Verification and Validation of Convex Optimization Algorithms for Model Predictive Control

TL;DR

This paper discusses the formal verification of the Ellipsoid method, a convex optimization algorithm used in model predictive control, addressing correctness, implementation, and numerical stability for safety-critical applications.

Contribution

It provides a detailed approach to verifying the Ellipsoid method's correctness, including code property encoding, proof strategies, and handling floating-point errors.

Findings

Verification methods for the Ellipsoid algorithm are developed.

Code properties and proof techniques are evaluated for applicability.

Modifications to improve numerical stability are proposed.

Abstract

Advanced embedded algorithms are growing in complexity and they are an essential contributor to the growth of autonomy in many areas. However, the promise held by these algorithms cannot be kept without proper attention to the considerably stronger design constraints that arise when the applications of interest, such as aerospace systems, are safety-critical. Formal verification is the process of proving or disproving the ''correctness'' of an algorithm with respect to a certain mathematical description of it by means of a computer. This article discusses the formal verification of the Ellipsoid method, a convex optimization algorithm, and its code implementation as it applies to receding horizon control. Options for encoding code properties and their proofs are detailed. The applicability and limitations of those code properties and proofs are presented as well. Finally, floating-point errors are taken into account in a numerical analysis of the Ellipsoid algorithm. Modifications to the algorithm are presented which can be used to control its numerical stability.