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Reliability-based linear parameter varying robust non-fragile control for hypersonic vehicles with disturbance observer

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Abstract

In this paper, the linear parameter varying (LPV) reliable non-fragile control for the hypersonic vehicle (HV) is studied in case of disturbance and controller gain variations. Due to the dramatic and complex change of the HV longitudinal dynamics, a polytopic LPV model is constructed for the HV system stability analysis and controller design in a large flight envelope. Then, a disturbance observer (DOB)-based non-fragile controller for HV system with disturbance and unknown controller uncertainty is designed to guarantee the closed-loop stability and control performance under an adequate level of reliability, which is formed with two parts. One part is a DOB to compensate the uncertain dynamics and disturbance. The other is a robust non-fragile controller, which is designed based on a novel robust reliability method to deal with controller uncertainties, and obtained by carrying out reliability-based linear matrix inequality optimization. The presented controller for HV can provide the robustness as well as an excellent performance under the condition that the prescribed reliability degree is satisfied. Finally, numerical simulation for an HV demonstrates the effectiveness of the proposed method.

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Acknowledgements

This work was supported in part by the National Nature Science Foundation of China (Grant Nos. 61473124 and 61573161).

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Correspondence to Lei Liu.

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Appendices

Appendix 1

$$A(V,h) = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } & 0 \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } & 0 \\ {a_{31} } & {a_{32} } & 0 & 0 & 0 \\ {a_{41} } & {a_{42} } & {a_{43} } & {a_{44} } & 1 \\ {a_{51} } & 0 & {a_{53} } & {a_{54} } & {a_{55} } \\ \end{array} } \right], \, B(V,h) = \left[ {\begin{array}{*{20}c} 0 & {b_{12} } \\ 0 & {b_{22} } \\ 0 & 0 \\ 0 & {b_{42} } \\ {b_{51} } & 0 \\ \end{array} } \right], \, C(V,h) = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \end{array} } \right],$$

where

$$\begin{aligned} a_{11} & = \frac{1}{m}\left( {\frac{\partial T}{\partial V}\cos \alpha (V,h) - \frac{\partial D}{\partial V}} \right), \, a_{12} = - g\cos \gamma (V,h), \\ a_{13} & = \frac{1}{m}\left( {\frac{\partial T}{\partial h}\cos \alpha (V,h) - \frac{\partial D}{\partial h}} \right) + \frac{2g\sin \gamma (V,h)}{r}, \, a_{14} = - \frac{1}{m}\left( {T\sin \alpha (V,h) + \frac{\partial D}{\partial \alpha }} \right), \\ a_{21} & = \frac{1}{mV}\left( {\frac{\partial L}{\partial V} + \frac{\partial T}{\partial V}\sin \alpha (V,h)} \right) - \frac{1}{{mV^{2} }}\left( {L + T\sin \alpha (V,h)} \right) + \left( {\frac{g}{{V^{2} }} + \frac{1}{r}} \right)\cos \gamma (V,h), \\ a_{22} & = \left( {\frac{g}{V} - \frac{V}{r}} \right)\sin \gamma (V,h), \, a_{23} = \frac{1}{mV}\left( {\frac{\partial L}{\partial h} + \frac{\partial T}{\partial h}\sin \alpha (V,h)} \right) + \left( {\frac{2g}{Vr} - \frac{V}{{r^{2} }}} \right)\cos \gamma (V,h), \\ a_{24} & = \frac{1}{mV}\left( {T\cos \alpha (V,h) + \frac{\partial L}{\partial \alpha }} \right), \, a_{31} = \sin \gamma (V,h), \, a_{32} = V\cos \gamma (V,h), \\ a_{41} & = - a_{21} , \, a_{42} = - a_{22} , \, a_{43} = - a_{23} , \, a_{44} = - a_{24} , \\ \end{aligned}$$
$$\begin{aligned} a_{51} & = \frac{1}{{I_{yy} }}\frac{{\partial M_{yy} }}{\partial V}, \, a_{53} = \frac{1}{{I_{yy} }}\frac{{\partial M_{yy} }}{\partial h}, \, a_{54} = \frac{1}{{I_{yy} }}\frac{{\partial M_{yy} }}{\partial \alpha }, \, a_{55} = \frac{1}{{I_{yy} }}\frac{{\partial M_{yy} }}{\partial q}, \\ b_{12} & = \frac{1}{m}\frac{\partial T}{\partial \beta }\cos \alpha (V,h), \, b_{22} = \frac{1}{mV}\frac{\partial T}{\partial \beta }\sin \alpha (V,h), \, b_{42} = - b_{22} , \, b_{51} = \frac{1}{{I_{yy} }}\frac{{\partial M_{yy} }}{{\partial \delta_{e} }}. \\ \end{aligned}$$

Appendix 2

Proof of Theorem 1

For \(Q = Q^{T} \ge 0\) and \(R = R^{T} \ge 0\), it can be seen from (26) that

$$P\tilde{A}_{c} (\omega ) + \tilde{A}_{c}^{T} (\omega )P + \varepsilon^{ - 1} P\tilde{G}\tilde{G}^{T} P < 0$$
(68)

For the control system (22), the Lyapunov function can be selected as follows

$$V(\tilde{x}) = \tilde{x}^{T} P\tilde{x}$$
(69)

then take the time derivative of \(V(\tilde{x})\), and we obtain

$$\begin{aligned} \dot{V}(\tilde{x}) = \dot{\tilde{x}}^{T} P\tilde{x} + \tilde{x}^{T} P\dot{\tilde{x}} \hfill \\ = \tilde{x}^{T} \left( {P\tilde{A}_{c} (\omega ) + \tilde{A}_{c}^{T} (\omega )P} \right)\tilde{x} + y_{r}^{T} \tilde{G}^{T} P\tilde{x} + \tilde{x}^{T} P\tilde{G}y_{r} . \hfill \\ \end{aligned}$$
(70)

According to Lemma 1 [39], we have

$$y_{r}^{T} \tilde{G}^{T} P\tilde{x} + \tilde{x}^{T} P\tilde{G}y_{r} \le \varepsilon^{ - 1} \tilde{x}^{T} P\tilde{G}\tilde{G}^{T} P\tilde{x} + \varepsilon y_{r}^{T} y_{r} ,$$
(71)

then

$$\dot{V}(\tilde{x}) \le \tilde{x}^{T} \left( {P\tilde{A}_{c} (\omega ) + \tilde{A}_{c}^{T} (\omega )P + \varepsilon^{ - 1} P\tilde{G}\tilde{G}^{T} P} \right)\tilde{x} + \varepsilon y_{r}^{T} y_{r} ,$$
(72)

from this and Eq. (68), we have \(\dot{V}(\tilde{x}) < 0\). Therefore, on the basis of the Lyapunov stability theory, the control system (22) is asymptotically stable with \(y_{r} (t) = 0\).Then, we consider the performance index \(J\) in (25)

$$\begin{aligned} J & = \int_{0}^{{t_{f} }} {\left( {\tilde{x}^{T} Q\tilde{x} + \bar{x}^{T} \bar{K}^{T} (\omega )R\bar{K}(\omega )\bar{x}} \right)d{\kern 1pt} t} \\ & = \int_{0}^{{t_{f} }} {\tilde{x}^{T} \left( {Q + \tilde{K}^{T} (\omega )R\tilde{K}(\omega )} \right)\tilde{x}{\kern 1pt} d{\kern 1pt} t} \\ & < - \int_{0}^{{t_{f} }} {\tilde{x}^{T} \left( {P\tilde{A}_{c} (\omega ) + \tilde{A}_{c}^{T} (\omega )P + \varepsilon^{ - 1} P\tilde{G}\tilde{G}^{T} P} \right)\tilde{x}{\kern 1pt} d{\kern 1pt} t} \\ & < - \int_{0}^{{t_{f} }} {\left( {\dot{V}(\tilde{x}) - \varepsilon y_{r}^{T} y_{r} } \right)d{\kern 1pt} t} \\ & = V(\tilde{x}(0)) - V(\tilde{x}(t_{f} )) + \varepsilon \int_{0}^{{t_{f} }} {y_{r}^{T} y_{r} {\kern 1pt} d{\kern 1pt} t} \\ & \le \tilde{x}^{T} (0)P\tilde{x}(0) + \varepsilon \int_{0}^{{t_{f} }} {y_{r}^{T} y_{r} {\kern 1pt} d{\kern 1pt} t} = J_{b} \\ \end{aligned}$$
(73)

The proof is completed.□

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Wei, X., Liu, L. & Wang, Y. Reliability-based linear parameter varying robust non-fragile control for hypersonic vehicles with disturbance observer. Cluster Comput 22 (Suppl 3), 6709–6728 (2019). https://doi.org/10.1007/s10586-018-2528-x

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