<p>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.</p>

<p>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.</p>

We investigate model reduction of large-scale linear time-invariant systems in generalized state-space form. We consider sparse state matrix pencils, including pencils with banded structure. The balancing-based methods employed here are composed of well-known linear algebra operations and have been recently shown to be applicable to large models by exploiting the structure of the matrices defining the dynamics of the system. In this paper we propose a modification of the LR-ADI iteration to solve large-scale generalized Lyapunov equations together with a practical convergence criterion, and several other implementation refinements. Using kernels from several serial and parallel linear algebra libraries, we have developed a parallel package for model reduction, SpaRed, extending the applicability of balanced truncation to sparse systems with up to $O(10^5)$ states. Experiments on an SMP parallel architecture consisting of Intel Itanium 2 processors illustrate the numerical performance of this approach and the potential of the parallel algorithms for model reduction of large-scale sparse systems.

Model reduction is a common theme within the simulation, control and optimization of complex dynamical systems. For instance, in control problems for partial differential equations, the associated large-scale systems have to be solved very often. To attack these problems in reasonable time it is absolutely necessary to reduce the dimension of the underlying system. We focus on model reduction by balanced truncation where a system theoretical background provides some desirable properties of the reduced-order system. The major computational task in balanced truncation is the solution of large-scale Lyapunov equations, thus the method is of limited use for really large-scale applications. We develop an effective implementation of balancing-related model reduction methods in exploiting the structure of the underlying problem. This is done by a data-sparse approximation of the large-scale state matrix A using the hierarchical matrix format. Furthermore, we integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the Lyapunov equations. This approach is well-suited for a class of practical relevant problems and allows the application of balanced truncation and related methods to systems coming from 2D and 3D FEM and BEM discretizations.

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order model. Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reduced-order model. We provide a systematic approach built on established interpolatory $\mathcal{H}_2$ optimal model reduction methods that will produce parameterized reduced-order models having high fidelity throughout a parameter range of interest. For single input/single output systems with parameters in the input/output maps, we provide reduced-order models that are \emph{optimal} with respect to an $\mathcal{H}_2\otimes\mathcal{L}_2$ joint error measure. The capabilities of these approaches are illustrated by several numerical examples from technical applications.

We discuss the numerical solution of large-scale sparse projected discrete-time periodic Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Tires serve as important components of wheeled vehicles and their analytical modeling has drawn the attention of many researches in the past decades. A high-resolution finite element (FE) tire model contains detailed structural and material characteristics of a tire that exhibit degrees-of-freedom (DoF) in the order of 10⁵ or greater. However, such high-resolution models in their full detail are not practically applicable in multibody dynamic analysis of vehicles and a reduction in their order becomes necessary. In this research different formulations to construct condensed FE tire models suitable for multibody simulations are developed and their characteristics are discussed. In addition, two new and novel forms of substructuring are presented that aim at isolating the contact region of a tire without the need for keeping the boundary DoF which otherwise remain in the reduced system in the standard substructuring procedures. The new substructuring methods provide a great tool in constructing condensed FE tire models with much less total number of DoF compared to cases where a standard substructuring is used. In order to increase the computational efficiency of the condensed FE tire models even further, the possibility of model condensation in the contact region is studied. This research also addresses the applicability of available friction models into the condensed FE tire models. Different formulations of a condensed tire model presented in this research are used to construct several computational models. These models are utilized to simulate certain scenarios and the results are discussed.

Methods of component mode synthesis, such as Craig and Bampton reduction, are known to generally yield more accurate results in deformable multibody dynamics. The main shortcoming of those methods is that they are intuitively based. Recently Nikravesh developed a reduction method called mode condensation which is derived from the equations of motion and yields the same results as Craig and Bampton reduction. In this dissertation, it is proven that these two methods span the same column space; therefore, they should yield identical results. We propose that mode condensation provides an analytical justification for Craig and Bampton reduction. Test results suggest that Craig and Bampton reduction and mode condensation are appropriate for a broader range of applications because their column space matches up well with the conditions under which the deformable body is restrained. Although Guyan reduction preserves exact solutions for static problems, its applications shall be limited to low frequency excitation because of raised eigen-frequencies. Modal truncation is not recommended for use in multibody dynamic settings because it lacks the ability to receive forces and displacements at the moving boundary. Another issue addressed in this dissertation is the misconception that if mean axes are adopted as the moving reference frame, only free-free modes should be used for model reduction. It was not clear how a restrained deformable body with mean axes can be condensed properly. We have shown that the conventional (nodal-fixed) mode shapes can be used with mean axes as long as the transformation matrix has full rank and contains complete rigid-body mode shapes.

No description available.

Good process control is often critical for the economic viability of large-scale production of several commercial products. In this work, the production of biodiesel from microalgae is investigated. Successful implementation of a model-based control strategy requires the identification of a model that properly captures the biochemical dynamics of microalgae, yet is simple enough to allow its implementation for controller design. For this purpose, two model reparameterization algorithms are proposed that partition the parameter space into estimable and inestimable subspaces. Both algorithms are applied using a first principles ODE model of a microalgal bioreactor, containing 6 states and 12 unknown parameters. Based on initial simulations, the non-linear algorithm achieved better degree of output prediction when compared to the linear one at a greatly decreased computational cost. Using the parameter estimates obtained through implementation of the non-linear algorithm on experimental data from a fed-batch bioreactor, the possible improvement in volumetric productivity was recognized. / Process Control