I have a model for which time delay is coming negative and I'm unable to find transfer function. How should I proceed to calculate tf?

I have a simulink model for a MIMO process model whose one of the pair on linearization giving parameters which have negative time delay and I'm unable to find transfer function for the model, which I need for my PID controller.

[Mathieu NOE]

how much is your negative time delay vs the decay time of your step response ?

that may be neglectable so consider it like 0

Answer 1

[Ayush]

Negative time delays in MIMO process can occur due to factors such as system dynamics, signal propagation delays, processing delays.

I understand that you are unable to find the transfer function required for PID controller due to negative time delays, in such a scenario you may consider following workarounds:

1. Using State-space representation as it can handle systems with time delays more effectively. Once you have state-space representation you can design PID controller using techniques such as pole placement of LQR. Read the documentation to convert transfer function to state-space models: Convert transfer function filter parameters to state-space form - MATLAB tf2ss - MathWorks India
2. In order to address negative time delays, use Delay compensation. You may approximate time delays with series of first-order or higher-order transfer function using Pade approximation.
3. Model Predictive Control (MPC) can be used instead of PID as MPC is well-suited for system with time-delays.
4. Consider non-linear technique if your system exhibits significant non-linearities or time delays causing complex behaviour.

Answer 2

[Sam Chak]

Hi @Shubham

Based on the data extracted from the step response, the system can be approximated as a first-order transfer function. With that, a PI controller can be designed.

x   = [0    2.9653    6.2281    8.8960   12.4562   15.2828   18.4562   20.9854   23.7719   26.9161   30.2281   33.3029   36.6843   40.1460   42.9124   46.0766   49.1405   51.7792   54.4562   57.4818   60.6843   66.0000   72.0000   78.0000   84.0000   90.0000   96.0000  102.0000  108.0000  114.0000  120.0000  126.0000  132.0000  138.0000  144.0000  150.0000  156.0000  162.0000  168.0000  174.0000  180.0000  186.0000  192.0000  198.0000  204.0000  210.0000  216.0000  222.0000  228.0000  234.0000  240.0000  246.0000  252.0000  258.0000  264.0000  270.2281  276.0000  282.0000  288.0000  294.0000  300.0000];
y   = [0   -0.0061   -0.0105   -0.0138   -0.0178   -0.0205   -0.0234   -0.0256   -0.0276   -0.0298   -0.0319   -0.0336   -0.0352   -0.0368   -0.0379   -0.0391   -0.0402   -0.0409   -0.0417   -0.0425   -0.0432   -0.0442   -0.0453   -0.0461   -0.0468   -0.0473   -0.0478   -0.0482   -0.0485   -0.0487   -0.0489   -0.0491   -0.0492   -0.0494   -0.0495   -0.0495   -0.0496   -0.0496   -0.0497   -0.0497   -0.0497   -0.0497   -0.0498   -0.0498   -0.0498   -0.0498   -0.0499   -0.0498   -0.0498   -0.0498   -0.0498   -0.0499   -0.0499   -0.0499   -0.0499   -0.0499   -0.0499   -0.0499   -0.0499   -0.0499   -0.0499];
   
% Identified transfer function model
s   = tf('s');
Gp  = -0.05/(30*s + 1)

Gp =
 
   -0.05
  --------
  30 s + 1
 
Continuous-time transfer function.

step(Gp, 300), grid on, hold on
plot(x, y, 'r.', 'markersize', 5), hold off
legend('model', 'data')

% PI Controller design
Gc  = pidtune(Gp, 'PI', 5)

Gc =
 
             1 
  Kp + Ki * ---
             s 

  with Kp = -2.59e+03, Ki = -7.59e+03
 
Continuous-time PI controller in parallel form.

% Closed-loop system
Gcl = minreal(feedback(Gc*Gp, 1))

Gcl =
 
     4.313 s + 12.64
  ---------------------
  s^2 + 4.347 s + 12.64
 
Continuous-time transfer function.

step(Gcl)