PEM grey box using merged data.

I am trying to understand exactly the underlying theory that pem uses working with merged data. Specifically, I am using a grey box model where the initial conditions and Kalman gain is parameterized by me. Is it just performing the identification separately and then combining the result or is it estimated in one run with two sets of initial conditions. Code is included as an example.

%%Define system

A = [0.8 0.1; 0.1 0.7];
B = [0.2; 0.7];
C = [1 0];
D = 0;
Q = 0.1;
R = 0.1;
Ts = 1;

%%Simulate the system twice
n = 200;
y = cell(2, 1);
y{1} = zeros(1, n);
y{2} = y{1};
u = [0*ones(1, 20), 1*ones(1, 30), 2*ones(1, 20), -1*ones(1, 20), 0*ones(1, 30), 2*ones(1, 20), 0*ones(1, 30), -2*ones(1, 30)];
x = zeros(2, n + 1);

for i = 1:2;
    if( i == 1 )
        x(:, 1) = [4; 4];
    else
        x(:, 1) = [-1; -1];
    end

      for k = 1:n;
          y{i}(k) = C*x(:, k) + D*u(:, k) + sqrt(R)*randn;
          x(:, k + 1) = A*x(:, k) + B*u(:, k) + sqrt(Q)*randn;
      end
  end

%%plot output
plot([y{1}', y{2}']);

%%gather data
data1 = iddata(y{1}', u', Ts);
data2 = iddata(y{2}', u', Ts);
data_all = merge(data1, data2);

%%identify
model1 = idgrey('sysmodel', zeros(1, 6), 'd');
options = greyestOptions('Display', 'On', 'Focus', 'Prediction');
options.SearchOption.MaxIter = 1000;
model_out = pem(data_all, model1, options);%, 'OutputWeight', [1 0; 0 0]);

MODEL FUNCTION

function [A, B, C, D, K, X0] = sysmodel(phi, Ts, extra)
%SYSMODEL Summary of this function goes here
%   Detailed explanation goes here

A = [phi(1) 0.1; 0.1 phi(2)];
B = [0.2; 0.7];
C = [1 0];
D = 0;
K = [phi(3); phi(4)];
X0 = [phi(5); phi(6)];

end

Bump.

[Walter Roberson]

Is "pem" the Prediction Error Method in this context?

[Mike]

Yes, I was using pem to specifically mean the function in MATLAB which uses the prediction error method to perform system identification.

Answer 1

[Rajiv Singh]

It is the latter - one combined estimation using both sets of data. If you parametrize initial states x0 (like you do in your sysmdoel file), then you are forcing a joint estimation of initial conditions that somehow do justice to both datasets (in an average sense). To avoid this, you can set the estimation option "InitialState" to "estimate" (options.InitialState = 'estimate'). This causes the X0 returned by sysmodel to be ignored and initial states are estimated separately for each data experiment. The estimated values are stored in model_out.Report.Parameters.X0 (and can also be returned as the second output argument of GREYEST).