SISO Controllability Rules

Skogestad & Postlethwaite, Multivariable Feedback Control, §5.14

Enter a SISO plant, disturbance, and tracking spec. The 8 input-output controllability rules tell you whether any controller can meet the requirements — before you design one.

Plant & Disturbance

G(s) numerator (space-separated, descending s)

G(s) denominator

G_d(s) numerator

G_d(s) denominator

Default: room heating example from Skogestad §5.16.2 — G = 1/(1000s+1), G_d = 1/(1000s+1). Scaling per §1.4: expected variations all of order 1.

Controller & Spec

K(s) numerator

Default: PID(0.4, 200, 60) → K = 0.4(60s² + s + 1/200) / s = 24s² + 0.4s + 0.002 / s

K(s) denominator

Tracking spec R (scaled)

Tracking bandwidth ω_r (rad/s)

Total loop delay θ (s)

Verdict

Open-Loop Bode Plot

|L(jω)|, |S(jω)|, |T(jω)| over the frequency grid. The grey band marks the achieved loop bandwidth ω_c.