非线性微分跟踪器

微分器描述

离散形式的非线性微分跟踪为

其中，h为采样周期；v(k)为第k时刻的输入信号，为决定跟踪快慢的系数；fst()为最速控制综合函数。描述如下

仿真分析

微分器测试

输入信号v(t)=sin2pit，采样周期h=0.001，δ=150；扰动信号为0.05的随机信号；

上图为带有噪声的正弦信号；下图为理想信号和微分跟踪器的输出信号；

上图为理想微分信号与差分求得的微分信号；

下图为理想微分信号和经过微分器输出的微分信号；

代码如下所示：

clear all;
close all;

h=0.001;  %Sampling time
delta=150;
r1_1=0;r2_1=0;
vn_1=0;
for k=1:1:10000
time(k)=k*h;

v(k)=sin(2*pi*k*h);
n(k)=0.05*rands(1);
vn(k)=v(k)+n(k);
dv(k)=2*pi*cos(2*pi*k*h);

r1(k)=r1_1+h*r2_1;
r2(k)=r2_1+h*fst(r1_1-v(k),r2_1,delta,h);

dvn(k)=(vn(k)-vn_1)/h;   %By difference

vn_1=vn(k);

r1_1=r1(k);
r2_1=r2(k);
end
figure(1);
subplot(211);
plot(time,v,'k:',time,vn,'r:','linewidth',2);
xlabel('time(s)'),ylabel('signal');
legend('ideal signal','signal with noise');
subplot(212);
plot(time,v,'k:',time,r1,'r:','linewidth',2);
xlabel('time(s)'),ylabel('signal');
legend('ideal signal','signal by TD');

figure(2);
subplot(211);
plot(time,dv,'k:',time,dvn,'r:','linewidth',2);
xlabel('time(s)'),ylabel('derivative signal');
legend('ideal derivative signal','derivative signal by difference');
subplot(212);
plot(time,dv,'k:',time,r2,'r:','linewidth',2);
xlabel('time(s)'),ylabel('derivative signal');
legend('ideal derivative signal','derivative signal by TD');

基于微分器的pd控制

输入信号v(t)=sint，采样周期h=0.001，δ=1000；扰动信号为0.05的随机信号；

控制对象:133/s2+25s

此时微分器的输入信号为对象的输出信号，采用微分器获取位置信息和速度信息；kp=10，kd=0.05；

红色为含有噪声的输入信号，蓝色为经过微分器处理的输入信号

黑色为理想的参考信号sin（t），红色为经过微分器处理后的反馈信号；

黑色为理想的微分参考信号cos（t），红色为经过微分器处理后的微分信号；

代码：

close all;
clear all;

h=0.001;               % 采样周期
y_1=0;yp_1=0;
dy_1=0;

%Plant
a=25;b=133;
sys=tf(b,[1,a,0]);
dsys=c2d(sys,h,'z');
[num,den]=tfdata(dsys,'v');
u_1=0;u_2=0;
p_1=0;p_2=0;
for k=1:1:5000
t=k*h;
time(k)=t;
    
p(k)=-den(2)*p_1-den(3)*p_2+num(2)*u_1+num(3)*u_2;
dp(k)=(p(k)-p_1)/h;

yd(k)=sin(t);
dyd(k)=cos(t);
d(k)=0.5*sign(rands(1)); %Noise
%d(k) = 0;
if mod(k,100)==1|mod(k,100)==2
    yp(k)=p(k)+d(k);     %Practical signal      
else
    yp(k)=p(k);
end

M=2;
if M==1         %By Difference
    y(k)=yp(k);
    dy(k)=(yp(k)-yp_1)/h;
elseif M==2     %By TD
    delta=1000;
    y(k)=y_1+h*dy_1;
    dy(k)=dy_1+h*fst(y_1-yp(k),dy_1,delta,h);
end    
kp=10;kd=0.5;
u(k)=kp*(yd(k)-y(k))+kd*(dyd(k)-dy(k));
%u(k) = yd(k);
y_1=y(k);
yp_1=yp(k);
dy_1=dy(k);

u_2=u_1;u_1=u(k);
p_2=p_1;p_1=p(k);
end	
figure(1);
plot(time,p,'k',time,yp,'r:',time,y,'b:','linewidth',2);
xlabel('time(s)');ylabel('position tracking');
legend('ideal position signal','position signal with noise','position signal by TD');
figure(2);
plot(time,yd,'k',time,p,'r:','linewidth',2);
xlabel('time(s)');ylabel('Position tracking');
legend('ideal position signal','tracking signal');