Krylov subspace techniques for model reduction and the solution of linear matrix equations

This thesis focuses on the model reduction of linear systems and the solution of large scale linear matrix equations using computationally efficient Krylov subspace techniques. Most approaches for model reduction involve the computation and factorization of large matrices. However Krylov subspace techniques have the advantage that they involve only matrix-vector multiplications in the large dimension, which makes them a better choice for model reduction of large scale systems. The standard Arnoldi/Lanczos algorithms are well-used Krylov techniques that compute orthogonal bases to Krylov subspaces and, by using a projection process on to the Krylov subspace, produce a reduced order model that interpolates the actual system and its derivatives at infinity. An extension is the rational Arnoldi/Lanczos algorithm which computes orthogonal bases to the union of Krylov subspaces and results in a reduced order model that interpolates the actual system and its derivatives at a predefined set of interpolation points. This thesis concentrates on the rational Krylov method for model reduction. In the rational Krylov method an important issue is the selection of interpolation points for which various techniques are available in the literature with different selection criteria. One of these techniques selects the interpolation points such that the approximation satisfies the necessary conditions for H2 optimal approximation. However it is possible to have more than one approximation for which the necessary optimality conditions are satisfied. In this thesis, some conditions on the interpolation points are derived, that enable us to compute all approximations that satisfy the necessary optimality conditions and hence identify the global minimizer to the H2 optimal model reduction problem. It is shown that for an H2 optimal approximation that interpolates at m interpolation points, the interpolation points are the simultaneous solution of m multivariate polynomial equations in m unknowns. This condition reduces to the computation of zeros of a linear system, for a first order approximation. In case of second order approximation the condition is to compute the simultaneous solution of two bivariate polynomial equations. These two cases are analyzed in detail and it is shown that a global minimizer to the H2 optimal model reduction problem can be identified. Furthermore, a computationally efficient iterative algorithm is also proposed for the H2 optimal model reduction problem that converges to a local minimizer. In addition to the effect of interpolation points on the accuracy of the rational interpolating approximation, an ordinary choice of interpolation points may result in a reduced order model that loses the useful properties such as stability, passivity, minimum-phase and bounded real character as well as structure of the actual system. Recently in the literature it is shown that the rational interpolating approximations can be parameterized in terms of a free low dimensional parameter in order to preserve the stability of the actual system in the reduced order approximation. This idea is extended in this thesis to preserve other properties and combinations of them. Also the concept of parameterization is applied to the minimal residual method, two-sided rational Arnoldi method and H2 optimal approximation in order to improve the accuracy of the interpolating approximation. The rational Krylov method has also been used in the literature to compute low rank approximate solutions of the Sylvester and Lyapunov equations, which are useful for model reduction. The approach involves the computation of two set of basis vectors in which each vector is orthogonalized with all previous vectors. This orthogonalization becomes computationally expensive and requires high storage capacity as the number of basis vectors increases. In this thesis, a restart scheme is proposed which restarts without requiring that the new vectors are orthogonal to the previous vectors. Instead, a set of two new orthogonal basis vectors are computed. This reduces the computational burden of orthogonalization and the requirement of storage capacity. It is shown that in case of Lyapunov equations, the approximate solution obtained through the restart scheme approaches monotonically to the actual solution

Implicit Higher-Order Moment Matching Technique for Model Reduction of Quadratic-bilinear Systems

We propose a projection based multi-moment matching method for model order reduction of quadratic-bilinear systems. The goal is to construct a reduced system that ensures higher-order moment matching for the multivariate transfer functions appearing in the input-output representation of the nonlinear system. An existing technique achieves this for the first two multivariate transfer functions, in what is called the symmetric form of the multivariate transfer functions. We extend this framework to an equivalent and simplified form, the regular form, which allows us to show moment matching for the first three multivariate transfer functions. Numerical results for three benchmark examples of quadratic-bilinear systems show that the proposed framework exhibits better performance with reduced computational cost in comparison to existing techniques.Comment: 19 pages, 11 subfigures in 6 figures, Journa

IoT Based Real Time Early Age Concrete Compressive Strength Monitoring

Concrete Strength determination has been an expensive and hectic job due to its orthodox methodology of measuring concrete strength where cylinders are filled with concrete. Its strength is measured using the crushing of concrete (Compression Test). A significant amount of waste is generated while performing this test multiple times during the execution of the project. The present study proposes a new IoT-based framework comprising a low-cost sensor and a window dashboard to estimate and monitor the real-time early-age concrete strength. This system will significantly help the construction industry to avoid the onsite laboratory testing of concrete for strength. In this study, a temperature sensor, along with an ESP32 microprocessor, is used to acquire and transmit the recorded temperature in real time to a cloud database. The window application developed load data from the cloud database and presented it as figures and graphs related to concrete strength with time. The strength calculated using the developed sensor was compared with the actual strength determined using a compression test for the same mix design, which showed a significant match. The project is a contribution toward the non-destructive testing of concrete. By knowing the concrete strength of any structural member in advance, the practitioners can make decisions well before time to avoid delays in the project

Study protocol of DIVERGE, the first genetic epidemiological study of major depressive disorder in Pakistan

INTRODUCTION: Globally, 80% of the burdenof major depressive disorder (MDD) pertains to low- and middle-income countries. Research into genetic and environmental risk factors has the potential to uncover disease mechanisms that may contribute to better diagnosis and treatment of mental illness, yet has so far been largely limited to participants with European ancestry from high-income countries. The DIVERGE study was established to help overcome this gap and investigate genetic and environmental risk factors for MDD in Pakistan. METHODS: DIVERGE aims to enrol 9000 cases and 4000 controls in hospitals across the country. Here, we provide the rationale for DIVERGE, describe the study protocol and characterise the sample using data from the first 500cases. Exploratory data analysis is performed to describe demographics, socioeconomic status, environmental risk factors, family history of mental illness and psychopathology. RESULTS AND DISCUSSION: Many participants had severe depression with 74% of patients who experienced multiple depressive episodes. It was a common practice to seek help for mental health struggles from faith healers and religious leaders. Socioeconomic variables reflected the local context with a large proportion of women not having access to any education and the majority of participants reporting no savings. CONCLUSION: DIVERGE is a carefully designed case-control study of MDD in Pakistan that captures diverse risk factors. As the largest genetic study in Pakistan, DIVERGE helps address the severe underrepresentation of people from South Asian countries in genetic as well as psychiatric research

A New Multilevel Inverter Topology for Grid-Connected Photovoltaic Systems

The demand for clean and sustainable energy has spurred research in all forms of renewable energy sources, including solar energy from photovoltaic systems. Grid-connected photovoltaic systems (GCPS) provide an effective solution to integrate solar energy into the existing grid. A key component of the GCPS is the inverter. The inverter can have a significant impact on the overall performance of the GCPS, including maximum power point (MPP) tracking, total harmonic distortion (THD), and efficiency. Multilevel inverters are one of the most promising classes of converters that offer a low THD. In this paper, we propose a new multilevel inverter topology with the motivation to improve all the three aforementioned aspects of performance. The proposed topology is controlled through direct model predictive control (DMPC), which is state of the art in control techniques. We compare the performance of the proposed topology with the topologies reported in literature. The proposed topology offers one of the best efficiency, MPP tracking, and voltage THD