DC Motor Dynamic Modeling & PID Speed Control using MATLAB Simulink

Project Overview

This project models the electromechanical dynamics of a DC motor using MATLAB Simulink, designs a PID controller for speed regulation, and compares numerical solvers (ode45 vs ode15s) using quantitative response metrics.

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1. Objective

• Build a physics-based model of a DC motor.
• Validate open-loop behavior using RL + rotational dynamics.
• Implement a PID controller for speed regulation.
• Tune gains manually for smooth and stable tracking.
• Compare solver performance and analyze step response metrics.

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2. Mathematical Model

The system is defined by two coupled differential equations representing the electrical and mechanical domains.

Electrical Model (Kirchhoff's Voltage Law)

$$L \frac{di}{dt} = V - Ri - K_e \omega$$

Mechanical Model (Newton's 2nd Law)

$$J \frac{d\omega}{dt} = K_t i - B\omega - T_L$$

Variable Definitions

Symbol	Meaning	Unit
$i$	Armature current	A
$\omega$	Angular speed	rad/s
$L$	Armature Inductance	H
$R$	Armature Resistance	$\Omega$
$K_e$	Back EMF Constant	V/(rad/s)
$K_t$	Torque Constant	Nm/A
$J$	Rotor Inertia	kg·m²
$B$	Viscous Damping Coeff.	Nms
$T_L$	Load Torque	Nm

System Parameters

R  = 1;      % Ohm
L  = 0.5;    % H
Ke = 0.01;   % V/(rad/s)
Kt = 0.01;   % Nm/A
J  = 0.01;   % kg*m^2
B  = 0.1;    % Nms
TL = 0.1;    % Nm
V  = 12;     % V

3. Simulink Model Overview

The Simulink model consists of:

1. Electrical Subsystem: Converts applied voltage ($V$) to current ($i$).
2. Mechanical Subsystem: Converts torque (from current) to speed ($\omega$).
3. Feedback Loop: Compares reference speed to actual speed to drive the PID controller.
4. Workspace Logging: Exports data for solver and performance analysis.

4. Open-Loop Results

Key observations from the open-loop simulation without a controller

Current $i(t)$: Rises exponentially to approx 12 A ($V/R$). Angular Speed: Stabilizes around 0.2 rad/s under load. Verification: Behavior matches theoretical steady-state calculations.🎛

5. PID Controller Design

Manual tuning was performed to reduce overshoot and achieve smooth settling.

Initial Gains

Values: $P = 10, I = 50, D = 0 Result: Very high overshoot (~80%) with significant oscillations.

Improved Attempt

Values: $P = 6, I = 40, D = 0.1 Result: Reduced oscillations but still unstable.

Final Tuned Gains

Values: $P = 5, I = 12, D = 0.2 Result: Smooth rise to reference. Near-zero overshoot. Fast settling time. Good stability under load torque.

6. Solver Comparison (ode45 vs ode15s)

Step response metrics were extracted using the stepinfo() command in MATLAB.

Metric	ode45 (Non-stiff)	ode15s (Stiff)
Rise Time	1.3031 s	1.3687 s
Settling Time	2.6784 s	2.8976 s
Overshoot	0 %	0.12 %
Peak Time	3.0 s	4.2058 s

Interpretation: ode45: Faster rise time, efficient for smooth systems. ode15s: Slightly slower but smoother response. Conclusion: Both solvers are stable and appropriate for this specific DC motor model.

7. Conclusions

The DC motor model accurately reflects electromechanical dynamics. PID control significantly improves speed tracking performance compared to open-loop. Final tuned gains ($P=5, I=12, D=0.2$) provide a stable, smooth, overshoot-free response.Closed-loop behavior demonstrates robust control against disturbances.

8. Future Scope

[ ] Implement Anti-windup PID. [ ] Test disturbance rejection (step change in load torque). [ ] Create State-space model of the motor. [ ] Implement advanced controllers (LQR, MPC). [ ] Model nonlinearities (saturation, dead-zone, friction). [ ] Hardware validation using Arduino + DC motor driver.

📂 9. Repository Contents

/MOTOR PROJECT
│── slprj/                   # Simulink project cache files
│── DC_Motor_Model.slx       # Main Simulink model
│── DC_Motor_Model.slxc      # Simulink cache file
│── DC_Motor_PID_Report.pdf  # Detailed project report
│── Matlab_model.png         # Screenshot of the Simulink model
│── motor_parameters.m       # MATLAB script for system parameters
│── README.md                # Project documentation

## 📄 License

This project is licensed under the MIT License - see the [LICENSE](LICENSE) file for details.