Computational methods for feedback control in structural systems

Numerical methods, LQR control, an abstract formulation and reduced basis techniques for a system consisting of a thin cylindrical shell with surface-mounted piezoceramic actuators are investigated. Donnell-Mushtari equations, modified to include Kelvin-Voigt damping and passive patch contributions, are used to model the system dynamics. The voltage-induced piezoceramic patch contributions, used as input in the control regime, enter the equations as external forces and moments. Existence and uniqueness of solutions are demonstrated through variational and semigroup formulations of the system equations. The semigroup formulation is also used to establish theoretical control results and illustrate convergence of the finite dimensional controls and Riccati operators. The spatial components of the state are discretized using a Galerkin expansion resulting in an ordinary differential equation that can be readily marched in time by existing ordinary differential equation solvers. Full order approximation methods which employ standard basis elements such as cubic or linear splines result in large matrix dimensions rendering the system computationally expensive for real-time simulations. To lessen on-line computational requirements, reduced basis methods employing snapshots of the full order model as basis functions are investigated. As a first step in validating the model, a shell with obtainable analytic natural frequencies and modes was considered. The derived frequencies and modes were then compared with numerical approximations using full order basis functions. Further testing on the static and dynamic performance of the full order model was carried out through the following steps: (i) choose true state solutions, (ii) solve for the forces in the equations that would lead to these known solutions, and (iii) compare numerical results excited by the derived forces with the true solutions. Reduced order methods employing the Lagrange and the Karhunen-Loeve proper orthogonal decomposition (POD) basis functions are implemented on the model. Finally, a state feedback method was developed and investigated computationally for both the full order and reduced order models.