step

Update model parameters and output online using recursive estimation algorithm

Syntax

[EstimatedParameters,EstimatedOutput] = step(obj,y,InputData)

Description

[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:

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

• recursiveARMA — step 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)

• recursiveARX — step 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).

• recursiveARMAX — step 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).

• recursiveOE — step 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).

• recursiveBJ — step 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).

• recursiveLS — step 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

Estimate an ARMAX Model Online

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.

load iddata1 z1;
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

Tune Recursive Estimation Algorithm Properties During Online Parameter Estimation

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.

load iddata1 z1;
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

Estimate Parameters of System Using Recursive Least Squares Algorithm

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.

load iddata3
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

obj — System object for online parameter estimation

recursiveAR object | recursiveARMA object | recursiveARX object | recursiveARMAX object | recursiveOE object | recursiveBJ object | recursiveLS object

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

• recursiveAR

• recursiveARMA

• recursiveARX

• recursiveARMAX

• recursiveOE

• recursiveBJ

• recursiveLS

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

y — Output data

real scalar

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

InputData — Input data

scalar | vector of real values

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

System object	Model Type	InputData
recursiveAR	Time-series	Not Applicable
recursiveARMA	Time-series	Not Applicable
recursiveARX	SISO ARX	Real scalar
	MISO ARX with N inputs	Column vector of length N, specified as real values
recursiveARMAX	SISO	Real scalar
recursiveOE	SISO	Real scalar
recursiveBJ	SISO	Real scalar
recursiveLS	Single output system with Np system parameters	Regressors, H, specified as a vector of real values of length Np

Output Arguments

EstimatedParameters — Estimated model parameters

vector of real values for each parameter

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 Object	Estimated 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

EstimatedOutput — Estimated output

real scalar

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.

Version History

Introduced in R2015b