Stability augmentation and autopilot basics

Why it matters

When the natural modes of an aircraft fail its flying-qualities requirements — a lightly damped Dutch roll at altitude, a sluggish short period, or a fighter deliberately made unstable for agility — the fix is feedback: sensors measure the motion and the control surfaces are moved automatically. Stability augmentation changes the aircraft's effective derivatives; autopilots wrap outer loops around it to hold attitude, altitude, heading or speed. Almost every modern aircraft depends on both.

Key ideas

Hierarchy of systems.

• Stability augmentation system (SAS): inner-loop feedback of angular rates (and sometimes α or β) to the control surfaces, adding damping or stiffness. The pilot still flies the aircraft; the SAS changes how it responds. Examples: pitch damper, yaw damper, roll damper.
• Control augmentation system (CAS): the pilot's stick commands a response (pitch rate, load factor, roll rate) and the system closes the loop to deliver it, mixing pilot command and feedback.
• Autopilot: outer loops that hold or track a reference — wing leveller, pitch-attitude hold, altitude hold, heading hold, Mach/airspeed hold (with auto-throttle), ILS approach coupling. Outer loops are made much slower than inner loops so that the inner loop looks like a fast, well-behaved actuator to them.

Feedback modifies derivatives. With a feedback law δ = K·x, the control derivative adds to the natural one:

• Pitch damper, δe = K_q·q: effective Mq′ = Mq + Mδe·K_q. Since Mδe < 0, a positive K_q makes Mq′ more negative — more short-period damping (and some extra frequency).
• Angle-of-attack feedback, δe = K_α·α: Mα′ = Mα + Mδe·K_α — adds stiffness; this is how relaxed-stability fighters with Mα > 0 are made flyable.
• Yaw damper, δr = K_r·r: Nr′ = Nr + Nδr·K_r — raises Dutch-roll damping. A washout (high-pass) filter is placed in the yaw-rate path so the damper does not oppose the steady yaw rate of a turn.
• Sign convention matters: with positive elevator and rudder trailing edge down/left, Mδe and Nδr are negative, so the damping gains are positive.

Autopilot loops (examples).

• Pitch-attitude hold: δe from θ error plus a q damping term.
• Altitude hold: an outer loop commanding θ (or flight-path angle) from altitude error, around the attitude loop.
• Heading hold: commands bank angle from heading error; a roll-attitude inner loop uses the ailerons; a yaw damper/turn coordinator keeps β ≈ 0.
• Speed hold: throttle (auto-throttle) or pitch (on climb/descent).

Design concerns. Gains too high cause actuator saturation, sensitivity to noise and structural-mode coupling; time delays and actuator rate limits erode stability margins and can cause pilot-induced oscillations. Flight-critical systems (fly-by-wire for unstable aircraft) need redundancy — typically triplex or quadruplex sensors, computers and actuators with voting — and dissimilar back-up.

Analysis tools. Root locus (how the closed-loop roots move with gain), Bode plots and gain/phase margins, and modern state-space methods; the simple derivative-substitution approach above gives first estimates.

Assumptions. Ideal sensors and actuators (no lag) in the simple formulas; real systems need actuator and filter dynamics included.

Formulas

Mq′ = Mq + Mδe·K_q (pitch damper, δe = K_q·q)

• Mq (s⁻¹), Mδe (s⁻²): dimensional derivatives; K_q: gain (rad of elevator per rad/s, i.e. s).

Mα′ = Mα + Mδe·K_α (α feedback, δe = K_α·α)

Nr′ = Nr + Nδr·K_r (yaw damper, δr = K_r·r)

Short-period estimate with damper: ω_sp² ≈ Zα·Mq′/u₀ − Mα′, 2·ζ_sp·ω_sp ≈ −(Mq′ + Mα̇ + Zα/u₀)

Dutch-roll estimate: ω_DR² ≈ (Yβ·Nr′ + u₀·Nβ)/u₀, 2·ζ_DR·ω_DR ≈ −(Yβ/u₀ + Nr′)

Washout filter: G(s) = τ_w·s/(τ_w·s + 1)

• τ_w: washout time constant (s); passes oscillations, blocks steady signals.

Worked examples

Example 1 (standard). At u₀ = 100 m/s an aircraft has Yβ = −20 m/s², Nβ = 3.0 s⁻², Nr = −0.3 s⁻¹ and Nδr = −2.0 s⁻². Find the Dutch-roll damping ratio without augmentation, and the yaw-damper gain K_r that gives Nr′ = −1.0 s⁻¹ and the new damping ratio.

1. Bare aircraft: ω² = ((−20)(−0.3) + 100 × 3.0)/100 = (6 + 300)/100 = 3.06, ω = 1.749 rad/s.
2. 2ζω = −(−0.2 − 0.3) = 0.5 → ζ = 0.5/(2 × 1.749) = 0.143.
3. Nr′ = Nr + Nδr·K_r → −1.0 = −0.3 − 2.0·K_r → K_r = 0.35 s.
4. Augmented: ω² = (20 + 300)/100 = 3.20, ω = 1.789 rad/s; 2ζω = −(−0.2 − 1.0) = 1.2 → ζ = 1.2/(2 × 1.789) = 0.335.

Answer: ζ_DR rises from 0.14 to 0.34 with K_r = 0.35 s.

Example 2 (GATE level). At high altitude an aircraft flies at u₀ = 200 m/s with Zα = −100 m/s², Mα = −6 s⁻², Mα̇ = −0.2 s⁻¹, Mq = −0.5 s⁻¹, Mδe = −8 s⁻². (a) Find the bare short-period frequency and damping. (b) Find the pitch-damper gain K_q that gives ζ_sp = 0.6. (c) A relaxed-stability version has Mα = +2 s⁻². What α-feedback gain K_α restores Mα′ = −10 s⁻²?

(a)

1. ω² = (−100)(−0.5)/200 − (−6) = 0.25 + 6 = 6.25 → ω_sp = 2.5 rad/s.
2. 2ζω = −(−0.5 − 0.2 − 0.5) = 1.2 → ζ_sp = 1.2/5.0 = 0.24.

(b)

1. Let x = −Mq′. Then ω² = 6 + 0.5x and 2ζω = x + 0.7.
2. For ζ = 0.6: x + 0.7 = 1.2·√(6 + 0.5x). Squaring: x² + 1.4x + 0.49 = 8.64 + 0.72x → x² + 0.68x − 8.15 = 0 → x = 2.535.
3. Mq′ = −2.535 = −0.5 + (−8)·K_q → K_q = 2.035/8 = 0.254 s.
4. Check: ω = √(6 + 1.268) = 2.696 rad/s, ζ = 3.235/(2 × 2.696) = 0.600 ✓.

(c)

1. Mα′ = Mα + Mδe·K_α → −10 = 2 − 8·K_α → K_α = 1.5 (rad of elevator per rad of α).

Answer: (a) ω_sp = 2.5 rad/s, ζ_sp = 0.24; (b) K_q ≈ 0.25 s (giving ω_sp ≈ 2.70 rad/s); (c) K_α = 1.5.

Common mistakes

• Getting the feedback sign wrong: with Mδe < 0, a positive K_q adds damping; the opposite sign destabilises.
• Forgetting the washout on a yaw damper, which makes it fight the pilot in steady turns.
• Assuming a pitch damper changes only damping — through the Zα·Mq/u₀ term it also raises ω_sp.
• Confusing SAS (changes response) with autopilot (holds a reference).
• Ignoring actuator rate limits and time delay, which can turn a good linear design into a PIO-prone one.
• Treating an unstable airframe's augmentation as optional: loss of the system means loss of control, so redundancy is mandatory.

For GATE AE

Expect conceptual questions on SAS versus autopilot, which feedback cures which mode (pitch-rate for short period, yaw-rate for Dutch roll, α for static instability), the purpose of washout filters and inner/outer loop structure, and short numericals finding an effective derivative or feedback gain.

Quick check

1. Which sensed variable does a yaw damper feed back, and to which surface?
2. Mq = −1.0 s⁻¹, Mδe = −5 s⁻², K_q = 0.2 s. Find Mq′.
3. Why is a washout filter used in a yaw damper?
4. Which loop should be faster: attitude hold or altitude hold?

Answers: 1. Yaw rate, to the rudder. 2. −1.0 − 1.0 = −2.0 s⁻¹. 3. To pass Dutch-roll oscillations but not the steady yaw rate of a turn. 4. Attitude hold (the inner loop).