F-16 Longitudinal State-Space Model

8 October 13, 2024

Here is one longitudinal state-space model from the literature with a single elevator input.[1] In this model the \(C\) matrix handles the conversion to degrees
$$A=\begin{bmatrix}-1.9311e-2 & 8.8157e0 & -3.2179e1&-5.7499e-1 \\
-2.5389e-4 & -1.0189e0 & 0.0e0 & 9.0506e-1\\
0.0e0& 0.0e0&0.0e0&1.0e0\\
2.9465e-12&8.222e-1&0.0e0&-1.0774e0\end{bmatrix}$$
$$B=\begin{bmatrix}1.7370e-1\\
-2.1499e-3\\
0.0e0\\
-1.755e0\end{bmatrix}$$
$$C=\begin{bmatrix}0.0e0 & 5.729578e1&0.0e0&0.0e0\\
0.0e0&0.0e0&0.0e0&5.729578e1\end{bmatrix}$$
This model gives the elevator to pitch-angle transfer function of.[1]
$$\frac{\alpha}{\delta_e}=\frac{-0.1232(s+75.00)(s+0.009820\pm j0.09379}{(s-0.09755)(s+1.912)(s+0.1507\pm j0.1153)}$$
This gives an unstable exponential mode with a time constant of around 10s for the pole at \( s\approx.098\).[1] The complex pole pair has a period of 33s with a damping ratio of 0.79.[1] With an actuator time constant of 1/20.2 and an AOA sensor time constant of 0.1, the new state-space representation including sensors is.[1]

This shows that the \(\delta_e\) input is still is connected to the actuator state \(x_a\) through a phase reversal.[1] The actuator is then driven by a new input \(u_e\).[1] Also a new filter state \(x_F\) for the AOA is included for feedback \(\alpha_F\).[1] The alpha feedback is used to pull the unstable pole \(s\approx.098\) back into the left-hand plane.[1]

This next models are linearized around fixed engine power as a percentage and a horizontal tail trim.[2]. The first condition is 14.85% engine power and an HT trim of -0.5708 deg.[2]
$$\begin{bmatrix}
\dot{\Delta V_T} \\
\dot{\Delta \alpha} \\
\dot{\Delta q} \\
\dot{\Delta \theta}
\end{bmatrix}=
\begin{bmatrix}
-0.0182 & 1.1250 & -0.02336 & -9.8043 \\
-0.0009 & -0.9145 & 0.9118 & 0 \\
0.0004 & 0.6531 & -0.9798 & 0\\
0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
{\Delta V_T} \\
{\Delta \alpha} \\
{\Delta q} \\
{\Delta \theta}
\end{bmatrix}+
\begin{bmatrix}
7.3609 & 0.0306\\
-0.0022 & -0.0022\\
0 & -0.1663\\
0 & 0
\end{bmatrix}
\begin{bmatrix}
\Delta \delta_{T1}\\
\Delta \delta_{ht1}
\end{bmatrix}$$
This next model uses 42.21% engine power and an HT trim of -0.5134 deg.[2]
$$\begin{bmatrix}
\dot{\Delta V_T} \\
\dot{\Delta \alpha} \\
\dot{\Delta q} \\
\dot{\Delta \theta}
\end{bmatrix}=
\begin{bmatrix}
-0.0189 & -2.2246 & -0.5453 & -9.7786 \\
-0.0010 & -0.3908 & 0.9630 & 0 \\
0.0002 & -0.1414 & -0.4208 & 0\\
0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
{\Delta V_T} \\
{\Delta \alpha} \\
{\Delta q} \\
{\Delta \theta}
\end{bmatrix}+
\begin{bmatrix}
2.2820 & -0.0105\\
-0.0029 & -0.0009\\
0 & -0.0540\\
0 & 0
\end{bmatrix}
\begin{bmatrix}
\Delta \delta_{T1}\\
\Delta \delta_{ht1}
\end{bmatrix}$$

[[A320 Lateral Directional Model]] – also shows a state-space model with sensor delays

Sources

Backlinks

[[Actuator Time Constant Reduction]]
[[Damping Ratio]]
[[F-16 Aerodynamic Model]]
[[Longitudinal Flight Dynamics]]
[[Sensor Modeling]]