04 / ONE INSTANT, THREE DIRECTIONS
Pitch and flight path are different angles
Actual angles: θ = α + γ. The autopilot controls pitch attitude. Airspeed and height can still change because thrust is fixed.
FLIGHT CONTROL & AUTOPILOT / CHAPTER 01
Pitch Attitude Hold
Set the intention. Follow the elevator. Watch the aircraft respond.
COMMAND → RESPONSEOriginal wing–tail teaching aircraft
At trim0.0 s
Body directionFlight pathLevel reference
3D is unavailable on this device. The pitch indicator, loop and calculated curves remain available.
Target change · Δθc—
Actual change · Δθ—
Tracking error · e—Target minus actual
Elevator change · Δδe—Positive = trailing edge down
0–1 s At trim1–2 s Command ramp8–9.5 s Optional disturbanceTo 30 s Observe response
Preparing the model
03 / THE CONTROL LOOP · LIVE CASE B
Follow one command around the loop
REFERENCE − MEASUREMENT
→Pitch error
e = Δθc − Δθ
ATTITUDE + RATE FEEDBACK
→Controller
FINITE RESPONSE TIME
→Actuator
τ Δδ̇e + Δδe = Δδcmd
COUPLED AIRCRAFT DYNAMICS
Pitch response
↶
Measured pitch and pitch rate return to the controller. Ideal sensors in this first version.
↵All loop values are changes from trim. A nose-up error commands trailing-edge-up elevator. Positive pitch rate contributes a trailing-edge-down command. The actuator determines the actual elevator motion.
05 / UNDERSTAND THE TRACKING ERROR
A stable response can retain an offset
Current error · B
Peak overshoot · 2–8 s
This controller uses proportional attitude feedback plus optional rate feedback. It has no integral action or automatic trim adjustment. The calculated response may therefore retain a pitch error; increasing gain also changes stability and control demand.
06 / THE SAME CLOCK, THE SAME EQUATIONS
Command, motion and consequences
Click a curve or use its arrow keys to scrub time.
A · dashed amber B · solid turquoise
Pitch change · Δθ / °
Elevator change · Δδe / °
Thick: actual elevator · thin dotted: controller command
Pitch rate · q / ° s⁻¹
Tracking error · e / °
Airspeed change · ΔV / m s⁻¹
Angle of attack change · Δα / °
Height change · Δh / m
External pitch moment · Md / N m
07 / HOLD A, EXPLORE B
Compare controller settings
| Quantity | A · saved | B · live |
|---|
A stays fixed for this page session. Reloading clears it. Export includes inputs, trim, system eigenvalues and full valid response histories.
How the model works · equations, limits & sources
Aircraft and controller
Reuse the original longitudinal aircraft: [ΔV, Δγ, Δθ, q]. At each initial flight condition, solve a level-flight trim, then calculate its force/moment derivatives. The fifth dynamic state is actual elevator change.
Pitch feedback and elevator lag
Autopilot OFF commands zero elevator change. This proportional-plus-rate controller has no integral action or automatic trim adjustment.
Symbols & units
- Change from the initial trim state. It does not indicate a different aircraft or an absolute angle.
- Kθ is dimensionless; Kq and τ are seconds. Rate feedback OFF sets Kq to zero.
- Pitch rate, rad/s; elevator change, rad. Positive elevator means trailing edge down.
Model context and notation
Δδcmd = −Kθ(Δθc − Δθ) + Kq q. Rate feedback can be switched off. The actuator obeys τ Δδ̇e = Δδcmd − Δδe. Autopilot OFF commands zero elevator change, while the same target remains visible for comparison.
Positive elevator means trailing edge down. The external disturbance enters q̇ as Md/Iyy; it is a prescribed moment, not a simulated gust.
Response and model boundaries
Pitch, angle of attack and flight-path angle remain distinct. Thrust is held at its initial trim value. This is a local pitch controller with ideal sensors, no integral action, no altitude hold and no speed hold.
Stop before speed change exceeds 10%; pitch, flight-path or angle-of-attack disturbance exceeds 5°; pitch rate exceeds 15°/s; height change exceeds 150 m; or absolute α leaves −6° to 12°. Also stop at elevator change 8°, total elevator 20°, command change 12°, or actuator rate 40°/s. No silent control clipping.
Interpret the evidence
All dynamic roots must have negative real parts for local asymptotic stability. A finite preview cannot prove stability by appearance alone. Height and forward-position integrators are excluded from the five-state spectrum.
Overshoot is measured from 2–8 s before the disturbance begins. It is unavailable for a zero target or if the history stops before 8 s. The error at 30 s is a finite-time result, not a claimed steady-state value. A/B use one elapsed-time clock and individual validity endpoints.
Teaching sources and applicability
Week 13, Review and Flight Control System and Flight Management System, 17 April 2026, slides 39–52; particularly slides 45–46 on aircraft dynamics and pitch autopilot.
MIT OCW · Basic Longitudinal Control.
Original illustrative parameters and diagrams. Shared 3D teaching geometry and local Three.js retain their existing provenance. Numerically verified within the stated equations; no real-aircraft flight-test validation or certified autopilot claim. Version .