线性二次最优控制-圆轨迹跟踪matlab

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clear all;
close all;
clc;
 
%% 参数
ts=0.1;
t=30;
d=1.45;
 
%% 初始状态
x(1)=0;
y(1)=0;
xita(1)=0;
v=3;
w=0.2;
u=[v;w];
Z(:,1)=[-3;-3;0.2];
for k=1:1:t/ts
    times(k+1)=k*ts;
    X(:,k)=[cos(xita(k)) -d*sin(xita(k));
                sin(xita(k)) d*cos(xita(k));
                0 1]*u;
    %% 状态更新
    x(k+1)=x(k)+X(1,k)*ts;
    y(k+1)=y(k)+X(2,k)*ts;
    xita(k+1)=xita(k)+X(3,k)*ts;
    
    
    A=[0 0 -v*sin(xita(k))-d*w*cos(xita(k));
        0 0 v*cos(xita(k))-d*w*sin(xita(k));
        0 0 0];
    B=[cos(xita(k)) -d*sin(xita(k));
        sin(xita(k)) d*cos(xita(k));
        0 1];
    Q=[2 0 0;
        0 2 0;
        0 0 2];
    R=[1 0; 0 1];
    K=lqr(A,B,Q,R);
    Z_=(A-B*K)*Z(:,k);%每秒变化率
    Z(:,k+1)=Z(:,k)+Z_*ts;%每步变化率
    xc(k)=Z(1,k)+x(k); %跟踪轨迹
    yc(k)=Z(2,k)+y(k);
end
 
 
figure(1)
for k=1:1:length(xc)
plot(x,y,'r','Linewidth',2);
hold on;
plot(xc(k),yc(k),'sb','MarkerSize',10);
pause(0.1);
end
figure(2)
plot(times,Z(1,:),'r');
figure(3)
plot(times,Z(2,:),'r');
figure(4);
plot(times,Z(3,:),'r');