# Create Function to Represent a Grey-Box Model

This example shows how to represent the structure of the following continuous-time model:

This equation represents an electrical motor, where is the angular position of the motor shaft, and is the angular velocity. The parameter is the inverse time constant of the motor, and is the static gain from the input to the angular velocity.

The motor is at rest at `t` = 0, but its angular position is unknown. Suppose that the approximate nominal values of the unknown parameters are , and . For more information about this example, see the section on state-space models in *System Identification: Theory for the User* , Second Edition, by Lennart Ljung, Prentice Hall PTR, 1999.

The continuous-time state-space model structure is defined by the following equation:

If you want to estimate the same model using a structured state-space representation, see `Estimating Structured Continuous-Time State-Space Models`.

To prepare this model for estimation:

- Create the following file to represent the model structure in this example:

```
function [A,B,C,D,K,x0] = myfunc(par,T)
A = [0 1; 0 par(1)];
B = [0;par(2)];
C = eye(2);
D = zeros(2,1);
K = zeros(2,2);
x0 = [par(3);0];
```

Save the file such that it is in the MATLAB® search path.

- Use the following syntax to define an
`idgrey`model object based on the`myfunc`file:

par = [-1; 0.25; 0]; aux = {}; T = 0; m = idgrey('myfunc',par,'c',aux,T);

where `par` represents a vector of all the user-defined parameters and contains their nominal (initial) values. In this example, all the scalar-valued parameters are grouped in the `par` vector. The scalar-valued parameters could also have been treated as independent input arguments to the ODE function `myfunc`. `'c'` specifies that the underlying parameterization is in continuous time. `aux` represents optional arguments. As `myfunc` does not have any optional arguments, use `aux = {}`. `T` specifies the sample time; `T = 0` indicates a continuous-time model.

Load the estimation data.

load(fullfile(matlabroot,'toolbox','ident','iddemos','data','dcmotordata')); data = iddata(y,u,0.1);

Use `greyest` to estimate the grey-box parameter values:

m_est = greyest(data,m);

where `data` is the estimation data and `m` is an estimation initialization `idgrey` model. `m_est` is the estimated `idgrey` model.