2.1 Position and velocity sensing

GPS provides position of a device, and accurate velocity can be determined by GPS with Doppler shift measurement or by a Doppler radar sensor.

Define the position measurement: ${p_m}(t) = {p_0}(t) + {d_1}(t)$ , where, ${p_0}(t)$ is the actual position; ${d_1}(t)$ is the disturbance in position measurement, and $\mathop {\sup }\nolimits_{t \in \left[ {0,\infty } \right)} \left| {{d_1}(t)} \right| \le {L_1} \lt \infty $ .

Define the velocity measurement: ${v_m}(t) = {v_0}(t) + {d_2}(t)$ , where, ${v_0}(t)$ is the actual velocity; ${d_2}(t)$ is the disturbance in velocity measurement, and $\mathop {\sup }\nolimits_{t \in \left[ {0,\infty } \right)} \left| {{d_2}(t)} \right| \le {L_2} \lt \infty $ .

Furthermore, due to the reliability of Doppler measurement, velocity accuracy usually remains unchanged. However, the position accuracy may become worse because of different environmental influences, i.e. ${L_1}$ may increase. Therefore, there exists a small constant $\varepsilon \gt 0$ , i.e. the upper-bound of sensor accuracy ratio, such that the sensor accuracy inequality $\max \left\{ {\frac{{{L_2}}}{{{L_1}}}} \right\} \le \varepsilon \ll 1$ holds.

2.2 Attitude sensing

The gyroscopes in IMU provide the relatively accurate angular velocities, e.g. their accuracy is about ${L_2}{ = 10^ \circ }$ /hr = 0.003º/s. The accuracy of attitude angles from IMU is relatively large, e.g. about ${L_1}{ = 1.0^ \circ }$ . Therefore, $\frac{{{L_2}}}{{{L_1}}} = \frac{{0.003}}{1} = 0.003 \ll 1$ , and we can select an upper-bound of sensor accuracy ratio $0.003 \le \varepsilon = 0.003 \ll 1$ to satisfy $\max \left\{ {\frac{{{L_2}}}{{{L_1}}}} \right\} \le \varepsilon \ll 1$ .

2.3 Effect of sensing disturbance on safe flight

Sensing disturbance has a serious impact on safe flight. Disturbance in position sensing may generate incorrect desired attitude angles, and they are mismatched to actual position trajectory. Therefore, the determined attitude is unwanted, and it may be dangerous.

In addition, in position measurement, the frequency bands of actual position ${p_0}(t)$ and sensing disturbance ${d_1}(t)$ may have intersections: the disturbance ${d_1}(t)$ may be in low-, mid- or high-frequency bands. It is impossible for the usual filters to separate ${d_1}(t)$ from the actual position signal ${p_0}(t)$ .

Questions: How to reject disturbance ${d_1}(t)$ in position measurement ${p_m}(t)$ using the relatively accurate velocity ${v_m}(t)$ ? Also, how to reject the disturbance in angle measurement using the relatively accurate angular velocity?

3.1 Preliminary

The related background is presented here.

Lemma 3.1 (sliding mode with prescribed convergence law) [Reference Perruquetti and Barbot26, Reference Levant27]: The following system is considered:

(1) \begin{align}\dot{e}_1 & = {e_2}\nonumber\\ \dot{e}_2 & = \varphi ( t ) + \gamma ( t )\dot{u} \end{align}

where, ${e_1}$ and ${e_2}$ are the states; u is the controller, and $\dot{u}$ in the system; $\left| {\varphi \left( t \right)} \right| \le \Phi $ , $0 \lt {\Gamma _m} \le \gamma \left( t \right) \le {\Gamma _M}$ , $\Phi \gt 0$ . A 2-sliding control algorithm is as follows:

(2) \begin{equation}\dot{u} = \begin{cases} - u,\ {\rm{if}}\left| u \right| \gt 1;\\ - {V_M}{\rm{sign}}\left[ {{e_2} + g({e_1})} \right]\!,\ {\rm{if}}| u | \le 1\end{cases}\end{equation}

where, $g({e_1})$ is smooth everywhere except on ${e_1} = 0$ , for example, $g({e_1}) = {k_1}{\left| {{e_1}} \right|^\alpha }{\rm{sign}}({e_1})$ , $\alpha \in \left[ {0.5,1} \right)$ ; ${V_M} \gt \frac{\Phi + \sup \left[ {{g^\prime}\left( {e_1} \right)g\left( {e_1} \right)} \right]}{\Gamma_m}$ . Then, we get the finite-time convergence law ${\dot{e}_1} = - g({e_1})$ (i.e. sliding surface ${e_2} + g({e_1}) = 0$ ), and there exists a finite time ${t_s} \gt 0$ , for $t \geqslant {t_s}$ , such that

(3) \begin{equation}{e_1} = 0\ {\rm{and}}\ {e_2} = 0\end{equation}

Remark 3.1: For system (1), when we select $\varphi \left( t \right) = 0$ and $\gamma \left( t \right) = 1$ , it becomes

(4) \begin{align}{\dot{e}_1} & = {e_2}\nonumber\\{\dot{e}_2} & = \dot{u}\end{align}

Then, the 2-sliding control algorithm (2) is expressed by

(5) \begin{equation}\dot{u} = {\dot{e}_2} = \begin{cases} - {e_2},\ {\rm{if}}\left| {{e_2}} \right| \gt 1;\\ - {V_M}{\rm{sign}}\left[ {{e_2} + g({e_1})} \right]\!,\ {\rm{if}}\left| {{e_2}} \right| \le 1\end{cases}\end{equation}

3.2 Corollary on 2-sliding mode system

Combining the system (4) and the 2-sliding control algorithm (5), we get the following corollary on a 2-sliding mode with prescribed convergence law.

Corollary 3.1 (sliding mode with prescribed finite-time convergence law): A 2-sliding mode system is as follows:

(6) \begin{align}{\dot e_1} & = {e_2}\nonumber\\{\dot e_2} & = \left\{ {\begin{array}{*{20}{l}}{ - {e_2},\ {\rm{if}}\left| {{e_2}} \right| \gt 1;}\\{ - {V_M} \cdot {\rm{sign}}\left[ {{e_2} + g({e_1})} \right]\!,\ {\rm{if}}\left| {{e_2}} \right| \le 1}\end{array}} \right.\end{align}

where, ${e_1}$ and ${e_2}$ are the sliding variables; $g({e_1})$ is smooth everywhere except on ${e_1} = 0$ , for example, $g({e_1}) = {k_1}{\left| {{e_1}} \right|^\alpha }{\rm{sign}}({e_1})$ , $\alpha \in \left[ {0.5,1} \right)$ ; ${V_M} \gt \sup \left[ {{g^\prime}\left( {{e_1}} \right)g\left( {{e_1}} \right)} \right]$ . Then, we get the finite-time convergence law ${\dot e_1} = - g({e_1})$ (i.e. sliding surface ${e_2} + g({e_1}) = 0$ ), and there exists a finite time ${t_s} \gt 0$ , for $t \geqslant {t_s}$ , such that

(7) \begin{equation}{e_1} = 0\ {\rm{and}}\ {e_2} = 0\end{equation}

3.3 Notation

“ $a( \omega )\;:\;{b_1} \to {b_2}$ as $\omega \;:\; {c_1} \to {c_2}$ ” means that function $a\left( \omega \right)$ varies monotonically increasing or decreasing from ${b_1}$ to ${b_2}$ as $\omega $ increases from ${c_1}$ to ${c_2}$ .

6.1 Modelling of jet UAV flight dynamics

For the UAV trajectory tracking control, the modelling is considered in the earth-fixed frame [Reference Çetinsoy, Dikyar, Hançer, Oner, Sirimoglu, Unel and Aksit23]. Furthermore, the constructed model is fit for observer design to estimate system uncertainties.

Let $\Gamma = ({E_x},{E_y},{E_z})$ denote the earth-fixed frame, and $\Lambda = (E_x^b,E_y^b,E_z^b)$ denote the body frame of the UAV. ${\Theta _\Gamma } = [\begin{array}{*{20}{c}}\phi \theta \psi \end{array}{]^T} \in {\Re ^3}$ describes the UAV roll, pitch and yaw angles (Euler angles), and ${\Omega _\Gamma } = [\begin{array}{*{20}{c}}{\dot \phi } {\dot \theta } {\dot \psi }\end{array}{]^T}$ . We use ${s_\theta }$ for $\sin \theta $ and ${c_\theta }$ for $\cos \theta $ . ${R_{\Gamma \Lambda }}$ is the transformation matrix representing the orientation of the body frame $\Lambda $ with respect to the earth-fixed frame $\Gamma $ :

(24) \begin{align}{R_{\Gamma \Lambda }} = \left[ \begin{matrix} {c_\theta }{c_\psi } &&& {{c_\psi }{s_\theta }{s_\phi } - {s_\psi }{c_\phi }} &&& {{c_\psi }{s_\theta }{c_\phi } + {s_\psi }{s_\phi }}\\{c_\theta }{s_\psi } &&& {{s_\psi }{s_\theta }{s_\phi } + {c_\psi }{c_\phi }} &&& {{s_\psi }{s_\theta }{c_\phi } - {c_\psi }{s_\phi }}\\ - {s_\theta } &&& {{s_\phi }{c_\theta }} &&& {{c_\phi }{c_\theta }}\end{matrix} \right]\end{align}

Let $\alpha $ and $\beta $ be the angle-of-attack and the sideslip angle, respectively, we can get

(25) \begin{equation}\alpha = \theta - \mathop {\arctan }\nolimits^{ - 1} (W/U),\beta = \mathop {\arcsin }\nolimits^{ - 1} (V/{V_T})\end{equation}

where, ${\upsilon _\Lambda } = (U,V,W)$ is the linear velocity in body frame $\Lambda $ , and ${V_T} = \sqrt {{U^2} + {V^2} + {W^3}} $ .

Define ${p_\Gamma } = (x,y,z)$ and ${\upsilon _\Gamma } = (\dot x,\dot y,\dot z)$ as the position and velocity vectors of centre of gravity, respectively, relative to the earth-fixed frame $\Gamma $ ; ${\Omega _\Lambda }$ is the angular rate vector of the airframe expressed in the body frame $\Lambda $ ; $m \in \Re $ is the UAV mass, and $J = {\rm{diag}}\{ {J_{xb}}$ , ${J_{yb}}$ , ${J_{zb}}\} \in {\Re ^{3 \times 3}}$ is the UAV inertia matrix. Then, the dynamic equations for the jet UAV subjected to force $F \in {\Re ^3}$ and torque $\tau \in {\Re ^3}$ are given by

(26) \begin{align}{\dot p_\Gamma }& = {\upsilon _\Gamma }\nonumber\\[3pt]m \cdot {\dot \upsilon _\Gamma } & = F + mg{E_z}\nonumber\\[3pt]J \cdot {\dot \Omega _\Lambda } & = - {\Omega _\Lambda } \times \left( {J{\Omega _\Lambda }} \right) + \tau \end{align}

where, ${E_z} = {\left[ {\begin{matrix}0 && 0&& 1\end{matrix}} \right]^T}$ . The relation between the angular rate vector ${\Omega _\Lambda } = {\left[ {\begin{matrix}{{p_\Lambda }}&& {{q_\Lambda }}&& {{r_\Lambda }}\end{matrix}} \right]^T}$ and the Euler angle derivative vector ${\Omega _\Gamma } = [\begin{matrix}{\dot \phi }&& {\dot \theta }&& {\dot \psi }\end{matrix}{]^T}$ is given by

(27) \begin{equation}{\Omega _\Lambda } = \mathbb{Z}{\Omega _\Gamma }\ {\rm{or}}\ {\Omega _\Gamma } = {\mathbb{Z}^{ - 1}}{\Omega _\Lambda }\end{equation}

where,

(28)

The total external force F consists of the thrust ${F_{jet}}$ generated by the jet engine, aerodynamic forces on the fixed wing ${F_w}$ , aerodynamic forces on the fuselage ${F_f}$ , the forces created by the rudders ${F_r}$ , the forces created by the elevators ${F_e}$ , and uncertainties and external disturbances ${F_d}$ . These forces are expressed in body frame $\Lambda $ , and they are transformed by ${R_{\Gamma \Lambda }}$ to be expressed in the earth-fixed frame $\Gamma $ as follows:

(29) \begin{equation}F = {R_{\Gamma \Lambda }}({F_{jet}} + {F_w} + {F_f} + {F_r} + {F_e} + {F_d})\end{equation}

The total moment $\tau $ consists of the moments created by the fixed wings ${\tau _w}$ , the moments created by the rudders ${\tau _r}$ , the moments created by the elevators ${\tau _e}$ , and moments due to the uncertainties and external disturbances ${\tau _d}$ :

(30) \begin{equation}\tau = {\tau _w} + {\tau _r} + {\tau _e} + {\tau _d}\end{equation}

The aerodynamic parametres of the UAV are from [Reference Wang and Cai17]. The CFD (computational fluid dynamics) simulation was performed, and the results from the wind tunnel tests were compared. For the boundary conditions, the method of free-stream boundary condition based on Riemann invariants was utilised [Reference Boelens28]: no-slip viscous flow condition; the linearised one-dimensional Euler equations; the free-stream values for determination of the value of the Riemann invariants; and the symmetrical boundary condition for the symmetrical UAV. The ANSYS Fluent was used for the CFD simulation, and the design steps included meshing fluid field, fluent simulation and post-processing [Reference Wang, Chen and Yuan29].

1. (1) Thrust by jet engine: The thrust of engine in body frame is expressed by

   (31) \begin{equation}{F_{jet}} = {\left[ {\begin{matrix}{{F_c}} && 0 &&0\end{matrix}} \right]^T}\end{equation}

2. (2) Aerodynamics of fixed wings

Define $q = 0.5\rho ({U^2} + {W^2})$ , where, $\rho $ is the air density. The lift and drag forces generated by the fixed wings are, respectively

(32) \begin{align}{L_i} & = q{S_w}{C_{Li}},{C_{Li}} = {C_{L0}} + {C_{L\alpha }}\alpha + {C_{L{\delta _i}}}{\delta _i}\nonumber\\[3pt]{D_i} & = q{S_w}{C_{Di}},{C_{Di}} = {C_{D0}} + C_{Li}^2/(\pi {A_w}{e_w})\nonumber\\[3pt]{e_w} & = 1.78(1 - 0.045A_w^{0.68}) - 0.46\end{align}

where $i = 1,2$ ; ${S_w}$ is the area of the half wing; ${C_{L0}}$ is the lift coefficient when the angle-of-attack $\alpha $ equals zero; ${C_{L\alpha }}$ is the lift coefficient due to the angle-of-attack $\alpha $ ; ${\delta _i}$ is the aileron deflection, and ${C_{L{\delta _i}}}$ is the lift coefficient due to the aileron deflection ${\delta _i}$ ; ${C_{D0}}$ is the drag coefficient when $\alpha = {\delta _i} = 0$ ; ${A_w}$ is the aspect ratio of the fixed wing; ${e_w}$ is the value of the Oswald’s efficiency factor. The expression of lift and drag coefficients is considered as valid for low angles of attack.

Then the aerodynamic force vector ${F_w}$ on the fixed wings in body frame can be written as

(33) \begin{equation}{F_w} = \left[ {\begin{matrix}{({L_1} + {L_2}){s_\alpha } - ({D_1} + {D_2}){c_\alpha }}\\[3pt]0\\[3pt]{ - ({L_1} + {L_2}){c_\alpha } - ({D_1} + {D_2}){s_\alpha }}\end{matrix}} \right]\end{equation}

The fixed-wing moment ${\tau _w}$ includes the aerodynamic moment ${\tau _{wa}}$ and control torque ${\tau _{wc}}$ around the body axis $E_x^b$ , i.e. ${\tau _w} = {\tau _{wa}} + {\tau _{wc}}$ , where,

(34) \begin{equation}{\tau _{wa}} = \left[ {\begin{matrix}{{l_w}({D_1} - {D_2}){s_\alpha }}\\[3pt]{{l_c}[({L_2} + {L_1}){c_\alpha } + ({D_2} + {D_1}){s_\alpha }]}\\[3pt]{{l_w}[({L_1} - {L_2}){s_\alpha } + ({D_2} - {D_1}){c_\alpha }]}\end{matrix}} \right]\end{equation}

and

(35) \begin{equation}{\tau _{wc}} = \left[ {\begin{array}{*{20}{c}}{{l_w}q{S_w}{C_{L{\delta _{1.2}}}}({\delta _1} - {\delta _2}){c_\alpha }}\\[3pt]0\\[3pt]0\end{array}} \right]\end{equation}

1. (3) Fuselage

The parametres of fuselage lift and drag are described as follows:

(36) \begin{align}{L_f}& = q{S_f}{C_{lf}},{C_{lf}} = {C_{lf\alpha }}\alpha ,\nonumber\\[3pt]{D_f} & = q{S_f}{C_{df}},{C_{df}} = {C_{df0}} + {C_{df\alpha }}\alpha \end{align}

where, ${S_f}$ is the fuselage equivalent cross-sectional area; ${L_f}$ and ${D_f}$ are the lift and drag forces generated by the fuselage, respectively; ${C_{lf}}$ is the lift coefficient; ${C_{df}}$ is the drag coefficient; ${C_{df0}}$ is the constant in the coefficient of drag force. Then the force vector ${F_f}$ on the fuselage in body frame is expressed by

(37) \begin{equation}{F_f} = \left[ {\begin{array}{*{20}{c}}{{L_f}{s_\alpha } - {D_f}{c_\alpha }}\\[3pt]0\\[3pt]{ - {L_f}{c_\alpha } - {D_f}{s_\alpha }}\end{array}} \right]\end{equation}

1. (4) Elevator

The parametres of elevator lift and drag are described as follows:

(38) \begin{align}{L_e} & = q{S_e}{C_{le}},{C_{le}} = {C_{le\alpha }}(\alpha + {\delta _e})\nonumber\\{D_e} & = q{S_e}{C_{de}},{C_{de}} = {C_{de0}} + C_{le}^2/(\pi {A_e}{e_e})\nonumber\\{e_e} & = 1.78(1 - 0.045A_e^{0.68}) - 0.46\end{align}

where ${S_e}$ is the area of the elevator, ${\delta _e}$ is the elevator deflection; ${C_{le\alpha }}$ is the lift coefficient due to the angle-of-attack $\alpha $ and the deflection ${\delta _e}$ ; ${C_{de0}}$ is the drag coefficient when $\alpha + {\delta _e} = 0$ ; ${A_e}$ is the aspect ratio of the elevator; ${e_e}$ is the value of the Oswald’s efficiency factor. Then the force vector ${F_e}$ on the elevator in body frame is expressed by

(39) \begin{equation}{F_e} = \left[ {\begin{array}{*{20}{c}}{{L_e}{s_\alpha } - {D_e}{c_\alpha }}\\0\\{ - {L_e}{c_\alpha } - {D_e}{s_\alpha }}\end{array}} \right]\end{equation}

The elevator moment ${\tau _e}$ includes the aerodynamic moment ${\tau _{ea}}$ and control torque ${\tau _{ec}}$ in the body axis $E_y^b$ , i.e. ${\tau _e} = {\tau _{ea}} + {\tau _{ec}}$ , where,

(40) \begin{equation}{\tau _{ea}} = \left[ {\begin{array}{*{20}{c}}0\\{ - {l_e}{D_e}{s_\alpha }}\\0\end{array}} \right]\end{equation}

and

(41) \begin{equation}{\tau _{ec}} = \left[ {\begin{array}{*{20}{c}}0\\{ - {l_e}q{S_e}{C_{le\alpha }}(\alpha + {\delta _e}){c_\alpha }}\\0\end{array}} \right]\end{equation}

1. (5) Rudders

Define $\bar q = 0.5\rho ({U^2} + {V^2})$ . The lift and drag forces generated by the rudders, respectively

(42) \begin{align}{L_r} & = \bar q{S_r}{C_{lr}},{C_{lr}} = {C_{lr\beta }}\beta + {C_{lr{\delta _r}}}{\delta _r}\nonumber\\{D_r} & = \bar q{S_r}{C_{dv}},{C_{dr}} = {C_{dr0}} + C_{lr}^2/(\pi {A_r}{e_r})\nonumber\\{e_r} & = 1.78(1 - 0.045A_r^{0.68}) - 0.46\end{align}

where, ${S_r}$ is the area of the rudders; ${C_{lr\beta }}$ is the lift coefficient due to the sideslip angle $\beta $ ; ${\delta _r}$ is the rudder deflection, and ${C_{lr{\delta _r}}}$ is the lift coefficient due to the deflection ${\delta _r}$ ; ${C_{dr0}}$ is the drag coefficient when $\beta = {\delta _r} = 0$ ; ${A_r}$ is the aspect ratio of the rudder; ${e_r}$ is the value of the Oswald’s efficiency factor. Then the aerodynamic force vector ${F_r}$ on the rudders in body frame can be expressed by

(43) \begin{equation}{F_r} = \left[ {\begin{array}{*{20}{c}}{{L_r}{s_\beta } - {D_r}{c_\beta }}\\{{L_r}{c_\beta } + {D_r}{s_\beta }}\\0\end{array}} \right]\end{equation}

The rudder moment ${\tau _r}$ includes the aerodynamic moment ${\tau _{ra}}$ and control torque ${\tau _{rc}}$ in body axis $E_z^b$ , i.e. ${\tau _r} = {\tau _{ra}} + {\tau _{rc}}$ , where,

(44) \begin{equation}{\tau _{ra}} = \left[ {\begin{array}{*{20}{c}}0\\0\\{{l_r}{D_r}{s_\beta }}\end{array}} \right]\end{equation}

and

(45) \begin{equation}{\tau _{rc}} = \left[ {\begin{array}{*{20}{c}}0\\0\\{{l_r}\bar q{S_r}({C_{lr\beta }}\beta + {C_{lr{\delta _r}}}{\delta _r}){c_\beta }}\end{array}} \right]\end{equation}

1. (6) UAV motion equations in the earth-fixed frame considering system uncertainties

According to [Reference Çetinsoy, Dikyar, Hançer, Oner, Sirimoglu, Unel and Aksit23], from (27) and (26), we get

(46) \begin{align}{\dot \Omega _\Gamma }& = {\left( {{\mathbb{Z}^{ - 1}}} \right)^{\prime}}{\Omega _\Lambda } + {\mathbb{Z}^{ - 1}}{\dot \Omega _\Lambda }\nonumber\\& = {\left( {{\mathbb{Z}^{ - 1}}} \right)^{\prime}}\mathbb{Z}{\Omega _\Gamma } - {\mathbb{Z}^{ - 1}}{J^{ - 1}}\left[ {\mathbb{Z}{\Omega _\Gamma } \times \left( {J\mathbb{Z}{\Omega _\Gamma }} \right)} \right] + {\mathbb{Z}^{ - 1}}{J^{ - 1}}\tau \end{align}

The total moment $\tau $ can be categorised into the control torque ${\tau _c}$ , uncertain moment ${\tau _d}$ and other moments ${\tau _{other}}$ , i.e.

(47) \begin{equation}\tau = {\tau _c} + {\tau _{other}} + {\tau _d}\end{equation}

From (35), (41) and (45), we get

(48) \begin{equation}{\tau _c} = {\tau _{wc}} + {\tau _{ec}} + {\tau _{rc}}\end{equation}

and from (34), (40) and (44), we get

(49) \begin{equation}{\tau _{other}} = {\tau _{wa}} + {\tau _{ea}} + {\tau _{ra}}\end{equation}

In system (46), considering (47), we define

(50) \begin{equation}{\left( {{\mathbb{Z}^{ - 1}}} \right)^{\prime}}\mathbb{Z}{\Omega _\Gamma } - {\mathbb{Z}^{ - 1}}{J^{ - 1}}\left[ {\mathbb{Z}{\Omega _\Gamma } \times \left( {J\mathbb{Z}{\Omega _\Gamma }} \right)} \right] + {\mathbb{Z}^{ - 1}}{J^{ - 1}}{\tau _{other}}\mathop = \limits^{{\rm{define}}} \left[ {\begin{array}{*{20}{c}}{{a_\phi }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right)}\\[3pt]{{a_\theta }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right)}\\[3pt]{{a_\psi }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right)}\end{array}} \right]\end{equation}

(51) \begin{equation}{\mathbb{Z}^{ - 1}}{J^{ - 1}}{\tau _c}\mathop = \limits^{{\rm{define}}} {\left[ {\begin{matrix}{{\tau _\phi }} &&& {{\tau _\theta }}&&& {{\tau _\psi }}\end{matrix}} \right]^T}\end{equation}

In (29), we define

(52) \begin{equation}{R_{\Gamma \Lambda }}({F_w} + {F_f} + {F_r} + {F_e})/m\mathop = \limits^{{\rm{define}}} \left[ {\begin{array}{*{20}{c}}{{F_{xa}}(\phi ,\theta ,\psi ,\alpha ,\beta )}\\[3pt]{{F_{ya}}(\phi ,\theta ,\psi ,\alpha ,\beta )}\\[3pt]{{F_{za}}(\phi ,\theta ,\psi ,\alpha ,\beta )}\end{array}} \right]\end{equation}

Then, the jet UAV motion equations written in terms of the centre of mass C in the earth-fixed frame are

(53) \begin{align}\ddot x & = {c_\theta }{c_\psi }{F_c}/m + {F_{xa}}(\phi ,\theta ,\psi ,\alpha ,\beta ) + {\Delta _x}\nonumber\\\ddot y & = {c_\theta }{s_\psi }{F_c}/m + {F_{ya}}(\phi ,\theta ,\psi ,\alpha ,\beta ) + {\Delta _y}\nonumber\\\ddot z & = - {s_\theta }{F_c}/m + {F_{za}}(\phi ,\theta ,\psi ,\alpha ,\beta ) + g + {\Delta _z}\end{align}

(54) \begin{align}\ddot \phi& = {a_\phi }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) + {\tau _\phi } + {\Delta _\phi }\nonumber\\\ddot \theta & = {a_\theta }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) + {\tau _\theta } + {\Delta _\theta }\nonumber\\\ddot \psi & = {a_\psi }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) + {\tau _\psi } + {\Delta _\psi }\end{align}

where, m is the mass of the UAV; g is the gravity acceleration; ( ${\Delta _x}$ , ${\Delta _y}$ , ${\Delta _z}$ ) and ( ${\Delta _\phi }$ , ${\Delta _\theta }$ , ${\Delta _\psi }$ ) are the bounded uncertainties in the position and attitude dynamics, respectively; ${\tau _\phi }$ , ${\tau _\theta }$ and ${\tau _\psi }$ are the control torques for roll, pitch and yaw dynamics, respectively, defined in (51).

6.2 Measurements

For the jet UAV, a GPS receiver provides the global position and the velocity. An IMU gives the attitude angle and angular velocity. The measurement outputs are expressed by

(55) \begin{equation}{y_{ * 1}} = * + {d_{ * 1}}(t),{y_{ * 2}} = \dot * + {d_{ * 2}}(t)\end{equation}

where, ${ * _i} = \{ x,y,z,\phi ,\theta ,\psi \} $ , and $\dot * = \{ \dot x,\dot y,\dot z,\dot \phi ,\dot \theta ,\dot \psi \} $ ; ${d_{ * 1}}(t)$ denotes the disturbances in position and angle measurements, and $\mathop {\sup }\nolimits_{t \in [0,\infty )} \left| {{d_{ * 1}}(t)} \right| \le {L_{ * 1}} \lt \infty $ ; ${d_{ * 2}}(t)$ denotes the disturbances in the measurements of flying velocity and angular velocity, and $\mathop {\sup }\nolimits_{t \in [0,\infty )} \left| {{d_{ * 2}}(t)} \right| \le {L_{ * 2}} \lt \infty $ ; ${L_{ * 2}} \ll 1$ , and $\max \left\{ {\frac{{{L_{ * 1}}}}{{{L_{ * 2}}}}} \right\} \le {\varepsilon _ * } \ll 1$ . The corrector (17) is used to reject the measurement disturbances and to estimate the actual (x, y, z, $\phi $ , $\theta $ , $\psi $ ).

6.3 Controller design

In this section, the control laws are derived for UAV trajectory tracking and attitude stabilisation. The position, attitude and system uncertainties are reconstructed by the presented corrector and an existing extended state observer.

The scheme of control system with correction and estimation is shown in Fig. 4: (1) the correctors estimate position and attitude angles, and the disturbances are rejected from the measurements; (2) the extended state observers estimate the uncertainties in position and attitude dynamics, respectively; (3) according to the reference trajectory and estimation from the correctors and observers, the position controller drives the position dynamics; (4) from the estimation of position, attitude, uncertainties and the reference trajectory, the desired attitude is determined; (5) according to the desired attitude and estimation from the correctors and observers, the attitude controller drives the attitude dynamics.

1. (1) Error systems

Figure 4. Scheme of control system.

Suppose the reference trajectory and its finite order derivatives are bounded, and can be generated directly. For the reference trajectory ${X_d} = ({x_d},{y_d},{z_d})$ , let ${e_x} = x - {x_d}$ , ${e_y} = y - {y_d}$ , and ${e_z} = z - {z_d}$ , then the system error for position dynamics (53) is

(56) \begin{equation}{\ddot e_p} = {u_p} + {\Xi _p} + {\Delta _p}\end{equation}

where,

(57) \begin{align}{e_p} & = \left[ {\begin{array}{*{20}{c}}{{e_x}}\\[3pt]{{e_y}}\\[3pt]{{e_z}}\end{array}} \right]\!,\ {u_p} = \left[ {\begin{array}{*{20}{c}}{{u_{px}}}\\[3pt]{{u_{py}}}\\[3pt]{{u_{pz}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{c_\theta }{c_\psi }}\\[3pt]{{c_\theta }{s_\psi }}\\[3pt]{ - {s_\theta }}\end{array}} \right]{F_c}/m,\nonumber\\{\Xi _p} & = \left[ {\begin{array}{*{20}{c}}{{F_{xa}}(\phi ,\theta ,\psi ,\alpha ,\beta ) - {{\ddot x}_d}}\\[3pt]{{F_{ya}}(\phi ,\theta ,\psi ,\alpha ,\beta ) - {{\ddot y}_d}}\\[3pt]{{F_{za}}(\phi ,\theta ,\psi ,\alpha ,\beta ) + g - {{\ddot z}_d}}\end{array}} \right]\!,\ {\Delta _p} = \left[ {\begin{array}{*{20}{c}}{{\Delta _x}}\\[3pt]{{\Delta _y}}\\[3pt]{{\Delta _z}}\end{array}} \right]\end{align}

For the desired attitude angle ${\Theta _d} = ({\phi _d},{\theta _d},{\psi _d})$ , let ${e_\phi } = \phi - {\phi _d}$ , ${e_\theta } = \theta - {\theta _d}$ , and ${e_\psi } = \psi - {\psi _d}$ , then the system error for attitude dynamics (54) is

(58) \begin{equation}{\ddot e_a} = {u_a} + {\Xi _a} + {\Delta _a}\end{equation}

where,

(59) \begin{align}{e_a} & = \left[ {\begin{array}{*{20}{c}}{{e_\phi }}\\[3pt]{{e_\theta }}\\[3pt]{{e_\psi }}\end{array}} \right]\!,\ {u_a} = \left[ {\begin{array}{*{20}{c}}{{\tau _\phi }}\\[3pt]{{\tau _\theta }}\\[3pt]{{\tau _\psi }}\end{array}} \right]\!,\ {\Delta _a} = \left[ {\begin{array}{*{20}{c}}{{\Delta _\phi }}\\[3pt]{{\Delta _\theta }}\\[3pt]{{\Delta _\psi }}\end{array}} \right]\!,\nonumber \\{\Xi _a} & = \left[ {\begin{array}{*{20}{c}}{{a_\phi }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) - {{\ddot \phi }_d}}\\[3pt]{{a_\theta }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) - {{\ddot \theta }_d}}\\[3pt]{{a_\psi }\left( {\phi ,\theta ,\psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) - {{\ddot \psi }_d}}\end{array}} \right]\end{align}

1. (2) Extended state observers for the uncertainties ${\Delta _p}$ and ${\Delta _a}$

The extended state observers [Reference Wang and Lin24] are used to estimate the uncertainties in position dynamics (53) and attitude dynamics (54). For the observers, the measurements of flying velocity and angular velocity ${y_{ * 2}} = \dot * + {d_{ * 2}}(t)$ (where, $\dot * = \{ \dot x,\dot y,\dot z,\dot \phi ,\dot \theta ,\dot \psi \} $ ) are the input signals. Then, the uncertainty ${\Delta _p} = {\left[ {\begin{array}{*{20}{c}}{{\Delta _x}} {{\Delta _y}} {{\Delta _z}}\end{array}} \right]^T}$ in the position dynamics and the uncertainty ${\Delta _a} = {\left[ {\begin{array}{*{20}{c}}{{\Delta _\phi }} {{\Delta _\theta }} {{\Delta _\psi }}\end{array}} \right]^T}$ in the attitude dynamics are estimated.

The continuous extended state observers can provide smooth and accurate estimations of the uncertainties in the system dynamics, reducing high-frequency vibrations.

The following extended state observers are used [Reference Wang and Lin24]:

(60) \begin{align}{\dot x_{ * 1}}& = {x_{ * 2}} - {\lambda _{ * 1}}{\left| {{x_{ * 1}} - {y_{ * 2}}} \right|^{\frac{{1 + {\alpha _ * }}}{2}}}sign({x_{ * 1}} - {y_{ * 2}}) + {H_ * }\nonumber\\{\dot x_{ * 2}}& = - {\lambda _{ * 2}}{\left| {{x_{ * 1}} - {y_{ * 2}}} \right|^{{\alpha _ * }}}sign({x_{ * 1}} - {y_{ * 2}})\end{align}

From Theorem 1 in, [Reference Wang and Lin24] there exist a finite time ${t_s} \gt 0$ such that, for $t \geqslant {t_s}$ ,

(61) \begin{equation}{x_{ * 1}} = \dot * ,{x_{ * 2}} = {\Delta _ * }\end{equation}

where, $ * = \left\{ {x,y,z,\psi ,\theta ,\phi } \right\}$ and $\dot * = \{ \dot x,\dot y,\dot z,\dot \phi ,\dot \theta ,\dot \psi \} $ ; ${\lambda _{ * 1}},{\lambda _{ * 2}} \gt 0$ , and ${\alpha _ * } \in (0,1)$ ; The measurement ${y_{ * 2}}$ defined in (55) is the observer input signal;

(62) \begin{align}{H_x} & = {u_{px}} + {F_{xa}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta )\nonumber\\{H_y} & = {u_{py}} + {F_{ya}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta )\nonumber\\{H_z} & = {u_{pz}} + {F_{za}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) - g\nonumber\\{H_\phi } & = {\tau _\phi } + {a_\phi }\left( {\widehat \phi ,\widehat \theta ,\widehat \psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right)\nonumber\\{H_\theta } & = {\tau _\theta } + {a_\theta }\left( {\widehat \phi ,\widehat \theta ,\widehat \psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right)\nonumber\\{H_\psi } & = {\tau _\psi } + {a_\psi }\left( {\widehat \phi ,\widehat \theta ,\widehat \psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right)\end{align}

and ( $\widehat \phi $ , $\widehat \theta $ , $\widehat \psi $ ) are from the outputs of the sliding mode correctors. From (60) and (61), we get

(63) \begin{equation}{\widehat \Delta _p} = {\left[ {\begin{array}{*{20}{c}}{{x_{x2}}} {{x_{y2}}} {{x_{z2}}}\end{array}} \right]^T}\end{equation}

and

(64) \begin{equation}{\widehat \Delta _a} = {\left[ {\begin{array}{*{20}{c}}{{x_{\psi 2}}} {{x_{\theta 2}}} {{x_{\phi 2}}}\end{array}} \right]^T}\end{equation}

1. (3) Controller design for position dynamics

For position dynamics (53), to track the reference trajectory ${X_d} = ({x_d},{y_d},{z_d})$ , when we select the following controller, the position system error (56) will converge asymptotically to zero:

(65) \begin{equation}{u_p} = - {\widehat \Xi _p} - {\widehat \Delta _p} - {k_{p1}}{\widehat e_p} - {k_{p2}}{\widehat {\dot e}_p}\end{equation}

where ${\widehat e_x} = \widehat x - {x_d}$ , ${\widehat {\dot e}_x} = \widehat {\dot x} - {\dot x_d}$ , ${\widehat e_y} = \widehat y - {y_d}$ , ${\widehat {\dot e}_y} = \widehat {\dot y} - {\dot y_d}$ , ${\widehat e_z} = \widehat z - {z_d}$ , ${\widehat {\dot e}_z} = \widehat {\dot z} - {\dot z_d}$ ; ${k_{p1}},{k_{p2}} \gt 0$ ; ( $\widehat x$ , $\widehat y$ , $\widehat z$ , $\widehat {\dot x}$ , $\widehat {\dot y}$ , $\widehat {\dot z}$ , $\widehat \phi $ , $\widehat \theta $ , $\widehat \psi $ ) are from the outputs of the correctors; ${\widehat \Delta _p}$ is from the outputs of the extended state observer; and

(66) \begin{align}{\widehat e_p} & = \left[ {\begin{array}{*{20}{c}}{{{\widehat e}_x}}\\[3pt]{{{\widehat e}_y}}\\[3pt]{{{\widehat e}_z}}\end{array}} \right]\!,\ {\widehat {\dot e}_p} = \left[ {\begin{array}{*{20}{c}}{{{\widehat {\dot e}}_x}}\\[3pt]{{{\widehat {\dot e}}_y}}\\[3pt]{{{\widehat {\dot e}}_z}}\end{array}} \right]\!,\ \nonumber\\[6pt]{\widehat \Xi _p} & = \left[ {\begin{array}{*{20}{c}}{{F_{xa}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) - {{\ddot x}_d}}\\[3pt]{{F_{ya}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) - {{\ddot y}_d}}\\[3pt]{{F_{za}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) - g - {{\ddot z}_d}}\end{array}} \right]\end{align}

From (57), we get the engine thrust

(67) \begin{equation}{F_c} = m{\left\| {{u_p}} \right\|_2} = m\sqrt {u_{px}^2 + u_{py}^2 + u_{pz}^2} \end{equation}

1. (4) Desired attitude angles

From (56) and (65), we get

(68) \begin{equation}{\ddot e_z} = - {s_\theta }{F_c}/m + {F_{za}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) + g - {\ddot z_d} + {\widehat \Delta _z} = - {k_{p1}}{\widehat e_z} - {k_{p2}}{\widehat {\dot e}_z}\end{equation}

Then, the desired pitch angle is

(69) \begin{equation}{\theta _d} = \arcsin \frac{{m({F_{za}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) + g - {{\ddot z}_d} + {{\widehat \Delta }_z} + {k_{p1}}{{\widehat e}_z} + {k_{p2}}{{\widehat {\dot e}}_z})}}{{{F_c}}}\end{equation}

Also, from (56) and (65), we get

(70) \begin{align}{\ddot e_x} & = {c_\theta }{c_\psi }{F_c}/m + {F_{xa}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) + {\widehat \Delta _x} = - {k_{p1}}{\widehat e_x} - {k_{p2}}{\widehat {\dot e}_x}\nonumber\\{\ddot e_y} & = {c_\theta }{s_\psi }{F_c}/m + {F_{ya}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) + {\widehat \Delta _y} = - {k_{p1}}{\widehat e_y} - {k_{p2}}{\widehat {\dot e}_y}\end{align}

Then, the desired yaw angle is

(71) \begin{equation}{\psi _d} = \arctan \frac{{{F_{ya}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) + {{\widehat \Delta }_y} + {k_{p1}}{{\widehat e}_y} + {k_{p2}}{{\widehat {\dot e}}_y}}}{{{F_{xa}}(\widehat \phi ,\widehat \theta ,\widehat \psi ,\alpha ,\beta ) + {{\widehat \Delta }_x} + {k_{p1}}{{\widehat e}_x} + {k_{p2}}{{\widehat {\dot e}}_x}}}\end{equation}

The combination of lift force from the wings, elevator and fuselage can provide the centripetal force. The radius of curvature is

(72) \begin{equation}r = \frac{{{{\dot x}^2} + {{\dot y}^2}}}{{\left| {\dot x\ddot y - \ddot x\dot y} \right|}}\end{equation}

and the centripetal force is

(73) \begin{equation}{f_{centri}} = \frac{{m({{\dot x}^2} + {{\dot y}^2})}}{r} = m\left| {\dot x\ddot y - \ddot x\dot y} \right|\end{equation}

Also, the centripetal force can be expressed by

(74) \begin{equation}{f_{centri}} = {L_{wef}}\cos {\phi _d}\end{equation}

where, ${L_{wef}} = - ({L_1} + {L_2}){c_\alpha } - ({D_1} + {D_2}){s_\alpha } - {L_e}{c_\alpha } - {D_e}{s_\alpha } - {L_f}{c_\alpha } - {D_f}{s_\alpha }$ . Then, we get the desired roll angle

(75) \begin{equation}{\phi _d} = \arccos \frac{{{f_{centri}}}}{{{L_{wef}}}}\end{equation}

1. (5) Controller design for attitude dynamics

For attitude dynamics (54), to track the desired attitude ${\Theta _d} = ({\psi _d},{\theta _d},{\phi _d})$ , when we select the following controller, the attitude system error (58) will converge asymptotically to zero:

(76) \begin{equation}{u_a} = - {\widehat \Xi _a} - {\widehat \Delta _a} - {k_{a1}}{\widehat e_a} - {k_{a2}}{\widehat {\dot e}_a}\end{equation}

where, ${k_{a1}},{k_{a2}}$ ; ${\widehat e_\phi } = \widehat \phi - {\phi _d}$ , ${\widehat {\dot e}_\phi } = \widehat {\dot \phi } - {\dot \phi _d}$ , ${\widehat e_\theta } = \widehat \theta - {\theta _d}$ , ${\widehat {\dot e}_\theta } = \widehat {\dot \theta } - {\dot \theta _d}$ , ${\widehat e_\psi } = \widehat \psi - {\psi _d}$ , ${\widehat {\dot e}_\psi } = \widehat {\dot \psi } - {\dot \psi _d}$ ; ${\widehat e_a} = {\left[ {\begin{matrix}{{{\widehat e}_\phi }}&& {{{\widehat e}_\theta }} &&{{{\widehat e}_\psi }}\end{matrix}} \right]^T}$ ; ${\dot e_a} = {\left[ {\begin{matrix}{{{\widehat {\dot e}}_\phi }}&& {{{\widehat {\dot e}}_\theta }}&& {{{\widehat {\dot e}}_\psi }}\end{matrix}} \right]^T}$ ; ( $\widehat \phi $ , $\widehat \theta $ , $\widehat \psi $ ) are from the outputs of the correctors; ${\widehat \Delta _a}$ is from the outputs of the extended state observers; and

(77) \begin{equation}{\widehat \Xi _a} = \left[ {\begin{array}{*{20}{c}}{{a_\phi }\left( {\widehat \phi ,\widehat \theta ,\widehat \psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) - {{\ddot \phi }_d}}\\[3pt]{{a_\theta }\left( {\widehat \phi ,\widehat \theta ,\widehat \psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) - {{\ddot \theta }_d}}\\[3pt]{{a_\psi }\left( {\widehat \phi ,\widehat \theta ,\widehat \psi ,\dot \phi ,\dot \theta ,\dot \psi ,\alpha ,\beta } \right) - {{\ddot \psi }_d}}\end{array}} \right]\end{equation}

8.1 Design of correctors

8.1.3 Rejection ratio and disturbance frequency bands

1. (1) Rejection ratios in the disturbance frequency bands for position

From ${\varepsilon _ * } = 0.034$ and $r = \frac{1}{2}$ , where, $ * = \left\{ {x,y,z} \right\}$ , the rejection ratio is expressed by

$$\rho ({\omega _{ * 1}}) = \frac{1}{{\sqrt {1 + \frac{1}{4}\varepsilon _ * ^{2r - 2}\omega _{ * 1}^2} }} + \frac{1}{2}\varepsilon _ * ^r = \frac{1}{{\sqrt {1 + 7.35\omega _{ * 1}^2} }} + 0.09$$

Therefore, the rejection ratios in the different frequency bands of $\left[ {0, + \infty } \right)$ can be described by:

1. (i) In $\left[ {3.98{\rm{rad/s}},\infty } \right)$ , $\rho ({\omega _{ * 1}})\;:\;0.18 \to 0.09$ as ${\omega _{ * 1}}\;:\;3.98 \to \infty \,({\rm{rad}}/{\rm{s}})$ ;

2. (ii) In $\left( {0.17,3.98{\rm{rad/s}}} \right)$ , $\rho ({\omega _{ * 1}})\;:\;1 \to 0.18$ as ${\omega _{ * 1}}\;:\;0.17 \to 3.98\,({\rm{rad}}/{\rm{s}})$ ;

3. (iii) In $\left[ {0,0.17{\rm{rad/s}}} \right]$ , $\rho ({\omega _{ * 1}})\;:\;1.09 \to 1$ (i.e. $\rho ({\omega _{ * 1}}) \approx 1$ ) as ${\omega _{ * 1}}\;:\;0 \to 0.17\,({\rm{rad}}/{\rm{s}})$ .

1. (2) Rejection ratios in the disturbance frequency bands for attitude ( $\phi $ , $\theta $ )

From ${\varepsilon _ * } = 0.005$ and $r = \frac{1}{2}$ , where, $ * = \left\{ {\phi ,\theta } \right\}$ , the rejection ratio is expressed by

$$\rho ({\omega _{ * 1}}) = \frac{1}{{\sqrt {1 + \frac{1}{4}\varepsilon _ * ^{2r - 2}\omega _1^2} }} + \frac{1}{2}\varepsilon _ * ^r = \frac{1}{{\sqrt {1 + 50\omega _{ * 1}^2} }} + 0.035$$

Therefore, the rejection ratios in the different frequency bands can be described by:

1. (i) In $\left[ {4{\rm{rad/s}},\infty } \right)$ , $\rho ({\omega _{ * 1}})\;:\;0.07 \to 0.035$ as ${\omega _{ * 1}}\;:\;4 \to \infty \,({\rm{rad}}/{\rm{s}})$ ;

2. (ii) In $\left( {0.04,4{\rm{rad/s}}} \right)$ , $\rho ({\omega _{ * 1}})\;:\;1 \to 0.07$ as ${\omega _{ * 1}}\;:\;0.04 \to 4\,({\rm{rad}}/{\rm{s}})$ ;

3. (iii) In $\left[ {0,0.04{\rm{rad/s}}} \right]\rho ({\omega _{ * 1}})\;:\;1.035 \to 1$ (i.e. $\rho ({\omega _{ * 1}}) \approx 1$ ) as ${\omega _{ * 1}}\;:\;0 \to 0.04\,({\rm{rad}}/{\rm{s}})$ .

1. (3) Rejection ratios in the disturbance frequency bands for attitude ( $\psi $ )

From ${\varepsilon _\psi } = 0.0025$ and $r = \frac{1}{2}$ , the rejection ratio is expressed by

$$\rho ({\omega _{\psi 1}}) = \frac{1}{{\sqrt {1 + \frac{1}{4}\varepsilon _\psi ^{2r - 2}\omega _{\psi 1}^2} }} + \frac{1}{2}\varepsilon _\psi ^r = \frac{1}{{\sqrt {1 + 100\omega _{\psi 1}^2} }} + 0.025$$

Therefore, the rejection ratios in the different frequency bands can be described by:

1. (i) In $\left[ {4{\rm{rad/s}},\infty } \right)$ , $\rho ({\omega _{\psi 1}})\;:\;0.05 \to 0.025$ as ${\omega _{\psi 1}}\;:\;4 \to \infty \,({\rm{rad}}/{\rm{s}})$ ;

2. (ii) In $\left( {0.023,4{\rm{rad/s}}} \right)$ , $\rho ({\omega _{\psi 1}})\;:\;1 \to 0.05$ as ${\omega _{\psi 1}}\;:\;0.023 \to 4\,({\rm{rad}}/{\rm{s}})$ ;

3. (iii) In $\left[ {0,0.023{\rm{rad/s}}} \right]$ , $\rho ({\omega _{\psi 1}})\;:\;1.025 \to 1$ (i.e. $\rho ({\omega _{\psi 1}}) \approx 1$ ) as ${\omega _{\psi 1}}\;:\;0 \to 0.023\,({\rm{rad}}/{\rm{s}})$ .

8.2 Parametres of observers and controllers

According to the selection rules of observer parametres [Reference Wang and Lin24], we select the extended state observer parametres: ${\lambda _{1 * }} = 4$ , ${\lambda _{ * 2}} = 20$ , ${\alpha _ * } = 0.6$ , where, $ * = \left\{ {x,y,z,\theta ,\phi ,\psi } \right\}$ . According to the properties and tests of engine and digital servos, we select the control law parametres: ${k_{p1}} = 16$ , ${k_{p2}} = 8$ , ${k_{a1}} = 25$ , ${k_{a2}} = 8$ .

8.3 Analysis of UAV navigation and control performance

Figure 8(a) shows the comparison of the flight trajectories, including the measured from GPS, the reference trajectory, and the estimations by the corrector and the EKF-based method. Meanwhile, the trajectory comparisons in the three directions are shown in Fig. 8(b): due to the effect of adverse conditions, e.g. engine vibration and communication, the actual measurement disturbances in position from GPS were about 10m. The estimate errors by the corrector were less than 1m, while the estimate errors by the KF were about 3m. From the estimate errors and the above numerical calculation, we can find that the position disturbances were mainly within the frequency band $\left[ {3.98{\rm{rad/s}}, + \infty } \right)$ . Therefore, the disturbances in position measurements were rejected sufficiently by the correctors, and the correctors provided the relatively accurate and smoothed correction outputs. The attitude angle comparisons in the three directions are shown in Fig. 9. During the UAV flight, the actual measurement disturbances in attitude from the IMU were about ${3^ \circ } \sim {4^ \circ }$ . The corrector estimate errors for attitude angle were less than ${0.2^ \circ }$ . Then, comparing the above calculation, we can find that the attitude disturbances were mainly within the frequency band $\left[ {4{\rm{rad/s}}, + \infty } \right)$ . Therefore, the disturbances in the attitude measurements were rejected sufficiently by the correctors due to the very small rejection ratio in this frequency band. From the flight test, we can find that the correctors also reduced the sensing disturbances from the effect of UAV vibrations, and the jet UAV remained in the safe flight condition throughout the flight.

Figure 9. Attitude correction.