A bilinear formulation is used for developing the time finite element method (TFM) to obtain transient responses of both linear, nonlinear, damped and undamped systems. Also the formulation, used in the h-, p- and hp-versions, is extended and found to be readily amenable to multi-degree-of-freedom systems. The resulting linear and nonlinear algebraic equations for the transient response are differentiated to obtain the sensitivity of the response with respect to various design parameters. The present developments were tested on a series of linear and nonlinear examples and were found to yield, when compared with other methods, excellent results for both the transient response and its sensitivity to system parameters. Mostly, the results were obtained using the Legendre polynomials as basis functions, though, in some cases other orthogonal polynomials namely, Hermite, Chebyshev, and integrated Legendre polynomials were also employed (but to no great advantage). A key advantage of TFM, and the one often overlooked in its past applications, is the ease in which the sensitivity of the transient response with respect to various design parameters can be obtained. Since a considerable effort is spent in determining the sensitivity of the response with respect to system parameters in many algorithms for parametric identification, an identification procedure based on the TFM is developed and tested for a number of nonlinear single-and two-degree-of-freedom system problems. An advantage of the TFM is the easy calculation of the sensitivity of the transient response with respect to various design parameters, a key requirement for gradient-based parameter identification schemes. The method is simple, since one obtains the sensitivity of the response to system parameters by differentiating the algebraic equations, not original differential equations. These sensitivities are used in Levenberg-Marquardt iterative direct method to identify parameters for nonlinear single- and two-degree-of-freedom systems. The measured response was simulated by integrating the example nonlinear systems using the given values of the system parameters. To study the influence of the measurement noise on parameter identification, random noise is added to the simulated response. The accuracy and the efficiency of the present method is compared to a previously available approach that employs a multistep method to integrate nonlinear differential equations. It is seen, for the same accuracy, the present approach requires fewer data points. Finally, the TFM for optimal control problems based on Hamiltonian weak formulation is proposed by adopting the p- and hp-versions as a finite element discretization process. The p-version can be used to improve the accuracy of the solution by adding more unknowns to each element without refining the mesh. The usage of hierarchical type of shape functions can lead to a significant saving in computational effort for a given accuracy. A set of Legendre polynomials are chosen as higher order shape functions and applied to two simple minimization problems for optimal control. The proposed formulation provides very accurate results for these problems. / Ph. D.

Mathematical models are useful for simulation, design, analysis, control, and optimization of complex systems. One important step necessary to create an effective model is designing an experiment from which the unknown model parameter can be accurately identified and then verified. The strategy which one approaches this problem is dependent on the amount of data that can be collected and the assumptions made about the behavior of the error in the statistical model. In this presentation we describe how to approach this problem using a combination of statistical and mathematical theory with reliable computation. More specifically, we present a new approach to bounded error parameter validation that approximates the membership set by solving an inverse problem rather than using the standard forward interval analysis methods. For our method we provide theoretical justification, apply this technique to several examples, and describe how it relates to designing experiments. We also address how to define infinite dimensional designs that can be used to create designs of any finite dimension. In general, finding a good design for an experiment requires a careful investigation of all available information and we provide an effective approach to dthe problem. / Ph. D.

This dissertation presents new fuzzy logic techniques for designing control systems for a wide class of complex systems. The methods are developed in detail for a crane system which contains one rigid-body and one oscillation mode. The crane problem is to transfer the rigid body a given distance such that the pendulation of the oscillation mode is regulated at the final time using a single control input. The investigations include in-depth studies of the time-optimal crane control problem as an integral part of the work. The main contributions of this study are: (1) Development of rule-based systems (both fuzzy and crisp) for the design of optimal controllers. This development involves control variable parametrization, rule derivation with parameter perturbation methods, and the design of rule based controllers, which can be combined with model-based feedback control methods. (2) A thorough investigation and analysis of the solutions for time-optimal control problems of oscillation mode systems, with particular emphasis on the use of phase-plane interpretation. (3) Development of fuzzy logic control system methodology using expert rules obtained through energy reducing considerations. In addition, dual mode control is a "spin-off" design method which, although no longer time optimal, can be viewed as a near-optimal control method which may be easier to implement. In both types of design optimization of the fuzzy logic controller can be used to improve performance. / Ph. D.

A method utilizing H2 control concepts and the numerical method of Polynomial Chaos was developed in order to create a novel robust probabilistically optimal control approach. This method was created for the practical reason that uncertainty in parameters tends to be inherent in system models. As such, the development of new methods utilizing probability density functions (PDFs) was desired. From a more theoretical viewpoint, the utilization of Polynomial Chaos for studying and designing control systems has not been very thoroughly investigated. The current work looks at expanding the H2 and related Linear Quadratic Regulator (LQR) control problems for systems with parametric uncertainty. This allows solving deterministic linear equations that represent probabilistic linear differential equations. The application of common LTI (Linear Time Invariant) tools to these expanded systems are theoretically justified and investigated. Examples demonstrating the utilized optimization process for minimizing the H2 norm and parallels to LQR design are presented. The dissertation begins with a thorough background section that reviews necessary probability theory. Also, the connection between Polynomial Chaos and dynamic systems is explained. Next, an overview of related control methods, as well as an in-depth review of current Polynomial Chaos literature is given. Following, formal analysis, related to the use of Polynomial Chaos, is provided. This lays the ground for the general method of control design using Polynomial Chaos and H2. Then an experimental section is included that demonstrates controller synthesis for a constructed probabilistic system. The experimental results lend support to the method. / Ph. D.

Falls from heights (FFH) are one of the leading causes of fatalities in skilled labor divisions such as construction, mining, agriculture/forestry, and manufacturing. Previous research has established that localized muscle fatigue (LMF) increases center of mass (COM)- and center of pressure (COP)-based measures of quiet stance. This is important because these increases have been linked to elevated risk of falls, and workers in the construction industry frequently engage in fatiguing activities while working at heights. In addition, the rate of fatality due to an occupational fall increases exponentially with age. Improved methods of fall prevention may be obtained through increased understanding of factors that have a deleterious effect on balance and postural control such as LMF and aging. An initial study was conducted to investigate the effects of LMF and aging on balance recovery from a postural perturbation without stepping. Sagittal plane postural perturbations were administered to young and older groups of participants before and after exercises to fatigue the lumbar extensors or ankle plantar flexors. Measures of balance recovery were based on the COM and COP trajectories and the maximum perturbation that could be withstood without stepping. Balance recovery measures were consistent with an LMF-induced decrement to recover from perturbations without stepping. Aging was also associated with an impaired ability to recover from the perturbations. The second study in the series investigated the effects of aging and LMF on the neural control of upright stance during small postural perturbations. Small magnitude postural perturbations were administered to young and older groups before and after fatiguing exercises. A single degree of freedom (DOF) model of the human body was developed that accurately simulated the experimentally collected kinematics during recovery from the perturbations. The model was controlled by invariant feedback gains that operated on the time-delayed kinematics. Feedback gains and time-delay were optimized for each participant, and a novel delay margin analysis was performed to assess system robustness toward instability. Results indicated that older individuals had a longer "effective" time-delay and exhibited greater reliance on afferent velocity information. No changes in feedback controller gains, time-delay, or delay margins were found with LMF in either age group. The final study investigated the use of a nonlinear controller to simulate responses to large magnitude postural perturbations. A three DOF model of the human body was developed and controlled with the state-dependent Riccati equation (SDRE). Parameters of the SDRE were optimized to fit the experimentally recorded kinematics. Unlike other forms of nonlinear control, the SDRE provides meaningful parameters for interpretation in the system identification. The SDRE approach was successful at stabilizing the dynamical system; however, accurate results were not obtained. Reasons for these errors are discussed, and an alternative formulation to the time-delayed optimal control problem using Roesser state space equations is presented. / Ph. D.

Accurately simulating human motions is a major function of and challenge to digital human models and integrating humans in computer-aided design systems. Numerous successful applications of human motion simulation have already demonstrated their ability to improve occupational efficiency, effectiveness, and safety. In this dissertation, a novel motion simulation model using fuzzy logic control is presented. This model was motivated by the fact that humans use linguistic terms to guide their behaviors while fuzzy logic provides mathematical representations of linguistic terms. Specifically in this model, fuzzy logic was used to specify a neural controller which was generally considered as the part in the postural control system that plans human motions. Fuzzy rules were generated according to certain trends observed from actual human motions. An optimization procedure was performed to specify the parameters of the membership functions by minimizing the differences between the simulated and actual final postures. This research contributed to the field of human movement science by providing a motion simulation model that can accurately predict novel human motions and provide interpretations of potential human motion planning strategies. Understanding balance control is another research focus in this dissertation. Investigating balance control may aid in preventing unnecessary fall-related incidents and understanding the postural control system. Since human behaviors are generally effective and efficient, balance control models (both two- and three-dimensional) based on an optimal control strategy were developed to aid in better understanding balance control. Specifically, the neural controller was considered as an optimal controller that minimizes a performance index defined by physical quantities relevant to sway. Free model parameters, such as weights of relevant physical quantities and sensory delay time, were determined by an optimization procedure whose objective was to minimize a scalar error between simulated and experimental center-of-pressure (COP) based measures. Many factors, such as aging, localized muscle fatigue, and external loads, have been found to adversely affect balance control. At the same time, behaviors during upright stance are commonly characterized by COP-based measures. Thus, changes in COP based measures with aging, LMF, and external loads were addressed by using the proposed models, and possible postural control mechanisms were identified by interpreting these changes. Findings from these studies demonstrated that the proposed models were able to accurately simulate human sway behaviors and provide plausible mechanisms regarding how the postural control system works when maintaining upright balance. / Ph. D.

In this thesis we develop a rigorous mathematical framework for analyzing and approximating optimal sensor placement problems for distributed parameter systems and apply these results to PDE problems defined by the convection-diffusion equations. The mathematical problem is formulated as a distributed parameter optimal control problem with integral Riccati equations as constraints. In order to prove existence of the optimal sensor network and to construct a framework in which to develop rigorous numerical integration of the Riccati equations, we develop a theory based on Bochner integrable solutions of the Riccati equations. In particular, we focus on ℐ_p-valued continuous solutions of the Bochner integral Riccati equation. We give new results concerning the smoothing effect achieved by multiplying a general strongly continuous mapping by operators in ℐ_p. These smoothing results are essential to the proofs of the existence of Bochner integrable solutions of the Riccati integral equations. We also establish that multiplication of continuous ℐ_p-valued functions improves convergence properties of strongly continuous approximating mappings and specifically approximating C₀-semigroups. We develop a Galerkin type numerical scheme for approximating the solutions of the integral Riccati equation and prove convergence of the approximating solutions in the ℐ_p-norm. Numerical examples are given to illustrate the theory. / Ph. D.

While the idea of critical break rate of water distribution pipeline (defined as the break rate after which it is no longer economical to continuously repair) has been accepted in the literature and among the practicing engineers, the formula to obtain the critical break rate has remained elusive. In this dissertation, an equation for identifying the threshold break rate of a pipe is developed. The threshold break rate equation gives a rule of thumb for pipe replacement decision. Input parameters to obtain the threshold break rate of a pipe are repair and replacement costs, interest rate, and the length of the pipe. In addition, a methodology that enables the use of threshold break rate with the failure intensity and hazard functions is developed. The methodology is drawn by considering the relationships of the definitions of the threshold break rate with intensity and hazard functions in the context of a repairable system's failure process modeling. As a result, the newly developed threshold break rate equation can be coupled with any appropriate intensity and hazard function to obtain economically optimal replacement time of a pipe. Also, practical usage of the threshold break rate is demonstrated with a number of numerical examples. Design aids in the form of charts and tables are provided. The threshold break rate can be easily obtained either graphically or with the aid of the tables. The methodology that links the threshold break rate and failure rate (intensity and hazard) functions is extended to accommodate stress multiplying environmental factors in the form of the proportional intensity and hazards model. The two models consist of an age dependent failure rate function and a covariate structure. They are applied to a case study area pipe system to obtain optimal replacement times for individual pipes in the system. As a result, important hazard characteristics of water distribution pipes are drawn, and implications on the optimal replacement analysis are discussed. A pipe break prediction model is also developed in this research. The model spans the space between the linear and exponential break trends. The model is applied to the case study area pipe system with various cost options. The results from this analysis are discussed in terms of practical implementation of the replacement strategies. / Ph. D.

Failure of pipes in water distribution systems is a common occurrence especially in large cities. The failure of a pipe results in: loss of water; property damage; interruption of service; decreased system performance; and the financial cost of restoring the failed pipe. The cost of replacement and rehabilitation in the United States is estimated at 23 plus billion dollars. It is virtually impossible to replace all vulnerable pipes at the same time. As a result, there is a need for methods that can help in progressive system rehabilitation and replacement subject to budgetary constraints. If delaying is considered a good strategy due to the time value of money then, the timing of preventive maintenance becomes a crucial element for system maintenance and operation. The central under pinning element in the decision process for scheduling preventive maintenance is the deteriorating nature of a pipe under a given surrounding. By planning to replace pipes before they fail, proper planning can be put in place for securing of finances and labor force needed to rehabilitate the pipes. With this approach, service interruptions are minimized as the loss of service time is limited to the time used in replacing the pipe. In this research, a mechanistic model for assessing the stage of deterioration of an underground pipe is developed. The developed model consists of three sub-models namely, the Pipe Load Model (PLM), the Pipe Deterioration Model (PDM), and the Pipe Break Model (PBM). The PLM simulates the loads and stresses exerted on a buried water main. These loads include the earth load, traffic load, internal pressure, expansive soil loads, thermal, and frost loads. The PDM simulates the deterioration of the pipe due to corrosion resulting from the physical characteristics of the pipe environment. The pipe deterioration effect is modeled in two stages. First, the thinning of the pipe wall is modeled using a corrosion model. Second, the localized pit growth is used to determine the residual strength of the pipe based on the fracture toughness and the initial design strength of the pipe. The PBM assesses the vulnerability of a pipe at any time in terms of a critical safety factor. The safety factor is defined as the ratio of residual strength to applied stress. For a conservative estimate the multiplier effect due to thermal and frost loads are considered. For a chosen analysis period, say 50 years, the pipes with safety factors less than the critical safety factor are selected and ordered by their rank. Aided by the prioritized list of failure prone pipes, utilities can organize a replacement schedule that minimizes cost over time. Additionally a physically based regression model for determining the optimal replacement time of pipe is also presented. A methodology for assessing the consequences of accelerated and delayed replacement is also provided. The methodologies developed in this dissertation will enable utilities to formulate future budgetary needs compatible with the intended level of service. An application of the model and results are included in the dissertation. / Ph. D.

In this work, we introduce the Sensitivity Equation Method (SEM) as a method for approximately solving infinite dimensional optimal design problems. The SEM couples a trust-region/quasi-Newton optimization algorithm with gradient information provided by apprOXimately solving the sensitivity equation for (design) sensitivities. The sensitivity equation is (in the problems considered here) a partial differential equation (POE) which describes the influence of a design parameter on the state of the system. It is shown that obtaining design sensitivities from the sensitivity equation has advantages over finite difference and semi-analytical methods in that there is no need to remesh or compute mesh sensitivities (even if the domain is parameter dependent), the sensitivity equation is a linear POE for the sensitivities and can be approximated in an efficient manner using the same approximation scheme used to approximate the states. The applicability of the SEM to shape optimization problems, where the state is described by the Euler equations, is studied in detail. In particular, we prove convergence of the method for a one dimensional test problem. These results are used to speculate on the applicability of the method for more complex problems. Finally. we solve a two dimensional forebody simulator design problem (for use in wind tunnel experiments) using the SEM, which is shown to be a very efficient method for this problem. / Ph. D.