Hot plate PID control according to measured graphs

I would like to control electric hot plate by PID control. Please can you advise me on how to identify system, calculate the PID parameters and let me know the equation by which I can control the temperature of the hotplate by reading the measured temperature in 1 s intervals and by means of MCU to control the SSR (PWM) accordingly to follow JEDEC reflow profile?

[Star Strider]

Identifying the system is straightforward. Coinverting the identified system to a PID representation does not appear to be possible.

THP = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/1338819/Esperanza_Hotplate.xlsx', 'VariableNamingRule','preserve')

THP = 490×2 table
    Time [s]    T [°C]
    ________    ______

        1       27.032
        2       26.966
        3       26.909
        4       26.859
        5       26.818
        6       26.784
        7       26.759
        8       26.742
        9       26.733
       10       26.731
       11       26.738
       12       26.752
       13       26.774
       14       26.804
       15       26.842
       16       26.887

VNTHP = THP.Properties.VariableNames;

figure
plot(THP{:,1}, THP{:,2})
grid
xlabel(VNTHP{1})
ylabel(VNTHP{2})

Ts = diff(THP{[1 2],1})                                     % Sampling Interval (Assumed Constant)

Ts = 1

HPData = iddata(THP{:,2},[],Ts)

HPData =

Time domain data set with 490 samples.
Sample time: 1 seconds                 
                                       
Outputs      Unit (if specified)       
   y1                                  
                                       

HPss = ssest(HPData,3)

HPss =
  Continuous-time identified state-space model:
      dx/dt = A x(t) + K e(t)
       y(t) = C x(t) + e(t)
 
  A = 
              x1         x2         x3
   x1  -0.009696    0.00484   0.005199
   x2   -0.01001    0.00952    0.01666
   x3   -0.01958   0.002297   -0.00606
 
  C = 
          x1     x2     x3
   y1  -2743  99.72  1.013
 
  K = 
               y1
   x1  -0.0002048
   x2     0.01029
   x3    -0.08902
 
Parameterization:
   FREE form (all coefficients in A, B, C free).
   Disturbance component: estimate
   Number of free coefficients: 15
   Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                            
Estimated using SSEST on time domain data "HPData".
Fit to estimation data: 100% (prediction focus)    
FPE: 1.1e-06, MSE: 1.061e-06                       
 

figure
compare(HPData, HPss)
grid

.

[Dan Richter]

Thank you very much for your advice. That's great.

What equation with PID coefficients should I use to control the SSR so that the resulting temperature matches the JEDEC temperature profile, see attached graph? When the peak of the curve is reached, I take the PCB off the hot plate.

I will measure the hot plate temperature each second.

Answer 1

[Joe Vinciguerra]

Control Systems - MATLAB & Simulink (mathworks.com)

[Dan Richter]

Is there any example? Although I am somewhat familiar with FFT calculation in MATLAB, I have no experience with PID control.