## Periodic Reporting for period 3 - Scale-FreeBack (Scale-Free Control for Complex Physical Network Systems)

Reporting period: 2019-09-01 to 2021-02-28

Scale-FreeBack proposes a new paradigm by approaching the problem of control of large-scale complex systems with a new holistic vision, and novel design methods ensuring the scalability of the whole chain (modelling, observation, and control). For this purpose, it is first proposed to investigate appropriate scale-free dynamic modeling approaches breaking down the network system complexity, and then to devise control algorithms specifically tailored for such models. Scale-free control refers here to the ability of a control algorithm to be easily scaled to system networks of arbitrary dimensions.

Scale-FreeBack proposes to take advantage of the new opportunities presented by the latest large-scale sensing technologies, while dealing with the demanding control issues which arise because of the need for cyber-physical complexity specific to large-scale interconnected networks. In addition to investigating the purely theoretical aspects, the Scale-FreeBack team also expects to come up with some innovative control solutions for improving traffic management systems.

Scale-FreeBack is intended to go beyond traditional control approaches by first focusing on developing appropriate mathematical scale-free dynamic modeling approaches which can be used to break down the complexity of network systems, and then on building estimation and control algorithms which will be specifically tailored to these models. It is also planned to apply, test and validate the findings obtained in the field of road traffic networks.

Scale-FreeBack proposes to take advantage of the new opportunities presented by the latest large-scale sensing technologies, while dealing with the demanding control issues which arise because of the need for cyber-physical complexity specific to large-scale interconnected networks. In addition to investigating the purely theoretical aspects, the Scale-FreeBack team also expects to come up with some innovative control solutions for improving traffic management systems.

Scale-FreeBack is intended to go beyond traditional control approaches by first focusing on developing appropriate mathematical scale-free dynamic modeling approaches which can be used to break down the complexity of network systems, and then on building estimation and control algorithms which will be specifically tailored to these models. It is also planned to apply, test and validate the findings obtained in the field of road traffic networks.

Scale-free modeling and aggregation. Here we deal with the problem of how to find a scale-free abstraction out of an arbitrary network and how to use this abstraction to benefit of the scale-free properties in term of control. We devised an algorithm, called MergeToScaleFree. It allows finding a partition of an arbitrary network resulting in a scale-free abstraction. This algorithm has been applied on large-scale networks such as the urban traffic network of Grenoble (20 000 nodes), and on a direct application of this algorithm in the context of epidemic spreading. Within the ERC, we have also explored several edges of the problem of modeling urban traffic networks: 2-D fluid macroscopic models for traffic models. Besides we developed a new boundary control design for large-scale urban traffic systems represented by an aggregated model PDE models. Right now, we are also validating of the 2D traffic models with real data coming from the GTL-ville experimental platform.

State-state estimation over scale-free networks. Here we deal with the problem of estimating the average state of certain sectors in a large-scale network, but also its variance. The method has been applied to the problem of thermal monitoring of large buildings. This technique along with a simple on/off control policy for regulation saves around 25.32% of the energy. We have devised a new method for on-line vehicle density reconstruction in large-scale traffic networks. For that, we have used flows and FCD speed measurements to jointly reconstruct density and flow in the entire network. A sensor radar networks has been installed in the city of Grenoble. It allows us to validate the proposed methodology, and to provide public information for the analysis of the city traffic conditions (road occupancy, energy vehicle consumption, vehicle emissions, and pollution diffusion). A demonstrator of these results are implemented in the GTL-Ville platform.

Control methods for scale-free network. We devised a very innovative solution to control large-scale systems: we have introduced the “continuation method” transforming spatially distributed ODE systems into continuous PDE. Most of the systems we encounter in real life consist of such a large number of particles that the direct analysis of their interaction is impossible. The method was illustrated by multiple examples including transport equations, Kuramoto equations and heat diffusion equations, an alternative solution to Hilbert's 6th problem. Currently several applications of the method are under consideration, including general linear networks, laser chains, traffic systems or rings of spintronic oscillators.

Proof-of-Concept: road networks. Proof-of-concept studies are conducted by performing field tests at our data collection center (GTL-Ville), and simulations are performed to test the validity of our models, using a large-scale micro-simulator. The equipment available at the GTL-Ville is used to test our findings on a representative network using real-life data. This project considers a 1 km x 1.4 km zone of the Grenoble downtown, in which different traffic data is to be recollected in real time. GTL-Ville platform is still a beta version.

See, http://gtlville.inrialpes.fr

See attached Fig1-6 showing screen shots from the web-platform

State-state estimation over scale-free networks. Here we deal with the problem of estimating the average state of certain sectors in a large-scale network, but also its variance. The method has been applied to the problem of thermal monitoring of large buildings. This technique along with a simple on/off control policy for regulation saves around 25.32% of the energy. We have devised a new method for on-line vehicle density reconstruction in large-scale traffic networks. For that, we have used flows and FCD speed measurements to jointly reconstruct density and flow in the entire network. A sensor radar networks has been installed in the city of Grenoble. It allows us to validate the proposed methodology, and to provide public information for the analysis of the city traffic conditions (road occupancy, energy vehicle consumption, vehicle emissions, and pollution diffusion). A demonstrator of these results are implemented in the GTL-Ville platform.

Control methods for scale-free network. We devised a very innovative solution to control large-scale systems: we have introduced the “continuation method” transforming spatially distributed ODE systems into continuous PDE. Most of the systems we encounter in real life consist of such a large number of particles that the direct analysis of their interaction is impossible. The method was illustrated by multiple examples including transport equations, Kuramoto equations and heat diffusion equations, an alternative solution to Hilbert's 6th problem. Currently several applications of the method are under consideration, including general linear networks, laser chains, traffic systems or rings of spintronic oscillators.

Proof-of-Concept: road networks. Proof-of-concept studies are conducted by performing field tests at our data collection center (GTL-Ville), and simulations are performed to test the validity of our models, using a large-scale micro-simulator. The equipment available at the GTL-Ville is used to test our findings on a representative network using real-life data. This project considers a 1 km x 1.4 km zone of the Grenoble downtown, in which different traffic data is to be recollected in real time. GTL-Ville platform is still a beta version.

See, http://gtlville.inrialpes.fr

See attached Fig1-6 showing screen shots from the web-platform

PWe devise an algorithm, called MergeToScaleFree, and allows finding a partition of an arbitrary network resulting in a scale-free abstraction. For the first time in the control arena an algorithm for such model aggregation is proposed. In a different approach we have proposed Aggregated Scale-Free Models for large-scale traffic system. The main progress beyond the state of the art is the derivation of a two-dimensional PDE describing how the flow along the network is propagate via a continuum distribution. The model then is able to predict shock, advection waves in the 2-D plane and multi-directions. For the first time, this type of model has been validated using real data from the GTL-Ville.

We have introduced a new concept of scale-free detectability, and the mathematical conditions for a network to have such that the average state of some area can be estimated. In simple terms, scale-free detectability means that the average states estimation becomes exact as the degree of the aggregated nodes grows. In addition to this, we have studied some resilience property when some of nodes in the network are broken, or the suited scale-free detectability condition is not satisfied.

Traffic density, traveling time and vehicle emission estimation in large-scale traffic networks. We have proposed a density and flow reconstruction. Experimental rea-time validation using the GLT platform has been realized in a selecte area of Grenoble Downtown.

Recently we devised a very innovative solution to control large-scale systems: we have introduced the “continuation method” transforming spatially distributed ODE systems into continuous PDE. As a main example a continuation of a Newtonian system of interacting particles was performed, thus obtaining the Euler equations for compressible fluids and thereby providing an original alternative solution to Hilbert's 6th problem. Using this derivation of the Euler equations to control multiagent systems it appeared to be possible to design a nonlinear control algorithm based on the continuous approximation to stabilize a robotic formation along the desired trajectory, performing a maneuver of passing through a window.

Images attached to the Summary for publication: PDF from the 5 min presentation at TRA.

We have introduced a new concept of scale-free detectability, and the mathematical conditions for a network to have such that the average state of some area can be estimated. In simple terms, scale-free detectability means that the average states estimation becomes exact as the degree of the aggregated nodes grows. In addition to this, we have studied some resilience property when some of nodes in the network are broken, or the suited scale-free detectability condition is not satisfied.

Traffic density, traveling time and vehicle emission estimation in large-scale traffic networks. We have proposed a density and flow reconstruction. Experimental rea-time validation using the GLT platform has been realized in a selecte area of Grenoble Downtown.

Recently we devised a very innovative solution to control large-scale systems: we have introduced the “continuation method” transforming spatially distributed ODE systems into continuous PDE. As a main example a continuation of a Newtonian system of interacting particles was performed, thus obtaining the Euler equations for compressible fluids and thereby providing an original alternative solution to Hilbert's 6th problem. Using this derivation of the Euler equations to control multiagent systems it appeared to be possible to design a nonlinear control algorithm based on the continuous approximation to stabilize a robotic formation along the desired trajectory, performing a maneuver of passing through a window.

Images attached to the Summary for publication: PDF from the 5 min presentation at TRA.