In this dissertation, a linear estimator for fault detection and isolation called the Game Theoretic Multiple-Fault Detection Filter is derived for both continuous and discrete systems. The detection filter uses a disturbance attenuation formulation to bound the transmission of disturbances to the output, approximately blocking all but one fault from each of a set of projected residuals. However, different from previous approximate methods for single-fault detection filters, the multiple-fault detection filter utilizes a secondary optimization problem to generate a solution for the estimator gain that achieves more advanced detection filter goals. Specifically, the current work examines an optimization that increases sensitivity of each projected residual to its target fault. For the continuous case, it is proven that the new detection filter approximates previous detection filters obtained from geometric and spectral theories and extends them to finite time-varying systems. Further, the detection filter is demonstrated via numerical examples.

To accommodate the growing air traffic demand, flights will need to be planned and navigated with a much higher level of precision than today's aircraft flight path. The Next Generation Air Transportation System (NextGen) stands to benefit significantly in safety and efficiency from such movement of aircraft along precisely defined paths. Air Traffic Operations (ATO) relying on such precision--the Precision Air Traffic Operations or PATO--are the foundation of high throughput capacity envisioned for the future airports. In PATO, the preferred method is to manage the air traffic by assigning a speed profile to each aircraft in a given fleet in a given airspace (in practice known as (speed control). In this research, an algorithm has been developed, set in the context of a Hybrid Control System (HCS) model, that determines whether a speed control solution exists for a given fleet of aircraft in a given airspace and if so, computes this solution as a collective speed profile that assures separation if executed without deviation. Uncertainties such as weather are not considered but the algorithm can be modified to include uncertainties. The algorithm first computes all feasible sequences (i.e., all sequences that allow the given fleet of aircraft to reach destinations without violating the FAA's separation requirement) by looking at all pairs of aircraft. Then, the most likely sequence is determined and the speed control solution is constructed by a backward trajectory generation, starting with the aircraft last out and proceeds to the first out. This computation can be done for different sequences in parallel which helps to reduce the computation time. If such a solution does not exist, then the algorithm calculates a minimal path modification (known as path control) that will allow separation-compliance speed control. We will also prove that the algorithm will modify the path without creating a new separation violation. The new path will be generated by adding new waypoints in the airspace. As a byproduct, instead of minimal path modification, one can use the aircraft arrival time schedule to generate the sequence in which the aircraft reach their destinations.

This paper presents an adaptive robust control strategy for a class of uncertain linear systems with disturbance input. The uncertain linear system contains unknown parameters in both the system and the input matrices. The system makes perfect partial measurement of the state. A disturbance attenuation function is transformed into a minimax differential game where the disturbance and the unknown parameters act as the cooperative players that maximize the cost function, whereas the controller is the opposing player that minimizes the cost function. Having shown the existence of a saddle point in the cost function, the optimization yields a minimax controller coupled with a stable nonlinear modified gain observer, which estimates the state and the unknown parameters. By maximizing the cost function with respect to the uncertain parameters, the minimax controller takes the parameter estimation confidence level into account to generate the worst case input for a given instance of time. This leads to two sets of Riccati equations, one for the controller and one for the observer.

In this class of parameter estimation problem, the measurement function and the augmented system, which is composed of the state and the uncertain parameters, are modifiable functions or can be transformed into modifiable functions using an appropriate change of coordinate system, such as the observable canonical form. The essence of a modifiable function is that although the observer dynamics are nonlinear, the error in the observer's estimation error is linear. The existence of the saddle point in the performance index is presented. Under certain conditions, such as the observability and controllability of the system, the existence of the solution to the two Riccati equation and finiteness of the value function, the close loop system is stable and the estimation error is bounded, which demonstrates the disturbance attenuation properties of the observer.

We present an unstable single-input single-output (SISO) four state vector case example with four unknown system parameters, subject to worst case disturbance. The simulation demonstrates the applicability of the disturbance attenuation controller coupled with the nonlinear modified gain observer, and assess its performance against the linear quadratic regulator (LQR) coupled with a modified gain extended Kalman observer (MGEKO).

Furthermore, the derived disturbance attenuation controller is applied to an air breathig hypersonic flight vehicle with full state measurement. The nonlinear longitudinal dynamics of the aircraft, which is subject to large aerodynamic uncertainty, is linearized around a nominal trim condition to derive a nominal linearized perturbation model. The nonlinear modified gain observer estimates four system parameters to yield an estimated linearized perturbation model and the corresponding estimation error weightings for the worst case controller. The simulation result demonstrates the applicability of the worst case controller coupled with the nonlinear modified gain observer and its superior performance when compared against the Sum-of-Squares method shown in the previous literature.

Game theory has application across multiple fields, spanning from economic strategy to optimal control of an aircraft and missile on an intercept trajectory. The idea of game theory is fascinating in that we can actually mathematically model real-world scenarios and determine optimal decision making. It may not always be easy to mathematically model certain real-world scenarios, nonetheless, game theory gives us an appreciation for the complexity involved in decision making. This complexity is especially apparent when the players involved have access to different information upon which to base their decision making (a nonclassical information pattern).

Here we will focus on the class of adversarial two-player games (sometimes referred to as pursuit-evasion games) with nonclassical information pattern. We present a two-sided (simultaneous) optimization solution method for the two-player linear quadratic Gaussian (LQG) multistage game. This direct solution method allows for further interpretation of each player's decision making (strategy) as compared to previously used formal solution methods. In addition to the optimal control strategies, we present a saddle point proof and we derive an expression for the optimal performance index value. We provide some numerical results in order to further interpret the optimal control strategies and to highlight real-world application of this game-theoretic optimal solution.

In this dissertation, the fundamental structure of a multivariate discrete-time state estimator with Cauchy distributed process noise and measurement noise is discussed in depth. The characteristic function (CF) of the unnormalized conditional probability density function (ucpdf) is found to be a sum of elements that increases at each update of the current measurement. Each term in this sum is composed of a coefficient term which contains the measurement history operating on an exponential term composed of a sum of absolute values whose argument is the inner product of a direction vector with the spectral variable. The objective is to understand the structure of the CF so as to simplify this sum. We shows that directions in the terms of the CF-s are co-aligned only along certain directions which are functions of a unique fundamental basis. Based on the knowledge of combining co-aligned directions, an indexing scheme, called "S" matrix, is developed to indicate which exponential terms can be combined without the necessity of numerical comparison. The S matrix is invariant for systems of the same dimension regardless of the system parameters. The coefficient terms are also restructured and simplified by eliminating all the redundant zero elements. For two-state systems, we show that there are no more than three non-zero elements in each layer of any new coefficient term. Furthermore, with these newly uncovered properties the Cauchy estimator is implemented efficiently using a pre-computational technique. The simulations of three-state and four-state systems illustrate the performance of Cauchy estimator compared with the Kalman Filter.

The spectral factorization solution for fractional order optimal control problems is provided. First, graphical tools are used to obtain stabilizing controllers as well as derive properties of fractional polynomials. Second, the spectral factorization solution to the output feedback H2 problem is extended to fractional systems, which are permitted to be unstable, non-minimum phase, or incommensurate order. Third, spectral factorization is used to solve the LQR problem of constructing the optimal full-state feedback law, which is shown to have strong connections to the rational LQR.

Millisecond pulsars are extremely stable and rapidly rotating neutron stars that emit electromagnetic radiation along their magnetic axes. Due to the misalignment between the rotational and the magnetic axes, the observed pulsar signals are analogous to the light beams of distant lighthouses. The predictable pulsing behavior is the fundamental mechanism that allows researchers to use pulsars as tools for science and engineering. This research focused on the analysis, simulation, estimation, and verification associated with pulsar-based navigation and clock calibration.

The autonomous pulsar-based navigation problem is formulated in terms of a nonlinear filtering problem where a single filter is used to estimate the spacecraft position and velocity. The positioning accuracy of a spacecraft traveling at known constant velocity was analyzed to build insights into the general navigation problem. An analytical comparison between the measurement noises of X-ray based and radio based pulsar timing/navigation system is discussed. A variation of the Extended Kalman Filter was developed and implemented to track simulated X-ray pulsar measurements collected by an orbiting spacecraft. This filter uses a multirate structure to more efficiently process pulsar measurements. The ephemeris of the DAWN spacecraft was used to investigate the performance of pulsar-based navigation in a more realistic mission scenario.

In order to show the feasibility of pulsar-based navigation, several existing pulsar tim- ing software packages and publicly available radio millisecond pulsar data were used to experimentally verify the concept. An Unscented Kalman Filter was used to process the time-of-arrival measurements from 5 isolated millisecond pulsars in order to estimate the position of the radio telescope in Earth fixed coordinate system.

This research also investigated the theoretical frequency stability of pulsar-aided atomic clocks from power spectral densities. Hadamard variance was used to analyze the unfiltered and the filtered clock systems. The result of the analysis shows that pulsars have the potential of enhancing the long-term frequency stability of stand-alone compact atomic clocks.

The last chapter of this dissertation discusses relative positioning using differential phase measurement. The proposed method can be used to eliminate common mode errors embed- ded in the pulsar measurements when the two observed signal frequencies are known.

Cyber-Physical Systems (CPSs) are complex systems resulting from intricate interaction of digital computational devices with the physical systems. With the recent dazzling advances in computational devices, CPSs have become ubiquitous in modern technology. The increasing presence of CPSs on one hand and the incapability of current methods to analyze them on the other hand, impel the development of novel approaches for analysis and design. In CPSs, embedded computers have the responsibility of monitoring and controlling the physical plants using feedback loops using which physical plants affect computations and vice versa. In these closed-loop fashions, controllers implemented in software are termed embedded control software.

Increasing use of embedded control software in life critical applications, such as aircraft flight control systems and automotive engine control systems, demands lots of efforts on software verifications and validations which are very costly. On the other hand, by changing the center of gravity from verification to design, it is possible to synthesize correct-by-design embedded control software while providing formal guarantees of correctness. The foundation of this proposed approach relies on some technical results showing how to construct equivalent finite state models for differential equation models describing physical plant. These finite state models are simpler descriptions of physical plant in which

each state of the finite model represents a collection or aggregate of states in the physical plant. Similar finite state models are used in software and hardware modeling, which enable the composition of such models with the finite models of the physical systems. The results of this composition are finite models capturing the behavior of the physical systems interacting with the digital computation devices. Once such models are

available, the methodologies and tools developed in computer science for verification and control synthesis purposes can be easily employed to physical systems, via these models. In the first part of this thesis I take an important step in my quest to synthesize correct-by-design embedded control software for CPSs by constructing finite state models for control systems. I propose a novel technique to compute bisimilar finite state models of incrementally stable nonlinear control systems. I show on practical examples that the finite state models computed by my procedure can be several orders of magnitude smaller than existing approaches. Moreover, I propose another technique to compute (not necessarily bisimilar) finite state models of any nonlinear control system as long as I am interested in its behavior in a compact set. In the second part of this thesis I will show some incremental properties under which nonlinear control systems admit finite state models. I propose some analysis tools to check those properties. Moreover, I provide some design techniques providing controllers enforcing those incremental properties for some special classes of nonlinear control systems.

State estimators are developed for discrete linear systems with scalar additive Laplace process and measurement noises and their properties are analyzed. For the scalar case, an analytic recursive estimator is presented, along with detailed analysis of its behavior with respect to noise parameters. In addition, a one-step model predictive controller is developed. Using an objective function with 1-norm control and terminal state costs, the expectation of the objective function with respect to the conditional probability density function is determined by using the computational structure developed for the estimator. Numerical simulations for both the estimator and one-step controller are presented to demonstrate their unique behavior, including robustness to noise spikes in the measurements. For the general vector system, update and propagation algorithms as well as a method for computing moments in closed form using characteristic functions are presented. An explicit state estimator is developed for the two-state case, and a numerical example is presented to demonstrate the algorithms and the unique properties of Laplace estimators.

An efficient recursive state estimator and an optimal model predictive controller are developed for two-state linear systems driven by Cauchy distributed process and measurement noises. For a general vector-state system, the estimator is based on recursively propagating the characteristic function (CF) of the conditional probability density function (cpdf), where the number of terms in the sum that expresses this CF grows with each measurement update. The proposed two-state estimator reduces substantially the number of terms needed to express the CF of the cpdf by taking advantage of relationships not yet developed in the general vector-state case. Further, by using a fixed sliding window of the most recent measurements, the improved efficiency of the proposed two state estimator allows an accurate approximation for real-time computation. For control of the general vector-state system, the conditional performance index is evaluated in the spectral domain using this CF. The expectation is of an objective function that is a product of functions resembling Cauchy pdfs. Using this method, the conditional performance index for a two-state system is obtained analytically in closed form by using Parseval's identity and integrating over the spectral variables. This forms a deterministic, non-convex function of the control signal and the measurement history that must be optimized numerically at each time step. Examples are given of both the estimator and the controller, to demonstrate their performance and expose their interesting robustness characteristics.