RLC Electrical System Modeling and PID Control (MATLAB & Simulink)

📖 Project Overview

This project focuses on the dynamic modeling, analysis, and control of a second-order RLC electrical system. The transfer function of the system is derived and analyzed using MATLAB. The system is also modeled using state-space representation in Simulink, and a PID controller is designed to improve transient and steady-state performance.

Full Project Report

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🎯 Project Objectives

• Derive and analyze the transfer function of an RLC circuit
• Analyze system time-domain response
• Design a state-space model
• Verify controllability and observability of the system
• Design and implement a PID controller
• Improve system dynamic performance

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⚙️ System Parameters

The circuit parameters used in this project:

Parameter	Value
Resistance (R)	9 Ω
Inductance (L)	1 H
Capacitance (C)	1 mF

Input Signal

Sinusoidal voltage input

Output Signal

Capacitor voltage

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📐 Transfer Function Analysis

The transfer function of the RLC system was derived using circuit analysis and Laplace transform techniques.

$$ H(s)=\frac{V_c(s)}{V_{in}(s)}=\frac{\frac{1}{LC}}{s^2+\frac{R}{L}s+\frac{1}{LC}} $$

$$ R=9\Omega,\quad L=1H,\quad C=1mF $$

$$ H(s)=\frac{1000}{s^2+9s+1000} $$

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System Characteristics

• The system has no zeros
• Poles are complex conjugate and located in the left half plane
• The system is stable

Pole values: s = -4.5 ± 31.3i

System Behavior

• Real part affects settling time
• Imaginary part determines oscillation frequency
• The system is underdamped

Damping ratio: ζ = 0.1423

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📊 Time Domain Analysis

System response was analyzed using impulse and step response simulations.

Key Transient Response Parameters

Parameter	Value
Rise Time	0.0252 s
Peak Time	0.0702 s
Maximum Overshoot	72.77 %
Settling Time	0.8572 s
Steady-State Value	1
Maximum Output	1.6366

MATLAB simulation results were validated using analytical calculations and final value theorem.

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🧮 State-Space Modeling

The RLC system was converted into state-space representation and implemented in Simulink.

State Variables

• Inductor current
• Capacitor voltage

System Properties

Controllability matrix rank: 2
Observability matrix rank: 2

These results confirm that the system is fully controllable and observable.

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🎛 PID Controller Design

A closed-loop control system was designed using a PID controller.

PID Parameters

Parameter	Value
Kp	4.1293
Ki	54.5229
Kd	0.0782

The step input final value was set to 5 for controller evaluation.

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📈 Performance Improvement with PID

Comparison Results

Parameter	Before PID	After PID
Rise Time	0.0252 s	0.0155 s
Peak Time	0.0702 s	0.0323 s
Overshoot	72.77 %	9.85 %
Settling Time	0.8572 s	0.2115 s
Steady-State Error	0.5	0

The PID controller significantly improved system stability and transient response.

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🛠 Tools and Technologies Used

• MATLAB
• Simulink
• Control System Toolbox

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▶️ How to Run the Project

1. Open MATLAB
2. Run MATLAB scripts for transfer function analysis
3. Open Simulink model
4. Run simulation and observe system response

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🎓 Additional Training

The following MATLAB training modules were completed during this project:

• Simulink Onramp
• Simulink Fundamentals
• Control System Modeling Essentials
• Linearization of Nonlinear Systems

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🚀 Future Work

• Hardware implementation
• Advanced control techniques (LQR, Adaptive Control)
• Real-time system simulation

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👨‍💻 Author

Kerem Danışık
Electrical and Electronics Engineering Student