Convert transfer function to state-space models
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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.
7 个评论
Star Strider
2019-10-11
Note that:
becomes:
in the state space realisation.
Diego Garcia
2019-10-11
编辑：Diego Garcia
2019-10-11
Thank you for your comment Star Strider.
Do you mean that the equation for 
would be something like the following?
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
2019-10-11
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
2019-10-11
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
2019-10-16
编辑：Walter Roberson
2019-10-16
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
2019-10-17
采纳的回答
Diego Garcia
2019-10-17
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.
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