DC-8 PID Controller Design | Anuranan Bharadwaj

Role

Team Member — Flight Dynamics & PID Design

AE 432: Flight Dynamics and Control · ERAU · Fall 2025

Tools

MATLAB · Transfer Function Analysis · Pole Placement
Characteristic Polynomial Matching · Final Value Theorem

Key Contributions

Derived short period, phugoid, and Dutch roll transfer functions from DC-8 stability derivatives
Characterized all three natural flight modes: ω_n, ζ, and poles at sea level standard
Designed AOA PID controller achieving 4% overshoot against a 5% specification
Verified zero steady-state tracking error analytically using the Final Value Theorem

This page walks through a full flight dynamics and control workflow for the DC-8: deriving modal transfer functions from stability derivatives, characterizing the aircraft's natural flight modes, and designing PID controllers for angle of attack and forward speed to meet specified handling-quality targets.

Project Context

Modern aircraft rely on automated flight control systems to maintain stability, track commanded states, and reject atmospheric disturbances — tasks too fast and too precise for a pilot to perform manually at all flight conditions. The PID controller is the foundational architecture behind elevator deflection systems, autothrottle, and autopilot modes on virtually every commercial aircraft flying today. This project applied that theory to the McDonnell Douglas DC-8: a classic long-range jet airliner whose well-documented stability derivatives made it an ideal subject for a full modal analysis and controller synthesis workflow.

Extract short period, phugoid, and Dutch roll transfer functions from DC-8 stability derivatives at sea level standard (Mach 0.218, V = 243 ft/s)
Compute natural frequencies and damping ratios for all three longitudinal and lateral-directional flight modes
Design an elevator-input PID controller to regulate angle of attack (Δα) to a 5° step command with M_p ≤ 5% and t_s < 10 s
Design a throttle-input PID controller to regulate forward speed (ΔU) with M_p ≤ 7% and t_s < 15 s
Verify steady-state tracking analytically with the Final Value Theorem for both control loops

DC-8 aircraft stability characteristics table

Transfer Functions & Modal Characterization

Three classical approximation transfer functions were derived from the DC-8’s published stability derivatives, one for each natural mode of motion. Each was obtained by applying Laplace transforms to the linearized equations of motion and isolating the ratio of output state to control surface deflection.

Short Period (elevator → angle of attack):

Δα(s) / Δδ_e(s) = (−0.04195s − 1.384) / (s² + 1.6815s + 2.616)

Phugoid / Long Period (throttle → forward speed):

ΔU(s) / Δδ_T(s) = 12.9s / (s² + 0.02915s + 0.03326)

Dutch Roll (rudder → roll rate):

ΔP(s) / Δδ_r(s) = (0.03379s − 0.3953) / (s² + 0.3794s + 0.7918)

Natural frequencies and damping ratios extracted from pole locations:

Mode	Poles	ω_n (rad/s)	ζ
Short Period	−0.8404 ± j1.3819	1.6144	0.5196
Phugoid	−0.01455 ± j0.18179	0.1823	0.0798
Dutch Roll	−0.1897 ± j0.8694	0.8899	0.2132

The phugoid mode’s low damping (ζ = 0.0798) is characteristic of commercial transport aircraft — it produces slow, lightly-damped speed-altitude oscillations that pilots typically correct through autothrottle engagement. The short period’s ζ = 0.5196 is within the acceptable MIL-SPEC range for Level 1 handling qualities.

Design Decision

Trade-off: The phugoid mode was left uncontrolled rather than designed against, despite its very low damping (ζ = 0.0798).

Why: This damping level is normal for transport aircraft and stable, just slow to settle — the project instead focused PID design effort on the AOA and forward-speed loops where closed-loop control directly addresses the performance specifications.

Angle of Attack PID Controller

The closed-loop transfer function C(s) for the AOA loop combines the short period plant G(s), an actuator model G_A(s) = 50/(s+50), and the PID controller G_C(s) = (k_ds² + k_ps + k_I)/s. To make the gain system solvable (3 unknowns, 3 equations), the actuator was treated as perfect (G_A = 1), reducing the closed-loop characteristic equation to 3rd order.

Performance specifications mapped to desired pole locations:

M_p ≤ 5% → ζ_d = 0.6901
t_s < 10 s → ω_nd = 3 / (t_s ζ_d) = 0.4347 rad/s
Third real pole: p₃ = −10 ζ_d ω_nd ≈ −3

Desired characteristic equation: s³ + 3.6s² + 1.989s + 0.567 = 0

Matching coefficients of the closed-loop denominator to the desired equation yielded:

k_P = 0.3718 k_I = −0.4366 k_d = −1.569

The Final Value Theorem confirmed zero steady-state error: Δα_ss = lim_s→0 s · C(s) · (5/s) = 5°, exactly tracking the commanded step.

AOA step response with designed PID controller — Fig. 1: AOA step response — 4% overshoot (spec met), settling time 15 s (spec not met), zero steady-state error

Design Decision

Trade-off: Treated the actuator as ideal (G_A = 1) rather than carrying its full 50/(s+50) dynamics into the design equations.

Why: Including the actuator pole would raise the characteristic equation above 3rd order, leaving 3 PID gains to satisfy more than 3 coefficient-matching equations — an unsolvable system. Assuming a perfect actuator kept the problem exactly determined while still meeting the overshoot specification.

Forward Speed PID Controller (Throttle Input)

The throttle-controlled forward speed loop used the phugoid transfer function as the plant, paired with an actuator G_A(s) = 75/(s+75). The closed-loop characteristic equation expanded to 4th order. Because the actuator introduced a 4th pole at s = −75 while only 3 PID gains are available, the system was underdetermined. The 4th pole was removed from the desired characteristic equation to make it solvable.

Performance specifications:

M_p ≤ 7% → ζ_d = 0.6461
t_s < 15 s → ω_nd = 0.3095 rad/s, p₃ ≈ −2

Solved PID gains: k_d = −0.075, k_p = −7.364×10⁻³, k_I = −2.381×10⁻³

The Final Value Theorem confirmed ΔU_ss = 0 — zero steady-state speed error. However, the extremely small gain magnitudes and the resulting large forward speed transients (visible in Fig. 2) indicated the plant model was not well-conditioned for this controller topology at this flight condition.

Forward speed response: uncontrolled vs PID controlled — Fig. 2: Forward speed response — uncontrolled (top) vs PID controlled (bottom); very small gains produce large-amplitude transients

Design Decision

Trade-off: Removed the actuator's 4th pole from the desired characteristic equation rather than leaving the 4th-order system underdetermined.

Why: With only 3 PID gains to satisfy 4 coefficient-matching equations, the system had no unique solution. Dropping the 4th pole made the problem solvable and preserved zero steady-state error, though the resulting near-zero gains and large transients suggest this plant/actuator pairing is not well-conditioned for a 3-gain PID at this flight condition.

Key Results

The AOA loop met the overshoot specification (4% ≤ 5%) but missed the settling time target (15 s vs. <10 s), while achieving exact zero steady-state error. The forward speed loop verified zero steady-state error analytically but produced large-amplitude transients, falling short of both transient specs.

Controller	Max Overshoot	Settling Time	Steady-State Error	Spec Met?
AOA (elevator)	4% (spec: ≤5%)	15 s (spec: <10 s)	0°	Overshoot ✓ t_s ✗
Forward Speed (throttle)	N/A — large transients	Not achieved	0 ft/s	Analytically ✗

PID Gain Summary:

Loop	k_P	k_I	k_d
AOA (elevator)	0.3718	−0.4366	−1.569
Forward Speed (throttle)	−7.364×10⁻³	−2.381×10⁻³	−0.075

Key Takeaways

PID gains can be solved analytically from characteristic polynomial matching.

Mapping performance specs (M_p, t_s) to desired poles, then equating denominator coefficients of the closed-loop TF to those desired poles, yields exact PID gain values — no trial and error. This coefficient-matching method is the standard workflow for any design-to-spec controller problem.

Actuator dynamics can make the system of equations unsolvable.

Including G_A(s) = 75/(s+75) raised the characteristic equation to 4th order with only 3 unknowns, creating an underdetermined system. Removing the actuator pole made it tractable, but the resulting gain magnitudes were near-zero — signaling the actuator may not meaningfully affect this loop at sea level.

The phugoid mode’s low damping is why autothrottle exists.

The DC-8’s phugoid damping ζ = 0.0798 means uncontrolled speed-altitude oscillations decay extremely slowly. This is normal for transport aircraft — the mode is stable but barely. Modern autothrottle and flight management systems exist precisely to regulate this otherwise sluggish response.

The integral gain guarantees zero steady-state error regardless of transient behavior.

The Final Value Theorem confirmed Δα_ss = 5° analytically — even though the settling time specification was missed, the error at steady state is guaranteed zero by the integrator. This distinction between transient and steady-state performance is fundamental: a controller can fail one spec while passing another.