Estimate Discrete-Time Grey-Box Model with Parameterized Disturbance

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 (Control System Toolbox) for computing the Kalman gain from the known and estimated noise variance.

Description of the SISO System

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

$\begin{array}{l} x (k T + T) = [\begin{matrix} p a r 1 & p a r 2 \\ 1 & 0 \end{matrix}] x (k T) + [\begin{matrix} 1 \\ 0 \end{matrix}] u (k T) + w (k T) \\ y (k T) = [\begin{matrix} p a r 3 & p a r 4 \end{matrix}] x (k T) + e (k T) \\ x (0) = x 0 \end{array}$

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.

Estimating the Parameters of an idgrey Model

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)

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