Thursday, 09:30 - 10:15 : Riccardo Sanfelice, Challenges and Recent Results on Contraction-type Properties for Hybrid Dynamical Systems
For a smooth dynamical system, the notion of (strict) contraction consists of the property that an appropriately defined distance between every pair of solutions decreases along them. For nonsmooth systems — in particular, for systems that have variables flowing continuously and, at times, jumping — a notion of (strict) contraction would require that the distance between every pair of solutions decreases along both regimes. The formulation of such type of notions, including incremental stability and convergence to time-varying solutions, is challenging in the hybrid systems setting since a pair of solutions for such a system may not have the same intervals of flow and instants at which jumps occur. Using variational analysis tools, we present a suitable notion for systems with variables that flow and jump, namely, hybrid dynamical systems. Necessary and sufficient conditions for such a notion are presented. Examples illustrating the ideas, challenges, and results will be given.
Thursday, 10:45 - 11:30 Paolo Forni, The Input-to-State Stability framework for Multistable Systems on Manifolds
We first consider a notion of global multistability for nonlinear systems on manifolds based on the existence of a finite number of compact, globally attractive, invariant sets satisfying a specific condition of acyclicity. We then generalize the classical ISS framework - Lyapunov characterizations, ISS, integral ISS, OSS - to such class of multistable systems. This novel ISS definition is shown to satisfy the following properties: conservation under cascade interconnection and input shifts; a characterization of the concept of recurrence in hybrid systems; robustness to perturbations under time-scale separation. Connections to incremental stability will be discussed throughout the talk.
Thursday, 11:45 - 12:30 Vincent Fromion, Some elements around incrementally stable systems on Lp
In 1995, I introduced in my PhD the weighted incremental norm approach as a natural framework for extending the well-known H∞ linear control concepts into the nonlinear context. More than twenty years later, the interest of the incremental type criteria seems rediscover, in particular for analyzing the behavior of nonlinear systems with respect to sets of specific inputs. This last point was already one of the essential bases of the proposed extension. In this talk, after after a short recall of the key features of the proposed extension, I will discuss three main interlaced issues:
Thursday, 13:45 - 14:30 Antoine Chaillet : Incremental stability of delayed dynamics: a useful tool for the analysis of brain oscillations
Brain oscillations, at various frequencies, play a central role in brain coordination. Several neurologic disorders are related to pathological brain oscillations. In the case of Parkinson’s disease, sustained low-frequency oscillations (especially in the beta-band, 13–30Hz) correlate with motor symptoms. A way to model the dynamics of cerebral structures relies on the Wilson-Cowan model, which rules the instantaneous average activity of a population depending on its afferent inputs. When the non-instantaneous communication between neurons is taken into account, this model takes the form of a Lurie system with delays. Furthermore, the spatiotemporal nature of brain oscillations can be modeled by neural fields, which then take the form of an integro-differential equation, possibly with space-dependant delays. In order to analyse these models, we propose tools to ensure incremental stability of such dynamics. More precisely, we propose a Lyapunov-Razumikhin condition for the incremental stability of delayed nonlinear systems, and a Lyapunov-Krasovskii approach to establish incremental stability of spatio-temporal delayed dynamics. Under mild conditions, we show that this property ensures that the system is entrained by its input meaning that, in response to any T-periodic input, its solution are also T-periodic after transients. This feature allow us to draw frequency profiles of neuronal populations. On a numerical example describing the brain circuitry involved in motor symptoms of Parkinson's disease, we show that a resonance takes place in the beta band, thus providing some insights on the mechanisms underlying pathological oscillations onset.
Thursday, 14:45 - 15:30 Yu Kawano : Nonlinear Eigenvalue Approach to Differential Riccati Equations for Contraction Analysis
In this talk, we extend the eigenvalue method of the algebraic Riccati equation to the differential Riccati equation (DRE) in contraction analysis by using recently introduced nonlinear eigenvalues and eigenvectors. One of the main results is showing that solutions to the DRE can be expressed as functions of nonlinear eigenvectors of the differential Hamiltonian matrix. Moreover, under an assumption for the differential Hamiltonian matrix, real symmetry, regularity, and positive semidefiniteness of solutions are characterized by nonlinear eigenvalues and eigenvectors.
Thursday, 16:00 - 16:45 Giovanni Russo, On Contraction Analysis and its applications to analyze and control dynamical systems
In this talk, we discuss the relevance of contraction analysis to the analysis and control of dynamical systems. Essentially, a system is contracting if all of its trajectories globally exponentially converge towards each other. We will start the talk with considering smooth dynamical systems. In particular, after introducing the main properties of contracting dynamics, we will show how such properties can be useful to study certain applications, including entrainment and synchronization/consensus, arising in biology and network control. After discussing an application arising in the context of automated vehicles, we will then conclude the talk by showing how contraction analysis can be extended to consider both Caratheodory and Filippov dynamics and we will discuss some applications of interest.
Thursday, 17:00 - 17:45 Laurent Praly and/or Ricardo Sanfelice, Convergence of Nonlinear Observers on R^n with a Riemannian Metric
We study how convergence of an observer whose state lives in a copy of the given system’s space can be established using a Riemannian metric. We propose necessary conditions and sufficient conditions for the convergence of an observer the state of which lives in a copy of the given system’s state space can be established using a Riemannian metric. The existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing implies the system is differentially detectable (i.e., the Lie derivative of the Riemannian metric along the system vector field is negative in the space tangent to the output function level sets). Also the differential detectability is related to the observability of the system’s linearization along its solutions. This relation is the starting point of techniques for designing a Riemannian metric exhibiting the differential detectability, assuming the system is strongly infinitesimally observable (i.e., each time-varying linear system resulting from the linearization along a solution to the system satisfies a uniform observability property) or is strongly differentially observable (i.e., the mapping state to output derivatives is an injective immersion). On the other hand, if the observer has an infinite gain margin then the level sets of the output function are geodesically convex. We give a full description of the family of Riemannian metrics giving this property and show how to express it when the system is strongly differentially observable of an order equal to the state dimension. We also discuss how it can be obtained via an immersion and a dynamic extension. Conversely, we establish that, if we have a complete Riemannian metric exhibiting the differential detectability property and making the level sets of the output function totally geodesic, then there exists an observer with an infinite gain margin. Actually, already without this last property, we can get a locally convergent observer.
Friday, 09:00 - 09:45 : Nathan Van de Wouw : The convergence property and its applications in systems and control
In this talk, the convergence property, which is a system-level stability property of nonlinear dynamical systems originally introduced in the 1960's in Russia, will be discussed in detail. A convergent system exhibits a unique, bounded globally asymptotically steady-state solution. Lyapunov characterisations of, sufficient conditions for and properties of convergent systems will be presented. Moreover, relations to notions such as e.g. incremental stability will be briefly addressed. In the second part of the talk it will be advocated that convergence is a useful property in the analysis and design of nonlinear control systems. Particular applications are: steady-state performance analysis through `nonlinear frequency response functions', output regulation (with tracking, synchronisation as particular sub-problems), observer design, model reduction and extremum seeking control.
Friday, 10:15 - 11:00 : Bjorn Rüffer : Contraction analysis, Extreme stability, and Convergent Dynamics
There are three system properties associated with incremental stability. The first is incremental stability itself, variations of which have been considered as "extreme" stability already almost a century ago. The second is the notion of contraction analysis, which originates from fluid mechanics ideas, and, like the first property, is about all trajectories of a system converging to each other. The third property, convergent dynamics, is intimately linked to the Demidovich condition. Again, if this condition holds, all trajectories converge to each other. Yet, the definitions of these three properties look very different. Naturally one asks, how related or different really are these properties? This talk intends to provide answers.
Friday, 11:15 - 12:00 : Fulvio Forni, Differentially passive systems beyond stability analysis
We discuss an interconnection theory for open nonlinear systems, revisiting passivity theory in the generalized framework of differential analysis. Taking advantage from contraction theory and monotone systems theory, we illustrate the potential of the theory for the analysis of multistable and oscillatory nonlinear systems.
Friday, 13:30 - 14:15 : Rodolfo Reyes-Baez, Trajectory tracking control of port-Hamiltonian mechanical systems via partial contraction
In this work, we propose a trajectory tracking control design for a class of mechanical systems in the port-Hamiltonian (pH) framework. This method is based on recent advances in contraction analysis and differential Lyapunov theory. The proposed controller renders a desired invariant sliding manifold attractive by making the closed-loop error system to be partially contracting with a pH-like structure. In order to show the performance of the controller, numerical simulations are presented.
Friday, 14:30 - 15:15 : Ian Manchester, Contraction and Convexity in Nonlinear System Design
The central problems in the design of nonlinear dynamic systems are highly challenging computionally. E.g., while the search for a Lyapunov (or storage) function to certify behavioural properties of a known dynamical system can be posed as a convex optimization problem, the search for a control Lyapunov function is non-convex. Similar problems arise in system identification, where the joint search for a model and a certificate of stability and model fidelity is highly non-convex. This talk will give an overview of some recent advances making use of differential dynamics, Riemannian geometry, contraction analysis, and Lagrangian relaxation. Well-known LMI methods for linear system design can be generalised to the nonlinear setting, and convex parameterizations and fidelity bounds can be derived for nonlinear system identification and reduction. Sum-of-squares relaxations yield effective finite-dimensional approximations of the resulting infinite-dimensional optimization problems. Extensions to scalable distributed algorithms for large-scale networked systems will be briefly discussed.
Friday, 15:45 - 16:30 : Cyrus Mostajeran, Invariant differential positivity
Invariant differential positivity is a property of dynamical systems on homogeneous spaces whose linearizations along trajectories are positive with respect to a cone field that is invariant with respect to the underlying geometry of the state space. The property can be viewed as a generalization of monotonicity in a linear space. In this talk, we will discuss the geometry of invariant cone fields and outline the mathematical framework for studying differential positivity of discrete and continuous-time dynamics with respect to invariant cone fields and motivate the use of this analysis framework through examples from nonlinear consensus theory on Lie groups and matrix function theory on the space of positive definite matrices.