lti_freq_io¶

MATLAB equivalent: sidLTIfreqIO

lti_freq_io ¶

LTI state-space realization from input-output data via Ho-Kalman.

lti_freq_io ¶

lti_freq_io(Y: ndarray | list, U: ndarray | list, H: ndarray, *, horizon: int | None = None, max_stabilize: float = 0.999) -> tuple[ndarray, ndarray]

Identify an LTI state-space model from input-output data.

Estimates a constant (LTI) state-space realization from input-output data, given a known observation matrix H:

.. math::

x(k+1) = A_0\,x(k) + B_0\,u(k)

y(k)   = H\,x(k)

Uses the Ho-Kalman realization algorithm applied to the frequency response estimated via :func:sid.freq_bt. The realization is transformed to the H-basis so that the observation equation y = H x holds exactly with the returned (A0, B0).

This is the Python port of sidLTIfreqIO.m.

Parameters:

Name	Type	Description	Default
`Y`	`ndarray or list of ndarray`	Output data. Accepted formats: `(N+1, py)` -- single trajectory `(N+1, py, L)` -- L trajectories, same horizon N `[Y_1, Y_2, ...]` -- list of arrays `(N_l+1, py)`. Trajectories shorter than `ceil(2/3 * max_horizon)` are discarded; the rest are trimmed to a common length.	required
`U`	`ndarray or list of ndarray`	Input data, matching format of Y: `(N, q)` -- single trajectory `(N, q, L)` -- L trajectories `[U_1, U_2, ...]` -- list of arrays `(N_l, q)`	required
`H`	ndarray, shape ``(py, n)``	Observation matrix.	required
`horizon`	`int or None`	Hankel matrix depth r. Default is `min(N_imp // 3, 50)` where N_imp is the number of impulse response coefficients.	`None`
`max_stabilize`	`float`	Maximum eigenvalue magnitude after stabilization. Unstable eigenvalues are reflected inside the unit circle, then clamped to this radius. Default is `0.999`.	`0.999`

Returns:

Name	Type	Description
`A0`	ndarray, shape ``(n, n)``	Estimated LTI dynamics matrix.
`B0`	ndarray, shape ``(n, q)``	Estimated LTI input matrix.

Raises:

Type	Description
`SidError`	If data is too short for Hankel construction (code: `'too_short'`).
`SidError`	If data dimensions are inconsistent (code: `'dim_mismatch'`).

Examples:

>>> import numpy as np
>>> import sid
>>> H = np.array([[1, 0]])
>>> A0, B0 = sid.lti_freq_io(Y, U, H)

Notes

Algorithm:

Estimate transfer function G(e^{jw}) via :func:sid.freq_bt.
Compute Markov parameters g(k) = H A^{k-1} B via IFFT.
Build block Hankel matrices H_0 and H_1 (shifted).
SVD of H_0, truncate to order n (Ho-Kalman realization).
Transform realization to H-basis: find T s.t. C_r T^{-1} = H.
Stabilize eigenvalues if needed.

Specification: SPEC.md section 8.12 -- Output-COSMIC (LTI initialization)

References

.. [1] Ho, B.L. and Kalman, R.E. "Effective construction of linear state-variable models from input/output functions." Regelungstechnik, 14(12):545-548, 1966.