Non-linear Dynamics Commons^™

Control Theory: The Double Pendulum Inverted On A Cart, Ian J P Crowe-Wright

Mathematics & Statistics ETDs

In this thesis the Double Pendulum Inverted on a Cart (DPIC) system is modeled using the Euler-Lagrange equation for the chosen Lagrangian, giving a second-order nonlinear system. This system can be approximated by a linear first-order system in which linear control theory can be used. The important definitions and theorems of linear control theory are stated and proved to allow them to be utilized on a linear version of the DPIC system. Controllability and eigenvalue placement for the linear system are shown using MATLAB. Linear Optimal control theory is likewise explained in this section and its uses are applied to …

Simulating The Electrical Properties Of Random Carbon Nanotube Networks Using A Simple Model Based On Percolation Theory, Roberto Abril Valenzuela

Physics

Carbon nanotubes (CNTs) have been subject to extensive research towards their possible applications in the world of nanoelectronics. The interest in carbon nanotubes originates from their unique variety of properties useful in nanoelectronic devices. One key feature of carbon nanotubes is that the chiral angle at which they are rolled determines whether the tube is metallic or semiconducting. Of main interest to this project are devices containing a thin film of randomly arranged carbon nanotubes, known as carbon nanotube networks. The presence of semiconducting tubes in a CNT network can lead to a switching effect when the film is electro-statically …

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

Near-Optimal Control Of Switched Systems With Continuous-Time Dynamics Using Approximate Dynamic Programming, Tohid Sardarmehni

Mechanical Engineering Research Theses and Dissertations

Optimal control is a control method which provides inputs that minimize a performance index subject to state or input constraints [58]. The existing solutions for finding the exact optimal control solution such as Pontryagin’s minimum principle and dynamic programming suffer from curse of dimensionality in high order dynamical systems. One remedy for this problem is finding near optimal solution instead of the exact optimal solution to avoid curse of dimensionality [31]. A method for finding the approximate optimal solution is through Approximate Dynamic Programming (ADP) methods which are discussed in the subsequent chapters.

In this dissertation, optimal switching in switched …

Theoretical Analysis Of Nonlinear Differential Equations, Emily Jean Weymier

Electronic Theses and Dissertations

Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, …