Cessna 172S Skyhawk — Longitudinal Flight Dynamics
A professional-grade MATLAB/Simulink implementation of the linearised longitudinal flight dynamics of the Cessna 172S Skyhawk — Developed as a personal aerospace engineering portfolio project by an Aerospace Engineering student at King Fahd University of Petroleum and Minerals (KFUPM).
The simulator implements the complete longitudinal analysis pipeline from first principles — no Optimization Toolbox, no Control System Toolbox — using only base MATLAB and Simulink.
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| Aerodynamic Polars | Eigenvalue Pole Map | Elevator Step Response (Pitch) |
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| Elevator Step Response (Velocity) | Linear vs. Nonlinear Comparison | Simulink Block Diagram |
Reference condition: V₀ = 55 m/s, sea level ISA (ρ = 1.225 kg/m³)
| Quantity | Symbol | Value | Unit |
|---|---|---|---|
| Trim angle of attack | α₀ | 0.397 | deg |
| Trim elevator deflection | δe₀ | 3.358 | deg |
| Trim lift coefficient | CL₀ | 0.3630 | — |
| Trim drag coefficient | CD₀ | 0.0329 | — |
| Lift-to-weight ratio | L/W | 1.0000 | — |
| Trim pitching moment | Cm | 0 (exact) | — |
| Thrust required | T | 988.4 | N |
| Mode | ωn [rad/s] | ζ [-] | Period [s] | T½ [s] | HQ Level |
|---|---|---|---|---|---|
| Short-period | 5.810 | 0.868 | 2.178 | 0.137 | MIL-SPEC Level 1 |
| Phugoid | 0.2007 | 0.0753 | 31.4 | 45.87 | MIL-SPEC Level 1 |
Both modes are stable. The short-period mode is well-damped (ζ ≈ 0.87) — pitch disturbances decay within ~0.5 s. The phugoid is lightly-damped (ζ ≈ 0.075), producing the characteristic slow speed-altitude exchange with a period of ~31 s.
Nonlinear longitudinal EOM in body-fixed axes:
u̇ = −q·w + (X_aero + T)/m − g·sin θ
ẇ = q·u + Z_aero/m + g·cos θ
q̇ = M_aero / Iyy
θ̇ = q
State vector: x = [Δu Δw Δq Δθ]ᵀ
ẋ = A·x + B·u_ctrl
y = C·x
Δu Δw Δq Δθ
Δu̇ [ Xu/m Xw/m 0 −g·cos θ₀ ]
Δẇ [ Zu/m Zw/m Zq/m + V₀ −g·sin θ₀ ]
Δq̇ [ Mu*/Iyy Mw*/Iyy Mq*/Iyy 0 ]
Δθ̇ [ 0 0 1 0 ]
Starred derivatives (Mu*, Mw*, Mq*) absorb the Mα̇·ẇ/V₀ coupling term analytically (Nelson 1998, §4.4).
Computed A matrix at trim (V₀ = 55 m/s, α₀ = 0.397°):
Δu Δw Δq Δθ
Δu̇ [ -0.0324 0.1071 0.0000 -9.8064 ]
Δẇ [ -0.3566 -2.1972 53.5689 -0.0680 ]
Δq̇ [ 0.0134 -0.3063 -7.8881 0.0000 ]
Δθ̇ [ 0.0000 0.0000 1.0000 0.0000 ]
The pipeline runs from a single parameter source through trim, state-space assembly, modal analysis, and simulation to validated engineering outputs. Each module is self-contained and unit-testable.
Running main.m orchestrates all six stages in sequence. Each canonical module accepts structured inputs and returns structured outputs — no global state.
A programmatic Simulink state-space block is built from the same A, B, C, D matrices produced by the MATLAB pipeline, providing an independent cross-validation path.
| Check | Simulink Workflow |
|---|---|
| Model construction | build_simulink_model.m builds C172_Longitudinal_SS.slx programmatically |
| State-space source | Same A, B, C, D from build_state_space.m |
| Cross-validation | run_simulink_comparison.m prints MATLAB vs Simulink error table |
| Toolbox required | Simulink only (no Control System Toolbox) |
| Feature | Implementation |
|---|---|
| No-toolbox trim solver | Exact 2×2 linear system (MATLAB \ operator) — no fsolve, no Optimization Toolbox |
| No-toolbox simulation | ode45 directly on ẋ = Ax + Bu with two-phase split — no lsim, no Control System Toolbox |
| Starred moment derivatives | Mα̇ coupling absorbed analytically before A-matrix assembly (Nelson 1998 §4.4) |
| Two-phase ODE integration | Step discontinuity handled by splitting integration at t_step — avoids event-detection overhead |
| Simulink cross-validation | Programmatic model built from same matrices — independent validation path |
| Quantitative validation | Trim residuals, MIL-SPEC-8785C Level 1 check, Lanchester approximation comparison |
| Engineering report | 10-section PDF report compiled from Markdown via Pandoc + XeLaTeX |
| Single parameter source | aircraft_parameters.m is the sole source of truth — no magic numbers anywhere else |
Aircraft-Flight-Dynamics-Simulator/
│
├── matlab/ MATLAB source code (13 modules)
│ ├── main.m Orchestrator — run this file
│ ├── aircraft_parameters.m Single parameter source of truth
│ ├── compute_derived_params.m Derived quantities (W, AR, q_bar, CL_trim)
│ ├── aero_forces.m Aerodynamic coefficients at any state
│ ├── compute_trim.m Trim solver — exact 2×2 linear system
│ ├── build_state_space.m A, B, C, D matrix assembly
│ ├── eom_longitudinal.m Nonlinear body-axis EOM for ode45
│ ├── analyze_modes.m Eigenvalue → SP and phugoid extraction
│ ├── simulate_response.m Linear + nonlinear ode45 integration
│ ├── plot_results.m Five engineering figures → figures/
│ ├── validate_model.m Quantitative validation report
│ └── [5 backward-compatibility wrappers]
│
├── simulink/ Simulink cross-validation
│ ├── build_simulink_model.m Builds C172_Longitudinal_SS.slx
│ ├── run_simulink_comparison.m MATLAB vs Simulink error table
│ └── C172_Longitudinal_SS.slx [generated — run build_simulink_model.m]
│
├── reports/ Engineering documentation
│ ├── Engineering_Report.pdf 10-section technical report (PDF)
│ └── Engineering_Report_PDFReady.md Pandoc + XeLaTeX source
│
├── docs/ Technical reference documentation
│ ├── Model_Reference.md Equations, module pipeline, assumptions
│ ├── State_Space_Model.md A, B, C, D matrix derivation
│ └── State_Variables.md State vector definitions and sign conventions
│
├── references/ Aircraft data (Nelson 1998 Appendix B)
├── figures/ Generated PNG figures (auto-populated by main.m)
├── README.md
├── RELEASES.md
├── .gitignore
└── .gitattributes
| Requirement | Version | Notes |
|---|---|---|
| MATLAB | R2019b or later | Base MATLAB only |
| Simulink | R2019b or later | Required only for simulink/ scripts |
| Optimization Toolbox | — | Not required |
| Control System Toolbox | — | Not required |
| Signal Processing Toolbox | — | Not required |
% 1. Clone the repository
% git clone https://github.com/yourusername/Aircraft-Flight-Dynamics-Simulator.git
% 2. In MATLAB, navigate to the matlab/ folder:
cd('path/to/Aircraft-Flight-Dynamics-Simulator/matlab')
% 3. Run the full longitudinal analysis pipeline:
mainThis executes all six pipeline stages and opens five engineering figures:
- Aerodynamic polars (lift curve, drag polar, L/D ratio)
- Eigenvalue pole map with constant-damping lines
- Step response — elevator command, pitch angle, pitch rate
- Step response — forward velocity and angle of attack
- Linear vs. nonlinear comparison (all four states)
Figures are saved as PNG files to the figures/ directory at 300 dpi.
cd('path/to/Aircraft-Flight-Dynamics-Simulator/matlab')
validate_modelPrints trim residuals, aerodynamic polar consistency, Lanchester approximation comparison, and MIL-SPEC-8785C Level 1 handling quality assessment.
cd('path/to/Aircraft-Flight-Dynamics-Simulator/simulink')
build_simulink_model % creates C172_Longitudinal_SS.slx
run_simulink_comparison % prints MATLAB vs Simulink error table| Check | Criterion | Result | Status |
|---|---|---|---|
| Trim: L/W residual | < 1e-10 | 5.5e-15 | PASS |
| Trim: Cm residual | < 1e-10 | 2.2e-16 | PASS |
| Trim CL > 0 | Physical | 0.3630 | PASS |
| Trim CD > 0 | Physical | 0.0329 | PASS |
| Elevator δe₀ in bounds | |δe| < 25° | 3.36° | PASS |
| SP: ωn (MIL-SPEC Level 1) | ≥ 1.0 rad/s | 5.810 rad/s | PASS |
| SP: ζ (MIL-SPEC Level 1) | 0.35 ≤ ζ ≤ 1.30 | 0.868 | PASS |
| SP: time to half amplitude | < 1 s (Level 1) | 0.137 s | PASS |
| PH: ζ (MIL-SPEC Level 1) | ≥ 0.04 | 0.0753 | PASS |
| PH: stability | ζ > 0 | 0.0753 | PASS |
| Lanchester ωn approx. | Within ~20% expected | 0.252 vs 0.201 rad/s | Expected |
| System stability | All eigenvalues LHP | 4/4 LHP | PASS |
All aerodynamic and stability data sourced from: R. C. Nelson, "Flight Stability and Automatic Control," 2nd ed., McGraw-Hill, 1998, Appendix B — Cessna 172 (modified).
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Wing area | S | 16.2 | m² |
| Wing span | b | 11.0 | m |
| Mean aerodynamic chord | c̄ | 1.494 | m |
| Aspect ratio | AR | 7.47 | — |
| Mass (gross weight) | m | 1111 | kg |
| Pitch moment of inertia | Iyy | 1285.3 | kg·m² |
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Lift-curve slope | CL_α | 4.44 | /rad |
| Zero-lift drag | CD0 | 0.0270 | — |
| Induced drag factor | k | 0.0450 | — |
| Static stability | Cm_α | −0.613 | /rad |
| Pitch damping | Cm_q | −12.40 | /rad |
| Elevator effectiveness | Cm_δe | −1.122 | /rad |
Static margin: 13.8% MAC (statically stable).
Trim solver — no Optimization Toolbox:
The aerodynamic model is linear in α and δe, so the two trim conditions (CL = CL_req and Cm = 0) form an exact 2×2 linear system. Solved with MATLAB's backslash operator — no fsolve, no optimoptions.
Simulation — no Control System Toolbox:
Uses ode45 directly on ẋ = Ax + Bu with two-phase integration split at the step discontinuity (t_step), instead of lsim. This guarantees correct input values on both sides of the discontinuity and avoids any dependency on the Control System Toolbox.
Starred moment derivatives:
The Mα̇ coupling term is absorbed analytically into starred derivatives (Mw*, Mq*, Mde*) before building the A matrix, following Nelson 1998, §4.4. This keeps the A matrix in standard 4×4 form without an augmented state.
Two-phase ODE integration:
The elevator step response is integrated in two sequential ode45 calls — one before the step and one after — with initial conditions for the second phase taken from the final state of the first. This avoids numerical artefacts at the discontinuity.
Single parameter source:
All aircraft constants are defined once in aircraft_parameters.m. No module contains hard-coded numerical values. Changing an aircraft parameter propagates automatically through the entire pipeline.
- R. C. Nelson, Flight Stability and Automatic Control, 2nd ed., McGraw-Hill, 1998.
- B. L. Stevens, F. L. Lewis, E. N. Johnson, Aircraft Control and Simulation, 3rd ed., Wiley, 2016.
- B. Etkin and L. D. Reid, Dynamics of Flight: Stability and Control, 3rd ed., Wiley, 1996.
- U.S. Department of Defense, MIL-SPEC-8785C: Flying Qualities of Piloted Airplanes, 1980.
- Cessna Aircraft Company, Cessna 172S Pilot's Operating Handbook, 2009.
- Lateral-directional dynamics — roll, yaw, sideslip modes (Dutch roll, spiral, roll subsidence)
- Coupled 6-DOF simulation — full nonlinear equations of motion
- PID autopilot design — pitch hold, altitude hold, airspeed hold
- Gust and turbulence response — Dryden turbulence model
- Multiple flight conditions — envelope analysis at different speeds and altitudes
- Aircraft parameter database
- Aerodynamic model (CL, CD, Cm, CX, CZ)
- Trim analysis (exact 2×2 linear solve — no toolbox)
- Linearised state-space model (A, B, C, D)
- Longitudinal modal analysis (short-period and phugoid)
- Nonlinear EOM simulation (ode45)
- Engineering figures (5 figures, 300 dpi PNG)
- Simulink state-space model (cross-validation)
- Quantitative validation report (MIL-SPEC-8785C)
- 10-section engineering report (PDF via Pandoc + XeLaTeX)
This project is licensed under the MIT License. See LICENSE for details.
- R. C. Nelson — the textbook that defines this course of study; the Cessna 172S aerodynamic dataset is from Appendix B.
- KFUPM Aerospace Engineering Department — for the academic foundation that made this project possible.
- MathWorks — for MATLAB and Simulink documentation and examples.
Osama AlFadel
Aerospace Engineering Student
King Fahd University of Petroleum and Minerals (KFUPM)