Symbolic Dynamic Filtering of Complex Systems

Open Access
Rajagopalan, Venkatesh
Graduate Program:
Electrical Engineering
Doctor of Philosophy
Document Type:
Date of Defense:
January 08, 2007
Committee Members:
  • Jeffrey Scott Mayer, Committee Chair
  • Asok Ray, Committee Chair
  • Constantino Manuel Lagoa, Committee Member
  • Marc Carpino, Committee Member
  • Markov Modeling
  • Anomaly Detection
  • Symbolic Dynamics
The study of complex systems has seen tremendous growth recently. Several techniques have been utilized, over the years, to study complex systems. This dissertation proposes a Symbolic Dynamic Filtering (SDF) methodology to analyze complex systems. Catastrophic failures in complex engineering systems can often be prevented if faults/anomalies are detected at their incipient stage. Anomalies, which are defined as deviations from nominal behavior, can be associated with parametric changes in a system. Hence, detection and estimation of such parametric variations are critical in identifying faults. Accurate and computationally tractable modeling of complex system dynamics, solely based on fundamentals of physics, is often infeasible. Hence, it might be necessary to learn the behavior of a system through times series data obtained from sensors. Symbolic dynamics provides a useful tool for time series analysis. Symbolic dynamics attempts to model a continuous time signal by a symbol sequence. Hence, the conversion of time series data to symbols is a crucial aspect of symbolic dynamics. In most symbolic time series analysis methods, time series data is directly converted to symbols. In this dissertation, symbols are generated from wavelet coefficients of time series data. Systematic procedures for selection of wavelet basis and scales are discussed. The advantages of wavelet preprocessing with regard to noise mitigation and robustness are illustrated with numerical experiments. A new procedure for designing customized wavelets for near periodic signals is also presented in this dissertation. The subsequent step to wavelet preprocessing is conversion of wavelet coefficients to symbol sequences. Systematic procedures for selecting number of symbols (alphabet size), partitioning scheme to generate symbols and modeling of symbol sequences with finite state automata are presented. The partitioning scheme along with the finite state machine model is referred as the Symbolic Dynamic Filter. A salient feature is that entropy is used as the quantity to determine all features of the Symbolic Dynamic Filter. A Maximum Entropy based partitioning technique is introduced. This partitioning technique seeks to create partitions based on information content in the data. Symbols are modeled by a special class of finite state automata called D-Markov machines. Depth of a D-Markov machine is chosen based on entropy rate. An algorithm for optimizing the structure of D-Markov Machine is also presented. Two applications of symbolic dynamic filtering are reported in this dissertation. Parameter estimation is investigated in two nonlinear systems described by the Duffing equation and Van der Pol equation. Application of SDF for anomaly detection is validated through experiments conducted in a Machine Condition Monitoring (MCM) test bed at Idaho National Laboratory.