1. 单摆模型

为使得单摆在  处保持平衡，力矩必须有一个稳态分量满足

【资料图】

选择状态变量为, 控制变量为,则状态方程变为

2. 线性化控制

将系统在原点线性化得：

取，容易验证当

-\frac{acos{\delta}}{c},k_2>\frac{b}{c}" class="mathcode" src="https://private.codecogs.com/gif.latex?k_1%3E-%5Cfrac%7Bacos%7B%5Cdelta%7D%7D%7Bc%7D%2Ck_2%3E%5Cfrac%7Bb%7D%7Bc%7D">

时，A-BK是赫尔维茨矩阵，力矩为

3. matlab仿真代码

%=========== 单摆控制-状态反馈 ===========%clear all;clc;close all;%% 参数设置g = 9.8; % 重力加速度l = 1;   % 摆长k = 0.5; % 摩擦系数m = 1;   % 摆球质量a = g/l; b = k/m; c = 1/(m*l^2);theta(1) = 0;delta = -pi/5;dtheta(1) = 0.2;interval = 0.05;t = 0:interval:40;k1 = -a*cos(delta)+5;k2 = -b/c + 5;A = [0 1; -a*cos(delta) -b]; B = [0;c];K = [k1 k2];eig(A - B*K)%% 状态变化for i = 2:1:length(t)   T = a*sin(delta)/c - k1*(theta(i-1)-delta) - k2*dtheta(i-1);%   T =  0;  ddtheta = -a*sin(theta(i-1)) - b*dtheta(i-1) +c*T;  dtheta(i) = dtheta(i-1) + ddtheta*interval;  theta(i) = theta(i-1) + dtheta(i)*interval;endfigure plot(t,theta,"r")figure%绘制横梁colordef blackplot([-0.2;0.2],[0;0],"-y","LineWidth",20);x0=l*sin(theta(1));% 初始 x 坐标y0=-l*cos(theta(1));% 初始 y 坐标axis([-0.75,0.75,-1.25,1.25]);axis off%创建摆锤%擦除模式为 xor% head=line(x0,y0,"color","r","linestyle",".",...% "erasemode","xor","markersize",40);hold on%创建摆杆body=line([0;x0],[-0.05;y0],"color","g","linestyle","-","erasemode","xor","LineWidth",2);head = [];for i = 2:1:length(t)    x=l*sin(theta(i));    y=-l*cos(theta(i));%     set(head,"xdata",x,"ydata",y);% 改变擦除对象的坐标数据    set(body,"xdata",[0;x],"ydata",[-0.05;y]);    delete(head);    head = plot(x,y,"m.","MarkerSize",40);        drawnow;% 刷新屏幕    pause(0.1)        F = getframe(gcf);    I = frame2im(F);    [I,map] = rgb2ind(I,256);    if (i == 2)       imwrite(I,map,"single.gif","gif","Loopcount",inf,"Delaytime",0.2);    else       imwrite(I,map,"single.gif","gif","WriteMode","append","DelayTime",0.2);     end    end

4. 控制效果

5. 积分控制

积分控制中，不用寻找计算为保持平衡位置所需要的稳态力矩。此时的反馈控制率为：

加入积分控制后，即不需要再寻找平衡力矩就可以实现稳态控制

6. 仿真结果

%=========== 单摆控制-线性化状态反馈 ===========%clear all;clc;close all;%% 参数设置g = 9.8; % 重力加速度l = 1;   % 摆长k = 0.5; % 摩擦系数m = 1;   % 摆球质量a = g/l; b = k/m; c = 1/(m*l^2);theta(1) = -pi/2;delta = pi/4;dtheta(1) = 0.2;interval = 0.05;t = 0:interval:40;k1 = -a*cos(delta)+5;k2 = -b/c + 5;A = [0 1; -a*cos(delta) -b]; B = [0;c];K = [k1 k2];k3 = 3;eig(A - B*K)alpha(1) = 0;%% 状态变化for i = 2:1:length(t)%    T = a*sin(delta)/c - k1*(theta(i-1)-delta) - k2*dtheta(i-1);%   T =  0;  dalpha = theta(i-1) - delta;  alpha(i) =  alpha(i-1) + dalpha*interval;  T = - k1*(theta(i-1)-delta) - k2*dtheta(i-1) -k3*alpha(i);  ddtheta = -a*sin(theta(i-1)) - b*dtheta(i-1) +c*T;  dtheta(i) = dtheta(i-1) + ddtheta*interval;  theta(i) = theta(i-1) + dtheta(i)*interval;endfigure plot(t,theta,"y","LineWidth",2)figure%绘制横梁colordef blackplot([-0.2;0.2],[0;0],"-y","LineWidth",20);x0=l*sin(theta(1));% 初始 x 坐标y0=-l*cos(theta(1));% 初始 y 坐标axis([-1,1,-1.25,1.25]);axis off%创建摆锤%擦除模式为 xor% head=line(x0,y0,"color","r","linestyle",".",...% "erasemode","xor","markersize",40);hold on%创建摆杆body=line([0;x0],[-0.05;y0],"color","g","linestyle","-","erasemode","xor","LineWidth",2);head = [];for i = 2:1:length(t)    x=l*sin(theta(i));    y=-l*cos(theta(i));%     set(head,"xdata",x,"ydata",y);% 改变擦除对象的坐标数据    set(body,"xdata",[0;x],"ydata",[-0.05;y]);    delete(head);    head = plot(x,y,"m.","MarkerSize",40);        drawnow;% 刷新屏幕    pause(0.1)    %     F = getframe(gcf);%     I = frame2im(F);%     [I,map] = rgb2ind(I,256);%     if (i == 2)%        imwrite(I,map,"single.gif","gif","Loopcount",inf,"Delaytime",0.2);%     else%        imwrite(I,map,"single.gif","gif","WriteMode","append","DelayTime",0.2); %     end    end