Prediction and control of uncertain system dynamics

Abstract

I investigate the importance of determining the exact dimensionality of a nonlinear system in time series prediction by comparing the effects of varying the embedding vector dimension of linear and nonlinear prediction algorithms. I use the logistic and Henon maps to demonstrate that when the embedding vector dimension of a prediction algorithm is less than the actual dimension of the underlying time series, then the prediction algorithm is unable to accurately capture the dynamics of the time series. On the other hand, when the embedding vector dimension is overestimated, then the prediction horizon collapses quickly, but systematically, with the predicted values filling a manifold bounded by the true dimensional time series attractor. I conclude that the prediction horizon and reconstruction of attractors is extremely sensitive to the accuracy of the estimation of the embedding vector dimension. This will be illustrated for a time series associated with a physical system that was part of the Santa Fe prediction competition [7]. I apply nonlinear chaos theory in modeling and forecasting variable-bitrate (VBR) video sequences. Nonlinear chaos modeling offers an alternative approach to stochastic (typically linear) approaches, with the advantages of lower dimensionality and more determinism. However, the goodness of its predictions strongly depends on the accuracy with which the dimensionality of a chaotic model is estimated from empirical data. The contributions of this study are twofold. First, I present a novel approach for estimating the embedding vector dimension of any chaotic time series that satisfies the functional relationship of Farmer and Sidorowich [12]. The proposed approach is applied to VBR video data and is used to show the existence of chaos in packetized video traffic. Second, I develop a chaos-theory-based model for VBR intra-coded video, which can be used to generate a rich set of synthetic traces that exhibit similar statistical structure to the original data. These traces are useful in performance evaluation and resource allocation in integrated computer networks. Chaotic systems and their control appear in many diverse situations and present challenging problems for control engineers. In this study, chaotic behavior is initiated in a nonlinear beam-balancing system by moving a mass back-and-forth in a particular move sequence. This mass movement could be considered a disturbance to the system. A predictive control strategy is created to counteract this chaotic disturbance and move the beam to a desired angular location. The control strategy does not depend on prior knowledge of the system dynamics. Instead, time series values from the system are used to predict future behavior of the necessary system variable(s).