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Modeling and robust decoupling control for hypersonic scramjet vehicle

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Abstract

In this paper, the modeling and the robust decoupling control for a generic hypersonic scramjet vehicle are studied. Firstly, the dynamics of the hypersonic vehicle are modeled by applying the Lagrangian approach, which captures the most primary characteristics such as elastic deformation, aerodynamics, aero-heating, variable mass, effect of spherical rotating earth and their inherent interactions. Then, a robust output decoupling controller is designed by using nonlinear dynamic inversion plus the desired proportional integral dynamics, and natural time-scale separation theorem between fast and slow variables. Finally, the nonlinear simulations confirm the effectiveness of the robust decoupling controller.

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Acknowledgements

This work was supported by the National Basic Research Program of China (973 Program, 2012CB821200, 2012CB82 1201) and the NSFC (61134005, 60921001, 90916024, 91116016).

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Correspondence to Xiaofeng Su.

Additional information

This work was presented in part at the 18th International Symposium on Artificial Life and Robotics, Daejeon, Korea, January 30–February 1, 2013.

Appendix

Appendix

Force equations:

$$ \begin{aligned} \dot{u}&=[\omega_{z}+\omega_{E}(T_{31}c\lambda-T_{33}s\lambda)]v\\ &-[\omega_{y}+\omega_{E}(T_{21}c\lambda-T_{23}s\lambda]w +gT_{13}+(F_{x}+\Uppsi_{F_{x}})/m\\ \dot{v}&=[\omega_{x}+\omega_{E}(T_{11}c\lambda-T_{13}s\lambda)]w\\ &-[\omega_{z}+\omega_{E}(T_{31}c\lambda-T_{33}s\lambda)]u +gT_{23}+(F_{y}+\Uppsi_{F_{y}})/m\\ \dot{w}&=[\omega_{y}+\omega_{E}(T_{21}c\lambda-T_{23}s\lambda)]u\\ &-[\omega_{x}+\omega_{E}(T_{11}c\lambda-T_{13}s\lambda)]v +gT_{33}+(F_{z}+\Uppsi_{F_{z}})/m \end{aligned} $$

Moment equations:

$$ \begin{aligned} &J_{xx}\dot{\omega}_{x}-J_{xz}\dot{\omega}_{z} =J_{xz}\omega_{x}\omega_{y}+(J_{yy}-J_{zz})\omega_{y}\omega_{z} +\bar{M}_{x}+\Uppsi_{\bar{M}_{x}}\\ &J_{yy}\dot{\omega}_{y}=(J_{zz}-J_{xx})\omega_{x}\omega_{z} +J_{xz}(\omega_{z}^{2}-\omega_{x}^{2}) +\bar{M}_{y}+\Uppsi_{\bar{M}_{y}}\\ &J_{zz}\dot{\omega}_{z}-J_{xz}\dot{\omega}_{x}= -J_{xz}\omega_{y}\omega_{z}+(J_{xx}-J_{yy})\omega_{x}\omega_{y} +\bar{M}_{z}+\Uppsi_{\bar{M}_{z}} \end{aligned} $$

Euler angle equations:

$$ \begin{aligned} \dot{\phi}&=\omega_{x}+(\omega_{z}c\phi+\omega_{y}s\phi)t\theta -(c\theta{c}\phi+s\theta{c}\phi{t}\theta)(v/R)\\ &\quad+(c\theta{s}\phi+s\theta{s}\phi{t}\theta)(w/R) -\omega_{E}(c\psi{s}\theta{t}\theta+c\psi{c}\theta)c\lambda\\ \dot{\theta}&=\omega_{y}c\phi-\omega_{z}s\phi +c\theta(u/R)+s\theta{s}\phi(v/R)+s\theta{c}\phi(w/R)\\ &\quad+\omega_{E}s\psi{c}\lambda\\ \dot{\psi}&=(1/c\theta)(\omega_{z}c\phi+\omega_{y}s\phi)-\omega_{E}(c\psi{t}\theta{c}\lambda-s\lambda)\\ &\quad+s\psi{c}\theta{t}\lambda(u/R) +[(s\psi{s}\theta{s}\phi+c\psi{c}\phi)t\lambda-c\phi{t}\theta](v/R)\\ &\quad+[(s\psi{s}\theta{c}\phi-c\psi{s}\phi)t\lambda+s\phi{t}\theta](w/R) \end{aligned} $$

Trajectory equations:

$$ \begin{aligned} \dot{x}=T_{11}u+T_{21}v+T_{31}w\\ \dot{y}=T_{12}u+T_{22}v+T_{32}w\\ -\dot{h}=T_{13}u+T_{23}v+T_{33}w \end{aligned} $$

where \(R=R_{E}+h, \lambda=\lambda_{0}+x/R_{E}, s=\sin, c=\cos, t=\tan,\)

$$ [T_{ij}]= \left[\begin{array}{lll} c\psi c\theta& s\psi c\theta & -s\theta\\ c\psi s\theta s\phi-s\psi c\phi & s\psi s\theta s\phi+c\psi c\phi & c\theta s\phi\\ c\psi s\theta c\phi+s\psi s\phi & s\psi s\theta c\phi-c\psi s\phi & c\theta c\phi \end{array}\right],$$

and uvw, ω x , ω y , ω z , ϕ, θ, ψ, xyh are the system states. Input control variables are \( \Upphi_{x}, \Upphi_{y}, \Upphi_{z}, \delta_{e}, \delta_{a}, \delta_{r}, \) which are included in the expressions of the force, moment and thrust.

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Su, X., Jiang, Y. & Jia, Y. Modeling and robust decoupling control for hypersonic scramjet vehicle. Artif Life Robotics 18, 58–63 (2013). https://doi.org/10.1007/s10015-013-0099-8

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