Transverse exponential stability and applications

We investigate how the following properties are related to each other: i)-A manifold is "transversally" exponentially stable; ii)-The "transverse" linearization along any solution in the manifold is exponentially stable; iii)-There exists a field of positive definite quadratic forms whose restrictions to the directions transversal to the manifold are decreasing along the flow. We illustrate their relevance with the study of exponential incremental stability. Finally, we apply these results to two control design problems, nonlinear observer design and synchronization. In particular, we provide necessary and sufficient conditions for the design of nonlinear observer and of nonlinear synchronizer with exponential convergence property

Contraction analysis of nonlinear systems and its application

The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents

Contraction analysis of nonlinear systems and its application

The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents

Contracting Nonlinear Observers: Convex Optimization and Learning from Data

A new approach to design of nonlinear observers (state estimators) is proposed. The main idea is to (i) construct a convex set of dynamical systems which are contracting observers for a particular system, and (ii) optimize over this set for one which minimizes a bound on state-estimation error on a simulated noisy data set. We construct convex sets of continuous-time and discrete-time observers, as well as contracting sampled-data observers for continuous-time systems. Convex bounds for learning are constructed using Lagrangian relaxation. The utility of the proposed methods are verified using numerical simulation.Comment: conference submissio

Cigarette Smoke Up-regulates PDE3 and PDE4 to Decrease cAMP in Airway Cells

BACKGROUND AND PURPOSE: 3', 5'-cyclic adenosine monophosphate (cAMP) is a central second messenger that broadly regulates cell function and can underpin pathophysiology. In chronic obstructive pulmonary disease (COPD), a lung disease primarily provoked by cigarette smoke (CS), the induction of cAMP-dependent pathways, via inhibition of hydrolyzing phosphodiesterases (PDEs), is a prime therapeutic strategy. Mechanisms that disrupt cAMP signaling in airway cells, in particular regulation of endogenous PDEs are poorly understood. EXPERIMENTAL APPROACH: We used a novel Förster resonance energy transfer (FRET) based cAMP biosensor in mouse in vivo, ex vivo precision cut lung slices (PCLS), and in human in vitro cell models to track the effects of CS exposure. KEY RESULTS: Under fenoterol stimulated conditions, FRET responses to cilostamide were significantly increased in in vivo, ex vivo PCLS exposed to CS and in human airway smooth muscle cells exposed to CS extract. FRET signals to rolipram were only increased in the in vivo CS model. Under basal conditions, FRET responses to cilostamide and rolipram were significantly increased in in vivo, ex vivo PCLS exposed to CS. Elevated FRET signals to rolipram correlated with a protein upregulation of PDE4 subtypes. In ex vivo PCLS exposed to CS extract, rolipram reversed downregulation of ciliary beating frequency, whereas only cilostamide significantly increased airway relaxation of methacholine pre-contracted airways. CONCLUSION AND IMPLICATIONS: We show that CS upregulates expression and activity of both PDE3 and PDE4, which regulate real-time cAMP dynamics. These mechanisms determine the availability of cAMP and can contribute to CS-induced pulmonary pathophysiology

Contraction analysis of switched systems with application to control and observer design

In many control problems, such as tracking and regulation, observer design, coordination and synchronization, it is more natural to describe the stability problem in terms of the asymptotic convergence of trajectories with respect to one another, a property known as incremental stability. Contraction analysis exploits the stability properties of the linearized dynamics to infer incremental stability properties of nonlinear systems. However, results available in the literature do not fully encompass the case of switched dynamical systems. To overcome these limitations, in this thesis we present a novel extension of contraction analysis to such systems based on matrix measures and differential Lyapunov functions. The analysis is conducted first regularizing the system, i.e. approximating it with a smooth dynamical system, and then applying standard contraction results. Based on our new conditions, we present design procedures to synthesize switching control inputs to incrementally stabilize a class of smooth nonlinear systems, and to design state observers for a large class of nonlinear switched systems including those exhibiting sliding motion. In addition, as further work, we present new conditions for the onset of synchronization and consensus patterns in complex networks. Specifically, we show that if network nodes exhibit some symmetry and if the network topology is properly balanced by an appropriate designed communication protocol, then symmetry of the nodes can be exploited to achieve a synchronization/consensus pattern

Integrated Application of Active Controls (IAAC) technology to an advanced subsonic transport project: Current and advanced act control system definition study. Volume 2: Appendices

The current status of the Active Controls Technology (ACT) for the advanced subsonic transport project is investigated through analysis of the systems technical data. Control systems technologies under examination include computerized reliability analysis, pitch axis fly by wire actuator, flaperon actuation system design trade study, control law synthesis and analysis, flutter mode control and gust load alleviation analysis, and implementation of alternative ACT systems. Extensive analysis of the computer techniques involved in each system is included