This dissertation advances tools for the certification of unmanned aircraft system (UAS) flight controllers. We develop two thrusts to this goal: (1) the validation and improvement of an uncertain UAS framework based on integral quadratic constraint (IQC) theory and (2) the development of novel IQC theorems which allow the analysis of uncertain systems having time-varying characteristics. Pertaining to the first thrust, this work improves and implements an IQC-based robustness analysis framework for UAS. The approach models the UAS using a linear fractional transformation on uncertainties and conducts robustness analysis on the uncertain system via IQC theory. By expressing the set of desired UAS flight paths with an uncertainty, the framework enables analysis of the uncertain UAS flying about any level path whose radius of curvature is bounded. To demonstrate the versatility of this technique, we use IQC analysis to tune trajectory-tracking and path-following controllers designed via H2 or H-infinity synthesis methods. IQC analysis is also used to tune path-following PID controllers. By employing a non-deterministic simulation environment and conducting numerous flight tests, we demonstrate the capability of the framework in predicting loss of control, comparing the robustness of different controllers, and tuning controllers. Finally, this work demonstrates that signal IQCs have an important role in obtaining IQC analysis results which are less conservative and more consistent with observations from flight test data. With regards to the second thrust, we prove a novel theorem which enables robustness analysis of uncertain systems where the nominal plant and the IQC multiplier are linear time-varying systems and the nominal plant may have a non-zero initial condition. When the nominal plant and the IQC multiplier are eventually periodic, robustness analysis can be accomplished by solving a finite-dimensional semidefinite program. Time-varying IQC multipliers are beneficial in analysis because they provide the possibility of reducing conservatism and are capable of expressing uncertainties that have unique time-domain characteristics. A number of time-varying IQC multipliers are introduced to better describe such uncertainties. The utility of this theorem is demonstrated with various examples, including one which produces bounds on the UAS position after an aggressive Split-S maneuver. / Doctor of Philosophy / This work develops tools to aid in the certification of unmanned aircraft system (UAS) flight controllers. The forthcoming results are founded on robust control theory, which allows the incorporation of a variety of uncertainties in the UAS mathematical model and provides tools to determine how robust the system is to these uncertainties. Such a foundation provides a complementary perspective to that obtained with simulations. Whereas simulation environments provide a probabilistic-type analysis and are oftentimes costly, the following results provide worst-case guarantees—for the allowable disturbances and uncertainties—and require far less computational resources. Here we take two approaches in our development of certification tools for UAS. First we validate and improve on an uncertain UAS framework that relies on integral quadratic constraint (IQC) theory to analyze the robustness of the UAS in the presence of uncertainties and disturbances. Our second approach develops novel IQC theorems that can aid in providing bounds on the UAS state during its flight trajectory. Though the applications in this dissertation are focused on UAS, the theory can be applied to a wide variety of physical and nonphysical problems wherein uncertainties in the mathematical model cannot be avoided.