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

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

`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

$$H(s)-\frac{Z(s)}{P(s)}-k\frac{(s-{z}_{1})(s-{z}_{2})\cdots (s-{z}_{n})}{(s-{p}_{1})(s-{p}_{2})\cdots (s-{p}_{n})}$$

of the continuous-time system

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

from the `i`

th input (using the `i`

th
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.

`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.

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

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