Transfer function derviation for a system

I plan to heat 10L of Palm Oil to 60C from room temperature and maintain that temperature for the transesterification of biodiesel. To maintain the temperature I am developing a PID controller with a 1000W immersion heater but I need to find a way to obtain the PID constants. So far I have developed this equation to model:

Where P is input power,

How to

• derive the system transfer function from matlab
• can I generate similar modelled equations as above for any adjustments to the system for example if I add a stirrer?

[DP]

A snip shot of the transfer function is attached

[Star Strider]

imshow(imread('Screenshot 202...17 012956.png'))

A transfer function is generally defined as:

with both being functions of time, transformed into the Laplace domain, so the assumptions of the Laplace transform would need to be taken into account. .

It would appear to me that the input should be the energy source that would be converted to heat, and the expression for that conversion, and the output would be temperature. A stirrer would help the system attain equilibrium faster. It would also add energy, however i doubt it would be significant.

A state space realisation of the control system when the input model is provided (assuming that it is not hidden in the expression provided) would likely be most appropriate.

There may also be other considerations.

.

Answer 1

[Sam Chak]

Hi @DP

From the differential equation, it can be rearranged to have the control term and the disturbance term.

In state-space matrix form, it looks like this:

.

This is a two-input, single-output system, so in theory, there should be two distinct transfer functions. However, with a little mathematical modification, you can make both transfer functions the same and have only one transfer function. The simple trick is to have the same element in the Input matrix B.

.

h   = 1;
A   = 3;
rho = 1;
V   = 0.5;
C   = 1;

A   = - (h*A)/(rho*V*C)

A = -6

B   =       1/(rho*V*C)

B = 2

%% state-space system
sys = ss(A, B, 1, 0);

%% transfer function
Gp  = tf(sys)

G =
 
    2
  -----
  s + 6
 
Continuous-time transfer function.

In Control System Block Diagram, it looks like this: