In this thesis we apply linear feedback control to spatially evolving flows in order to minimize disturbance growth. The dynamics is assumed to be described by the linearized Navier--Stokes equations. Actuators and sensor are designed and a Kalman filtering technique is used to reconstruct the unknown flow state from noisy measurements. This reconstructed flow state is used to determine the control feedback which is applied to the Navier--Stokes equations through properly designed actuators. Since the control and estimation gains are obtained through an optimization process, and the Navier--Stokes equations typically forms a very high-dimensional system when discretized there is an interest in reducing the complexity of the equations. One possible approach is to perform Fourier decomposition along (almost) homogeneous spatial directions and another is by constructing a reduced order model by Galerkin projection on a suitable set of vectors. The first strategy is used to control the evolution of a range of instabilities in the classical family of Falkner--Skan--Cooke flows whereas the second is applied to a more complex cavity type of geometry. / QC 20101122

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Bilinear systems are a class of nonlinear dynamical systems that arise in a variety of applications. In order to obtain a sufficiently accurate representation of the underlying physical phenomenon, these models frequently have state-spaces of very large dimension, resulting in the need for model reduction. In this work, we introduce two new methods for the model reduction of bilinear systems in an interpolation framework. Our first approach is to construct reduced models that satisfy multipoint interpolation constraints defined on the Volterra kernels of the full model. We show that this approach can be used to develop an asymptotically optimal solution to the H_2 model reduction problem for bilinear systems. In our second approach, we construct a solution to a bilinear system realization problem posed in terms of constructing a bilinear realization whose kth-order transfer functions satisfy interpolation conditions in k complex variables. The solution to this realization problem can be used to construct a bilinear system realization directly from sampling data on the kth-order transfer functions, without requiring the formation of the realization matrices for the full bilinear system. / Ph. D.

To meet the requirements for the next generation of space missions, a paradigm shift is required from current structures that are static, heavy and stiff, to innovative structures that are adaptive, lightweight, versatile, and intelligent. This work proposes the use of a novel morphing structure, the thermally actuated anisogrid morphing boom, to meet the design requirements by making the primary structure actively adapt to the on-orbit environment. The proposed concept achieves the morphing capability by applying local and global thermal gradients and using the resulting thermal strains to introduce a 6 Degree of Freedom (DOF) morphing control. To address the key technical challenges associated with implementing this concept, the work is broken into four sections. First, the capability to develop and reduce large dynamic models using the Data Based Loewner-SVD method is demonstrated. This reduction method provides the computationally efficient dynamic models required for evaluation of the concept and the assessment of a vast number of loading cases. Secondly, a sensitivity analysis based parameter ranking methodology is developed to define parameter importance. A five parameter model correlation effort is used to demonstrate the ability to simplify complex coupled problems. By reducing the parameters to only the most critical, the resulting morphing optimization computation and engineering time is greatly reduced. The third piece builds the foundation for the thermal morphing anisogrid structure by describing the concept, defining the modeling assumptions, evaluating the design space, and building the performance metrics. The final piece takes the parameter ranking methodology, developed in part two, and the modeling capability of part three, and performs a trust-region optimization to define optimal morphing geometric configuration. The resulting geometry, optimized for minimum morphing capability, is evaluated to determine the morphing workspace, the frequency response capability, and the minimum and maximum morphing capability in 6 DOF. This work has demonstrated the potential and provided the technical tools required to model and optimize this novel smart structural concept for a variety of applications. / Ph. D.

This dissertation studies the model reduction and distributed control problems for interconnected systems, i.e., systems that consist of multiple interacting agents/subsystems. The study of the analysis and synthesis problems for interconnected systems is motivated by the multiple applications that can benefit from the design and implementation of distributed controllers. These applications include automated highway systems and formation flight of unmanned aircraft systems. The systems of interest are modeled using arbitrary directed graphs, where the subsystems correspond to the nodes, and the interconnections between the subsystems are described using the directed edges. In addition to the states of the subsystems, the adopted frameworks also model the interconnections between the subsystems as spatial states. Each agent/subsystem is assumed to have its own actuating and sensing capabilities. These capabilities are leveraged in order to design a controller subsystem for each plant subsystem. In the distributed control paradigm, the controller subsystems interact over the same interconnection structure as the plant subsystems. The models assumed for the subsystems are linear time-varying or linear parameter-varying. Linear time-varying models are useful for describing nonlinear equations that are linearized about prespecified trajectories, and linear parameter-varying models allow for capturing the nonlinearities of the agents, while still being amenable to control using linear techniques. It is clear from the above description that the size of the model for an interconnected system increases with the number of subsystems and the complexity of the interconnection structure. This motivates the development of model reduction techniques to rigorously reduce the size of the given model. In particular, this dissertation presents structure-preserving techniques for model reduction, i.e., techniques that guarantee that the interpretation of each state is retained in the reduced order system. Namely, the sought reduced order system is an interconnected system formed by reduced order subsystems that are interconnected over the same interconnection structure as that of the full order system. Model reduction is important for reducing the computational complexity of the system analysis and control synthesis problems. In this dissertation, interior point methods are extensively used for solving the semidefinite programming problems that arise in analysis and synthesis. / Ph. D. / The work in this dissertation is motivated by the numerous applications in which multiple agents interact and cooperate to perform a coordinated task. Examples of such applications include automated highway systems and formation flight of unmanned aircraft systems. For instance, one can think of the hazardous conditions created by a fire in a building and the benefits of using multiple interacting multirotors to deal with this emergency situation and reduce the risks on humans. This dissertation develops mathematical tools for studying and dealing with these complex systems. Namely, it is shown how controllers can be designed to ensure that such systems perform in the desired way, and how the models that describe the systems of interest can be systematically simplified to facilitate performing the tasks of mathematical analysis and control design.

The optimal estimation theory of the Wiener-Kalman filter is extended to cover the situation in which the number of memory elements in the estimator is restricted. A method, based on the simultaneous diagonalisation of two symmetric positive definite matrices, is given which allows the weighted least square estimation error to be minimised. A control system design method is developed utilising this estimator, and this allows the dynamic controller in the feedback path to have a low order. A 12-order once-through boiler model is constructed and the performance of controllers of various orders generated by the design method is investigated. Little cost penalty is found even for the one-order controller when compared with the optimal Kalman filter system. Whereas in the Kalman filter all information from past observations is stored, the given method results in an estimate of the state variables which is a weighted sum of the selected information held in the storage elements. For the once-through boiler these weighting coefficients are found to be smooth functions of position, their form illustrating the implicit model reduction properties of the design method. Minimal-order estimators of the Luenberger type also generate low order controllers and the relation between the two design methods is examined. It is concluded that the design method developed in this thesis gives better plant estimates than the Luenberger system and, more fundamentally, allows a lower order control system to be constructed. Finally some possible extensions of the theory are indicated. An immediate application is to multivariable control systems, while the existence of a plant state estimate even in control systems of very low order allows a certain adaptive structure to be considered for systems with time-varying parameters.

L’objectif est de construire un modèle d’écoulement qui se prête bien à des problèmes de contrôle, en associant un faible nombre de degrés de liberté à la possibilité de décrire la dynamique d’un écoulement relativement complexe. Dans ce travail nous considérons un écoulement bidimensionnel laminaire autour d’un cylindre carré. Des actionneurs placés sur le cylindre permettent un contrôle actif par sou?age et aspiration. Ce contrôle peut être dé?ni par rétroaction, exploitant des mesures de la vitesse dans le sillage du cylindre. Nous construisons un modèle d’ordre réduit (ROM) des équations de Navier-Stokes incompressibles, basé sur la technique de décomposition orthogonale aux valeurs propres (POD). Une façon classique de construire un tel modèle est de réaliser une projection Galerkin des équations sur le sous-espace réduit obtenu par POD. Un tel modèle peut cependant être peu précis, voire instable. Une technique de calibration est alors mise en place pour assurer la bonne représentativité dynamique du modèle. Nous dé?nissons ensuite une stratégie pour mettre à jour le modèle au cours d’un processus d’optimisation. La méthode est en?n appliquée pour réduire la di?érence entre l’écoulement contrôlé et la solution stationnaire instable à Re = 150. / The aim is to build a ?ow model adapted for control applications combining a low number of degrees of freedom with the possibility of describing relatively complex ?ows. In this work a two-dimensional laminar ?ow past a square cylinder is considered. Actuators placed on the cylinder enable active control by blowing and suction. Proportional feedback control can then be applied using velocity measurements taken in the cylinder wake. The proper orthogonal decom- position (POD) approach is used to build a low order model of the incompressible Navier-Stokes equations. A classical way of obtaining a Reduced-Order Model (ROM) is to perform a Galerkin projection of the equations onto the subspace spanned by the POD modes. Such a model can however be inaccurate, even unstable. A calibration technique is therefore applied, leading to a model that is accurate and robust to variations of the control parameters. A strategy is then de?ned to update the model within an optimisation loop. The method is tested at Re = 150 for reducing the di?erence between the actuated ?ow ?eld and the steady unstable solution.

La dynamique d'instabilité d'un écoulement laminaire décollé est étudiée expérimentalement et son contrôle par le biais de la simulation numérique. La configuration étudiée est une couche limite laminaire décollée au dessus d'une géométrie de type bosse.Pour une certaine gamme de paramètres, l'écoulement de recirculation en aval de la bosse est caractérisé par un battement basse fréquence. L'étude expérimentale de cette dynamique a permis de retrouver les différents régimes d'instabilité mis a jour par voie numérique. Ces résultats prouvent notamment que les instabilités basse fréquence, dont l'existence a été surtout mise en évidence dans des configurations d'écoulements compressibles, sont un phénomène générique pour des bulles de recirculations allongées. Le contrôle du battement basse fréquence est ensuite étudié par voie numérique suivant deux approches complémentaires. Un asservissement en boucle fermée de la dynamique de perturbation linéaire est tout d'abord proposé. Les modes d'instabilité linéaires sont utilisés afin de construire des modèles réduits de la dynamique de perturbation. Cette réduction de modèle donne lieu à des estimateurs de faible dimension capables d'estimer la dynamique et de la contrôler. Ainsi la dynamique d'instabilité linéaire peut être supprimée en couplant le système de Navier-Stokes linéarisé avec le contrôleur.Le contrôle de la dynamique non linéaire est ensuite étudié en utilisant une méthode d'optimisation Lagrangienne. Cette méthode permet de calculer les lois de contrôle à partir de la dynamique non linéaire des équations de Navier-Stokes. / The dynamics and control of a separated boundary-layer flow have been investigated. Separation is triggered by a bump mounted on a flat plate and the transition dynamics has been investigated experimentally. For a certain parameter range, the recirculation region is subject to self-sustained low-frequency oscillations, and results from the numerical simulation for the same geometry are recovered. These results show that low frequency oscillations, observed mainly in compressible flow regimes, are inherent to elongated recirculation bubbles.The control of this low-frequency instability has been investigated using modern control theory based on two complementary approaches. Feedback control of the linear perturbation dynamics is first considered. Global instability modes are used to build reduced-order estimators. This model reduction gives rise to low-dimensional compensators capable of controlling the unstable dynamics. Once coupled to the unstable linearised Navier-Stokes system, the compensator is seen to succesfully control the unstable dynamics. The control of the nonlinear dynamics is then investigated using adjoint-based optimisation procedures. This method is used to compute control laws based on a complete knowledge of the nonlinear dynamics. Although the low-frequency instability is clearly attenuated, it seems difficult to control the flow towards its steady state, using only a few blowing/suction actuators localized on the wall.

Model order reduction is necessary even in a time where the parallel processing is usual in almost any personal computer. The recent Model Reduction Methods are useful tools nowadays on reducing the problem processing. This work intends to describe a combination between POD (Proper Orthogonal Decomposition) and Ritz vectors that achieve an efficient Galerkin projection that changes during the processing, comparing the development of the error and the convergence rate between the full space and the projection space, in addition to check the stability of the projection space, leading to an adaptive model order reduction for nonlinear dynamical problems more efficient. This model reduction is supported by a secant formulation, which is updated by BFGS (Broyden - Fletcher - Goldfarb - Shanno) method to accelerate convergence of the model, and a tangent formulation to correct the projection space. Furthermore, this research shows that this method permits a correction of the reduced model at low cost, especially when the classical POD is no more efficient to represent accurately the solution. / A Redução de ordem de modelo é necessária, mesmo em uma época onde o processamento paralelo é usado em praticamente qualquer computador pessoal. Os recentes métodos de redução de modelo são ferramentas úteis nos dias de hoje para a redução de processamento de um problema. Este trabalho pretende descrever uma combinação entre POD (Proper Orthogonal Decomposition) e vetores de Ritz para uma projecção de Galerkin eficiente que sofre alterações durante o processamento, comparando o desenvolvimento do erro e a taxa de convergência entre o espaço total e o espaço de projeção, além da verificação de estabilidade do espaço de projeção, levando a uma redução de ordem do modelo adaptativo mais eficiente para problemas dinâmicos não-lineares. Esta redução de modelo é assistida por uma formulação secante, que é atualizado pela formula de BFGS (Broyden - Fletcher- Goldfarb - Shanno) com o intuito de acelerar a convergência do modelo, e uma formulação tangente para a correção do espaço de projeção. Além disso, esta pesquisa mostra que este método permite a correção do modelo reduzido com baixo custo, especialmente quando o clássico POD não é mais eficiente para representar com precisão a solução.

Hybrid neural-microelectronic systems, systems composed of biological neural networks and neuronal models, have great potential for the treatment of neural injury and disease. The utility of such systems will be ultimately determined by the ability of the engineered component to correctly replicate the function of biological neural networks. These models can take the form of mechanistic models, which reproduce neural function by describing the physiologic mechanisms that produce neural activity, and empirical models, which reproduce neural function through more simplified mathematical expressions. We present our research into the role of model complexity in creating robust and flexible behaviors in hybrid systems. Beginning with a complex mechanistic model of a leech heartbeat interneuron, we create a series of three systematically reduced models that incorporate both mechanistic and empirical components. We then evaluate the robustness of these models to parameter variation, and assess the flexibility of the models activities. The modeling studies are validated by incorporating both mechanistic and semi-empirical models in hybrid systems with a living leech heartbeat interneuron. Our results indicate that model complexity serves to increase both the robustness of the system and the ability of the system to produce flexible outputs.