MATLAB Implementation for Inverted Pendulum Model Development

The inverted pendulum system represents a classical stability control problem in control theory, where its nonlinear characteristics make it an ideal platform for validating various control algorithms. Establishing an inverted pendulum model in MATLAB and implementing control typically involves three core stages: dynamic modeling, controller design, and simulation verification. Model Establishment The dynamic model of the inverted pendulum can be derived using Lagrange equations or Newtonian mechanics, resulting in coupled differential equations for pendulum angle and cart position. The model must consider parameters such as pendulum mass, cart mass, pendulum length, and friction. In MATLAB implementation, these equations are typically converted to state-space form for subsequent control design. State variables generally include cart displacement, velocity, pendulum angle, and angular velocity. Key MATLAB functions for this stage include symbolic math toolbox for equation derivation and ss() for state-space representation. Variable Structure Control Design Variable structure control (e.g., sliding mode control) employs switching logic to drive system states toward convergence along a sliding surface, demonstrating strong robustness against parameter disturbances. For inverted pendulum problems, the sliding surface can be designed as a linear combination of angle and angular velocity, with switching gains adjusted to ensure rapid response. The chattering issue requires attention and can be mitigated using saturation functions or boundary layer methods. MATLAB implementation typically involves defining switching conditions using sign() or sat() functions and tuning control gains through simulation. Optimal Control Strategy Linear Quadratic Regulator (LQR) serves as a common optimal control method. By solving the Riccati equation to obtain state feedback matrices, it minimizes a cost function incorporating state errors and control energy. LQR suits the stabilization phase of inverted pendulums but requires model linearization around equilibrium points. MATLAB implementation involves using lqr() function with properly weighted Q and R matrices, and linmod() for linearization. Swing-up and Stabilization Switching The swing-up phase requires swinging the pendulum from hanging position to inverted position, achievable through energy control or Bang-Bang control. When the pendulum approaches equilibrium, switching to LQR or sliding mode control enables stabilization. MATLAB simulation can automatically trigger controller switching through event detection (e.g., angle thresholds) using odeset events or Simulink triggered subsystems. Implementation Key Points - Using ODE45 solver for nonlinear dynamic simulation with appropriate error tolerances - Encapsulating controller logic through S-functions or Simulink subsystems - Comparing response curves and disturbance rejection performance across different control strategies This framework can be extended to multi-stage inverted pendulums or combined with intelligent control algorithms for further performance optimization.