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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.

Linear systems are good approximations of the input-output relationship for many real systems. In practice, ensembles of nominally identical systems can seldom be adequately represented by a single linear system. Some accommodation needs to be made for representing inherent ensemble randomness. One approach would be to use a parameterized model of the system, with the random nature captured in discrete parameters of the parameterized model. Another approach is to use data measured from each system in the ensemble to represent the random nature of the ensemble in a non-parametric form. This is the approach described in this paper. There are several different types of data that could be collected from each system in the ensemble, each type capable of capturing the input-output behavior of the linear system: input-output time domain data, perhaps with specific excitation sequences [1,2]; Markov parameters [2,3]; and Frequency Response Functions (FRFs) [2] to name a few. We choose to work with FRF data because many system attributes can be easily interpreted by inspection of the FRF [4]. FRF data can also be used directly for control design [5,6,7]. This paper develops a Karhunen-Loeve expansion [8] representation for linear system behavior based on FRF data to develop a compact representation of the uncertainty inherent in an ensemble of systems. 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 [5,6,7].