Documentation

# idtf

Transfer function model with identifiable parameters

## Syntax

``sys = idtf(num,den)``
``sys = idtf(num,den,Ts)``
``sys = idtf(___,Name,Value)``
``sys = idtf(sys0)``

## Description

````sys = idtf(num,den)` creates a continuous-time transfer function with identifiable parameters (an `idtf` model). `num` specifies the current values of the transfer function numerator coefficients. `den` specifies the current values of the transfer function denominator coefficients.```
````sys = idtf(num,den,Ts)` creates a discrete-time transfer function with identifiable parameters. `Ts` is the sample time.```
````sys = idtf(___,Name,Value)` creates a transfer function with properties specified by one or more `Name,Value` pair arguments.```
````sys = idtf(sys0)` converts any dynamic system model, `sys0`, to `idtf` model form.```

## Object Description

An `idtf` model represents a system as a continuous-time or discrete-time transfer function with identifiable (estimable) coefficients.

A SISO transfer function is a ratio of polynomials with an exponential term. In continuous time,

`$G\left(s\right)={e}^{-\tau s}\frac{{b}_{n}{s}^{n}+{b}_{n-1}{s}^{n-1}+...+{b}_{0}}{{s}^{m}+{a}_{m-1}{s}^{m-1}+...+{a}_{0}}.$`

In discrete time,

`$G\left({z}^{-1}\right)={z}^{-k}\frac{{b}_{n}{z}^{-n}+{b}_{n-1}{z}^{-n+1}+...+{b}_{0}}{{z}^{-m}+{a}_{m-1}{z}^{-m+1}+...+{a}_{0}}.$`

In discrete time, zk represents a time delay of kTs, where Ts is the sample time.

For `idtf` models, the denominator coefficients a0,...,am–1 and the numerator coefficients b0,...,bn can be estimable parameters. (The leading denominator coefficient is always fixed to 1.) The time delay τ (or kin discrete time) can also be an estimable parameter. The `idtf` model stores the polynomial coefficients a0,...,am–1 and b0,...,bn in the `Denominator` and `Numerator` properties of the model, respectively. The time delay τ or k is stored in the `IODelay` property of the model.

A MIMO transfer function contains a SISO transfer function corresponding to each input-output pair in the system. For `idtf` models, the polynomial coefficients and transport delays of each input-output pair are independently estimable parameters.

There are three ways to obtain an `idtf` model.

• Estimate the `idtf` model based on input-output measurements of a system, using `tfest`. The `tfest` command estimates the values of the transfer function coefficients and transport delays. The estimated values are stored in the `Numerator`, `Denominator`, and `IODelay` properties of the resulting `idtf` model. The `Report` property of the resulting model stores information about the estimation, such as handling of initial conditions and options used in estimation.

When you obtain an `idtf` model by estimation, you can extract estimated coefficients and their uncertainties from the model. To do so, use commands such as `tfdata`, `getpar`, or `getcov`.

• Create an `idtf` model using the `idtf` command.

You can create an `idtf` model to configure an initial parameterization for estimation of a transfer function to fit measured response data. When you do so, you can specify constraints on such values as the numerator and denominator coefficients and transport delays. For example, you can fix the values of some parameters, or specify minimum or maximum values for the free parameters. You can then use the configured model as an input argument to `tfest` to estimate parameter values with those constraints.

• Convert an existing dynamic system model to an `idtf` model using the `idtf` command.

### Note

Unlike `idss` and `idpoly`, `idtf` uses a trivial noise model and does not parameterize the noise.

So, H = 1 in $y=Gu+He$.

## Examples

collapse all

Specify a continuous-time, single-input, single-output (SISO) transfer function with estimable parameters. The initial values of the transfer function are:

`$G\left(s\right)=\frac{s+4}{{s}^{2}+20s+5}$`

```num = [1 4]; den = [1 20 5]; G = idtf(num,den);```

`G` is an `idtf` model. `num` and `den` specify the initial values of the numerator and denominator polynomial coefficients in descending powers of $s$. The numerator coefficients having initial values 1 and 4 are estimable parameters. The denominator coefficient having initial values 20 and 5 are also estimable parameters. The leading denominator coefficient is always fixed to 1.

You can use `G` to specify an initial parameterization for estimation with `tfest`.

Specify a continuous-time, SISO transfer function with known input delay. The transfer function initial values are given by:

`$G\left(s\right)={e}^{-5.8s}\frac{5}{s+5}$`

Label the input of the transfer function with the name `'Voltage'` and specify the input units as `volt`.

Use `Name,Value` input pairs to specify the delay, input name, and input unit.

```num = 5; den = [1 5]; input_delay = 5.8; input_name = 'Voltage'; input_unit = 'volt'; G = idtf(num,den,'InputDelay',input_delay,... 'InputName',input_name,'InputUnit',input_unit);```

$G$ is an `idtf` model. You can use `G` to specify an initial parameterization for estimation with `tfest`. If you do so, model properties such as `InputDelay`, `InputName`, and `InputUnit` are applied to the estimated model. The estimation process treats `InputDelay` as a fixed value. If you want to estimate the delay and specify an initial value of 5.8 s, use the `IODelay` property instead.

Specify a discrete-time SISO transfer function with estimable parameters. The initial values of the transfer function are:

`$H\left(z\right)=\frac{z-0.1}{z+0.8}$`

Specify the sample time as 0.2 seconds.

```num = [1 -0.1]; den = [1 0.8]; Ts = 0.2; H = idtf(num,den,Ts);```

`num` and `den` are the initial values of the numerator and denominator polynomial coefficients. For discrete-time systems, specify the coefficients in ascending powers of ${z}^{-1}$.

`Ts` specifies the sample time for the transfer function as 0.2 seconds.

`H` is an `idtf` model. The numerator and denominator coefficients are estimable parameters (except for the leading denominator coefficient, which is fixed to 1).

Specify a discrete-time, two-input, two-output transfer function. The initial values of the MIMO transfer function are:

`$H\left(z\right)=\left[\begin{array}{cc}\frac{1}{z+0.2}& \frac{z}{z+0.7}\\ \frac{-z+2}{z-0.3}& \frac{3}{z+0.3}\end{array}\right]$`

Specify the sample time as 0.2 seconds.

```nums = {1,[1,0];[-1,2],3}; dens = {[1,0.2],[1,0.7];[1,-0.3],[1,0.3]}; Ts = 0.2; H = idtf(nums,dens,Ts);```

`nums` and `dens` specify the initial values of the coefficients in cell arrays. Each entry in the cell array corresponds to the numerator or denominator of the transfer function of one input-output pair. For example, the first row of `nums` is `{1,[1,0]}`. This cell array specifies the numerators across the first row of transfer functions in `H`. Likewise, the first row of `dens`, `{[1,0.2],[1,0.7]}`, specifies the denominators across the first row of `H`.

`Ts` specifies the sample time for the transfer function as 0.2 seconds.

`H` is an `idtf` model. All of the polynomial coefficients are estimable parameters, except for the leading coefficient of each denominator polynomial. These coefficients are always fixed to 1.

Specify the following discrete-time transfer function in terms of `q^-1`:

`$H\left({q}^{-1}\right)=\frac{1+0.4{q}^{-1}}{1+0.1{q}^{-1}-0.3{q}^{-2}}$`

Specify the sample time as 0.1 seconds.

```num = [1 0.4]; den = [1 0.1 -0.3]; Ts = 0.1; convention_variable = 'q^-1'; H = idtf(num,den,Ts,'Variable',convention_variable);```

Use a `Name,Value` pair argument to specify the variable `q^-1`.

`num` and `den` are the numerator and denominator polynomial coefficients in ascending powers of ${q}^{-1}$.

`Ts` specifies the sample time for the transfer function as 0.1 seconds.

`H` is an `idtf` model.

Specify a transfer function with estimable coefficients whose initial value is the static gain matrix:

`$H\left(s\right)=\left[\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ 3& 0& 2\end{array}\right]$`

```M = [1 0 1; 1 1 0; 3 0 2]; H = idtf(M);```

`H` is an `idtf` model that describes a three input (`Nu=3`), three output (`Ny=3`) transfer function. Each input/output channel is an estimable static gain. The initial values of the gains are given by the values in the matrix `M`.

Convert a state-space model with identifiable parameters to a transfer function with identifiable parameters.

Convert the following identifiable state-space model to an identifiable transfer function.

`$\begin{array}{l}\underset{}{\overset{\sim }{x}}\left(t\right)=\left[\begin{array}{cc}-0.2& 0\\ 0& -0.3\end{array}\right]x\left(t\right)+\left[\begin{array}{c}-2\\ 4\end{array}\right]u\left(t\right)+\left[\begin{array}{c}0.1\\ 0.2\end{array}\right]e\left(t\right)\\ y\left(t\right)=\left[\begin{array}{cc}1& 1\end{array}\right]x\left(t\right)\end{array}$`

```A = [-0.2, 0; 0, -0.3]; B = [2;4]; C = [1, 1]; D = 0; K = [0.1; 0.2]; sys0 = idss(A,B,C,D,K,'NoiseVariance',0.1); sys = idtf(sys0);```

`A`, `B`, `C`, `D` and `K` are matrices that specify `sys0`, an identifiable state-space model with a noise variance of 0.1.

`sys = idtf(sys0)` creates an `idtf` model, `sys`.

Load time-domain system response data and use it to estimate a transfer function for the system.

```load iddata1 z1; np = 2; sys = tfest(z1,np);```

`z1` is an `iddata` object that contains time-domain, input-output data.

`np` specifies the number of poles in the estimated transfer function.

`sys` is an `idtf` model containing the estimated transfer function.

To see the numerator and denominator coefficients of the resulting estimated model `sys`, enter:

`sys.Numerator`
```ans = 1×2 2.4554 176.9856 ```
`sys.Denominator`
```ans = 1×3 1.0000 3.1625 23.1631 ```

To view the uncertainty in the estimates of the numerator and denominator and other information, use `tfdata`.

Create an array of transfer function models with identifiable coefficients. Each transfer function in the array is of the form:

`$H\left(s\right)=\frac{a}{s+a}.$`

The initial value of the coefficient $a$ varies across the array, from 0.1 to 1.0, in increments of 0.1.

```H = idtf(zeros(1,1,10)); for k = 1:10 num = k/10; den = [1 k/10]; H(:,:,k) = idtf(num,den); end```

The first command preallocates a one-dimensional, 10-element array, `H`, and fills it with empty `idtf` models.

The first two dimensions of a model array are the output and input dimensions. The remaining dimensions are the array dimensions. `H(:,:,k)` represents the ${k}^{th}$ model in the array. Thus, the `for` loop replaces the ${k}^{th}$ entry in the array with a transfer function whose coefficients are initialized with $a=k/10$.

## Input Arguments

 `num` Initial values of transfer function numerator coefficients. For SISO transfer functions, specify the initial values of the numerator coefficients `num` as a row vector. Specify the coefficients in order of: Descending powers of s or p (for continuous-time transfer functions)Ascending powers of z–1 or q–1 (for discrete-time transfer functions) Use `NaN` for any coefficient whose initial value is not known. For MIMO transfer functions with `Ny` outputs and `Nu` inputs, `num` is a `Ny`-by-`Nu` cell array of numerator coefficients for each input/output pair. `den` Initial values of transfer function denominator coefficients. For SISO transfer functions, specify the initial values of the denominator coefficients `den` as a row vector. Specify the coefficients in order of: Descending powers of s or p (for continuous-time transfer functions)Ascending powers of z–1 or q–1 (for discrete-time transfer functions) The leading coefficient in `den` must be 1. Use `NaN` for any coefficient whose initial value is not known. For MIMO transfer functions with `Ny` outputs and `Nu` inputs, `den` is a `Ny`-by-`Nu` cell array of denominator coefficients for each input/output pair. `Ts` Sample time. For continuous-time models, `Ts = 0`. For discrete-time models, `Ts` is a positive scalar representing the sampling period. This value is expressed in the unit specified by the `TimeUnit` property of the model. To denote a discrete-time model with unspecified sample time, set ```Ts = -1```. Changing this property does not discretize or resample the model. Use `c2d` and `d2c` to convert between continuous- and discrete-time representations. Use `d2d` to change the sample time of a discrete-time system. Default: `0` (continuous time) `sys0` Dynamic system. Any dynamic system to convert to an `idtf` model. When `sys0` is an identified model, its estimated parameter covariance is lost during conversion. If you want to translate the estimated parameter covariance during the conversion, use `translatecov`.

### Name-Value Pair 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 additional properties of `idtf` models during model creation. For example, `idtf(num,den,'InputName','Voltage')` creates an `idtf` model with the `InputName` property set to `Voltage`.

## Properties

`idtf` object properties include:

 `Numerator` Values of transfer function numerator coefficients. If you create an `idtf` model `sys` using the `idtf` command, `sys.Numerator` contains the initial values of numerator coefficients that you specify with the `num` input argument. If you obtain an `idtf` model by identification using `tfest`, then `sys.Numerator` contains the estimated values of the numerator coefficients. For an `idtf` model `sys`, the property `sys.Numerator` is an alias for the value of the property `sys.Structure.Numerator.Value`. For SISO transfer functions, the values of the numerator coefficients are stored as a row vector in order of: Descending powers of s or p (for continuous-time transfer functions)Ascending powers of z–1 or q–1 (for discrete-time transfer functions) Any coefficient whose initial value is not known is stored as `NaN`. For MIMO transfer functions with `Ny` outputs and `Nu` inputs, `Numerator` is a `Ny`-by-`Nu` cell array of numerator coefficients for each input/output pair. `Denominator` Values of transfer function denominator coefficients. If you create an `idtf` model `sys` using the `idtf` command, `sys.Denominator` contains the initial values of denominator coefficients that you specify with the `den` input argument. If you obtain an `idtf` model `sys` by identification using `tfest`, then `sys.Denominator` contains the estimated values of the denominator coefficients. For an `idtf` model `sys`, the property `sys.Denominator` is an alias for the value of the property `sys.Structure.Denominator.Value`. For SISO transfer functions, the values of the denominator coefficients are stored as a row vector in order of: Descending powers of s or p (for continuous-time transfer functions)Ascending powers of z–1 or q–1 (for discrete-time transfer functions) The leading coefficient in `Denominator` is fixed to 1. Any coefficient whose initial value is not known is stored as `NaN`. For MIMO transfer functions with `Ny` outputs and `Nu` inputs, `Denominator` is a `Ny`-by-`Nu` cell array of denominator coefficients for each input/output pair. `Variable` Transfer function display variable, specified as one of the following values: `'s'` — Default for continuous-time models`'p'` — Equivalent to `'s'``'z^-1'` — Default for discrete-time models`'q^-1'` — Equivalent to `'z^-1'` The value of `Variable` is reflected in the display, and also affects the interpretation of the `num` and `den` coefficient vectors for discrete-time models. For `Variable = 'z^-1'` or `'q^-1'`, the coefficient vectors are ordered as ascending powers of the variable. `IODelay` Transport delays. `IODelay` is a numeric array specifying a separate transport delay for each input/output pair. If you create an `idtf` model `sys` using the `idtf` command, `sys.IODelay` contains the initial values of the transport delay that you specify with a `Name,Value` argument pair. If you obtain an `idtf` model `sys` by identification using `tfest`, then `sys.IODelay` contains the estimated values of the transport delay. For an `idtf` model `sys`, the property `sys.IODelay` is an alias for the value of the property `sys.Structure.IODelay.Value`. For continuous-time systems, transport delays are expressed in the time unit stored in the `TimeUnit` property. For discrete-time systems, specify transport are expressed as integers denoting delay of a multiple of the sample time `Ts`. For a MIMO system with `Ny` outputs and `Nu` inputs, set `IODelay` as a `Ny`-by-`Nu` array. Each entry of this array is a numerical value representing the transport delay for the corresponding input/output pair. You can set `IODelay` to a scalar value to apply the same delay to all input/output pairs. Default: `0` for all input/output pairs `Structure` Information about the estimable parameters of the `idtf` model. `Structure.Numerator`, `Structure.Denominator`, and `Structure.IODelay` contain information about the numerator coefficients, denominator coefficients, and transport delay, respectively. Each contains the following fields: `Value` — Parameter values. For example, `sys.Structure.Numerator.Value` contains the initial or estimated values of the numerator coefficients. `NaN` represents unknown parameter values. For denominators, the value of the leading coefficient, specified by `sys.Structure.Denominator.Value(1)` is fixed to 1.For SISO models, `sys.Numerator`, `sys.Denominator`, and `sys.IODelay` are aliases for `sys.Structure.Numerator.Value`, `sys.Structure.Denominator.Value`, and `sys.Structure.IODelay.Value`, respectively.For MIMO models, `sys.Numerator{i,j}` is an alias for `sys.Structure(i,j).Numerator.Value`, and `sys.Denominator{i,j}` is an alias for `sys.Structure(i,j).Denominator.Value`. Additionally, `sys.IODelay(i,j)` is an alias for `sys.Structure(i,j).IODelay.Value``Minimum` — Minimum value that the parameter can assume during estimation. For example, ```sys.Structure.IODelay.Minimum = 0.1``` constrains the transport delay to values greater than or equal to 0.1. `sys.Structure.IODelay.Minimum` must be greater than or equal to zero.`Maximum` — Maximum value that the parameter can assume during estimation.`Free` — Boolean specifying whether the parameter is a free estimation variable. If you want to fix the value of a parameter during estimation, set the corresponding ```Free = false```. For example, ```sys.Structure.Denominator.Free = false``` fixes all of the denominator coefficients in `sys` to the values specified in `sys.Structure.Denominator.Value`.For denominators, the value of `Free` for the leading coefficient, specified by `sys.Structure.Denominator.Free(1)`, is always `false` (the leading denominator coefficient is always fixed to 1).`Scale` — Scale of the parameter’s value. `Scale` is not used in estimation.`Info` — Structure array for storing parameter units and labels. The structure has `Label` and `Unit` fields.Specify parameter units and labels as character vectors. For example, `'Time'`. For a MIMO model with `Ny` outputs and `Nu` input, `Structure` is an `Ny`-by-`Nu` array. The element `Structure(i,j)` contains information corresponding to the transfer function for the `(i,j)` input-output pair. `NoiseVariance` The variance (covariance matrix) of the model innovations e. An identified model includes a white, Gaussian noise component e(t). `NoiseVariance` is the variance of this noise component. Typically, the model estimation function (such as `tfest`) determines this variance. For SISO models, `NoiseVariance` is a scalar. For MIMO models, `NoiseVariance` is a Ny-by-Ny matrix, where Ny is the number of outputs in the system. `Report` Summary report that contains information about the estimation options and results when the transfer function model is obtained using estimation commands, such as `tfest` and `impulseest`. Use `Report` to query a model for how it was estimated, including its: Estimation methodEstimation optionsSearch termination conditionsEstimation data fit and other quality metrics The contents of `Report` are irrelevant if the model was created by construction. ```m = idtf([1 4],[1 20 5]); m.Report.OptionsUsed``` ```ans = []``` If you obtain the transfer function model using estimation commands, the fields of `Report` contain information on the estimation data, options, and results. ```load iddata2 z2; m = tfest(z2,3); m.Report.OptionsUsed``` ``` InitializeMethod: 'iv' InitializeOptions: [1x1 struct] InitialCondition: 'auto' Focus: 'simulation' EstimateCovariance: 1 Display: 'off' InputOffset: [] OutputOffset: [] Regularization: [1x1 struct] SearchMethod: 'auto' SearchOptions: [1x1 idoptions.search.identsolver] OutputWeight: [] Advanced: [1x1 struct]``` `Report` is a read-only property. For more information on this property and how to use it, see the Output Arguments section of the corresponding estimation command reference page and Estimation Report. `InputDelay` Input delays. `InputDelay` is a numeric vector specifying a time delay for each input channel. For continuous-time systems, specify input delays in the time unit stored in the `TimeUnit` property. For discrete-time systems, specify input delays in integer multiples of the sample time `Ts`. For example, ```InputDelay = 3``` means a delay of three sample times. For a system with `Nu` inputs, set `InputDelay` to an `Nu`-by-1 vector. Each entry of this vector is a numerical value representing the input delay for the corresponding input channel. You can also set `InputDelay` to a scalar value to apply the same delay to all channels. Estimation treats `InputDelay` as a fixed constant of the model. Estimation uses the `IODelay` property for estimating time delays. To specify initial values and constraints for estimation of time delays, use `sys.Structure.IODelay`. Default: `0` for all input channels `OutputDelay` Output delays. For identified systems, like `idtf`, `OutputDelay` is fixed to zero. `Ts` Sample time. For continuous-time models, `Ts = 0`. For discrete-time models, `Ts` is a positive scalar representing the sampling period. This value is expressed in the unit specified by the `TimeUnit` property of the model. To denote a discrete-time model with unspecified sample time, set ```Ts = -1```. Changing this property does not discretize or resample the model. Use `c2d` and `d2c` to convert between continuous- and discrete-time representations. Use `d2d` to change the sample time of a discrete-time system. Default: `0` (continuous time) `TimeUnit` Units for the time variable, the sample time `Ts`, and any time delays in the model, specified as one of the following values:`'nanoseconds'``'microseconds'``'milliseconds'``'seconds'` `'minutes'``'hours'``'days'``'weeks'``'months'``'years'` Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use `chgTimeUnit` to convert between time units without modifying system behavior. Default: `'seconds'` `InputName` Input channel names, specified as one of the following: Character vector — For single-input models, for example, `'controls'`.Cell array of character vectors — For multi-input models. Alternatively, use automatic vector expansion to assign input names for multi-input models. For example, if `sys` is a two-input model, enter: `sys.InputName = 'controls';` The input names automatically expand to `{'controls(1)';'controls(2)'}`. When you estimate a model using an `iddata` object, `data`, the software automatically sets `InputName` to `data.InputName`. You can use the shorthand notation `u` to refer to the `InputName` property. For example, `sys.u` is equivalent to `sys.InputName`. Input channel names have several uses, including: Identifying channels on model display and plotsExtracting subsystems of MIMO systemsSpecifying connection points when interconnecting models Default: `''` for all input channels `InputUnit` Input channel units, specified as one of the following: Character vector — For single-input models, for example, `'seconds'`.Cell array of character vectors — For multi-input models. Use `InputUnit` to keep track of input signal units. `InputUnit` has no effect on system behavior. Default: `''` for all input channels `InputGroup` Input channel groups. The `InputGroup` property lets you assign the input channels of MIMO systems into groups and refer to each group by name. Specify input groups as a structure. In this structure, field names are the group names, and field values are the input channels belonging to each group. For example: ```sys.InputGroup.controls = [1 2]; sys.InputGroup.noise = [3 5];``` creates input groups named `controls` and `noise` that include input channels 1, 2 and 3, 5, respectively. You can then extract the subsystem from the `controls` inputs to all outputs using: `sys(:,'controls')` Default: Struct with no fields `OutputName` Output channel names, specified as one of the following: Character vector — For single-output models. For example, `'measurements'`.Cell array of character vectors — For multi-output models. Alternatively, use automatic vector expansion to assign output names for multi-output models. For example, if `sys` is a two-output model, enter: `sys.OutputName = 'measurements';` The output names automatically expand to `{'measurements(1)';'measurements(2)'}`. When you estimate a model using an `iddata` object, `data`, the software automatically sets `OutputName` to `data.OutputName`. You can use the shorthand notation `y` to refer to the `OutputName` property. For example, `sys.y` is equivalent to `sys.OutputName`. Output channel names have several uses, including: Identifying channels on model display and plotsExtracting subsystems of MIMO systemsSpecifying connection points when interconnecting models Default: `''` for all output channels `OutputUnit` Output channel units, specified as one of the following: Character vector — For single-output models. For example, `'seconds'`.Cell array of character vectors — For multi-output models. Use `OutputUnit` to keep track of output signal units. `OutputUnit` has no effect on system behavior. Default: `''` for all output channels `OutputGroup` Output channel groups. The `OutputGroup` property lets you assign the output channels of MIMO systems into groups and refer to each group by name. Specify output groups as a structure. In this structure, field names are the group names, and field values are the output channels belonging to each group. For example: ```sys.OutputGroup.temperature = [1]; sys.InputGroup.measurement = [3 5];``` creates output groups named `temperature` and `measurement` that include output channels 1, and 3, 5, respectively. You can then extract the subsystem from all inputs to the `measurement` outputs using: `sys('measurement',:)` Default: Struct with no fields `Name` System name, specified as a character vector. For example, `'system_1'`. Default: `''` `Notes` Any text that you want to associate with the system, stored as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if `sys1` and `sys2` are dynamic system models, you can set their `Notes` properties as follows: ```sys1.Notes = "sys1 has a string."; sys2.Notes = 'sys2 has a character vector.'; sys1.Notes sys2.Notes``` ```ans = "sys1 has a string." ans = 'sys2 has a character vector.' ``` Default: `[0×1 string]` `UserData` Any type of data you want to associate with system, specified as any MATLAB® data type. Default: `[]` `SamplingGrid` Sampling grid for model arrays, specified as a data structure. For arrays of identified linear (IDLTI) models that are derived by sampling one or more independent variables, this property tracks the variable values associated with each model. This information appears when you display or plot the model array. Use this information to trace results back to the independent variables. Set the field names of the data structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables should be numeric and scalar valued, and all arrays of sampled values should match the dimensions of the model array. For example, if you collect data at various operating points of a system, you can identify a model for each operating point separately and then stack the results together into a single system array. You can tag the individual models in the array with information regarding the operating point: ```nominal_engine_rpm = [1000 5000 10000]; sys.SamplingGrid = struct('rpm', nominal_engine_rpm)``` where `sys` is an array containing three identified models obtained at rpms 1000, 5000 and 10000, respectively. For model arrays generated by linearizing a Simulink® model at multiple parameter values or operating points, the software populates `SamplingGrid` automatically with the variable values that correspond to each entry in the array. For example, the Simulink Control Design™ commands `linearize` and `slLinearizer` populate `SamplingGrid` in this way. Default: `[]`