# greyest

Estimate ODE parameters of linear grey-box model

## Description

incorporates a `sys`

= greyest(`data`

,`init_sys`

,`opt`

)`greyestOptions`

option set
`opt`

that specifies options such as handling of initial states and
regularization options.

## Examples

### Estimate Grey-Box Model

Estimate the parameters of a DC motor using the linear grey-box framework.

Load the measured data.

load(fullfile(matlabroot, 'toolbox', 'ident', 'iddemos', 'data', 'dcmotordata')); data = iddata(y, u, 0.1, 'Name', 'DC-motor'); data.InputName = 'Voltage'; data.InputUnit = 'V'; data.OutputName = {'Angular position', 'Angular velocity'}; data.OutputUnit = {'rad', 'rad/s'}; data.Tstart = 0; data.TimeUnit = 's';

`data`

is an `iddata`

object containing the measured data for the outputs, the angular position, the angular velocity. It also contains the input, the driving voltage.

Create a grey-box model representing the system dynamics.

For the DC motor, choose the angular position (rad) and the angular velocity (rad/s) as the outputs and the driving voltage (V) as the input. Set up a linear state-space structure of the following form:

$$\underset{}{\overset{\dot{}}{x}}(t)=\left[\begin{array}{cc}0& 1\\ 0& -\frac{1}{\tau}\end{array}\right]x(t)+\left[\begin{array}{c}0\\ \frac{G}{\tau}\end{array}\right]u(t)$$

$$y(t)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]x(t).$$

$$\tau $$ is the time constant of the motor in seconds, and $$G$$ is the static gain from the input to the angular velocity in rad/(V*s) .

G = 0.25; tau = 1; init_sys = idgrey('motorDynamics',tau,'cd',G,0);

The governing equations in state-space form are represented in the MATLAB® file `motorDynamics.m`

. To view the contents of this file, enter `edit motorDynamics.m`

at the MATLAB command prompt.

$$G$$ is a known quantity that is provided to `motorDynamics.m`

as an optional argument.

$$\tau $$ is a free estimation parameter.

`init_sys`

is an `idgrey`

model associated with `motor.m`

.

Estimate $$\tau $$.

sys = greyest(data,init_sys);

`sys`

is an `idgrey`

model containing the estimated value of $$\tau $$.

To obtain the estimated parameter values associated with `sys`

, use `getpvec(sys)`

.

Analyze the result.

opt = compareOptions('InitialCondition','zero'); compare(data,sys,Inf,opt)

`sys`

provides a 98.35% fit for the angular position and an 84.42% fit for the angular velocity.

### Estimate Grey-Box Model Using Regularization

Estimate the parameters of a DC motor by incorporating prior information about the parameters when using regularization constants.

The model is parameterized by static gain `G`

and time constant $$\tau $$. From prior knowledge, it is known that `G`

is about 4 and $$\tau $$ is about 1. Also, you have more confidence in the value of $$\tau $$ than `G`

and would like to guide the estimation to remain close to the initial guess.

Load estimation data.

load regularizationExampleData.mat motorData

The data contains measurements of motor's angular position and velocity at given input voltages.

Create an `idgrey`

model for DC motor dynamics. Use the function `DCMotorODE`

that represents the structure of the grey-box model.

mi = idgrey(@DCMotorODE,{'G', 4; 'Tau', 1},'cd',{}, 0); mi = setpar(mi, 'label', 'default');

If you want to view the `DCMotorODE`

function, type:

`type DCMotorODE.m`

function [A,B,C,D] = DCMotorODE(G,Tau,Ts) %DCMOTORODE ODE file representing the dynamics of a DC motor parameterized %by gain G and time constant Tau. % % [A,B,C,D,K,X0] = DCMOTORODE(G,Tau,Ts) returns the state space matrices % of the DC-motor with time-constant Tau and static gain G. The sample % time is Ts. % % This file returns continuous-time representation if input argument Ts % is zero. If Ts>0, a discrete-time representation is returned. % % See also IDGREY, GREYEST. % Copyright 2013 The MathWorks, Inc. A = [0 1;0 -1/Tau]; B = [0; G/Tau]; C = eye(2); D = [0;0]; if Ts>0 % Sample the model with sample time Ts s = expm([[A B]*Ts; zeros(1,3)]); A = s(1:2,1:2); B = s(1:2,3); end

Specify regularization options Lambda.

opt = greyestOptions; opt.Regularization.Lambda = 100;

Specify regularization options R.

opt.Regularization.R = [1, 1000];

You specify more weighting on the second parameter because you have more confidence in the value of $$\tau $$ than `G`

.

Specify the initial values of the parameters as regularization option $$\theta $$*.

`opt.Regularization.Nominal = 'model';`

Estimate the regularized grey-box model.

sys = greyest(motorData, mi, opt);

## Input Arguments

`data`

— Estimation data

timetable | numeric matrix pair | `iddata`

object | `frd`

object | `idfrd`

object

Uniformly sampled estimation data, specified as a timetable, a comma-separated matrix pair, or a time-domain or frequency-domain data object, as the following sections describe.

`data`

has the same input and output dimensions as
`init_sys`

.

By default, the software sets the sample time of the model to the sample time of the estimation data.

#### Timetable

Specify `data`

as a `timetable`

that uses a regularly spaced time vector.
`data`

contains variables representing input and output
channels.

#### Comma-Separated Matrix Pair

Specify `data`

as a comma-separated pair of numeric matrices
that contain input and output time-domain signal values. Use this
`data`

specification only for discrete-time systems.

#### Data Object

Specify data as an `iddata`

, `idfrd`

, or
`frd`

object.

For time-domain estimation,

`data`

must be a time-domain`iddata`

object containing the input and output signal values.For frequency-domain estimation,

`data`

can be one of the following:

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

`opt`

— Estimation options

`greyestOptions`

option set

Estimation options, specified as a `greyestOptions`

option set. Options specified by `opt`

include:

Handling of initial states

Regularization

Numerical search method used for estimation

## Output Arguments

`sys`

— Identified linear grey-box model

`idgrey`

model

Estimated grey-box model, returned as an `idgrey`

model. This model is created using the specified initial system,
and estimation options.

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. | ||||||||||||||||||

`InitialState` | Handling of initial states during estimation, returned as one of the following: `'model'` — The initial state is parameterized by the ODE file used by the`idgrey` model.`'zero'` — The initial state is set to zero.`'estimate'` — The initial state is treated as an independent estimation parameter.`'backcast'` — The initial state is estimated using the best least squares fit.Vector of doubles of length *Nx*, where*Nx*is the number of states. For multiexperiment data, a matrix with*Ne*columns, where*Ne*is the number of experiments.
This field is especially useful to view how the initial
states were handled when the | ||||||||||||||||||

`DisturbanceModel` | Handling of the disturbance component ( `'model'` —*K*values are parameterized by the ODE file used by the`idgrey` model.`'fixed'` — The value of the`K` property of the`idgrey` model is fixed to its original value.`'none'` —*K*is fixed to zero.`'estimate'` —*K*is treated as an independent estimation parameter.
This field is especially useful to view the how the
disturbance component was handled when the | ||||||||||||||||||

`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 the following fields:
| ||||||||||||||||||

`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 | ||||||||||||||||||

`RandState` | State of the random number stream at the start of estimation. Empty,
| ||||||||||||||||||

`DataUsed` | Attributes of the data used for estimation, returned as a structure with the following fields.
| ||||||||||||||||||

`Termination` | Termination conditions for the iterative search used for prediction error minimization, returned as a structure with the following fields:
For estimation methods that do not require numerical search optimization,
the |

For more information on using `Report`

, see Estimation Report.

`x0`

— Initial states

matrix

Initial states computed during the estimation, returned as a matrix containing a column vector corresponding to each experiment.

This array is also stored in the `Parameters`

field of the model
`Report`

property.

## Version History

**Introduced in R2012a**

### R2022b: Time-domain estimation data is accepted in the form of timetables and matrices

Most estimation, validation, analysis, and utility functions now accept time-domain
input/output data in the form of a single timetable that contains both input and output data
or a pair of matrices that contain the input and output data separately. These functions
continue to accept `iddata`

objects as a data source as well, for
both time-domain and frequency-domain data.

### R2018a: Advanced Options are deprecated for `SearchOptions`

when `SearchMethod`

is `'lsqnonlin'`

Specification of `lsqnonlin`

- related advanced options are deprecated,
including the option to invoke parallel processing when estimating using the
`lsqnonlin`

search method, or solver, in Optimization Toolbox™.

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