LMIs in Control/Applications/F16 Longitudinal Stabilitzation

F-16 Longitudinal Stabilization using Hinf Output Feedback

The linearized longitudinal dynamics of the F-16 aircraft are considered in this example. The nonlinear 6DOF model is linearized using straight-and-level flight conditions of:

${\begin{aligned}V=502ft/s\\{\bar {q}}=300psf\\X_{cg}=0.35{\bar {c}}\\\delta _{trim}=-0.7588deg\end{aligned}}$

See chapter 3 of [] for further details.

The System

The state-space representation for the linearized longitudinal dynamics can be written as follows:

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}w(t)+B_{2}u(t)\\y(t)&=C_{1}x(t)+D_{11}w(t)+D_{12}u(t)\\z(t)&=C_{2}x(t)+D_{21}w(t)+D_{22}u(t)\\u(t)=Ky(t)\\\end{aligned}}$

where $x=[V\quad \alpha \quad \theta \quad q]^{\text{T}}$, $u=\delta$ , $w=\delta _{cmd}$, and $z=[a_{n}\quad (\delta _{cmd}-\delta )]^{\text{T}}$ are the state variable, control input, disturbance, and regulated output vectors, respectively. In this case, $V$ is velocity (ft/s), $\alpha$ is angle of attack (rad), $\theta$ is pitch (rad), $q$ is pitch rate (rad/s), $\delta$ is elevator deflection with respect to trimmed deflection (rad), $\delta _{cmd}$ is commanded elevator deflection with respect to trimmed deflection (rad), and $a_{n}$ is normal acceleration at the pilot location (ft/s^2).

The Data

For the linearization used, the above matrices are:

${\begin{aligned}A={\begin{bmatrix}-1.9311e-2&&8.8157&&-3.217e1&&-5.7499e-1\\-2.5389e-4&&-1.0189&&0&&9.0506e-1\\0&&0&&0&&1\\2.9465e-12&&8.2225e-1&&0&&-1.0774\end{bmatrix}}\end{aligned}}$

$B_{1}={\begin{bmatrix}1.737e-1\\-2.1499e-3\\0\\-1.7555e-1\end{bmatrix}}$

$B_{2}={\begin{bmatrix}1.737e-1\\-2.1499e-3\\0\\-1.7555e-1\end{bmatrix}}$

${\begin{aligned}C_{1}={\begin{bmatrix}0&&5.729578e1&&0&&0\\0&&0&&0&&5.729578e1\end{bmatrix}}\end{aligned}}$

${\begin{aligned}C_{2}={\begin{bmatrix}0.003981&&15.88&&0&&1.481\\0&&0&&0&&0\\\end{bmatrix}}\end{aligned}}$

$D_{11}=D_{12}=0$

$D_{21}={\begin{bmatrix}0\\1\end{bmatrix}}$

$D_{21}={\begin{bmatrix}0.03333\\-1\end{bmatrix}}$

The LMI: Optimal Output Feedback $H_{\infty }$ Control LMI

The following are equivalent.

1) There exists a ${\hat {K}}={\begin{bmatrix}A_{K}&B_{K}\\C_{K}&D_{K}\end{bmatrix}}$ such that $||S(K,P)||_{H_{\infty }}<\gamma$

2) There exists $X_{1}$, $Y_{1}$, $Z$, $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$ such that

${\begin{bmatrix}X_{1}&I\\I&Y_{1}\end{bmatrix}}>0$

${\begin{bmatrix}AY_{1}+Y_{1}A^{\text{T}}+B_{2}C_{n}+C_{n}B_{2}^{\text{T}}&*^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\A^{\text{T}}+A_{n}+(B_{2}D_{n}C_{2})^{\text{T}}&X_{1}A+A^{\text{T}}+B_{n}C_{2}+C_{2}^{\text{T}}B_{n}^{\text{T}}&*^{\text{T}}&*^{\text{T}}\\(B_{1}+B_{2}D_{n}D_{21})^{\text{T}}&(X_{1}B_{1}+B_{n}D_{21})^{\text{T}}&-\gamma I&*^{\text{T}}\\C_{1}Y_{1}+D_{12}C_{n}&C_{1}+D_{12}D_{n}C_{2}&D_{11}+D_{12}D_{n}D_{21}&-\gamma I\\\end{bmatrix}}<0$

Conclusion:

Using this problem formulation, an output controller ${\hat {K}}$ can be found that attenuates normal acceleration and tracks reference elevator deflection command.

References

Stevens, Brian L., et al. Aircraft Control and Simulation. Third ed., Wiley, 2016.