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Exact and Approximate Convex Reformulation of Linear Stochastic Optimal Control with Chance Constraints

Tanmay Dokania, Yashwanth Kumar Nakka

Abstract

In this paper, we present an equivalent convex optimization formulation for discrete-time stochastic linear systems subject to linear chance constraints, alongside a tight convex relaxation for quadratic chance constraints. By lifting the state vector to encode moment information explicitly, the formulation captures linear chance constraints on states and controls across multiple time steps exactly, without conservatism, yielding strict improvements in both feasibility and optimality. For quadratic chance constraints, we derive convex approximations that are provably less conservative than existing methods. We validate the framework on minimum-snap trajectory generation for a quadrotor, demonstrating that the proposed approach remains feasible at noise levels an order of magnitude beyond the operating range of prior formulations.

Exact and Approximate Convex Reformulation of Linear Stochastic Optimal Control with Chance Constraints

Abstract

In this paper, we present an equivalent convex optimization formulation for discrete-time stochastic linear systems subject to linear chance constraints, alongside a tight convex relaxation for quadratic chance constraints. By lifting the state vector to encode moment information explicitly, the formulation captures linear chance constraints on states and controls across multiple time steps exactly, without conservatism, yielding strict improvements in both feasibility and optimality. For quadratic chance constraints, we derive convex approximations that are provably less conservative than existing methods. We validate the framework on minimum-snap trajectory generation for a quadrotor, demonstrating that the proposed approach remains feasible at noise levels an order of magnitude beyond the operating range of prior formulations.
Paper Structure (16 sections, 5 theorems, 34 equations, 1 figure, 2 tables)

This paper contains 16 sections, 5 theorems, 34 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Convex Reformulation
  4. Stochastic basis
  5. State representation
  6. Control representation
  7. Linear Chance Constraint
  8. Terminal Covariance Constraint
  9. Conservative Approximation of Quadratic Chance Constraint to Covariance Constraint
  10. Conservative Approximation of Quadratic Chance Constraint to Quadratic Constraints
  11. Main Theorem: Deterministic Convex Problem
  12. Numerical Simulations
  13. Circle Arena
  14. Funnel Corridor
  15. Terminal Quadratic Chance Constraint
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\mathbf{y}$ be a scalar R.V. If it is representable in the basis $\mathbf e^{(k)}$, i.e., $\mathbf y = y_0 + y \mathbf e^{(k)}$, where $y_0 \in \mathbb{R}, y \in \mathbb{R}^{1 \times n_e(k)}$ then, a chance constraint $\mathbb{P}\!\left(\mathbf y \le b\right)\ge 1-\epsilon$ can be exactly repre

Figures (1)

  • Figure 1: Minimum Snap Path Planning for the linearized lateral dynamics of a quadrotor under the two different setups under stochastic perturbation forces ($\sigma_1=\sigma_2 = 0.05$). The proposed approach leads to a cost reduction as compared to lew_chance-constrained_2020, as it can shape the covariance.

Theorems & Definitions (14)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 2
  • Remark 3
  • Theorem 1: Exact Convex Reformulation
  • ...and 4 more