Use of the Adjoint Method for Controlling the Mechanical Vibrations of Nonlinear Systems

Carmine Pappalardo,Domenico Guida

doi:10.3390/machines6020019

Abstract

In this work, the analytical derivation and the computer implementation of the adjoint method are described. The adjoint method can be effectively used for solving the optimal control problem associated with a large class of nonlinear mechanical systems. As discussed in this investigation, the adjoint method represents a broad computational framework, rather than a single numerical algorithm, in which the control problem for nonlinear dynamical systems can be effectively formulated and implemented employing a set of advanced analytical methods as well as an array of well-established numerical procedures. A detailed theoretical derivation and a comprehensive description of the numerical algorithm suitable for the computer implementation of the methodology used for performing the adjoint analysis are provided in the paper. For this purpose, two important cases are analyzed in this work, namely the design of a feedforward control scheme and the development of a feedback control architecture. In this investigation, the control problem relative to the mechanical vibrations of a nonlinear oscillator characterized by a generalized Van der Pol damping model is considered in order to illustrate the effectiveness of the computational algorithm based on the adjoint method by means of numerical experiments.

Highlights

This paper is focused on the development of an adjoint-based analytical and computational framework for the optimal design of open-loop and closed-loop control laws suitable for controlling nonlinear mechanical systems
The research of the authors is mainly focused on the development and the improvement of analytical methods and computational procedures for the nonlinear dynamic analysis of mechanical systems, for performing the experimental parameter identification of structural systems, and for the design of effective control strategies applicable to multibody systems [87–90]
This work represents a study on the analytical derivation and the computer implementation of the adjoint method

Summary

Introduction

This paper is focused on the development of an adjoint-based analytical and computational framework for the optimal design of open-loop and closed-loop control laws suitable for controlling nonlinear mechanical systems. The classical control methods are based on the linearization of the dynamic equations of the mechanical system under examination Such methods lead to a loss of information on the complex nonlinear dynamic behavior and work properly only when the time response of the mechanical system at hand evolves around a fixed point of the state space or in the proximity of a prescribed trajectory. This is not the case of several mechanical systems employed in engineering applications in which a fully nonlinear dynamic behavior is found and, more complex control strategies are required

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