step

Update model parameters and output online using recursive estimation algorithm

Description

example

[EstimatedParameters,EstimatedOutput] = step(obj,y,InputData) updates parameters and output of the model specified in System object™, obj, using measured output, y, and input data.

step puts the object into a locked state. In a locked state you cannot change any nontunable properties of the object, such as model order, data type, or estimation algorithm.

The EstimatedParameters and InputData depend on the online estimation System object:

• recursiveARstep returns the estimated polynomial A(q) coefficients of a single output AR model using time-series output data.
[A,EstimatedOutput] = step(obj,y)

• recursiveARMAstep returns the estimated polynomial A(q) and C(q) coefficients of a single output ARMA model using time-series output data, y.
[A,C,EstimatedOutput] = step(obj,y)

• recursiveARXstep returns the estimated polynomial A(q) and B(q) coefficients of a SISO or MISO ARX model using measured input and output data, u and y, respectively.
[A,B,EstimatedOutput] = step(obj,y,u).

• recursiveARMAXstep returns the estimated polynomial A(q), B(q), and C(q) coefficients of a SISO ARMAX model using measured input and output data, u and y, respectively.
[A,B,C,EstimatedOutput] = step(obj,y,u).

• recursiveOEstep returns the estimated polynomial B(q), and F(q) coefficients of a SISO Output-Error polynomial model using measured input and output data, u and y, respectively.
[B,F,EstimatedOutput] = step(obj,y,u).

• recursiveBJstep returns the estimated polynomial B(q), C(q), D(q), and F(q) coefficients of a SISO Box-Jenkins polynomial model using measured input and output data, u and y, respectively.
[B,C,D,F,EstimatedOutput] = step(obj,y,u).

• recursiveLSstep returns the estimated system parameters, θ, of a single output system that is linear in estimated parameters, using regressors H and output data y.
[theta,EstimatedOutput] = step(obj,y,H).

Examples

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Create a System object for online parameter estimation of an ARMAX model.

obj = recursiveARMAX;

The ARMAX model has a default structure with polynomials of order 1 and initial polynomial coefficient values, eps.

Load the estimation data. In this example, use a static data set for illustration.

output = z1.y;
input = z1.u;

Estimate ARMAX model parameters online using step.

for i = 1:numel(input)
[A,B,C,EstimatedOutput] = step(obj,output(i),input(i));
end

View the current estimated values of polynomial A coefficients.

obj.A
ans = 1×2

1.0000   -0.8298

View the current covariance estimate of the parameters.

obj.ParameterCovariance
ans = 3×3

0.0001    0.0001    0.0001
0.0001    0.0032    0.0000
0.0001    0.0000    0.0001

View the current estimated output.

EstimatedOutput
EstimatedOutput = -4.5595

Create a System object for online parameter estimation of an ARMAX model.

obj = recursiveARMAX;

The ARMAX model has a default structure with polynomials of order 1 and initial polynomial coefficient values, eps.

Load the estimation data. In this example, use a static data set for illustration.

output = z1.y;
input = z1.u;
dataSize = numel(input);

Estimate ARMAX model parameters online using the default recursive estimation algorithm, Forgetting Factor. Change the ForgettingFactor property during online parameter estimation.

for i = 1:dataSize
if i == dataSize/2
obj.ForgettingFactor = 0.98;
end
[A,B,C,EstimatedOutput] = step(obj,output(i),input(i));
end

The system has two parameters and is represented as:

$\mathit{y}\left(\mathit{t}\right)={\mathit{a}}_{1}\mathit{u}\left(\mathit{t}\right)+{\mathit{a}}_{2}\mathit{u}\left(\mathit{t}-1\right)$

Here,

• $\mathit{u}$ and $y$ are the real-time input and output data, respectively.

• $\mathit{u}\left(\mathit{t}\right)$ and $\mathit{u}\left(\mathit{t}-1\right)$ are the regressors, H, of the system.

• ${\mathit{a}}_{1}$ and ${\mathit{a}}_{2}$ are the parameters, theta, of the system.

Create a System object for online estimation using the recursive least squares algorithm.

obj = recursiveLS(2);

Load the estimation data, which for this example is a static data set.

input = z3.u;
output = z3.y;

Create a variable to store u(t-1). This variable is updated at each time step.

oldInput = 0;

Estimate the parameters and output using step and input-output data, maintaining the current regressor pair in H. Invoke the step function implicitly by calling the obj system object with input arguments.

for i = 1:numel(input)
H = [input(i) oldInput];
[theta, EstimatedOutput] = obj(output(i),H);
estimatedOut(i)= EstimatedOutput;
theta_est(i,:) = theta;
oldInput = input(i);
end

Plot the measured and estimated output data.

numSample = 1:numel(input);
plot(numSample,output,'b',numSample,estimatedOut,'r--');
legend('Measured Output','Estimated Output'); Plot the parameters.

plot(numSample,theta_est(:,1),numSample,theta_est(:,2))
title('Parameter Estimates for Recursive Least Squares Estimation')
legend("theta1","theta2") View the final estimates.

theta_final = theta
theta_final = 2×1

-1.5322
-0.0235

Input Arguments

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System object for online parameter estimation, created using one of the following commands:

The step command updates parameters of the model using the recursive estimation algorithm specified in obj and the incoming input-output data.

Output data acquired in real time, specified as a real scalar.

Input data acquired in real time, specified as a scalar or vector of real values depending on the type of System object.

System objectModel TypeInputData
recursiveARTime-seriesNot Applicable
recursiveARMATime-seriesNot Applicable
recursiveARXSISO ARXReal scalar
MISO ARX with N inputsColumn vector of length N, specified as real values
recursiveARMAXSISOReal scalar
recursiveOESISOReal scalar
recursiveBJSISOReal scalar
recursiveLSSingle output system with Np system parametersRegressors, H, specified as a vector of real values of length Np

Output Arguments

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Estimated model parameters, returned as vectors of real values. The number of estimated parameters, and so the step syntax, depend on the type of System object:

Online Estimation System ObjectEstimated Parameters
recursiveAR

Polynomial A(q) coefficients

recursiveARMA

Polynomials A(q) and C(q) coefficients

recursiveARX

Polynomials A(q) and B(q) coefficients

recursiveARMAX

Polynomials A(q), B(q), and C(q) coefficients

recursiveOE

Polynomials B(q) and F(q)

recursiveBJ

Polynomials B(q), C(q), D(q), and F(q) coefficients

recursiveLS

System parameters, θ

Estimated output, returned as a real scalar. The output is estimated using input-output estimation data, current parameter values, and recursive estimation algorithm specified in obj.

Tips

• Starting in R2016b, instead of using the step command to update model parameter estimates, you can call the System object with input arguments, as if it were a function. For example, [A,EstimatedOutput] = step(obj,y) and [A,EstimatedOutput] = obj(y) perform equivalent operations.

Introduced in R2015b

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