Structural Analysis by Modified Signature Matrix for Integro-differential-algebraic Equations

TL;DR

This paper introduces a modified signature matrix and new structural analysis techniques for integro-differential-algebraic equations, addressing issues like hidden constraints, singular Jacobians, and overestimation, to improve solution regularization and analysis termination.

Contribution

It redefines the signature matrix, proposes a degree of freedom concept, and develops detection and embedding methods to enhance structural analysis of IDAEs, including those with symbolic cancellation and numerical degeneration.

Findings

Redefinition of signature matrix and degree of freedom guarantees analysis termination.

Detection method effectively addresses overestimation issues.

Embedding method successfully handles structurally unamenable IDAEs.

Abstract

Integro-differential-algebraic equations (IDAE)s are widely used in applications of engineering and analysis. When there are hidden constraints in an IDAE, structural analysis is necessary. But if derivatives of dependent variables appear in their integrals, the existing definition of the signature matrix for an IDAE cannot be satisfied. Moreover, if an IDAE has a singular Jacobian matrix after structural analysis by the Sigma-method, improved structural analysis methods are proposed to regularize it. However, the optimal value of an IDAE may be negative which can not ensure the termination of the regularization. Furthermore, overestimation of the signature matrix may also lead to failure of its structural analysis. In this paper, firstly, we redefine the signature matrix and introduce a definition of the degree of freedom for IDAEs. Thus, the termination of improved structural analysis methods can be guaranteed. Secondly, the detection method by points is proposed to deal with the problem of overestimation of signature matrix. Thirdly, the embedding method has proved to suitable for structural unamenable IDAEs, including those types that arise from symbolic cancellation and numerical degeneration. Finally, the global numerical method is applied to an example of two-stage drive system which can help to find all solutions for IDAEs by witness points. Hopefully, through the example of pendulum curtain, the approach for IDAEs proposed in this paper can be applied to integro-partial-differential-algebraic equations (IPDAE)s.