Data Informativity and Parameter Sloppiness of an LFT-Parameterized System

In this article, we investigate conditions for persistent excitation of input sequences for a discrete-time dynamical system, as well as quantitative measurements of its sloppy parameter variations. We adopt a system model that depends on the parameters through a linear fractional transformation, which can include a networked dynamic system as a special case. When an associated transfer function matrix has a full normal column rank, we establish an explicit relation between the parameters and the input–output sequence, from which a matrix rank-based necessary and sufficient condition for persistent excitation of an input sequence is derived. Based on this result, the absolute and relative sloppiness metrics that describe the sloppiness of parameter variations with frequency-domain data are extended to time-domain data, under the situation that the input sequence is persistently exciting and prescribed. Explicit formulas for computing these two metrics are also obtained. Hence, the effects of different input sequences on the sloppiness of parameter variations can be conveniently analyzed with the suggested two metrics. Finally, we illustrate our results with a discrete-time model of a typical mass-spring-damper system, where we show that the choice of input signals significantly influences the absolute and relative sloppiness metrics.