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Nonlinear fuzzy fault-tolerant control of hypersonic flight vehicle with parametric uncertainty and actuator fault

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Abstract

This study presents a nonlinear fuzzy fault-tolerant control (FTC) and a fault observer for longitudinal dynamics of hypersonic flight vehicle (HFV) with parameter uncertainty and actuator gain loss fault via sliding-mode and backstepping theory. An affine nonlinear dynamic model of HFV with parameter uncertainty and actuator fault is established based on feedback linearization technology. A nominal sliding-mode control is developed to track the command of altitude and velocity. Unknown nonlinear functions in the controller are approximated by fuzzy logic system through updating the weight parameters online. In view of the occurrence of actuator fault, a backstepping sliding-mode observer is constructed to estimate the fault. A nonlinear fuzzy FTC is then designed with the estimate fault obtained from the observer to address the problem of actuator fault and parameter uncertainty. The stability of the controller is analyzed utilizing Lyapunov theory. Numerical simulation results demonstrate the validity and robustness of the proposed controller and observer.

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Abbreviations

\(\alpha \) :

Angle of attack, rad

\(\bar{c}\) :

Reference length, 80 ft

\(\beta _{\mathrm{Tc}}\) :

Desirable throttle setting, \(\%/100\)

\(\beta _{\mathrm{T}}\) :

Throttle setting, \(\%/100\)

\(\delta _{\mathrm{e}}\) :

Elevator deflection, rad

\(\gamma \) :

Flight-path angle, rad

\(\mu \) :

Gravitational constant, ft \(^3\)/s\(^2\)

\(\omega _{\mathrm{n}}\) :

Frequency of engine system, rad/s

\(\rho \) :

Density of air, slug/ft\(^3\)

\(\xi _\mathrm{n}\) :

Damping of engine system

\(c_\mathrm{e}\) :

Constant, 0.0292

\(C_{\mathrm{D}}\) :

Drag coefficient

\(C_{\mathrm{L}}\) :

Lift coefficient

\(C_{\mathrm{M}}(\alpha )\) :

Moment coefficient due to angle of attack

\(C_{\mathrm{M}}(\delta _{\mathrm{e}})\) :

Moment coefficient due to elevator deflection

\(C_{\mathrm{M}}(q)\) :

Moment coefficient due to pitch rate

\(C_{\mathrm{T}}\) :

Thrust coefficient

D :

Drag, lbf

h :

Altitude, ft

\(h_\mathrm{d}\) :

Reference altitude, ft

\(I_{\mathrm{yy}}\) :

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

L :

Lift, lbf

m :

Mass, slug

\(M_{\mathrm{yy}}\) :

Pitching moment, lbf-ft

q :

Pitch rate, rad/s

r :

Radial distance from Earth’s center, ft

\(R_\mathrm{e}\) :

Earth radius, ft

S :

Reference aerodynamic area, ft\(^2\)

T :

Thrust, lbf

V :

Velocity, ft/s

\(V_\mathrm{d}\) :

Reference velocity, ft/s

HFV:

Hypersonic flight vehicle

FTC:

Fault-tolerant control

SMC:

Sliding-mode control

SMO:

Sliding-mode observer

FLS:

Fuzzy logic system

NN:

Neural networks

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Acknowledgements

The project was supported by the National Natural Science Foundation of China (61533009, 61473146), a project funded by the Priority Academic Programme Development of Jiangsu Higher Education Institutions.

Author information

Authors and Affiliations

  1. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Juan Niu & Fuyang Chen

  2. Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, 22903, USA

    Gang Tao

Authors
  1. Juan Niu
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  2. Fuyang Chen
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  3. Gang Tao
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Corresponding author

Correspondence to Fuyang Chen.

Appendix

Appendix

The detailed expressions of the vectors about feedback linearization matrices are as follows:

$$\begin{aligned} \ddot{\gamma }= & {} {\pi _1}\dot{z}\\ \dddot{\gamma }= & {} {\pi _1}\ddot{z} + {\dot{z}^T}{\pi _2}\dot{z}\\ {{\ddot{\alpha }}_0}= & {} - \,{\pi _1}\dot{z} + \frac{{0.5\rho {V^2}S\bar{c}}}{{{I_\mathrm{yy}}}}({C_\mathrm{M}}(\alpha ) + {C_\mathrm{M}}(q) - 0.0292\alpha ) \\ {{\ddot{\alpha }}_{{\delta _\mathrm{e}}}}= & {} 0.0292\frac{{0.5\rho {V^2}S\bar{c}}}{{{I_\mathrm{yy}}}} \\ \ddot{\alpha }= & {} {\ddot{\alpha }_0} + {\ddot{\alpha }_{{\delta _\mathrm{e}}}}{\delta _\mathrm{e}} \\ {\ddot{\beta }_\mathrm{T}}= & {} -\, 2{\xi _\mathrm{n}}{\omega _\mathrm{n}}{\dot{\beta }_\mathrm{T}} - \omega _\mathrm{n}^2{\beta _\mathrm{T}} + \omega _\mathrm{n}^2{\beta _{\mathrm{Tc}}} = {\ddot{\beta }_0} + \omega _\mathrm{n}^2{\beta _{\mathrm{Tc}}} \\ {z_0}= & {} \left[ {\begin{array}{*{20}{c}} {\ddot{V}}\quad {\ddot{\gamma }} \quad {{{\ddot{\alpha }}_0}} \quad {{{\ddot{\beta }}_0}} \quad {\ddot{h}} \end{array}} \right] \\ \ddot{z}= & {} {\ddot{z}_0} + {u^T}{\ddot{z}_u}\\ {F_V}= & {} \frac{{{\omega _1}{{\ddot{z}}_0} + {{\dot{z}}^T}{\omega _2}\dot{z}}}{m}\\ {F_h}= & {} 3\ddot{V}\dot{\gamma }\cos \gamma - 3\dot{V}{{\dot{\gamma }}^2}\sin \gamma + 3\dot{V}\ddot{\gamma }\cos \gamma \\&-\,3V\dot{\gamma }\ddot{\gamma }\sin \gamma - V{{\dot{\gamma }}^3}\cos \gamma + {F_v}\sin \gamma \\&+\, V({\pi _1}{{\ddot{z}}_0} +{{\dot{z}}^T}{\pi _2}\dot{z})cos\gamma \\ {g_{11}}= & {} \frac{{\frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\cos \alpha }}{m}\omega _\mathrm{n}^2\\ {g_{12}}= & {} \frac{{\frac{{\partial T}}{{\partial \alpha }}\cos \alpha - T\sin \alpha - \frac{{\partial D}}{{\partial \alpha }}}}{m}{{\ddot{\alpha }}_{{\delta _\mathrm{e}}}}\\ {g_{21}}= & {} \frac{{\frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\sin (\alpha + \gamma )}}{m}\omega _\mathrm{n}^2\\ {g_{22}}= & {} \frac{{\ddot{\alpha }}_{{\delta _\mathrm{e}}}}{m}\left( {{\frac{{\partial T}}{{\partial \alpha }}}}\sin (\alpha + \gamma ) + T\cos (\alpha + \gamma )\right. \\&+ \left. {{\frac{{\partial L}}{{\partial \alpha }}} }\cos \gamma - {{\frac{{\partial D}}{{\partial \alpha }}} }\sin \gamma \right) \\ {\omega _1}= & {} {\left[ {\begin{array}{*{20}{c}} {\frac{{\partial T}}{{\partial V}}\cos \alpha - \frac{{\partial D}}{{\partial V}}}\\ { - \,\frac{{m\mu \cos \gamma }}{{{r^2}}}}\\ { -\, T\sin \alpha - \frac{{\partial D}}{{\partial \alpha }}}\\ {\frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\cos \alpha }\\ {\frac{{2m\mu \sin \gamma }}{{{r^3}}}} \end{array}} \right] ^T}\\ {\omega _2}= & {} \left[ {\begin{array}{*{20}{c}} {{\omega _{21}}}&{{\omega _{22}}}&{{\omega _{23}}}&{{\omega _{24}}}&{{\omega _{25}}} \end{array}} \right] \\ {\omega _{21}}= & {} \left[ {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}T}}{{{\partial ^2}V}}\cos \alpha - \frac{{{\partial ^2}D}}{{{\partial ^2}V}}}\\ 0\\ { - \,\frac{{\partial T}}{{\partial V}}\sin \alpha - \frac{{{\partial ^2}D}}{{\partial V\partial \alpha }}}\\ {\frac{{{\partial ^2}T}}{{\partial V\partial {\beta _\mathrm{T}}}}\cos \alpha }\\ 0 \end{array}} \right] \\ {\omega _{22}}= & {} \left[ {\begin{array}{*{20}{c}} 0\\ {\frac{{m\mu \sin \gamma }}{{{r^2}}}}\\ 0\\ 0\\ {\frac{{2m\mu \cos \gamma }}{{{r^3}}}} \end{array}} \right] \\ \end{aligned}$$
$$\begin{aligned} {\omega _{23}}= & {} \left[ {\begin{array}{*{20}{c}} { - \,\frac{{\partial T}}{{\partial V}}\sin \alpha - \frac{{{\partial ^2}D}}{{\partial V\partial \alpha }}}\\ 0\\ { -\, T\cos \alpha - \frac{{{\partial ^2}D}}{{{\partial ^2}\alpha }}}\\ { -\, \frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\sin \alpha }\\ 0 \end{array}} \right] \\ {\omega _{24}}= & {} \left[ {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}T}}{{\partial V\partial {\beta _\mathrm{T}}}}\cos \alpha }\\ 0\\ { -\, \frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\sin \alpha }\\ 0\\ 0 \end{array}} \right] \\ {\omega _{25}}= & {} \left[ {\begin{array}{*{20}{c}} 0\\ {\frac{{2m\mu \cos \mathrm{{\gamma }}}}{{{r^3}}}}\\ 0\\ 0\\ { -\, \frac{{6m\mu \mathrm{{cos\gamma }}}}{{{r^4}}}} \end{array}} \right] \\ {\pi _1}= & {} {\left[ {\begin{array}{*{20}{c}} {\frac{{\frac{{\partial L}}{{\partial V}} + \frac{{\partial T}}{{\partial V}}\sin \alpha }}{{mV}} - \frac{{L + T\sin \alpha }}{{m{V^2}}} + \frac{{\mu \cos \gamma }}{{{V^2}{r^2}}} + \frac{{\cos \gamma }}{r}}\\ {\frac{{\mu \sin \gamma }}{{V{r^2}}} - \frac{{V\sin \gamma }}{r}}\\ {\frac{{\frac{{\partial L}}{{\partial \alpha }} + T\cos \alpha }}{{mV}}}\\ {\frac{{\frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\sin \alpha }}{{mV}}}\\ {\frac{{2\mu \cos \gamma }}{{V{r^3}}} - \frac{{V\cos \gamma }}{{{r^2}}}} \end{array}} \right] ^T}\\ {\pi _2}= & {} \left[ {\begin{array}{*{20}{c}} {{\pi _{21}}}&{{\pi _{22}}}&{{\pi _{23}}}&{{\pi _{24}}}&{{\pi _{25}}} \end{array}} \right] \\ {\pi _{22}}= & {} \left[ {\begin{array}{*{20}{c}} { -\, \frac{{\mu \sin \gamma }}{{{V^2}{r^2}}} - \frac{{\sin \gamma }}{r}}\\ {\frac{{\mu \cos \gamma }}{{V{r^2}}} - \frac{{V\cos \gamma }}{r}}\\ 0\\ 0\\ { - \,\frac{{2\mu \sin \gamma }}{{V{r^3}}} + \frac{{V\sin \gamma }}{{{r^2}}}} \end{array}} \right] \\ {\pi _{23}}= & {} \left[ {\begin{array}{*{20}{c}} {\frac{{\frac{{{\partial ^2}L}}{{\partial \alpha \partial V}} + \left( \frac{{\partial T}}{{\partial V}}\right) \cos \alpha }}{{mV}} - \frac{{\frac{{\partial L}}{{\partial \alpha }} + T\cos \alpha }}{{m{V^2}}}}\\ 0\\ {\frac{{\frac{{{\partial ^2}L}}{{{\partial ^2}\alpha }} - T\sin \alpha }}{{mV}}}\\ {\frac{{\left( \frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\right) \cos \alpha }}{{mV}}}\\ 0 \end{array}} \right] \\ {\pi _{24}}= & {} \left[ {\begin{array}{*{20}{c}} {\frac{{\left( \frac{{{\partial ^2}T}}{{\partial {\beta _\mathrm{T}}\partial V}}\right) \sin \alpha }}{{mV}} - \frac{{\left( \frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\right) \sin \alpha }}{{m{V^2}}}}\\ 0\\ {\frac{{\left( \frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}}\right) \cos \alpha }}{{mV}}}\\ 0\\ 0 \end{array}} \right] \\ \end{aligned}$$
$$\begin{aligned} {\pi _{25}}= & {} \left[ {\begin{array}{*{20}{c}} { -\, \frac{{2\mu \cos \gamma }}{{{V^2}{r^3}}} - \frac{{\cos \gamma }}{{{r^2}}}}\\ { -\, \frac{{2\mu \sin \gamma }}{{V{r^3}}} + \frac{{V\sin \gamma }}{{{r^2}}}}\\ 0\\ 0\\ { -\, \frac{{6\mu \cos \gamma }}{{V{r^4}}} + \frac{{2V\cos \gamma }}{{{r^3}}}} \end{array}} \right] \\ {\pi _{21}}= & {} \left[ {\begin{array}{*{20}{c}} \begin{array}{c} \frac{{\frac{{{\partial ^2}L}}{{{\partial ^2}V}} \,+\, \left( \frac{{{\partial ^2}T}}{{{\partial ^2}V}}\right) \sin \alpha }}{{mV}} - \frac{{2\left( \frac{{\partial L}}{{\partial V}} \,+\, \left( \frac{{\partial T}}{{\partial V}}\right) \sin \alpha \right) }}{{m{V^2}}} \\ +\, \frac{{2(L \,+\, T\sin \alpha )}}{{m{V^3}}} - \frac{{2\rho \cos \gamma }}{{{V^3}{r^2}}} \\ \end{array} \\ { -\, \frac{{\rho \sin \gamma }}{{{V^2}{r^2}}} - \frac{{\sin \gamma }}{r}} \\ {\frac{{\frac{{{\partial ^2}L}}{{\partial \alpha \partial V}} \,+\, \left( \frac{{\partial T}}{{\partial V}}\right) \cos \alpha }}{{mV}} - \frac{{\frac{{\partial L}}{{\partial \alpha }} \,+\, T\cos \alpha }}{{m{V^2}}}} \\ {\frac{{\left( \frac{{{\partial ^2}T}}{{\partial {\beta _\mathrm{T}}\partial V}}\right) \sin \alpha }}{{mV}} - \frac{{(\frac{{\partial T}}{{\partial {\beta _\mathrm{T}}}})\sin \alpha }}{{m{V^2}}}} \\ { -\, \frac{{2\rho \cos \gamma }}{{{V^2}{r^3}}} - \frac{{\cos \gamma }}{{{r^2}}}} \\ \end{array}} \right] \end{aligned}$$

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Niu, J., Chen, F. & Tao, G. Nonlinear fuzzy fault-tolerant control of hypersonic flight vehicle with parametric uncertainty and actuator fault. Nonlinear Dyn 92, 1299–1315 (2018). https://doi.org/10.1007/s11071-018-4127-z

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