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Center for Nonlinear Studies |
Classical control theory does not scale well for large systems like traffic networks, power networks and chemical reaction networks. However, many of these applications can be handled efficiently using the concept of positive system, exploiting that the set of positive states is left invariant by the dynamics. Positive systems, and the nonlinear counterpart monotone systems, are common in many branches of science and engineering. In this presentation, we will highlight several fundamental advantages of positive control systems: Verification and synthesis can be done with a complexity that scales linearly with the number of states and interconnections. Distributed controllers can be designed by convex optimization. Lyapunov functions and storage functions for nonlinear monotone systems can be built from scalar functions of the states, with dramatic simplifications as a result. In spite of a rich set of existing results, several fundamental questions in control of positive systems remain open. For example, negative feedback can easily destroy positivity of the closed loop system. On the other hand, intuition tells us that something is wrong with a traffic control system where fewer cars leads to more congestion. Hence, we need to better understand the limitations and potential of closed loop positive systems.
Host: Misha Chertkov