LMIs in Control/Click here to continue/Applications of Linear systems/Mixed H2-H∞ LMI Satellite Attitude Control

LMIs in Control/Click here to continue/Applications of Linear systems/Mixed H2-H∞ LMI 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 Mixed $H_{2}$-$H_{\infty }$ Satellite Attitude Control is given.

The System

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

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

$where$

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

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

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

$where:$

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

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

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

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

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

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

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

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 earth. 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 $\mathbb {D} =\{s|s\in \mathbb {C} ,L+sM+{\bar {s}}M^{T}<0\}$, 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

$u=Kx$

such that

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

The LMI: Mixed H₂-H_∞ Satellite Attitude Control

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

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

Conclusion

Once the solutions are calculated, the state feedback gain matrix can be constructed as $K=WX^{-1}$, and $\gamma _{2}$ = ${\sqrt {\rho }}$

Implementation

This LMI can be translated into MATLAB code that uses YALMIP and an LMI solver of choice such as MOSEK or CPLEX.

Related LMIs

• H2 LMI for Satellite Attitude Control
• Hinf LMI for Satellite Attitude Control

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• [1] - LMIs in Control Systems: Analysis, Design, and Applications - Duan and Yu