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Estimate input-output and time-series polynomial model coefficients

**Library:**System Identification Toolbox / Estimators

Use the Recursive Polynomial Model Estimator block to estimate discrete-time input-output polynomial and time-series models.

These model structures are:

AR —

*A*(*q*)*y*(*t*) =*e*(*t*)ARMA —

*A*(*q*)*y*(*t*) =*C*(*q*)*e*(*t*)ARX —

*A*(*q*)*y*(*t*) =*B*(*q*)*u*(*t*–*n*) +_{k}*e*(*t*)ARMAX —

*A*(*q*)*y*(*t*) =*B*(*q*)*u*(*t*–*n*) +_{k}*C*(*q*)*e*(*t*)OE — $$y(t)=\frac{B(q)}{F(q)}u(t-{n}_{k})+e(t)$$

BJ — $$y(t)=\frac{B(q)}{F(q)}u(t-{n}_{k})+\text{}\frac{C(q)}{D(q)}e(t)$$

*q* is the time-shift operator and *nk* is the input
delay. *u*(*t*) is the input,
*y*(*t*) is the output, and
*e*(*t*) is the error. For MISO models, there are as
many *B*(*q*) polynomials as the number of inputs.

The orders of these models correspond to the maximum number of time shifts, as
represented by the exponent of *q*. For instance, the order
*na* is represented in the *A*(*q*)
polynomial by:

1 +
*a _{1}*

An equivalent representation applies to the *C*(*q*),
*D*(*q*), and *F*(*q*)
polynomials and their corresponding orders *nc*, *nd*, and
*nf*.

The *B*(*q*) polynomial is unique with respect to the
others, because this polynomial operates on the input and contains the system zeros. For
*B*(*q*), the order *nb* is the order
of the polynomial *B*(*q*) + 1:

*b*_{1} +
*b _{2}*

The orders *na*, *nb*, *nc*,
*nd*, *nf*, and input delay *nk* are
known ahead of time. Specify these values as block parameters. Provide
*u*(*t*) and *y*(*t*)
through the **Inputs** and **Outputs** inports,
respectively. The block estimates the set of *A*(*q*),
*B*(*q*), *C*(*q*),
*D*(*q*), and *F*(*q*)
coefficients that the model structure uses and outputs them in the
**Parameters** outport. During the estimation, the block constrains the
estimated *C*, *D*, and *F* polynomials
to a stable region with roots in the unit disk, while allowing the estimated
*A* and *B* polynomials to be unstable. The
**Parameters** outport provides a bus signal with the following
elements:

A — Vector containing [1

*a*(_{1}*t*) ...*a*(_{na}*t*)].B — Vector containing [0

_{1}… 0_{nk},*b*(_{1}*t*) ...*b*(_{nb}*t*)]. For MISO data,*B*is a matrix where the*i*-th row parameters correspond to the*i*-th input.C — Vector containing [1

*c*(_{1}*t*) ...*c*(_{nc}*t*)].D — Vector containing [1

*d*(_{1}*t*) ...*d*(_{nd}*t*)].F — Vector containing [1

*f*(_{1}*t*) ...*f*(_{nf}*t*)].

For example, suppose that you want to estimate the coefficients for the following SISO ARMAX model:

*y*(*t*) +
*a _{1}*

*y*, *u*, *na*,
*nb*, *nc*, and *nk* are known quantities
that you provide to the block. For each time step, *t*, the block estimates
the *A*, *B*, and *C* parameter values,
constraining only the *C* polynomial to a stable region. The block then
outputs these estimated values using the **Parameters** outport.

The block supports several estimation methods and data input formats. The block can
provide both infinite-history [1] and finite-history [2] (also known as
sliding-window) estimates for *θ*. Configurable options in the block include:

Multiple inputs (ARX model structure only) — See the

**Inputs**port.Sample-based or frame-based data format — See the

**Input Processing**parameter.Multiple infinite-history estimation methods [1] — See the

**Estimation Method**parameter.Infinite-history (all model structures) or finite-history (AR, ARX, or OE model structures only) — See the

**History**parameter.Initial conditions, enable flag, and reset trigger — See the

**Initial Estimate**,**Add enable port**, and**External Reset**parameters.

For more information on the estimation methods, see Recursive Algorithms for Online Parameter Estimation.

`Inputs`

— Input signal vector | matrix

Input signal *u*(*t*). The **Input
Processing** parameter and the number of inputs *nu*
define the dimensions of the signal. Only the ARX model structure can have multiple
inputs, with *nu* greater than 1.

Sample-based input processing and

*nu*inputs —*nu*-by-1 vectorFrame-based input processing with

*M*samples per frame and*nu*inputs —*M*-by-*nu*matrix

To enable this port, set the **Model Structure** parameter to
`ARX`

, `ARMAX`

,
`BJ`

, or `OE`

.

**Data Types: **`single`

| `double`

`Output`

— Measured output signalscalar | vector

Measured output signal *y*(*t*). The
**Input Processing** parameter defines the dimensions of the signal:

Sample-based input processing — Scalar

Frame-based input processing with

*M*samples per frame —*M*-by-1 vector

**Data Types: **`single`

| `double`

`Enable`

— Enable estimation updates`true`

(default) | `false`

External signal that allows you to enable and disable estimation updates. If the signal value is:

`true`

— Estimate and output the parameter values for the time step.`false`

— Do not estimate the parameter values, and output the most recent previously estimated value.

To enable this port, select the **Add enable port**
parameter.

**Data Types: **`single`

| `double`

| `Boolean`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

`Reset`

— Reset triggerscalar

Reset parameter estimation to its initial conditions. The value of the
**External reset** parameter determines the trigger type. The
trigger type dictates whether the reset occurs on a signal that is rising, falling,
either rising or falling, level, or on level hold.

To enable this port, select any option other than
`None`

in the **External reset**
dropdown.

**Data Types: **`single`

| `double`

| `Boolean`

| `int8`

| `int16`

| `int32`

| `uint8`

| `uint16`

| `uint32`

`InitialParameters`

— Initial parameter estimates`bus`

objectInitial parameter estimates, supplied from a source external to the block. The
block uses this inport at the beginning of the simulation or when you trigger an
algorithm reset using the **Reset** signal.

For information on the contents of the **InitialParameters** bus
object, see the **Parameters** port description.

To enable this port, set **History** to
`Infinite`

and **Initial Estimate** to
`External`

.

**Data Types: **`single`

| `double`

`InitialCovariance`

— Initial covariance of parameterspositive scalar (default) | vector of positive scalars | symmetric positive-definite matrix

Initial parameter covariances, supplied from a source external to the block. For
details, see the **Parameter Covariance Matrix** parameter. The block
uses this inport at the beginning of the simulation or when you trigger an algorithm
reset using the **Reset** signal.

To enable this port, set the following parameters:

**History**to`Infinite`

**Estimation Method**to`Forgetting Factor`

or`Kalman Filter`

**Initial Estimate**to`External`

**Data Types: **`single`

| `double`

`InitialInputs`

— Initial values of the inputsmatrix

Initial set of inputs when using finite-history (sliding-window) estimation, supplied from an external source.

If

**Model Structure**is`ARX`

, then the signal to this port must be a (*W*–1+max(*nb*)+max(*nk*))-by-*nu*matrix, where*W*is the window length and*nu*is the number of inputs.*nb*is the vector of*B*(*q*) polynomial orders and*nk*is the vector of input delays.If

**Model Structure**is`OE`

, then the signal to this port must be a (*W*–1+*nb*+*nk*)-by-1 vector, where*W*is the window length.*nb*is the vector of*B*(*q*) polynomial orders and*nk*is the vector of input delays.

The block uses this inport at the beginning of the simulation or whenever the
**Reset** signal triggers.

If the initial buffer is set to `0`

or does not contain enough
information, you see a warning message during the initial phase of your estimation.
The warning should clear after a few cycles. The number of cycles it takes for
sufficient information to be buffered depends upon the order of your polynomials and
your input delays. If the warning persists, you should evaluate the content of your
signals.

To enable this port, set:

**History**to`Finite`

**Model Structure**to`ARX`

or`OE`

**Initial Estimate**to`External`

**Data Types: **`single`

| `double`

`InitialOutputs`

— Initial values of the measured outputs buffervector

Initial set of output measurements when using finite-history (sliding-window) estimation, supplied from an external source.

If

**Model Structure**is`AR`

or`ARX`

, then the signal to this port must be a (*W*+*na*)-by-1 vector, where*W*is the window length and*na*is the polynomial order of*A*(*q*).If

**Model Structure**is`OE`

, then the signal to this port must be a (*W*+*nf*)-by-1 vector, where*W*is the window length and*nf*is the polynomial order of*F*(*q*).

The block uses this inport at the beginning of the simulation or whenever the
**Reset** signal triggers.

If the initial buffer is set to `0`

or does not contain enough
information, you see a warning message during the initial phase of your estimation.
The warning should clear after a few cycles. The number of cycles it takes for
sufficient information to be buffered depends upon the order of your polynomials and
your input delays. If the warning persists, you should evaluate the content of your
signals.

To enable this port, set:

**History**to`Finite`

**Model Structure**to`AR`

,`ARX`

, or`OE`

**Initial Estimate**to`External`

**Data Types: **`single`

| `double`

`Parameters`

— Estimated parameters`bus`

objectEstimated polynomial coefficients, returned as a bus. The bus contains an element
for each of the *A*, *B*, *C*,
*D*, and *F* polynomials that correspond to the
structure that you specify in **Model Structure** (see Model Structures ).

Each bus element is a vector signal containing the associated polynomial
coefficients. For example, the *A* element contains [1
*a _{1}*(

Estimated *C*, *D*, and *F*
values are constrained to be stable discrete-time polynomials. That is, these
polynomials all have roots within the unit circle. Estimated *A* and
*B* polynomials are allowed to be unstable.

**Data Types: **`single`

| `double`

`Error`

— Estimation errorscalar | vector

Estimation error, returned as:

Scalar — Sample-based input processing

*M*-by-1 vector — Frame-based input processing with*M*samples per frame

To enable this port, select the **Output estimation error**
parameter.

**Data Types: **`single`

| `double`

`Covariance`

— Parameter estimation error covariance Pmatrix

Parameter estimation error covariance *P*, returned as an
*N*-by-*N* matrix, where *N* is
the number of parameters. For details, see the **Output Parameter Covariance
Matrix** parameter.

To enable this port:

Select the

**Output parameter covariance matrix**parameter.If

**History**is`Infinite`

, set**Estimation Method**to`Forgetting Factor`

or`Kalman Filter`

.

**Data Types: **`single`

| `double`

Estimated model structure, specified as one of the following:

`ARX`

— SISO or MISO ARX model`ARMAX`

— SISO ARMAX model`OE`

— SISO OE model`BJ`

— SISO BJ model`AR`

— Time-series AR model`ARMA`

— Time-series ARMA model

`Initial Estimate`

— Source of initial parameter estimates`None`

(default) | `Internal`

| `External`

Specify how to provide initial parameter estimates to the block:

`None`

— Do not specify initial estimates.The block uses

`0`

as the initial parameter estimate.Specify the parameters that the block enables based on your choice of model structure and estimation method.

Specify the set of

**Number of Parameters ()**parameters that the block enables based on your**Model Structure**. For instance, if your setting for**Model Structure**is`AR`

, specify the**Number of Parameters in A(q) (na)**parameter.Specify the

**Input Delay (nk)**parameter that the block enables when your model structure uses a*B*(*q*) element.Specify the

**Parameter Covariance Matrix**if**Estimation Method**is`Forgetting Factor`

or`Kalman Filter`

.

`Internal`

— Specify initial parameter estimates internally to the block.Specify the initial parameter values

**Initial ()**parameters that the block enables based on your**Model Structure**and**History**. For instance, if your setting for**Model Structure**is`AR`

and**History**is`Infinite`

, specify the**Initial A(q)**parameter.Specify the

**Input Delay (nk)**parameter that the block enables when your model structure uses a*B*(*q*) element.Specify the

**Parameter Covariance Matrix**parameter if**Estimation Method**is`Forgetting Factor`

or`Kalman Filter`

.Specify the

**Initial Inputs**parameter (`ARX`

and`OE`

only) and the**Initial Outputs**parameter (`ARX`

,`AR`

, and`OE`

) if**History**is`Finite`

.

`External`

— Specify initial parameter estimates as an input signal to the block.Specify the

**Number of Parameters ()**parameters that the block enables based on your**Model Structure**. Your setting for**Model Structure**and for the**History**parameter determines which signals to connect to the relevant ports:If

**History**is`Infinite`

—**InitialParameters**and**InitialCovariance**If

**History**is`Finite`

—**InitialOutputs**for the`AR`

,`ARX`

, and`OE`

model structures, and**InitialInputs**for the`ARX`

and`OE`

model structures

Block Parameter:
`InitialEstimateSource` |

Type: character vector, string |

Values: `'None'` ,
`'Internal'` , `'External'` |

Default: `'None'` |

`Number of Parameters in A(q) (na)`

— Number of estimated parameters in the `1`

(default) | non-negative integerSpecify the number of estimated parameters *na* in the
*A*(*q*) polynomial.

To enable this parameter, either:

Set

**History**to`Infinite`

,**Model Structure**to`AR`

,`ARX`

,`ARMA`

, or`ARMAX`

, and**Initial Estimate**to`None`

or`External`

.Set

**History**to`Finite`

,**Model Structure**to`AR`

or`ARX`

, and**Initial Estimate**to`None`

or`External`

.

Block Parameter:
`A0` |

Type: non-negative integer |

Default: `1` |

`Number of Parameters in B(q) (nb)`

— Number of estimated parameters in the `1`

(default) | vector of positive integersSpecify the number of estimated parameters *nb* in the
*B*(*q*) polynomial.

For MISO systems using an ARX model structure, specify *nb* as a
vector with as many elements as there are inputs. Each element of this vector
represents the order of the *B*(*q*) polynomial
associated with the corresponding input. For example, suppose that you have a
two-input MISO system whose *B*(*q*) elements are: $$\left[\begin{array}{l}{B}_{1}\\ {B}_{2}\end{array}\right]=\left[\begin{array}{ccc}0& {b}_{11}& 0\\ 0& {b}_{21}& {b}_{22}\end{array}\right]$$. The zero at the beginning of each polynomial represents a single
input delay for each input (see the **Initial B(q)** parameter
description). The trailing zero in *B _{1}* is for
equalizing the length of the polynomials and has no impact on estimation.

`[1 2]`

,
and `[1 1]`

. To enable this parameter, either:

Set

**History**to`Infinite`

,**Model Structure**to`ARX`

,`ARMAX`

,`BJ`

, or`OE`

, and**Initial Estimate**to`None`

or`External`

.Set

**History**to`Finite`

with**Model Structure**of`ARX`

or`OE`

and**Initial Estimate**to`None`

or`External`

.

Block Parameter:
`B0` |

Type: positive integer |

Default: `1` |

`Number of Parameters in C(q) (nc)`

— Number of estimated parameters in the `1`

(default) | positive integerSpecify the number of estimated parameters *nc* in the
*C*(*q*) polynomial.

To enable this parameter, set **History** to
`Infinite`

, **Model Structure** to
`ARMA`

, `ARMAX`

, or
`BJ`

, and **Initial Estimate** to
`None`

or `External`

Block Parameter:
`C0` |

Type: positive integer |

Default: `1` |

`Number of Parameters in D(q) (nd)`

— Number of estimated parameters in the `1`

(default) | positive integerSpecify the number of estimated parameters *nd* in the
*D*(*q*) polynomial.

To enable this parameter, set **History** to
`Infinite`

, **Model Structure** to
`BJ`

, and **Initial Estimate** to
`None`

or `External`

.

Block Parameter:
`D0` |

Type: positive integer |

Default: `1` |

`Number of Parameters in F(q) (nf)`

— Number of estimated parameters in the `1`

(default) | positive integerSpecify the number of estimated parameters *nf* in the
*F*(*q*) polynomial.

To enable this parameter, set **Initial Estimate** to
`None`

or `External`

and
either:

**History**to`Infinite`

,**Model Structure**to`OE`

or`BJ`

, and**Initial Estimate**to`None`

or`External`

**History**to`Finite`

,**Model Structure**to`OE`

, and**Initial Estimate**to`None`

or`External`

.

Block Parameter:
`F0` |

Type: positive integer |

Default: `1` |

`Input Delay (nk)`

— Input delay`1`

(default) | vector of non-negative integersSpecify the input delay as an integer representing the number of time steps that
occur before the input affects the output. This delay is also called the
*dead time* in the system. The block encodes the input delay as
fixed leading zeros of the *B*(*q*) polynomial. For
more information, see the *B*(*q*) parameter
description.

For MISO systems with ARX model structure, specify *nk* as a
vector with elements specifying the delay for each input. This vector is of length
*nu*, where *nu* is the number of inputs.

For example, suppose that you have a two-input MISO system whose
*B*(*q*) elements are: $$\left[\begin{array}{l}{B}_{1}\\ {B}_{2}\end{array}\right]=\left[\begin{array}{ccc}0& {b}_{11}& {b}_{12}\\ 0& 0& {b}_{21}\end{array}\right]$$. The zeros at the beginning of the polynomials represent a
single-sample delay for the first input, and a two-sample delay for the second input
(see the **Initial B(q)** parameter description).
*nb* for each polynomial is equal to the number of estimated
parameters following the initial zeros, or 2 for input 1 and 1 for input 2. Specify
**Input Delay (nk)** as `[1 2]`

, and
**Number of Parameters in B(q) (nb)** as ```
[2
1]
```

.

To enable this parameter, either:

Set

**History**to`Infinite`

,**Model Structure**to`ARX`

,`ARMAX`

,`OE`

, or`BJ`

, and**Initial Estimate**to`None`

or`External`

.Set

**History**to`Finite`

,**Model Structure**to`ARX`

or`OE`

, and**Initial Estimate**to`None`

or`External`

.

Block Parameter:
`nk` |

Type: non-negative integer vector |

Default: `1` |

`Parameter Covariance Matrix`

— Initial parameter covariance`1e4`

(default) | scalar | vector | matrixReal positive scalar,

*α*— Covariance matrix is an*N*-by-*N*diagonal matrix, with α as the diagonal elements.Vector of real positive scalars, [

*α*(*a*),*α*(*b*),*α*(*c*),*α*(*d*),*α*(*f*)] — Covariance matrix is an*N*-by-*N*diagonal matrix, with [*α*(*a*),*α*(*b*),*α*(*c*),*α*(*d*),*α*(*f*)] as the diagonal elements.*α*(*a*) is a vector of the covariance for each coefficient of the*A*polynomial. Similarly,*α*(*b*),*α*(*c*),*α*(*d*) and*α*(*f*) are vectors containing the covariance of the coefficients of the*B*,*C*,*D*and*F*polynomials, respectively.*N*-by-*N*symmetric positive-definite matrix.*N*can be one of the following:`AR`

—*N*=*na*`ARX`

— $$\begin{array}{l}N\text{}=na+{\displaystyle \sum _{i=1}^{{N}_{u}}n}{b}_{i}\\ \end{array}$$`ARMA`

—*N*=*na*+*nc*`ARMAX`

—*N*=*na*+*nb*+*nc*`OE`

—*N*=*nb*+*nf*`BJ`

—*N*=*nb*+*nc*+*nd*+*nf*

To enable this parameter, set

**History**to`Infinite`

**Initial Estimate**to`None`

or`Internal`

**Estimation Method**to`Forgetting Factor`

or`Kalman Filter`

The block uses this parameter at the beginning of the simulation or whenever the
**Reset** signal triggers.

Block Parameter:
`P0` |

Type: scalar, vector, or matrix |

Default: `1e4` |

`Initial A(q)`

— Initial values of the `[1 eps]`

(default) | vectorSpecify the initial estimate of the *A*(*q*)
polynomial coefficients as a row vector of length *na*+1.

The leading coefficient of *A* must be
`1`

.

To enable this parameter, set:

**Model Structure**to`AR`

,`ARX`

,`ARMA`

, or`ARMAX`

**Initial Estimate**to`Internal`

Block Parameter:
`A0` |

Type: real vector |

Default: ```
[1
eps]
``` |

`Initial B(q)`

— Initial values of the `[0 eps]`

(default) | vector | matrixSpecify the initial estimate of the *B*(*q*)
polynomial coefficients as a row vector of length
*nb*+*nk*. For multiple-input models, specify a
matrix where the *i*th row corresponds to the *i*th
input.

The block counts the leading zeros in *B*(*q*)
and interprets them as input delay *nk*. Those zeros are fixed
throughout the estimation. *nb* is the number of elements after the
first nonzero element in *B*(*q*). The block
estimates the value of these *nb* elements.

For example:

`[0 eps]`

corresponds to*nk*=1 and*nb*=1.`[0 0 eps]`

corresponds to*nk*=2 and*nb*=1.`[0 0 eps 0 eps]`

corresponds to*nk*=2 and*nb*=3.

The default value is `[0 eps]`

.

To enable this parameter, set:

**Model Structure**to`ARX`

,`ARMAX`

,`OE`

, or`BJ`

**Initial Estimate**to`Internal`

Block Parameter:
`B0` |

Type: real vector or matrix |

Default: ```
[0
eps]
``` |

`Initial C(q)`

— Initial values of the `[1 eps]`

(default) | vectorSpecify the initial estimate of the *C*(*q*)
polynomial coefficients as a row vector of length *nc*+1.

The leading coefficient of *C*(*q*) must be
1.

The coefficients must define a stable discrete-time polynomial, that is, have all polynomial roots within the unit circle.

To enable this parameter, set:

**History**to`Infinite`

**Model Structure**to`ARMA`

,`ARMAX`

, or`BJ`

**Initial Estimate**to`Internal`

Block Parameter:
`C0` |

Type: real vector |

Default: ```
[1
eps]
``` |

`Initial D(q)`

— Initial values of the `[1 eps]`

(default) | vectorSpecify the initial estimate of the *D*(*q*)
polynomial coefficients as a row vector of length *nd*+1.

The leading coefficient of *D*(*q*) must be
1.

The coefficients must define a stable discrete-time polynomial, that is, have all polynomial roots within the unit circle.

To enable this parameter, set:

**History**to`Infinite`

**Model Structure**to`BJ`

**Initial Estimate**to`Internal`

Block Parameter:
`D0` |

Type: real vector |

Default: ```
[1
eps]
``` |

`Initial F(q)`

— Initial values of the `[1 eps]`

(default) | vectorSpecify the initial estimate of the *F*(*q*)
polynomial coefficients as a row vector of length *nf*+1.

The leading coefficient of *F*(*q*) must be
1.

The coefficients must define a stable discrete-time polynomial, that is, have all polynomial roots within the unit circle.

To enable this parameter, set:

**Model Structure**to`OE`

or to`BJ`

**Initial Estimate**to`Internal`

Block Parameter:
`F0` |

Type: real vector |

Default: ```
[1
eps]
``` |

`Initial Inputs`

— Initial values of the inputs buffer`0`

(default) | matrixSpecify initial values of the inputs buffer when using finite-history (sliding window) estimation. The buffer dimensions accommodate the specified window length, the regressors associated with polynomials within that window, the input delays, and the number of inputs. These elements drive a matrix size of:

ARX model structure — (

*W*–1+max(*nb*)+max(*nk*))-by-*nu*OE model structure — (

*W*–1+*nb*+*nk*)-by-1

where *W* is the window length and
*nu* is the number of inputs. *nb* is the vector
of *B*(*q*) polynomial orders and
*nk* is the vector of input delays.

When the initial value is set to `0`

, the block populates the
buffer with zeros.

If the initial buffer is set to `0`

or does not contain enough
information, you see a warning message during the initial phase of your estimation.
The warning should clear after a few cycles. The number of cycles it takes for
sufficient information to be buffered depends upon the order of your polynomials and
your input delays. If the warning persists, you should evaluate the content of your
signals.

The block uses this parameter at the beginning of the simulation or whenever the
**Reset** signal triggers.

To enable this parameter, set

**History**to`Finite`

**Model Structure**to`ARX`

or`OE`

**Initial Estimate**to`Internal`

.

Block Parameter:
`InitialInputs` |

Type: real matrix |

Default: `0` |

`Initial Outputs`

— Initial values of the measured outputs buffer`0`

(default) | vectorSpecify initial values of the measured outputs buffer when using finite-history (sliding-window) estimation. The buffer dimensions accommodate the specified window length and the regressors associated with the polynomials within that window.

AR or ARX model structure — (

*W*+*na*)-by-1 vector, where*W*is the window length and*na*is the polynomial order of*A*(*q*).OE model structure — (

*W*+*nf*)-by-1 vector, where*W*is the window length and*nf*is the polynomial order of*F*(*q*).

When the initial value is set to `0`

, the block populates the
buffer with zeros.

`0`

or does not contain enough
information, you see a warning message during the initial phase of your estimation.
The warning should clear after a few cycles. The number of cycles it takes for
sufficient information to be buffered depends upon the order of your polynomials and
your input delays. If the warning persists, you should evaluate the content of your
signals.

The block uses this parameter at the beginning of the simulation or whenever the
**Reset** signal triggers.

To enable this parameter, set:

**History**to`Finite`

**Model Structure**to`AR`

,`ARX`

, or`OE`

**Initial Estimate**to`Internal`

Block Parameter:
`InitialOutputs` |

Type: real vector |

Default: `0` |

`Input Processing`

— Choose sample-based or frame-based processing`Sample-based`

(default) | `Frame-based`

`Sample-based`

processing operates on signals streamed one sample at a time.`Frame-based`

processing operates on signals containing samples from multiple time steps. Many machine sensor interfaces package multiple samples and transmit these samples together in frames.`Frame-based`

processing allows you to input this data directly without having to first unpack it.

Specifying frame-based data adds an extra dimension of *M* to
some of your data inports and outports, where *M* is the number of
time steps in a frame. These ports are:

**Inputs****Output****Error**

For more information, see the port descriptions in Ports.

Block Parameter:
`InputProcessing` |

Type: character vector, string |

Values: `'Sample-based'` ,
`'Frame-based'` |

Default:
`'Sample-based'` |

`Sample Time`

— Block sample time `–1`

(default) | positive scalarSpecify the data sample time, whether by individual samples for sample-based
processing (*t _{s}*), or by frames for
frame-based processing (

Specify **Sample Time** as a positive scalar to override the
inheritance.

Block Parameter:
`Ts` |

Type: real scalar |

Values: `–1` , positive
scalar |

Default: `–1` |

`History`

— Choose infinite or finite data history `Infinite`

(default) | `Finite`

The **History** parameter determines what type of recursive
algorithm you use:

`Infinite`

— Algorithms in this category aim to produce parameter estimates that explain all data since the start of the simulation. These algorithms retain the history in a data summary. The block maintains this summary within a fixed amount of memory that does not grow over time.The block provides multiple algorithms of the

`Infinite`

type. Selecting this option enables the**Estimation Method**parameter with which you specify the algorithm.`Finite`

— Algorithms in this category aim to produce parameter estimates that explain only a finite number of past data samples. The block uses all of the data within a finite window, and discards data once that data is no longer within the window bounds. This method is also called sliding-window estimation.The block provides one algorithm of the

`Finite`

type. You can use this option only with the`AR`

,`ARX`

, and`OE`

model structures.Selecting this option enables the

**Window Length**parameter.

For more information on recursive estimation methods, see Recursive Algorithms for Online Parameter Estimation

Block Parameter:
`History` |

Type: character vector, string |

Values: `'Infinite'` ,
`'Finite'` |

Default:
`'Infinite'` |

`Window Length`

— Window size for finite sliding-window estimation`200`

(default) | positive integerThe **Window Length** parameter determines the number of time
samples to use for the finite-history (sliding-window) estimation method. Choose a
window size that balances estimation performance with computational and memory burden.
Sizing factors include the number and time variance of the parameters in your model.
Always specify **Window Length** in samples, even if you are using
frame-based input processing.

**Window Length** must be greater than or equal to the number of
estimated parameters.

Suitable window length is independent of whether you are using sample-based or
frame-based input processing. However, when using frame-based processing,
**Window Length** must be greater than or equal to the number of
samples (time steps) contained in the frame.

To enable this parameter, set **History** to
`Finite`

.

Block Parameter:
`WindowLength` |

Type: positive integer |

Default: `200` |

`Estimation Method`

— Recursive estimation algorithm`Forgetting Factor`

(default) | `Kalman Filter`

| `Normalized Gradient`

| `Gradient`

Specify the estimation algorithm when performing infinite-history estimation. When you select any of these methods, the block enables additional related parameters.

Forgetting factor and Kalman filter algorithms are more computationally intensive than gradient and normalized gradient methods. However, these more intensive methods have better convergence properties than the gradient methods. For more information about these algorithms, see Recursive Algorithms for Online Parameter Estimation.

Block Parameter:
`EstimationMethod` |

Type: character vector, string |

Values: ```
'Forgetting
Factor'
``` ,`'Kalman Filter'` ,```
'Normalized
Gradient'
``` ,`'Gradient'` |

Default: ```
'Forgetting
Factor'
``` |

`Forgetting Factor`

— Discount old data using forgetting factor`1`

(default) | positive scalar in (0 1] rangeThe forgetting factor *λ* specifies if and how much old data is
discounted in the estimation. Suppose that the system remains approximately constant
over *T _{0}* samples. You can choose

$${T}_{0}=\frac{1}{1-\lambda}$$

Setting

*λ*= 1 corresponds to “no forgetting” and estimating constant coefficients.Setting

*λ*< 1 implies that past measurements are less significant for parameter estimation and can be “forgotten.” Set*λ*< 1 to estimate time-varying coefficients.

Typical choices of *λ* are in the [0.98 0.995]
range.

To enable this parameter, set **History** to
`Infinite`

and **Estimation Method** to
`Forgetting Factor`

.

Block Parameter:
`AdaptationParameter` |

Type: scalar |

Values: (0 1] range |

Default: `1` |

`Process Noise Covariance`

— Process noise covariance for Kalman filter estimation method`1`

(default) | nonnegative scalar | vector of nonnegative scalars | symmetric positive semidefinite matrix**Process Noise Covariance** prescribes the elements and
structure of the noise covariance matrix for the Kalman filter estimation. Using
*N* as the number of parameters to estimate, specify the
**Process Noise Covariance** as one of the following:

Real nonnegative scalar,

*α*— Covariance matrix is an*N*-by-*N*diagonal matrix, with*α*as the diagonal elements.Vector of real nonnegative scalars, [

*α*_{1},...,*α*_{N}] — Covariance matrix is an*N*-by-*N*diagonal matrix, with [*α*_{1},...,*α*_{N}] as the diagonal elements.*N*-by-*N*symmetric positive semidefinite matrix.

The Kalman filter algorithm treats the parameters as states of a dynamic system
and estimates these parameters using a Kalman filter. **Process Noise
Covariance** is the covariance of the process noise acting on these
parameters. Zero values in the noise covariance matrix correspond to constant
coefficients, or parameters. Values larger than 0 correspond to time-varying
parameters. Use large values for rapidly changing parameters. However, expect the
larger values to result in noisier parameter estimates. The default value is 1.

To enable this parameter, set **History** to
`Infinite`

and **Estimation Method** to
`Kalman Filter`

.

Block Parameter:
`AdaptationParameter` |

Type: scalar, vector, matrix |

Default: `1` |

`Adaptation Gain`

— Adaptation gain specification for gradient estimation methods`1`

(default) | positive scalarThe adaptation gain *γ* scales the influence of new measurement
data on the estimation results for the gradient and normalized gradient methods. When
your measurements are trustworthy, or in other words have a high signal-to-noise
ratio, specify a larger value for *γ*. However, setting
*γ* too high can cause the parameter estimates to diverge. This
divergence is possible even if the measurements are noise free.

When **Estimation Method** is
`NormalizedGradient`

, **Adaptation Gain**
should be less than 2. With either gradient method, if errors are growing in time (in
other words, estimation is diverging), or parameter estimates are jumping around
frequently, consider reducing **Adaptation Gain**.

To enable this parameter, set **History** to
`Infinite`

and **Estimation Method** to
`Normalized Gradient`

or to
`Gradient`

.

Block Parameter:
`AdaptationParameter` |

Type: scalar |

Default: `1` |

`Normalization Bias`

— Bias for adaptation gain scaling for normalized gradient estimation method`eps`

(default) | nonnegative scalar The normalized gradient algorithm scales the adaptation gain at each step by the
square of the two-norm of the gradient vector. If the gradient is close to zero, the
near-zero denominator can cause jumps in the estimated parameters.
**Normalization Bias** is the term introduced to the denominator to
prevent these jumps. Increase **Normalization Bias** if you observe
jumps in estimated parameters.

To enable this parameter, set **History** to
`Infinite`

and **Estimation Method** to
`Normalized Gradient`

.

Block Parameter:
`NormalizationBias` |

Type: scalar |

Default: `eps` |

`Output estimation error`

— Add Error outport to block`off`

(default) | onUse the **Error** outport signal to validate the estimation. For
a given time step *t*, the estimation error
*e*(*t*) is calculated as:

$$e(t)=y(t)-{y}_{est}(t),$$

where *y*(*t*) is the measured output that you
provide, and *y _{est}*(

Block Parameter:
`OutputError` |

Type: character vector, string |

Values:
`'off'` ,`'on'` , |

Default: `'off'` |

`Output parameter covariance matrix`

— Add covariance outport to block`off`

(default) | onUse the **Covariance** outport signal to examine parameter
estimation uncertainty. The software computes parameter covariance
`P`

assuming that the residuals,
*e*(*t*), are white noise, and the variance of
these residuals is 1.

The interpretation of `P`

depends on the estimation approach you
specify in **History** and **Estimation Method** as follows:

If

**History**is`Infinite`

, then your**Estimation Method**selection results in:`Forgetting Factor`

— (*R*/2)_{2}`P`

is approximately equal to the covariance matrix of the estimated parameters, where*R*is the true variance of the residuals. The block returns these residuals through the_{2}**Error**port.`Kalman Filter`

—*R*_{2}`P`

is the covariance matrix of the estimated parameters, and*R*/_{1}*R*is the covariance matrix of the parameter changes. Here,_{2}*R*is the covariance matrix that you specify in_{1}**Parameter Covariance Matrix**.`Normalized Gradient`

or`Gradient`

— Covariance`P`

is not available.

If

**History**is`Finite`

(sliding-window estimation) —*R*_{2}`P`

is the covariance of the estimated parameters. The sliding-window algorithm does not use this covariance in the parameter-estimation process. However, the algorithm does compute the covariance for output so that you can use it for statistical evaluation.

Block Parameter:
`OutputP` |

Type: character vector, string |

Values:
`'off'` ,`'on'` |

Default: `'off'` |

`Add enable port`

— Add Enable inport to block`off`

(default) | onUse the **Enable** signal to provide a control signal that
enables or disables parameter estimation. The block estimates the parameter values for
each time step that parameter estimation is enabled. If you disable parameter
estimation at a given step, *t*, then the software does not update
the parameters for that time step. Instead, the block output contains the last
estimated parameter values.

You can use this option, for example, when or if:

Your regressors or output signal become too noisy, or do not contain information at some time steps

Your system enters a mode where the parameter values do not change in time

Block Parameter:
`AddEnablePort` |

Type: character vector, string |

Values:
`'off'` ,`'on'` |

Default: `'off'` |

`External reset`

— Specify trigger for external reset`None`

(default) | `Rising`

| `Falling`

| `Either`

| `Level`

| `Level hold`

Set the **External reset** parameter to both add a
**Reset** inport and specify the inport signal condition that
triggers a reset of algorithm states to their specified initial values. Reset the
estimation, for example, if parameter covariance is becoming too large because of lack
of either sufficient excitation or information in the measured signals. The
**External reset** parameter determines the timing for the reset.

Suppose that you reset the block at a time step, *t*. If the
block is enabled at *t*, the software uses the initial parameter
values specified in **Initial Estimate** to estimate the parameter
values. In other words, at *t*, the block performs a parameter update
using the initial estimate and the current values of the inports.

If the block is disabled at *t* and you reset the block, the
block output contains the values specified in **Initial
Estimate**.

Specify this option as one of the following:

`None`

— Algorithm states and estimated parameters are not reset.`Rising`

— Trigger reset when the control signal rises from a negative or zero value to a positive value. If the initial value is negative, rising to zero triggers reset.`Falling`

— Trigger reset when the control signal falls from a positive or a zero value to a negative value. If the initial value is positive, falling to zero triggers reset.`Either`

— Trigger reset when the control signal is either rising or falling.`Level`

— Trigger reset in either of these cases:Control signal is nonzero at the current time step.

Control signal changes from nonzero at the previous time step to zero at the current time step.

`Level hold`

— Trigger reset when the control signal is nonzero at the current time step.

When you choose any option other than `None`

, the
software adds a Reset inport to the block. You provide the reset control input signal
to this inport.

Block Parameter:
`ExternalReset` |

Type: character vector, string |

Values:
`'None'` ,`'Rising'` ,`'Falling'` ,
`'Either'` , `'Level'` , ```
'Level
hold'
``` |

Default: `'None'` |

[1] Ljung, L. *System
Identification: Theory for the User*. Upper Saddle River, NJ: Prentice-Hall PTR,
1999, pp. 363–369.

[2] Zhang, Q. "Some Implementation
Aspects of Sliding Window Least Squares Algorithms." *IFAC Proceedings*.
Vol. 33, Issue 15, 2000, pp. 763–768.

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Recursive Least Squares Estimator | Kalman Filter

- Estimate Parameters of System Using Simulink Recursive Estimator Block
- Online Recursive Least Squares Estimation
- Preprocess Online Parameter Estimation Data in Simulink
- Validate Online Parameter Estimation Results in Simulink
- Generate Online Parameter Estimation Code in Simulink
- Recursive Algorithms for Online Parameter Estimation

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