MATLAB: PEM grey box using merged data.

idgreymergepemsystem identificationSystem Identification Toolbox

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.

Best Answer

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).
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