PID Control

Wikipedia · PID controller · CC BY-SA 4.0

The PID controller computes three terms from the error signal: proportional (present), integral (past), derivative (future). Tuning the three gains trades off speed, stability, and steady-state accuracy. Most industrial controllers are PID.

The three terms

P (proportional): output proportional to the current error. Large error, large correction. But P alone leaves a steady-state offset: the system settles near but not at the setpoint. I (integral): output proportional to the accumulated error over time. Eliminates the offset but can cause overshoot and oscillation. D (derivative): output proportional to the rate of change of error. Damps oscillation by anticipating where the error is heading. Too much D amplifies noise.

Ziegler-Nichols tuning

Set I and D gains to zero. Increase the P gain until the system oscillates with constant amplitude. Record this ultimate gain Ku and the oscillation period Tu. Then set: Kp = 0.6*Ku, Ki = 2*Kp/Tu, Kd = Kp*Tu/8. This is a starting point, not an optimal solution. It often produces aggressive response that needs manual refinement.

Simulate a PID controller

Scheme

; PID controller simulation
; Plant: simple integrator (velocity -> position)
; Setpoint: 10.0

(define setpoint 10.0)
(define kp 0.5)   ; proportional gain
(define ki 0.05)  ; integral gain
(define kd 0.3)   ; derivative gain
(define dt 0.1)

(define (pid-step temp integral prev-error steps)
  (if (= steps 0)
    (begin (display "Done. Final: ") (display temp) (newline))
    (let* ((error (- setpoint temp))
           (new-integral (+ integral (* error dt)))
           (derivative (/ (- error prev-error) dt))
           (output (+ (* kp error)
                      (* ki new-integral)
                      (* kd derivative)))
           (new-temp (+ temp (* output dt))))
      (if (= 0 (modulo steps 5))
        (begin
          (display "  temp=") (display new-temp)
          (display "  error=") (display error)
          (newline))
        'skip)
      (pid-step new-temp new-integral error (- steps 1)))))

(display "PID controller (Kp=0.5, Ki=0.05, Kd=0.3):") (newline)
(pid-step 0.0 0.0 0.0 100)

; PID controller simulation
; Plant: simple integrator (velocity -> position)
; Setpoint: 10.0

(define setpoint 10.0)
(define kp 0.5)   ; proportional gain
(define ki 0.05)  ; integral gain
(define kd 0.3)   ; derivative gain
(define dt 0.1)

(define (pid-step temp integral prev-error steps)
  (if (= steps 0)
    (begin (display "Done. Final: ") (display temp) (newline))
    (let* ((error (- setpoint temp))
           (new-integral (+ integral (* error dt)))
           (derivative (/ (- error prev-error) dt))
           (output (+ (* kp error)
                      (* ki new-integral)
                      (* kd derivative)))
           (new-temp (+ temp (* output dt))))
      (if (= 0 (modulo steps 5))
        (begin
          (display "  temp=") (display new-temp)
          (display "  error=") (display error)
          (newline))
        'skip)
      (pid-step new-temp new-integral error (- steps 1)))))

(display "PID controller (Kp=0.5, Ki=0.05, Kd=0.3):") (newline)
(pid-step 0.0 0.0 0.0 100)

Neighbors

Ziegler-Nichols method — the classic PID tuning procedure

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