A Procedure for Solving Under-actuated Optimal Attenuation of Vibrations by means of Symbolic Operators

The problem of optimal attenuation of vibrations with the number of controlled modes greater than the number of independent discrete actuators is analyzed. The modal variables are coupled via non-holonomic constraints that are satisfied with the help of time-dependent Lagrange's multipliers. A given configuration of actuators is characterized by a normalized matrix of constraints with constant coefficients. The coefficients are used to evaluate controllability and overall effectiveness of the actuators as configured. For the quadratic performance index the optimality equations are derived from Pontryagin's principle in a compact form containing time derivatives of the controlled modal variables and Lagrange's multipliers. The problem is formulated as a two-points-boundary-value problem for the modes involved. The equations are solved automatically by applying symbolic differential operators and standard mathematical software. The proposed procedure consists of structural and control phases, and permits determining the optimal histograms of forces in each actuator and the expected response of the structure, including the active damping ratios. It also provides hints for adjusting the configuration of actuators to improve their performance in the case where some of the modes are too difficult to control. Numerical examples illustrate the procedure.