\[
\Phi(z)=\frac{G(z)}{1+G(z)}=\frac{(T+\mathrm{e}^{-T}-1)z+(1-\mathrm{e}^{-T}-T\mathrm{e}^{-T})}{z^{2}+(T-2)z+(1-T\mathrm{e}^{-T})}
\]
\[
C(z)=\Phi(z)R(z)=\frac{z}{z-1}\cdot\frac{(T+\mathrm{e}^{-T}-1)z+(1-\mathrm{e}^{-T}-T\mathrm{e}^{-T})}{z^{2}+(T-2)z+(1-T\mathrm{e}^{-T})}
\]
\(T=1.0\) 时
\[
C(z)=\frac{z}{z-1}\cdot\frac{0.3679z+0.2642}{z^{2}-z+0.6321}=\frac{0.3679z^{2}+0.2642z}{z^{3}-2z^{2}+1.6321z-0.6321}
\]
\[
=0.3679z^{-1}+z^{-2}+1.3996z^{-3}+\cdots
\]
\(T=2.0\) 时
\[
C(z)=\frac{z}{z-1}\cdot\frac{1.1353z+0.594}{z^{2}+0.7293}=\frac{1.1353z^{2}+0.594z}{z^{3}-z^{2}+0.7293z-0.7293}
\]
\[
=1.1353z^{-1}+1.7293z^{-2}+0.9014z^{-3}+\cdots
\]
\(T=4.0\) 时
\[
C(z)=\frac{z}{z-1}\cdot\frac{3.0183z+0.9084}{z^{2}+2z+0.9267}=\frac{3.0183z^{2}+0.9084z}{z^{3}+z^{2}-1.0733z-0.9267}
\]
\[
=3.0183z^{-1}-2.1099z^{-2}+5.3494z^{-3}+\cdots
\]
MATLAB 验证:系统在不同 \(T\) 值下的单位阶跃响应如图 7-23 所示,动态性能如表 7-1所示。分析与仿真表明:采样周期 \(T\) 增大,会恶化采样系统的稳定性及动态性能。
MATLAB 文本:exe721.m
figure(1)
T1=1;t1=0:1:20;
sys1=tf([0.3679,0.2642],[1,-1,0.6321],T1);
step(sys1,t1);grid;
figure(2)
T2=2;t2=0:2:40;
sys2=tf([1.1353,0.594],[1,0,0.7293],T2);
step(sys2,t2);grid;
figure(3)
T3=4;t3=0:4:80;
sys3=tf([3.0183,0.9084],[1,1,-1.0733,-0.9267],T3);
step(sys3,t3);grid;
图 7-23 采样系统单位阶跃响应(MATLAB)
(a) \(T=1.0\) (b) \(T=2.0\) (c) \(T=4.0\)
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