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零极点对系统性能的影响分析

1任务步骤

1）分析原开环传递函数G0（s）的性能，绘制系统的阶跃响应曲线得到系统的暂态性能（包括上升时间，超调时间，超调量，调节时间）；

2）在G0（s）上增加零点，使开环传递函数为G1（s），绘制系统的根轨迹,分析系统的稳定性。绘制系统的阶跃响应曲线得到系统的暂态性能（包括上升时间，超调时间，超调量，调节时间）；

3）在G0（s）上增加极点，使开环传递函数为G2（s），绘制系统的根轨迹，分析系统的稳定性；绘制系统的阶跃响应曲线得到系统的暂态性能（包括上升时间，超调时间，超调量，调节时间）；

4）同时在G0（s）上增加零极点，使开环传递函数为G3（s），绘制系统的根轨迹，分析系统的稳定性；绘制系统的阶跃响应曲线得到系统的暂态性能（包括上升时间，超调时间，超调量，调节时间）；

5）综合数据，分析零极点对系统性能的影响。

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2开环传递函数G0（s）的性能分析

2.1 原G0（s）性能分析

取原开环传函数为：G0(s)?1(s?0.25)(s?0.52)

2,1.1分析系统稳定性

在M指令中输入:

g0=zpk([],[-0.25,-0.52],[1]) g=feedback(g0,1); [z,p,k]=zpkdata(g,'v') 按Enter回车得： Zero/pole/gain: 1 ----------------- (s+0.25) (s+0.52) z =

Empty matrix: 0-by-1 p = -0.2500 -0.5200 k = 1

即闭环系统两个极点均具有负实部，系统稳定2.1.2系统的根轨迹

Matlab指令：

G0=zpk([],[-0.25,-0.52],[1]) g=feedback(g0,1);

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rlocus(g) 得到图形：

根据原函数的根轨迹可得：

Root Locus543-1Imaginary Axis (seconds)210-1-2-3-4-5-0.9System: gGain: 0.101Pole: -0.385 + 1.04iDamping: 0.347Overshoot (%): 31.3Frequency (rad/s): 1.11System: gGain: 0.0369Pole: -0.385 - 1.01iDamping: 0.356Overshoot (%): 30.2Frequency (rad/s): 1.08-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.1Real Axis (seconds-1)

图1 原函数G0(s)的根轨迹

系统的两个极点分别是-0.52和-0.25，零点在无限远处。

2.1.3G0（s）的阶跃响应

Matlab指令：

g0=zpk([],[-0.25,-0.52],[1]) g=feedback(g0,1); step(g) 得到图形：

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Step Response1.4System: gTime (seconds): 3.18Amplitude: 1.151.2System: gTime (seconds): 1.971Amplitude: 0.8880.8System: gTime (seconds): 11.3Amplitude: 0.887Amplitude0.60.40.2005Time (seconds)1015

图2 原函数的阶跃响应曲线

由阶跃响应曲线分析系统暂态性能： 曲线最大峰值为1.15，稳态值为0.887， 上升时间tr=1.97s 超调时间tp=3.18s 调节时间ts=11.3s，??2

?%超调量p=29.505%

2.2 增加零点后的开环传递函数G1（s）的性能分析

传递函数的表达式上单独增加一个零点S=-a,并改变a值大小，即离虚轴的距离，分析比较系统性能的变化。所以增加零点后为了分析开环传递函数的零点对系统性能的影响，现在在原开环的开环传递函数为：

s?a

(s?0.25)(s?0.52)2.2.1 当a分别为0.01,0.1,1,10,100时，G1（s）的根轨与阶跃响应如下excel

G(s)?.

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a传递函数1magnaryAxs(seconds)1s?1s?0.1s?0.01s?10G(s)?G(s)?G(s)?G(s)?(s?0.25)(s?0.52)(s?0.25)(s?0.52)(s?0.25)(s?0.52)(s?0.25)(s?0.52) g0=zpk([-0.1],[-0.25,-0.52],[1])g=feedback(g0,1);[z,p,k]=zpkdata(g,'v') Zero/pole/gain: (s+0.1)-----------------(s+0.25) (s+0.52)z = -0.1000p = -1.6288 -0.1412k = 1.00000.010.110G(s)?100s?100(s?0.25)(s?0.52) g0=zpk([-0.01],[-0.25,-0.52],[1])M指令g=feedback(g0,1);[z,p,k]=zpkdata(g,'v') Zero/pole/gain: (s+0.01)-----------------(s+0.25) (s+0.52)z = -0.0100p = -1.6870 -0.0830k = 1.0000 g0=zpk([-1],[-0.25,-0.52],[1])g=feedback(g0,1);[z,p,k]=zpkdata(g,'v')Zero/pole/gain: (s+1)-----------------(s+0.25) (s+0.52)z = -1.0000p = -0.8850 + 0.5889i -0.8850 - 0.5889ik = 1.0000 g0=zpk([-10],[-0.25,-0.52],[1])g=feedback(g0,1);[z,p,k]=zpkdata(g,'v') Zero/pole/gain: (s+10)-----------------(s+0.25) (s+0.52)z = -10.0000p = -0.8850 + 3.0572i -0.8850 - 3.0572ik = 1.0000 g0=zpk([-100],[-0.25,-0.52],[1])g=feedback(g0,1);[z,p,k]=zpkdata(g,'v') Zero/pole/gain: (s+100)-----------------(s+0.25) (s+0.52)z = -100.0000p = -0.8850 + 9.9673i -0.8850 - 9.9673ik = 1.00001magnaryAxs(seconds))0.1 0-0.1-0.2-0.30System: gGain: 0.0165Pole: -1.65Damping: 1Overshoot (%): 0Frequency (rad/s): 1.65Imagnary Axs (seconds0.25magnaryAxs(seconds10.1magnaryAxs(second)s1)1AmpudeAmpudeAmpude 0.40.40.50.40.3AmpudeAmpudeσ%tr系统稳定性根轨迹阶跃响应暂态性能结果是否稳定M指令是g0=zpk([-0.01],[-0.25,-0.52],[1])g=feedback(g0,1);rlocus(g)Root Locus0.4是g0=zpk([-0.1],[-0.25,-0.52],[1])g=feedback(g0,1);rlocus(g)Root Locus0.40.3System: gGain: InfPole: -0.1Damping: 1Overshoot (%): 0Frequency (rad/s): 0.1是g0=zpk([-1],[-0.25,-0.52],[1])g=feedback(g0,1);rlocus(g)Root Locus0.8System: gGain: 0.0102Pole: -0.89 + 0.589iDamping: 0.834Overshoot (%): 0.87Frequency (rad/s): 1.07是g0=zpk([-10],[-0.25,-0.52],[1])g=feedback(g0,1);rlocus(g)Root Locus15是g0=zpk([-100],[-0.25,-0.52],[1])g=feedback(g0,1);rlocus(g)Root Locus1500.3System: gGain: InfPole: -0.01Damping: 1Overshoot (%): 0Frequency (rad/s): 0.01System: gGain: 0.00812Pole: -1.7Damping: 1Overshoot (%): 0Frequency (rad/s): 1.70.60.20.2100.4System: gGain: 1.43Pole: -1.6 - 0.00255iDamping: 1Overshoot (%): 0Frequency (rad/s): 1.6System: gGain: 152Pole: -1Damping: 1Overshoot (%): 0Frequency (rad/s): 10-0.1 System: gSystem: gGain: 37.5Gain: 1.06e+003Pole: -19.6 - 0.0474iPole: -10.1Damping: 1Damping: 1Overshoot (%): 0Overshoot (%): 0Frequency (rad/s): 19.6Frequency (rad/s): 10.1System: gGain: 0Pole: -0.885 + 3.06iDamping: 0.278Overshoot (%): 40.3Frequency (rad/s): 3.18100System: gSystem: gGain: 397Gain: InfPole: -200Pole: -100Damping: 1Damping: 1Overshoot (%): 0Overshoot (%): 0Frequency (rad/s): 200Frequency (rad/s): 10050System: gGain: 0.358Pole: -1.07 + 11.6iDamping: 0.0915Overshoot (%): 74.9Frequency (rad/s): 11.700-0.2-50-0.2-5-100-0.4-0.3-0.4-6-0.4-6-0.6-5-4-3-2-101Real Axis (seconds-1-10-150-400-350-300-250-200-150-100-50050Real Axis (seconds-1-5-4-3-2-101Real Axis (seconds-1-0.8-3.5-3-2.5-2-1.5-1-0.500.5Real Axis (seconds-1-15-40-35-30-25-20-15-10-505Real Axis (seconds-1) g0=zpk([-0.01],[-0.25,-0.52],[1])M指令g=feedback(g0,1);step(g)Step Response0.7g0=zpk([-0.1],[-0.25,-0.52],[1])g=feedback(g0,1);step(g)Step Response0.7System: gTime (seconds): 2.32Amplitude: 0.562System: gTime (seconds): 26.9Amplitude: 0.439g0=zpk([-1],[-0.25,-0.52],[1])g=feedback(g0,1);step(g)10.9System: g0.8Time (seconds): 1.98Amplitude: 0.8830.70.6System: gTime (seconds): 6.03Amplitude: 0.888System: gStep ResponseTime (seconds): 2.94Amplitude: 0.922g0=zpk([-10],[-0.25,-0.52],[1])g=feedback(g0,1);step(g)1.5System: gTime (seconds): 0.921Step ResponseAmplitude: 1.41g0=zpk([-100],[-0.25,-0.52],[1])g=feedback(g0,1);step(g)1.81.6System: gTime (seconds): 0.308Amplitude: 1.75Step Response0.6System: gTime (seconds): 2.02Amplitude: 0.5150.6System: gTime (seconds): 0.7820.5Amplitude: 0.4340.5System: gTime (seconds): 0.502Amplitude: 0.991System: gTime (seconds): 5.53Amplitude: 0.9891.4System: g1.2Time (seconds): 0.156Amplitude: 0.99810.80.6System: gTime (seconds): 5.56Amplitude: 0.9950.30.30.50.40.200.2System: gTime (seconds): 0.0757Amplitude: 0.0707System: gTime (seconds): 65.3Amplitude: 0.07390.20.20.1000.10.10123456700204060Time (seconds)8010012000123456701234567Time (seconds)051015Time (seconds)202530Time (seconds)Time (seconds)σ% tp(s)ts(s)tr(s)lga 596.89%1.9265.30.087-228.02%2.3927.10.79-13.83%2.996.01204.27%0.9215.60.5175%0.3165.520.1572 700.00`0.00P0.00@0.0000.00 0.000.00%0.00%图表标题系列1增加零点对系统影响 2.521.510.50系列1.