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# Recursive Polynomial Model Estimator

Estimate input-output and time-series polynomial model coefficients

Estimators

## Description

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(tnk)+e(t)

• ARMAX — A(q)y(t) = B(q)u(tnk)+C(q)e(t)

• OE — $y\left(t\right)=\frac{B\left(q\right)}{F\left(q\right)}u\left(t-{n}_{k}\right)+e\left(t\right)$

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

q is the time-shift operator and nk is the 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 are:

• na — 1+a1q-1+a2q-2+...+anaq-na

• nbb1+b2q-1+b3q-2+...+bnbq-(nb-1)

• nc — 1+c1q-1+c2q-2+...+cncq-nc

• nd — 1+d1q-1+d2q-2+...+dndq-nd

• nf — 1+f1q-1+f2q-2+...+fnfq-nf

The orders na, nb, nc, nd, nf and delay, nk, are known a priori and fixed. These are provided through the Model Parameters tab of the block dialog. u(t) and y(t) are provided through the Inputs and Outputs inports, respectively. The block estimates the A(q), B(q), C(q), D(q) and F(q) coefficients and outputs them in the Parameters outport. This outport provides a bus signal with the following elements:

• A — Vector containing [1 a1(t) ... ana(t)].

• B — Vector containing [`zeros(1,nk)` b1(t) ... bnb(t)]. For MISO data, B is a matrix where the i-th row parameters correspond to the i-th input.

• C — Vector containing [1 c1(t) ... cnc(t)].

• D — Vector containing [1 d1(t) ... dnd(t)].

• F — Vector containing [1 f1(t) ... fnf(t)].

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

y(t)+a1y(t–1)+...+anay(tna) = b1u(tnk)+...+bnbu(tnbnk+1)+c1e(t–1)+...+cnce(tnc)

y, u, na, nb, nc, nd, nf, and nk are known quantities that you provide to the block. The block estimates the A, B, C, D, and F parameter values. Estimated C, D, and F polynomials are enforced to be stable, that is, having roots in the unit disk. Estimated A and B polynomials can be unstable.

For a given time step, t, specify y and u as inputs to the Output and Inputs inports, respectively. Specify the na, nb, nc, and nk values in the Model Parameters tab of the block dialog. The block estimates the A, B, C, D and F parameter values and outputs these estimated values using the Parameters outport.

## Parameters

### Model Structure

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.

### Model Parameters

Initial Estimate

Initial guess of the values of the parameters to be estimated, specified as one of the following options:

Default: `'None'`

#### Sample Time

Block sample time, specified as -1 or a positive scalar.

The default value is -1. The block inherits its sample time based on the context of the block within the model.

### Algorithm and Block Options

#### Algorithm Options

Estimation Method

Recursive estimation algorithm, specified as one of the following (each option can change the block dialog):

• `Forgetting Factor` — (Default) Forgetting factor algorithm

If you select this option, you must specify the Forgetting Factor, λ, as a scalar in the (0 1] range. λ specifies the measurement window relevant for parameter estimation. Suppose the system remains approximately constant over T0 samples. You can choose λ such that:

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

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.

The default value is 1.

• `Kalman Filter` — Kalman filter algorithm

If you select this option, you must specify the Noise Covariance Matrix 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, [α1,...,αN] — Covariance matrix is an N-by-N diagonal matrix, with [α1,...,αN] as the diagonal elements.

• N-by-N symmetric positive semidefinite matrix.

N is the number of parameters to be estimated.

0 values in the noise covariance matrix correspond to estimating constant coefficients. Values larger than 0 correspond to time-varying parameters. Large values correspond to rapidly changing parameters.

The default value is 1.

• `Normalized Gradient` — Normalized gradient adaptation algorithm

If you select this option, you must specify the following:

• Adaptation Gain — Adaptation gain, γ, specified as a real positive scalar. γ is directly proportional to the relative information content in the measurements. That is, when your measurements are trustworthy, specify a large value for γ.

The default value is 1.

• Normalization Bias — Bias in adaptation gain scaling, Bias, specified as a real 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, this can cause jumps in the estimated parameters. Bias is the term introduced in the denominator to prevent these jumps. Increase Bias if you observe jumps in estimated parameters.

The default value is `eps`.

• `Gradient` — Unnormalized gradient adaptation algorithm

If you select this option, you must specify the Adaptation Gain, γ, as a real, positive scalar. γ is directly proportional to the relative information content in the measurements. That is, when your measurements are trustworthy, specify a large value for γ, and vice versa.

The default value is 1.

#### Block Options

Output estimation error

Add Error outport to the block. Use this signal to validate the estimation.

For a given time step, t, the estimation error is calculated as:

`$e\left(t\right)=y\left(t\right)-yest\left(t\right).$`
On

Off

(Default) Do not add Error outport.

Output parameter covariance matrix

Add Covariance outport to the block. Use this signal to examine parameter estimation uncertainty.

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

This option is not available when Estimation Method is either `Normalized Gradient` or `Gradient`.

On

Off

(Default) Do not add Covariance outport.

Add Enable inport to the block. Use this input signal to specify 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 outputs the last estimated parameter values. Use this option, for example, to disable parameter estimation when the system enters a mode where the parameter values do not vary with time.

On

Off

(Default) Do not add Enable inport.

External reset

Option to reset estimated parameters and parameter covariance matrix using specified initial values.

Suppose 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. The block outputs these updated parameter value estimates using the Parameters outport.

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

Use this option, for example, when you reset the input because it did not excite the system as needed, resulting in poor estimation results.

Specify this option as one of the following:

• `None` — (Default) Estimated parameters and covariance matrix values are not reset.

• `Rising` — Triggers 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` — Triggers 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` — Triggers reset when the control signal is either rising or falling.

• `Level` — Triggers 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` — Triggers 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.

## Ports

Port Name

Port Type

(In/Out)

Description
InputsInu(t), specified as a real scalar or vector. The port is available when the Model Structure is `ARX`, `ARMAX`, `BJ` or `OE`.
OutputIny(t), specified as a real scalar signal.
ParametersOut

Estimated polynomial coefficients, returned as a bus. The bus contains an element each for the A, B, C, D and F polynomials. The following signals are available for the corresponding model:

• `AR`

• `ARX`

• `ARMA`

• `ARMAX`

• `OE`

• `BJ`

Each bus element contains a vector of the associated polynomial coefficients. For example, the A element contains [1 a1(t) ... ana(t)].

Estimated C, D, and F polynomials are enforced to be stable, that is, have all roots in the unit disk. Estimated A and B polynomials can be unstable.
Enable (Optional)InControl signal to enable parameter estimation, specified as a scalar.
Reset (Optional)InControl signal to reset parameter estimation, specified as a scalar.
InitialParameters (Optional)InInitial guess of the values of the parameters to be estimated, specified as a bus.
InitialCovariance (Optional)InInitial covariance of parameters, specified as a real nonnegative scalar, vector of real nonnegative scalars, or positive semi-definite matrix.
Error (Optional)OutEstimation error, returned as a scalar.
Covariance (Optional)OutCovariance of estimated parameters, returned as an N-by-N matrix. N is the number of parameters.

## Supported Data Types

• Double-precision floating point

• Single-precision floating point

 Note:   The Inputs and Output inports must have matching data types.

## References

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