canon

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

• obsv_mat — observability matrix
• cont_mat — controllability matrix
• ctr_gram — controllability gramian
• contrss — controllable part
• ppol — pole placement
• contr — controllability, controllable subspace, staircase
• stabil — stabilization