ivx

ARX model estimation using instrumental variable method with arbitrary instruments

Syntax

sys = ivx(tt,[na nb nk],x)

sys = ivx(u,y,[na nb nk],x)

sys = ivx(data,[na nb nk],x)

sys = ivx(___,max_size)

sys = ivx(___,Name,Value)

[sys,ic] = ivx(___)

Description

Estimate ARX Polynomial Model

sys = ivx(tt,[na nb nk],x) estimates an ARX polynomial model sys using the time-domain data in the timetable tt. [na nb nk] specifies the ARX structure orders of the A and B polynomials and the input-to-output delay, expressed in the number of samples. x specifies the instrument variable matrix.

The software estimates sys using the instrumental variable method with arbitrary instruments.

An ARX model is represented as:

$$A(q)y(t)=B(q)u(t-nk)+v(t)$$

For more details on the ARX model structure, see arx.

sys = ivx(u,y,[na nb nk],x) uses the time-domain input and output signals in the comma-separated matrices u,y. The software assumes that the data sample time is one second. To change the sample time, set Ts using name-value syntax.

sys = ivx(data,[na nb nk],x) uses the time-domain or frequency-domain data in the data object data. Use this syntax especially when you want to estimate a model using frequency-domain or frequency-response data, or when you want to take advantage of the additional information, such as data sample time or experiment labeling, that data objects provide.

sys = ivx(___,max_size) specifies the maximum size of matrices formed during estimation.

You can use this syntax with any of the previous input-argument combinations.

Specify Additional Model Options

sys = ivx(___,Name,Value) specifies additional options using one or more name-value arguments.

Return Estimated Initial Conditions

[sys,ic] = ivx(___) returns the estimated initial conditions as an initialCondition object. For more information on ic, see the ic argument description.

Use this syntax if you plan to simulate or predict the model response using the same estimation input data and then compare the response with the same estimation output data. Incorporating the initial conditions yields a better match during the first part of the simulation.

Input Arguments

tt — Timetable-based estimation data

timetable | cell array of timetables

Timetable-based estimation data, specified as a timetable that uses a regularly spaced time vector. tt contains variables representing input and output channels.

For multiexperiment data, specify tt as an Ne-by-1 cell array of timetables, where Ne is the number of experiments. The sample times of all the experiments must match.

To select individual input and output channels to use for estimation, use the InputName and OutputName name-value arguments.

For example, use the following command to select timetable variables "u1" and "u3" as inputs and the variables "y2" and "y4" as outputs.

sys = ivx(tt,__,'InputName',["u1" "u3"],'OutputName',["y2" "y4"])

For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.

u, y — Matrix-based estimation data

matrices | cell array of matrices

Estimation data, specified for SISO systems as a comma-separated pair of N_s-element numeric column vectors that contain uniformly sampled input and output time-domain signal values. Here, N_s is the number of samples.

For MIMO systems, specify u,y as an input/output matrix pair with the following dimensions:

• u — N_s-by-N_u, where N_u is the number of inputs

• y — N_s-by-N_y, where N_y is the number of outputs

For multiexperiment data, specify u,y as a pair of 1-by-N_e cell arrays, where N_e is the number of experiments. The sample times of all the experiments must match.

For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.

data — Estimation data

iddata object | frd object | idfrd object

Estimation data, specified as an iddata, idfrd, or frd (Control System Toolbox) object.

When using frequency-domain data, there must be only one output.

For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.

[na nb nk] — ARX polynomial orders and delay

row vector of nonnegative integers | row vector of nonnegative integer matrices

ARX polynomial orders and delay, specified as a row vector of integers or integer matrices.

• For SISO models, [na nb nk] is a 1-by-3 row vector of nonnegative integers that correspond to the orders of the A(q) and B(q) polynomials and the input-to-output delays, respectively. The polynomial order is equal to the number of coefficients to estimate in that polynomial.

• For a model with N_y outputs and N_u inputs:

  • na is the order of polynomial A(q), specified as an N_y-by-N_y matrix of nonnegative integers.

  • nb is the order of polynomial B(q) + 1, specified as an N_y-by-N_u matrix of nonnegative integers.

  • nk is the input-output delay, also known as the transport delay, specified as an N_y-by-N_u matrix of nonnegative integers.

  .

For more details on the ARX model structure, see arx.

x — Instrument variable matrix

matrix | cell array of matrices

Instrument variable matrix that contains the arbitrary instruments for use in the instrumental variable method, specified as a matrix that is the same size as the output data, data.y. For multi-experiment data, specify x as a cell array with one entry for each experiment.

The instruments used are analogous to the regression vector, with y replaced by x. For more information on choosing instruments, see Picking Instrumental Variables for System Identification.

max_size — Maximum matrix size

250000 | positive integer

Maximum matrix size for any matrix formed by the algorithm for estimation, specified as a positive integer. Specify max_size as a reasonably large value.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: InputName=["u1" "u3"] specifies timetable variables u1 u3 as the input variables. Alternatively, you can use 'InputName',["u1","u3"].

InputName — Input channel names

string | character vector | string array | cell array of character vectors

Input channel names, specified as a string, character vector, string array, or cell array of character vectors.

If you are using a timetable for the data source, the names in InputName must be a subset of the timetable variables.

Example: sys = ivx(tt,__,'InputName',["u1" "u2"]) selects the variables u1 and u2 as the input channels from the timetable tt to use for the estimation.

OutputName — Output channel names

string | character vector | string array | cell array of character vectors

Output channel names, specified as a string, character vector, string array, or cell array of character vectors.

If you are using a timetable for the data source, the names in OutputName must be a subset of the timetable variables.

Example: sys = ivx(tt,__,'OutputName',["y1" "y3"]) selects the variables y1 and y3 as the output channels from the timetable tt to use for the estimation.

Output Arguments

sys — Identified polynomial model

idpoly object

ARX model that fits the estimation data, returned as a discrete-time idpoly object. This model is created using the specified model orders, delays, and estimation options. ivx does not return any estimated covariance information for sys.

Information about the estimation results and options used is stored in the Report property of the model. Report has the following fields:

Report Field	Description
Status	Summary of the model status, which indicates whether the model was created by construction or obtained by estimation
Method	Estimation command used
InitialCondition	Handling of initial conditions during model estimation, returned as one of the following values: 'zero' — The initial conditions were set to zero. 'estimate' — The initial conditions were treated as independent estimation parameters. 'backcast' — The initial conditions were estimated using the best least squares fit. This field is especially useful to view how the initial conditions were handled when the InitialCondition option in the estimation option set is 'auto'.
Fit	Quantitative assessment of the estimation, returned as a structure. See Loss Function and Model Quality Metrics for more information on these quality metrics. The structure has these fields. FitPercent — Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as the percentage fitpercent = 100(1-NRMSE) LossFcn — Value of the loss function when the estimation completes MSE — Mean squared error (MSE) measure of how well the response of the model fits the estimation data FPE — Final prediction error for the model AIC — Raw Akaike Information Criteria (AIC) measure of model quality AICc — Small-sample-size corrected AIC nAIC — Normalized AIC BIC — Bayesian Information Criteria (BIC)
Parameters	Estimated values of model parameters
OptionsUsed	Option set used for estimation. If no custom options were configured, this is a set of default options. See arxOptions for more information.
RandState	State of the random number stream at the start of estimation. Empty, [], if randomization was not used during estimation. For more information, see rng.
DataUsed	Attributes of the data used for estimation, returned as a structure with the following fields. Name — Name of the data set Type — Data type Length — Number of data samples Ts — Sample time InterSample — Input intersample behavior, returned as one of the following values: 'zoh' — A zero-order hold maintains a piecewise-constant input signal between samples. 'foh' — A first-order hold maintains a piecewise-linear input signal between samples. 'bl' — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency. InputOffset — Offset removed from time-domain input data during estimation. For nonlinear models, it is []. OutputOffset — Offset removed from time-domain output data during estimation. For nonlinear models, it is [].

For more information on using Report, see Estimation Report.

ic — Initial conditions

initialCondition object | object array of initialCondition values

Estimated initial conditions, returned as an initialCondition object or an object array of initialCondition values.

• For a single-experiment data set, ic represents, in state-space form, the free response of the transfer function model (A and C matrices) to the estimated initial states (x₀).

• For a multiple-experiment data set with N_e experiments, ic is an object array of length N_e that contains one set of initialCondition values for each experiment.

For more information, see initialCondition.

Tips

• Use iv4 first for IV estimation to identify ARX polynomial models where the instruments x are chosen automatically. Use ivx for nonstandard situations. For example, use ivx when there is feedback present in the data or when instruments other than those used in iv4 need to be tried. You can also use ivx to automatically generate instruments from certain custom defined filters.

References

[1] Ljung, Lennart. System Identification: Theory for the User, 2nd ed. Prentice Hall Information and System Sciences Series. Upper Saddle River, NJ: Prentice Hall PTR, 1999.

Version History

Introduced before R2006a