Sankalp Tiwari, Junaidvali Shaik, C. P. Vyasarayani
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Data Driven Approach to Determine Linear Stability of Delay Differential Equations Using Orthonormal History Functions
Delay differential equations (DDEs) appear in many applications, and determining their stability is a challenging task that has received considerable attention. Numerous methods for stability determination of a given DDE exist in the literature. However, in practical scenarios it may be beneficial to be able to determine the stability of a delayed system based solely on its response to given inputs, without the need to consider the underlying governing DDE. In this work we propose such a data-driven method, assuming only three things about the underlying DDE: (i) it is linear, (ii) its coefficients are either constant or time-periodic with a known fundamental period, and (iii) the largest delay is known. Our approach involves giving the first few functions of an orthonormal polynomial basis as input, and measuring/computing the corresponding responses to generate a state transition matrix M, whose largest eigenvalue determines the stability. We demonstrate the correctness, efficacy and convergence of our method by studying four candidate DDEs with differing features. We show that our approach is robust to noise, thereby establishing its suitability for practical applications, wherein measurement errors are unavoidable.
期刊介绍:
The purpose of the Journal of Computational and Nonlinear Dynamics is to provide a medium for rapid dissemination of original research results in theoretical as well as applied computational and nonlinear dynamics. The journal serves as a forum for the exchange of new ideas and applications in computational, rigid and flexible multi-body system dynamics and all aspects (analytical, numerical, and experimental) of dynamics associated with nonlinear systems. The broad scope of the journal encompasses all computational and nonlinear problems occurring in aeronautical, biological, electrical, mechanical, physical, and structural systems.