UC Irvine

Differential-geometric control theory exploits \textit{differential geometry} in the analysis of dynamical control systems. Differential geometry is a mathematical discipline that is concerned with studying calculus on curved (non-Euclidean) spaces. The main focus of this Dissertation is to formulate modern aeronautical engineering applications in a differential-geometric control framework.

Nonlinear controllability analysis is one of the most important aspects of differential-geometric control theory. Its importance emanates from the fact that linear controllability conditions for linearized systems are not necessary. That is, there exists a class of nonlinear systems that are linearly uncontrollable but nonlinearly controllable. In particular, it allows for identification of the ability to generate motions along unactuated (nonintuitive) directions through specific interactions between the system dynamics and control inputs.

In this Dissertation, differential-geometric control theory is utilized to analyze the nonlinear controllability of airplane flight dynamics. This study lead to several discoveries. First, it is found that an airplane in an upset situation where all control surfaces are blocked (inoperative) and linear controllability is lost, nonlinear controllability can be recovered using engine control only. More importantly, the study reveals unconventional mechanisms to generate motion along different directions in the state space. In particular, a novel roll mechanism that relies on nonlinear interactions between the elevator and aileron control surfaces is discovered. This novel roll mechanism is found to be superior in comparison to the conventional one (using ailerons only) near stall. Using differential-geometric control tools (e.g., non-holonomic motion planning and Fliess functional expansion), it is shown that the discovered roll mechanism can provide four times rolling motion near stall in comparison to the conventional roll control using ailerons. This result suggests that the discovered roll mechanism will provide a significant solution to the loss of control problem near stall, which is the leading cause of fatal accidents in general aviation airplanes.

Combined with \textit{chronological calculus}, differential-geometric control theory provides rigorous analysis tools for time-varying vector fields, such as higher-order averaging of time-periodic systems and decomposition of multi-scale time-varying vector fields. These tools are applied, in this Dissertation, to the multi-body, time-periodic, flapping-wing flight dynamics. The rigorous analysis using these combined geometric-control-averaging techniques revealed unconventional balance and stability characteristics in the rich flapping flight dynamics of insects/birds. In particular, in contrast to the prevailing belief in the flapping flight dynamics community that insects are unstable at hover due to the lack of pitch stiffness, the current analysis revealed that flapping species enjoy a vibrational stabilization mechanism. That is, the natural wing periodic forcing induces a passive stabilization mechanism in the form of pitch stiffness, similar to the Kapitza pendulum.