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Qualitative properties and stability analysis of the mathematical model for a DC-DC electric circuit

E. V. Chistyakova, D. N. Sidorov, A. V. Domyshev, V. F. Chistyakov

TL;DR

The paper addresses solvability and stability analysis of a DC-DC converter model with a PID regulator described by a system of integral-differential equations featuring an identically singular leading coefficient. It introduces a reduction framework based on a left regularizing operator to transform the problem into a finite-dimensional kernel, enabling rigorous solvability analysis. A reduced two-state model is derived, yielding a second-order characteristic equation and stability conditions via the Routh-Hurwitz criteria, with parameter dependencies clarified for circuit elements and controller gains. The results provide guidelines for parameter selection (including inertia terms $T_d$, $T_{dd}$ and gains $K_p$, $K_d$, $K_i$, $K_{dd}$) to ensure a piecewise differentiable, bounded response on $[0,T]$, contributing to robust DC-DC converter design under large, fast load changes.

Abstract

This paper describes a simplified model of an electric circuit with a DC-DC converter and a PID-regulator as a system of integral differential equations with an identically singular matrix multiplying the higher derivative of the desired vector-function. We use theoretical results on integral and differential equations and their systems to prove solvability of such a model and analyze its stability.

Qualitative properties and stability analysis of the mathematical model for a DC-DC electric circuit

TL;DR

The paper addresses solvability and stability analysis of a DC-DC converter model with a PID regulator described by a system of integral-differential equations featuring an identically singular leading coefficient. It introduces a reduction framework based on a left regularizing operator to transform the problem into a finite-dimensional kernel, enabling rigorous solvability analysis. A reduced two-state model is derived, yielding a second-order characteristic equation and stability conditions via the Routh-Hurwitz criteria, with parameter dependencies clarified for circuit elements and controller gains. The results provide guidelines for parameter selection (including inertia terms , and gains , , , ) to ensure a piecewise differentiable, bounded response on , contributing to robust DC-DC converter design under large, fast load changes.

Abstract

This paper describes a simplified model of an electric circuit with a DC-DC converter and a PID-regulator as a system of integral differential equations with an identically singular matrix multiplying the higher derivative of the desired vector-function. We use theoretical results on integral and differential equations and their systems to prove solvability of such a model and analyze its stability.
Paper Structure (3 sections, 7 theorems, 48 equations, 1 figure)

This paper contains 3 sections, 7 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reduction to a system of singular integral differential equations
  3. Qualitative properties of the mathematical model for the DC-DC electric circuit

Key Result

Theorem 1

The solution space of system (eqdaeK11) is finite-dimensional $({\rm dim}\; {\rm ker}(\Lambda_k+{\cal V})<\infty)$, if and only if there exists an LRO for the operator $(\Lambda_k+{\cal V})$.

Figures (1)

  • Figure 1: An idealized model of a DC-DC converter

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Theorem 5