﻿ 高超声速飞行器弹性自适应控制方法
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (6): 1026-1031  DOI: 10.11990/jheu.201611043 0

### 引用本文

YU Chaojun, JIANG Ju, ZHEN Ziyang, et al. A novel resilient adaptive control scheme for hypersonic vehicles[J]. Journal of Harbin Engineering University, 2018, 39(6), 1026-1031. DOI: 10.11990/jheu.201611043.

### 文章历史

A novel resilient adaptive control scheme for hypersonic vehicles
YU Chaojun, JIANG Ju, ZHEN Ziyang, ZHOU Jun
College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Abstract: This paper proposes a novel resilient adaptive control scheme for controlling hypersonic flying vehicles under parametric uncertainties. First, a typically longitudinal dynamic model on a hypersonic flying vehicle was provided, and feedback linearization was implemented. The effect of parametric uncertainties on the linearized model was analyzed. On the basis of the abovementioned actions, the resilient adaptive control law was designed in two steps. A nominal controller was designed using the adaptive backstepping control scheme, and the constructed Lyapunov function was analyzed. The final controller was obtained, and the system stability was verified. Finally, a simulative experiment was carried out, and the results were compared with those of the standard backstepping control method. The proposed control method presented better adaptive ability for parametric uncertainties than the standard control method.
Key words: hypersonic vehicles    parameter uncertainties    backstepping    adaptive control    resilient control

1 高超声速飞行器控制问题描述 1.1 高超声速飞行器模型

 $\left\{ \begin{array}{l} \dot V = \frac{{T\cos \alpha - D}}{m} - g\sin \gamma \\ \dot h = V\sin \gamma \\ \dot \gamma = \frac{{L + T\sin \alpha }}{{mV}} - \frac{{g\cos \gamma }}{V}\\ \dot \alpha = q - \dot \gamma \\ \dot q = \frac{{{M_{yy}}}}{{{I_{yy}}}} \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} L = \bar qS{C_L}\left( \alpha \right)\\ D = \bar qS{C_D}\left( \alpha \right)\\ T = \bar qS{C_T}\left( \beta \right)\\ {M_{yy}} = \bar qS\bar c[{C_{M\alpha }}\left( \alpha \right) + {C_{M{\delta _e}}}(\alpha , {\delta _e}) + {C_{Mq}}\left( {V, q, \alpha } \right)] \end{array} \right.$ (2)

 $\ddot \beta = - 2\xi {\omega _n}\dot \beta - \omega _n^2\beta + \omega _n^2{\beta _c}$ (3)

1.2 高超声速飞行器参数不确定问题

 $\left\{ \begin{array}{l} m = {m_0}(1 + {\mathit{\Delta }_m}), {I_{yy}} = {I_{yy0}}(1 + {\mathit{\Delta }_{{I_{yy}}}})\\ S = {S_0}(1 + {\mathit{\Delta }_S}), \bar c = {{\bar c}_0}(1 + {\mathit{\Delta }_{\bar c}})\\ {c_e} = {c_{e0}}(1 + {\mathit{\Delta }_{{c_e}}}), \rho = {\rho _0}(1 + {\mathit{\Delta }_\rho }) \end{array} \right.$ (4)

2 弹性自适应控制律设计

2.1 反馈线性化

 $\left[ \begin{gathered} {\dddot V} \hfill \\ {h^{(4)}} \hfill \\ \end{gathered} \right] = \left[ \begin{gathered} {f_V} \hfill \\ {f_h} \hfill \\ \end{gathered} \right] + \left[ {\begin{array}{*{20}{c}} {{b_{11}}}&{{b_{12}}} \\ {{b_{21}}}&{{b_{22}}} \end{array}} \right]\left[ \begin{gathered} {\delta _e} \hfill \\ {\beta _c} \hfill \\ \end{gathered} \right] = \mathit{\boldsymbol{f}} + \mathit{\boldsymbol{Gu}}$ (5)

 $\left[ \begin{gathered} {\dddot V} \hfill \\ {h^{(4)}} \hfill \\ \end{gathered} \right] = \left[ \begin{gathered} {f_V} \hfill \\ {f_h} \hfill \\ \end{gathered} \right] + \left[ \begin{gathered} \Delta {f_V} \hfill \\ \Delta {f_h} \hfill \\ \end{gathered} \right] + \left[ {\begin{array}{*{20}{c}} {{b_{11}}}&{{b_{12}}} \\ {{b_{21}}}&{{b_{22}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\mathit{\Lambda }_{11}}}&0 \\ 0&{{\mathit{\Lambda }_{22}}} \end{array}} \right]\left[ \begin{gathered} {\delta _e} \hfill \\ {\beta _c} \hfill \\ \end{gathered} \right]$ (6)

2.2 自适应反步法控制律设计

 $\left\{ \begin{array}{l} \dot h = {h_1}\\ {{\dot h}_1} = {h_2}\\ {{\dot h}_2} = {h_3}\\ {{\dot h}_3} = {f_h} + \Delta {f_h} + {b_{11}}{\delta _e} + {b_{12}}{\beta _c}\\ \dot V = {V_1}\\ {{\dot V}_1} = V\\ {{\dot V}_2} = {f_V} + \Delta {f_V} + {b_{21}}{\delta _e} + {b_{22}}{\beta _c}_2 \end{array} \right.$ (7)

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{u}}_n} = {\mathit{\boldsymbol{G}}^{ - 1}}\left( { - \left[ \begin{array}{l} {k_{{V_2}}}{z_{{V_2}}}\\ {k_{{h_3}}}{z_{{V_2}}} \end{array} \right] - \left[ \begin{array}{l} {z_{{V_1}}}\\ {z_{{h_2}}} \end{array} \right] - \mathit{\boldsymbol{f}} - \Delta \mathit{\boldsymbol{\hat f}} + \left[ \begin{array}{l} {{\dot \alpha }_{{V_2}}}\\ {{\dot \alpha }_{{h_3}}} \end{array} \right]} \right)\\ {\alpha _{{h_3}}} = - {k_{{h_2}}}{z_{{h_2}}} + {{\dot \alpha }_{{h_2}}} - {z_{{h_1}}}\\ {\alpha _{{h_2}}} = - {k_{{h_1}}}{z_{{h_1}}} + {{\dot \alpha }_{{h_1}}} - {z_h}\\ {\alpha _{{h_1}}} = - {k_h}{z_h} + {{\dot h}_c}\\ {\alpha _{{V_2}}} = - {k_{{V_1}}}{z_{{V_1}}} + {{\dot \alpha }_{{V_1}}} - {z_V}\\ {\alpha _{{V_1}}} = - {k_V}{z_V} + {{\dot V}_c} \end{array} \right.$ (8)

 $\mathit{\Delta }\dot{\hat{\mathit{\boldsymbol{f}}}} = {\left[ {\frac{1}{{{k_{\mathit{\Delta }{f_V}}}}}{z_{{V_2}}}\;\;\frac{1}{{{k_{\mathit{\Delta }{f_h}}}}}{z_{{h_3}}}} \right]^{\rm{T}}}$ (9)

 $\begin{array}{l} {V_{{\rm{all}}}} = \frac{1}{2}(z_h^2 + z_{{h_1}}^2 + z_{{h_2}}^2 + z_{{h_3}}^2 + z_V^2 + z_{{V_1}}^2 + \\ \;\;\;\;\;\;z_{{V_2}}^2 + \frac{1}{{{k_{\Delta {f_V}}}}}\Delta \mathit{\boldsymbol{\tilde f}}_V^2 + \frac{1}{{{k_{\Delta {f_h}}}}}\Delta \mathit{\boldsymbol{\tilde f}}_h^2) \end{array}$ (10)

 $\begin{array}{l} {{\dot V}_{{\rm{all}}}} = - {k_h}z_h^2 - {k_{{h_1}}}z_{{h_1}}^2 - {k_{{h_2}}}z_{{h_2}}^2 - {k_{{h_3}}}z_{{h_3}}^2 - \\ \;\;\;\;\;\;\;{k_V}z_V^2 - {k_{{V_1}}}z_{{V_1}}^2 - {k_{{V_2}}}z_{{V_2}}^2 \end{array}$ (11)

2.3 弹性控制律设计

 $\mathit{\boldsymbol{u}} = {\mathit{\boldsymbol{u}}_n} - {\mathop{\rm sgn}} {\left( {\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {z_3}}}\mathit{\boldsymbol{G}}} \right)^{\rm{T}}}\left( {\left\| {{\mathit{\boldsymbol{u}}_n}} \right\| + \frac{{\left\| {{\mathit{\boldsymbol{u}}_n}} \right\|\beta }}{\varepsilon }} \right)$ (12)

$\mathit{\boldsymbol{\tilde u}} = \mathit{\boldsymbol{u}} - {\mathit{\boldsymbol{u}}_n}, \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\Lambda }_{11}}}&0\\ 0&{{\mathit{\Lambda }_{22}}} \end{array}} \right] - \mathit{\boldsymbol{I}}$，则有|Ωii| < 1-εi=1, 2。

 $\begin{array}{l} {V_{{\rm{res}}}} = \frac{1}{2}(z_h^2 + z_{{h_1}}^2 + z_{{h_2}}^2 + z_{{h_3}}^2 + z_V^2 + z_{{V_1}}^2 + \\ \;\;\;\;\;\;z_{{V_2}}^2 + \frac{1}{{{k_{\Delta {f_V}}}}}\Delta \mathit{\boldsymbol{\tilde f}}_V^2 + \frac{1}{{{k_{\Delta {f_h}}}}}\Delta \mathit{\boldsymbol{\tilde f}}_h^2) \end{array}$ (13)

 ${{\dot V}_{{\rm{res}}}} = {{\dot V}_{{\rm{all}}}} + \frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G \boldsymbol{\varOmega} u}} + \frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G\tilde u}}$ (14)

 $\begin{array}{l} \;\;\;\;\;\;\;\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G \boldsymbol{\varOmega} u}} = \frac{{\partial {V_{{\rm{res}}}}}}{{\partial {z_3}}}\mathit{\boldsymbol{G \boldsymbol{\varOmega} }}{\mathit{\boldsymbol{u}}_n} - \\ \frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G \boldsymbol{\varOmega} }}{\mathop{\rm sgn}} {\left( {\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G}}} \right)^{\rm{T}}}\left( {\left\| {{\mathit{\boldsymbol{u}}_n}} \right\| + \frac{{\left\| {{\mathit{\boldsymbol{u}}_n}} \right\|\beta }}{\varepsilon }} \right) \le \\ \;\;\;\;\;\;\;\;\;\;\left\| {\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G}}} \right\|\left\| \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\|\left\| {{\mathit{\boldsymbol{u}}_n}} \right\|\left( {2 + \frac{\beta }{\varepsilon }} \right) \le \\ \;\;\;\;\;\;\;\left\| {\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_n}} \right\|\left( {2 + \frac{\beta }{\varepsilon }} \right)\left( {1 - \varepsilon } \right) \end{array}$ (15)

 $\begin{array}{l} \frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G\tilde u}} = - \frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G}}{\mathop{\rm sgn}} {\left( {\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G}}} \right)^{\rm{T}}}\left( {\left\| {{\mathit{\boldsymbol{u}}_n}} \right\| + \frac{{\left\| {{\mathit{\boldsymbol{u}}_n}} \right\|\beta }}{\varepsilon }} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\; - \left\| {\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G}}} \right\|\left\| {{\mathit{\boldsymbol{u}}_n}} \right\|\left( {1 + \frac{\beta }{\varepsilon }} \right) \end{array}$ (16)

 $\begin{array}{l} {{\dot V}_{{\rm{res}}}} \le {{\dot V}_{{\rm{all}}}} + \left\| {\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G}}} \right\|\left\| \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\|\left\| {{\mathit{\boldsymbol{u}}_n}} \right\|\left( {\left( {2 + \frac{\beta }{\varepsilon }} \right) \cdot } \right.\\ \left. {\;\;\;\;\;\;\;\;\;\;\left( {1 - \varepsilon } \right) - \left( {1 + \frac{\beta }{\varepsilon }} \right)} \right) = \\ {{\dot V}_{{\rm{all}}}}\left\| {\left\| {\frac{{\partial {V_{{\rm{res}}}}}}{{\partial {\mathit{\boldsymbol{z}}_3}}}\mathit{\boldsymbol{G}}} \right.} \right\|\left\| \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right\|\left\| {{\mathit{\boldsymbol{u}}_n}} \right\|\left( {\beta + 2\varepsilon - 1} \right) \le \\ \;\; - {k_h}z_h^2 - {k_{{h_1}}}z_{{h_1}}^2 - {k_{{h_2}}}z_{{h_2}}^2 - {k_{{h_3}}}z_{{h_3}}^2 - \\ \;\;\;\;\;\;\;\;{k_V}z_V^2 - {k_{{V_1}}}z_{{V_1}}^2 - {k_{{V_2}}}z_{{V_2}}^2 \end{array}$ (17)

3 仿真验证与分析

3.1 标准Backstepping控制仿真实验

 Download: 图 1 backstepping控制作用下高度和速度响应曲线 Fig. 1 Response curves of velocity and altitude under backstepping control
 Download: 图 2 backstepping控制作用下迎角响应曲线 Fig. 2 Response curve of attack angle under backstepping control

3.2 弹性自适应Backstepping控制仿真实验

 Download: 图 3 高度和速度响应曲线 Fig. 3 Response curves of velocity and altitude
 Download: 图 4 航迹倾斜角、迎角和俯仰角速度响应曲线 Fig. 4 Response curves of flight path angle, attack angle and pitch angle rate