基于运动学模型的无人机模型预测控制(MPC)-3

基于差分模型的无人机模型预测控制(MPC)-无约束情况

1. 模型建立

无人机运动学模型：
$\left\{ \begin{aligned} \dot x & = v_x \qquad \dot{v_x}=u_x\\ y & = v_y \qquad \dot{v_y}=u_y \\ \end{aligned} \right.${x˙y​=vx​vx​˙​=ux​=vy​vy​˙​=uy​​
其中 $n_x：状态变量量个数，n_u:控制变量个数，n_m:输出变量个数$nx​：状态变量量个数，nu​:控制变量个数，nm​:输出变量个数，我们得到如下状态空间：
$\begin{bmatrix} \dot{x}\\ \dot v_x \\ \dot y\\ \dot v_y \end{bmatrix}= \begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ \end{bmatrix} \begin{bmatrix} x\\ v_x \\ y\\ v_y \end{bmatrix}+ \begin{bmatrix} 0 & 0\\ 1 & 0 \\ 0 & 0\\ 0 & 1\\ \end{bmatrix} \begin{bmatrix} u_x \\ u_y\\ \end{bmatrix}$⎣⎢⎢⎡​x˙v˙x​y˙​v˙y​​⎦⎥⎥⎤​=⎣⎢⎢⎡​0000​1000​0000​0010​⎦⎥⎥⎤​⎣⎢⎢⎡​xvx​yvy​​⎦⎥⎥⎤​+⎣⎢⎢⎡​0100​0001​⎦⎥⎥⎤​[ux​uy​​]
$\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 1&0&0&0\\ 0&0&1&0\\ \end{bmatrix} \begin{bmatrix} x\\ v_x\\ y\\ v_y \end{bmatrix}$[xy​]=[10​00​01​00​]⎣⎢⎢⎡​xvx​yvy​​⎦⎥⎥⎤​
其中
$A = \begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ \end{bmatrix} \quad B = \begin{bmatrix} 0 & 0\\ 1 & 0 \\ 0 & 0\\ 0 & 1\\ \end{bmatrix} \quad C=\begin{bmatrix} 1&0&0&0\\ 0&0&1&0\\ \end{bmatrix}\quad x(k)=\begin{bmatrix} x\\ v_x \\ y\\ v_y \end{bmatrix} \quad u(k)=\begin{bmatrix} u_x \\ u_y\\ \end{bmatrix} \quad$A=⎣⎢⎢⎡​0000​1000​0000​0010​⎦⎥⎥⎤​B=⎣⎢⎢⎡​0100​0001​⎦⎥⎥⎤​C=[10​00​01​00​]x(k)=⎣⎢⎢⎡​xvx​yvy​​⎦⎥⎥⎤​u(k)=[ux​uy​​]

令：
$\begin{aligned} &x_m(k)= \begin{bmatrix} x(k) \quad v_x(k)\quad y(k)\quad v_y(k) \end{bmatrix}^{T}\\ &u_m(k)=[u_x(k)\quad u_y(k)]^{T} \end{aligned}$​xm​(k)=[x(k)vx​(k)y(k)vy​(k)​]Tum​(k)=[ux​(k)uy​(k)]T​
将上述模型离散化，我们得到：
$\begin{aligned} &A_k=A*\Delta t+I\\ &B_k=B*\Delta t \end{aligned}$​Ak​=A∗Δt+IBk​=B∗Δt​

即我们得到系统方程：
$\begin{aligned} &x_m(k+1)=A_kx_m(k)+B_ku_m(k)\\ &y_m(k+1) = Cx_m(k+1)=CA_kx_m(k)+CB_ku_m(k)\\ &x_m(k+1)\in{n_{x}\times1}\quad A_k\in{n_{x}\times n_{x}}\quad B_k\in{n_{x}\times n_{u}} \end{aligned}$​xm​(k+1)=Ak​xm​(k)+Bk​um​(k)ym​(k+1)=Cxm​(k+1)=CAk​xm​(k)+CBk​um​(k)xm​(k+1)∈nx​×1Ak​∈nx​×nx​Bk​∈nx​×nu​​

构建差分系统方程：
$\Delta x_m(k+1)=x_m(k+1)-x_m(k)=A_k\Delta x_m(k)+B_k\Delta u_m(k)$Δxm​(k+1)=xm​(k+1)−xm​(k)=Ak​Δxm​(k)+Bk​Δum​(k)
即得到：
$\begin{aligned} \begin{bmatrix} \Delta x_m(k+1)\\ y_m(k+1) \end{bmatrix}= \begin{bmatrix} &(A_k)_{{n_x\times n_x}}&0_{n_x\times n_m}\\ &(CA_k)_{{n_m\times n_x}}&I_{n_m\times n_m} \end{bmatrix} \begin{bmatrix} &\Delta x_m(k)_{n_x\times 1}\\ &y_m(k)_{n_m\times 1} \end{bmatrix}+ \begin{bmatrix} (B_k)_{{n_x\times n_u}}\\ (CB_k)_{{n_m\times n_u}} \end{bmatrix} \Delta u_m(k)_{n_u\times 1} \end{aligned}$[Δxm​(k+1)ym​(k+1)​]=[​(Ak​)nx​×nx​​(CAk​)nm​×nx​​​0nx​×nm​​Inm​×nm​​​][​Δxm​(k)nx​×1​ym​(k)nm​×1​​]+[(Bk​)nx​×nu​​(CBk​)nm​×nu​​​]Δum​(k)nu​×1​​

$y_m(k+1)-y_m(k)=C(x_m(k+1)-x_m(k))=C\Delta x_m(k+1)=CA_k\Delta x_m(k)+CB_k\Delta u_m(k)$ym​(k+1)−ym​(k)=C(xm​(k+1)−xm​(k))=CΔxm​(k+1)=CAk​Δxm​(k)+CBk​Δum​(k)

我们得到如下差分系统方程：
$\begin{aligned} \Delta x(k+1)&=A_u\Delta x(k)+B_u\Delta u(k)\\ \Delta y(k)&=C_u\Delta x(k) \end{aligned}$Δx(k+1)Δy(k)​=Au​Δx(k)+Bu​Δu(k)=Cu​Δx(k)​

其中：
$\begin{aligned} &\Delta x(k+1)=[\Delta x_m(k+1);y_m(k+1)]_{(n_x+n_m)\times 1}\\ &\Delta u(k)_{n_u\times 1}=\Delta u_m(k)\\ &\Delta y(k)_{n_m\times 1}\\ &A_u= \begin{bmatrix} &(A_k)_{{n_x\times n_x}}&0_{n_x\times n_m}\\ &(CA_k)_{{n_m\times n_x}}&I_{n_m\times n_m} \end{bmatrix}\quad B_u= \begin{bmatrix} (B_k)_{{n_x\times n_u}}\\ (CB_k)_{{n_m\times n_u}} \end{bmatrix} \quad C_u= \begin{bmatrix} 0_{n_m\times n_x}\quad I_{n_m\times n_m} \end{bmatrix} \end{aligned}$​Δx(k+1)=[Δxm​(k+1);ym​(k+1)](nx​+nm​)×1​Δu(k)nu​×1​=Δum​(k)Δy(k)nm​×1​Au​=[​(Ak​)nx​×nx​​(CAk​)nm​×nx​​​0nx​×nm​​Inm​×nm​​​]Bu​=[(Bk​)nx​×nu​​(CBk​)nm​×nu​​​]Cu​=[0nm​×nx​​Inm​×nm​​​]​

递推公式推导：
$\left\{ \begin{aligned} \Delta x(k_i+1|k_i)&=A_u\Delta x(k_i)+B_u\Delta u(k_i)\\ \Delta x(k_i+2|k_i)&=A_u^{2}\Delta x(k_i)+A_uB_u \Delta u(k_i)+B_u\Delta u(k_i+1)\\ \Delta x(k_i+3|k_i)&=A_u^{3}\Delta x(k_i)+A_u^{2}B_u\Delta u(k_i)+A_uB_u\Delta u(k_i+1)+B_u\Delta u(k_i+2)\\ \quad\vdots\\ \Delta x(k_i+N_p|k_i)&=A_u^{N_p}\Delta x(k_i)+A_u^{N_p-1}B_u\Delta u(k_i)+A_u^{N_p-2}B_u\Delta u(k_i+1)+\cdots +A_u^{N_p-N_c}B_u\Delta u(k_i+N_c-1)\\ \end{aligned} \right.$⎩⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎧​Δx(ki​+1∣ki​)Δx(ki​+2∣ki​)Δx(ki​+3∣ki​)⋮Δx(ki​+Np​∣ki​)​=Au​Δx(ki​)+Bu​Δu(ki​)=Au2​Δx(ki​)+Au​Bu​Δu(ki​)+Bu​Δu(ki​+1)=Au3​Δx(ki​)+Au2​Bu​Δu(ki​)+Au​Bu​Δu(ki​+1)+Bu​Δu(ki​+2)=AuNp​​Δx(ki​)+AuNp​−1​Bu​Δu(ki​)+AuNp​−2​Bu​Δu(ki​+1)+⋯+AuNp​−Nc​​Bu​Δu(ki​+Nc​−1)​

$\left\{ \begin{aligned} \Delta y(k_i+1|k_i)&=C_uA_u\Delta x(k_i)+C_uB_u\Delta u(k_i)\\ \Delta y(k_i+2|k_i)&=C_uA_u^{2}\Delta x(k_i)+C_uA_uB_u\Delta u(k_i)+C_uB_u\Delta u(k_i+1)\\ \Delta y(k_i+3|k_i)&=C_uA_u^{3}\Delta x(k_i)+C_uA_u^{2}B_u\Delta u(k_i)+C_uA_uB_u\Delta u(k_i+1)+C_uB_u\Delta u(k_i+2)\\ \quad\vdots\\ \Delta y(k_i+N_p|k_i)&=C_uA_u^{N_p}\Delta x(k_i)+C_uA_u^{N_p-1}B_u\Delta u(k_i)+C_uA_u^{N_p-2}B_u\Delta u(k_i+1)+\cdots \\ &+C_uA_u^{N_p-N_c}B_u\Delta u(k_i+N_c-1)\\ \end{aligned} \right.$⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧​Δy(ki​+1∣ki​)Δy(ki​+2∣ki​)Δy(ki​+3∣ki​)⋮Δy(ki​+Np​∣ki​)​=Cu​Au​Δx(ki​)+Cu​Bu​Δu(ki​)=Cu​Au2​Δx(ki​)+Cu​Au​Bu​Δu(ki​)+Cu​Bu​Δu(ki​+1)=Cu​Au3​Δx(ki​)+Cu​Au2​Bu​Δu(ki​)+Cu​Au​Bu​Δu(ki​+1)+Cu​Bu​Δu(ki​+2)=Cu​AuNp​​Δx(ki​)+Cu​AuNp​−1​Bu​Δu(ki​)+Cu​AuNp​−2​Bu​Δu(ki​+1)+⋯+Cu​AuNp​−Nc​​Bu​Δu(ki​+Nc​−1)​

即得到如下递推方程：
$Y = F\Delta x(k_i)+\Phi U$Y=FΔx(ki​)+ΦU

性能指标：
$\begin{aligned} J&=(R_s-Y)^{T}(R_s-Y)+U^{T}RU\\ &=(R_s-Fx(k_i)-\Phi U)^{T}(R_s-Fx(k_i)-\Phi U)+U^{T}RU\\ &=(R_s-Fx(k_i))^{T}(R_s-Fx(k_i))-2U^{T}\Phi (R_s-Fx(k_i))+U^{T}(\Phi^{T}\Phi+R)U \end{aligned}$J​=(Rs​−Y)T(Rs​−Y)+UTRU=(Rs​−Fx(ki​)−ΦU)T(Rs​−Fx(ki​)−ΦU)+UTRU=(Rs​−Fx(ki​))T(Rs​−Fx(ki​))−2UTΦ(Rs​−Fx(ki​))+UT(ΦTΦ+R)U​

求$\frac{\partial J}{\partial U}$∂U∂J​得：
$\begin{aligned} \frac{\partial J}{\partial U}&=-2\Phi^{T}(R_s-Fx(k_i))+2(\Phi^{T}\Phi+R)U=0\\ U&=(\Phi^{T}\Phi+R)^{-1}\Phi^{T}(R_s-Fx(k_i)) \end{aligned}$∂U∂J​U​=−2ΦT(Rs​−Fx(ki​))+2(ΦTΦ+R)U=0=(ΦTΦ+R)−1ΦT(Rs​−Fx(ki​))​

即差分方程迭代：
$\begin{aligned} \Delta x(k_i+1)&=A_k\Delta x(:,k_i)+B_kU(1:n_m)\\ U &= U(1:n_m)+oldU\\ X(:,i+1)&=A_kX(:,k_i)+B_kU \end{aligned}$Δx(ki​+1)UX(:,i+1)​=Ak​Δx(:,ki​)+Bk​U(1:nm​)=U(1:nm​)+oldU=Ak​X(:,ki​)+Bk​U​

2. matlab 仿真代码

%================无人机模型预测控制-基于差分模型的模型预测================%
clear all;clc;close all;
%% 无人机参数设定--采用运动学模型进行轨迹跟踪
x0 = 10; y0 = 5; x1 = 11; y1 = 6;
vx0 = 0; vy0 = 0; vx1 = 1; vy1 = 1;
x(1) = x0; y(1) = y0;vx(1) = vx0;vy(1) = vy0;
%% 领航者参数设定
inter = 0.05;  % 采样周期
time = 60;  % 总时长
R = 2;
omega = 2;
t = 0:inter:time;
%% 八字形
for i = 1:1:length(t)
   if (mod(floor(omega*t(i)/(2*pi)),2) == 0)
    Xr(i) = R*cos(omega*t(i))-R;
    Yr(i) = R*sin(omega*t(i));
    Vxr(i) = -R*sin(omega*t(i))*omega;
    Vyr(i) = R*cos(omega*t(i))*omega;
    Uxr(i) = -R*cos(omega*t(i))*omega^2;
    Uyr(i) = -R*sin(omega*t(i))*omega^2;
   else
    Xr(i) = -R*cos(omega*t(i))+R;
    Yr(i) = R*sin(omega*t(i));   
    Vxr(i) = R*sin(omega*t(i))*omega;
    Vyr(i) = R*cos(omega*t(i))*omega;
    Uxr(i) = R*cos(omega*t(i))*omega^2;
    Uyr(i) = -R*sin(omega*t(i))*omega^2;
   end

end
%% 直线
% Xr = (2*t)';
% Yr = 3*ones(length(t),1);
% Vxr = 2*ones(length(t),1);
% Vyr = 2*zeros(length(t),1);
% Uxr = zeros(length(t),1);
% Uyr = zeros(length(t),1);
%% 圆形
% Xr = -R*cos(t);
% Yr = R*sin(t);
% Vxr = R*sin(t);
% Vyr = R*cos(t);
% Uxr = R*cos(t);
% Uyr = -R*sin(t);
%%
% EX(:,1) = [x0 - Xr(1);vx0 - Vxr(1);y0 - Yr(1);vy0 - Vyr(1)];

X(:,1) = [x0;vx0;y0;vy0];
deltaX(:,1) = [0;0;0;0;x0;y0];
%% 领航者轨迹
% figure
% grid minor
% l1 = [];
% axis([-7 7 -7 7]);
% axis equal
% for i = 2:1:length(t)
%   hold on
%   plot([Xr(i) Xr(i-1)],[Yr(i) Yr(i-1)],'b');
%   hold on
%   delete(l1);
%   l1 =  plot(Xr(i),Yr(i),'r.','MarkerSize',20);
%   pause(0.1);
%   
% end
%% 模型预测控制参数设定
Np = 20;     % 预测步长
Nc = 10;      % 控制步长
A = [0 1 0 0;0 0 0 0;0 0 0 1;0 0 0 0];  B = [0 0;1 0;0 0;0 1]; 
C = [1 0 0 0;0 0 1 0];
nx = size(A);
nx = nx(1);
nu = size(B);
nu = nu(2);
nm = 2;
 R = 0.002*eye(Nc*nu);
Ak = A*inter + eye(nx);
Bk = B*inter;
Au = [Ak zeros(nx,nm);C*Ak eye(nm)];
Bu = [Bk;C*Bk];
Cu = [zeros(nm,nx) eye(nm)];

F = cell(Np,1);
PHI = cell(Np,Nc);
for i = 1:1:Np         % 计算预测方程矩阵
  F{i,1} = Cu*Au^i;
end
F = cell2mat(F);

for i = 1:1:Np
   for j = 1:1:Nc
       if (j<=i)
           PHI{i,j} = Cu*Au^(i-j)*Bu;
       else
           PHI{i,j} = zeros(nm,nu);
       end
   end
end
PHI = cell2mat(PHI);
k1 =2;k2 =2;
XX = [];
%% 迭代计算
k = 1;
oldU = [0;0];
for i = 1:1:length(t)-1
   for j = i:1:(Np+i-1)
     if j >= length(Xr)
         j = length(Xr);
     end
     XX = [XX;[Xr(j);Yr(j)]]; 
   end
  U = inv(PHI'*PHI + R)*PHI'*(XX- F*deltaX(:,i));
  XX = [];
  u = U(1:2,1) + oldU;
  oldU = u;
  X(:,i+1) = Ak*X(:,i) + Bk*u;
  deltaX(:,i+1) = Au*deltaX(:,i) + Bu*U(1:2,1);
  
%   err =[X(:,i+1) - [Xr(i+1);Vxr(i+1);Yr(i+1);Vyr(i+1)]] ;
end
x = (X(1,:))';
vx = (X(2,:))';
y = (X(3,:))';
vy = (X(4,:))';
% VV = vecnorm([Vxr;Vyr]);
% VX = vecnorm([vx;vy]);
% plot(t,VV,'r')
% hold on
% plot(t,VX(1:length(t)),'b')
figure
thetr = atan2(Yr,Xr);
thet = atan2(y,x);
plot(t,thetr(1:length(t)),'r');
hold on
plot(t,thet(1:length(t)),'k');
legend('Leader','follower1')

l1 = [];
l2 = [];
pic_num = 1;
figure
 grid minor
%  axis([-5 5 -5 5])
axis equal
Tag1 = animatedline('Color','r');
for i = 1:1:length(Xr)-1
    
    hold on
    delete(l1);
   delete(l2);

    plot([x(i) x(i+1)],[y(i) y(i+1)],'b');
   hold on
   plot([Xr(i) Xr(i+1)],[Yr(i) Yr(i+1)],'r');
   hold on
   l1 = plot(x(i+1),y(i+1),'b.','MarkerSize',20);
   hold on
   l2 = plot(Xr(i+1),Yr(i+1),'r.','MarkerSize',20);
   pause(0.1);
%    addpoints(Tag1,t(i),x(i));
%    drawnow;
%     F=getframe(gcf);
%     I=frame2im(F);
%     [I,map]=rgb2ind(I,256);
%     if pic_num == 1
%         imwrite(I,map,'test.gif','gif', 'Loopcount',inf,'DelayTime',0.2);
%     else
%         imwrite(I,map,'test.gif','gif','WriteMode','append','DelayTime',0.2);
%     end
%     pic_num = pic_num + 1;
    F = getframe(gcf);
    I = frame2im(F);
    [I,map] = rgb2ind(I,256);
    if pic_num == 1
        imwrite(I,map,'test.gif','gif','Loopcount',inf,'DelayTime',0.2);
    else
        imwrite(I,map,'test.gif','gif','WriteMode','append','DelayTime',0.2);
    end
    pic_num = pic_num + 1;
     
end