[Paper Review] Weighted Orthogonal Polynomials-Based Generalization of Wirtinger-Type Integral Inequalities for Delayed Continuous-Time Systems

Xian Zhang, Yuanyuan Han+2 more|arXiv (Cornell University)|Sep 16, 2015

Stability and Control of Uncertain Systems13 references3 citations

TL;DR

This paper introduces a novel class of weighted orthogonal polynomials-based integral inequalities that generalize and improve upon existing Wirtinger-type and Jensen's inequalities for stability analysis of delayed continuous-time systems. By leveraging orthogonal polynomials with respect to a weighted inner product, the proposed inequalities provide tighter lower bounds for integral terms in Lyapunov-Krasovskii functional derivatives, leading to less conservative stability criteria, with special cases recovering or refining prior results from the literature.

ABSTRACT

In the past three years, many researchers have proven and/or employed some Wirtinger-type integral inequalities to establish less conservative stability criteria for delayed continu\-ous-time systems. In this present paper, we will investigate weighted orthogonal polynomials-based integral inequalities which is a generalization of the existing Jensen's inequalities and Wirtinger-type integral inequalities.

Motivation & Objective

• To develop tighter integral inequalities for bounding Lyapunov-Krasovskii functional derivatives in delayed continuous-time systems.
• To reduce the conservatism inherent in existing stability criteria derived via Jensen’s inequality or Wirtinger-type inequalities.
• To unify and generalize existing integral inequalities, including Jensen’s and Wirtinger-type inequalities, as special cases of a broader framework.
• To provide a systematic method based on weighted orthogonal polynomials for deriving improved lower bounds on integral terms involving time-delayed states and their derivatives.

Proposed method

• Derives a general integral inequality using weighted orthogonal polynomials (WOPs) defined on the interval $[a,b]$ with weight function $(s-a)^m$.
• Constructs an orthogonal basis $\{p_{km}(s)\}$ in the space of polynomials of degree $\leq \mathcal{N}$ with respect to the inner product $(p,q)_m = \int_a^b (s-a)^m p(s)q(s) \, ds$.
• Applies the orthogonality property to derive a lower bound for the integral $\mathcal{I}_m(w_t) = \int_a^b \cdots \int_{\theta_m}^b w_t^T(s) R w_t(s) \, ds \cdots d\theta_1$.
• Establishes the inequality $\mathcal{I}_m(w_t) \geq \frac{(m+1)!}{(b-a)^{m+1}} \left( \Theta_m^T R \Theta_m + \text{higher-order terms} \right)$, where $\Theta_m$ involves weighted averages of $w_t(b)$ and integrals of $w_t(s)$.
• Demonstrates that the inequality reduces to known results (e.g., Jensen’s, Wirtinger-type) when $\mathcal{N} = 0$ or $\mathcal{N} = 1$, and improves upon them for $\mathcal{N} \geq 2$.
• Uses the framework to derive specific corollaries for $\mathcal{N} = 0,1$ and compares the resulting bounds with those in [11, 12, 14, 15, 16, 23, 29, 30], showing improved tightness.

Experimental results

Research questions

• RQ1 Can weighted orthogonal polynomials be used to construct a general class of integral inequalities that subsumes and improves upon existing Wirtinger-type and Jensen’s inequalities?
• RQ2 How does the choice of polynomial degree $\mathcal{N}$ affect the tightness of the lower bound for $\mathcal{I}_m(w_t)$?
• RQ3 To what extent do the new inequalities reduce conservatism in stability criteria for delayed continuous-time systems compared to existing methods?
• RQ4 Are the proposed inequalities equivalent to or strictly less conservative than known results such as those in [14, 15, 16, 29, 30]?

Key findings

• The proposed WOPs-based inequality generalizes Jensen’s inequality (when $\mathcal{N} = 0$) and Wirtinger-type inequalities (when $\mathcal{N} = 1$) as special cases.
• For $\mathcal{N} = 1$, the inequality provides a tighter lower bound than [8, Lemma 5 and 6], with a larger coefficient $\frac{m!(m+1)^2(m+3)}{(b-a)^{m+1}}$ compared to $\frac{m!(m+3)}{(b-a)^{m+1}}$.
• When $\mathcal{N} = 1$ and $m=1$, the inequality matches [15, (25)] and is shown to be less conservative than [11, Lemma 2.4] and [12, (13)].
• The inequality derived for $\mathcal{N} = 1$ and $m=1$ is equivalent to [29, Lemma 4] and [30, Lemma 1], but without requiring free-weighting matrices.
• The bound for $\mathcal{I}_1(\dot{w}_t)$ in Corollary 13 is tighter than [14, Corollary 1], as the coefficient of the second term is larger.
• The framework is extendable by replacing $(s-a)^k$ with $(b-s)^k$, yielding new inequalities that generalize [18, Corollary 4], [23, (3.1) and (3.8)], and [15, (18) and (26)].