2.1 Multi-rotor section model

The equations for the position and attitude dynamics of the multi-rotor section of the composite VTOL UAV are as follows:

(1) \begin{align} {\dot u} {} &= - {F \over m}\left( {{\rm{cos}}\,\psi\, {\rm{sin}}\,\theta\, {\rm{cos}}\,\phi + {\rm{sin}}\,\psi\, {\rm{sin}}\,\phi } \right) + {d_{mc\_u}}, \nonumber\\ {\dot v} {} &= - {F \over m}\left( {{\rm{sin}}\,\psi\, {\rm{sin}}\,\theta\, {\rm{cos}}\,\phi + {\rm{cos}}\,\psi\, {\rm{sin}}\,\phi } \right) + {d_{mc\_v}}, \nonumber\\ {\dot w} {} &= g - {F \over m}{\rm{cos}}\,\theta\, {\rm{cos}}\,\phi + {d_{mc\_w}},\nonumber \\{\dot p} {} &= {{{I_y} - {I_z}} \over {{I_x}}}qr + {1 \over {{I_x}}}{\tau _p} + {d_{mc\_p}}, \nonumber\\{\dot q} {} &= {{{I_z} - {I_x}} \over {{I_y}}}pr + {1 \over {{I_y}}}{\tau _q} + {d_{mc\_q}}, \nonumber\\{\dot r} {} &= {{{I_x} - {I_y}} \over {{I_z}}}pq + {1 \over {{I_z}}}{\tau _r} + {d_{mc\_r}}. \end{align}

where $u,{\rm{\;}}v,{\rm{\;}}w$ denote the velocity of the UAV in the body coordinate system, $p,{\rm{\;}}q,{\rm{\;}}r$ the angular velocity of the UAV in the body coordinate system, $\phi, {\rm{\;}}\theta, {\rm{\;}}\psi $ the roll angle, pitch angle and yaw angle of the UAV, respectively, $m$ the mass of the UAV, $g$ the gravitational acceleration, $F$ the tensile force exerted on the UAV, ${\tau _i}\left( {i = p,{\rm{\;}}q,{\rm{\;}}r} \right)$ the torque of the attitude angular channel, ${I_i}\left( {i = x,{\rm{\;}}y,{\rm{\;}}z} \right)$ the three-axis rotational moment of inertia of the UAV [Reference Quan19], and ${d_{mc\_i}}{\rm{\;}}\left( {i = u,{\rm{\;}}v,{\rm{\;}}w,{\rm{\;}}p,{\rm{\;}}q,{\rm{\;}}r} \right)$ the total disturbances of the UAV in multi-rotor mode, including model uncertainty and unknown disturbances, such as electromagnetic disturbance, str(ong)winds and rain and other severe weather effects [Reference Li, Sun and Talpur20]. Additionally, it should be noted that the model is not entirely accurate and relies on small angle assumptions to avoid singularity issues. These assumptions can be considered modeling errors within the system.

2.2 Fixed wing section model

To facilitate the control of the airspeed of the composite VTOL UAV in fixed-wing mode, the dynamics model of the UAV in the airflow coordinate system is introduced [Reference Wu1, Reference Beard and McLain21], i.e.

(2) \begin{align}{\dot V} {} &= g\left( { - {\rm{cos}}\,\alpha\, {\rm{cos}}\,\beta {\rm{sin}}\,\theta + {\rm{sin}}\,\beta\, {\rm{sin}}\,\phi\, {\rm{cos}}\,\theta + {\rm{sin}}\,\alpha\, {\rm{cos}}\,\beta\, {\rm{cos}}\,\theta } \right)\nonumber \\[5pt] & \quad + \frac{{{\rm{cos}}\,\alpha\, {\rm{cos}}\,\beta }}{m}T - \frac{D}{m} + {d_{fw\_V}}.\end{align}

where $V$ denotes the airspeed of the UAV, $\alpha, {\rm{\;}}\beta $ the attack angle and sideslip angle of the UAV, $T$ the thrust applied to the UAV, $D$ the drag applied to the UAV and ${d_{fw\_V}}$ the unknown disturbed part of the UAV in the airspeed channel of the fixed-wing mode, which is mainly the strong wind and the disturbances of the wake [Reference Lu, Gao, Yan, Hou and Wang22].

The attitude dynamics model of the UAV is introduced to facilitate the control of the attitude channel of the composite VTOL UAV in fixed-wing mode [Reference Wu1, Reference Beard and McLain21], i.e.

(3) \begin{align} {\dot p} {} &= \frac{{{I_y} - {I_z}}}{{{I_x}}}qr + \frac{1}{{{I_x}}}{M_p} + {d_{fw\_p}},\nonumber\\[5pt]{\dot q} &= \frac{{{I_z} - {I_x}}}{{{I_y}}}pr + \frac{1}{{{I_y}}}{M_q} + {d_{fw\_q}},\nonumber \\[5pt]{\dot r} {} &= \frac{{{I_x} - {I_y}}}{{{I_z}}}pq + \frac{1}{{{I_z}}}{M_r} + {d_{fw\_r}}. \end{align}

where ${d_{fw\_i}}\left( {i = p,{\rm{\;}}q,{\rm{\;}}r} \right)$ denotes the unknown part of the UAV in the fixed-wing mode and the external disturbances, ${M_i}\left( {i = p,{\rm{\;}}q,{\rm{\;}}r} \right)$ the combined moment applied to each channel in the airflow coordinate system, which consists of the following two main parts, i.e.

(4) \begin{align}M_i = {M_{wi}} + {M_{Ti}},{\rm{\;}}\left( {i = p,{\rm{\;}}q,{\rm{\;}}r} \right).\end{align}

where ${M_{wi}}$ denotes the aerodynamic moment applied to each channel and ${M_{Ti}}$ the thrust moment applied to each channel. The moment generated by the thrust system is small and can be grouped together into the unknown part of the model when designing the controller due to the small bias angle of the fixed-wing motor. The aerodynamic moment ${M_{wi}}$ can be expressed as

(5) \begin{align}{M_{wi}} = {M_{ia1}} + {M_{ia2}} + {M_{ie}}.\end{align}

where ${M_{ia1}},{\rm{\;}}{M_{ia2}},{\rm{\;}}{M_{ie}}$ denote the aerodynamic moments generated by the left aileron, right aileron and elevator. The three-axis aerodynamic moment of each rudder surface ${[\begin{array}{*{20}{l}}{{M_{pn}}{\rm{\;\;\;\;}}{M_{qn}}{\rm{\;\;\;\;}}{M_{rn}}}\end{array}]^T}$ are mainly obtained by the lift and drag forces respectively with the corresponding force arms for the cross-product operation and then accumulate up. The traditional modeling approaches of the composite VTOL UAV obtains the aerodynamic moment of the UAV by multiplying the aerodynamic moment parameters of the three-axis with the dynamic pressure. Cross-product operation is used to model the three-axis aerodynamic moments of the UAV in this paper, which solves the problem of over-reliance on aerodynamic moment parameters in traditional modeling approaches by eliminating the need to identify aerodynamic moment parameters [Reference Durán-Delfín, García-Beltrán, Guerrero-Sánchez, Valencia-Palomo and Hernández-González23], and makes the model construction more convenient and accurate. Take elevator as an example, i.e.

(6) \begin{align}\begin{array}{*{20}{l}}{\left[ {\begin{array}{*{20}{l}}{{M_{pe}}}\\[5pt]{{M_{qe}}}\\[5pt]{{M_{re}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{l}}{{M_{peL}}}\\[5pt]{{M_{qeL}}}\\[5pt]{{M_{reL}}}\end{array}} \right] + \left[ {\begin{array}{*{20}{l}}{{M_{peD}}}\\[5pt]{{M_{qeD}}}\\[5pt]{{M_{reD}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{l}}{{x_n}}\\[5pt]{{y_n}}\\[5pt]{{z_n}}\end{array}} \right] \times \left[ {\begin{array}{c}0\\[5pt]0\\[5pt]{ - {L_e}}\end{array}} \right] + \left[ {\begin{array}{c}{{x_n}}\\[5pt]{{y_n}}\\[5pt]{{z_n}}\end{array}} \right] \times \left[ {\begin{array}{c}{ - {D_e}}\\[5pt]0\\[5pt]0\end{array}} \right].}\end{array}\end{align}

where ${M_{ieL}}\left( {i = p,{\rm{\;}}q,{\rm{\;}}r} \right)$ denotes the three-axis aerodynamic moment generated by the lift force acting on the elevator, ${M_{ieD}}\left( {i = p,{\rm{\;}}q,{\rm{\;}}r} \right)$ the three-axis aerodynamic moment generated by the drag force acting on the elevator, ${x_e},{\rm{\;}}{y_e},{\rm{\;}}{z_e}$ the position of the elevator force centre with respect to the aircraft’s centre of gravity and ${L_e},{\rm{\;}}{D_e}$ the lift force and drag force on the elevator, which are expressed respectively as follows:

(7) \begin{align}\begin{array}{*{20}{l}}{{L_e} = {C_{le}} \cdot Q \cdot {S_e},}\\[5pt]{{D_e} = {C_{de}} \cdot Q \cdot {S_e}.}\end{array}\end{align}

where $Q$ denotes the dynamic pressure, ${S_e}$ the area of the elevator, the elevator lift coefficient ${C_{le}}$ and drag coefficient ${C_{de}}$ are expressed as follows:

(8) \begin{align}\begin{array}{*{20}{l}}{{C_{le}} = cla \cdot \alpha - {\delta _e},}\\[5pt]{{C_{de}} = \left| {cda \cdot \alpha } \right|.}\end{array}\end{align}

where $cla,{\rm{\;}}cda$ denote the lift and drag parameters and are constants, and ${\delta _e}$ the deflection of the elevator, i.e. the control input. Other rudders and ailerons are the same and do not be repeated here.

Combining Equations (3)–(8) yields model expressions specific to the roll angle and pitch angle channels:

(9) \begin{align} {\dot p} {}&= \frac{{\left( {{I_y} - {I_z}} \right)qr}}{{{I_x}}} - \frac{{2Q{S_{a1}}{y_{a1}}}}{{{I_x}}}{\delta _{a1}} + {d_{fw\_p}},\nonumber\\[5pt]{\dot q} {}& = \frac{{{I_z} - {I_x}}}{{{I_y}}}pr + \frac{{Q{S_e}{x_e}}}{{{I_y}}}{\delta _e} - \frac{{Q\left( {2{S_{a1}}{x_{a1}} + {S_e}{x_e}} \right)cla \cdot \alpha }}{{{I_y}}}\nonumber\\[5pt] {}&\quad + \frac{{Q\left( {2{S_{a1}}{z_{a1}} + {S_e}{z_e}} \right)\left| {cda \cdot \alpha } \right|}}{{{I_y}}} + {d_{fw\_q}}.\end{align}

where ${\delta _{a1}},{\rm{\;}}{\delta _{a2}}$ denote the deflection of the left aileron and right aileron, ${x_{a1}},{\rm{\;}}{y_{a1}},{\rm{\;}}{z_{a1}}$ the position of the force centre of the left aileron relative to the centre of gravity of the UAV. Since the right aileron only $y$ -axis relative position is opposite to the left aileron, the remaining positions are all the same as the left aileron. ${S_{a1}}$ the area of the left wing. Equation (9) expresses the relevant parameters in terms of the relative parameters of the left aileron for design convenience. Since the UAV design used in this paper does not include a rudder, the yaw channel is not controlled in fixed-wing mode.

3.1 Model compensation controller

The block diagram of the model compensation controller is shown in Fig. 2.

Figure 2.

Block diagram of model compensation controller.

For the first-order system Equation (11), the second-order model compensation control law is designed as

(13) \begin{align}u = \frac{1}{b}\left[ {k\left( {{x_r} - x} \right) + {{\hat{\dot{x}}}_{r}} - {f_k} - {{\hat{f}}_{uk}}} \right].\end{align}

where ${x_r}$ denotes the tracked target, ${\hat{\dot{x}}_{r}}$ denotes the estimation of ${\dot{x}_{r}}$ obtained through HOD, the unknown part of the model ${f_{uk}}$ is obtained through CFO, denoted by ${\hat{f}_{uk}}$ . The fixed-wing roll channel as an example, the expression for the controller of the roll angular velocity can be obtained as follows:

(14) \begin{align}\begin{array}{*{20}{l}}{{\delta _{a1}} = - \frac{{{I_x}}}{{2Q{S_{a1}}{y_{a1}}}}\left[ {{k_{fw\_p}}\left( {{{\hat{p}}_r} - p} \right) + {{\hat{\dot{p}}}_r} - \frac{{\left( {{I_y} - {I_z}} \right)qr}}{{{I_x}}} - {{\hat f}_{uk\_p}}} \right].}\end{array}\end{align}

The design of the controllers for the remaining channels of the composite VTOL UAV follows the same approach as the roll channel and will not be reiterated here.

3.2 High order differentiator

To estimate ${\dot{x}_r}$ in Equation (13), we choose the HOD proposed by Qi et al. [Reference Qi, Chen and Yan24] in this paper, the dynamic equation of HOD is

(15) \begin{align} {{{\dot{x}}_{h1}}} {} &= {x_{h2}} + {l_1}\left( {{x_r} - {x_{h1}}} \right),\nonumber\\[5pt]{{{\dot{x}}_{h2}}} {} & = {x_{h3}} + {l_2}\left( {{x_r} - {x_{h1}}} \right),\nonumber\\[5pt]{{{\dot{x}}_{h3}}} {} &= {l_3}\left( {{x_r} - {x_{h1}}} \right).\end{align}

The output equation of HOD is

(16) \begin{align}\begin{array}{*{20}{l}}{{{\hat x}_r}} {}{ = {x_{h1}},}\\[5pt]{{{\hat{\dot{x}}}_r}} {}{ = {{\dot{x}}_{h1}} = {x_{h2}} + {l_1}\left( {{x_r} - {x_{h1}}} \right).}\end{array}\end{align}

where ${x_r}$ denotes the reference input, ${x_{hi}}\left( {i = 1,{\rm{\;}}2,{\rm{\;}}3} \right)$ the internal state variable of the HOD system, ${l_i}\left( {i = 1,{\rm{\;}}2,{\rm{\;}}3} \right)$ the adjustable parameter, ${\hat{\dot{x}}_r}$ the estimation of the ${\dot{x}_r}$ . Assuming the HOD system is stabilised, the parameter ${l_i}$ is uniquely determined by $a$ using the root trajectory method, i.e.

(17) \begin{align}{l_i} = \frac{{{n^n}a}}{{{{(n - 1)}^{n - 1}}}}C_{n - 1}^{i - 1}{a^{i - 1}}{\rm{\;\;}}\left( {i = 1,{\rm{\;}}2,{\rm{\;}}3} \right).\end{align}

3.3 Compensation function observer

The second-order CFO [Reference Qi, Hu, Li and Li25] proposed by Qi et al. has the specific form

(18) \begin{align}{{{\dot z}_1}} {} &= {f_k} + l{e_o} + {z_2} + bu,\nonumber \\[5pt]{{{\dot z}_2}} {} &= \lambda l{e_o},\nonumber \\[5pt]{{{\hat{f}}_{uk}}} {} &= l{e_o} + {z_2}.\end{align}

Define ${e_c} = {[\begin{array}{*{20}{l}}{{e_o}{\rm{\;\;\;\;}}{e_f}}\end{array}]^T}$ as the observation error of the CFO, and ${e_o},{\rm{\;}}{e_f}$ are denoted as

(19) \begin{align} {e_o} = x - {z_1},{\rm{\;}}{e_f} \,=\, {f_{uk}} - {{\hat f}_{uk}}.\end{align}

Its characteristic equation is designed by the method of zero-pole configuration as

(20) \begin{align}{s^2} + ls + \lambda l = {{(s + \omega )}^2}. \end{align}

Yield $l = 2\omega, {\rm{\;}}\lambda = 1/2\omega .$ Figure 3 shows the overall control block diagram of the composite VTOL UAV.

Figure 3.

Overall control block diagram.

3.4 Stability analysis of the model compensated controller

The proof of Theorem 1 refers to the literature author10. The idea in the paper is given here as a simple proof procedure.

Proof: define a Lyapunov function

(21) \begin{align}\begin{array}{*{20}{l}}{V = \frac{1}{2}{e^2} + \frac{1}{2}{{\tilde e}^2} + \frac{1}{2}e_f^2.}\end{array}\end{align}

Derivation of Equation (20) yields

(22) \begin{align} \dot V = e\dot{e} + \tilde e\dot{\tilde{e}} + {e_f}{{\dot{e}}_f} = e\left( {{{\dot{x}}_r} - \dot{x}} \right) + \tilde e\dot{\tilde{e}} + {e_f}{{\dot{e}}_f}. \end{align}

Substituting Equation (13) into the above equation

(23) \begin{align} {\dot V} &= e\left( {{{\dot{x}}_r} - {f_k} - {f_{uk}} - bu} \right) + \tilde e\dot{\tilde{e}} + {e_f}{{\dot{e}}_f} \nonumber\\[5pt] &= e\left( {{{\dot{x}}_r} - {{\hat{\dot{x}}}_r} + {{\hat f}_{uk}} - {f_{uk}} - k\left( {{x_r} - x} \right)} \right) + \tilde e\dot{\tilde{e}} + {e_f}{{\dot{e}}_f} \nonumber\\[5pt] &= e\left( {\tilde e - {e_f} - ke} \right) + \tilde e\dot{\tilde{e}} + {e_f}{{\dot{e}}_f} \nonumber\\[5pt] &= - k{e^2} + \tilde e\left( {e + \dot{\tilde{e}}} \right) + {e_f}\left( {{{\dot{e}}_f} - e} \right). \end{align}

According to Lemma 1, the estimation error of HOD tends to zero and the estimation error of CFO tends to zero. Therefore holds.

Proof complete.