Documentation

idgrey

Linear ODE (grey-box model) with identifiable parameters

Syntax

```sys = idgrey(odefun,parameters,fcn_type) sys = idgrey(odefun,parameters,fcn_type,optional_args) sys = idgrey(odefun,parameters,fcn_type,optional_args,Ts) sys = idgrey(odefun,parameters,fcn_type,optional_args,Ts,Name,Value) ```

Description

`sys = idgrey(odefun,parameters,fcn_type)` creates a linear grey-box model with identifiable parameters, `sys`. `odefun` specifies the user-defined function that relates the model parameters, `parameters`, to its state-space representation.

`sys = idgrey(odefun,parameters,fcn_type,optional_args)` creates a linear grey-box model with identifiable parameters using the optional arguments required by `odefun`.

`sys = idgrey(odefun,parameters,fcn_type,optional_args,Ts)` creates a linear grey-box model with identifiable parameters with the specified sample time, `Ts`.

`sys = idgrey(odefun,parameters,fcn_type,optional_args,Ts,Name,Value)` creates a linear grey-box model with identifiable parameters with additional options specified by one or more `Name,Value` pair arguments.

Object Description

An `idgrey` model represents a system as a continuous-time or discrete-time state-space model with identifiable (estimable) coefficients.

A state-space model of a system with input vector, u, output vector, y, and disturbance, e, takes the following form in continuous time:

`$\begin{array}{l}\stackrel{˙}{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+Ke\left(t\right)\\ y\left(t\right)=Cx\left(t\right)+Du\left(t\right)+e\left(t\right)\end{array}$`

In discrete time, the state-space model takes the form:

`$\begin{array}{l}x\left[k+1\right]=Ax\left[k\right]+Bu\left[k\right]+Ke\left[k\right]\\ y\left[k\right]=Cx\left[k\right]+Du\left[k\right]+e\left[k\right]\end{array}$`

For `idgrey` models, the state-space matrices A, B, C, and D are expressed as a function of user-defined parameters using a MATLAB® function. You access estimated parameters using `sys.Structures.Parameters`, where `sys` is an `idgrey` model.

Use an `idgrey` model when you know the system of equations governing the system dynamics explicitly. You should be able to express these dynamics in the form of ordinary differential or difference equations. You specify complex relationships and constraints among the parameters that cannot be done through structured state-space models (`idss`).

You can create an `idgrey` model using the `idgrey` command. To do so, write a MATLAB function that returns the A, B, C, and D matrices for given values of the estimable parameters and sample time. The MATLAB function can also return the K matrix and accept optional input arguments. The matrices returned may represent a continuous-time or discrete-time model, as indicated by the sample time.

Use the estimating functions `pem` or `greyest` to obtain estimated values for the unknown parameters of an `idgrey` model.

You can convert an `idgrey` model into other dynamic systems, such as `idpoly`, `idss`, `tf`, `ss` etc. You cannot convert a dynamic system into an `idgrey` model.

Examples

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Create an `idgrey` model to represent a DC motor. Specify the motor time-constant as an estimable parameter and that the ODE function can return continuous- or discrete-time state-space matrices.

Create the `idgrey` model.

```odefun = 'motorDynamics'; parameters = 1; fcn_type = 'cd'; optional_args = 0.25; Ts = 0; sys = idgrey(odefun,parameters,fcn_type,optional_args,Ts);```

`sys` is an `idgrey` model that is configured to use the shipped file `motorDynamics.m` to return the $A$, $B$, $C$, $D$, and $K$ matrices. `motorDynamics.m` also returns the initial conditions, $X0$. The motor constant, $\tau$, is defined in `motorDynamics.m` as an estimable parameter, and `parameters = 1` specifies its initial value as 1.

You can use `pem` or `greyest` to refine the estimate for $\tau$.

Specify the known parameters of a grey-box model as fixed for estimation. Also specify a minimum bound for an estimable parameter.

Create an ODE file that relates the pendulum model coefficients to its state-space representation. Save this function as `LinearPendulum.m` such that it is in the MATLAB® search path.

```function [A,B,C,D] = LinearPendulum(m,g,l,b,Ts) A = [0 1; -g/l, -b/m/l^2]; B = zeros(2,0); C = [1 0]; D = zeros(1,0); end ```

In this function:

• `m` is the pendulum mass.

• `g` is the gravitational acceleration.

• `l` is the pendulum length.

• `b` is the viscous friction coefficient.

• `Ts` is the model sample time.

Create a linear grey-box model associated with the ODE function.

```odefun = 'LinearPendulum'; m = 1; g = 9.81; l = 1; b = 0.2; parameters = {'mass',m;'gravity',g;'length',l;'friction',b}; fcn_type = 'c'; sys = idgrey(odefun,parameters,fcn_type); ```

`sys` has four parameters.

Specify the known parameters, `m`, `g`, and `l`, as fixed for estimation.

```sys.Structure.Parameters(1).Free = false; sys.Structure.Parameters(2).Free = false; sys.Structure.Parameters(3).Free = false; ```

`m`, `g`, and `l` are the first three parameters of `sys`.

Specify a zero lower bound for `b`, the fourth parameter of `sys`.

```sys.Structure.Parameters(4).Minimum = 0; ```

Similarly, to specify an upper bound for an estimable parameter, use the `Maximum` field of the parameter.

Create a grey-box model with identifiable parameters. Name the input and output channels of the model, and specify seconds for the model time units.

Use `Name,Value` pair arguments to specify additional model properties on model creation.

```odefun = 'motorDynamics'; parameters = 1; fcn_type = 'cd'; optional_args = 0.25; Ts = 0; sys = idgrey(odefun,parameters,fcn_type,optional_args,Ts,'InputName','Voltage',... 'OutputName',{'Angular Position','Angular Velocity'});```

To change or specify more attributes of an existing model, you can use dot notation. For example:

`sys.TimeUnit = 'seconds'; `

Use the `stack` command to create an array of linear grey-box models.

```odefun1 = @motorDynamics; parameters1 = [1 2]; fcn_type = 'cd'; optional_args1 = 1; sys1 = idgrey(odefun1,parameters1,fcn_type,optional_args1); odefun2 = 'motorDynamics'; parameters2 = {[1 2]}; optional_args2 = 0.5; sys2 = idgrey(odefun2,parameters2,fcn_type,optional_args2); sysarr = stack(1,sys1,sys2);```

`stack` creates a 2-by-1 array of `idgrey` models, `sysarr`.

Input Arguments

 `odefun` MATLAB function that relates the model parameters to its state-space representation. `odefun` specifies the name of a MATLAB function (.m, .p, a function handle or .mex* file). This function establishes the relationship between the model parameters, `parameters`, and its state-space representation. The function may optionally relate the model parameters to the disturbance matrix and initial states. If the function is not on the MATLAB path, then specify the full file name, including the path. The syntax for `odefun` must be as follows: `[A,B,C,D] = odefun(par1,par2,...,parN,Ts,optional_arg1,optional_arg2,...)` The function outputs describe the model in the following linear state-space innovations form: `$\begin{array}{c}xn\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+Ke\left(t\right);x\left(0\right)={x}_{0}\\ y\left(t\right)=Cx\left(t\right)+Du\left(t\right)+e\left(t\right)\end{array}$` In discrete time xn(t)=x(t+Ts) and in continuous time, $xn\left(t\right)=\stackrel{˙}{x}\left(t\right)$. `par1,par2,...,parN` are model parameters. Each entry may be a scalar, vector or matrix. `Ts` is the sample time. `optional_arg1,optional_arg2,...` are the optional inputs that `odefun` may require. The values of the optional input arguments are unchanged through the estimation process. However, the values of `par1,par2,...,parN` are updated during estimation to fit the data. Use optional input arguments to vary the constants and coefficients used by your model without editing `odefun`. The disturbance matrix, K, and the initial state values, x0, are not parametrized. Instead, these values are determined separately, using the `DisturbanceModel` and `InitialState` estimation options, respectively. For more information regarding the estimation options, see `greyestOptions`. A good choice for achieving the best simulation results is to set the `DisturbanceModel` option to `'none'`, which fixes K to zero. (Optional) Parameterizing Disturbance: `odefun` can also return the disturbance component, K, using the syntax: `[A,B,C,D,K] = odefun(par1,par2,...,parN,Ts,optional_arg1,optional_arg2,...)` If `odefun` returns a value for K that contains `NaN` values, then the estimating function assumes that K is not parameterized. In this case, the value of the `DisturbanceModel` estimation option determines how K is handled. (Optional) Parameterizing Initial State Values: To make the model initial states, X0, dependent on the model parameters, use the following syntax for `odefun`: `[A,B,C,D,K,X0] = odefun(par1,par2,...,parN,Ts,optional_arg1,optional_arg2,...)` If `odefun` returns a value for X0 that contains `NaN` values, then the estimating function assumes that X0 is not parameterized. In this case, X0 may be fixed to zero or estimated separately, using the `InitialStates` estimation option. `parameters` Initial values of the parameters required by `odefun`. Specify `parameters` as a cell array containing the parameter initial values. If your model requires only one parameter, which may itself be a vector or a matrix, you may specify `parameters` as a matrix. You may also specify parameter names using an N-by-2 cell array, where N is the number of parameters. The first column specifies the names, and the second column specifies the values of the parameters. For example: `parameters = {'mass',par1;'stiffness',par2;'damping',par3}` `fcn_type` Indicates whether the model is parameterized in continuous-time, discrete-time, or both. `fcn_type` requires one of the following values: `'c'` — `odefun` returns matrices corresponding to a continuous-time system, regardless of the value of `Ts`.`'d'` — `odefun` returns matrices corresponding to a discrete-time system, whose values may or may not depend on the value of `Ts`.`'cd'` — `odefun` returns matrices corresponding to a continuous-time system, if `Ts=0`. Otherwise, if `Ts>0`, `odefun` returns matrices corresponding to a discrete-time system. Select this option to sample your model using the values returned by `odefun`, rather than using the software’s internal sample time conversion routines. `optional_args` Optional input arguments required by `odefun`. Specify `optional_args` as a cell array. If `odefun` does not require optional input arguments, specify `optional_args` as `{}`. `Ts` Model sample time. If `Ts` is unspecified, it is assumed to be: `-1` — If `fcn_type` is `'d'` or `'cd'`.`Ts = -1` indicates a discrete-time model with unknown sample time.`0` — If `fcn_type` is `'c'`.`Ts = 0` indicates a continuous-time model. `Name,Value` Specify optional comma-separated pairs of `Name,Value` arguments, where `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single 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 `idgrey` models during model creation. For example, `idgrey(odefun,parameters,fcn_type,'InputName','Voltage')` creates an `idgrey` model with the `InputName` property set to `Voltage`.

Properties

`idgrey` object properties include:

 `A,B,C,D` Values of state-space matrices. `A` — State matrix A, an Nx-by-Nx matrix, as returned by the ODE function associated with the `idgrey` model. Nx is the number of states.`B` — Input-to-state matrix B, an Nx-by-Nu matrix, as returned by the ODE function associated with the `idgrey` model. Nu is the number of inputs and Nx is the number of states.`C` — State-to-output matrix C, an Ny-by-Nx matrix, as returned by the ODE function associated with the `idgrey` model. Nx is the number of states and Ny is the number of outputs.`D` — Feedthrough matrix D, an Ny-by-Nu matrix, as returned by the ODE function associated with the `idgrey` model. Ny is the number of outputs and Nu is the number of inputs. The values `A,B,C,D` are returned by the ODE function associated with the `idgrey` model. Thus, you can only read these matrices; you cannot set their values. `K` Value of state disturbance matrix, K `K` is Nx-by-Ny matrix, where Nx is the number of states and Ny is the number of outputs. If `odefun` parameterizes the K matrix, then `K` has the value returned by `odefun`. `odefun` parameterizes the K matrix if it returns at least five outputs and the value of the fifth output does not contain `NaN` values.If `odefun` does not parameterize the K matrix, then `K` is a zero matrix of size Nx-by-Ny. Nx is the number of states and Ny is the number of outputs. The value is treated as a fixed value of the K matrix during estimation. To make the value estimable, use the `DisturbanceModel` estimation option.Regardless of whether the K matrix is parameterized by `odefun` or not, you can set the value of the `K` property explicitly as an Nx-by-Ny matrix. Nx is the number of states and Ny is the number of outputs. The specified value is treated as a fixed value of the K matrix during estimation. To make the value estimable, use the `DisturbanceModel` estimation option. To create an estimation option set for `idgrey` models, use `greyestOptions`. `StateName` State names, specified as one of the following: Character vector — For first-order models, for example, `'velocity'`.Cell array of character vectors — For models with two or more states`''` — For unnamed states. Default: `''` for all states `StateUnit` State units, specified as one of the following: Character vector — For first-order models, for example, `'velocity'`Cell array of character vectors — For models with two or more states`''` — For states without specified units Use `StateUnit` to keep track of the units each state is expressed in. `StateUnit` has no effect on system behavior. Default: `''` for all states `Structure` Information about the estimable parameters of the `idgrey` model. `Structure` stores information regarding the MATLAB function that parameterizes the `idgrey` model. `Structure.Function` — Name or function handle of the MATLAB function used to create the `idgrey` model.`Structure.FunctionType` — Indicates whether the model is parameterized in continuous-time, discrete-time, or both.`Structure.Parameters` — Information about the estimated parameters. `Structure.Parameters` contains the following fields:`Value` — Parameter values. For example, `sys.Structure.Parameters(2).Value` contains the initial or estimated values of the second parameter.`NaN` represents unknown parameter values.`Minimum` — Minimum value that the parameter can assume during estimation. For example, ```sys.Structure.Parameters(1).Minimum = 0``` constrains the first parameter to be greater than or equal to zero.`Maximum` — Maximum value that the parameter can assume during estimation.`Free` — Boolean value specifying whether the parameter is estimable. If you want to fix the value of a parameter during estimation, set `Free = false` for the corresponding entry.`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'`.`Structure.ExtraArguments` — Optional input arguments required by the ODE function.`Structure.StateName` — Names of the model states.`Structure.StateUnit` — Units of the model states. `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 greyest or pem) 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 grey-box model is obtained using the `greyest` estimation command. 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. ```odefun = 'motorDynamics'; m = idgrey(odefun,1,'cd',0.25,0); m.Report.OptionsUsed``` ```ans = []``` If you obtain the grey-box model using estimation commands, the fields of `Report` contain information on the estimation data, options, and results. ```load(fullfile(matlabroot,'toolbox','ident','iddemos','data','dcmotordata')); data = iddata(y,u,0.1,'Name','DC-motor'); odefun = 'motorDynamics'; init_sys = idgrey('motorDynamics',1,'cd',0.25,0); m = greyest(data,init_sys); m.Report.OptionsUsed``` ```InitialState: 'auto' DisturbanceModel: 'auto' Focus: 'prediction' EstimateCovariance: 1 Display: 'off' InputOffset: [] OutputOffset: [] Regularization: [1x1 struct] OutputWeight: [] SearchMethod: 'auto' SearchOptions: [1x1 idoptions.search.identsolver] 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 delay for each input channel, specified as a scalar value or numeric vector. 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 that represents 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. Default: 0 `OutputDelay` Output delays. For identified systems, like `idgrey`, `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 sample time 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. For `idgrey` models, there is no unique default value for `Ts`. `Ts` depends on the value of `fcn_type`. `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: `[]`

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