trzeros

Arguments

Sl

linear system (syslin list)

nt

complex vectors

dt

real vector

rk

integer (normal rank of Sl)

Description

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.

Examples

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

See also

• spec — eigenvalues of matrices and pencils
• kroneck — Kronecker form of matrix pencil

History