Convert transfer function to state-space models

I'm trying to convert a transfer function to state space model. In order to avoid the control toolbox functions since I don't have it's license.

The transfer function that I'm using is quite simple. Is the model for a heatsink temperature.

For this transfer function I have the next values for A,B,C and D.

I've tried to compare the output from the lsim() function and the equations from above but the output differs

Here's the code that I've used

P = rand(1,10)*1000;
t = 0:1:length(P)-1;

A = -400;
B = 0.5;
C = 0.6658;
D = 0;

%%%%%%%%%%%%%%%%%%%%%%%%%
%%%    State space    %%%
%%%%%%%%%%%%%%%%%%%%%%%%%

state_space_sys = ss(A,B,C,D);
state_space = lsim(state_space_sys,P,t);

figure
plot(t,state_space)
grid on,grid minor, title('State space')

%%%%%%%%%%%%%%%%%%%%%%%%%
%%%   My function     %%%
%%%%%%%%%%%%%%%%%%%%%%%%%

x(1:length(P)) = 0;
y(1:length(P)) = 0;
u = P;

for k = 1:length(u)
    x(k+1) = A*x(k) + B*u(k);
    y(k) = C*x(k) + D*u(k);
end

figure
plot(t,y)
grid on,grid minor, title('My function')

Is there any mistakes on the approach?

It should be quite simple, but I can't manage to find the solution.

[Star Strider]

Note that:

becomes:

in the state space realisation.

[Diego Garcia]

Thank you for your comment Star Strider.

Do you mean that the equation for would be something like the following?

x(k+1) = exp(400*t(k))*x(k) + B*u(k);

With this modification the result is still too different.

[Star Strider]

No.

The state space equaiton you wrote would be integrated into:

if I integrated it correctly, or here:

that you can easily simplify.

[Diego Garcia]

I think that I'm missing something, the output doesn't match. Is the implementation right?

P = rand(1,10)*1000;
t = 0:1:length(P)-1;
A = -400;
B = 0.5;
C = 0.6658;
D = 0;

%%%%%%%%%%%%%%%%%%%%%%%%%
%%%   My function     %%%
%%%%%%%%%%%%%%%%%%%%%%%%%
x(1:length(P)) = 0;
y(1:length(P)) = 0;
u = P;
for k = 1:length(u)
    y(k) = C*exp(-A*t(k))*B*u(k);
end
figure
plot(t,y)
grid on,grid minor, title('My function')

All the information that I've found about state space models is in discrete. Is there any reason why it can't be used in this case?

[Akbar]

What is Rth? What does th stand for in Rth?

Maybe you should use:

state_space_sys = ss(A,B,C,D,[]);

command to have your system as a discrete model and then compare, i got similar plots in that case.

[Walter Roberson]

Is there a reason not to use tf() and ss() ? That is, if you pass a tf into ss() then it will be converted.

[Diego Garcia]

Hello @Akbar. Rth is the thermal resistance of the heatsink. th stands for thermal.

@Akbar & @Walter Roberson , I'm trying to aboid the system control toolbox, since I don't have the license and I can only use the trial for 30 days.

Answer 1

[Diego Garcia]

At the end, I found another solution. I'm posting this here, so if someone has the same problem, can use this approach.

If you take the transfer function, and develop it, we reach to the following expresion

Now, we know that 1/s is equal to integrate. So in the end, we have to integrate the right side of the equation. The code would be like this.

Cth = Tau/Rth;
deltaT = zeros(size(P));

for i = 2:length(P)
    deltaT(i) = (1/Cth * (P(i)-deltaT(i-1)/Rth))*(time(i)-time(i-1)) + deltaT(i-1);
end

This integral has the same output as the function tf() and the function lsim() with the A,B,C and D values.