Gilbert Realization

pathsim.utils.gilbert.gilbert_realization(Poles=[], Residues=[], Const=0.0, tolerance=1e-09)[source]

Build real valued statespace model from transfer function in pole residue form by Gilbert’s method and an additional similarity transformation to get fully real valued matrices.

pole residue form:

\[\mathbf{H}(s) = \mathmf{D} + \sum_{n=1}^N \frac{\mathbf{R}_n}{s - p_n} )\]

statespace form:

\[\mathbf{H}(s) = \mathbf{C} (s \mathbf{I} - \mathbf{A})^{-1} * \mathbf{B} + \mathbf{H}\]

Notes

The resulting system is identical to the so-called ‘Modal Form’ and is a minimal realization.

Parameters:

  • Poles (array) – real and complex poles

  • Residues (array) – array of real and complex residue matrices

  • Const (array) – matrix for constant term

  • tolerance (float) – relative tolerance for checking real poles

Returns:

  • A (array) – state matrix

  • B (array) – input mapping matrix

  • C (array) – state to output projection matrix

  • D (array, float) – direct passthrough

Note

If some poles are complex-valued, their conjugate-values are automatically added if missing, to enforce the model realness and stability.