The active vibration attenuation of linearly elastic structures modeled by the finite element method, with a possibly large number of degrees of freedom, is considered. The approach, formulated in modal space, applies mathematical optimization to obtain exact solutions to systems that may involve any number of modes to be controlled by an equal or smaller number of discrete actuators. Such systems are under-actuated and generally involve second-order non-holonomic constraints that impose limitations on the dynamically admissible motions that the system can be made to follow. The approach presented in this thesis has value as a tool for the designing and analyzing active vibration attenuation in structures under idealized conditions, but does not replace traditional control approaches are necessary for practical implementation of such systems.
The optimal attenuation of the structure subject to any initial disturbance is obtained by applying Pontryagin’s principle to solve for the minimum solution to a quadratic performance index subject to additional under-actuated constraints that are satisfied by the introduction of time-dependant Lagrange multipliers. The optimality conditions are derived in a compact form and solved by applying symbolic differential operators. The approach uses commercial finite element analysis software and symbolic mathematical software to obtain the optimal actuation forces required by each discrete actuator and the trajectory that the system will undergo.
The approach, which is called the constrained modal space optimal control method involves three primary stages in the solution process. The first stage –the structural stage – involves the transformation of any system modeled by finite elements into a sufficient number of modal variables and selection of the number and positioning of potential actuator locations. In this stage any problems with poor controllability can be quickly assessed and mitigated prior to proceeding with the next solution stage – the control stage. In the control stage the optimal control problem is solved and all unknown system forces and trajectories are obtained. System gains for the closed loop system can also be obtained in this stage. In the third stage – the verification stage – the actuation forces obtained in the control stage are tested on a transient time-integrated finite element model to evaluate if the system will respond as expected. Any potential spillover effects on higher modes of vibration not considered in the control can be observed in the verification stage
