Extremum seeking in the presence of large delays via time-delay approach to averaging

1 Introduction

• The introduction situates extremum seeking within a broader literature that includes averaging-based stability analysis and non-local, Newton-based, stochastic, and application-specific methods.The first rigorous stability analysis using averaging and singular perturbations is attributed to Krstic and Wang.Source: §1 · p. 1
• Earlier delayed-ES analyses were qualitative and did not provide quantitative dither-frequency bounds that preserve stability.The cited prior work addressed long delays or varying delays under different assumptions.Source: §1 · p. 1
• The paper establishes quantitative conditions for practical stability with large distinct input delays and large output delay with small, fast-varying measurement-delay uncertainty.It also presents sampled-data ES with large distinct input delays.Source: §1 · p. 2
• The analysis yields explicit inequalities for practical stability and quantitative bounds on the dither period and ultimate seeking-error bound.For any large delays and initial box, suitable gains can ensure practical stability with sufficiently fast dithers and sufficiently small uncertainties.Source: §1 · p. 2
• The method diagonalizes the nominal Hessian part, then uses variation of constants and positivity of fundamental solutions to derive tighter, simpler stability conditions.The authors say this reduces conservatism relative to existing time-delay ES analyses.Source: §1 · pp. 1–2

2 Notation and Preliminaries

• The paper uses Euclidean vector norms and induced matrix 2-norms, with positive or negative definiteness denoted for symmetric matrices.It also defines Z+ as the nonnegative integers, N as the positive integers, and I_n as the identity matrix.Source: §2 · p. 2
• The preliminaries invoke the variation-of-constants formula and properties of fundamental solutions for delay differential equations.These results are brought from.Source: §2 · p. 2
• For a scalar delay differential equation with measurable inputs and bounded piecewise-continuous history, the stated framework gives a unique solution in the specified form.The history term is applied when its delayed argument lies before the initial time, and X(t, s) is the fundamental solution.Source: §2 · p. 2

3.1 Multi-variable static map

• The analysis assumes a known nominal positive Hessian with norm-bounded uncertainty, a bounded unknown extremum value, and fast-varying output-delay uncertainty bounded by ρ.The extremum point is also assumed to lie in a known box; this is a grey-box setting that enables quantitative analysis.Source: §3.1 · p. 3
• An orthogonal transformation diagonalizes the nominal Hessian, while commensurable nominal input delays and specially chosen dither frequencies align the delayed dither signals.The delay treatment retains a common delay for each subsystem’s full error state, unlike the simplified componentwise-delay model discussed in prior work.Source: §3.1 · pp. 3–4
• The proof averages the transformed system and bounds its delayed disturbance by O(max{ε, µ, κ}) using the linear delay system’s solution representation.Gain choices ensure exponential stability of the nominal delayed dynamics, yielding bounds on the estimation error.Source: §3.1 · pp. 4–5
• For any large output and distinct input delays, the multi-variable ES algorithm converges when the gains satisfy a stability condition and the dither period and uncertainties are sufficiently small.Theorem 1 gives exponentially attracting error bounds independent of the dither parameter and time-varying input delays; the corollary allows arbitrary positive delays with sufficiently small ε, µ, and κ.Source: §3.1 · p. 6
• For fixed tuning parameters and any positive error margin, the estimation error enters a neighborhood of its ultimate bound in finite time.The stated entry time depends on the delays, ε, the gain, and the initial error bound.Source: §3.1 · p. 6

3.2 Single-variable static map

• The single-variable subsection introduces a static map with output delay D + ∆D(t), where D is known and positive, |∆D(t)| ≤ ρ, and the scalar curvature h is unknown.The passage states that A3 is assumed to hold.Source: §3.2 · p. 6
• Corollary 2 guarantees that, for ε ≤ ε* under its stated conditions, the estimation error is exponentially attracted to an interval.The result applies when the initial error at DM is bounded by σ0.Source: §3.2 · p. 7

3.3 Examples

• For matched σ0, σ, and µ, Corollary 2 permits larger upper bounds κ and ε* (hence smaller ω*) and a smaller UB than Theorem 1.The Example 3.1 simulation confirms the UB reported in Table 1.Source: §3.3 · pp. 7–8, Table 1 · p. 8
• In the autonomous-vehicle example without GPS orientation, the goal is to reach the origin, and simulations with ε = 1/4 confirm the UB values shown in Table 2.The listed UB corresponds to ε = 1/4 (< ε* = 0.3).Source: §3.3 · p. 8
• For the two-dimensional map with known and uncertain Hessians, simulations confirm the calculated upper bounds for the estimation errors.The cases use κ = 0 and κ = 0.2, respectively.Source: §3.3 · pp. 8–9

4.1 Multi-variable static map

• The analysis assumes rational delay ratios, expressed by integers mi satisfying miDj = mjDi, and chooses the corresponding mi as small as possible.The delays are known constants, and the sampled-data scheme uses periodic fast sampling and square-wave dithers.Source: §4.1 · p. 9
• The algorithm uses square-wave dithers and selects the adaptation gain so that K H̄ is Hurwitz.The method builds the sampled-data ES dynamics around a map measured at sampling instants.Source: §4.1 · p. 9
• The method transforms the sampled-data dynamics into a retarded time-delay system and uses a solution representation formula to bound estimation error.The authors say this simplifies stability analysis and improves quantitative bounds on ε and the input delays compared with.Source: §4.1 · p. 10
• Theorem 2 provides practical-stability conditions for sampled-data ES with distinct input delays, using gain choices and bounds on the sampling parameter.The theorem's conditions include ε∗ and delay-dependent requirements; the supplied passages do not expose their full inequalities.Source: §4.1 · p. 10

These solutions are exponentially attracted to the box

• Solutions are exponentially attracted to a box, with a decay rate given by the minimum of the component decay rates.The box bounds are formed from the component bounds Θj and coefficients bji.Source: These solutions are exponentially… · pp. 10–11
• With κ = 0 and ai, ki of order O(1), the estimation-error upper bound is O(ε), compared with O(√ε) in.Theorem 2 also allows distinct, arbitrarily large constant input delays, unlike the D = O(ε) restriction described for.Source: These solutions are exponentially… · p. 11
• The presented approach covers uncertain H, whereas has no result for that uncertainty case.The source says this uncertainty case can be handled using the time-delay approach.Source: These solutions are exponentially… · p. 11

4.2 Single-variable static map

• For the single-variable map, choosing k by inequality (48) ensures exponential stability of the averaged system with decay rate δ.The section also specifies an ES algorithm using y(t) = Q(θ(t)) and θ(t) = θ̂(t) + S(t).Source: §4.2 · p. 11

4.3 Examples

• The Table 4 comparison reports larger δ and ε∗, larger delay D, and smaller UB for the proposed results than.The prose also reports that the method allows much larger uncertainties in Q∗ and h than.Source: Table 4 · p. 12, §4.3 · p. 12
• The Table 5 comparison reports larger δ and ε∗, larger delay D, and smaller UB than.The table caption identifies the comparison measures as δ, D, ε∗, and ω∗ = 2π/ε∗.Source: Table 5 · p. 12, §4.3 · p. 12
• In Example 4.3, the reported UB values are B1 = 2.6 · 10^-2 and B2 = 1.82 · 10^-2.For known and uncertain H, respectively, the example reports ε = 1/2 (< ε∗ = 0.79) and ε = 1/4 (< ε∗ = 0.36).Source: Table 6 · p. 12, §4.3 · p. 12

5 Conclusion

• The approach addresses uncertain quadratic maps with large measurement delays and distinct input delays, including sampled-data ES with distinct input delays.Source: §5 · p. 12
• Choosing gains to stabilize averaged time-delay systems yields explicit inequalities that guarantee practical stability.Source: §5 · p. 12
• The method provides quantitative bounds on control parameters and the ultimate bound of seeking error.Source: §5 · p. 12
• Compared with recent quantitative sampled-data results, the method simplifies stability conditions, improves results, and allows distinct and large delays.Source: §5 · p. 12

• The dither frequencies satisfy i · ωj = j · ωi for all i, j = 1, . . . , n.Source: A1: Proof of Lemma 2 · p. 12
• The delay-frequency relation is ωi · D̄j = 2π · qimj, with q a natural number for all i, j = 1, . . . , n.Source: A1: Proof of Lemma 2 · p. 13
• Because qimj are positive integers, the preceding relation implies (18), completing the proof.Source: A1: Proof of Lemma 2 · p. 13

A2: Proof of Theorem 1

• The proof divides part (i) into deriving upper bounds under an assumed component bound, proving practical stability of each zi-system, and verifying that assumption by contradiction.Source: A2: Proof of Theorem 1 · p. 13
• Under condition (40), the component bounds |˜ϑi(t)| < σ̄i hold for all t ≥ 0, established successively over time intervals by contradiction.Source: A2: Proof of Theorem 1 · pp. 13, 15
• For each component, the proof defines Xi(t, s) using the homogeneous delayed system żi(t) = ki h̄i z(t − Di(t)).Source: A2: Proof of Theorem 1 · p. 14