ANR PANOPLY

General Objective

The general objective of the JCJC research project PANOPLY is to develop a systematic framework for the practical control of networks of linear hyperbolic systems. The proposed control strategies are said to be practical in the sense that they are constructive, easily implementable, and fulfill a given set of performance specifications. 

Context

Networks of hyperbolic systems, possibly coupled with ordinary differential equations (ODEs), constitute an essential paradigm to describe a wide variety of large complex systems, including wave propagation, traffic network systems, multiscale and multiphysics systems. Controlling and monitoring networks of hyperbolic systems are difficult control engineering problems due to the distributed nature of the different subsystems composing the network (time and space dependency), the possibly involved graph structure of the network, and the physical/economic infeasibility of placing sensors and actuators everywhere along the spatial domain. The stringent operating, environmental and economical requirements and the high mathematical complexity of these systems explain why traditional control methods exhibit a limited range of applicability and have not been successful at high technology readiness levels. Thus, the theory of control of distributed parameter systems needs substantial advancements to achieve control and estimation objectives for such network structures. 

Methodology

A network of hyperbolic systems can be described as a graph: each edge corresponds to an elementary hyperbolic subsystem, and interactions between the subsystems composing the network occur at the graph's vertices. This graph representation will be a cornerstone of the methodology we present in PANOPLY.

The first objective of PANOPLY is to understand better the links between the network structure (nb of cycles, incidence matrix) and its controllability/observability (C/O) properties. We aim to characterize the configurations for a given graph structure that guarantee C/O. To identify reflections of graph-theoretic notions on the system properties, we will use the concept of structural controllability as a starting point. The second objective is to develop generic analytical techniques to quantify closed-loop performances w.r.t industry-inspired performance indices (e.g., sensitivity, robustness margins, data sampling, convergence rate, computational effort). This objective is crucial to optimize actuator placement and to tune the controllers we will design. Finally, we aim to design modular, scalable, and numerically implementable feedback controllers for an admissible configuration of actuators and sensors. The design will introduce degrees of freedom to guarantee potential trade-offs w.r.t implementation constraints. 

The proposed methodology will be based on a graph representation of the network and its systematic structural analysis. It also relies on the theory of integral equations. We will use recent results obtained by the P.I. for simple networks showing strong relations between spectral controllability and the existence of a solution for a set of appropriate integral Fredholm equations, from which it is possible to derive explicit controllers. We will consider two realistic case studies to validate our theoretical results and compare them to conventional strategies. The first case study corresponds to the problem of traffic control regulation on interconnected freeway segments. It is an academic example we will use to validate our results in simulation for several realistic graph configurations. The second experimental test case is the active control of vibrations in mechanical structures equipped with piezoelectric actuators. This test case will help us analyze the closed-loop performance and the effect of actuator/sensor placement.