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canon Scilab Group Scilab Function canon
NAME
canon - canonical controllable form
CALLING SEQUENCE
[Ac,Bc,U,ind]=canon(A,B)
PARAMETERS
Ac,Bc : canonical form
U : current basis (square nonsingular matrix)
ind : vector of integers, controllability indices
DESCRIPTION
gives the canonical controllable form of the pair (A,B).
Ac=inv(U)*A*U, Bc=inv(U)*B
The vector ind is made of the epsilon_i's indices of the pencil [sI - A ,
B] (decreasing order). For example with ind=[3,2], Ac and Bc are as
follows:
[*,*,*,*,*] [*]
[1,0,0,0,0] [0]
Ac= [0,1,0,0,0] Bc=[0]
[*,*,*,*,*] [*]
[0,0,0,1,0] [0]
If (A,B) is controllable, by an appropriate choice of F the * entries of
Ac+Bc*F can be arbitrarily set to desired values (pole placement).
EXAMPLE
A=[1,2,3,4,5;
1,0,0,0,0;
0,1,0,0,0;
6,7,8,9,0;
0,0,0,1,0];
B=[1,2;
0,0;
0,0;
2,1;
0,0];
X=rand(5,5);A=X*A*inv(X);B=X*B; //Controllable pair
[Ac,Bc,U,ind]=canon(A,B); //Two indices --> ind=[3.2];
index=1;for k=1:size(ind,'*')-1,index=[index,1+sum(ind(1:k))];end
Acstar=Ac(index,:);Bcstar=Bc(index,:);
s=poly(0,'s');
p1=s^3+2*s^2-5*s+3;p2=(s-5)*(s-3);
//p1 and p2 are desired closed-loop polynomials with degrees 3,2
c1=coeff(p1);c1=c1($-1:-1:1);c2=coeff(p2);c2=c2($-1:-1:1);
Acstardesired=[-c1,0,0;0,0,0,-c2];
//Acstardesired(index,:) is companion matrix with char. pol=p1*p2
F=Bcstar\(Acstardesired-Acstar); //Feedbak gain
Ac+Bc*F // Companion form
spec(A+B*F/U) // F/U is the gain matrix in original basis.
SEE ALSO
obsv_mat, cont_mat, ctr_gram, contrss, ppol, contr, stabil
AUTHOR
F.D.
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