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Constrained Multi-Modal Density Control of Linear Systems via Covariance Steering Theory

Isin M Balci, Efstathios Bakolas

Abstract

In this paper, we investigate finite-horizon optimal density steering problems for discrete-time stochastic linear dynamical systems whose state probability densities can be represented as Gaussian Mixture Models (GMMs). Our goal is to compute optimal controllers that can ensure that the terminal state distribution will match the desired distribution exactly (hard-constrained version) or closely (soft-constrained version) where in the latter case we employ a Wasserstein like metric that can measure the distance between different GMMs. Our approach relies on a class of randomized control policies which allow us to reformulate the proposed density steering problems as finite-dimensional optimization problems, and in particular, linear and bilinear programs. Additionally, we explore more general density steering problems based on the approximation of general distributions by GMMs and characterize bounds for the error between the terminal distribution under our policy and the approximated GMM terminal state distribution. Finally, we demonstrate the effectiveness of our approach through non-trivial numerical experiments.

Constrained Multi-Modal Density Control of Linear Systems via Covariance Steering Theory

Abstract

In this paper, we investigate finite-horizon optimal density steering problems for discrete-time stochastic linear dynamical systems whose state probability densities can be represented as Gaussian Mixture Models (GMMs). Our goal is to compute optimal controllers that can ensure that the terminal state distribution will match the desired distribution exactly (hard-constrained version) or closely (soft-constrained version) where in the latter case we employ a Wasserstein like metric that can measure the distance between different GMMs. Our approach relies on a class of randomized control policies which allow us to reformulate the proposed density steering problems as finite-dimensional optimization problems, and in particular, linear and bilinear programs. Additionally, we explore more general density steering problems based on the approximation of general distributions by GMMs and characterize bounds for the error between the terminal distribution under our policy and the approximated GMM terminal state distribution. Finally, we demonstrate the effectiveness of our approach through non-trivial numerical experiments.
Paper Structure (17 sections, 13 theorems, 75 equations, 13 figures, 1 table)

This paper contains 17 sections, 13 theorems, 75 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Optimal Covariance Steering for Linear Systems
  4. GMM Steering Policies
  5. Finite-Dimensional Optimization Problem Formulations of GMM Steering Problems
  6. Reduction of Unconstrained Problem to a Linear Program
  7. Reduction of the Soft Constrained Problem to an NLP
  8. Reduction of Total Cost Constrained Problem to an NLP
  9. Reduction of Step Cost Constrained Problem to NLP
  10. Block Coordinate Descent Solution Procedure
  11. Error bounds on GMM Approximations
  12. Numerical Simulations
  13. 2D Single Integrator - Problem \ref{['prob:hard-GMM-density-steering']}
  14. 2D Single Integrator - Problems \ref{['prob:soft-GMM-density-steering']} and \ref{['prob:total-cost-constrained-GMM-density-steering']}
  15. Control of Drone Swarm - Problem 4
  16. ...and 2 more sections

Key Result

Proposition 1

Under Assumption assumption:linear-controllable, the optimal control sequence $\mathbf{\Bar{U}}^{\star}$ that solves problem eq:Mean-Steering-Problem is given by: where $M = \mathbf{R} + \mathbf{H_u Q H_u}^{\mathrm{T}}$, $Y = \mathbf{\Gamma} \mu_0 - \mathbf{X'}$. Furthermore, the optimal value of the objective function is given as:

Figures (13)

  • Figure 1: Graphical illustration of the multi-modal density control problem. In this problem, a feedback control policy is sought to steer the uncertain initial state $x_0$ drawn from a (blue) GMM distribution to a terminal state $x_N$ drawn from another (red) GMM distribution in finite time.
  • Figure 2:
  • Figure 3: Actual densities.
  • Figure 4: Approximated densities.
  • Figure 5: Evolution of state density (PDF).
  • ...and 8 more figures

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Remark 2
  • Proposition 3
  • ...and 13 more