Fulvio Forni

A Differential Lyapunov Framework for Contraction Analysis

Lyapunovs second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves.

Event-triggered transmission for linear control over communication channels

We consider an exponentially stable closed loop interconnection between a continuous-time linear plant and a continuous-time linear controller, and we study the problem of interconnecting the plant output to the controller input through a digital channel. We propose an event-triggered transmission policy whose goal is to transmit the measured plant output information as little as possible while preserving closed-loop stability. Global asymptotic stability is guaranteed when the plant state is available or when an estimate of the state is available (provided by a classical continuous-time linear observer). Under further assumptions, the transmission policy guarantees global exponential stability of the origin.

Follow the Bouncing Ball: Global Results on Tracking and State Estimation With Impacts

In this paper, we formulate tracking and state-estimation problems of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous trajectories arising from the impacts, we use hybrid systems stability analysis tools to establish the results. Using a novel concept of mirrored images of the target mass we prove that 1) a tracking control algorithm, and 2) an observer algorithm guarantee global exponential stability results for specific classes of polyhedral billiards, including rectangles. Moreover, we combine these two algorithms within dynamic controllers that guarantee global output feedback tracking. The results are illustrated via simulations.

On Differentially Dissipative Dynamical Systems

Dissipativity is an essential concept of systems theory. The paper provides an extension of dissipativity, named differential dissipativity, by lifting storage functions and supply rates to the tangent bundle. Differential dissipativity is connected to incremental stability in the same way as dissipativity is connected to stability. It leads to a natural formulation of differential passivity when restricting to quadratic supply rates. The paper also shows that the interconnection of differentially passive systems is differentially passive, and provides preliminary examples of differentially passive electrical systems.

Lyapunov-Based Sufficient Conditions for Exponential Stability in Hybrid Systems

Lyapunov-based sufficient conditions for exponential stability in hybrid systems are presented. The focus is on converting non-strict Lyapunov conditions, having certain observability properties, into strict Lyapunov conditions for exponential stability. Both local and global results are considered. The utility of the results is illustrated through an example.

Technical communique: Gain-scheduled, model-based anti-windup for LPV systems

The aim of this paper is to show that a recently proposed technique for anti-windup control of exponentially unstable plants can be easily extended to solve the corresponding robust anti-windup problem for linear parameter varying systems, for which the time varying parameters are measured online. The proposed technique is minimally conservative with respect to the size of the resulting operating region (which coincides, up to an arbitrarily small quantity, with the largest set on which asymptotic stability can be guaranteed for the considered plant with the given saturation level and uncertainty characteristics), and is not limited to plants having only small uncertainties or being open-loop stable.

Differentially Positive Systems

The paper introduces and studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. A generalization of Perron-Frobenius theory is developed in this differential framework to show that the property induces a conal order that strongly constrains the asymptotic behavior of solutions. The results illustrate that behaviors constrained by local order properties extend beyond the well-studied class of linear positive systems and monotone systems, which both require a constant cone field and a linear state space.

Tracking control in billiards using mirrors without smoke, Part I: Lyapunov-based local tracking in polyhedral regions

In this paper we formulate the tracking problem of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous dynamics arising from nonsmooth impacts, the tracking problem is formulated within a hybrid systems framework and a Lyapunov function is given, which decreases during flow (continuous motion) and remains constant across jumps (impacts of the masses). To guarantee non-increase of the Lyapunov function at jumps, we introduce a novel concept of mirrored images of the target mass and prove that, with this concept, local tracking is achieved. Several simulations illustrate the effectiveness of the proposed approach as compared to alternative solutions. In a companion paper [6] we address global results and generalize the local approach to curved billiards.

On differential passivity of physical systems

Differential passivity is a property that allows to check with a pointwise criterion that a system is incrementally passive, a property that is relevant to study interconnected systems in the context of regulation, synchronization, and estimation. The paper investigates how restrictive is the property, focusing on a class of open gradient systems encountered in the coenergy modeling framework of physical systems, in particular the Brayton-Moser formalism for nonlinear electrical circuits.

Tracking control in billiards using mirrors without smoke, Part II: Additional Lyapunov-based local and global results

Two control results are described: 1) local tracking control for convex billiards with piecewise locally Lipschitz boundary, and 2) global tracking control for special polyhedral billiards, including rectangles and equilateral triangles. The controllers are based on Lyapunov functions and a mirroring concept introduced in a companion paper. The local results require the impacts to satisfy an average dwell-time condition with parameters that depend on the Lipschitz constant of the function that characterizes the boundary. For piecewise constant boundary, and for the global results, the average dwell-time parameters are arbitrary. Tools from stability analysis for hybrid systems are used to establish the results.