Control of nonlinear vibrations using the adjoint method

In this paper, a new methodology is proposed to address the problems of suppressing structural vibrations and attenuating contact forces in nonlinear mechanical systems. The computational algorithms developed in this work are based on the mathematical framework of the calculus of variation and take advantage of the numerical implementation of the adjoint method. To this end, the principal aspects of the optimal control theory are reviewed and employed to derive the adjoint equations which form a nonlinear differential-algebraic two-point boundary value problem that defines the mathematical solution of the optimal control problem. The adjoint equations are obtained and solved numerically for the optimal design of control strategies considering a twofold control structure: a feedforward (open-loop) control architecture and a feedback (closed-loop) control scheme. While the feedforward control strategy can be implemented using only the active control paradigm, the feedback control method can be realized employing both the active and the passive control approaches. For this purpose, two dual numerical procedures are developed to numerically compute a set of optimal control policies, namely the adjoint-based control optimization method for feedforward control actions and the adjoint-based parameter optimization method for feedback control actions. The computational methods developed in this work are suitable for controlling nonlinear nonautonomous dynamical systems and feature a broad scope of application. In particular, it is shown in this paper that by setting an appropriate mathematical form of the cost functional, the proposed methods allow for simultaneously solving the problems of suppressing vibrations and attenuating interaction forces for a general class of nonlinear mechanical systems. The numerical example described in the paper illustrates the key features of the adjoint method and demonstrates the feasibility and the effectiveness of the proposed adjoint-based computational procedures.