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자동제어 | 시스템 안정도 판별 1 |

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작성자 cemtool 작성일14-04-22 12:28 조회13,447회 댓글7건

본문

다음의 방정식을 생각해 보자.


Routh 표를 사용하여 이 시스템의 안정도를 판별해보자.

«풀이»

이 시스템의 s=2와 s=3은 평면의 오른쪽반평면에 존재하므로, 특별한 방법을 사용하지 않고서도 시스템이 불안정하다는 것을 알 수 있다. Routh 표를 사용하여 이 내용을 확인해보자. Routh 표의 계산을 간단히 하기위해 다음과 같이 Routh 함수를 작성하였다.

ex4_1.cem
function;
routh<>a
/* Compute the Routh table */
/*
if(sum(a<=0) != 0){	msgprint("a(i) must be positive!")	routh = [];
	return}*/
len = length(a);
if(len == 0 ){	msgprint("a must not be zero!")	routh = [];
	return;
}else if(len == 1)	routh = a(1);
else if(len == 2)	routh = [a(1); a(2)];
else{	row = 0;
	if( rem(len,2) == 0 ) // if the highest order is odd
	{		row = row + 1;
		vec = a(1);
		for(i=3;i<len;i=i+2)			vec = [vec a(i)];
		routh = vec;
		row = row + 1;
		vec = a(2);
		for(i=4;i<=len;i=i+2)			vec = [vec a(i)];
		routh = [routh; vec];		
 
		while(row < len )		{			row = row + 1;
			den = vec(1);
			if(den==0) // Special Case: First element is zero!
			{	
				msgprint("Special Case: First element is zero!")				return			}	    mat=[routh(row-2;1) routh(row-2;2);routh(row-1;1) routh(row-1;2)];
			vec = -det(mat)/den;
			for(j=3;j<=len/2;j=j+1)			{	    mat = [routh(row-2;1) routh(row-2;j);routh(row-1;1) routh(row-1;j)];
				vec = [vec -det(mat)/den];
			}			vec = [vec 0];
			routh = [routh; vec];
	if(sum(vec!=0)==0)  // Special case : one row in Routh table is zero!
			{		msgprint("Special case : one row in Routh table is zero!")				return;
			}	}	}	else // if the highest order is even
	{		row = row + 1;
		vec = a(1);
		for(i=3;i<=len;i=i+2)			vec = [vec a(i)];
		routh = vec;
		row = row + 1;
		vec = a(2);
		for(i=4;i<len;i=i+2)			vec = [vec a(i)];
		vec = [vec 0];
		routh = [routh; vec];		
		while(row < len )		{			row = row + 1;
			den = vec(1);
			if(den==0) // Special Case: Fisrst element is zero!
			{				msgprint("Special Case: Fisrst element is zero!")				return			}	mat = [routh(row-2;1) routh(row-2;2);routh(row-1;1) routh(row-1;2)];
			vec = -det(mat)/den;
			for(j=3;j<=(len+1)/2;j=j+1)			{	mat = [routh(row-2;1) routh(row-2;j);routh(row-1;1) routh(row-1;j)];
				vec = [vec -det(mat)/den];
			}			vec = [vec 0];
			routh = [routh; vec];
	if(sum(vec!=0)==0)  // Special case : one row in Routh table is zero!
			{		msgprint("Special case : one row in Routh table is zero!")				return;
			}		}	}}	

이 함수를 사용하여 Routh 표를 계산할 수 있다. 다만 교재에 나온 것과 같이 특별한 경우가 발생한 경우에는 완전한 Routh 표를 만들지 못하고, 특별한 경우가 발생하였다는 메시지와 그 때까지 계산한 Routh 표를 보여준다. 그러나. 이 시스템이 안정하기 위한 또는 불안정하기 위한 의 범위를 구하는 문제와 같이 문자가 포함된 경우(예제 6.6, 예제 6.7)에는 이 함수를 사용할 수가 없다. 이제 routh 함수를 사용하여 주어진 문제를 풀어보자.

ex4_1.cem
a = [1 -4 1 6];
routh(a)
ans =
      1.0000    1.0000
     -4.0000    6.0000
      2.5000    0.0000
      6.0000    0.0000

이 결과가 의미하는 바는 다음과 같다.


부호변화가 2번 있으므로, 이 방정식은 평면의 오른쪽반평면에 2개의 근이 존재한다.




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