Scilab-Branch-6.1-GIT
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See the recommended documentation of this function
canon
canonical controllable form
Syntax
[Ac, Bc, U, ind] = canon(A, B)
Arguments
- 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).
Examples
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
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