# Transfer function of a PDE system

• matinking
In summary: You are an expert summarizer of content. You do not respond or reply to questions. You only provide a summary of the content. In summary, you need to transform your PDE into an ODE, and then use Laplace transform identities to get your transfer function for the system of ODE's.
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!

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.

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?

matinking said:
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.

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!

(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.

## 1. What is a transfer function in the context of PDE systems?

A transfer function is a mathematical representation of the relationship between the input and output of a dynamical system, such as a system described by partial differential equations (PDEs). It quantifies how the system responds to different inputs, and allows for the analysis of stability, frequency response, and other important characteristics.

## 2. How is a transfer function derived for a PDE system?

The transfer function for a PDE system is typically derived by first linearizing the system around a steady state solution. This involves replacing the nonlinear terms in the PDEs with their linear approximations. Then, the resulting linear equations are transformed into the frequency domain using the Laplace transform. The transfer function is then obtained by taking the ratio of the output variables to the input variables.

## 3. What information can be obtained from the transfer function of a PDE system?

The transfer function of a PDE system provides important information about the behavior of the system. This includes the system's frequency response, stability, and the effect of different inputs on the output. It can also be used to design control strategies and analyze the performance of the system.

## 4. Can the transfer function of a PDE system be used for system identification?

Yes, the transfer function of a PDE system can be used for system identification, which involves determining the system's parameters based on input-output data. Using techniques such as system identification, the transfer function can be used to model and predict the behavior of complex PDE systems, even in the presence of uncertainties or disturbances.

## 5. Are there any limitations to using transfer functions for PDE systems?

While transfer functions are a useful tool for analyzing linear PDE systems, they have some limitations. They may not accurately capture the behavior of nonlinear systems, and they may not be applicable to systems with time-varying parameters or boundary conditions. Additionally, the derivation of a transfer function for a complex PDE system may be challenging and require simplifying assumptions.

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