Improved stochastic process models for linear structure behavior

Linear mathematical models frequently provide good approximations to the input-output relations for real systems. However, ensembles of systems that are nominally identical cannot, usually, be adequately represented with a single model because real systems are stochastic. The randomness in real systems must be modeled if the randomness bears on critical behaviors of the system. The behavior of linear systems can be represented in parametric or non-parametric form; the latter framework is used, here. Among the frameworks available for characterization of system behavior, we choose the frequency response function (FRF). We choose to work with the FRF because many system attributes can be interpreted by inspection of the FRF, and it can be used directly for control design. This paper improves a previously developed Karhunen-Loeve expansion (KLE) representation for linear system behavior based on FRF data. The improvement yields a compact representation of the uncertainty inherent in an ensemble of systems and avoids the introduction of unwanted features in the system representation. This non-parametric, compact representation of the distribution of linear systems can then be used to characterize the performance and stability of a given feedback control law, as well as for control law design.