控制系统仿真（MATLAB版）（六）
控制系统时域运动响应分析
>> A = [-2 -2.5 -2.5;1 0 0; 0 1 0];
>> B = [1;0;0];
>> C = [0 1.5 1];
>> D = 0;
>> x0 = [2 2 0];
>> G = ss(A,B,C,D);
>> initial(G,x0)
>> A = [-0.2 0.5 0 0 0 ;0 -1.5 1.6 0 0 ;0 0 -14.3 85.8 0;0 0 0 -33.3 100;0 0 0 0 -10];
>> B = [0;0;0;0;30];
>> C = [1,0,0,0,0];
>> D = 0;
>> G = ss(A,B,C,D)
G =
A =
x1 x2 x3 x4 x5
x1 -0.2 0.5 0 0 0
x2 0 -1.5 1.6 0 0
x3 0 0 -14.3 85.8 0
x4 0 0 0 -33.3 100
x5 0 0 0 0 -10
B =
u1
x1 0
x2 0
x3 0
x4 0
x5 30
C =
x1 x2 x3 x4 x5
y1 1 0 0 0 0
D =
u1
y1 0
Continuous-time state-space model.
>> impulse(G)
>> G = tf([5 8],[1 4 6 3 3]);
>> step(G)
>> s = tf('s');
>> G = 143.7*(s+1.5)/((s^2+2*s+5)*(s^2+10*s+26)*(s+1.7))
G =
143.7 s + 215.5
-----------------------------------------------------
s^5 + 13.7 s^4 + 71.4 s^3 + 188.7 s^2 + 303.4 s + 221
Continuous-time transfer function.
>> zpk(G)
ans =
143.7 (s+1.5)
---------------------------------------
(s+1.7) (s^2 + 10s + 26) (s^2 + 2s + 5)
Continuous-time zero/pole/gain model.
>> pzmap(G)
>> [p,z] = pzmap(G)
p =
-5.0000 + 1.0000i
-5.0000 - 1.0000i
-1.0000 + 2.0000i
-1.0000 - 2.0000i
-1.7000 + 0.0000i
z =
-1.5000
判断出主导极点
p =
-5.0000 + 1.0000i
-5.0000 - 1.0000i
-1.0000 + 2.0000i
-1.0000 - 2.0000i
-1.7000 + 0.0000i
用主导极点简化原系统（高阶系统降阶），注意开环增益的变化
转换后如下
>> G1 = 147.3/26*(s+1.5)/((s+1.7)*(s^2+2*s+5))
G1 =
5.665 s + 8.498
---------------------------
s^3 + 3.7 s^2 + 8.4 s + 8.5
Continuous-time transfer function.
>> zpk(G1)
ans =
5.6654 (s+1.5)
----------------------
(s+1.7) (s^2 + 2s + 5)
Continuous-time zero/pole/gain model.
勉强能接受
>> G = tf(500,conv([1 10 50],[1 10]))
G =
500
--------------------------
s^3 + 20 s^2 + 150 s + 500
Continuous-time transfer function.
>> zpk(G)
ans =
500
-----------------------
(s+10) (s^2 + 10s + 50)
Continuous-time zero/pole/gain model.
>> pzmap(G)
>>
>> [p,z] = pzmap(G)
p =
-10.0000 + 0.0000i
-5.0000 + 5.0000i
-5.0000 - 5.0000i
z =
0×1 empty double column vector
主导极点
p =
-10.0000 + 0.0000i
-5.0000 + 5.0000i
-5.0000 - 5.0000i
降阶
>> G1 = tf(50,[1 10 50])
G1 =
50
---------------
s^2 + 10 s + 50
>> step(G,G1)
>> t = 0:0.01:4*pi;
>> ut1 = sin(t+30/180*pi);
>> ut2 = 2*cos(5*t+30/180*pi);
>> sys = 5*tf([1 1],conv([1 0],[1 4 2 3]));
>> lsim(sys,ut1,t)
lsimplot(sys,ut2,t)
>> G1 = tf([1],conv([1 0],conv([1 2 2],[1 6 13])));
>> G2 = tf([1 12],conv([1 0],conv([1 12 100],[1 10])));
>> G3 = tf([0.05 1],conv([1 0],conv([0.0714 1],[0.0125 0.1 1])));
>> G4 = tf([1 -1 -20],conv([1 0],conv([1 1],[1 3])));
>> rlocus(G1)
>> rlocus(G2)
>> rlocus(G3)
>> rlocus(G4)
>> G = tf([1 2 4],conv([1 0],conv([1 4],conv([1 6],[1 1.4 1]))))
G =
s^2 + 2 s + 4
-----------------------------------------
s^5 + 11.4 s^4 + 39 s^3 + 43.6 s^2 + 24 s
Continuous-time transfer function.
>> zpk(G)
ans =
(s^2 + 2s + 4)
------------------------------
s (s+6) (s+4) (s^2 + 1.4s + 1)
Continuous-time zero/pole/gain model.
>> rlocus(G)
>>sgrid
>> G = tf([1 2],conv([1 0],conv([1 4],conv([1 8],[1 2 5]))));
>> zpk(G)
ans =
(s+2)
----------------------------
s (s+8) (s+4) (s^2 + 2s + 5)
Continuous-time zero/pole/gain model.
>> rlocus(G) %负反馈
>> rlocus(-G) %正反馈
>> G = zpk([],[0;-1;-2],1)
G =
1
-------------
s (s+1) (s+2)
Continuous-time zero/pole/gain model.
>> rlocus(G,1.5)
幅值裕量和相位裕量
>> G = zpk([],[0 -1 -2],1.5)
G =
1.5
-------------
s (s+1) (s+2)
Continuous-time zero/pole/gain model.
>> [gm,pm,wg,wp] = margin(G)
gm =
4.0000
pm =
41.5340
wg =
1.4142
wp =
0.6118
>> margin(G)
>> wn = 0.7;
>> s = tf('s');
>> kesi = [0.1,0.4,1.0,1.6,2.0];
>> for ii = kesi
figure;
G = wn^2/(s^2+2*ii*wn*s+wn^2);
bode(G);
end
>>
>> G = 500*tf([0.0167 1],conv([1 0],conv([0.0025 1],conv([0.05 1],[0.001 1])))) ;
>> bode(G)
>> [gm,pm,wg,wp] = margin(G)
gm =
7.1968
pm =
45.5298
wg =
586.6697
wp =
161.7414
>> margin(G)
>> nyquist(zpk([-1],[-0.8+1.6*j -0.8-1.6*j],3))
kosi = [0.4,0.7,1.0,1.3];
for ii = kosi
figure;
G = tf(1,[1 2*kosi 1]);
nyquist(G);
end
>> G1 = zpk([],[0 0 1/5 -5],2);
>> G2 = 8*tf([1 1],conv([1 0 0],conv([1 15],[1 6 10])));
>> G3 = zpk([-3],[0 -50 -20 -10],4);
>> A = [0 2 1;-3 -2 0;1 3 4];
>> B = [4;3;2];
>> C = [1 2 3];
>> D = 0;
>> G4 = ss(A,B,C,D);
>> bode(G1)
>> nyquist(G1)
>> bode(G2)
>> nyquist(G2)
>> bode(G3)
>> nyquist(G3)
>> bode(G4)
>> nyquist(G4)
