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Integrated guidance and control for reusable launch vehicle in reentry phase

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Abstract

An integrated guidance and control scheme is developed for next generation of reusable launch vehicle (RLV) with the aim to improve the flexibility, safety and autonomy. Firstly, an outer-loop optimal feedback reentry guidance law with online trajectory reshaping capability is designed. Then, a novel reentry attitude control strategy is proposed based on multivariables smooth second-order sliding mode controller and disturbance observer. The proposed control scheme is able to guarantee that the guidance commands generated from the guidance system can be tracked in finite time. Furthermore, a control allocation is integrated in the system in order to transform the control moments to control surface deflection. Finally, some representative simulation tests are conducted to demonstrate the effectiveness of the proposed integrated guidance and control strategy for six-degree-of-freedom RLV.

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Acknowledgments

This work has been supported by National Natural Science Foundation of China (61203012, 61304018, 61273092), Key Grant Project of Chinese Ministry of Education (311012) and Tianjin Research Program of Application Foundation and Advanced Technology (12JCZDJC30300). Independent Innovation Fund of Tianjin University (2013XQ-0022),.

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Authors and Affiliations

  1. School of Electric and Automation Engineering, Tianjin University, Tianjin, 300072, China

    Bailing Tian & Qun Zong

  2. Aeronautical Automation College, Civil Aviation University of China, Tianjin, 300300, China

    Wenru Fan

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  1. Bailing Tian
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Correspondence to Bailing Tian.

Appendix: Proof of Theorem 1

Appendix: Proof of Theorem 1

Proof

For brevity, we present system (16) in a form convenient for Lyapunov analysis. To this end, a new state vector is introduced

$$\begin{aligned} \mathbf{z}_1 =\left\| {\mathbf{x}_1 } \right\| ^{p-1}\mathbf{x}_1 ,\quad \mathbf{z}_2 =\mathbf{x}_2 \end{aligned}$$
(36)

Furthermore, system (16) can be rewritten as

$$\begin{aligned} \dot{\mathbf{z}}_1&= \left( {\mathbf{I}_m +\left( {p-1} \right) \frac{\mathbf{x}_1 \mathbf{x}_1 ^{T}}{\left\| {\mathbf{x}_1 } \right\| ^{2}}} \right) \left\| {\mathbf{x}_1 } \right\| ^{\left( {p-1} \right) }\dot{\mathbf{x}}_1,\nonumber \\ \dot{\mathbf{z}}_2&= -k_2 p\left\| {\mathbf{x}_1 } \right\| ^{p-1}\mathbf{z}_1 +\varvec{\Delta }_2 \end{aligned}$$
(37)

where \(\mathbf{I}_m \) denotes m-dimensional unit matrix. Taking into account the definition in (36), we obtain \(\left\| {\mathbf{z}_1 } \right\| =\left\| {\mathbf{x}_1 } \right\| ^{p}\) and \(\mathbf{x}_1 =\mathbf{z}_1 /\left\| {\mathbf{x}_1 } \right\| ^{p-1}=\mathbf{z}_1 /\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\). Therefore, system (37) can be equivalently written as

$$\begin{aligned} \dot{\mathbf{z}}_1&= \left( {\mathbf{I}_m +\left( {p-1} \right) \frac{\mathbf{z}_1 \mathbf{z}_1 ^{T}}{\left\| {\mathbf{z}_1 } \right\| ^{2}}} \right) \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\nonumber \\&\times \left( {-k_1 \mathbf{z}_1 +\mathbf{z}_2 +\varvec{\Delta }_1 } \right) ,\nonumber \\ \dot{\mathbf{z}}_2&= -k_2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1 +\varvec{\Delta }_2 \end{aligned}$$
(38)

It follows from the definition in (36) that \(\mathbf{z}_1,\,\mathbf{z}_2 \rightarrow 0\) in finite time implies that \(\mathbf{x}_1,\,\mathbf{x}_2 \rightarrow 0\) in finite time. Next, a Lyapunov function candidate is proposed as

$$\begin{aligned} V\left( {\mathbf{z}_1,\mathbf{z}_2 } \right) =\left( {k_2 +\frac{1}{2}k_1^2 } \right) \mathbf{z}_1^T \mathbf{z}_1 +\mathbf{z}_2^T \mathbf{z}_2 -k_1 \mathbf{z}_1^T \mathbf{z}_2\nonumber \\ \end{aligned}$$
(39)

It is easy to verify that \(V\left( {\mathbf{z}_1,\mathbf{z}_2 } \right) \) is positive definite and radially unbounded for arbitrary positive constants \(k_1 >0\,\hbox {and}\,k_2 >0\). The derivative of \(V\left( {\mathbf{z}_1,\mathbf{z}_2 } \right) \) is given by

$$\begin{aligned} \dot{V}\left( {\mathbf{z}_1,\mathbf{z}_2 } \right)&= \left( {2k_2 +k_1^2 } \right) \mathbf{z}_1^T \dot{\mathbf{z}}_1 +2\mathbf{z}_2^T \dot{\mathbf{z}}_2\nonumber \\&\quad -\,k_1 \left( {\dot{\mathbf{z}}_1^T \mathbf{z}_2 +\mathbf{z}_1^T \dot{\mathbf{z}}_2 } \right) \end{aligned}$$
(40)

Substituting (38) into (40) yields

$$\begin{aligned}&V=\left( {2k_2 +k_1^2 } \right) \mathbf{z}_1^T \left[ \left( {\mathbf{I}_m +\left( {p-1} \right) \frac{\mathbf{z}_1 \mathbf{z}_1 ^{T}}{\left\| {\mathbf{z}_1 } \right\| ^{2}}} \right) \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\right. \nonumber \\&\left. \quad \times \left( {-k_1 \mathbf{z}_1 +\mathbf{z}_2 +\varvec{\Delta }_1 } \right) \right] +2\mathbf{z}_2^T \left( {-k_2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1 +\varvec{\Delta }_2 } \right) \nonumber \\&\quad -\,k_1 \left[ \left( {\mathbf{I}_m +\left( {p-1} \right) \frac{\mathbf{z}_1 \mathbf{z}_1 ^{T}}{\left\| {\mathbf{z}_1 } \right\| ^{2}}} \right) \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\right. \nonumber \\&\qquad \left. \left( {-k_1 \mathbf{z}_1 +\mathbf{z}_2 +\varvec{\Delta }_1 } \right) \right] ^{T}\mathbf{z}_2 \nonumber \\&\quad -\,k_1 \mathbf{z}_1^T \left( {-k_2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1 +\varvec{\Delta }_2 } \right) \nonumber \\&\quad =\left( {2k_2 +k_1^2 } \right) p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left( {-k_1 \left\| {\mathbf{z}_1 } \right\| ^{2}+\mathbf{z}_1^T \mathbf{z}_2 +\mathbf{z}_1^T \varvec{\Delta }_1 } \right) \nonumber \\&\quad -\,2k_2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1^T \mathbf{z}_2 +2\mathbf{z}_2^T \varvec{\Delta }_2 +k_1 \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\nonumber \\&\quad \left[ {\left( {k_1 \mathbf{z}_1^T -\mathbf{z}_2^T -\varvec{\Delta }_1^T } \right) \left( {\mathbf{I}_m +\left( {p-1} \right) \frac{\mathbf{z}_1 \mathbf{z}_1^T }{\left\| {\mathbf{z}_1 } \right\| ^{2}}} \right) \mathbf{z}_2 } \right] \nonumber \\&\quad +\,k_1 \mathbf{z}_1 ^{T}\left[ {k_2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1 -\varvec{\Delta }_2 } \right] \nonumber \\&\quad =\left( {2k_2 +k_1^2 } \right) p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left( {-k_1 \left\| {\mathbf{z}_1 } \right\| ^{2}+\mathbf{z}_1^T \mathbf{z}_2 +\mathbf{z}_1^T \varvec{\Delta }_1 } \right) \nonumber \\&\quad -\,2k_2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1^T \mathbf{z}_2 +2\mathbf{z}_2^T \varvec{\Delta }_2 +k_1 \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\nonumber \\&\quad \left[ k_1 p\mathbf{z}_1^T \mathbf{z}_2 -\left( {\left\| {\mathbf{z}_2 } \right\| ^{2}+\left( {p-1} \right) \frac{\left( {\mathbf{z}_2^T \mathbf{z}_1 } \right) \left( {\mathbf{z}_1^T \mathbf{z}_2 } \right) }{\left\| {\mathbf{z}_1 } \right\| ^{2}}} \right) \right. \nonumber \\&\left. \quad -\varvec{\Delta }_1^T \left( {\mathbf{I}_m +\left( {p-1} \right) \frac{\mathbf{z}_1 \mathbf{z}_1^T }{\left\| {\mathbf{z}_1 } \right\| ^{2}}} \right) \mathbf{z}_2\right] \nonumber \\&\quad +\,k_1 k_2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left\| {\mathbf{z}_1 } \right\| ^{2}-k_1 \mathbf{z}_1^T \varvec{\Delta }_2 \nonumber \\&\quad =\left[ -\left( {k_2 +k_1^2 } \right) k_1 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left\| {\mathbf{z}_1 } \right\| ^{2}\right. \nonumber \\&\quad \left. +\,2k_1^2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1^T \mathbf{z}_2-\,k_1 \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\right. \nonumber \\&\quad \left. \left( {\left\| {\mathbf{z}_2 } \right\| ^{2}+\left( {p-1} \right) \frac{\left( {\mathbf{z}_2^T \mathbf{z}_1 } \right) \left( {\mathbf{z}_1^T \mathbf{z}_2 } \right) }{\left\| {\mathbf{z}_1 } \right\| ^{2}}} \right) \right] \nonumber \\&\quad +\,\left[ -k_1 \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left( {\varvec{\Delta }_1^T \mathbf{z}_2 +\left( {p-1} \right) \frac{\left( {\Delta _1^T \mathbf{z}_1 } \right) \left( {\mathbf{z}_1^T \mathbf{z}_2 } \right) }{\left\| {\mathbf{z}_1 } \right\| ^{2}}} \right) \right. \nonumber \\&\quad \left. +\,2\mathbf{z}_2^T \varvec{\Delta }_2 -k_1 \mathbf{z}_1^T \varvec{\Delta }_2 \right. \nonumber \\&\quad +\,\left. {\left( {2k_2 +k_1^2 } \right) p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1^T \varvec{\Delta }_1 } \right] \nonumber \\&\quad =\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left[ -\left( {k_2 +k_1^2 } \right) k_1 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left\| {\mathbf{z}_1 } \right\| ^{2}\right. \nonumber \\&\quad \left. +\,2k_1^2 p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\mathbf{z}_1^T \mathbf{z}_2 -k_1 \left\| {\mathbf{z}_2 } \right\| ^{2}\right. \nonumber \\&\quad \left. +k_1 \left( {1-p} \right) \frac{\left( {\mathbf{z}_2^T \mathbf{z}_1 } \right) \left( {\mathbf{z}_1^T \mathbf{z}_2 } \right) }{\left\| {\mathbf{z}_1 } \right\| ^{2}} \right] \nonumber \end{aligned}$$
$$\begin{aligned}&\quad +\,\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left[ \left( {2k_2 +k_1^2 } \right) p\mathbf{z}_1^T \varvec{\Delta }_1 -k_1 \mathbf{z}_1^T \varvec{\Delta }_2\right. \nonumber \\&\quad \left. +\,k_1 \left( {1-p} \right) \frac{\left( {\Delta _1^T \mathbf{z}_1 } \right) \left( {\mathbf{z}_1^T \mathbf{z}_2 } \right) }{\left\| {\mathbf{z}_1 } \right\| ^{2}} \right] \nonumber \\&\quad +\,\left( {2\mathbf{z}_2 -k_1 \mathbf{z}_2 } \right) ^{T}\varvec{\Delta }_2 \end{aligned}$$
(41)

Taking into account \(p\in \left( {0.5,1} \right) \) and using Cauchy–Schwarz inequality on the inner product terms, we have

$$\begin{aligned} V&\le \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left[ -\left( {k_2 +k_1^2 } \right) k_1 p\left\| {\mathbf{z}_1 } \right\| ^{2}+2k_1^2 p\left\| {\mathbf{z}_1 } \right\| \right. \nonumber \\&\left. \quad \times \left\| {\mathbf{z}_2 } \right\| -\,k_1\left\| {\mathbf{z}_2 } \right\| ^{2}+k_1 \left( {1-p} \right) \left\| {\mathbf{z}_2 } \right\| ^{2}\right] \nonumber \\&\quad +\,\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left[ \left( {2k_2 +k_1^2 } \right) p\left\| {\mathbf{z}_1 } \right\| \left\| {\varvec{\Delta }_1 } \right\| \right. \nonumber \\&\left. \quad +\,k_1 \left\| {\varvec{\Delta }_1 } \right\| \left\| {\mathbf{z}_2 } \right\| +k_1 \left( {1-p} \right) \left\| {\varvec{\Delta }_1 } \right\| \left\| {\mathbf{z}_2 } \right\| \right] \nonumber \\&\quad +\,\left\| {\left( {2\mathbf{z}_2 -k_1 \mathbf{z}_2 } \right) } \right\| \left\| {\varvec{\Delta }_2 } \right\| \nonumber \\&\le p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left[ -\left( {k_2 +k_1^2 } \right) k_1 \left\| {\mathbf{z}_1 } \right\| ^{2}\right. \nonumber \\&\left. \quad +\,2k_1^2 \left\| {\mathbf{z}_1 } \right\| \left\| {\mathbf{z}_2 } \right\| -k_1 \left\| {\mathbf{z}_2 } \right\| ^{2} \right] \nonumber \\&\quad +\,\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left[ \left( {2k_2 +k_1^2 } \right) p\left\| {\mathbf{z}_1 } \right\| \right. \nonumber \\&\left. \quad +\,\left( {2k_1 -k_1 p}\right) \left\| {\mathbf{z}_2 } \right\| \right] \left\| {\varvec{\Delta }_1 } \right\| \nonumber \\&\quad +\,\sqrt{4+k_1^2 }\sqrt{\left\| {\mathbf{z}_1 } \right\| ^{2}+\left\| {\mathbf{z}_2 } \right\| ^{2}}\left\| {\varvec{\Delta }_2 } \right\| \nonumber \\&\le -p\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left[ \left( {k_2 +k_1^2 } \right) k_1 \left\| {\mathbf{z}_1 } \right\| ^{2}-2k_1^2 \right. \nonumber \\&\left. \quad \,\left\| {\mathbf{z}_1 } \right\| \left\| {\mathbf{z}_2 } \right\| +k_1 \left\| {\mathbf{z}_2 } \right\| ^{2} \right] \nonumber \\&\quad +\,\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\sqrt{\left( {2k_2 +k_1^2 } \right) ^{2}p^{2}+\left( {2k_1 -k_1 p} \right) ^{2}}\nonumber \\&\quad \,\sqrt{\left\| {\mathbf{z}_1 } \right\| ^{2}+\left\| {\mathbf{z}_2 } \right\| ^{2}}\left\| {\varvec{\Delta }_1 } \right\| \nonumber \\&\quad +\,\sqrt{4+k_1^2 }\sqrt{\left\| {\mathbf{z}_1 } \right\| ^{2}+\left\| {\mathbf{z}_2 } \right\| ^{2}}\left\| {\varvec{\Delta }_2 } \right\| \end{aligned}$$
(42)

Let \(\mathbf{z}=\left[ {\left\| {\mathbf{z}_1 } \right\| \,\left\| {\mathbf{z}_2 } \right\| } \right] ^{T}\), it is obvious that the first item on the right-hand side of (42) can be rewritten as \(-p\left\| {\mathbf{z}_\mathbf{1} } \right\| ^{\frac{p-1}{p}}\mathbf{z}^{T}\mathbf{Qz}\) with positive definite matrix \(\mathbf{Q}=\left[ {{\begin{array}{ll} {\left( {k_2 +k_1^2 } \right) k_1 }&{} {-k_1^2 } \\ {-k_1^2 }&{} {k_1 } \\ \end{array} }} \right] \) for any positive constants \(k_1 >0\,\hbox {and}\,k_2 >0\). In addition, it is easily verified that \(\lambda _\mathrm{mim} \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| ^{2}\le \mathbf{z}^{T}\mathbf{Qz}\le \lambda _{\max } \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| ^{2}\) with \(\lambda _\mathrm{mim} \left( \mathbf{Q} \right) \) and \(\lambda _{\max } \left( \mathbf{Q} \right) \) being the minimum and maximum eigenvalue of matrix \(\mathbf{Q}\), respectively. Furthermore, in view of bounds on the terms \(\left\| {\varvec{\Delta }_1 } \right\| \) and\(\left\| {\varvec{\Delta }_2 } \right\| \), we have

$$\begin{aligned} \dot{V}&\le -p\left\| {\mathbf{z}_\mathbf{1} } \right\| ^{\frac{p-1}{p}}\lambda _{\min } \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| ^{2}\nonumber \\&\quad +\,\left\| {\mathbf{z}_\mathbf{1} } \right\| ^{\frac{p-1}{p}}\sqrt{\left( {2k_2 +k_1^2 } \right) ^{2}p^{2}+\left( {2k_1 -k_1 p} \right) ^{2}}\nonumber \\&\quad \times \left\| \mathbf{z} \right\| \delta _1+\,\sqrt{4+k_1^2 }\left\| \mathbf{z} \right\| \delta _2 \end{aligned}$$
(43)

Next, we want to use part of \(-p\left\| {\mathbf{z}_\mathbf{1} } \right\| ^{\frac{p-1}{p}}\lambda _{mim} \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| ^{2}\) to dominate the last two items on the right-hand side of (43) for large \(\left\| \mathbf{z} \right\| \). To this end, we rewrite the inequality (43) as

$$\begin{aligned} \dot{V}&\le -p\left( {1-\theta _1 -\theta _2 } \right) \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\lambda _{\min } \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| ^{2}\nonumber \\&\quad +\,\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\left( \sqrt{\left( {2k_2 +k_1^2 } \right) ^{2}p^{2}+\left( {2k_1 -k_1 p} \right) ^{2}}\delta _1\right. \nonumber \\&\quad \left. -\,\theta _1 \lambda _{\min } \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| \right) \left\| \mathbf{z} \right\| \nonumber \\&\quad +\,\left( {\sqrt{4+k_1^2 }\delta _2 -\theta _2 \left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\lambda _{\min } \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| } \right) \left\| \mathbf{z} \right\| \nonumber \\ \end{aligned}$$
(44)

where \(\theta _1\,\hbox {and}\,\theta _2 \) are positive constants and satisfy \(0<\theta _1+\theta _2 <1\). Furthermore, let \(\mathbf{Z}=\left[ {\mathbf{z}_1;\,\mathbf{z}_2 } \right] \). It follows from the definition of Lyapunov function in (39) that it can be rewritten as \(V\left( {\mathbf{z}_1,\,\mathbf{z}_2 } \right) =\mathbf{Z}^{T}\mathbf{PZ}\) for an appropriate symmetric positive definite matrix \(\mathbf{P}\in R^{2m\times 2m}\). Therefore, the following inequality is satisfied for any vector \(\mathbf{Z}\)

$$\begin{aligned} \lambda _{\min } \left( \mathbf{P} \right) \left\| \mathbf{Z} \right\| ^{2}\le V\le \lambda _{\max } \left( \mathbf{P} \right) \left\| \mathbf{Z} \right\| ^{2} \end{aligned}$$
(45)

where \(\lambda _{\min } \left( \mathbf{P} \right) \) and \(\lambda _{\max } \left( \mathbf{P} \right) \) represent the minimum and maximum eigenvalue of matrix \(\mathbf{P}\), respectively. In addition, it can be observed from the definition of \(\mathbf{z}\,\hbox {and}\,\mathbf{Z}\) that \(\left\| {\mathbf{z}} \right\| =\left\| \mathbf{Z} \right\| \). Therefore, based on (45), we have inequality \(\frac{V^{1/2}}{\left[ {\lambda _{\mathrm{max}} \left( \mathbf{P} \right) } \right] ^{1/2}}\le \left\| \mathbf{z} \right\| =\left\| \mathbf{Z} \right\| =\le \frac{V^{1/2}}{\left[ {\lambda _{\min } \left( \mathbf{P} \right) } \right] ^{1/2}}\). Since \(0.5<p<1\), it can be concluded that \(\left\| {\mathbf{z}_\mathbf{1} } \right\| ^{\frac{p-1}{p}}\ge \left\| \mathbf{z} \right\| ^{\frac{p-1}{p}}\ge \frac{V^{\left( {p-1} \right) /\left( {2p} \right) }}{\left[ {\lambda _{\max } \left( \mathbf{P} \right) } \right] ^{\left( {p-1} \right) /\left( {2p} \right) }}\). Then, inequality (44) satisfies

$$\begin{aligned} \dot{V}&\le -p\left( {1-\theta _1 -\theta _2 } \right) \frac{\lambda _{\mathrm{min}} \left( \mathbf{Q} \right) }{\left[ {\lambda _{\max } \left( \mathbf{P} \right) } \right] \left[ {\lambda _{\min } \left( \mathbf{P} \right) } \right] ^{\left( {p-1} \right) /\left( {2p} \right) }}\nonumber \\&\quad \times V^{\left( {3p-1} \right) /\left( {2p} \right) }+\left\| {\mathbf{z}_1 } \right\| ^{\frac{p-1}{p}}\nonumber \\&\quad \times \left( \sqrt{\left( {2k_2 +k_1^2 } \right) ^{2}p^{2}+\left( {2k_1 -k_1 p} \right) ^{2}}\delta _1\right. \nonumber \\&\quad \quad \left. -\theta _1 \lambda _{\min } \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| \right) \left\| \mathbf{z}\right\| \nonumber \\&\quad +\,\left( {\sqrt{4+k_1^2 }\delta _2 -\theta _2 \lambda _{\min } \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| ^{\frac{2p-1}{p}}} \right) \left\| \mathbf{z} \right\| \nonumber \\ \end{aligned}$$
(46)

Then,

$$\begin{aligned}&\dot{V}\le -p\left( {1-\theta _1 -\theta _2 } \right) \frac{\lambda _{\mathrm{min}} \left( \mathbf{Q} \right) }{\left[ {\lambda _\mathrm{max} \left( \mathbf{P} \right) } \right] \left[ {\lambda _{\mathrm{min}} \left( \mathbf{P} \right) } \right] ^{\left( {p-1} \right) /\left( {2p} \right) }}V^{\left( {3p-1} \right) /\left( {2p} \right) },\forall \left\| \mathbf{z} \right\| \in \Xi _1 \nonumber \\&\quad =\left\{ {\left\| \mathbf{z} \right\| \ge \underbrace{\hbox {max}\left( {\frac{\sqrt{\left( {2k_2 +k_1^2 } \right) ^{2}p^{2}+\left( {2k_1 -k_1 p} \right) ^{2}}\delta _1 }{\theta _1 \lambda _{\mathrm{min}} \left( \mathbf{Q} \right) },\left( {\frac{\sqrt{4+k_1^2 }\delta _2 }{\theta _2 \lambda _{\mathrm{min}} \left( \mathbf{Q} \right) }} \right) ^{p/\left( {2p-1} \right) }} \right) }_\Upsilon } \right\} \end{aligned}$$
(47)

It follows from (47) and theorem 4.18 in [49] that the solution for [A3] is globally bounded. Furthermore, inequality (A8) is reduced to

$$\begin{aligned}&\dot{V}\le -p\left\| {\mathbf{z}_\mathbf{1} } \right\| ^{\frac{p-1}{p}}\lambda _{\min } \left( \mathbf{Q} \right) \left\| \mathbf{z} \right\| ^{2}\nonumber \\&\quad \le \frac{\lambda _{\min } \left( \mathbf{Q} \right) }{\left[ {\lambda _{\min } \left( \mathbf{P} \right) \lambda _{\min } \left( \mathbf{P} \right) } \right] ^{\frac{p-1}{2p}}}V^{\frac{3p-1}{2p}} \end{aligned}$$
(48)

if \(\varvec{\Delta }_1 =\varvec{\Delta }_2 =0\). It follows from \(0.5<p<1\) that we obtain \(\frac{3p-1}{2p}\in \left( {0.5,1} \right) \). According to Lemma 1, it can be concluded that \(\mathbf{z}_1 \), \(\mathbf{z}_2 \rightarrow 0\) in finite time. This completes the proof.\(\square \)

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Tian, B., Fan, W. & Zong, Q. Integrated guidance and control for reusable launch vehicle in reentry phase. Nonlinear Dyn 80, 397–412 (2015). https://doi.org/10.1007/s11071-014-1877-0

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