2.1 Dynamic model of hypersonic morphing vehicle

This study examines an HMV featuring a fixed fuselage coupled with a pair of symmetric variable-sweep wings. Figure 1 is a simplified schematic diagram of the HMV. ${O_b}{x_b}{y_b}{z_b}$ represents the body coordinate system. $\eta $ is the back-swept angle. The left swept-back wing can rotate relative to the fuselage about the axis passing through point A. The extended length of the swept wing is $l$ . The distances from the fuselage centre of mass ${O_b}$ to the connection point A between the fuselage and the wings in the ${x_b}$ and ${z_b}$ directions are denoted as $d$ and $c$ .

Figure 1.

Diagram of a simplified HMV model.

In order to accurately depict the changes in aerodynamic characteristics of the HMV resulting from variations in the sweep angle, including additional aerodynamic forces and moments, this paper introduces a six-degree-of-freedom model of the HMV using multi-body dynamics principles [Reference Zhang, Huang and Nie13, Reference Bao, Wang and Tang42]. The HMV attitude model can be established as:

\begin{align*}&\!\!\left\{ \begin{array}{l}\dot \alpha = {\omega _z} + ({\omega _y}\sin \alpha - {\omega _x}\cos \alpha )\tan \beta + \dfrac{1}{{\cos \beta }}(\dot \chi \sin \mu \cos \gamma - \dot \gamma \cos \mu )\\[8pt]\dot \beta = {\omega _x}\sin \alpha + {\omega _y}\cos \alpha - (\dot \gamma \sin \mu + \dot \chi \cos \mu \cos \gamma )\\[8pt]\dot \mu = \dfrac{1}{{\cos \beta }}({\omega _x}\cos \alpha - {\omega _y}\sin \alpha ) - \dot \chi (\sin \gamma + \cos \gamma \sin \mu \tan \beta ) + \dot \gamma \cos \mu \tan \beta \\[18pt]{{\dot \omega }_x} = \dfrac{{{c_1}{M_x} + {c_3}{M_y} + {c_1}{c_2} + {c_3}{c_4}}}{{{c_5}{c_1} - {c_3}^2}}\\[18pt]{{\dot \omega }_y} = \dfrac{{{c_3}{M_x} + {c_5}{M_y} + {c_3}{c_7} + {c_5}{c_6}}}{{{c_5}{c_1} - {c_3}^2}}\\[18pt]{{\dot \omega }_z} = \dfrac{{{M_z} + {c_8}}}{{{c_9}}}\\[10pt]\dot V = \dfrac{{ - D}}{{{m_T}}} - g\sin \gamma - \omega _{\rm{e}}^2r({\rm{cos}}\phi {\rm{sin}}\phi {\rm{cos}}\chi {\rm{cos}}\gamma - {\rm{co}}{{\rm{s}}^2}\phi {\rm{sin}}\gamma )\\[10pt]\dot \gamma = \dfrac{1}{{{m_T}V}}(Z\cos \mu - C\sin \mu - {m_T}g\cos \gamma ) + \dfrac{{\omega _{\rm{e}}^2}}{V}r{\rm{cos}}\phi ({\rm{sin}}\phi {\rm{cos}}\chi {\rm{sin}}\gamma + {\rm{cos}}\phi {\rm{cos}}\gamma ) - 2{\omega _{\rm{e}}}{\rm{cos}}\phi {\rm{sin}}\chi \\[10pt]\dot \chi = \dfrac{1}{{ - {m_T}V\cos \gamma }}(Z\sin \mu + C\cos \mu ) + \dfrac{{\omega _{\rm{e}}^2}}{{V\cos \gamma }}r{\rm{sin}}\phi {\rm{cos}}\phi {\rm{sin}}\chi\\[15pt]\quad + 2\dfrac{{{\omega _{\rm{e}}}}}{{\cos \gamma }}({\rm{cos}}\phi {\rm{cos}}\chi {\rm{sin}}\gamma - {\rm{sin}}\phi {\rm{cos}}\gamma )\end{array} \right.\nonumber\\[5pt]{c_1} & = {J_{By}} + 2{J_{Wy}} - 2{m_W}\left( {{d^2} + {c^2} + {l^2} - 2dl\sin \eta + 2cl\cos \eta } \right)\nonumber\end{align*}

(1) \begin{align}{c_2} & = - {J_{Bxy}}{\omega _x}{\omega _z} - \left( {{J_{Bz}} + 2{J_{Wz}} - {J_{By}} - 2{J_{Wy}}} \right){\omega _y}{\omega _z} - {\rm{2}}{m_W}\left( {c + l\cos \eta } \right)\left( {{\omega _y}{\omega _z}\left( {c + l\cos \eta } \right) + 2{\omega _x}l\dot \eta \sin \eta } \right)\nonumber\\{c_3} & = {J_{Bxy}}\nonumber\\{c_4} & = {J_{Bxy}}{\omega _y}{\omega _z} - \left( {{J_{Bx}} + 2{J_{Wx}} - {J_{Bz}} - 2{J_{Wz}}} \right){\omega _x}{\omega _z} - {\rm{2}}{m_W}\left( {c + l\cos \eta } \right)\left( {2{\omega _y}l\dot \eta \sin \eta - {\omega _x}{\omega _z}\left( {c + l\cos \eta } \right)} \right)\nonumber\\ & - {\rm{2}}{m_W}\left( {d - l\sin \eta } \right)\left( {{{\dot V}_z} + {\omega _y}\left( { - {V_x} + 2l\dot \eta \cos \eta } \right) + {\omega _x}\left( {{V_y} + {\omega _z}\left( {d - l\sin \eta } \right)} \right)} \right)\nonumber\\{c_5} & = {J_{Bx}}{\rm{ + 2}}{J_{Wx}} - 2{m_W}{\left( {c + l\cos \eta } \right)^2}\nonumber\\{c_6} & = {J_{Bxy}}{\omega _y}{\omega _z} - \left( {{J_{Bx}} + 2{J_{Wx}} - {J_{Bz}} - 2{J_{Wz}}} \right){\omega _x}{\omega _z} - {\rm{2}}{m_W}\left( {d - l\sin \eta } \right)\left( \begin{array}{l}{{\dot V}_z} + {\omega _y}\left( { - {V_x} + 2l\dot \eta \cos \eta } \right)\nonumber\\[5pt] + {\omega _x}\left( {{V_y} + {\omega _z}\left( {d - l\sin \eta } \right)} \right)\end{array} \right)\nonumber\\[5pt]& - 2{m_W}\left( {c{\rm{ + }}l\cos \eta } \right)\left( {2{\omega _y}l\dot \eta \sin \eta - {\omega _x}{\omega _z}\left( {c + l\cos \eta } \right)} \right)\nonumber\\{c_7} & = - {J_{Bxy}}{\omega _x}{\omega _z} - \left( {{J_{Bz}} + 2{J_{Wz}} - {J_{By}} - 2{J_{Wy}}} \right){\omega _y}{\omega _z} - {\rm{2}}{m_W}\left( {c + l\cos \eta } \right)\left( {{\omega _y}{\omega _z}\left( {i + l\cos \eta } \right) + 2{\omega _x}l\dot \eta \sin \eta } \right)\nonumber\\[5pt]{c_8} & = {J_{Bxy}}( {{\omega _x}^2 - {\omega _y}^2} ) - ( {{J_{By}} + 2{J_{Wy}} - {J_{Bx}} - 2{J_{Wx}}} ){\omega _x}{\omega _y} + 2{m_W}\left( {d - l\sin \eta } \right)\left(\! \begin{array}{l}{{\dot V}_y} + {\omega _z}\left( {{V_x} - 2l\dot \eta \cos \eta } \right)\nonumber\\[5pt] - {\omega _x}( {{V_z} - {\omega _y}( {d - l\sin \eta } )} )\end{array}\! \right)\nonumber\\{c_9} & = {J_{Bz}} + 2{J_{Wz}} - 2{m_W}{\left( {d - l\sin \eta } \right)^2}\end{align}

where $\alpha $ , $\beta $ and $\mu $ are the angle-of-attack, sideslip angle and bank angle, respectively; $\gamma $ and $\chi $ are trajectory inclination angle and trajectory angle, respectively; ${\omega _x}$ , ${\omega _y}$ , ${\omega _z}$ are the roll, yaw and pitch angular velocities, respectively; $V$ , ${V_x}$ , ${V_y}$ and ${V_z}$ are the body centroid velocity and body centroid velocity about the $x$ , $y$ , $z$ axes, respectively; $g$ denotes the acceleration of gravity, $\phi $ denotes the current geocentric latitude, $r$ denotes the radial distance from the earth’s centre to the HMV’s centroid, and ${\omega _{\rm{e}}}$ represents the angular velocity of earth’s rotation; $D$ , $C$ , $Z$ are the drag force, lateral force, lift force; ${M_x}$ , ${M_y}$ , ${M_z}$ are the roll torque, yaw torque and pitch torque of the vehicle. ${m_T}$ and ${m_W}$ denote the total mass and swept-back wing mass, respectively; ${J_{Bx}}$ , ${J_{By}}$ , ${J_{Bz}}$ , ${J_{Bxy}}$ represent the moment inertia of the fuselage part. ${J_{Wx}}$ , ${J_{Wy}}$ , ${J_{Wz}}$ are the moment inertia of swept-back wing.

The aerodynamic moment can be described as Equation (2), where $Q$ , $L$ , ${S_{ref}}$ represent the dynamic pressure, reference length and reference area, respectively. ${\delta _x}$ , ${\delta _y}$ and ${\delta _z}$ represent actual three-axis control deflections. ${m_{x0}}$ , ${m_{y0}}$ , ${m_{z0}}$ represent the moment coefficients, which are functions of $\alpha $ , $\beta $ , $\eta $ and Mach number $Ma$ . $m_x^{{\delta _x}}$ , $m_x^{{\delta _y}}$ , $m_y^{{\delta _x}}$ , $m_y^{{\delta _y}}$ and $m_z^{{\delta _z}}$ denote the moment coefficients with respect to ${\delta _x}$ , ${\delta _y}$ and ${\delta _z}$ .

(2) \begin{align}{M_x} & = QL{S_{ref}}\left( {{m_{x0}} + m_x^{{\delta _x}}{\delta _x} + m_x^{{\delta _y}}{\delta _y}} \right)\nonumber\\[5pt]{M_y} & = QL{S_{ref}}\left( {{m_{y0}} + m_y^{{\delta _x}}{\delta _x} + m_y^{{\delta _y}}{\delta _y}} \right)\nonumber\\[5pt]{M_z} & = QL{S_{ref}}\left( {{m_{z0}} + m_z^{{\delta _z}}{\delta _z}} \right)\end{align}

Thus, combining Equations (1) and (2), the HMV model can be formulated as a state-space representation with a strict feedback structure, as shown below. Where $\unicode{x1D734} = {\left[ {\alpha ,\beta ,\mu } \right]^{\mathrm{T}}}$ , $\boldsymbol{\omega} = {\left[ {{\omega _x},{\omega _y},{\omega _z}} \right]^{\mathrm{T}}}$ , $\boldsymbol{u} = {\left[ {{\delta _x},{\delta _y},{\delta _z}} \right]^{\mathrm{T}}}$ , ${\boldsymbol{d}_1}$ and ${\boldsymbol{d}_2}$ represent the model uncertainties and external disturbances of the inner and outer loops, respectively.

(3) \begin{equation}\left\{ \begin{array}{l}\dot{\unicode{x1D734}} = {\boldsymbol{f}_1} + {\boldsymbol{g}_1}\boldsymbol\omega + {\boldsymbol{d}_1}\\[5pt]\dot{\boldsymbol\omega} = {\boldsymbol{f}_2} + {\boldsymbol{g}_2}\boldsymbol{u} + {\boldsymbol{d}_2}\end{array} \right.\end{equation}

\begin{align*}{\boldsymbol{f}_1} = \left[ {\begin{array}{*{20}{c}}{\dfrac{1}{{\cos \beta }}(\dot \chi \sin \mu \cos \gamma - \dot \gamma \cos \mu )}\\[9pt]{ - (\dot \gamma \sin \mu + \dot \chi \cos \mu \cos \gamma )}\\[5pt]{ - \dot \chi (\sin \gamma + \cos \gamma \sin \mu \tan \beta ) + \dot \gamma \cos \mu \tan \beta }\end{array}} \right],{\boldsymbol{g}_1} = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{ - \cos \alpha \tan \beta } & {}{\sin \alpha \tan \beta } & {} {}1\\[5pt]{\sin \alpha } & {} {}{\cos \alpha } & {} {}0\\[5pt]{\dfrac{{\cos \alpha }}{{\cos \beta }}} {} & {}{ - \dfrac{{\sin \alpha }}{{\cos \beta }}} & 0\end{array}} \right]\\\end{align*}

(4) \begin{align}{\boldsymbol{f}_2} & = \left[ {\begin{array}{c}{\dfrac{{{c_1}{c_2} + {c_3}{c_4}}}{{{c_5}{c_1} - {c_3}^2}}}\\[15pt]{\dfrac{{{c_3}{c_7} + {c_5}{c_6}}}{{{c_5}{c_1} - {c_3}^2}}}\\[15pt]{\dfrac{{{c_8}}}{{{c_9}}}}\end{array}} \right] + QL{S_{ref}}\left[ {\begin{array}{c@{\quad}c@{\quad}c}{\dfrac{{{c_1}}}{{{c_5}{c_1} - {c_3}^2}}} & {\dfrac{{{c_3}}}{{{c_5}{c_1} - {c_3}^2}}} & 0\\[15pt]{\dfrac{{{c_3}}}{{{c_5}{c_1} - {c_3}^2}}} & {\dfrac{{{c_5}}}{{{c_5}{c_1} - {c_3}^2}}} & 0\\[15pt]0 & {}0 & {}{\dfrac{1}{{{c_9}}}}\end{array}} \right]\left[ {\begin{array}{c}{{m_{x0}}}\\[5pt]{{m_{y0}}}\\[5pt]{{m_{z0}}}\end{array}} \right]\nonumber\\[10pt]{\boldsymbol{g}_2} & = QL{S_{ref}}\left[ {\begin{array}{c@{\quad}c@{\quad}c}{\dfrac{{{c_1}}}{{{c_5}{c_1} - {c_3}^2}}} & {\dfrac{{{c_3}}}{{{c_5}{c_1} - {c_3}^2}}} {} & 0\\[15pt]{\dfrac{{{c_3}}}{{{c_5}{c_1} - {c_3}^2}}} {} & {\dfrac{{{c_5}}}{{{c_5}{c_1} - {c_3}^2}}} & {}0\\[15pt]0 {} & 0 & {}{\dfrac{1}{{{c_9}}}}\end{array}} \right]\left[ {\begin{array}{c@{\quad}c@{\quad}c}{m_x^{{\delta _x}}} & {}{m_x^{{\delta _y}}} & {}0\\[5pt]{m_y^{{\delta _x}}} & {}{m_y^{{\delta _y}}} & {}0\\[5pt]0 & {}0 & {}{m_z^{{\delta _z}}}\end{array}} \right]\end{align}

2.2 Control-oriented model

Selecting state tracking error vector as ${\boldsymbol{x}_1} = \unicode{x1D734} - {\unicode{x1D734} _d} = {\left[ {\Delta \alpha , \Delta \beta , \Delta \mu } \right]^{\mathrm{T}}}$ , ${\boldsymbol{x}_2} = \dot{\unicode{x1D734}} - {\dot{\unicode{x1D734}}_d} = {\left[ {\Delta \dot \alpha , \Delta \dot \beta , \Delta \dot \mu } \right]^{\mathrm{T}}}$ , where ${\unicode{x1D734} _d} = {\left[ {{\alpha _d}, {\beta _d}, {\mu _d}} \right]^{\mathrm{T}}}$ and ${\dot{\unicode{x1D734}}_d} = {\left[ {{{\dot \alpha }_d}, {{\dot \beta }_d}, {{\dot \mu }_d}} \right]^{\mathrm{T}}}$ represent the command of the attitude angle and its rate of change, respectively. Therefore, the state tracking error model of the hypersonic morphing vehicle is given as the following form:

(5) \begin{align}& \quad \left\{ \begin{array}{l}{{\dot{\boldsymbol{x}}}_1} = {\boldsymbol{x}_2}\\[3pt]{{\dot{\boldsymbol{x}}}_2} = \boldsymbol{F} + \boldsymbol{Gsat}\left( {{\boldsymbol{u}_c}} \right) + \boldsymbol{d}\end{array} \right.\nonumber\\[6pt]\boldsymbol{sat}\left( {{u_{ci}}} \right) & = \left\{ \begin{array}{l}{u_{\max }},{u_{ci}} \gt {u_{\max }}\\[3pt]{u_{ci}},{u_{\min }} \lt {u_{ci}} \lt {u_{\max }}\\[3pt] - {u_{\min }},{u_{ci}} \lt - {u_{\min }}\end{array} \right.\end{align}

where $\boldsymbol{F} = {\boldsymbol{g}_1}\,{\boldsymbol{f}_2}$ , $\boldsymbol{G} = {\boldsymbol{g}_1}{\boldsymbol{g}_2}$ and $\boldsymbol{d} = {\dot{\boldsymbol{f}}_1} + {\dot{\boldsymbol{g}}_1}\boldsymbol\omega + {\boldsymbol{g}_1}{\boldsymbol{d}_2} + {\dot{\boldsymbol{d}}_1} - {\ddot{\unicode{x1D734}}_d}$ , while ${u_{\max }}$ and ${u_{\min }}$ denote the minimum and maximum limits of fin deflection, ${\boldsymbol{u}_c} = {\left[ {{u_{c1}},{u_{c2}},{u_{c3}}} \right]^{\mathrm{T}}}$ denote the desired fin deflection. Define $\Delta \boldsymbol{u} = \boldsymbol{u} - {\boldsymbol{u}_c}$ , which will be used in the anti-saturation auxiliary system.

3.1 Definitions and lemmas

Lemma 1. For the given system [Reference Riaz, Li, Qi and Zhang43] $\dot{\boldsymbol{x}} = \boldsymbol{f}({\boldsymbol{x}(t)}),\boldsymbol{x}(0) = {\boldsymbol{x}_{\rm{0}}} \in {{\mathbb R}^n},\boldsymbol{x} \in {{\mathbb R}^n}$ , where $\boldsymbol{f}$ is a continuous nonlinear function on ${\mathbb R}^n$ . If there exists a positive-definite Lyapunov function $V(\boldsymbol{x})$ such that

(6) \begin{equation}\dot V(\boldsymbol{x}) \le - \frac{\boldsymbol{\pi} }{{rT\sqrt {ab} }}\left( {aV{{(\boldsymbol{x})}^{1 + \frac{r}{2}}} + bV{{(\boldsymbol{x})}^{1 - \frac{r}{2}}}} \right)\end{equation}

where $0 \lt r \lt 1$ and $\mathrm{T} \gt 0$ , $a \gt 0$ , $b \gt 0$ . Then, the origin serves as a globally predefined-time stable equilibrium for the system under consideration, with $T$ representing the predefined time, namely, $\boldsymbol{x}$ converge to origin within $t \le \mathrm{T}$ .

(7) \begin{equation}{\left( {\sum\limits_{i = 1}^n {\left| {{x_1}} \right|} } \right)^a} \le \sum\limits_{i = 1}^n {{{\left| {{x_1}} \right|}^a}} ,{\rm{ }}{\left( {\sum\limits_{i = 1}^n {\left| {{x_1}} \right|} } \right)^b} \le {n^{b - 1}}\sum\limits_{i = 1}^n {{{\left| {{x_1}} \right|}^b}} \end{equation}

3.2 Predefined-time disturbance observer

In this section, we will introduce a predefined-time disturbance observer combined with an RBF neural network. The observer can accurately estimate the lumped disturbances during flight, including model uncertainties and external disturbances, thereby enhancing the robustness of the control system.

The observer system is constructed as follows

(8) \begin{equation}\dot{\boldsymbol{Z}} = {\boldsymbol{F}} + {\boldsymbol{Gu}} + \hat{{\boldsymbol{d}}} + {k_1}{{\boldsymbol{e}}_z} + {k_2}\mathrm{sgn}( {{{\boldsymbol{e}}_z}}) + \frac{\boldsymbol{\pi} }{{{\rm{2}}{r_1}{\mathrm{T}_d}}}\left( {1 + {{\left\| {{{\boldsymbol{e}}_z}} \right\|}^{2{r_1}}}} \right)\frac{{{{\boldsymbol{e}}_z}}}{{{{\left\| {{{\boldsymbol{e}}_z}} \right\|}^{{r_1}}}}}\end{equation}

Noting that ${k_1}$ and ${k_2}$ are positive constant, $0 \lt {r_1} \lt 1$ , ${T_d}$ is the user-defined time. The states ${\boldsymbol{Z}}$ denote the estimation of states ${{\boldsymbol{x}}_2}$ , ${{\boldsymbol{e}}_z} = {{\boldsymbol{x}}_2} - {\boldsymbol{Z}}$ , $\hat{{\boldsymbol{d}}}$ is the approximation of lumped disturbance.

Then, we have

(9) \begin{equation}\hat{{\boldsymbol{d}}} = {\hat{{\boldsymbol{W}}}^{\mathrm{T}}}{\boldsymbol{h}}( {{{\boldsymbol{e}}_z}})\end{equation}

(10) \begin{align}{h_i}( {{{\boldsymbol{e}}_z}}) & = \exp \left( { - \frac{{{{\left\| {{{\boldsymbol{e}}_z} - {{\boldsymbol{c}}_i}} \right\|}^2}}}{{2{b^2}}}} \right),0 \le i \le N\nonumber\\{\boldsymbol{h}}( {{{\boldsymbol{e}}_z}}) & = {\left[ {{h_1}( {{{\boldsymbol{e}}_z}}),{h_2}( {{{\boldsymbol{e}}_z}}), \ldots ,{h_N}( {{{\boldsymbol{e}}_z}})} \right]^{\mathrm{T}}}\end{align}

where $\hat{\boldsymbol{W}} \in {{\boldsymbol{R}}^{N \times 3}}$ is the weight matrix and ${{\boldsymbol{c}}_i} \in {\boldsymbol{R}^{3 \times 1}}$ is the $i$ -th column of ${\boldsymbol{c}} \in {\boldsymbol{R}^{3 \times N}}$ . According to Ref. [Reference Cheng, Zhang, Jiang and Simani36], there is an optimal weight matrix available. Suppose that ${\left\| {{{\boldsymbol{W}}^*}} \right\|_F} \le \bar W$ and $\left\| \boldsymbol\varepsilon \right\| \le \bar \varepsilon $ , we can get:

(11) \begin{equation}{\boldsymbol{d}} = {{\boldsymbol{W}}^*}^{\mathrm{T}}{\boldsymbol{h}}( {{{\boldsymbol{e}}_z}}) + \boldsymbol\varepsilon \end{equation}

Denote $\tilde{{\boldsymbol{W}}} = {{\boldsymbol{W}}^ * } - \hat{{\boldsymbol{W}}}$ as the estimation error of optimal weight matrix.

Theorem 1. Taking into account the following adaptive law:

(12) \begin{equation}\dot{\hat{{\boldsymbol{W}}}} = \unicode{x1D71E} \big( {{\boldsymbol{h}}( {{{\boldsymbol{e}}_z}}){{\boldsymbol{e}}_z}^{\rm{T}} - {\mu _w}\hat{{\boldsymbol{W}}}} \big)\end{equation}

where $\unicode{x1D71E} \in {{\boldsymbol{R}}^{N \times N}}$ is the constant symmetric positive definite matrices. $\mu \gt 0$ is the constant parameters. We can conclude that the state and disturbance estimation errors $( {{{\boldsymbol{e}}_z},\tilde{\boldsymbol{d}}})$ will converge to the zero accurately within ${T_d}$ , if parameters are chosen reasonably.

Proof. For brevity, the proof of Theorem 1 is broken down into the following two steps.

Step 1: Initially, the uniform stability is examined. Let the candidate Lyapunov function be defined as follows.

(13) \begin{equation}{V_1} = \frac{1}{2}{\boldsymbol{e}_z}^{\mathrm{T}}{\boldsymbol{e}_z} + \frac{1}{2}tr\big({\tilde{\boldsymbol{W}}^{\mathrm{T}}}{\unicode{x1D71E} ^{ - 1}}\tilde{\boldsymbol{W}}\big)\end{equation}

According to the adaptive law designed in Equation (12), taking the derivative of ${V_1}$ , we have

(14) \begin{align}{{\dot V}_1} & = {\boldsymbol{e}_z}^{\mathrm{T}}{{\dot{\boldsymbol{e}}}_z} - tr\Big( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}{\unicode{x1D71E} ^{ - 1}}\dot{\hat{\boldsymbol{W}}}} \Big)\nonumber\\[5pt]& = {\boldsymbol{e}_z}^{\mathrm{T}}\left( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\boldsymbol{h}\left( {{\boldsymbol{e}_z}} \right) + \boldsymbol\varepsilon - {k_1}{\boldsymbol{e}_z} - {k_2}\mathrm{sgn}\left( {{\boldsymbol{e}_z}} \right) - \frac{\boldsymbol{\pi} }{{{\rm{2}}{r_1}{\mathrm{T}_d}}}\left( {1 + {{\left\| {{\boldsymbol{e}_z}} \right\|}^{2{r_1}}}} \right)\frac{{{\boldsymbol{e}_z}}}{{{{\left\| {{\boldsymbol{e}_z}} \right\|}^{{r_1}}}}}} \right)\nonumber\\[5pt]& - \mathrm{tr}\Big( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\boldsymbol{h}\left( {{\boldsymbol{e}_z}} \right){\boldsymbol{e}_z}^{\mathrm{T}}} \Big) + {\mu _w}\mathrm{tr}\big( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\hat{\boldsymbol{W}}} \big)\end{align}

Note that ${\boldsymbol{e}_z}^{\mathrm{T}}{\tilde{\boldsymbol{W}}^{\mathrm{T}}}\boldsymbol{h}( {{\boldsymbol{e}_z}} ) = \mathrm{tr}( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\boldsymbol{h}( {{\boldsymbol{e}_z}} ){\boldsymbol{e}_z}^{\mathrm{T}}} )$ and ${\boldsymbol{e}_z}^{\mathrm{T}}\left( {{k_2}sgn( {{\boldsymbol{e}_z}} ) + \frac{\boldsymbol{\pi} }{{{\rm{2}}{r_1}{T_d}}}( {1 + {{\| {{\boldsymbol{e}_z}} \|}^{2{r_1}}}} )\frac{{{\boldsymbol{e}_z}}}{{{{\| {{\boldsymbol{e}_z}} \|}^{{r_1}}}}}} \right) \ge 0$ always holds. By using Young’s inequality and $\mathrm{tr}( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\hat{\boldsymbol{W}}} ) \le \frac{1}{2}\mathrm{tr}( {{\boldsymbol{W}^{ * T}}{\boldsymbol{W}^ * }} ) - \frac{1}{2}\mathrm{tr}( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\tilde{\boldsymbol{W}}} )$ , we can obtain that

(15) \begin{align}{{\dot V}_1} & \le \frac{1}{2}{\| {{\boldsymbol{e}_z}} \|^2} + \frac{1}{2}{\| \boldsymbol\varepsilon \|^2} - {k_1}{\| {{e_z}} \|^2} + {\mu _w}\left( {\frac{1}{2}\mathrm{tr}( {{\boldsymbol{W}^{ * T}}{\boldsymbol{W}^ * }} ) - \frac{1}{2}\mathrm{tr}( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\tilde{\boldsymbol{W}}} )} \right)\nonumber\\[5pt] & \le - {k_1}{\| {{\boldsymbol{e}_z}} \|^2} - \frac{{{\mu _w}}}{2}\mathrm{tr} \big( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\tilde{\boldsymbol{W}}} \big) + \frac{1}{2}{\| \boldsymbol\varepsilon \|^2} + \frac{{{\mu _w}}}{2}\mathrm{tr}( {{\boldsymbol{W}^{ * T}}{\boldsymbol{W}^ * }} )\nonumber\\[5pt] & \le - {k_1}{\| {{\boldsymbol{e}_z}} \|^2} - \frac{{{\mu _w}}}{2}\mathrm{tr} \big( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\tilde{\boldsymbol{W}}} \big) + {\kappa _1}\end{align}

where ${\lambda _1} = {k_1} - \frac{1}{2}$ , ${\kappa _1} = \frac{1}{2}{\bar \varepsilon ^2} + \frac{{{\mu _w}}}{2}{\bar W^2}$ . By choosing ${k_1} \gt \frac{1}{2}$ and ${\mu _w} \gt {\frac{{2{k_1}}}{{\left\| {{{\unicode{x1D71E}}^{ - 1}}} \right\|}}_F}$ , we have

(16) \begin{equation}{\dot V_1} \le - 2{\lambda _1}{V_1} + {\kappa _1}\end{equation}

Therefore, we can conclude that the estimation deviations ${\boldsymbol{e}_z}$ and $\tilde{\boldsymbol{W}}$ are ultimately uniformly bounded. Hence, $\tilde{\boldsymbol{d}} = {\tilde{\boldsymbol{W}}^{\mathrm{T}}}\boldsymbol{h}\left( {{\boldsymbol{e}_\boldsymbol{z}}} \right) + {\boldsymbol{\varepsilon}}$ is also bounded.

Step 2: In the following, let us prove that the estimation errors ${\boldsymbol{e}_z}$ and $\tilde{\boldsymbol{d}}$ can converge to the zero within ${T_d}$ . Consider the following Lyapunov function:

(17) \begin{equation}{V_2} = {\boldsymbol{e}_{\boldsymbol{z}}}^{\mathrm{T}}{\boldsymbol{e}_\boldsymbol{z}}\end{equation}

Based on the Equation (16), we conclude that the ${\tilde{\boldsymbol{W}}^{\mathrm{T}}}\boldsymbol{h}\left( {{\boldsymbol{e}_\boldsymbol{z}}} \right) + {\boldsymbol{\varepsilon}}$ is bounded. Therefore, by choosing proper ${\lambda _2}$ satisfying ${\boldsymbol{e}_{\boldsymbol{z}}}^{\mathrm{T}} ( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\boldsymbol{h}\left( {{\boldsymbol{e}_z}} \right) + \boldsymbol\varepsilon - {k_2}\mathrm{sgn}\left( {{\boldsymbol{e}_z}} \right)} ) \le 0$ , we can obtain that

(18) \begin{align}{{\dot V}_2} & = 2{\boldsymbol{e}_\boldsymbol{z}}^{\mathrm{T}}\left( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\boldsymbol{h}\left( {{\boldsymbol{e}_z}} \right) + \boldsymbol\varepsilon - {k_1}{\boldsymbol{e}_z} - {k_2}\mathrm{sgn}\left( {{\boldsymbol{e}_{\boldsymbol{z}}}} \right) - \frac{\boldsymbol{\pi} }{{2{r_1}{T_d}}}\left( {1 + {{\left\| {{\boldsymbol{e}_{\boldsymbol{z}}}} \right\|}^{2{r_1}}}} \right)\frac{{{\boldsymbol{e}_{\boldsymbol{z}}}}}{{{{\left\| {{\boldsymbol{e}_{\boldsymbol{z}}}} \right\|}^{{r_1}}}}}} \right)\nonumber\\ & \le 2{\boldsymbol{e}_\boldsymbol{z}}^{\mathrm{T}}\left( {{{\tilde{\boldsymbol{W}}}^{\mathrm{T}}}\boldsymbol{h}\left( {{\boldsymbol{e}_z}} \right) + \boldsymbol\varepsilon - {k_2}\mathrm{sgn}\left( {{\boldsymbol{e}_z}} \right) - \frac{\boldsymbol{\pi} }{{2{r_1}{T_d}}}\left( {1 + {{\left\| {{\boldsymbol{e}_{\boldsymbol{z}}}} \right\|}^{2{r_1}}}} \right)\frac{{{\boldsymbol{e}_{\boldsymbol{z}}}}}{{{{\left\| {{\boldsymbol{e}_{\boldsymbol{z}}}} \right\|}^{{r_1}}}}}} \right)\nonumber\\ & \le - \frac{\boldsymbol{\pi} }{{{r_1}{T_d}}}\left( {{{\left\| {{\boldsymbol{e}_\boldsymbol{z}}} \right\|}^{2 + {r_1}}} + {{\left\| {{\boldsymbol{e}_\boldsymbol{z}}} \right\|}^{2 - {r_1}}}} \right)\nonumber\\ & \le - \frac{\boldsymbol{\pi} }{{{r_1}{T_d}}}\left( {{V_2}^{1 - \frac{{{r_1}}}{2}} + {V_2}^{1 + \frac{{{r_1}}}{2}}} \right)\end{align}

Following Lemma 1, it is inferred that the state errors ${\boldsymbol{e}_\boldsymbol{z}}$ can regulate to the origin within predefined time ${T_d}$ . Combined with Equation (8), there is:

(19) \begin{equation}{\dot{\boldsymbol{e}}_\boldsymbol{z}} = \tilde{\boldsymbol{d}} - {k_1}{\boldsymbol{e}_{\boldsymbol{z}}} - {k_2}\mathrm{sgn}\left( {{\boldsymbol{e}_z}} \right) - \frac{\boldsymbol{\pi} }{{{\rm{2}}{r_1}{T_d}}}\left( {1 + {{\left\| {{\boldsymbol{e}_z}} \right\|}^{2{r_1}}}} \right)\frac{{{\boldsymbol{e}_z}}}{{{{\left\| {{\boldsymbol{e}_z}} \right\|}^{{r_1}}}}}\end{equation}

Since ${\boldsymbol{e}_\boldsymbol{z}}$ will converge to the origin, combining with Equation (19), we get $\tilde{\boldsymbol{d}} = 0$ . The proof is complete.

3.3 Predefined-time anti-saturation controller

In order to ensure that the attitude tracking error converges to zero with input constraint and address singularity issues, a nonsingular predefined-time anti-saturation sliding mode controller has been developed.

Firstly, the nonsingular predefined-time sliding mode (NPTSM) manifold is established as

(20) \begin{align}\boldsymbol{s} & = {\left[ {{s_{\rm{1}}},{s_{\rm{2}}},{s_{\rm{3}}}} \right]^{\rm{T}}},\bar{\boldsymbol{s}} = {\left[ {{{\bar s}_{\rm{1}}},{{\bar s}_{\rm{2}}},{{\bar s}_{\rm{3}}}} \right]^{\rm{T}}}\nonumber\\[5pt]{s_i} & = \left\{ \begin{array}{l@{\quad}l}{{\bar s}_i} & {{\bar s}_i} = 0\ and\ {{\bar s}_i} \ne 0,\left| {{x_{1i}}} \right| \ge \varepsilon \\[5pt]{x_{2i}} + \dfrac{{{e^{{{\left| {{x_{1i}}} \right|}^{{r_2}}}}}}}{{{r_2}{T_s}}}\left( {{l_1}{x_{1i}} + {l_2}\mathrm{sig}{{\left( {{x_{1i}}} \right)}^2}} \right) & {{\bar s}_i} \ne 0,\left| {{x_{1i}}} \right| \lt \varepsilon \end{array} \right.\nonumber\\[5pt]{{\bar s}_i} & = {x_{2i}} + \frac{{{e^{{{\left| {{x_{1i}}} \right|}^{{r_2}}}}}}}{{{r_2}{\mathrm{T}_s}}}\mathrm{sig}{\left( {{x_{1i}}} \right)^{1 - {r_2}}}\end{align}

where ${T_s} \gt 0$ and $0 \lt {r_2} \lt 1$ , $\varepsilon \gt 0$ is a small positive constant, ${l_1} = \left( {1 + {r_2}} \right){\varepsilon ^{ - {r_2}}}$ , ${l_2} = - {r_2}{\varepsilon ^{ - 1 - {r_2}}}$ .

Proof. In the sliding mode, we have $s \equiv 0$ . Firstly, when $\boldsymbol{s} = \bar{\boldsymbol{s}} = 0$ , we have

(21) \begin{equation}{x_{2i}} = - \frac{{{e^{\left| {{x_{1i}}} \right|}}^{^{{r_2}}}}}{{{r_2}{T_s}}}\mathrm{sig}{\left( {{x_{1i}}} \right)^{1 - {r_2}}}\end{equation}

Choose the Lyapunov function candidate: ${V_{31}} = \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} {V_{31i}} = \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} {x_{1i}}^2$ , and take the time-derivative of ${V_{{\rm{31}}}}$ , it yields to

(22) \begin{equation}{\dot V_{31}} = \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} {\dot V_{31i}} = - \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} \frac{{{e^{{{\left( {{V_{31i}}} \right)}^{\frac{{2 - {r_2}}}{2}}}}}}}{{{r_2}{T_s}}}{e^{{{\left( {{V_{31i}}} \right)}^{\frac{{2 - {r_2}}}{2}}}}}\end{equation}

For the $i$ th joint, there is ${\dot V_{31i}} = - \frac{{{e^{{{\left( {{V_{31i}}} \right)}^{\frac{{{r_2}}}{2}}}}}}}{{{r_2}{T_s}}}{e^{{{\left( {{V_{31i}}} \right)}^{\frac{{2 - {r_2}}}{2}}}}}$ . Suppose that at time ${t_i}$ , ${V_{31i}}\left( {{t_i}} \right) = 0$ , then the convergence time of tracking errors is calculated as

(23) \begin{equation}{t_i}{\rm{ = }}{{\rm{T}}_s}\Bigg( {1 - {e^{\Big( { - {{\left( {{V_{31i}}} \right)}^{\frac{{{r_2}}}{2}}}} \Big)}}} \Bigg) \le {{\rm{T}}_s}\end{equation}

Therefore, at time $t = \max \left( {{t_i}} \right)$ , ${V_{31}}(t) = 0$ . Now we can conclude that the states $\left( {{\boldsymbol{x}_1},{\boldsymbol{x}_2}} \right)$ converge to the origin in no more than the predefined time ${\mathrm{T}_{\mathrm{s}}}$ while on the sliding manifold $\boldsymbol{s} = \bar{\boldsymbol{s}} = 0$ .

Secondly, in the case that $\boldsymbol{s} = {\rm{0}}, \bar s \ne 0$ , according to the Equation (20), it is inferred that ${\boldsymbol{x}_1}$ and ${\boldsymbol{x}_2}$ are within the neighbourhoods of the origin ${\Omega^{\prime}_{1}}$ , ${\Omega _2}$ . Then, we choose the Lyapunov function candidate: ${V_{32}} = \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} {V_{32i}} = \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} \frac{{\rm{1}}}{{\rm{2}}}{x_{1i}}^2$ and take the time-derivative of ${V_{{\rm{32}}}}$ , it yields to

(24) \begin{equation}{\dot V_{32}} = \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} {\dot V_{32i}} = \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} {x_{1i}}{\dot x_{1i}} = - \mathop \sum \limits_{i = {\rm{1}}}^{\rm{3}} \frac{{{e^{\left| {{x_{1i}}} \right|}}^{^{{r_2}}}}}{{{r_2}{T_s}}}\left( {{l_1}{x_{1i}}^2 + {l_2}{{\left| {{x_{1i}}} \right|}^3}} \right)\end{equation}

It is evident that when $\left| {{x_{1i}}} \right| \in \left[ {\left. {0,\frac{{2\left( {1 + {r_2}} \right)\varepsilon }}{{3{r_2}}}} \right)} \right.$ , the function $ - \left( {{l_1}{x_{1i}}^2 + {l_2}{{\left| {{x_{1i}}} \right|}^3}} \right)$ is monotonically decreasing. Notice that $\varepsilon \lt \frac{{2\left( {1 + {r_2}} \right)\varepsilon }}{{3{r_2}}}$ , hence, it can be obtained that $ - \left( {{l_1}{x_{1i}}^2 + {l_2}{{\left| {{x_{1i}}} \right|}^3}} \right) \le 0$ always holds when $\left| {{x_{1i}}} \right| \le \varepsilon $ . Furthermore, we can get that ${\dot V_{32}} \le 0\ \forall \left| {{x_{1i}}} \right| \le \varepsilon $ , which will drive ${\boldsymbol{x}_1}$ and ${\boldsymbol{x}_2}$ to the origin asymptotially.

From the results in the two cases, we can conclude that once the sliding mode surface $\boldsymbol{s} \equiv 0$ is reached, ${\boldsymbol{x}_1}$ and ${\boldsymbol{x}_2}$ will converge to the neighbourhoods of the origin, ${\Omega^{\prime}_{1}} = \left\{ {{\boldsymbol{x}_1}|\left| {{x_{1i}}} \right| \le \varepsilon } \right\}$ , ${\Omega _2} = \left\{ {{x_2}|\left| {{x_{{\rm{2}}i}}} \right| \le \frac{{{e^{\left| \varepsilon \right|}}^{^{{r_2}}}}}{{{r_2}{T_s}}}\mathrm{sig}{{(\varepsilon)}^{1 - {r_2}}}} \right\}$ within ${\mathrm{T}_s}$ . The proof is complete.

Remark 2. In this paper, we construct a nonsingular predefined-time sliding mode manifold, a piecewise function is constructed to solve singularity problems and get faster convergence rate. According to Theorem 2, it is inferred that when the user-defined time ${T_s}$ is given, the final tracking errors will decrease by choosing proper ${r_2}$ and a small positive number $\varepsilon $ .

Based on the NPTSM surface, we can design the following disturbance-based predfined-time anti-saturation controller:

(25) \begin{align}{\boldsymbol{u}_c} & = {\boldsymbol{G}^{{\rm{ - 1}}}}\left( \begin{array}{l} - \boldsymbol{F} - \hat{\boldsymbol{d}} - \boldsymbol{R} - \dfrac{\boldsymbol{\pi} }{{2{r_3}{{\rm{T}}_c}}}\left( {{a_1}{3^{\frac{{{r_3}}}{2}}}{\rm{sig}}{{(\boldsymbol\chi)}^{1 + {r_3}}} + {b_1}{\rm{sig}}{{(\boldsymbol\chi)}^{1 - {r_3}}}} \right)\\[15pt] - \dfrac{1}{2}\boldsymbol\chi - \dfrac{\boldsymbol{\pi} }{{2{r_4}{{\rm{T}}_\xi }}}\left( {{a_2}{3^{\frac{{{r_4}}}{2}}}{\rm{sig}}{{\left( {\boldsymbol{\xi }} \right)}^{1 + {r_4}}} + {b_2}{\rm{sig}}{{\left( {\boldsymbol{\xi }} \right)}^{1 - {r_4}}}} \right)\end{array} \right)\\[10pt] {R_i} & = \left\{ \begin{array}{l@{\quad}l}\dfrac{{1 - {r_2}}}{{{r_2}{{\rm{T}}_s}}}{e^{{{\left| {{x_{1i}}} \right|}^{{r_2}}}}}{\left| {{x_{1i}}} \right|^{ - {r_2}}}{x_{2i}} + \dfrac{1}{{{{\rm{T}}_s}}}{e^{{{\left| {{x_{1i}}} \right|}^{{r_2}}}}}{x_{2i}} & {{\bar s}_i} = 0\ and\ {{\bar s}_i} \ne 0,\left| {{x_{1i}}} \right| \ge \varepsilon \\[15pt] \dfrac{{\mathrm{sig}{{\left( {{x_{1i}}} \right)}^{{r_2} - 1}}{x_{2i}}{e^{{{\left| {{x_{1i}}} \right|}^{{r_2}}}}}}}{{{\mathrm{T}_s}}}\left( {{l_1}{x_{1i}} + {l_2}\mathrm{sig}{{\left( {{x_{1i}}} \right)}^2}} \right) + \dfrac{{{e^{{{\left| {{x_{1i}}} \right|}^{{r_2}}}}}}}{{{r_2}{\mathrm{T}_s}}}\left( {{l_1}{x_{2i}} + 2{l_2}\left| {{x_{1i}}} \right|{x_{2i}}} \right) & {{\bar s}_i} \ne 0,\left| {{x_{1i}}} \right| \lt \varepsilon \end{array} \right.\nonumber\end{align}

(26) \begin{equation}\dot{\boldsymbol\xi} = - \frac{\boldsymbol{\pi} }{{2{r_4}{{\rm{T}}_\xi }\sqrt {{a_2}{b_2}} }}\left( {{a_2}{3^{\frac{{{r_4}}}{2}}}{\rm{sig}}{{\left( {\boldsymbol{\xi }} \right)}^{1 + {r_4}}} + {b_2}{\rm{sig}}{{\left( {\boldsymbol{\xi }} \right)}^{1 - {r_4}}}} \right) + \boldsymbol{G}\Delta \boldsymbol{u}\end{equation}

where $0 \lt {r_3} \lt 1$ , $0 \lt {r_4} \lt 1$ , and ${{\rm{T}}_c} \gt 0$ , ${{\rm{T}}_\xi } \gt 0$ , $\boldsymbol\chi = \boldsymbol{s} - {\boldsymbol{\xi }}$ and ${\boldsymbol{\xi }}$ is the auxiliary variable.

Proof. For brevity, the proof of Theorem 3 is broken down into the following two steps.

Step 1: Firstly, the convergence of $\boldsymbol\chi $ will be proved. Choosing the Lyapunov function ${V_4} = {{\boldsymbol{\chi}}^{\rm{T}}}{\boldsymbol{\chi}}$ and combining with Equation (25), the derivative of ${V_4}$ is satisfied with:

(27) \begin{align}{{\dot V}_4} & = 2{{\boldsymbol{\chi}}^{\rm{T}}}\dot{\boldsymbol{\chi}} = 2{{\boldsymbol{\chi}}^{\rm{T}}}\left( {\dot{\boldsymbol{s}} - \dot{\boldsymbol{\xi}}} \right)\nonumber\\[5pt]& = 2{{\boldsymbol{\chi}}^{\rm{T}}}\left( { - \frac{\boldsymbol{\pi} }{{2{r_3}{{\rm{T}}_c}\sqrt {{a_1}{b_1}} }}\left( {{a_1}{3^{\frac{{{r_3}}}{2}}}{\rm{sig}}{{(\boldsymbol\chi)}^{1 + {r_3}}} + {b_1}{\rm{sig}}{{(\boldsymbol\chi)}^{1 - {r_3}}}} \right) - \frac{1}{2}\boldsymbol\chi + \tilde{\boldsymbol{d}}} \right) \end{align}

By using Lemma 2, we can obtain

(28) \begin{align}{{\dot V}_4} & \le 2{{\boldsymbol{\chi}}^{\rm{T}}}\left( { - \frac{\boldsymbol{\pi} }{{2{r_3}{{\rm{T}}_c}\sqrt {{a_1}{b_1}} }}\big( {{3^{\frac{{{r_3}}}{2}}}{a_1}{\rm{sig}}{{(\chi)}^{1 + {r_3}}} + {b_1}{\rm{sig}}{{(\chi)}^{1 - {r_3}}}} \big) - \frac{1}{2}\chi } \right) + \big( {{{\left\| \chi \right\|}^2} + {{\| {\tilde{\boldsymbol{d}}} \|}^2}} \big)\nonumber\\[5pt] & \le - \frac{\boldsymbol{\pi} }{{{r_3}{{\rm{T}}_c}\sqrt {{a_1}{b_1}} }}\big( {{a_1}{V_4}^{1 + \frac{{{r_3}}}{2}} + {b_1}{V_4}^{1 - \frac{{{r_3}}}{2}}} \big) + {\| {\tilde{\boldsymbol{d}}} \|^2}\nonumber\\[5pt] & \le - \frac{\boldsymbol{\pi} }{{{r_3}{{\rm{T}}_c}\sqrt {{a_1}{b_1}} }}\big( {{a_1}{V_4}^{1 + \frac{{{r_3}}}{2}} + {b_1}{V_4}^{1 - \frac{{{r_3}}}{2}}} \big) + \nu \end{align}

From the demonstration of Theorem 1, it can be known the estimation error $\tilde{\boldsymbol{d}}$ is bounded during $t \in \left( {0,{\mathrm{T}_d}} \right)$ , $\nu = {\| {\tilde{\boldsymbol{d}}} \|^2}$ is bounded. According to Ref. [Reference Chen, Chen, Zhang, Xiao and Cao27] and Lemma 1, it can be proved that ${V_4}$ is able to converge into the region ${V_4} \le \min \bigg\{ {{{\left( {\dfrac{{\theta {r_3}{{\rm{T}}_c}\sqrt {{a_1}{b_1}} \nu }}{{\left( {\theta - 1} \right)\boldsymbol{\pi} {b_1}}}} \right)}^{\frac{2}{{2 - {r_3}}}}},{{\left( {\dfrac{{\theta {r_3}{{\rm{T}}_c}\sqrt {{a_1}{b_1}} \nu }}{{\left( {\theta - 1} \right)\boldsymbol{\pi} {a_1}}}} \right)}^{\frac{2}{{2 + {r_3}}}}}} \bigg\},\theta \gt 1$ within $\sqrt \theta {{\rm{T}}_c}$ , which means $\boldsymbol\chi $ is bounded before the disturbance estimation error $\tilde{\boldsymbol{d}}$ converges to the zero.

When the disturbance estimation error $\tilde{\boldsymbol{d}}$ converges to the zero after ${{\rm{T}}_d}$ , we can obtain that ${\dot V_4} \le - \dfrac{\boldsymbol{\pi} }{{{r_3}{{\rm{T}}_c}\sqrt {{a_1}{b_1}} }}\big( {{a_1}{V_4}^{1 + \frac{{{r_3}}}{2}} + {b_1}{V_4}^{1 - \frac{{{r_3}}}{2}}} \big)$ , by using Lemma 1, it can be known $\boldsymbol\chi $ will converge to the origin exactly wihin ${{\rm{T}}_c}$ .

Step 2: The auxiliary variable ${\boldsymbol{\xi }}$ can ensure actuator exit saturated region. When the control input exits the saturation state, namely, $\Delta \boldsymbol{u} = 0$ , the convergence of auxiliary variable ${\boldsymbol{\xi }}$ can be proved as:

Choosing the Lyapunov function ${V_5} = {{\boldsymbol{\xi }}^{\mathrm{T}}}{\boldsymbol{\xi }}$ , combining with Equation (26) and Lemma 2, it satisfies:

(29) \begin{align}{{\dot V}_5} = 2{{\boldsymbol{\xi }}^{\mathrm{T}}}\dot{\boldsymbol{\xi}} & = 2{{\boldsymbol{\xi }}^{\mathrm{T}}}\left( { - \frac{\boldsymbol{\pi} }{{2{r_4}{{\rm{T}}_\xi }\sqrt {{a_2}{b_2}} }}\left( {{3^{\frac{{{r_4}}}{2}}}{a_2}{\rm{sig}}{{\left( {\boldsymbol{\xi }} \right)}^{1 + {r_4}}} + {b_2}{\rm{sig}}{{\left( {\boldsymbol{\xi }} \right)}^{1 - {r_4}}}} \right)} \right)\nonumber\\[5pt]& \le - \frac{\boldsymbol{\pi} }{{{r_4}{{\rm{T}}_\xi }\sqrt {{a_2}{b_2}} }}\left( {{a_2}{V_4}^{1 + \frac{{{r_4}}}{2}} + {b_2}{V_4}^{1 - \frac{{{r_4}}}{2}}} \right)\end{align}

By using Lemma 1, it can be concluded that ${\boldsymbol{\xi }}$ will converge to the origin exactly within ${{\rm{T}}_\xi }$ .

From the above conclusions, we can obtain that after the convergence of ${\boldsymbol{\xi }}$ , $\boldsymbol\chi $ equals $\boldsymbol{s}$ , which means $\boldsymbol{s} = 0$ can be reached within the predefined time. According to Theorem 2, it can be known that the ${\boldsymbol{x}_1}$ and ${\boldsymbol{x}_2}$ will reach the vicinity of the origin within the set time $\mathrm{T} = {\mathrm{T}_d} + {\mathrm{T}_s} + {\mathrm{T}_c} + {{\rm{T}}_\xi }$ . The proof is complete.

4.1 Control characteristic verification

To confirm the predefined-time convergence of the proposed method, different settling times are set as $\mathrm{T} = 3$ , $\mathrm{T}= 5$ and $\mathrm{T} = 8$ s, while the initial states and commands are specified as ${\alpha _0} = {0^ \circ }$ , ${\beta _0} = {3^ \circ }$ , ${\mu _0} = {0^ \circ }$ , ${\alpha _c} = {3^ \circ }$ , ${\beta _c} = {0^ \circ }$ , ${\mu _c} = {5^ \circ }$ . The controller settings corresponding to several settling times are presented in Table 1. Figure 3 shows the simulation outcomes.

Table 1.

Controller parameters

Figure 3.

Tracking results with different predefined time.

To further demonstrate the characteristics of the predefined-time structure, simulations under various initial-error conditions are tested, as detailed in Table 2. Control commands are uniformly configured to: ${\alpha _c} = {3^ \circ }$ , ${\beta _c} = {0^ \circ }$ , ${\mu _c} = {5^ \circ }$ . Furthermore, the settling time is set to T = 3 s, and the controller parameters are chosen to match the first row of Table 1. The simulation results are shown in Fig. 4.

Table 2.

Initial condition configurations

Figure 4.

Tracking results under various initial errors with settling time T=3 s.

As can be seen from the simulation results, consistent with Theorem 3, the attitude angle tracking error can converge to a vicinity of the origin within the user-defined time, and the settling time is unaffected by the initial attitude states.

4.2 Control effectiveness verification

In this subsection, we will further verify the tracking performance and anti-saturation characteristics of the developed controller during the deformation process under nominal case. The initial attitude states are set as ${\alpha _0} = {0^ \circ }$ , ${\beta _0} = {3^ \circ }$ , ${\mu _0} = {0^ \circ }$ . The reference commands and sweep angle variation process are shown in the simulation results. The controller parameters are specified as: ${\mathrm{T}_s} = 2$ , ${\mathrm{T}_c} = 5$ , ${\mathrm{T}_\xi } = 2$ , ${\mathrm{T}_d} = 1$ , ${k_1} = 10$ , ${k_2} = 0.1$ , ${a_1} = 5$ , ${b_1} = 1$ , ${a_2} = 5$ , ${b_2} = 5$ , $\unicode{x1D71E} = 2000{\boldsymbol{I}_{17 \times 17}}$ , ${\mu _w} = 0.01$ .

Figure 5 shows the response of the attitude angle to the given command as well as the variation of the sweep angle. The simulation results indicate that the designed control scheme has excellent tracking accuracy for the attitude angle command. When the deformation occurs at 30 s, due to the effect of the additional moment, the angle-of-attack attitude tracking error increases. Figure 6 illustrates the control input in the different scenarios. It can be seen from the figure that deformation leads to changes in the elevator angle and the attitude system quickly adjusts to re-establish a balanced state. Moreover, compared with the control method without the anti-saturation compensator, the proposed approach enables faster recovery from input saturation during transients and reduces rudder usage. Figure 7 shows the auxiliary variables in the anti-saturation system. The results indicate that the auxiliary variables converge to zero after driving the control input to exit the saturation state within a predefined time.

Figure 5.

Angle responses and sweep angle.

Figure 6.

Control inputs.

Figure 7.

Anti-saturation variables.

4.3 Comparative analysis

In this subsection, we will validate the robustness of the developed controller under external disturbances; the disturbances are set as Equation (30). Moreover, −30% bias for aerodynamic parameters is considered during the morphing phase to simulate the parameters uncertainties. The controller parameters and the variation of the sweep angle are consistent with Section 4.2.

(30) \begin{align}{\boldsymbol{d}_1} & = 0.03 \times {\left[ {\sin \left( {\frac{\boldsymbol{\pi} }{2}t} \right) + \sin \left( {\frac{\boldsymbol{\pi} }{4}t} \right),\cos \left( {\frac{\boldsymbol{\pi} }{2}t} \right) + \cos \left( {\frac{\boldsymbol{\pi} }{4}t} \right),\sin \left( {\frac{\boldsymbol{\pi} }{2}t} \right) + \cos \left( {\frac{\boldsymbol{\pi} }{4}t} \right)} \right]^{\mathrm{T}}},10 \lt t \lt 18\nonumber\\[5pt]{\boldsymbol{d}_2} & = 0.4 \times {\left[ {\sin \left( {\frac{\boldsymbol{\pi} }{2}t} \right) + \sin \left( {\frac{\boldsymbol{\pi} }{4}t} \right),\cos \left( {\frac{\boldsymbol{\pi} }{2}t} \right) + \cos \left( {\frac{\boldsymbol{\pi} }{4}t} \right),\sin \left( {\frac{\boldsymbol{\pi} }{2}t} \right) + \cos \left( {\frac{\boldsymbol{\pi} }{4}t} \right)} \right]^{\mathrm{T}}},10 \lt t \lt 18\end{align}

To demonstrate the superiority of our approach, a predefined-time controller in this paper without its disturbance observer (PTC) and a finite-time controller with a fixed-time observer (FxTO-FTC) [Reference Liang, Wang, Hu and Dong44] are employed to compare with the proposed controller. The parameters of the FxTO-FTC are carefully selected as $\kappa = 0.2$ , $m = 4.5$ , $n = 2.5$ , $\eta = 0.002$ , $\mu = 1$ and ${l_1} = {l_2} = {l_3} = 4$ .

Figure 8.

Angle responses with different methods.

Figure 9.

Tracking errors with different methods.

Table 3.

Root mean square tracking error

Figure 10.

Control inputs with different methods.

To evaluate the performance of the proposed controller more quantitatively and comprehensively, the root mean square error (RMSE) is introduced as Equation (31) where ${t_1}$ and ${t_2}$ indicate the beginning and end times of the simulation, respectively. $e$ means tracking error.

(31) \begin{equation}\sigma = \sqrt {\frac{1}{{{t_2} - {t_1}}}\int_{{t_1}}^{{t_2}} {{{\left| e \right|}^2}dt} } \end{equation}

The simulation results are shown in Figs 8–10 and Table 3. Figures 8 and 9 illustrate the attitude tracking performance and state errors, respectively. It is demonstrated that in the nominal phase, all three methods can achieve convergence of the tracking error to the small residual sets of the origin. However, owing to the absence of compensation for unknown disturbances, the tracking performance of the FTC is shown to deteriorate dramatically during 10–18 s. The FxTO-FTC shows better tolerance to disturbances, but the vibration occurs at 10 and 18 s. In this case, the proposed method can accurately estimate the disturbances, and thus, it keeps a decent performance under external disturbances. Moreover, the magnified inset of the pitch channel shows that the proposed method yields the lowest attitude-tracking error during the deformation phase. Notably, unlike the finite-time control scheme, the controller in this paper does not rely on the initial error and shows improved transient response. As summarised in Table 3, it achieves the smallest root mean square error in each channel. Figure 10 depicts the control input with different methods, which demonstrates that the suggested method can drive the control input away from the saturation state after entering it.

Figures 11 and 12, respectively, show the estimation of the attitude rate tracking errors by the disturbance observer from FxTO-FTC and the observer proposed in this paper. The figures indicate that both estimation errors quickly converge to the small residual sets of the origin within a fixed time. On the other hand, in the initial stage, the fixed-time observer experiences difficulties in observing the rapidly changing variables, while the designed observer shows favorable performance. When external disturbances occur at 10 s, the proposed observer ensures the estimation error converges to a small residual set of the origin more rapidly compared to the fixed-time observer. Owing to the introduction of the neural network and the predefined-time technique, the proposed observer can approximate the lumped disturbances in the high nonlinear morphing vehicle system shortly. Figure 13 shows the proposed observer’s estimation of the disturbances in each channel.

Figure 11.

The observed states with the fixed-time observer.

Figure 12.

The observed states with the proposed observer.

Figure 13.

Estimation of the lumped disturbances.