Beam observer-based adaptive fuzzy control of uncertain nonlinear coupled rigid-flexible systems

This paper investigates the beam observer-based adaptive fuzzy control (AFC) for uncertain coupled rigid-flexible systems, which are modeled by nonlinear ordinary differential equations (ODEs) and Euler-Bernoulli beam equations. Particularly, a nonlinear uncertainty in the ODE-beam state coupling is taken into account within the dynamics of the ODE subsystem. Initially, a beam observer is constructed to estimate the unavailable spatiotemporal state through boundary measurements. On the basis of the beam observer, a fuzzy logic system is adopted to approximate the unknown coupling nonlinearity and a nonlinear robust compensator is also presented to eliminate fuzzy modeling errors. Then, a beam observer-based AFC scheme including an ODE state feedback controller, a beam observer-based adaptive fuzzy approximator, a projection-operator-based adaptive law, and a nonlinear robust compensator is incorporated into the ODE subsystem to counteract the adverse effects induced by beam vibrations. Sufficient conditions for existence of the beam observer-based AFC is developed with the help of space-dependent bilinear matrix inequalities, ensuring the asymptotic stability of the closed-loop augmented system. Furthermore, the closed-loop well-posedness analysis is established via semigroup theory. Using the linear matrix inequality technique, a two-step procedure is also provided to solve the feasibility problem. Finally, simulation results on a flexible spacecraft illustrate the performance of the proposed approach.