IFAC ACNDC 2024

Elena Panteley, CNRS and CentraleSupélec

Title: In search of a hollistic approach to understand complex networked systems; the plausible emergence of dynamic consensus under strong and weak coupling

Abstract:

Networked systems are ubiquituous, so motivating their study without appealing to platitudes seems futile. But how can we understand the synchronised behaviour of heterogeneous networked systems? There are two apparently antagonistic trains of thought that address this question in a broad sense: reductionism and emergentism.  The first maintains that any whole can be reduced to its constituents—as is the case of networked linear systems, while the tenet of emergentism is that a new behaviour appears as a consequence of the interaction of the constituent parts.

Are these views truly opposed or is there a gray area where both approaches can coexist? The latent assertion of this talk is that there is.

We present some of the  work we have carried out in the  past years on complex networked systems. In general, primarily the following four aspects determine collective behaviour. The first is the nature of the individual systems’ dynamics in the network.  We focus on nonlinear herterogeneous systems. The second aspect corresponds to the network’s topology; we focus on general connected-graph networks. The third pertains to the nature of the interconnections; we concentrate on static diffusive coupling, but we devote a great deal of attention to the fourth: the strength of the coupling.

It is well known that if the coupling is sufficiently strong, a coherent, synchronised behavior appears in the network. As we show, this behavior is dychotomic; it consists on one hand, of the synchronisation error dynamics and  a so-called emergent/reduced-order dynamics that has the  dimension of a singe system.  In this view, some type of synchronisation is achieved if and only if all the systems enter in a state of what we call dynamic consensus. This is a consensual dynamic behaviour per which all the systems may attain a common equilibrium, adopt  a periodic oscillatory behaviour or, more generally, simply acquire the dynamics, similar to that of a reduced-order system, inherent to the network.

We give an insight to the mathematical tools to study dynamic consensus.

We show that strong coupling leads to practical asymptotic synchronisation of heterogeneous networks of nonlinear systems. Then, we demonstrate how more refined results can be obtained using singularly-perturbations theory. In that setting the slow dynamics corresponds to that of a reduced-order system, while the synchronisation errors evolve in a faster time-scale. Finally, we present an extension of the proposed approach to the analysis of networks with weak coupling.

It turns out that in this case the reduced-order dynamics can be represented by a (reduced-order) network.

We wrap up the talk with a discussion on open problems.

Biography: Elena Panteley is a Senior Researcher at CNRS and a member of the Laboratory of Signals and Systems. She received her PhD. degree in Applied Mathematics from the State University of St. Petersburg, Russia, in 1997. From 1986 to 1998, she held a research position with the Institute for Problem of Mechanical Engineering, Russian Academy of Science, St. Petersburg. She is co-chair of the International Graduate School of Control of the European Embedded Control Institute (EECI-IGSC). Elena Panteley is the Book-reviews Editor for Automatica and Associate Editor for IEEE Control Systems Letters. Her research interests include stability and control of nonlinear dynamical systems and networked systems with applications to multi-agent systems.

Andrew R. Teel, University of California, Santa Barbara

Title: Stochastic approximation of hybrid dynamical systems with optimization applications

Abstract: Stochastic approximation methods for numerical optimization have a long history dating back to the middle of the previous century. In these optimization methods, noisy or deliberately stochastic perturbations of the gradient of a function are used in iterative schemes with small step sizes. The methods are effective because the average of a large number of iterations approximately reproduces samples of the solutions of a gradient descent differential equation. Recently, there has been interest in developing optimization algorithms that employ hybrid dynamical systems. In these algorithms, the flows may correspond to accelerated gradient descent, while the jumps may correspond to resetting the momentum variable in accelerated gradient descent to improve the rate of convergence to an optimal point. Implementing the flows via small step size iterations involving stochastic gradients is also of interest. To be confident that such an approach will be effective, a general theory for stochastic approximation of hybrid dynamical systems is needed. This new theory will be discussed in this talk and applications of the theory to optimization problems will be presented.

Biography: Andrew R. Teel received his A.B. degree in Engineering Sciences from Dartmouth College in Hanover, New Hampshire, in 1987, and his M.S. and Ph.D. degrees in Electrical Engineering from the University of California, Berkeley, in 1989 and 1992, respectively. After receiving his Ph.D., he was a postdoctoral fellow at the École des Mines de Paris in Fontainebleau, France. In 1992 he joined the faculty of the Electrical Engineering Department at the University of Minnesota, where he was an assistant professor until 1997. Subsequently, he joined the faculty of the Electrical and Computer Engineering Department at the University of California, Santa Barbara, where he is currently a Distinguished Professor and director of the Center for Control, Dynamical systems, and Computation.  His research interests are in nonlinear and hybrid dynamical systems, with a focus on stability analysis and control design. He has received NSF Research Initiation and CAREER Awards, the 1998 IEEE Leon K. Kirchmayer Prize Paper Award, the 1998 George S. Axelby Outstanding Paper Award, and was the recipient of the first SIAM Control and Systems Theory Prize in 1998. He was the recipient of the 1999 Donald P. Eckman Award and the 2001 O. Hugo Schuck Best Paper Award, both given by the American Automatic Control Council, and also received the 2010 IEEE Control Systems Magazine Outstanding Paper Award.  In 2016, he received the Certificate of Excellent Achievements from the IFAC Technical Committee on Nonlinear Control Systems.  He is Editor-in-Chief for Automatica, and a Fellow of the IEEE and of IFAC.

Nathan van de Wouw, Eindhoven University of Technology

Title: Taming complexity of nonlinear and interconnected dynamical systems

Abstract: Engineering systems are becoming more and more intricate, with examples being traffic systems, energy networks, high-tech equipment, and many more. Consequently, the complexity of dynamic models for such systems often scales to proportions that obstruct their usage for system design, monitoring and control. Therefore, model complexity reduction of dynamical systems is increasingly important. In this talk, two challenges in complexity management of large-scale dynamical systems will be addressed.

Firstly, how can we reduce the complexity of nonlinear dynamical systems, while guaranteeing the preservation of non-local stability properties in the reduction process? In this scope, system-level incremental stability properties will prove to be key. In addition, such properties also allow to formulate computable bounds on the error induced by reduction. Such error bounds can serve to guide the reduction process in trading off model complexity vs. model accuracy. This approach is broadly applicable in the scope of different reduction methodologies, such as balancing methods and time-domain moment matching techniques.

Another challenge is how to effectively pursue model reduction for systems consisting of (networks of) interconnected sub-systems. In the scope of networks of linear systems, a methodology will be presented to quantitatively relate accuracy requirements for the (networked) system as a whole to accuracy requirements on the sub-system models. This facilitates an entirely modular reduction approach, thereby promoting scalability to truly complex systems. Next, a balancing-based reduction approach for interconnected systems is presented that preserves stability and passivity properties of the interconnected system in the reduction process. This reduction approach also takes the interconnected nature of the system into account in the reduction, thereby warranting superior accuracy of the resulting models. It will be shown how these methods can be used to manage complexity in interconnected structural dynamics models of complex high-tech semi-conductor equipment.

Biography: Nathan van de Wouw obtained his Ph.D.-degree in Mechanical Engineering from the Eindhoven University of Technology, the Netherlands, in 1999. He currently holds a full professor position at the Mechanical Engineering Department of the Eindhoven University of Technology, and chairs the Dynamics and Control section. He has held a (part-time) full professor position the Delft University of Technology, the Netherlands, from 2015-2019. He has also held an adjunct full professor position at the University of Minnesota, U.S.A, from 2014-2021. He has published the books ‘Uniform Output Regulation of Nonlinear Systems: A convergent Dynamics Approach’ with A.V. Pavlov and H. Nijmeijer (Birkhäuser, 2005) and `Stability and Convergence of Mechanical Systems with Unilateral Constraints’ with R.I. Leine (Springer-Verlag, 2008). In 2015, he received the IEEE Control Systems Technology Award “For the development and application of variable-gain control techniques for high-performance motion systems”. He is an IEEE Fellow for his contributions to hybrid, data-based and networked control.