  The vertical stabilizer is the key aerodynamic surface that provides an
aircraft with its directional stability characteristic while ailerons and
rudder are the primary control surfaces that give pilots control authority of
the yawing and banking maneuvers. Losing the vertical stabilizer will,
therefore, result in the consequential loss of lateral/directional stability
and control, which is likely to cause a fatal crash. In this paper, we
construct a scenario of a damaged aircraft model which has no physical rudder
control surface, and then a strategy based on differential thrust is proposed
to be utilized as a control input to act as a "virtual" rudder to help maintain
stability and control of the damaged aircraft. The $H_{\infty}$ loop-shaping
approach based robust control system design is implemented to achieve a stable
and robust flight envelope, which is aimed to provide a safe landing.
Investigation results demonstrate successful application of such robust
differential thrust methodology as the damaged aircraft can achieve stability
within feasible control limits. Finally, the robustness analysis results
conclude that the stability and performance of the damaged aircraft in the
presence of uncertainty remain within desirable limits, and demonstrate not
only a robust, but a safe flight mission through the proposed $H_{\infty}$
loop-shaping robust differential thrust control methodology.
