2. 复杂系统控制与智能协同技术重点实验室, 北京 100074;
3. 北京航空航天大学 飞行器控制一体化技术国防科技重点实验室, 北京 100083;
4. 北京航空航天大学 大数据科学与脑机智能高精尖创新中心, 北京 100083
2. Science and Technology on Complex System Control and Intelligent Agent Cooperation Laboratory, Beijing 100074, China;
3. Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100083, China;
4. Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100083, China
近年来多飞行器协同制导吸引了越来越多的关注,多个飞行器通过空间以及时间上的相互配合能够实现更高的作战效率,同时实现更高的性价比。针对具有良好防御能力的重点目标来说,多飞行器饱和攻击是对其实施有效攻击的最有效的方式之一。
实现多飞行器同时攻击的主要方式有2种。第一种为个体命中时间控制[1-7],该方法通过对每枚导弹的命中时间进行预设来实现多飞行器同时命中目标的效果。由于飞行器在飞行过程中会收到外界干扰因素影响,并且在飞行过程中容易对航迹进行调整,因此如何设定合理的命中时间具有一定的难度。同时该方法对多飞行器命中时间偏差的控制属于开环控制,受外界干扰以及飞行器自身状态变化影响较大,控制鲁棒性较差。第二种为协同制导,该方法不需要预先设定固定的命中时间,飞行器在飞向目标的过程中通过相互通讯对飞行器的飞行状态进行调整来保证命中时间的同步。
协同制导分为集中式协同制导[8-13]和分布式协同制导。集中式协同制导需要对全局信息进行实时更新,而分布式协同制导不需要全局信息,因此在实际应用中分布式协同制导更容易实现,同时其鲁棒性更强。
孙雪娇等[14]基于分布式通讯及网络同步原理,在增广比例导引的基础上设计了一种分布式协同制导律。在此基础上周锐等[15]通过将目标视为“领弹”,采用“领-从”模式设计了一种分布式时间协同制导律,实现多飞行器同时收敛到目标位置。Wang等[16]提出了一种两段式制导策略,第1步基于二阶一致性的分布式协同制导律实现剩余飞行时间的一致,第2步采用比例导引的方式实现对目标的同时攻击。Zhao等[17-18]基于最优控制及非线性模型预测控制理论,提出了一种仅在特定时间进行制导指令更新的协同制导律。Zhao等[19]提出了两阶段的协同制导律,第1阶段首先对非线性追踪方程进行反馈线性化处理,再此基础上基于二阶一致性设计时间协同制导律。第2阶段利用比例导引针对固定目标不改变剩余飞行时间变化率的特性保持多飞行器剩余飞行时间的一致。
协同制导工作主要集中在针对固定目标的协同制导,针对机动目标的协同制导研究相对较少。文献[20-27]研究了针对机动目标的协同制导问题,但文献[20]需要假设追踪方程能够满足小扰动线性化条件;文献[20-25]需要假设能够获得目标加速度的直接量测信息,这通常难以在工程实践中实现;文献[26]虽然研究了机动目标的协同制导问题,但是其采用的方法是集中式的。文献[27]中采用的方法需要通讯拓扑是无向的,这通常会导致更多的通讯量以及能量消耗。
本文提出了一种适用于机动目标的分布式协同制导律。包括分布式扩张状态观测器以及协同制导律2部分。分布式扩张状态观测器用于同时观测追踪方程的系统状态以及干扰,协同制导律将非线性追踪方程线性化为二阶线性系统,通过一致性分析以及干扰补偿的方式实现飞行器对目标的同时攻击。
本文的创新之处主要有以下3点:
1) 本文所提出的协同制导律能够应用于机动目标,而文献[1-19]中的方法仅能应用于固定目标。由于目标随机机动带来的问题使得用于固定目标的协同制导律设计会变得更加复杂,同时机动目标相对于固定目标来说适用范围更加广泛。
2) 文献[20-27]研究了针对机动目标的协同制导问题,但是文献[20-25]中需要目标加速度信息的测量值,这在工程中通常难以直接获得。本文的协同制导律设计中没有用到目标的加速度信息。通过分布式扩张状态观测器同时对追踪方程的状态以及由目标加速度带来的干扰进行估计,同时通过极点分析的方式给出了协同制导律收敛的充分必要条件。
3) 虽然文献[26-27]没有用到目标加速度信息,但在通讯拓扑方面,文献[26]中的方法是集中式的,文献[27]中需要的通讯拓扑是无向的,而本文中的通讯拓扑是无向的并且仅需要满足拓扑中有生成树的条件。另外文献[27]需要飞行器具备轴向速度调整能力,而本文则不需要。
1 预备知识及问题描述 1.1 代数图论相关知识假设有N个飞行器参与了一次协同攻击,多飞行器之间的信息交互关系可以用有向图来描述。每一个导弹都可以被看作有向图中的一个节点。
假设G={M, E, W}为有向图,M={m1, m2, …, mN}为多导弹集合; E⊆{(mi, mj):mi, mj∈M}为多弹之间的边的集合,E中的一条边可以由eij=(mi, mj) (i≠j)来表示; W=[ωij]∈R为边的权值矩阵。当且仅当有信息从mj流向mi时边的权值ωij≠0。
值得注意的是存在边eij意味着mi可以接收到mj的信息,但是反之不一定成立。如果至少存在一个节点具有指向其他节点的有向路径则称该有向图包含生成树。
引理1[28] 假设L∈RN×N为有向拓扑M的拉普拉斯矩阵,那么L至少含有一个零特征值,1N为该特征值的特征向量,也就是说L1N=0。
如果通讯拓扑M含有生成树,那么0是拉普拉斯矩阵L的单特征值,同时L的其他N-1个特征值都有正实部。
1.2 问题描述工程实践中,飞行器的飞行轨迹通常被分解为横向及纵向平面,本文主要关注飞行器在横向平面中的运动,考虑多飞行器协同制导问题时的标准假设:①飞行器和目标都被视为平面上的质点;②与制导回路相比,飞行器导引头和自动驾驶仪的动力学速度足够快;③飞行器的轴向速度为常值。
制导过程的几何图形如图 1所示。
图 1中:mi(i∈{1, 2, …, N}为i个导弹;qi、θi和ηi分别为视线角、航向角以及前置角。
由图 1中可以得到
$ {\eta _i}(t) = {q_i}(t) - {\theta _i}(t) $ | (1) |
制导过程的动态方程为
$ \left\{ {\begin{array}{*{20}{l}} {{{\dot R}_i}(t) = {V_T}{\rm{cos}}{\eta _{{\rm{T}}i}}(t) - {V_i}{\rm{cos}}{\eta _i}(t)}\\ {{{\dot q}_i}(t) = \frac{{{V_i}}}{{{R_i}(t)}}{\rm{sin}}{\eta _i}(t) - \frac{{{V_T}}}{{{R_i}(t)}}{\rm{sin}}{\eta _{{\rm{T}}i}}(t)}\\ {{{\dot \theta }_i}(t) = {n_i}(t)g/{V_i}} \end{array}} \right. $ | (2) |
式中:Ri(t)为从mi到目标的距离;g为重力加速度;VT和ηTi(t)分别为目标的速度和前置角;Vi、ηi(t)、ni(t)为飞行器的轴向速度、前置角以及法向加速度。Vi的方向由ni(t)来控制,其中ni(t)方向垂直于Vi。
令ri(t)=Ri(t)/Vi,由式(1)和式(2)可得
$ \left\{ {\begin{array}{*{20}{l}} {{{\dot r}_i}(t) = \frac{{{V_{\rm{T}}}}}{{{V_i}}}{\rm{cos}}{\eta _{Ti}}(t) - {\rm{cos}}{\eta _i}(t)}\\ {{{\dot r}_i}(t) = - \frac{{{V_{\rm{T}}}}}{{{V_i}}}{\rm{sin}}{\eta _{{\rm{T}}i}}(t){{\dot \eta }_{{\rm{T}}i}}(t) + {\rm{sin}}{\eta _i}(t)}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {\frac{{{\rm{sin}}{\eta _i}(t)}}{{{r_i}(t)}} - \frac{{{V_{\rm{T}}}}}{{{V_i}{r_i}(t)}}{\rm{sin}}{\eta _{{\rm{T}}i}}(t) - {n_i}(t)g/{V_i}} \right)} \end{array}} \right. $ | (3) |
如果Vi是固定的,那么导弹mi的命中剩余时间由ri(t)以及
定义1 对于任意mi, mj,若ri(t), rj(t)以及
$ \left\{ {\begin{array}{*{20}{l}} {\mathop {{\rm{lim}}}\limits_{t \to \infty } |{r_i}(t) - {r_j}(t)| = 0}\\ {\mathop {{\rm{lim}}}\limits_{t \to \infty } |{{\dot r}_i}(t) - {{\dot r}_j}(t)| = 0} \end{array}} \right. $ | (4) |
通过上述分析,将针对机动目标的多飞行器协同制导问题转化为具有非线性动力学特性的多智能体系统一致性问题。本文中将重点关注以下问题:①如何基于邻居信息设计协同制导律。②如何设计制导律的参数多飞行器能够实现在有向通讯拓扑的条件下针对机动目标的同时攻击。
2 分布式协同制导律设计首先设计了分布式扩张状态观测器,同时对追踪系统的状态及干扰进行估计。然后基于分布式状态观测器设计了分布式协同制导律,同时基于一致性理论及极点分析的方法得出制导律收敛的充分必要条件。
2.1 分布式扩张状态观测器设计记
$ \left\{ \begin{array}{l} {d_i}(t) = - \frac{{{V_{\rm{T}}}}}{{{V_i}}}{\rm{sin}}{\eta _{{\rm{T}}i}}(t){{\dot \eta }_{{\rm{T}}i}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{V_{\rm{T}}}{\rm{sin}}{\eta _i}(t){\rm{sin}}{\eta _{{\rm{T}}i}}(t)}}{{{V_i}{r_i}(t)}}\\ {u_i}(t) = \frac{{{V_i}{\rm{sin}}{\eta _i}{{(t)}^2}}}{{{r_i}(t)}} - \frac{{{\rm{sin}}{\eta _i}(t){n_i}(t)g}}{{{V_i}}} \end{array} \right. $ | (5) |
以及x1i(t)=ri(t)-rref(t),
式(3)可以被改写为
$ \left\{ {\begin{array}{*{20}{l}} {{{\dot x}_{1i}}(t) = {x_{2i}}(t)}\\ {{{\dot x}_{2i}}(t) = {u_i}(t) + {d_i}(t)} \end{array}} \right. $ | (6) |
式中:x1i(t)、x2i(t)为系统(5)的状态;ui(t)和di(t)分别为系统的控制输入以及干扰。
定义1中的条件可以转化为
$ \left\{ {\begin{array}{*{20}{l}} {\mathop {{\rm{lim}}}\limits_{t \to \infty } |{x_{1i}} - {x_{1j}}| = 0}\\ {\mathop {{\rm{lim}}}\limits_{t \to \infty } |{x_{2i}} - {x_{2j}}| = 0} \end{array}} \right. $ | (7) |
引理2 [29]考虑以下高阶非线性系统:
$ \left\{ {\begin{array}{*{20}{l}} {{{\dot x}_k}(t) = {x_{k + 1}}(t),k = 1,2, \cdots ,n - 1}\\ {{{\dot x}_n}(t) = u(t) + f(\mathit{\boldsymbol{x}}(t),t)}\\ {y(t) = {x_1}(t)} \end{array}} \right. $ | (8) |
式中:x(t)=[x1(t), x2(t), …, xn(t)]T∈Rn为系统的状态;u(t)∈R为系统控制输入;f(x(t), t)为广义上的不确定性其中包括系统的非线性部分以及外部的干扰。
如果d(f(x(t), t))/dt是有界的,那么可以将xn+1(t)=f(x(t), t)作为系统的扩张状态。扩张状态观测器可以表示为
$ \left\{ {\begin{array}{*{20}{l}} {{{\dot {\hat x}}_k}(t) = {{\hat x}_{k + 1}}(t) + {\beta _{0k}}({x_1}(t) - {{\hat x}_1}(t))}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \cdots ,n - 1}\\ {{{\dot {\hat x}}_n}(t) = u(t) + {{\hat x}_{n + 1}}(t) + {\beta _{0n}}({x_1}(t) - {{\hat x}_1}(t))}\\ {y(t) = {x_1}(t)} \end{array}} \right. $ | (9) |
式中:
根据引理2设计分布式扩张状态观测器,令
$ \left\{ {\begin{array}{*{20}{l}} {{{\dot {\hat x}}_{1i}}(t) = {{\hat x}_{2i}}(t) + {\beta _1}{\rho _i}(t)}\\ {{{\dot {\hat x}}_{2i}}(t) = {u_i}(t) + {{\hat x}_{3i}}(t) + {\beta _2}{\rho _i}(t)}\\ {{{\dot {\hat x}}_{3i}}(t) = {\beta _3}{\rho _i}(t)} \end{array}} \right. $ | (10) |
记估计误差为
$ \left\{ {\begin{array}{*{20}{l}} {{{\dot {\tilde x}}_{1i}}(t) = {{\dot {\tilde x}}_{2i}}(t) - {\beta _1}\sum\limits_{j = 1}^{N - 1} {({{\dot {\tilde x}}_{1i}}(} t) - {{\dot {\tilde x}}_{1j}}(t))}\\ {{{\dot {\tilde x}}_{2i}}(t) = {{\dot {\tilde x}}_{3i}}(t) - {\beta _2}\sum\limits_{j = 1}^{N - 1} {({{\dot {\tilde x}}_{1i}}(} t) - {{\dot {\tilde x}}_{1j}}(t))}\\ {{{\dot {\tilde x}}_{3i}}(t) = - {\beta _3}\mathop \sum \limits^{N - 1} ({{\dot {\tilde x}}_{1i}}(t) - {{\dot {\tilde x}}_{1j}}(t))} \end{array}} \right. $ | (11) |
基于分布式状态观测器设计的分布式协同制导律为
$ \begin{array}{*{20}{c}} {{u_i}(t) = - {k_1}{x_{1i}}(t) - {k_2}\sum\limits_{j = 1}^{n - 1} {{a_{ij}}} (({x_{1i}}(t) - {x_{1j}}(t)) + }\\ {({{\hat x}_{2i}}(t) - {{\hat x}_{2j}}(t))) - {{\hat x}_3}(t)} \end{array} $ | (12) |
在协同制导律(12)的作用下,系统(6)可以改写为
$ \left\{ {\begin{array}{*{20}{l}} {{x_{1i}} = {x_{2i}}}\\ {{{\dot x}_{2i}} = - {k_1}{x_{1i}}(t) - {k_2}\sum\limits_{j = 1}^{n - 1} {{a_{ij}}} (({x_{1i}} - {x_{1j}}) + }\\ {\;\:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({x_{2i}} - {x_{2j}}) - ({{\tilde x}_{2i}} - {{\tilde x}_{2j}})) + {{\tilde x}_{3i}}} \end{array}} \right. $ | (13) |
记
$ {{\xi _{1i}}(t) = {x_{1i}}(t),{\xi _{2i}}(t) = {x_{2i}}(t),{\xi _{3i}}(t) = {{\tilde x}_{1i}}(t)} $ |
$ {{\xi _{4i}}(t) = {{\tilde x}_{2i}}(t),{\xi _{5i}}(t) = {{\tilde x}_{3i}}(t)} $ |
$ {{\mathit{\boldsymbol{\xi }}_i}(t) = {{[{\xi _{1i}}(t),{\xi _{2i}}(t),{\xi _{3i}}(t),{\xi _{4i}}(t),{\xi _{5i}}(t)]}^{\rm{T}}}} $ |
$ {\mathit{\boldsymbol{\xi }}(t) = {{[{\mathit{\boldsymbol{\xi }}_1}(t),{\mathit{\boldsymbol{\xi }}_2}(t), \cdots ,{\mathit{\boldsymbol{\xi }}_n}(t)]}^{\rm{T}}}} $ |
那么式(13)可以被改写为
$ \begin{array}{l} \mathit{\boldsymbol{\dot \xi }}(t) = {\mathit{\boldsymbol{I}}_N} \otimes \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0\\ { - {k_1}}&0&0&0&1\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}} \right]\mathit{\boldsymbol{\xi }}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{L}} \otimes \left[ {\begin{array}{*{20}{c}} 0&0&0&0&0\\ { - {k_2}}&{ - {k_2}}&0&{{k_2}}&0\\ 0&0&{ - {\beta _1}}&0&0\\ 0&0&{ - {\beta _2}}&0&0\\ 0&0&{ - {\beta _3}}&0&0 \end{array}} \right]\mathit{\boldsymbol{\xi }}(t) \end{array} $ | (14) |
根据定义1可以直接得到引理3。
引理3 多飞行器协同实现协同攻击当且仅当系统(14)实现状态一致。
记λi(i=1, 2, …, N)为拉普拉斯矩阵L的特征值,其中特征值λ1=0对应的特征向量记为u1=1N,记0 < Re(λ2)≤…≤Re(λN)。
令U-1LU=J,其中:U=[u1, u2, …, uN],
令c1∈R5, c2∈R5, c3∈R5, c4∈R5, c5∈R5是线性无关的向量,同时记pj=ui⊗cq(j=5(i-1)+q; i=1, 2, …, N; q=1, 2, 3, 4, 5),那么一致子空间可以定义为C(U), 其由p1=u1⊗c1=1N⊗c1,p2=u1⊗c2=1N⊗c2,…, p5=u1⊗c5=1N⊗c5扩张而成。非一致子空间可以相应定义为C(U), 其由p6, p7, …, p5N扩张而成。由于pj(j=1, 2, …, 5N)是线性无关的,那么可以得到引理4。
引理4 C(U)⊕C(U)=C
由于有向拓扑的拉普拉斯矩阵特征值有可能是复数因此需要引入以下2个引理。
引理5[30] 考虑系统
引理6[30] 考虑系统
1) Re(a1)>0
2) Re(a1)·Re(a1a2-a3)-(lma2)2>0
3) [Re(a1)Re(a2a3)-(Re(a3))2]·[Re(a1)Re(a1a2-a3)-(lm(a2))2]- [Re(a1)lm(a1a3)+(Re(a3)lm(a2))]2>0
定理1 多飞行器实现对机动目标的协同攻击当且仅当以下条件同时成立:
对于任意i∈{1, 2, …, N}
条件1
$ \begin{array}{*{20}{l}} {{k_2} {\rm{Re}} ({\lambda _i}) > 0}\\ {{k_2} {\rm{Re}} ({\lambda _i})({k_1}{k_2} {\rm{Re}} ({\lambda _i}) + {k_2} {\rm{Re}} {{({\lambda _i})}^2} + {\rm{Im}} {{({\lambda _i})}^2})) - }\\ {k_2^2 {\rm{Im}} {{({\lambda _i})}^2} > 0} \end{array} $ | (15) |
条件2
$ \begin{array}{*{20}{l}} {{\beta _1} {\rm{Re}} ({\lambda _i}) > 0}\\ {{\beta _1} {\rm{Re}} ({\lambda _i})({\beta _1}{\beta _2}( {\rm{Re}} {{({\lambda _i})}^2} + {\rm{Im}} {{({\lambda _i})}^2}) - {\beta _3} {\rm{Re}} ({\lambda _i})) - }\\ {\beta _2^2 {\rm{Im}} {{({\lambda _i})}^2} > 0}\\ {[{\beta _1}{\beta _2}{\beta _3} {\rm{Re}} ({\lambda _i})( {\rm{Re}} {{({\lambda _i})}^2} + {\rm{Im}} {{({\lambda _i})}^2}) - \beta _3^2 {\rm{Re}} {{({\lambda _i})}^2}] \cdot }\\ {[{\beta _1} {\rm{Re}} ({\lambda _i})({\beta _1}{\beta _2}( {\rm{Re}} {{({\lambda _i})}^2} + {\rm{Im}} {{({\lambda _i})}^2}) - {\beta _3} {\rm{Re}} ({\lambda _i})) - }\\ {\beta _2^2 {\rm{Im}} {{({\lambda _i})}^2}] - {{[{\beta _2}{\beta _3} {\rm{Re}} ({\lambda _i}) {\rm{Im}} ({\lambda _i})]}^2} > 0} \end{array} $ | (16) |
证明 由引理4,可以得到J=diag{0, J},其中J包括特征值λi(i=2, 3, …, N)对应的约旦块。
令
$ \begin{array}{l} \mathit{\boldsymbol{\dot \zeta }}(t) = \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0\\ { - {k_1}}&0&0&0&1\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}} \right]\mathit{\boldsymbol{\zeta }}(t)\\ \mathit{\boldsymbol{ \boldsymbol{\dot \varsigma} }}(t) = \left\{ {{\mathit{\boldsymbol{I}}_{n - 1}} \otimes \left[ {\begin{array}{*{20}{l}} 0&1&0&0&0\\ { - {k_1}}&0&0&0&1\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}} \right] + } \right.\\ \left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{J}} \otimes \left[ {\begin{array}{*{20}{c}} 0&0&0&0&0\\ { - {k_2}}&{ - {k_2}}&0&{{k_2}}&0\\ 0&0&{ - {\beta _1}}&0&0\\ 0&0&{ - {\beta _2}}&0&0\\ 0&0&{ - {\beta _3}}&0&0 \end{array}} \right]} \right\}\mathit{\boldsymbol{\zeta }}(t) \end{array} $ | (17) |
令
$ {{\mathit{\boldsymbol{\xi }}_\mathit{\boldsymbol{C}}}(t) = (\mathit{\boldsymbol{U}} \otimes {\mathit{\boldsymbol{I}}_5}){{[{\mathit{\boldsymbol{\zeta }}^{\rm{H}}}(t),0]}^{\rm{H}}}} $ |
$ {{\mathit{\boldsymbol{\xi }}_{\mathit{\boldsymbol{\bar C}}}}(t) = (\mathit{\boldsymbol{U}} \otimes {\mathit{\boldsymbol{I}}_5}){{[0,{\mathit{\boldsymbol{ \boldsymbol{\varsigma} }}^{\rm{H}}}(t)]}^{\rm{H}}}} $ |
以及ei∈RN为第i个元素为1其他元素为0的向量。
因c1, c2, c3, c4, c5为线性无关的向量,所以存在α1(t), α2(t), α3(t), α4(t), α5(t)以及α5k+j(t)(k=1, 2, …, N-1;j=1, 2, 3, 4, 5)使得
$ \begin{array}{l} \mathit{\boldsymbol{\zeta }}(t) = {\alpha _1}(t){\mathit{\boldsymbol{c}}_1} + {\alpha _2}(t){\mathit{\boldsymbol{c}}_2} + {\alpha _3}(t){\mathit{\boldsymbol{c}}_3} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\alpha _4}(t){\mathit{\boldsymbol{c}}_4} + {\alpha _5}(t){\mathit{\boldsymbol{c}}_5} \end{array} $ |
$ \begin{array}{*{20}{l}} {\mathit{\boldsymbol{ \boldsymbol{\varsigma} }}(t) = [{\alpha _6}(t)\mathit{\boldsymbol{c}}_1^{\rm{H}} + {\alpha _7}(t)\mathit{\boldsymbol{c}}_2^{\rm{H}} + {\alpha _8}(t)\mathit{\boldsymbol{c}}_3^{\rm{H}} + {\alpha _9}(t)\mathit{\boldsymbol{c}}_4^{\rm{H}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\alpha _{10}}(t)\mathit{\boldsymbol{c}}_5^{\rm{H}}, \cdots ,{\alpha _{5N - 4}}(t)\mathit{\boldsymbol{c}}_1^{\rm{H}} + {\alpha _{5N - 3}}(t)\mathit{\boldsymbol{c}}_2^{\rm{H}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\alpha _{5N - 2}}(t)\mathit{\boldsymbol{c}}_3^{\rm{H}} + {\alpha _{5N - 1}}(t)\mathit{\boldsymbol{c}}_4^{\rm{H}} + {\alpha _{5N}}(t)\mathit{\boldsymbol{c}}_5^{\rm{H}}{]^{\rm{H}}}} \end{array} $ |
由于[ζH(t), 0]H=e1⊗ζ(t),可以得到
$ \begin{array}{*{20}{c}} {{\mathit{\boldsymbol{\xi }}_\mathit{\boldsymbol{C}}}(t) = (\mathit{\boldsymbol{U}} \otimes {\mathit{\boldsymbol{I}}_5})({\mathit{\boldsymbol{e}}_1} \otimes \mathit{\boldsymbol{\zeta }}(t)) = {{\mathit{\boldsymbol{\bar u}}}_1} \otimes \mathit{\boldsymbol{\zeta }}(t) = }\\ {{\alpha _1}(t){\mathit{\boldsymbol{p}}_1} + {\alpha _2}(t){\mathit{\boldsymbol{p}}_2} + \cdots + {\alpha _5}(t){\mathit{\boldsymbol{p}}_5} \in \mathit{\boldsymbol{C}}(\mathit{\boldsymbol{U}})} \end{array} $ | (18) |
根据pj(j=6, 7, …, 5N),可以得到
$ \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{\xi }}_{\mathit{\boldsymbol{\bar C}}}}(t) = \sum\limits_{i = 2}^N {({\alpha _{5i - 4}}(t)(} {{\mathit{\boldsymbol{\bar u}}}_i} \otimes {\mathit{\boldsymbol{c}}_1}) + {\alpha _{5i - 3}}(t)({{\mathit{\boldsymbol{\bar u}}}_i} \otimes {\mathit{\boldsymbol{c}}_2}) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdots + {\alpha _{5i}}(t)({{\mathit{\boldsymbol{\bar u}}}_i} \otimes {\mathit{\boldsymbol{c}}_5})) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{j = 6}^N {{\alpha _j}} (t){\mathit{\boldsymbol{p}}_j} \in \mathit{\boldsymbol{\bar C}}(\mathit{\boldsymbol{U}})} \end{array} $ | (19) |
由于[ζH(t), ςH(t)]H=(U-1⊗I5)ξ(t),可以得到ξ(t)=ξC(t)+ξC(t)。由引理3和引理4多飞行器实现协同攻击当且仅当
$ \mathop {{\rm{lim}}}\limits_{t \to \infty } \mathit{\boldsymbol{ \boldsymbol{\varsigma} }}(t) = 0 $ | (20) |
从J的结构可知,方程(20)等价于以下N-1个子系统的稳定性:
$ {\mathit{\boldsymbol{ \boldsymbol{\dot \varsigma} }}_i}(t) = \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0\\ { - {k_1} - {k_2}{\lambda _i}}&{ - {k_2}{\lambda _i}}&0&{{k_2}{\lambda _i}}&1\\ 0&0&{ - {\beta _1}{\lambda _i}}&1&0\\ 0&0&{ - {\beta _2}{\lambda _i}}&0&1\\ 0&0&{ - {\beta _3}{\lambda _i}}&0&0 \end{array}} \right]{\mathit{\boldsymbol{ \boldsymbol{\varsigma} }}_i}(t) $ | (21) |
该系统的特征方程为
$ \begin{array}{*{20}{l}} {{f_i}(s) = ({s^2} + {k_2}{\lambda _i}s + {k_1} + {k_2}{\lambda _i}) \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({s^3} + {\lambda _i}{\beta _1}{s^2} + {\lambda _i}{\beta _2}s + {\lambda _i}{\beta _3})} \end{array} $ | (22) |
根据引理5和6可知当且仅当条件式(15), 式(16)成立时式(20)成立,即多飞行器系统实现对机动目标的协同攻击。
在定理1的基础上,联立式(4)和式(11)可以得到制导律的数学表达式为
$ \begin{array}{l} {n_i}(t) = \frac{{{V_i}{\rm{sin}}({\eta _i}(t))}}{{g{r_i}(t)}} - \frac{{{V_i}}}{{g{\rm{sin}}({\eta _i}(t))}}[ - {k_1}{x_{1i}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_2}\sum\limits_{j = 1}^{n - 1} {{a_{ij}}} (({x_{1i}}(t) - {x_{1j}}(t)) + ({{\hat x}_{2i}}(t) - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\hat x}_{2j}}(t))) - {{\hat x}_3}(t)] \end{array} $ | (23) |
为验证本文所提出的针对机动目标的分布式协同制导律的有效性,以4个飞行器对机动目标进行协同攻击的情形为例,在通讯拓扑以及过载限幅的条件下,进行仿真分析。其通讯拓扑如图 2所示。
该拓扑的拉普拉斯矩阵为
$ \mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0\\ { - 1}&1&0&0\\ { - 1}&0&2&{ - 1}\\ 0&{ - 1}&0&1 \end{array}} \right] $ |
其特征值为λ1=0, λ2=1, λ3=λ4=2,根据定理1参数选取范围为k1>-k2, k2>0且β1>0, β2>0, β3>0, β1β2-β3>0。取k1=0.1, k2=0.7, β1=50, β2=7 500, β3=125 000进行仿真。初始条件如表 1所示。
目标初始位置为(10 000, 0), 速度大小为20 m/s2,其过载为0.1 m/s2。在飞行器距离目标大于500 m时采用本文中所提出的协同制导律,小于500 m时采用比例导引,仿真结果如图 3~图 6所示。
图 3为飞行过程中的飞行轨迹,图 4为Ri/Vi在飞行过程中的变化情况,图 5为飞行器在飞行过程中的过载指令变化情况,图 6为飞行器飞行剩余飞行时间偏差收敛的情况。
从图 4和图 6中可以看出在一段时间之后飞行器剩余飞行时间偏差收敛到一致的状态,同时从图 5中可以看出在整个飞行过程中飞行器的过载被限制在20g以内。需要指出的是图 6中,最后命中时间出现了一定的偏差,这是由于在切换成比例导引之后命中时间不再受控制因此导致命中时间出现了一定偏差,但是该偏差量相比命中时间初始偏差小一个数量级。
4 结论1) 基于反馈线性化方法及一致性理论,提出了针对机动目标的协同制导律,并通过极点分析给出了制导律参数的选取方法。
2) 所提出的协同制导律能够在有向拓扑、不需要目标加速度测量信息、不需要飞行器具备轴向机动的条件下实现对机动目标的协同攻击。
3) 在制导末端存在制导指令过大的问题,目前采用切换成比例导引的方式避免该问题,但是会导致命中时间在制导末端产生一定的误差,针对该问题仍需进一步研究。
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