Convergence and Equivalence results for the Jensen's inequality - Application to time-delay and sampled-data systems

Corentin Briat

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Convergence and Equivalence results for the Jensen's inequality - Application to time-delay and sampled-data systems

Corentin Briat

arXiv:1204.1069v1Apr 4, 2012eess.SYmath.DSmath.OC

TL;DR

The paper addresses conservatism in Jensen-based analysis of time-delay and sampled-data systems and studies whether fragmentation reduces the Jensen gap. It uses the Grüss inequality and characterizes equivalent bounds, showing that uniform fragmentation can make the gap arbitrarily small while alternative bounds preserve useful affine and well-posed structures.

Problem
Jensen’s inequality is useful in time-delay and sampled-data analysis, but its conservatism and the theoretical effect of fragmentation require study.
Method
The paper uses the Grüss inequality to bound Jensen’s gap, analyzes uniform and non-uniform fragmentation, and characterizes a family of bounds equivalent to Jensen’s.
Results
Uniform fragmentation makes the Jensen gap arbitrarily small as its order increases, while non-uniform schemes can accelerate convergence in certain cases.
Takeaways & Limitations
Equivalent bounds can provide affine dependence on the integration measure and remain well-posed as the measure tends to 0, supporting LMI-oriented numerical analysis.

Abstract

from arXiv · show

The Jensen's inequality plays a crucial role in the analysis of time-delay and sampled-data systems. Its conservatism is studied through the use of the Grüss Inequality. It has been reported in the literature that fragmentation (or partitioning) schemes allow to empirically improve the results. We prove here that the Jensen's gap can be made arbitrarily small provided that the order of uniform fragmentation is chosen sufficiently large. Non-uniform fragmentation schemes are also shown to speed up the convergence in certain cases. Finally, a family of bounds is characterized and a comparison with other bounds of the literature is provided. It is shown that the other bounds are equivalent to Jensen's and that they exhibit interesting well-posedness and linearity properties which can be exploited to obtain better numerical results.

I. INTRODUCTION

The paper studies Jensen’s inequality in time-delay and sampled-data systems, focusing on its conservatism and on fragmentation-based reduction of the Jensen gap. It also characterizes equivalent bounds with structural properties useful for numerical analysis.

Jensen’s inequality bounds convex functions of integrals or sums and is widely used across analysis, probability, information, statistics, control, and systems theory.
Its applications in systems theory include bounding L2-gains of integral operators and integral quadratic terms in time-delay systems.
The paper uses the Grüss inequality to study Jensen conservatism, derive gap bounds, and assess the effect of differentiability.
Uniform fragmentation is proved to reduce the Jensen gap, with the upper bound converging sublinearly to 0 as fragmentation order increases.
A characterized family of equivalent bounds can depend affinely on the integration-interval measure and remain well-posed as that measure tends to 0.

II. CONSERVATISM OF THE JENSEN’S INEQUALITY

The paper introduces a Grüss inequality for bounded measurable functions in an inner product space and uses it to bound Jensen-type deviations. Its constant 1/4 is sharp under the stated assumptions.

The Grüss inequality applies to bounded measurable f and g constrained between lower and upper bounds almost everywhere.
The bound depends on the ranges δf = f+ − f− and δg = g+ − g− of the two functions.
The coefficient 1/4 is sharp and is attained by signum functions switching at the midpoint of the interval.
More general Grüss variants extend to complex functions, broader measure spaces, and other inner product spaces.

B. Conservatism of the Jensen’s inequality

For vector-valued functions and the quadratic convex function φ(z) = zTz, the paper derives upper bounds on Jensen’s gap. A differentiability-based bound refines the general result.

The continuous function space consists of bounded measurable functions on a connected bounded subset U of R, equipped with an inner product.
Using the Grüss inequality, the paper obtains a bound on the Jensen gap for f ∈ Lc(U, Rn) and φ(z) = zTz.
The general quadratic Jensen-gap bound has a sharp 1/4 constant and is attained componentwise by signum functions.
For continuous functions differentiable almost everywhere, Corollary 2.1 provides a refined upper bound involving the measure of U.
The differentiability-based Grüss coefficient is not sharp because differentiability is not incorporated into that derivation.

III. JENSEN’S BOUND GAP REDUCTION

The paper analyzes fragmentation by partitioning the integration domain and summing bounds over disjoint fragments. This establishes gap reduction results for general and differentiable functions.

General results on gap reduction by fragmentation: Fragmentation partitions U into N disjoint connected parts whose measures add to the measure of U.
General results on gap reduction by fragmentation: The fragmented bound is obtained by bounding each fragment’s contribution separately and adding the resulting bounds.
General results on gap reduction by fragmentation: The analysis can use weak derivatives at nonsmooth points, taking the supremum over all possible derivative values there.
General results on gap reduction by fragmentation: Theorem 3.1 bounds the Jensen gap for φ(z) = zTz when U is partitioned into N parts.
General results on gap reduction by fragmentation: A differentiability-based fragmented upper bound is also established for continuous functions differentiable almost everywhere.

B. Equidistant fragmentation

Equidistant fragmentation partitions the integration domain into N equal-measure parts and yields Jensen-gap bounds that converge sublinearly to zero as N increases. The convergence is faster for slowly varying functions and slower as function variability grows.

B. Equidistant fragmentation: Uniform fragmentation divides U into N parts of identical Lebesgue measure and produces an explicit upper bound for the Jensen’s gap.The bound is given under both the general assumptions and the additional continuous, almost-everywhere differentiable setting.
B. Equidistant fragmentation: The upper bounds e1(N) and e2(N) converge asymptotically to zero, with sublinear convergence as the fragmentation order increases.Proposition 3.1 treats the non-continuous case through e1(N) and the continuous differentiable case through e2(N).
B. Equidistant fragmentation: The section establishes asymptotic convergence by summing Jensen bounds over the equal subintervals and analyzing the resulting N-dependent expression.The derivation considers intervals [ih, (i + 1)h] with h = 1/N before establishing the convergence behavior.
B. Equidistant fragmentation: The convergence rate depends on function variability: slowly varying functions converge very quickly, whereas increasing α makes convergence slower.Figure 1 compares the normalized bound JN(α)/J(α) for different α values; the stated explanation links the rate to the quadratic variability dependence of the Jensen gap.
B. Equidistant fragmentation: For α = 1, Figure 2 compares the bounds e1(N) and e2(N) with the actual Jensen gap.This comparison evaluates both upper-bound families against the actual gap for the specified example.

C. Nonuniform fragmentation

Nonuniform fragmentation can accelerate convergence by allocating smaller fragments where the function varies more, although this advantage may not transfer across time-delay trajectories.

C. Nonuniform fragmentation: Adaptive fragmentation assigns smaller fragments where function variability is higher, exploiting Jensen equality on constant-function regions.The paper motivates this scheme because the Jensen gap is zero for constant functions.
C. Nonuniform fragmentation: For a discontinuous function, concentrating the discontinuity in a suitably chosen interval can reduce the Jensen gap to an arbitrarily small value.The other fragments contain constant portions, so their Jensen bounds are exact.
C. Nonuniform fragmentation: Uniform fragmentation of the discontinuous function need not converge monotonically, but its gap remains bounded by monotonic bounds e1 and e2.Increasing the number of fragments can locally increase the measure of the interval containing the discontinuity.
C. Nonuniform fragmentation: For an increasing-slope exponential function, the interval is partitioned nonuniformly with fragment sizes decreasing toward 1.The construction uses delimitating points t_i to create N nonuniform fragments.
C. Nonuniform fragmentation: For α = 100 and ε = 10^-4, nonuniform fragmentation substantially increases convergence speed compared with uniform fragmentation.This comparison is reported for the exponential example in Fig. 3.
C. Nonuniform fragmentation: A trajectory-dependent nonuniform partition may fail across different intervals for oscillating time-delay solutions, making a common partition generally unavailable.The paper therefore motivates choosing a uniform fragmentation for time-delay and sampled-data analysis.

IV. EQUIVALENCE BETWEEN JENSEN’S BOUND AND SOME

The paper characterizes a family of bounds equivalent to Jensen’s in tightness and uses this framework to explain their computational and structural advantages.

BOUNDS OF THE LITERATURE: The section derives a complete family of bounds equivalent to Jensen’s in tightness, although the family is computationally more complex.The additional complexity comes from introducing additional variables.
BOUNDS OF THE LITERATURE: The family includes bounds affine in the integration-interval measure and LMIs that remain well-posed as that measure tends to zero.These properties are useful when the interval measure is time-varying or uncertain.
BOUNDS OF THE LITERATURE: The affine bounds are not worse than Jensen’s in tightness and can support LMI-based numerical analysis.Their structural properties motivate their use when numerical tools are sought.
BOUNDS OF THE LITERATURE: The proof relies on a lemma whose matrix inequality is convex in N and has a unique global minimizer N* = -C^-1B.The minimizer follows by completing the squares.
BOUNDS OF THE LITERATURE: Theorem 4.1 states equivalence between an integral inequality and the existence of a matrix N satisfying a corresponding matrix inequality.The theorem assumes an integrable vector function, a positive-definite symmetric matrix R, and an integral relation involving M and w(·).
BOUNDS OF THE LITERATURE: The framework is applied to establish equivalence between different bounds reported in the literature.A discrete-time formulation is noted but omitted because of space limitations.

A. A first Integral inequality

The paper examines an integral bound involving Jensen’s inequality and an auxiliary matrix formulation. The bound is equivalent to Jensen’s, but an affine formulation can be preferable in some cases.

A differentiable function x(t) and the stacked vector w(·) = col(x(t), x(tk)) are used to formulate the integral bound.
The bound introduces an additional matrix N that must be determined.Its matrix R depends on N and the interval length t − tk.
Theorem 4.1 establishes equivalence between this bound and Jensen’s inequality.
The affine formulation can sometimes be better suited than the rational formulation.

B. A reason for using the affine formulation rather the rational one

The affine formulation is motivated by ill-posedness and numerical tractability when the integration interval can shrink to zero. In the sampled-data example, it yields a less conservative LMI condition than the rational formulation.

When the integral support varies and may vanish, the rational formulation becomes ill-posed, whereas the affine formulation remains well-posed.This makes affine bounds more appropriate for numerical tools such as LMIs.
The sampled-data setting uses t ∈ [tk, tk+1], with tk+1 − tk ≤ τm and tk denoting sampling instants.
τm = 1.6894 is obtained with the affine formulation, compared with τm = 0.8691 for the rational formulation.The comparison is made on the system from [18, Example 4] and [19, Example 1].
Although the two bounds are initially equivalent, enforcing tractable well-posed LMIs introduces considerable conservatism for the rational formulation.

C. A second Integral inequality

A bound from [23] is shown to be equivalent to Jensen’s inequality through direct application of Theorem 4.1.

The paper considers the bound introduced in [23, equation (7)].
Simple calculations show that Theorem 4.1 applies, establishing equivalence between this bound and Jensen’s inequality.
The result places the bound from [23] among alternative formulations with the same tightness as Jensen’s inequality.

D. A sum inequality

The paper shows that a bound from [9] is equivalent to discrete-time Jensen’s inequality and concludes that uniform fragmentation offers the best supported tradeoff among the examined schemes. The characterized family also provides affine, well-posed alternatives with higher computational complexity.

The bound introduced in [9, equation (7)] is equivalent to the discrete-time Jensen’s inequality.
Uniform fragmentation offers the best tradeoff because nonuniform fragmentation accelerates convergence only in specific cases.
The characterized family has Jensen-equivalent tightness but higher computational complexity.
Its bounds are affine in the integration-interval measure and remain well-posed as that measure tends to zero, which is important for LMIs.
The paper suggests combining adaptive fragmentation with this affine structure to obtain asymptotically exact well-posed approximants.