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The goal of an optimal control algorithm is to improve the performance of a system. For a stochastic system, a typical optimal control method minimizes the mean (first cumulant) of the cost function. However, there are other statistical properties of the cost function, such as variance (second cumulant) and skewness (third cumulant), which will affect the system performance. In this dissertation, the work on the statistical optimal control are presented, which extends the traditional optimal control method using cost cumulants to shape the system performance. Statistical optimal control will allow more design freedom to achieve better performance. The solutions of statistical control involve solving partial differential equations known as Hamilton-Jacobi-Bellman equation. A numerical method based on neural networks is employed to find the solutions of the Hamilton-Jacobi-Bellman partial differential equation. Furthermore, a complex problem such as multiple satellite control, has both continuous and discrete dynamics. Thus, a hierarchical hybrid architecture is developed in this dissertation where the discrete event system is applied to discrete dynamics, and the statistical control is applied to continuous dynamics. Then, the application of a multiple satellite navigation system is analyzed using the hierarchical hybrid architecture. Through this dissertation, it is shown that statistical control theory is a flexible optimal control method which improves the performance; and hierarchical hybrid architecture allows control and navigation of a complex system which contains continuous and discrete dynamics. / Electrical and Computer Engineering

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.

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.

The use of autonomous underwater vehicles (AUVs) for oceanic observation and research is becoming more common. Underwater gliders are a specific class of AUV that do not use conventional propulsion. Instead they change their buoyancy and center of mass location to control attitude and trajectory. The vehicles spend most of their time in long, steady glides, so even minor improvements in glide range can be magnified over multiple dives. This dissertation presents a rigid-body dynamic system for a generic vehicle operating in a moving fluid (ocean current or wind). The model is then reduced to apply to underwater gliders. A reduced-order point-mass model is analyzed for optimal gliding in the presence of a current. Different numerical method solutions are compared while attempting to achieve maximum glide range. The result, although approximate, provides good insight into how the vehicles may be operated more effectively. At the end of each dive, the gliders must change their buoyancy and pitch to transition to a climb. Improper scheduling of the buoyancy and pitch change may cause the vehicle to stall and lose directional stability. Optimal control theory is applied to the buoyancy and angle of attack scheduling of a point-mass model. A rigid-body model is analyzed on a singular arc steady glide. An analytical solution for the control required to stay on the arc is calculated. The model is linearized to calculate possible perturbation directions while remaining on the arc. The nonlinear model is then propagated in forward and reverse time with the perturbations and analyzed. Lastly, one of the numerical solutions is analyzed using the singular arc equations for verification. This work received support from the Office of Naval Research under Grant Number N00014-08-1-0012. / Ph. D.