A Unified Approach to the Modeling and Control of Multi-Scale Dynamical Systems

摘要

In this paper, a unified methodology for modeling and control of complex multi-scale mechanical systems is presented. The method utilizes recent results in analytical dynamics adapted to general complex multibody systems. The term multi-scale is employed to denote those mechanical systems whose equations of motion are highly nonlinear, non-autonomous, and can yield motions at multiple time and distance scales. These types of problems can easily become difficult to model and control and the complexity of the equations of motion grows rapidly as the number of component bodies in the system increases. The approach considered herein opens up new routes for modeling and control of general multibody systems by explicitly developing closed form expressions in terms of any desirable number of generalized coordinates that may appropriately describe the configuration of the multi-scale system. The approach is simple in implementation and it poses no restrictions on the total number and nature of modeling constraints to construct the explicit equations of motion of the system. Likewise, no restrictions are placed on the nature and number of the feasible tracking control requirements to obtain explicit expressions for the requisite generalized control forces. Conceptually, the method relies on a simple six-step procedure. It utilizes in a pivotal manner the the explicit equations of motion for constrained mechanical systems with singular mass matrices. The simplicity of the unified method is highlighted by illustrating its use in the modeling and control of a dumbbell made of two unequal masses which is required to be controlled so that one of the masses follows a prescribed time-varying trajectory.