Improving Tractability of Real-Time Control Schemes via Simplified $\mathcal{S}$-Lemma

TL;DR

This paper introduces a reformulation of robust quadratic constraints in control schemes, replacing semidefinite programs with linear and second-order cone constraints to enable faster real-time solutions.

Contribution

It presents a novel reformulation technique that significantly reduces computational complexity in real-time control applications involving the $ ext{S}$-lemma.

Findings

Achieves substantial speedup in solving control optimization problems

Replaces semidefinite programs with more efficient linear and second-order cone constraints

Demonstrates practical benefits through a numerical example

Abstract

Various control schemes rely on a solution of a convex optimization problem involving a particular robust quadratic constraint, which can be reformulated as a linear matrix inequality using the well-known $\mathcal{S}$-lemma. However, the computational effort required to solve the resulting semidefinite program may be prohibitively large for real-time applications requiring a repeated solution of such a problem. We use some recent advances in robust optimization that allow us to reformulate such a robust constraint as a set of linear and second-order cone constraints, which are computationally better suited to real-time applications. A numerical example demonstrates a huge speedup that can be obtained using the proposed reformulation.