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Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations With Time Periodic Coefficients

[+] Author Affiliations
Venkatesh Deshmukh

Villanova University, Villanova, PA

Paper No. DETC2007-35263, pp. 771-781; 11 pages
doi:10.1115/DETC2007-35263
From:
  • ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C
  • Las Vegas, Nevada, USA, September 4–7, 2007
  • Conference Sponsors: Design Engineering Division and Computers and Information in Engineering Division
  • ISBN: 0-7918-4806-X | eISBN: 0-7918-3806-4
  • Copyright © 2007 by ASME

abstract

Stability theory of Nonlinear Delay Differential Algebraic Equations (DDAE) with periodic coefficients is proposed with a geometric interpretation of the evolution of the linearized system. First, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a finite dimensional approximation or a “monodromy matrix” of a compact infinite dimensional operator. The monodromy operator is essentially a map of the Chebyshev coefficients of the state form the delay interval to the next adjacent interval of time. The monodromy matrix is obtained by a similarity transformation of the momodromy matrix of the associated semi-explicit system. The computations are entirely performed in the original system form to avoid cumbersome transformations associated with the semi-explicit system. Next, two computational algorithms are detailed for obtaining solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

Copyright © 2007 by ASME

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