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Nonlinear Model Predictive Control for Near-Space Interceptor Based on Finite Time Disturbance Observer

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Abstract

This paper proposes a novel nonlinear model predictive control design method based on the finite time disturbance observer (FTDO) and dynamic control allocation (DCA) algorithm for the near-space interceptor (NSI). First, the FTDO is introduced to estimate the mismatched system disturbance. The nonlinear model predictive control scheme based on the estimation value is proposed to obtain the virtual control command, and the mismatched disturbance can be effectively removed from the output channels of the NSI by designing a feed-forward disturbance compensation term with an appropriate compensation gain matrix. Moreover, a new DCA method is given to distribute the above virtual control command among the corresponding actuators (reaction jets and aerodynamic fins). Finally, numerical simulations illustrate that the proposed control scheme can track the command signal with high precision and have strong robustness against the mismatched disturbance.

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Acknowledgements

This work was supported by Aeronautical Science Foundation of China under Grants 2016ZC12005.

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Authors and Affiliations

  1. Luoyang Optoelectro Technology Development Center, Luoyang, 471009, China

    Chao Guo, Chen Song & Yu-Jie Zhao

  2. School of Automation and Electrical Engineering, University of Science and Technology, Beijing, Beijing, 100083, China

    Jun-Wei Wang

Authors
  1. Chao Guo
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  2. Chen Song
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  3. Yu-Jie Zhao
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Corresponding author

Correspondence to Chao Guo.

Appendix 1

Appendix 1

1.1 Lie Derivative Expressions of Subsystem (17) ~ (19)

The Lie derivative expressions of the roll angle subsystem (17) are given by:

$$ \begin{aligned} L_{\varvec{f}} h_{1} &= \frac{{\partial h_{1} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{f}(\varvec{x}) = f_{\gamma } ,\\ L_{{\varvec{g}_{1} }} h_{1} &= \frac{{\partial h_{1} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{1} (\varvec{x}) = [0,\;0,\;0] \hfill \\ L_{{\varvec{g}_{2} }} h_{1} &= \frac{{\partial h_{1} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{2} (\varvec{x}) = [1,\;0,\;0,\;0,\;0,\;0], \hfill \\ L_{\varvec{f}}^{2} h_{1} &= L_{\varvec{f}} (L_{\varvec{f}} h_{1} ) = \frac{{\partial L_{\varvec{f}} h_{1} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{f}(\varvec{x}) \hfill \\ &= \frac{{\partial f_{\gamma } }}{\partial \gamma }f_{\gamma } + \frac{{\partial f_{\gamma } }}{\partial \beta }f_{\beta } + \frac{{\partial f_{\gamma } }}{\partial \alpha }f_{\alpha } + \frac{{\partial f_{\gamma } }}{{\partial \omega_{x} }}f_{{\omega_{x} }} \\ &\quad + \frac{{\partial f_{\gamma } }}{{\partial \omega_{y} }}f_{{\omega_{y} }} + \frac{{\partial f_{\gamma } }}{{\partial \omega_{z} }}f_{{\omega_{z} }} \hfill \\ L_{{\varvec{g}_{1} }} L_{\varvec{f}} h_{1} &= \frac{{\partial L_{\varvec{f}} h_{1} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{1} (\varvec{x}) \\ &\quad= \left[ {\frac{1}{{J_{x} }}\frac{{\partial f_{\gamma } }}{{\partial \omega_{x} }},\;\frac{1}{{J_{y} }}\frac{{\partial f_{\gamma } }}{{\partial \omega_{y} }},\;\frac{1}{{J_{z} }}\frac{{\partial f_{\gamma } }}{{\partial \omega_{z} }}} \right] \hfill \\ L_{{\varvec{g}_{2} }} L_{\varvec{f}} h_{1} &= \frac{{\partial L_{\varvec{f}} h_{1} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{2} (\varvec{x}) \\ &\quad= \left[ {\frac{{\partial f_{\gamma } }}{\partial \gamma },\;\frac{{\partial f_{\gamma } }}{\partial \beta },\;\frac{{\partial f_{\gamma } }}{\partial \alpha },\;\frac{{\partial f_{\gamma } }}{{\partial \omega_{x} }},\;\frac{{\partial f_{\gamma } }}{{\partial \omega_{y} }},\;\frac{{\partial f_{\gamma } }}{{\partial \omega_{z} }}} \right] \hfill \\ \frac{{\partial f_{\gamma } }}{\partial \gamma } &= \omega_{y} \sin \gamma \tan \vartheta + \omega_{z} \cos \gamma \tan \vartheta \hfill \\ \frac{{\partial f_{\gamma } }}{\partial \beta } &= 0,\frac{{\partial f_{\gamma } }}{\partial \alpha } = - \frac{{\omega_{y} \cos \gamma }}{{\cos^{2} \alpha }} + \frac{{\omega_{z} \sin \gamma }}{{\cos^{2} \alpha }} \hfill \\ \frac{{\partial f_{\gamma } }}{{\partial \omega_{x} }} &= 1,\frac{{\partial f_{\gamma } }}{{\partial \omega_{y} }} = - \cos \gamma \tan \vartheta ,\frac{{\partial f_{\gamma } }}{{\partial \omega_{z} }} = \sin \gamma \tan \vartheta \hfill \\ \end{aligned} $$

The Lie derivative expressions of the vertical overload subsystem (18) are given by:

$$ \begin{aligned} L_{\varvec{f}} h_{2} &= \frac{{\partial h_{2} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{f}(\varvec{x}) = \frac{{q{\text{SC}}_{z}^{\beta } }}{\text{mg}}f_{\beta } ,\\ L_{{\varvec{g}_{1} }} h_{2} &= \frac{{\partial h_{2} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{1} (\varvec{x}) = [0,\;0,\;0] \hfill \\ L_{{\varvec{g}_{2} }} h_{2} &= \frac{{\partial h_{2} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{2} (\varvec{x}) = \left[ {0,\;\frac{{qSC_{z}^{\beta } }}{mg},\;0,\;0,\;0,\;0} \right] \hfill \\ L_{\varvec{f}}^{2} h_{2} &= L_{\varvec{f}} (L_{\varvec{f}} h_{2} ) = \frac{{\partial L_{\varvec{f}} h_{2} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{f}(\varvec{x}) \hfill \\ &= \frac{{qSC_{z}^{\beta } }}{mg}\left( {\frac{{\partial f_{\beta } }}{\partial \gamma }f_{\gamma } + \frac{{\partial f_{\beta } }}{\partial \beta }f_{\beta } + \frac{{\partial f_{\beta } }}{\partial \alpha }f_{\alpha } + \frac{{\partial f_{\beta } }}{{\partial \omega_{x} }}f_{{\omega_{x} }} }\right.\\ &\quad\left.{+ \frac{{\partial f_{\beta } }}{{\partial \omega_{y} }}f_{{\omega_{y} }} + \frac{{\partial f_{\beta } }}{{\partial \omega_{z} }}f_{{\omega_{z} }} } \right) \hfill \\ L_{{\varvec{g}_{1} }} L_{\varvec{f}} h_{2} &= \frac{{\partial L_{\varvec{f}} h_{2} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{1} (\varvec{x}) \\ & = \frac{{qSC_{z}^{\beta } }}{mg}\left[ {\frac{1}{{J_{x} }}\frac{{\partial f_{\beta } }}{{\partial \omega_{x} }},\;\frac{1}{{J_{y} }}\frac{{\partial f_{\beta } }}{{\partial \omega_{y} }},\;\frac{1}{{J_{z} }}\frac{{\partial f_{\beta } }}{{\partial \omega_{z} }}} \right] \hfill \\ L_{{\varvec{g}_{2} }} L_{\varvec{f}} h_{2} &= \frac{{\partial L_{\varvec{f}} h_{2} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{2} (\varvec{x}) \\ & = \frac{{qSC_{z}^{\beta } }}{mg}\left[ {\frac{{\partial f_{\beta } }}{\partial \gamma },\;\frac{{\partial f_{\beta } }}{\partial \beta },\;\frac{{\partial f_{\beta } }}{\partial \alpha },\;\frac{{\partial f_{\beta } }}{{\partial \omega_{x} }},\;\frac{{\partial f_{\beta } }}{{\partial \omega_{y} }},\;\frac{{\partial f_{\beta } }}{{\partial \omega_{z} }}} \right] \hfill \\ \end{aligned} $$
$$\begin{aligned} \frac{{\partial f_{\beta } }}{\partial \gamma } &= 0,\frac{{\partial f_{\beta } }}{\partial \beta } = \frac{{qSC_{y}^{\alpha } \alpha \sin \alpha \cos \beta }}{{mV_{m} }} \\ & \quad+ \frac{{qSC_{z}^{\beta } (\cos \beta - \beta \sin \beta )}}{{mV_{m} }} \end{aligned}$$

\( \frac{{\partial f_{\beta } }}{\partial \alpha } = \omega_{x} \cos \alpha - \omega_{y} \sin \alpha + \frac{{qSC_{y}^{\alpha } (\sin \alpha + \alpha \cos \alpha )\sin \beta }}{{mV_{m} }} \),\( \frac{{\partial f_{\beta } }}{{\partial \omega_{x} }} = \sin \alpha ,\frac{{\partial f_{\beta } }}{{\partial \omega_{y} }} = \cos \alpha ,\frac{{\partial f_{\beta } }}{{\partial \omega_{z} }} = 0 \), \( \frac{{\partial f_{\beta } }}{{\partial \omega_{y} }} = \cos \alpha \), \( \frac{{\partial f_{\beta } }}{{\partial \omega_{z} }} = 0 \).

The Lie derivative expressions of the lateral overload subsystem (19) are given by:

$$ \begin{aligned}L_{\varvec{f}} h_{3} &= \frac{{\partial h_{3} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{f}(\varvec{x}) = \frac{{qSC_{y}^{\alpha } }}{mg}f_{\alpha } ,\\ L_{{\varvec{g}_{1} }} h_{3} &= \frac{{\partial h_{3} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{1} (\varvec{x}) = [0,\;0,\;0] \end{aligned}$$

\( L_{{\varvec{g}_{2} }} h_{3} = \frac{{\partial h_{3} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{2} (\varvec{x}) = \left[ {0,\;0,\;\frac{{qSC_{y}^{\alpha } }}{mg},\;0,\;0,\;0} \right], \)

$$ \begin{aligned} L_{\varvec{f}}^{2} h_{3} &= L_{\varvec{f}} (L_{\varvec{f}} h_{3} ) = \frac{{\partial L_{\varvec{f}} h_{3} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{f}({\varvec{x}}) \hfill \\ &= \frac{{qSC_{y}^{\alpha } }}{mg}\left( {\frac{{\partial f_{\alpha } }}{\partial \gamma }f_{\gamma } + \frac{{\partial f_{\alpha } }}{\partial \beta } f_{\beta } + \frac{{\partial f_{\alpha } }}{\partial \alpha }f_{\alpha } }\right.\\ &\quad\left.{+ \frac{{\partial f_{\alpha } }}{{\partial \omega_{x} }} f_{{\omega_{x} }} + \frac{{\partial f_{\alpha } }}{{\partial \omega_{y} }} f_{{\omega_{y} }} + \frac{{\partial f_{\alpha } }}{{\partial \omega_{z} }}f_{{\omega_{z} }} } \right) \end{aligned} $$
$$\begin{aligned} L_{{\varvec{g}_{1} }} L_{\varvec{f}} h_{3} &= \frac{{\partial L_{\varvec{f}} h_{3} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{1} (\varvec{x}) = \frac{{qSC_{y}^{\alpha } }}{mg}\\ &\quad\left[ {\frac{1}{{J_{x} }}\frac{{\partial f_{\alpha } }}{{\partial \omega_{x} }},\;\frac{1}{{J_{y} }}\frac{{\partial f_{\alpha } }}{{\partial \omega_{y} }},\;\frac{1}{{J_{z} }}\frac{{\partial f_{\alpha } }}{{\partial \omega_{z} }}} \right] \end{aligned}$$
$$\begin{aligned} L_{{\varvec{g}_{2} }} L_{\varvec{f}} h_{3} &= \frac{{\partial L_{\varvec{f}} h_{3} (\varvec{x})}}{{\partial \varvec{x}}}\varvec{g}_{2} (\varvec{x}) = \frac{{qSC_{y}^{\alpha } }}{mg}\\ &\quad\left[ {\frac{{\partial f_{\alpha } }}{\partial \gamma },\;\frac{{\partial f_{\alpha } }}{\partial \beta },\;\frac{{\partial f_{\alpha } }}{\partial \alpha },\;\frac{{\partial f_{\alpha } }}{{\partial \omega_{x} }},\;\frac{{\partial f_{\alpha } }}{{\partial \omega_{y} }},\;\frac{{\partial f_{\alpha } }}{{\partial \omega_{z} }}} \right] \end{aligned}$$
$$\begin{aligned} \frac{{\partial f_{\alpha } }}{\partial \gamma } &= 0,\frac{{\partial f_{\alpha } }}{\partial \beta } = \frac{{\omega_{y} \sin \alpha }}{{\cos^{2} \beta }}\\ &\quad - \frac{{\omega_{x} \cos \alpha }}{{\cos^{2} \beta }} - \frac{{qSC_{y}^{\alpha } \alpha \cos \alpha \sin \beta }}{{mV_{m} \cos^{2} \beta }} \end{aligned}$$
$$ \begin{aligned}\frac{{\partial f_{\alpha } }}{\partial \alpha } &= \omega_{y} \tan \beta \cos \alpha + \omega_{x} \tan \beta \sin \alpha \\ & \quad- \frac{{qSC_{y}^{\alpha } (\cos \alpha - \alpha \sin \alpha )}}{{mV_{m} \cos \beta }} \end{aligned}$$
$$\begin{aligned} \frac{{\partial f_{\alpha } }}{{\partial \omega_{x} }} &= - \tan \beta \cos \alpha ,\frac{{\partial f_{\alpha } }}{{\partial \omega_{y} }} = \tan \beta \sin \alpha ,\\ &\quad \frac{{\partial f_{\alpha } }}{{\partial \omega_{z} }} = 1 \end{aligned}$$

Based on the above computation, we can obtain that the input relative degrees (IRD) of the NSI system are \( (\sigma_{1} ,\;\sigma_{2} ,\;\sigma_{3} ) = (2,\;2,\;2) \), and the corresponding disturbance relative degrees (DRD) are \( (\bar{\tau }_{1} ,\;\bar{\tau }_{2} ,\;\bar{\tau }_{3} ) = (1,\;1,\;1) \).

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Guo, C., Song, C., Zhao, YJ. et al. Nonlinear Model Predictive Control for Near-Space Interceptor Based on Finite Time Disturbance Observer. Int. J. Aeronaut. Space Sci. 19, 945–961 (2018). https://doi.org/10.1007/s42405-018-0077-4

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