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# pole

Compute poles of dynamic system

pole(sys)

## Description

pole(sys) computes the poles p of the SISO or MIMO dynamic system model sys.

If sys has internal delays, poles are obtained by first setting all internal delays to zero (creating a zero-order Padé approximation) so that the system has a finite number of zeros. For some systems, setting delays to 0 creates singular algebraic loops, which result in either improper or ill-defined, zero-delay approximations. For these systems, pole returns an error. This error does not imply a problem with the model sys itself.

## Limitations

Multiple poles are numerically sensitive and cannot be computed to high accuracy. A pole λ with multiplicity m typically gives rise to a cluster of computed poles distributed on a circle with center λ and radius of order

$\rho \approx {\epsilon }^{1/m}$

where ε is the relative machine precision (eps).

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### Algorithms

For state-space models, the poles are the eigenvalues of the A matrix, or the generalized eigenvalues of AλE in the descriptor case.

For SISO transfer functions or zero-pole-gain models, the poles are simply the denominator roots (see roots).

For MIMO transfer functions (or zero-pole-gain models), the poles are computed as the union of the poles for each SISO entry. If some columns or rows have a common denominator, the roots of this denominator are counted only once.