2023 ACC Workshop on “Contraction Theory for Systems, Control, and Learning”

Full-Day Workshop, in conjunction with the 2023 American Control Conference in San Diego, California on Tuesday, May 30, 2023

Organizer: Francesco Bullo, UC Santa Barbara

Location: Hilton San Diego Bayfront Hotel, Aqua 310B

Schedule. Tuesday May 30, 2023

The workshop includes 8 talks from invited speakers, a tutorial lecture, a rapid presentation session, and a panel discussion. All times are listed in Pacific Daylight Time (PDT), UTC -7:00. Each talk is 30 minutes + 10 minutes for Q&A and open discussion. Morning and afternoon coffee breaks: 10:00-10:30am and 3:00-3:30pm.

8:00am - 9:00am: Tutorial session: Basics of contraction theory (PDF), Francesco Bullo, UC Santa Barbara, USA (1h)

9:00am - 9:20am: Introductions and Welcome (20 min)

9:20am - 10:00am: Flow-dependent Lyapunov functions for contraction analysis, Eduardo Sontag, Northeastern University, USA (40 min)

10:00am - 10:30am: Coffee break (30 min)

10:30am - 11:10am: A contraction-based approach to study synchronization properties of complex networks, Zahra Aminzare, University of Iowa, USA (40 min)

11:10am - 11:50pm: Contraction-guided reachability analysis of neural network controlled systems (PDF), Samuel Coogan, GeorgiaTech, USA (40 min)

11:50pm - 12:30pm: Numerical computation of contraction metrics (PDF), Sigurdur F. Hafstein, University of Iceland, Iceland (40 min)

12:30pm - 1:40pm: Lunch break (1h 10min)

1:40pm - 2:20pm: Robust learning of dynamics and feedback policies via contracting neural models (PDF), Ian R. Manchester, University of Sydney, Australia (40 min)

2:20pm - 3:00pm: Non-quadratic S-Lemma and contractivity of Lur'e-type systems (PDF), Anton Proskurnikov, Politecnico di Torino, Italy (40 min, remote presentation)

3:00pm - 3:30pm: Coffee break (30 min)

3:30pm - 3:40pm: Towards certifiable robot localization & mapping with contraction theory, Brett Lopez, UC Los Angeles, USA, (10min, recorded presentation)

3:40pm - 4:10pm: Rapid presentation session, (all attendees are invited to give a 5min research presentation):

If you are an attendee and would like to give a 5-minute presentation during the Rapid Session, please email Francesco (bullo at ucsb.edu) ASAP (with name, URL, affiliation title).

4:10pm - 4:50pm: On the design of scalable network systems: a contraction theory approach, Giovanni Russo, Universita di Salerno, Italy (40 min)

4:50pm - 5:30pm: Advances in contraction theory: Ergodic coefficients and convex optimization (PDF), Francesco Bullo, UC Santa Barbara, USA (40 min)

5:30pm - 6:00pm: Panel discussion with all speakers. Conclusion and wrap-up

(6:30pm – 8:30: ACC Opening Reception, Promenade Plaza)

Links to recent related educational and research events:

Workshop Abstract

Much recent research has focused on the application of the Banach contraction principle to control and dynamical systems. Similarly, key problems in machine learning and dynamical neuroscience can be addressed with these tools. Contracting dynamical systems automatically enjoy numerous safety and stability guarantees. Moreover, an important complement to these theoretical tools is given by the increasingly-applied theory of monotone operators. The workshop will present an extensive list of presentations by leading scientists worldwide on (1) the foundations of contraction theory, (2) theoretical developments for complex networks, including progress on synchronization and scalability, (3) computational advances in the design of contraction metrics and contracting dynamical systems solving optimization problems, and (4) applications to machine learning, planning and robust control. Of special interest to the AAC audience will be results on robust stability analysis and control design for deterministic and stochastic systems as well as formal robustness and stability guarantees for various learning-based control problems.

This workshop will bring together experts of diverse backgrounds to discuss recent theoretical and computational advances, identify emerging challenges, and discuss rapidly-developing application opportunities. The workshop should be of interest to both junior and senior researchers interested in theoretical and computational advances in systems, control, and learning. A recent successful tutorial session at the 2021 IEEE CDC confirms the interest of the control community in these topics.

Speakers Abstracts

Speaker: Zahra Aminzare, University of Iowa
Title: A contraction-based approach to study synchronization properties of complex networks
Abstract: Contraction theory provides an elegant way to analyze the behavior of certain nonlinear dynamical systems. In this lecture, we discuss the application of contraction to network synchronization. First, we review conditions that guarantee synchronization in networks of homogeneous and deterministic systems which are diffusively coupled. Motivated by neural networks, we allow heterogeneity, stochasticity, and non-diffusive coupling across the network and show how these factors may affect synchronization properties.

Speaker: Sam Coogan, Georgia Tech
Title: Contraction-guided reachability analysis of neural network controlled systems
Abstract: In this talk, we consider computing reachable sets of a dynamical system in feedback with a neural network controller. We first embed the closed-loop dynamics into a larger system using an inclusion function of the neural network. By leveraging the theory of monotone dynamical systems, we show that hyper-rectangular over-approximations of the reachable sets are efficiently computed using a single trajectory of the embedding system. Moreover, we show that, if this embedding system is constructed in a certain way, the contraction rate of the embedding system is the same as the original closed-loop system. Thus, this embedding provides a scalable approach for reachability analysis of the neural network control loop while preserving the nonlinear structure of the system. We design an algorithm to leverage this computational advantage through partitioning strategies, improving our reachable set estimates while balancing its runtime with tunable parameters.

Speaker: Sigurður F. Hafstein, University of Iceland
Title: Numerical computation of contraction metrics
Abstract: We discuss numerical methods for the computation of contraction metrics for ODE systems with stable equilibria or periodic orbits. We present approximation using generalized interpolation in reproducing kernel Hilbert spaces or integration-quadrature formulas from converse theorems. Further, we present a semidefinite optimization problem, whose feasible solutions deliver true contraction metrics (not approximations). Finally, we show how approximations can be used to deliver feasible solutions to the optimization problems, providing a numerically efficient method to compute true contraction metrics.

Speaker: Brett Lopez, UC Los Angeles
Title: Towards Certifiable Robot Localization & Mapping with Contraction Theory
Abstract: We will examine the current and future role of contraction theory within the context of developing certifiable localization and mapping algorithms for autonomous mobile systems. In particular, we will present a new contracting hierarchical observer that possesses strong convergence and robustness properties. We will then discuss the potential of modeling these algorithms as a feedback combination, and how contraction theory should be used to guide the development and analysis of these algorithms in the future.

Speaker: Ian R. Manchester, University of Sydney
Title: Robust learning of dynamics and feedback policies via contracting neural models
Abstract: In this tutorial, we will introduce a new approach to building nonlinear dynamical models with built-in behavioural guarantees. We show how to construct smooth & unconstrained parameterisations of neural model architectures which are guaranteed to satisfy prescribed incremental quadratic constraints. These can include l2 Lipschitz bounds, incremental passivity, and other constraint types. These “direct” parameterizations enable learning of robust models and control policies via simple first-order methods, without any auxiliary constraints or projections. We will illustrate the approach in the context of system identification, observer design, and reinforcement learning.

Speaker: Anton Proskurnikov, Politecnico di Torino
Coworkers: Alexander Davydov and Francesco Bullo
Title: Non-quadratic S-Lemma and contractivity of Lur'e-type systems
Abstract: The S-Lemma was proposed by Yakubovich as a criterion of the existence of a quadratic Lyapunov function in the Lur’e problem of absolute stability. A natural question arises: can non-Euclidean norms (or squared norms) serve as Lyapunov functions in stability problems? This talk presents a novel non-polynomial S-Lemma that leads to constructive criteria for the existence of Lyapunov functions of the type \(V(x)=\|Rx\|_pˆ2\) (squared weighted \(\ell_p\)-norm).

Speaker: Giovanni Russo, Universita’ di Salerno
Title: On the design of scalable network systems: a contraction theory approach
Abstract: Over the last few years, network systems have considerably increased their size and complexity. For these systems, a key challenge is then that of designing control protocols for the network that do not only guarantee the fulfilment of some desired behavior, but also satisfy the following key requirements: (i) rejection of certain classes of disturbances; (ii) non-amplification of the disturbances that are not rejected. In this talk, the fulfilment of these requirements is captured via a scalability property. Subsequently, it is shown how non-Euclidean contraction can be leveraged to design protocols ensuring network scalability. Application examples are leveraged to illustrate the results.

Speaker: Eduardo Sontag, Northeastern University
Coworkers: M. Ali Al-Radhawi and David Angeli
Title: Flow-dependent Lyapunov functions for contraction analysis
Abstract: In previous work, we have developed an approach to understanding the dynamics of chemical reaction networks, based on “flow-dependent Lyapunov functions” (FDLf’s). Such functions depend on state variables (species) only through the flows among species, and more specifically reaction rates. They can be equivalently defined as common Lyapunov functions for finite families of linear systems, which leads to an effective computational package. We have shown the power of FDLf’s in establishing stability and ensuring safety constraints in several biological applications. In this work, we will explore the implications of FDLf’s in contraction analysis.

Speaker: Francesco Bullo, UC Santa Barbara
Coworkers: Alexander Davydov, Anton Proskurnikov, Veronica Centorrino, Giovanni Russo, Anand Gokhale
Title: On contraction theory and monotone operators for control and learning
Abstract: I will present recent results on contraction theory and its twin theory of monotone operators, motivated by the study of neural networks. We will review problems inspired by neuroscience and by machine learning. We will study the interplay between discrete and continuous time dynamics. We will draw examples from implicit models in machine learning, biologically plausible learning in neuroscience, and numerical optimal control problems.


Partial funding for this work is provided by the Air Force Office of Scientific Research through grant FA9550-22-1-0059.