% ex8_21
% 已知雙輸入雙輸出系統傳遞函數矩陣G，計算關聯系統的相對增益矩陣
G=tf({[1],[1];[-0.2],[0.8]},{[1 1],[0.1 1];[0.5 1],[1 1]});
syms x u1 u2 y1 y2
for i=1:2;
    for j=1:2;
        ki=poly2sym(G(i,j).num{1})/poly2sym(G(i,j).den{1});
        k(i,j)=limit(ki,x,0);
    end;
end;
y=k*[u1;u2];
y1=y(1,1);
y2=y(2,1); 
la=jacobian([y1 y2],[u1 u2]);
la11u=la(1,1);la12u=la(1,2);
la21u=la(2,1);la22u=la(2,2);
y1=y(1,1);y2=y(2,1);
u11=solve(y2,u2);y1=subs(y1,{u2},{u11});
la11y=jacobian(y1,u1);lam11=la11u/la11y;
y1=y(1,1);y2=y(2,1);
u12=solve(y2,u1);y1=subs(y1,{u1},{u12});
la12y=jacobian(y1,u2);lam12=la12u/la12y;
y1=y(1,1);y2=y(2,1);
u21=solve(y1,u2);y2=subs(y2,{u2},{u21});
la21y=jacobian(y2,u1);lam21=la21u/la21y;
y1=y(1,1);y2=y(2,1);
u22=solve(y1,u1);y2=subs(y2,{u1},{u22});
la22y=jacobian(y2,u2);lam22=la22u/la22y;
Lamda=[lam11 lam12;lam21 lam22]