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Stability Analysis of Nonlinear Dynamical Systems With Lévy Typeperturbations
(01. Izmir Institute of Technology, 2023) Tamcı, Ege; Batal, Ahmet; Savacı, Ferit Acar
In order to model the noise in power networks, generally, normal distribution is used. However, normal distribution is not convenient in modelling noise which has sudden peaks. Instead, combination of a continuous process and a jump processes is much more suitable. With this idea in mind, in this thesis, the stability and control of two equations used in modeling power grids is analyzed, under the assumption that they are exposed to Lévy process noise which includes jumps. These equations are the swing equation and the Kuramoto Model. The swing equation is used to model the single machine infinite bus system (SMIBS). Kuramoto Model is used when a large number of generators are considered as a network of coupled oscillators with their own natural frequencies. In our stability control study in the SMIBS, the noise in the system has sudden and finite changes is assumed and therefore should be modelled with a modified tempered α-stable process obtained by adding a finite jump condition on the tempered α-stable process when α < 1. The control functions depending on the mechanical power input and damping parameters are designed in order to make the system stable in probability and exponential stable at its equilibrium point. These theoretical results are supported by numerical studies. For Kuromato model, assuming that the power network consists of two layers, namely oscillator, and control layers and that is affected with a general Lévy process which has finite jumps, functions which provide the stability of phase and frequencies are obtained, depending on oscillator and coupling strengths. In the light of the numerical studies, the control of frequency and phase synchronization up to a certain noise intensity level can be evaluated, but it is not possible beyond that level is concluded.
Boundary Feedback Stabilization of Some Evolutionary Partial Differential Equations
(01. Izmir Institute of Technology, 2022) Yılmaz, Kemal Cem; Batal, Ahmet; Özsarı, Türker; Özsarı, Türker; Batal, Ahmet
The purpose of this study is to control long time behaviour of solutions to some evolutionary partial differential equations posed on a finite interval by backstepping type controllers. At first we consider right endpoint feedback controller design problem for higher-order Schrödinger equation. The second problem is observer design problem, which has particular importance when measurement across the domain is not available. In this case, the sought after right endpoint control inputs involve state of the observer model. However, it is known that classical backstepping strategy fails for designing right endpoint controllers to higher order evolutionary equations. So regarding these controller and observer design problems, we modify the backstepping strategy in such a way that, the zero equilibrium to the associated closed-loop systems become exponentially stable. From the well-posedness point of view, this modification forces us to obtain a time-space regularity estimate which also requires to reveal some smoothing properties for some associated Cauchy problems and an initial-boundary value problem with inhomogeneous boundary conditions. As a third problem, we introduce a finite dimensional version of backstepping controller design for stabilizing infinite dimensional dissipative systems. More precisely, we design a boundary control input involving projection of the state onto a finite dimensional space, which is still capable of stabilizing zero equilibrium to the associated closed-loop system. Our approach is based on defining the backstepping transformation and introducing the associated target model in a novel way, which is inspired from the finite dimensional long time behaviour of dissipative systems. We apply our strategy in the case of reaction-diffusion equation. However, it serves only as a canonical example and our strategy can be applied to various kind of dissipative evolutionary PDEs and system of evolutionary PDEs. We also present several numerical simulations that support our theoretical results.

Phd Degree / Doktora

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