DPIA demands differentiated rate accelerations in orthogonal axes. These rate accelerations are then transferred to the control surface as rates, which are then integrated into deflections of the control surfaces. This is accomplished by inverting the control power matrix.
$${P}=
\begin{bmatrix}
N_{e\xi} & N_{e\zeta} \\
L_{e\xi} & L_{e\zeta}
\end{bmatrix}^{-1}=
\begin{bmatrix}
H_{\xi\dot{p}e} & H{\xi\dot{r}e} \\ H{\zeta\dot{p}e} & H{\zeta\dot{r}e} \end{bmatrix} $$ The integration must be performed on intermediate orthogonal axes to avoid the roll and yaw integrators winding up. The control power matrix varies based on the AOA. The digital update equations is $$\begin{bmatrix} \xi \\ \eta \end{bmatrix}_K=P_K \begin{bmatrix} \dot{p_e} \\ \dot{r_e} \end{bmatrix}_K$$ The integration is a simple Euler Integration $$\begin{bmatrix} \dot{p}_e \\ \dot{r}_e \end{bmatrix}_K= \begin{bmatrix} \dot{p}_e \\ \dot{r}_e \end{bmatrix}_{K-1}+T_S\begin{bmatrix}
\ddot{p}e \\ \ddot{r}_e \end{bmatrix}_K $$ This is transformed into the orthogonal axes using 2 transformations. $$\Delta \begin{bmatrix} \xi \\ \zeta \end{bmatrix}_{K,1}=
T_SP_K
\begin{bmatrix}
\ddot{p}e \\ \ddot{r}{e1}
\end{bmatrix}_K$$ $$\Delta \begin{bmatrix} \xi \\ \zeta \end{bmatrix}_{K,2}=
\begin{bmatrix}
P_KP_{K-1}^{-1}-I
\end{bmatrix}
\begin{bmatrix}
\xi\\ \zeta
\end{bmatrix}_{K-1}$$ Which then makes the integrator update the equation in the new orthogonal axis $$\begin{bmatrix} \xi \\ \zeta \end{bmatrix}_K= \begin{bmatrix} \xi \\ \zeta \end{bmatrix}_{K-1}+
\Delta
\begin{bmatrix}
\xi \\ \zeta
\end{bmatrix}_{K,1}+ \Delta \begin{bmatrix} \xi \\ \zeta \end{bmatrix}_{K,2}$$
Limiting the surface increments of this equation dependent on each other improves coordination under saturation. \(\xi\) is the differential Flaperon deflection, and \(\zeta\) is the rudder.
MFA Control Law – uses cascading inverse to allocate control surface positions
Sources
- [1] “Realization of the Eurofighter 2000 Primary Lateral/Directional Flight Control Laws with Differential PI-Algorithm.” Accessed: Feb. 10, 2024. [Online]. Available: https://arc.aiaa.org/doi/epdf/10.2514/6.2004-4751
Backlinks
[[Angle of Attack]]
[[Anti-Windup Scheme]
Eurofighter Flight Control Laws
Forward Euler
[[Integral Windup
Rate Saturation Equations
[[State-Space Model]]