Linear First-Order Inverted Pendulum LQR Control

This comprehensive MATLAB program implements Linear Quadratic Regulator (LQR) control for a first-order inverted pendulum system. The solution employs state-space modeling where the system dynamics are linearized around the unstable equilibrium point. Key implementation features include: - System matrix formulation using physical parameters (mass, length, friction coefficients) - LQR weight matrix optimization for balancing state regulation and control effort - Riccati equation solution via MATLAB's built-in lqr() function for optimal gain calculation - Real-time state feedback implementation with stability margin verification The program structure includes: 1. Parameter initialization section for pendulum physical properties 2. State-space model construction using linearized equations 3. Q and R matrix configuration for performance tuning 4. Control law computation and closed-loop simulation 5. Visualization modules for time-domain response and phase portrait analysis This verified implementation provides accurate results for both academic study and practical control applications, with detailed comments guiding users through each computational step. The code demonstrates fundamental LQR design principles while maintaining numerical robustness through proper discretization and error handling.