Doctor of Philosophy (PhD), Ohio University, 2024, Mathematics (Arts and Sciences)

The Lyapunov stability of equilibria in dynamical systems is determined by the interplay between the linearization and the nonlinear terms. In this work, we study the case when the spectrum of the linearization is diffusively stable with high-order spectral degeneracy at the origin. In particular, spatially periodic solutions called roll solutions at the zigzag boundary of the Swift-Hohenberg equation (SHE), typically selected by patterns and defects in numerical simulations, are shown to be nonlinearly stable. This also serves as an example where linear decay weaker than classical diffusive decay, together with quadratic nonlinearity, still gives nonlinear stability of spatially periodic patterns. The study is conducted on two physical domains: the 2D plane, $\R^2$, and the cylinder, $T_{2\pi}\times \R$. Linear analysis reveals that instead of the classical $t^{-1}$ diffusive decay rate, small localized perturbation of roll solutions with zigzag wavenumbers decay with slower algebraic rates ($t^{-\frac{3}{4}}$ for the 2D plane; $t^{-\frac{1}{4}}$ for the cylindrical domain) due to the high order degeneracy of the translational mode at the origin of the Bloch-Fourier spaces. The nonlinear stability proofs are based on decompositions of the neutral translational mode and the faster decaying modes, and fixed-point arguments, demonstrating the irrelevancy of the nonlinear terms.

Committee: Qiliang Wu (Advisor); Alexander Neiman (Committee Member); Todd Young (Committee Member); Tatiana Savin (Committee Member) Subjects: Mathematics

Doctor of Philosophy, University of Akron, 2020, Engineering-Applied Mathematics

A model is developed for predicting long-wavelength film thickness nonuniformities in drying latex paint films. After applying the lubrication approximation to the model equations, both linear and nonlinear stability analyses are performed. For the linear stability analysis, spatially independent base state solutions are found. These equations are solved numerically and a linear stability analysis of these base state solutions is conducted. In this case, the base state solution of the height represents a uniformly drying latex paint film with respect to time. For the nonlinear stability analysis, the leading order equations are solved numerically. The stability of the film is dependent on temperature, latex particle volume fraction, titanium dioxide particle volume fraction, extender particle volume fraction, surface surfactant density, bulk surfactant density, and several material and environmental factors. Slow evaporation, temperature gradients, surfactant desorption, surface tension gradients, low initial surface tension are identified as destabilizing mechanisms while fast evaporation, fast surfactant adsorption, high initial surface tension, high particle volume fractions, and viscosity are identified as stabilizing mechanisms.

Committee: Patrick Wilber (Advisor); Kevin Kreider (Committee Member); Curtis Clemons (Committee Member); George Chase (Committee Member); Qixin Zhou (Committee Member); Ali Dhinojwala (Committee Member) Subjects: Applied Mathematics; Chemical Engineering

Doctor of Philosophy, The Ohio State University, 2017, Electrical and Computer Engineering

Tail-controlled missiles, hard-disk drives and multi-link flexible manipulators all share a common trait - nonminimum phase (NMP) property of their dynamical models. The NMP property can be described as the presence of unobservable or internal dynamics which are unstable when the output is identically zero, by a suitable choice of input and initial conditions. The NMP property poses a significant challenge for control systems design by imposing limitations on the controller structure. Specifically, it prohibits the use of typical inversion based control techniques. In this dissertation, the NMP property is considered as it appears in aerospace applications, specifically air-breathing hypersonic vehicles (HSV). Air-breathing hypersonic vehicles, powered by scramjet engines, are intended to achieve long-duration hypersonic flight at airspeed exceeding Mach 5. HSVs have extensive military applications and have the potential for use as atmospheric reentry vehicles, which would improve access to space travel. The NMP property is seen in HSVs due to vehicle frame being tightly integrated with the propulsion system, causing an elevator-to-lift coupling. In addition to NMP behavior, hypersonic vehicles experience flexible effects from the fuselage. Furthermore, the vehicle models available for control design are not fully known, with uncertainty in the aerodynamic forces and moment. A suitable controller for the HSV must stabilize the internal dynamics, obtain a desirable response for the vehicle outputs and be robust to model uncertainties. In this study, the longitudinal dynamics of the HSV are considered, with the objective of achieving a desired trim condition in vehicle airspeed and altitude. Two solutions that address the issues with hypersonic vehicles are presented here. First, is a modular adaptive control method with input-to-state analysis of internal dynamics. In this case, the output is redefined with pitch angle as a dummy output such that the inter (open full item for complete abstract)

Committee: Andrea Serrani Prof. (Advisor); Kevin Passino Prof. (Committee Member); Abhishek Gupta Prof. (Committee Member) Subjects: Electrical Engineering

PhD, University of Cincinnati, 2017, Engineering and Applied Science: Mechanical Engineering

As noises that reach the passenger cabin are substantially reduced by recent advances in noise, vibration and harshness (NVH) engineering, the noise generated inside the passenger cabin, such as squeak and rattle (S&R), stands out to form detrimental perception of the quality of vehicles. S&R noises are unwanted, annoying noises generated by friction-induced vibration or impact between surfaces. Highly nonlinear and random nature of the underlying mechanism, S&R noises pose one of most difficult problems for automotive NVH engineers. A systematic procedure that integrates analytical model, numerical simulations and experiments is required and proposed to effectively handle S&R problems in the design stage. A computerized procedure for automatic detection and rating of S&R noises will be highly useful for automotive engineers by enabling consistent, repeatable inspection and quantitative assessment of the S&R problems. An algorithm developed in the past was enhanced in this work, which utilizes the perceived transient loudness (PTL) that approximates the human perception of transient noises. Various signal processing techniques and psychoacoustic models, such as analytic wavelet transform and Zwicker's loudness, were applied to enable S&R detection based on the characteristics of human perception. A new algorithm to differentiate S&R noises was developed by utilizing sound quality metrics to enhance the automatic detection and rating algorithm.

Committee: Jay Kim Ph.D. (Committee Chair); Yijun Liu Ph.D. (Committee Member); Allyn Phillips Ph.D. (Committee Member); David Thompson Ph.D. (Committee Member) Subjects: Mechanical Engineering; Mechanics

PhD, University of Cincinnati, 2003, Engineering : Electrical Engineering

Closed loop system stability is the first requirement of any control system design. Nonlinearity and uncertainty in the dynamics of systems are two issues that can make the design of stable control systems difficult. Nonlinearities can reduce, or eliminate our ability to use tractable linear mathematics and uncertainties can cause us to sacrifice performance in order to insure adequate control over a range of plant behavior. Early application of neural networks to control system design focused on the learning capabilities without formal regard for stability of the closed loop system. This dissertation investigates the use of neural networks in a control system design framework that guarantees stability. The design method is able to address plants with nonlinearities and bounded uncertainties. Stability is guaranteed as a part of the design process using Lyapunov's second (direct) method. Feed-forward neural networks are used to provide approximations to the uncertain dynamics. Both linear-in-the-parameters and nonlinear-in-the-parameters network structures are used. The uncertainties in the dynamics are considered to be both parametric and non-parametric, and they are allowed to originate in the state dependent quantities of model representations which are affine in the controls.

Committee: Dr. Marios Polycarpou (Advisor) Subjects:

PhD, University of Cincinnati, 2002, Engineering : Mechanical Engineering

Design for robust stability is one of the most important issues in nonlinear systems theory. The validity of linear system design in a small neighborhood is not a sufficient criterion for systems that undergo parametric variations and have strong nonlinear characteristics. With rapid growth in the systems theory, the design of nonlinear systems using bifurcation theory-based procedures has been one of the key developments. Servo-hydraulic systems are one of the most commonly used actuation and control devices, due to their force to weight ratio. They also are highly nonlinear in nature and hence provide considerable difficulty in the design and analysis of these systems and their control algorithms.The goal of this dissertation is to tackle some of the issues of the nonlinear systems theory with applications to servo-hydraulic systems. The use of bifurcation theory for the design and analysis of a nonlinear system is illustrated, and a detailed investigation into the dynamics associated with the servo-hydraulic systems is done. Further, the model decomposition/reduction strategy for parametric study in the nonlinear system is suggested. The idea of control-induced bifurcation is introduced and explained in light of servo-hydraulic systems. The servo-hydraulic system nonlinearities are explained and their effects on the robust stability are highlighted. This numerical work is also complemented with the experimental results on the servo-hydraulic circuits. This general procedure for robust stability design and control design, under the influence of nonlinearities, presented in this work can be used for any nonlinear system. The limitations of bifurcation theory based tools are also highlighted.

Committee: Dr. D. Thompson (Advisor) Subjects: Engineering, Mechanical

Doctor of Philosophy, The Ohio State University, 2005, Mathematics

We consider a model system, consisting of two nonlinearly coupled partial differential equations, to investigate nonlinear convective instabilities of travelling waves. The system exhibits front solutions which are travelling waves that asymptotically connect two different spatially homogeneous rest states. In the coordinate frame that moves with the front, the rest state ahead of the front is asymptotically stable, while the rest state behind the front experiences a Turing instability: upon increasing a bifurcation parameter, spatially periodic patterns arise. We show that the front is nonlinearly convectively unstable with respect to perturbations that are exponentially localized at positive infinity. More precisely, in the co-moving coordinate frame, the corresponding solution converges pointwise to a translate of the front, and the perturbation to the front is transported toward negative infinity. The proof of this stability result (which appears to be the first of this kind for dissipative systems) is based on the interplay of various norms, including norms for ordinary, uniformly local, and exponentially weighted Sobolev spaces. Among the methods employed are energy and semigroup estimates derived using multiplier theory in uniformly local spaces.

Committee: Bjorn Sandstede (Advisor) Subjects: Mathematics

Master of Science (MS), Ohio University, 2005, Electrical Engineering & Computer Science (Engineering and Technology)

This thesis presents the development of nonlinear tracking by Trajectory Regulation Control using the Backstepping method. The purpose of this thesis is to perform a feasibility study on the Backstepping method. A brief background of the need for nonlinear tracking control techniques is presented with an overview on stabilization, based on control Lyapunov functions. The Backstepping method employed the design of a tracking controller for a benchmark nonlinear plant that is unstable and nominimum phase. The design is implemented and tested in MATLAB/SIMULINK. The Backstepping controller is compared against the Trajectory Linearization Control and Sliding Mode in tracking performance and robustness testing. Results of the Backstepping model show an increase tracking performance and similar robustness when compared to Trajectory Linearization Control and Sliding Mode Control.

Committee: J. Zhu (Advisor) Subjects:

Doctor of Philosophy (PhD), Ohio University, 2013, Electrical Engineering (Engineering and Technology)

In this dissertation, Singular Perturbation Margin (SPM) and Generalized Gain Margin (GGM) are proposed as stability metrics for Linear Time-Invariant (LTI) systems, Linear Time-Varying (LTV) systems, Nonlinear Time-Invariant (NLTI) systems and Nonlinear Time-Varying (NLTV) systems, which are: (i) theoretically based, (ii) practically measurable, (iii) backward compatible in the sense that when applied to LTI systems, these metrics have a bijective correspondence with the Gain Margin (GM) and Phase Margin (PM). Based on eigenvalue perturbation theory, Lyapunov's first and second methods, and algebraic spectral theory, the SPM and GGM assessment methods are established for LTI and LTV systems. The SPM and GGM equivalences between LTI and NLTI, LTV and NLTV systems are then established, which make it possible to estimate the SPM and GGM for NLTI and NLTV systems with the corresponding assessment methods for LTI and LTV systems, respectively. In the analysis and design of nonlinear systems, the determination of the Domain of Attraction (DOA) is a problem of fundamental importance, and through bifurcation analysis and Lyapunov's methods, the relationship between the estimated DOA and the perturbation parameters are established. The concepts SPM, GGM and the corresponding assessment methods are helpful to analyze the advanced nonlinear and time-varying control techniques, thereby fully exploiting the performance potentials and improving the safety of the systems in the near future.

Committee: Jim Zhu (Committee Chair); Douglas Lawrence (Committee Member) Subjects: Electrical Engineering