Optimal Control Algorithms for Inverted Pendulum Systems

Optimal Control Algorithms for Inverted Pendulum Systems

The inverted pendulum represents a classic underactuated system in control theory, widely used for validating various control algorithms. Optimal control employs mathematical optimization techniques to determine control strategies that satisfy system constraints while achieving predefined performance metrics.

System Modeling Inverted pendulums are typically modeled as nonlinear systems, but linearization around equilibrium points enables the application of linear optimal control theory. Using Newtonian mechanics or Lagrangian equations, we derive state-space equations capturing key variables (pendulum angle, angular velocity, cart position, and velocity). In code implementation, this involves defining system matrices A and B using physical parameters like mass and length.

Optimal Control Problem Formulation The core objective involves designing control inputs that minimize a cost function. For inverted pendulums, this typically employs quadratic forms combining state deviations and control efforts, balancing response speed against energy consumption. Programmatically, this translates to designing weighting matrices Q and R that penalize state errors and control inputs respectively.

LQR Algorithm Application Linear Quadratic Regulator (LQR) stands as a fundamental optimal control method. By solving the Riccati equation, we obtain optimal feedback gain matrices where control inputs become linear combinations of states. In MATLAB, this can be implemented using the lqr() function with syntax: K = lqr(A,B,Q,R). The algorithm guarantees system stability and superior dynamic performance.

Implementation and Validation Practical applications require real-time state feedback from sensors and control computation via microcontrollers. During simulation phases, engineers adjust weighting matrices in the cost function to observe parameter effects on pendulum stability and cart movement. Code validation often involves plotting phase portraits and time-domain responses to verify performance.

Optimal control provides systematic design methodology for inverted pendulums, though model inaccuracies and external disturbances require consideration. Advanced research may integrate robust control or adaptive strategies to enhance disturbance rejection capabilities through techniques like H-infinity control or model reference adaptive systems.