Variable input observer for nonstationary high-rate dynamic systems

  • Jonathan HongEmail author
  • Simon Laflamme
  • Liang Cao
  • Jacob Dodson
  • Bryan Joyce
Original Article


Engineering systems experiencing events of amplitudes higher than 100 gn for a duration under 100 ms, here termed high-rate dynamics, can undergo rapid damaging effects. If the structural health of such systems could be accurately estimated in a timely manner, preventative measures could be employed to minimize adverse effects. For complex high-rate problems, adaptive observers have shown promise due to their capability to deal with nonstationary, noisy, and uncertain systems. However, adaptive observers have slow convergence rates, which impede their applicability to the high-rate problems. To improve on the convergence rate, we propose a variable input space concept for optimizing the use of data history of high-rate dynamics, with the objective to produce an optimal representation of the system of interest. Using the embedding theory, the algorithm sequentially selects and adapts a vector of inputs that preserves the essential dynamics of the high-rate system. In this paper, the variable input space is integrated in a wavelet neural network, which constitutes a variable input observer. The observer is simulated using experimental data from a high-rate system. Different input space adaptation methods are studied, and the performance is also compared against an optimized fixed input strategy. It is found that a smooth transition of the input space eliminates error spikes and yields faster convergence. The variable input observer is further studied in a hybrid model-/data-driven formulation, and results demonstrate significant improvement in performance gained from the added physical knowledge.


High-rate dynamics Input space Embedding Adaptive observer Neural network Structural health monitoring 



The authors would like to acknowledge the financial support from the Air Force Office of Scientific Research (AFOSR) award number FA9550-17-1-0131, and AFRL/RWK contract number FA8651-17-D-0002. Additionally, the authors would like to acknowledge Dr. Janet Wolfson for providing the experimental data. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the US Air Force.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil, Construction, and Environmental EngineeringIowa State UniversityAmesUSA
  2. 2.ATLSS Engineering Research CenterLehigh UniversityBethlehemUSA
  3. 3.Air Force Research LaboratoryEglin AFBUSA
  4. 4.University of Dayton Research InstituteEglin AFBUSA

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