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Asymptotic Analysis of Discrete-Time Models for Linear Control Systems with Fast Random Sampling

In this paper, we study the dynamics of a linear feedback control system where control is effected via a sample-and-hold implementation of a state-feedback control law, with samples taken at the random event times of a renewal process. Our primary interest is in quantifying, using limit theorems of probability, fluctuations of the system with fast—but finite rate—sampling from its idealized continuously sampled counterpart. Exploiting the linearity and explicit solvability of the system in between samples, questions about the original continuous-time system can be studied through the investigation of an embedded discrete-time stochastic process. The latter records the system state at just the sampling instants, and can be represented in terms of a product of random matrices. We now use limit theorems of the Law of Large Numbers (LLN) and Central Limit Theorem (CLT) type for random matrix products to obtain information about the mean behavior and the typical fluctuations about the mean for the discrete-time process in the limit as the temporal sampling rate goes to infinity.