1 Introduction

With the increase of high requirements on safety and reliability of automation and control systems, the topic of fault tolerance (or reliable control) has attracted the interest of many researchers worldwide (see, e.g., [3, 6, 7, 17, 22, 25]). In general, repairing and maintenance services cannot be provided instantly, which makes the reliable control to be very important. The objective of reliable control is to design an appropriate controller such that the resulting closed-loop system can tolerate abnormal operations of specific control components and retain the overall system stability with acceptable system performance. Among the existing results of reliable control, several approaches have been reported. These approaches include the algebraic Riccatti equation-based approach [19], the coprime factorization approach [20], the Hamilton–Jacobi (HJ)-based approach [12], the sliding-mode control (SMC)-based approach [13], the linear matrix inequality (LMI)-based approach [14]. Among the mentioned reliable control studies, the LMI-based approach is relatively simpler and easier to be implemented, which has increasingly been the most popular approach in the field of fault-tolerant control (for example, [8, 16, 18, 22, 26] and the references therein).

Near-space hypersonic vehicle (NSHV) is one kind of new aerospace vehicles, which not only can make the supersonic speed cruising flight in the atmosphere, but also can pass through the atmosphere and make the cruising flight [9]. Compared with the existing aerospace vehicles, NSHV has many advantages, for instance, in launch cost, reusability, rapidity, maintainability, and so on. It can be seen that NSHV has very high value in military and civil applications, so it has the broad prospects for development [5]. Different from the traditional flight vehicles, owing to the engine-airframe integration design and flight conditions of high altitude and Mach number, NSHV is sensitive to variations in the flight conditions as well as physical and aerodynamics parameters. Furthermore, under the complex flight conditions, NSHV may be subject to actuator faults that result in an unacceptable performance or even destabilize the system [5]. In order to maintain the desired performance of the vehicle, it is essential that the actuator faults should be taken into account in the controller design of NSHV. In [1], Buschek et al. designed a fixed-order controller for a linearized NSHV dynamic model using \(\mathcal{H}_{\infty}\) and μ-synthesis techniques. In [24], Xue et al. proposed a trajectory linearization control approach for an NSHV dynamic system based on RBF neural network technique. In [4], Dong et al. presented a model reference switching control scheme for a linearized NSHV model with actuator saturation by using the adaptive technique. To the best of our knowledge, although considerable effort has been made on the control design for an NSHV dynamical system, the important issue of fault-tolerant (or reliable) control of an NSHV dynamical system has not been fully investigated yet, which remains challenging and motivates us to do this study.

In this paper, we design an observer-based passive FTC approach for a linearized NSHV dynamical system subject to parameter uncertainty and actuator faults. Firstly, the longitudinal dynamics of an NSHV is introduced, which can be linearized at nominal hypersonic cruise flight (M=15, h=110000 ft) condition. By considering the parameter uncertainty of system matrices and the loss of effectiveness type of actuator faults, a new NSHV faulty model is introduced. Based on the uncertain dynamic model established, the robust reliability of NSHV dynamics is taken into consideration in the design. The existence conditions for admissible controller are formulated in the form of LMIs, and the controller design problem is cast into a convex optimization problem subject to LMIs constraints. If the optimization problem is solvable, a desired reliable controller can be readily constructed. The simulation results are provided to show the effectiveness of the proposed FTC approach.

2 Near-Space Hypersonic Vehicle Model

A model for the longitudinal dynamics of a generic near-space hypersonic vehicle developed at NASA Langley Research Center is presented in [21, 23]. The equations of motion include an inverse-square-law gravitational model and the centripetal acceleration for the nonrotating Earth. The aerodynamic coefficients are simplified around the nominal cruising flight. The nominal flight of the vehicle is at a trimmed cruise condition: Mach number 15, Velocity 15060 ft/s, flight altitude 110000 ft, flight-path angle 0 deg, pitch rate 0 deg/s. The longitudinal dynamics of NSHV can be described by a set of differential equations for velocity, flight-path angle, altitude, angle of attack, and pitch rate as

$$ \left\{\begin{array}{l}\dot{V}=\frac{T\cos\alpha-D}{m}-\frac{\mu \sin \gamma}{r^2},\\[5pt]\dot{\gamma}=\frac{L+T\sin \alpha}{mV}-\frac{(\mu-V^2r)\cos\gamma}{V r^2},\\[5pt]\dot{h}=V \sin \gamma,\\[5pt]\dot{\alpha}= q-\dot{\gamma},\\[5pt]\dot{q}=M_{yy}/I_{yy},\end{array} \right.$$
(1)

where V(ft/s) is velocity, γ(deg) is flight-path angle, h(ft) is altitude, α(deg) is angle of attack, and q(deg/s) is pitch rate. The equations for lift L(lbf), drag D(lbf), thrust T(lbf), pitching moment M yy (lbf⋅ft), and radial distance from Earth’s center r(ft) can be computed as follows:

(2)
(3)

The lift coefficient C L , drag coefficient C D , thrust coefficient C T , and pitching moment coefficients C M (α), C M (δ e ), and C M (q) can be considered approximately as follows:

(4)
(5)
(6)
(7)
(8)
(9)
(10)

The control inputs are the throttle setting β and the elevator deflection δ e . The values of the inertial and aerodynamic parameters can be found in Table 1.

Table 1 Value of the inertial and aerodynamic parameters
Full size table

The longitudinal model of the generic near-space hypersonic vehicle described by (1) can be transformed into a general MIMO nonlinear system form:

(11)

where X is the state vector [V γ h α q]T, and U k (k=1,2) is the control input vector (including β and δ e ). In order to facilitate the control design, the longitudinal model of near-space hypersonic vehicle (1) can be linearized at nominal hypersonic cruise flight (M=15, h=110000 ft) before around its straight-and-level flight trim condition

(12)

satisfying F(X 0,U 0)=0 in order to obtain the following linearized model [1]:

$$\dot{x}(t)=Ax(t)+Bu(t), \qquad y(t)=Cx(t), $$
(13)

where

with x=X−X 0, u=U−U 0, and the dynamic output matrix C. Because the linearized model is defined for a trimmed flight condition, the linear state vector x contains the perturbation state [ΔV Δγ Δh Δα Δq]T and the control input perturbation vector [Δβ Δδ e ]T. From the system matrices A and B it can be seen that the pair (A,B) is controllable and the matrix B has full rank.

As the aforementioned description, the design of NSHV resulting in the integration of the engine with the airframe can cause an increased sensitivity to variations in angle of attack (Δα) which can be addressed as parametric uncertainty in the pitching moment C M (α), with the modification of (7) given by

$$C_M(\alpha)=10^{-4}\bigl(0.06-e^{-M/3}\bigr)\bigl[-6565(\alpha+\Delta \alpha)^2+6875(\alpha+\Delta \alpha)+1\bigr].$$
(14)

In order to approximate the dynamics of NSHV, the parameter uncertainty is considered in model (13). Meanwhile, to increase the robustness of FTC approach, the external disturbance is also considered in model (13). Therefore, the uncertain NSHV dynamical system can be expressed as

(15)

For the convenience of this study, the uncertain matrix ΔA is assumed to be norm-bounded, namely, ΔA=MH(t)N, where M and N are known constant matrices of appropriate dimensions, H(t) is the unknown time varying matrix satisfying H T(t)H(t)<I. ω(t) and B ω are the unknown bounded external disturbance input vector and the known distribute matrix, respectively.

When an actuator fault occurs, we use u F(t) to describe the control signal sent from actuators. Here, the actuator fault model with fault parameter matrix F is introduced [22]:

$$u^F(t)=Fu(t), $$
(16)

where \(0\leq \underline{F}=\mathrm{diag}\{\underline{f_{1}},\underline{f_{2}}\}\leq F=\mathrm{diag}\{f_{1}, f_{2}\} \leq \overline{F}=\mathrm{diag}\{\overline{f_{1}}, \overline{f_{2}}\}\), in which the variables f i (i=1,2) quantify the actuator control effectiveness.

Let

(17)
(18)

Apparently, \(\widetilde{f_{i}}=\{\overline{f_{i}}-\underline{f_{i}}\}/2 \ (i=1, 2)\), and we can rewrite F as follows:

$$F=F_0+\Delta\triangleq F_0+\mathrm{diag}\{\delta_1, \delta_2\}, $$
(19)

where \(\|\delta_{i}\|\leq \widetilde{f_{i}} \ (i=1, 2)\).

Based on the above analysis, the uncertain NSHV faulty model is given by

$$\dot{x}(t)=(A+\Delta A)x(t)+BFu(t)+B_\omega\omega(t),\qquad y(t)=Cx(t). $$
(20)

Remark 1

Flight control actuator failures can be broadly divided into two categories: (i) failures that result in a total loss of control effectiveness; (ii) failures that cause partial loss of control effectiveness. The former includes lock-in-place (LIP), float, and hard-over failure (HOF), while the latter describes general loss-of-effectiveness (LOE) types of failure. Note that the fault type (16) considered in this paper is LOE, which often occurs in actual flight operation and is the focus of our study in this paper.

3 Observer-Based FTC Design

Owing to the complex property of NSHV (such as hypersonic flight, high temperature, coupling of airframe, and engine) in flight, some state information of NSHV dynamics is not available for measurement, so the conventional state feedback control approach is not applicable in designing a stable flight control system for NSHV. Based on the above analysis, an observer-based passive FTC control scheme is proposed in this study to stabilize the uncertain faulty system (20):

(21)

where \(\hat{x}(t)\) is the estimated state vector, \(\hat{y}(t)\) is the observer output vector, and K and L are the controller gain and the observer gain, respectively. The design parameter matrices K, L, and A c will be determined.

Now, we use the observer-based output feedback control (21) to analysis the robust reliable stability of system (20) without disturbance input w(t).

From (20) and (21) with w(t)=0 we have

$$ \left[\begin{array}{c}\dot{\hat{x}}(t)\\\dot{e}(t)\end{array}\right]=\left[\begin{array}{c@{\quad}c}A_c-BK&LC\\A+\Delta A-A_c-B(F-I)K&A-LC+ \Delta A\end{array}\right]\left[\begin{array}{c}\hat{x}(t)\\e(t)\end{array}\right],$$
(22)

where \(e(t)=x(t)-\hat{x}(t)\) is the observer error.

Now, we are ready to present our first result concerning with the design matrices K, L, and A c in (21).

Theorem 1

Consider the uncertain NSHV dynamics (20) with w(t)=0 and known actuator fault parameter matrix F. Suppose that there exist two positive definite matrices \(\overline{P}_{1}\), \(\overline{P}_{2}\), real matrices \(\overline{A}_{c}\), \(\hat{P}\), \(\overline{K}\), \(\overline{L}\), and a positive scalar ε>0 such that the following conditions hold:

(23)
(24)

where \(\mathbb{H}_{11}=\overline{A}_{c}+\overline{A}_{c}^{T}-B\overline{K}-\overline{K}^{T} B^{T}\), \(\mathbb{H}_{12}=\mathbb{H}_{21}^{T}=\overline{L}C+\overline{P}_{1}A^{T}-\overline{A}^{T}_{c}- \overline{K}^{T}(F-I)B^{T}\), and \(\mathbb{H}_{22}=A\overline{P}_{2}+\overline{P}_{2}A^{T}-\overline{L}C-C^{T}\overline{L}^{T}\). Then NSHV dynamics (20) is robustly reliably stabilizable by the observer-based output feedback control (21) with \(K=\overline{K}\ \overline{P}_{1}^{-1}\), \(L=\overline{L}\hat{P}^{-1}\), and \(A_{c}=\overline{A}_{c}\overline{P}_{1}^{-1}\).

Proof

Using Schur complement, (24) can be transformed into the following form:

(25)

Pre- and post-multiplying ∏ by

$$\left[\begin{array}{c@{\quad}c}P_1&0\\0&P_2\end{array}\right]=\left[\begin{array}{c@{\quad}c}\overline{P}_1^ {-1}&0\\0&\overline{P}_2 ^{-1}\end{array}\right]>0$$

and using the equalities \(C\overline{P}_{2}=\hat{P}C\), \(K=\overline{K} \ \overline{P}_{1}^{-1}\), \(L=\overline{L}\hat{P}^{-1}\), \(A_{c}=\overline{A}_{c}\overline{P}_{1}^{-1}\), we have:

(26)

Define a Lyapunov function as

$$V(t)=\hat{x}^T(t)P_1\hat{x}(t)+e^T(t)P_2e(t),$$

where \(P_{1}=\overline{P}_{1}^{-1}>0\) and \(P_{2}=\overline{P}_{2}^{-1}>0\). Taking the time derivative of V(t) along the trajectory of (22), we have

(27)

On the other hand, it can be easily seen that

(28)

Combining (27) with (29), we have

$$\dot{V}(t)\leq\left[\begin{array}{c}\hat{x}(t)\\e(t)\end{array}\right]^T\coprod \left[\begin{array}{c}\hat{x}(t)\\e(t)\end{array}\right]. $$
(29)

From (26) it can be seen that the NSHV dynamics (20) with w(t)=0 is robustly reliably stabilizable for actuator faults (16) by observer-based output feedback control (21), which completes the proof. □

Remark 2

From (13) it can be seen that the system output matrix C is of full row rank and the matrix \(\hat{P}\) in (23) must be nonsingular for \(\overline{P}_{2}>0\). Here, we consider how to solve the conditions in Theorem 1. It is easy to solve inequality (24) by Matlab LMI toolbox, but a difficulty is added because of the equality constraint (23). Actually, there is no effective tool to solve (23) and (24) simultaneously; however, the equality constraint (23) in Theorem 1 can be transformed into an optimization problem to solve [2]: Minimize β subject to (24) and

(30)

Remark 3

In [15], Lien et al. presented a modified observer-based output feedback control, which can be borrowed to solve the robust reliable control problem of NSHV dynamics in this study. It is worth mentioning that there are two major differences between the work [15] and ours. First, the effect of actuator faults is not considered in [15] but is taken into account in this study. Second, there exists a condition of strict equality constraints in the main results of [15], which is relaxed by a linear matrix inequality using the optimization technique in this study.

Note that, when F=I in Theorem 1, our results cover the corresponding results for the “no-fault” case as a special one, which has been investigated extensively in the past. Furthermore, when F is unknown but satisfies the constraints (17)–(19), we have the following result.

Corollary 1

Consider the uncertain NSHV dynamics (20) with w(t)=0 and unknown actuator fault parameter matrix F. Suppose that there exist two positive definite matrices \(\overline{P}_{1}\), \(\overline{P}_{2}\), a diagonal matrix S>0, real matrices \(\overline{A}_{c}\), \(\hat{P}\), \(\overline{K}\), \(\overline{L}\), and a positive scalar ε>0 such that the following conditions hold:

(31)
(32)

where \(\widehat{\mathbb{H}}_{11}=\overline{A}_{c}+\overline{A}_{c}^{T}-B\overline{K}-\overline{K}^{T}B^{T},\ \widehat{\mathbb{H}}_{12}=\widehat{\mathbb{H}}_{21}^{T}=\overline{L}C+\overline{P}_{1}A^{T}-\overline{A}^{T}_{c} -\overline{K}^{T}(F_{0}-I)B^{T}\), and \(\widehat{\mathbb{H}}_{22}=A\overline{P}_{2}+\overline{P}_{2}A^{T}-\overline{L}C-C^{T}\overline{L}^{T}+B^{T}SB\). Then the NSHV dynamics (20) is robustly reliably stabilizable by the observer-based output feedback control (21) with \(K=\overline{K}\ \overline{P}_{1}^{-1}\), \(L=\overline{L}\hat{P}^{-1}\), and \(A_{c}=\overline{A}_{c}\overline{P}_{1}^{-1}\).

Proof

By (19), the LMl (24) in Theorem 1 can be rewritten as

(33)

where

From (33) we can easily obtain the following inequality for all S=S T>0:

(34)

where \(\mathbb{N}_{11}=\overline{A}_{c}+\overline{A}_{c}^{T}-B\overline{K}-\overline{K}^{T}B^{T}+\overline{K}^{T}S^{-1}\widetilde{F}^{2}\overline{K}\), \(\mathbb{N}_{12}=\mathbb{N}_{21}^{T}=\overline{L}C+\overline{P}_{1}A^{T}-\overline{A}^{T}_{c}-\overline{K}^{T}(F_{0}-I)B^{T}\), and \(\mathbb{N}_{22}=A\overline{P}_{2}+\overline{P}_{2}A^{T}-\overline{L}C-C^{T}\overline{L}^{T}+BSB^{T}\).

Utilizing Schur complement to (32), we have ℕ<0. By Theorem 1, it can be easily shown that the NSHV dynamics (8) with w(t)=0 is robustly reliably stabilizable under the observer-based output feedback control (9). □

Remark 4

Note that the result obtained in Theorem 1 must rely on an exactly known fault parameter F, which leads to some conservatism for the designed reliable controller in stabilizing the faulty systems. But the result obtained in Corollary 1 relies only on a given interval for fault parameter F such that the proposed reliable controller can tolerate a wider range of actuator faults.

In the following, we shall solve the passive FTC problem for NSHV dynamics (20) with disturbance input ω(t) using the observer-based output feedback control (21).

Definition 1

Considering the NSHV dynamics (20) with the observer-based output feedback control (21), suppose that the following conditions are satisfied:

  1. (i)

    When w(t)=0, the closed-loop system (20) with actuator fault model (16) is asymptotically stable.

  2. (ii)

    Under zero initial condition, dynamic output y(t) satisfies the following \(\mathcal {H}_{\infty}\) performance index:

    where γ is the disturbance attenuation level. Then the dynamic system (20) is said to be robustly reliably stabilizable with disturbance attenuation γ, and the control (21) is said to be an observer-based passive FTC for dynamic system (20).

After some manipulations, systems (20) and (21) can be rewritten as

(35)

The unknown parameter matrices K, L, and A c for the observer-based passive fault-tolerant control of system (8) could be designed based on the following result.

Theorem 2

Consider the NSHV dynamics (20) with known actuator fault parameter matrix F for a given positive scalar γ>0. If there exist positive definite matrices \(\overline{P}_{1}\), \(\overline{P}_{2}\), real matrices \(\overline{A}_{c}\), \(\hat{P}\), \(\overline{K}\), \(\overline{L}\), and a positive scalar ε>0 such that the following conditions hold:

(36)
(37)

where ℍ11, ℍ12, ℍ21, and ℍ22 are the same lines as in (24). Then system (20) is robustly reliably stabilizable with disturbance attenuation level γ by the observer-based output feedback control (21) with \(K=\overline{K}\ \overline{P}_{1}^{-1}\), \(L=\overline{L}\hat{P}^{-1}\), and \(A_{c}=\overline{A}_{c}\overline{P}_{1}^{-1}\).

Proof

Define a Lyapunov function as follows:

$$V(t)=\hat{x}^T(t)P_1\hat{x}(t)+e^T(t)P_2e(t),$$

where \(P_{1}=\overline{P}_{1}^{-1}>0\) and \(P_{2}=\overline{P}_{2}^{-1}>0\). The time derivative of V(t) along the trajectory of (35) is given by

(38)

where \(\varXi_{11}=P_{1}A_{c}+A_{c}^{T}P_{1}-P_{1}BK-K^{T}B^{T}P_{1}\), \(\varXi_{12}=P_{1}LC+A^{T}P_{2}-A^{T}_{c}P_{2}-B^{T}(F-I)P_{2}^{T}\), Ξ 21=C T L T P 1+P 2 A−P 2 A c −P 2(F−I)B, and Ξ 22=P 2 A+A T P 2−P 2 LC−C T L T P 2.

Along the same line as the development in (34), we have the following inequality:

(39)

Here, we introduce an \(\mathcal {H}_{\infty}\) performance index function described by

(40)

Notice that \(x(t)=\hat{x}(t)+e(t)\), so the above equality can be rewritten as follows:

(41)

where \(\varPi_{11}=P_{1}A_{c}+A_{c}^{T}P_{1}-P_{1}BK-K^{T}B^{T}P_{1}+C^{T}C+\frac{1}{\varepsilon}N^{T}N\), \(\varPi_{12}=P_{1}LC+A^{T}P_{2}-A^{T}_{c}P_{2}-B^{T}(F-I)P_{2}^{T}+C^{T}C+\frac{1}{\varepsilon}N^{T}N\), \(\varPi_{21}=C^{T}L^{T}P_{1}+P_{2}A-P_{2}A_{c}-P_{2}(F-I)B+C^{T}C+\frac{1}{\varepsilon}N^{T}N\), and \(\varPi_{22}=P_{2}A+A^{T}P_{2}-P_{2}LC-C^{T}L^{T}P_{2}+C^{T}C+\varepsilon P_{2}MM^{T}P_{2}+\frac{1}{\varepsilon}N^{T}N\). □

From Theorem 1 it can be seen that \(\dot{V}(t)|_{w(t)=0}<0\) for nonzero \(\hat{x}(t)\) and e(t). Hence, the closed-loop system (20) with (21) when w(t)=0 is asymptotically stable.

By employing the same techniques used in Theorem 1, using Schur complement to (37) together with (36), we can obtain that J(t)<0. Integrating (41) from 0 to ∞, we have

Under the zero initial condition, we have

By Definition 1, it can be seen that system (20) is robustly reliably stabilizable with disturbance attenuation level γ by the feedback control (21), and thus we have the desired result.

When actuator fault parameter matrix F is unknown but satisfies constraints (17)–(19), similarly to Corollary 1, we have the following result.

Corollary 2

Consider the NSHV dynamics (20) with unknown actuator fault parameter matrix F for a given positive scalar γ>0. Suppose that there exist two positive definite matrices \(\overline{P}_{1}\), \(\overline{P}_{2}\), a diagonal matrix S>0, real matrices \(\overline{A}_{c}\), \(\hat{P}\), \(\overline{K}\), \(\overline{L}\), and a positive scalar ε>0 such that the following conditions hold:

(42)
(43)

where \(\widehat{\mathbb{H}}_{11}\), \(\widehat{\mathbb{H}}_{12}\), \(\widehat{\mathbb{H}}_{21}\), and \(\widehat{\mathbb{H}}_{22}\) are the same as in (32). Then the NSHV dynamics (20) is robustly reliably stabilizable with disturbance attenuation level γ by the observer-based output feedback control (21) with \(K=\overline{K}\ \overline{P}_{1}^{-1}\), \(L=\overline{L}\hat{P}^{-1}\), and \(A_{c}=\overline{A}_{c}\overline{P}_{1}^{-1}\).

Remark 5

As for the equality constraints in Corollaries 1–2 and Theorem 2, we utilize the optimization technique provided in Remark 2 to deal with them. Thus, the main results obtained in this study are in terms of strictly linear matrix inequalities, which can be readily solved by Matlab LMI toolbox.

Remark 6

In [1, 4, 24], the state-feedback control problem is investigated for the linearized NSHV dynamical system under the assumption that the real-time state signals can be measured directly. Note that the output feedback control problem is more important in actual aerospace engineering applications. Based on a modified observer design approach, the output-feedback control problem of NSHV dynamic system is investigated in this study.

Remark 7

It should be mentioned that the study in this paper is motivated by the work of [22], where an \(\mathcal {H}_{\infty}\) passive FTC control scheme was proposed for a class of linear discrete-time systems with time delays, but the parameter uncertainty (robustness) was not considered. In this paper, an observer-based \(\mathcal {H}_{\infty}\) passive FTC approach is presented for a near-space hypersonic vehicle longitudinal dynamical system using both robust control and LMI techniques.

Remark 8

In [10], different kinds of stability for linear continuous-time system with constant coefficients are discussed, and the fundamental definitions of controllability both for finite-dimensional and infinite-dimensional systems are given. Meanwhile, the necessary and sufficient conditions for different kinds of controllability are formulated. But the controllability issue in actuator/sensor fault case is not further discussed in [10], while this study is concerned with the controllability for a linearized NSHV dynamical system with actuator faults by using the observer-based output feedback control scheme, and the results obtained can be regarded as the extension and supplement of [10].

4 Simulation Example

In this section, we provide an example to illustrate the usefulness and advantage of the observer-based passive FTC scheme proposed in this study.

The parameter matrices of linearized NSHV dynamical system have been given in (13). The parameter uncertainty in C M (α) is set to be varying in the angle of attack as |Δα|≤0.1. In this paper, we assume that the admissible lower and upper bounds of actuator faults are \(\underline{f}_{i}=0.4\) and \(\overline{f}_{i}=1\) (i=1,2), respectively. In this study, it is assumed that the fifth state vector pitch rate q is not available for feedback, and the NSHV dynamics output matrix C is chosen as [I 4 0]. Setting disturbance attenuation level γ=1.5, by solving conditions (42) and (43) in Corollary 2 using Matlab toolbox, the unknown matrix parameters for the observer-based control (21) can be obtained as follows:

In simulation, we assume that f 1=0.9 and f 2=0.6, namely, the throttle setting β loses 10% control effectiveness, and the elevator deflection δ e loses 40% control effectiveness. Dryden turbulence model is used to introduce external disturbances, and the turbulence spectrum is defined by the weight

The disturbance distributed matrix is determined by B w =[0 0.5 0 0.5 0]T. For comparison, we give different simulation results using the observer-based passive FTC scheme developed in [11] and that developed in this study, respectively. From Figs. 1–2 it can be seen that the passive FTC scheme proposed in [11] cannot maintain the asymptotical stability of the NSHV closed-loop system in actuator fault case. Therefore, the FTC approach presented in [11] is not suitable for the NSHV dynamical system considered in this study. However, it can be seen from Figs. 3–4 that the observer-based passive FTC approach presented in this study can maintain the asymptotical stability of the NSHV closed-loop system in actuator fault case, which demonstrates the feasibility and potential of our proposed approach for application in aerospace engineering.

Fig. 1
figure 1

NSHV output responses using the FTC approach developed in [11]

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Fig. 2
figure 2

NSHV control input responses using the FTC approach developed in [11]

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Fig. 3
figure 3

NSHV output responses using the FTC approach developed in this paper

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Fig. 4
figure 4

NSHV control responses using the FTC approach developed in this paper

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5 Conclusions

In this paper, an observer-based passive fault tolerant control strategy is proposed for a linearized near-space hypersonic vehicle dynamical system with parameter uncertainty and actuator faults. By considering the parameter uncertainty in the pitching moment C M (α) and the actuator fault described by (16)–(19), an uncertain NSHV faulty model is established. Based on a modified observer design approach, the passive FTC problem is then studied using both robust control and LMI techniques. Finally, simulation results are given to illustrate the usefulness of the proposed FTC scheme.