Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Causal Tracking of Distributions in Wasserstein Space: A Model Predictive Control Scheme

Max Emerick, Jared Jonas, Bassam Bamieh

TL;DR

This work addresses tracking a moving reference distribution $D_t$ with a swarm represented by a resource distribution $R_t$ in the Wasserstein space, minimizing $\int_0^T \mathcal{W}_2^2(R_t,D_t) + \alpha^2 \|V_t\|^2_{L^2(R_t)} dt$ under the advection dynamics $\partial_t R_t = - \nabla \cdot (V_t R_t)$. It introduces a model-predictive control scheme that leverages noncausal optimal transport-based conditions by using a predictive model for $D_t$ and reformulates in a Lagrangian particle framework, enabling real-time, causal tracking even when future demand is unknown. Computation relies on a particle discretization and discrete OT (via Sinkhorn) to approximate OT maps, with a time-stepping update $Q_{k+1} = (1-e^{-\

Abstract

We consider a problem of optimal swarm tracking which can be formulated as a tracking problem for distributions in the Wasserstein space. Optimal solutions to this problem are non-causal and require knowing the time-trajectory of the reference distribution in advance. We propose a scheme where these non-causal solutions can be used together with a predictive model for the reference to achieve causal tracking of a priori-unknown references. We develop a model-predictive control scheme built around the simple case where the reference is constant-in-time. A computational algorithm based on particle methods and discrete optimal mass transport is presented, and numerical simulations are provided for various classes of reference signals. The results demonstrate that the proposed control algorithm achieves reasonable performance even when using simple predictive models.

Causal Tracking of Distributions in Wasserstein Space: A Model Predictive Control Scheme

TL;DR

This work addresses tracking a moving reference distribution with a swarm represented by a resource distribution in the Wasserstein space, minimizing under the advection dynamics . It introduces a model-predictive control scheme that leverages noncausal optimal transport-based conditions by using a predictive model for and reformulates in a Lagrangian particle framework, enabling real-time, causal tracking even when future demand is unknown. Computation relies on a particle discretization and discrete OT (via Sinkhorn) to approximate OT maps, with a time-stepping update $Q_{k+1} = (1-e^{-\

Abstract

We consider a problem of optimal swarm tracking which can be formulated as a tracking problem for distributions in the Wasserstein space. Optimal solutions to this problem are non-causal and require knowing the time-trajectory of the reference distribution in advance. We propose a scheme where these non-causal solutions can be used together with a predictive model for the reference to achieve causal tracking of a priori-unknown references. We develop a model-predictive control scheme built around the simple case where the reference is constant-in-time. A computational algorithm based on particle methods and discrete optimal mass transport is presented, and numerical simulations are provided for various classes of reference signals. The results demonstrate that the proposed control algorithm achieves reasonable performance even when using simple predictive models.
Paper Structure (14 sections, 1 theorem, 17 equations, 5 figures, 2 algorithms)

This paper contains 14 sections, 1 theorem, 17 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Problem Formulation
  3. Notation and Preliminaries
  4. Problem Setting and Motivation
  5. Structure of Noncausal Solutions
  6. Model-Predictive Control Scheme
  7. Description of Controller
  8. From Eulerian to Lagrangian Coordinates
  9. Computational Algorithm
  10. Simulations
  11. Static Demand
  12. Fading Demand
  13. Constant-Velocity Demand
  14. Conclusion

Key Result

Lemma 2

Under the dynamics Q_dynamics, with control transformed_control, $Q$ evolves such that $\tilde{M}_{R_{t+\tau} \to D_t}$ remains constant. In particular,

Figures (5)

  • Figure 1: Pictorial representation of Problem \ref{['orig_prob']}. The resource $R$ aims to track the demand $D$ via transport by the velocity field $V$. The total cost is minimized over the maneuver.
  • Figure 2: Block diagram depiction of proposed control scheme. The controller is composed of two components: a predictive model which forecasts the demand trajectory $\hat{D}$ from its current state $D_t$, and a noncausal controller which determines the control input $V_t$ by solving the necessary conditions for optimality \ref{['necc_cond_3']} - \ref{['necc_cond_4']} supposing $D = \hat{D}$.
  • Figure 3: Timeseries for the static demand case. Here, 20 resource particles in white track the demand, which is given by a Gaussian mixture. The colored image represents the density of the demand distribution at the given time, where blue is less dense and yellow is more dense. The arrows attached to each particle show their respective assigned positions as determined by the optimal transport map.
  • Figure 4: Timeseries for a time-varying demand. At $t=0$ and $t=3$, the demand is a Gaussian placed to the left or right side of the image respectively. When $0<t<3$, one fades out while the other fades in.
  • Figure 5: Timeseries for a time-varying demand distribution composed of three Gaussians. At first, all Gaussians start in the center. At $t=0.5$, each begins to move away from the center at a constant velocity.

Theorems & Definitions (1)

  • Lemma 2