Gain-Scheduled Pitch Autopilot

Autopilot

Commanded pitch θ_cmd 10°

Flight condition

Airspeed V 250 kt

Altitude h 0 m

Live state

Pitch angle θ

0.0°

Pitch rate q

0.0°/s

Elevator δe

0.0°

Dyn. pressure Q̂

1.00×

Effective gains (Kp / Kd)

1.00 / 0.70

How it works

The aircraft's short-period pitch dynamics are integrated from a linearized 2-state model in angle of attack α and pitch rate q (all angles in radians internally):

α̇ = q − Ẑα·α
q̇ = M̂α·α + M̂q·q + M̂δe·δe
θ̇ = q

The aerodynamic derivatives M̂α, M̂q, Ẑα (stability/damping) and M̂δe (elevator control power) all scale with dynamic pressure, Q = ½ρ(h)V², normalized to a reference condition as Q̂ = Q/Q_ref — using an exponential atmosphere ρ(h) = ρ₀·e^−h/8500m:

M̂α = M̂α0·Q̂    M̂q = M̂q0·Q̂    M̂δe = M̂δe0·Q̂    Ẑα = Ẑα0·Q̂

A pitch-attitude-hold autopilot commands the elevator as δe = Kp·(θ_cmd − θ) − Kd·q. Because the plant's control power M̂δe grows linearly with Q̂, a fixed Kp, Kd tuned at the reference condition (Q̂ = 1) produces a loop gain that scales with Q̂ too — at high speed / low altitude (large Q̂) the loop over-amplifies and the pitch response overshoots and oscillates; at low speed / high altitude (small Q̂) the same gains barely move the elevator and the response goes sluggish.

Gain scheduling cancels this by dividing the gains by the measured Q̂: Kp = Kp0/Q̂, Kd = Kd0/Q̂, so the loop gain Kp·M̂δe = Kp0·M̂δe0 stays constant across the whole flight envelope — the classic dynamic-pressure gain-scheduling design used on real fly-by-wire pitch autopilots.

Gain scheduling toggle — switch between the fixed reference-point gains and the Q̂-scheduled gains.
Commanded pitch — the attitude setpoint θ_cmd the autopilot tries to hold.
Airspeed / Altitude — set the flight condition; together they determine Q̂ and hence how badly a fixed-gain controller mistunes itself.

θ = 0.0° θcmd = 10.0° Q̂ = 1.00× δe = 0.0°

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