Dong X, Hu G. Time-varying formation tracking for linear multiagent systems with multiple leaders[J]. IEEE Transactions on Automatic Control, 2017, 62(7): 3658-3664.
Dong X, Yu B, Shi Z, et al. Time-varying formation control for unmanned aerial vehicles: Theories and applications[J]. IEEE Transactions on Control Systems Technology, 2014, 23(1): 340-348.
Dong X, Zhou Y, Ren Z, et al. Time-varying formation control for unmanned aerial vehicles with switching interaction topologies[J]. Control Engineering Practice, 2016, 46: 26-36.
Time-varying formation tracking for linear multiagent systems with multiple leaders
- Theorem 1
- Theorem 2
- Codes: TimevaryingFormation
- Codes: TimevaryingFormation2
控制协议为：
u i ( t ) = K ∑ j = 1 M w i j ( ( x i ( t ) − h i ( t ) ) − ( x j ( t ) − h j ( t ) ) ) + K ∑ k = M + 1 N w i k ( ( x i ( t ) − h i ( t ) ) − x k ( t ) ) + v i ( t ) = K ∑ j = 1 M w i j ( ( x i − h i ) − ( x j − h j ) ) + K ∑ k = M + 1 N w i k ( ( x i − h i ) − x k ) + v i = K ∑ j = 1 M w i j ( δ i − δ j ) + K ∑ k = M + 1 N w i k ( δ i − x k ) + v i \begin{aligned} u_i(t) =& K \sum_{j=1}^{M} w_{ij} ((x_i(t) - h_i(t)) - (x_j(t)-h_j(t))) \\ &+ K \sum_{k=M+1}^{N} w_{ik} ((x_i(t) - h_i(t) )-x_k(t)) + v_i(t) \\ =& K \sum_{j=1}^{M} w_{ij} ((x_i - h_i) - (x_j-h_j)) \\ &+ K \sum_{k=M+1}^{N} w_{ik} ((x_i - h_i )-x_k) + v_i \\ =& K \sum_{j=1}^{M} w_{ij} (\delta_i - \delta_j) \\ &+ K \sum_{k=M+1}^{N} w_{ik} (\delta_i - x_k) + v_i \\ \end{aligned} ui(t)===Kj=1∑Mwij((xi(t)−hi(t))−(xj(t)−hj(t)))+Kk=M+1∑Nwik((xi(t)−hi(t))−xk(t))+vi(t)Kj=1∑Mwij((xi−hi)−(xj−hj))+Kk=M+1∑Nwik((xi−hi)−xk)+viKj=1∑Mwij(δi−δj)+Kk=M+1∑Nwik(δi−xk)+vi
Theorem 1
Theorem 2
Codes: TimevaryingFormation
%% Time-varying formation tracking for linear multiagent systems with multiple leaders
% Author: Zhao-Jichao
% Date: 2021-07-08
clear
clc
%% System Description
global A B L1 L2 K
A = [0 1 1
1 2 1
-2 -6 -3];
B = [0 1
-1 0
0 0];
Bbar = [0 -1 0
1 0 0];
Btilde=[0 0 1];
delta = 0.55;
Q = eye(3);
R = eye(2);
P = care(A,B,Q,R);
L = [1 -1 0 0 0 0 0 0 0 0 0 0 0
0 3 0 0 0 0 0 0 0 0 -1 -1 -1
-1 0 1 0 0 0 0 0 0 0 0 0 0
0 0 -1 1 0 0 0 0 0 0 0 0 0
0 0 0 -1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 -1 0 0 0 0 0 0
0 0 0 0 0 0 1 -1 0 0 0 0 0
0 0 0 0 0 0 -1 5 0 -1 -1 -1 -1
0 0 0 0 0 0 0 -1 1 0 0 0 0
0 0 0 0 0 0 0 0 -1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0];
L1 = L(1:10, 1:10);
L2 = L(1:10, 11:13);
lambda1 = min(eig(L1));
K = -delta*1/(lambda1)*inv(R)*B'*P;
% Initial states of followers
x1 = [03 05 07]';
x2 = [04 06 08]';
x3 = [05 07 09]';
x4 = [06 08 10]';
x5 = [07 09 11]';
x6 = [08 10 12]';
x7 = [09 11 13]';
x8 = [10 12 14]';
x9 = [11 13 15]';
x10= [12 14 16]';
% Initial states of leaders
x11= [13 15 17]';
x12= [14 16 18]';
x13= [15 17 19]';
XF = [x1' x2' x3' x4' x5' x6' x7' x8' x9' x10']';
XE = [x11' x12' x13']';
in = [XF' XE']';
syms t real
h1 = [15*sin(t+(1-1)*pi/5) -15*cos(t+(1-1)*pi/5) 30*cos(t+(1-1)*pi/5)]';
h2 = [15*sin(t+(2-1)*pi/5) -15*cos(t+(2-1)*pi/5) 30*cos(t+(2-1)*pi/5)]';
h3 = [15*sin(t+(3-1)*pi/5) -15*cos(t+(3-1)*pi/5) 30*cos(t+(3-1)*pi/5)]';
h4 = [15*sin(t+(4-1)*pi/5) -15*cos(t+(4-1)*pi/5) 30*cos(t+(4-1)*pi/5)]';
h5 = [15*sin(t+(5-1)*pi/5) -15*cos(t+(5-1)*pi/5) 30*cos(t+(5-1)*pi/5)]';
h6 = [15*sin(t+(6-1)*pi/5) -15*cos(t+(6-1)*pi/5) 30*cos(t+(6-1)*pi/5)]';
h7 = [15*sin(t+(7-1)*pi/5) -15*cos(t+(7-1)*pi/5) 30*cos(t+(7-1)*pi/5)]';
h8 = [15*sin(t+(8-1)*pi/5) -15*cos(t+(8-1)*pi/5) 30*cos(t+(8-1)*pi/5)]';
h9 = [15*sin(t+(9-1)*pi/5) -15*cos(t+(9-1)*pi/5) 30*cos(t+(9-1)*pi/5)]';
h10= [15*sin(t+(10-1)*pi/5) -15*cos(t+(10-1)*pi/5) 30*cos(t+(10-1)*pi/5)]';
H = [h1' h2' h3' h4' h5' h6' h7' h8' h9' h10']';
%%
tBegin = 0;
tFinal = 100;
[t, out] = ode45(@odeFun, [tBegin, tFinal], in);
%% Draw Results
figure(1)
plot3(out(:,1), out(:,2), out(:,3), 'linewidth',1.5); hold on; grid on;
plot3(out(:,4), out(:,5), out(:,6), 'linewidth',1.5); hold on; grid on;
plot3(out(:,7), out(:,8), out(:,9), 'linewidth',1.5); hold on; grid on;
plot3(out(:,10),out(:,11),out(:,12), 'linewidth',1.5); hold on; grid on;
plot3(out(:,13),out(:,14),out(:,15), 'linewidth',1.5); hold on; grid on;
plot3(out(:,16),out(:,17),out(:,18), 'linewidth',1.5); hold on; grid on;
plot3(out(:,19),out(:,20),out(:,21), 'linewidth',1.5); hold on; grid on;
plot3(out(:,22),out(:,23),out(:,24), 'linewidth',1.5); hold on; grid on;
plot3(out(:,25),out(:,26),out(:,27), 'linewidth',1.5); hold on; grid on;
plot3(out(:,28),out(:,29),out(:,30), 'linewidth',1.5); hold on; grid on;
plot3(out(:,31),out(:,32),out(:,33), 'linewidth',1.5); hold on; grid on;
plot3(out(:,34),out(:,35),out(:,36), 'linewidth',1.5); hold on; grid on;
plot3(out(:,37),out(:,38),out(:,39), 'linewidth',1.5); hold on; grid on;
legend('f1','f2','f3','f4','f5','f6','f7','f8','f9','f10','l11','l12','l13')
figure(2) % Plot snapshots
t = 1000;
scatter3(out(t,1), out(t,2), out(t,3), '*', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,4), out(t,5), out(t,6), '*', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,7), out(t,8), out(t,9), '*', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,10),out(t,11),out(t,12),'o', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,13),out(t,14),out(t,15),'o', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,16),out(t,17),out(t,18),'d', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,19),out(t,20),out(t,21),'d', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,22),out(t,23),out(t,24),'d', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,25),out(t,26),out(t,27),'^', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,28),out(t,29),out(t,30),'^', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,31),out(t,32),out(t,33),'p', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,34),out(t,35),out(t,36),'p', 'linewidth',1.5); hold on; grid on;
scatter3(out(t,37),out(t,38),out(t,39),'p', 'linewidth',1.5); hold on; grid on;
legend('f1','f2','f3','f4','f5','f6','f7','f8','f9','f10','l11','l12','l13')
%%
function out = odeFun(t, in)
global A B L1 L2 K
XF = in(1:30,:);
XE = in(31:39,:);
h1 = [15*sin(t+(1-1)*pi/5) -15*cos(t+(1-1)*pi/5) 30*cos(t+(1-1)*pi/5)]';
h2 = [15*sin(t+(2-1)*pi/5) -15*cos(t+(2-1)*pi/5) 30*cos(t+(2-1)*pi/5)]';
h3 = [15*sin(t+(3-1)*pi/5) -15*cos(t+(3-1)*pi/5) 30*cos(t+(3-1)*pi/5)]';
h4 = [15*sin(t+(4-1)*pi/5) -15*cos(t+(4-1)*pi/5) 30*cos(t+(4-1)*pi/5)]';
h5 = [15*sin(t+(5-1)*pi/5) -15*cos(t+(5-1)*pi/5) 30*cos(t+(5-1)*pi/5)]';
h6 = [15*sin(t+(6-1)*pi/5) -15*cos(t+(6-1)*pi/5) 30*cos(t+(6-1)*pi/5)]';
h7 = [15*sin(t+(7-1)*pi/5) -15*cos(t+(7-1)*pi/5) 30*cos(t+(7-1)*pi/5)]';
h8 = [15*sin(t+(8-1)*pi/5) -15*cos(t+(8-1)*pi/5) 30*cos(t+(8-1)*pi/5)]';
h9 = [15*sin(t+(9-1)*pi/5) -15*cos(t+(9-1)*pi/5) 30*cos(t+(9-1)*pi/5)]';
h10= [15*sin(t+(10-1)*pi/5) -15*cos(t+(10-1)*pi/5) 30*cos(t+(10-1)*pi/5)]';
H = [h1' h2' h3' h4' h5' h6' h7' h8' h9' h10']';
dXE = kron(eye(3),A)*XE;
U = kron(L1,eye(2))*kron(eye(10),K)*(XF-H)...
+ kron(L2,eye(2))*kron(eye(3),K)*XE;
dXF = kron(eye(10),A)*XF + kron(eye(10),B)*U;
out = [dXF; dXE];
endCodes: TimevaryingFormation2
%% Time-varying formation tracking for linear multiagent systems with multiple leaders
% Author: Zhao-Jichao
% Date: 2021-07-08
clear
clc
%% System Description
global A B L1 L2 K
A = [0 1 1
1 2 1
-2 -6 -3];
B = [0 1
-1 0
0 0];
Bbar = [0 -1 0
1 0 0];
Btilde=[0 0 1];
delta = 0.55;
Q = eye(3);
R = eye(2);
P = care(A,B,Q,R);
L = [1 -1 0 0 0 0 0 0 0 0 0 0 0
0 3 0 0 0 0 0 0 0 0 -1 -1 -1
-1 0 1 0 0 0 0 0 0 0 0 0 0
0 0 -1 1 0 0 0 0 0 0 0 0 0
0 0 0 -1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 -1 0 0 0 0 0 0
0 0 0 0 0 0 1 -1 0 0 0 0 0
0 0 0 0 0 0 -1 5 0 -1 -1 -1 -1
0 0 0 0 0 0 0 -1 1 0 0 0 0
0 0 0 0 0 0 0 0 -1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0];
L1 = L(1:10, 1:10);
L2 = L(1:10, 11:13);
lambda1 = min(eig(L1));
K = -delta*1/(lambda1)*inv(R)*B'*P;
% Initial states of followers
x1 = [03 05 07]';
x2 = [04 06 08]';
x3 = [05 07 09]';
x4 = [06 08 10]';
x5 = [07 09 11]';
x6 = [08 10 12]';
x7 = [09 11 13]';
x8 = [10 12 14]';
x9 = [11 13 15]';
x10= [12 14 16]';
% Initial states of leaders
x11= [13 15 17]';
x12= [14 16 18]';
x13= [15 17 19]';
XF = [x1' x2' x3' x4' x5' x6' x7' x8' x9' x10']';
XE = [x11' x12' x13']';
in = [XF' XE']';
syms t real
h1 = [15*sin(t+(1-1)*pi/5) -15*cos(t+(1-1)*pi/5) 30*cos(t+(1-1)*pi/5)]';
h2 = [15*sin(t+(2-1)*pi/5) -15*cos(t+(2-1)*pi/5) 30*cos(t+(2-1)*pi/5)]';
h3 = [15*sin(t+(3-1)*pi/5) -15*cos(t+(3-1)*pi/5) 30*cos(t+(3-1)*pi/5)]';
h4 = [15*sin(t+(4-1)*pi/5) -15*cos(t+(4-1)*pi/5) 30*cos(t+(4-1)*pi/5)]';
h5 = [15*sin(t+(5-1)*pi/5) -15*cos(t+(5-1)*pi/5) 30*cos(t+(5-1)*pi/5)]';
h6 = [15*sin(t+(6-1)*pi/5) -15*cos(t+(6-1)*pi/5) 30*cos(t+(6-1)*pi/5)]';
h7 = [15*sin(t+(7-1)*pi/5) -15*cos(t+(7-1)*pi/5) 30*cos(t+(7-1)*pi/5)]';
h8 = [15*sin(t+(8-1)*pi/5) -15*cos(t+(8-1)*pi/5) 30*cos(t+(8-1)*pi/5)]';
h9 = [15*sin(t+(9-1)*pi/5) -15*cos(t+(9-1)*pi/5) 30*cos(t+(9-1)*pi/5)]';
h10= [15*sin(t+(10-1)*pi/5) -15*cos(t+(10-1)*pi/5) 30*cos(t+(10-1)*pi/5)]';
H = [h1' h2' h3' h4' h5' h6' h7' h8' h9' h10']';
%%
tBegin = 0;
tFinal = 100;
[t, out] = ode45(@odeFun, [tBegin, tFinal], in);
%% Draw Results
% figure(1)
% plot3(out(:,1), out(:,2), out(:,3), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,4), out(:,5), out(:,6), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,7), out(:,8), out(:,9), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,10),out(:,11),out(:,12), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,13),out(:,14),out(:,15), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,16),out(:,17),out(:,18), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,19),out(:,20),out(:,21), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,22),out(:,23),out(:,24), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,25),out(:,26),out(:,27), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,28),out(:,29),out(:,30), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,31),out(:,32),out(:,33), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,34),out(:,35),out(:,36), 'linewidth',1.5); hold on; grid on;
% plot3(out(:,37),out(:,38),out(:,39), 'linewidth',1.5); hold on; grid on;
% legend('f1','f2','f3','f4','f5','f6','f7','f8','f9','f10','l11','l12','l13')
% figure(2) % Plot snapshots
% t = 1000;
% scatter3(out(t,1), out(t,2), out(t,3), '*', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,4), out(t,5), out(t,6), '*', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,7), out(t,8), out(t,9), '*', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,10),out(t,11),out(t,12),'o', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,13),out(t,14),out(t,15),'o', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,16),out(t,17),out(t,18),'d', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,19),out(t,20),out(t,21),'d', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,22),out(t,23),out(t,24),'d', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,25),out(t,26),out(t,27),'^', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,28),out(t,29),out(t,30),'^', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,31),out(t,32),out(t,33),'p', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,34),out(t,35),out(t,36),'p', 'linewidth',1.5); hold on; grid on;
% scatter3(out(t,37),out(t,38),out(t,39),'p', 'linewidth',1.5); hold on; grid on;
% legend('f1','f2','f3','f4','f5','f6','f7','f8','f9','f10','l11','l12','l13')
figure(3)
for time = 1:1:length(t)-00
out1 = [out(:,1) out(:,4) out(:,7) out(:,10) out(:,13) out(:,16) out(:,19) out(:,22) out(:,25) out(:,28) out(:,31) out(:,34) out(:,37)];
out2 = [out(:,2) out(:,5) out(:,8) out(:,11) out(:,14) out(:,17) out(:,20) out(:,23) out(:,26) out(:,29) out(:,32) out(:,35) out(:,38)];
out3 = [out(:,3) out(:,6) out(:,9) out(:,12) out(:,15) out(:,18) out(:,21) out(:,24) out(:,27) out(:,30) out(:,33) out(:,36) out(:,39)];
scatter3(out3(time,:), out2(time,:), out1(time,:), 'linewidth',1.5);
axis([-300, 300, -100,100, -100,100]);
tt = kron(ones(1,size(out1,2)),t);
pause(0.1)
end
disp('end')
%%
function out = odeFun(t, in)
global A B L1 L2 K
XF = in(1:30,:);
XE = in(31:39,:);
h1 = [15*sin(t+(1-1)*pi/5) -15*cos(t+(1-1)*pi/5) 30*cos(t+(1-1)*pi/5)]';
h2 = [15*sin(t+(2-1)*pi/5) -15*cos(t+(2-1)*pi/5) 30*cos(t+(2-1)*pi/5)]';
h3 = [15*sin(t+(3-1)*pi/5) -15*cos(t+(3-1)*pi/5) 30*cos(t+(3-1)*pi/5)]';
h4 = [15*sin(t+(4-1)*pi/5) -15*cos(t+(4-1)*pi/5) 30*cos(t+(4-1)*pi/5)]';
h5 = [15*sin(t+(5-1)*pi/5) -15*cos(t+(5-1)*pi/5) 30*cos(t+(5-1)*pi/5)]';
h6 = [15*sin(t+(6-1)*pi/5) -15*cos(t+(6-1)*pi/5) 30*cos(t+(6-1)*pi/5)]';
h7 = [15*sin(t+(7-1)*pi/5) -15*cos(t+(7-1)*pi/5) 30*cos(t+(7-1)*pi/5)]';
h8 = [15*sin(t+(8-1)*pi/5) -15*cos(t+(8-1)*pi/5) 30*cos(t+(8-1)*pi/5)]';
h9 = [15*sin(t+(9-1)*pi/5) -15*cos(t+(9-1)*pi/5) 30*cos(t+(9-1)*pi/5)]';
h10= [15*sin(t+(10-1)*pi/5) -15*cos(t+(10-1)*pi/5) 30*cos(t+(10-1)*pi/5)]';
H = [h1' h2' h3' h4' h5' h6' h7' h8' h9' h10']';
dXE = kron(eye(3),A)*XE;
U = kron(L1,eye(2))*kron(eye(10),K)*(XF-H)...
+ kron(L2,eye(2))*kron(eye(3),K)*XE;
dXF = kron(eye(10),A)*XF + kron(eye(10),B)*U;
out = [dXF; dXE];
end
