Lead–Lag Compensation of an Aircraft Pitch Control System

Continues from → Project 06: Lead Compensator Design

Classical Control Systems | Lead–Lag Compensator | Steady-State Error | Control System Designer | MATLAB | Aerospace Engineering

This repository contains my seventh independent control systems project — a systematic investigation into whether a Lead–Lead–Lag cascade compensator can succeed where a single lead compensator (Project 06) failed on the aircraft pitch control system.

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Engineering Question

"Can a Lead–Lead–Lag cascade satisfy overshoot < 10% and settling time < 10 seconds — with meaningful steady-state tracking — on a non-minimum phase plant?"

Answer: Not simultaneously. Transient specs and steady-state accuracy trade off against each other, and the trade-off is structural, not a tuning problem.

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Overview

Project 06 proved that a single lead compensator cannot meet the specifications — the required 86.75° of phase lead exceeds the ~65° practical limit of one stage. Project 07 asks the natural follow-up: does cascading two lead stages with a lag stage close the gap?

Four sequential investigation stages were used:

1. Lead–Lead cascade — combine two lead stages for more phase authority
2. Lag compensator study — eight configurations (Tests A–H) to recover DC gain
3. Gain sweep — test whether raw gain increase can resolve the remaining deficit
4. Automated optimization — let MATLAB search the full Lead–Lead–Lag space

All four stages point to the same root cause.

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Plant

G(s) = (−1.282s + 1.282) / (s³ + 1.935s² + 0.987s + 0.179)

Property	Value
RHP Zero	s = +1 (non-minimum phase)
Phugoid Poles	−0.3336 ± 0.1730j (dominant, lightly damped)
Short Period Pole	−1.2679 (fast, heavily damped)
Gain Margin (from P05)	K = 0.46

Performance Specifications:

• Overshoot < 10% → damping ratio ζ ≥ 0.59
• Settling time < 10 s → real part σ ≥ 0.4
• Desired dominant pole: s_d = −0.5 + 0.6j

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Stage 1 — Lead–Lead Investigation

C_LL(s) = K (s + 0.5)(s + 0.8) / [(s + 2)(s + 4)]

Parameter	Result	Requirement	Status
Settling Time	9.51 s	< 10 s	✅
Overshoot	0.625%	< 10%	✅
Closed-Loop DC Gain	5.26×10⁻⁵	≈ 1.0	❌
Steady-State Error	≈ 100%	0%	❌

The catch: rlocfind() selected K ≈ 1.47×10⁻⁴ — a gain so small the loop barely closes. The transient specs pass on paper, but the closed-loop poles are nearly identical to the open-loop poles. The system is operating near open-loop conditions with negligible control authority.

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Stage 2 — Lag Compensator Study (Tests A–H)

Eight lag zero/pole pairs were swept, holding the Lead–Lead base fixed, to boost low-frequency gain.

Test	z_lag	p_lag	Settling Time	Overshoot	DC Gain	Verdict
A	−0.1	−0.01	122.63 s	0%	0.714	❌ too slow
D	−0.8	−0.20	15.18 s	4.18%	0.464	❌ ts high
E	−1.0	−0.25	14.46 s	7.41%	0.489	🏆 best balance
F	−1.2	−0.30	12.81 s	13.13%	0.596	❌ OS too high
H	−2.0	−0.50	10.66 s	1.40%	0.189	❌ tracking collapse

Trend: moving the lag closer to the lead frequencies speeds up the response but reintroduces overshoot. Test E is the best compromise — and it still misses the settling-time spec while only tracking 48.9% of the commanded pitch angle.

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Stage 3 — Gain Sweep on Test E

K	DC Gain	Settling Time	Overshoot
0.279	0.500	201.2 s	0%
1.500	0.843	75.6 s	0%
4.000	0.935	41.3 s	2.08%

Pushing gain improves tracking but the settling time never approaches spec, and overshoot begins creeping in at the high end. Gain is not a free parameter on this plant.

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Stage 4 — Automated Optimization

MATLAB's Response Optimizer searched the full Lead–Lead–Lag parameter space:

C_Optimizer(s) = 1.2599 · (s+0.6056)(s+0.5187)(s+0.4230) / [(s+2.166)(s+4.045)(s+0.4004)]

Parameter	Result	Requirement	Status
Settling Time	5.71 s	< 10 s	✅
Overshoot	1.66%	< 10%	✅
Closed-Loop DC Gain	0.2547	≈ 1.0	—
Steady-State Error	74.5%	≈ 0%	❌

The optimizer proved the transient specs are achievable with Lead–Lead–Lag — but only by placing the lag pole almost on top of the lag zero, effectively cancelling the lag stage and sacrificing steady-state tracking to get there.

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Complete Controller Comparison — Projects 04–07

Controller	Settling Time	Overshoot	DC Gain	SS Error	Both Specs?
P only	13.23 s	18.15%	Low	58%	❌
PID	19.49 s	3.39%	High	0%	❌ (ts)
Best Lead (P06)	13.31 s	28.61%	Low	Large	❌
Lead–Lead	9.51 s	0.625%	5.26×10⁻⁵	≈100%	❌ (tracking)
Lead–Lead–Lag (Test E)	14.46 s	7.41%	0.489	51%	❌ (ts)
Optimizer (Best)	5.71 s	1.66%	0.255	74.5%	❌ (SSE)

No classical controller across four projects has met OS < 10%, ts < 10s, and meaningful steady-state tracking at the same time.

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Key Engineering Conclusions

1. Lead–Lead transient compliance is a mathematical artifact of a near-zero loop gain, not genuine control performance.

2. The lag stage successfully recovers DC gain (5.26×10⁻⁵ → 0.489) but reintroduces overshoot as it approaches the lead frequency range — a direct speed-stability trade-off.

3. Gain adjustment cannot resolve the trade-off: settling time stays roughly 4× over spec at every tested gain, while overshoot grows at high gain.

4. The automated optimizer confirms Lead–Lead–Lag can meet both transient specs — at the cost of 74.5% steady-state error.

5. The RHP zero at s = +1 is the common root cause across all seven projects: it caps achievable bandwidth, produces the characteristic undershoot in every step response, and prevents simultaneous optimization of transient speed and DC tracking.

6. The trade-off is structural, not a tuning failure. State-space methods (Projects 09–11) are needed to place all closed-loop poles simultaneously and break this limitation.

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Aerospace Applications

• Commercial autopilots (777/A350-class): use inner pitch-rate + outer pitch-angle loops specifically to separate non-minimum phase dynamics from the tracking integrator.
• Military fighters (F-16/F-22-class): deliberately statically unstable, relying on state-space control with Kalman filtering rather than classical cascades.
• UAV pitch control: gain scheduling is used in practice to work around the same speed-stability trade-off found here.
• Teknofest VLR Rocket: the plant's non-minimum phase behavior worsens as thrust and center of mass shift during burn — motivating the LQR/Kalman filter work in Projects 10–11 for the VLR attitude controller.

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Project Roadmap

✅ Project 01 — Mass-Spring-Damper Analysis
✅ Project 02 — DC Motor Modeling
✅ Project 03 — PID Speed Control
✅ Project 04 — Aircraft Pitch Control
✅ Project 05 — Root Locus Design
✅ Project 06 — Lead Compensator Investigation
✅ Project 07 — Lead–Lag Compensator Design ← YOU ARE HERE

→ Project 08 — Frequency Response Analysis
→ Project 09 — State-Space Modeling
→ Project 10 — Pole Placement Control
→ Project 11 — LQR Optimal Control
→ Project 12 — Kalman Filter Design
→ Project 13 — UAV Attitude Control
→ Project 14 — Rocket Attitude Control
→ Project 15 — Satellite Attitude Control
→ Project 16 — Missile Guidance and Control
→ Project 17 — Integrated Flight Control System

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Software Used

• MATLAB R2024b
• Control System Toolbox
• Control System Designer (SISO Tool)

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Author

Zohaib Imtiaz Aerospace Engineering Student | Teknofest VLR Team — Flight Control

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License

This project is released under the MIT License.

Project Cover

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Optimizer Working

Optimizer.2.mp4