Date of Award
8-2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mechanical Engineering
Committee Chair/Advisor
Dr. Atul Kelkar
Committee Member
Dr. Umesh Vaidya
Committee Member
Dr. Ardalan Vahidi
Committee Member
Dr. Xin Zhao
Abstract
Uncertainty analysis plays an essential role in system design due to the impact on the accuracy and reliability of system performance predictions. The analysis helps improve the robustness of the design, optimize performance, and ensure that the system meets safety and reliability requirements. In this context, this research focuses on developing the framework for analyzing the response due to uncertainties in the dynamical systems, with results obtained for ground vehicles as they operate under uncertain operating conditions. The uncertainty can be attributed to the inherent characteristics of system components, random inputs, external forces, parameters at the interfaces between different systems, or incomplete knowledge of the dynamic processes. It is essential to ensure the safe operation of vehicles under all conditions.
Over the decades, various approaches have been developed to enhance the accuracy and computational efficiency of uncertainty analyses of ground vehicles. These include Monte Carlo simulation, polynomial chaos, first-order linearization, and moment-based methods. However, due to the complex nonlinear dynamics of the vehicle, the existing approaches often fall short of accurately predicting the response to uncertainties. Many of these methods rely on linearized models that cannot capture the interactions between vehicle subsystems. In this work, Koopman theory-based uncertainty analysis approaches are presented. The research work comprises three distinct approaches based on the Koopman theory to analyze the response due to uncertainties in vehicle dynamics.
First, this work analyzes the safe operating limits of a vehicle by performing the reachability analysis for uncertain initial states and parameters using the Koopman-spectrum approach. Reachable sets provide information on the set of states starting from a given initial set at any given time and give definitive bounds for the trajectories. The reachable sets are computed using the Koopman principal eigenfunctions, and the simulation results are obtained for the two-degree-of-freedom nonlinear quarter-car model. Further, using the obtained reachable sets, a framework for statistical analysis is developed for the uncertainty in the vehicle system.
Second, a data-driven model-based uncertainty analysis is performed on a tracked-wheel vehicle using the Koopman operator. The model is obtained using the data from an extensive Simulink model of the tracked vehicle. The predicted states of the tracked vehicle are obtained using the data-driven model and compared with the prediction of actual dynamics. This shows the efficacy of the proposed Koopman framework. Further, the equations are derived for the mean and variance propagation using the data-driven model and considering uncertain states and parameters of the vehicle.
Lastly, a non-Lyapunov-based perspective using the Koopman theory is discussed to analyze the stability boundary of the lateral dynamics of the vehicle for uncertain model parameters. The stability boundary of the vehicle dynamics is obtained using the Koopman eigenfunctions. The eigenfunctions are obtained from the path-integral formulation. Here, the zero-level set of the unstable Koopman eigenfunctions characterizes the lateral stability boundary for the vehicle dynamics. The simulation results are provided to obtain the lateral stability boundary for the two-degree-of-freedom planar nonlinear vehicle dynamics for uncertain parameters and compared with the results of the phase portrait.
In summary, this research work presents a Koopman theory perspective to analyze the response due to uncertainties in the dynamical systems. The simulation results are obtained for three distinct vehicle models: nonlinear quarter car, lateral dynamics, and tracked-wheel vehicle. The results demonstrate the computational efficiency of the proposed Koopman theory approaches for uncertainty analysis of complex and nonlinear vehicle dynamics. It will help identify potential risks and ensure that vehicles remain within safe operating parameters, enhancing overall vehicle safety and performance.
Recommended Citation
Kumar, Alok, "Uncertainty Propagation in Dynamical Systems using the Koopman Spectrum: Application to Vehicle Dynamics" (2024). All Dissertations. 3676.
https://open.clemson.edu/all_dissertations/3676
Author ORCID Identifier
0009-0001-8261-5302
Included in
Controls and Control Theory Commons, Electro-Mechanical Systems Commons, Other Mechanical Engineering Commons