Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

This example shows how to create a single-input and single-output
grey-box model structure when you know the variance of the measurement
noise. The code in this example uses the Control System
Toolbox™ command `kalman`

for computing the Kalman gain
from the known and estimated noise variance.

This example is based on a discrete, single-input and single-output (SISO) system represented by the following state-space equations:

$$\begin{array}{l}x(kT+T)=\left[\begin{array}{cc}par1& par2\\ 1& 0\end{array}\right]x(kT)+\left[\begin{array}{c}1\\ 0\end{array}\right]u(kT)+w(kT)\\ y(kT)=\left[\begin{array}{cc}par3& par4\end{array}\right]x(kT)+e(kT)\\ x(0)=x0\end{array}$$

where *w* and *e* are
independent white-noise terms with covariance matrices *R1* and *R2*,
respectively. *R1*=E{ww'} is a 2–by-2 matrix
and *R2*=E{ee'} is a scalar. *par1*, *par2*, *par3*,
and *par4* represent the unknown parameter values
to be estimated.

Assume
that you know the variance of the measurement noise *R2* to
be 1. *R1*(1,1) is unknown and is treated as an
additional parameter *par5*. The remaining elements
of *R1* are known to be zero.

You can represent the system described in Description of the SISO System as an `idgrey`

(grey-box)
model using a function. Then, you can use this file and the `greyest`

command
to estimate the model parameters based on initial parameter guesses.

To run this example, you must load an input-output data set
and represent it as an `iddata`

or `idfrd`

object
called `data`

. For more information about this operation,
see Representing Time- and Frequency-Domain Data Using iddata Objects or Representing Frequency-Response Data Using idfrd Objects.

To estimate the parameters of a grey-box model:

Create the file

`mynoise`

that computes the state-space matrices as a function of the five unknown parameters and the auxiliary variable that represents the known variance`R2`

. The initial conditions are not parameterized; they are assumed to be zero during this estimation.### Note

`R2`

is treated as an auxiliary variable rather than assigned a value in the file to let you change this value directly at the command line and avoid editing the file.function [A,B,C,D,K] = mynoise(par,T,aux) R2 = aux(1); % Known measurement noise variance A = [par(1) par(2);1 0]; B = [1;0]; C = [par(3) par(4)]; D = 0; R1 = [par(5) 0;0 0]; [~,K] = kalman(ss(A,eye(2),C,0,T),R1,R2);

Specify initial guesses for the unknown parameter values and the auxiliary parameter value

`R2`

:par1 = 0.1; % Initial guess for A(1,1) par2 = -2; % Initial guess for A(1,2) par3 = 1; % Initial guess for C(1,1) par4 = 3; % Initial guess for C(1,2) par5 = 0.2; % Initial guess for R1(1,1) Pvec = [par1; par2; par3; par4; par5] auxVal = 1; % R2=1

Construct an

`idgrey`

model using the`mynoise`

file:Minit = idgrey('mynoise',Pvec,'d',auxVal);

The third input argument

`'d'`

specifies a discrete-time system.Estimate the model parameter values from data:

opt = greyestOptions; opt.InitialState = 'zero'; opt.Display = 'full'; Model = greyest(data,Minit,opt)

- Estimate Linear Grey-Box Models
- Estimate Continuous-Time Grey-Box Model for Heat Diffusion
- Estimate Coefficients of ODEs to Fit Given Solution
- Estimate Model Using Zero/Pole/Gain Parameters
- Estimate Nonlinear Grey-Box Models

Was this topic helpful?