Optimal control design and implementation for the double inverted pendulum system in a graduate control course

This paper presents an optimal control design methodology used in a graduate electrical engineering course. The proposed methodology was applied and used on the double inverted pendulum system. The regulators presented are based on the state-variable feedback with Linear Quadratic Regulator (LQR) design. An asymptotic observer was designed to estimate the state-variables of the system. A Kalman Filter was designed as an alternate estimator to include sensor and process Gaussian noise. The Loop Transfer Recovery technique was used to recover some of the robustness properties that can be lost when implementing the Kalman filter. The Kalman Filter resulted in a better state estimator with less estimation noise on the velocities state variables, compared to the velocities state variables that were estimated with the asymptotic observer. Furthermore, Loop-Transfer Recovery allowed for a better tuning of the Kalman filter based controller and resulted in a better time response of the system. Theoretical aspects were experienced and confirmed in a laboratory setup as each design technique was applied and implemented sequentially in the real system. This is an innovative practice because the students learn by doing in a deductive way, as they build upon their design.