{s_{12}}/\sigma } \right)\end{align}

Figure 7. Attitude angle tracking error response curves.

To ensure fairness, the selection of parameters adheres to three principles [Reference Sha, Wang and Liu34]: effective control, uniform variables and moderate control variable values. Taking the pitch channel as an example, the simulation parameters for the remaining two channels are consistent with the pitch channel.

The observer parameter values are as follows:

1. (1) ESO: ${a_1} = 1$ , ${a_2} = 0.5$ , ${a_7} = 0.25$ , ${b_1} = 100$ , ${b_2} = 1,000$ , ${b_3} = 6,000$ , $\delta = 0.01$ .

2. (2) RESO: ${\bar a_2} = 1$ , ${\bar a_7} = 0.5$ , ${\bar a_2} = 100$ , ${\bar a_7} = 200$ , $\delta = 0.01$ .

The simulation parameter values for the controllers are as follows:

1. (1) CNFTSMC: $b = 0.03$ , ${\alpha _1} = 0.2$ , ${\beta _1} = 1.5$ , ${\alpha _2} = 0.2$ , ${\beta _2} = 1.5$ , ${p_1} = 5$ , ${q_1} = 3$ , ${m_1} = 57$ , ${n_1} = 55$ , ${w_1} = 2$ , $\sigma = 1$ , ${\eta _1} = 0.08$ , ${k_1} = 2$ .

2. (2) NFTSMC: ${\alpha _2} = 0.2$ , ${\beta _2} = 1.5$ , ${w_1} = 2$ , $\sigma = 1$ , ${\eta _1} = 0.08$ , ${k_1} = 2$ .

Table 3. Maximum absolute error ( ${\bf{max}}\left| e \right|$ )

Table 4. Mean absolute error ( $\frac{\sum |e|}{N}$ )

Table 5. Absolute error standard deviation ( $\sqrt{\frac1N\sum(|e|-\frac1N\sum|e|)^2}$ )

Table 6. Integral absolute error ( $\int_0^t|e|dt$ )

5.2 Simulation results analysis

Figure 3 depicts the estimation response of the two observers to the total disturbances under the CNFTSMC. It is evident that the proposed improved RESO exhibits a more rapid and accurate estimation of the disturbances, thereby mitigating the phase lag induced by the observer. In addition, the tracking error and control input of the two comparative observers are illustrated in Figs. 4 and 5. It is apparent that, for the three channels with uniform control input, the tracking error based on the improved RESO is inferior to that based on the ESO, verifying the effectiveness of the proposed observer.

Figure 8. Control moment response curves.

Figure 6 presents the attitude angle tracking response curves for three control methods. It is apparent that the proposed CNFTSMC+RESO, NFTSMC+RESO, and the method in (30)+RESO adeptly compensate for the disturbances and parameter perturbations by utilising estimation information. Consequently, they achieve precise tracking of the pitch, yaw and roll angle commands in a multiple-disturbances environment.

Figure 7 illustrates the attitude angle tracking error response curves for three control methods. We conduct a comparative analysis with regard to their convergence speed and disturbance rejection capability. It is evident that prior to 150s, in the absence of external disturbances, CNFTSMC+RESO demonstrates a faster convergence speed in the pitch and yaw channels compared to NFTSMC+RESO and the method in (30)+RESO. In the roll channel, when the tracking error is greater than or equal to 0.03, CNFTSMC+RESO and NFTSMC+RESO exhibit the same convergence speed, while the method in (30)+RESO achieves a faster convergence speed compared to the other two methods. When the tracking error falls below 0.03, the convergence speed of CNFTSMC+RESO surpasses that of NFTSMC+RESO and the method in (30)+RESO.

It is noteworthy that the method in (30)+RESO exhibits the advantage of a faster convergence speed when the tracking error is larger, and additionally, this study also evaluates the disturbance rejection capability. Between 150s and 180s, when exposed to constant disturbances, or at 180s when confronted with time-varying disturbances, the attitude angle tracking response under CNFTSMC+RESO exhibits a more rapid recovery, resulting in a shorter adjustment time compared to the other two control methods.

Furthermore, in order to comprehensively compare the performance of the control methods, this section introduces performance indicators for tracking errors, which include the maximum absolute error, average absolute error, absolute error standard deviation and integral absolute error [Reference Zhang, Guo, Gong, Zhu and Tian35]. The control performance indicators for attitude tracking errors under the three control methods are shown in Tables 3, 4, 5 and 6. It can be observed that the proposed CNFTSMC+RESO method attains superior control performance indicators in terms of tracking errors, validating the effectiveness and superiority of the proposed control algorithm.

In conclusion, the CNFTSMC+RESO method demonstrates smaller maximum absolute error, average absolute error, absolute error standard deviation and integral absolute error in comparison to NFTSMC+RESO and the method in (30)+RESO. Additionally, it showcases faster convergence speed and stronger capability for suppressing disturbances.

Figure 8 displays the control torque response curves for the three control methods. By utilising hyperbolic tangent function instead of the sign function in the design of the sliding mode controller, all three control algorithms ensure the continuity of the control quantity. From the figure, it can be observed that the control quantities of the three control algorithms are of equivalent magnitude, indicating that they provide comparable control energy required to achieve the control objectives.