Efficient Chance Constrained Covariance Control with Output Feedback Study

Core Concepts

Developing computationally efficient solutions for covariance control with chance constraints using output feedback.

Abstract

This paper explores the development of efficient solutions for steering state distribution between Gaussian distributions with chance constraints. It introduces a method using Kalman filters and DC constraints, showcasing its effectiveness on a double integrator example. Chance-constrained covariance steering problems are discussed, along with practical applications and data-driven control methods.

I. Introduction

• Covariance control aims to steer system states' distribution.
• Historical context from the 1980s to recent years.
• Practical applications involve constraints on state and control.

II. Problem Statement

• Notation conventions for vectors, matrices, and random variables.
• Formulation of the stochastic linear system dynamics and measurement model.

III. Kalman Filter

• Optimal observer for linear dynamics with a measurement model.
• State estimation updates using Kalman filter equations.

IV. Filtered State Control Design

• Affine filtered state feedback control design formulation.
• Separable cost function into mean and covariance costs.

V. Chance Constraints

• Imposing probabilistic constraints on state and input domains.
• Decomposition of joint chance constraints into individual ones.

VI. Numerical Example

• Illustration of OFCS algorithm on a double integrator system.
• Enforcement of state chance constraints with polytope definitions.

VII. Conclusion

• Extension of efficient CC-CS approach to partial state information cases.
• Introduction of DC constraints handling for chance constraints.

Stats

The proposed method is illustrated on a double integrator example with varying time horizons compared to other methods.

Quotes

"The efficiency of the proposed method is illustrated on a double integrator example."
"The proposed method introduces a novel approach to make the state and control chance constraints tractable."

Key Insights Distilled From

Computationally Efficient Chance Constrained Covariance Control with Output Feedback

by Joshua Pilip... at arxiv.org 03-25-2024

https://arxiv.org/pdf/2310.02485.pdf

Deeper Inquiries

How can this approach be extended to handle more complex systems

To extend this approach to handle more complex systems, one could consider incorporating non-Gaussian distributions for the state and control variables. By utilizing concentration inequalities or other probabilistic tools, the chance constraints can be approximated conservatively for non-Gaussian systems. Additionally, introducing more sophisticated filtering techniques beyond the Kalman filter, such as particle filters or unscented Kalman filters, could improve state estimation accuracy in complex systems with nonlinear dynamics or significant disturbances. Furthermore, integrating model predictive control (MPC) strategies into the optimization framework can enhance performance by considering future predictions and constraints.

What are potential drawbacks or limitations of reformulating chance constraints as DC constraints

One potential drawback of reformulating chance constraints as DC constraints is that it may lead to a loss of optimality compared to directly solving the original non-convex problem. The linearization process around reference values introduces conservatism in handling uncertainties and may result in suboptimal solutions. Moreover, dealing with DC programming involves additional computational complexity due to iterative convex-concave procedures required for convergence. This iterative nature might increase computation time and make real-time implementation challenging for certain applications.

How might data-driven control methods impact the future application of this methodology

Data-driven control methods have the potential to revolutionize how this methodology is applied in practice. By leveraging data collected from system operation instead of relying solely on accurate models, data-driven approaches offer flexibility in adapting to unknown dynamics or disturbances present in real-world scenarios. These methods enable learning controllers directly from raw data without explicit knowledge of system matrices or parameters. Incorporating data-driven techniques into covariance steering with output feedback can enhance adaptability and robustness against uncertainties while reducing reliance on precise modeling assumptions.