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자동제어 | 가관측성표준형 |

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작성자 cemtool 작성일14-04-22 12:27 조회13,951회 댓글1건

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상태방정식의 계수행렬이 다음과 같다.


이 상태방정식을 가관측성표준형(observability canonical form)으로 변환하여라.

«풀이»

상태방정식을 가관측성표준형으로 변환하려면, 먼저 행렬 Q을 구한 후 공식을 이용하여 를 구한다. 행렬 Q는 을 사용하여 구한다.

ex3_7.cem
A = [1 2 1; 0 1 3; 1 1 1];
B = [1; 0; 1];
C = [1 1 0];
D = [0];
coeff = poly(A);
a1 = coeff(3); a2 = coeff(2);
M = [a1 a2 1; a2 1 0; 1 0 0]
M  =  
     -1.0000   -3.0000    1.0000
     -3.0000    1.0000    0.0000
      1.0000    0.0000    0.0000
ex3_7.cem
V = [C; C*A; C*A^2]
V  =  
           1         1         0
           1         3         4
           5         9        14
ex3_7.cem
Q = inv(M*V)
Q  =  
      0.3333   -0.1667    0.3333
     -0.3333    0.1667    0.6667
      0.1667    0.1667    0.1667
ex3_7.cem
Abar = inv(Q)*A*Q
Abar  =  
      0.0000    0.0000    3.0000
      1.0000    0.0000    1.0000
      0.0000    1.0000    3.0000
ex3_7.cem
Bbar = inv(Q)*B
Bbar  =  
      3.0000
      2.0000
      1.0000
ex3_7.cem
Cbar = C*Q
Cbar  =  
           0         0         1

앞의 예제와 마찬가지로 tr2ocf 라는 함수를 만들어 사용할 수 있다.

ex3_7.cem
function;
Q, Abar, Bbar, Cbar<>A, B, C, D
a = poly(A);
if( a != 1 )	a = a/a(1);
order = length(a)-1;
V = C; M = a(order);
for(i=1;i<order;i=i+1)	M = [M; a(order-i)];
for(i=1;i<order;i=i+1){	V = [V; C*A^i];
	vec = a(order-i);
	for(j=1;j<order-i;j=j+1)		vec = [vec; a(order-i-j)];
	for(j=order-i;j<order;j=j+1)		vec = [vec; 0];
	M = [M vec];
}Q = inv(M*V);
Abar = inv(Q)*A*Q;
Bbar = inv(Q)*B;
Cbar = C*Q;
Dbar = [];
if(nargin == 4)	Dbar = D;

tr2ocf를 사용하여 가관측성표준형을 구해보자.

ex3_7.cem
A = [1 2 1; 0 1 3; 1 1 1];
B = [1; 0; 1];
C = [1 1 0];
D = [0];
[Q, Abar, Bbar, Cbar] = tr2ocf(A,B,C)
Q  =  
      0.3333   -0.1667    0.3333
     -0.3333    0.1667    0.6667
      0.1667    0.1667    0.1667

Abar  =  
      0.0000    0.0000    3.0000
      1.0000    0.0000    1.0000
      0.0000    1.0000    3.0000

Bbar  =  
      3.0000
      2.0000
      1.0000

Cbar  =  
           0         0         1




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