Equivalent Representations of Vibration Reduction Control Problems

Control strategies for vibration reduction problems are discussed. The problem is represented as an optimal control problem, and a generalized vibration cost functional is given. Two formulations are presented where either the cost objective is minimized with a given settling time, or the settling time is minimized with a constraint on the required objective value. The selection criterion depends upon the designer´s prior knowledge of the acceptable settling time, or the allowable objective limits. The current work shows the equivalence of these two formulations under some conditions on the cost functional. The resulting optimal solution is the same for both fixed-time and the free-time control problems. Discretization of the generalized vibration reduction problem is shown to be a convex problem for the fixed-time formulation and a quasi-convex problem for the free-time formulation. The two formulations are compared in terms of the solution method complexity and practical implementation.