Basic Concepts¶

This page summarizes the data and model conventions used by pyvrft.

Signals¶

Input and output signals are represented as NumPy matrices shaped (N, n):

N is the number of samples.
n is the number of inputs or outputs.

For a SISO system, use one-column arrays shaped (N, 1) rather than one-dimensional vectors.

Transfer functions¶

pyvrft uses SciPy discrete-time transfer functions, usually created with dt=1:

from scipy import signal

G = signal.TransferFunction([1], [1, -0.9], dt=1)

When a numerator is one, use [1] rather than [0, 1] to avoid SciPy warnings.

SISO systems¶

For SISO systems, some APIs accept a transfer function directly. Internally, pyvrft converts it to the same nested-list format used for MIMO systems.

Td = signal.TransferFunction([0.2], [1, -0.8], dt=1)
L = signal.TransferFunction([0.25], [1, -0.75], dt=1)

MIMO transfer matrices¶

MIMO transfer functions are represented as nested Python lists. The first index is the output and the second index is the input.

G = [
    [G11, G12],
    [G21, G22],
]

Zero entries are written as the literal value 0:

Td = [
    [Td11, 0],
    [0, Td22],
]

Controller structure¶

The controller structure C specifies the basis transfer functions used to estimate the controller parameters.

Each C[i][j] entry is a list of transfer functions for the subcontroller from output j to input i.

Cpi = [
    [signal.TransferFunction([1, 0], [1, -1], dt=1)],
    [signal.TransferFunction([1], [1, -1], dt=1)],
]

C = [
    [Cpi, Cpi],
    [Cpi, Cpi],
]

Use an empty list [] when a subcontroller is absent.

The parameter vector returned by vrft.design is stacked by controller block, following the order documented in vrft/control.py.

Instrumental variables¶

The third argument of vrft.design is y_iv, the output data used for instrumental variables.

Use the same output data for least-squares VRFT:

p = vrft.design(u, y, y, Td, C, L)

Use a second output data set for instrumental-variable VRFT:

p = vrft.design(u, ya, yb, Td, C, L)

VRFT design objects¶

The main design call is:

p = vrft.design(u, y, y_iv, Td, C, L)

where:

Td is the desired closed-loop reference model.
C is the controller structure.
L is the VRFT pre-filter.