# recursiveLS

Create System object for online parameter estimation using recursive least squares algorithm

## Syntax

obj = recursiveLS
obj = recursiveLS(Np)
obj = recursiveLS(Np,theta0)
obj = recursiveLS(___,Name,Value)

## Description

Use the recursiveLS command for parameter estimation with real-time data. If all data necessary for estimation is available at once and you are estimating a time-invariant model, use mldivide, \.

obj = recursiveLS creates a System object™ for online parameter estimation of a default single output system that is linear in estimated parameters. Such a system can be represented as:

y(t) = H(t)θ(t)+e(t).

Here, y is the output, θ are the parameters, H are the regressors, and e is the white-noise disturbance. The default system has one parameter with initial parameter value 1.

After creating the object, use the step command to update model parameter estimates using recursive least squares algorithms and real-time data. Alternatively, you can call the object directly. For more information, see Tips.

obj = recursiveLS(Np) also specifies the number of parameters to be estimated.

obj = recursiveLS(Np,theta0) also specifies the number of parameters and initial values of the parameters.

obj = recursiveLS(___,Name,Value) specifies additional attributes of the system and recursive estimation algorithm using one or more Name,Value pair arguments.

## Object Description

recursiveLS creates a System object for online parameter estimation of a single output system that is linear in its parameters.

A System object is a specialized MATLAB® object designed specifically for implementing and simulating dynamic systems with inputs that change over time. System objects use internal states to store past behavior, which is used in the next computational step.

After you create a System object, you use commands to process data or obtain information from or about the object. System objects use a minimum of two commands to process data — a constructor to create the object and the step command to update object parameters using real-time data. This separation of declaration from execution lets you create multiple, persistent, reusable objects, each with different settings.

You can use the following commands with the online estimation System objects in System Identification Toolbox™:

CommandDescription
step

Update model parameter estimates using recursive estimation algorithms and real-time data.

step puts the object into a locked state. In a locked state, you cannot change any nontunable properties or input specifications, such as model order, data type, or estimation algorithm. During execution, you can only change tunable properties.

release

Unlock the System object. Use this command to enable setting of nontunable parameters.

reset

Reset the internal states of a locked System object to the initial values, and leave the object locked.

clone

Create another System object with the same object property values.

Do not create additional objects using syntax obj2 = obj. Any changes made to the properties of the new object created this way (obj2) also change the properties of the original object (obj).

isLocked

Query locked status for input attributes and nontunable properties of the System object.

Use the recursiveLS command to create an online estimation System object. Then estimate the system parameters (theta) and output using the step command with regressors and incoming output data, H and y.

[theta,EstimatedOutput] = step(obj,y,H)

For recursiveLS object properties, see Properties.

## Examples

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obj = recursiveLS
obj =
recursiveLS with properties:

NumberOfParameters: 1
Parameters: []
InitialParameters: 1
ParameterCovariance: []
InitialParameterCovariance: 10000
EstimationMethod: 'ForgettingFactor'
ForgettingFactor: 1
History: 'Infinite'
InputProcessing: 'Sample-based'
DataType: 'double'

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

Use frame-based signals with the recursiveLS command. Machine interfaces often provide sensor data in frames containing multiple samples, rather than in individual samples. The recursiveLS object accepts these frames directly when you set InputProcessing to Frame-based.

The object uses the same estimation algorithms for sample-based and frame-based input processing. The estimation results are identical. There are some special considerations, however, for working with frame-based inputs.

This example is the frame-based version of the sample-based recursiveLS example in 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_f = recursiveLS(2,'InputProcessing','Frame-Based');

Load the data, which contains input and output time series signals. Each signal consists of 30 frames and each frame contains ten individual time samples.

input = input_sig_frame.data;
output = output_sig_frame.data;
numframes = size(input,3)
numframes = 30
mframe = size(input,1)
mframe = 10

Initialize the regressor frame, which for a given frame, is of the form

${H}_{f}=\left[\begin{array}{cc}{u}_{1}& {u}_{0}\\ {u}_{2}& {u}_{1}\\ ⋮& ⋮\\ {u}_{10}& {u}_{9}\end{array}\right]$,

where the most recent point in the frame is ${\mathit{u}}_{10}^{}$.

Hframe = zeros(10,2);

For this first-order example, the regressor frame includes one point from the previous frame. Initialize this point.

oldInput = 0;

Estimate the parameters and output using step and input-output data, maintaining the current regressor frame in Hframe.

• The input and output arrays have three dimensions. The third dimension is the frame index, and the first two dimensions represent the contents of individual frames.

• Use the circshift function to populate the second column of Hframe with the past input value for each regressor pair by shifting the input vector by one position.

• Populate the Hframe element holding the oldest value, Hframe(1,2), with the regressor value stored from the previous frame.

• Invoke the step function implicitly by calling the obj system object with input arguments. The step function is compatible with frames, so no loop function within the frame is necessary.

• Save the most recent input value to use for the next frame calculation.

EstimatedOutput = zeros(10,1,30);
theta = zeros(2,30);
for i = 1:numframes
Hframe = [input(:,:,i) circshift(input(:,:,i),1)];
Hframe(1,2) = oldInput;
[theta(:,i), EstimatedOutput(:,:,i)] = obj_f(output(:,:,i),Hframe);
oldInput = input(10,:,i);
end

Plot the parameters.

theta1 = theta(1,:);
theta2 = theta(2,:);
iframe = 1:numframes;
plot(iframe,theta1,iframe,theta2)
title('Frame-Based Recursive Least Squares Estimation')
legend('theta1','theta2','location','best')

View the final estimates.

theta_final = theta(:,numframes)
theta_final = 2×1

-1.5322
-0.0235

The final estimates are identical to the sample-based estimation.

Create System object for online parameter estimation using recursive least squares algorithm of a system with two parameters and known initial parameter values.

obj = recursiveLS(2,[0.8 1],'InitialParameterCovariance',0.1);

InitialParameterCovariance represents the uncertainty in your guess for the initial parameters. Typically, the default InitialParameterCovariance (10000) is too large relative to the parameter values. This results in initial guesses being given less importance during estimation. If you have confidence in the initial parameter guesses, specify a smaller initial parameter covariance.

## Input Arguments

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Number of parameters in the system, specified as a positive integer.

Initial value of parameters, specified as one of the following:

• Scalar — All the parameters have the same initial value.

• Vector of real values of length Np— The ith parameter has initial value theta0(i).

The default initial value for all parameters is 1.

Note

If the initial parameter values are much smaller than InitialParameterCovariance, these initial values are given less importance during estimation. Specify a smaller initial parameter covariance if you have high confidence in the initial parameter values. This statement applies only for infinite-history estimation. Finite-history estimation does not use InitialParameterCovariance.

### Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Use Name,Value arguments to specify writable properties of recursiveLS System object during object creation. For example, obj = recursiveLS(2,'EstimationMethod','Gradient') creates a System object to estimate the system parameters using the 'Gradient' recursive estimation algorithm.

## Properties

recursiveLS System object properties consist of read-only and writable properties. The writable properties are tunable and nontunable properties. The nontunable properties cannot be changed when the object is locked, that is, after you use the step command.

Use Name,Value arguments to specify writable properties of recursiveLS objects during object creation. After object creation, use dot notation to modify the tunable properties.

obj = recursiveLS;
obj.ForgettingFactor = 0.99;

 NumberOfParameters Number of parameters to be estimated, returned as a positive integer. NumberOfParameters is a read-only property. If Np is specified during object construction, NumberOfParameters takes the value assigned to Np. Default: 1 Parameters Estimated parameters, returned as a column vector of real values. Parameters is a read-only property and is initially empty after you create the object. It is populated after you use the step command for online parameter estimation. InitialParameters Initial values of parameters, specified as one of the following: Scalar — All the parameters have the same initial value.Vector of real values of length Np— The ith parameter has initial value InitialParameters(i). If the initial parameter values are much smaller than InitialParameterCovariance, these initial values are given less importance during estimation. Specify a smaller initial parameter covariance if you have high confidence in initial parameter values. This statement applies only for infinite-history estimation. Finite-history estimation does not use InitialParameterCovariance. InitialParameters is a tunable property. You can change InitialParameters when the object is in a locked state. Default: 1 InitialOutputs Initial values of the outputs buffer in finite-history estimation, specified as 0 or as a W-by-1 vector, where W is the window length. The InitialOutputs property provides a means of controlling the initial behavior of the algorithm. When InitialOutputs is set to 0, the object 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. Specify InitialOutputs only when History is Finite. InitialOutputs is a tunable property. You can change InitialOutputs when the object is in a locked state. Default: 0 InitialRegressors Initial values of the regressors buffer in finite-history estimation, specified as 0 or as a W-by-Np matrix, where W is the window length and Np is the number of parameters. The InitialRegressors property provides a means of controlling the initial behavior of the algorithm. When the InitialRegressors is set to 0, the object 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. Specify InitialRegressors only when History is Finite. InitialRegressors is a tunable property. You can change InitialRegressors when the object is in a locked state. Default: 0 ParameterCovariance Estimated covariance P of the parameters, returned as an N-by-N symmetric positive-definite matrix. N is the number of parameters to be estimated. The software computes P assuming that the residuals (difference between estimated and measured outputs) are white noise, and the variance of these residuals is 1. ParameterCovariance is applicable only when EstimationMethod is 'ForgettingFactor' or 'KalmanFilter' or when History is Finite. The interpretation of P depends on your settings for the History and EstimationMethod properties. If History is Infinite, then your EstimationMethod selection results in one of the following:'ForgettingFactor' — (R2/2)P is approximately equal to the covariance matrix of the estimated parameters, where R2 is the true variance of the residuals.'KalmanFilter' — R2P is the covariance matrix of the estimated parameters, and R1 /R2 is the covariance matrix of the parameter changes. Here, R1 is the covariance matrix that you specify in ProcessNoiseCovariance.If History is Finite (sliding-window estimation) — R2P 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. ParameterCovariance is a read-only property and is initially empty after you create the object. It is populated after you use the step command for online parameter estimation. InitialParameterCovariance Covariance of the initial parameter estimates, specified as one of the following: Real positive scalar, α — Covariance matrix is an N-by-N diagonal matrix, with α as the diagonal elements. N is the number of parameters to be estimated.Vector of real positive scalars, [α1,...,αN] — Covariance matrix is an N-by-N diagonal matrix, with [α1,...,αN] as the diagonal elements. N-by-N symmetric positive-definite matrix. InitialParameterCovariance represents the uncertainty in the initial parameter estimates. For large values of InitialParameterCovariance, less importance is placed on the initial parameter values and more on the measured data during beginning of estimation using step. Use only when EstimationMethod is 'ForgettingFactor' or 'KalmanFilter'. InitialParameterCovariance is a tunable property. You can change it when the object is in a locked state. Default: 10000 EstimationMethod Recursive least squares estimation algorithm used for online estimation of model parameters, specified as one of the following values: 'ForgettingFactor' — Algorithm used for parameter estimation'KalmanFilter' — Algorithm used for parameter estimation'NormalizedGradient' — Algorithm used for parameter estimation'Gradient' — Unnormalized gradient algorithm used for parameter estimation Forgetting factor and Kalman filter algorithms are more computationally intensive than gradient and unnormalized gradient methods. However, they have better convergence properties. For information about these algorithms, see Recursive Algorithms for Online Parameter Estimation. These methods all use an infinite data history, and are available only when History is 'Infinite'. EstimationMethod is a nontunable property. You cannot change it during execution, that is, after the object is locked using the step command. Default: Forgetting Factor ForgettingFactor Forgetting factor, λ, relevant for parameter estimation, specified as a scalar in the range (0,1]. Suppose that 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 range [0.98 0.995]. Use only when EstimationMethod is 'ForgettingFactor'. ForgettingFactor is a tunable property. You can change it when the object is in a locked state. Default: 1 EnableAdapation Enable or disable parameter estimation, specified as one of the following: true or 1— The step command estimates the parameter values for that time step and updates the parameter values.false or 0 — The step command does not update the parameters for that time step and instead outputs the last estimated value. You can use this option when your system enters a mode where the parameter values do not vary with time.NoteIf you set EnableAdapation to false, you must still execute the step command. Do not skip step to keep parameter values constant, because parameter estimation depends on current and past I/O measurements. step ensures past I/O data is stored, even when it does not update the parameters. EnableAdapation is a tunable property. You can change it when the object is in a locked state. Default: true DataType Floating point precision of parameters, specified as one of the following values: 'double' — Double-precision floating point'single' — Single-precision floating point Setting DataType to 'single' saves memory, but leads to loss of precision. Specify DataType based on the precision required by the target processor where you will deploy generated code. DataType is a nontunable property. It can only be set during object construction using Name,Value arguments and cannot be changed afterward. Default: 'double' ProcessNoiseCovariance Covariance matrix of parameter variations, specified 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. ProcessNoiseCovariance is applicable when EstimationMethod is 'KalmanFilter'. Kalman filter algorithm treats the parameters as states of a dynamic system and estimates these parameters using a Kalman filter. ProcessNoiseCovariance is the covariance of the process noise acting on these parameters. Zero values in the noise covariance matrix correspond to estimating constant coefficients. Values larger than 0 correspond to time-varying parameters. Use large values for rapidly changing parameters. However, the larger values result in noisier parameter estimates. ProcessNoiseCovariance is a tunable property. You can change it when the object is in a locked state. Default: 0.1 AdaptationGain Adaptation gain, γ, used in gradient recursive estimation algorithms, specified as a positive scalar. AdaptationGain is applicable when EstimationMethod is 'Gradient' or 'NormalizedGradient'. Specify a large value for AdaptationGain when your measurements have a high signal-to-noise ratio. AdaptationGain is a tunable property. You can change it when the object is in a locked state. Default: 1 NormalizationBias Bias in adaptation gain scaling used in the 'NormalizedGradient' method, specified as a nonnegative scalar. NormalizationBias is applicable when EstimationMethod is 'NormalizedGradient'. The normalized gradient algorithm divides 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. NormalizationBias is the term introduced in the denominator to prevent these jumps. Increase NormalizationBias if you observe jumps in estimated parameters. NormalizationBias is a tunable property. You can change it when the object is in a locked state. Default: eps History Data history type defining which type of recursive algorithm you use, specified as: 'Infinite' — Use an algorithm that aims to minimize the error between the observed and predicted outputs for all time steps from the beginning of the simulation.'Finite' — Use an algorithm that aims to minimize the error between the observed and predicted outputs for a finite number of past time steps. Algorithms with infinite history aim to produce parameter estimates that explain all data since the start of the simulation. These algorithms still use a fixed amount of memory that does not grow over time. The object provides multiple algorithms of the 'Infinite' History type. Specifying this option activates the EstimationMethod property with which you specify an algorithm. Algorithms with finite history aim to produce parameter estimates that explain only a finite number of past data samples. This method is also called sliding-window estimation. The object provides one algorithm of the 'Finite' type. Specifying this option activates the WindowLength property that sizes the window. For more information on recursive estimation methods, see Recursive Algorithms for Online Parameter Estimation. History is a nontunable property. It can be set only during object construction using Name,Value arguments and cannot be changed afterward. Default: 'Infinite' WindowLength Window size determining the number of time samples to use for the sliding-window estimation method, specified as a positive integer. Specify WindowLength only when History is Finite. 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. WindowLength 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 (see InputProcessing). However, when using frame-based processing, your window length must be greater than or equal to the number of samples (time steps) contained in the frame. WindowLength is a nontunable property. It can be set only during object construction using Name,Value arguments and cannot be changed afterward. Default: 200 InputProcessing Option for sample-based or frame-based input processing, specified as a character vector or string. 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. Your InputProcessing specification impacts the dimensions for the input and output signals when using the step command: [theta,EstimatedOutput] = step(obj,y,H) Sample-based y and EstimatedOutput are scalars.H is a 1-by-Np vector, where Np is the number of parameters. Frame-based with M samples per framey and EstimatedOutput are M-by-1 vectors.H is an M-by-Np matrix. InputProcessing is a nontunable property. It can be set only during object construction using Name,Value arguments and cannot be changed afterward. Default: 'Sample-based'

## Output Arguments

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System object for online parameter estimation, returned as a recursiveLS System object. Use step command to estimate the parameters of the system. You can then access the estimated parameters and parameter covariance using dot notation. For example, type obj.Parameters to view the estimated parameters.

## 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, [theta,EstimatedOutput] = step(obj,y,H) and [theta,EstimatedOutput] = obj(y,H) perform equivalent operations.

## Extended Capabilities

Introduced in R2015b

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