An Optimization Approach to Verifying and Synthesizing K-Cooperative Systems

TL;DR

This paper introduces an optimization-based heuristic method for verifying and synthesizing K-cooperative systems, leveraging geometric insights and linear programming to handle nonlinear systems with multiple equilibria.

Contribution

It reformulates strict K-cooperativity conditions as an LP-based optimization problem and develops a unique cone-finding algorithm for system verification and synthesis.

Findings

Developed a heuristic cone-finding algorithm using LP.

Demonstrated the approach on nonlinear systems with multiple equilibria.

Extended the method to compute polyhedral Lyapunov functions.

Abstract

Differential positivity and K-cooperativity, a special case of differential positivity, extend differential approaches to control to nonlinear systems with multiple equilibria, such as switches or multi-agent consensus. To apply this theory, we reframe conditions for strict K-cooperativity as an optimization problem. Geometrically, the conditions correspond to finding a cone that a set of linear operators leave invariant. Even though solving the optimization problem is hard, we combine the optimization perspective with the geometric intuition to construct a heuristic cone-finding algorithm centered around Linear Programming (LP). The algorithm we obtain is unique in that it modifies existing rays of a candidate cone instead of adding new ones. This enables us to also take a first step in tackling the synthesis problem for K-cooperative systems. We demonstrate our approach on some examples, including one in which we repurpose our algorithm to obtain a novel alternative tool for computing polyhedral Lyapunov functions of bounded complexity.