Hybrid Regressor and Approximation-Based Adaptive Control of Piezoelectric Flexible Beams

One of the inherent limitations of adaptive regressor control is that the control system would degrade if the host system has unmodeled dynamics. Whereas, adaptive approximation control has the potential for controlling the host system in the presence of external/internal disturbances. Accordingly, this paper is concerned with a combination of both the regressor and approximation techniques for adaptive vibration control of smart beam. The idea behind this paper is to partition the equation of motion for multi-modes into two terms: one term for estimation of the unknown parameters associated with the known modeling using regressor technique, while the disturbance related to unmodeled dynamics can be approximated as a linear combination of orthogonal basis functions. The partial differential equation of the host beam is simplified to multi-modes ordinary differential equations using the assumed method. Then the control law and the updating adaptive laws for the uncertain parameters and the constant weighting coefficients are designed based on Lyapunov's stability theory. The modeling error is carefully considered and compensated using a robust sliding mode term. A simulation study is performed for a simply supported beam with piezoelectric patches to prove the validity of the proposed controller.