text:
\section*{Nomenclature}

\begin{tabbing}
  XXX \= \kil
$A/C$ 		\> aircraft \\
$A_d$ 		\> state matrix of the damaged aircraft \\
$B_d$ 		\> input matrix of the damaged aircraft \\
$b$ 		\> aircraft wing span, $ft$ \\
$C_d$ 		\> output matrix of the damaged aircraft\\
${C_L}_{i}$		\> dimensionless derivative of rolling moment, $i=p, r, \beta, \delta_a, \delta_r$\\
${C_N}_{i}$		\> dimensionless derivative of yawing moment, $i=p, r, \beta, \delta_a, \delta_r$\\
${C_Y}_{i}$		\> dimensionless derivative of side force, $i=p, r, \beta, \delta_a, \delta_r$\\
$D_d$ 		\> state transition matrix of the damaged aircraft \\
$\frac{d\sigma}{d\beta}$ 		\> change in side wash angle with respect to change in side slip angle\\
$g$ 		\> gravitational acceleration, $ft/s^2$ \\
$\overline{I_{xx}}$ 		\> normalized mass moment of inertia about the x axis, $slug*ft^2$ \\
$\overline{I_{xz}}$ 		\> normalized product of inertia about the xz axis, $slug*ft^2$\\
$\overline{I_{zz}}$ 		\> normalized mass moment of inertia about the z axis, $slug*ft^2$ \\
$L_{i}$		\> dimensional derivative of rolling moment, $i=p, r, \beta, \delta_a, \delta_r$\\
$L_v$ 		\>  vertical stabilizer lift force, $lbf$\\
$l_v$ 		\> distance from the vertical stabilizer aerodynamic center to the aircraft center of gravity, $ft$\\
$m$ 		\> aircraft mass, $slugs$ \\
$N_{i}$		\> dimensional derivative of yawing moment, $i=p, r, \beta, \delta_a, \delta_r, \delta T$\\
$p$ 		\> roll rate, $deg/s$ \\
$r$ 		\> yaw rate, $deg/s$ \\
$S$ 		\> aircraft wing area, $ft^2$ \\
$S_v$ 		\> vertical stabilizer area, $ft^2$ \\
$T$ 		\> engine thrust, $lbf$ \\
$T_c$ 		\> engine thrust command, $lbf$ \\
$t$		\> time, $s$ \\
$t_d$		\> time delay, $s$\\
$V_v$ 		\> vertical stabilizer volume ratio, $ft^3$\\
$v$ 		\> airspeed, $ft/s$ \\
$W$ 		\> aircraft weight, $lbs$ \\
$Y_{i}$		\> dimensional derivative of side force, $i=p, r, \beta, \delta_a, \delta_r$\\
$y_e$ 		\> distance from the outermost engine to the aircraft center of gravity, $ft$\\
$z_v$ 		\> distance from the vertical stabilizer center of pressure to the fuselage center line, $ft$\\
$\alpha$ 		\> angle of attack, $deg$ \\
$\beta$ 		\> side slip angle, $deg$ \\
$\gamma$  	\> flight path angle, $deg$ \\
$\delta_a$		\> aileron deflection, $deg$ \\
$\delta_r$		\> rudder deflection, $deg$ \\
$\Delta T$		\> collective thrust, $lbf$ \\
$\delta T$		\> differential thrust, $lbf$ \\
$\zeta$		\> damping ratio \\
$\eta$  	\> efficiency factor \\
$\theta$  	\> pitch angle, $deg$ \\
$\rho$		\> air density, $slug/ft^3$ \\
$\tau$   	\> time constant, $s$ \\
$\phi$    	\> roll angle, $deg$ \\
$\omega$		\> bandwidth frequency , $1/s$ \\
$\dot{( \ )}$ 	\>  first order time derivative \\
$\ddot{( \ )}$ 	\>  second order time derivative \\
$\bar{( \ )}$ 	\>  trimmed value \\
${( \ )}_d$ 	\>  damaged aircraft dynamics \\
 \end{tabbing}

\section{Introduction} \label{Introduction}

The vertical stabilizer of an aircraft is an essential element in providing the aircraft with its directional stability characteristic while ailerons and rudder serve as the primary control surfaces of the yawing and banking maneuvers. In the event of an aircraft losing its vertical stabilizer, the sustained damage will cause lateral/directional stability to be compromised and lack of control is likely to result in a fatal crash. Notable examples of such a scenario are the crash of the American Airline 587 in 2001 when an Airbus A300-600 lost its vertical stabilizer in wake turbulence, killing all passengers and crew members \cite{NTSB_flight587} and the crash of Japan Airlines Flight 123 in 1985 when a Boeing 747-SR100 lost its vertical stabilizer leading to an uncontrollable aircraft, resulting in 520 casualties \cite{faa_flight123}.

However, not all situations of losing the vertical stabilizer resulted in a total disaster. In one of those cases, the United Airlines Flight 232 in 1989 \cite{NTSB_flight_232}, differential thrust was proved to be able to make the aircraft controllable. Another remarkable endeavor is the landing of the Boeing 52-H even though the aircraft lost most of its vertical stabilizer in 1964 \cite{Hartnett06}.

In literature, research on this topic has been conducted with two main goals: to understand the response characteristics of the damaged aircraft such as the work of Bacon and Gregory \cite{Bacon_Gregory_07}, Nguyen and Stepanyan \cite{NguyenStepanyan10}, and Shah \cite{Shah08}, as well as to come up with an automatic control algorithm to save the aircraft from disasters where the work of Burcham et al. \cite{Burcham_et_al_98}, Guo et al. \cite{Guo_et_al_11}, Liu et al. \cite{Liu_et_al_09}, Tao and Ioanou \cite{Tao_Ioannou_88}, and Urnes and Nielsen \cite{UrnesNielsen10} hold the detailed analysis.

There are many valuable efforts in literature, such as the work of Shah \cite{Shah08}, where a wind tunnel study was performed to evaluate the aerodynamic effects of damages to lifting and stability/control surfaces of a commercial transport aircraft. In his work, Shah \cite{Shah08} studied this phenomenon in the form of partial or total loss of the wing, horizontal, or vertical stabilizers for the development of flight control systems to recover the damaged aircraft from adverse events. The work of Nguyen and Stepanyan \cite{NguyenStepanyan10} investigates the effect of the engine response time requirements of a generic transport aircraft in severe damage situations associated with the vertical stabilizer. They carried out a study which concentrates on assessing the requirements for engine design for fast responses in an emergency situation. In addition, the use of differential thrust as a propulsion command for the control of directional stability of a damaged transport aircraft was studied by Urnes and Nielsen \cite{UrnesNielsen10} to identify the effect of the change in aircraft performance due to the loss of the vertical stabilizer and to make an improvement in stability utilizing engine thrust as an emergency yaw control mode with feedback from the aircraft motion sensors.

Existing valuable research in literature provides very unique insight into the dynamics of such an extreme scenario whereas in this paper a unique extension to the existing works is provided. Here, we provide a $H_{\infty}$ loop-shaping based robust differential thrust control methodology to control such a damaged aircraft and land safely. This research is motivated to improve air travel safety in commercial aviation by incorporating the utilization of differential thrust to regain lateral/directional stability for a commercial aircraft (in this case, a Boeing 747-100) with a damaged vertical stabilizer. For this purpose, the damaged aircraft model is constructed in Section \ref{AC Model}, where its plant dynamics is investigated. In Section \ref{Differential Thrust}, the engine dynamics of the aircraft is modeled as a system of differential equations with corresponding time constant and time delay terms to study the engine response characteristic with respect to a differential thrust input, and the novel differential thrust control module is developed to map the rudder input to differential thrust input. In Section \ref{Open Loop}, the aircraft's open loop system response is investigated. In Section \ref{Robust}, the robust control system design based on $H_{\infty}$ loop-shaping approach is implemented as a means to stabilize the damaged aircraft. Investigation provides remarkable results of such robust differential thrust methodology as the damaged aircraft can reach steady state stability relatively fast, under feasible control efforts. In Section \ref{Robustness}, robustness and uncertainty analysis is conducted to test the guaranteed stability bounds and to validate the overall performance of the system in the presence of given ball of uncertainty. With Section \ref{Conclusion}, the paper is finalized, and future work is discussed.

\section{The Damaged Aircraft Model} \label{AC Model}
In this paper, the flight scenario is chosen to be a steady, level cruise flight for the Boeing 747-100 at Mach 0.65 and 20,000 feet. It is assumed that at one point during the flight, the vertical stabilizer is completely damaged, and in the followings the means to control such aircraft is investigated in such an extreme case scenario. For this purpose, first, the damaged aircraft model is developed for analysis, where the flight conditions for the damaged model are summarized in Table \ref{table:Flight conditions}.
\begin{table}[htbp!]
\centering
\caption{Flight conditions}
\label{table:Flight conditions}
 \begin{tabular}{|c|c|}
 \hline
 \textbf{Parameter} & \textbf{Value}\\ \hline
 Altitude $(ft)$ & 20,000 \\ \hline
Air Density $(slug/ft^3)$ & 0.001268 \\ \hline
Airspeed $(ft/s)$ & 673 \\ \hline
\end{tabular}
\end{table}

\subsection{The Damaged Aircraft Model} \label{AC model}

The Boeing 747-100 was chosen for this research due to its widely available technical specification, aerodynamics, and stability derivative data. The data for the nominal (undamaged) Boeing 747-100 are summarized in Table \ref{table:The undamaged aircraft data}.

\begin{table}[htbp!]
\centering
\caption{The nominal (undamaged) aircraft data \cite{747_ac_charac, roskam87}}
\label{table:The undamaged aircraft data}
 \begin{tabular}{|c|c|}
 \hline
 \textbf{Parameter} & \textbf{Value}\\ \hline
$S$ $(ft^2)$ & 5500  \\ \hline
$b$ $(ft)$ & 196  \\ \hline
$\bar{c}$ $(ft)$ & 27.3  \\ \hline
$y_e$ $(ft)$ & 69.83  \\ \hline
$W$ $(lbs)$ &  $6.3663*10^5$\\ \hline
$m$ $(slugs)$ &  $19786.46$\\ \hline
$I_{xx}$ $(slug*ft^2)$ & $18.2*10^6$  \\ \hline
$I_{yy}$ $(slug*ft^2)$ & $33.1*10^6$  \\ \hline
$I_{zz}$ $(slug*ft^2)$ & $49.7*10^6$  \\ \hline
$I_{xz}$ $(slug*ft^2)$ & $0.97*10^6$  \\ \hline
${C_L}_{\beta}$	& -0.160 \\ \hline
${C_L}_{p}$	& -0.340 \\ \hline
${C_L}_{r}$	& 0.130 \\ \hline
${C_L}_{\delta_a}$	& 0.013 \\ \hline
${C_L}_{\delta_r}$	& 0.008 \\ \hline
${C_N}_{\beta}$	& 0.160 \\ \hline
${C_N}_{p}$	& -0.026 \\ \hline
${C_N}_{r}$	& -0.28 \\ \hline
${C_N}_{\delta_a}$	& 0.0018 \\ \hline
${C_N}_{\delta_r}$	& -0.100 \\ \hline
${C_Y}_{\beta}$	& -0.90 \\ \hline
${C_Y}_{p}$	& 0 \\ \hline
${C_Y}_{r}$	& 0 \\ \hline
${C_Y}_{\delta_a}$	& 0 \\ \hline
${C_Y}_{\delta_r}$	& 0.120 \\ \hline
\end{tabular}
\end{table}

For the modeling of the damaged aircraft, in case of the loss of the vertical stabilizer, lateral/directional stability derivatives need to be re-examined and recalculated due to the resulting significant changes in the geometry and the aerodynamic behavior. Since the whole aerodynamic structure is affected, the new corresponding stability derivatives have to be calculated and studied. The lateral/directional dimensionless derivatives that depend on the vertical stabilizer include \cite{nelson98}:

\begin{equation}
 {C_Y}_{\beta}=-\eta \frac{S_v}{S}{{C_L}_\alpha}_v\left(1+\frac{d\sigma}{d\beta}\right)
\end{equation}

\begin{equation}
  {C_Y}_r=-2\left(\frac{l_v}{b}\right){{C_Y}_\beta}_{tail}
\end{equation}

\begin{equation}
 {C_N}_{\beta}={{C_N}_{\beta}}_{wf}+\eta_v V_v{{C_L}_\alpha}_v\left(1+\frac{d\sigma}{d\beta}\right)
\end{equation}

\begin{equation}
  {C_N}_r=-2\eta_v V_v\left(\frac{l_v}{b}\right){{C_L}_{\alpha}}_v
\end{equation}

\begin{equation}
  {C_L}_r=\frac{C_L}{4}-2\left(\frac{l_v}{b}\right)\left(\frac{z_v}{b}\right){{C_Y}_\beta}_{tail}
\end{equation}

When the vertical stabilizer is completely lost in an aircraft, the vertical tail area, vertical tail volume, and associated efficiency factor will all be zero, resulting in ${C_Y}_{\beta}={C_Y}_r={C_N}_r=0$. If the vertical stabilizer is assumed to be the primary aerodynamic surface responsible for the weathercock stability, then ${C_N}_{\beta}=0$ and finally, we have ${C_L}_r=\frac{C_L}{4}$.

In addition, without the vertical stabilizer, the mass and inertia data of the damaged aircraft are going to change as well, where the values that reflect such a scenario (for the damaged aircraft) are calculated and listed in Table \ref{table:The damaged aircraft mass and inertia data}.

\begin{table}[htbp!]
\centering
\caption{The damaged aircraft mass and inertia data}
\label{table:The damaged aircraft mass and inertia data}
 \begin{tabular}{|c|c|}
 \hline
 \textbf{Parameter} & \textbf{Value}\\ \hline
$W$ $(lbs)$ &  $6.2954*10^5$\\ \hline
$m$ $(slugs)$ &  $19566.10$\\ \hline
$I_{xx}$ $(slug*ft^2)$ & $17.893*10^6$  \\ \hline
$I_{yy}$ $(slug*ft^2)$ & $30.925*10^6$  \\ \hline
$I_{zz}$ $(slug*ft^2)$ & $47.352*10^6$  \\ \hline
$I_{xz}$ $(slug*ft^2)$ & $0.3736*10^6$  \\ \hline
\end{tabular}
\end{table}

In this study, during an event of the loss of the vertical stabilizer, it is proposed that the differential thrust component of aircraft dynamics be utilized as an alternate control input replacing the rudder control to regain stability and control of lateral/directional flight dynamics. Next, the lateral-directional linear equations of motion of the damaged aircraft are presented, with the ailerons $(\delta_a)$, differential thrust $(\delta T)$, and collective thrust $(\Delta T)$ as control inputs \cite{NguyenStepanyan10}, as

\begin{equation}
\left[
\begin{array}{c}
\dot{\phi}\\
\dot{p}\\
\dot{\beta}\\
\dot{r}
\end{array}\right]
= \left[
\begin{array}{cccc}
0 & 1 & 0 & \bar{\theta} \\
0 & L_p & L_{\beta} & L_r \\
\frac{g}{\bar{V}} & \frac{Y_p}{\bar{V}} & \frac{Y_{\beta}+g\bar{\gamma}}{\bar{V}} & \frac{Y_p}{\bar{V}}-1 \\
0 & N_p & N_{\beta} & N_r
\end{array}\right]
\left[
\begin{array}{c}
{\phi}\\
{p}\\
{\beta}\\
{r}
\end{array}\right]
+
\left[
\begin{array}{ccc}
0 & 0 & 0\\
L_{\delta_a} & \frac{\overline{I_{xz}}y_e}{\overline{I_{xx}}\overline{I_{zz}}-\overline{I_{xz}}^2} & 0\\
\frac{Y_{\delta_a}}{\bar{V}} & 0 & \frac{-\bar{\beta}}{m\bar{V}}\\
N_{\delta_a} & \frac{\overline{I_{xx}}y_e}{\overline{I_{xx}}\overline{I_{zz}}-\overline{I_{xz}}^2} & 0
\end{array}\right]
\left[
\begin{array}{c}
{\delta_a}\\
{\delta T} \\
{\Delta T} \\
\end{array}\right]
\end{equation}

In this case, if the initial trim side-slip angle is zero, then $\Delta T$ does not have any significance in the control effectiveness for a small perturbation around the trim condition \cite{NguyenStepanyan10}, which means that the above equations can be reduced to the final form as:

\begin{equation}
\left[
\begin{array}{c}
\dot{\phi}\\
\dot{p}\\
\dot{\beta}\\
\dot{r}
\end{array}\right]
= \left[
\begin{array}{cccc}
0 & 1 & 0 & \bar{\theta} \\
0 & L_p & L_{\beta} & L_r \\
\frac{g}{\bar{V}} & \frac{Y_p}{\bar{V}} & \frac{Y_{\beta}+g\bar{\gamma}}{\bar{V}} & \frac{Y_p}{\bar{V}}-1 \\
0 & N_p & N_{\beta} & N_r
\end{array}\right]
\left[
\begin{array}{c}
{\phi}\\
{p}\\
{\beta}\\
{r}
\end{array}\right]
+
\left[
\begin{array}{cc}
0 & 0\\
L_{\delta_a} & \frac{\overline{I_{xz}}y_e}{\overline{I_{xx}}\overline{I_{zz}}-\overline{I_{xz}}^2}\\
\frac{Y_{\delta_a}}{\bar{V}} & 0 \\
N_{\delta_a} & \frac{\overline{I_{xx}}y_e}{\overline{I_{xx}}\overline{I_{zz}}-\overline{I_{xz}}^2}
\end{array}\right]
\left[
\begin{array}{c}
{\delta_a}\\
{\delta T} \\
\end{array}\right]
\end{equation}

\subsection{Plant Dynamics} \label{Plant Dynamics}

Based on the calculated data for the lateral/directional stability derivatives, mass and inertia of the aircraft without its vertical stabilizer, the lateral/directional state-space representation of the damaged aircraft is obtained as the following

\begin{equation} \label{eq:Ad}
{A_d}= \left[
\begin{array}{cccc}
0 & 1 & 0 & 0\\
0 & -0.8566 & -2.7681 &  0.1008\\
0.0478 & 0 & 0 & -1 \\
0 & -0.0248 & 0 & 0
\end{array}\right]
\end{equation}

\begin{equation} \label{eq:Bd}
{B_d}= \left[
\begin{array}{cc}
0 & 0\\
0.2249 & 0.0142\\
0 & 0\\
0.0118 & 0.6784
\end{array}\right]
\end{equation}

\begin{equation} \label{eq:Cd}
C_d= \left[
\begin{array}{cccc}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0& 1
\end{array}\right]
\end{equation}

\begin{equation} \label{eq:Dd}
D_d= \left[
\begin{array}{cc}
0 & 0\\
0 & 0\\
0 & 0\\
0 & 0
\end{array}\right]
\end{equation}
where the subscript $(~~)_{d}$ stands for the $damaged$ aircraft dynamics.

Here, ${A_d}$ represents the state matrix of the damaged aircraft, and ${B_d}$ stands for the input matrix of the scenario where the ailerons $(\delta_a)$ and differential thrust $(\delta T)$ are control inputs of the damaged aircraft. ${C_d}$ is the output matrix, and ${D_d}$ is the direct (state transition) matrix of the damaged aircraft.

In addition, for more in-depth intuition, the damping characteristics of the damaged aircraft are also provided in Table \ref{table:damaged_damping}.

\begin{table}[htbp!]
\centering
\caption{Damping characteristics of the damaged aircraft}
\label{table:damaged_damping}
 \begin{tabular}{|c|c|c|c|c|}
 \hline
\textbf{Mode} & \textbf{Pole Location} & \textbf{Damping} & \textbf{Frequency $(1/s)$} & \textbf{Period $(s)$}\\ \hline
Dutch Roll & $0.0917\pm i0.43$ & $-0.209$ & $0.439$ & $14.2969$\\ \hline
Spiral & $6.32*10^{-18}$ & $-1$ & $6.32*10^{-18}$ & $9.9486*10^{17}$\\ \hline
Roll & $-1.04$ & $1$ & $1.04$ & $6.0422$\\
\hline
\end{tabular}
\end{table}

Table \ref{table:damaged_damping} clearly indicates the unstable nature of the damaged aircraft in the Dutch roll mode by the obvious Right Half Plane (RHP) pole locations. Furthermore, the pole of the spiral mode lies at the origin. The only stable mode of the damaged aircraft remains to be the roll mode.

Following the aerodynamic and stability analysis of the damaged aircraft, we next concentrate on the differential thrust methodology.

\section{Differential Thrust as a Control Mechanism} \label{Differential Thrust}

\subsection{Propulsion Dynamics} \label{Propulsion}

With emerging advancements in manufacturing processes, structures, and materials, it is a well known fact that aircraft engines have become highly complex systems and include numerous nonlinear processes, which affect the overall performance (and stability) of the aircraft. From the force-balance point of view, this is usually due to the existing coupled and complex dynamics between engine components and their relationships in generating thrust. However, in order to utilize the differential thrust generated by the jet engines as a control input for lateral/directional stability, the dynamics of the engine need to be modeled in order to gain an insight into the response characteristics of the engines.

Engine response, generally speaking, depends on its time constant and time delay characteristics. Time constant dictates how fast the thrust is generated by the engine while time delay (which is inversely proportional to the initial thrust level) is due to the lag in engine fluid transport and the inertia of the mechanical systems such as rotors and turbo-machinery blades \cite{NguyenStepanyan10}.

It is also suggested \cite{NguyenStepanyan10} that the non-linear engine dynamics model can be simplified as a second-order time-delayed linear model as
\begin{equation}
\ddot{T}+2\zeta\omega\dot{T}+\omega^2T= \omega^2T_c(t-t_d)
\end{equation}
where $\zeta$ and $\omega$ are the damping ratio and bandwidth frequency of the closed-loop engine dynamics, respectively; $t_d$ is the time delay factor, and $T_c$ is the thrust command prescribed by the engine throttle resolver angle.

With the time constant defined as the inverse of the bandwidth frequency $(\tau=\frac{1}{\omega})$, and $\zeta$ chosen to be 1 representing a critically damped engine response (to be comparable to existing studies), the engine dynamics can be represented as

\begin{equation}
\left[
\begin{array}{c}
\dot{T}\\
\ddot{T}
\end{array}\right]
= \left[
\begin{array}{cc}
0 & 1\\
\frac{-1}{\tau^2} & \frac{-2}{\tau}
\end{array}\right]
\left[
\begin{array}{c}
T\\
\dot{T}
\end{array}\right]
+
\left[
\begin{array}{c}
0 \\
\frac{1}{\tau^2}
\end{array}\right]
T_c(t-t_d)
\end{equation}

For this study, the Pratt and Whitney JT9D-7A engine is chosen for the application in the Boeing 747-100, where the engine itself produces a maximum thrust of 46,500 lbf \cite{boeing747_tech_spec}. At Mach 0.65 and 20,000 feet flight conditions, the engine time constant is 1.25 seconds, and the time delay is 0.4 second \cite{NguyenStepanyan10}.

\begin{figure}[h!]
 \centering
 \includegraphics[width=4truein]{Figures/engine_response.png}
 \caption{Engine thrust response at Mach 0.65 and 20,000 feet}
 \label{fig:engine_response}
 \end{figure}

The engine thrust response curve at Mach 0.65 and 20,000 feet is, therefore, obtained as shown in Fig. \ref{fig:engine_response}, which provides a useful insight into how the time constant and time delay factors affect the generation of thrust for the JT9D-7A jet engine. At Mach 0.65 and 20,000 feet, with the engine time constant of 1.25 seconds, and the time delay of 0.4 second, it takes approximately ten seconds for the engine to reach steady state and generate its maximum thrust capacity at 46,500 lbf from the trim thrust of 3221 lbf. The increase in thrust generation follows a relatively linear fashion with the engine response characteristic of approximately 12,726 lbf/s during the first two seconds, and then the thrust curve becomes nonlinear until it reaches its steady state at maximum thrust capacity after about ten seconds. This represents one major difference between the rudder and differential thrust as a control input. Due to the lag in engine fluid transport and turbo-machinery inertia, differential thrust (as a control input) cannot respond as instantaneously as the rudder, which has to be taken into account very seriously in control system design.

\subsection{Thrust Dynamics and Configuration}
In order to utilize differential thrust as a control input for the four-engined Boeing 747-100 aircraft, a differential thrust control module must be developed. Here, the differential thrust input is defined as the difference between the thrust generated by engine number 1 and engine number 4 while the amounts of thrust generated by engine number 2 and 3 are kept equal to each other as shown in Eqs. (\ref{eq:d_th}-\ref{eq:th23}). This concept is illustrated in further details in Fig. \ref{fig:model}.

\begin{equation}\label{eq:d_th}
\delta T=T_1-T_4
\end{equation}
\begin{equation}\label{eq:th23}
T_2=T_3
\end{equation}

\begin{figure}[htbp!]
 \centering
 \includegraphics[width=6truein]{Figures/model.png}
 \caption{The free body diagram}
 \label{fig:model}
 \end{figure}

Engine number 1 and 4 are employed to generate the differential thrust due to the longer moment arm $(y_e)$, which makes the differential thrust more effective as a control for yawing moment. This brings into the picture the need of developing a logic that maps the rudder input to differential thrust input, which is further explained in the following section.

\subsection{Rudder Input to Differential Thrust Input Mapping Logic}

When the vertical stabilizer of the aircraft is intact (i.e. with nominal plant dynamics), the pilot has the ailerons and rudder as major control surfaces. However, when the vertical stabilizer is damaged, most probably, the pilot will keep on demanding the control effort from the rudder until it is clear that there is no response from the rudder. To eliminate this mishap, but to still be able to use the rudder demand, a differential thrust control module is introduced in the control logic, as shown in Fig. \ref{fig:control_logics} and Fig. \ref{fig:dT_control_module}, respectively. This differential thrust control module maps  corresponding input/output dynamics from the rudder pedals to the aircraft response, so that when the rudder is lost, the rudder input (from the pilot) will still be utilized but switched to the differential thrust input, which will act as a \emph{rudder-like input} for lateral/directional controls. This logic constitutes one of the novel approaches introduced in this paper.

\begin{figure}[htbp!]
 \centering
 \includegraphics[width=5truein]{Figures/control_logics.png}
 \caption{Aircraft control logic diagram}
 \label{fig:control_logics}
 \end{figure}

 \begin{figure}[htbp]
 \centering
 \includegraphics[width=5truein]{Figures/dT_control_module.png}
 \caption{Differential thrust control module}
 \label{fig:dT_control_module}
 \end{figure}

As it can be also seen from Fig. \ref{fig:control_logics} and Fig. \ref{fig:dT_control_module}, the differential thrust control module's function is to convert the rudder pedal input from the pilot to the differential thrust input. In order to achieve that, the rudder pedal input (in radians) is converted to the differential thrust input (in pounds-force) which is then provided into the engine dynamics, as discussed previously in Section \ref{Propulsion}. With this modification, the engine dynamics will dictate how differential thrust is generated, which is then provided as a "virtual rudder" input into the aircraft dynamics. The radian to pound-force mapping is provided in the next section, for the completeness of analysis.

\subsection{Radian to Pound-Force Conversion Factor}
Using Fig. \ref{fig:model} and with the steady, level flight assumption, the following relationship can be obtained:
\begin{equation}
N_{\delta_r}= N_{\delta T}
\end{equation}

\begin{equation}
qSb{C_N}_{\delta_r}\delta_r=(\delta T) y_e
\end{equation}
which means the yawing moment by deflecting the rudder and by using differential thrust have to be the same. Therefore, the relationship between the differential thrust control input $(\delta T)$ and the rudder control input $(\delta_r)$ can be obtained as

\begin{equation}
\delta T= \left(\frac{qSb{C_N}_{\delta_r}}{y_e}\right)\delta_r
\end{equation}

Based on the flight conditions at Mach 0.65 and 20,000 feet, and the data for the Boeing 747-100 summarized in Table \ref{table:Flight conditions} and Table \ref{table:The undamaged aircraft data}, the conversion factor for the rudder control input to the differential thrust input is calculated to be

\begin{equation}
\frac{\delta T}{\delta_r}= -4.43*10^5 \> \frac{lbf}{rad}
\end{equation}

Due to the sign convention of rudder deflection and the free body diagram in Fig. \ref{fig:model}, $\delta_r$ here is negative. Therefore, for the Boeing 747-100, in this study, the conversion factor for the mapping of a rudder input to a differential thrust input is found to be

\begin{equation}
\frac{\delta T}{\delta_r}= 4.43*10^5 \> \frac{lbf}{rad}
\end{equation}

\subsection{Commanded vs. Available Differential Thrust}
At this point, the worst case scenario is considered, and it is assumed that the aircraft has lost its vertical stabilizer so that the rudder input is converted to the differential thrust input according to the logic discussed previously in this section.

Unlike the rudder, due to delayed engine dynamics with time constant, there is a major difference in the commanded differential thrust and the available differential thrust as shown in Fig. \ref{fig:dT}.

\begin{figure}[htbp!]
 \centering
 \includegraphics[width=4truein]{Figures/dT.png}
 \caption{Commanded vs. available differential thrust}
 \label{fig:dT}
 \end{figure}

It can be seen from Fig. \ref{fig:dT} that compared to the commanded differential thrust, the available differential thrust is equal in amount but longer in the time delivery. For a one degree step input on the rudder, the corresponding equivalent commanded and available differential thrust are 7737 lbf,  which is deliverable in ten second duration. Unlike the instantaneous control of the rudder input, there is a lag associated with the use of differential thrust as a control input. This is due to the lag in engine fluid transport and the inertia of the mechanical systems such as rotors and turbo-machinery blades \cite{NguyenStepanyan10}. This is a major constraint of plant dynamics and will be taken into account during the robust control system design phase in the following sections.

\section{Open Loop System Response Analysis}\label{Open Loop}

Following to this, the open loop response characteristics of the aircraft with a damaged vertical stabilizer to one degree step inputs from the ailerons and differential thrust are presented in Fig. \ref{fig:open_loop_response}. Here, it can be clearly seen that when the aircraft is damaged and the vertical stabilizer is lost, the aircraft response to the pilot's inputs is absolutely unstable in all four states (as it was also obvious (and expected) from the pole locations). This shows the pilot will not have much chance to stabilize the aircraft in time, which calls for a novel approach to save the damaged aircraft. This is another point where the second novel contribution of the paper is introduced: automatic control strategy to stabilize the aircraft,  which allows safe (i.e. intact) landing of the aircraft.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=5truein]{Figures/open_loop_response.png}
 \caption{Open loop system response of the damaged aircraft}
 \label{fig:open_loop_response}
 \end{figure}

\section{Robust Control System Design based on $H_{\infty}$ Loop-Shaping Approach} \label{Robust}

\subsection{$H_{\infty}$ Loop-Shaping Design}

As having been witnessed in Section \ref{Open Loop}, the open loop responses of the damaged aircraft are unstable in all four lateral/directional states. This means the pilot will not have much chance to save the aircraft, which calls for a novel approach to save the damaged aircraft and to provide a safe landing. Therefore, in this section, the robust control system design based on $H_{\infty}$ loop-shaping approach is chosen as a means to stabilize the damaged aircraft due to its ability to suppress external disturbances and noise, and overall system robustness.

The damaged aircraft plant's open loop singular values are shaped by the pre- and post-compensation \cite{McFarlane_Glover90} as illustrated in Fig. \ref{fig:shaped_plant}.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=5truein]{Figures/shaped_plant.png}
 \caption{The damaged aircraft's shaped plant and controller}
 \label{fig:shaped_plant}
 \end{figure}

As it can be seen from Fig. \ref{fig:shaped_plant}, $G$ is the open loop plant of the damaged aircraft while $W_1$ and $W_2$ are the pre- and post-compensation, respectively. The shaped plant is, therefore,

\begin{equation} \label{eq:G_s}
G_s=W_2GW_1
\end{equation}

In addition, the controller $(K_s)$ is synthesized by solving the robust stabilization problem for the shaped plant $(G_s)$ with a normalized left coprime factorization of $G_s={M_s}^{-1}N_s$, and the feedback controller for plant $G$ is, therefore, $K=W_1K_sW_2$ \cite{Skogestad_Postlethwaite96}.

Next, the implemented $H_{\infty}$ loop-shaping diagram is shown in Fig. \ref{fig:hinf_loop}.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=6truein]{Figures/hinf_loop.png}
 \caption{$H_{\infty}$ loop-shaping diagram}
 \label{fig:hinf_loop}
 \end{figure}

Figure \ref{fig:hinf_loop} provides an insight into the robust controller design based on the $H_{\infty}$ loop-shaping approach. The main goal of this controller is to stabilize all four lateral/directional states of the damaged aircraft, which are roll rate $(p)$, roll angle $(\phi)$, side-slip angle $(\beta)$, and yaw rate $(r)$. The control inputs of the damaged aircraft are ailerons $(\delta_a)$ and differential thrust $(\delta T)$. As it is obvious from Fig. \ref{fig:hinf_loop}, the pilot's inputs, which are one degree step inputs for both ailerons and rudder, will go through the pre-filter gain $K_s(0)W_2(0)$ and then the \emph{Input Control Module}, where the aileron control signal is routed through the saturation limits of $\pm$ 26 degrees \cite{faa_747} and the rudder control input is routed through the \emph{Differential Thrust Control Module}, where the rudder input is converted to differential thrust input as discussed previously in Section \ref{Differential Thrust}. It is also worth noting that there are also a differential thrust saturation limit of 43,729 lbf and a thrust generation rate limiter of 12,726 lbf/s imposed on the differential thrust control as discussed in Section \ref{Differential Thrust}. This is to make sure the control inputs are within the limits of both the ailerons and differential thrust. Furthermore, in order for the control efforts to be feasible in a real-life scenario, the control effort signals are also routed through the \emph{Actuation Dynamics}, where the same saturation and rate limits are imposed on the ailerons and differential thrust as discussed previously.

The next step is to select the weighting functions, which should be taken into careful consideration due to the dimensions of the system.

After an iterative process, the selected weighting functions for $W_1$ are as

\begin{equation}
{W_1}_{(11)}=\frac{4s + 1}{4s + 10}
\end{equation}

\begin{equation}
{W_1}_{(22)}=\frac{50s + 5}{18s + 25}
\end{equation}

\noindent We can then construct the \emph{system matrix} $W_1$ as

\begin{equation}
W_1= \left[
\begin{array}{ccccc}
-2.5000 & 0 & 2.5981 & 0 & 2.0000\\
0 & -1.3889 & 0 & 1.8922 & 0\\
-2.5981 & 0 & 3.0000 & 0 & 0\\
0 & -1.8922 & 0 & 2.7778 & 0\\
0 & 0 & 0 & 0 & -Inf
\end{array}\right]
\end{equation}

The selected weighting functions for $W_2$ are as

\begin{equation}
{W_2}_{(11)}=\frac{16}{s + 16}
\end{equation}

\begin{equation}
{W_2}_{(22)}={W_2}_{(33)}={W_2}_{(44)}=\frac{120}{s + 120}
\end{equation}

where the \emph{system matrix} of $W_2$ becomes

\begin{equation}
W_2= \left[
\begin{array}{ccccccccc}
-16.0000    &     0     &    0    &     0  &  4.0000   &      0   &      0    &     0  &  4.0000\\
0 & -120.0000    &     0    &     0   &      0  &  10.9545   &      0   &      0   &      0\\
0   &      0  & -120.0000   &      0   &      0   &      0 &   10.9545  &       0  &       0\\
0   &      0   &      0 & -120.0000   &      0   &      0   &      0  & 10.9545   &      0\\
4.0000   &      0   &      0    &     0   &      0   &      0   &      0    &     0    &     0\\
0 &  10.9545  &       0   &      0   &      0   &      0   &      0   &      0   &      0\\
0   &      0 &  10.9545   &      0   &      0   &      0   &      0   &      0   &      0\\
0   &      0  &       0 &  10.9545   &      0   &      0   &      0     &    0   &      0\\
0    &     0  &       0   &      0    &     0   &      0   &      0    &     0   &   -Inf
\end{array}\right]
\end{equation}

As it is customary in general loop-shaping formulation, the maximum stability margin $(e_{max})$ is defined as a performance criterion
\begin{equation}
e_{max}=1/\gamma_{min}
\end{equation}

\noindent where $\gamma_{min}$ is the $H_{\infty}$ optimal cost. For our analysis, the maximum stability margin is then obtained as

\begin{equation}
e_{max}= 0.2763
\end{equation}

\noindent which is within the suggested optimal value bounds: $0.25<e_{max}<0.30$.

Next, the Sensitivity (S) and Co-sensitivity (T) plots are constructed.
Figure \ref{fig:Si_So} illustrates the input and output sensitivity functions.

 \begin{figure}[htbp]
   \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1\linewidth]{Figures/Si.png}\\
    (a) Input sensitivity functions
    \vspace{2ex}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1.06\linewidth]{Figures/So.png}\\
    (b) Output sensitivity functions
    \vspace{2ex}
  \end{minipage}
  \caption{Input and output sensitivity functions}
  \label{fig:Si_So}
 \end{figure}

Next, we can investigate the shaped plant behaviors for inputs and outputs in Fig. \ref{fig:Li_Lo}.

 \begin{figure}[htbp]
   \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1\linewidth]{Figures/Li.png}\\
    (a) Input responses of shaped plant
    \vspace{2ex}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1.03\linewidth]{Figures/Lo.png}\\
    (b) Output responses of shaped plant
    \vspace{2ex}
  \end{minipage}
  \caption{Input and output responses of shaped plant}
  \label{fig:Li_Lo}
 \end{figure}

\noindent From Fig. \ref{fig:Li_Lo}, we can see that for the aileron control input $(\delta_a)$, the shaped plant has very good response characteristics of having high gain at low frequencies for tracking and low gain at high frequencies for disturbance rejection. For the differential thrust control input $(\delta T)$, although the gain at low frequencies is not as high as that of the aileron control input, but the gain is relatively linear, which makes the differential thrust control quite predictable for the pilots. At high frequencies, the gain roll-off is also linear and quite similar to that of the aileron control input, which is helpful at rejecting disturbances. In addition, Fig. \ref{fig:Li_Lo} also shows the output responses of the shaped plant in all four lateral/directional states, from which we can see that the implemented controller based on $H_{\infty}$ loop-shaping approach can augment the system in two groups: roll angle and roll rate ($\phi$ and $p$) and side-slip angle and yaw rate ($\beta$ and $r$). It is also obvious that roll angle and roll rate ($\phi$ and $p$) have higher gains than side-slip angle and yaw rate ($\beta$ and $r$), which is expected when the aircraft loses its vertical stabilizer.

In addition, it is possible to look at the augmented control action by the $H_{\infty}$ loop-shaping controller in Fig. \ref{fig:K_aug}. Furthermore, the Co-sensitivity plots related to the shaped plant behaviors for the inputs and outputs are presented in Fig. \ref{fig:Ti_To}. Fig \ref{fig:K_aug} and Fig. \ref{fig:Ti_To} prove that the controller based on $H_{\infty}$ loop-shaping approach is able to achieve desirable closed loop response characteristics in the frequency domain in terms of robustness and stability.

 \begin{figure}[htbp]
 \centering
 \includegraphics[width=4truein]{Figures/K_aug.png}
 \caption{Augmented control action by the $H_{\infty}$ loop-shaping controller}
 \label{fig:K_aug}
 \end{figure}

 \begin{figure}[htbp]
   \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1.03\linewidth]{Figures/Ti.png}\\
    (a) Co-sensitivity plot for input responses of shaped plant
    \vspace{2ex}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1.02\linewidth]{Figures/To.png}\\
    (b) Co-sensitivity plot for output responses of shaped plant
    \vspace{2ex}
  \end{minipage}
  \caption{Co-sensitivity plot for input and output responses of shaped plant}
  \label{fig:Ti_To}
 \end{figure}

Next, the comparison of the open-loop, shaped, and robustified plant response is carried out to investigate the effect of the $H_{\infty}$ loop-shaping on the plant dynamics as illustrated in Fig.  \ref{fig:loop_da} and Fig. \ref{fig:loop_dT}.

As seen in Fig.  \ref{fig:loop_da} and Fig. \ref{fig:loop_dT}, compared to the open-loop plant response, the performance of the damaged aircraft is further improved by controller.

\begin{figure}[htbp]
   \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1\linewidth]{Figures/loop_da_phi.png}\\
    (a) Transfer function $\frac{\phi}{\delta_a}$
    \vspace{2ex}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1.01\linewidth]{Figures/loop_da_p.png} \\
    (b) Transfer function $\frac{p}{\delta_a}$
    \vspace{2ex}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1.008\linewidth]{Figures/loop_da_beta.png}\\
    (c) Transfer function $\frac{\beta}{\delta_a}$
    \vspace{2ex}
  \end{minipage
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1\linewidth]{Figures/loop_da_r.png}\\
    (d) Transfer function $\frac{r}{\delta_a}$
    \vspace{2ex}
  \end{minipage}
  \caption{Open-loop $(G)$ vs. shaped $(G_s)$ vs. robustified $(G_sK_s)$ plant for aileron input}
  \label{fig:loop_da}
\end{figure}

\begin{figure}[htbp]
    \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1\linewidth]{Figures/loop_dT_phi.png}\\
    (a) Transfer function $\frac{\phi}{\delta T}$
    \vspace{2ex}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1\linewidth]{Figures/loop_dT_p.png}\\
    (b) Transfer function $\frac{p}{\delta T}$
    \vspace{2ex}
  \end{minipage}
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1.01\linewidth]{Figures/loop_dT_beta.png}\\
    (c) Transfer function $\frac{\beta}{\delta T}$
    \vspace{2ex}
  \end{minipage
  \begin{minipage}[b]{0.5\linewidth}
    \centering
    \includegraphics[width=1\linewidth]{Figures/loop_dT_r.png} \\
    (d) Transfer function $\frac{r}{\delta T}$
    \vspace{2ex}
  \end{minipage}
    \caption{Open-loop $(G)$ vs. shaped $(G_s)$ vs. robustified $(G_sK_s)$ plant for differential thrust input}
  \label{fig:loop_dT}
\end{figure}

\subsection{Allowable Gain/Phase Variations: Disk Margin Analysis}

Furthermore, it is also worth checking the allowable gain and phase margin variations (i.e. associated uncertainty balls in gain and phase margins for a robust response and guaranteed stability). Obtained results are described in Fig. \ref{fig:diskmargin_input} and Fig. \ref{fig:diskmargin_output}.

From Fig. \ref{fig:diskmargin_input} and Fig. \ref{fig:diskmargin_output}, it can be clearly seen that with the $H_{\infty}$ loop-shaping control system design, both the inputs and outputs can achieve desirable and safe stability margins.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=4truein]{Figures/diskmargin_input.png}
 \caption{Disk gain and phase margins for inputs}
 \label{fig:diskmargin_input}
 \end{figure}

 \begin{figure}[htbp]
 \centering
 \includegraphics[width=4truein]{Figures/diskmargin_output.png}
 \caption{Disk gain and phase margins for outputs}
 \label{fig:diskmargin_output}
 \end{figure}

\subsection{Simulation Results}

Now it is time to look at the time domain behaviors of our system, which are shown in Fig. \ref{fig:hinf_state}. It is worth mentioning that the control inputs for both plants are one degree step inputs for both the ailerons and differential thrust to simulate an extreme scenario test to see whether the damaged aircraft utilizing differential thrust can hold itself in a continuous yawing and banking maneuver without becoming unstable and losing control. From Fig. \ref{fig:hinf_state}, it is obvious that after only 15 seconds, all four states of the aircraft's lateral/directional dynamics reach steady state values, which means that the controller can stabilize the damaged aircraft in only 15 seconds. However, this comes at the cost of the control efforts as shown in Fig. \ref{fig:hinf_control_efforts}, which are still under the control limits and without any saturation of the actuators.

\begin{figure}[htbp]
 \centering
 \includegraphics[width=6truein]{Figures/hinf_state.png}
 \caption{Closed loop time domain responses of the damaged aircraft}
 \label{fig:hinf_state}
 \end{figure}

\begin{figure}[htbp]
 \centering
 \includegraphics[width=4truein]{Figures/hinf_control_efforts.png}
 \caption{Control efforts}
 \label{fig:hinf_control_efforts}
 \end{figure}

As discussed previously in this section, in order to have a feasible control strategy in real-life situation, limiting factors are imposed on the aileron and differential thrust control efforts. The aileron deflection is limited at $\pm$26 degrees \cite{faa_747}. For differential thrust, a differential thrust saturation is set at 43,729 lbf, which is the difference of the maximum thrust and trimmed thrust values of the JT9D-7A engine. In addition, a rate limiter is also imposed on the thrust response characteristic at 12,726 lbf/s as discussed in Section \ref{Differential Thrust}.

The aileron control effort, as indicated by Fig. \ref{fig:hinf_control_efforts}, calls for the maximum deflection of approximately 2.4 degrees and reaches steady state at approximately -0.1 degree of deflection after 15 seconds. This aileron control effort is very reasonable and achievable if the ailerons are assumed to have instantaneous response characteristics by neglecting the lag from actuators or hydraulic systems. The differential thrust control effort demands a maximum differential thrust of approximately 3350 lbf, which is well within the thrust capability of the JT9D-7A engine, and the differential thrust control effort reaches steady state at around 15 lbf after 15 seconds. Therefore, it can be concluded that the robust control system design based on the $H_{\infty}$ loop-shaping approach is proven to be able to stabilize and save the damaged aircraft.

\section{Robustness and Uncertainty Analysis} \label{Robustness}

The robustness of the $H_{\infty}$ loop-shaping robust differential thrust control system design presented in this paper is investigated by the introduction of 30\% of full block, additive uncertainty into the plant dynamics of the damaged aircraft to test the performance of the aircraft in the presence of uncertainty. Fig. \ref{fig:robust_hinf_loop} shows the logic behind the robust control system design in the presence of uncertainty.

\begin{figure}[htbp!]
 \centering
 \includegraphics[scale=0.5]{Figures/robust_hinf_loop.png}
 \caption{$H_{\infty}$ loop-shaping diagram in the presence of uncertainty}
 \label{fig:robust_hinf_loop}
 \end{figure}

One thousand Monte-Carlo simulations were conducted to test the robustness of the damaged plant in the presence of uncertainty. The state responses in the presence of 30\% of uncertainty are shown in Fig. \ref{fig:robust_output_U_30}, where it is obvious that the robust control system design is able to perform well under given uncertain conditions and the damaged aircraft has stable steady state responses within only 15 seconds. In that sense, the uncertain plant dynamics are well within the expected bounds.

 \begin{figure}[htbp!]
 \centering
 \includegraphics[width=6truein]{Figures/robust_output_U_30.png}
 \caption{Closed loop time domain responses of the damaged aircraft in the presence of 30\% uncertainty}
 \label{fig:robust_output_U_30}
 \end{figure}

However, these favorable characteristics come at the expense of the control effort from the ailerons and differential thrust as shown in Fig. \ref{fig:robust_control_effort_U_30}.

  \begin{figure}[htbp!]
 \centering
 \includegraphics[width=4truein]{Figures/robust_control_effort_U_30.png}
 \caption{Control efforts in the presence of 30\% uncertainty}
 \label{fig:robust_control_effort_U_30}
 \end{figure}

According to Fig. \ref{fig:robust_control_effort_U_30}, when there is 30\% full block, additive uncertainty, the aileron control demands the maximum deflection of approximately 2.4 degrees and reaches steady state between -0.12 and -0.09 degree after 15 seconds, which is quite similar to the required aileron control effort when there is no uncertainty. Therefore, the aileron control effort demands are reasonable and feasible due to the limiting factor of $\pm$ 26 degrees of the aileron deflection \cite{faa_747} and the assumption that ailerons have instantaneous response characteristics by neglecting the lag from actuators or hydraulic systems.

As for differential thrust, when there is 30\% uncertainty, the differential thrust control demands at maximum approximately 6500 lbf, which is within the thrust capability of the JT9D-7A engine, and the differential thrust control effort reaches steady state between -100 lbf (negative differential thrust means $T1<T4$) and 100 lbf (positive differential thrust means $T1>T4$) after 15 seconds. Compared to the case when there is no uncertainty, the demanded differential thrust associated with uncertain plant dynamics is higher in both magnitude and rate. It is also obvious from Fig. \ref{fig:robust_control_effort_U_30} that in a few cases, the differential thrust control effort demand hit the thrust generation rate limiter, which is set at 12,726 lbf/s for the JT9D-7A engine, but fortunately, the control system is so robust that throughout 1000 Monte-Carlo simulations, it can stabilize the aircraft's uncertain plant dynamics. Again, due to the differential thrust saturation set at 43,729 lbf and the thrust response limiter set at 12,726 lbf/s, this control effort of differential thrust in the presence of uncertainty is achievable in real life situation.

\section{Conclusion and Future Work} \label{Conclusion}
This paper studied the utilization of differential thrust as a control input to help a Boeing 747-100 aircraft with a damaged vertical stabilizer regain its lateral/directional stability. The motivation of this research study is to improve the safety of air travel in the event of losing the vertical stabilizer by providing control means to safely control and/or land the aircraft.

Throughout this paper, the necessary damaged aircraft model was constructed, where lateral/directional equations of motion were revisited to incorporate differential thrust as a control input for the damaged aircraft. Then the open loop plant dynamics of the damaged aircraft was investigated. The engine dynamics of the aircraft was modeled as a system of differential equations with engine time constant and time delay terms to study the engine response time with respect to a commanded thrust input. Next, the novel differential thrust control module was presented to map the rudder input to differential thrust input. The $H_{\infty}$ loop-shaping approach based robust control system design's ability to stabilize the damaged aircraft was proven as investigation results demonstrated that the damaged aircraft was able to reach steady state stability within only 15 seconds under feasible control efforts. Further analysis on robustness showed that the $H_{\infty}$ loop-shaping approach based robust differential thrust methodology was also able to stabilize the uncertain plant dynamics in the presence of 30\% full block, additive uncertainty associated with the damaged aircraft dynamics.

Through listed analyses above, the ability to save the damaged aircraft by the robust differential thrust control strategy  has been demonstrated in this paper. This framework provides an automatic control methodology to save the damaged aircraft and avoid the dangerous coupling of the aircraft and pilots, which led to crashes in a great number of commercial airline incidents. Furthermore, it has also been concluded that due to the heavy dependence of the differential thrust generation on the engine response, in order to better incorporate the differential thrust as an effective control input in a life-saving scenario, major developments in engine response characteristics are also desired to better assist such algorithms.

Future work includes the implementation of such control algorithms into a small scaled unmanned aerial vehicle (UAV) for real-life testing.

redpajama_set_name: RedPajamaArXiv


Champaign County Soil & Water Conservation District | S.T.A.R.
The Stewardship Committee of the Champaign County Soil & Water Conservation District developed a FREE tool to assist farm operators and land owners in evaluating their nutrient and soil loss management practices on individual fields. The purpose of S.T.A.R. is to motivate those making cropping decisions to use the "best management practices" that will ultimately meet the goals of the Illinois Nutrient Loss Reduction Strategy. To meet those goals for the agriculture sector, farmer's need to reduce the nitrogen losses by applying nitrogen when the plants need it and by preventing soil losses by reducing tillage.
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Nutrient applications are intended to have minimal negative effect on water quality with minimal negative effect on yields.
12. Section 11: If NH3 (anhydrous ammonium = 82-0-0) is used, it should be applied when the temperature is below 50 degrees and you should use an inhibitor. It is recommended that NH3 will be no more than 50% of the total Nitrogen Program as it is risky to lose some of the N. From an environmental perspective, it is best if NO Nitrogen or Phosphorus are applied at all. However, if Nitrogen is applied, it is best done in the spring and/or summer.
13. Section 11: It is okay to use MAP (11-52-0) or DAP (18-46-0) in the fall before December 1st as it actually is good IF there is also a "winter hardy" cover crop being used. The amount of N in the MAP or DAP is rather insignificant but will help the cover crop become established and grow.
14. Section 11: Manure applications are used to replace other forms of nutrients. It is much better to apply in the spring when there is less likelihood of any leaching or runoff and management of such applications should be balanced with soil tests and what exactly is being added by the manure. Use of manure counts as adding a somewhat significant level of Nitrogen.
15. Section 12: As stated earlier, Nitrogen applications are best done in the spring and/or summer. It is a best management practice to side-dress some of the nitrogen on corn.
16. Section 13: The amount of Nitrogen identified here is based on the maintenance needs for normal yield goals in east central Illinois. The recommendation is to apply less than 200 pounds on a field with a corn and soybean rotation. If those guidelines are followed, there would be limited denitrification and leaching losses. The continuous corn rotation allows more Nitrogen (less than 225 pounds per acre) because soybeans were not used to "credit" some Nitrogen to that field.
It would be even better to follow the guidelines of the "Corn N-Rate Calculator" that is a part of the NRCS 590 Nutrient Management standards and specifications, found at this link to Iowa State University: http://cnrc.agron.iastate.edu . That system uses current corn and nitrogen prices to calculate the MRTN (Maximum Return To Nitrogen).
18. Section 13: If Nitrogen or Phosphorus (including manure) is broadcast on either frozen ground OR on snow covered ground, that would be VERY BAD. Even worse would be application of nutrients on frozen ground that also has snow on it.
19. Section 14: Crop rotation is better than a continuous crop of any kind, but continuous soybeans is better than continuous corn. Better yet, include a forage, such as alfalfa or clover, or a small grain, such as wheat.
20. Section 15: A major concern about soil loss, and therefore nutrient loss, is the amount of residue on the soil at any time. The type of tillage used is a simple method to estimate the amount of residue on a field and much easier than trying to ask the operator to measure it or do an estimate. Fewer passes and shallower tillage are best. However, if a cover crop is planted or manure is applied in the fall, a shallow tillage operation to incorporate has some benefit. Ideally, everyone would use no-till or strip-till!
21. Section 16: This section includes many very positive conservation practices. Most of the items on the list are self-explanatory if they are pertinent, but should only be checked if it fits the field being evaluated. The first six items on the list should only be checked if they are still functioning as intended.
"On-site agronomic trial" = Any study done on THAT field that may include variety comparisons, use of N-TRACK, use of N-WATCH, taking tissue and plant samples and having them analyzed, or any other study done to improve your cropping decisions.
"Attended soil or nutrient management meeting" may have been any meeting that includes some discussion or recommendations related to soil, nutrient use, or cover crops, including field days, no matter the length of time. It should have been within the past year at the time of completing the form.


Our company has mastered the art in offering Avaya EPABX System to the clients. Offered range is known for attributes such as low power consumption, high efficiency and easy installation. This product is offered in different models and can be availed in varied surface finishes. Avaya EPABX System is made using optimum quality raw material and advanced technologies at vendors' end. Offered range is stringently tested and analyzed on well-defined quality parameters..


text:
Q: Can suddenly only boot with "nomodeset" (radeon, 16.04) As of yesterday, I can only boot with the kernel option nomodeset. The only thing that happened before normal shutdown was an upgrade:
Start-Date: 2016-11-26  00:09:51
Commandline: apt-get -y dist-upgrade
Requested-By: XXXXX (1000)
Upgrade: imagemagick:amd64 (8:6.8.9.9-7ubuntu5.1, 8:6.8.9.9-7ubuntu5.2),     libpulsedsp:amd64 (1:8.0-0ubuntu3, 1:8.0-0ubuntu3.1), pulseaudio:amd64 (1:8.0-0ubuntu3, 1:8.0-0ubuntu3.1), gstreamer1.0-plugins-good:amd64 (1.8.2-1ubuntu0.1, 1.8.2-1ubuntu0.2), python2.7-minimal:amd64 (2.7.12-1~16.04, 2.7.12-1ubuntu0~16.04.1), libmagickwand-6.q16-2:amd64 (8:6.8.9.9-7ubuntu5.1, 8:6.8.9.9-7ubuntu5.2), libpython2.7:amd64 (2.7.12-1~16.04, 2.7.12-1ubuntu0~16.04.1), python2.7:amd64 (2.7.12-1~16.04, 2.7.12-1ubuntu0~16.04.1), libpython3.5:amd64 (3.5.2-2~16.04, 3.5.2-2ubuntu0~16.04.1), python3.5:amd64 (3.5.2-2~16.04, 3.5.2-2ubuntu0~16.04.1), python3.5-minimal:amd64 (3.5.2-2~16.04, 3.5.2-2ubuntu0~16.04.1), gstreamer1.0-plugins-base:amd64 (1.8.2-1ubuntu0.1, 1.8.2-1ubuntu0.2), gstreamer1.0-plugins-base:i386 (1.8.2-1ubuntu0.1, 1.8.2-1ubuntu0.2), libgstreamer-plugins-good1.0-0:amd64 (1.8.2-1ubuntu0.1, 1.8.2-1ubuntu0.2), imagemagick-6.q16:amd64 (8:6.8.9.9-7ubuntu5.1, 8:6.8.9.9-7ubuntu5.2), libmagickcore-6.q16-2-extra:amd64 (8:6.8.9.9-7ubuntu5.1, 8:6.8.9.9-7ubuntu5.2), libpulse0:amd64 (1:8.0-0ubuntu3, 1:8.0-0ubuntu3.1), libpulse0:i386 (1:8.0-0ubuntu3, 1:8.0-0ubuntu3.1), libpulse-mainloop-glib0:amd64 (1:8.0-0ubuntu3, 1:8.0-0ubuntu3.1), tar:amd64 (1.28-2.1, 1.28-2.1ubuntu0.1), libgstreamer-plugins-base1.0-0:amd64 (1.8.2-1ubuntu0.1, 1.8.2-1ubuntu0.2), libgstreamer-plugins-base1.0-0:i386 (1.8.2-1ubuntu0.1, 1.8.2-1ubuntu0.2), libmagickcore-6.q16-2:amd64 (8:6.8.9.9-7ubuntu5.1, 8:6.8.9.9-7ubuntu5.2), pulseaudio-module-x11:amd64 (1:8.0-0ubuntu3, 1:8.0-0ubuntu3.1), libpython2.7-minimal:amd64 (2.7.12-1~16.04, 2.7.12-1ubuntu0~16.04.1), imagemagick-common:amd64 (8:6.8.9.9-7ubuntu5.1, 8:6.8.9.9-7ubuntu5.2), libpython3.5-stdlib:amd64 (3.5.2-2~16.04, 3.5.2-2ubuntu0~16.04.1), libpython2.7-stdlib:amd64 (2.7.12-1~16.04, 2.7.12-1ubuntu0~16.04.1), libpython3.5-minimal:amd64 (3.5.2-2~16.04, 3.5.2-2ubuntu0~16.04.1), pulseaudio-utils:amd64 (1:8.0-0ubuntu3, 1:8.0-0ubuntu3.1)
End-Date: 2016-11-26  00:10:03

Lubuntu has been running for months now without problems. Now, if I try to boot normally, the screen just stays black. If I boot without quiet and splash, then I can see a lot of text rushing by (normal boot messages) and then the screen turns off and stays off. Normally, cryptsetup should ask me for my password at that point (my system partition is encrypted).
If I try to enter the password blindly, nothing happens (no HDD activity either). Interestingly, after some time, the green led on my DVD-Drive (???) starts flashing. There is nothing in there and I haven't used the drive in months.
:~$ sudo lshw -class display
  *-display UNCLAIMED
       description: VGA compatible controller
       product: Cayman PRO [Radeon HD 6950]
       vendor: Advanced Micro Devices, Inc. [AMD/ATI]
       physical id: 0
       bus info: pci@0000:01:00.0
       version: 00
       width: 64 bits
       clock: 33MHz
       capabilities: pm pciexpress msi vga_controller bus_master cap_list
       configuration: latency=0
       resources: memory:d0000000-dfffffff memory:fddc0000-fdddffff ioport:ee00(size=256) memory:fdd00000-fdd1ffff

When booting with nomodeset, Lubuntu detects only one of my three screens and all menu icons and most text (including lxterminal text) are really small and unreadable. The "x" to close windows is only a single pixel in size.
If it matters: This system doesn't have UEFI and is not dual boot. I tried using an older kernel, with no success.
:~$ dmesg | grep 'drm'
[    1.789860] [drm] Initialized drm 1.1.0 20060810
[    1.818735] [drm] VGACON disable radeon kernel modesetting.
[    1.818756] [drm:radeon_init [radeon]] *ERROR* No UMS support in radeon module!
[   11.759756] [drm] VGACON disable radeon kernel modesetting.
[   11.759781] [drm:radeon_init [radeon]] *ERROR* No UMS support in radeon module!

modprobe radeon modeset=1 will turn the screen off and I am not able to do anything anymore until I hard-reset the computer.
I can't really find anything else on Google and I'm out of ideas how to fix this.

redpajama_set_name: RedPajamaStackExchange


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5 honored at Women Tech Awards
SALT LAKE CITY — Carol Fineagan grew up in an alcoholic and emotionally abusive household in Baltimore which, if she had let it, could have sabotaged her entire future.
Instead, Fineagan drew strength from her participation in the Girl Scouts and from her mother's best friend, Joyce Britton, who provided encouragement, help and a positive role model.
Fineagan went on to earn several degrees, changed careers a number of times and now is senior vice president and CIO at EnergySolutions.
Fineagan on Thursday was one of five women recognized for their accomplishments and contributions to the technology world and the community at the Women Tech Awards luncheon sponsored by the Women Tech Council.
The event at the Grand America Hotel packed one of the ballrooms and Fineagan's award for Leadership Excellence drew hearty applause. She arrived in Utah four years ago to join EnergySolutions and, as part of her work, integrated all of the IT and communication processes in the company in only 10 months.
(See full article)
By Linda Thomson, Deseret News

redpajama_set_name: RedPajamaCommonCrawl


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> Mon. May 22, 2017
[EJ] Poster
S (Solid Earth Sciences) » S-SS Seismology
[S-SS10] [EJ] Crustal Deformation
Mon. May 22, 2017 5:15 PM - 6:30 PM Poster Hall (International Exhibition Hall HALL7)
View the poster hall map
[SSS10-P12] Cluster Analysis of the GNSS Velocity Field in the Tohoku Area, northeasten Japan
*Yoshiaki Ito1, Satoshi Miura1, Yusaku Ohta1 (1.Graduate School of Science, Tohoku University)
Recently, the cluster analysis technique has been applied to GNSS velocity data by Simpson et al. (2012) to reveal tectonic boundaries around the western USA. Assuming a number of clusters, the technique classifies GNSS velocities into some clusters, which shows similar characteristics. The optimal number of clusters can be determined by a statistical test (e.g. the gap statistic by Tibshirani et al., 2001). The advantage of this technique is to extract block-like behavior and to identify tectonic boundaries without considering geological and/or geographical informations. Loveless and Meade (2010) constructed a model composed of 20 blocks in Japan (JB1 model). This model in Tohoku area has two block boundaries, one along the Ou backbone range and the other along the eastern margin of the Japan Sea. The purpose of this research is to apply this clustering technique to the GNSS velocity field of the Tohoku Area, and compare with the JB1 model and the known fault system. GNSS data obtained from 298 continuous GNSS stations operated by the Geospatial Information Authority of Japan and Tohoku University are analyzed using the precise point positioning strategy of the GIPSY/OASIS-II software. We obtain the site velocities by fitting a linear function into coordinate time series from 1 January 2010 to 8 March 2011. We performed the cluster analysis for the horizontal components of the GNSS velocity field with the k-means clustering method. First, assuming the number of clusters we label every site with a cluster randomly. Then, we calculate the centroids of each cluster, and relabel each site with the closest centroid. This procedure is repeated until no more relabeling occurs. This method, however, has a disadvantage, namely the result sometimes depends on the initial random labeling. To avoid this problem, we carry out a thousand of clustering, and calculate the sum of L2 norm of every site pair in each cluster. Then the case of minimum sum is adopted as the optimal clustering. There is ambiguity in assuming the number of clusters. We decide the optimal cluster number by the gap statistic, which compares an ensemble mean of the logarithm of the sum of the L2 norm calculated from a random data, and the logarithm of the L2 norm calculated from the observed data. The gap statistic usually increases with the number of clusters and become stable around the optimal cluster number. The optimal number is 2 for our data set. The result demonstrates a cluster boundary, which runs along the Ou backbone range, and roughly coincides with the boundary of the JB1 model. There are two regions where our clustering result mismatches the boundary in the JB1 model. The reasons may be the ambiguity in the clustering method and/or the possible failure in estimating the site velocities.


Within Nigeria's bubbling fintech space, micro lending is arguably one of the hottest verticals at the moment. There are a variety of products/solutions all aiming to do one thing: make it easier for the average literate Nigerian with an internet connection to access small-bill credit.
If you have seen any marketing from these startups, you've likely seen something along the lines of "No collateral needed!", "Get a loan in minutes!", and so on. While those lines may not necessarily be false, they don't necessarily tell the entire story either – at least not in my case. I tried to get a 'medium' sized loan of N650,000 recently and this is what happened.
Perhaps the first thing I discovered was just how different almost every micro-lending platform is from the jump. Paylater, for example, only has an Android app and you can't apply for loans through its website. RenMoney on the other hand will let you get to a certain stage online but the rest of the process is offline (you have to scan documents and fill forms) more on this later.
The first thing I did when I decided to take a loan was look for the best possible deal. This deal would involve three things – interest rate, payment period and monthly deductible. In other words, the best deal would be one that had a decent interest rate, monthly deductible rates I could live with and repayment period long enough to accommodate surviving as a bachelor in Lagos (before I end up under a bridge begging for money).
It is interesting how Nigeria's micro-lending platforms approach information around these three metrics I've listed. Some had more detailed information about their process than others and others were just vague. Nevertheless, a few Google searches will give you decent information about any service (unless you're not as smart as I am).
After window shopping, the problem became deciding which micro-lending platform had the easiest process. I don't exactly have a lot of time on my hands – I needed something that would not require me to leave the office in the middle of a work day or handle/transmit physical copies of anything.
Now this is unique to me because the dynamics of my personal life vis-a-vis work will most likely be different from that of any other person. That said, I was able to find all the preliminary information I needed relatively easily.
Back to this thing about micro-lending startups in Nigeria employing wildly different models in how they position their product. Some of them have loan tenures that do not go beyond a month, others have caps on the size of loan principals and the rest have some kind of differentiation in terms of the documents they require. Even payment after your loan has been approved is different.
For example, at the point of disbursement, RenMoney will pay the full principal into your account and calculate its fee and interest into the total sum you will pay back while FINT will deduct its charges and send you the balance. So, if you earn N200k/month and want to borrow N600K for 6 months from FINT, you'll receive N545K and pay back N709K (18% interest) over the time period. In comparison, RenMoney will pay the full N600K into your account but you'll be paying back around N756K (26%) over the same period.
Also, there is the processing methodology. Paylater, for example, uses different metrics from RenMoney in determining who it grants loans to. Also, using the same data, I only got a loan offer of N20K from Paylater while RenMoney (and other platforms I tried) offered between N500K and N580K on the desired N650K I requested for.
Caveat: I tried Paylater using a colleague's phone (I don't have an Android device) so that could have skewed Paylater's algorithm somewhat.
With all the many requirement, steps and processes out of the way, I settled for RenMoney because they seemed to tick all the boxes I mentioned earlier. This is where things got interesting.
I started by filling the online application on web (to the best of my knowledge RenMoney does not have an app) which was no hassle at all although it was incredibly slow (could be internet or RenMoney's backend). Until I got stuck at the point where I had to upload a utility bill. The problem with this is this: we use prepaid meters in my house and pay for electricity online. Also, my neighbour who handles the waste bill had travelled with his family and wasn't going to be back for some time.
I abandoned the application that day and in the evening I got an email from a RenMoney representative – Bola (not her real name). I responded to the mail to explain my predicament and she called to get a better understanding of the problem. The most interesting thing she told me (apart from the solution to my problem) was that the process didn't end online and there was some offline element to completing the loan application process. I found that quite odd.
Moving on, Bola then sent me a form to print out, fill and then scan back to her (I hated this). She also warned me that the information I was putting on the form must match what I had already uploaded in the online application (before I knew about this form). After filling and scanning the form back to her, I got another email with a loan offer of N580K, which I printed out and filled/signed before scanning back to Bola and her colleague. Later that night, I got a payment alert and that was it.
So what have we learnt here today?
Micro-lending is still quite fragmented in Nigeria, mostly because of the information bottlenecks (no central database, BVN is still a silo system of some sort, address system is outdated etc).
Perhaps, with better access to customer information, I suspect that most platforms will be able to design better processes around their products that drastically cut down the loan application and processing time as well as a much more harmonised approach for the entire segment.
That said, cultural/societal nuances also have to be taken into consideration in designing these processes to create a quick, seamless way for the average literate Nigerian to access small-bill credit. As it stands, Nigeria's micro-lenders are doing the best they can with what they have.
The post The Absence Of Customer Data Is Fragmenting The Micro-Lending Experience in Nigeria appeared first on TechCabal.


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text:
La moschea di Fenâri Isa (nome completo ), nell'età bizantina conosciuta anche come il monastero di Lips (), è una moschea di Istanbul, costituita da due ex-chiese ortodosse orientali.

Posizione
Il complesso è situato ad Istanbul, nel distretto di Fatih (corrispondente alla città murata), lungo Adnan Menderes Caddesi (già Vatan Caddesi), in un contesto moderno.

Storia

Periodo bizantino

Nel 908, l'ammiraglio bizantino Costantino Lips inaugurò un monastero alla presenza dell'imperatore Leone VI il Saggio (r. 886-912). Il monastero, dedicato alla Vergine Theotókos Panachrantos ("Immacolata Madre di Dio"), si trovava nella valle del Lykos (il fiumiciattolo - ora interrato -  che attraversava Costantinopoli) in un luogo chiamato "Merdosangaris" (in greco: ). Il monastero era conosciuto anche con il nome del fondatore (mone tou Libos), e divenne uno dei più grandi di Costantinopoli.

La chiesa fu costruita sui resti di un altro santuario del VI secolo, riutilizzando fra l'altro le lapidi di un vicino cimitero romano. Le reliquie di santa Irene erano conservate qui. Questa chiesa è generalmente conosciuta nella letteratura specialistica come "Chiesa Nord".

Dopo l'invasione latina e la restaurazione dell'impero bizantino, tra il 1286 e il 1304, l'imperatrice Teodora, vedova dell'imperatore Michele VIII Paleologo (r. 1259 - 1282), eresse un'altra chiesa dedicata a san Giovanni Battista () a sud della prima chiesa. Oltre Teodora, diversi esponenti della dinastia imperiale dei Paleologi furono qui sepolti: suo figlio Costantino, l'imperatrice Irene di Monferrato e suo marito l'imperatore Andronico II (r. 1282-1328)
L'imperatrice restaurò anche il convento, che a quel tempo era probabilmente in stato di abbandono.  Secondo il suo typikon, il complesso monastico in quel momento ospitava in totale cinquanta donne e comprendeva anche uno Xénon per donne laiche con 15 posti letto annessi.

Nel corso del XIV secolo un nartece ed un parekklesion vennero aggiunti alla chiesa. L'usanza di seppellire i membri della famiglia imperiale nel complesso continuo' nel XV secolo con Anna, prima moglie dell'imperatore Giovanni VIII Paleologo (r. 1425-1448), qui inumata nel 1417. Probabilmente la chiesa venne utilizzata come cimitero anche dopo il 1453.
Questa chiesa è generalmente nota fra gli specialisti come "Chiesa Sud".

Periodo ottomano
Nel 1497-1498, poco dopo la caduta di Costantinopoli, durante il regno del sultano Bayezid II (1481-1512), la chiesa sud venne trasformata in una mescit (una piccola moschea) dal dignitario Ottomano Fenarizade Alaeddin Ali ben Yusuf Effendi, Qadi'asker di Rumeli, e nipote di Molla Şemseddin Fenari, la cui famiglia apparteneva alla classe religiosa degli ʿulamāʾ. Egli eresse un minareto nell'angolo sud-est, ed un miḥrāb nell'abside.  Dal momento che uno dei predicatori capo della madrasa era chiamato Isa ("Gesù" in arabo e turco), il suo nome venne aggiunto a quello della moschea. L'edificio, bruciato nel 1633, venne restaurato nel 1636 dal Gran Visir Bayram Pascià, il quale elevò l'edificio al rango di Cami ("moschea") e convertì la chiesa nord in una Tekke (loggia dei dervisci). In questa occasione le colonne della chiesa nord vennero sostituite con pilastri, Le due cupole furono restaurate, e la decorazione musiva venne rimossa. Dopo un altro incendio nel 1782, il complesso fu restaurato di nuovo nel 1847/48. In questa occasione anche le colonne della chiesa sud vennero sostituite con pilastri, e venne anche rimossa la balaustra del nartece. L'edificio bruciò ancora una volta nel 1918, e venne quindi abbandonato. Durante gli scavi effettuati nel 1929, vennero trovati ventidue sarcofagi.  Il complesso è stato completamente restaurato tra gli anni 1970 e 1980 dalla Byzantine Society of America, e da allora è di nuovo aperto al culto come moschea.

Architettura e decorazione

Chiesa Nord
La chiesa nord ha un'insolita pianta a quinconce (detta anche a croce inscritta), ed è uno dei primi santuari di Costantinopoli ad adottare questa tipologia, il cui prototipo è forse la Nea Ekklesia ("Chiesa Nuova"), eretta a Costantinopoli nell'anno 880 ed oggi completamente scomparsa. Durante il periodo ottomano le quattro colonne al centro della croce sono state sostituite da una volta a costoloni la quale copre tutta la navata.

L'edificio ha piccole dimensioni: la naos misura 13 metri di lunghezza per 9,5 metri di larghezza, e venne dimensionata in base al numero di occupanti del monastero al momento della costruzione. La muratura della chiesa nord fu eretta alternando file di mattoni a file di piccoli blocchi di pietra grezza. In questa tecnica, che è tipica dell'architettura bizantina del X secolo, i mattoni affondano in uno spesso letto di malta. La chiesa nord è sormontata da una cupola di tipo ottomano traforata da otto finestre.

Lo spazio interno è concluso verso est da tre alte absidi: quella centrale è poligonale, ed è affiancata dalle altre due, che servivano come pastophoria: la prothesis e il diakonikon.

Le absidi ricevono luce da finestre archiacute triple (in quello centrale) e singole. Le pareti dei bracci centrali della croce della naos hanno due ordini di finestre: l'ordine inferiore è composto da trifore, mentre le finestre superiori sono semicircolari. Due lunghi parekklesia, ognuno concluso da un basso abside, fiancheggiano il presbiterio della naos. Le navate angolari e centrali sono molto strette. Al livello del tetto, ai quattro lati dell'edificio insistono quattro cappelle, ciascuna sormontata da una cupola.

I resti della decorazione originale di questa chiesa consistono nelle basi di tre delle quattro colonne della campata centrale, ed in molti elementi che sopravvivono sui pilastri delle finestre e sulla struttura della cupola. La decorazione consisteva in origine in pannelli di marmo e piastrelle colorate: le volte erano decorate con mosaici. Solo tracce di essi sono ora visibili.

Nel complesso, la chiesa nord presenta forti analogie con la moschea Bodrum (già chiesa del Myrelaion).

Chiesa Sud

La chiesa sud è sormontata da una cupola ottomana del tipo ad elmetto e consiste in un'aula quadrata circondata da due deambulatoria, un endonartece e un parekklesion (aggiunto in seguito). Il deambulatorium della chiesa nord coincide con il parekklesion della chiesa sud.  Questa moltiplicazione degli spazi intorno alla parte centrale della chiesa è un motivo tipico della tarda architettura paleologa: la ragione di esso è la necessità di ricavare uno spazio maggiore per tombe, monumenti eretti ai benefattori della chiesa, ecc La sala centrale è divisa dalle navate da una galleria tripla. Durante la messa i fedeli erano confinati nei deambulatoria, che erano poco profondi e bui, e riuscivano a malapena a vedere cosa accadeva nella parte centrale della chiesa. La chiesa è conclusa da tre absidi, orientate verso est.
La muratura è costituita da file alternate di mattoni e pietre, un motivo tipico dell'architettura tardo bizantina a Costantinopoli.

La decorazione lussuosa delle absidi meridionale e principale (quest'ultimo di forma eptagonale), è costituita da un triplice ordine di nicchie: nell'ordine centrale queste si alternano con finestre triple. I mattoni sono disposti in modo da formare motivi ornamentali come archi, ganci, greche, croci solari, svastiche e ventagli. Tra questi motivi appaiono bande rosse bianche e scure, ottenute alternando una fila di pietre con più file (da due sino a cinque) di mattoni. Questa è la prima apparizione di questo importantissimo motivo decorativo dell'architettura paleologa a Costantinopoli.

La chiesa possiede un nartece sormontato da una galleria, il quale è stato ampliato in modo da raggiungere anche la chiesa nord. Il parekklesion fu eretto lungo il lato sud della chiesa meridionale, e venne collegato al nartece, in modo che l'ambiente circondasse l'intero complesso ad ovest e sud. Al suo interno sono esposti diversi sarcofagi marmorei.

Nell'insieme, questo complesso rappresenta un esempio notevole della media e tarda architettura bizantina a Costantinopoli.

Note

Bibliografia

Voci correlate
 Moschee di Istanbul

Altri progetti

Chiese di Costantinopoli
Moschee ottomane di Istanbul

redpajama_set_name: RedPajamaWikipedia


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Corsikomeris remyi är en mångfotingart som beskrevs av Karl Wilhelm Verhoeff 1943. Corsikomeris remyi ingår i släktet Corsikomeris och familjen klotdubbelfotingar. Inga underarter finns listade i Catalogue of Life.

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remyi


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Catholic schools ready for Distance Learning
Our Lady of the Assumption School teacher Tish Godsy interacts with her class through Zoom
With the closing of schools across the state and nation to prevent the spread of COVID-19, virtual learning has gone from an aspirational idea to the only game in town.
Beginning in mid- March, Catholic schools in the Diocese began closing their campuses as their corresponding public school districts did the same. On Apr. 2 Superintendent of Catholic Schools Sam Torres announced, in step with the directive from the State of California, that Catholic school campuses in the Diocese would be closed for the remainder of the academic year.
Catholic schools take top honors at robotics competition
Ontario—Students from Sacred Heart School, Rancho Cucamonga and St. Joseph School, Upland secured top honors at the first annual Inland Empire VEX IQ Robotics Middle School League Championshipthat was held at St. George School in Ontario.
Golf classic draws 200 to the desert to support Catholic schools
LA QUINTA—The 18th annual Bishop's Golf Classic took place on December 5th at PGA West in La Quinta. About 200 golfers and 45 volunteers enjoyed beautiful weather as they played on two of the nations premier golf courses. This tournament supports the efforts of the Catholic Education Foundation.
Marines visit St. James School as a thank you for care packages
St. Francis de Sales School receives items from late Archbishop Quinn
RIVERSIDE—St. Francis de Sales School got a surprise package from its most famous alum this year, care of the Archdiocese of San Francisco, on the Feast of Our Lady of Guadalupe.
Aquinas captures second CIF title in three years, plays for state championship
SAN BERNARDINO—The Aquinas High School football program continues to cement its status as a football powerhouse, after capping the season one victory away from a state championship.


Some of Hollywood's biggest stars have joined with the anti-nuclear group, Global Zero, in a new You Tube video to demand more action on the elimination of nuclear weapons.
Among the stars making a cameo in the video are Whoopi Goldberg, Morgan Freeman, Naomi Watts, Alex Baldwin, Michael Douglas, Martin Sheen, Danny DeVito, Robert DeNiro, Matt Damon, Zoe Kravitz, and Christoph Waltz.
The stars then take turns explaining the importance of eliminating nuclear weapons—including denouncing the notion that the spread of nuclear weapons cannot be stopped. While admitting the enormity of the task, the stars contend in the video that if the world decides it can't stop the spread of nuclear weapons then it is in some ways admitting that nuclear war is inevitable.
In the second half of the video, the stars introduce themselves and one-by-one say that they demand zero, a reference to Global Zero's goal of a world without nuclear weapons.


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Hiram C. "Pete" Cartee
October 17, 1923 - June 22, 2021
Hiram C. "Pete" Cartee Augusta, GA – On June 22, 2021, Hiram C. "Pete" Cartee fell into the arms of our Lord, after almost five months with kidney failure, sepsis, and pneumonia. Pete was a native of Metter, Georgia but spent his years growing up in Lakeland and Orlando, Florida. He joined the Navy at age 17 to eagerly serve his county in WWII. While his ship was docked in Boston Harbor, he and a couple of his buddies, while looking for some entertainment in the city, decided to go roller skating. It was in that skating rink that he met Gwen Peach, who was to become his wife for the next fifty-five years. After the war, he decided to leave the Navy and join the Army so he could spend time with his new wife and begin his family. He was one of the oldest military survivors of WWII and as late as 2018, still participated in veterans' affairs, visiting the war monuments in Washington, DC. He is survived by his son and daughter: Jim Cartee and Carol Hall (Albert), both born in Orlando, Florida; granddaughters: Melissa Castle (Dale) and Bridget Houck (Gary); great grandchildren: Caleb Burke (Jayden) and Halei Burke; great great granddaughter: Lainey Burke, born last year. Jim and Carol would like to acknowledge and say thank you to Ms. Martha Jean Culpepper, who offered Pete love and care since the passing of our mother in 1998. Mr. Cartee's career included retiring from the US Army, owning and operating several service stations for Standard Oil Company, British Petroleum (BP); serving as a law enforcement officer for Waynesboro, Metter, Wrens and Harlem, Georgia; managing a package shop for his son and perfecting his love of golf. He was baptized into the Baptist faith in 1998 at First Baptist Church of Augusta by Dr. Timothy Owings, where he remained an active member until his death. Services will be held at Storey Chapel of First Baptist Church, Saturday, June 26, 2021, at 2:00 P.M. with family visitation being an hour preceding the service. Loving and devoted family friend, Dr. Greg DeLoach, will be officiating. Entombment will be private at Westover Mausoleum. In lieu of flowers or food, contributions may be made to the First Baptist Church of Augusta and That's What Friends Are For, animal rescue. Thomas Poteet & Son Funeral Directors, 214 Davis Rd., Augusta, GA 30907 (706) 364-8484. Please sign the guestbook at www.thomaspoteet.com
Hiram C. "Pete" Cartee Augusta, GA – On June 22, 2021, Hiram C. "Pete" Cartee fell into the arms of our Lord, after almost five months with kidney failure, sepsis, and pneumonia. Pete was a native of Metter, Georgia... View Obituary & Service Information
The family of Hiram C. "Pete" Cartee created this Life Tributes page to make it easy to share your memories.
Hiram C. "Pete" Cartee
Augusta, GA – On June 22, 2021, Hiram...
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Damien Dempsey will be among those playing The Galway Spiegeltent Sessions 2018.
THE SPIEGELTENT returns to Eyre Square to house another week of raucous, riotous, comedy, but this year it will not just be comedy that takes place within - some of the very best of Irish music from Declan O'Rourke to The Undertones will be heard also.
This 380 capacity venue, featuring ballooning velvet canopies, circular teak dance floor, and stained cut-glass windows, and a lush velvet and brocade finish, will be the main hub of the Vodafone Comedy Carnival Galway, but in the run up to that event it hosts a series of concerts, entitled The Galway Spiegeltent Sessions.
The sessions begin on Wednesday October 17 with a headline set from the acclaimed singer-songwriter Declan O'Rourke, with support from left-field singer Paddy Hanna. After that comes the great Damien Dempsey and his band, with support from Junior Brother on Thursday 18; Bell X1 front-man Paul Noonan plays a solo show on Friday 19, with support from We Cut Corners. On Saturday 20, the brilliant Cathy Davey makes a welcome return to Galway, with support from David Keenan.
The sessions come to a close on Sunday 21 when Irish punk legends, The Undertones, the band behind such classic songs as 'Teenage Kicks, 'Here Comes The Summer', and 'My Perfect Cousin' take to the stage. Support is from Tuam pop-punk/garage rock trio Oh Boland.


Bus tickets from San Salvador de Jujuy to Caleta Olivia are being sold from Andesmar, Tramat.
The bus travel from San Salvador de Jujuy and Caleta Olivia takes approximately 2 days.
The distance between San Salvador de Jujuy to Caleta Olivia is (N/A) and you can choose between the services Semi Cama, Salón Cama; dependent on the bus company you travel with (Andesmar, Tramat).


Feb. 13, 2019 1:05 p.m.
Clayton's latest — and littlest — libraries will open on Family Day with a fun celebration.
United Way has been working with a team of volunteers to set up book nooks throughout Clayton. On Monday, Feb. 18, locals are invited to come down, have a pancake breakfast, and then tour the neighbourhood to check out the new libraries set up throughout the community.
The idea behind the project is to promote literacy, and also to create connections between neighbours.
The little libraries are part of a larger initiative to foster local love in Clayton Heights. In October last year, United Way made $12,000 in grants available for projects that boost neighbourhood morale on a local level.
At an October town hall, it was noted that although the Clayton area was built as an intentional community, many people who live in the area feel isolated or disconnected from their neighbours.
The event will take place rain, snow or shine on Monday, Feb. 18, from 10 a.m. to 12 p.m. The pancakes are free, but attendees are asked to please bring a book to share with the libraries.
Meet in front of Katzie Elementary, located at 6887 194A St., for breakfast before heading out to explore the neighbourhood.