Issues tuning pid using simulink

I'm trying to evaluate a closed loop system using Zigler nichols table for a Pid with G(s)= and k_p=5,16,k_i= 3,85 and k_d =1,72

But everytime I set a K_d > 1 i have the following error log by simulink

An error occurred while running the simulation and the simulation was terminated

Caused by: Derivative of state '1' in block 'untitled1/Sistema_analogico/Gs1' at time 11.225798821660552 is not finite. The simulation will be stopped. There may be a singularity in the solution. If not, try reducing the step size (either by reducing the fixed step size or by tightening the error tolerances)

Where the simularion stops depends ow how large I set the k_d, i don t know how to fix this.

Thanks in advance for your help

Answer 1

[Sam Chak]

Hi @Paolo

Your question on the control problem went unnoticed for more than 1 month. The plant system has a zero (s = 0.1) in the right-half plane. Therefore, it is not easy to design a satisfactory ideal PID controller. You may have noticed that as the derivative gain approaches 1, the undershoot of the closed-loop system also grows infinitely large in the negative direction.

The example below illustrates four cases, with the first being the uncompensated system (stable but with unacceptably large undershoot), and the other three involving 'ideal' PID controllers. If a smaller undershoot is desired, you will need to explore other configurations of controllers.

%% Non-minimum Phase Plant
s   = tf('s');
Gp  = tf((- s + 0.1)/((s + 0.6)*(s + 8)))

Gp =
 
      -s + 0.1
  -----------------
  s^2 + 8.6 s + 4.8
 
Continuous-time transfer function.

%% OP's manually-tuned PID Controller (Ziegler–Nichols)
Gc1 = pid(5.16, 3.85, 0.6172)     % with kd < 1 is selected

Gc1 =
 
             1          
  Kp + Ki * --- + Kd * s
             s          

  with Kp = 5.16, Ki = 3.85, Kd = 0.617
 
Continuous-time PID controller in parallel form.

Gcl1= minreal(feedback(Gc1*Gp, 1))

Gcl1 =
 
  -1.612 s^3 - 13.32 s^2 - 8.71 s + 1.006
  ---------------------------------------
     s^3 + 9.148 s^2 + 3.83 s + 1.006
 
Continuous-time transfer function.

%% Auto-tuned PID Controller (Unspecified requirements)
Gc2 = pidtune(Gp, 'PID')

Gc2 =
 
             1 
  Kp + Ki * ---
             s 

  with Kp = 2.18, Ki = 2.23
 
Continuous-time PI controller in parallel form.

Gcl2= minreal(feedback(Gc2*Gp, 1));

%% Optimized 'ideal' PID Controller
kp  = 0.0290818775069017;   % P-gain
ki  = 2.45979506934827;     % I-gain
kd  = 0.216673683005054;    % D-gain
Gc3 = pid(kp, ki, kd)

Gc3 =
 
             1          
  Kp + Ki * --- + Kd * s
             s          

  with Kp = 0.0291, Ki = 2.46, Kd = 0.217
 
Continuous-time PID controller in parallel form.

Gcl3= minreal(feedback(Gc3*Gp, 1));

%% Plot Step response and data collection
Gcl = [Gp/dcgain(Gp), Gcl1, Gcl2, Gcl3];

for j = 1:4
    sNFO(j) = stepinfo(Gcl(j)); % stored in sNFO structure
    step(Gcl(j), 60)            % simulate up to 60 sec
    hold on
end
hold off
grid on
legend('Case 1: Uncompensated', 'Case 2: Z–N tuned PID', 'Case 3: Auto-tuned PI', 'Case 4: Optimized PID', 'location', 'E')

%% Table of Step-response Characteristics
myCell  = [fieldnames(sNFO), permute(struct2cell(sNFO), [1 3 2])];
myTable = cell2table(myCell, "VariableNames", ["Performance Aspect", "Case 1", "Case 2", "Case 3", "Case 4"])

myTable = 9×5 table
    Performance Aspect    Case 1     Case 2     Case 3     Case 4 
    __________________    _______    _______    _______    _______

    {'RiseTime'     }       3.662     2.8299       21.4     11.753
    {'TransientTime'}      6.9809     18.727     42.795     26.376
    {'SettlingTime' }      9.8932     20.148     46.693     27.065
    {'SettlingMin'  }     0.90047    0.91668    0.90014    0.90376
    {'SettlingMax'  }     0.99978     1.2083    0.99981       1.02
    {'Overshoot'    }           0     20.832          0          2
    {'Undershoot'   }      473.94     161.23     50.338     49.091
    {'Peak'         }      4.7394     1.6123    0.99981       1.02
    {'PeakTime'     }     0.33387          0     91.299     35.244