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

Abstract

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.

Appendix

Proof of the disturbance observer in Eq. (32)

Proof

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

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

(56)

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

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

(57)

where

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

(58)

The derivative of $\xi$ results in

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

(59)

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

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

(60)

where

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

(61)

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.