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Robustness of Nonlinear Predictor Feedback Laws to Time- and State-Dependent Delay Perturbations

Nikolaos Bekiaris-Liberis, Miroslav Krstic

arXiv:1209.1729v1math.OC

TL;DR

The robustness of nominal constant-delay predictor feedback to unknown time- and state-dependent delay variations had remained unexplored. The paper analyzes this feedback using a nonlinear backstepping transformation and Lyapunov functional, and proves local asymptotic stability for nonlinear systems under sufficiently small perturbations and rates, with global exponential stability results for linear systems under time-varying perturbations.

  • Problem

    Robustness of nominal constant-delay predictor feedback to unknown time- and state-dependent delay perturbations remained unexplored, despite network delays potentially degrading performance or destabilizing control.

    Source: §1 · p. 1

  • Method

    The paper analyzes nonlinear predictor feedback with a backstepping transformation and Lyapunov functional, and treats linear systems with time-varying delay perturbations separately.

    Source: §1 · pp. 1–2

  • Results

    Nonlinear closed-loop local asymptotic stability is preserved for sufficiently small delay perturbations and rates; linear systems retain global exponential stability under four alternative smallness or convergence conditions.

    Source: §1 · p. 2

  • Takeaways & Limitations

    For linear systems with time-varying perturbations, global exponential stability can be established under conditions beyond restricting the perturbation and its rate to small magnitudes.

    Source: §3 · p. 7

    For nonlinear systems with state-dependent perturbations, achievable region of attraction trades off against perturbation size and its rate at the origin.

    Source: §5 · p. 10

Abstract

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Much recent progress has been achieved for stabilization of linear and nonlinear systems with input delays that are long and dependent on either time or the plant state---provided the dependence is known. In this paper we consider the delay variations as unknown and study robustness of nominal constant-delay predictor feedbacks under delay variations that depend on time and the state. We show that when the delay perturbation and its rate have sufficiently small magnitude, the local asymptotic stability of the closed-loop system, under the nominal predictor-based design, is preserved. For the special case of linear systems, and under only time-varying delay perturbations, we prove robustness of global exponential stability of the predictor feedback when the delay perturbation and its rate are small in any one of four different metrics. We present two examples, one that is concerned with the control of a DC motor through a network and one of a bilateral teleoperation between two robotic systems.

1 Introduction

The paper addresses the unexplored robustness of nominal constant-delay predictor feedback to unknown time- and state-dependent delay perturbations. For nonlinear systems, it establishes local asymptotic stability under sufficiently small perturbations and their rates, using a Lyapunov analysis based on a nonlinear backstepping transformation.

1 Introduction

  • The robustness question arises because unknown network-induced delay variations can degrade control performance or destabilize the network, while robustness of nominal constant-delay predictor feedback remained unexplored.Source: §1 · p. 1

2 Robustness to time- and state-dependent delay perturbations for nonlinear systems

  • A nonlinear infinite-dimensional backstepping transformation yields a Lyapunov functional for the plant, predictor feedback, and actuator-state observer.Source: §1 · p. 1
  • The design targets forward-complete nonlinear plants that are locally exponentially stabilizable without delay, with sufficiently restricted perturbations and initial conditions.Strong forward completeness ensures solutions exist for all times under the stated inputs.Source: §1 · p. 1, §2 · p. 3
  • The nonlinear closed loop remains locally asymptotically stable when the delay perturbation and its rate are sufficiently small.The theorem also requires initial conditions satisfying its stated conditions.Source: §1 · p. 2, §2 · p. 3

3 Robustness to time-varying delay perturbation for linear systems

For linear systems, the section treats the delay perturbation as time-dependent and presents an explicit predictor controller. It establishes exponential stability under small-perturbation or alternative conditions, subject to the stated assumptions.

3 Robustness to time-varying delay perturbation for linear systems

  • The linear-case analysis uses an explicit predictor and globally well-defined backstepping transformations; time-only δ removes the actuator-state supremum restriction.Source: §3 · p. 6
  • For the small-perturbation case, Theorem 2 yields an exponentially decaying Lyapunov bound for the closed loop.Source: §3 · pp. 6, 8
  • When condition (65) is not used, alternative perturbation tests still require propagation speed π to be positive and uniformly bounded above and below.Source: §3 · p. 6
  • Under Assumptions 3 and 4, any of the theorem’s listed delay-perturbation conditions yields exponential stability of the closed loop.Source: §3 · p. 6

4 Examples

The examples apply nominal predictor feedback to a networked DC motor and a bilateral teleoperation setup under delay perturbations. The motor is locally stabilized, while the teleoperation slave tracks the master, though perturbed torques oscillate.

4.1 Control of a DC motor over a network

  • Despite a time-varying perturbation that increases with armature current, nominal predictor feedback locally stabilizes the networked DC motor at its desired equilibrium.The displayed currents and motor variables compare the perturbed case with the zero-perturbation case.Source: §4.1 · p. 9, Figure 2 · p. 9, Figure 3 · p. 10

4.2 Bilateral teleoperation

  • The teleoperation test uses a perturbation that is neither L1, vanishing at infinity, nor small in magnitude, but whose mean becomes small after a long time.Source: §4.2 · p. 10
  • Under nominal predictor feedback, the slave position tracks the master with and without perturbation, while perturbed control torques oscillate with the disturbance.Source: §4.2 · p. 10

5 Conclusions

For nonlinear systems with state-dependent perturbations, the achievable region of attraction trades off against perturbation size and rate. For linear systems with time-varying perturbations, global exponential stability holds subject to restrictions on perturbation size and rate.

5 Conclusions

  • For nonlinear systems with state-dependent perturbations, the achievable region of attraction trades off against perturbation size and rate at the origin.Source: §5 · p. 10
  • Figures 5 and 6 depict master–slave position error under zero or network-induced delay perturbation and the manipulators’ input torques under perturbation.Source: Figure 5 · p. 11, Figure 6 · p. 11
  • For linear systems under time-varying perturbations, global exponential stability holds when perturbation size and rate are appropriately restricted.Source: §5 · p. 11
  • The analysis restricts both perturbation magnitude and its derivative, which guarantees invertibility of φ = t−ˆD−δ; these quantities can be unbounded for white-noise or low-pass white-noise delays.Source: §5 · p. 11

A The perturbation signals of Lemma 1

This appendix introduces perturbation signals and represents second derivatives of the vector field with a matrix of predictor-state derivatives.

A The perturbation signals of Lemma 1

B Technical Lemmas

The technical lemmas provide bounds on predictor-related quantities and perturbation terms for solutions satisfying the stated conditions.

B Technical Lemmas

  • The transformed state ˆw and its spatial derivative ˆwx are bounded using ˆu and Ω(t).Source: B Technical Lemmas · p. 12
  • Lemma 14 establishes positive constants M∗ and c∗ for solutions satisfying (51), with an estimate stated over x ∈ [0, 1].Source: B Technical Lemmas · p. 13
  • Lemmas 15 and 16 bound ˆu and its spatial derivative, as well as several perturbation terms using functions of Y(t) and constants μ1 through μ5.Source: B Technical Lemmas · p. 13

C Proof of Lemma 2

The proof uses Fact 1 and Lemma 5 to establish conditions for the perturbation and solutions, including a sufficient relation over the interval from φ(t) to t.

C Proof of Lemma 2

  • Fact 1 provides a class KC∞ function ζ̂1 under conditions on the perturbation and solutions, for φ(t) ≤ θ ≤ t.Source: C Proof of Lemma 2 · p. 14
  • The proof states that conditions (17) and (50) are satisfied if a sufficient relation holds for every φ(t) ≤ θ ≤ t.Source: C Proof of Lemma 2 · p. 14
  • The proof applies Fact 1 and Lemma 5 to establish the required conditions.Source: C Proof of Lemma 2 · p. 14

D Proof of Lemma 3

The proof estimates the derivative of a Lyapunov function and selects parameters so a comparison principle yields relation (55).

D Proof of Lemma 3

  • Integration by parts and inequalities combine the derivative terms, with the terms multiplying |f~| bounded by a gain-weighted expression times κ5(R).The stated bound is (g7e^g4 + g9e^g10 + g8e^g5 + g12e^g3)κ5(R).Source: D Proof of Lemma 3 · pp. 14–15
  • The functions αr and α1,rx are locally Lipschitz in their first argument, supporting a bound on α2,rx for bounded s.The passage gives α2,rx(s, R) ≤ L2(R)α26(R)s.Source: D Proof of Lemma 3 · p. 16
  • The proof chooses gain parameters and a sufficiently small ĉ, then applies the comparison principle to obtain relation (55).The parameter selection uses bounds involving B1, B2, and B3 + B.Source: D Proof of Lemma 3 · p. 16

E Proof of Lemma 4

These proof passages use derivative and integral estimates to establish the stated lemma bounds, including a bound on r in terms of X and the transformed state variables.

E Proof of Lemma 4

  • For Lemma 4, the bound on |ũ(x, t)| by the integral of |ũx|, followed by Cauchy–Schwarz, makes the proof immediate..Source: E Proof of Lemma 4 · p. 16

F Proof of Lemma 7

  • The proof of Lemma 7 differentiates V along the stated equations and uses integration by parts to derive an estimate.The displayed estimate includes a term involving ˆwx and ˜u(0, t).Source: F Proof of Lemma 7 · p. 16
  • For Lemma 7, Young’s and Cauchy–Schwarz inequalities bound r(x, t) in terms of X, ˆw, and ˆwx, yielding relation (99).The passage says this follows using (F.2), (F.3), and (F.5).Source: F Proof of Lemma 7 · pp. 16–17

G Proof of Lemma 8

The proof of Lemma 8 bounds the delay-perturbation term and establishes a sufficient smallness condition; the next proof passage estimates terms using Young’s and Cauchy–Schwarz inequalities.

G Proof of Lemma 8

  • In the Lemma 8 argument, γ(t) equals |δ(σ(t))|, and the lemma follows if δ satisfies (65) with δ1r2 < r1..Source: G Proof of Lemma 8 · p. 17
  • The Lemma 8 condition δ1r2 < r1 is achievable by choosing δ1 sufficiently small.The passage states r1 can be chosen independent of δ1 or very large, while r2 is bounded.Source: G Proof of Lemma 8 · p. 17

H Proof of Lemma 9

  • The Lemma 9 proof applies Young’s and Cauchy–Schwarz inequalities to the relations (77)–(78) to derive bounds.The remaining passages display terms involving ˆD, K, A, B, and A + BK.Source: H Proof of Lemma 9 · p. 17
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