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Feedforward Density-Driven Optimal Control for Tracking Time-Varying Distributions with Guaranteed Stability

Julian Martinez, Kooktae Lee

Abstract

This paper addresses the spatiotemporal mismatch in multi-agent distribution tracking within time-varying environments. While recent advancements in Density-Driven Optimal Control (D$^2$OC) have enabled finite-time distribution matching using Optimal Transport theory, existing formulations primarily assume a stationary reference density. In dynamic scenarios, such as tracking evolving wildfires or moving plumes, this assumption leads to a structural tracking lag where the agent configuration inevitably falls behind the shifting reference flow. To resolve this, we propose a feedforward-augmented D$^2$OC framework that explicitly incorporates the reference velocity field, modeled via the continuity equation, into the control law. We provide a formal mathematical quantification of the induced tracking lag and analytically prove that the proposed predictive mechanism effectively reduces the cumulative tracking error. Furthermore, an analytical ultimate bound for the local Wasserstein distance is established under discretization errors and transport jitter. Theoretical analysis and numerical results demonstrate that our approach significantly mitigates tracking latency, ensuring robust and high-fidelity tracking performance in rapidly changing environments.

Feedforward Density-Driven Optimal Control for Tracking Time-Varying Distributions with Guaranteed Stability

Abstract

This paper addresses the spatiotemporal mismatch in multi-agent distribution tracking within time-varying environments. While recent advancements in Density-Driven Optimal Control (DOC) have enabled finite-time distribution matching using Optimal Transport theory, existing formulations primarily assume a stationary reference density. In dynamic scenarios, such as tracking evolving wildfires or moving plumes, this assumption leads to a structural tracking lag where the agent configuration inevitably falls behind the shifting reference flow. To resolve this, we propose a feedforward-augmented DOC framework that explicitly incorporates the reference velocity field, modeled via the continuity equation, into the control law. We provide a formal mathematical quantification of the induced tracking lag and analytically prove that the proposed predictive mechanism effectively reduces the cumulative tracking error. Furthermore, an analytical ultimate bound for the local Wasserstein distance is established under discretization errors and transport jitter. Theoretical analysis and numerical results demonstrate that our approach significantly mitigates tracking latency, ensuring robust and high-fidelity tracking performance in rapidly changing environments.

Paper Structure

This paper contains 22 sections, 5 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal Transport and Wasserstein Metric
  4. Standard D$^2$OC Protocol and its Limitations
  5. Time-Varying Reference Samples for D$^2$OC
  6. Discrete-Time Continuity Equation Approximation
  7. Impact on D$^2$OC Barycenter Computation
  8. Formulation of the D$^2$OC Cost Function via Quadratic Programming
  9. Predictive Compensation for Density-Flow Induced Lag
  10. Explicit Derivation of the Predicted Barycenter Drift
  11. Feedforward-Augmented QP Input
  12. Lag Reduction Analysis
  13. Boundedness and Robustness Analysis
  14. Stability Analysis under Total Perturbation
  15. Stability of the Wasserstein Distance
  16. ...and 7 more sections

Key Result

Proposition 1

lee2025connectivity Consider the index set $\mathcal{S}_i(k+h)$ of local samples and the corresponding transport weights $\pi_j(k+h) \ge 0$ computed over the prediction window $h \in \{r, \dots, r+H-1\}$. Let the time-varying local barycenter be defined as: Furthermore, define the augmented output vector $Y_i$, the barycenter trajectory $\bar{Q}_i$, and the weighting matrix $\boldsymbol{\Omega}_i

Figures (2)

  • Figure F1: Simulation results for multi-agent plume tracking using nominal and feedforward (FF) D$^2$OC controllers: Snapshot comparisons of agent trajectories at selected step indices ($k=0, 100, ...400$), showing the evolution of agent positions relative to the drifting plume distribution.
  • Figure F2: Simulation results for multi-agent plume tracking using nominal and feedforward (FF) D$^2$OC controllers: (a and b) Horizon error-norm ratio under different control penalties $R$. (c) Evolution of the local Wasserstein-based error metric.

Theorems & Definitions (14)

  • Remark 1: Discretization and Total Perturbation
  • Definition 1: Output Relative Degree
  • Proposition 1
  • Theorem 1: Uniqueness of Optimal Input lee2025connectivity
  • Remark 2: Structural Lag in Reactive Predictive Control
  • Remark 3: Frozen Transport Plan Assumption
  • Lemma 1: QP Solution with Feedforward Compensation
  • proof
  • Theorem 2: Lag Reduction via Feedforward
  • proof
  • ...and 4 more