control design for marginally stable system

Hello dears

this system is a marginly stable system, I am trying to design a PID controller for it but the problem is that it has a big pole value that causes the system keep osilating for ever. please any body has an idea how to fix the pole location to design the controller

thank you

Mohamed

A =[        0            0      -2.2444            0      0.01995
            0            0      0.81081      -1.3514   -0.0054054
       29.508      -19.672   -0.0018769            0            0
            0       222.22            0      -4444.4            0
      -0.2623      0.13115            0            0            0]

B =[      104.74
            0
        -2.56
            0
            0]



C =[ 0     0     1     0     0]
D=0
sys1=ss(A,B,C,D)

egv=eig(sys1)
egv=[-4444.4 +          0i
   -0.0074954 +     9.0655i
   -0.0074954 -     9.0655i
    -0.053263 +          0i
   -0.0011919 +          0i]

Answer 1

[Sam Chak]

Hi @Mohamed

Since no performance requirements are specified, the pidtune() command can be used as the initial step in designing the PID controller. Additionally, it is possible to cancel out the undesired pole in the closed-loop transfer function, as demonstrated in this case.

%% Original system in State-space
A   = [ 0        0       -2.2444        0       0.01995
        0        0        0.81081      -1.3514 -0.0054054
       29.508  -19.672   -0.0018769     0       0
        0      222.22     0         -4444.4     0
       -0.2623   0.13115  0             0       0];
B   = [104.74
         0
        -2.56
         0
         0];
C   = [0 0 1 0 0];
D   = 0*C*B;
sys = ss(A, B, C, D);

%% Plant
P   = tf(sys)

P =
 
     -2.56 s^4 - 8287 s^3 + 1.374e07 s^2 + 9.281e05 s - 3250
  --------------------------------------------------------------
  s^5 + 4444 s^4 + 390.8 s^3 + 3.653e05 s^2 + 1.989e04 s + 23.19
 
Continuous-time transfer function.

%% Controller
[C, info] = pidtune(P, 'PIDF')

C =
 
             1            s    
  Kp + Ki * --- + Kd * --------
             s          Tf*s+1 

  with Kp = -0.00745, Ki = -6.76e-06, Kd = 1.55, Tf = 209
 
Continuous-time PIDF controller in parallel form.

info = struct with fields:
                Stable: 1
    CrossoverFrequency: 0.0013
           PhaseMargin: 64.8902

%% Closed-loop system
% Gcl = minreal(feedback(C*sys, 1));
Gcl = minreal(feedback(C*P, 1));
zcl = zero(Gcl)     % closed-loop zeros

zcl = 4×1

1.0e+03 * 1.2073 -0.0001 0.0000 -0.0000

pcl = pole(Gcl)     % closed-loop poles

pcl = 
  -0.0083 + 9.0655i
  -0.0083 - 9.0655i
  -0.0539 + 0.0000i
  -0.0016 + 0.0026i
  -0.0016 - 0.0026i
  -0.0006 + 0.0000i

%% Plot the results
subplot(2,1,1)
step(sys),       grid on, legend('Original system',    'location', 'east')
subplot(2,1,2)
step(Gcl, 6000), grid on, legend('Compensated system', 'location', 'east')