Two methods for estimating optimal solutions to nonlinear-programming problems (NLP), with perturbed constraints are developed. The first is a second order Classical Perturbation Analysis. The second is a labelling technique which involves the assumption of low constraint curvature near the optimal solution. These estimation procedures require the Hessians of the cost and constraint functions, and so, procedures for estimating the Hessian of a function are also discussed. Both estimation procedures are compared to an exact solution for a simple problem. For this simple example the labelling technique provided a better estimate than the perturbation technique. What might be termed conventional methods for solving complex NLP problems with perturbed constraints include using the initial guess from the unperturbed problem as the initial guess for the perturbed problem, and using the final solution to the unperturbed problem as the initial guess for the perturbed problem. The estimated solutions can also be used as an initial guess in an iterative NLP problem solution. The results using these estimation procedures are compared to the conventional methods. The labelling technique, being limited to small curvature, was not capable of estimating a solution to the more complex example. Results for this complex example show that the Classical Perturbation Method provides a factor of seven improvement in computational effort over the use of the unperturbed initial guess, and a factor of four improvement over the use of the unperturbed optimal solution as an initial guess. / M. S.

Thesis (B.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1978. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 110-111. / B.S.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERONAUTICS. / Vita. / Includes bibliographical references. / by Woodrow Whitlow, Jr. / Ph.D.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / by Stephen Paul Taylor. / M.S.

The Lyapunov direct method is utilized to determine the robustness bounds for nonlinear, time-variant uncertainies p[subscript i]. Determination of the robustness bounds consists of two principal steps: (i) generation of a Lyapunov function and (ii) determination of the bounds based on the generated Lyapunov function. Presently in robustness investigations, a Lyapunov function is generated by inserting the nominal matrix to the Lyapunov equation and setting Q as identity matrix. The objective of this study is to utilize structural features of the uncertainties to develop a recursive algorithm for the generation of the globally optimal quadratic Lyapunov function. The proposed method is seemingly an improvement with respect to those reported in recent literature in three senses: i) ease of application, given an interactive program which requires only system matrices as inputs; ii) provision of improved estimates of the robustness bounds; and iii) extendability of the procedure to the design of robust controllers. The algorithm and the program prepared (in MATLAB) are presented. Several examples are considered for purposes of the comparison of robustness bounds estimates. Examples are demonstrated to show the superiority of the robustness bounds estimated by the proposed method over those obtained by small gain theorem. In a number of cases, the estimated robustness bounds are proven to be the exact robustness bounds. / Graduation date: 1993

In this thesis, we study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be non-autonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative per- turbations. We apply Semi-group Theory and Lyapunov-Perron Integral Equations with some careful estimates to handle the O(1) driving force in the system so that we can approximate the full system through some simpler limiting system. In the investigation of homoclinics, a diagonalization procedure and some normal form transformation should be first carried out. Such diagonalization procedure is not trivial at all. We discuss this issue in the appendix. We use Melnikov type analysis to study the weakly dissipative case, while the conservative case is based on some energy methods. As a concrete example, we have shown rigrously the persistence of homoclinic solutions of an elastic pendulum model which may be affected by damping, external forcing and other potential fields.

Noncovalent interactions play a vital role throughout much of chemistry. The understanding and characterization of these interactions is an area where theoretical chemistry can provide unique insight. While many methods have been developed to study noncovalent interactions, symmetry-adapted perturbation theory (SAPT) stands out as one of the most robust. In addition to providing energetic information about an interaction, it provides insight into the underlying physics of the interaction by decomposing the energy into electrostatics, exchange, induction and dispersion. Therefore, SAPT is capable of not only answering questions about how strongly a complex is bound, but also why it is bound. This proves to be an invaluable tool for the understanding of noncovalent interactions in complex systems. The wavefunction-based formulation of SAPT can provide qualitative results for large systems as well as quantitative results for smaller systems. In order to extend the applicability of this method, approximations to the two-electron integrals must be introduced. At low-order, the introduction of density fitting approximations allows SAPT computations to be performed on systems with up to 220 atoms and 2850 basis functions. Higher-orders of SAPT, which boasts accuracy rivaling the best theoretical methods, can be applied to systems with over 40 atoms. Higher-order SAPT also benefits from approximations that attempt to truncate unneccesary unoccupied orbitals.

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The research effort in this thesis is devoted to develop techniques to accurately and rapidly identify the location, orientation, and magnitude of the defects by using structural health monitoring concepts that use Laser Doppler Vibrometer as a non-contact sensor with multi-point sensing capability. The first research area addresses the formulation and validation of an innovative Damage Measure that is based on the ratios of the strain energy distributions of the damaged and undamaged structure. The innovations include use of a single set of actuator/sensor pair to excite and detect the responses of a structure for low frequency vibrations as well as guided wave propagation studies. A second new capability is the estimation of the Damage Measure without requiring any knowledge of the undamaged baseline structure. This method is made possible because of the development of these new technologies: Spatial Decimation and Wavenumber/Frequency filtering. The third contribution is to develop analytical models for the structural dynamics of damaged structure and seek solutions that use perturbation methods to detect damage in a plate structure. The fourth contribution is the development of a comprehensive damage detection technique over a wide frequency dynamic range. The fifth topic of research involves automation in Structural Health Monitoring based on the comprehensive Damage Measure formulation. Under the control of software the Scanning Laser Doppler Vibrometer is used to acquire the low frequency vibration mode data for a coarse identification of all the suspect regions of damage using a threshold criterion on the Damage Measure. Each suspect region of damage is further investigated using the high frequency elastic wave propagation to clearly identify the location, orientation, and extent of the damage. The computer control of the Laser Doppler Vibrometer and a quantitative assessment of the damage provide the enabling technologies for the automation proof of concept. Finally the developed techniques of damage detection are successfully demonstrated on practical structures such as a turbine blade in the laboratory and an F-15 vertical tail in field maintenance conditions

Thesis (M. S.)--Aerospace Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Mavris, Dimitri; Committee Member: Bauchau, Olivier; Committee Member: Schrage, Daniel; Committee Member: Volovoi, Vitali; Committee Member: Yu, Wenbin.