s平面與z平面映射分析（ Analysis of Mapping from?S-plane to Z-plane）

Mapping horizontal s-plane line to z-plane

xx=[0:0.05:1]'; %設(shè)定采樣時間范圍

N=length(xx); %求采樣點(diǎn)個數(shù)

s0=-xx*35;

ts = 0.1;

s=s0*[1 1 1 1 1]+1j*ones(N,1)*[0,0.25,0.5,0.75,1]*pi/ts;

plot(real(s(:,1)),imag(s(:,1)),'-o',...

? ? real(s(:,2)),imag(s(:,2)),'-s',...

? ? real(s(:,3)),imag(s(:,3)),'-^',...

? ? real(s(:,4)),imag(s(:,4)),'-*',...

? ? real(s(:,5)),imag(s(:,5)),'-v')

sgrid %畫S平面域水平線的極點(diǎn)軌跡

figure

z=exp(s*ts);

plot(real(z(:,1)),imag(z(:,1)),'-o',...

? ? real(z(:,2)),imag(z(:,2)),'-s',...

? ? real(z(:,3)),imag(z(:,3)),'-^',...

? ? real(z(:,4)),imag(z(:,4)),'-*',...

? ? real(z(:,5)),imag(z(:,5)),'-v'),

zgrid %對應(yīng)S 平面域畫Z 平面域的極點(diǎn)軌跡

Mapping of vertical s-plane line to z-plane

s0 = 1i*xx*pi/ts;

ts = 0.1;

s = ones(N,1)*[0,-5,-10,-20,-30]+s0*[1 1 1 1 1];

? ? plot(real(s(:,1)),imag(s(:,1)),'-o',real(s(:,2)),imag(s(:,2)),'-s',...

? ? ? ? ?real(s(:,3)),imag(s(:,3)),'-^',real(s(:,4)),imag(s(:,4)),'-*',...

? ? ? ? ?real(s(:,5)),imag(s(:,5)),'-v'),sgrid

figure

z=exp(s*ts) ;

plot(real(z(:,1)),imag(z(:,1)),'-o',real(z(:,2)),imag(z(:,2)),'-s',...

? ? ?real(z(:,3)),imag(z(:,3)),'-^',real(z(:,4)),imag(z(:,4)),'-*',...

? ? ?real(z(:,5)),imag(z(:,5)),'-v'),zgrid

Mapping of constant damping ratio s-plane lines into z-plane

s = s0*[1 1 1 1]-imag(s0)*[0,1/tan(67.5*pi/180),...

1/tan(45*pi/180),1/tan(22.5*pi/180)];

s=[s,real(s(:,4))];

plot(real(s(:,1)),imag(s(:,1)),'-o',real(s(:,2)),imag(s(:,2)),'-s',...

? ? real(s(:,3)),imag(s(:,3)),'-^',real(s(:,4)),imag(s(:,4)),'-*',...

? ? real(s(:,5)),imag(s(:,5)),'-v'),sgrid

figure

z=exp(s*ts);

plot(real(z(:,1)),imag(z(:,1)),'-o',real(z(:,2)),imag(z(:,2)),'-s',...

real(z(:,3)),imag(z(:,3)),'-^',real(z(:,4)),imag(z(:,4)),'-*',...

real(z(:,5)),imag(z(:,5)),'-v'),zgrid

Mapping of circle s-plane line to z-plane

phi=xx*pi/2 ;

s0=(pi/ts)*(-cos(phi)+1i*sin(phi)) ;

s=s0*[1,0.75,0.5,0.25,0] ;

plot(real(s(:,1)),imag(s(:,1)),'-o',real(s(:,2)),imag(s(:,2)),'-s',...

real(s(:,3)),imag(s(:,3)),'-^',real(s(:,4)),imag(s(:,4)),'-*',...

real(s(:,5)),imag(s(:,5)),'-v'),sgrid

figure

z=exp(s*ts);

plot(real(z(:,1)),imag(z(:,1)),'-o',real(z(:,2)),imag(z(:,2)),'-s',...

real(z(:,3)),imag(z(:,3)),'-^',real(z(:,4)),imag(z(:,4)),'-*',...

real(z(:,5)),imag(z(:,5)),'-v'),zgrid

參考教材北交大《計算機(jī)控制技術(shù)實驗指導(dǎo)書》