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 $H_2$-$H_{\mathrm{\infty }}$ Satellite Attitude Control is given.

The System

The system described below for Mixed H${}_2-$$H_{\mathrm{\infty }}$ Satellite Attitude Control is the same as the one used for separate $H_2$ and $H_{\mathrm{\infty }}$ Satellite Attitude controls.

$\begin{array}{cc}& \{\begin{array}{c}{I}_x\ddot{\varphi }+4(I_y-I_z){\omega }_0^2\varphi +(I_y-I_z-I_z){\omega }_0\dot{\psi }=T_{cx}+T_{dx}\\ I_y\ddot{\theta }+3(I_x-I_z){\omega }_0^2\theta =T_{cy}+T_{dy}\\ I_z\ddot{\psi }+(I_x+I_z-I_y){\omega }_0\dot{\psi }=T_{cz}+T_{dz}\end{array}\end{array}$

$where$

• $T_c$ 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 $\theta$, $\varphi$, and $\psi$ are the three Euler angles.

The state space representation of The Mixed $H_2-H_{\mathrm{\infty }}$ Satellite Attitude Control system is given below, which is the same as the one described on the $H_2$ and $H_{\mathrm{\infty }}$ Satellite Attitude Control pages.

$\begin{array}{cc}& \{\begin{array}{c}\dot{x}=Ax+B_{1}u+B_{2}d\\ z_{\mathrm{\infty }}=C_{1}x+D_{1}u+D_{2}d\\ z_2=C_{2}x\end{array}\end{array}$

$where:$

$A=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ \frac{-4{\omega }_0^2{I}_{yz}}{I_x}& 0& 0& 0& 0& \frac{-{\omega }_0{I}_{yzx}}{I_x}\\ 0& \frac{-3{\omega }_0^2{I}_{xz}}{I_y}& 0& 0& 0& 0\\ 0& 0& \frac{-{\omega }_0^2{I}_{yx}}{I_z}& \frac{{\omega }_0{I}_{yzx}}{I_x}& 0& 0\end{array}\right]$

$B_1=B_2=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ \frac{1}{I_x}& 0& 0\\ 0& \frac{1}{I_y}& 0\\ 0& 0& \frac{1}{I_z}\end{array}\right]$

$C_1=10^{-3}\times \left[\begin{array}{ccccc}-4{\omega }_0^2{I}_{yz}& 0& 0& 0& -{\omega }_0{I}_{yxz}\\ 0& -3{\omega }_0^2{I}_{xz}& 0& 0& 0& 0\\ 0& 0& -{\omega }_0^2{I}_{yx}& {\omega }_0{I}_{yxz}& 0& 0\end{array}\right]$

$C_2=\left[\begin{array}{cc}{I}_{3x3}& 0_{3x3}\end{array}\right]$

$D_1=10^{-3}\times L_1,D_2=10^{-3}\times L_2$

$I_{ab}=I_a-I_b,I_{abc}=I_a-I_b-I_c$

$q=\left[\begin{array}{c}\varphi \\ \theta \\ \psi \end{array}\right],x={\left[\begin{array}{cc}q& \dot{q}\end{array}\right]}^T,M=diag(I_x,I_y,I_z),$ $z_{\mathrm{\infty }}=10^{-3}M\ddot{q},$ $z_2=q$

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 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 $𝔻=\{s|s\in ℂ,L+sM+\overline{s}{M}^T<0\}$, 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 $𝔻$,
   • $\lambda (A+BK)$$\subset 𝔻$
2. That the H2 and Hinf performance conditions below are satisfied with a small ${\gamma }_{\mathrm{\infty }}$ and ${\gamma }_2$:
   • $\Vert G_{z_{\mathrm{\infty }}d}{\Vert }_{\mathrm{\infty }}=\Vert (C_1+N_{2}K)(sI-(A+B_{1}K){)}^{-1}{B}_2+N_1{\Vert }_{\mathrm{\infty }}\le {\gamma }_{\mathrm{\infty }}$
   • $\Vert G_{z_{2}d}{\Vert }_2=\Vert C_2(sI-(A+B_{1}K){)}^{-1}{B}_2{\Vert }_{\mathrm{\infty }}\le {\gamma }_2$

The LMI: Mixed H₂-H_∞ Satellite Attitude Control

$\begin{array}{c}\{\begin{array}{cc}& \text{min}{c}_{\mathrm{\infty }}{\gamma }_{\mathrm{\infty }}+c_2\rho \\ & \text{s.t.}\left[\begin{array}{cc}-Z& C_{2}X\\ X{C}_2^T& -X\end{array}\right]<0\\ \\ & trace(Z)<\rho \\ \\ & AX+B_{1}W+(AX+B_{1}W{)}^T+B{B}^T<0\\ \\ & L\otimes X+M\otimes (AX+B_{1}W)+M^T\otimes (AX+B_{1}W{)}^T<0\\ \\ & \left[\begin{array}{ccc}(AX+B_{1}W{)}^T+AX+B_{1}W& B_1& (C_{1}X+D_{2}W{)}^T\\ B& -{\gamma }_{\mathrm{\infty }}I& D_1^T\\ (C_{1}X+D_{2}W)& Do{s}_1& -{\gamma }_{\mathrm{\infty }}I\end{array}\right]<0\\ \end{array}\\ \end{array}$

Solving the above LMI gives the value Op ${\gamma }_{\mathrm{\infty }}$, $\rho$, 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 construções as $K=W{X}^{-1}$, and Failed to parse (unknown function "\quarto"): {\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.

Related LMIs

• [[LMIs in Control/Applications/H2 LMI SatelliteAttitLMIs in Control/Applications/Mixed H2-H∞ Satellite Attitude Control falo
  udeControl|H2 LMI for Satellite Attitude Control]]
• Infelizmente LMI for Satellite Attitude Control

External Links

• LMI Methods in Opcional and Robust Control - A course on Luis in Control by Matthew Peet.
• [1] - Luis in Control Systemjg.
• só um
• Analysis, Design, and Applications - Duan and Yuri

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