Two-criteria problems for optimal protection of elastic structures from vibration

In a multi-objective formulation with criteria such as the maximal deformation of the elastic object to be protected and maximal deformations of the protection devices, a new class of optimal vibration protection problems is considered. The mathematical problem is to find a linear feedback control minimizing the above criteria in Pareto sense. A general approach to solving these problems based on results of modern control theory using linear matrix inequalities technique is presented. A system of linear matrix inequalities for obtaining the desired gain matrix is derived. An example of a solution of two-criteria problem for a multistorey building under seismic disturbances is given. Pareto set on the plane of the criteria is constructed. The «ideal» Pareto optimal isolator and optimal isolators of active and passive types are compared as well. It is shown that the «active» vibration isolators are not much better than the passive one, but all these isolators are noticeably inferior to the «ideal» vibration isolator.