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Optimal Covariance Steering of Linear Stochastic Systems with Hybrid Transitions

Hongzhe Yu, Diana Frias Franco, Aaron M. Johnson, Yongxin Chen

Abstract

This work addresses the problem of optimally steering the state covariance of a linear stochastic system from an initial to a target, subject to hybrid transitions. The nonlinear and discontinuous jump dynamics complicate the control design for hybrid systems. Under uncertainties, stochastic jump timing and state variations further intensify this challenge. This work aims to regulate the hybrid system's state trajectory to stay close to a nominal deterministic one, despite uncertainties and noises. We address this problem by directly controlling state covariances around a mean trajectory, and this problem is termed the Hybrid Covariance Steering (H-CS) problem. The jump dynamics are approximated to the first order by leveraging the Saltation Matrix. When the jump dynamics are nonsingular, we derive an analytical closed-form solution to the H-CS problem. For general jump dynamics with possible singularity and changes in the state dimensions, we reformulate the problem into a convex optimization over path distributions by leveraging Schrodinger's Bridge duality to the smooth covariance control problem. The covariance propagation at hybrid events is enforced as equality constraints to handle singularity issues. The proposed convex framework scales linearly with the number of jump events, ensuring efficient, optimal solutions. This work thus provides a computationally efficient solution to the general H-CS problem. Numerical experiments are conducted to validate the proposed method.

Optimal Covariance Steering of Linear Stochastic Systems with Hybrid Transitions

Abstract

This work addresses the problem of optimally steering the state covariance of a linear stochastic system from an initial to a target, subject to hybrid transitions. The nonlinear and discontinuous jump dynamics complicate the control design for hybrid systems. Under uncertainties, stochastic jump timing and state variations further intensify this challenge. This work aims to regulate the hybrid system's state trajectory to stay close to a nominal deterministic one, despite uncertainties and noises. We address this problem by directly controlling state covariances around a mean trajectory, and this problem is termed the Hybrid Covariance Steering (H-CS) problem. The jump dynamics are approximated to the first order by leveraging the Saltation Matrix. When the jump dynamics are nonsingular, we derive an analytical closed-form solution to the H-CS problem. For general jump dynamics with possible singularity and changes in the state dimensions, we reformulate the problem into a convex optimization over path distributions by leveraging Schrodinger's Bridge duality to the smooth covariance control problem. The covariance propagation at hybrid events is enforced as equality constraints to handle singularity issues. The proposed convex framework scales linearly with the number of jump events, ensuring efficient, optimal solutions. This work thus provides a computationally efficient solution to the general H-CS problem. Numerical experiments are conducted to validate the proposed method.

Paper Structure

This paper contains 20 sections, 9 theorems, 103 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smooth Linear Covariance Steering
  4. Schrödinger Bridge Problem
  5. Problem Formulation
  6. Hybrid Systems and Saltation Approximation
  7. Covariance Steering with Hybrid Transitions
  8. Results for Invertible Jump Dynamics
  9. Results for Rectangular Jump Dynamics
  10. The Optimal Distribution of Schrödinger Bridge
  11. A Finite Dimensional Convex Program for H-CS
  12. An SDP formulation
  13. Numerical Example
  14. Bouncing Ball with Elastic Impact
  15. Linearized Spring-Loaded Inverted Pendulum (SLIP): Jumping to a higher dimensional state space.
  16. ...and 5 more sections

Key Result

Lemma 1

(CheGeoPav17a, Lemma 3 ). The entries of the state transition kernel $\Phi(t,s)$ satisfy and the entries $\Phi_{12}(t,s)$ and $\Phi_{11}(t,s)$ are invertible for all $t$, and $(\Phi_{12}(t,t_0)^{-1}\Phi_{11}(t,t_0))^{-1}$ is monotonically decreasing in the positive definite sense with left limit $0$ as $t \searrow t_0$.

Figures (3)

  • Figure 1: Covariance steering for a bouncing ball dynamics with elastic impacts. The H-CS controller guarantees the terminal covariance constraint.
  • Figure 2: Controlled covariance tube and sampled trajectories for the bouncing ball dynamics.
  • Figure 3: Deterministic nominal trajectory under H-iLQR controller and stochastic trajectories under the H-CS controller for the SLIP model. The transparent legs with springs are the nominal trajectory, and the dots in cyan color are the SLIP body's position trajectories of the samples starting from the initial distribution, marked by red diamond shapes. We draw the $3-\sigma$ boundary of the terminal-time target covariance.

Theorems & Definitions (11)

  • Lemma 1
  • Proposition 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1: Main results for invertible jump dynamics
  • Lemma 5
  • Lemma 6
  • Theorem 2: Main results for general jump dynamics
  • ...and 1 more