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Structural Preprocessing Method for Nonlinear Differential-Algebraic Equations Using Linear Symbolic Matrices

Taihei Oki, Yujin Song

TL;DR

A new structural regularization method is proposed extending the idea of the IOT method, which approximates the system Jacobian by more expressive symbolic matrices, called rank-1 coefficient mixed (1CM) matrices, which makes the method more widely applicable than the existing method.

Abstract

Differential-algebraic equations (DAEs) have been used in modeling various dynamical systems in science and engineering. Several preprocessing methods for DAEs, such as consistent initialization and index reduction, use structural information on DAEs. Unfortunately, these methods may fail when the system Jacobian, which is a functional matrix, derived from the DAE is singular. To transform a DAE with a singular system Jacobian into a nonsingular system, several regularization methods have been proposed. Most of all existing regularization methods rely on symbolic computation to eliminate the system Jacobian for finding a certificate of singularity, resulting in much computational time. Iwata--Oki--Takamatsu (2019) proposed a method (IOT-method) to find a certificate without symbolic computations. The IOT method approximates the system Jacobian by a simpler symbolic matrix, called a layered mixed matrix, which admits a fast combinatorial algorithm for singularity testing. However, it often overlooks the singularity of the system Jacobian since the approximation largely discards algebraic relationships among entries in the original system Jacobian. In this study, we propose a new regularization method extending the idea of the IOT method. Instead of layered mixed matrices, our method approximates the system Jacobian by more expressive symbolic matrices, called rank-1 coefficient mixed (1CM) matrices. This makes our method more widely applicable. We give a fast combinatorial algorithm for finding a singularity certificate of 1CM-matrices, which is free from symbolic elimination. Our method is also advantageous in that it globally preserves the solution set to the DAE. Through numerical experiments, we confirmed that our method runs fast for large-scale DAEs from real instances.

Structural Preprocessing Method for Nonlinear Differential-Algebraic Equations Using Linear Symbolic Matrices

TL;DR

A new structural regularization method is proposed extending the idea of the IOT method, which approximates the system Jacobian by more expressive symbolic matrices, called rank-1 coefficient mixed (1CM) matrices, which makes the method more widely applicable than the existing method.

Abstract

Differential-algebraic equations (DAEs) have been used in modeling various dynamical systems in science and engineering. Several preprocessing methods for DAEs, such as consistent initialization and index reduction, use structural information on DAEs. Unfortunately, these methods may fail when the system Jacobian, which is a functional matrix, derived from the DAE is singular. To transform a DAE with a singular system Jacobian into a nonsingular system, several regularization methods have been proposed. Most of all existing regularization methods rely on symbolic computation to eliminate the system Jacobian for finding a certificate of singularity, resulting in much computational time. Iwata--Oki--Takamatsu (2019) proposed a method (IOT-method) to find a certificate without symbolic computations. The IOT method approximates the system Jacobian by a simpler symbolic matrix, called a layered mixed matrix, which admits a fast combinatorial algorithm for singularity testing. However, it often overlooks the singularity of the system Jacobian since the approximation largely discards algebraic relationships among entries in the original system Jacobian. In this study, we propose a new regularization method extending the idea of the IOT method. Instead of layered mixed matrices, our method approximates the system Jacobian by more expressive symbolic matrices, called rank-1 coefficient mixed (1CM) matrices. This makes our method more widely applicable. We give a fast combinatorial algorithm for finding a singularity certificate of 1CM-matrices, which is free from symbolic elimination. Our method is also advantageous in that it globally preserves the solution set to the DAE. Through numerical experiments, we confirmed that our method runs fast for large-scale DAEs from real instances.
Paper Structure (30 sections, 9 theorems, 54 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 30 sections, 9 theorems, 54 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Related Work.
  4. Organization.
  5. Notations.
  6. Structural Methods for DAEs
  7. Regularization Methods
  8. Combinatorial Relaxation Framework
  9. Regularizing Linear Time-Invariant DAEs
  10. Regularizing Nonlinear DAEs
  11. Proposed Method
  12. Overview
  13. Step 1: Constructing 1CM-matrix
  14. Step 2: Finding Vanishing Pair
  15. Step 3: Modifying DAE Using Vanishing Pair
  16. ...and 15 more sections

Key Result

Theorem 3.2

Let $A=\binom{Q}{T}$ be a layered mixed matrix with columns $C$ and $m_Q$ be the number of rows of $Q$. Then, we have

Figures (5)

  • Figure 1: Robotic arm whose behavior is described by \ref{['roboteq1']}.
  • Figure 2: Structure of the matrix $M$. Grayed out blocks represent zero matrices.
  • Figure 3: The robotic arm whose behavior is described by \ref{['roboteq']}.
  • Figure 4: Running time for the robotic arm DAE \ref{['roboteq']}
  • Figure 5: Electrical circuit described by \ref{['mna']}.

Theorems & Definitions (23)

  • Example 2.1: robotic arm deluca88
  • Definition 3.1: layered mixed matrix murota85
  • Theorem 3.2: murotatext
  • Example 3.3
  • Remark 3.4
  • Definition 4.1
  • Example 4.2
  • Example 4.3: Continued from Example \ref{['robotex']}
  • Remark 4.4
  • Theorem 4.5: soma14
  • ...and 13 more