transmission zeros and normal rank
[tr]=trzeros(Sl)
[nt,dt,rk]=trzeros(Sl)
:Sl linear system ( syslin list) : :nt complex vectors : :dt real vector : :rk integer (normal rank of Sl) :
Called with one output argument, trzeros(Sl) returns the transmission zeros of the linear system Sl.
Sl may have a polynomial (but square) D matrix.
Called with 2 output arguments, trzeros returns the transmission zeros of the linear system Sl as tr=nt./dt;
(Note that some components of dt may be zeros)
Called with 3 output arguments, rk is the normal rank of Sl
Transfer matrices are converted to state-space.
If Sl is a (square) polynomial matrix trzeros returns the roots of its determinant.
For usual state-space system trzeros uses the state-space algorithm of Emami-Naeni and Van Dooren.
If D is invertible the transmission zeros are the eigenvalues of the ” A matrix” of the inverse system : A - B*inv(D)*C;
If C*B is invertible the transmission zeros are the eigenvalues of N*A*M where M*N is a full rank factorization of eye(A)-B*inv(C*B)*C;
For systems with a polynomial D matrix zeros are calculated as the roots of the determinant of the system matrix.
Caution: the computed zeros are not always reliable, in particular in case of repeated zeros.
W1=`ssrand`_(2,2,5);trzeros(W1) //call trzeros
`roots`_(`det`_(`systmat`_(W1))) //roots of det(system matrix)
s=`poly`_(0,'s');W=[1/(s+1);1/(s-2)];W2=(s-3)*W*W';[nt,dt,rk]=trzeros(W2);
St=`systmat`_(`tf2ss`_(W2));[Q,Z,Qd,Zd,numbeps,numbeta]=`kroneck`_(St);
St1=Q*St*Z;rowf=(Qd(1)+Qd(2)+1):(Qd(1)+Qd(2)+Qd(3));
colf=(Zd(1)+Zd(2)+1):(Zd(1)+Zd(2)+Zd(3));
`roots`_(St1(rowf,colf)), nt./dt //By Kronecker form