Control of (highly) flexible Reissner beam by energy shaping of distributed port-Hamiltonian system
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The main focus of this work is control of large overall motion of (highly) flexible Reissner beam that can represent in geometrically exact manner large displacements, large rotations and large strains. This nonlinear control problem is cast in distributed port-Hamiltonian framework, extending the previous works [1] dealing with finite dimensional systems (such as rigid components of multibody system interconnected with flexible joints) to infinite dimensional system with Hamiltonian density [2]. The nonlinear control is performed by defining the desired state of the flexible system through energy-shaping of distributed port- Hamiltonian, previously introduced for lumped-parameter system [3,4]. We first show how to perform the energy shaping for internal energy density by bringing the flexible beam with large overall motion into deformed configuration that is in static equilibrium, which is also the closest to the desired configuration of Reissner beam after large overall motion. We then show how to perform the energy shaping for kinetic energy, with the illustration provided for the choice of uniform rotational motion, which also requires the corresponding choice of internal energy defined by Casimir functional [5]. We finally discuss different procedures for damping injection that will stabilize the system to configurations with desired Hamiltonian, including viscous damping, frictional damping and energy decaying of high frequency modes for Reissner nonlinear beam. The results of illustrative numerical simulations of large overall motion of (very) flexible geometrically exact beam confirm an excellent performance of the proposed approach
