【控制】高增益扩张状态观测器的设计（9.1）

1 理论框架

2 一阶系统描述

3 二阶系统描述

对于上述状态及扰动项的估计，参考​​状态观测器的设计​​内容。

4 仿真实例

5 仿真结果

相关参数matlab中均已给出。

仿真分别采用了线性扩张状态观测器(LESO)和非线性扩张状态观测器（NLESO）进行的。结果如下：

Ⅰ：NLESO仿真结果

参数：。

Ⅱ：LESO仿真结果

参数：

6 代码

LESO

%ESO观测器程序

clear all;

h=0.01; %采样时间
T=0.01;
time = 20;
N = time/T; %仿真步数
n=0:N-1;
% x = sin(n*T);

for k=1:1:N
    % original state
    x1(1) = 0.3;
    x2(1) = 0.3;
    x3(1) = 0.3;%新的扰动初始状态
    
    x1(k+1) = x1(k) + h*x2(k);
    x2(k+1) = x2(k) + h*(-(1+0.5*cos(k*T))*x1(k)-(1+sin(k*T/3))*x2(k)+sign(sin(1.5*k*T)));
    x3(k+1) = -(1+0.5*cos(k*T))*x1(k)-(1+sin(k*T/3))*x2(k)+sign(sin(1.5*k*T));%新的扰动状态
    y(k) = x1(k);
    
    % state observer
    z1(1) = 0.2;
    z2(1) = 0.2;
    z3(1) = 0.2;
    e(k) = z1(k) - y(k);
    
    %LESO
    z1(k+1) = z1(k) + h*(z2(k)-100*e(k));
    z2(k+1) = z2(k) + h*(z3(k)-2500*e(k));
    z3(k+1) = - h*2500*e(k);    
    
end

% 估计值与实际值对比
figure
plot(n*T,x1(1,1:N),'b+',n*T,z1(1,1:N),'r.');
legend('x1','z1');

figure
plot(n*T,x2(1,1:N),'b+',n*T,z2(1,1:N),'r.');
legend('x2','z2');

figure
plot(n*T,x2(1,1:N),'b+',n*T,z2(1,1:N),'r.');
legend('x3','z3');

% 估计值与实际值的误差分析
figure
plot(n*T,x1(1,1:N)-z1(1,1:N),'b--');
legend('error1=x1-z1');

figure
plot(n*T,x2(1,1:N)-z2(1,1:N),'b--');
legend('error2=x2-z2');

figure
plot(n*T,x3(1,1:N)-z3(1,1:N),'b--');
legend('error3=x3-z3');

NLESO

%ESO观测器程序

clear all;

h=0.01; %采样时间
T=0.01;
time = 20;
N = time/T; %仿真步数
n=0:N-1;
% x = sin(n*T);

for k=1:1:N
    % original state
    x1(1) = 0.3;
    x2(1) = 0.3;
    x3(1) = 0.3;%新的扰动初始状态
    
    x1(k+1) = x1(k) + h*x2(k);
    x2(k+1) = x2(k) + h*(-(1+0.5*cos(k*T))*x1(k)-(1+sin(k*T/3))*x2(k)+sign(sin(1.5*k*T)));
    x3(k+1) = -(1+0.5*cos(k*T))*x1(k)-(1+sin(k*T/3))*x2(k)+sign(sin(1.5*k*T));%新的扰动状态
    y(k) = x1(k);
    
    % state observer
    z1(1) = 0.2;
    z2(1) = 0.2;
    z3(1) = 0.2;
    e(k) = z1(k) - y(k);
    
    %NLESO
    fe = fal(e(k), 0.5, 0.01);
    z1(k+1) = z1(k) + h*(z2(k)-100*e(k));
    z2(k+1) = z2(k) + h*(z3(k)-500*fe);
    z3(k+1) = - h*1000*fe;
    
end

% 估计值与实际值对比
figure
plot(n*T,x1(1,1:N),'b+',n*T,z1(1,1:N),'r.');
legend('x1','z1');

figure
plot(n*T,x2(1,1:N),'b+',n*T,z2(1,1:N),'r.');
legend('x2','z2');

figure
plot(n*T,x2(1,1:N),'b+',n*T,z2(1,1:N),'r.');
legend('x3','z3');

% 估计值与实际值的误差分析
figure
plot(n*T,x1(1,1:N)-z1(1,1:N),'b--');
legend('error1=x1-z1');

figure
plot(n*T,x2(1,1:N)-z2(1,1:N),'b--');
legend('error2=x2-z2');

figure
plot(n*T,x3(1,1:N)-z3(1,1:N),'b--');
legend('error3=x3-z3');