Stability

Wikipedia · BIBO stability · CC BY-SA 4.0

The characteristic equation

The denominator of the transfer function H(s) is the characteristic polynomial. Setting it to zero gives the characteristic equation. Its roots are the poles. If all poles have negative real parts (left half-plane), the system is stable. If any pole has a positive real part (right half-plane), the system is unstable. Poles on the imaginary axis are marginally stable: the output oscillates forever without growing or decaying.

Routh-Hurwitz criterion

You don't need to find the roots to check stability. The Routh-Hurwitz criterion builds a table from the coefficients of the characteristic polynomial. If all entries in the first column are positive, all poles are in the left half-plane and the system is stable. A sign change in the first column means an unstable pole.

; Check stability by examining poles (roots of characteristic equation)
; For a second-order system: s^2 + 2*zeta*wn*s + wn^2 = 0
; Poles: s = -zeta*wn +/- wn*sqrt(zeta^2 - 1)

(define (check-stability zeta wn)
  (let* ((real-part (* (- zeta) wn)))
    (display "  zeta=") (display zeta)
    (display "  wn=") (display wn)
    (display "  Re(pole)=") (display real-part)
    (display (cond
      ((< real-part 0) " -> STABLE")
      ((= real-part 0) " -> MARGINALLY STABLE")
      (else " -> UNSTABLE")))
    (newline)))

(display "Second-order system stability:") (newline)
(check-stability 0.5 2.0)    ; underdamped, stable
(check-stability 1.0 2.0)    ; critically damped, stable
(check-stability 0.0 2.0)    ; undamped, marginally stable
(check-stability -0.3 2.0)   ; negative damping, unstable

# Check stability from damping ratio and natural frequency
def check_stability(zeta, wn):
    real_part = -zeta * wn
    if real_part < 0:
        status = "STABLE"
    elif real_part == 0:
        status = "MARGINALLY STABLE"
    else:
        status = "UNSTABLE"
    print(f"  zeta={zeta}  wn={wn}  Re(pole)={real_part:+.1f} -> {status}")

print("Second-order system stability:")
check_stability(0.5, 2.0)   # underdamped, stable
check_stability(1.0, 2.0)   # critically damped, stable
check_stability(0.0, 2.0)   # undamped, marginally stable
check_stability(-0.3, 2.0)  # negative damping, unstable