How do I determine whether a PID controller I've designed is stable?

How do I determine its stability due to the transfer function it's being applied to, and how well it rejects disturbance and noise?

  • \$\begingroup\$ You can simulate a simple system (in Octave, for instance). Otherwise it's highly system dependent. Do you only have a simple system model? Or are you building a real thing? \$\endgroup\$ – Samuel May 25 '14 at 19:07
  • \$\begingroup\$ It is a PID or a P only? in on answer you commented that is't as P only. The answers won't be the same. Then is it a analog controller(using op amps?) or a digital system with some sampling? \$\endgroup\$ – Blup1980 May 26 '14 at 8:48
  • \$\begingroup\$ I Have 2 controller one which is a P and another one which is a PID \$\endgroup\$ – Carlton Banks May 26 '14 at 9:21

Using classical methods (non state-space) for a SISO system, I suggest take a look on frequency response methods (bode plot), where the goal is to reach specified phase and margin gains for the open-loop transfer function (before close the loop). With the increase of the frequency, the magnitude must cross unity gain before the phase be -180 degrees. By default Matlab tuning algorithms are designed for a 60 degree phase margin. In other hand, true robust control is concerned with \$S\$ functions (sensitivity) with respect to varying parameters affecting the system. Simulation tools are helpful here (eg Sisotool).

  • \$\begingroup\$ but the my PID controller consist of only a P value.. \$\endgroup\$ – Carlton Banks May 26 '14 at 8:17
  • \$\begingroup\$ It says that my gain margin is infinite.. which doesn't not make that much sense \$\endgroup\$ – Carlton Banks May 26 '14 at 8:17
  • \$\begingroup\$ Then, your controller is not a PID. Gain margin infinite: Probably, in bode plot, the phase shift tends to exactly 180° as the frequency tends to infinity. \$\endgroup\$ – Dirceu Rodrigues Jr May 26 '14 at 15:06
  • \$\begingroup\$ P control features a relatively high overshoot, a long settling time as mainly a steady-state error on output. Disadvantages minimized using a PI or PID controller. A naturally unstable open loop process may require a PD controller. \$\endgroup\$ – Dirceu Rodrigues Jr May 26 '14 at 17:52

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