Flatness-based Deformation Control of an Euler-Bernoulli Beam with In-domain Actuation

This paper addresses the problem of deformation control of an Euler-Bernoulli beam with in-domain actuation. The proposed control scheme consists in first relating the system model described by an inhomogeneous partial differential equation to a target system under a standard boundary control form. Then, a combination of closed-loop feedback control and flatness-based motion planning is used for stabilizing the closed-loop system around reference trajectories. The validity of the proposed method is assessed through well-posedness and stability analysis of the considered systems. The performance of the developed control scheme is demonstrated through numerical simulations of a representative micro-beam.Comment: Preprint of an original research wor

Infinite-Dimensional Modelling and Control of a MEMS Deformable Mirror with Applications in Adaptive Optics

RÉSUMÉ Le contrôle de déformation est un problème émergent dans les micro structures intelligentes. Une des applications type est le contrôle de la déformation de miroirs dans l’optique adaptative dans laquelle on oriente la face du miroir selon une géométrie précise en utilisant une gamme de micro-vérins afin d’éliminer la distortion lumineuse. Dans cette thèse, le problème de la conception du contrôle du suivi est considéré directement avec les modèles décrits par des équations aux dérivées partielles définies dans l’espace de dimension infinie. L’architecture du contrôleur proposée se base sur la stabilisation par retour des variables et le suivi des trajectoires utilisant la théorie des systèmes différentiellement plats. La combinaison de la commande par rétroaction et la planification des trajectoires permet de réduire la complexité de la structure du contrôleur pour que ce dernier puisse être implémentée dans les microsystèmes avec les techniques disponibles de nos jours. Pour aboutir à une architecture implémentable dans les applications en temps réel, la fonction de Green est considérée comme une fonction de test pour concevoir le contrôleur et pour représenter les trajectoires de référence dans la planification de mouvements.----------ABSTRACT Deformation control is an emerging problem for micro-smart structures. One of its exciting applications is the control of deformable mirrors in adaptive optics systems, in which the mirror face-sheet is steered to a desired shape using an array of micro-actuators in order to remove light distortions. This technology is an enabling key for the forthcoming extremely large ground-based telescopes. Large-scale deformable mirrors typically exhibit complex dynamical behaviors mostly due to micro-actuators distributed in the domain of the system which in particular complicates control design. A model of this device may be described by a fourth-order in space/second-order in time partial differential equation for the mirror face-sheet with Dirac delta functions located in the domain of the system to represent the micro-actuators. Most of control design methods dealing with partial differential equations are performed on lumped models, which often leads to high-dimensional and complex feedback control structures. Furthermore, control designs achieved based on partial differential equation models correspond to boundary control problems. In this thesis, a tracking control scheme is designed directly based on the infinite-dimensional model of the system. The control scheme is introduced based on establishing a relationship between the original nonhomogeneous model and a target system in a standard boundary control form. Thereby, the existing boundary control methods may be applicable. For the control design, we apply the tool of differential flatness to a partial differential equation system controlled by multiple actuators, which is essentially a multiple-input multiple-output partial differential equation problem. To avoid early lumping in the motion planning, we use the properties of the Green’s function of the system to represent the reference trajectories. A finite set of these functions is considered to establish a one-to-one map between the input space and output space. This allows an implementable scheme for real-time applications. Since pure feedforward control is only applicable for perfectly known, and stable systems, feedback control is required to account for instability, model uncertainties, and disturbances. Hence, a stabilizing feedback is designed to stabilize the system around the reference trajectories. The combination of differential flatness for motion planning and stabilizing feedback provides a systematic control scheme suitable for the real-time applications of large-scale deformable mirrors

In-Domain Control of Partial Differential Equations

RÉSUMÉ Cette thèse porte sur la commande des systèmes à dimension infinie décrit par les équa-tions aux dérivées partielles (EDP). La commande d’EDP peut être divisée approximative-ment en deux catégories en fonction de l’emplacement des actionneurs: la commande à la frontière, où les actionnements sont appliqués à la frontière des systèmes d’EDP, et la com-mande dans le domaine, où les actionneurs pénètrent à l’intérieur du domaine des systèmes d’EDP. Dans cette thèse, nous étudierons la commande dans le domaine de l’équation d’Euler-Bernoulli, de l’équation de Fisher, l’équation de Chafee-Infante et de l’équation de Burgers. L’équation d’Euler-Bernoulli est un modèle classique d’EDP linéaire décrivant la flexion pure des structures flexibles. L’équation de Fisher et l’équation de Chafee-Infante sont des EDP paraboliques semi-linéaires, qui peuvent être utilisées pour modéliser certains phénomènes physiques, chimiques ou biologiques. L’équation de Burgers peut être considérée comme une simplification d’équations de Navier-Stokes en mécanique des fluides, en dynamique des gaz, en fluidité de la circulation, etc. Ces systèmes jouent des rôles très importants en mathéma-tiques, en physique et dans d’autres domaines. Dans cette thèse, de nouvelles méthodes qui se basent sur la dynamique des zéros et le compensateur dynamique ont été développées pour la conception et l’implémentation de lois de commande pour la commande des EDP avec des actionnements dans le domaine. Tout d’abord, nous étudions le contrôle de l’équation d’Euler-Bernoulli avec plusieurs actionneurs internes. L’inverse de la dynamique des zéros a été utilisé dans la conception de la loi de commande, ce qui permet de suivre la trajectoire prescrit souhaitée. Afin de concevoir la trajectoire souhaitée, la fonction de Green est utilisée pour déterminer la commande sta-tique. La planification de mouvement est générée par des contrôleurs dynamiques basés sur la méthode de platitude di˙érentielle. Pour les équations paraboliques non linéaires, la dy-namique des zéros est régie par une EDP non linéaire. Par conséquent, nous avons recours à la méthode de décomposition d’Adomian (ADM) pour générer la commande dynamique afin de suivre les références désirées. Dans le cas de l’équation de Burgers, un compensateur dynamique a été utilisé. Pour obtenir la stabilité globale de l’équation de Burgers contrôlée, une rétroaction non linéaire a été appliquée à la frontière. La méthode d’ADM et la platitude ont été utilisées dans l’implémentation du compensateur dynamique.----------ABSTRACT This thesis addresses in-domain control of partial di˙erential equation (PDE) systems. PDE control can in general be classified into two categories according to the location of the ac-tuators: boundary control, where the actuators are assigned to the boundary of the PDE systems, and in-domain control, where the actuation penetrates inside the domain of the PDE systems. This thesis investigates the in-domain control of some well-known PDEs, including the Euler-Bernoulli equation, the Fisher’s equation, the Chafee-Infante equation, and Burgers’ equation. Euler-Bernoulli equation is a classical linear PDE used to describe the pure bending of flexible structures. Fisher’s equation and the Chafee-Infante equation are semi-linear parabolic PDEs that can be used to model physical, chemical, and biolog-ical phenomena. Burgers’ equation can be viewed as simplified Navier-Stokes equations in lower dimensions in applied mathematics, and it has been widely adopted in fluid mechan-ics, gas dynamics, traÿc flow modeling, etc. These PDE systems play important roles in mathematics, physics, and other fields. In this work, in-domain control of linear and semi-linear parabolic equations are treated based on dynamic compensators. First, we consider the in-domain control of an Euler-Bernoulli equation with multiple internal actuators. The method of zero dynamics inverse is adopted to derive the in-domain control to allow an asymptotic tracking of the prescribed desired outputs. A linear proportional boundary feedback control is employed to stabilize the Euler-Bernoulli equation around its zero dynamics. To design the desired trajectory, the Green’s function is employed to determine the static control, and then motion planning is generated by dynamic control based on di˙erential flatness. For the semi-linear parabolic equations, zero dynamics are governed by nonlinear PDEs. Therefore, the implementation of the in-domain control of linear PDEs cannot be directly applied. We resort then to the Adomian decomposition method (ADM) to implement the dynamic control in order to track the desired set-points. Finally, the in-domain control of a Burgers’ equation is addressed based on dynamic compensator. A nonlinear boundary feedback control is used to achieve the global stability of the controlled Burgers’ equation, and the ADM as well as the flatness are used in the implementation of the proposed in-domain control scheme

Inverse dynamics of underactuated flexible mechanical systems governed by quasi-linear hyperbolic partial differential equations

Diese Arbeit befasst sich mit der inversen Dynamik unteraktuierter, flexibler, mechanischer Systeme, welche durch quasi-lineare hyperbolische partielle Differentialgleichungen beschrieben werden können. Diese Gleichungnen, sind zeitlich veränderlichen Dirichlet-Randbedingungen unterworfen, welche durch unbekannte, räumlich disjunkte, also nicht kollokierte Neumann-Randbedingungen erzwungen werden. Die zugrundeliegenden Gleichungen werden zunächst abstrakt hergeleitet, bevor verschiedene mechanische Systeme vorgestellt werden können, die mit der eingangs postulierten Formulierung übereinstimmen. Hierzu werden geometrisch exakte Theorien hergeleitet, welche in der Lage sind große Bewegungen schlanker Strukturen wie Seile und Balken, aber auch ganz allgemein, dreidimensionaler Festkörper zu beschreiben. In der Regel werden Anfangs-Randwertprobleme, die in der nichtlinearen Strukturdynamik auftreten, durch Anwendung einer sequentiellen Diskretisierung in Raum und Zeit gelöst. Diese Verfahren basieren für gewöhnlich auf einer räumlichen Diskretisierung mit finiten Elementen, gefolgt von einer geeigneten zeitlichen Diskretisierung, welche meist auf finiten Differenzen beruht. Ein kurzer Überblick über derartige sequentielle Integrationsverfahren für das vorliegende Anfangs-Randwertproblem wird zunächst anhand der direkten Formulierung des Problems gegeben werden. D.h. es wird zunächst das reine Neumann-Randproblem betrachtet, bevor anschließend ganz allgemein, verschiedene Möglichkeiten zur Einbindung etwaiger Dirichlet-Randbedingungen diskutiert werden. Darauf aufbauend wird das Problem der inversen Dynamik im Kontext räumlich diskreter mechanischer Systeme, welche rheonom-holonomen Servo-Bindungen unterliegen, eingeführt. Eine ausführliche Untersuchung dieser Art von gebundenen Systemen soll die grundlegenden Unterschiede zwischen Servo-Bindungen und klassischen Kontakt-Bindungen herausarbeiten. Die daraus resultierenden Folgen für die Entwicklung geeigneter numerisch stabiler Integrationsverfahren können dabei ebenfalls angesprochen werden, bevor zahlreich ausgewählte Beispiele vorgestellt werden können. Aufgrund der sehr eingeschränkten Anwendbarkeit der sequentiellen Lösung der inversen Dynamik in Raum und Zeit, wird eine eingehende Analyse des vorliegenden Anfangs-Randwertproblems unternommen. Vor allem durch die Freilegung der hyperbolischen Struktur der zugrundeliegenden partiellen Differentialgleichungen werden sich weitere Einblicke in das vorliegende Problem erhofft. Die Erforschung der daraus resultierenden Mechanismen der Wellenausbreitung in kontinuierlichen Strukturen öffnet die Tür zur Entwicklung numerisch stabiler Integrationsverfahren für die inverse Dynamik. So kann unter anderem eine Methode vorgestellt werden, die auf der Integration der partiellen Differentialgleichungen entlang charakteristischer Mannigfaltigkeiten beruht. Dies regt zu der Entwicklung neuartiger Galerkinverfahren an, die ebenfalls in dieser Arbeit vorgestellt werden können. Diese neu entwickelten Methoden können anschlie\ss end auf die Steuerung verschiedener mechanischer Systeme angewendet werden. Darüber hinaus können die neuartigen Integrationsverfahren auch auf flexible Mehrkörpersysteme übertragen werden. Angeführt seien hier beispielsweise die kooperative Steuerung eines an mehreren flexiblen Seilen aufgehängten starren Körpers oder die Steuerung des Endeffektors eines flexiblen mehrgliedrigen Schwenkarms. Ausgewählte numerische Beispiele verdeutlichen die Relevanz der hier vorgeschlagenen, in Raum und Zeit simultanen Integration des vorliegenden Anfangs-Randwertproblems

On the Use of Piezoelectric Sensors in Structural Mechanics: Some Novel Strategies

In the present paper, a review on piezoelectric sensing of mechanical deformations and vibrations of so-called smart or intelligent structures is given. After a short introduction into piezoelectric sensing and actuation of such controlled structures, we pay special emphasis on the description of some own work, which has been performed at the Institute of Technical Mechanics of the Johannes Kepler University of Linz (JKU) in the last years. Among other aspects, this work has been motivated by the fact that collocated control of smart structures requires a sensor output that is work-conjugated to the input by the actuator. This fact in turn brings into the play the more general question of how to measure mechanically meaningful structural quantities, such as displacements, slopes, or other quantities, which form the work-conjugated quantities of the actuation, by means piezoelectric sensors. At least in the range of small strains, there is confidence that distributed piezoelectric sensors or sensor patches in smart structures do measure weighted integrals over their domain. Therefore, there is a need of distributing or shaping the sensor activity in order to be able to re-interpret the sensor signals in the desired mechanical sense. We sketch a general strategy that is based on a special application of work principles, more generally on displacement virials. We also review our work in the past on bringing this concept to application in smart structures, such as beams, rods and plates

Mathematical Modeling, Motion Planning and Control of Elastic Structures with Piezoelectric Elements

The objective of this work is the development of a motion planning and tracking control approach for elastic structures. Motivated by the morphing wing concept of the field of aerospace engineering a so-called “smart wingsail” defines the center of the presented research. The motion planning and tracking control approach has to achieve different rest-to-rest motions of the wingsail’s transversal displacement. The design of the mechanical structure as well as the control concept of the wingsail relies on the results of proof of concept studies. For this purpose, different systems of interconnected bending beams are considered which emulates parts of the wingsail. The development of the model based control approaches requires an accurate system description. The modeling itself is done by an analytic energy based approach for the beams’ systems, where for the wingsail the finite elements method is used due to the risen complexity of the curved structure. To achieve a precise description of the governing dynamics different parameter identification concepts are discussed and applied. This leads to a precise but rather complex system description which covers the measured behavior of the experimental setups. Considering the objective of a real time capable control approach the complexity has to be reduced without a significant loss of accuracy. For this purpose different model order reduction techniques are discussed and applied. The resulting systems models are the bases of the control designs. Two different control concepts are presented and evaluated. At first the two-degrees-of-freedom control approach is introduced which combines a flatness-based feedforward control approach with a feedback controller. On the other hand, the so-called model predictive control approach is presented which is based on the solution of an optimization problem. Both concepts are evaluated by numeric analyses and by experiments

Linear-Quadratic Control of a MEMS Micromirror Using Kalman Filtering

The deflection limitations of electrostatic flexure-beam actuators are well known. Specifically, as the beam is actuated and the gap traversed, the restoring force necessary for equilibrium increases proportionally with the displacement to first order, while the electrostatic actuating force increases with the inverse square of the gap. Equilibrium, and thus stable open-loop voltage control, ceases at one-third the total gap distance, leading to actuator snap-in. A Kalman Filter is designed with an appropriately complex state dynamics model to accurately estimate actuator deflection given voltage input and capacitance measurements, which are then used by a Linear Quadratic controller to generate a closed-loop voltage control signal. The constraints of the latter are designed to maximize stable control over the entire gap. The design and simulation of the Kalman Filter and controller are presented and discussed, with static and dynamic responses analyzed, as applied to basic, 100 micrometer by 100 micrometer square, flexure-beam-actuated micromirrors fabricated by PolyMUMPs. Successful application of these techniques enables demonstration of smooth, stable deflections of 50% and 75% of the gap

SPECIFIED MOTION AND FEEDBACK CONTROL OF ENGINEERING STRUCTURES WITH DISTRIBUTED SENSORS AND ACTUATORS

This dissertation addresses the control of flexible structures using distributed sensors and actuators. The objective to determine the required distributed actuation inputs such that the desired output is obtained. Two interrelated facets of this problem are considered. First, we develop a dynamic-inversion solution method for determining the distributed actuation inputs, as a function of time, that yield a specified motion. The solution is shown to be useful for intelligent structure design, in particular, for sizing actuators and choosing their placement. Secondly, we develop a new feedback control method, which is based on dynamic inversion. In particular, filtered dynamic inversion combines dynamic inversion with a low-pass filter, resulting in a high-parameter-stabilizing controller, where the parameter gain is the filter cutoff frequency. For sufficiently large parameter gain, the controller stabilizes the closed-loop system and makes the L2-gain of the performance arbitrarily small, despite unknown-and-unmeasured disturbances. The controller is considered for both linear and nonlinear structural models

Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft

Nano-optomechanical measurement in the photon counting regime

Optically measuring in the photon counting regime is a recurrent challenge in modern physics and a guarantee to develop weakly invasive probes. Here we investigate this idea on a hybrid nano-optomechanical system composed of a nanowire hybridized to a single Nitrogen-Vacancy (NV) defect. The vibrations of the nanoresonator grant a spatial degree of freedom to the quantum emitter and the photon emission event can now vary in space and time. We investigate how the nanomotion is encoded on the detected photon statistics and explore their spatio-temporal correlation properties. This allows a quantitative measurement of the vibrations of the nanomechanical oscillator at unprecedentedly low light intensities in the photon counting regime when less than one photon is detected per oscillation period, where standard detectors are dark-noise-limited. These results have implications for probing weakly interacting nanoresonators, for low temperature experiments and for investigating single moving markers