2024 CDC Workshop on “Contraction Theory for Systems, Control, Optimization, and Learning”

Full-Day Workshop, in conjunction with the 2024 Conference on Decision and Control in Milano, Italy

Organizer: Francesco Bullo, UC Santa Barbara

Date: Sunday, December 15, 2024
Time: 8:30-17:45, Central European Time
Location: Amber 1 (Level 2), Allianz MiCo, Milan Convention Centre, Italy

Registration: link to conference website

Pictures from the workshop: available upon request

Speaker: Angeli, David, Imperial College, UK
Title: On 2-Contraction and Non-Oscillatory Systems: Some Theory and Applications
Abstract: The notion of 2-contraction arises when considering pairs of infinitesimal perturbations in initial conditions with respect to a nominal trajectory, and following the propagation in time of the surface area spanned by these in all projected two-dimensional quadrants. If these propagations contracts in some norm, then the system is 2-contractive and this prevents existence of oscillatory behaviours in the original nonlinear dynamics. We will review some of the definitions and main results, including non-existence of periodic or almost periodic solutions, a priori bound on Lyapunov exponents (no chaos), and applications of such results to convergence analysis.

Speaker: Astolfi, Daniele, Université de Lyon, France
Title: Contraction Theory of Output Regulation
Abstract: The problem of output regulation consists in tracking desired references while rejecting disturbances affecting the system dynamics. A possible manner to solve such a problem relies on the internal model principle, i.e. by incorporating in the controller a copy of the model generating such referencesdisturbances. For instance, in the context of constant referencesperturbations, an integral action is needed. Once the integral action is added, then one needs to stabilize the extended cascade system in an appropriate manner so that closed-loop trajectories possess an equilibrium in the presence of external constant signals. One of the main difficulties in the nonlinear context is precisely such a step. In this talk we will show how contraction theory plays a fundamental role and we propose an extension of the forwarding method used to stabilize cascade forms in such an incremental context. The proposed paradigm is applied in the context of integral action and harmonic regulation for the rejection of periodic signals.

Speaker: Francesco Bullo, UC Santa Barbara, USA
Coworkers: Alexander Davydov, Veronica Centorrino, Giovanni Russo, Anand Gokhale
Title: Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach
Abstract: This talk presents a novel and broadly-applicable contraction-theoretic approach to continuous-time time-varying convex optimization. For any parameter-dependent contracting dynamics, we show that the tracking error is asymptotically proportional to the rate of change of the parameter with proportionality constant upper bounded by Lipschitz constant in which the parameter appears divided by the contraction rate of the dynamics squared. We additionally establish that any parameter-dependent contracting dynamics can be augmented with a feedforward prediction term to ensure that the tracking error converges to zero exponentially quickly. To apply these results to time-varying convex optimization problems, we establish the strong infinitesimal contractivity of dynamics solving three canonical problems, namely monotone inclusions, linear equality-constrained problems, and composite minimization problems. For each of these problems, we prove the sharpest-known rates of contraction and provide explicit tracking error bounds between solution trajectories and minimizing trajectories. We validate our theoretical results on numerical examples including an application to control-barrier function based controller design.

Speaker: Dall'Anese, Emiliano, Boston University, USA
Title: Contractivity of Interconnected Continuous and Discrete-time Systems
Coworkers: Yiting Chen, Emilio Benenati, and Francesco Bullo
Abstract: The small gain theorem and singular perturbation analysis offer foundational principles and tools for analyzing interconnected continuous-time dynamical systems. This talk aims to extend these principles to the interconnection of a continuous-time system with a discrete-time system (via a sample-and-hold device), providing new perspectives and insights into the stability and contractivity of these hybrid interconnections. The talk begins by presenting a sufficient condition for the interconnection to be discrete-time contracting, using a framework analogous to classical small-gain arguments. It then advances toward mirroring the concepts of “two-time-scale system” and “reduced model”, focusing on cases where discrete-time dynamics emerge from the early termination of an iterative algorithm used to compute a state feedback policy. This approach aligns with emerging trends in optimization-based and model predictive control, where first-order optimization algorithms are often interrupted before full convergence due to computational and timing constraints. Here, the “reduced model” is the feedback policy and the discrete-time system represents the optimization algorithm with an early termination. In this setup, conditions on the continuous-time dynamics and on the reduced model (i.e., the feedback policy) for the interconnected system to render an equilibrium asymptotically stable are presented; these conditions specify bounds on the number of steps embedded in the discrete-time system used to approximate the feedback policy.

Speaker: Giesl, Peter, University of Sussex, UK
Title: Computation of Contraction Metrics with Meshfree Collocation
Abstract: In this talk, we consider a general dynamical system given by solutions of an autonomous ordinary differential equation. The long-term behaviour can be characterised by attractors and their corresponding basins of attraction. Examples of attractors are equilibria or periodic orbits, and their respective basins of attraction consist of all initial points such that the corresponding solutions converge towards them.

A contraction metric is a metric such that the distance between adjacent solutions decreases with respect to the metric, and thus they share the same long-term behaviour. Hence, a contraction metric is a tool to determine the basin of attraction of an attractor without the need to know its location. A contraction metric can be described by a matrix-valued function, which is a symmetric and positive definite matrix at every point.

We will discuss the numerical construction of contraction metrics for the case of an equilibrium and a periodic orbit using meshfree collocation with Radial Basis Functions. First, we will establish existence results for contraction metrics which satisfy a certain partial differential equation (PDE), and then we will use meshfree collocation to compute an approximate solution of the PDE. Error estimates ensure that the approximation, if sufficiently close, is itself a contraction metric.

Much of this is joint work with Sigurdur Hafstein and Holger Wendland.

Speaker: Kawano, Yu, Hiroshima University, Japan
Title: Youla-Kucera Parametrization in the Contraction Framework
Coworkers: Arjan J. van der Schaft (University of Groningen), Jacquelien M.A. Scherpen (University of Groningen) Abstract: In this talk, we study incrementally exponentially stable (IES) image and kernel representations for nonlinear systems with the aim of generalizing the Youla-Kucera parametrization in the contraction framework. We first construct them and their stable inverses in the contraction framework and then provide a parametrization of stabilizing controllers by additionally assuming incremental input-to-state stability for the image representation. Focusing on constant metrics results in a parametrization of all stabilizing controllers rendering the closed-loop systems IES with respect to constant metrics.

Speaker: Ian R. Manchester, University of Sydney, Australia
Title: A Robust Learning Framework built on Contraction and Monotonicity
Abstract: This talk will discuss an emerging set of tools allowing precise control of behavioural properties of neural network models. Neural networks have enormous representational capacity, but in many applications it is important to ensure properties such as robustness, dynamical stability, energy dissipation, invertibility of models, solvability of optimization models, and others. We will describe a new framework that builds on notions of contracting systems and monotone operators, and present novel links to the concepts of Koopman operators for dynamical systems theory, KKL observers, Polyak-Lojasiewicz functions in optimization theory, and Hamiltonian mechanics.

Speaker: Margaliot, Michael, Tel Aviv University, Israel
Title: Compound Matrices and Dynamical Systems
Abstract: Compound matrices have found applications in combinatorics, graph theory, multilinear algebra, and ordinary differential equations. More recently, they were used to generalize important classes of dynamical systems including contractive systems and cooperative systems. We describe these applications as well as several promising research directions.

Speaker: Anton Proskurnikov, Politecnico di Torino, Italy
Coworkers: Alexander Davydov and Francesco Bullo
Title: Towards Non-quadratic Absolute Stability Theory
Abstract: Classical absolute stability and contraction theory, originating from Lur'e and Postnikov, relies on quadratic Lyapunov functions. Their existence is typically ensured by Yakubovich's S-lemma and LMI feasibility, traditionally validated via semidefinite programming or the Kalman-Yakubovich-Popov lemma. A quadratic Lyapunov function is essentially a squared Euclidean norm (associated with an inner product). Thus, the classical absolute stability/contractivity problem requires finding a robust estimate for some Euclidean norm of a solution (or the deviation between two solutions) in the presence of uncertain nonlinearities.

While the Euclidean norm is standard in control theory, it may be inconvenient for higher-dimensional systems like recurrent neural networks. In some machine learning problems, estimating the Manhattan (L1) or L-inf norm is necessary. Although formally equivalent, different norms differ by multiplicative constants that grow with dimensionality. Consequently, estimates based solely on Euclidean norms and quadratic Lyapunov functions can be overly conservative.

New mathematical tools introduced by Davydov et al. within the framework of contraction analysis open up the possibility for a “non-quadratic” absolute stability theory, providing less conservative estimates for non-Euclidean norms of the solutions. This talk gives a brief introduction to this new direction of research and presents some open problems.

Speaker: Giovanni Russo, Universita’ di Salerno, Italy
Title: Towards Contracting Biologically Plausible Neural Networks
Coworkers: Veronica Centorrino, Francesco Bullo, Alexander Davydov, Anand Gokhale
Abstract: In this talk, we will present a normative framework for the design of continuous-time biologically plausible neural networks to tackle sparse reconstruction problems. These problems, which naturally arise in the context of e.g., signal processing and neuroscience, can be formalized as regularized least squares optimization and involve approximating (e.g., sensory) inputs with just a few active neurons given an available dictionary. First, we link the optimal solution of these problems to the equilibria of a suitably defined proximal gradient dynamics. Then, using contraction theory, we show that this dynamics exhibits a linear-exponential convergence property under a standard assumption on the dictionary. We conclude by discussing the integration of Hebbian learning. Throughout the talk, the results are illustrated via numerical examples.

Speaker: Rodolphe Sepulchre, Cambridge University, UK
Title: Regulation Without Calibration
Abstract: The internal model principle suggests that perfect regulation requires an exact internal copy of the exosystem. How to reconcile this calibration principle with systems made of uncertain and variable components? The talk will suggest that machines that synchronize events rather than trajectories considerably release the calibration requirements of regulation. I will highlight the role of excitability and synaptic coupling to synchronize the events of uncertain trajectories.

Speaker: Jean-Jacques Slotine, MIT, USA
Title: A Quarter Century of Contraction Analysis
Abstract: It has been more than a quarter century since the paper in Automatica by Lohmiller and Slotine introduced contraction analysis to the nonlinear control community, outlining the role of differential analysis using state-dependent Riemannian metrics and its many potential applications. Research in this domain is now very active, and we will review basics as well as some recent work in our group on applications to machine learning and to non-autonomous partial differential equations.

Speaker: Eduardo Sontag, Northeastern University, USA
Title: Contractions for Interaction Networks
Abstract: This talk will focus on the dynamics of “interaction networks” which follow the formalism of chemical reaction networks (CRN’s). We will present results on the construction of polyhedral norms that provide contraction, as well as study relations to monotone systems. Some of these norms provide only non-expansivity, as opposed to strict contraction, so we also study omega-limit sets of non-expansive systems, showing that they tend to be well-behaved. Much of this work has been jointly performed with M. Ali Al-Radhawi, David Angeli, and Alon Duvall.