Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.Comment: submitted to EP

Regulation of inhomogeneous drilling model with a P-I controller

International audienceIn this paper, we demonstrate that a Proportional Integral controller allows the regulation of the angular velocity of a drill-string despite unknown frictional torque and measuring only the angular velocity at the surface. Our model is an one dimensional damped inhomogeneous wave equation subject to an unknown dynamic at one side while the control and the measurement are in the other side. After writing this system of balance laws into the Riemann coordinates, we design a Lyapunov functional to prove the exponential stability of the closed-loop and show how it implies the regulation of the angular velocity

Stability analysis of coupled ordinary differential systems with a string equation: application to a drilling mechanism

Cette thèse porte sur l'analyse de stabilité de couplage entre deux systèmes, l'un de dimension finie et l'autre infinie. Ce type de systèmes apparait en physique car il est intimement lié aux modèles de structures. L'analyse générique de tels systèmes est complexe à cause des natures très différentes de chacun des sous-systèmes. Ici, l'analyse est conduite en utilisant deux méthodologies. Tout d'abord, la séparation quadratique est utilisée pour traiter le côté fréquentiel de ce système couplé. L'autre méthode est basée sur la théorie de Lyapunov pour prouver la stabilité asymptotique de l'interconnexion. Tous ces résultats sont obtenus en utilisant la méthode de projection de l'état de dimension infinie sur une base polynomiale. Il est alors possible de prendre en compte le couplage entre les deux systèmes et ainsi d'obtenir des tests numériques fiables, rapides et peu conservatifs. De plus, une hiérarchie de conditions est établie dans le cas de Lyapunov. L'application au cas concret du forage pétrolier est proposée pour illustrer l'efficacité de la méthode et les nouvelles perspectives qu'elle offre. Par exemple, en utilisant la notion de stabilité pratique, nous avons montré qu'une tige de forage controlée à l'aide d'un PI est sujette à un cycle limite et qu'il est possible d'estimer son amplitude.This thesis is about the stability analysis of a coupled finite dimensional system and an infinite dimensional one. This kind of systems emerges in the physics since it is related to the modeling of structures for instance. The generic analysis of such systems is complex, mainly because of their different nature. Here, the analysis is conducted using different methodologies. First, the recent Quadratic Separation framework is used to deal with the frequency aspect of such systems. Then, a second result is derived using a Lyapunov-based argument. All the results are obtained considering the projections of the infinite dimensional state on a basis of polynomials. It is then possible to take into account the coupling between the two systems. That results in tractable and reliable numerical tests with a moderate conservatism. Moreover, a hierarchy on the stability conditions is shown in the Lyapunov case. The real application to a drilling mechanism is proposed to illustrate the efficiency of the method and it opens new perspectives. For instance, using the notion of practical stability, we show that a PI-controlled drillstring is subject to a limit cycle and that it is possible to estimate its amplitude

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

Enhancing vibration control in cable-tip-mass systems using asymmetric peak detector boundary control

In this study, a boundary controller based on a peak detector system has been designed to reduce vibrations in the cable–tip–mass system. The control procedure is built upon a recent modification of the controller, incorporating a non-symmetric peak detector mechanism to enhance the robustness of the control design. The crucial element lies in the identification of peaks within the boundary input signal, which are then utilized to formulate the control law. Its mathematical representation relies on just two tunable parameters. Numerical experiments have been conducted to assess the performance of this novel approach, demonstrating superior efficacy compared to the boundary damper control, which has been included for comparative purposes"This work has been funded by the Generalitat de Catalunya through the research projects 2021-SGR-01044."Peer ReviewedPostprint (published version

Suppression of drill-string stick–slip vibration by sliding mode control : Numerical and experimental studies

Peer reviewedPostprin

Exponential Stabilization of a Class of Nonlinear Neutral Type Time-Delay Systems, an Oilwell Drilling Model Example

International audienceThis paper deals with exponential stabilization of the class of nonlinear neutral type time-delay systems that can be transformed into a multi-model system. The approach is based on Lyapunov-Krasovskii techniques and uses a descriptor representation. The exponential stability properties are proved using an appropriate change of variables associated with a polytopic representation. The results are given in terms of LMIs. As an application example, we determine an e ective stabilizing controller for an oilwell drilling system

A Method of Drilling a Ground Using a Robotic Arm

Underground tunnel face bolting and pipe umbrella reinforcement are one of the most challenging tasks in construction whether industrial or not, and infrastructures such as roads or pipelines. It is one of the first sectors of economic activity in the world. Through a variety of soil and rock, a cyclic Conventional Tunneling Method (CTM) remains the best one for projects with highly variable ground conditions or shapes. CTM is the only alternative for the renovation of existing tunnels and creating emergency exit. During the drilling process, a wide variety of non-desired vibrations may arise, and a method using a robot arm is proposed. The main kinds of drilling through vibration here is the bit-bouncing phenomenon (resonant axial vibration). Hence, assisting the task by a robot arm may play an important role on drilling performances and security. We propose to control the axial-vibration phenomenon along the drillstring at a practical resonant frequency, and embed a Resonant Sonic Drilling Head (RSDH) as a robot end effector for drilling. Many questionable industry drilling criteria and stability are discussed in this paper

Delay-Adaptive Boundary Control of Coupled Hyperbolic PDE-ODE Cascade Systems

This paper presents a delay-adaptive boundary control scheme for a $2\times 2$ coupled linear hyperbolic PDE-ODE cascade system with an unknown and arbitrarily long input delay. To construct a nominal delay-compensated control law, assuming a known input delay, a three-step backstepping design is used. Based on the certainty equivalence principle, the nominal control action is fed with the estimate of the unknown delay, which is generated from a batch least-squares identifier that is updated by an event-triggering mechanism that evaluates the growth of the norm of the system states. As a result of the closed-loop system, the actuator and plant states can be regulated exponentially while avoiding Zeno occurrences. A finite-time exact identification of the unknown delay is also achieved except for the case that all initial states of the plant are zero. As far as we know, this is the first delay-adaptive control result for systems governed by heterodirectional hyperbolic PDEs. The effectiveness of the proposed design is demonstrated in the control application of a deep-sea construction vessel with cable-payload oscillations and subject to input delay