Modal cost analysis of flexible structures: Modeling flexible structures for control design

The integration issues of structural modeling and control design for large flexible structures are very important for developing sound models in a closed loop environment. Consideration of these issues leads one to conclude that the modeling problem and control problem are not independent. The connection between these two problems must be understood to develop reliable control algorithms and reduce the cost of extensive laboratory and flight testing. For many large space structures, the primary consideration is the quality of the response at specific locations due to excitation at other locations. It is this input-output type phenomenon which must be reflected in our structural modeling. Classically, modal frequencies have been of primary concern. We will show that it is more important to obtain accurate modal costs in our structural models. The modal costs represent the contribution of a vibration mode in the system response for given input and output locations. This research provides a complete modal cost analysis for certain distributed parameter systems and shows that structural modeling and model reduction methods should be influenced by the specific control objectives. The analysis proceeds via the following steps: (1) Convergence properties of modal costs are discussed for vibration of various simple continua. Explicit formulas for the total cost of the system are derived. These formulas are useful for determining the modes that must be retained to provide a specified accuracy in the structural model with respect to the control objectives. (2) Open loop modal cost analysis is applied to finite element models of beam-like structures. These results are compared with exact modal costs in order to evaluate the accuracy of the modal costs determined using finite element methods. (3) Finite element models are used to develop Linear Quadratic Gaussian control laws while exact models are used to evaluate performance of the resulting closed loop systems. These control laws are necessarily suboptimal. The closed loop performance is then related to the open loop modal cost errors. This research presents a comprehensive modal cost analysis for simple continua and discusses its implications in subsequent control system design. (Abstract shortened with permission of author.