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Data/moment-driven approaches for fast predictive control of collective dynamics

Giacomo Albi, Sara Bicego, Michael Herty, Yuyang Huang, Dante Kalise, Chiara Segala

TL;DR

The paper tackles the challenge of fast predictive control for large-scale collective dynamics by analyzing two complementary strategies to avoid expensive online optimization. It first develops offline supervised-learning MPC to approximate optimal feedback laws from trajectories generated via PMP or SDRE, enabling real-time control with surrogates. It then introduces moment-driven predictive control (MdPC), which linearizes around macroscopic ensemble moments and applies Riccati-based control with adaptive re-linearization driven by moment decay estimates. Through numerical tests on Cucker-Smale-type models, the work demonstrates that learned controllers can achieve near-optimal behavior with substantial speedups, while MdPC provides fast, robust suboptimal control informed by mean-field dynamics.

Abstract

Feedback control synthesis for large-scale particle systems is reviewed in the framework of model predictive control (MPC). The high-dimensional character of collective dynamics hampers the performance of traditional MPC algorithms based on fast online dynamic optimization at every time step. Two alternatives to MPC are proposed. First, the use of supervised learning techniques for the offline approximation of optimal feedback laws is discussed. Then, a procedure based on sequential linearization of the dynamics based on macroscopic quantities of the particle ensemble is reviewed. Both approaches circumvent the online solution of optimal control problems enabling fast, real-time, feedback synthesis for large-scale particle systems. Numerical experiments assess the performance of the proposed algorithms.

Data/moment-driven approaches for fast predictive control of collective dynamics

TL;DR

The paper tackles the challenge of fast predictive control for large-scale collective dynamics by analyzing two complementary strategies to avoid expensive online optimization. It first develops offline supervised-learning MPC to approximate optimal feedback laws from trajectories generated via PMP or SDRE, enabling real-time control with surrogates. It then introduces moment-driven predictive control (MdPC), which linearizes around macroscopic ensemble moments and applies Riccati-based control with adaptive re-linearization driven by moment decay estimates. Through numerical tests on Cucker-Smale-type models, the work demonstrates that learned controllers can achieve near-optimal behavior with substantial speedups, while MdPC provides fast, robust suboptimal control informed by mean-field dynamics.

Abstract

Feedback control synthesis for large-scale particle systems is reviewed in the framework of model predictive control (MPC). The high-dimensional character of collective dynamics hampers the performance of traditional MPC algorithms based on fast online dynamic optimization at every time step. Two alternatives to MPC are proposed. First, the use of supervised learning techniques for the offline approximation of optimal feedback laws is discussed. Then, a procedure based on sequential linearization of the dynamics based on macroscopic quantities of the particle ensemble is reviewed. Both approaches circumvent the online solution of optimal control problems enabling fast, real-time, feedback synthesis for large-scale particle systems. Numerical experiments assess the performance of the proposed algorithms.
Paper Structure (19 sections, 2 theorems, 47 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 19 sections, 2 theorems, 47 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Optimal control for collective dynamics
  3. Dynamic programming and the HJB PDE
  4. The link between PMP and HJB
  5. State Dependent Riccati Equation
  6. Frozen Riccati Approach
  7. Semilinearization of the Cucker-Smale consensus problem
  8. Supervised learning MPC approach
  9. Gradient-augmented regression and data generation
  10. Feedforward Neural Networks
  11. MPC with the learned feedback control
  12. Moment-driven predictive control
  13. Riccati-based open-loop control
  14. Moments estimates for error analysis
  15. MdPC approach
  16. ...and 4 more sections

Key Result

Proposition 1

For the linearized dynamics, the solution of the Riccati equation eq:Riccati reduces to the solution of with terminal conditions $k_d(T)=k_o(T)=0$, and where $\alpha(N) = \frac{N-1}{N}$.

Figures (4)

  • Figure 1: Test 1: Comparison of learned and target trajectories for a random initial condition in $[0,1]^{2dN}$. All the models improve consensus with respect to the uncontrolled dynamics. In terms of learned models, the $\mathbf{u}_V^{PMP}$ model can be considered the optimal one, with an overall cost (at final time horizon $T$) $\mathcal{J}\approx1.44$.
  • Figure 2: Test 1: System configuration at time $t=0,1,10$ seconds respectively, under the control action provided by the feedback map $\mathbf{u}_V^{PMP}$.
  • Figure 3: Test 2: On the left, comparison of MdPC-controlled and uncontrolled trajectories in terms of particles' target distance. Variance decay together with updates and variance bounds in the middle ($\delta_{tol}=1$) and on the right ($\delta_{tol}=0.1$).
  • Figure 4: Test 2: System configuration at time $t=0,1,10$ seconds respectively, under the control action provided by $\mathbf{u}^{MdPC}$ with tolerance $\delta_{tol} = 0.1$.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2