﻿ 近空间可变翼飞行器小翼切换自适应控制方法
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (5): 886-891  DOI: 10.11990/jheu.201801014 0

### 引用本文

YANG Zheng, ZHEN Ziyang, JIANG Shuoying, et al. Winglet switching adaptive control for a near space morphing vehicle[J]. Journal of Harbin Engineering University, 2019, 40(5), 886-891. DOI: 10.11990/jheu.201801014.

### 文章历史

1. 南京航空航天大学 自动化学院, 江苏 南京 210016;
2. 弗吉尼亚大学 工程与应用科学学院, 美国 夏洛茨维尔 22904-4743

Winglet switching adaptive control for a near space morphing vehicle
YANG Zheng 1, ZHEN Ziyang 1, JIANG Shuoying 1, TAO Gang 2
1. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
2. School of Engineering and Applied Science, University of Virginia, Charlottesville 22904-4743, USA
Abstract: Considering the problem of parameter uncertainties in a near space morphing vehicle's (NMV) winglet telescopic mode switching process, a winglet switching adaptive controller based on matrix lower triangular-diagonal symmetric(LDS) factorization is designed to realize the smooth switching between different modes. First, the NMV nonlinear model is linearized to establish an uncertain linear model with winglet switching. Based on the linearized model of the NMV, the state feedback controller is then designed and the reference model is determined. Next, the adaptive compensation controller is designed to eliminate the influence of parameter uncertainty. The problem of adaptive control for winglet switching is transformed into the problem of parameter uncertainties, and the stability of the winglet switching process is proven theoretically. Results show that the proposed adaptive control method based on matrix decomposition can effectively solve the parameters uncertainty of a NMV. This method has strong robustness and provides a reliable solution to the problem of parameter uncertainty of NVM.
Keywords: near space morphing vehicle (NMV)    winglet switch    state feedback control    adaptive update law    reference model    parameter uncertainties    matrix factorization

1 近空间可变翼飞行器模型

 $\left\{\begin{array}{l}{L=\frac{1}{2} \rho V^{2} S_{\mathrm{w}} C_{\mathrm{L}}} \\ {D=\frac{1}{2} \rho V^{2} S_{\mathrm{w}} C_{\mathrm{D}}} \\ {M_{y}=\frac{1}{2} \rho V^{2} S_{\mathrm{w}} c_{\mathrm{A}} C_{\mathrm{m}}}\end{array}\right.$ (1)

 $\ddot{\beta}=-2 \zeta \omega \dot{\beta}-\omega^{2} \beta+\omega^{2} \beta_{\mathrm{c}}$ (2)

 $C_{\mathrm{T}}=\left\{\begin{array}{ll}{0.02576 \beta, } ~~~ {\beta<1} \\ {0.0224+0.00336 \beta, } ~~~ {\beta \geqslant 1}\end{array}\right.$ (3)
 $T=\frac{1}{2} \rho V^{2} S_{w} C_{\mathrm{T}}$ (4)

 $\begin{array}{C} {S_{\rm{w}}} = \\ \left\{\begin{array}{ll}{389, } & {t \leqslant T_{1}} \\ {389 \mathrm{e}^{-a\left(t-T_{1}\right)}+369\left(1-\mathrm{e}^{-a\left(t-T_{1}\right)}\right), } & {T_{1}T_{4}}\end{array}\right. \end{array}$ (5)

 $\left[ \begin{array}{l}{\dot{V}=\frac{T \cos \alpha-D}{m}-\frac{\mu}{r^{2}} \sin \gamma} \\ {\dot{\gamma}=\frac{L+T \sin \alpha}{m V}-\frac{\left(\mu-V^{2} r\right) \cos \gamma}{V_{r^{2}}}} \\ {q=\frac{M_{y}}{I_{y}}} \\ {\dot{\alpha}=q-\dot{\gamma}} \\ {\dot{h}=V \sin \gamma}\end{array}\right.$ (6)

2 自适应控制系统设计 2.1 控制问题描述

 $\left\{\begin{array}{l}{\dot{\boldsymbol{x}}(t)=A \boldsymbol{x}(t)+B \boldsymbol{u}(t)+f} \\ {\boldsymbol{y}(t)=C \boldsymbol{x}(t)}\end{array}\right.$ (7)

 Download: 图 1 近空间可变翼飞行器MRAC结构图 Fig. 1 Near space morphing vehicle MRAC structure
2.2 控制器设计

2.2.1 参考模型设计

 $\left\{\begin{array}{l}{\boldsymbol{y}_{m}(t)=\boldsymbol{W}_{m}(s) \boldsymbol{r}(t)} \\ {\boldsymbol{W}_{m}(s)=\boldsymbol{\xi}_{m}^{-1}(s)}\end{array}\right.$ (8)

 $\boldsymbol{y}_{m}(t)=\left[ \begin{array}{cc}{\frac{1}{\left(s+p_{1}\right)}} & {0} \\ {0} & {\frac{1}{\left(s+p_{2}\right)\left(s+p_{3}\right)}}\end{array}\right] \boldsymbol{r}(t)$ (9)

2.2.2 状态反馈控制器设计

 $\boldsymbol{u}(t)=\boldsymbol{K}_{1}^{* \mathrm{T}} x(t)+\boldsymbol{K}_{2}^{*} \boldsymbol{r}(t)+\boldsymbol{K}_{3}^{*}$ (10)

 $\boldsymbol{C}\left(s \boldsymbol{I}-\boldsymbol{A}-\boldsymbol{B} \boldsymbol{K}_{1}^{* \mathrm{T}}\right)^{-1} \boldsymbol{B} \boldsymbol{K}_{2}^{*}=\boldsymbol{W}_{m}(s), \boldsymbol{W}_{m}(s)=\boldsymbol{\xi}_{m}^{-1}(s)$ (11)

 $\boldsymbol{y}(s)=\boldsymbol{C}\left(s \boldsymbol{I}-\boldsymbol{A}-\boldsymbol{B} \boldsymbol{K}_{1}^{* \mathrm{T}}\right)^{-1} \boldsymbol{B} \boldsymbol{K}_{2}^{*} r(s)+\Delta(s)$ (12)

 $\boldsymbol{e}(s)=\boldsymbol{y}(s)-\boldsymbol{y}_{m}(s)$ (13)

 $\lim\limits_{t \rightarrow \infty} e(t)=\lim _{s \rightarrow 0} s e(s)=\boldsymbol{D} \boldsymbol{K}_{3}^{*}+d$ (14)

 $\boldsymbol{K}_{3}^{*}=-\boldsymbol{D}^{-1} d$ (15)

 $\lim\limits_{t \rightarrow \infty}\left(\boldsymbol{y}(t)-\boldsymbol{y}_{m}(t)\right)=\lim\limits_{t \rightarrow \infty} \boldsymbol{\delta}(t)=0。$

2.2.3 自适应律设计

 $\boldsymbol{u}(t)=\boldsymbol{K}_{1}^{\mathrm{T}} x(t)+\boldsymbol{K}_{2} r(t)+\boldsymbol{K}_{3}(t)$ (16)

 \left\{\begin{aligned} \dot{\boldsymbol{x}}(t)=&\left(\boldsymbol{A}+\boldsymbol{B} \boldsymbol{K}_{1}^{* \mathrm{T}}\right) x(t)+\boldsymbol{B} \boldsymbol{K}_{2}^{*} \boldsymbol{r}(t)+\boldsymbol{B K}_{3}^{*}+f+\\ & \boldsymbol{B}\left(\tilde{\boldsymbol{K}}_{1}^{\mathrm{T}}(t) x(t)+\tilde{\boldsymbol{K}}_{2}(t) \boldsymbol{r}(t)+\tilde{\boldsymbol{K}}_{3}(t)\right) \\ \boldsymbol{y}(t) &=\boldsymbol{C x}(t) \end{aligned}\right. (17)

3 仿真验证 3.1 小翼切换控制

 $V_{c}=\left\{\begin{array}{ll}{\frac{1}{6.25 s^{2}+4 s+1} \times 40+4590, } & {0 $ \begin{array}{l} {h_c} = \\ \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{6.25{s^2} + 4\;s + 1}} \times 60 + 33528, \quad 0 < t \le 50\;{\rm{s}}}\\ {\frac{1}{{6.25\;{s^2} + 4\;s + 1}} \times 60 + 33588, \quad 50\;{\rm{s}} < t \le 80\;{\rm{s}}} \end{array}} \right. \end{array} $(32) 仿真中，小翼先由伸出到收回，再从收回到伸出。在0~10 s，小翼处于伸出状态；在10~15 s，小翼由伸出状态切换到收回状态；在15~50 s，小翼处于收回状态；在50~55 s小翼由收回状态切换到伸出状态；在55~80 s，小翼处于伸出状态。 自适应参数设置为Γ=diag(10, 5)，Ds=diag(10, 0.001)，$\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{\theta _2}}^{ - 1} = 10 $。计算得Kp的初值为$\boldsymbol{K}_{p}=\left[ \begin{array}{cc}{330} & {-118} \\ {0} & {5.23}\end{array}\right] $。参考模型仿真参数选择为p1=-100, p2=p3=-60。选取自适应律K1, K2, K3的初值为K1(0)=0.9K1*, K2=0.9K2*K3=0.9K3* 为了验证本文设计的小翼切换自适应控制器的有效性，仿真实验以不加小翼增量线性模型的常规自适应控制器和本章设计的考虑小翼伸缩切换控制器作对比。近空间可变翼飞行器指令跟踪仿真结果如图 2~6  Download: 图 2 小翼切换时油门、升降舵角度响应曲线 Fig. 2 Response curves of throttle and elevator for winglet switching  Download: 图 3 小翼切换时速度、高度跟踪曲线 Fig. 3 Response curve of velocity and height for winglet switching  Download: 图 4 俯仰角速率、迎角响应曲线 Fig. 4 Response curves of pitch angle rate and angle of attack  Download: 图 5 速度、高度跟踪曲线 Fig. 5 Response curves of velocity and height with parameter perturbation  Download: 图 6 参数摄动时油门、升降舵响应曲线 Fig. 6 Response curves of throttle and elevator with parameted pertubation 图 2~4可以看出，基于小翼伸缩不确定模型设计的自适应小翼切换控制器有良好的控制效果。常规自适应控制方法在50 s小翼伸出后不能准确跟踪速度信号，飞行器的油门在小翼切换时处于满油门状态，且抖动明显。然而本文设计的小翼切换控制器能有效跟踪飞行速度与高度，飞行器很快能跟踪指令信号，说明控制器对小翼伸缩时参数不确定有很好的抑制作用，油门和升降舵变化在合理范围，且无抖动。飞行器的其余状态量都在合理变化范围内，迎角和俯仰角速率变化平稳，整个过程平滑稳定。从整个过程可以看出，小翼由伸出到收回切换过程比小翼由收回到伸出切换过程的抖动小，这是因为小翼伸出时，飞行器处于加速爬升状态，需要能量更大，变化相对明显。 仿真结果表明：采用基于矩阵分解的自适应小翼切换控制器相比于常规自适应控制器，控制效果有明显的改善，该方法可以使小翼伸缩模态切换过程平稳过渡，具有良好的控制效果。 3.2 参数摄动下小翼切换控制 近空间可变翼飞行器在小翼切换过程中具有参数不确定性，为了验证自适应控制器的鲁棒性，可表示为： $ \left\{\begin{array}{l}{\rho=\rho_{0}(1+\Delta \rho)} \\ {I_{y y}=I_{y 0}\left(1+\Delta I_{y y}\right)} \\ {c_{e}=c_{e 0}\left(1+\Delta c_{e}\right)} \\ {c_{a}=c_{a 0}\left(1+\Delta c_{A}\right)}\end{array}\right. \$ (33)

4 结论

1) 本文设计的自适应小翼切换控制器，对参数存在摄动的情况有良好的控制效果，具有良好的稳定性和较强的鲁棒性。

2) 将小翼伸缩的不确定性引入飞行器状态方程表达式，考虑了小翼伸缩的不同过程，针对新的含有不确定值的状态方程表达式设计基于矩阵分解思想的自适应控制系统。

3) 从理论上证明在小翼切换过程中控制系统的稳定性及自适应律的有界性，实验结果证明了理论推导的正确性。

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