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Methods in hydrodynamic stability, systems and control theory are applied to spatially developing flows, where the flow is not required to vary slowly in the streamwise direction. A substantial part of the thesis presents a theoretical framework for the stability analysis, input-output behavior, model reduction and control design for fluid dynamical systems using examples on the linear complex Ginzburg-Landau equation. The framework is then applied to high dimensional systems arising from the discretized Navier–Stokes equations. In particular, global stability analysis of the three-dimensional jet in cross flow and control design of two-dimensional disturbances in the flat-plate boundary layer are performed. Finally, a parametric study of the passive control of two-dimensional disturbances in a flat-plate boundary layer using streamwise streaks is presented. / QC 20101103

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.

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.

Simulation, design, and control of dynamical systems play an important role in numerous scientific and industrial tasks. The need for detailed models leads to large-scale dynamical systems, posing tremendous computational difficulties when employed in numerical simulations. In order to overcome these challenges, we perform model reduction, replacing the large-scale dynamics with high-fidelity reduced representations. There exist a plethora of methods for reduced order modeling of linear systems, including the Iterative Rational Krylov Algorithm (IRKA), Balanced Truncation (BT), and Hankel Norm Approximation. However, these methods generally target stable systems and the approximation is performed over an infinite time horizon. If we are interested in a finite horizon reduced model, we utilize techniques such as Time-limited Balanced Truncation (TLBT) and Proper Orthogonal Decomposition (POD). In this dissertation we establish interpolation-based optimality conditions over a finite horizon and develop an algorithm, Finite Horizon IRKA (FHIRKA), that produces a locally optimal reduced model on a specified time-interval. Nonetheless, the quantities being interpolated and the interpolant are not the same as in the infinite horizon case. Numerical experiments comparing FHIRKA to other algorithms further support our theoretical results. Next, we discuss model reduction for nonlinear dynamical systems. For models with unstructured nonlinearities, POD is the method of choice. However, POD is input dependent and not optimal with respect to the output. Thus, we use operator splitting to integrate the best features of system theoretic approaches with trajectory based methods such as POD in order to mitigate the effect of the control inputs for the approximation of nonlinear dynamical systems. We reduce the linear terms with system theoretic methods and the nonlinear terms terms via POD. Evolving the linear and nonlinear terms separately yields the reduced operator splitting solution. We present an error analysis for this method, as well as numerical results that illustrate the effectiveness of our approach. While in this dissertation we only pursue the splitting of linear and nonlinear terms, this approach can be implemented with Quadratic Bilinear IRKA or Balanced Truncation for Quadratic Bilinear systems to further diminish the input dependence of the reduced order modeling. / Doctor of Philosophy / Simulation, design, and control of dynamical systems play an important role in numerous scientific and industrial tasks such as signal propagation in the nervous system, heat dissipation, electrical circuits and semiconductor devices, synthesis of interconnects, prediction of major weather events, spread of fires, fluid dynamics, machine learning, and many other applications. The need for detailed models leads to large-scale dynamical systems, posing tremendous computational difficulties when applied in numerical simulations. In order to overcome these challenges, we perform model reduction, replacing the large-scale dynamics with high-fidelity reduced representations. Reduced order modeling helps us to avoid the outstanding burden on computational resources. Numerous model reduction techniques exist for linear models over an infinite horizon. However, in practice we usually are interested in reducing a model over a specific time interval. In this dissertation, given a reduced order, we present a method that finds the best local approximation of a dynamical system over a finite horizon. We present both theoretical and numerical evidence that supports the proposed method. We also develop an algorithm that integrates operator splitting with model reduction to solve nonlinear models more efficiently while preserving a high level of accuracy.

Dynamical systems are a commonly used and studied tool for simulation, optimization and design. In many applications such as inverse problem, optimal control, shape optimization and uncertainty quantification, those systems typically depend on a parameter. The need for high fidelity in the modeling stage leads to large-scale parametric dynamical systems. Since these models need to be simulated for a variety of parameter values, the computational burden they incur becomes increasingly difficult. To address these issues, parametric reduced models have encountered increased popularity in recent years. We are interested in constructing parametric reduced models that represent the full-order system accurately over a range of parameters. First, we define a global joint error mea- sure in the frequency and parameter domain to assess the accuracy of the reduced model. Then, by assuming a rational form for the reduced model with poles both in the frequency and parameter domain, we derive necessary conditions for an optimal parametric reduced model in this joint error measure. Similar to the nonparametric case, Hermite interpolation conditions at the reflected images of the poles characterize the optimal parametric approxi- mant. This result extends the well-known interpolatory H2 optimality conditions by Meier and Luenberger to the parametric case. We also develop a numerical algorithm to construct locally optimal reduced models. The theory and algorithm are data-driven, in the sense that only function evaluations of the parametric transfer function are required, not access to the internal dynamics of the full model. While this first framework operates on the continuous function level, assuming repeated transfer function evaluations are available, in some cases merely frequency samples might be given without an option to re-evaluate the transfer function at desired points; in other words, the function samples in parameter and frequency are fixed. In this case, we construct a parametric reduced model that minimizes a discretized least-squares error in the finite set of measurements. Towards this goal, we extend Vector Fitting (VF) to the parametric case, solving a global least-squares problem in both frequency and parameter. The output of this approach might lead to a moderate size reduced model. In this case, we perform a post- processing step to reduce the output of the parametric VF approach using H2 optimal model reduction for a special parametrization. The final model inherits the parametric dependence of the intermediate model, but is of smaller order. A special case of a parameter in a dynamical system is a delay in the model equation, e.g., arising from a feedback loop, reaction time, delayed response and various other physical phenomena. Modeling such a delay comes with several challenges for the mathematical formulation, analysis, and solution. We address the issue of transient behavior for scalar delay equations. Besides the choice of an appropriate measure, we analyze the impact of the coefficients of the delay equation on the finite time growth, which can be arbitrary large purely by the influence of the delay. / Ph. D. / Mathematical models play an increasingly important role in the sciences for experimental design, optimization and control. These high fidelity models are often computationally expensive and may require large resources, especially for repeated evaluation. Parametric model reduction offers a remedy by constructing models that are accurate over a range of parameters, and yet are much cheaper to evaluate. An appropriate choice of quality measure and form of the reduced model enable us to characterize these high quality reduced models. Our first contribution is a characterization of optimal parametric reduced models and an efficient implementation to construct them. While this first framework assumes we have access to repeated evaluations of the full model, in some cases merely measurement data might be available. In this case, we construct a parametric model that fits the measurements in a least squares sense. The output of this approach might lead to a moderate size reduced model, which we address with a post-processing step that reduces the model size while maintaining important properties. A special case of a parameter is a delay in the model equation, e.g., arising from a feedback loop, reaction time, delayed response and various other physical phenomena. While asymptotically stable solutions eventually vanish, they might grow large before asymptotic behavior takes over; this leads to the notion of transient behavior, which is our main focus for a simple class of delay equations. Besides the choice of an appropriate measure, we analyze the impact of the structure of the delay equation on the transient growth, which can be arbitrary large purely by the influence of the delay.

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. / 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 combination of a unique structure with a new control method to provide a novel way to achieve smart structural adaptive (morphing) capability. This concept takes advantage of the natural expansion and contraction of materials caused by temperature changes (Coefficient of Thermal Expansion) to bend and twist the structure of interest into the desired configuration. By applying local heat (thermal energy) to induce temperature differences in the structure, the temperature induced material expansion (thermal strain) can be used to control and direct the structure into all different planes and rotations(X, Y, and Z displacements as well as the rotations in the X, Y, and Z axis). This structure is called an anisogrid boom. The potential applications of model range from passive thermal mitigation system to an active precision low frequency vibration control system. The anisogrid boom is a low mass and efficient structural configuration with a long history in aerospace applications. The thermally morphing anisogrid structure is made up of helical members (these members are heated and cooled to actively bend and twist the structure) and circumferential members (these members provide stability and rigidity to the structure). This work details the methods used to move and control the structure as well as the development of the technical capabilities required to effectively demonstrate the limits of the concept. To address the key technical challenges associated with implementing this concept, the work is broken into four sections. First, the method is detailed that is used to generate the large dynamic computer models and the method to reduce them using a model reduction method, Data Based Loewner-SVD. This reduction method provides the computationally-efficient dynamic models required for evaluation of the concept. Secondly, a novel methodology is discussed identifying the impact of an input parameter change on the resulting output, such that the parameters of importance can be ranked. This ranking provides insight into what parameters should be the focus for further evaluation and optimization. The last two sections address how these tools are used to demonstrate the performance of this novel morphing structure as a function of the geometric input parameters relative to multiple developed performance metrics. Finally, the structure is optimized to achieve the most accurate morphing possible so that the limits of the capability can be better understood. 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.

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.