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Predictor-Based Exponential Tracking of Turbulent Solutions for the Kuramoto--Sivashinsky Equation with Input Delay
Amadou Cissé
TL;DR
The paper addresses tracking nonstationary Kuramoto–Sivashinsky trajectories when actuation is delayed, a setting where delay can misalign feedback with the evolving reference. It combines predictor and backstepping designs; numerical experiments illustrate recovery of tracking with predictor compensation.
Problem
Existing PDE predictor designs primarily address equilibrium or set-point stabilization, leaving tracking of nonlinear, nonautonomous Kuramoto–Sivashinsky trajectories with delayed actuation unresolved.
Source: §1 · p. 2Method
Results
Numerical experiments illustrate loss and recovery of tracking, terminal predictor consistency, and extinction of the transformed actuator state after one delay horizon.
Source: §1 · p. 3, §4.2 · p. 22Takeaways & Limitations
Predictor compensation restores temporal alignment without removing or shortening the physical delay, and the stated objective is uniform tracking across the attractor reference family.
Source: §1 · pp. 2–3The delay must be constant and exactly known, and the reference must be available over the prediction horizon.
Source: §1 · p. 2, §2.1 · p. 4
Abstract
from arXiv · showhide
Uniform exponential tracking is addressed for nonstationary trajectories of the nonlinear Kuramoto--Sivashinsky equation subject to a constant input delay. The reference belongs to a family of complete trajectories contained in the global attractor and need not be stationary, periodic, slowly varying, or generated by a finite-dimensional exosystem. The delayed input is represented by a first-order transport equation coupled with the fourth-order tracking-error dynamics. A predictor--backstepping transformation compensates for the temporal mismatch between command generation and actuation by mapping the augmented closed-loop system into the nominal delay-free error dynamics driven by the outgoing trace of a homogeneous transport subsystem. This subsystem vanishes after one delay interval. Uniform attractor bounds permit the feedback parameters and stability constants to be selected independently of the initial time and the reference trajectory. Global well-posedness and uniform exponential stability are established in the augmented state space. Numerical results show that, for the considered configuration, uncompensated delayed feedback amplifies the tracking error, whereas predictor compensation restores sustained decay. Predictor-consistency, finite-time-extinction, and discretization-refinement diagnostics support the numerical implementation.
1. Introduction.
Delayed actuation complicates uniform tracking of nonstationary Kuramoto–Sivashinsky trajectories: prior results largely address invariant states or assume instantaneous feedback. The paper combines predictor compensation with a trajectory-uniform stability analysis and reports numerical recovery of tracking under delay.
1. Introduction.
- Existing KS control results mainly concern equilibria or invariant states; fewer address prescribed nonstationary trajectories, and those cited do not incorporate command-to-actuation delay.Prior work established exact controllability or semiglobal feedback stabilization for time-dependent trajectories.Source: §1 · p. 1
- Existing predictor designs for PDEs primarily address equilibrium or set-point stabilization, whereas this problem involves nonlinear, nonautonomous tracking of an infinite-dimensional reference family.The required transformation estimates must hold uniformly over complete trajectories rather than rely on a finite-dimensional exosystem.Source: §1 · p. 2
- The target references are complete KS trajectories in the global attractor, without requirements of stationarity, periodicity, slow variation, or generation by a finite-dimensional exosystem.The prescribed reference is assumed available over the prediction horizon, while the feedback uses no future plant state.Source: §1 · p. 2
- The predictor–backstepping design yields global well-posedness and uniform exponential stability in the augmented state space, with design and stability constants independent of initial time and reference trajectory.The transformed system is a cascade of nominal stabilized tracking dynamics and a homogeneous transport subsystem whose outgoing trace vanishes after one delay interval.Source: §1 · p. 3
- Numerical experiments illustrate tracking loss with uncompensated delayed feedback and recovery with predictor compensation, alongside predictor-consistency and finite-time actuator-state extinction diagnostics.The predictor does not remove or shorten the physical delay; it restores temporal alignment between command generation and the future state on which the command acts.Source: §1 · p. 3
2. Problem Formulation.
The problem formulation models finite-dimensional delayed actuation through a transport state coupled to the KS tracking error. It specifies admissible attractor references and seeks uniform exponential stability of the augmented system.
2.1. Delayed Kuramoto–Sivashinsky Model and Actuation.
- The finite-dimensional actuation uses the first N nonzero Fourier wavenumbers, including cosine and sine components, and N must satisfy a uniform stabilizability condition.Covering the nonnegative eigenvalue modes alone does not ensure uniform stabilizability of the reference-dependent linearized tracking dynamics.Source: §2.1 · p. 5
2.2. Transport Representation and Tracking Dynamics.
- The input delay is represented by a transport state whose characteristic solution shifts the control signal along the delay line, with its outgoing trace supplying the delayed input.The delay is a known positive constant, and the initial delay-line state is set by the prescribed control history.Source: §2.1 · p. 4, §2.2 · p. 6
2.3. Control Objective.
- The admissible reference family comprises complete global-attractor trajectories, while the control objective requires a globally exponentially stable augmented system.The stability constants are independent of the initial time and selected reference trajectory.Source: §2.2 · p. 7, §2.3 · p. 9
2.2. Transport Representation and Tracking Dynamics.
- Tracking error is defined as the plant state minus the reference, and subtracting their dynamics produces the reference-dependent error equation coupled to the delay-line state.The reference is a complete uncontrolled KS trajectory in the global attractor, and its relevant bound is uniform over admissible trajectories.Source: §2.2 · p. 7
2.3. Control Objective.
- The controller objective is global exponential stability of the augmented tracking origin, with decay uniform over initial times and prescribed complete reference trajectories in the global attractor.The constants may depend on system parameters, delay, actuator configuration, and the uniform attractor bound, but not on initial time or the selected reference.Source: §2.3 · p. 9
- The feedback design first stabilizes the delay-free reference-dependent tracking dynamics, then uses a nonlinear predictor over the delay horizon and backstepping to form a stable cascade.The predictor uses reference preview over the horizon; the plant-dependent feedback arguments remain restricted to the current augmented state.Source: §2.3 · p. 8, §3 · p. 9
3. Predictor–Backstepping Design.
The section develops a predictor–backstepping design that recasts delayed tracking dynamics as nominal dynamics coupled to a transport component that vanishes after one delay horizon, with uniform attractor bounds supporting stability independent of initial time and reference trajectory. Under condition (3.3), it establishes unique global mild solutions and uniform exponential stability for every delay, complete reference trajectory, and initial state, and presents numerical experiments illustrating delay-induced tracking loss and its recovery through predictor compensation.
3.1. Nominal Feedback and Uniform Stability.
- The selected nominal feedback yields a unique global solution and global exponential stability, uniformly over initial time and complete reference trajectory.The constants depend only on the attractor bound, domain, and fixed design parameters.Source: §3.1 · pp. 10–11
3.2. Nonlinear Predictor and Delay Compensation.
- A causal predictor computed from the current augmented state and reference segment ensures the command’s action at the plant exactly matches nominal feedback when it arrives.The identity holds along an existing closed-loop trajectory; the implemented law itself remains causal.Source: §3.2 · pp. 12–13
3.3. Predictor Transformation and Target Dynamics.
- The predictor transformation is a bijection with uniformly equivalent state norms and a one-to-one correspondence between delayed-system and target-system mild solutions.The norm-equivalence constants are independent of time and the selected complete reference trajectory.Source: §3.3 · pp. 14, 16
- Target dynamics are nominal tracking dynamics forced by the transport trace r(t,0); the transport component vanishes after one delay horizon, and the target system has a unique global mild solution.After extinction, the tracking equation reduces exactly to the nominal feedback dynamics.Source: §3.3 · pp. 15–17
3.4. Exponential Stability of the Delayed Closed-Loop System
- For every τ ≥ 0, complete reference trajectory in A, and initial state in X, the delayed closed-loop system has a unique global mild solution and a globally uniformly exponentially stable origin.The bound constant MD is independent of τ, the selected reference trajectory, and the initial data.Source: §3.4 · pp. 18–19
- Transport characteristics show that the transformed actuator state and its outgoing trace vanish after one delay interval, enabling nominal tracking stability to establish the delayed-system estimate.The tracking component is treated with d(t) = Br(t, 0).Source: §3.4 · p. 18
- Numerical experiments show delay-induced tracking loss and recovery through predictor compensation, while assessing the terminal predictor relation and finite-time extinction of the transformed actuator state.Observed tracking decay and actuator behavior are consistent with the mechanisms established in Theorem 3.7.Source: §4 · p. 19
4. Numerical Illustration.
The numerical illustration compares uncontrolled dynamics, uncompensated delayed feedback, and predictor compensation. In the considered configuration, uncompensated feedback amplifies tracking error, while predictor compensation restores sustained decay and synchronizes evolving spatial patterns.
4.1. Numerical Setting.
- The simulations use a Fourier pseudospectral discretization with Nx = 128, ∆t = 10−2, and fourth-order exponential time-differencing Runge–Kutta integration.The quadratic term is computed in physical space with 2/3 de-aliasing, and the zero Fourier coefficient is reset after each update.Source: §4.1 · p. 19
- The predictor is integrated over the delay interval, and its terminal state sets the command; matching the transport-grid step to ∆t permits a discrete characteristic shift without interpolation.The predictor-compensated command is U(t) = −µB∗p(t, D).Source: §4.1 · p. 20
4.2. Tracking and Predictor Performance.
- At T = 40, Eol(T) = 1.1162 × 101, Edel(T) = 4.3974 × 102, and Epred(T) = 2.1111 × 10−2; uncompensated delayed feedback amplifies error, while predictor compensation restores sustained decay.The final error is approximately 2.1% of its initial value.Source: Figure 1 · p. 20, §4.2 · p. 20
- Error attenuation is not pointwise monotone, but nonlinear mode coupling and fourth-order dissipation attenuate residual discrepancies as the controlled pattern aligns with the moving reference.The feedback directly damps the first two nonzero Fourier wave numbers; final-window profiles are nearly indistinguishable at the plotted scale while continuing to evolve.Source: §4.2 · pp. 21–22
- The terminal predictor residual reaches 7.8904 × 10−10, while the transformed actuator state falls to 8.4001 × 10−14 after one delay horizon.The residual compares separately integrated terms; the actuator-state result is consistent with finite-time evacuation of the initial delay-channel content.Source: §4.2 · pp. 21–22
- Under the considered refinement, Epred(T) changes from 2.111135 × 10−2 to 2.113984 × 10−2 and the fitted rate from 0.119830 to 0.119831; the conclusion assumes a known constant delay and full-state access.The relative variation in Epred(T) is approximately 1.35×10−3; delay mismatch, time-varying or unknown delays, and output-based feedback remain for future investigation.Source: §4.2 · pp. 22–23, §5 · p. 23