From Matrix s-domain to state space and Z-domain

I'd like to advice on correct and moste effective matrix s-domain use:

I have a that my LTI system should represented as

Zij transfer function is represented as

where n is order and is known (n=4 in my my case) and as well Rij_m, tauij_m are known

How to form and use such matrices (T(s), Z(s), P(s)) in Matlab where considering Tc(s) constant value per some period suppied by 2D lookup table?

Secondly, how to obtain the state-space equations/matrices from the Laplace s-domain?

Thirdly I'd like to convert these LTI system ( (T(s), Z(s), P(s)), Tc(s)) to z-domain (discretization)?

Thank you for your advice and help.

Answer 1

[Paul]

Hi JD,

What does "Tc(s) constant value per some period suppied by 2D lookup table?" mean?

If you have the R_ij and tau_ij for each element of Z(s), then each element of Z(s) can be formed, for example, as follows:

R_11 = 1:4;
tau_11 = 0.1:0.1:0.4;
Zfun11 = zpk([],[],0);
for ii = 1:numel(R_11)
    Zfun11 = parallel(Zfun11,zpk([],-1/tau_11(ii),R_11(ii)/tau_11(ii)));
end
Zfun11

Zfun11 =
 
  40 (s+8.464) (s+4.296) (s+2.865)
  --------------------------------
   (s+10) (s+5) (s+3.333) (s+2.5)
 
Continuous-time zero/pole/gain model.

The details of how to form the full Z(s) will depend on how the R and tau vectors are stored.

Conversion to state space can be acomplished with @doc:ss ss. Will also need @doc:series series to multiply Z(s) and P(s), after both are converted to ss.

If still having trouble, post your code and show/explain where the hangup is.

[Jack Daniels]

I'd like to realize a thermal model similarly as "SwitchingLossCalculation" example

In my case the thermal circuirty is more complex than just one-cell RC network :)

My thermal circuitry in SimScape is here where (for simplicity) is cosnidered Z11,Z12,Z13,Z14 - first row of the Z-matrix

I want to convert this circuitry into Simulink S-domain which has coresponding transfer function 1,2,3,4 and then to continous state and discrete state space...

So first Simscape circuitry works and now as I mentioned I need to form out of it - s-domain transfer functions to calculate the temp.

fcn1 = Numerator: [R1/tau1], Denumerator: [1 1/tau1]

fcn2 = Numerator: [R2/tau2], Denumerator: [1 1/tau2]

fcn3 = Numerator: [R3/tau3], Denumerator: [1 1/tau3]

fcn3 = Numerator: [R4/tau4], Denumerator: [1 1/tau4]

the functions are entred into Simulink continous transfer functions.

How to correctly simulate circuit behaviour of that initial circuitry in these domains?

[Paul]

Now I'm confused.

In the question, Z(s) is 4x4 transfer function matrix, and each of the 16 elements of Z(s) is defined as the sum of four, first-order Laplace transforms. But this Simulink diagram shows the product of four blocks, labeled 1-4 (I think that leftmost block is labeled as "1"), each with a first order transform. But the text of the comment suggests that each of those four blocks is an element of the first row of Z(s), but that's not what's implemented in the Simulink diagram.