Recursive terminal sliding mode control for hypersonic flight vehicle with sliding mode disturbance observer

Abstract

A recursive terminal sliding mode controller (RTSMC) based on sliding mode disturbance observer (SMDOB) is proposed for the longitudinal dynamics of a generic hypersonic flight vehicles (HFVs) in the presence of parametric uncertainties, measurement noises and external disturbances. First, a sliding mode tracking controller is presented by introducing recursive terminal sliding mode manifolds, in which each manifold will reach zero subsequently in finite time as well as the usual singularity problem will not occur. The RTSMC embraces advantages of both nonsingular terminal sliding mode control and high-order sliding mode control. Next, for the sake of enhancing the robustness of controller for uncertainties, a SMDOB is proposed to estimate and compensate the disturbances. Then, a composite controller that is composed of RTSMC and SMDOB is designed, and its stability is analyzed utilizing Lyapunov function method. Finally, numerical simulation is conducted for cruise flight condition of HFV. Simulation results show the expected control performance.

1 Introduction

As a cost-effective and reliable space aircraft, HFVs have attracted more and more interests from civil and military due to its high-speed and prompt global responses. HFV is a class of aircrafts that flying in the near space which is 30–70 km from the ground. The atmospheric environment of near space is unstable, and the flight condition is complex. In addition, there are strong interactions in HFV between the engine dynamics and airframe. It is sensitive to any tiny uncertainty change due to the peculiar aerodynamic structure of HFV. Hence, it is a significant challenging task for modeling and control of HFV. The research of HFV started about 1960s, so fruitful results had been achieved up to date. In 2004, the flight tests of X43-A HFV were held successfully, which motivate the further research for modern control and application of HFV dynamic system. In the last decade, a variety of modern methods and techniques have been applied to the area of flight control design of HFV, and considerable research achievements have sprung up.

Aeropropulsive and aeroelastic of HFV were analyzed in detail in reference [1], which laid the foundation for establishing HFV model. Bolender and Doman [2, 3] built a nonlinear model for the longitudinal dynamics of air-breathing HFV, which captured the complex interactions between the aerodynamics, propulsion system and structural dynamics, and its motion equations were derived using Lagrange’s equations. In terms of nonlinear model, robust flight control systems were synthesized [4] that stability and performance robustness was estimated by Monte Carlo evaluation. And stochastic robust nonlinear dynamic inversion (NDI) control law was proposed for longitudinal motion of HFV containing 28 uncertain parameters [5]. For the sake of utilizing linear control method, feedback linearization model and control-oriented linearized model were derived in [6, 7]. Xu [8] designed an adaptive sliding mode controller for longitudinal dynamics of input–output linearization model of HFV, where the input–output linearization model is proved to be an effective and extensively used model hereafter. Obaid proposed mini–max linear quadratic regulator (LQR) optimal controller and mini–max optimal linear quadratic Gaussian (LQG) controller by the virtue of mini–max linear quadratic theory for feedback linearization model of HFV, in which the optimal control minimizes the maximum value of the quadratic cost function and gives an optimal solution for the selected cost function [9–11]. The optimal controller can achieve good performance. However, the construction of Riccati equation parameter matrix in LQR is a difficulty work. Because of the peculiarity of many uncertainties for HFV model, sliding mode control is frequently adopted, which is not only robust to uncertainty but also easily designed.

The basic of sliding mode control is the designing of sliding mode surface. Different sliding surfaces can lead to various convergence performances. Integral sliding surface can eliminate the static error when the system is stable. Terminal sliding surface results in finite-time convergence for system states. Global sliding surface has global robustness due to its initial states on the sliding manifold. In reference [12], an integral-type sliding manifold is designed and then a nonfragile $H_\infty$ controller based on that is investigated for a flexible air-breathing HFV. Taking into account adaptive sliding control, Burak putted forward a varying sliding surface with varying slopes and offsets, whose parameters were updated by solving state-dependent Riccati equations [13]. A robust sliding manifold-based adaptive sliding mode controller is presented in [14] for air-breathing HFV, where the system error dynamics can be driven onto the predefined sliding surface in finite time. Nevertheless, the system states do not always convergence to the equilibrium point in finite time once the system states are driven onto the sliding manifold. But the terminal sliding mode control can realize it.

For terminal sliding mode control, system states arrive at the sliding manifold from their initial conditions in finite time; then, the states slide along the sliding manifold to the equilibrium point in finite time. Therefore, the terminal sliding mode control is a kind of finite-time convergence control. So far, there are numerous research results about terminal sliding mode control, such as fast terminal sliding mode control design [15], continuous terminal sliding mode control [16], et al. Terminal sliding mode control usually has singularity problem which limits its application. For this, Yong Feng proposed nonsingular terminal sliding mode control and applied it to the rigid manipulators successfully [17]. Combining the above methods, Li [18] proposed an adaptive nonsingular fast terminal sliding mode control scheme for electromechanical actuator.

It is worth noting that HFV is a high-order nonlinear system. Hence, the low-order sliding mode controller for HFV has chattering phenomenon and slow converge speed. Albeit references [19, 20] have presented high-order sliding mode control for HFV, their first sliding surfaces remain linear form, namely asymptotic convergence. Therefore, above both controllers do not converge in finite time. Aiming at the finite-time convergence problem, a recursive high-order terminal sliding mode control is proposed for HFV in this paper. The proposed scheme has robustness to uncertainty parameters.

In addition to the parameters uncertainty, there are external disturbance during the HFV flight. A small number of references consider external disturbance [19]. In this paper, a novel sliding mode disturbance observer (SMDOB) is presented, which can estimate the disturbance and further compensate it in the controller. As a result, the composite control in this paper is RTSMC+SMDOB, where RTSMC achieves command tracking control meanwhile eliminate chattering, and SMDOB estimates external disturbance and compensates it. The major contributions of this paper are as follows

• (I) A novel recursive terminal sliding mode controller is proposed for hypersonic flight vehicles, which is the first time that the method is applied in hypersonic aircraft. The hypersonic flight vehicle is highly nonlinear, and the recursive sliding mode controller can meet its control requirements. The recursive terminal sliding mode controller is nonlinear controller, which is the superior than linear controller for nonlinear systems.

• (II) The novel sliding mode control has merits of both nonsingular terminal sliding mode control (NTSMC) and high-order sliding mode control (HOSMC). The system states under this controller are finite-time convergent and have no singularity problem, and at the same time, the chattering problem existed in common sliding mode control is attenuated.

• (III) A SMDOB is introduced for the external disturbance of HFV. The SMDOB is simple and not relay on the plant and has shorter estimated time than other methods. This disturbance observer cannot affect the controller designing and application, which is relatively independent in the whole system. The disturbance observer is as compensation part for the controller, by which the external disturbance is eliminated.

• (IV) The engineering application of NTSMC and SMDOB is considered. In the process of designing, the methods are considered for implementation by hardware, not only in mathematical deduction. So these methods have a certain value of engineering application.

The remainder of the paper is organized as follows. Section 2 formulates the HFV model and control problem. Section 3 shows the design process of RTSMC and SMDOB in detail. The composite controller is presented in Sect. 4, and the stability analysis for controller is given at the same time. Numerical simulations are conducted in Sect. 5. Finally, Sect. 6 gives the conclusions.

2 Model and problem formulation

2.1 Hypersonic flight vehicle model

The model of longitudinal dynamics for a generic HFV is developed by NASA Langley Research Center, which consists of differential equations described by velocity, altitude, angle of attack, flight path angle and pitch rate [8, 21]. The motion equations are as follows:

$$\begin{aligned} \left\{ {\begin{array}{l} \dot{V}=\frac{T\cos \alpha -D}{m}-\frac{\mu \sin \gamma }{r^{2}} \\ \dot{h}=V\sin \gamma \\ \dot{\gamma }=\frac{L+T\sin \alpha }{mV}-\frac{\left( {\mu -V^{2}r} \right) \cos \gamma }{Vr^{2}} \\ \dot{\alpha }=q-\dot{\gamma } \\ \dot{q}=\frac{M_{yy} }{I_{yy} } \\ \end{array}} \right. \end{aligned}$$

(1)

where $V,h,\gamma ,\alpha ,q$ are velocity, altitude, flight path angle, angle of attack and pitch rate of HFV, respectively; $m$ is mass and $I_{yy}$ is moment of inertia of the aircraft; $L,D,T,M_{yy}$ are lift, drag, thrust, pitching moment, respectively, that acting on the aircraft and $r$ is radial distance from Earth’s center. The expression details of $L,D,T,M_{yy},r$ are, respectively, as follows

$$\begin{aligned} L= & {} \frac{1}{2}\rho V^{2}\text {SC}_L \end{aligned}$$

(2)

$$\begin{aligned} D= & {} \frac{1}{2}\rho V^{2}\textit{SC}_D \end{aligned}$$

(3)

$$\begin{aligned} T= & {} \frac{1}{2}\rho V^{2}\textit{SC}_T \end{aligned}$$

(4)

$$\begin{aligned} M_{yy}= & {} \frac{1}{2}\rho V^{2}S\bar{{c}}\left[ {C_M (\alpha )\,+\,C_M (\alpha ,\delta _e )\,+\,C_M (\alpha ,q)} \right] \nonumber \\ \end{aligned}$$

(5)

$$\begin{aligned} r= & {} h+R_E \end{aligned}$$

(6)

where $\rho ,S,\bar{{c}},R_E$ denote density of air, reference area, mean aerodynamic chord and radius of the Earth, respectively; $\delta _e$ is the elevator deflection angle; $C_L ,C_D ,C_T ,C_M (\alpha ),C_M (\alpha ,\delta _e ),C_M (\alpha ,q)$ are relevant aerodynamic coefficient parameters, respectively, as

$$\begin{aligned}&C_L =0.6203\alpha \end{aligned}$$

(7)

$$\begin{aligned}&C_D =0.6450\alpha ^{2}+0.0043378\alpha +0.003772 \end{aligned}$$

(8)

$$\begin{aligned}&C_T =\left\{ {{\begin{array}{ll} 0.02576\beta ,&{}\quad \beta <1 \\ 0.0224+0.00336\beta ,&{}\quad \beta >1 \\ \end{array} }} \right. \end{aligned}$$

(9)

$$\begin{aligned}&C_M (\alpha )=-0.035\alpha ^{2}+0.036617(1+\Delta C_{M\alpha } )\alpha \nonumber \\&\qquad \quad \qquad +\,5.3261\times 10^{-6} \end{aligned}$$

(10)

$$\begin{aligned}&C_M (\alpha ,q)=\frac{\bar{{c}}q}{2V}\left( {-6.796\alpha ^{2}+0.3015\alpha -0.2289} \right) \nonumber \\ \end{aligned}$$

(11)

$$\begin{aligned}&C_M (\alpha ,\delta _e )=c_e \left( {\delta _e +d_2 (t)-\alpha } \right) \end{aligned}$$

(12)

where $d_2 (t)$ means the external disturbance reflected on the elevator.

A second-order system is used to represent the engine dynamics as

$$\begin{aligned} \ddot{\beta }=-2\zeta \omega _n \dot{\beta }-\omega _n^2 \beta +\omega _n^2 \left( {\beta _c +d_1 (t)} \right) \end{aligned}$$

(13)

where $\beta$ and $\beta _c$ are throttle setting and throttle setting command, respectively; $\xi$ is damping ratio and $\omega _n$ is natural frequency; $d_1 (t)$ is the external disturbance on behalf of torques and generalized elastic forces.

In order to certify the robustness and immunity of the proposed controller, some certain parameter uncertainties are added in the HFV model that

$$\begin{aligned} \left\{ {\begin{array}{l} m=m_0 \left( {1+\Delta m} \right) \\ I_{yy} =I_0 \left( {1+\Delta I} \right) \\ S=S_0 \left( {1+\Delta S} \right) \\ \bar{{c}}=\bar{{c}}_0 \left( {1+\Delta \bar{{c}}} \right) \\ \rho =\rho _0 \left( {1+\Delta \rho } \right) \\ c_e =c_{e0} (1+\Delta c_e ) \\ C_{M\alpha } =C_{M\alpha 0} (1+\Delta C_{M\alpha } ) \\ \end{array}} \right. \end{aligned}$$

(14)

where $\left( {{\bullet }_0 } \right)$ represents the nominal value of the parameter $\left( {\bullet } \right)$ and $\left( {\Delta {\bullet }} \right)$ denotes the parameter uncertainties.

Assumption 1

The external disturbance $d_i (t)$ and its first time derivative are assumed to be upper bounded [21, 22], i.e., $\left| {d_i (t)} \right| \le g_i, \left| {\dot{d}_i (t)} \right| \le \bar{{g}}_i$, here $g_i$ and $\bar{{g}}_i$ are known positive constants for $i=1, 2$.

Assumption 2

The parameter uncertainties in Eq. (14) are assumed to be bounded. And they satisfy the following conditions:

$$\begin{aligned}&\left| {\Delta m} \right| \le \Delta m^{*}, \left| {\Delta I} \right| \le \Delta I^{*}, \left| {\Delta S} \right| \le \Delta S^{*}, \left| {\Delta \bar{{c}}} \right| \le \Delta \bar{{c}}^{*},\\&\left| {\Delta \rho } \right| \le \Delta \rho ^{*}, \left| {\Delta c_e } \right| \le \Delta c_e^*, \left| {\Delta C_{M\alpha } } \right| \le \Delta C_{M\alpha }^*\end{aligned}$$

where $\Delta m^{*},\Delta I^{*},\Delta S^{*},\Delta \bar{{c}}^{*},\Delta \rho ^{*},\Delta c_e^*,\Delta C_{M\alpha }^*$ are all positive real constants.

Remark 1

According to model (1), velocity $V$ and altitude $h$ are regarded as output variables, while the input variables are selected as engine throttle setting command $\beta _c$ and elevator deflection $\delta _e$. The control objective is designing a suitable controller such that the velocity $V$ and altitude $h$ track the command $V_d$ and $h_d$ in finite time in the presence of uncertainty, noises and disturbance, respectively.

2.2 Input–output linearization

The model of HFV in Eq. (1) is highly nonlinear and strong coupling. For the sake of designing control law expediently, input–output linearization method needs to be applied to linearize the model. In terms of Remark 1, the linearization target is that the linearized results can apparently express the relationship between output variables $\left( {V,h} \right)$ and input variables $\left( {\beta _c ,\delta _e } \right)$.

In line with the nonlinear system theory and taking advantage of tools for Lie derivative, it is concluded that input variables $\left( {\beta _c ,\delta _e } \right)$ can appear in the motion equations by differentiating $V$ three times and $h$ four times [8], respectively. So the relative degree of the HFV nonlinear system is $r=3+4=7$. Due to the order of system in Eq. (1) is five, a second-order engine dynamics system is introduced. Therefore, the order of whole system is seven, i.e., $n=5+2=7$. In terms of $r=n$, the closed-loop system has no zero dynamics [23] and the nonlinear longitudinal model can be linearized completely [5]. The linearization process is as follows

$$\begin{aligned}&\left\{ \begin{array}{l} \dot{V}=f_V (\mathbf{x}) \\ \ddot{V}={\varvec{\Omega }}_1 {\dot{\mathbf{x}}}/m \\ {\dddot{V}}={({\varvec{\Omega }}_1 {\ddot{\mathbf{x}}}+\dot{\mathbf{x}}^\mathrm{T}{\varvec{\Omega }}_2 \dot{\mathbf{x}})}/m \\ \end{array} \right. \end{aligned}$$

(15)

$$\begin{aligned}&\left\{ {\begin{array}{l} \ddot{h}=\dot{V}\sin \gamma +V\dot{\gamma }\cos \gamma \\ \dddot{h}=\ddot{V}\sin \gamma +2\dot{V}\dot{\gamma }\cos \gamma -V\dot{\gamma }^{2}\sin \gamma +V\ddot{\gamma }\cos \gamma \\ {h}^{(4)}={\dddot{V}}\sin \gamma \,+\,3\ddot{V}\dot{\gamma }\cos \gamma -3\dot{V}\dot{\gamma }^{2}\sin \gamma \,+\,3\dot{V}\ddot{\gamma }\cos \gamma \\ \quad \qquad -\,3V\dot{\gamma }\ddot{\gamma }\sin \gamma -V\dot{\gamma }^{3}\cos \gamma +V{\dddot{\gamma }}\cos \gamma \\ \end{array}} \right. \end{aligned}$$

(16)

$$\begin{aligned}&\dot{\gamma }=f_h (\mathbf{x}),\ddot{\gamma }=\varvec{\Pi }_1 \dot{\mathbf{x}},{\dddot{\gamma }}=\varvec{\Pi }_1 \ddot{\mathbf{x}}+\dot{\mathbf{x}}^\mathrm{T}\varvec{\Pi }_2 \dot{\mathbf{x}} \end{aligned}$$

(17)

where $\mathbf{x}=[{\begin{array}{lllll} V&{} \gamma &{} \alpha &{} \beta &{} h \\ \end{array} }]^\mathrm{T}$ is the state vector, ${\varvec{\varOmega }}_{1} ,\varvec{\varOmega }_{2}, \varvec{\varPi }_{1}, \varvec{\varPi }_{2}$ are the system equations’ first-order and second-order partial derivatives regarding to state variables, respectively, whose detailed expressions are shown in “Appendix 1.”

The expressions of $\ddot{\alpha }$ and $\ddot{\beta }$ can be written two parts as

$$\begin{aligned} \ddot{\alpha }= & {} \ddot{\alpha }_0 +\frac{\rho V^{2}S\bar{{c}}c_e }{2I_{yy} }\left( {\delta _e +d_2 (t)} \right) \nonumber \\ \ddot{\beta }= & {} \ddot{\beta }_0 +\omega _n^2 \left( {\beta _c +d_1 (t)} \right) \end{aligned}$$

(18)

where

$$\begin{aligned} \ddot{\alpha }_0= & {} \frac{1}{2I_{yy} }\rho V^{2}S\bar{{c}}\left[ {C_M (\alpha )+C_M (\alpha ,q)-c_e \alpha )} \right] -\ddot{\gamma } \\ \ddot{\beta }_0= & {} -2\xi \omega _n \dot{\beta }-\omega _n^2 \beta \end{aligned}$$

Given ${\ddot{\mathbf{x}}}_0 =[{\begin{array}{lllll} {\ddot{V}}&{} {\ddot{\gamma }}&{} {\ddot{\alpha }_0 }&{} {\ddot{\beta }_0 }&{} {\ddot{h}} \\ \end{array} }]^\mathrm{T}$, then

$$\begin{aligned} \left\{ {\begin{array}{l} \dddot{V}=F_V +b_{11} \left( {\beta _c +d_1 (t)} \right) +b_{12} \left( {\delta _e +d_2 (t)} \right) \\ h^{(4)}=F_h +b_{21} \left( {\beta _c +d_1 (t)} \right) +b_{22} \left( {\delta _e +d_2 (t)} \right) \\ \end{array}} \right. \end{aligned}$$

(19)

where

$$\begin{aligned} F_V= & {} \frac{\left( {{\varvec{\varOmega }}_{1} \varvec{\ddot{x}}_0 +\varvec{\dot{x}}^\mathrm{T}{\varvec{\varOmega }}_{2} \varvec{\dot{x}}} \right) }{m}\\ F_h= & {} 3\ddot{V}\dot{\gamma }\cos \gamma -3\dot{V}\dot{\gamma }^{2}\sin \gamma +3\dot{V}\ddot{\gamma }\cos \gamma \\&-\,3V\dot{\gamma }\ddot{\gamma }\sin \gamma -V\dot{\gamma }^{3}\cos \gamma +F_V \sin \gamma \\&+\,V\cos \gamma \left( \varvec{\varPi }_{1} {\varvec{\ddot{x}}}_0 +{\varvec{\dot{x}}}^\mathrm{T}\varvec{\varPi }_{2} \varvec{\dot{x}} \right) \\ b_{11}= & {} \left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \cos \alpha \\ b_{12}= & {} -\left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left( {T\sin \alpha +D_\alpha } \right) \\ b_{21}= & {} \left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \sin (\alpha +\gamma )\\ b_{22}= & {} \left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left[ T\cos (\alpha +\gamma )\right. \\&\left. +\,L_\alpha \cos \gamma -D_\alpha \sin \gamma \right] \\ c_\beta= & {} \frac{\partial C_T }{\partial \beta }, D_\alpha =\frac{\partial D}{\partial \alpha }, L_\alpha =\frac{\partial L}{\partial \alpha } \end{aligned}$$

Then, Eq. (19) can be rewritten as

$$\begin{aligned} \left[ {{\begin{array}{l} {\dddot{V}} \\ {h^{\left( 4 \right) }} \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {F_V } \\ {F_h } \\ \end{array} }} \right] +\mathbf{B}\left[ {{\begin{array}{l} {\beta _c } \\ {\delta _e } \\ \end{array} }} \right] +\mathbf{B}\left[ {{\begin{array}{l} {d_1 (t)} \\ {d_2 (t)} \\ \end{array} }} \right] \end{aligned}$$

(20)

where $\mathbf{B}=\left[ {{\begin{array}{ll} {b_{11} }&{} {b_{12} } \\ {b_{21} }&{} {b_{22} } \\ \end{array} }} \right]$.

Assumption 3

The matrix $\mathbf{B}$ is assumed to be invertible.

Remark 2

For the input–output combination, the matrix $\mathbf{B}$ is nonsingular for the entire flight envelope except on a vertical flight path [5]. Hence, the Assumption 3 is reasonable to be assumed. When the flight condition is near the singularity, the measures taken are shown in “Appendix 2.”

Remark 3

The nonlinear system in Eq. (1) can be expressed as affine nonlinear system of form

$$\begin{aligned} \begin{array}{l} \varvec{\dot{x}}(t)={{\mathbf {f}}}({{\mathbf {x}}})+\sum \limits _{k=1}^m {{{\mathbf {g}}}_k ({{\mathbf {x}}})u_k } \\ y_i =h_i (\mathbf{x}),\quad i=1,2,\ldots ,m \\ \end{array} \end{aligned}$$

where ${{\mathbf {f}},{\mathbf {g}},{\mathbf {h}}}$ are smooth functions in ${{\mathbf {R}}}^{n}$. In terms of above contents, relative degree of the system, $r=n$, the order of the system, there is no zero dynamics. Therefore, the control input $u=\left[ {{\begin{array}{ll} {\beta _c }&{} {\delta _e } \\ \end{array} }} \right] ^\mathrm{T}$ in linearized system (20) is affine control input.

Remark 4

Given $x_1 =v,x_2 =\dot{v},x_3 =\ddot{v}$, the velocity subsystem is changed to the form of state space without considering the external disturbance as

$$\begin{aligned} \begin{array}{l} \left[ {{\begin{array}{l} {\dot{x}_1 } \\ {\dot{x}_2 } \\ {\dot{x}_3 } \\ \end{array} }} \right] =\left[ {{\begin{array}{lll} 0&{} 1&{} 0 \\ 0&{} 0&{} 1 \\ {a_{11} }&{} {a_{12} }&{} {a_{13} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {x_1 } \\ {x_2 } \\ {x_3 } \\ \end{array} }} \right] +\left[ {{\begin{array}{ll} 0&{} 0 \\ 0&{} 0 \\ {b_{11} }&{} {b_{12} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\beta _c } \\ {\delta _e } \\ \end{array} }} \right] \\ y=\left[ {{\begin{array}{lll} 1&{} 0&{} 0 \\ \end{array} }} \right] \left[ {{\begin{array}{l} {x_1 } \\ {x_2 } \\ {x_3 } \\ \end{array} }} \right] +\left[ {{\begin{array}{ll} 0&{} 0 \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\beta _c } \\ {\delta _e } \\ \end{array} }} \right] \\ \end{array} \end{aligned}$$

where $a_{1i}$ is related to $F_V$ and $a_{1i} \ne 0, i=1,2,3$. The system matrixes are, respectively,

$$\begin{aligned} {{\mathbf {A}}}= & {} \left[ {{\begin{array}{lll} 0&{} 1&{} 0 \\ 0&{} 0&{} 1 \\ {a_{11} }&{} {a_{12} }&{} {a_{13} } \\ \end{array} }} \right] , \quad {{\mathbf {B}}}=\left[ {{\begin{array}{ll} 0&{} 0 \\ 0&{} 0 \\ {b_{11} }&{} {b_{12} } \\ \end{array} }} \right] ,\\ {{\mathbf {C}}}= & {} \left[ {{\begin{array}{lll} 1&{} 0&{} 0 \\ \end{array} }} \right] , \quad {{\mathbf {D}}}=\left[ {{\begin{array}{ll} 0&{} 0 \\ \end{array} }} \right] . \end{aligned}$$

According to the output controllability criterion [24], the output $V$ is controllable.

Similarly, the output variable $h$ is also controllable.

In one word, the input–output linearized model is controllable, and the system can be stable if the controller is appropriate.

3 Controller and observer design

In this section, a recursive terminal sliding mode controller is firstly designed to track the reference command, and then, a sliding mode disturbance observer is presented for compensating the external disturbance.

3.1 Tracking controller design

Terminal sliding mode control usually has the singularity problem. An approximately terminal sliding mode control can be introduced according to the sliding mode twisting algorithm, which can avoid the singularity problem.

Lemma 1

[25, 26] Define a derivative–integral terminal sliding function for a second-order system as

$$\begin{aligned} s=\dot{e}+\beta e+\alpha e_I \end{aligned}$$

(21)

where $\alpha ,\beta >0, \dot{e}_I (t)+T_s e_I (t)=sign\left( {e(t)} \right)$ with the system tracking error $e(t)$.

When the initial condition is chosen as $e_I (0)=-{\left( {\dot{e}(0)+\beta e(0)} \right) }/\alpha$, then the system tracking error $e$ will converge to zero in finite time, i.e., the convergence time $t_\mathrm{reach}$ has an upper bound.

Proof

Substituting the initial condition into Eq. (21) yields $s=0$. In other words, the system tracking error trajectory is on the sliding mode manifold that

$$\begin{aligned} s=\dot{e}+\beta e+\alpha e_I =0 \end{aligned}$$

(22)

Define

$$\begin{aligned} z=-\alpha e_I \end{aligned}$$

(23)

Then

$$\begin{aligned} \dot{e}=z-\beta e \end{aligned}$$

(24)

According to [25], an arbitrary absolutely continuous function $V(e,z)$ can be found satisfying

$$\begin{aligned} \dot{e}\cdot \frac{\partial V}{\partial e}-\alpha \cdot \frac{\partial V}{\partial z}=k\cdot V^{1/2} \end{aligned}$$

(25)

Choose $s(e,z)$ and $m(e,z)$, respectively, as

$$\begin{aligned}&s(e,z)=2\alpha \left| e \right| -\beta ez\,+\,z^{2}>0 \quad \hbox {for} \, e^{2}+z^{2}\ne 0 \end{aligned}$$

(26)

$$\begin{aligned}&m(e,z)=\frac{1}{\sqrt{g-1}}arc\tan \frac{\beta ge-2z}{2\sqrt{g-1}z}\qquad \end{aligned}$$

(27)

where $g={8\alpha }/{\beta ^{2}}$.

Then, the Lyapunov function candidate $V(e,z)$ can be written as

$$\begin{aligned} V(e,z)=\frac{k^{2}}{4}\left( {-e_I +k_0 e^{m(e,z)}\sqrt{s(e,z)}} \right) ^{2} \end{aligned}$$

(28)

where $k_0 >-z\cdot \left( {\alpha e^{m(e,z)}\sqrt{s(e,z)}} \right) ^{-1}$.

In accordance with Theorem 1 in [25], the reaching time has an upper bound, i.e.,

$$\begin{aligned} t_\mathrm{reach} \le 2k_{\min }^{-1} \max \left\{ {\sqrt{V\left( {e(0),z(0)} \right) }} \right\} \end{aligned}$$

(29)

where

$$\begin{aligned} k_{\min }= & {} \frac{\beta }{\sqrt{8}}\mathop {\min }\limits _{g\in \left\{ {g^{-},g^{+}} \right\} } \left\{ {g\bar{{k}}-\sqrt{g}e^{{\arctan \left( {\frac{-1}{\sqrt{g-1}}} \right) }/{\sqrt{g-1}}}} \right\} ,\\&\bar{{k}}>0. \end{aligned}$$

Therefore, the system tracking error $e$ will converge to zero in finite time $t_\mathrm{reach}$. The proof is completed. $\square$

Remark 5

The above Lemma is generated in terms of sliding mode twisting algorithm. In other words, the tracking error $e$ and its integration state $e_I$ converge to zero along sliding mode twisting trajectory in finite time. The merit is that fewer nonlinear operations are employed.

Theorem 1

Consider the HFV model (1) and its linearization form (20) based on Assumption 3 and Remark 3, the system output variables $V$ and $h$ can track the reference command $V_d$ and $h_d$ in finite time, respectively, if the affine controller is chosen as

$$\begin{aligned} u=\mathbf{B}^{-1}\left[ {{\begin{array}{l} {-\Phi _1 -F_V +{\dddot{V}}_d -k_1 \left| {s_1 } \right| ^{\gamma _1 }\hbox {sgn}(s_1 )} \\ {-\Phi _2 -F_h +h_d^{(4)} -k_2 \left| {s_2 } \right| ^{\gamma _2 }\hbox {sgn}(s_2 )} \\ \end{array} }} \right] \end{aligned}$$

(30)

where $k_1 >0,k_2 >0, 0<\gamma _1 <1,0<\gamma _2 <1$. The recursive sliding mode manifolds, respectively, as

$$\begin{aligned} e_{11}= & {} \dot{e}_{01} +\beta _{11} e_{01} +\alpha _{11} e_{I11} \end{aligned}$$

(31a)

$$\begin{aligned} e_{21}= & {} \dot{e}_{11} +\beta _{21} e_{11} +\alpha _{21} e_{I21} \end{aligned}$$

(31b)

$$\begin{aligned} s_1= & {} e_{21} \end{aligned}$$

(31c)

$$\begin{aligned} e_{02}= & {} \dot{e}_2 +\lambda _2 e_2 \end{aligned}$$

(32a)

$$\begin{aligned} e_{12}= & {} \dot{e}_{02} +\beta _{12} e_{02} +\alpha _{12} e_{I12} \end{aligned}$$

(32b)

$$\begin{aligned} e_{22}= & {} \dot{e}_{12} +\beta _{22} e_{12} +\alpha _{22} e_{I22} \end{aligned}$$

(32c)

$$\begin{aligned} s_2= & {} e_{22} \end{aligned}$$

(32d)

where $e_{01} =V-V_d, e_2 =h-h_d$ are tracking errors, $\lambda _2$ is strictly positive constant, $\dot{e}_{I1i} +T_s e_{I1i} =sat\left( {{e_{0i} }/{\mu _{1i} }} \right) , \dot{e}_{I2i} +T_s e_{I2i} =sat\left( {{e_{1i} }/{\mu _{2i} }} \right) , T_s >0, \mu _{1i} ,\mu _{2i} >0$ for $i=1,2$.

Proof

Taking time derivative for Eq. (31) yields

$$\begin{aligned} \dot{e}_{11}= & {} \dddot{e}_{01} +\beta _{11} \dot{e}_{01} +\alpha _{11} \dot{e}_{I11} \nonumber \\ \ddot{e}_{11}= & {} {e}_{01} +\beta _{11} \ddot{e}_{01} +\alpha _{11} \ddot{e}_{I11} \nonumber \\ \dot{e}_{21}= & {} \ddot{e}_{11} +\beta _{21} \dot{e}_{11} +\alpha _{21} \dot{e}_{I21} \end{aligned}$$

(33)

$$\begin{aligned} \dot{s}_1= & {} \dot{e}_{21} ={\dddot{e}}_{01} \,+\,\beta _{11} \ddot{e}_{01} \,+\,\alpha _{11} \ddot{e}_{I11} \,+\,\beta _{21} \dot{e}_{11} \,+\,\alpha _{21} \dot{e}_{I21}\nonumber \\ \end{aligned}$$

(34)

where

$$\begin{aligned} \ddot{e}_{I11}= & {} M_{s1} -T_s \dot{e}_{I11}\\ M_{s1}= & {} \left\{ {{\begin{array}{ll} {{\begin{array}{ll} {{\dot{e}_{01} }/{\mu _{11} ,}}&{}\quad {\left| {{e_{01} }/{\mu _{11} }} \right| \le 1} \\ {0,}&{}\quad {otherwise} \\ \end{array} }} \\ \end{array} }} \right. . \end{aligned}$$

Due to ${\dddot{e}}_{01} ={\dddot{V}}-{\dddot{V}}_d$, Eq. (34) is changed as

$$\begin{aligned} \dot{s}_1 ={\dddot{V}}-{\dddot{V}}_d +\Phi _1 \end{aligned}$$

(35)

where

$$\begin{aligned} \Phi _1 =\beta _{11} \ddot{e}_{01} +\alpha _{11} \ddot{e}_{I11} +\beta _{21} \dot{e}_{11} +\alpha _{21} \dot{e}_{I21} \end{aligned}$$

Similarly, taking first time derivative for Eq. (32) yields

$$\begin{aligned} \dot{e}_{02}= & {} \ddot{e}_2 +\lambda _2 \dot{e}_2 \nonumber \\ \ddot{e}_{02}= & {} {\dddot{e}}_2 +\lambda _2 \ddot{e}_2 \nonumber \\ {\dddot{e}}_{02}= & {} e_2^{(4)} +\lambda _2 {\dddot{e}}_2 \end{aligned}$$

(36)

$$\begin{aligned} \dot{e}_{12}= & {} \ddot{e}_{02} +\beta _{12} \dot{e}_{02} +\alpha _{12} \dot{e}_{I12} \nonumber \\ \ddot{e}_{12}= & {} e_2^{(4)} +\lambda _2 {\dddot{e}}_2 +\beta _{12} \ddot{e}_{02} +\alpha _{12} \ddot{e}_{I12} \nonumber \\ \dot{e}_{22}= & {} \ddot{e}_{12} +\beta _{22} \dot{e}_{12} +\alpha _{22} \dot{e}_{I22} \end{aligned}$$

(37)

where

$$\begin{aligned} \ddot{e}_{I12}= & {} M_{s2} -T_s \dot{e}_{I12}\nonumber \\ M_{s2}= & {} \left\{ {{\begin{array}{l} {{\begin{array}{ll} {{\dot{e}_{02} }/{\mu _{12} ,}}&{}\quad {\left| {{e_{02} }/{\mu _{12} }}\right| \le 1} \\ {0,}&{}\quad \hbox {otherwise} \\ \end{array} }} \end{array} }} \right. .\nonumber \\ \dot{s}_2= & {} \dot{e}_{22} =e_2^{(4)} \,+\,\lambda _2 {\dddot{e}}_2 \,+\,\beta _{12} \left( {{\dddot{e}}_2 \,+\,\lambda _2 \ddot{e}_2 } \right) \nonumber \\&+\,\alpha _{12}\ddot{e}_{I12}+\,\beta _{22} \dot{e}_{12} +\alpha _{22} \dot{e}_{I22} \end{aligned}$$

(38)

Substituting $e_2^{(4)} =h^{(4)}-h_d^{(4)}$ into Eq. (38), yields

$$\begin{aligned} \dot{s}_2 =h^{(4)}-h_d^{(4)} +\Phi _2 \end{aligned}$$

(39)

where

$$\begin{aligned} \Phi _2= & {} \lambda _2 {\dddot{e}}_2 +\beta _{12} \left( {{\dddot{e}}_2 +\lambda _2 \ddot{e}_2 } \right) +\alpha _{12} \ddot{e}_{I12} \\&+\,\beta _{22} \dot{e}_{12} +\alpha _{22} \dot{e}_{I22} \end{aligned}$$

Then, Eq. (35) and Eq. (39) are expressed together as

$$\begin{aligned} \left[ {{\begin{array}{l} {\dot{s}_1 } \\ {\dot{s}_2 } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {\dddot{V}} \\ {h^{(4)}} \\ \end{array} }} \right] -\left[ {{\begin{array}{l} {{\dddot{V}}_d } \\ {h_d^{(4)} } \\ \end{array} }} \right] +\left[ {{\begin{array}{l} {\Phi _1 } \\ {\Phi _2 } \\ \end{array} }} \right] \end{aligned}$$

(40)

Substituting Eq. (20) in the absence of external disturbance into Eq. (40) acquires

$$\begin{aligned} \left[ {{\begin{array}{l} {\dot{s}_1 } \\ {\dot{s}_2 } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {F_V } \\ {F_h } \\ \end{array} }} \right] -\left[ {{\begin{array}{l} {{\dddot{V}}_d } \\ {h_d^{(4)} } \\ \end{array} }} \right] +\left[ {{\begin{array}{l} {\Phi _1 } \\ {\Phi _2 } \\ \end{array} }} \right] +\mathbf{B}\cdot u \end{aligned}$$

(41)

Choose a Lyapunov function candidate as

$$\begin{aligned} L_y =\frac{1}{2}s^\mathrm{T}s \end{aligned}$$

(42)

where $s=\left[ {{\begin{array}{ll} {s_1 }&{} {s_2 } \\ \end{array} }} \right] ^\mathrm{T}$.

The derivative of $L_y$ satisfies

$$\begin{aligned} \dot{L}_y =s^\mathrm{T}\dot{s}=s^\mathrm{T}\left( {\left[ {{\begin{array}{l} {F_V } \\ {F_h } \\ \end{array} }} \right] -\left[ {{\begin{array}{l} {{\dddot{V}}_d } \\ {h_d^{(4)} } \\ \end{array} }} \right] +\left[ {{\begin{array}{l} {\Phi _1 } \\ {\Phi _2 } \\ \end{array} }} \right] +\mathbf{B}\cdot u} \right) \nonumber \\ \end{aligned}$$

(43)

Substituting Eq. (30) into Eq. (43) acquires

$$\begin{aligned} \dot{L}_y= & {} s^\mathrm{T}\left[ {{\begin{array}{l} {-k_1 \left| {s_1 } \right| ^{\gamma _1 }\hbox {sgn}(s_1 )} \\ {-k_2 \left| {s_2 } \right| ^{\gamma _2 }\hbox {sgn}(s_2 )} \\ \end{array} }} \right] \nonumber \\= & {} -k_1 \left| {s_1 } \right| ^{\gamma _1 +1}-k_2 \left| {s_2 } \right| ^{\gamma _2 +1}\le 0 \end{aligned}$$

(44)

Hence, the system states arrive at sliding manifolds $s_1$ and $s_2$ in finite time, i.e., the reaching time is $t_{s_i } ={\left| {s_i (0)} \right| ^{1-\gamma _i }}/{(k_i }(1-\gamma _i))$ for $i=1,2$. When system states slides along the sliding manifolds, then it has $s_i =e_{2i} =0$ for $i=1,2$. In terms of Eqs. (31b) and (32c), the results are acquired

$$\begin{aligned} e_{2i} =0=\dot{e}_{1i} +\beta _{2i} e_{1i} +\alpha _{2i} e_{I2i},\quad i=1,2 \end{aligned}$$

(45)

According to Lemma 1, $e_{1i}$ will converge to zero in finite time, namely the convergence time $t_\mathrm{reach}^{e_{1i}}$ has an upper bound. Then, substituting $e_{1i} =0$ into Eqs. (31a) and (32b) yields

$$\begin{aligned} \dot{e}_{0i} +\beta _{1i} e_{0i} +\alpha _{1i} e_{I1i} =0,\quad i=1,2 \end{aligned}$$

(46)

Similarly, $e_{0i}$ will also converge to zero in finite time, and the convergence time $t_\mathrm{reach}^{e_{0i} }$ has an upper bound. Then consider Eq. (32a) in accordance with $e_{02} =0$ that

$$\begin{aligned} \dot{e}_2 +\lambda _2 e_2 =0 \end{aligned}$$

(47)

It is worth noting that Eq. (47) is asymptotic convergence. Define an arbitrary small neighborhood for $e_2$, i.e., $Z_{e_2 } \in \left[ {-\delta ,+\delta } \right] ,\delta >0$. Once the altitude tracking error converging to the neighborhood, it is regarded that the altitude $h$ had tracked the command $h_d$. Then, the convergence time is

$$\begin{aligned} t_{e_2 } =\frac{1}{\lambda _2 }\left( {\ln \left( {e_2 (0)} \right) -\ln \delta } \right) \end{aligned}$$

(48)

Therefore, the convergence time of the velocity tracking error $e_{01}$ is

$$\begin{aligned} t_1 =t_{s_1 } +t_\mathrm{reach}^{e_{11} } +t_\mathrm{reach}^{e_{01} } \end{aligned}$$

(49)

For the altitude subsystem, the arriving time to the neighborhood $Z_{e_2}$ is

$$\begin{aligned} t_2 =t_{s_2 } +t_\mathrm{reach}^{e_{12} } +t_\mathrm{reach}^{e_{02} } +t_{e_2 } \end{aligned}$$

(50)

In brief, the system output variables $V$ and $h$ can track the reference command $V_d$ and $h_d$, respectively, in finite time. This completes the proof. $\square$

Remark 6

For the velocity subsystem, there are three layers for sliding mode manifolds and four layers for the altitude subsystem. In each subsystem, these sliding manifolds are recursive. The last-layer sliding manifold is first arrived, and at this time, the finite-time arriving condition for the second layer is met. After a period of time, the second-layer sliding manifold is also arrived. In this order, each sliding manifold is arrived successively. Finally, the system tracking error can converge to zero in limited time.

Remark 7

Due to no singular terms appearing on Theorem 1 and all derivative terms are well defined, there is no singularity problem that usually arising in terminal sliding mode control.

3.2 Sliding mode disturbance observer design

The above proposed controller has robust for parameter uncertainties. However, for external disturbances, an additional disturbance observer is needed to compensate its influence to controller. In this section, a sliding mode disturbance observer is presented for the bounded external disturbance. Before designing, the equation in the presence of external disturbance is displayed. Substituting Eq. (20) into Eq. (40) acquires

(51)

where

Lemma 2

[27] Assume an perturbed nonlinear differential equation

$$\begin{aligned} \dot{x}+\omega _1 \left| x \right| ^{1/2}\hbox {sgn}(x)+\omega _2 P(x)=\xi (t) \end{aligned}$$

(52)

where $\dot{P}(x)+T_s P(x)=\hbox {sgn}(x), \left| {\dot{\xi }(t)} \right| \le D, D$ is an positive constant.

If $\omega _1 =1.5\sqrt{D}, \omega _2 =1.1D$, then the solution $x(t)$ of above equation and its first time derivative $\dot{x}(t)$ converge to zero in finite time, i.e.,

$$\begin{aligned} T_r \le \frac{7.6x(0)}{\omega _2 -D} \end{aligned}$$

(53)

where $T_r$ is the convergence time and $x(0)$ is the initial value of $x(t)$ at $t=0$.

Theorem 2

Consider the linearization system (20) with bounded external disturbance $d_i (t)$, which satisfy Assumption 1. Then, the external disturbance $d_i (t)$ can be exactly estimated in finite time, namely

$$\begin{aligned} T_{ri} \le \frac{7.6\sigma _i (0)}{\omega _{2i} -\tilde{g}_i } \end{aligned}$$

(54)

If the auxiliary sliding mode control variables are chosen as

$$\begin{aligned} v_i =\omega _{1i} \left| {\sigma _i } \right| ^{1/2}\hbox {sgn}(\sigma _i )+\omega _{2i} P(\sigma _i ) \end{aligned}$$

(55)

where $\dot{P}(\sigma _i )+T_s P(\sigma _i )=\hbox {sgn}(\sigma _i ), \omega _{1i} =1.5\sqrt{\tilde{g}_i }, \omega _{2i} =1.1\tilde{g}_i$ for $i=1,2$.

Proof

Choose an auxiliary sliding mode variable as

$$\begin{aligned} \sigma _i= & {} s_i +z_i \end{aligned}$$

(56)

(57)

where $i=1,2$.

According to Eqs. (56) and (57), Eq. (51) can be rewritten as

(58)

The $\sigma _i$ dynamics is derived in terms of Eq. (58) as

(59)

Substituting Eq. (55) into Eq. (59), then the $\sigma _i$ dynamics is changed as

(60)

By means of transposing, Eq. (60) changes for

(61)

where .

In accordance with Lemma 2, $\sigma _i$ and $\dot{\sigma }_i$ will converge to zero in finite time, which satisfy $T_{ri} \le {7.6\sigma _i (0)}/{\left( {\omega _{2i} -\tilde{g}_i } \right) }$. It means that

$$\begin{aligned} \sigma _i =\dot{\sigma }_i =0\quad \forall t>T_{ri} \end{aligned}$$

(62)

Substituting Eq. (62) into Eq. (59), it is obtained that

(63)

Hence, the disturbance is estimated as

(64)

Therefore, the external disturbance can be exactly estimated in finite time. The theorem is proven completely. $\square$

Remark 8

The parameters $\omega _{1i}$ and $\omega _{2i}$ have a relationship with the disturbance boundary and its derivative boundary $\tilde{g}_i$. So the proposed SMDOB estimates only the bounded disturbance.

Remark 9

The auxiliary sliding mode control variable $v_i$ is continuous due to the discontinuous term $\hbox {sgn}\left( {\sigma _i } \right)$ is integrated as well as in the presence of the fractional power term $\left| {\sigma _i } \right| ^{1/2}$.

On the basis of Theorem 2, the terminal sliding mode controller with SMDOB is designed as

$$\begin{aligned} u=\left[ {{\begin{array}{l} {\beta _c } \\ {\delta _e } \\ \end{array} }} \right]= & {} \mathbf{B}^{-1}\left[ {{\begin{array}{l} {-\Phi _1 -F_V \,+\,{\dddot{V}}_d -k_1 \left| {s_1 } \right| ^{\gamma _1 }\hbox {sgn}(s_1 )} \\ {-\Phi _2 -F_h \,+\,h_d^{(4)} -k_2 \left| {s_2 } \right| ^{\gamma _2 }\hbox {sgn}(s_2 )} \\ \end{array} }} \right] \nonumber \\&-\,\left[ {{\begin{array}{l} {\hat{{d}}_1 (t)} \\ {\hat{{d}}_2 (t)} \\ \end{array} }} \right] \end{aligned}$$

(65)

Remark 10

The controller of Eq. (65) is improved with respect to the tracking controller of Eq. (30). The former has the rejection for external disturbance.

Remark 11

The chattering phenomenon usually in sliding mode control is significantly reduced, implying the features of recursive terminal sliding mode controller with proposed disturbance observer in Eq. (65). The reason is the sign function item is continuous.

4 Composite controller and stability analysis

In this section, the stability and robustness of the composite controller are analyzed. The composite controller is given in Eq. (65), which consists of RTSMC tracking controller and SMDOB.

Theorem 3

For the longitudinal dynamics of HFV model (1), the system is stable and robust for the reference command in the presence of parameter uncertainties and external disturbances, if the controller is chosen to Eq. (65).

Proof

Choosing the Lyapunov function candidate $L_a =\frac{1}{2}s^\mathrm{T}s$ and taking its time derivative along the sliding mode manifold dynamics in Eq. (51) yields

$$\begin{aligned} \dot{L}_a = s^\mathrm{T}\dot{s}= & {} s^\mathrm{T}\left( \left[ {{\begin{array}{l} {F_V -{\dddot{V}}_d +\Phi _1 } \\ {F_h -h_d^{(4)} +\Phi _2 } \\ \end{array} }} \right] \right. \nonumber \\&\left. +\,\mathbf{B}\cdot u+\mathbf{B}\cdot \left[ {{\begin{array}{l} {d_1 (t)} \\ {d_2 (t)} \\ \end{array} }} \right] \right) \end{aligned}$$

(66)

Substituting Eq. (65) into Eq. (66), the function $L_a$ dynamics is changed as

$$\begin{aligned} \dot{L}_a =s^\mathrm{T}\left( {\left[ {{\begin{array}{l} {-k_1 \left| {s_1 } \right| ^{\gamma _1 }\hbox {sgn}(s_1 )} \\ {-k_2 \left| {s_2 } \right| ^{\gamma _2 }\hbox {sgn}(s_2 )} \\ \end{array} }} \right] +\mathbf{B}\cdot \left[ {{\begin{array}{l} {\tilde{d}_1 (t)} \\ {\tilde{d}_2 (t)} \\ \end{array} }} \right] } \right) \end{aligned}$$

(67)

where $\tilde{d}_i (t)=d_i (t)-\hat{{d}}_i (t), i=1,2$ is the error between the actual and estimated disturbance.

In accordance with Theorem 2, the external disturbance can be exactly estimated in finite time. The disturbance error $\tilde{d}_i (t)$ will converge to zero in finite time, i.e., $\tilde{d}_i (t)\rightarrow 0,\forall t>T_{ri}$. So when $t>T_{ri}$, Eq. (67) is changed as

$$\begin{aligned} \dot{L}_a= & {} s^\mathrm{T}\left[ {{\begin{array}{l} {-k_1 \left| {s_1 } \right| ^{\gamma _1 }\hbox {sgn}(s_1 )} \\ {-k_2 \left| {s_2 } \right| ^{\gamma _2 }\hbox {sgn}(s_2 )} \\ \end{array} }} \right] \nonumber \\= & {} -k_1 \left| {s_1 } \right| ^{\gamma _1 +1}-k_2 \left| {s_2 } \right| ^{\gamma _2 +1}\le 0 \end{aligned}$$

(68)

Hence, the sliding mode manifolds $s_i\, (i=1,2)$ equal to zero in finite time, and then, the each layer sliding surface will converge to zero recursively by means of proof process of Theorem 1. Finally, the system can track the reference command. Therefore, the system is stable and robust for uncertainty and disturbance in finite time. This completes the proof. $\square$

Remark 12

In Theorem 3, the finite time is on the basis of $t_i (i=1,2)$ in Theorem 1, which also consists of the estimated time for disturbance, namely

$$\begin{aligned} t_{fi} =t_i +T_{ri},\quad i=1,2 \end{aligned}$$

(69)

where $t_{fi}$ is the overall time in Theorem 3.

Remark 13

Theorem 3 gives the stability proof from the theory point of view. In the practical engineering, the sampling time may affect the system stability. In order to verify the effect of the sampling time on the convergence of the composed controller (65), simulations with different sampling time are conducted. The results are displayed in Table 1.

Table 1 The effect of sampling time on the system

It is obviously seen from Table 1 that the sampling time affects the convergence of the system. The shorter the sampling time, the more stable the system. When the sampling time is larger than some value, the system becomes unstable. Therefore, in this paper, the sampling time is selected for 0.001 s.

The structure diagram of composite control scheme for HFV in this paper is shown in Fig. 1.

Fig. 1

Structure diagram of composite control scheme

5 Numerical simulations

The numerical simulation of composite controller that comprises RTSMC and SMDOB proposed in previous section is conducted in this section. The simulation is built in the trimmed cruise flight condition of HFV. In the cruise flight condition, the initial simulation parameters are altitude 110,000 ft, velocity 15,060 ft/s, Mach 15, angle of attack 0 rad and pitch rate 0 rad/s, respectively. The damping ratio and natural frequency of the engine dynamics model are taken 0.7 and 5 rad/s, respectively. In the simulation, the reference command for velocity subsystem is step signal $100\,\text {ft}/\text {s}$ and a step command of 500 ft for altitude subsystem. The nominal dynamic parameters [8] for HFV are presented in Table 2.

Table 2 Nominal dynamic parameters of HFV

The external disturbances in the simulation are selected as

$$\begin{aligned} d_1 (t)=0.1\sin \left( {0.5t} \right) + 0.15\cos (0.3t) + 0.2\sin (0.6t)\nonumber \\ \end{aligned}$$

(70)

$$\begin{aligned} d_2 (t)=0.2\sin \left( {0.5t} \right) + 0.1\sin (0.3t) + 0.2\cos (0.6t)\nonumber \\ \end{aligned}$$

(71)

The disturbances above are added to the system at 70 s in simulation.

The simulation results of nominal HFV model in the presence of parameter uncertainties controlled by RTSMC tracking controller are shown in Figs. 2 and 3. The results of velocity subsystem are displayed in Figs. 2 and 3 for altitude subsystem.

Fig. 2

The velocity subsystem response for nominal model. a Velocity, b altitude, c flight path angle, d elevator deflection angle, e throttle setting

Fig. 3

The altitude subsystem response for nominal model. a Velocity, b altitude, c flight path angle, d elevator deflection angle, e throttle setting

Figure 2a displays the velocity tracking performance. It is seen from the figure that the aircraft velocity tracks the reference command in about 25 s, and the tracking speed holds constant. Due to the constant speed, the system is influenced when the command is tracked, which can be seen in Fig. 2b–e. The influence of throttle setting $\beta _c$ is greater in Fig. 2e. But the effects vanish immediately, and the system becomes stable in no time. The other variables also converge to constant value after a limited time in Fig. 2b–e.

The altitude tracking performance is displayed in Fig. 3b. The tracking time to altitude reference command is about 45 s. The tracking speed is not constant in this subsystem. Because the fourth-layer sliding mode manifold is asymptotic convergence, whose speed is not constant. In about 32.5 s, there is a switch from the finite-time convergence sliding surface to the asymptotic convergence sliding surface. So the system is influenced at this time which is obviously seen in Fig. 3a, c–e. Likewise, the effects are small and vanish immediately.

From the Fig. 2d, e and Fig. 3d, e, there is no chattering in the actuators, which verify the stable and robust performances of proposed RTSMC in the presence of parameter uncertainties. The finite-time convergence performance for Theorem 1 is demonstrated in Figs. 2 and 3. The trajectories of sliding variables are shown in Fig. 4.

Fig. 4

Trajectories of sliding variables. a Variable $s_1$, b variable $s_2$, c variable $\dot{s}_1$, d variable $\dot{s}_2$, e variable $e_{11}$, f variable $e_{02}$

In Fig. 4a, b are the last-layer sliding manifolds, which are first arrived. Figure 4c, d are the time derivatives of variables $s_1$ and $s_2$, respectively. Fig. 4e, f are the first-layer sliding mode manifolds, which are related to the system tracking error.

For further demonstrating the validity of the proposed method RTSMC, this method and the traditional SMC in reference [8] which has linear sliding mode surfaces are compared for simulation. The simulation results are displayed in Fig. 5.

Fig. 5

Comparing results between RTSMC and SMC. a Velocity subsystem comparison, b altitude subsystem comparison

By the mean of comparison, the convergence time of RTSMC is smaller than that of SMC and the robust of RTSMC is stronger than SMC. The altitude variable in Fig. 5a and velocity variable in Fig. 5b of SMC have larger variation than that of RTSMC. In addition, the actuators for SMC are chattering during the simulation, while the RTSMC are not.

When the external disturbances in Eqs. (70) and (71) are considered, the simulation results controlled by RTSMC are shown in Figs. 6 and 7.

From Figs. 6 to 7, the system responses are impacted by external disturbances after 70 s. Comparing Fig. 2 and 6, Fig. 3 and 7, the stable and tracking performances of the system are not affected before adding into the external disturbances.

Fig. 6

The velocity subsystem response for disturbance. a Velocity, b altitude, c flight path angle, d elevator deflection angle, e throttle setting

Fig. 7

The altitude subsystem response for disturbance. a Velocity, b altitude, c flight path angle, d elevator deflection angle, e throttle setting

When the proposed SMDOB is implied to estimate the disturbance and compensate the controller, the system responses are shown in Figs. 8 and 9.

Fig. 8

The velocity subsystem response with SMDOB. a Velocity, b altitude, c flight path angle, d elevator deflection angle, e throttle setting

Fig. 9

The altitude subsystem response with SMDOB. a Velocity, b altitude, c flight path angle, d elevator deflection angle, e throttle setting

Comparing Figs. 6a–c and 8a–c, Fig. 7a–c and 9a–c, respectively, it is obviously seen that perfect performance is obtained for SMDOB. The disturbance in velocity $V$, altitude $h$, flight path angle $\gamma$ is eliminated by SMDOB, which is reflected in Figs. 8a–c and 9a–c. The external disturbance is estimated by SMDOB, and then, it is compensated in the controller, which is revealed in Figs. 8d, e and 9d, e. The estimation for external disturbance is shown in Fig. 10.

Fig. 10

The external disturbance and its estimation by SMDOB

The external disturbance is well estimated by SMDOB from Fig. 10. This implies that the method of SMDOB is valid and available.

In order to verify the influence of measurement noises on the composed controller (65), some random noises are added in the system. Then, the simulation results with random noises are shown in Fig. 11.

Fig. 11

Simulation results with measurement noises. a Velocity, b altitude

In Fig. 11, the system output variables are convergent in spite of having small deviation. So the measurement noises do not affect the convergence of the composed controller, which demonstrates the strong robustness of the proposed controller.

In terms of above simulation, the perfect tracking and disturbance rejection performance are obtained for the longitudinal dynamic system of HFV controlled by RTSMC with SMDOB. The proposed method is stable in finite time. With the method, the system is robust with respect to parameter uncertainties and random noises and the chattering is eliminated in actuator. The external disturbance is estimated in finite time through SMDOB, which has high-precision estimation.

6 Conclusions

The recursive terminal sliding mode controller with sliding mode disturbance observer for longitudinal dynamics of generic hypersonic flight vehicle is proposed in this paper. First, a recursive terminal sliding mode tracking controller is designed by means of recursive terminal sliding mode theory. This controller is finite-time convergence and robust for parameter uncertainties. Next, a sliding mode disturbance observer is designed for external disturbance, whose estimation process is also finite time. Then, the composite controller embracing above tracking controller and disturbance observer is suggested. Finally, the numerical simulation for trimmed cruise flight condition of hypersonic flight vehicle is conducted. The simulation results show that the proposed scheme satisfies the expected performance.

Appendices

Appendix 1

The detailed expressions of the vectors ${\varvec{\Omega }}_{1}, \varvec{\Pi }_{1}$ and matrices ${\varvec{\Omega }}_{2}, \varvec{\Pi }_{2}$ are as follows:

$$\begin{aligned}&{\varvec{\Omega }}_1 =\left[ {{\begin{array}{c} {\left( {\frac{\partial T}{\partial V}} \right) \cos \alpha -\frac{\partial D}{\partial V}} \\ {-\frac{m\mu \cos \gamma }{r^{2}}} \\ {-T\sin \alpha -\left( {\frac{\partial D}{\partial \alpha }} \right) } \\ {\left( {\frac{\partial T}{\partial \beta }} \right) \cos \alpha } \\ {\frac{2m\mu \sin \gamma }{r^{3}}} \\ \end{array} }} \right] ^\mathrm{T}\\&{\varvec{\Omega }}_2 =\left[ {{\begin{array}{ccccc} {\varvec{\omega }_{21} }&{} {\varvec{\omega }_{22} }&{} {\varvec{\omega }_{23} }&{} {\varvec{\omega }_{24} }&{} {\varvec{\omega }_{25} } \\ \end{array} }} \right] \end{aligned}$$

where

$$\begin{aligned}&\varvec{\omega }_{21} =\left[ {{\begin{array}{c} {\left( {\frac{\partial ^{2}T}{\partial V^{2}}} \right) \cos \alpha -\frac{\partial ^{2}D}{\partial V^{2}}} \\ 0 \\ {-\left( {\frac{\partial T}{\partial V}} \right) \sin \alpha -\frac{\partial ^{2}D}{\partial V\partial \alpha }} \\ {\left( {\frac{\partial ^{2}T}{\partial V\partial \beta }} \right) \cos \alpha } \\ 0 \\ \end{array} }} \right] , \varvec{\omega }_{22} =\left[ {{\begin{array}{c} 0 \\ {\frac{m\mu \sin \gamma }{r^{2}}} \\ 0 \\ 0 \\ {\frac{2m\mu \cos \gamma }{r^{3}}} \\ \end{array} }} \right] ,\\&\varvec{\omega }_{23} =\left[ {{\begin{array}{c} {-\left( {\frac{\partial T}{\partial V}} \right) \sin \alpha -\left( {\frac{\partial ^{2}D}{\partial V\partial \alpha }} \right) } \\ 0 \\ {-T\cos \alpha -\left( {\frac{\partial ^{2}D}{\partial \alpha ^{2}}} \right) } \\ {-\left( {\frac{\partial T}{\partial \beta }} \right) \sin \alpha } \\ 0 \\ \end{array} }} \right] \\&\varvec{\omega }_{24} =\left[ {{\begin{array}{c} {\frac{\partial ^{2}T}{\partial V\partial \beta }\cos \alpha } \\ 0 \\ {-\left( {\frac{\partial T}{\partial \beta }} \right) \sin \alpha } \\ 0 \\ 0 \\ \end{array} }} \right] , \varvec{\omega }_{25} =\left[ {{\begin{array}{c} 0 \\ {\frac{2m\mu \cos \gamma }{r^{3}}} \\ 0 \\ 0 \\ {-\frac{6m\mu \sin \gamma }{r^{4}}} \\ \end{array} }} \right] \\&\varvec{\Pi }_1 =\left[ {{\begin{array}{c} {\frac{{\partial L}/{\partial V+\left( {{\partial T}/{\partial V}} \right) \sin \alpha }}{mV}-\frac{L+T\sin \alpha }{mV^{2}}+\frac{\mu \cos \gamma }{V^{2}r^{2}}+\frac{\cos \gamma }{r}} \\ {\frac{\mu \sin \gamma }{Vr^{2}}-\frac{V\sin \gamma }{r}} \\ {\frac{{\partial L}/{\partial \alpha +T\cos \alpha }}{mV}} \\ {\frac{\left( {{\partial T}/{\partial \beta }} \right) \sin \alpha }{mV}} \\ {\frac{2\mu \cos \gamma }{Vr^{3}}-\frac{V\cos \gamma }{r^{2}}} \\ \end{array} }} \right] ^\mathrm{T}\\&\varvec{\Pi }_2 =\left[ {{\begin{array}{ccccc} {\varvec{\pi }_{21} }&{} {\varvec{\pi }_{22} }&{} {\varvec{\pi }_{23} }&{} {\varvec{\pi }_{24} }&{} {\varvec{\pi }_{25} } \\ \end{array} }} \right] \end{aligned}$$

$$\begin{aligned}&\varvec{\pi }_{21} =\left[ {{\begin{array}{c} {\frac{{\partial ^{2}L}/{\partial V^{2}+\left( {{\partial ^{2}T}/{\partial V^{2}}} \right) \sin \alpha }}{mV}-\frac{2\left[ {{\partial L}/{\partial V+\left( {{\partial T}/{\partial V}} \right) \sin \alpha }} \right] }{mV^{2}}} \\ +\frac{2\left( {L+T\sin \alpha } \right) }{mV^{3}}-\frac{2\mu \cos \gamma }{V^{3}r^{2}}{-\frac{\mu \sin \gamma }{V^{2}r^{2}}-\frac{\sin \gamma }{r}} \\ {\frac{\left( {{\partial ^{2}L}/{\partial \alpha \partial V}} \right) +\left( {{\partial T}/{\partial V}} \right) \cos \alpha }{mV}-\frac{{\partial L}/{\partial \alpha +T\cos \alpha }}{mV^{2}}} \\ {\frac{\left( {{\partial ^{2}T}/{\partial \beta \partial V}} \right) \sin \alpha }{mV}-\frac{\left( {{\partial T}/{\partial \beta }} \right) \sin \alpha }{mV^{2}}} \\ {-\frac{2\mu \cos \gamma }{V^{2}r^{3}}-\frac{\cos \gamma }{r^{2}}} \\ \end{array} }} \right] \\&\varvec{\pi }_{22} =\left[ {{\begin{array}{c} {-\frac{\mu \sin \gamma }{V^{2}r^{2}}-\frac{\sin \gamma }{r}} \\ {\frac{\mu \cos \gamma }{Vr^{2}}-\frac{V\cos \gamma }{r}} \\ 0 \\ 0 \\ {-\frac{2\mu \sin \gamma }{Vr^{3}}+\frac{V\sin \gamma }{r^{2}}} \\ \end{array} }} \right] ,\\&\varvec{\pi }_{23} =\left[ {{\begin{array}{c} {\frac{\left( {{\partial ^{2}L}/{\partial V\partial \alpha }} \right) +\left( {{\partial T}/{\partial V}} \right) \cos \alpha }{mV}-\frac{{\partial L}/{\partial \alpha +T\cos \alpha }}{mV^{2}}} \\ 0 \\ {\frac{{\partial ^{2}L}/{\partial \alpha ^{2}-T\sin \alpha }}{mV}} \\ {\frac{\left( {{\partial T}/{\partial \beta }} \right) \cos \alpha }{mV}} \\ 0 \\ \end{array} }} \right] \\&\varvec{\pi }_{24} =\left[ {{\begin{array}{c} {\frac{\left( {{\partial ^{2}T}/{\partial V\partial \beta }} \right) \sin \alpha }{mV}-\frac{\left( {{\partial T}/{\partial \beta }} \right) \sin \alpha }{mV^{2}}} \\ 0 \\ {\frac{\left( {{\partial T}/{\partial \beta }} \right) \cos \alpha }{mV}} \\ 0 \\ 0 \\ \end{array} }} \right] ,\\&\varvec{\pi }_{25} =\left[ {{\begin{array}{c} {-\frac{2\mu \cos \gamma }{V^{2}r^{3}}-\frac{\cos \gamma }{r^{2}}} \\ {-\frac{2\mu \sin \gamma }{Vr^{3}}+\frac{V\sin \gamma }{r^{2}}} \\ 0 \\ 0 \\ {-\frac{6\mu \cos \gamma }{Vr^{4}}+\frac{2V\cos \gamma }{r^{3}}} \\ \end{array} }} \right] \end{aligned}$$

Appendix 2

Without loss of generality, the angle of attack and flight path angle are assumed near this singularity to be

$$\begin{aligned} \alpha =\varepsilon ,\gamma =\frac{\pi }{2} \end{aligned}$$

(72)

where $\varepsilon$ is a small positive number.

Then,

$$\begin{aligned} b_{11}= & {} \left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \cos \alpha =\left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \cos \varepsilon \nonumber \\\approx & {} \frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m} \end{aligned}$$

(73)

$$\begin{aligned} b_{12}= & {} -\left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left( {T\sin \alpha +D_\alpha } \right) \nonumber \\\approx & {} -\left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left( {T\varepsilon +\sigma } \right) \end{aligned}$$

(74)

where $\sigma =\frac{\partial \left( {\frac{1}{2}\rho V^{2}SC_D } \right) }{\partial \alpha }$ and which is a small real number.

$$\begin{aligned} b_{21}= & {} \left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \sin (\alpha +\gamma )\nonumber \\= & {} \left( {\frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m}} \right) \sin (\varepsilon \,+\,\frac{\pi }{2})\approx \frac{\rho V^{2}Sc_\beta \omega _n^2 }{2m} \end{aligned}$$

(75)

$$\begin{aligned} b_{22}= & {} \left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left[ T\cos (\alpha +\gamma )\right. \nonumber \\&\left. +\,L_\alpha \cos \gamma -D_\alpha \sin \gamma \right] \nonumber \\\approx & {} \left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \left( {0+0-\sigma } \right) \nonumber \\= & {} -\sigma \left( {\frac{c_e \rho V^{2}S\bar{{c}}}{2mI_{yy} }} \right) \end{aligned}$$

(76)

In the above case, $b_{11}$ is equal to $b_{21}$, and the matrix $\mathbf{B}$ will be singular if $b_{12} =b_{22} =0$. Therefore, we give $b_{12}$ and $b_{22}$ some restrictions near the singularity, namely

$$\begin{aligned} b_{12} =\left\{ {{\begin{array}{l} {\delta _1^*,\left| {b_{12} } \right| \le \delta _1^*} \\ {b_{12} ,\left| {b_{12} } \right| >\delta _1^*} \\ \end{array} }} \right. \end{aligned}$$

(77)

$$\begin{aligned} b_{22} =\left\{ {{\begin{array}{l} {\delta _2^*,\left| {b_{22} } \right| \le \delta _2^*} \\ {b_{22} ,\left| {b_{22} } \right| >\delta _2^*} \\ \end{array} }} \right. \end{aligned}$$

(78)

moreover, $\delta _1^*\ne \delta _2^*, b_{12} \ne b_{22}$.

In that way, the matrix

$$\begin{aligned} \mathbf{B}=\left[ {{\begin{array}{ll} {b_{11} }&{}\quad {b_{12} } \\ {b_{21} }&{}\quad {b_{22} } \\ \end{array} }} \right] \ne \mathbf{0} \end{aligned}$$

(79)

And the controller in the paper can be used.