所以,系统(10)闭环不稳定。

MATLAB 验证:设系统为单位反馈系统,作系统的单位阶跃响应,如图 5-90 所示,表明闭环系统确是不稳定的。

MATLAB 文本:exe543.m

G1=tf(100,conv([1,0],[0.2,1]));

G2=tf(100*0.2*[1,0,0,0],conv(conv([1,0],[0.25,1]),conv([0.0625,1],[0.8,1])));

G4=tf(100,conv(conv([1,0],[0.8,1]),[0.25,1]));

G3=tf(50,conv(conv([0.2,1],[1,2]),[1,0.5]));

G5=tf(100*[1,1],conv(conv([1,0],[0.1,1]),conv([0.5,1],[0.8,1])));

G6=tf(-10,2*conv([1,0],[-20,1]));

G7=tf(10,conv(conv([1,0],[0.1,1]),[0.25,1]));

G8=tf(10,conv(conv([1,0],[0.2,1]),[1,-1]));

G9=tf(1000,conv(conv([1,0],[1,2,0]),[0.2,1]));

G10=tf(5*[-0.5,1],conv(conv([1,0],[0.1,1]),[-0.2,1]));

close=feedback(G10,1);

figure(1);nyquist(G1);figure(2);margin(G1);grid;

figure(3);nyquist(G2);figure(4);margin(G2);grid;

figure(5);nyquist(G3);figure(6);margin(G3);grid;

figure(7);nyquist(G4);figure(8);margin(G4);grid;

figure(9);nyquist(G5);figure(10);margin(G5);grid;

figure(11);nyquist(G6);figure(12);margin(G6);grid;

figure(13);nyquist(G7);figure(14);margin(G7);grid;

figure(15);nyquist(G8);figure(16);margin(G8);grid;

figure(17);nyquist(G9);figure(18);margin(G9);grid;

figure(19);nyquist(G10);figure(20);margin(G10);grid;

figure(21);step(close);grid;

5-44 设单位反馈系统的开环幅相特性曲线如图 5-91 所示。当 \(K=50\) 时,系统幅值裕度 \(h=1\),穿越频率 \(\omega_x=1\),试求输入为 \(r(t)=t^2+5\sin\omega_x t\),幅值裕度为下述值时,系统的稳态误差。

(1) \(h=0.5\); (2) \(h=3\)。

图 5-91 单位反馈系统的开环幅相特性曲线

解 设系统的开环传递函数为 \(KG(s)\),

因 \(h = \dfrac{1}{|50G(j\omega_x)|}\Big|_{\omega_x=1} = 1\),故

(1) 若 \(h=\dfrac{1}{|KG(j1)|}=0.5<1\),有 \(|KG(j1)|>2\),则系统闭环不稳定。

(2) 若 \(h=\dfrac{1}{|KG(j1)|}=3\),有 \(|KG(j1)|=\dfrac{1}{3}<1\),则系统闭环稳定。

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