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Distributionally Robust Density Control with Wasserstein Ambiguity Sets

Joshua Pilipovsky, Panagiotis Tsiotras

TL;DR

This paper model the distributional uncertainty of the noise process in terms of Wasserstein ambiguity sets, which have been shown to be an effective means of capturing and propagating uncertainty through stochastic LTI systems, and proposes a distributionally-robust framework.

Abstract

Precise control under uncertainty requires a good understanding and characterization of the noise affecting the system. This paper studies the problem of steering state distributions of dynamical systems subject to partially known uncertainties. We model the distributional uncertainty of the noise process in terms of Wasserstein ambiguity sets, which, based on recent results, have been shown to be an effective means of capturing and propagating uncertainty through stochastic LTI systems. To this end, we propagate the distributional uncertainty of the state through the dynamical system, and, using an affine feedback control law, we steer the ambiguity set of the state to a prescribed, terminal ambiguity set. We also enforce distributionally robust CVaR constraints for the transient motion of the state so as to reside within a prescribed constraint space. The resulting optimization problem is formulated as a semi-definite program, which can be solved efficiently using standard off-the-shelf solvers. We illustrate the proposed distributionally-robust framework on a quadrotor landing problem subject to wind turbulence.

Distributionally Robust Density Control with Wasserstein Ambiguity Sets

TL;DR

This paper model the distributional uncertainty of the noise process in terms of Wasserstein ambiguity sets, which have been shown to be an effective means of capturing and propagating uncertainty through stochastic LTI systems, and proposes a distributionally-robust framework.

Abstract

Precise control under uncertainty requires a good understanding and characterization of the noise affecting the system. This paper studies the problem of steering state distributions of dynamical systems subject to partially known uncertainties. We model the distributional uncertainty of the noise process in terms of Wasserstein ambiguity sets, which, based on recent results, have been shown to be an effective means of capturing and propagating uncertainty through stochastic LTI systems. To this end, we propagate the distributional uncertainty of the state through the dynamical system, and, using an affine feedback control law, we steer the ambiguity set of the state to a prescribed, terminal ambiguity set. We also enforce distributionally robust CVaR constraints for the transient motion of the state so as to reside within a prescribed constraint space. The resulting optimization problem is formulated as a semi-definite program, which can be solved efficiently using standard off-the-shelf solvers. We illustrate the proposed distributionally-robust framework on a quadrotor landing problem subject to wind turbulence.
Paper Structure (12 sections, 6 theorems, 68 equations, 5 figures, 2 tables)

This paper contains 12 sections, 6 theorems, 68 equations, 5 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. NOTATION
  3. PROBLEM STATEMENT
  4. PROBLEM REFORMULATION
  5. DR-CVaR Constraints
  6. DR Objective Reformulation
  7. Terminal Constraints
  8. NUMERICAL EXAMPLES
  9. Double Integrator Path Planning
  10. Quadrotor Landing with Wind Turbulence
  11. CONCLUSION
  12. ACKNOWLEDGMENTS

Key Result

Theorem 1

Let $\mathbb{P}\in\mathcal{P}(\mathbb{R}^{d})$, and consider the linear transformation defined by the matrix $A\in\mathbb{R}^{m\times d}$. Moreover, let $c:\mathbb{R}^{d}\rightarrow\mathbb{R}_{\geq 0}$ be orthomonotoneThat is, $c(x_1 + x_2) \geq c(x_1)$ for all $x_1,x_2\in\mathbb{R}^{d}$ satisfying Moreover, if the matrix $A$ is full row-rank, then with $A^\dagger = A^\intercal (AA^\intercal)^{-

Figures (5)

  • Figure 1: Optimal trajectories for (left) DR-DS solution with $\varepsilon = 15$, and (right) baseline CS solution, subject to nominal disturbance $\mathbb{P}_w$.
  • Figure 2: Optimal trajectories for (left) DR-DS solution with $\varepsilon = 15$, and (right) baseline CS solution, subject to maximal disturbance $\mathbb{P}_w$ in disturbance ambiguity set $\mathbb{B}_{\varepsilon}^{\|\cdot\|}(\hat{\mathbb{P}}_{w})$.
  • Figure 3: Optimal trajectories for (left) DR-DS solution with $\varepsilon = 15$, and (right) baseline CS solution, subject to non-Gaussian t-distribution disturbance.
  • Figure 4: Terminal position distribution and propagated Monte Carlo samples for DR-DS (black) and baseline CS (blue) under nominal turbulence model.
  • Figure 5: Monte Carlo trajectories and terminal splashpoints for (a) DR-DS and (b) CS solutions with severe wind turbulence.

Theorems & Definitions (18)

  • Definition 1: DRO_Kuhn_risk
  • Definition 2
  • Remark 1
  • Remark 2
  • Definition 3
  • Theorem 1
  • Remark 3
  • Definition 4
  • Theorem 2: DRO_Kuhn_risk
  • Theorem 3
  • ...and 8 more