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자동제어 | 시스템의 안정한 범위구하기 |

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작성자 cemtool 작성일14-04-22 13:28 조회9,397회 댓글7건

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다음과 같은 루프전달함수를 가진 제어시스템을 생각하자.

시스템이 안정하기 위한 K의 범위를 구해보자.

«풀이»

먼저 근궤적선도를 통해 시스템이 안정하기 위한 K의 임계값을 구하고, 실제로 몇 개의 K에 대해서 Nyquist 선도 또는 Routh-Hurwitz 판별법을 통해 결과를 확인해보도록 하자. 다음은 주어진 시스템에 대한 근궤적선도이다.

ex7_5.cem
num = [10 20];
den = [1 3 0 10];
rlocus(num, den);
title("Root-Locus Plot of G(s) = 10K(s + 2 )/[(s^3 + 3s^2 + 10)]")



ex7_5-1.png



그림 7-의 근궤적선도를 보면 K가 어떤 K값일 때 허수축을 지나고 있다. 이 때의 값이 시스템이 안정하기 위한 임계값이 된다. CEMTool을 사용하여 K의 임계값을 찾기는 쉽지 않다. 따라서, 손으로 직접 계산하거나 여러 번의 시행착오를 통해 그 값을 구해야 한다. 이 예제의 경우에는 K=1일 때 근궤적선도상의 허수축을 지난다. 즉, 0<K<1일 때는 불안정, K=1일 때는 임계안정, K>1일 때는 안정하다. 이 사실을 K=0.5, 1, 2의 예를 통해 확인해보자. 각 K의 값에 대하여 안정도를 판별할 때에는 Nyquist 선도나 Routh 표를 이용하면 된다. 


(1) K=0.5일 때

ex7_5.cem
// Nyquist plotnum = 5*[1 2];
den = [1 3 0 10];
nyquist(num, den);
title("Nyquist Plot of L(s) = [5(s + 2)]/(s^3 + 3s^2 + 10)")// Routh table
routh([0 0 num] + den)
      1.0000    5.0000
      3.0000   20.0000
     -1.6667    0.0000
     20.0000    0.0000


프로그램을 실행시키면 그림 7-과 같은 Nyquist 선도를 얻을 수 있다. 시스템이 안정하기 위해서는 이 이어야 하나 실제 값은 이므로 시스템은 불안정하다. 또한 Routh 표에서도 두 번의 부호변화가 있으므로 시스템이 불안정함을 알 수 있다. 


ex7_5-2.png


(2) K=1일 때

ex7_5.cem
// Nyquist plotnum = 10*[1 2];
den = [1 3 0 10];
nyquist(num, den);
title("Nyquist Plot of L(s) = [10(s + 2)]/(s^3 + 3s^2 + 10)")// Routh table
routh([0 0 num] + den)
  Special case : one row in Routh table is zero!
           1        10
           3        30
           0         0

그림 7-는 결과로 나온 그래프의 x축 범위를 [-2,0]으로, y축의 범위를 [-1,1]로 조정한 것이다.


ex7_5-3.png


Nyquist 선도가 임계점 (-1,j0)을 지나므로 시스템은 임계안정하다. 이 때에는 Routh 표에서 특별한 경우가 나오므로 CEMTool을 통해서 완전한 Routh 표를 구할 수 없고 교재에서 설명한 방법 등을 사용하여 다시 계산해야 한다. 


(3) K=2일 때

ex7_5.cem
// Nyquist plotnum = 20*[1 2];
den = [1 3 0 10];
nyquist(num, den);
title("Nyquist Plot of L(s) = [20(s + 2)]/(s^3 + 3s^2 + 10)")// Routh table
routh([0 0 num] + den)
      1.0000   20.0000
      3.0000   50.0000
      3.3333    0.0000
     50.0000    0.0000
ex7_5-4.png

그림 7-의 Nyquist 선도에서는 (-1,j0)근방에서의 모양을 알 수 없다. 그림 7-은 (-1,j0)근방에서의 정확한 모양을 보기 위해 x축 범위를 [-2,0]으로, y축의 범위를 [-1,1]로 조정한 그래프이다.


ex7_5-5.png

 그림 7-와 그림 7-의 Nyquist 선도로부터 이 임을 알 수 있다. 따라서, 시스템은 안정하다. 또한 Routh 표에 부호변화가 생기지 않으므로 같은 결론을 내릴 수 있다.

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