Anti-Lock Braking System (ABS) Controller Design and Analysis
This repository contains comprehensive analysis and implementation of observer-based state feedback control for automotive anti-lock braking systems using quarter-vehicle dynamics models.
Kiran Sairam Bethi Balagangadaran
MS Robotics, Northeastern University
Professor: Dr. Nivii Kalavakonda Chandrasekar
Course: ME5659 - Control Systems Engineering
This project demonstrates advanced control systems design techniques applied to automotive safety systems. Two comprehensive reports are included:
Complete system analysis and controller design for ABS
- Nonlinear quarter-vehicle model with tire-road friction dynamics
- Linearization around optimal slip ratio (λ* = 0.2, Vx* = 20 m/s)
- Stability, controllability, and observability analysis
- Standard form decomposition revealing unobservable position state
- State feedback controller design using pole placement
- Luenberger observer design for state estimation from slip measurements
- Validation on nonlinear plant with disturbance rejection
- Key Results: 50% overshoot, 0.039s settling time, excellent disturbance rejection
Minimum energy control and animated visualization
- Minimum energy open-loop steering to equilibrium
- Observer convergence demonstration with poor initial estimates
- Real-time animated simulation showing ABS operation
- Interactive visualization with heads-up display (HUD)
- Key Results: 0.5s observer convergence, successful regulation with animation
- Plant: Nonlinear quarter-vehicle ABS model
- States: Stopping distance (Sx), vehicle velocity (Vx), slip ratio (λ)
- Control Input: Brake torque (0-1200 Nm)
- Output: Slip ratio from wheel speed sensors
- Operating Point: λ* = 0.2, Vx* = 20 m/s (maximum friction coefficient)
Wheel Speed Sensors → Observer → State Feedback Controller → Brake Modulator → Vehicle Dynamics
↑ ↓
└─────────── Slip Ratio Measurement ────────┘
- Stability Analysis: Identified unstable open-loop system with eigenvalues at {0, 0.0203, 2.9059}
- Controllability: Full rank controllability matrix - all states controllable via brake torque
- Observability: Rank-2 observable subsystem [Vx, λ] sufficient for ABS control
- Controller Design: Pole placement with ts = 2s, ζ = 0.9 specifications
- Observer Design: 5× faster poles than controller for rapid convergence (0.446s)
- Performance: 50% overshoot, 0.039s settling time on nonlinear plant
- Disturbance Rejection: Maintains performance under 45% friction reduction (dry→wet)
- Minimum Energy Control: Optimal open-loop steering within actuator limits
- Animation: Real-time visualization demonstrating control effectiveness
Control-Systems-Engineering-Project/
├── Matlab Code/ # MATLAB simulation files
│ ├── sec41.m # Section 4.1: State feedback controller
│ ├── sec42.m # Section 4.2: Luenberger observer
│ ├── sec43.m # Section 4.3: Observer-based control (linear)
│ ├── sec44.m # Section 4.4: Nonlinear plant application
│ ├── sec45.m # Section 4.5: Disturbance rejection
│ ├── sec51.m # Section 5.1: Minimum energy control
│ ├── sec52.m # Section 5.2: Observer convergence demo
│ └── sec53.m # Section 5.3: Animated ABS simulation
│ └── sec53.gif # Animation output (6.74 MB)
├── Extra Credit Control Systems.pdf # Extra credit report
├── Final Report Controls Systems.pdf # Main project report
└── README.md # This file
The project includes an animated simulation (sec53.gif) showing:
- Vehicle position along roadway
- Real-time velocity tracking (true vs. estimated)
- Real-time slip ratio tracking (true vs. estimated)
- Heads-up display with system metrics
- Observer convergence from poor initial estimates
All sections include comprehensive plots:
- State trajectories with equilibrium tracking
- Control effort within actuator limits
- Phase portraits showing system dynamics
- Estimation error decay (exponential convergence)
- Disturbance rejection response
- MATLAB R2020b or later
- Control System Toolbox
- (Optional) Symbolic Math Toolbox for analytical verification
Each MATLAB file is self-contained and can be run independently:
% State feedback controller
run('sec41.m')
% Luenberger observer
run('sec42.m')
% Observer-based control on linear plant
run('sec43.m')
% Nonlinear plant validation
run('sec44.m')
% Disturbance rejection
run('sec45.m')
% Minimum energy control
run('sec51.m')
% Observer convergence demonstration
run('sec52.m')
% Animated ABS simulation
run('sec53.m')To view the ABS animation:
- Open
sec53.gifin any image viewer - Or run
sec53.min MATLAB to generate a new animation - Online version available at: Google Drive Link
Nonlinear State Equations:
ẋ₁ = x₂
ẋ₂ = -μ(x₃,x₂)Fₙ/m
ẋ₃ = -μ(x₃,x₂)Fₙ/x₂[(1-x₃)/m + R²/Jw] + R/(Jwx₂)u
Friction Model (Pacejka-style):
μ(λ,Vx) = c₁(1-e^(-c₂λ)) - c₃λ · e^(-c₄Vx)
Linearized System:
A = [0 1.0000 0 ] B = [0 ]
[0 0.1883 1.4362] [0 ]
[0 0.3178 2.7380] [0.0146 ]
C = [0 0 1]
Controller Design:
- Desired poles: p = -2.000 ± 0.969j
- Feedback gains: K = [294.887, 474.404]
Observer Design:
- Observer poles: pobs = -10.000 ± 4.843j (5× faster)
- Observer gains: L = [401.871; 22.926]
| Metric | Linear Plant | Nonlinear Plant |
|---|---|---|
| Overshoot | 171.36% | 50.17% |
| Settling Time | 2.907s | 0.039s |
| Steady-State Error | 0.000128 | 0.000000 |
| Control Effort | 656-765 Nm | 0-725 Nm |
| Parameter | Symbol | Value |
|---|---|---|
| Vehicle Mass | m | 342 kg |
| Wheel Inertia | Jw | 1.13 kg·m² |
| Wheel Radius | R | 0.33 m |
| Normal Force | Fₙ | 3354 N |
| Gravity | g | 9.81 m/s² |
- Open-loop system is unstable with two positive eigenvalues
- Instability arises from operating on downslope of friction curve (∂μ/∂λ < 0)
- Active feedback control is necessary for ABS operation
- System is fully controllable via brake torque
- Position state is unobservable from slip measurements
- Observable subsystem [Vx, λ] is sufficient for ABS objectives
- Separation principle validated: Observer-based control nearly identical to perfect state feedback
- Nonlinear improvement: 121% overshoot reduction due to friction saturation
- Disturbance rejection: Maintains performance under 45% friction drop
- Controller uses only wheel speed sensors (no direct velocity measurement needed)
- Bounded control effort within actuator limits (0-1200 Nm)
- Fast convergence: Observer estimates ready within 0.5s
- Robust to model mismatch: Performs better on nonlinear plant than predicted
This control strategy is applicable to:
- Automotive Safety Systems: Modern ABS implementations
- Brake-by-Wire Systems: Electronic brake force distribution
- Autonomous Vehicles: Emergency braking controllers
- Motorcycle ABS: Similar slip ratio control objectives
- Aircraft Landing Systems: Anti-skid braking control
- LQR Optimization: Replace pole placement with Linear Quadratic Regulator for optimal trade-off between performance and control effort
- Adaptive Observer: Extended Kalman Filter to handle model mismatch on nonlinear plant
- Full Vehicle Model: Extend to 4-wheel coordination with load transfer dynamics
- Road Surface Estimation: Adaptive slip reference based on detected friction coefficient
- Multi-Objective Control: Simultaneously optimize stopping distance and ride comfort
- Load transfer (front/rear weight distribution during braking)
- Road grade compensation (uphill/downhill operation)
- Tire temperature and wear dynamics
- Aerodynamic drag forces
- Suspension dynamics coupling
- K. Ogata, Modern Control Engineering, 5th ed. Prentice Hall, 2010
- H. K. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall, 2002
- B. Friedland, Control System Design: An Introduction to State-Space Methods, Dover, 2005
- U. Kiencke, L. Nielsen, Automotive Control Systems, 2nd ed. Springer, 2004
- R. Isermann, Automotive Control, 1st ed. Springer, 2021
- H. Pacejka, Tire and Vehicle Dynamics, 3rd ed. Elsevier, 2012
- R. Rajamani, Vehicle Dynamics and Control, 2nd ed. Springer, 2012
- P. B. Bhivate, "Modelling & development of antilock braking system," B.Tech Thesis, NIT Rourkela, 2011
- A. B. Sharkawy, "Genetic fuzzy self-tuning PID controllers for ABS," Eng. App. of AI, vol. 23, pp. 1041-1052, 2010
- H. Mirzaeinejad, M. Mirzaei, "Non-linear control of wheel slip in ABS," Control Eng. Practice, vol. 18, pp. 918-926, 2010
- S. Ç. Başlamişli et al., "Robust control of ABS," Vehicle System Dynamics, vol. 45, no. 3, pp. 217-232, 2007
- S. B. Choi, "ABS with continuous wheel slip control," IEEE Trans. Control Syst. Tech., vol. 16, no. 5, 2008
- MATLAB/Simulink - All analysis, simulations, and computations
- LaTeX/Overleaf - Professional technical report preparation
- Claude AI - Document formatting and structuring assistance
- Microsoft Copilot - MATLAB debugging and code optimization
This project is licensed under the MIT License - see the LICENSE file for details.
Special thanks to Professor Dr. Nivii Kalavakonda Chandrasekar for guidance throughout this project and for her expertise in control systems engineering at Northeastern University.
Kiran Sairam Bethi Balagangadaran
MS Robotics, Northeastern University
Email: bethi.k@northeastern.edu
GitHub: @Kiran1510
This project demonstrates advanced control system design techniques including linearization, stability analysis, controllability/observability decomposition, state feedback control, Luenberger observer design, and validation on nonlinear dynamics - all applied to a practical automotive safety system.