This thesis explores the connection between Crouzeix's conjecture and the convergence of the GMRES algorithm. GMRES is a popular iterative method for solving linear systems and is one of the many Krylov methods. Despite its popularity, the convergence of GMRES is not completely understood. While the spectrum can in some cases be a good indicator of convergence, it has been shown that in general, the spectrum does not provide sufficient information to fully explain the behavior of GMRES iterations. Other sets associated with a matrix that can also help predict convergence are the pseudospectrum and the numerical range. This work focuses on convergence bounds obtained by considering the latter. In particular, it focuses on the application of Crouzeix's conjecture, which relates the norm of a matrix polynomial to the size of that polynomial over the numerical range, to describing GMRES convergence.

In the modern dynamic engineering field, experimental dynamics is an important area of study. This area includes structural dynamics, structural control, and structural health monitoring. In experimental dynamics, methods to obtain measured data have seen a great influx of research efforts to develop an accurate and reliable experimental analysis result. A technical challenge is the procurement of informative data that exhibits the desired system information. In many cases, the number of sensors is limited by cost and difficulty of data archive. Furthermore, some informative data has technical difficulty when measuring input force and, even if obtaining the desired data were possible, it could include a lot of noise in the measuring data. As a result, researchers have developed many analytical tools with limited informative data. Subspace identification method is used one of tools in these achievements. Subspace identification method includes three different approaches: Deterministic Subspace Identification (DSI), Stochastic Subspace Identification (SSI), and Deterministic-Stochastic Subspace Identification (DSSI). The subspace identification method is widely used for fast computational speed and its accuracy. Based on the given information, such as output only, input/output, and input/output with noises, DSI, SSI, and DSSI are differently applied under specific assumptions, which could affect the analytical results. The objective of this study is to observe the effect of assumptions on subspace identification with various data conditions. Firstly, an analytical simulation study is performed using a sixdegree-of-freedom mass-damper-spring system which is created using MATLAB. Various conditions of excitation insert to the simulation test model, and its excitation and response are iv analyzed using the subspace identification method. For stochastic problems, artificial noise is contained to the excitation and followed the same steps. Through this simulation test, the effects of assumption on subspace identification are quantified. Once the effects of the assumptions are studied using the simulation model, the subspace identification method is applied to dynamic response data collected from large-scale 12-story buildings with different foundation types that are tested at Tongji University, Shanghai, China. Noise effects are verified using three different excitation types. Furthermore, using the DSSI, which has the most accurate result, the effect of different foundations on the superstructure are analyzed.

A formulation of a subspace identification method for multi-phase processes with applications to rotational moulding and suggestions for improvements and experimental applications. / This thesis focuses on the implementation of subspace identification (SSID) for nonlinear, chemical batch processes by introducing a model identification method for multi-phase processes. In this thesis, a multi-phase process refers to chemical or biological batch-like processes with properties that cause a change in the dynamics during the evolution of the process. This can occur, for example, when a process undergoes a change of state upon reaching a melting point. Existing SSID techniques are not designed to utilize any known, multiphase nature of a process in the model identification stage. The proposed approach, Multiphase Subspace Identification (MPSSID), is conducted by first splitting historical data into phases during the identification step and then building a subspace model for each phase. The phases are then connected via a partial least squares (PLS) model that transforms the states from one phase to the next. This approach makes use of existing SSID techniques that allow for model construction using batches of nonunifrom length. Here, MPSSID is applied to a uniaxial rotational moulding process. In rotational moulding, the dynamics switch as the process undergoes heating, melting, and sintering stages that are visibly distinct and recognizable upon a certain temperature (not time) being reached. Results demonstrate the ability of multiphase models to better predict the temperature trajectories and final product quality of validation batches. As an extension to this rotational moulding analysis, additional MPSSID methods of implementation are proposed and the results are compared. A MPSSID mixed integer linear program is then introduced for implementation within model predictive control. The applications to rotational moulding are presented within the context of plastics manufacturing and the impact of plastic on the global climate crisis, with suggestions for future work. / Thesis / Master of Applied Science (MASc) / The control of chemical processes is an important factor in achieving high quality products. To control a process well, the mathematical model of the system must be accurate. In the past, mathematical models for process control were designed based on engineering approximations. Now, with major advances in computing and sensor technology, it is possible to design a simulation of the entire process. These simulations can be designed using first-principles or black box approaches. First-principles approaches utilize rigorous models that are based on the complex chemical and physical formulas that govern a system. Black box approaches do not look at the first-principles dynamics. They only utilize the measured process inputs and outputs to form a model of the system. They are widely used because of their ease of implementation in comparison to first-principles approaches. In this thesis, a new black box process control model is proposed and is found to yield better theoretical results than existing techniques. This model is tested on data from a plastics manufacturing process called rotational moulding, which involves loading polymer powders into a mould that is simultaneously rotated and heated to yield seamless plastic parts. Lastly, a control framework that is compatible with the new black box model is proposed to be used for future experimental tests.

Data-driven model predictive control framework of semicontinuous distillation process / Distillation technology is one of the most sought-after operations in the chemical process industries. Countless research has been done in the past to reduce the cost associated with distillation technology. As a result of process intensification, a semicontinuous distillation system is proposed as an alternative for purifying the n-component mixture (n>=3) which has the advantage over both batch and continuous process for low to medium production rates. A traditional distillation setup requires n-1 columns to separate the components to the desired purity. However, a semicontinuous system performs the same task by integrating a distillation column with n-2 middle vessel (storage tank). Consequently, with lower capital cost, the total annualized cost (TAC) per tonne of feed processed is less for a semicontinuous system compared to a traditional setup for low to medium throughput. Yet, the operating cost of a semicontinuous system exceed those of the conventional continuous setup. Semicontinuous system exhibits a non-linear dynamic behavior with a cyclic steady state and has three modes of operation. The main goal of this thesis is to reduce the operating cost per tonne of feed processed which leads to lower TAC per tonne of feed processed using a model predictive control (MPC) scheme compared to the existing PI configuration This work proposes a novel multi-model technique using subspace identification to identify a linear model for each mode of operation without attaining discontinuity. Subsequently, the developed multi-model framework was implemented in a shrinking horizon MPC architecture to reduce the TAC/tonne of feed processed while maintaining the desired product purities at the end of each cycle. The work uses Aspen Plus Dynamics simulation as a test bed to simulate the semicontinuous system and the shrinking horizon MPC scheme is formulated in MATLAB. VBA is used to communicate the inputs from MPC in MATLAB to the process in Aspen Plus Dynamics. / Thesis / Master of Science in Chemical Engineering (MSChE)

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