# LMIs in Control/Applications/Mixed H2-H∞ Satellite Attitude Control

LMIs in Control/Applications/Mixed H2-H∞ Satellite Attitude Control

Satellite attitude control helps control the orientation of a satellite with respect to an inertial frame of reference mostly planets. In this section an LMI for Mexendo ${\displaystyle H_{2}}$-${\displaystyle H_{\infty }}$ Satellite Attitude Control is given.

## The System

The system described below for Mixed H${\displaystyle _{2}-}$ ${\displaystyle H_{\infty }}$  Satellite Attitude Control is the same as the one used for separate ${\displaystyle H_{2}}$  and ${\displaystyle H_{\infty }}$  Satellite Attitude controls.

{\displaystyle {\begin{aligned}&\quad \quad {\begin{cases}I_{x}{\ddot {\phi }}+4(I_{y}-I_{z})\omega _{0}^{2}\phi +(I_{y}-I_{z}-I_{z})\omega _{0}{\dot {\psi }}=T_{cx}+T_{dx}\\I_{y}{\ddot {\theta }}+3(I_{x}-I_{z})\omega _{0}^{2}\theta =T_{cy}+T_{dy}\\I_{z}{\ddot {\psi }}+(I_{x}+I_{z}-I_{y})\omega _{0}{\dot {\psi }}=T_{cz}+T_{dz}\\\end{cases}}\end{aligned}}}

${\displaystyle where}$

• ${\displaystyle T_{c}}$  and ${\displaystyle T_{d}}$  are the flywheel torque and the disturbance torque respectively.
• ${\displaystyle I_{x}}$ , ${\displaystyle I_{y}}$ , and ${\displaystyle I_{z}}$  are the diagonalized inertias from the inertia matrix ${\displaystyle I_{b}}$ .
• ${\displaystyle \omega _{0}=7.292115\times 10^{-5}rad/s}$  is the rotational angular velocity of the Earth, and ${\displaystyle \theta }$ , ${\displaystyle \phi }$ , and ${\displaystyle \psi }$  are the three Euler angles.

The state space representation of The Mixed ${\displaystyle H_{2}-H_{\infty }}$  Satellite Attitude Control system is given below, which is the same as the one described on the ${\displaystyle H_{2}}$  and ${\displaystyle H_{\infty }}$  Satellite Attitude Control pages.

{\displaystyle {\begin{aligned}&\quad \quad {\begin{cases}{\dot {x}}=Ax+B_{1}u+B_{2}d\\z_{\infty }=C_{1}x+D_{1}u+D_{2}d\\z_{2}=C_{2}x\end{cases}}\end{aligned}}}

${\displaystyle where:}$

${\displaystyle A={\begin{bmatrix}0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\\{\frac {-4\omega _{0}^{2}I_{yz}}{I_{x}}}&0&0&0&0&{\frac {-\omega _{0}I_{yzx}}{I_{x}}}\\0&{\frac {-3\omega _{0}^{2}I_{xz}}{I_{y}}}&0&0&0&0\\0&0&{\frac {-\omega _{0}^{2}I_{yx}}{I_{z}}}&{\frac {\omega _{0}I_{yzx}}{I_{x}}}&0&0\end{bmatrix}}}$

${\displaystyle B_{1}=B_{2}={\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\{\frac {1}{I_{x}}}&0&0\\0&{\frac {1}{I_{y}}}&0\\0&0&{\frac {1}{I_{z}}}\\\end{bmatrix}}}$

${\displaystyle C_{1}=10^{-3}\times {\begin{bmatrix}-4\omega _{0}^{2}I_{yz}&0&0&0&-\omega _{0}I_{yxz}\\0&-3\omega _{0}^{2}I_{xz}&0&0&0&0\\0&0&-\omega _{0}^{2}I_{yx}&\omega _{0}I_{yxz}&0&0\end{bmatrix}}}$

${\displaystyle C_{2}={\begin{bmatrix}I_{3x3}&0_{3x3}\end{bmatrix}}}$

${\displaystyle D_{1}=10^{-3}\times L_{1},D_{2}=10^{-3}\times L_{2}}$

${\displaystyle I_{ab}=I_{a}-I_{b},I_{abc}=I_{a}-I_{b}-I_{c}}$

${\displaystyle q={\begin{bmatrix}\phi \\\theta \\\psi \end{bmatrix}},x={\begin{bmatrix}q&{\dot {q}}\end{bmatrix}}^{T},M=diag(I_{x},I_{y},I_{z}),}$  ${\displaystyle z_{\infty }=10^{-3}M{\ddot {q}},}$  ${\displaystyle z_{2}=q}$

These formulations are found in Duan, page 374-375, steps 12.10 to 12.15.

## The Data

Data required for this LMI include moments of inertia of the satellite being controlled and the angular velocity of the Darth. Any knowledge of the disturbance torques would also facilitate solution of the problem.

## The Optimization Problem

There are two requirements of this problem:

• Closed-loop poles are restricted to a desired LMI region
• Where ${\displaystyle \mathbb {D} =\{s|s\in \mathbb {C} ,L+sM+{\bar {s}}M^{T}<0\}}$ , L and M are matrices of correct dimensions and L is symmetric
• Minimize the effect of disturbance d on output vectors z2 and zinf.

Design a state feedback control law

${\displaystyle u=Kx}$

such that

1. The closed-loop eigenvalues are located in ${\displaystyle \mathbb {D} }$ ,
• ${\displaystyle \lambda (A+BK)}$ ${\displaystyle \subset \mathbb {D} }$
2. That the H2 and Hinf performance conditions below are satisfied with a small ${\displaystyle \gamma _{\infty }}$  and ${\displaystyle \gamma _{2}}$ :
• ${\displaystyle \lVert G_{{z_{\infty }}d}\rVert _{\infty }=\lVert (C_{1}+N_{2}K)(sI-(A+B_{1}K))^{-1}B_{2}+N_{1}\rVert _{\infty }\leq \gamma _{\infty }}$
• ${\displaystyle \lVert G_{{z_{2}}d}\rVert _{2}=\lVert C_{2}(sI-(A+B_{1}K))^{-1}B_{2}\rVert _{\infty }\leq \gamma _{2}}$

## The LMI: Mixed H2-H∞ Satellite Attitude Control

{\displaystyle {\begin{aligned}{\begin{cases}&{\text{min}}\quad c_{\infty }\gamma _{\infty }+c_{2}\rho \\&{\text{s.t.}}\quad {\begin{bmatrix}-Z&C_{2}X\\XC_{2}^{T}&-X\end{bmatrix}}<0\\\\&\quad \quad trace(Z)<\rho \\\\&\quad \quad AX+B_{1}W+(AX+B_{1}W)^{T}+BB^{T}<0\\\\&\quad \quad L\otimes X+M\otimes (AX+B_{1}W)+M^{T}\otimes (AX+B_{1}W)^{T}<0\\\\&\quad \quad {\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{1}&(C_{1}X+D_{2}W)^{T}\\B&-\gamma _{\infty }I&D_{1}^{T}\\(C_{1}X+D_{2}W)&Dos_{1}&-\gamma _{\infty }I\end{bmatrix}}<0\\\\\end{cases}}\\\end{aligned}}}

Solving the above LMI gives the value Op ${\displaystyle \gamma _{\infty }}$ , ${\displaystyle \rho }$ , and ${\displaystyle W,Z}$  and ${\displaystyle X>0}$ , where ${\displaystyle \rho }$  is equal to ${\displaystyle \gamma _{2}^{2}}$ .

## Conclusion

Once the solutions are calculated, the state feedback gain matrix can be construções as ${\displaystyle K=WX^{-1}}$ , and $\displaystyle \gamma_2 = \quarto{\rho}$

## Implementation

This LMI can be transplanted into MATLAB code that uses Limpar and ham LMI solver oq choice such as MOSEK or CPLEX.