Real-Time Monitoring of Chaotic Systems With Known Dynamical Equations

Chaotic systems have attracted considerable attention due to their potential applications in aerodynamics, robotics, and power systems. Although there has been a lot of work on the state estimation of linear systems, how to efficiently reconstruct a nonlinear system, especially a chaotic system still remains open. In this paper, we are interested in the real-time monitoring of chaotic systems. In particular, we present a prediction-based transmission strategy, in which the predictor is based on the known dynamics of the observed chaotic process. In contrast to nonchaotic system, the real-time monitoring of chaotic systems may suffer from large prediction residuals that grow dramatically even without external noise. To overcome this issue, we present a finite-time Lyapunov exponent-based approach to bridge the probability distributions of the prediction and quantization errors. To strike an optimal rate-distortion tradeoff, we first develop a time-domain statistical model of the reconstruction error based on the finite-time Lyapunov exponent. Then we present the distortion-outage probability that the instantaneous reconstruction error exceeds a threshold, for both periodic and threshold-based sampling methods, also referred to as Riemann and Lebesgue sampling. It is interestingly shown that the average data rate approaches the intrinsic entropy asymptotically for a one-dimensional chaotic system with both sampling methods. Further, the average sampling rate is shown to be inversely proportional to the number of quantization bits per sample after lossless compression given an upper bound on the reconstruction error. Our theoretical results are validated by extensive simulations.