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ss2zp - Convert state-space filter parameters to zero-pole-gain form

Syntax

[z,p,k] = ss2zp(A,B,C,D,i)

Description

ss2zp converts a state-space representation of a given system to an equivalent zero-pole-gain representation. The zeros, poles, and gains of state-space systems represent the transfer function in factored form.

[z,p,k] = ss2zp(A,B,C,D,i) calculates the transfer function in factored form

of the continuous-time system

from the ith input (using the ith columns of B and D). The column vector p contains the pole locations of the denominator coefficients of the transfer function. The matrix z contains the numerator zeros in its columns, with as many columns as there are outputs y (rows in C). The column vector k contains the gains for each numerator transfer function.

ss2zp also works for discrete time systems. The input state-space system must be real.

The ss2zp function is part of the standard MATLAB language.

Examples

Here are two ways of finding the zeros, poles, and gains of a discrete-time transfer function:

b = [2 3 0]; 
a = [1 0.4 1];
[z,p,k] = tf2zp(b,a)
z =
    0.0000
   -1.5000
p =
    -0.2000 + 0.9798i
    -0.2000 - 0.9798i
k =
    2
[A,B,C,D] = tf2ss(b,a);
[z,p,k] = ss2zp(A,B,C,D,1)
z =
    0.0000
   -1.5000
p =
    -0.2000 + 0.9798i
    -0.2000 - 0.9798i
k =
    2

Algorithm

ss2zp finds the poles from the eigenvalues of the A array. The zeros are the finite solutions to a generalized eigenvalue problem:

z = eig([A B;C D], diag([ones(1,n) 0]);

In many situations this algorithm produces spurious large, but finite, zeros. ss2zp interprets these large zeros as infinite.

ss2zp finds the gains by solving for the first nonzero Markov parameters.

References

[1] Laub, A.J., and B.C. Moore, "Calculation of Transmission Zeros Using QZ Techniques," Automatica 14 (1978), p.557.