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arXiv · 1911.07796

An Optimization Approach to Verifying and Synthesizing K-Cooperative Systems

Dimitris Kousoulidis·Fulvio Forni
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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.

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Dimitris Kousoulidis, Fulvio Forni. 2019-11-18. An Optimization Approach to Verifying and Synthesizing K-Cooperative Systems. https://arxiv.org/abs/1911.07796

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