1 Introduction

Air-breathing hypersonic vehicles (AHVs) present a cost efficient way to make access to space routine. But it uses scramjet engines integrated with the airframe, so the vehicle dynamics display strong interactions between the elastic airframe and the propulsion system (see [1–3] and references therein). In addition, significant flexible effects cannot be neglected in the control design due to the slender geometries and light structures of flexible air-breathing hypersonic vehicles (FAHVs) (see [4–6] and references therein). Thus, flight control of AHVs is very important and difficult especially for FAHVs.

In the past few years, many effective modeling and control techniques have been presented for AHVs (see e.g. [6–31]). The available control methods can be divided into two subcategories: linear model based and nonlinear model based. The linear approaches are simple and easily performed, but the capability of the linear models to represent the dynamics and the coupling effects is limited (see e.g. [11–17]). As such, nonlinear models that carry more information instead of linear ones have been investigated (see e.g. [18–31]). Many nonlinear control methods were successfully applied for a model of AHVs developed at NASA Langley Research Center (see e.g. [18–28]). For example, an effective neural control method via time-scale decomposition with throttle setting constraint was proposed for this AHV model in [26]. But rigid-body dynamics were only contained in the nonlinear AHV model.

Therefore, a nonlinear model of FAHVs was developed by Bolender and Doman in [6], which includes not only the interactions between the propulsion system and the airframe dynamics, but also the strong flexible effects. Based on this nonlinear model of FAHVs, a control-design model in closed form was obtained by replacing complex force and moment functions with curve-fitted approximations in [29]. Many nonlinear controllers were applied effectively for the curve-fitted model (see e.g. [29–31]). However, the nonlinear controllers proposed in [29] and [30] were constructed based on control-design models without including the rigid-flexible coupling dynamics. It has been shown that the undesired flexible effects may cause degradation of the performance of flight control systems in [29]. So the rigid-flexible coupling dynamics should be considered and studied to design nonlinear controllers for FAHVs. In [31], the flexible effects were transformed into some terms that depended on the flexible modes and appeared in forces and moment. Since many coefficients used for approximating these effects have to be estimated by using adaptive methods, it may lead to a large amount of calculations online. To overcome these problems, the flexible effects on pitch rate produced by rigid-flexible coupling dynamics are considered as a kind of unknown disturbance and modeled using known information in this paper. Then a coupling-observer-based nonlinear controller, which is a novel nonlinear composite controller, is proposed to reject the flexible effects and track the desired trajectories.

Recently, the disturbance-observer-based control strategy has attracted considerable attention. In most cases, principle of disturbance observer design in the time domain is similar to the one of state observer presented in [32], which can be concluded as follows: with suitable selection of disturbance observer gain, the dynamics of disturbance observer must be faster than the ones of the actual system, so that the estimate of disturbance can approach the actual disturbance as quickly as possible. In [34], it has been shown that this kind of composite hierarchical anti-disturbance control scheme can be applied to solve many control problems with multiple disturbances. Moreover, it has been applied effectively in many practical systems (see e.g. [12, 22, 27, 35]). For example, a disturbance-observer-based dynamic inversion controller was designed first for missile systems in [35], and several effective composite control methods based on disturbance observer have been presented for the AHV model in [12, 22, 27]. However, the study for the nonlinear FAHV model with rigid-flexible coupling dynamics using composite control strategy is still insufficient due to its special structure.

In conclusion, the main contributions in this paper can be summarized as follows:

  1. (1)

    The flexible effects on pitch rate are formulated as a kind of unknown disturbance, which is considered in the control-design model and estimated by a coupling observer.

  2. (2)

    Combining a dynamic-inversion-based controller with a coupling-observer-based compensator, a novel nonlinear composite controller is proposed to make velocity and flight-path angle track desired signals and reject the flexible effects.

  3. (3)

    The stability of composite closed-loop system which combines two tracking-error equations and a coupling estimation error equation is analyzed.

In simulation studies, it will be demonstrated that the rigid-flexible coupling dynamics may cause degradation of control performance or high-frequency oscillations of control input and flexible state using the nonlinear controller presented in [29], while the proposed coupling-observer-based nonlinear controller can avoid these problems successfully.

The rest of the paper is organized as follows. In Sect. 2, the FAHV model is introduced and the control objective is defined. In Sect. 3, a novel nonlinear composite controller is proposed. Simulation results are given in Sect. 4. Finally, some conclusions are drawn in Sect. 5.

Notation The notations used throughout the paper are standard. The superscript “\(T\)” and “\(-1\)” stand for transpose and inversion of a matrix or vector, respectively. \(R^{n}\) denotes the \(n\)-dimensional Euclidean space, the notation \(P>0\) means that \(P\) is a real symmetric and positive definite matrix, \(I\) and \(0\) represent the identity matrix and a zero matrix, respectively. \(\mathrm {diag}\{\ldots \}\) stands for a block-diagonal matrix, \(\left\| \cdot \right\| \) denotes the Euclidean norm of a vector and its induced norm of a matrix.

2 Problem formulation

A nonlinear model for the longitudinal dynamics of flexible air-breathing hypersonic vehicles (FAHVs), as given by Bolender and Doman in [6], is described by the following nonlinear equations:

$$\begin{aligned}&\dot{h}=V\sin \gamma ,\end{aligned}$$
(1)
$$\begin{aligned}&\dot{V}=\frac{T\cos \alpha -D}{m}-g\sin \gamma , \end{aligned}$$
(2)
$$\begin{aligned}&\dot{\alpha }=-\frac{L+T\sin \alpha }{mV}+Q+\frac{g}{V}\cos \gamma , \end{aligned}$$
(3)
$$\begin{aligned}&\dot{\gamma }=\frac{L+T\sin \alpha }{mV}-\frac{g}{V}\cos \gamma , \end{aligned}$$
(4)
$$\begin{aligned}&I_{yy}\dot{Q}=M+\tilde{\psi }_1\ddot{\eta }_1+\tilde{\psi }_2\ddot{\eta }_2, \end{aligned}$$
(5)
$$\begin{aligned}&\ddot{\eta }_i=-2\zeta _i\omega _i\dot{\eta }_i-\omega _i^2\eta _i+N_i +\tilde{\psi }_i\dot{Q}, \; i=1,2. \end{aligned}$$
(6)

The model is composed of rigid-body state \(x_r=[h \;V \; \alpha \;\gamma \; Q]^{ T}\), control input \(u=[\delta _e \; \varPhi ]^{T}\) and flexible modes \(\eta _1\) and \(\eta _2\), whereas the output to be controlled is selected as \(y=[V \; \gamma ]^{ T}\). The nomenclature is given in Table 1. The fuel equivalence ratio affects the thrust directly and the pitching moment indirectly via coupling between the engine and the airframe. The aerodynamic forces/moment and the generalized elastic forces are influenced by the aerodynamic control surfaces.

Table 1 Nomenclature
Full size table

Remark 1

The FAHV nonlinear model described by (1)–(6) has strong interactions among rigid-body state, control input, and flexible modes due to rigid-flexible coupling dynamics (that is, \(\tilde{\psi }_1\ddot{\eta }_1+\tilde{\psi }_2\ddot{\eta }_2\) and \(\tilde{\psi }_i\dot{Q}\)). It may cause the performance degradation to design nonlinear controller without considering these couplings (see [6] and [29]). The flexible effects were studied and added in the approximations of forces and moment in [31]. However, it may lead to a large amount of calculations online because many coefficients (such as \(C_L^{\eta },\,C_D^{\eta }, \,C_M^{\eta }\) and \(C_{N_i}^{\eta }\)) have to be estimated using the adaptive methods. To avoid these problems, the flexible effects on pitch rate are modeled as a kind of unknown disturbance in this paper, which is estimated by a coupling observer. And then, a feedforward compensator based on the coupling observer is designed to reject these flexible effects.

The approximations of the forces and moment adopted in this paper are given by Parker and Fiorentini in [29], which are described as follows:

$$\begin{aligned}&L= q S(C_L^{\alpha }\alpha +C_L^0+C_L^{\delta _e}\delta _e), \\&D= q S(C_D^{\alpha ^2}\alpha ^2+C_D^{\alpha }\alpha +C_D^0+C_D^{\delta _e^2}\delta _e^2 +C_D^{\delta _e}\delta _e),\\&T= (C_{T\varPhi }^{\alpha ^3}\alpha ^3+C_{T\varPhi }^{\alpha ^2}\alpha ^2+C_{T\varPhi }^{\alpha } \alpha +C_{T\varPhi }^0)\varPhi \\&\quad +C_T^{\alpha ^3}\alpha ^3+C_T^{\alpha ^2}\alpha ^2+C_T^{\alpha }\alpha +C_T^0,\\&M= z_TT+q S\bar{c}(C_M^{\alpha ^2}\alpha ^2+C_M^{\alpha }\alpha +C_M^0+C_M^{\delta _e} \delta _e),\\&N_1= C_{N_1}^{\alpha ^2}\alpha ^2+C_{N_1}^{\alpha }\alpha +C_{N_1}^0, \\&N_2= C_{N_2}^{\alpha ^2}\alpha ^2+C_{N_2}^{\alpha }\alpha +C_{N_2}^0 +C_{N_2}^{\delta _e}\delta _e, \end{aligned}$$

where \(C_{M}^{(\cdot )},\, C_{T}^{(\cdot )},\, C_{T \varPhi }^{(\cdot )}\), \(C_{N_i}^{(\cdot )},\, C_{L}^{(\cdot )}\), and \(C_{D}^{(\cdot )}\) are the coefficients of the curve-fit approximations. \(S\) denotes the reference area, \(\bar{c}\) the mean aerodynamic chord and \(z_T\) the thrust moment arm. Dynamic pressure \(q=(\rho V^2)/2\) and air density \(\rho =\rho _0 \exp (-(h-h_0)/h_s)\), in which \(h_s^{-1}\) is the air density decay rate, \(\rho _0\) and \(h_0\) are the values of air density and altitude at trimmed cruise condition.

Similarly to [30] and [31], the hypersonic cruising regimes are only considered for the above nonlinear model in this paper. As consequence of the physical constraints that characterize hypersonic flight and operability of scramjet engines, the rigid-body state \(x_r\) is bound to remain within the admissible range shown in Table 2, which determine the flight envelope, together with the admissible range for the control input \(u\). And the admissible ranges for variables in Table 2 are same as the ones of Table 1 in [31].

Table 2 Admissible ranges for variables
Full size table

In this paper, our objective is to design a nonlinear control law such that the output \(y\) can track the desired signals \([V_d \;\;\gamma _d]^{\ T}\) for the nonlinear model of FAHVs, where the desired trajectories are assumed to be bounded and with bounded derivatives of any order. The next section is to provide a novel nonlinear composite controller to achieve the control objective.

3 Coupling-observer-based nonlinear controller design

This section presents a novel composite controller with a hierarchical architecture for the nonlinear model of FAHVs. First, a control-design model with rigid-flexible coupling dynamics is provided. Second, the coupling observer is constructed to estimate the flexible effects on pitch rate. Finally, the nonlinear composite controller, which combines a coupling-observer-based compensator and a dynamic-inversion-based controller, is proposed to reject the flexible effects and track the desired trajectories. In addition, stability of the composite closed-loop system involving two tracking-error equations and a coupling observation error equation is analyzed.

3.1 Control-design model

Similarly to [29], the control-design model for rigid-body variables is obtained from the FAHV nonlinear model (1)-(6) by removing the altitude and flexible modes and setting to zero the weak elevator couplings, which is given by following equations:

$$\begin{aligned} \dot{V}&= f_1(x_r, \varPhi ),\end{aligned}$$
(7)
$$\begin{aligned} \dot{\alpha }&= f_2(x_r, \varPhi ), \end{aligned}$$
(8)
$$\begin{aligned} \dot{\gamma }&= f_3(x_r, \varPhi ),\end{aligned}$$
(9)
$$\begin{aligned} \dot{Q}&= f_4(x_r, \varPhi )+g_4(x_r, \varPhi )\delta _e+d_c, \end{aligned}$$
(10)

where nonlinear functions \(f_1(x_r, \varPhi ),\, f_2(x_r, \varPhi )\), \(f_3(x_r, \varPhi ),\, f_4(x_r, \varPhi )\), and \(g_4(x_r, \varPhi )\) are sufficiently smooth with respect to \(x_r\) and \(\varPhi \), disturbance \(d_c\) is produced by the rigid-flexible coupling dynamics. The concrete forms of \(f_1(x_r, \varPhi ),\, f_2(x_r, \varPhi ), f_3(x_r, \varPhi ),\, f_4(x_r, \varPhi ),\, g_4(x_r, \varPhi )\), and \(d_c\) are described as \(f_1(x_r, \varPhi )=(T\cos \alpha -\bar{D})/m-g\sin \gamma \), \(f_2(x_r, \varPhi )=-(\bar{L}+T\sin \alpha )/(mV)+Q+(g/V)\cos \gamma ,\) \(f_3(x_r, \varPhi )=(\bar{L}+T\sin \alpha )/(mV)-(g/V)\cos \gamma \), \(f_4(x_r, \varPhi )=\bar{I}_{yy}^{-1}[z_TT+ q S\bar{c}(C_{M}^{\alpha ^2}\alpha ^2+C_{M}^{\alpha }\alpha +C_{M}^0) +\tilde{\psi }_1N_1 + \tilde{\psi }_2 (C_{N_2}^{\alpha ^2}\alpha ^2+C_{N_2}^{\alpha }\alpha +C_{N_2}^0)]\), \(g_4(x_r, \varPhi )=\bar{I}_{yy}^{-1}(q S\bar{c}C_{M}^{\delta _e}+\tilde{\psi }_2 C_{N_2}^{\delta _e})\) and \(d_c=C_f [\eta _1 \;\;\eta _2 \;\;\dot{\eta }_1 \;\;\dot{\eta }_2]^{ T}\), where \(\bar{L}=qS(C_{L}^{\alpha }\alpha +C_{L}^0)\), \(\bar{D}=qS(C_{D}^{\alpha ^2}\alpha ^2+C_{D}^{\alpha }\alpha +C_{D}^0) \), \(\bar{I}_{yy}=I_{yy}-\tilde{\psi }_1^2-\tilde{\psi }_2^2\) and \(C_f=\bar{I}_{yy}^{-1}[-\tilde{\psi }_1\omega _1^2 \;\; -\tilde{\psi }_2\omega _2^2 \;\; -2\tilde{\psi }_1\zeta _1\omega _1 \;\;-2\tilde{\psi }_2\zeta _2\omega _2]\).

Remark 2

It is different from [29] that the flexible effects produced by the rigid-flexible coupling dynamics are considered and added in the control-design model (7)–(10).

Using (5) and (6), the model for flexible modes \(\eta _1\) and \(\eta _2\), which is provided to construct the coupling observer in next subsection, can be written as

$$\begin{aligned} \dot{x}_f = A_fx_f+B_f(x_r)\delta _e+F_f(x_r, \varPhi ), \end{aligned}$$
(11)

where \(x_f=[\eta _1 \;\;\eta _2 \;\;\dot{\eta }_1 \;\;\dot{\eta }_2]^{ T}\) is the flexible state, and

$$\begin{aligned}&A_f= \left[ \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1\\ -p_1 \omega _1^2 &{} -p_3 \omega _2^2&{} -2p_1\zeta _1\omega _1 &{} -2p_3\zeta _2\omega _2\\ -p_3 \omega _1^2 &{} -p_2 \omega _2^2&{} -2p_3\zeta _1\omega _1 &{} -2p_2\zeta _2\omega _2\\ \end{array} \right] ,\\&B_f(x_r)= \left[ \begin{array}{c} 0\\ 0\\ b_{f1}\\ b_{f2}\\ \end{array} \right] ,\; F_f(x_r, \varPhi )= \left[ \begin{array}{c} 0\\ 0\\ f_{f1}\\ f_{f2}\\ \end{array} \right] , \end{aligned}$$

in which \(p_1=1+\bar{I}_{yy}^{-1}\tilde{\psi }_1^2\), \(p_2=1+\bar{I}_{yy}^{-1}\tilde{\psi }_2^2\), \(p_3=\bar{I}_{yy}^{-1} \tilde{\psi }_1 \tilde{\psi }_2\), \(b_{f1}=\bar{I}_{yy}^{-1}\tilde{\psi }_1 q S\bar{c}C_{M}^{\delta _e}+p_3C_{N_2}^{\delta _e}\), \(b_{f2}=\bar{I}_{yy}^{-1}\tilde{\psi }_2 q S\bar{c}C_{M}^{\delta _e}+p_2C_{N_2}^{\delta _e}\), \(f_{f1}= \bar{I}_{yy}^{-1}\tilde{\psi }_1[z_TT+ q S\bar{c}(C_{M}^{\alpha ^2}\alpha ^2+C_{M}^{\alpha }\alpha +C_{M}^0)] +p_3(C_{N_2}^{\alpha ^2}\alpha ^2+C_{N_2}^{\alpha }\alpha +C_{N_2}^0) +p_1N_1\), \(f_{f2}= \bar{I}_{yy}^{-1}\tilde{\psi }_2[z_TT+ q S\bar{c}(C_{M}^{\alpha ^2}\alpha ^2+C_{M}^{\alpha }\alpha +C_{M}^0)]+p_3N_1 +p_2(C_{N_2}^{\alpha ^2}\alpha ^2+C_{N_2}^{\alpha }\alpha +C_{N_2}^0)\).

3.2 Coupling observer design

We can construct a nonlinear Luenberger observer to estimate the flexible state \(x_f\) based on (10) and (11). So a coupling observer to estimate the unknown disturbance \(d_c\) can be given by

$$\begin{aligned} \left\{ \begin{array}{lll} \hat{d}_c &{}=&{} C_f\hat{x}_f\\ \hat{x}_f &{}=&{} v_c + L_cQ\\ \dot{v}_c &{}=&{} (A_f-L_cC_f)(v_c + L_c Q)+B_f(x_r) \delta _e\\ &{} &{}+F_f(x_r, \varPhi )-L_c(f_4(x_r, \varPhi )+g_4(x_r, \varPhi )\delta _e), \end{array}\right. \end{aligned}$$
(12)

where \(\hat{d}_c\) and \(\hat{x}_f\) are estimates of \(d_c\) and \(x_f\), respectively. Besides, \(v_c\) is the internal state of observer (12) and the observer gain, \(L_c{\in }\mathcal {R}^{4\times 1}\), is to be designed.

Define flexible state estimation error \(e_f(t)=x_f-\hat{x}_f\) and coupling estimation error \(e_c(t)=d_c-\hat{d}_c\). Obviously, \(e_c(t)=C_fe_f(t)\). Using coupling observer (12), the first differential of \(e_f(t)\) can be given by

$$\begin{aligned} \dot{e}_f(t)&= \dot{x}_f-\dot{\hat{x}}_f\\&= \dot{x}_f - \dot{v}_c - L_c \dot{Q}\\&= A_f x_f+B_f(x_r)\delta _e+F_f(x_r, \varPhi )\\&-(A_f-L_c C_f)(v_c + L_c Q)-B_f(x_r) \delta _e\\&-F_f(x_r,\varPhi )+L_c(f_4(x_r, \varPhi )+g_4(x_r, \varPhi )\delta _e)\\&-L_c(f_4(x_r, \varPhi )+g_4(x_r, \varPhi )\delta _e + C_f e_f(t))\\&= (A_f-L_c C_f) e_f(t). \end{aligned}$$

Thus, the differential equation for \(e_f(t)\) is described as

$$\begin{aligned} \dot{e}_f(t)=\bar{A}_fe_f(t), \end{aligned}$$
(13)

where \(\bar{A}_f=A_f-L_cC_f\). The system (13) is asymptotically stable, if an observer gain \(L_c\) can be found such that \(\bar{A}_f\) is a Hurwitz matrix.

When \(A_f=0,\, C_f=I,\, B_f(x_r)=0\) and \(F_f(x_r, \varPhi )=0\) (that is, \(\dot{d}_c \equiv 0\)), a simpler observer can be designed to estimate \(d_c\) as

$$\begin{aligned} \left\{ \begin{array}{lll} \hat{d}_c &{}=&{} z_d + L_d Q\\ \dot{z}_d &{}=&{} -L_d(z_d + L_d Q)-L_d(f_4(x_r, \varPhi )\\ &{}&{}+g_4(x_r, \varPhi )\delta _e), \end{array}\right. \end{aligned}$$
(14)

where \(z_d\) and \(L_d\) are the internal state and the gain of observer (14), respectively. The observer (14) is similar to the disturbance observer presented in [22, 27, 35, 36]. The estimation error using observer (14) can be reduced by increasing the observer gain \(L_d\), if the disturbance is slowly time-varying. Another kind of disturbance observer presented in [12, 37, 38] were designed to estimate the disturbances that are generated by exogenous systems which are different from the disturbance \(d_c\) under consideration in this paper.

Remark 3

The moment of inertia \(\bar{I}_{yy}\), interplay coupling parameter \(\tilde{\psi }_i\), natural frequency \(\omega _i\), and damping ratio \(\zeta _i\) can be worked out, if we know the mass, length, and rigidity of the vehicle. So the parameter matrices \(A_f\) and \(C_f\) can be obtained. In fact, the mass, length, and rigidity may have uncertainty in practice. Thus, a work is being done to design a robust coupling observer in order to estimate the disturbance \(d_c\) when the matrices \(A_f,\, B_f(x_r),\, C_f\) and \(F_f(x_r, \varPhi )\) are uncertain.

Remark 4

In order to reject the disturbance \(d_c\) by using coupling-observer-based compensator later, the boundedness of \(d_c\) needs to be analyzed. Recall that \(d_c=C_f x_f\), this problem turns to the boundedness analysis of \(x_f\). First, Let us rewrite the model (11) as

$$\begin{aligned} \dot{x}_f = A_fx_f+w_f, \end{aligned}$$
(15)

where \(w_f=B_f(x_r)\delta _e+F_f(x_r, \varPhi )\) is considered as an input of flexible system (15). Then the characteristic equation of \(A_f\) is

$$\begin{aligned} {\det } (\lambda I-A_f) = a_0 \lambda ^4 + a_1 \lambda ^3+ a_2 \lambda ^2+ a_3 \lambda + a_4=0, \end{aligned}$$
(16)

where \(a_0=1,\, a_1=2(\bar{\zeta }_1 \bar{\omega }_1 + \bar{\zeta }_2 \bar{\omega }_2)\), \(a_2=\bar{\omega }_1^2 + \bar{\omega }_2^2 +4\bar{p} \bar{\zeta }_1 \bar{\omega }_1 \bar{\zeta }_2 \bar{\omega }_2\), \(a_3=2\bar{p} \bar{\omega }_1 \bar{\omega }_2(\bar{\zeta }_1 \bar{\omega }_2 + \bar{\zeta }_2 \bar{\omega }_1)\), \(a_4= \bar{p} \bar{\omega }_1^2 \bar{\omega }_2^2\), in which \(\bar{\zeta }_1=\sqrt{p_1} \zeta _1\), \(\bar{\omega }_1=\sqrt{p_1}\omega _1\), \(\bar{\zeta }_2=\sqrt{p_2}\zeta _2\), \(\bar{\omega }_2=\sqrt{p_2} \omega _2\), \(\bar{p}=1-(p_3^2/(p_1p_2))\). Obviously, \(0 < \bar{p} <1\). The Routh Table is described as

$$\begin{aligned} \begin{array}{lccc} s^4 &{} a_0 \;\;&{} a_2 \;\;\;\;\;\;\; &{} a_4 \\ s^3 &{} a_1 \;\;&{} a_3 \;\;\;\;\;\;\;&{} 0 \\ s^2 &{} (a_1a_2-a_0a_3)/a_1 \;\;&{} a_4 \;\;\;\;\;\;\; &{} 0 \\ s^1 &{} (a_1a_2a_3-a_0a_3^2-a_4a_1^2)/(a_1a_2-a_0a_3) \;\; &{} 0\;\;\;\;\;\;\; &{} \\ s^0 &{} a_4 &{} &{} \\ \end{array} \end{aligned}$$

where

$$\begin{aligned}&a_1a_2-a_0a_3 = 2(1-\bar{p})(\bar{\zeta }_1 \bar{\omega }_1\bar{\omega }_2^2 + \bar{\zeta }_2 \bar{\omega }_2 \bar{\omega }_1^2) + 2(\bar{\zeta }_1 \bar{\omega }_1^3\\&\quad \qquad \qquad \qquad \bar{\zeta }_2 \bar{\omega }_2^3) + 8\bar{p}\bar{\zeta }_1 \bar{\omega }_1 \bar{\zeta }_2 \bar{\omega }_2(\bar{\zeta }_1 \bar{\omega }_1 + \bar{\zeta }_2 \bar{\omega }_2),\\&a_1a_2a_3-a_0a_3^2-a_4a_1^2=4\bar{p}(1-\bar{p})(\bar{\zeta }_1 \bar{\omega }_2 + \bar{\zeta }_2 \bar{\omega }_1)^2 \\&+ 4\bar{p}\bar{\zeta }_1 \bar{\omega }_1 \bar{\zeta }_2 \bar{\omega }_2 (\bar{\omega }_1^2-\bar{\omega }_2^2)^2 +8\bar{\zeta }_1 \bar{\omega }_1 \bar{\zeta }_2 \bar{\omega }_2(\bar{\zeta }_1 \bar{\omega }_1 + \bar{\zeta }_2 \bar{\omega }_2). \end{aligned}$$

Observe that coefficients of the first column in Routh Table are all positive, when \(\zeta _1>0,\, \zeta _2>0,\, \omega _1>0\) and \(\omega _2>0\). Thus \(A_f\) is Hurwitz stable. The solution of state-space equation (15) is given by \(x_f(t)=e^{A_ft}x_f(0)+\int _0^t e^{A_f(t-\tau )}w_f(\tau )\mathrm{d}\tau \), where \(w_f\) is bounded because the angle of attack \(\alpha \), the dynamic pressure \(q\) and the control inputs \(\varPhi \) and \(\delta _e\) are all bounded for the flight control of cruising regimes. Therefore, the flexible state \(x_f\) and the disturbance \(d_c\) are bounded.

3.3 Nonlinear composite controller design

With the coupling observer (12), a nonlinear composite controller which combines a coupling-observer-based feedforward compensator and a dynamic-inversion-based feedback controller is constructed in this subsection. Let us consider the following second-order actuator model, which is appended to the input \(\varPhi \).

$$\begin{aligned} \ddot{\varPhi }=-2\zeta \omega \dot{\varPhi }-\omega ^2\varPhi +\omega ^2\varPhi _c. \end{aligned}$$
(17)

By combing with (7)–(10) and selecting \(\bar{u}=[\delta _e \;\; \varPhi _c]^{ T}\) as the new control input, a six dimension model with full vector relative degree can be obtained to design a dynamic-inversion-based feedback controller. From (7)-(10) and (17), the third differential of \(V\) and \(\gamma \) can be obtained as follows:

$$\begin{aligned} \dddot{V} \!&= \! (\partial {f_1(x_r, \varPhi )}/\partial {x}) \ddot{x}\!+\!\dot{x}^{ T}(\partial {^2 f_1(x_r, \varPhi )}/ \partial {x}^2)\dot{x}\end{aligned}$$
(18)
$$\begin{aligned} \dddot{\gamma } \!&= \! (\partial {f_3(x_r, \varPhi )}/\partial {x}) \ddot{x}\!+\!\dot{x}^{ T}(\partial {^2 f_3(x_r, \varPhi )}/\partial {x}^2)\dot{x}, \end{aligned}$$
(19)

where \(x=[V \;\; \alpha \;\; \gamma \;\; \varPhi ]^{ T}\), \(\dot{x}=[\dot{V} \;\; \dot{\alpha } \;\; \dot{\gamma } \;\; \dot{\varPhi }]^{ T}\), and \(\ddot{x}=[\ddot{V} \;\; \ddot{\alpha } \;\; \ddot{\gamma } \;\; \ddot{\varPhi }]^{ T}\). Because \(\dot{x}\) and \(\ddot{x}\) contain the functions about control input \(\bar{u}\) and disturbance \(d_c\), the expressions of \(\dddot{V}\) and \(\dddot{\gamma }\) can be rewritten as

$$\begin{aligned} \begin{array}{lll} \left[ \begin{array}{c} \dddot{V} \\ \dddot{\gamma }\\ \end{array} \right] \!&{}=&{}\! \left[ \begin{array}{c} f_V(x_r, \varPhi ) \\ f_{\gamma }(x_r, \varPhi )\\ \end{array} \!\!\right] + \left[ \begin{array}{cc} g_{V1}(x_r, \varPhi )\;\;&{}g_{V2}(x_r, \varPhi ) \\ g_{\gamma 1}(x_r, \varPhi )\;\;&{}g_{\gamma 2}(x_r, \varPhi )\\ \end{array} \right] \bar{u}\\ &{} &{}+\left[ \begin{array}{c} g_{V3}(x_r, \varPhi ) \\ g_{\gamma 3}(x_r, \varPhi )\\ \end{array} \right] d_c, \end{array} \end{aligned}$$
(20)

where

$$\begin{aligned}&f_V(x_r, \varPhi )= \pi ^V \ddot{x}_0+\dot{x}_0^{ T}\varPi ^V\dot{x}_0, \\&g_{V1}(x_r, \varPhi )\!=\!\frac{g_4(x_r, \varPhi )}{m} \left( \frac{\partial {T}}{\partial \alpha } \cos \alpha \!-\!T \sin \alpha \!-\!\frac{\partial {\bar{D}}}{\partial \alpha }\right) \!, \\&g_{V2}(x_r, \varPhi )=\frac{\omega ^2}{m}\left( \frac{\partial T}{\partial \varPhi } \cos \alpha \right) ,\\&g_{V3}(x_r, \varPhi )=\frac{1}{m} \left( \frac{\partial {T}}{\partial \alpha }\cos \alpha \!-\!T \sin \alpha \!-\! \frac{\partial {\bar{D}}}{\partial \alpha }\right) \!,\\&f_{\gamma }(x_r, \varPhi )= \pi ^{\gamma } \ddot{x}_0+\dot{x}_0^{ T} \varPi ^{\gamma } \dot{x}_0,\\&g_{\gamma 1}(x_r, \varPhi )=\frac{g_4(x_r, \varPhi )}{mV} \left( \frac{\partial {T}}{\partial \alpha }\sin \alpha \!+\!T \cos \alpha \!+\! \frac{\partial {\bar{L}}}{\partial \alpha }\right) , \\&g_{\gamma 2}(x_r, \varPhi )=\frac{\omega ^2}{mV} \left( \frac{\partial T}{\partial \varPhi }\sin \alpha \right) ,\\&g_{\gamma 3}(x_r, \varPhi )=\frac{1}{mV} \left( \frac{\partial {T}}{\partial \alpha }\sin \alpha + T \cos \alpha + \frac{\partial {\bar{L}}}{\partial \alpha }\right) , \end{aligned}$$

in which

$$\begin{aligned} \begin{array}{l} \dot{x}_0= \left[ \begin{array}{c} f_1(x_r, \varPhi )\\ f_2(x_r, \varPhi )\\ f_3(x_r, \varPhi )\\ \dot{\varPhi }\\ \end{array} \right] , \; \ddot{x}_0= \left[ \begin{array}{c} \pi ^V \dot{x}_0\\ -\pi ^{\gamma } \dot{x}_0+f_4(x_r, \varPhi )\\ \pi ^{\gamma } \dot{x}_0\\ -2\zeta \omega \dot{\varPhi }-\omega ^2\varPhi \\ \end{array} \right] ,\\ \pi ^V=\partial {f_1(x_r, \varPhi )}/\partial {x}, \; \pi ^{\gamma }=\partial {f_3(x_r, \varPhi )}/\partial {x}, \\ \varPi ^V=\partial {^2 f_1(x_r, \varPhi )}/\partial {x}^2, \; \varPi ^{\gamma }=\partial {^2 f_3(x_r, \varPhi )}/\partial {x}^2. \end{array} \end{aligned}$$

The detailed expressions of \(\pi ^V,\, \pi ^{\gamma },\, \varPi ^V\), and \(\varPi ^{\gamma }\) are given in the Appendix. It is observed that the nonlinear functions \(f_V(x_r, \varPhi ),\, f_{\gamma }(x_r, \varPhi )\), \(g_{Vj}(x_r, \varPhi )\), and \(g_{\gamma j}(x_r, \varPhi )\), \(j=1,2,3\) are sufficiently smooth with respect to \(x_r\) and \(\varPhi \).

Next, we begin to design the nonlinear composite controller based on the coupling observer. Using the estimate \(\hat{d}_c\) produced by observer (12), the composite controller can be designed as

$$\begin{aligned} \bar{u}= u_1 + u_2 + u_3, \end{aligned}$$
(21)

where

$$\begin{aligned} \begin{array}{l} u_1=- G(x_r, \varPhi )^{-1}\; \left[ \begin{array}{c} f_V(x_r, \varPhi ) \\ f_{\gamma }(x_r, \varPhi )\\ \end{array} \right] ,\\ u_2 =- G(x_r, \varPhi )^{-1}\; \left[ \begin{array}{l} -\dddot{V}_d+\upsilon _1 \\ -\dddot{\gamma }_d+\upsilon _2 \\ \end{array} \right] ,\\ u_3 =- G(x_r, \varPhi )^{-1}\; \left[ \begin{array}{c} g_{V3}(x_r, \varPhi )\\ g_{\gamma 3}(x_r, \varPhi )\\ \end{array} \right] \hat{d}_c,\\ G(x_r, \varPhi )= \left[ \begin{array}{cc} g_{V1}(x_r, \varPhi )\;\;&{}g_{V2}(x_r, \varPhi ) \\ g_{\gamma 1}(x_r, \varPhi )\;\;&{}g_{\gamma 2}(x_r, \varPhi )\\ \end{array} \right] ,\\ \upsilon _1= k_{V4}\ddot{e}_V(t)+k_{V3}\dot{e}_V(t)+k_{V2}e_V(t)\\ \qquad +k_{V1}\int _0^te_V(\tau )\mathrm{d}\tau ,\\ \upsilon _2=k_{\gamma 4}\ddot{e}_{\gamma }(t)+k_{\gamma 3}\dot{e}_{\gamma }(t) +k_{\gamma 2}e_{\gamma }(t)\\ \qquad +k_{\gamma 1}\int _0^te_{\gamma }(\tau )\mathrm{d}\tau , \end{array} \end{aligned}$$

in which \(e_V(t)=V-V_d\) and \(e_{\gamma }(t)=\gamma -\gamma _d\) are tracking errors of velocity and flight-path angle, respectively. The control gains \(K_V=-[k_{V1} \;\; k_{V2}\;\; k_{V3}\;\; k_{V4}]\) and \(K_{\gamma }=-[k_{\gamma 1} \;\; k_{\gamma 2}\;\; k_{\gamma 3}\;\; k_{\gamma 4}]\) are to be obtained by solving Riccati equations in the following.

Remark 5

The control law (21) includes three parts: the first term is designed to compensate the nonlinear functions \(f_V(x_r, \varPhi )\) and \(f_{\gamma }(x_r, \varPhi )\), the second term is designed to track the desired signals \(V_d\) and \(\gamma _d\), and the third term is designed to reject the disturbance \(d_c\). The dynamic-inversion-based feedback controller is composed of the first and the second parts, while the coupling-observer-based feedforward compensator is the third part.

Remark 6

When \(\hat{d}_c \equiv 0\), that is, there is no coupling compensation, the nonlinear composite controller (21) obviously becomes the dynamic-inversion-based nonlinear controller presented in [29], which is written as

$$\begin{aligned} \bar{u}=\bar{u}_1 + \bar{u}_2, \end{aligned}$$
(22)

where

$$\begin{aligned} \begin{array}{l} \bar{u}_1=- \bar{G}(x_r, \varPhi )^{-1}\; \left[ \begin{array}{c} \bar{f}_V(x_r, \varPhi ) \\ \bar{f}_{\gamma }(x_r, \varPhi )\\ \end{array} \right] ,\\ \bar{u}_2=- \bar{G}(x_r, \varPhi )^{-1}\; \left[ \begin{array}{l} -\dddot{V}_d+\upsilon _1 \\ -\dddot{\gamma }_d+\upsilon _2 \\ \end{array} \right] , \end{array} \end{aligned}$$

in which

$$\begin{aligned}&\bar{G}(x_r, \varPhi )=\left[ \begin{array}{cc} \bar{g}_{V1}(x_r, \varPhi )\;\;&{}g_{V2}(x_r, \varPhi ) \\ \bar{g}_{\gamma 1}(x_r, \varPhi )\;\;&{}g_{\gamma 2}(x_r, \varPhi )\\ \end{array} \right] ,\\&\ddot{\bar{x}}_0= \left[ \begin{array}{c} \pi ^V \dot{x}_0\\ -\pi ^{\gamma } \dot{x}_0+\bar{f}_4(x_r, \varPhi )\\ \pi ^{\gamma } \dot{x}_0\\ -2\zeta \omega \dot{\varPhi }-\omega ^2\varPhi \\ \end{array} \right] ,\\&\bar{f}_V(x_r, \varPhi )= \pi ^V \ddot{\bar{x}}_0+\dot{x}_0^{ T}\varPi ^V\dot{x}_0, \\&\bar{g}_{V1}(x_r, \varPhi )\!=\!\frac{\bar{g}_4(x_r, \varPhi )}{m} \left( \frac{\partial {T}}{\partial \alpha } \cos \alpha \!-\!T \sin \alpha \!-\!\frac{\partial {\bar{D}}}{\partial \alpha }\right) , \\&\bar{f}_{\gamma }(x_r, \varPhi )= \pi ^{\gamma } \ddot{\bar{x}}_0+\dot{x}_0^{ T} \varPi ^{\gamma } \dot{x}_0,\\&\bar{g}_{\gamma 1}(x_r, \varPhi )=\frac{\bar{g}_4(x_r, \varPhi )}{mV} \left( \frac{\partial {T}}{\partial \alpha }\sin \alpha \!+\!T \cos \alpha \!+\! \frac{\partial {\bar{L}}}{\partial \alpha }\right) ,\nonumber \\&\bar{f}_4(x_r, \varPhi )=\bar{I}_{yy}^{-1}[z_TT\!+\! q S\bar{c}(C_{M}^{\alpha ^2}\alpha ^2 \!+\!C_{M}^{\alpha }\alpha \!+\!C_{M}^0)],\\&\bar{g}_4(x_r, \varPhi )=\bar{I}_{yy}^{-1} q S\bar{c}C_{M}^{\delta _e}. \end{aligned}$$

Remark 7

Note that the non-singular conditions for \(G(x_r, \varPhi )\) and \(\bar{G}(x_r, \varPhi )\) can be represented as

$$\begin{aligned} \begin{array}{l} \det [G(x_r, \varPhi )] \ne 0 \Longleftrightarrow \\ \frac{\omega ^2}{m^2 V} \frac{\partial {T}}{\partial \varPhi } g_4(x_r, \varPhi ) (T + \frac{\partial {\bar{D}}}{\partial \alpha } \sin \alpha + \frac{\partial {\bar{L}}}{\partial \alpha } \cos \alpha )\ne 0, \end{array}\\ \begin{array}{l} \det [\bar{G}(x_r, \varPhi )] \ne 0 \Longleftrightarrow \\ \frac{\omega ^2}{m^2 V} \frac{\partial {T}}{\partial \varPhi } \bar{g}_4(x_r, \varPhi ) (T + \frac{\partial {\bar{D}}}{\partial \alpha } \sin \alpha + \frac{\partial {\bar{L}}}{\partial \alpha } \cos \alpha )\ne 0. \end{array} \end{aligned}$$

In [29], the matrix \(\bar{G}(x_r, \varPhi )\) has been verified to be nonsingular over the operating range of interest, which is the range of parameter values used in the curve fits. Thus \((T + \frac{\partial {\bar{D}}}{\partial \alpha } \sin \alpha + \frac{\partial {\bar{L}}}{\partial \alpha } \cos \alpha )\ne 0\) holds due to \(\frac{\omega ^2}{m^2 V} \frac{\partial {T}}{\partial \varPhi } \bar{g}_4(x_r, \varPhi )\ne 0\). Moreover, \(\frac{\omega ^2}{m^2 V} \frac{\partial {T}}{\partial \varPhi } g_4(x_r, \varPhi )\ne 0\) also holds for the dynamic pressure \(q\) considered in this paper. Finally, the \(\frac{\omega ^2}{m^2 V} \frac{\partial {T}}{\partial \varPhi } g_4 (x_r, \varPhi ) (T + \frac{\partial {\bar{D}}}{\partial \alpha } \sin \alpha + \frac{\partial {\bar{L}}}{\partial \alpha } \cos \alpha )\ne 0\) holds, that is, the matrix \(G(x_r, \varPhi )\) is also nonsingular.

Remark 8

The gains (\(K_V\) and \(K_{\gamma }\)) of controller (22) are designed using LQR in [29], which obtained by solving two Riccati equations. In order to compare controller (21) with controller (22) clearly, the gains (\(L_c,\, K_V\) and \(K_{\gamma }\)) of controller (21) are also related with the solutions of Riccati equations detailed in the following.

Considering (13) and substituting (21) into (20), a composite closed-loop system can be obtained as follows:

$$\begin{aligned} \left\{ \begin{array}{lll} \dot{e}_f(t) &{}=&{} \bar{A}_f e_f(t)\\ \dot{\bar{e}}_V(t) &{}=&{} \bar{A}_V\bar{e}_V(t) + \bar{B}_{V}C_fe_f(t)\\ \dot{\bar{e}}_{\gamma }(t) &{}=&{} \bar{A}_{\gamma }\bar{e}_{\gamma }(t) + \bar{B}_{\gamma }C_fe_f(t),\\ \end{array}\right. \end{aligned}$$
(23)

where \(\bar{A}_V=A_0+B_0K_V\), \(\bar{A}_{\gamma }=A_0+B_0K_{\gamma }\), \(\bar{B}_V=g_{V3}(x_r, \varPhi )B_0\), \(\bar{B}_{\gamma }=g_{\gamma 3}(x_r, \varPhi )B_0\),

$$\begin{aligned} \begin{array}{l} \bar{e}_V(t)= \left[ \begin{array}{c} \int _0^te_V(\tau )\mathrm{d}\tau \\ e_V(t)\\ \dot{e}_V(t)\\ \ddot{e}_V(t)\\ \end{array} \right] , \; \bar{e}_{\gamma }(t)= \left[ \begin{array}{c} \int _0^te_{\gamma }(\tau )\mathrm{d}\tau \\ e_{\gamma }(t)\\ \dot{e}_{\gamma }(t)\\ \ddot{e}_{\gamma }(t)\\ \end{array} \right] ,\\ A_0= \left[ \begin{array}{cccc} 0 &{}1 &{} 0&{} 0\\ 0 &{}0 &{} 1&{} 0\\ 0 &{}0 &{} 0&{} 1\\ 0 &{}0 &{} 0&{} 0\\ \end{array} \right] , \; B_0= \left[ \begin{array}{c} 0\\ 0\\ 0\\ 1\\ \end{array} \right] . \end{array} \end{aligned}$$

The composite closed-loop system (23) can be described compactly as

$$\begin{aligned} \dot{e}(t)&= A e(t), \end{aligned}$$
(24)

where

$$\begin{aligned} e(t)= \left[ \begin{array}{c} e_f(t) \\ \bar{e}_V(t) \\ \bar{e}_{\gamma }(t) \\ \end{array} \right] , \; A= \left[ \begin{array}{ccc} \bar{A}_f &{}0 &{} 0\\ \bar{B}_VC_f &{}\bar{A}_V &{} 0\\ \bar{B}_{\gamma }C_f &{}0 &{}\bar{A}_{\gamma } \\ \end{array} \right] . \end{aligned}$$

To design the gains of controller (21) and analyze the stability of composite system (24), the following assumption is needed in this paper. The assumption can be satisfied for the hypersonic cruising regimes.

Assumption 1

Nonlinear functions \(g_{V3}(x_r, \; \varPhi )\) and \(g_{\gamma 3}(x_r, \varPhi )\) are bounded, that is, \(||g_{V3}(x_r, \varPhi )|| \le \sigma _{V3} \), \(||g_{\gamma 3}(x_r, \varPhi )|| \le \sigma _{\gamma 3}\), where \(\sigma _{V3}\) and \(\sigma _{\gamma 3}\) are all known positive constants.

Theorem 1

Considering the closed-loop system (24), for \(\beta _V > 0\) and \(\beta _{\gamma } > 0\), if there exist three positive definite matrices, \(P_1{\in }\mathcal {R}^{4\times 4}\), \(P_2{\in }\mathcal {R}^{4\times 4}\) and \(P_3{\in }\mathcal {R}^{4\times 4}\), satisfying the following Riccati equations:

$$\begin{aligned}&P_1A_f^{ T}+ A_f P_1-P_1C_f^{ T}R_f^{-1}C_f P_1+Q_f=0,\end{aligned}$$
(25)
$$\begin{aligned}&P_2A_0+ A_0^{ T} P_2-P_2B_0R_V^{-1}B_0^{ T} P_2+Q_V=0,\end{aligned}$$
(26)
$$\begin{aligned}&P_3A_0+ A_0^{ T} P_3-P_3B_0R_{\gamma }^{-1}B_0^{ T} P_3+Q_{\gamma }=0, \end{aligned}$$
(27)

and the weights \(Q_f{\in }\mathcal {R}^{4\times 4}\), \(Q_V{\in }\mathcal {R}^{4\times 4}\), \(Q_{\gamma }{\in }\mathcal {R}^{4\times 4}\), \(R_f{\in }\mathcal {R}\), \(R_V{\in }\mathcal {R}\) and \(R_{\gamma }{\in }\mathcal {R}\), satisfying

$$\begin{aligned}&Q_f \ge (\beta _V^{-2}\sigma _{V3}^2+\beta _{\gamma }^{-2}\sigma _{\gamma 3}^2)C_f^{ T}C_f,\;\; Q_V \ge 0,\\&Q_{\gamma } \ge 0 0 < R_f,\;\;0 < R_V \le \beta _V^{-2},\;\; 0 < R_{\gamma } \le \beta _{\gamma }^{-2} \end{aligned}$$

then by selecting \(L_c^{ T}=R_f^{-1}C_fP_1\), \(K_V=-R_V^{-1}B_0^{ T}P_2\) and \(K_{\gamma }=-R_{\gamma }^{-1}B_0^{ T}P_3\), the composite closed-loop system (24) is asymptotically stable.

Proof

A Lyapunov function candidate for the system (24) is chosen as

$$\begin{aligned} W(t)=e^{ T}(t)Pe(t), \end{aligned}$$

where \(P=\mathrm{diag}\{\bar{P}_1, \; P_2, \;P_3 \}\) and \(\bar{P}_1=P_1^{-1}\).

The derivative of \(W(t)\) along trajectories of system (24) can be written as

$$\begin{aligned} \dot{W}(t)&= e^{ T}(t)(PA+A^{ T}P)e(t)\\&= e^{ T}(t)\bar{\varGamma } e(t)\!-\!e^{ T}(t)\chi _1^{ T}\!\chi _1 e(t)\!-\!e^{ T}(t)\chi _2^{ T}\chi _2 e(t)\\&\le e^{ T}(t)\bar{\varGamma } e(t), \end{aligned}$$

where \( \bar{\varGamma }= \mathrm{diag}\{\bar{\varGamma }_1,\; \bar{\varGamma }_2,\; \bar{\varGamma }_3 \}\), \(\chi _1=[\beta _V^{-1} g_{V3}(x_r, \varPhi ) C_f \;\;-\beta _V B_0^{ T} P_2\;\;0]\), \(\chi _2=[\beta _{\gamma }^{-1} g_{\gamma 3}(x_r, \varPhi ) C_f \;\;0\;\;-\beta _{\gamma } B_0^{ T} P_3] \), in which

$$\begin{aligned} \begin{array}{ll} \bar{\varGamma }_1=&{}\bar{A}_f^{ T} \bar{P}_1+ \bar{P}_1\bar{A}_f+ [\beta _V^{-2} g_{V3}^2(x_r, \varPhi )\\ &{}+ \beta _{\gamma }^{-2} g_{\gamma 3}^2(x_r, \varPhi )]C_f^{ T} C_f, \\ \bar{\varGamma }_2=&{}\bar{A}_V^{ T} P_2+ P_2\bar{A}_V+P_2 B_0 \beta _V^{2} B_0^{ T} P_2,\\ \bar{\varGamma }_3=&{}\bar{A}_{\gamma }^{ T} P_3+ P_3\bar{A}_{\gamma }+P_3 B_0 \beta _{\gamma }^{2} B_0^{ T} P_3. \end{array} \end{aligned}$$

According to Assumption 1, we know that

$$\begin{aligned} \beta _V^{-2} g_{V3}^2(x_r, \varPhi )\!+\!\beta _{\gamma }^{-2} g_{\gamma 3}^2(x_r, \varPhi ) \!\le \! \beta _{V}^{-2}\sigma _{V3}^2\!+\!\beta _{\gamma }^{-2}\sigma _{\gamma 3}^2. \end{aligned}$$

Therefore

$$\begin{aligned} \dot{W}(t)\le e^{ T}(t)\varGamma e(t), \end{aligned}$$
(28)

where the matrix \(\varGamma = \mathrm{diag}\{\bar{A}_f^{ T} \bar{P}_1+ \bar{P}_1\bar{A}_f+ (\beta _{V}^{-2}\sigma _{V3}^2+\beta _{\gamma }^{-2}\sigma _{\gamma 3}^2)C_f^{ T}C_f,\;\bar{A}_V^{ T} P_2+ P_2\bar{A}_V+P_2 B_0 \beta _V^{2} B_0^{ T} P_2, \bar{A}_{\gamma }^{ T} P_3+ P_3\bar{A}_{\gamma }+P_3 B_0 \beta _{\gamma }^{2} B_0^{ T} P_3 \}\). Pre- and post-multiply the inequality (28) by \(\mathrm{diag}\{P_1,\;I,\;I\}\) and substituting matrices \(\varGamma \), \(\bar{A}_f\), \(\bar{A}_V\) and \(\bar{A}_{\gamma }\) into the inequality, we obtain

$$\begin{aligned} \dot{W}(t)&\le e_f^{ T}(t)(P_1 A_f^{ T}+A_f P_1-P_1 C_f^{ T} L_c^{ T} - L_c C_f P_1\\&+ (\beta _{V}^{-2}\sigma _{V3}^2+\beta _{\gamma }^{-2}\sigma _{\gamma 3}^2)C_f^{ T}C_f) e_f(t) + \bar{e}_{V}^{ T}(t)\\&(A_V^{ T} P_2+ P_2A_V + K_V^{ T} B_0^{ T} P_2 + P_2 B_0 K_V\\&+ P_2 B_0 \beta _V^{2} B_0^{ T} P_2) \bar{e}_{V}(t)\!+\! \bar{e}_{\gamma }^{ T}(t)(A_{\gamma }^{ T} P_3 \!+\! P_3 A_{\gamma }\\&+ K_{\gamma }^{ T} B_0^{ T} P_3 \!+\! P_3 B_0 K_{\gamma } \!+\! P_3 B_0 \beta _{\gamma }^{2} B_0^{ T} P_3) \bar{e}_{\gamma }(t). \end{aligned}$$

If the matrices \(P_1,\, P_2\) and \(P_3\) are the solutions of the Riccati equations (25)–(27) and the gains of controller (21) are chosen as \(L_c^{ T}=R_f^{-1}C_fP_1\), \(K_V=-R_V^{-1}B_0^{ T}P_2\) and \(K_{\gamma }=-R_{\gamma }^{-1}B_0^{ T}P_3\), we obtain

$$\begin{aligned} \dot{W}(t)&\le e_f^{ T}\mathbf (t)(-P_1C_f^{ T}R_f^{-1}C_f P_1-Q_f+ (\beta _{V}^{-2}\sigma _{V3}^2\\&\!\!+\beta _{\gamma }^{-2}\sigma _{\gamma 3}^2)C_f^{ T}C_f) e_f(t)\!+\! \bar{e}_{V}^{ T}(t)(\!-\!P_2B_0R_V^{-1}\\&\!\!B_0^{ T} P_2 -Q_V+ P_2 B_0 \beta _V^{2} B_0^{ T} P_2) \bar{e}_{V}(t)+ \bar{e}_{\gamma }^{ T}(t)\\&\!\!(\!-P_3B_0R_{\gamma }^{-1}B_0^{ T} P_3\!-\!Q_{\gamma } \!\!+\! P_3 B_0 \beta _{\gamma }^{2} B_0^{ T} P_3) \bar{e}_{\gamma }(t). \end{aligned}$$

When the weights \(Q_f,\, Q_V,\, Q_{\gamma }\, R_f,\, R_V\) and \(R_{\gamma }\) satisfy the following conditions:

$$\begin{aligned}&Q_f \ge (\beta _V^{-2}\sigma _{V3}^2+\beta _{\gamma }^{-2}\sigma _{\gamma 3}^2)C_f^{ T}C_f,\;\; Q_V \ge 0,\\&Q_{\gamma } \ge 0, 0 < R_f,\;\;0 < R_V \le \beta _V^{-2},\;\; 0 < R_{\gamma } \le \beta _{\gamma }^{-2}, \end{aligned}$$

the \(\dot{W}(t)<0\), that is, the composite closed-loop system (24) is asymptotically stable. \(\square \)

Remark 9

The gains (\(L_c,\, K_V\) and \(K_{\gamma }\)) of controller (21) can be adjusted to improve the estimation performance and control performance by choosing appropriate values of weights (\(Q_f,\, Q_V,\, Q_{\gamma },\, R_f,\, R_V\) and \(R_{\gamma }\)) and positive parameters (\(\beta _{V}\) and \(\beta _{\gamma }\)).

4 Simulations

In this section, the effectiveness of the proposed method is demonstrated by numerical simulations. To compare with the controller (21), the controller (22) will be also applied for the flight control of FAHVs. For simplification, the controllers (21) and (22) are called as controller 1 and controller 2, respectively. The simulation results are obtained for the full nonlinear model of FAHVs with the elevator-to-lift coupling and rigid-flexible coupling. The values of rigid-body state and flexible state at trimmed cruise condition, the vehicle parameters (that is, \(I_{yy},\, m,\, \rho ,\, S,\, \bar{c}\)) and the coefficients that appear in the forces and moment (that is, \(C_{M}^{(\cdot )}\), \(C_{T}^{(\cdot )}\), \(C_{N_i}^{(\cdot )}\), \(C_{L}^{(\cdot )}\), \(C_{D}^{(\cdot )}\)) are given in [29].

In the simulation, the FAHV needs to complete the same maneuver tasks as [29]. The selected reference trajectory begin at \(q=2,000 \) lb ft\(^{-2}\) and \(h=85,000\)  ft. The aircraft climbs at a steady \(50\, \mathrm{ft}\,\mathrm{s}^{-1}\), while maintaining a constant dynamic pressure. Once the FAHV reaches Mach 10 at time \(220\) s, the Mach number is held constant at climb rate increase to \(139\, \mathrm{ft}\,\mathrm{s}^{-1}\) until leveling off at \(115{,}000\)  ft. With above maneuver tasks, the desired signals \(V_d(t)\) and \(\gamma _d(t)\) can be obtained shown in Fig. 1a, b by using the relationships \(q_d(t)=0.5 \rho _0 V_d^2(t) \exp (-(h_d(t)-h_0)/h_s)\), \(M_a=V_d(t)/(20\sqrt{T_e})\) and \(\dot{h}_d(t)=V_d(t)\mathrm{sin}(\gamma _d(t))\), where \(q_d(t)\) and \(h_d(t)\) are the desired signals of dynamic pressure and altitude, \(M_a\) and \(T_e\) denote Mach number and air temperature, respectively.

Fig. 1
figure 1

Case study 1: a velocity \(V(t)\) and desired trajectory \(V_d(t)\), b flight path angle \(\gamma (t)\) and desired trajectory \(\gamma _d(t)\), c velocity tracking error \(e_V(t)\), d flight-path angle tracking error \(e_{\gamma }(t)\)

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In order to fully analyze the performances of controller 1 and controller 2, two case studies will be considered. In the first case, controller 1 and controller 2 choose the same gains \(K_V\) and \(K_{\gamma }\) to analyze the effects caused by the disturbance \(d_c\). In the second case, the different gain \(K_{\gamma }\) is chosen to compare the performance of the two controllers.

The first case sets weights \(Q_f= 4.225\times 10^5C_f^{ T} C_f\), \(Q_V=\mathrm{diag} \{ 1,\;10,\;1,\;1 \}\), \(Q_{\gamma }=I_{4\times 4}\), \(R_f=200\), \(R_V=1\), \(R_{\gamma }=0.1\), and parameters \(\beta _{V}=1\), \(\beta _{\gamma }=1\). Then we have \(K_V=-[3.4849 \;\;5.5722 \;\;4.5983\;\;1]\), \(K_{\gamma }=-[5.4399 \;\;9.7962 \;\;8.4827\;\;3.1623]\) and \(L_c=[11.6038\;\; -7.2228\;\; -1.4551\;\; -1.6087]^{ T}\) according to Theorem 1.

The simulation results of rigid-body state, control input, flexible state and disturbance \(d_c\) under controller 1 and controller 2 with the same gains \(K_V\) and \(K_{\gamma }\) are shown in Figs. 1, 2, 3, and 4. It can be seen from Fig. 1 that the tracking performances of controller 1 are similar to the ones of controller 2 due to choosing the same gains \(K_V\) and \(K_{\gamma }\). However, there are oscillations in the simulation results of tracking errors, angle of attack, pitch rate, control input and flexible state using controller 2 shown in Figs. 1, 2, and 3. It can be observed from Fig. 4a that the simulation results of disturbance \(d_c\) under controller 2 also has high-frequency oscillation. Thus, the oscillations that appear in rigid-body variables, control input and flexible state are all caused by the disturbance \(d_c\). Furthermore, the oscillations will be intensified, if the smaller tracking error of flight-path angle is needed for controller 2. Meanwhile, it also can be seen from Figs. 1, 2, 3, and 4 that the controller 1 with the same tracking performance as controller 2 can avoid these oscillations by using the coupling-observer-based compensator which is able to reject the disturbance \(d_c\).

Fig. 2
figure 2

Case study 1: a angle of attack \(\alpha (t)\), b pitch rate \(Q(t)\), c elevator deflection \(\delta _e(t)\), d fuel equivalence ratio \(\varPhi (t)\)

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Fig. 3
figure 3

Case study 1: a flexible mode \(\eta _1(t)\), b flexible mode \(\eta _2(t)\), c \(\dot{\eta }_1(t)\), d \(\dot{\eta }_2(t)\)

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Fig. 4
figure 4

Case study 1: a disturbance \(d_c\), b estimation value and error of \(d_c\)

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The vehicle may be out of control due to the high-frequency oscillation of elevator deflection \(\delta _e\). The durable oscillations appeared in flexible state may cause vehicle damage. Therefore, the oscillations produced by using controller 2 in the first case are undesired in practice. Thus the bigger weight \(R_{\gamma }\) of controller 2 (that is, \(R_{\gamma }=1\)) is chosen in the second case study to reduce the oscillations, while the smaller weight \(R_{\gamma }\) (that is, \(R_{\gamma }=0.08\)) is chosen to further improve the tracking performance of controller 1. So the new gain \(K_{\gamma }\) can be worked out as \(K_{\gamma }=-[5.8258 \;\;10.7200 \;\;9.3979\;\;3.5355]\) for controller 1 and \(K_{\gamma }=-[3.0777 \;\;4.2361 \;\;3.0777\;\;1]\) for controller 2. Other gains of controller 1 and controller 2 are same as the first case.

The simulation results of rigid-body state, control input, flexible state and disturbance \(d_c\) under controller 1 and controller 2 with different gain \(K_{\gamma }\) are shown in Figs. 5, 6, 7, and 8. It can be seen from Figs. 5, 6, 7, and 8 that the oscillations of tracking errors, angle of attack, pitch rate, control input, flexible state and disturbance are reduced in the simulation results using controller 2, but the tracking performance of controller 2 is degraded (that is, the tracking error of fight-path angle using controller 2 is bigger than the one using controller 1). Moreover, even if the smaller weight \(R_{\gamma }\) is chosen, there are no durable or high-frequency oscillations in the simulation results under controller 1. In addition, it can be seen from Figs. 4b and 8b that the disturbance \(d_c\) using controller 1 can be estimated very well by the coupling observer in the two case studies.

Fig. 5
figure 5

Case study 2: a velocity \(V(t)\) and desired trajectory \(V_d(t)\), b flight path angle \(\gamma (t)\) and desired trajectory \(\gamma _d(t)\), c velocity tracking error \(e_V(t)\), d flight-path angle tracking error \(e_{\gamma }(t)\)

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Fig. 6
figure 6

Case study 2: a angle of attack \(\alpha (t)\), b pitch rate \(Q(t)\), c elevator deflection \(\delta _e(t)\), d fuel equivalence ratio \(\varPhi (t)\)

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Fig. 7
figure 7

Case study 2: a flexible mode \(\eta _1(t)\), b flexible mode \(\eta _2(t)\), c \(\dot{\eta }_1(t)\), d \(\dot{\eta }_2(t)\)

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Fig. 8
figure 8

Case study 2: a disturbance \(d_c\), b estimation value and error of \(d_c\)

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In a word, it may cause degradation of control performance or oscillations of control input and flexible state using controller 2 without considering the rigid-flexible couplings, while the controller 1 proposed in this paper can avoid these problems. That is, the tracking performances can be improved by using controller 1 compared with controller 2.

5 Conclusion

In this paper, the flexible effects produced by rigid-flexible coupling dynamics between flexible and rigid-body variables of FAHVs are formulated as a kind of disturbance. First, a control-design model with rigid-flexible coupling dynamics is given. And then, the nonlinear composite controller which combines a coupling-observer-based feedforward compensator and a dynamic-inversion-based feedback controller is presented, where the feedforward compensator is constructed to reject the flexible effects on pitch rate and the feedback controller is designed to guarantee velocity and flight-path angle track the desired signals. Simulation results show that it may cause degradation of control performance or high-frequency oscillations of control input and flexible state using the traditional nonlinear controller without considering the rigid-flexible couplings, while the coupling-observer-based controller can avoid these problems. In other words, the tracking performances can be improved by using the proposed composite controller compared with the previous dynamic-inversion-based nonlinear controller. One of the future research topics is to extend the result developed in this paper to FAHVs with uncertainties.