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:

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

See chapter 3 of [] for further details.

The System

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

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

where $x=[V\alpha \theta q{]}^{\text{T}}$, $u=\delta$ , $w={\delta }_{cmd}$, and $z=[a_n({\delta }_{cmd}-\delta ){]}^{\text{T}}$ 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:

$\begin{array}{c}A=\left[\begin{array}{ccccccc}-1.9311e-2& & 8.8157& & -3.217e1& & -5.7499e-1\\ -2.5389e-4& & -1.0189& & 0& & 9.0506e-1\\ 0& & 0& & 0& & 1\\ 2.9465e-12& & 8.2225e-1& & 0& & -1.0774\end{array}\right]\end{array}$

$B_1=\left[\begin{array}{c}1.737e-1\\ -2.1499e-3\\ 0\\ -1.7555e-1\end{array}\right]$

$B_2=\left[\begin{array}{c}1.737e-1\\ -2.1499e-3\\ 0\\ -1.7555e-1\end{array}\right]$

$\begin{array}{c}{C}_1=\left[\begin{array}{ccccccc}0& & 5.729578e1& & 0& & 0\\ 0& & 0& & 0& & 5.729578e1\end{array}\right]\end{array}$

$\begin{array}{c}{C}_2=\left[\begin{array}{ccccccc}0.003981& & 15.88& & 0& & 1.481\\ 0& & 0& & 0& & 0\end{array}\right]\end{array}$

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

$D_{21}=\left[\begin{array}{c}0\\ 1\end{array}\right]$

$D_{21}=\left[\begin{array}{c}0.03333\\ -1\end{array}\right]$

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

The following are equivalent.

1) There exists a $\hat{K}=\left[\begin{array}{cc}{A}_K& B_K\\ C_K& D_K\end{array}\right]$ such that $||S(K,P)|{|}_{H_{\mathrm{\infty }}}<\gamma$

2) There exists $X_1$, $Y_1$, $Z$, $A_n$, $B_n$, $C_n$, $D_n$ such that

$\left[\begin{array}{cc}{X}_1& I\\ I& Y_1\end{array}\right]>0$

$\left[\begin{array}{cccc}A{Y}_1+Y_1{A}^{\text{T}}+B_2{C}_n+C_n{B}_2^{\text{T}}& {*}^{\text{T}}& {*}^{\text{T}}& {*}^{\text{T}}\\ A^{\text{T}}+A_n+(B_2{D}_n{C}_2{)}^{\text{T}}& X_{1}A+A^{\text{T}}+B_n{C}_2+C_2^{\text{T}}{B}_n^{\text{T}}& {*}^{\text{T}}& {*}^{\text{T}}\\ (B_1+B_2{D}_n{D}_{21}{)}^{\text{T}}& (X_1{B}_1+B_n{D}_{21}{)}^{\text{T}}& -\gamma I& {*}^{\text{T}}\\ C_1{Y}_1+D_{12}{C}_n& C_1+D_{12}{D}_n{C}_2& D_{11}+D_{12}{D}_n{D}_{21}& -\gamma I\end{array}\right]<0$

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.

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