Model Structures¶

pysib estimates discrete-time polynomial input/output models for SISO systems.

The model structure determines how the input signal, output dynamics, and noise dynamics are represented. Different estimators in pysib target different structures.

Polynomial convention¶

The polynomial arrays returned in model are stored in powers of the delay operator:

A(z) = 1 + a1 z^-1 + ... + a_na z^-na

The same convention is used for C, D, and F. The B polynomial contains the input delay through leading zeros in model["B"].

ARX¶

The ARX structure is:

       B(z)            1
y(t) = ---- u(t-nz) + ---- e(t)
       A(z)           A(z)

ARX uses the same polynomial A for the deterministic dynamics and the noise model. It is usually the simplest structure to estimate.

Available ARX estimators:

Function	Method
`pysib.arx`	Least squares
`pysib.iv`	Instrumental variables
`pysib.correlation`	Correlation error minimization

Use ARX as a simple first model or as an initial estimate for more detailed structures.

Output Error¶

The Output Error structure is:

       B(z)
y(t) = ---- u(t-nz) + e(t)
       F(z)

OE separates the deterministic input/output dynamics from the additive output error. It does not estimate a separate colored-noise model.

Available OE estimators:

Function	Method
`pysib.sm`	Stieglitz-McBride iteration
`pysib.oe`	Prediction-error method with C/LAPACK optimizer
`pysib.oe_filtered`	Filtered continuation variant

Use OE when the main goal is to estimate the deterministic transfer from input to output.

ARMAX¶

The ARMAX structure is:

       B(z)           C(z)
y(t) = ---- u(t-nz) + ---- e(t)
       A(z)           A(z)

ARMAX extends ARX with a moving-average noise polynomial C. The A polynomial still appears in both the deterministic and noise paths.

Available ARMAX estimators:

Function	Method
`pysib.armax`	Prediction-error method with C/LAPACK optimizer
`pysib.armax_filtered`	Filtered continuation variant

Use ARMAX when the noise dynamics matter but a shared A polynomial is acceptable.

Box-Jenkins¶

The Box-Jenkins structure is:

       B(z)           C(z)
y(t) = ---- u(t-nz) + ---- e(t)
       F(z)           D(z)

Box-Jenkins separates the deterministic dynamics from the noise dynamics. The deterministic path uses B/F, while the noise path uses C/D.

Available Box-Jenkins estimators:

Function	Method
`pysib.bj`	Prediction-error method with C/LAPACK optimizer
`pysib.bj_filtered`	Filtered continuation variant

Use Box-Jenkins when the process dynamics and noise dynamics should be modeled independently.

Returned model polynomials¶

Every estimator returns a model dictionary with the same keys, even when a polynomial is not used by a specific structure.

Structure	`A`	`B`	`C`	`D`	`F`
ARX	estimated	estimated	`[1]`	`[1]`	`[1]`
OE	`[1]`	estimated	`[1]`	`[1]`	estimated
ARMAX	estimated	estimated	estimated	`[1]`	`[1]`
Box-Jenkins	`[1]`	estimated	estimated	estimated	estimated

This shared convention allows predict and simulate to work with models from all estimators.