arx - Estimate parameters of ARX or AR model using least squares

Syntax

Description

Note   arx does not support continuous-time estimations. Use tfest instead.

sys = arx(data,[na nb nk]) returns an ARX structure polynomial model, sys, with estimated parameters and covariances (parameter uncertainties) using the least-squares method and specified orders.

sys = arx(data,[na nb nk],Name,Value) estimates a polynomial model with additional options specified by one or more Name,Value pair arguments.

sys = arx(data,[na nb nk],___,opt) specifies estimation options that configure the estimation objective, initial conditions and handle input/output data offsets.

Input Arguments

Name-Value Pair Arguments

Output Arguments

Definitions

ARX structure

arx estimates the parameters of the ARX model structure:

The parameters na and nb are the orders of the ARX model, and nk is the delay.

• — Output at time .

• — Number of poles.

• — Number of zeroes plus 1.

• — Number of input samples that occur before the input affects the output, also called the dead time in the system.

• — Previous outputs on which the current output depends.

• — Previous and delayed inputs on which the current output depends.

• — White-noise disturbance value.

A more compact way to write the difference equation is

q is the delay operator. Specifically,

Time Series Models

For time-series data that contains no inputs, one output and orders = na, the model has AR structure of order na.

The AR model structure is

Multiple Inputs and Single-Output Models

For multiple-input systems, nb and nk are row vectors where the ith element corresponds to the order and delay associated with the ith input.

Multi-Output Models

For models with multiple inputs and multiple outputs, na, nb, and nk contain one row for each output signal.

In the multiple-output case, arx minimizes the trace of the prediction error covariance matrix, or the norm

To transform this to an arbitrary quadratic norm using a weighting matrix Lambda

use the syntax

opt = arxOptions('OutputWeight', inv(lambda))
m = arx(data, orders, opt)

Estimating Initial Conditions

For time-domain data, the signals are shifted such that unmeasured signals are never required in the predictors. Therefore, there is no need to estimate initial conditions.

For frequency-domain data, it might be necessary to adjust the data by initial conditions that support circular convolution.

Set the InitialCondition estimation option (see arxOptions) to one the following values:

• 'zero' — No adjustment.

• 'estimate' — Perform adjustment to the data by initial conditions that support circular convolution.

• 'auto' — Automatically choose between 'zero' and 'estimate' based on the data.

Examples

This example generates input data based on a specified ARX model, and then uses this data to estimate an ARX model.

A = [1  -1.5  0.7]; B = [0 1 0.5];
m0 = idpoly(A,B);
u = iddata([],idinput(300,'rbs'));
e = iddata([],randn(300,1));
y = sim(m0, [u e]);
z = [y,u];
m = arx(z,[2 2 1]);

Algorithms

QR factorization solves the overdetermined set of linear equations that constitutes the least-squares estimation problem.

The regression matrix is formed so that only measured quantities are used (no fill-out with zeros). When the regression matrix is larger than MaxSize, data is segmented and QR factorization is performed iteratively on these data segments.

See Also

ar | armax | arxOptions | arxstruc | bj | iv4 | n4sid | nlarx | oe

How To

• Using Linear Model for Nonlinear ARX Estimation

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