【控制】多智能体系统总结。4.控制协议。
3. 控制协议3.1 动态一致性控制协议控制协议为ui=α∑j∈Niaij(pj−pi)+β∑j∈Niaij(vj−vi)()u_i = \red{\alpha} \sum_{j \in N_i}a_{ij} (p_j - p_i) + \blue{\beta} \sum_{j \in N_i}a_{ij} (v_j - v_i)\tag{}ui=αj∈Ni∑aij(pj−pi)+βj
【控制】多智能体系统总结。1. 系统模型。2.控制目标。3.模型转换。
【控制】多智能体系统总结。4.控制协议。
【控制】多智能体系统总结。5.系统合并。
文章目录
4. 控制协议
4.1 一阶一维系统
u i = α ∑ j ∈ N i a i j ( p j − p i ) ( ) u_i = \alpha \sum_{j \in N_i} a_{ij} (p_j - p_i) \tag{} ui=αj∈Ni∑aij(pj−pi)()
u 1 = α a 11 ( p 1 − p 1 ) + α a 12 ( p 2 − p 1 ) + α a 13 ( p 3 − p 1 ) + ⋯ u 2 = α a 21 ( p 1 − p 2 ) + α a 22 ( p 2 − p 2 ) + α a 23 ( p 3 − p 2 ) + ⋯ u 3 = α a 31 ( p 1 − p 3 ) + α a 32 ( p 2 − p 3 ) + α a 33 ( p 3 − p 3 ) + ⋯ ( ) \begin{aligned} u_1 &= {\alpha} a_{11} (p_1 - p_1) &+ {\alpha} a_{12} (p_2 - p_1) &+ {\alpha} a_{13} (p_3 - p_1) &+ \cdots \\ u_2 &= {\alpha} a_{21} (p_1 - p_2) &+ {\alpha} a_{22} (p_2 - p_2) &+ {\alpha} a_{23} (p_3 - p_2) &+ \cdots \\ u_3 &= {\alpha} a_{31} (p_1 - p_3) &+ {\alpha} a_{32} (p_2 - p_3) &+ {\alpha} a_{33} (p_3 - p_3) &+ \cdots \\ \end{aligned} \tag{} u1u2u3=αa11(p1−p1)=αa21(p1−p2)=αa31(p1−p3)+αa12(p2−p1)+αa22(p2−p2)+αa32(p2−p3)+αa13(p3−p1)+αa23(p3−p2)+αa33(p3−p3)+⋯+⋯+⋯()
[ u 1 u 2 u 3 ] = [ − α d 1 α a 12 α a 13 α a 21 − α d 2 α a 23 α a 31 α a 32 − α d 3 ] [ p 1 p 2 p 3 ] = − α L ⋅ X ( ) \begin{aligned} \left[\begin{matrix} u_1 \\ u_2 \\ u_3 \\ \end{matrix}\right] &= \left[\begin{matrix} -\alpha d_{ 1} & \alpha a_{12} & \alpha a_{13} \\ \alpha a_{21} & -\alpha d_{ 2} & \alpha a_{23} \\ \alpha a_{31} & \alpha a_{32} & -\alpha d_{ 3} \\ \end{matrix}\right] \left[\begin{matrix} p_1 \\ p_2 \\ p_3 \\ \end{matrix}\right] \\ &= \red{-\alpha L \cdot X} \end{aligned} \tag{} ⎣⎡u1u2u3⎦⎤=⎣⎡−αd1αa21αa31αa12−αd2αa32αa13αa23−αd3⎦⎤⎣⎡p1p2p3⎦⎤=−αL⋅X()
4.2 一阶二维系统
u i x = α ∑ j ∈ N i a i j ( p j x − p i x ) u i y = α ∑ j ∈ N i a i j ( p j y − p i y ) ( ) \begin{aligned} u_i^x &= \alpha \sum_{j \in N_i} a_{ij} (p_j^x - p_i^x) \\ u_i^y &= \alpha \sum_{j \in N_i} a_{ij} (p_j^y - p_i^y) \\ \end{aligned}\tag{} uixuiy=αj∈Ni∑aij(pjx−pix)=αj∈Ni∑aij(pjy−piy)()
u 1 x = α a 11 ( p 1 x − p 1 x ) + α a 12 ( p 2 x − p 1 x ) + α a 13 ( p 3 x − p 1 x ) + ⋯ u 1 y = α a 11 ( p 1 y − p 1 y ) + α a 12 ( p 2 y − p 1 y ) + α a 13 ( p 3 y − p 1 y ) + ⋯ u 2 x = α a 21 ( p 1 x − p 2 x ) + α a 22 ( p 2 x − p 2 x ) + α a 23 ( p 3 x − p 2 x ) + ⋯ u 2 y = α a 21 ( p 1 y − p 2 y ) + α a 22 ( p 2 y − p 2 y ) + α a 23 ( p 3 y − p 2 y ) + ⋯ u 3 x = α a 31 ( p 1 x − p 3 x ) + α a 32 ( p 2 x − p 3 x ) + α a 33 ( p 3 x − p 3 x ) + ⋯ u 3 y = α a 31 ( p 1 y − p 3 y ) + α a 32 ( p 2 y − p 3 y ) + α a 33 ( p 3 y − p 3 y ) + ⋯ ( ) \begin{aligned} u_1^x &= {\alpha} a_{11} (p_1^x - p_1^x) &+ {\alpha} a_{12} (p_2^x - p_1^x) &+ {\alpha} a_{13} (p_3^x - p_1^x) &+ \cdots \\ u_1^y &= {\alpha} a_{11} (p_1^y - p_1^y) &+ {\alpha} a_{12} (p_2^y - p_1^y) &+ {\alpha} a_{13} (p_3^y - p_1^y) &+ \cdots \\ u_2^x &= {\alpha} a_{21} (p_1^x - p_2^x) &+ {\alpha} a_{22} (p_2^x - p_2^x) &+ {\alpha} a_{23} (p_3^x - p_2^x) &+ \cdots \\ u_2^y &= {\alpha} a_{21} (p_1^y - p_2^y) &+ {\alpha} a_{22} (p_2^y - p_2^y) &+ {\alpha} a_{23} (p_3^y - p_2^y) &+ \cdots \\ u_3^x &= {\alpha} a_{31} (p_1^x - p_3^x) &+ {\alpha} a_{32} (p_2^x - p_3^x) &+ {\alpha} a_{33} (p_3^x - p_3^x) &+ \cdots \\ u_3^y &= {\alpha} a_{31} (p_1^y - p_3^y) &+ {\alpha} a_{32} (p_2^y - p_3^y) &+ {\alpha} a_{33} (p_3^y - p_3^y) &+ \cdots \\ \end{aligned} \tag{} u1xu1yu2xu2yu3xu3y=αa11(p1x−p1x)=αa11(p1y−p1y)=αa21(p1x−p2x)=αa21(p1y−p2y)=αa31(p1x−p3x)=αa31(p1y−p3y)+αa12(p2x−p1x)+αa12(p2y−p1y)+αa22(p2x−p2x)+αa22(p2y−p2y)+αa32(p2x−p3x)+αa32(p2y−p3y)+αa13(p3x−p1x)+αa13(p3y−p1y)+αa23(p3x−p2x)+αa23(p3y−p2y)+αa33(p3x−p3x)+αa33(p3y−p3y)+⋯+⋯+⋯+⋯+⋯+⋯()
4.2.1 方式一
[ u 1 x u 1 y u 2 x u 2 y u 3 x u 3 y ] = [ − α d 1 0 α a 12 0 α a 13 0 0 − α d 1 0 α a 12 0 α a 13 α a 21 0 − α d 2 0 α a 23 0 0 α a 21 0 − α d 2 0 α a 23 α a 31 0 α a 32 0 − α d 3 0 0 α a 31 0 α a 32 0 − α d 3 ] [ p 1 x p 1 y p 2 x p 2 y p 3 x p 3 y ] = − α L ⊗ I 2 ⋅ X ( ) \begin{aligned} \left[\begin{matrix} u_1^x \\ u_1^y \\ u_2^x \\ u_2^y \\ u_3^x \\ u_3^y \\ \end{matrix}\right] &= \left[\begin{matrix} -\alpha d_{ 1} & 0 & \alpha a_{12} & 0 & \alpha a_{13} & 0 \\ 0 & -\alpha d_{ 1} & 0 & \alpha a_{12} & 0 & \alpha a_{13} \\ \alpha a_{21} & 0 & -\alpha d_{ 2} & 0 & \alpha a_{23} & 0 \\ 0 & \alpha a_{21} & 0 & -\alpha d_{ 2} & 0 & \alpha a_{23} \\ \alpha a_{31} & 0 & \alpha a_{32} & 0 & -\alpha d_{ 3} & 0 \\ 0 & \alpha a_{31} & 0 & \alpha a_{32} & 0 & -\alpha d_{ 3} \\ \end{matrix}\right] \left[\begin{matrix} p_1^x \\ p_1^y \\ p_2^x \\ p_2^y \\ p_3^x \\ p_3^y \\ \end{matrix}\right] \\ &= \red{-\alpha L \otimes I_2 \cdot X} \end{aligned} \tag{} ⎣⎢⎢⎢⎢⎢⎢⎡u1xu1yu2xu2yu3xu3y⎦⎥⎥⎥⎥⎥⎥⎤=⎣⎢⎢⎢⎢⎢⎢⎡−αd10αa210αa3100−αd10αa210αa31αa120−αd20αa3200αa120−αd20αa32αa130αa230−αd300αa130αa230−αd3⎦⎥⎥⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎢⎢⎡p1xp1yp2xp2yp3xp3y⎦⎥⎥⎥⎥⎥⎥⎤=−αL⊗I2⋅X()
4.2.2 方式二
[ u 1 x u 2 x u 3 x u 1 y u 2 y u 3 y ] = [ − α d 1 α a 12 α a 13 0 0 0 α a 21 − α d 2 α a 23 0 0 0 α a 31 α a 32 − α d 3 0 0 0 0 0 0 − α d 1 α a 12 α a 13 0 0 0 α a 21 − α d 2 α a 23 0 0 0 α a 31 α a 32 − α d 3 ] [ p 1 x p 2 x p 3 x p 1 y p 2 y p 3 y ] = I 2 ⊗ − α L ⋅ X ( ) \begin{aligned} \left[\begin{matrix} u_1^x \\ u_2^x \\ u_3^x \\ u_1^y \\ u_2^y \\ u_3^y \\ \end{matrix}\right] &= \left[\begin{matrix} -\alpha d_{ 1} & \alpha a_{12} & \alpha a_{13} & 0 & 0 & 0 \\ \alpha a_{21} & -\alpha d_{ 2} & \alpha a_{23} & 0 & 0 & 0 \\ \alpha a_{31} & \alpha a_{32} & -\alpha d_{ 3} & 0 & 0 & 0 \\ 0 & 0 & 0 & -\alpha d_{ 1} & \alpha a_{12} & \alpha a_{13} \\ 0 & 0 & 0 & \alpha a_{21} & -\alpha d_{ 2} & \alpha a_{23} \\ 0 & 0 & 0 & \alpha a_{31} & \alpha a_{32} & -\alpha d_{ 3} \\ \end{matrix}\right] \left[\begin{matrix} p_1^x \\ p_2^x \\ p_3^x \\ p_1^y \\ p_2^y \\ p_3^y \\ \end{matrix}\right] \\ &= \red{I_2 \otimes -\alpha L \cdot X} \end{aligned} \tag{} ⎣⎢⎢⎢⎢⎢⎢⎡u1xu2xu3xu1yu2yu3y⎦⎥⎥⎥⎥⎥⎥⎤=⎣⎢⎢⎢⎢⎢⎢⎡−αd1αa21αa31000αa12−αd2αa32000αa13αa23−αd3000000−αd1αa21αa31000αa12−αd2αa32000αa13αa23−αd3⎦⎥⎥⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎢⎢⎡p1xp2xp3xp1yp2yp3y⎦⎥⎥⎥⎥⎥⎥⎤=I2⊗−αL⋅X()
4.3 二阶一维系统
u i = α ∑ j ∈ N i a i j ( p j − p i ) + β ∑ j ∈ N i a i j ( v j − v i ) ( ) u_i = \alpha \sum_{j \in N_i} a_{ij} (p_j - p_i) + \beta \sum_{j \in N_i} a_{ij} (v_j - v_i) \tag{} ui=αj∈Ni∑aij(pj−pi)+βj∈Ni∑aij(vj−vi)()
u 1 = α a 11 ( p 1 − p 1 ) + β a 11 ( v 1 − v 1 ) + α a 12 ( p 2 − p 1 ) + β a 12 ( v 2 − v 1 ) + α a 13 ( p 3 − p 1 ) + β a 13 ( v 3 − v 1 ) + ⋯ u 2 = α a 21 ( p 1 − p 2 ) + β a 21 ( v 1 − v 2 ) + α a 22 ( p 2 − p 2 ) + β a 22 ( v 2 − v 2 ) + α a 23 ( p 3 − p 2 ) + β a 23 ( v 3 − v 2 ) + ⋯ u 3 = α a 31 ( p 1 − p 3 ) + β a 31 ( v 1 − v 3 ) + α a 32 ( p 2 − p 3 ) + β a 32 ( v 2 − v 3 ) + α a 33 ( p 3 − p 3 ) + β a 33 ( v 3 − v 3 ) + ⋯ ( ) \begin{aligned} u_1 &= {\alpha} a_{11} (p_1 - p_1) + {\beta} a_{11} (v_1 - v_1) &+ {\alpha} a_{12} (p_2 - p_1) + {\beta} a_{12} (v_2 - v_1) &+ {\alpha} a_{13} (p_3 - p_1) + {\beta} a_{13} (v_3 - v_1) &+ \cdots \\ u_2 &= {\alpha} a_{21} (p_1 - p_2) + {\beta} a_{21} (v_1 - v_2) &+ {\alpha} a_{22} (p_2 - p_2) + {\beta} a_{22} (v_2 - v_2) &+ {\alpha} a_{23} (p_3 - p_2) + {\beta} a_{23} (v_3 - v_2) &+ \cdots \\ u_3 &= {\alpha} a_{31} (p_1 - p_3) + {\beta} a_{31} (v_1 - v_3) &+ {\alpha} a_{32} (p_2 - p_3) + {\beta} a_{32} (v_2 - v_3) &+ {\alpha} a_{33} (p_3 - p_3) + {\beta} a_{33} (v_3 - v_3) &+ \cdots \\ \end{aligned} \tag{} u1u2u3=αa11(p1−p1)+βa11(v1−v1)=αa21(p1−p2)+βa21(v1−v2)=αa31(p1−p3)+βa31(v1−v3)+αa12(p2−p1)+βa12(v2−v1)+αa22(p2−p2)+βa22(v2−v2)+αa32(p2−p3)+βa32(v2−v3)+αa13(p3−p1)+βa13(v3−v1)+αa23(p3−p2)+βa23(v3−v2)+αa33(p3−p3)+βa33(v3−v3)+⋯+⋯+⋯()
4.3.1 方式一
[ u 1 u 2 u 3 ] = [ − α d 1 − β d 1 α a 12 β a 12 α a 13 β a 13 α a 21 β a 21 − α d 2 − β d 2 α a 23 β a 23 α a 31 β a 31 α a 32 β a 32 − α d 3 − β d 3 ] [ p 1 v 1 p 2 v 2 p 3 v 3 ] = ( − L ) ⊗ [ α β ] ⋅ X ( ) \begin{aligned} \left[\begin{matrix} u_1 \\ u_2 \\ u_3 \\ \end{matrix}\right] &= \left[\begin{matrix} -\alpha d_{ 1} & -\beta d_{ 1} & \alpha a_{12} & \beta a_{12} & \alpha a_{13} & \beta a_{13} \\ \alpha a_{21} & \beta a_{21} & -\alpha d_{ 2} & -\beta d_{ 2} & \alpha a_{23} & \beta a_{23} \\ \alpha a_{31} & \beta a_{31} & \alpha a_{32} & \beta a_{32} & -\alpha d_{ 3} & -\beta d_{ 3} \\ \end{matrix}\right] \left[\begin{matrix} p_1 \\ v_1 \\ p_2 \\ v_2 \\ p_3 \\ v_3 \\ \end{matrix}\right] \\ &= \red{(-L) \otimes \left[\begin{matrix} \alpha & \beta \end{matrix}\right] \cdot X} \end{aligned} \tag{} ⎣⎡u1u2u3⎦⎤=⎣⎡−αd1αa21αa31−βd1βa21βa31αa12−αd2αa32βa12−βd2βa32αa13αa23−αd3βa13βa23−βd3⎦⎤⎣⎢⎢⎢⎢⎢⎢⎡p1v1p2v2p3v3⎦⎥⎥⎥⎥⎥⎥⎤=(−L)⊗[αβ]⋅X()
4.3.2 方式二
[ u 1 u 2 u 3 ] = [ − α d 1 α a 12 α a 13 − β d 1 β a 12 β a 13 α a 21 − α d 2 α a 23 β a 21 − β d 2 β a 23 α a 31 α a 32 − α d 3 β a 31 β a 32 − β d 3 ] [ p 1 p 2 p 3 v 1 v 2 v 3 ] = [ α β ] ⊗ ( − L ) ⋅ X ( ) \begin{aligned} \left[\begin{matrix} u_1 \\ u_2 \\ u_3 \\ \end{matrix}\right] &= \left[\begin{matrix} -\alpha d_{ 1} & \alpha a_{12} & \alpha a_{13} & -\beta d_{ 1} & \beta a_{12} & \beta a_{13} \\ \alpha a_{21} & -\alpha d_{ 2} & \alpha a_{23} & \beta a_{21} & -\beta d_{ 2} & \beta a_{23} \\ \alpha a_{31} & \alpha a_{32} & -\alpha d_{ 3} & \beta a_{31} & \beta a_{32} & -\beta d_{ 3} \\ \end{matrix}\right] \left[\begin{matrix} p_1 \\ p_2 \\ p_3 \\ v_1 \\ v_2 \\ v_3 \\ \end{matrix}\right] \\ &= \red{ \left[\begin{matrix} \alpha & \beta \end{matrix}\right] \otimes (-L) \cdot X} \end{aligned} \tag{} ⎣⎡u1u2u3⎦⎤=⎣⎡−αd1αa21αa31αa12−αd2αa32αa13αa23−αd3−βd1βa21βa31βa12−βd2βa32βa13βa23−βd3⎦⎤⎣⎢⎢⎢⎢⎢⎢⎡p1p2p3v1v2v3⎦⎥⎥⎥⎥⎥⎥⎤=[αβ]⊗(−L)⋅X()
4.4 二阶二维系统
u i x = α ∑ j ∈ N i a i j ( p j x − p i x ) + β ∑ j ∈ N i a i j ( v j x − v i x ) u i y = α ∑ j ∈ N i a i j ( p j y − p i y ) + β ∑ j ∈ N i a i j ( v j y − v i y ) ( ) \begin{aligned} u_i^x &= \alpha \sum_{j \in N_i} a_{ij} (p_j^x - p_i^x) + \beta \sum_{j \in N_i} a_{ij} (v_j^x - v_i^x) \\ u_i^y &= \alpha \sum_{j \in N_i} a_{ij} (p_j^y - p_i^y) + \beta \sum_{j \in N_i} a_{ij} (v_j^y - v_i^y) \\ \end{aligned}\tag{} uixuiy=αj∈Ni∑aij(pjx−pix)+βj∈Ni∑aij(vjx−vix)=αj∈Ni∑aij(pjy−piy)+βj∈Ni∑aij(vjy−viy)()
u 1 = α a 11 ( p 1 x − p 1 x ) + β a 11 ( v 1 x − v 1 x ) + α a 12 ( p 2 x − p 1 x ) + β a 12 ( v 2 x − v 1 x ) + α a 13 ( p 3 x − p 1 x ) + β a 13 ( v 3 x − v 1 x ) + ⋯ u 2 = α a 21 ( p 1 − p 2 ) + β a 21 ( v 1 − v 2 ) + α a 22 ( p 2 − p 2 ) + β a 22 ( v 2 − v 2 ) + α a 23 ( p 3 − p 2 ) + β a 23 ( v 3 − v 2 ) + ⋯ u 3 = α a 31 ( p 1 − p 3 ) + β a 31 ( v 1 − v 3 ) + α a 32 ( p 2 − p 3 ) + β a 32 ( v 2 − v 3 ) + α a 33 ( p 3 − p 3 ) + β a 33 ( v 3 − v 3 ) + ⋯ ( ) \begin{aligned} u_1 &= {\alpha} a_{11} (p_1^x - p_1^x) + {\beta} a_{11} (v_1^x - v_1^x) &+ {\alpha} a_{12} (p_2^x - p_1^x) + {\beta} a_{12} (v_2^x - v_1^x) &+ {\alpha} a_{13} (p_3^x - p_1^x) + {\beta} a_{13} (v_3^x - v_1^x) &+ \cdots \\ u_2 &= {\alpha} a_{21} (p_1 - p_2) + {\beta} a_{21} (v_1 - v_2) &+ {\alpha} a_{22} (p_2 - p_2) + {\beta} a_{22} (v_2 - v_2) &+ {\alpha} a_{23} (p_3 - p_2) + {\beta} a_{23} (v_3 - v_2) &+ \cdots \\ u_3 &= {\alpha} a_{31} (p_1 - p_3) + {\beta} a_{31} (v_1 - v_3) &+ {\alpha} a_{32} (p_2 - p_3) + {\beta} a_{32} (v_2 - v_3) &+ {\alpha} a_{33} (p_3 - p_3) + {\beta} a_{33} (v_3 - v_3) &+ \cdots \\ \end{aligned} \tag{} u1u2u3=αa11(p1x−p1x)+βa11(v1x−v1x)=αa21(p1−p2)+βa21(v1−v2)=αa31(p1−p3)+βa31(v1−v3)+αa12(p2x−p1x)+βa12(v2x−v1x)+αa22(p2−p2)+βa22(v2−v2)+αa32(p2−p3)+βa32(v2−v3)+αa13(p3x−p1x)+βa13(v3x−v1x)+αa23(p3−p2)+βa23(v3−v2)+αa33(p3−p3)+βa33(v3−v3)+⋯+⋯+⋯()
4.4.1 方式一
[ u 1 x u 1 y u 2 x u 2 y u 3 x u 3 y ] = [ − α d 1 0 − β d 1 0 α a 12 0 β a 12 0 α a 13 0 β a 13 0 0 − α d 1 0 − β d 1 0 α a 12 0 β a 12 0 α a 13 0 β a 13 α a 21 0 β a 21 0 − α d 2 0 − β d 2 0 α a 23 0 β a 23 0 0 α a 21 0 β a 21 0 − α d 2 0 − β d 2 0 α a 23 0 β a 23 α a 31 0 β a 31 0 α a 32 0 β a 32 0 − α d 3 0 − β d 3 0 0 α a 31 0 β a 31 0 α a 32 0 β a 32 0 − α d 3 0 − β d 3 ] [ p 1 x p 1 y v 1 x v 1 y p 2 x p 2 y v 2 x v 2 y p 3 x p 3 y v 3 x v 3 y ] = − L ⊗ [ α β ] ⊗ I 2 ⋅ X ( ) \begin{aligned} \left[\begin{matrix} u_1^x \\ u_1^y \\ u_2^x \\ u_2^y \\ u_3^x \\ u_3^y \\ \end{matrix}\right] &= \left[\begin{matrix} -\alpha d_{ 1} & 0 & -\beta d_{ 1} & 0 & \alpha a_{12} & 0 & \beta a_{12} & 0 & \alpha a_{13} & 0 & \beta a_{13} & 0 \\ 0 & -\alpha d_{ 1} & 0 & -\beta d_{ 1} & 0 & \alpha a_{12} & 0 & \beta a_{12} & 0 & \alpha a_{13} & 0 & \beta a_{13} \\ \alpha a_{21} & 0 & \beta a_{21} & 0 & -\alpha d_{ 2} & 0 & -\beta d_{ 2} & 0 & \alpha a_{23} & 0 & \beta a_{23} & 0 \\ 0 & \alpha a_{21} & 0 & \beta a_{21} & 0 & -\alpha d_{ 2} & 0 & -\beta d_{ 2} & 0 & \alpha a_{23} & 0 & \beta a_{23} \\ \alpha a_{31} & 0 & \beta a_{31} & 0 & \alpha a_{32} & 0 & \beta a_{32} & 0 & -\alpha d_{ 3} & 0 & -\beta d_{ 3} & 0 \\ 0 & \alpha a_{31} & 0 & \beta a_{31} & 0 & \alpha a_{32} & 0 & \beta a_{32} & 0 & -\alpha d_{ 3} & 0 & -\beta d_{ 3} \\ \end{matrix}\right] \left[\begin{matrix} p_1^x \\ p_1^y \\ v_1^x \\ v_1^y \\ p_2^x \\ p_2^y \\ v_2^x \\ v_2^y \\ p_3^x \\ p_3^y \\ v_3^x \\ v_3^y \\ \end{matrix}\right] \\ &= \red{ -L \otimes \left[\begin{matrix} \alpha & \beta \end{matrix}\right] \otimes I_2 \cdot X} \end{aligned} \tag{} ⎣⎢⎢⎢⎢⎢⎢⎡u1xu1yu2xu2yu3xu3y⎦⎥⎥⎥⎥⎥⎥⎤=⎣⎢⎢⎢⎢⎢⎢⎡−αd10αa210αa3100−αd10αa210αa31−βd10βa210βa3100−βd10βa210βa31αa120−αd20αa3200αa120−αd20αa32βa120−βd20βa3200βa120−βd20βa32αa130αa230−αd300αa130αa230−αd3βa130βa230−βd300βa130βa230−βd3⎦⎥⎥⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡p1xp1yv1xv1yp2xp2yv2xv2yp3xp3yv3xv3y⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤=−L⊗[αβ]⊗I2⋅X()
4.4.2 方式二
[ u 1 x u 2 x u 3 x u 1 y u 2 y u 3 y ] = [ − α d 1 α a 12 α a 13 0 0 0 − β d 1 β a 12 β a 13 0 0 0 α a 21 − α d 2 α a 23 0 0 0 β a 21 − β d 2 β a 23 0 0 0 α a 31 α a 32 − α d 3 0 0 0 β a 31 β a 32 − β d 3 0 0 0 0 0 0 − α d 1 α a 12 α a 13 0 0 0 − β d 1 β a 12 β a 13 0 0 0 α a 21 − α d 2 α a 23 0 0 0 β a 21 − β d 2 β a 23 0 0 0 α a 31 α a 32 − α d 3 0 0 0 β a 31 β a 32 − β d 3 ] [ p 1 x p 2 x p 3 x p 1 y p 2 y p 3 y v 1 x v 2 x v 3 x v 1 y v 2 y v 3 y ] = [ α β ] ⊗ I 2 ⊗ ( − L ) ⋅ X ( ) \begin{aligned} \left[\begin{matrix} u_1^x \\ u_2^x \\ u_3^x \\ u_1^y \\ u_2^y \\ u_3^y \\ \end{matrix}\right] &= \left[\begin{matrix} -\alpha d_{ 1} & \alpha a_{12} & \alpha a_{13} & 0 & 0 & 0 &-\beta d_{ 1} & \beta a_{12} & \beta a_{13} & 0 & 0 & 0 \\ \alpha a_{21} & -\alpha d_{ 2} & \alpha a_{23} & 0 & 0 & 0 & \beta a_{21} & -\beta d_{ 2} & \beta a_{23} & 0 & 0 & 0 \\ \alpha a_{31} & \alpha a_{32} & -\alpha d_{ 3} & 0 & 0 & 0 & \beta a_{31} & \beta a_{32} & -\beta d_{ 3} & 0 & 0 & 0 \\ 0 & 0 & 0 & -\alpha d_{ 1} & \alpha a_{12} & \alpha a_{13} & 0 & 0 & 0 &-\beta d_{ 1} & \beta a_{12} & \beta a_{13} \\ 0 & 0 & 0 & \alpha a_{21} & -\alpha d_{ 2} & \alpha a_{23} & 0 & 0 & 0 & \beta a_{21} & -\beta d_{ 2} & \beta a_{23} \\ 0 & 0 & 0 & \alpha a_{31} & \alpha a_{32} & -\alpha d_{ 3} & 0 & 0 & 0 & \beta a_{31} & \beta a_{32} & -\beta d_{ 3} \\ \end{matrix}\right] \left[\begin{matrix} p_1^x \\ p_2^x \\ p_3^x \\ p_1^y \\ p_2^y \\ p_3^y \\ v_1^x \\ v_2^x \\ v_3^x \\ v_1^y \\ v_2^y \\ v_3^y \\ \end{matrix}\right] \\ &= \red{ \left[\begin{matrix} \alpha & \beta \end{matrix}\right] \otimes I_2 \otimes (-L) \cdot X} \end{aligned} \tag{} ⎣⎢⎢⎢⎢⎢⎢⎡u1xu2xu3xu1yu2yu3y⎦⎥⎥⎥⎥⎥⎥⎤=⎣⎢⎢⎢⎢⎢⎢⎡−αd1αa21αa31000αa12−αd2αa32000αa13αa23−αd3000000−αd1αa21αa31000αa12−αd2αa32000αa13αa23−αd3−βd1βa21βa31000βa12−βd2βa32000βa13βa23−βd3000000−βd1βa21βa31000βa12−βd2βa32000βa13βa23−βd3⎦⎥⎥⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡p1xp2xp3xp1yp2yp3yv1xv2xv3xv1yv2yv3y⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤=[αβ]⊗I2⊗(−L)⋅X()
3.1 动态一致性控制协议
控制协议为
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u_i = \red{\alpha} \sum_{j \in N_i} a_{ij} (p_j - p_i) + \blue{\beta} \sum_{j \in N_i} a_{ij} (v_j - v_i) \tag{}
ui=αj∈Ni∑aij(pj−pi)+βj∈Ni∑aij(vj−vi)()
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通过分析控制协议：
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\begin{aligned} u_1 &= \red{\alpha} a_{11} (p_1 - p_1) + \blue{\beta} a_{11} (v_1 - v_1) &+ \red{\alpha} a_{12} (p_2 - p_1) + \blue{\beta} a_{12} (v_2 - v_1) &+ \red{\alpha} a_{13} (p_3 - p_1) + \blue{\beta} a_{13} (v_3 - v_1) &+ \cdots \\ u_2 &= \red{\alpha} a_{21} (p_1 - p_2) + \blue{\beta} a_{21} (v_1 - v_2) &+ \red{\alpha} a_{22} (p_2 - p_2) + \blue{\beta} a_{22} (v_2 - v_2) &+ \red{\alpha} a_{23} (p_3 - p_2) + \blue{\beta} a_{23} (v_3 - v_2) &+ \cdots \\ u_3 &= \red{\alpha} a_{31} (p_1 - p_3) + \blue{\beta} a_{31} (v_1 - v_3) &+ \red{\alpha} a_{32} (p_2 - p_3) + \blue{\beta} a_{32} (v_2 - v_3) &+ \red{\alpha} a_{33} (p_3 - p_3) + \blue{\beta} a_{33} (v_3 - v_3) &+ \cdots \\ \end{aligned} \tag{}
u1u2u3=αa11(p1−p1)+βa11(v1−v1)=αa21(p1−p2)+βa21(v1−v2)=αa31(p1−p3)+βa31(v1−v3)+αa12(p2−p1)+βa12(v2−v1)+αa22(p2−p2)+βa22(v2−v2)+αa32(p2−p3)+βa32(v2−v3)+αa13(p3−p1)+βa13(v3−v1)+αa23(p3−p2)+βa23(v3−v2)+αa33(p3−p3)+βa33(v3−v3)+⋯+⋯+⋯()
可以得到以下控制协议的矩阵形式：
[ u 1 x u 1 y u 2 x u 2 y u 3 x u 3 y ] = [ − α d 1 0 − β d 1 0 α a 12 0 β a 12 0 α a 13 0 β a 13 0 0 − α d 1 0 − β d 1 0 α a 12 0 β a 12 0 α a 13 0 β a 13 α a 21 0 β a 21 0 − α d 2 0 − β d 2 0 α a 23 0 β a 23 0 0 α a 21 0 β a 21 0 − α d 2 0 − β d 2 0 α a 23 0 β a 23 α a 31 0 β a 31 0 α a 32 0 β a 32 0 − α d 3 0 − β d 3 0 0 α a 31 0 β a 31 0 α a 32 0 β a 32 0 − α d 3 0 − β d 3 ] [ p 1 x p 1 y v 1 x v 1 y p 2 x p 2 y v 2 x v 2 y p 3 x p 3 y v 3 x v 3 y ] = [ − d 1 a 12 a 13 a 21 − d 2 a 23 a 31 a 32 − d 3 ] ⊗ [ α β ] ⊗ I 2 ⋅ X i = − L ⊗ [ α β ] ⊗ I 2 ⋅ X i \begin{aligned} \left[\begin{matrix} u_1^x \\ u_1^y \\ u_2^x \\ u_2^y \\ u_3^x \\ u_3^y \\ \end{matrix}\right]&= \left[\begin{matrix} -\red{\alpha}d_1 & 0 & -\blue{\beta}d_1 & 0 & \red{\alpha}a_{12} & 0 & \blue{\beta}a_{12} & 0 & \red{\alpha}a_{13} & 0 & \blue{\beta}a_{13} & 0 \\ 0 & -\red{\alpha}d_1 & 0 & -\blue{\beta}d_1 & 0 & \red{\alpha}a_{12} & 0 & \blue{\beta}a_{12} & 0 & \red{\alpha}a_{13} & 0 & \blue{\beta}a_{13} \\ \red{\alpha}a_{21} & 0 & \blue{\beta}a_{21} & 0 & -\red{\alpha}d_{2} & 0 & -\blue{\beta}d_{2} & 0 & \red{\alpha}a_{23} & 0 & \blue{\beta}a_{23} & 0 \\ 0 & \red{\alpha}a_{21} & 0 & \blue{\beta}a_{21} & 0 & -\red{\alpha}d_{2} & 0 & -\blue{\beta}d_{2} & 0 & \red{\alpha}a_{23} & 0 & \blue{\beta}a_{23} \\ \red{\alpha}a_{31} & 0 & \blue{\beta}a_{31} & 0 & \red{\alpha}a_{32} & 0 & \blue{\beta}a_{32} & 0 & -\red{\alpha}d_{3} & 0 & -\blue{\beta}d_{3} & 0 \\ 0 & \red{\alpha}a_{31} & 0 & \blue{\beta}a_{31} & 0 & \red{\alpha}a_{32} & 0 & \blue{\beta}a_{32} & 0 & -\red{\alpha}d_{3} & 0 & -\blue{\beta}d_{3} \\ \end{matrix}\right] \left[\begin{matrix} p_1^x \\ p_1^y \\ v_1^x \\ v_1^y \\ p_2^x \\ p_2^y \\ v_2^x \\ v_2^y \\ p_3^x \\ p_3^y \\ v_3^x \\ v_3^y \\ \end{matrix}\right] \\ &= \left[\begin{matrix} -d_{1} & a_{12} & a_{13} \\ a_{21} & -d_{2} & a_{23} \\ a_{31} & a_{32} & -d_{3} \\ \end{matrix}\right] \otimes \left[\begin{matrix} \red{\alpha} & \blue{\beta} \end{matrix}\right] \otimes I_2 \cdot X_i \\ &= \red{-L \otimes \left[\begin{matrix} \red{\alpha} & \blue{\beta} \end{matrix}\right] \otimes I_2} \cdot X_i \end{aligned} ⎣⎢⎢⎢⎢⎢⎢⎡u1xu1yu2xu2yu3xu3y⎦⎥⎥⎥⎥⎥⎥⎤=⎣⎢⎢⎢⎢⎢⎢⎡−αd10αa210αa3100−αd10αa210αa31−βd10βa210βa3100−βd10βa210βa31αa120−αd20αa3200αa120−αd20αa32βa120−βd20βa3200βa120−βd20βa32αa130αa230−αd300αa130αa230−αd3βa130βa230−βd300βa130βa230−βd3⎦⎥⎥⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡p1xp1yv1xv1yp2xp2yv2xv2yp3xp3yv3xv3y⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤=⎣⎡−d1a21a31a12−d2a32a13a23−d3⎦⎤⊗[αβ]⊗I2⋅Xi=−L⊗[αβ]⊗I2⋅Xi
其中
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3.2 静态一致性控制协议
控制协议为
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u_i = \red{\alpha} \sum_{j \in N_i} a_{ij} (p_j - p_i) - \blue{\beta} v_i \tag{}
ui=αj∈Ni∑aij(pj−pi)−βvi()
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α：表示控制参数
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β：表示控制参数
通过分析控制协议：
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\begin{aligned} u_1 &= \red{\alpha} a_{11} (p_1 - p_1) + \red{\alpha} a_{12} (p_2 - p_1) + \red{\alpha} a_{13} (p_3 - p_1) + \cdots + \blue{\beta} v_1 \\ u_2 &= \red{\alpha} a_{21} (p_1 - p_2) + \red{\alpha} a_{22} (p_2 - p_2) + \red{\alpha} a_{23} (p_3 - p_2) + \cdots + \blue{\beta} v_2 \\ u_3 &= \red{\alpha} a_{31} (p_1 - p_3) + \red{\alpha} a_{32} (p_2 - p_3) + \red{\alpha} a_{33} (p_3 - p_3) + \cdots + \blue{\beta} v_3 \\ \end{aligned} \tag{}
u1u2u3=αa11(p1−p1)+αa12(p2−p1)+αa13(p3−p1)+⋯+βv1=αa21(p1−p2)+αa22(p2−p2)+αa23(p3−p2)+⋯+βv2=αa31(p1−p3)+αa32(p2−p3)+αa33(p3−p3)+⋯+βv3()
可以得到以下控制协议的矩阵形式：
[ u 1 x u 1 y u 2 x u 2 y u 3 x u 3 y ] = [ − α d 1 0 − β 0 α a 12 0 0 0 α a 13 0 0 0 0 − α d 1 0 − β 0 α a 12 0 0 0 α a 13 0 0 α a 21 0 0 0 − α d 2 0 − β 0 α a 23 0 0 0 0 α a 21 0 0 0 − α d 2 0 − β 0 α a 23 0 0 α a 31 0 0 0 α a 32 0 0 0 − α d 3 0 − β 0 0 α a 31 0 0 0 α a 32 0 0 0 − α d 3 0 − β ] [ p 1 x p 1 y v 1 x v 1 y p 2 x p 2 y v 2 x v 2 y p 3 x p 3 y v 3 x v 3 y ] = ( [ − d 1 a 12 a 13 a 21 − d 2 a 23 a 31 a 32 − d 3 ] ⊗ [ α 0 ] ⊗ I 2 + [ − 1 0 0 0 − 1 0 0 0 − 1 ] ⊗ [ 0 β ] ⊗ I 2 ) ⋅ X i = ( − L ⊗ [ α 0 ] ⊗ I 2 + − I N ⊗ [ 0 β ] ⊗ I 2 ) ⋅ X i \begin{aligned} \left[\begin{matrix} u_1^x \\ u_1^y \\ u_2^x \\ u_2^y \\ u_3^x \\ u_3^y \\ \end{matrix}\right]&= \left[\begin{matrix} -\red{\alpha}d_1 & 0 & -\blue{\beta} & 0 & \red{\alpha}a_{12} & 0 & 0 & 0 & \red{\alpha}a_{13} & 0 & 0 & 0 \\ 0 & -\red{\alpha}d_1 & 0 & -\blue{\beta} & 0 & \red{\alpha}a_{12} & 0 & 0 & 0 & \red{\alpha}a_{13} & 0 & 0 \\ \red{\alpha}a_{21} & 0 & 0 & 0 & -\red{\alpha}d_{2} & 0 & -\blue{\beta} & 0 & \red{\alpha}a_{23} & 0 & 0 & 0 \\ 0 & \red{\alpha}a_{21} & 0 & 0 & 0 & -\red{\alpha}d_{2} & 0 & -\blue{\beta} & 0 & \red{\alpha}a_{23} & 0 & 0 \\ \red{\alpha}a_{31} & 0 & 0 & 0 & \red{\alpha}a_{32} & 0 & 0 & 0 & -\red{\alpha}d_{3} & 0 & -\blue{\beta} & 0 \\ 0 & \red{\alpha}a_{31} & 0 & 0 & 0 & \red{\alpha}a_{32} & 0 & 0 & 0 & -\red{\alpha}d_{3} & 0 & -\blue{\beta} \\ \end{matrix}\right] \left[\begin{matrix} p_1^x \\ p_1^y \\ v_1^x \\ v_1^y \\ p_2^x \\ p_2^y \\ v_2^x \\ v_2^y \\ p_3^x \\ p_3^y \\ v_3^x \\ v_3^y \\ \end{matrix}\right] \\ &=( \left[\begin{matrix} -d_{1} & a_{12} & a_{13} \\ a_{21} & -d_{2} & a_{23} \\ a_{31} & a_{32} & -d_{3} \\ \end{matrix}\right] \otimes \left[\begin{matrix} \red{\alpha} & 0 \end{matrix}\right] \otimes I_2 + \left[\begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ \end{matrix}\right] \otimes \left[\begin{matrix} 0 & \blue{\beta} \end{matrix}\right] \otimes I_2) \cdot X_i \\ &= (\red{-L \otimes \left[\begin{matrix} \red{\alpha} & 0 \end{matrix}\right] \otimes I_2} + \red{-I_N \otimes \left[\begin{matrix} 0 & \blue{\beta} \end{matrix}\right] \otimes I_2}) \cdot X_i \end{aligned} ⎣⎢⎢⎢⎢⎢⎢⎡u1xu1yu2xu2yu3xu3y⎦⎥⎥⎥⎥⎥⎥⎤=⎣⎢⎢⎢⎢⎢⎢⎡−αd10αa210αa3100−αd10αa210αa31−β000000−β0000αa120−αd20αa3200αa120−αd20αa3200−β000000−β00αa130αa230−αd300αa130αa230−αd30000−β000000−β⎦⎥⎥⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡p1xp1yv1xv1yp2xp2yv2xv2yp3xp3yv3xv3y⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤=(⎣⎡−d1a21a31a12−d2a32a13a23−d3⎦⎤⊗[α0]⊗I2+⎣⎡−1000−1000−1⎦⎤⊗[0β]⊗I2)⋅Xi=(−L⊗[α0]⊗I2+−IN⊗[0β]⊗I2)⋅Xi
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d_2 = a_{21} + a_{23} + \cdots
d2=a21+a23+⋯,
d
3
=
a
31
+
a
32
+
⋯
d_3 = a_{31} + a_{32} + \cdots
d3=a31+a32+⋯,
⋯
\cdots
⋯,
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N
_N
N 表示智能体的个数。
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