连续时间系统的s域分析(Matlab)

简单

• 系统函数
  $H(s) = \frac{s^2+5}{s^2+2s+5}$H(s)=s2+2s+5s2+5​
• 起始状态及输入信号
  $y(0_-)=0,y^{(1)}(0_-)=-2 \\ x(t)=u(t)$y(0−​)=0,y(1)(0−​)=−2x(t)=u(t)
• Matlab求响应
  $Y(s)=\frac{-2}{s^2+2s+5}+\frac{s^2+5}{s^2+2s+5}X(s)$Y(s)=s2+2s+5−2​+s2+2s+5s2+5​X(s)

syms t s
Yzi = -2/(s^2+2*s+5); % zero input
yzi = ilaplace(Yzi);
xt = heaviside(t);
X = laplace(xt);
Yzs = X * (s^2+5) / (s^2+2*s+5); % zero state
yzs = ilaplace(Yzs);
yt = simplify(yzi+yzs);

$y_{zi}(t)=-e^{-t}sin(2t)u(t) \\ y_{zi}(t)=(1-e^{-t}sin(2t))u(t) \\ y(t)=(1-2e^{-t}sin(2t))u(t)$yzi​(t)=−e−tsin(2t)u(t)yzi​(t)=(1−e−tsin(2t))u(t)y(t)=(1−2e−tsin(2t))u(t)

零极点分布

• 系统函数
  $H(s) = \frac{s+1}{s^2+2s+5}$H(s)=s2+2s+5s+1​

b = [1 1];
a = [1 2 5];
sys = tf(b,a);
[p,z] = pzmap(sys);
pzmap(sys)
% p = pole(sys)
% z = zero(sys)

$p_1 = -1+2i \\ p_2=-1-2i \\ z = -1$p1​=−1+2ip2​=−1−2iz=−1

单位冲激响应

• $H(s)=\frac{1}{s+2}$H(s)=s+21​

b = [1];
a = [1 2];
sys = tf(b,a);
[p,z] = pzmap(sys);
subplot(121);
pzmap(sys)
subplot(122);
impulse(sys);grid on

左半平面，衰减，稳定

• $H(s)=\frac{1}{s-3}$H(s)=s−31​

b = [1];
a = [1 -3];
sys = tf(b,a);
[p,z] = pzmap(sys);
subplot(121);
pzmap(sys)
subplot(122);
impulse(sys);grid on

右半平面，增长，不稳定

• $H(s)=\frac{2}{s^2+4}$H(s)=s2+42​

    b = [2];
    a = [1 0 4];
    sys = tf(b,a);
    [p,z] = pzmap(sys);
    subplot(121);
    pzmap(sys)
    subplot(122);
    impulse(sys);grid on

虚轴上两个一阶共轭极点，等幅正弦震荡，临界稳定

• $H(s)=\frac{2s}{(s^2+1)^2}$H(s)=(s2+1)22s​

    b = [2 0];
    a = [1 0 2 0 1];
    sys = tf(b,a);
    [p,z] = pzmap(sys);
    subplot(121);
    pzmap(sys)
    subplot(122);
    impulse(sys);grid on

虚轴上两个高阶极点，增长正弦震荡，不稳定

• $H(s)=\frac{1}{s}$H(s)=s1​

    b = [1];
    a = [1 0];
    sys = tf(b,a);
    [p,z] = pzmap(sys);
    subplot(121);
    pzmap(sys)
    subplot(122);
    impulse(sys);grid on

临界稳定

• $H(s)=\frac{1}{s^2}$H(s)=s21​

    b = [1];
    a = [1 0 0];
    sys = tf(b,a);
    [p,z] = pzmap(sys);
    subplot(121);
    pzmap(sys)
    subplot(122);
    impulse(sys);grid on

系统不稳定

• $H(s)=\frac{s+1}{(s+1)^2+4}$H(s)=(s+1)2+4s+1​

    b = [1 1];
    a = [1 2 5];
    sys = tf(b,a);
    [p,z] = pzmap(sys);
    subplot(121);
    pzmap(sys)
    subplot(122);
    impulse(sys);grid on

• $H(s)=\frac{s}{(s+1)^2+4}$H(s)=(s+1)2+4s​

    b = [1 0];
    a = [1 2 5];
    sys = tf(b,a);
    [p,z] = pzmap(sys);
    subplot(121);
    pzmap(sys)
    subplot(122);
    impulse(sys);grid on

• $H(s)=\frac{(s+1)^2}{(s+1)^2+4}$H(s)=(s+1)2+4(s+1)2​

    b = [1 2 1];
    a = [1 2 5];
    sys = tf(b,a);
    [p,z] = pzmap(sys);
    subplot(121);
    pzmap(sys)
    subplot(122);
    impulse(sys);grid on

零点从-1移到0，波形的幅度和相位发生变化；-1处从一阶变到二阶，波形的幅度和相位发生变化，还出现冲激

抄自图书ISBN：978-7-307-19477-9