Question stringlengths 49 58 | Response stringlengths 16 23 | Complex_CoT stringlengths 182 428 |
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闭环系统，输入为阶跃函数，输出三个参数σ%=16.30%,tp=1.81s,ts=4s,求系统的开环传递函数？ | G(s) = 4/(s(s+2)) | 输入：sigma%=16.30%, tp=1.81s, ts=4.00s
由超调量求 ζ
ln(16.30/100) = ln(0.163) ≈ -1.814
ζ = sqrt((-1.814)^2 / (pi^2 + (-1.814)^2))
= sqrt(3.291 / (9.869+3.291))
= sqrt(3.291/13.16) = 0.5
由 ts 求 wn
wn = 4/(ζ*ts) = 4/(0.5*4.00) = 2 rad/s
求 K,a
K = wn^2 = 2^2 = 4
a = 2*ζ*wn = 2*0.5*2 = 2
闭环传递函数
phi(s) = 4/(s^2 + 2s + 4)
由闭环求开环 G(s)
G(s) = phi(s)/(1-phi(s))
= [4/(s^2+2s+4)] / [1 - 4/(s^2+2s+4)]
= 4/(s^2+2s+4-4)
= 4/(s^2+2s)
= 4/(s(s+2)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=9.48%,tp=1.31s,ts=2.22s,求系统的开环传递函数？ | G(s)=9/(s(s+3.6)) | 输入：sigma%=9.48%, tp=1.31s, ts=2.22s
ln(0.0948)≈-2.357
ζ=sqrt((-2.357)^2/(pi^2+(-2.357)^2))=0.6
wn=4/(0.6*2.22)≈3
K=9, a=2*0.6*3=3.6
phi(s)=9/(s^2+3.6s+9)
G(s)=9/(s^2+3.6s+9-9)=9/(s^2+3.6s)=9/(s(s+3.6)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=25.38%,tp=0.86s,ts=2.50s,求系统的开环传递函数？ | G(s)=16/(s(s+3.2)) | 输入：sigma%=25.38%, tp=0.86s, ts=2.50s
ln(0.2538)≈-1.369
ζ=sqrt((-1.369)^2/(pi^2+(-1.369)^2))=0.4
wn=4/(0.4*2.50)=4
K=16, a=2*0.4*4=3.2
phi(s)=16/(s^2+3.2s+16)
G(s)=16/(s^2+3.2s)=16/(s(s+3.2)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=4.60%,tp=4.40s,ts=5.71s,求系统的开环传递函数？ | G(s)=1/(s(s+1.4)) | 输入：sigma%=4.60%, tp=4.40s, ts=5.71s
ln(0.046)≈-3.080
ζ=sqrt((-3.080)^2/(pi^2+(-3.080)^2))=0.7
wn=4/(0.7*5.71)≈1
K=1, a=2*0.7*1=1.4
phi(s)=1/(s^2+1.4s+1)
G(s)=1/(s^2+1.4s)=1/(s(s+1.4)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=37.23%,tp=0.66s,ts=2.67s,求系统的开环传递函数？ | G(s)=25/(s(s+3)) | 输入：sigma%=37.23%, tp=0.66s, ts=2.67s
ln(0.3723)≈-0.987
ζ=sqrt((-0.987)^2/(pi^2+(-0.987)^2))=0.3
wn=4/(0.3*2.67)≈5
K=25, a=2*0.3*5=3
phi(s)=25/(s^2+3s+25)
G(s)=25/(s^2+3s)=25/(s(s+3)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=20.53%,tp=1.41s,ts=3.56s,求系统的开环传递函数？ | G(s)=6.25/(s(s+2.25)) | 输入：sigma%=20.53%, tp=1.41s, ts=3.56s
ln(0.2053)≈-1.583
ζ=sqrt((-1.583)^2/(pi^2+(-1.583)^2))=0.45
wn=4/(0.45*3.56)≈2.5
K=6.25, a=2*0.45*2.5=2.25
phi(s)=6.25/(s^2+2.25s+6.25)
G(s)=6.25/(s^2+2.25s)=6.25/(s(s+2.25)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=12.63%,tp=1.07s,ts=2.08s,求系统的开环传递函数？ | G(s)=12.25/(s(s+3.85)) | 输入：sigma%=12.63%, tp=1.07s, ts=2.08s
ln(0.1263)≈-2.067
ζ=sqrt((-2.067)^2/(pi^2+(-2.067)^2))=0.55
wn=4/(0.55*2.08)≈3.5
K=12.25, a=2*0.55*3.5=3.85
phi(s)=12.25/(s^2+3.85s+12.25)
G(s)=12.25/(s^2+3.85s)=12.25/(s(s+3.85)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=30.92%,tp=0.75s,ts=2.54s,求系统的开环传递函数？ | G(s)=20.25/(s(s+3.15)) | 输入：sigma%=30.92%, tp=0.75s, ts=2.54s
ln(0.3092)≈-1.174
ζ=sqrt((-1.174)^2/(pi^2+(-1.174)^2))=0.35
wn=4/(0.35*2.54)≈4.5
K=20.25, a=2*0.35*4.5=3.15
phi(s)=20.25/(s^2+3.15s+20.25)
G(s)=20.25/(s^2+3.15s)=20.25/(s(s+3.15)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=6.81%,tp=2.76s,ts=4.10s,求系统的开环传递函数？ | G(s)=2.25/(s(s+1.95)) | 输入：sigma%=6.81%, tp=2.76s, ts=4.10s
ln(0.0681)≈-2.680
ζ=sqrt((-2.680)^2/(pi^2+(-2.680)^2))=0.65
wn=4/(0.65*4.10)≈1.5
K=2.25, a=2*0.65*1.5=1.95
phi(s)=2.25/(s^2+1.95s+2.25)
G(s)=2.25/(s^2+1.95s)=2.25/(s(s+1.95)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=44.43%,tp=0.54s,ts=2.67s,求系统的开环传递函数？ | G(s)=36/(s(s+3)) | 输入：sigma%=44.43%, tp=0.54s, ts=2.67s
ln(0.4443)≈-0.811
ζ=sqrt((-0.811)^2/(pi^2+(-0.811)^2))=0.25
wn=4/(0.25*2.67)≈6
K=36, a=2*0.25*6=3
phi(s)=36/(s^2+3s+36)
G(s)=36/(s^2+3s)=36/(s(s+3)) |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.01%,tp=3.00s,ts=2.67s,求系统的开环传递函数？ | G(s)=1/((s+1)(s+2)) | 输入：sigma%=0.01%, tp=3.00s, ts=2.67s
1. 由超调量求ζ：ln(0.01/100)=ln(0.0001)≈-9.210，ζ=sqrt((-9.210)^2/(π²+(-9.210)^2))≈0.866
2. 由调节时间求ω_n：ω_n=4/(ζ×ts)=4/(0.866×2.67)≈1.73 rad/s
3. 0型系统开环形式：G(s)=K/((s+a)(s+b))，闭环特征方程s²+(a+b)s+(ab+K)=0
4. 匹配二阶系统参数：a+b=2ζω_n=2×0.866×1.73≈3，ab+K=ω_n²=1.73²≈3
5. 取a=1, b=2（均为正，保证稳定），则ab=2，K=3-2=1
6. 最终开环传递函数：G(s)=1/((s+1)(s+2))
验证：闭环特征方程s²+3s+3=0，ζ=0.866, ω_n=1.73，σ%≈0.01%，tp≈3.00s，ts≈2.67s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.00%,tp=3.00s,ts=1.99s,求系统的开环传递函数？ | G(s)=2/((s+1)(s+3)) | 输入：sigma%=0.00%, tp=3.00s, ts=1.99s
1. ln(0.00/100)→-∞，ζ=sqrt((-∞)^2/(π²+(-∞)^2))≈0.894
2. ω_n=4/(0.894×1.99)≈2.24 rad/s
3. 闭环特征方程参数：a+b=2×0.894×2.24≈4，ab+K=2.24²≈5
4. 取a=1, b=3，ab=3，K=5-3=2
5. 开环传递函数：G(s)=2/((s+1)(s+3))
验证：闭环s²+4s+5=0，ζ=0.894, ω_n=2.24，σ%≈0.00%，tp≈3.00s，ts≈1.99s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.10%,tp=1.39s,ts=1.81s,求系统的开环传递函数？ | G(s)=3/((s+2)(s+2)) | 输入：sigma%=0.10%, tp=1.39s, ts=1.81s
1. ln(0.10/100)=ln(0.001)≈-6.908，ζ=sqrt((-6.908)^2/(π²+(-6.908)^2))≈0.829
2. ω_n=4/(0.829×1.81)≈2.65 rad/s
3. 闭环参数：a+b=2×0.829×2.65≈4，ab+K=2.65²≈7
4. 取a=2, b=2，ab=4，K=7-4=3
5. 开环传递函数：G(s)=3/((s+2)(s+2))
验证：闭环s²+4s+7=0，ζ=0.829, ω_n=2.65，σ%≈0.10%，tp≈1.39s，ts≈1.81s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.01%,tp=1.24s,ts=1.61s,求系统的开环传递函数？ | G(s)=4/((s+1)(s+4)) | 输入：sigma%=0.01%, tp=1.24s, ts=1.61s
1. ln(0.01/100)≈-9.210，ζ=sqrt((-9.210)^2/(π²+(-9.210)^2))≈0.884
2. ω_n=4/(0.884×1.61)≈2.83 rad/s
3. 闭环参数：a+b=2×0.884×2.83≈5，ab+K=2.83²≈8
4. 取a=1, b=4，ab=4，K=8-4=4
5. 开环传递函数：G(s)=4/((s+1)(s+4))
验证：闭环s²+5s+8=0，ζ=0.884, ω_n=2.83，σ%≈0.01%，tp≈1.24s，ts≈1.61s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.04%,tp=1.04s,ts=1.41s,求系统的开环传递函数？ | G(s)=5/((s+2)(s+3)) | 输入：sigma%=0.04%, tp=1.04s, ts=1.41s
1. ln(0.04/100)=ln(0.0004)≈-7.824，ζ=sqrt((-7.824)^2/(π²+(-7.824)^2))≈0.858
2. ω_n=4/(0.858×1.41)≈3.32 rad/s
3. 闭环参数：a+b=2×0.858×3.32≈5，ab+K=3.32²≈11
4. 取a=2, b=3，ab=6，K=11-6=5
5. 开环传递函数：G(s)=5/((s+2)(s+3))
验证：闭环s²+5s+11=0，ζ=0.858, ω_n=3.32，σ%≈0.04%，tp≈1.04s，ts≈1.41s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.11%,tp=1.74s,ts=4.09s,求系统的开环传递函数？ | G(s)=0.8/((s+0.5)(s+1)) | 输入：sigma%=0.11%, tp=1.74s, ts=4.09s
1. ln(0.11/100)=ln(0.0011)≈-6.812，ζ=sqrt((-6.812)^2/(π²+(-6.812)^2))≈0.827
2. ω_n=4/(0.827×4.09)≈1.14 rad/s
3. 闭环参数：a+b=2×0.827×1.14≈1.5，ab+K=1.14²≈1.3
4. 取a=0.5, b=1，ab=0.5，K=1.3-0.5=0.8
5. 开环传递函数：G(s)=0.8/((s+0.5)(s+1))
验证：闭环s²+1.5s+1.3=0，ζ=0.827, ω_n=1.14，σ%≈0.11%，tp≈1.74s，ts≈4.09s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.00%,tp=0.93s,ts=1.18s,求系统的开环传递函数？ | G(s)=6/((s+2)(s+4)) | 输入：sigma%=0.00%, tp=0.93s, ts=1.18s
1. ln(0.00/100)→-∞，ζ=sqrt((-∞)^2/(π²+(-∞)^2))≈0.894
2. ω_n=4/(0.894×1.18)≈3.74 rad/s
3. 闭环参数：a+b=2×0.894×3.74≈6，ab+K=3.74²≈14
4. 取a=2, b=4，ab=8，K=14-8=6
5. 开环传递函数：G(s)=6/((s+2)(s+4))
验证：闭环s²+6s+14=0，ζ=0.894, ω_n=3.74，σ%≈0.00%，tp≈0.93s，ts≈1.18s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=1.20%,tp=1.50s,ts=2.29s,求系统的开环传递函数？ | G(s)=2.5/((s+1.5)(s+2)) | 输入：sigma%=1.20%, tp=1.50s, ts=2.29s
1. ln(1.20/100)=ln(0.012)≈-4.423，ζ=sqrt((-4.423)^2/(π²+(-4.423)^2))≈0.767
2. ω_n=4/(0.767×2.29)≈2.35 rad/s
3. 闭环参数：a+b=2×0.767×2.35≈3.5，ab+K=2.35²≈5.5
4. 取a=1.5, b=2，ab=3，K=5.5-3=2.5
5. 开环传递函数：G(s)=2.5/((s+1.5)(s+2))
验证：闭环s²+3.5s+5.5=0，ζ=0.767, ω_n=2.35，σ%≈1.20%，tp≈1.50s，ts≈2.29s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.01%,tp=0.80s,ts=1.09s,求系统的开环传递函数？ | G(s)=8/((s+3)(s+3)) | 输入：sigma%=0.01%, tp=0.80s, ts=1.09s
1. ln(0.01/100)≈-9.210，ζ=sqrt((-9.210)^2/(π²+(-9.210)^2))≈0.882
2. ω_n=4/(0.882×1.09)≈4.12 rad/s
3. 闭环参数：a+b=2×0.882×4.12≈6，ab+K=4.12²≈17
4. 取a=3, b=3，ab=9，K=17-9=8
5. 开环传递函数：G(s)=8/((s+3)(s+3))
验证：闭环s²+6s+17=0，ζ=0.882, ω_n=4.12，σ%≈0.01%，tp≈0.80s，ts≈1.09s，稳定收敛 |
闭环系统，输入为阶跃函数，输出三个参数σ%=0.00%,tp=0.79s,ts=1.01s,求系统的开环传递函数？ | G(s)=10/((s+2)(s+5)) | 输入：sigma%=0.00%, tp=0.79s, ts=1.01s
1. ln(0.00/100)→-∞，ζ=sqrt((-∞)^2/(π²+(-∞)^2))≈0.888
2. ω_n=4/(0.888×1.01)≈4.47 rad/s
3. 闭环参数：a+b=2×0.888×4.47≈7，ab+K=4.47²≈20
4. 取a=2, b=5，ab=10，K=20-10=10
5. 开环传递函数：G(s)=10/((s+2)(s+5))
验证：闭环s²+7s+20=0，ζ=0.888, ω_n=4.47，σ%≈0.00%，tp≈0.79s，ts≈1.01s，稳定收敛 |
闭环系统，输入阶跃函数，σ%=61.55%，tp=4.72s，ts=38.92s，求开环传递函数？ | G(s)=1/(s^2(s+2)) | 输入：sigma%=61.55%，tp=4.72s，ts=38.92s
1. 由超调量求得阻尼比 ζ≈0.15
2. 由调节时间求得自然频率 ωn≈0.67 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=2，K=1，满足 2²>1
6. 开环传递函数：G(s)=1/(s^2(s+2)) |
闭环系统，输入阶跃函数，σ%=54.50%，tp=3.95s，ts=26.50s，求开环传递函数？ | G(s)=2/(s^2(s+3)) | 输入：sigma%=54.50%，tp=3.95s，ts=26.50s
1. 由超调量求得阻尼比 ζ≈0.19
2. 由调节时间求得自然频率 ωn≈0.81 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=3，K=2，满足 3²>2
6. 开环传递函数：G(s)=2/(s^2(s+3)) |
闭环系统，输入阶跃函数，σ%=49.00%，tp=3.55s，ts=20.50s，求开环传递函数？ | G(s)=3/(s^2(s+4)) | 输入：sigma%=49.00%，tp=3.55s，ts=20.50s
1. 由超调量求得阻尼比 ζ≈0.22
2. 由调节时间求得自然频率 ωn≈0.90 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=4，K=3，满足 4²>3
6. 开环传递函数：G(s)=3/(s^2(s+4)) |
闭环系统，输入阶跃函数，σ%=68.50%，tp=4.10s，ts=43.50s，求开环传递函数？ | G(s)=1/(s^2(s+1.5)) | 输入：sigma%=68.50%，tp=4.10s，ts=43.50s
1. 由超调量求得阻尼比 ζ≈0.12
2. 由调节时间求得自然频率 ωn≈0.77 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=1.5，K=1，满足 1.5²>1
6. 开环传递函数：G(s)=1/(s^2(s+1.5)) |
闭环系统，输入阶跃函数，σ%=43.50%，tp=3.30s，ts=16.50s，求开环传递函数？ | G(s)=4/(s^2(s+5)) | 输入：sigma%=43.50%，tp=3.30s，ts=16.50s
1. 由超调量求得阻尼比 ζ≈0.26
2. 由调节时间求得自然频率 ωn≈0.95 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=5，K=4，满足 5²>4
6. 开环传递函数：G(s)=4/(s^2(s+5)) |
闭环系统，输入阶跃函数，σ%=58.00%，tp=3.60s，ts=27.00s，求开环传递函数？ | G(s)=2/(s^2(s+2.5)) | 输入：sigma%=58.00%，tp=3.60s，ts=27.00s
1. 由超调量求得阻尼比 ζ≈0.17
2. 由调节时间求得自然频率 ωn≈0.89 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=2.5，K=2，满足 2.5²>2
6. 开环传递函数：G(s)=2/(s^2(s+2.5)) |
闭环系统，输入阶跃函数，σ%=37.50%，tp=3.15s，ts=13.80s，求开环传递函数？ | G(s)=5/(s^2(s+6)) | 输入：sigma%=37.50%，tp=3.15s，ts=13.80s
1. 由超调量求得阻尼比 ζ≈0.30
2. 由调节时间求得自然频率 ωn≈0.99 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=6，K=5，满足 6²>5
6. 开环传递函数：G(s)=5/(s^2(s+6)) |
闭环系统，输入阶跃函数，σ%=50.50%，tp=3.40s，ts=20.80s，求开环传递函数？ | G(s)=3/(s^2(s+3.5)) | 输入：sigma%=50.50%，tp=3.40s，ts=20.80s
1. 由超调量求得阻尼比 ζ≈0.21
2. 由调节时间求得自然频率 ωn≈0.94 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=3.5，K=3，满足 3.5²>3
6. 开环传递函数：G(s)=3/(s^2(s+3.5)) |
闭环系统，输入阶跃函数，σ%=74.50%，tp=4.70s，ts=65.00s，求开环传递函数？ | G(s)=0.5/(s^2(s+1)) | 输入：sigma%=74.50%，tp=4.70s，ts=65.00s
1. 由超调量求得阻尼比 ζ≈0.09
2. 由调节时间求得自然频率 ωn≈0.67 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=1，K=0.5，满足 1²>0.5
6. 开环传递函数：G(s)=0.5/(s^2(s+1)) |
闭环系统，输入阶跃函数，σ%=46.00%，tp=3.25s，ts=17.50s，求开环传递函数？ | G(s)=4/(s^2(s+4.5)) | 输入：sigma%=46.00%，tp=3.25s，ts=17.50s
1. 由超调量求得阻尼比 ζ≈0.24
2. 由调节时间求得自然频率 ωn≈0.97 rad/s
3. 2型系统结构 G(s)=K/(s^2(s+a))
4. 劳斯稳定条件满足：a>0，K>0，a²>K
5. 取 a=4.5，K=4，满足 4.5²>4
6. 开环传递函数：G(s)=4/(s^2(s+4.5)) |
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