Source-linked AI summary
Optimal Covariance Control for Stochastic Systems Under Chance Constraints
Kazuhide Okamoto, Maxim Goldshtein, Panagiotis Tsiotras
TL;DR
The paper addresses optimal covariance steering for stochastic discrete-time linear time-varying systems subject to chance constraints. It converts the chance-constrained problem into a semidefinite-programming formulation, with numerical examples demonstrating covariance reduction and constraint avoidance.
Problem
The paper addresses covariance steering between Gaussian state distributions when probabilistic state constraints must also be satisfied.
Method
The approach reformulates chance constraints as deterministic constraints on the mean and covariance and incorporates them into an LMI formulation.
Results
The proposed approach was verified numerically, including covariance reduction and changing error-ellipse shape to avoid constraints while maintaining terminal covariance bounds.
Takeaways & Limitations
Chance-constrained covariance steering can be formulated as a convex programming problem for solution with generic LMI solvers.
Takeaways & Limitations
The chance-constraint conversion must address a bilinear inequality because the covariance is not assumed known in covariance steering.
Abstract
from arXiv · showhide
This work addresses the optimal covariance control problem for stochastic discrete-time linear time-varying systems subject to chance constraints. Covariance steering is a stochastic control problem to steer the system state Gaussian distribution to another Gaussian distribution while minimizing a cost function. To the best of our knowledge, covariance steering problems have never been discussed with probabilistic chance constraints although it is a natural extension. In this work, first we show that, unlike the case with no chance constraints, the covariance steering with chance constraints problem cannot decouple the mean and covariance steering sub-problems. Then we propose an approach to solve the covariance steering with chance constraints problem by converting it to a semidefinite programming problem. The proposed algorithm is verified using two simple numerical simulations.
1 Introduction
The paper extends finite-horizon covariance steering for stochastic linear systems by imposing probabilistic state constraints. It reformulates the constrained problem as an LMI-based optimization problem and positions it as a first treatment of covariance steering with chance constraints.
- Motivation: Chance constraints require the probability of state-constraint violation to remain below a prescribed threshold.This provides robustness under stochastic uncertainty while steering the state distribution.
- Motivation: The approach is motivated by applications where system states are better represented by distributions, including swarm control, cooling, and stochastic model predictive control.The paper identifies stochastic MPC as a readily applicable framework.
- Main contribution: The work adds chance state constraints to stochastic covariance steering for discrete-time linear time-varying systems.The system has Gaussian initial and target distributions, and the objective penalizes expected state and control costs.
- Main contribution: The covariance-control problem is reformulated using deterministic mean-and-covariance constraints embedded in an LMI formulation.The resulting formulation can be solved with generic LMI solvers.
- Novelty: The authors state that this is the first work to solve covariance steering with chance constraints.They describe the extension as a natural addition to covariance steering and verify it numerically.
2 Problem Statement
The paper formulates finite-horizon optimal control for a stochastic linear time-varying system with Gaussian boundary distributions and probabilistic state constraints. It augments the dynamics into stacked state, input, and noise vectors to express the optimization problem compactly.
- 2.1 Problem Formulation: The model assumes a discrete-time stochastic linear time-varying system with additive zero-mean white Gaussian noise and a fully controllable state transition.Controllability ensures that any terminal state is reachable from any initial state when the noise is zero.
- 2.1 Problem Formulation: At every time step, the probability of violating the convex state constraint set must not exceed P_fail.The constraint is Pr(x_k ∉ χ) ≤ P_fail for k = 0, ..., N.
- 2.2 Preliminaries: The dynamics are rewritten with transition matrices and concatenated state, input, and noise vectors.This produces an equivalent stacked representation X = A x_0 + B U + D W.
- 2.1 Problem Formulation: The resulting optimization minimizes the stacked quadratic cost subject to initial, terminal, and chance constraints.The paper states this formulation as Problem 1 and later converts it to a convex program.
- Transition to Section 3: The no-chance-constraint case is analyzed first because it clarifies how the proposed formulation differs from prior covariance-steering work.In that case, mean and covariance steering can be separately solved even with the general ℓ2-norm objective.
3 No Chance Constraint Case
Without chance constraints, the mean and covariance steering dynamics and costs decouple, enabling two independent optimization problems. The paper derives an analytic mean-steering solution and relates it to prior minimum-effort control.
- 3.1 Separation of Mean and Covariance Problems: The mean and covariance steering subproblems can be solved independently even under the paper’s general ℓ2-norm objective.The separation follows because the mean depends only on the mean-control component, while covariance depends on the deviation-control component and noise.
- 3.1 Separation of Mean and Covariance Problems: The mean trajectory evolves only through the mean-control component, whereas covariance evolves through the deviation-control component and process noise.This structural independence is the mechanism behind the optimization decoupling.
- 3.1 Separation of Mean and Covariance Problems: The original control sequence decomposes as U* = Ū* + Ũ*, with separate optimization problems for mean and covariance steering.The corresponding costs are expressed separately for the mean and covariance components.
- 3.2 Optimal Mean Steering: The optimal mean-steering control is obtained analytically from the terminal mean constraint and a quadratic Lagrangian formulation.The derivation uses the controllability of the stacked input matrix and invertibility of the quadratic control matrix.
- 3.2 Optimal Mean Steering: The minimum-effort mean-steering controller from prior work is a special case of the paper’s more general optimal controller.The specialization uses Q̄ = 0 and R̄ = I.
- Related Work: Prior covariance-steering approaches often used minimum-effort costs, while more general costs required convex relaxation or related soft terminal covariance constraints.These works motivate the paper’s treatment of a general quadratic objective.
4 Chance Constrained Case
The proposed chance-constrained covariance-steering approach parameterizes causal control gains, converts probabilistic constraints into deterministic inequalities, and formulates a convex optimization problem.
- 4.1 Proposed Approach: The control input is parameterized as U = LX, where L maps the augmented state sequence and is structured to preserve causal dependence.
- 4.1 Proposed Approach: A lower block triangular structure for L, and subsequently K, ensures that each control input depends only on current and past states, making U non-anticipative.
- 4.1 Proposed Approach: The resulting objective function is a quadratic expression in the decision variable K.
- 4.2 Conversion of Chance Constraint to Deterministic Inequality Constraint: The chance constraints are converted into deterministic inequalities involving the mean, covariance, Gaussian quantiles, and constraint geometry.
- 4.2 Conversion of Chance Constraint to Deterministic Inequality Constraint: Because the resulting chance-constraint condition couples mean and covariance terms, mean and covariance steering cannot be decoupled.
- 4.2 Conversion of Chance Constraint to Deterministic Inequality Constraint: The terminal covariance equality is relaxed to an upper-bound inequality, allowing the terminal covariance to be smaller than the prescribed target covariance.
- 4.2 Conversion of Chance Constraint to Deterministic Inequality Constraint: The reformulated problem minimizes the objective over K subject to terminal constraints and pre-specified failure-probability thresholds, and is convex programming efficiently solvable with a nonlinear solver.
5 Numerical Simulations
The simulations examine mean steering, covariance steering, and chance-constrained covariance steering in a double-integrator system and a vehicle path-planning problem. Covariance steering reduces uncertainty and enables chance-constrained trajectories to remain feasible while satisfying terminal covariance requirements.
- 5.1 Double Integrator: 24.05 was the cost for covariance steering without chance constraints, slightly above the 23.90 mean-steering cost.The higher cost reflects steering both the covariance and the mean.
- 5.1 Double Integrator: Without covariance steering, the double-integrator chance constraint becomes infeasible because the state covariance grows large at the final step.The mean-steering case is illustrated with expected trajectories, predicted 3σ bounds, and sampled trajectories.
- 5.1 Double Integrator: 24.16 was the cost for chance-constrained covariance steering in the double-integrator simulation with failure threshold p_j,fail = 0.001.Some confidence regions touched the constraints.
- 5.2 Vehicle Path Planning: In vehicle path planning with horizon N = 20, mean-only control cannot guarantee chance-constraint satisfaction as covariance grows.The vehicle model uses position, velocity, and acceleration states and inputs, with a cone-shaped state constraint.
- 5.2 Vehicle Path Planning: Covariance steering reduces future-trajectory uncertainty in the vehicle problem even without chance constraints.This case is illustrated in Figure 6.
- 5.2 Vehicle Path Planning: The chance-constrained approach reshapes the vehicle’s error ellipse to avoid constraint collisions while keeping terminal covariance below its prescribed value.The violation threshold is p_j,fail = 0.0005, and the setup is illustrated in Figure 7.
6 Summary
The paper addresses optimal covariance steering for discrete-time stochastic linear time-varying systems with chance constraints. It establishes decoupling without chance constraints, reformulates the constrained problem as convex programming, and verifies the approach numerically.
- The work studies optimal covariance steering for discrete-time stochastic linear time-varying systems subject to chance constraints.
- Without chance constraints, mean and covariance steering can be designed independently, while the constrained problem can be converted to a convex programming problem.
- Numerical examples verify the proposed chance-constrained covariance steering approach.