Transfer function of a PDE system

[matinking]

Hi everyone!

I want to design a robust controller for a system which is driven by a PDE. I need to acquire its transfer function in 's' parameter which means it should be transferred by Laplace transformation. I know that the result transfer function will be an infinite series of transfer functions but i don't know how to calculate it!

would you mind introducing me a reference which help me to learn that?

thank you!

 

[chiro]

Hey matinking and welcome to the forums.

It would help if you gave us some information firstly about the PDE itself if you don't mind.

 

[matinking]

Hi chiro
it is a dynamic model of an electrical system which could be described as below equations.
except t,tetha,y and i are constants.

f1 - f2 = f3 + f4

f1 = \frac{4 I\ddot{\theta}{\beta}
f2 = \dot{\theta} {M^{2}\ddot{y}^{2} + \tilde{M}^2\tilde{g}^2 + 2 M\tilde{M}\tilde{g}\ \ddot{y}}{4 \tau_{0}^{2}}}
f3 = 4F
f4 = A i^{2} t^2

the laplace transform should be applied on this equation in order to transform all of the non-constant parameters(tetha,y,i,t) to 's' parameter!

have you any idea?

 

[chiro]

Hi chiro
it is a dynamic model of an electrical system which could be described as below equations.
except t,tetha,y and i are constants.

f1 - f2 = f3 + f4

f1 = \frac{4 I\ddot{\theta}{\beta}
f2 = \dot{\theta} {M^{2}\ddot{y}^{2} + \tilde{M}^2\tilde{g}^2 + 2 M\tilde{M}\tilde{g}\ \ddot{y}}{4 \tau_{0}^{2}}}
f3 = 4F
f4 = A i^{2} t^2

the laplace transform should be applied on this equation in order to transform all of the non-constant parameters(tetha,y,i,t) to 's' parameter!

have you any idea?

It would help if you (or a moderator) fixed up the latex as I can't interpret it from what is being written.

However I can offer some advice. If you can transform your PDE into an ODE, then you can use normal Laplace transform to get your transfer function F(s) and depending on the function you could get an analytic expression using tables and transform identities, or you could use the direct approach and use residue theorems to extract the function in the time domain.

If you need to use the direct approach, then you will need to use residue theory which means you will need to be familiar with complex analysis at a basic level.

When you fix up the latex, I can give you more specific help but unfortunately I can't really make sense of relationships: but yeah see if you can transform your PDE into a system of ODE's and from there you can use standard Laplace transform identities to get your transfer functions for the system of ODE's and then solve for the function using indirect or direct method.

 

[matinking]

I've written the equation. in fact i,tetha and y are functions of the t! which means i should coup with a equation which is constructed from different derivatives from different functions based on t!

please check it up!


(4Iθ{dotdot})/β-{(M^2*y{dotdot}^2+ N^2*g^2+2MNgy{dotdot})/(4τ_0^2 )} θ{dot}= 4F+Z*i^2* t^2
t: independent parameter which must be mapped by laplace transform
θ= θ(t)
i=i(t)
y=y(t)
The other variabes are constant.