控制系統(tǒng)仿真（MATLAB版）（七）
根軌跡漸近線的繪制
>> G = tf([1 2 4],conv([1 4 0],conv([1 6],[1 1.4 1])));
>> rlocus(G)
>> hold on
>> thga = -9.4/3;
>> Gc = zpk([],[thga thga thga],1);
>> rlocus(Gc)
>>?
判斷系統(tǒng)的穩(wěn)定性
一、閉環(huán)特征根的大小
1+GH = 0(不夠準(zhǔn)確，指閉環(huán)特征根)
>> G = zpk([-2],[0 -0.5 -0.8 -3],0.2);
>> sys = feedback(tf(G),1);? %注意是閉環(huán)
>> roots(sys.den{1})
ans =
? -3.0121 + 0.0000i
? -1.0000 + 0.0000i
? -0.1440 + 0.3348i
? -0.1440 - 0.3348i
? ? ?或者
>> eig(sys)
ans =
? -3.0121 + 0.0000i
? -1.0000 + 0.0000i
? -0.1440 + 0.3348i
? -0.1440 - 0.3348i
所有閉環(huán)極點(diǎn)位于s平面（虛軸）左側(cè)，系統(tǒng)穩(wěn)定
>> pzmap(sys)
在MATLAB中，有更直接的判斷方法
>> isstable(sys)
ans =
? logical
? ?1
>> allmargin(G)? %這里用的是開環(huán)傳遞函數(shù)
ans =?
? struct with fields:
? ? ?GainMargin: 4.7503
? ? GMFrequency: 0.7071
? ? PhaseMargin: 44.5277
? ? PMFrequency: 0.2770
? ? DelayMargin: 2.8058
? ? DMFrequency: 0.2770
? ? ? ? ?Stable: 1
4.2
求動(dòng)態(tài)誤差系數(shù)
>> syms s
>> G = 50*(s+2)/(s^3+2*s^2+51*s+100);
>> kp = limit(G,s,0)
kp =
1
>> kv = limit(s*G,s,0)
kv =
0
>> ka = limit(s^2*G,s,0)
ka =
0
?4.4
>> G = tf([-0.5 1],conv([0.5 1],conv([0.2 1],[0.1 1])));
>> rlocus(G)
閉環(huán)系統(tǒng)穩(wěn)定的K
0 < K <1.33
>> G = 1.33*tf([-0.5 1],conv([0.5 1],conv([0.2 1],[0.1 1])));isstable(feedback(G,1))
ans =
? logical
? ?0
>> G = 1.32*tf([-0.5 1],conv([0.5 1],conv([0.2 1],[0.1 1])));isstable(feedback(G,1))
ans =
? logical
? ?1
4.5求動(dòng)態(tài)誤差系數(shù)
>> syms s
>> G = 10/(s^2+s+2)
G =
10/(s^2 + s + 2)?
>> kp = limit(G,s,0)?
kp =?
5?
>> kv = limit(s*G,s,0)
kv =
0?
>> ka = limit(s^2*G,s,0)
ka =
0
4.6
>> G1 = tf(5,conv([1 1 0],[0.1 1]));
>> sys1 = feedback(G1,1);
>> G2 = tf(20,conv([1 1 0],[0.1 1]));
>> sys2 = feedback(G2,1);
>> pzmap(sys1)
pzmap(sys2)
>> [mag,phase,w] = bode(G1);
>> margin(mag,phase,w)
>> [mag,phase,w] = bode(G2);
>> margin(mag,phase,w)
4.7
>> s = tf('s')
s =
?
? s
?
Continuous-time transfer function.
>> G = 16*(19*s+1)*(0.44*s+1)/((0.625*s+1)*(0.676*s-1)*(43.5*s-1)*(0.033*s+1)*(0.0004*s^2+0.015*s+1))
G =
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 133.8 s^2 + 311 s + 16
? ------------------------------------------------------------------------------
? 0.0002426 s^6 + 0.01647 s^5 + 0.8832 s^4 + 18.43 s^3 - 0.2936 s^2 - 43.5 s + 1
?
Continuous-time transfer function.
>> [Gm,Pm,Wcg,Wcp] = margin(G)
Gm =
? ? 4.8847
Pm =
? ?53.1362
Wcg =
? ?32.1207
Wcp =
? ? 7.2063
>> margin(G)
4.8
>> G = tf([1 1],conv([1 0 0],[0.1 1]));
>> [Gm,Pm,Wcg,Wcp] = margin(G)
Gm =
? ? ?0
Pm =
? ?44.4594
Wcg =
? ? ?0
Wcp =
? ? 1.2647
%% 該系統(tǒng)穩(wěn)定
>> margin(G)
>> figure
>> bode(G)
4.10
>> G1 = zpk([],[-1;-2;-5],2*5*10)
G1 =
?
? ? ? ? ?100
? -----------------
? (s+1) (s+2) (s+5)
?
Continuous-time zero/pole/gain model.
>> nyquist(G1)
穩(wěn)定
p = 0
Nyquist曲線不包含（-1，j 0）
k = 50
>> G2 = zpk([],[-1;-2;-5],2*5*50)
G2 =
?
? ? ? ? ?500
? -----------------
? (s+1) (s+2) (s+5)
?
Continuous-time zero/pole/gain model.
>> nyquist(G2)
不穩(wěn)定
p = 0
Nyquist圖包圍（-1，j0）一次
