Controllability and observability measures for Craig-Bampton substructure representations

Measures of controllabili ty and observability are derived for a Craig-Bampton substructure representation for which fixed interface modes and constraint modes are computed relative to control actuator locations. The measures facilitate modal ordering and act as an index of dynamic completeness for modal truncation. Example problems are given to demonstrate that the controllable and observable reduced substructure models have a greater accuracy in predicting closed-loop pole locations, transfer functions, and actuation commands than conventional normal-mode models. UE to the intimate relationship between the control and struc- tural dynamics of proposed large space structures (LSS), there is a need for unification of the two disciplines. In general, the first step in the analysis of an LSS is the generation of a finite element representation. The resulting finite element model (FEM) is usu- ally much too large for the efficient application of modern methods of control dynamics. In contrast, the structural dynamics commu- nity has had a great deal of success performing dynamic analysis with very large FEMs by using substructure representations. Over the past few years, researchers in the control dynamics community have sought to take advantage of this methodology in order to de- velop decentralized control strategies that reduce the computational complexity of controlling LSS.1'2 However, the work to date has not taken full advantage of the substructure representations available in the context of control design. This paper specifically addresses the advantages of using the Craig-Bampton (CB) substructure representation3 in control system design. The CB representation is the basis of a well-known compo- nent mode synthesis technique developed to analyze linear structural dynamic problems of very high order. It is the most frequently used format in the aerospace industry for structural-load analysis. As a preliminary step toward future work in decentralized control, this investigation considers the complete LSS as a single CB represen- tation. Instead of the usual interface degrees of freedom connecting to other substructures, there are now actuator locations that are con- strained during the computation of fixed interface mode shapes used to describe the dynamics of the LSS internal to the actuator degrees of freedom. The use of the CB representatio n in this fashion has been in- vestigated to a small extent. Reference 4 demonstrated that fixed interface mode representations provide more accurate closed-loop pole locations and transfer functions for a smaller number of re- tained modes than the more conventional approach of using models based solely upon free-free modes. This is due to the static com- pleteness of the CB mode set and the improved accuracy obtained in representing boundary conditions at actuator locations. However, in order for the CB substructure representation to be a truly useful tool in control system design, conditions for controllability and ob- servability must be available. This investigation extends the work in Ref. 4 by deriving conditions and measures of controllability and observability corresponding to the CB state space.