Robust fault accommodation strategy of the reentry vehicle: a disturbance estimate-triggered approach


This study proposes a novel fault accommodation scheme for the strong coupled attitude system of the hypersonic reentry vehicle (HRV) with both actuator drift and loss of efficiency. A general coupling/fault/uncertainty effect-triggered control concept is first introduced for the HRV attitude tracking system to improve its robustness and dynamic performance, which can be derived easily via Lyapunov stability. The design of such a control approach is based on an improved adaptive disturbance observer (ADO) to estimate the lumped uncertainties and actuator faults. The proposed scheme can achieve graceful degradation in tracking performance for the fault-tolerant control system by eliminating the detrimental uncertainty and actuator fault while keeping the beneficial uncertainty and actuator fault. A detailed design procedure has been presented with consideration of the implementation problem. Simulation results obtained on the HRV have demonstrated the effectiveness of the approach proposed.

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m :

Mass, slugs

Ma :

Mach number

V :

Velocity, ft/s

H :

Altitude, ft

\(\theta \) :

Flight-path angle, rad

\(\psi _v\) :

Azimuth angle

\(\alpha \) :

Angle of attack, rad

\(\beta \) :

Sideslip angle, rad

\(\gamma _v\) :

Bank angle, rad

\(\omega _x\) :

Roll rate, rad/s

\(\omega _y\) :

Yaw rate, rad/s

\(\omega _z\) :

Pitch rate, rad/s

\(\delta _e\) :

Elevator deflection, rad

\(\delta _a\) :

Aileron deflection, rad

\(\delta _r\) :

Rudder deflection, rad

\(\delta _{bf}\) :

Body flap deflection, rad

\(C_D\) :

Drag coefficient

\(C_L\) :

Lift coefficient

\(I_{xx}\) :

Moment of inertia around x axis, \( slug-ft^2\)

\(M_x\) :

Rolling moment, lbf-ft

\(I_{yy}\) :

Moment of inertia around y axis, \(slug-ft^2\)

\(M_{y}\) :

Yawing moment, lbf-ft

\(I_{zz}\) :

Moment of inertia around z axis, \(slug-ft^2\)

\(M_z\) :

Pitching moment, lbf-ft

\(I_{xy,xz,yz}\) :

Product of inertia, \(slug-ft^2\)

Y :

Side force, lbf

L :

Lift force, lbf

D :

Drag force, lbf

\(\rho \) :

Density of air, slugs \(ft^3\)

T :

Thrust, lbf

\(\mu \) :

Gravitational constant, \( 1.39 \times 10^{16} ft^3/s^2\)

\(S_{ref}\) :

Reference area, \(ft^2\)

\(\phi \) :

Latitude, rad

R :

The radius of the earth, ft

a :

Speed of sound, ft/s

\(\bar{c}\) :

Reference length, ft


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This study was supported by National Natural Science Foundation of China (Grant Number 61803308, 62003252).

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Correspondence to Zongyi Guo.

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Proof of the disturbance observer in Eq. (32)


Subtracting (9) from (35) yields the error system as

$$\begin{aligned} \left\{ \begin{array} {l} \dot{e}_i = -\sqrt{2L_{i}}(t)|e_i|^{1/2} \mathrm {sign}(e_i) + \tilde{d}_i\\ \dot{\tilde{d}}_i = -4L_i(t) \mathrm {sign}(e_i) - \dot{\zeta }_i\\ \end{array} \right. \end{aligned}$$

where \(\tilde{d}_i = \hat{\zeta }_i - \zeta _i\). Consider the following Lyapunov function candidate for the error system (56)

$$\begin{aligned} V(t) = \frac{1}{L_i^{2/3}(t)}\xi ^T P(t) \xi \end{aligned}$$


$$\begin{aligned} P(t) = \frac{1}{2}\begin{bmatrix} 18L_i(t) &{} \quad - \sqrt{2L_{i}}(t)\\ -\sqrt{2L_{i}}(t)&{} \quad 2 \end{bmatrix}, \xi = \begin{bmatrix} |e_i|^{1/2} \mathrm {sign}(e_i) \\ \tilde{d}_i \end{bmatrix} \end{aligned}$$

The derivative of \(\xi \) results in

$$\begin{aligned} \begin{aligned} \dot{\xi }&= \begin{bmatrix} \frac{1}{ 2|e_i|^{1/2}} \left( - \sqrt{2L_{i}(t)}|e_i|^{1/2} \mathrm {sign}(e_i) + \tilde{d}_i \right) \\ -4L_i(t) \mathrm {sign}(e_i)- \dot{\zeta }_i \end{bmatrix} \\&= - \frac{1}{ 2|e|^{1/2}} A_1(t) \xi + A_2(t) \end{aligned} \end{aligned}$$

where \(A_1 = \begin{bmatrix} \sqrt{2L_{i}(t)} &{} -1 \\ 8L_i(t) &{} 0\end{bmatrix}\) and \(A_2 = \begin{bmatrix} 0 \\ -\dot{\zeta }_i \end{bmatrix}\). Taking the derivative of V yields

$$\begin{aligned} \dot{V}&= -\frac{L_0}{2} \xi ^T \underbrace{ \begin{bmatrix} 9L_i^{-3/2}(t) &{} \quad -\sqrt{2} L_i^{-2}(t) \nonumber \\ -\sqrt{2} L_i^{-2}(t) &{} \quad 3L_i^{-5/2} (t) \end{bmatrix} }_{Q_1(t)} \xi \nonumber \\&+ \frac{\xi ^T P(t)A_2(t)}{L_i^{3/2}(t)} \nonumber \\&- \frac{1}{2L_i^{3/2}(t) |e_i|^{1/2}}{\xi }^T \underbrace{\left( A_1(t)^T P(t) + P(t)A_1(t)\right) }_{Q_2(t)} \xi \nonumber \\&=-\frac{1}{2}L_0\xi ^TQ_1(t)\xi -\frac{1}{2L_i^{3/2}(t) |e_i|^{1/2}} \xi ^T Q_2(t) \xi \nonumber \\&+ \frac{\xi ^T P(t)A_2(t)}{L_i^{3/2}(t)} \nonumber \\&\le -\frac{1}{2}L_0\xi ^TQ_1(t)\xi -\frac{1}{2L_i^{3/2}(t) |e_i|^{1/2}} \xi ^T \tilde{Q}(t) \xi \nonumber \\&\le - \underbrace{\frac{1}{2}L_0\frac{\lambda _{\mathrm{min}}(Q_1)}{ \lambda _{\mathrm{max}}(P)}}_{\eta _1} V \nonumber \\&- \underbrace{ \frac{1}{\sqrt{2} L_i(t)} \frac{\sqrt{\lambda _{\mathrm{min}}(P)} \lambda _{\mathrm{min}}(\tilde{Q})}{ \lambda _{\mathrm{max}}(P)}}_{\eta _2} V^{1/2} \end{aligned}$$


$$\begin{aligned}&\tilde{Q}(t) \nonumber \\&\quad = \sqrt{2L_{i}(t)} \begin{bmatrix} 10 L_i(t) - 2\Delta _i -\frac{2\Delta _i^2}{\sqrt{2L_{i}(t) }}&{} \quad -\sqrt{2L_{i}(t)} \\ -\sqrt{2L_{i}(t)} &{} \quad 2-\frac{2}{\sqrt{2L_{i}(t)}} \end{bmatrix} \end{aligned}$$

Since \(\dot{L}_i = L_0>0\), it is easily found that \(\tilde{Q}(t)\) will be positive in finite time. After that, we have \(\dot{V}+ \eta _1 V + \eta _2 V^{1/2} \le 0\). In view of Lemma 2 in [29], the vector \(\xi \rightarrow 0\) in finite time. Obviously, the identity \(\xi \equiv 0\) implies \(e_i \equiv 0\) and \(\tilde{d}_i \equiv 0\). Thus, the estimates of disturbances are available in finite time.

This completes the proof.

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Chang, J., Guo, Z., Cieslak, J. et al. Robust fault accommodation strategy of the reentry vehicle: a disturbance estimate-triggered approach. Nonlinear Dyn 103, 2605–2625 (2021).

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  • Fault tolerant control
  • Adaptive disturbance observer
  • Coupling/fault/uncertainty effect
  • Reentry vehicle