how to use state space model?

Hi everybody,

I'm trying to use state-space model to do prediction at runtime.

I tryed the following code just to understand how to achieve that.

However, the manual calculation is way off of the predict command.

Note that I'm new to all of this, not sure if I really understood

state-space models, what did I miss in my calculation?

Many thanks in advance.

load iddata1 z1;
z1e = z1(1:155);
z1v = z1(156:300);

order = 1;

opt = n4sidOptions("InitialState", "zero", "Focus", "prediction");
n4sid_pf = n4sid(z1e, order, opt, "DisturbanceModel", "none");

test = n4sid_pf;
r = 42;

truth_43 = z1e.OutputData(r + 1,:)';

pred = predict(z1e(r:r+1), n4sid_pf, 1);
predict_43 = pred.OutputData(1,:);

manual_43 = test.C * (test.A * z1e.OutputData(r,:)' + test.B * z1e.InputData(r,:)');

truth_43, predict_43, manual_43

truth_43 =

    1.8925


predict_43 =

    4.4982


manual_43 =

 -255.7701
 

Best Regards,

Willy

Answer 1

[Emma Wilson]

Hi,

I think the issue is you are using the output data as the state in the manual estimation. Instead you need to use the estimate of the state. A quick very quick example is below - note you'd probably get closer still if you used a kalman filter for your state estimation.

X(1)=0 ; %initial state,

for k=1:43 ;

X(k+1)= test.A*X(k) + test.B*z1e.InputData(k) ;

y(k)=test.C*X(k) ;

end ;

manual_43 = y(43) ;

Answer 2

[Rajiv Singh]

If you want to reproduce the response of "predict" by (manual) simulation, you will need to generate the right prediction model first for the chosen prediction horizon. Note that the prediction model uses both measured inputs and measured outputs as its inputs. That is, if your original model has 2 inputs and 1 output, the predictive model will have 2+1=3 inputs and 1 output.

For 1-step ahead prediction, the prediction model is:

 x(t+1) = (A-K*C)x(t) + (B-KD)u_m(t) + Ky_m(t)
 y(t) = Cx(t) + Du(t)  

Here u_m is measured input signal and y_m is measured output signal. x(t) is the predicted sate sequence and y(t) is the predicted response.

For prediction to horizons N> 1 steps, you can treat is as a sequence of N 1-step ahead predictions, where the results from the previous step are treated as measured output for the next step. You can also do this with a single simulation (as in the case of 1-step ahead prediction), but the model you will need to use will be more complicated. Note that the 3rd output argument of the PREDICT command returns this model. In the help content for PREDICT, you will see:

%{
[YP, X0E, MPRED] = predict(MODEL, DATA,...)
returns the estimated values of initial conditions X0E and a predictor system MPRED. 
MPRED is a dynamic system whose simulation using [DATA.OutputData, Data.InputData] as input signal 
yields YP.OutputData as the response ... For discrete-time data (time domain data 
or frequency domain data with Ts>0), MPRED is a discrete-time system, even if 
MODEL is continuous-time. If DATA is multiexperiment, MPRED is an array of Ne systems, 
where Ne = number of data experiments.
%}