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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

🎯 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

⚙️ 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

📐 Transfer Function Analysis

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

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

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

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

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

📊 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.

🧮 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.

🎛 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.

📈 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.

🛠 Tools and Technologies Used

  • MATLAB
  • Simulink
  • Control System Toolbox

▶️ 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

🎓 Additional Training

The following MATLAB training modules were completed during this project:

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

🚀 Future Work

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

👨‍💻 Author

Kerem Danışık
Electrical and Electronics Engineering Student