Kharitonov's Theorem with Degree Drop: a Wronskian Approach

TL;DR

The paper tackles robust Hurwitz stability of interval polynomials by reducing verification to the four Kharitonov polynomials $k_j$. It presents a simplified, self-contained proof using a Wronskian-based framework, built on a root-continuity lemma and an extended Boundary Crossing Theorem that accommodates degree drop. Key contributions include an elementary derivation of Kharitonov's Theorem via $W[h,g](\omega)$ and the Kharitonov rectangle $K(i\omega)$, avoiding Bezoutian methods. The approach yields a shorter, more accessible path with practical implications for robust stability analysis in control and systems theory, and suggests avenues for generalizations.

Abstract

In this paper, we present a simplified proof of Kharitonov's Theorem, an important result on determining the Hurwitz stability of interval polynomials. Our new approach to the proof, which is based on the Wronskian of a pair of polynomials, is not only more elementary in comparison to known methods, but is able to handle the degree drop case with ease.

Figures (2)

• Figure 1: Plots of all the roots in the complex plane of the Kharitonov polynomials $k_j, j=1,2,3,4$, given by (\ref{['DefKharPolysExample']}) in Example \ref{['example2']} as generated by MATLAB. Here the roots are shown as the red dots for $k_1(z)$ on the top left, $k_2(z)$ on the bottom left, $k_3(z)$ on the top right, and $k_4(z)$ on the bottom right. Using the $\operatorname{roots}(\cdot)$ command in MATLAB, one obtains the approximation for the roots of $k_1(z)$ as $z=-0.28, -1.53\pm0.81i$, the roots of $k_2(z)$ as $z=-7.87, -2.13, -1.00\pm0.51i$, the roots of $k_3(z)$ as $z=-2.23\pm5.05i, -0.77\pm0.31i$, and the roots of $k_4(z)$ as $z=-5.10, -1.32, -0.25$. As the real parts of all these roots are negative (which can also be shown using the methods described in Footnote \ref{['AFootnote']}), Kharinotov's Theorem (i.e., Theorem \ref{['ThmKhar']}) implies that every polynomial $p$ in the interval polynomial $P$, as given by (\ref{['ExInvervalPolyPartI']}) and (\ref{['ExInvervalPolyPartII']}) in this example, is a Hurwitz stable polynomial.
• Figure 2: Under the assumptions in Corollary \ref{['CorKharRectUnDirected']}, this figure depicts the Kharitonov rectangle $K(i\omega)=\{p(i\omega):p\in P\}=\{x+iy:h_-(\omega)\leq x\leq h_+(\omega), g_-(\omega)\leq y\leq g_+(\omega)\}$ in the complex plane with vertices the Kharitonov polynomials $k_j(i\omega)$, $j=1,2,3,4$, evaluated at $z=i\omega$ for a fixed $\omega\geq 0$.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1: Kharitonov’s Theorem
• Example 1
• Lemma 1
• Lemma 2
• proof
• Example 2
• Example 3