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Abstract A technique based on the Wiener path integral (WPI) is developed for determining the stochastic response of diverse nonlinear systems with fractional derivative elements. Specifically, a reduced-order WPI formulation is proposed, which can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. In fact, the herein developed technique can determine, directly, any lower-dimensional joint response probability density function corresponding to a subset only of the response vector components. This is done by utilizing an appropriate combination of fixed and free boundary conditions in the related variational, functional minimization, problem. Notably, the reduced-order WPI formulation is particularly advantageous for problems where the interest lies in few only specific degrees-of-freedom whose stochastic response is critical for the design and optimization of the overall system. An indicative numerical example is considered pertaining to a stochastically excited tuned mass-damper-inerter nonlinear system with a fractional derivative element. Comparisons with relevant Monte Carlo simulation data demonstrate the accuracy and computational efficiency of the technique.
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An algorithm for computing robust forward invariant sets of two dimensional nonlinear systems
Summary Robustness of nonlinear systems can be analyzed by computing robust forward invariant sets (RFISs). Knowledge of the smallest RFIS of a system, can help analyze system performance under perturbations. A novel algorithm is developed to compute an approximation of the smallest RFIS for two‐dimensional nonlinear systems subjected to a bounded additive disturbance. The problem of computing an RFIS is formulated as a path planning problem, and the algorithm developed plans a path which iteratively converges to the boundary of an RFIS. Rigorous mathematical analysis shows that the proposed algorithm terminates in a finite number of iterations, and that the output of the proposed algorithm is an RFIS. Simulations are presented to illustrate the proposed algorithm, and to support the mathematical results. This work may aid future development, for use with higher dimensional systems.
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- PAR ID:
- 10450067
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Asian Journal of Control
- Volume:
- 23
- Issue:
- 5
- ISSN:
- 1561-8625
- Page Range / eLocation ID:
- p. 2403-2419
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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