1 Introduction

The design of a control system for an airbreathing hypersonic vehicle (AHV) is a challenging task due to the inherently nonlinear dynamics, such as strong couplings between the engine and flight dynamics [1, 2], variation of atmospheric conditions as well as physical and aerodynamic parameters [2, 3], nonminimum phase behavior [4, 5], etc.

In the past decades, the study of AHV mainly concentrates on altitude, velocity, and attitude control. The attitude control problem is discussed in [6–10]. In this paper, the velocity and altitude control problem is considered, which will be presented as follows.

Up to date, many meaningful results have been reported for the longitudinal model of an AHV. Based on the model obtained by approximate linearization in given flight conditions, some control methods are available in the literature, for example, linear stochastic robust control [11], guaranteed cost control [12], and multiobject fault-tolerant control [13]. Although these methods can improve the performance of system from different aspects, thus designed controllers may cause unsatisfactory performance when operating states are far from selected operating points. To further improve the control performance of system, many nonlinear control strategies based on a nonlinear model have been proposed for the longitudinal mode in the last decades, including adaptive control [14–16], robust control [17, 18], H ∞ control [19], sliding mode control [3, 20], fuzzy-logic control [21, 22], neural-network control [23–25], and so on.

It is well known that sliding mode control is an efficient method to deal with nonlinear perturbations, external disturbances and parametric uncertainties. Thus, such method can make the closed-loop system has a good robust and disturbance rejection performance. Generally, in most of the sliding mode control schemes for nonlinear systems, a linear sliding mode surface is employed to design a controller [26, 27]. However, such linear sliding mode only guarantees system states converge to an equilibrium point in infinite time.

In order to enhance the convergence speed of a sliding mode control system, a solution is to employ nonlinear sliding mode manifolds. One of such manifolds is the terminal sliding mode (TSM). TSM control can guarantee system state convergence to an equilibrium point in finite time, which is different from conventional sliding mode control. Besides an advantage of finite time convergence, TSM control also has a better disturbance rejection performance. Hence, TSM has been widely applied to kinds of fields, for instance, spacecraft control systems [28], robotic control systems [29], missile control systems [30, 31], etc. However, the TSM controller may cause the closed-loop system to be singular. To avoid a singularity problem, a switching control technique is employed in [32]. This control scheme can drive the trajectory to a specified open region in which the terminal sliding mode control is not singular. However, it does not fundamentally solve the singular problem. To this end, a nonsingularity TSM is designed in [29], but this result is only used in a second-order system. For a kind of higher order systems, a kind of finite time integral sliding mode is presented in [33].

The above mention results face an unavoidable application problem-chattering. To alleviate the chattering phenomenon, one of the solutions is to employ a saturation function instead of sign function in the control input [3]. However, to do this, the disturbance rejection performance is sacrificed to some extent. Another efficient method for alleviating the chattering problem is to employ a disturbance observer to estimate the disturbances. The disturbance estimation is used for compensation. The disturbance observer technique is first proposed in [34], and up to now, disturbance observer based control (DOBC) schemes for linear and nonlinear systems have been studied and applied in many control fields, e.g., robotic systems [35], grinding systems [36], hard disk drive systems [37, 38], nonlinear MAGnetic LEViation (MAGLEV) suspension systems [39] and general systems [40–42], and the references therein.

In this paper, the tracking control problem for the longitudinal model of an AHV is considered in the presence of external disturbances. By introducing finite time integral sliding mode manifolds, a novel finite time sliding mode controller is designed for the longitudinal model of an AHV, which guarantees that the closed-loop system has good disturbance rejection and tracking performance. However, such a signum function is used in the controller; it will lead to chattering phenomenon in a closed-loop system. In addition, the switching gain of the sliding mode controller should be larger than the bound of the disturbances. If there is no precise estimation on the bound of disturbances, the switching gain will be conservatively selected to be high enough, which worsens the chattering phenomenon. Then, to reduce the chattering, an efficient disturbance estimation technique, nonlinear disturbance observer (NDOB), is employed to estimate the disturbances. After disturbance compensation based on NDOB, the switching gain only needs to be larger than the bound of the disturbance compensation error, which is usually much smaller than that of the disturbances. Hence, the chattering will be reduced while the disturbance rejection performance of the closed-loop system can be maintained. Thus, an improved sliding mode control scheme is proposed by combination of finite time integral sliding mode control method and NDOB technique. Finally, simulation results are given to confirm the effectiveness of the proposed approach.

The rest of the paper is organized as follows. In the next section, model description is given. The input-output linearization model of the longitudinal dynamics of the hypersonic vehicle is presented in Sect. 3. The main results are shown in Sect. 4, where a tracking controller is developed by combining the finite time integral sliding mode control method and the disturbance observer technique. In Sect. 5, simulation results are discussed. Conclusions are presented in Sect. 6.

2 Model and problem formulation

The longitudinal model of an AHV is described as [3]

(1)
(2)
(3)
(4)
(5)

where

\(\bar{q}\) is the dynamic pressure and satisfies \(\bar{q}=\frac{1}{2}\rho V^{2}\). V, γ, h, α, and q are velocity, flight path angle, altitude, angle of attack, pitch rate of an AHV, respectively. T, D, L, M yy , δ E , and β mean thrust, drag, lift, pitching moment, the elevator deflection angle, the throttle setting, respectively. d 2(t) represents external disturbances. m, ρ, I yy , S, μ, R e denote mass of hypersonic vehicle, density of air, moment of inertia, reference area, gravitational constant, radius of the earth, respectively. \(\bar{c}\) and c e are constants.

The engine dynamics are modeled by a second-order system [3]

(6)

where β c denotes the demand of the control input, and d 1(t) means external disturbance torques and generalized elastic forces. Moreover, the outputs are the velocity V and the altitude h.

Assumption 1

[43, 44]

The external disturbances d i (t) and its first time derivative are assumed to be bounded by known constants, i.e., |d i (t)|≤c i , \(|\dot{d}_{i}(t)|\leq \bar{c}_{i}\), where \(c_{i},\bar{c}_{i}\) are known constants, i=1,2.

Control object

In this paper, we aim at designing a controller such that the velocity V and the altitude h track the reference signals V d and h d in finite time in the presence of external disturbances, respectively.

3 Input–output linearization

The longitudinal model of an AHV described by (1)–(6) can be rewritten as

(7)

where f(x), g k (x) are smooth functions in R n.

Using the Lie derivative notation, the derivative of y i in system (7) can be described as

(8)

where the Lie derivatives are defined as

and \(L_{g_{k}}(L_{f}^{j-1}(h_{i}))\) satisfies the following conditions:

(9)

Then (r 1,…,r m ) is the relative degree of the nonlinear system. If r=r 1+⋯+r m =n, the nonlinear system can be completely transformed to a linear system. If r<n, the nonlinear system can only partially linearized. In this case, the stability of the system (7) relies on the stability of zero dynamics.

Employing the aforesaid technique to the longitudinal model of an AHV, we can calculate the differential of the system outputs V and h until the control inputs come in the final functions. Based on this goal, the output dynamics V and h can be obtained by differentiating three times and four times as below. So, r=4+3=7=n, the longitudinal model can be completely feedback linearized as the following form:

(10)
(11)

In (11),

(12)

where x T=[V γ α β h], f 1(x) and f 2(x) are the expression of (1) and (2), respectively, \(\varpi_{1}=\frac{\partial f_{1}(x)}{\partial x}\), \(\varpi_{2}=\frac{\partial \varpi_{1}(x)}{\partial x}\), \(\varphi_{1}=\frac{\partial f_{2}(x)}{\partial x}\), \(\varphi_{2}=\frac{\partial \varphi_{1}(x)}{\partial x}\). The detailed expressions for ϖ 1, ϖ 2, φ 1, φ 2 are listed in the Appendix. The expression of second derivatives for α and β can be regarded as

(13)
(14)

where

(15)
(16)

Defining \(\ddot{x}_{0}^{T}= [ \ddot{V} \ \ \ddot{\gamma} \ \ \ddot{\alpha}_{0} \ \ \ddot{\beta}_{0} \ \ \ddot{h} ] \), the output dynamics of V and h can be rewritten as

(17)
(18)

where

Systems (17) and (18) can be expressed as

(19)

where

Assumption 2

The input matrix B is nonsingular.

Remark 1

The matrix B turns to be singular in the case of a vertical flight path, i.e., \(\gamma=\pm\frac{\pi}{2}\) [3, 45]. Since γ is quite small during the cruise phrase [16, 46], this assumption is reasonable.

4 Tracking controller design

In this section, two different controllers are developed to ensure the disturbance rejection performance of system. Before giving the controller design, some lemmas are presented which will be used in the subsequent control development and analysis.

Lemma 1

[47]

Consider the following system:

(20)

Suppose there exists a continuous function V(x):U→R such that

  1. 1.

    V(x) is positive definite.

  2. 2.

    There exist real numbers c>0 and α∈(0,1) and an open neighborhood U 0⊂U of the origin such that \(\dot{V}(x)+cV^{\alpha}(x)\leq0, x\in U_{0}\setminus \{0\}\).

Then the origin is a finite-time stable equilibrium of system (20). If U=U 0=R n, the origin is a globally finite time stable equilibrium of (20).

Lemma 2

[48]

Consider the following system:

(21)

If the controller is designed as

(22)

the closed-loop system (21) and (22) is globally finite time stable, where \(\alpha_{i-1}=\frac{\alpha_{i}\alpha_{i+1}}{2\alpha_{i+1}-\alpha_{i}}\), i=2,3,…,n, α n =α, α n+1=1, α∈(1−ε,1), ε∈(0,1), and k 1,…,k n ensure that s n+k n s n−1+⋯+k 1=0 is Hurwitz.

Lemma 3

[49]

For x i ∈R, i=1,…,n, 0<p≤1 is a real number, then the following inequality holds:

4.1 Sliding mode tracking controller design

Theorem 1

Consider system (19) with Assumptions 1 and 2, velocity V, and altitude h track the reference signals V d and h d in finite time, respectively, if the controller is designed as

(23)

where η i ≥a i +c 1|b i1|+c 2|b i2|, a i >0 is a constant, i=1,2.

Proof

Inspired by [33], two decoupled sliding mode manifolds s 1 and s 2 are chosen as

(24)
(25)

Computing the derivatives of s 1 and s 2 along system (19), we obtain

(26)
(27)

Equations (26) and (27) can be rewritten as

(28)

where

Choose the Lyapunov function as

(29)

Computing the first order derivative of V 1 along system (28), one obtains

(30)

Substituting control law (23) into system (30), yields

(31)

According to Assumption 1, |d i (t)|≤c i , hence (31) can be expressed as

(32)

Choose η i ≥a i +c 1|b i1|+c 2|b i2|, a i >0, i=1,2, (32) can be rewritten as

(33)

From Lemma 3, the following inequality is true:

(34)

where a=min{a 1,a 2}. According to Lemma 1, system signals can reach the sliding mode manifolds in finite time, i.e., s i =0,i=1,2. When s i =0, we have

(35)
(36)

Furthermore, the systems (35) and (36) can be rewritten as

(37)
(38)

Based on Lemma 2, e v , e h can converge to zero in finite time, i.e., V and h track the reference signals V d and h d in finite time, respectively. □

Remark 2

The external disturbances are the main problems to be handled. To guarantee that the closed-loop system has a good disturbance rejection performance, the switching gain needs to be chosen larger than the bound of the external disturbances. If there is no precise information on the bound of the disturbances, the switching gain will be taken to be large enough. This inaccuracy worsens the chattering caused by sliding mode control.

4.2 An improved sliding mode tracking controller design based on NDOB

An efficient disturbance estimation technique, i.e., NDOB, is introduced to estimate the disturbances of the system. The disturbance estimation will be used as a compensation term in the controller. After disturbance compensation based on NDOB, the switching gain only needs to be taken larger than the bound of the disturbance compensation error, which is usually much smaller than that of the external disturbances.

4.2.1 Nonlinear disturbance observer design

A NDOB is provided as follows.

Theorem 2

Consider system (19) with Assumption 1. The disturbance estimation given by

(39)

converges to a neighborhood Ω 0 of the real value of the disturbances, where \(\varOmega_{0}= \{ {\hat{{d}}} \mid \|{\hat{{d}}}\|\leq\|{d}\|+\frac{\bar{c}}{\theta_{1}\varGamma_{{\min}}} \}\), , Γ min=min{Γ 1,Γ 2}, 0<θ 1<1, \(\bar{c}=\sqrt{\bar{c}^{2}_{1}+\bar{c}^{2}_{2}}\), (⋅)′ denotes the derivative of (⋅).

Proof

The disturbance estimation dynamics are presented as

(40)

where \(e_{1}(t)={d}_{1}(t)-\hat{d}_{1}(t)\), \(e_{2}(t)={d}_{2}(t)-\hat{d}_{2}(t)\). The disturbance estimation error dynamics are described as

(41)

Choose the Lyapunov function as

(42)

where e T=[e 1 e 2]. Taking the derivative of (42) along the error dynamics (41), we obtain

(43)
(44)

According to Assumption 1, \(|\dot{ d}_{i}|\leq\bar{c}_{i}\), one has

(45)

Equation (45) is rewritten as

(46)

Let \(\varOmega_{1}= \{e \mid \|{e}\|\leq\frac{{\bar{c}}}{\theta_{1}\varGamma_{\min}} \}\). If e∉Ω 1, we have

(47)

There are two cases for all initial states. One case is that they are outside of Ω 1. Since \(\dot{V}_{e}<0\), there exists a t 2 such that e(t 2)∈bdΩ 1, where bdΩ 1 denotes the boundary of the Ω 1. Another case is that they are in Ω 1. If e remains in Ω 1 and never goes across it, we do not need to prove it. We only consider the case that e will escape from Ω 1. For this case, there also exists a t 2>0 such that e∈bdΩ 1. Now let us prove that e∈Ω 1 for every t∈[t 2,∞). The proof is inspired by [47].

Let \(p=\inf_{{e}_{\hat{\bar{d}}}\in bd\varOmega_{1}}\|{e}\|^{2}\), then \(p=\frac{\varsigma^{2}}{(\theta_{1} \varGamma_{\min})^{2}} \).

Furthermore, we obtain

Since V e is continuous, it obtains that there exists a n>0 such that e∈Ω 1 for all t∈[t 2,t 2+n). Suppose that there exists a h 0∈[t 2,∞) such that e∉Ω 1. Then there exists a δ∈(t 2,h 0) such that e(δ)∈bdΩ 1.

Noting that \(\dot{V}_{e}\leq-(1-\theta_{1})\varGamma_{\min}p<0\), and according to the continuity of V e , there exists an arbitrarily small n 1>0 such that V e is monotonously decreasing on [δ−n 1,δ).

Now we have

which is a contradiction. Hence, we reach a conclusion e∈Ω 1, ∀t≥t 2. □

Remark 3

On the basis of above analysis, it can be seen that the estimation errors depend on parameter Γ. By choosing appropriate parameter, the estimation errors can be forced to be small enough such that the disturbances are effectively estimated by a NDOB.

Remark 4

It is worth pointing out that, although the upper bound of the derivative of the disturbances, \(\bar{c}_{i}\), is employed to analyze the region of disturbance estimation errors, it is not needed to estimate this parameter because the design of disturbance observer and control law does not refer to such a parameter.

4.2.2 Composite controller design

Theorem 3

Consider system (19). If Assumption 2 holds, the improved sliding mode controller given by

(48)

guarantees that velocity V and altitude h track the reference signals V d and h d in finite time with the sliding mode manifolds presented by (24) and (25), where \(\hat{{d}}_{i}\) is obtained by (39) with Γ i >0, and \(\bar{\eta}_{i}\geq a_{i}+ \kappa_{1}|b_{i1}|+\kappa_{2}|b_{i2}|\), a i >0 is a constant, κ i is the bound of disturbance estimation error, i=1,2.

Proof

Choose the Lyapunov function as

(49)

Taking the derivative of (49) along the system (28) and (41), we have

(50)

Substituting control law (48) into (50), leads to

(51)

According to Theorem 2, we have |e i |≤κ i , hence, (51) can be rewritten as

(52)

Similar to the proof of Theorem 1, we obtain that V and h track the reference signals V d and h d in finite time, respectively. □

Remark 5

The difference between Theorems 1 and 3 is that a feedforward compensation term is introduced into controller in Theorem 3. When the parameter of the disturbance observer is appropriately chosen, d i (t) will be well estimated. Usually, the bound of disturbance compensation errors can be less than the disturbances. In this case, the composite controller may select a smaller value for the switching gain without sacrificing disturbance rejection performance, which helps to reduce large chattering caused by high gain.

Remark 6

In the proof process of Theorem 3, as shown in (51), it contains the compensated disturbance \({d}_{i}(t)-\hat {{d}}_{i}(t)\). According to Theorem 2, the disturbance estimation error will be bounded. So the signum function \(\bar{\eta}_{i}\rm{sign}(s_{i})\) in the controller (48) can suppress the compensated disturbance error by choosing \(\bar{\eta}_{i}\) bigger than the bound of \(b_{i1}({d}_{i}(t)-\hat {{d}}_{i}(t))\). This can guarantee the stability of integrated controller and observer.

5 Simulation results

In this section, simulation results are provided to confirm the effectiveness of the proposed method in the above section. As a representative case study, the vehicle is initially trimmed at \(V=15060~\mathrm{ft/s}\) and h=110000 ft. The model parameters are given as [45]

To show the robustness of the proposed controller, parameter uncertainties are chosen as △m=−0.03, △I=−0.03, while △S=0.03, \(\triangle \bar{c}=0.03\), △ρ=0.03, △c e =0.03. The external disturbances are given as d 1(t)=sin(0.2t), d 2(t)=0.1sin(0.2t), which are imposed on the system at t=30 s.

The reference commands for h d and V d are chosen as 112000 ft and \(15160~\mathrm{ft/s}\), respectively.

5.1 sliding mode tracking controller

The tracking control problem for the longitudinal model of an AHV is simulated in this subsection to testify the performance of the sliding mode tracking controller (23).

The controller parameters are chosen as

(53)

The response curves of altitude and velocity are shown in Fig. 1(a) and (b), which demonstrate that the sliding mode controller can obtain a good tracking performance. Figure 1(f) depicts the curves of sliding mode manifolds. From these, we can see that the sliding mode manifolds are stable in spite of external disturbances and parametric uncertainties. Curve of the throttle setting of engine β is presented in Fig. 1(d).

Fig. 1
figure 1figure 1figure 1

Response curves of hypersonic vehicles variables under controller (23) (a) altitude h, (b) velocity V, (c) elevator deflection angle δ E , (d) throttle setting of engine β, (e) the demand of control input β c , (f) sliding mode manifolds S h and S V

Full size image

In order to suppress external disturbances and parameter uncertainties, the controller employs a standard sliding mode control method, by which external disturbances and parameter uncertainties are assumed to be bounded. However, this approach faces an unavoidable application problem: chattering, which is presented in Fig. 1(c) and (e).

5.2 Comparison results

In order to demonstrate the effectiveness of the proposed composite control strategy (48), the controller (23) is used for comparison. The parameters of two controllers are listed as

(54)

The difference between (53) and (54) is that switching gains in (54) are smaller than those in (53).

As presented in Fig. 2(a) and (b), the proposed composite controller (48) works effectively and achieves the control target in the existence of the external disturbances and parametric uncertainties, but the controller (23) with small switching gains does not obtain good control performance.

Fig. 2
figure 2figure 2

Response curves of hypersonic vehicles variables (a) altitude h, (b) velocity V, (c) elevator deflection angle δ E , (d) throttle setting of engine β, (e) the demand of control input β c , (f) sliding mode manifolds S h and S V , (g) d 1 and \(\hat{{d}}_{1}\), (h) d 2 and \(\hat{{d}}_{2}\)

Full size image

Figure 2(c) and (e) depict curves of the control inputs δ E and β c , respectively. It can be seen that the undesired chattering is reduced under these two controllers. Note that the disturbance rejection performance of the closed-loop system cannot be maintained under controller (23) with small switching gains. However, the composite controller (48) can select small value for the switching gain without sacrificing disturbance rejection performance. This is because disturbance observer technique is employed to estimate the parametric uncertainties and external disturbances. Thus, the external disturbances and parameter uncertainties can be compensated by the estimated value in the control input.

Figure 2(d) describes the curve of the throttle setting of engine β. The disturbance estimation results illustrated in Fig. 2(h) and (i) demonstrate the effectiveness of proposed composite control method. These figures show that the NDOBs can effectively estimate the disturbances. Curves of the sliding mode manifolds under controller (48) are demonstrated in Fig. 2(f) to confirm the effectiveness of the proposed composite controller.

6 Conclusions

In this paper, the tracking control problem for the longitudinal model of an AHV subject to external disturbances has been discussed. First, by employing finite time integral sliding mode manifolds, a finite time sliding mode control method has been proposed. This control method can make the velocity and altitude track the reference signals in finite time. Then to reduce the chattering without sacrificing the disturbance rejection performance of system, an improved finite time sliding mode control method based on NDOB technique has been developed. Finally, simulation results have been presented to confirm the effectiveness of the proposed methods.