0 Introduction

Attitude control is critical to stability and trajectory control for hypersonic aircraft^[1-3], but it is challenged by many factors, including strong couplings, complex nonlinearities, uncertainties, and limited control surface deflections and thrusts^[4-8].

Recent studies concentrate on three aspects. First, accurate control model caught researchers' attentions. Refs.[9-12] designed online neural network adaptive control laws based on feedback linearization. Feedback linearization methods, however, heavily depend on model accuracy and can hardly guarantee good control performance with model uncertainties and external disturbances. Subsequently, linear robust methods were introduced. Ref.[13] proposed an intelligent controller based on the fuzzy dynamic characteristic modeling. Ref.[14] presented a robust control system featuring fixed feedback gains and applied muti-model eigen structure assignment with respect to the specific tasks of a hypersonic vehicle. Although these two methods could effectively achieve robustness, their linearization steps may cause considerable modeling errors and uncertainties. Thus, nonlinear robust control methods emerged. Sliding mode control has good adaptability and high robustness for system disturbances and parameter perturbations, and has been successfully applied to industrial control^[15-17]. Ref.[18] designed sliding mode controller to improve the robustness. Then, Ref.[19] presented a disturbance observer to estimate the unknown disturbance of a hypersonic vehicle.

Although coupling problems are frequently mentioned in the above studies, none of them has proposed theories to directly solve them. Strong couplings can cause complicated interactions between variables which consequently lead to more complex behaviors of aircraft with more variations and uncertainties. Some scholars have noticed this and started exploratory studies on coordinated controllers^[20-24]. Ref.[20] presented a nonlinear longitudinal model for a hypersonic vehicle, which could describe the inertial coupling effects between the pitch dynamics. Ref.[21] proposed an assessment of the interactions among the airframe, engine, and structural dynamics and designed a highly integrated airframe engine control system to deal with the strong couplings. These studies, however, focused on analysis of strong couplings without any expressions. Therefore, inspired by previous studies, we present a nonlinear robust coordinated control scheme for hypersonic aircraft, which highlights certain privileges as follows.

(1) Inspired by the ideas in Refs.[25-26], we categorize the coupling relationships of the variables in three types:Attitude coupling, inertia coupling and aerodynamic coupling.

(2) Based on the three coupling types, a series of novel coordinated factors are obtained.

(3) Uncertain parameters are integrated into a vector for the convenience of online estimation, so that we can construct an adaptive estimator and enhance the robustness of the controller. We also introduce a nonlinear observer to estimate the disturbance in the control-coefficient matrix.

(4) We introduce designed coordinated factors to deduce the coordinated moment of force.

1 Hypersonic Aircraft Model

Ignoring the flexibility effects of the structure, the wind, the earth rotation, and the earth curvature, the attitude dynamic model of a hypersonic aircraft^[3] containing aerodynamic parameters' CFD (computational fluid dynamics) data can be described by the following nonlinear equations^{[3, 27-28]}

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }} = {\mathit{\boldsymbol{f}}_s} + {\mathit{\boldsymbol{g}}_s}\omega \\ \mathit{\boldsymbol{\dot \omega }} = {\mathit{\boldsymbol{f}}_f} + {\mathit{\boldsymbol{g}}_f}{\mathit{\boldsymbol{M}}_c} \end{array} \right. $

(1)

where Ω=[α, β, μ]^T, f_s=[f_α, f_β, f_μ]^T, ω=[p, q, r]^T, f_f=[f_p, f_q, f_r]^T, and

$ \begin{array}{l} {f_\alpha } = \frac{1}{{MV\cos \beta }}\left( { - \bar qS{C_{L,\alpha }} + Mg\cos \gamma \cos \mu } \right)\\ {f_\beta } = \frac{1}{{MV}}\left( { - \bar qS{C_{Y,\beta }}\beta + Mg\cos \gamma \sin \mu } \right)\\ \begin{array}{*{20}{c}} {{f_\mu } = \frac{1}{{MV}}\bar qS{C_{Y,\beta }}\beta \tan \gamma \cos \mu + \frac{1}{{MV}}\bar qS{C_{L,\alpha }} \cdot }\\ {\left( {\tan \gamma \sin \mu + \tan \beta } \right) - \frac{g}{V}\cos \gamma \cos \mu \tan \beta } \end{array} \end{array} $

(2)

$ \mathit{\boldsymbol{g}_s} = \left[ {\begin{array}{*{20}{c}} { - \cos \alpha \tan \beta }&1&{ - \sin \alpha \tan \beta }\\ {\sin \alpha }&0&{ - \cos \alpha }\\ {\cos \alpha \sec \beta }&0&{\sin \alpha \sec \beta } \end{array}} \right] $

(3)

$ \begin{array}{l} {f_p} = \frac{1}{{{I_{xx}}}}\left( {{l_{{\rm{aero}}}} + \left( {{I_{yy}} - {I_{zz}}} \right)qr - {{\dot I}_{xx}}p} \right)\\ {f_q} = \frac{1}{{{I_{yy}}}}\left( {{m_{{\rm{aero}}}} + \left( {{I_{zz}} - {I_{xx}}} \right)pr - {{\dot I}_{yy}}q} \right)\\ {f_r} = \frac{1}{{{I_{zz}}}}\left( {{n_{{\rm{aero}}}} + \left( {{I_{xx}} - {I_{yy}}} \right)pq - {{\dot I}_{zz}}r} \right) \end{array} $

(4)

$ {\mathit{\boldsymbol{g}}_f} = \left[ {\begin{array}{*{20}{c}} {g_l^p}&0&0\\ 0&{g_m^q}&0\\ 0&0&{g_n^r} \end{array}} \right] $

(5)

$ \begin{array}{l} {l_{{\rm{aero}}}} = \bar qSb\left[ {{C_{l,\beta }}\beta + {C_{l,p}}pb/\left( {2V} \right) + {C_{l,r}}rb/\left( {2V} \right)} \right]\\ {m_{{\rm{aero}}}} = \bar qSc\left[ {{C_{m,\alpha }} + {C_{m,q}}qc/\left( {2V} \right)} \right) + \\ \;\;\;\;\;\;\;\;\;{X_{{\rm{cg}}}}\bar qS\left( {{C_{D,\alpha }}\sin \alpha + {C_{L,\alpha }}\cos \alpha } \right)\\ {n_{{\rm{aero}}}} = \bar qSb\left[ {{C_{n,\beta }}\beta + {C_{n,p}}pb/\left( {2V} \right) + {C_{n,r}}rb/\left( {2V} \right)} \right] + \\ \;\;\;\;\;\;\;\;\;{X_{{\rm{cg}}}}\bar qS{C_{Y,\beta }}\beta \end{array} $

(6)

where α, β, μ are the angle of attack, the sideslip angle, and the bank angle, respectively; p, q, r the roll rate, the pitch rate, and the yaw rate, respectively; M_c=g_fδu, where g_fδ∈ R^3×3 is the fast loop allocation matrix, u=[δ_e, δ_a, δ_r]^T, where δ_e, δ_a, δ_r are the deflection angles of elevator, aileron, and rudder, respectively; M is the vehicle mass; V the velocity; q the dynamic pressure; S, c, b are the reference area, the reference length, and reference width, respectively; I_xx, I_yy, I_zz the roll, the yaw, and pitch moments of inertia, respectively; and X_cg is the longitudinal distance from the momentum reference to the vehicle. The basic parameters and aerodynamic coefficients of the vehicle can be obtained in Refs.[3, 27-28].

Remark 1 Due to the complicated nonlinear dynamics of hypersonic aircraft, it is difficult to study directly the rigid-flexible coupling problem without the knowledge of the rigid couplings. Ref.[29] investigated a double-loop coordinated decoupling control to deal with the couplings, but it lacks explicit investigation on the effect of the flexible dynamics on the attitude system. By using statistical sampling method, a series of coupling degrees were provided based on rigid-body dynamics^[30-31]. If considering the flexible coupling dynamics, the flexibility variables should be involved in the method^[30-31], and the coupling degrees must be calculated online. But the statistical sampling method^[30-31] does not support online computing. In Refs.[32-34], the flexible coupling effects were modeled as a kind of unknown disturbance, then a coupling-observer-based compensator was proposed to restrict the flexible effects on pitch rate. If considering all flexible dynamics, the control method may cause high-frequency oscillations. Based on a rigid-body dynamic model, Refs.[35-36] introduced a novel coupling analysis index to describe the coupling relationships among the variables. But if the flexible coupling dynamics was considered, the method^[35-36] would not be directly applied to design the controller, because the couplings of the attitude channels have positive and negative polarities. To analyze the strong coupling characteristics of a hypersonic aircrafts Ref.[25] defined the nonlinear degree based on rigid-body dynamics, and then introduced it to characterize the rigid-flexible couplings. Ref.[25] provides a new direction for further study. Therefore, dealing with the couplings of the flight state variables based on the rigid-body dynamics is the first step of our research.

2 Coupling Analysis

Obviously, strong nonlinear couplings exist in the above mentioned attitude systems (1)-(6), and they can be described as the following coupling types.

(1) Attitude coupling

To consider the effects of the angular rates on the attitude angles while neglecting f_s in the first sub-equation of Eq.(1), we describe the attitude coupling as

$ \begin{array}{l} \dot \alpha = - \tan \beta \left( {p\cos \alpha + r\sin \alpha } \right)\\ \dot \beta = p\sin \alpha - r\cos \alpha \\ \dot \mu = \sec \beta \left( {p\cos \alpha + r\sin \alpha } \right) \end{array} $

(7)

(2) Inertia coupling

If the effects of l_aero, m_aero, n_aero on the angular rates are ignored, the inertia coupling which among the channels of the roll, the pitch, and the yaw in Eq.(4) can be described as

$ \begin{array}{l} {f_p} = \left( {{I_{yy}} - {I_{zz}}} \right)qr - {{\dot I}_{xx}}p\\ {f_q} = \left( {{I_{zz}} - {I_{xx}}} \right)pr - {{\dot I}_{yy}}q\\ {f_r} = \left( {{I_{xx}} - {I_{yy}}} \right)pq - {{\dot I}_{xz}}p \end{array} $

(8)

(3) Surface deflection coupling

According to Refs.[27-28], it can be known that there always exists surface deflection coupling due to the complicated aerodynamics of hypersonic aircraft, then combing the expression of g_fδ in Refs.[23, 28], the surface deflection coupling can be described as

$ \begin{array}{l} {g_l} = \bar qSb\left( {{C_{l,{\delta _a}}} + {C_{l,{\delta _r}}}} \right)\\ {g_m} = \bar qSc\left( {{C_{m,{\delta _a}}} + {C_{m,{\delta _r}}}} \right)\\ {g_n} = \bar qSb\left( {{C_{n,{\delta _a}}} + {C_{n,{\delta _r}}}} \right) \end{array} $

(9)

Obviously, the above equations show that there exist complex coupling relationships within hypersonic aircraft. In hypersonic speed, the above couplings are more complex, resulting in aileron reversal operation phenomenon and system uncertainties, or even the system instability. Some scholars have put forward decoupling methods^[29]. But they are not an effective way for hypersonic aircraft, because the decoupling methods may change the intrinsic nonlinear characteristics of the vehicles. Therefore, Ref.[30] presented a new scheme to describe the coupling relationships in mathematics, and the linear coupling degree could be obtained by using statistical sampling method. However, the coupling matrices provided in Ref.[30] are linear, which is not accurate enough to describe the complex couplings of the hypersonic flight vehicles. Therefore, we propose a novel method to describe the nonlinear couplings of the aircraft and design a coordinated attitude controller to coordinate the attitude dynamics.

3 Design of Coordinated Factors 3.1 Coordinated factors for attitude coupling

For flight dynamics, the sideslip angle may cause unstable flight. Therefore, $\dot \beta $ is set as zero (Eq. (7)), that is, p sina-rcosα=0. Then r=p tan α can be obtained. Thus, the coordinated factor can be designed as

$ {\lambda _{{r_1}}} = {k_1}p\tan \alpha $

(10)

where k₁>0 is a designed parameter.

In contrast to the unavailable coupling between α and β, the coupling between α and μ can be regarded as an available coupling when the aircraft is flying at a very small angle of attack. Then we have

$ \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} p\cos \alpha + r\sin \alpha = p\\ p\cos \alpha + r\sin \alpha \le p + \rho r\\ p - \sigma r < p\cos \alpha + r\sin \alpha \end{array}&\begin{array}{l} \alpha = 0\\ \alpha > 0\\ \alpha < 0 \end{array} \end{array}} \right. $

(11)

where ρ is a positive constant and sinα < ρ, and σ a positive constant, σ>|sinα|. Then the strong attitude coupling relationships in Eq.(7) can be described as p-σr < p ≤ pcosα+rsinα ≤ p+ρr. In order to increase the damping torque, β is introduced as feedback to the rudder channel. Thus, the coordinated factor is selected as

$ {\lambda _{{r_2}}} = {k_2}\beta + {k_3}\frac{{p\left( {1 - \cos \alpha } \right)}}{{\sin \alpha + {\mathit{\Delta} _\alpha }}} $

(12)

where Δ_α is a designed parameter to avoid the singularity of Eq.(12), |sinα| ≠Δ_α, and k₂, k₃>0 are designed parameters.

Combining Eq.(10) and Eq.(12), the attitude couplingcoordinated factor can be described as

$ \begin{array}{*{20}{c}} {{\lambda _r} = {\lambda _{{r_1}}} + {\lambda _{{r_2}}} = }\\ {{k_1}p\tan \alpha + {k_2}\beta + {k_3}\frac{{p\left( {1 - \cos \alpha } \right)}}{{\sin + {\mathit{\Delta} _\alpha }}}} \end{array} $

(13)

3.2 Coordinated factors for inertia coupling

The sideslip angle tends to diverge because of the inertial coupling torques, so the terms (I_yy-I_zz)qr, (I_zz-I_xx)pr, and (I_xx-I_yy)pq in Eq.(8) cannot be ignored. Taking the yaw inertia as an example, when the inertia is in critical state, f_r+f'_r+f'_β=0, where f_r is the yaw inertia coupling torque; f'_r the yaw damping torque, and f'_β the yaw steady torque. With the accumulation of inertia coupling, the inertia coupling torque will be greater than the damping torque and the steady torque, which will lead to the sideslip angle unstable. To reduce the negative impacts of the inertia coupling, the damping torque and steady torque should be increased to restrain the trend. The basic idea of the coordinated control is to introduce β and r into the aileron loop to increase the damping torque, β and q into the rudder loop to increase the steady torque, and α and p into the elevator loop to increase the damping torque. Thus, the inertia coupling coordinated factors are designed as

$ \left\{ \begin{array}{l} {\lambda _e} = {k_4}\alpha + {k_5}p\\ {\lambda _a} = {k_6}\beta + {k_7}r\\ {\lambda _{{r_3}}} = {k_8}\beta + {k_9}q \end{array} \right. $

(14)

where k₄, k₅, k₆, k₇, k₈>0 are designed parameters.

3.3 Coordinated factors for surface deflection

With consideration of the flight dynamics of hypersonic aircraft, Ref.[30] provided a method to describe the coupling degree in mathematics.But by analysis, the coupling degree should reasonably describe the couplings between the rudder deflection and aileron deflection^[31].

Thus, the surface deflection coupling degree is chosen as^[31]

$ E = \frac{{C_x^{{\delta _r}}}}{{C_x^{{\delta _a}}}} \times 100\% $

(15)

where E represents the surface deflection degree between the aileron deflection and rudder deflection. It indicates that the surface deflection degree is determined by the aerodynamic shape and flight state. And

$ C_x^{{\delta _r}} = \frac{{{S_{{\rm{cwr}}}}{S_{{\rm{cw}}}}}}{{{S^2}}}\frac{{{y_{{\rm{cwr}}}}}}{L}\xi \cos \chi $

(16)

$ C_x^{{\delta _a}} = \frac{1}{2}\left\{ {n\frac{{\left( {{\eta _k} + 1} \right){{\left( {1 - \bar D} \right)}^2}}}{{{\eta _k} + 1 - 2\bar D}}\left[ {\bar D + \left( {1 - \bar D} \right)f} \right]} \right\} $

(17)

where S_cwr, S_cw are the rudder area and the vertical area, respectively; S, L are the reference area and reference length; y_cwr is the distance from the rudder surface center to the vertical axis; χ the sweep back angle; ξ the correction factor; n the relative efficiency of the aileron; η_k, D are the exposed wing ratio and diameter ratio, respectively; and f is the ratio between the distance from exposed wing root profile to center pressure and wing length. In Eq.(17), S_cwr ≤ S, S_cw ≤ S, y_cwr ≤ L, 0 < ξ ≤ 1, 0 < cos_χ ≤ 1, and C_x^δ_a ≥ 1. Thus, the surface coordinated defection strategy is chosen as

$ \begin{array}{*{20}{c}} {{\delta _a} = E{\delta _r}}&{0 < E \le 1} \end{array} $

(18)

Based on Eqs.(13), (14), the hypersonic vehicle attitude coordinated factors are given as

$ \left\{ \begin{array}{l} {\lambda _e} = {k_4}\alpha + {k_5}q\\ {\lambda _r} = {\lambda _{r1}} + {\lambda _{r2}} + {\lambda _{r3}}\\ {\lambda _a} = {k_6}\beta + {k_7}p \end{array} \right. $

(19)

where k_i>0, i=1, …, 7 is designed parameter.

4 Coordinated Attitude Controller Design

By introducing the uncertain parameters, a hypersonic aircraft attitude dynamics can be rewritten as

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }} = {\mathit{\boldsymbol{f}}_1} + {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\Omega } + {\mathit{\boldsymbol{g}}_s}\mathit{\boldsymbol{\omega }}}\\ {\mathit{\boldsymbol{\dot \omega }} = {\mathit{\boldsymbol{ \boldsymbol{\varXi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\omega } + {\mathit{\boldsymbol{g}}_f}{\mathit{\boldsymbol{M}}_c} + \mathit{\boldsymbol{d}}} \end{array} $

(20)

where d is the external perturbation, and

$ {\mathit{\boldsymbol{\theta }}_\Omega } = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\theta }}_1^{\rm{T}}}&{\mathit{\boldsymbol{\theta }}_2^{\rm{T}}}&{\mathit{\boldsymbol{\theta }}_3^{\rm{T}}} \end{array}} \right]_{4 \times 1}^{\rm{T}},{\mathit{\boldsymbol{\theta }}_\omega } = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\theta }}_4^{\rm{T}}}&{\mathit{\boldsymbol{\theta }}_5^{\rm{T}}}&{\mathit{\boldsymbol{\theta }}_6^{\rm{T}}} \end{array}} \right]_{17 \times 1}^{\rm{T}} $

(21)

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{\theta }}_1} = {C_{L,\alpha }},{\mathit{\boldsymbol{\theta }}_2} = {C_{Y,\beta }},{\mathit{\boldsymbol{\theta }}_3} = {{\left[ {\begin{array}{*{20}{c}} {{C_{Y,\beta }},}&{{C_{L,\alpha }}} \end{array}} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{\theta }}_4} = {{\left[ {\begin{array}{*{20}{c}} {\frac{{{I_{xx}} - {I_{yy}}}}{{{I_{zz}}}}}&{\frac{{{{\dot I}_{zz}}}}{{{I_{zz}}}}}&{\frac{{{C_{n,\beta }}}}{{{I_{zz}}}}}&{\frac{{{C_{n,p}}}}{{{I_{zz}}}}}&{\frac{{{C_{n,r}}}}{{{I_{zz}}}}}&{\frac{{{C_{\gamma ,\beta }}}}{{{I_{zz}}}}} \end{array}} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{\theta }}_5} = {{\left[ {\begin{array}{*{20}{c}} {\frac{{{I_{zz}} - {I_{xx}}}}{{{I_{yy}}}}}&{\frac{{{{\dot I}_{yy}}}}{{{I_{yy}}}}}&{\frac{{{C_{m,\alpha }}}}{{{I_{yy}}}}}&{\frac{{{C_{m,q}}}}{{{I_{yy}}}}}&{\frac{{{C_{D,\alpha }}}}{{{I_{yy}}}}}&{\frac{{{C_{L,\alpha }}}}{{{I_{yy}}}}} \end{array}} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{\theta }}_6} = {{\left[ {\begin{array}{*{20}{c}} {\frac{{{I_{yy}} - {I_{zz}}}}{{{I_{xx}}}}}&{\frac{{{{\dot I}_{xx}}}}{{{I_{xx}}}}}&{\frac{{{C_{l,\beta }}}}{{{I_{xx}}}}}&{\frac{{{C_{l,p}}}}{{{I_{xx}}}}}&{\frac{{{C_{l,r}}}}{{{I_{xx}}}}} \end{array}} \right]}^{\rm{T}}}} \end{array} $

(22)

$ \begin{array}{*{20}{c}} {{\mathit{\Psi }_1}\left( x \right) = - \frac{{\bar qS}}{{MV\cos \beta }},{\mathit{\Psi }_2} = - \frac{{\bar qS\beta \cos \beta }}{{MV}}}\\ {{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_3} = {{\left[ {\begin{array}{*{20}{c}} {\frac{{\bar qS\beta \tan \gamma \cos \mu \cos \beta }}{{MV}}}&{\frac{{\bar qS\left( {\tan \gamma \sin \mu + \tan \beta } \right)}}{{MV}}} \end{array}} \right]}^{\rm{T}}}} \end{array} $

(23)

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}_1}\left( x \right) = {{\left[ {\begin{array}{*{20}{c}} {qr}&{ - p}&{\bar qSb\beta }&{\frac{{\bar qS{b^2}p}}{{2V}}}&{\frac{{\bar qS{b^2}r}}{{2V}}} \end{array}} \right]}^{\rm{T}}}}\\ {{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}_2}\left( x \right) = \left[ {\begin{array}{*{20}{c}} {pr}&{ - q}&{\bar qSc}&{\frac{{qS{c^2}q}}{{2V}}} \end{array}} \right.}\\ {{{\left. {\begin{array}{*{20}{c}} {{X_{{\rm{cg}}}}\bar qS\sin \alpha }&{{X_{{\rm{cg}}}}\bar qS\cos \alpha } \end{array}} \right]}^{\rm{T}}}} \end{array} $

(24)

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}_3}\left( x \right) = {{\left[ {\begin{array}{*{20}{c}} {pq}&{ - r}&{\bar qSb\beta }&{\frac{{\bar qS{b^2}p}}{{2V}}}&{\frac{{\bar qS{b^2}r}}{{2V}}}&{{X_{{\rm{cg}}}}\bar qS\beta } \end{array}} \right]}^{\rm{T}}}}\\ {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }} = {{\left[ {\begin{array}{*{20}{c}} {{\mathit{\Psi }_1}}&0&0\\ 0&{{\mathit{\Psi }_2}}&0\\ {{{\bf{0}}_{2 \times 1}}}&{{{\bf{0}}_{2 \times 1}}}&{{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}_3}} \end{array}} \right]}_{4 \times 3}}}\\ {\mathit{\boldsymbol{ \boldsymbol{\varXi} }} = {{\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}_1}}&{{{\bf{0}}_{5 \times 1}}}&{{{\bf{0}}_{5 \times 1}}}\\ {{{\bf{0}}_{6 \times 1}}}&{{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}_2}}&{{{\bf{0}}_{6 \times 1}}}\\ {{{\bf{0}}_{6 \times 1}}}&{{{\bf{0}}_{6 \times 1}}}&{{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}_3}} \end{array}} \right]}_{17 \times 3}}} \end{array} $

(25)

$ {\mathit{\boldsymbol{f}}_1} = \left[ {\begin{array}{*{20}{c}} {\frac{{g\cos \gamma \cos \mu }}{{V\cos \beta }}}\\ {\frac{{g\cos \gamma \sin \mu }}{V}}\\ { - \frac{{g\cos \gamma \cos \mu \tan \beta }}{V}} \end{array}} \right] $

(26)

Then, the control objective is that the attitude angle Ω can track the desired signal Ω_c, then the tracking error vector can be defined as e₁=Ω - Ω_c. In addition, the tracking error of the angular rates is also defined as e₂=ω-ω_c, where the desired angular rates ω_c can be obtained from the controller of the attitude angles. According to the above, a coordinated scheme based on the coordinated factors is given in Fig. 1.

4.1 Slow loop controller design

The sliding mode surface is given as

$ \mathit{\boldsymbol{\sigma }} = {\mathit{\boldsymbol{e}}_1} + \mathit{\boldsymbol{K}}\int_0^t {{\mathit{\boldsymbol{e}}_1}{\rm{d}}t} $

(27)

where K=diag{K₁, K₂, K₃}, K_i>0, i=1, 2, 3.

The derivative of the sliding mode function can be obtained as follows

$ \begin{array}{*{20}{c}} {\dot \sigma = {{\mathit{\boldsymbol{\dot e}}}_1} + \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{e}}_1} = }\\ {{\mathit{\boldsymbol{f}}_1} + {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\Omega } + {\mathit{\boldsymbol{g}}_s}\mathit{\boldsymbol{\omega }} - {{\mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }}}_c} + \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{e}}_1} = }\\ {{\mathit{\boldsymbol{f}}_1} + {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\Omega } + {\mathit{\boldsymbol{g}}_s}\left( {{\mathit{\boldsymbol{e}}_2} + {\mathit{\boldsymbol{\omega }}_c}} \right) - {{\mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }}}_c} + \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{e}}_1}} \end{array} $

(28)

And, we can obtain

$ {\mathit{\boldsymbol{\omega }}_c} = - \mathit{\boldsymbol{g}}_s^{ - 1}\left( {{\mathit{\boldsymbol{\kappa }}_1}\mathit{\boldsymbol{\sigma }} + {\mathit{\boldsymbol{f}}_1} + {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}{{\mathit{\boldsymbol{\hat \theta }}}_\Omega } - {{\mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }}}_c} + \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{e}}_1}} \right) $

(29)

Similar to Ref.[32], the adaptation law of ${{\mathit{\boldsymbol{\hat \theta }}}_\mathit{\Omega }}$ is designed as

$ {{\mathit{\boldsymbol{\dot {\hat \theta} }}}_\mathit{\Omega }} = {\rm{Pro}}{{\rm{j}}_{{{\mathit{\boldsymbol{\hat \theta }}}_\Omega }}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} \sigma }} - {\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }}} \right)} \right) $

(30)

where κ₁>0, Γ₁∈ R^4×4, λ_Ω=diag{λ₁, λ₂, λ₃, λ₄}>0.

Consider a Lyapunov function candidate as

$ {V_1} = \frac{1}{2}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} + \frac{1}{2}\mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1^{ - 1}\mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}} $

(31)

The time derivative of Eq.(31) is

$ \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\dot V}}}_1} = {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\left( {{\mathit{\boldsymbol{f}}_1} + {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\Omega } + {\mathit{\boldsymbol{g}}_s}\left( {{\mathit{\boldsymbol{e}}_2} + {\mathit{\boldsymbol{\omega }}_c}} \right) - {{\mathit{\boldsymbol{ \boldsymbol{\dot \varOmega} }}}_c} + \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{e}}_1}} \right) + }\\ {\mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1^{ - 1}{\rm{Pro}}{{\rm{j}}_{{{\mathit{\boldsymbol{\hat \theta }}}_\mathit{\Omega }}}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} \sigma }} - {\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }}} \right)} \right)} \end{array} $

(32)

From Eqs.(29), (30), (32), we can obtain

$ \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\dot V}}}_1} = - {\kappa _1}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} + {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{g}}_s}{\mathit{\boldsymbol{e}}_2} + {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\Omega } - }\\ {{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}{{\mathit{\boldsymbol{\hat \theta }}}_\Omega } + \mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1^{ - 1} \cdot {\rm{Pro}}{{\rm{j}}_{{{\mathit{\boldsymbol{\dot {\hat \theta} }}}_\mathit{\Omega }}}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} \sigma }} - {\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }}} \right)} \right) = }\\ { - {\kappa _1}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} + {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{g}}_s}{\mathit{\boldsymbol{e}}_2} - {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}^{\rm{T}}}{{\mathit{\boldsymbol{\tilde \theta }}}_\Omega } + }\\ {\mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1^{ - 1}{\rm{Pro}}{{\rm{j}}_{{{\mathit{\boldsymbol{\dot {\hat \theta} }}}_\mathit{\Omega }}}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_1}\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} \sigma }} - {\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }}} \right)} \right) \le }\\ { - {\kappa _1}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} + {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{g}}_s}{\mathit{\boldsymbol{e}}_2} - \mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}{\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }}} \end{array} $

(33)

4.2 Fast loop controller design

Differentiating e₂ yields

$ {{\mathit{\boldsymbol{\dot e}}}_2} = \mathit{\boldsymbol{\dot \omega }} - {{\mathit{\boldsymbol{\dot \omega }}}_c} = {\mathit{\boldsymbol{ \boldsymbol{\varXi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\omega } + {\mathit{\boldsymbol{g}}_f}{\mathit{\boldsymbol{M}}_c} + \mathit{\boldsymbol{d}} - {{\mathit{\boldsymbol{\dot \omega }}}_c} $

(34)

Then, the control law M_c is designed as

$ {\mathit{\boldsymbol{M}}_c} = - \mathit{\boldsymbol{g}}_f^{ - 1}\left( {{\mathit{\boldsymbol{\kappa }}_2}{\mathit{\boldsymbol{e}}_2} + {\mathit{\boldsymbol{ \boldsymbol{\varXi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\omega } + \mathit{\boldsymbol{\hat d}} + \mathit{\boldsymbol{g}}_s^{\rm{T}}\mathit{\boldsymbol{\sigma }} - {{\mathit{\boldsymbol{\dot \omega }}}_c}} \right) $

(35)

where ${\mathit{\boldsymbol{\hat d}}}$ is the estimate output of the disturbance observer. The disturbance estimation error ${\mathit{\boldsymbol{\tilde d}}}$ is defined as $\mathit{\boldsymbol{\tilde d}} = \mathit{\boldsymbol{d}} - \mathit{\boldsymbol{\hat d}}$. And the adaptation law of ${{\mathit{\boldsymbol{\hat \theta }}}_\mathit{\Omega }}$ is chosen as

$ {{\mathit{\boldsymbol{\dot {\hat \theta} }}}_\omega } = {\rm{Pro}}{{\rm{j}}_{{{\mathit{\boldsymbol{\hat \theta }}}_\omega }}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}\left( {\mathit{\boldsymbol{ \boldsymbol{\varXi} }}{e_2} - {\mathit{\boldsymbol{\lambda }}_\omega }{{\mathit{\boldsymbol{\tilde \theta }}}_\omega }} \right)} \right) $

(36)

where κ₂>0, Γ₂∈ R^17×17>0, and λ_ω∈ R^17×17>0.

According to Refs.[32-36], the disturbance observer is designed as

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{\hat d}} = \mathit{\boldsymbol{z}} + \mathit{\boldsymbol{Q}}\left( \mathit{\boldsymbol{e}} \right)\\ \mathit{\boldsymbol{\dot z}} = - \mathit{\boldsymbol{L}}\left( \mathit{\boldsymbol{e}} \right)\mathit{\boldsymbol{z}} - \mathit{\boldsymbol{L}}\left( \mathit{\boldsymbol{e}} \right)\left( {{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}^{\rm{T}}}{{\mathit{\boldsymbol{\hat \theta }}}_\omega } + {\mathit{\boldsymbol{g}}_f}\mathit{\boldsymbol{M}} + \mathit{\boldsymbol{Q}}\left( \mathit{\boldsymbol{e}} \right)} \right) \end{array} \right. $

(37)

where $\mathit{\boldsymbol{z}} = \mathit{\boldsymbol{\hat d}} - \mathit{\boldsymbol{Q}}\left( \mathit{\boldsymbol{e}} \right), \mathit{\boldsymbol{Q}}\left( \mathit{\boldsymbol{e}} \right) = \left[ {{q_1}\left( e \right), {q_2}\left( e \right), \cdots , } \right.$${\left. {{q_n}\left( e \right)} \right]^{\rm{T}}} \in {\mathit{\boldsymbol{R}}^n}, \mathit{\boldsymbol{L}}\left( \mathit{\boldsymbol{e}} \right) = \frac{{\partial \mathit{\boldsymbol{Q}}\left( \mathit{\boldsymbol{e}} \right)}}{{\partial \mathit{\boldsymbol{e}}}}$.

Remark 2 Different from Ref.[33], the designed observer exists the uncertain parameters estimation vector ${{\mathit{\boldsymbol{\hat \theta }}}_\omega }$, so the convergence of the observer should be reconsidered. At the origin, ${{\mathit{\boldsymbol{\tilde \theta }}}_\omega }$ is asymptotic stable if θ_ω is convergent. In Eq.(37), ${{\mathit{\boldsymbol{\hat \theta }}}_\omega }$ can be written as ${\mathit{\boldsymbol{\theta }}_\omega } + {{\mathit{\boldsymbol{\tilde \theta }}}_\omega }$, then ${{\mathit{\boldsymbol{\tilde \theta }}}_\omega }{\rm{ = }}0$, and Eq.(37) is simplified as the form of Ref.[33]. So the convergence of Eq.(37) depends on the convergence of the adaptive estimator, and the asymptotic stability of the adaptive estimator will be proved in the next.

Theorem 1 For the attitude dynamics in Eq.(20), under the control laws (29) and (35), with the adaptive laws (30) and (36), the designed scheme guarantees that all signals of the closed-loop system are uniformly and fully bounded and the tracking errors are asymptotically stable.

Proof Consider a Lyapunov function candidate as follows

$ \mathit{\boldsymbol{V}} = {\mathit{\boldsymbol{V}}_1} + \frac{1}{2}\left( {\mathit{\boldsymbol{e}}_2^{\rm{T}}{\mathit{\boldsymbol{e}}_2} + \mathit{\boldsymbol{\tilde \theta }}_\omega ^{\rm{T}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2^{ - 1}{{\mathit{\boldsymbol{\tilde \theta }}}_\omega } + {{\mathit{\boldsymbol{\tilde d}}}^{\rm{T}}}\mathit{\boldsymbol{\tilde d}}} \right) $

(38)

Then, the time derivative of V is

$ \mathit{\boldsymbol{\dot V}} = {{\mathit{\boldsymbol{\dot V}}}_1} + \mathit{\boldsymbol{e}}_2^{\rm{T}}{{\mathit{\boldsymbol{\dot e}}}_2} + \mathit{\boldsymbol{\tilde \theta }}_\omega ^{\rm{T}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2^{ - 1}{{\mathit{\boldsymbol{\dot {\hat \theta} }}}_\omega } + {{\mathit{\boldsymbol{\tilde d}}}^{\rm{T}}}\mathit{\boldsymbol{\dot {\hat d}}} $

(39)

Substituting Eqs. (33)-(35), (36) into Eq.(39) and considering that the disturbance's change frequency is very small, that is $\dot d \approx 0$, we have

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{\dot V}} = - {\mathit{\boldsymbol{\kappa }}_1}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} + {\mathit{\boldsymbol{\sigma }}^{\rm{T}}}{\mathit{\boldsymbol{g}}_s}{\mathit{\boldsymbol{e}}_2} - \mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}{\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }} + }\\ {\mathit{\boldsymbol{e}}_2^{\rm{T}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varXi} }}^{\rm{T}}}{\mathit{\boldsymbol{\theta }}_\omega } + {\mathit{\boldsymbol{g}}_f}{\mathit{\boldsymbol{M}}_c} + } \right.}\\ {\left. {\mathit{\boldsymbol{d}} - {{\mathit{\boldsymbol{\dot \omega }}}_c}} \right) + \mathit{\boldsymbol{\tilde \theta }}_\omega ^{\rm{T}}\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2^{ - 1}{\rm{Pro}}{{\rm{j}}_{{{\mathit{\boldsymbol{\hat \theta }}}_\omega }}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_2}\left( {\mathit{\boldsymbol{ \boldsymbol{\varXi} }}{e_2} - {\mathit{\boldsymbol{\lambda }}_\omega }{{\mathit{\boldsymbol{\tilde \theta }}}_\omega }} \right)} \right) + {{\mathit{\boldsymbol{\tilde d}}}^{\rm{T}}}\mathit{\boldsymbol{\dot d}} \le }\\ { - {\mathit{\boldsymbol{\kappa }}_1}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} - \mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}{\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }} - {\mathit{\boldsymbol{\kappa }}_2}\mathit{\boldsymbol{e}}_2^{\rm{T}}{\mathit{\boldsymbol{e}}_2} - \mathit{\boldsymbol{\tilde \theta }}_\omega ^{\rm{T}}{\mathit{\boldsymbol{\lambda }}_\omega }{{\mathit{\boldsymbol{\tilde \theta }}}_\omega } + {{\mathit{\boldsymbol{\tilde d}}}^{\rm{T}}}\mathit{\boldsymbol{\dot {\hat d}}} \le }\\ { - {\mathit{\boldsymbol{\kappa }}_1}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} - \mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}{\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }} - {\mathit{\boldsymbol{\kappa }}_2}\mathit{\boldsymbol{e}}_2^{\rm{T}}{\mathit{\boldsymbol{e}}_2} - \mathit{\boldsymbol{\tilde \theta }}_\omega ^{\rm{T}}{\mathit{\boldsymbol{\lambda }}_\omega }\mathit{\boldsymbol{\tilde \theta }} - {{\mathit{\boldsymbol{\tilde d}}}^{\rm{T}}}\mathit{\boldsymbol{L\tilde d}}} \end{array} $

(40)

Denote L(e)=c>1 and Q(e)=ce, it can be obtained that

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{\dot V}} \le - {\mathit{\boldsymbol{\kappa }}_1}{\mathit{\boldsymbol{\sigma }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }} - \mathit{\boldsymbol{\tilde \theta }}_\mathit{\Omega }^{\rm{T}}{\mathit{\boldsymbol{\lambda }}_\mathit{\Omega }}{{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }} - {\mathit{\boldsymbol{\kappa }}_2}\mathit{\boldsymbol{e}}_2^{\rm{T}}{\mathit{\boldsymbol{e}}_2} - \mathit{\boldsymbol{\tilde \theta }}_\omega ^{\rm{T}}{\mathit{\boldsymbol{\lambda }}_\omega }{{\mathit{\boldsymbol{\tilde \theta }}}_\omega } - }\\ {\left( {c - 1} \right){{\left\| {\mathit{\boldsymbol{\tilde d}}} \right\|}^2} \le 0} \end{array} $

(41)

Eqs.(39), (41) confirm that the signals σ, ${{\mathit{\boldsymbol{\tilde \theta }}}_\mathit{\Omega }}, {{\mathit{\boldsymbol{\tilde \theta }}}_\omega }, \mathit{\boldsymbol{\tilde d}}$, e₂ are bounded since ${\mathit{\boldsymbol{\dot V}}}$ is negative semi-definite. From Eq.(27), we know that e₁ is bounded. Then Eq.(41) can be written as

$ {V_i}\left( t \right) = - {\zeta _i}e_i^2\left( t \right)\;\;\;\;\;i = 1,2 $

(42)

where ζ_i>0, and e_i(t) is the tracking error. By integrating (42) from 0 to ∞, we have

$ \int_0^\infty {{\zeta _i}e_i^2\left( t \right)} \le - \int_0^\infty {{{\dot V}_i}\left( t \right)} = {V_i}\left( 0 \right) - {V_i}\left( \infty \right) $

(43)

This implies that e_i(t)∈L₂, since V_i(0) and V_i(∞) are bounded. Since e_i(t)∈L_∞, ${{\dot e}_i}\left( t \right) \in {L_\infty }$, e_i(t)∈L₂, we can conclude that $\mathop {\lim }\limits_{t \to \infty } {e_i}\left( t \right) = 0$ by using Barbalat's lemma. Therefore, the tracking error e_i(t) is ultimately and asymptotically stable, i.e., $\mathop {\lim }\limits_{t \to \infty } {e_i}\left( t \right) = 0$.

4.3 Coordinated implement

The objective of the coordinated control is first to calculate the coordinated moment by using the coordinated factors and the control moment M_c, and then to allocate the coordinated moment to the deflection angles of the elevator, the aileron, and the rudder, respectively.

Considering the coordinated factor in Eq.(19), the attitude control moment M_c, and the allocation matrix g_fδ of the fast-loop, the coordinated moment can be obtained as follows

$ {\mathit{\boldsymbol{M}}_c} = {\mathit{\boldsymbol{g}}_{\mathit{\Gamma }f\delta }} \cdot \mathit{\boldsymbol{\delta }} $

(44)

where

$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{g}}_{\mathit{\Gamma }f\delta }} = {\mathit{\boldsymbol{g}}_{f\delta }} + \mathit{\boldsymbol{ \boldsymbol{\varGamma} }} = }\\ {\left[ {\begin{array}{*{20}{c}} {{g_{p,{\delta _e}}} + {\lambda _e}}&{{g_{p,{\delta _a}}}}&{{g_{p,{\delta _r}}}}\\ {{g_{q,{\delta _e}}}}&{{g_{q,{\delta _a}}} + {\lambda _a}}&{{g_{q,{\delta _r}}}}\\ {{g_{r,{\delta _e}}}}&{{g_{r,{\delta _a}}}}&{{g_{r,{\delta _r}}} + {\lambda _r}} \end{array}} \right]} \end{array} $

(45)

By using Eqs.(44), (45), we have

$ \mathit{\boldsymbol{\delta }} = \mathit{\boldsymbol{g}}_{\mathit{\Gamma }f\delta }^{ - 1} \cdot {\mathit{\boldsymbol{M}}_c} $

(46)

According to Eq.(18), when the control strategy is using the elevator deflection to implement the pitch moment and using the coordinated deflections of the aileron and rudder to implement the yaw and roll moments, Eq.(47) can be obtained

$ {\delta _{{\rm{coe}}}} = {\delta _e},{\delta _{{\rm{coa}}}} = E{\delta _r},{\delta _{{\rm{cor}}}} = {\delta _r} $

(47)

5 Simulation and Analysis

The initial values of the simulation study are chosen as V₀=2 200 m/s, H₀=21 000 m, α=1°, β=5°, μ=0, p=q=r=0 °/s, and the desired flight attitude angles are chosen as α_c=5°, β_c=0°, μ_c=4°.

Suppose that there are +20% and -20% uncertainties on the aerodynamic coefficients and the aerodynamic moment coefficients, respectively. And the unknown external disturbance D is given as

$ \begin{array}{*{20}{c}} {\mathit{\boldsymbol{D}}\left( t \right) = 10 \times \left[ {\begin{array}{*{20}{c}} {0.5\sin \left( {1.5t} \right)}&{0.5\cos \left( {2t} \right)} \end{array}} \right.}\\ {{{\left. {0.5\sin \left( {2t} \right)} \right]}^{\rm{T}}}} \end{array} $

The design parameters of the attitude controller are chosen as k_i=1, i=1, …, 7, K₁=K₂=K₃=4, and κ₁=κ₂=5.

The simulation results of the attitude angles are shown in Fig. 2. The overshoot of uncoordinated α is 35% and that of coordinated α is 10%. The overshoot of the coordinated β is less than the uncoordinated β. The simulation result of μ is similar to β, which demonstrates that the performance of the coordinated attitude controller is superior to the uncoordinated one. It is obvious that the advantages of the coordinated factor in Eq.(13) lie in the smaller overshoot, less jitter, faster response, and quicker stabilization process. From Fig. 3, we can find that the angular rates can approach the equilibrium more quickly, resulted from the coordinated controller, which shows that the coordinated factor in Eq.(14) ensures that the jitter times are kept in a small range. In Fig. 4, by using the coordinated strategy of Eq.(47), it promotes the efficiency of the control surface deflection. And this performance improvement can enhance the control-ability of the attitude system and the maneuverability of the hypersonic flight vehicle.

6 Conclusions

This study indicates that, although the flight condition of a type of hypersonic aircraft contains strong couplings and uncertain disturbances, the design controllers can effectively compensate the effect of the compound disturbance and suppress the coupling phenomena. The definition of the coordinated factors provides a new solution to deal with the complex nonlinear relationships of the coupling variables. The proposed method provides the coupling analysis based on the flight dynamics, as well as analytical expressions, which can help to achieve satisfied tracking performance. In further studies, the robust coordinated control of the longitudinal dynamics of hypersonic aircraft will be discussed.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 61773204, 61374212).