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# rarx

Estimate parameters of ARX or AR models recursively

## Description

thm = rarx(z,nn,adm,adg) estimates the parameters thm of single-output ARX model from input-output data z and model orders nn using the algorithm specified by adm and adg. If z is a time series y and nn = na, rarx estimates the parameters of a single-output AR model.

[thm,yhat,P,phi] = rarx(z,nn,adm,adg,th0,P0,phi0) estimates the parameters thm, the predicted output yhat, final values of the scaled covariance matrix of the parameters P, and final values of the data vector phi of single-output ARX model from input-output data z and model orders nn using the algorithm specified by adm and adg. If z is a time series y and nn = na, rarx estimates the parameters of a single-output AR model.

## Definitions

The general ARX model structure is:

$A\left(q\right)y\left(t\right)=B\left(q\right)u\left(t-nk\right)+e\left(t\right)$

The orders of the ARX model are:

Models with several inputs are defined, as follows:

A(q)y(t) = B1(q)u1(tnk1)+...+Bnuunu(tnknu)+e(t)

## Input Arguments

z

Name of the matrix iddata object that represents the input-output data or a matrix z = [y u], where y and u are column vectors.

For multiple-input models, the u matrix contains each input as a column vector:

```u = [u1 ... unu]
```
nn

For input-output models, specifies the structure of the ARX model as:

```nn = [na nb nk]
```

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

For multiple-input models, nb and nk are row vectors that define orders and delays for each input.

For time-series models, nn = na, where na is the order of the AR model.

 Note:   The delay nk must be larger than 0. If you want nk = 0, shift the input sequence appropriately and use nk = 1 (see nkshift).

adm = 'ff' and adg = lam specify the forgetting factor algorithm with the forgetting factor λ=lam. This algorithm is also known as recursive least squares (RLS). In this case, the matrix P has the following interpretation: R2/2 * P is approximately equal to the covariance matrix of the estimated parameters.R2 is the variance of the innovations (the true prediction errors e(t)).

adm ='ug' and adg = gam specify the unnormalized gradient algorithm with gain gamma = gam. This algorithm is also known as the normalized least mean squares (LMS).

adm ='ng' and adg = gam specify the normalized gradient or normalized least mean squares (NLMS) algorithm. In these cases, P is not applicable.

adm ='kf' and adg =R1 specify the Kalman filter based algorithm with R2=1 and R1 = R1. If the variance of the innovations e(t) is not unity but R2; then R2* P is the covariance matrix of the parameter estimates, while R1 = R1 /R2 is the covariance matrix of the parameter changes.

th0

Initial value of the parameters in a row vector, consistent with the rows of thm.

Default: All zeros.

P0

Initial values of the scaled covariance matrix of the parameters.

Default: 104 times the identity matrix.

phi0

The argument phi0 contains the initial values of the data vector:

φ(t) = [y(t–1),...,y(tna),u(t–1),...,u(tnbnk+1)]

If z = [y(1),u(1); ... ;y(N),u(N)], phi0 = φ(1) and phi = φ(N). For online use of rarx, use phi0, th0, and P0 as the previous outputs phi, thm (last row), and P.

Default: All zeros.

## Output Arguments

thm

Estimated parameters of the model. The kth row of thm contains the parameters associated with time k; that is, the estimate parameters are based on the data in rows up to and including row k in z. Each row of thm contains the estimated parameters in the following order:

```thm(k,:) = [a1,a2,...,ana,b1,...,bnb]
```

For a multiple-input model, the b are grouped by input. For example, the b parameters associated with the first input are listed first, and the b parameters associated with the second input are listed next.

yhat

Predicted value of the output, according to the current model; that is, row k of yhat contains the predicted value of y(k) based on all past data.

P

Final values of the scaled covariance matrix of the parameters.

phi

phi contains the final values of the data vector:

φ(t) = [y(t–1),...,y(tna),u(t–1),...,u(tnbnk+1)]

## Examples

Adaptive noise canceling: The signal y contains a component that originates from a known signal r. Remove this component by recursively estimating the system that relates r to y using a sixth-order FIR model and the NLMS algorithm.

```z = [y r];
[thm,noise] = rarx(z,[0 6 1],'ng',0.1);
% noise is the adaptive estimate of the noise
% component of y
plot(y-noise)
```

If this is an online application, you can plot the best estimate of the signal y - noise at the same time as the data y and u become available, use the following code:

```phi = zeros(6,1);
P=1000*eye(6);
th = zeros(1,6);
axis([0 100 -2 2]);
plot(0,0,'*'), hold on
% Use a while loop
while ~abort
[th,ns,P,phi] = rarx([y r],'ff',0.98,th,P,phi);
plot(time,y-ns,'*')
time = time + Dt
end
```

This example uses a forgetting factor algorithm with a forgetting factor of 0.98. readAD is a function that reads the value of an A/D converter at the indicated time instant.