How can i plot PI controller step respone

I used the following steps to plot the step response using (step) function:

-> num2_PI=conv(395.5*[1 1/8],[0 -0.0005854 -0.01121 0.0003176]);

den2_PI=[ 1 1.117 10.18 0.0005226 0]+num2_PI;

TF2_PI=tf(num2_PI,den2_PI);

step(TF2_PI)

the plot was :

So , how can I read the overshoot, final value and other readings from this plot

[Sam Chak]

You have successfully plotted the step response of an unknown system that doesn't look like a Pure PI Controller. Unfortunately, the response indicates instability and those characteristics {overshoot, final value, etc.} are undefined for unstable systems.

[Mathieu NOE]

hello

fyi, your "plant" is unstable in open loop and cannot be stabilized with this PI correction , neither with negative or positive feedback

% PI 
num1 = 395.5*[1 1/8]; % P , I gains
den1 = [1 0];
C = tf(num1,den1);

% Plant 
num2=[0 -0.0005854 -0.01121 0.0003176];
den2=[ 1 1.117 10.18 0.0005226 0];
G=tf(num2,den2);
step(G)

sys = feedback(G,C,-1) % standard negative feedback 

sys =
 
          -0.0005854 s^3 - 0.01121 s^2 + 0.0003176 s
  -----------------------------------------------------------
  s^5 + 1.117 s^4 + 9.948 s^3 - 4.462 s^2 - 0.4286 s + 0.0157
 
Continuous-time transfer function.

step(sys)

sys = feedback(G,C,+1)%  positive feedback 

sys =
 
          -0.0005854 s^3 - 0.01121 s^2 + 0.0003176 s
  -----------------------------------------------------------
  s^5 + 1.117 s^4 + 10.41 s^3 + 4.463 s^2 + 0.4286 s - 0.0157
 
Continuous-time transfer function.

step(sys)

Answer 1

[Sam Chak]

Hi @Lina

The 4th-order plant under consideration appears to be unstable and cannot be effectively stabilized using a low-order, simple 2-parameter PI controller. Additionally, the unstable plant also exhibits positive zeros, which can lead to initial undershoot in the system response.

In an effort to address these challenges, I have attempted to strategically place the poles and zeros of a matching 4th-order controller (on the feedback path) and the pre-filter, with the goal of limiting the undershoot percentage to less than 20% and ensuring the compensated system converges within 5 minutes. The MATLAB stepinfo() command can be utilized to analyze the step-response characteristics of the stabilized system.

%% Plant
Gp  = tf([0, -0.0005854, -0.01121, 0.0003176], [1, 1.117, 10.18, 0.0005226, 0])

Gp =
 
   -0.0005854 s^2 - 0.01121 s + 0.0003176
  -----------------------------------------
  s^4 + 1.117 s^3 + 10.18 s^2 + 0.0005226 s
 
Continuous-time transfer function.

%% Feedback Compensator
cz  = [ 1.8729742918689                             % zeros
       -5.13296170895781e-05
       -6.69459607654879e-09];
cp  = [-3.27517464154048   +                 0i     % poles
        2.1086089615566    + 0.506178267136755i
        2.1086089615566    - 0.506178267136755i
        0.0282900517606163 +                 0i];
ck  =  -7456.33220301306;                           % gain
Gc  = tf(zpk(cz, cp, ck))

Gc =
 
  -7456 s^3 + 1.397e04 s^2 + 0.7169 s + 4.799e-09
  -----------------------------------------------
  s^4 - 0.9703 s^3 - 9.083 s^2 + 15.66 s - 0.4357
 
Continuous-time transfer function.

%% Pre-filter
fz  = [];                       % zeros
fp  = [-19.1775896766893        % poles
       -3.27517464154048];
fk  =  -6.9180823798271e-07;    % gain
Gf  = tf(zpk(fz, fp, fk))

Gf =
 
       -6.918e-07
  ---------------------
  s^2 + 22.45 s + 62.81
 
Continuous-time transfer function.

%% Filtered Closed-loop
Gcl = feedback(Gp, Gc);     % put Gc on the feedback path
Fcl = minreal(series(Gf, Gcl))

Fcl =
 
                          4.05e-10 s^4 - 1.731e-09 s^3 + 2.001e-09 s^2 - 1.091e-10 s + 1.524e-12
  -----------------------------------------------------------------------------------------------------------------------
  s^8 + 0.1467 s^7 + 0.01311 s^6 + 0.0007778 s^5 + 3.284e-05 s^4 + 9.877e-07 s^3 + 2.03e-08 s^2 + 2.561e-10 s + 1.524e-12
 
Continuous-time transfer function.

%% Step response
step(Fcl, 600), grid on

stepinfo(Fcl)

ans = struct with fields:
         RiseTime: 68.8473
    TransientTime: 287.5175
     SettlingTime: 289.4285
      SettlingMin: 0.9002
      SettlingMax: 1.0118
        Overshoot: 1.1755
       Undershoot: 17.9067
             Peak: 1.0118
         PeakTime: 420.8477