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 $$H_2$$-$$H_{\infty}$$ 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.

$$ \begin{align} &\quad \quad \begin{cases} I_x\ddot{\phi}+4(I_y-I_z)\omega _0^2 \phi + (I_y - I_z - I_z)\omega _0 \dot{\psi} = T_{cx} + T_{dx}\\ I_y\ddot{\theta}+3(I_x-I_z)\omega _0^2 \theta = T_{cy} + T_{dy}\\ I_z\ddot{\psi}+(I_x+I_z-I_y)\omega _0 \dot{\psi} = T_{cz} + T_{dz}\\ \end{cases} \end{align}$$

$$where$$


 * $$T_c$$ 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.

$$ \begin{align} &\quad \quad \begin{cases} \dot x =Ax+B_1u+B_2 d \\ z_\infty=C_1x+D_1u+D_2d \\ z_2=C_2x \end{cases}\end{align} $$

$$where: $$

$$A = \begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\               0 & 0 & 0 & 0 & 0 & 1 \\                \frac{-4\omega_0^2 I_{yz}}{I_x}  & 0 & 0 & 0 & 0 & \frac{-\omega_0 I_{yzx}}{I_x} \\ 0 & \frac{-3\omega_0^2 I_{xz}}{I_y} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{-\omega_0^2 I_{yx}}{I_z} & \frac{\omega_0 I_{yzx}}{I_x} & 0 & 0 \end{bmatrix} $$

$$ B_1 = B_2 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\                           0 & 0 & 0 \\                            \frac{1}{I_x} & 0 & 0 \\ 0 & \frac{1}{I_y} & 0 \\ 0 & 0 & \frac{1}{I_z} \\ \end{bmatrix}$$

$$C_1 = 10^{-3} \times \begin{bmatrix} -4\omega_0^2I_{yz} & 0 & 0 & 0 & -\omega_0 I_{yxz}\\ 0 & -3\omega_0^2 I_{xz} & 0 & 0 & 0 & 0\\ 0 & 0 & -\omega_0^2 I_{yx} & \omega_0 I_{yxz} & 0 & 0 \end{bmatrix}$$

$$C_2 = \begin{bmatrix} I_{3x3} & 0_{3x3} \end{bmatrix} $$

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

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

$$q = \begin{bmatrix} \phi \\ \theta \\ \psi \end{bmatrix},x = \begin{bmatrix} q & \dot q\end{bmatrix}^T, M = diag(I_x , I_y, I_z),$$ $$z_{\infty} = 10^{-3} M \ddot q ,$$ $$z_2 = q$$

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 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 $$\mathbb{D} = \{s|s \in \mathbb{C}, L + sM + \bar{s}M^T < 0\}$$, 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 $$\mathbb{D}$$,
 * 2) * $$ \lambda (A+BK) $$$$\subset \mathbb{D}$$
 * 3) That the H2 and Hinf performance conditions below are satisfied with a small $$\gamma_\infty  $$ and $$\gamma_2 $$:
 * 4) * $$\lVert G_{{z_\infty}d} \rVert_\infty = \lVert (C_1+N_2 K)(sI-(A+B_1K))^{-1}B_2+N_1 \rVert_\infty \leq\gamma_\infty$$
 * 5) * $$\lVert G_{{z_2}d} \rVert_2 = \lVert C_2(sI-(A+B_1 K))^{-1} B_2 \rVert_\infty \leq\gamma_2$$

The LMI: Mixed H2-H∞ Satellite Attitude Control
$$\begin{align} \begin{cases} &\text{min} \quad c_\infin \gamma_\infin + c_2\rho \\ &\text{s.t.} \quad \begin{bmatrix} -Z & C_2X \\ XC_2^T & -X \end{bmatrix} < 0 \\ \\ &\quad \quad trace(Z) < \rho \\ \\ &\quad \quad AX+B_1W + (AX+B_1W)^T + BB^T< 0 \\ \\ &\quad \quad L \otimes X + M\otimes (AX+B_1W) + M^T \otimes (AX+B_1W)^T < 0 \\ \\ &\quad \quad \begin{bmatrix} (AX+B_1W)^T +AX + B_1W & B_1 & (C_1X+D_2W)^T \\ B & -\gamma_\infin I & D_1^T \\ (C_1X+D_2W) & Dos_1 & -\gamma_\infin I \end{bmatrix} < 0 \\ \\ \end{cases} \\ \end{align}$$

Solving the above LMI gives the value Op $$\gamma_\infin$$, $$\rho$$, 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 $$K = WX^{-1}$$, and $$\gamma_2 = \quarto{\rho}$$

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

 * H2 LMI for Satellite Attitude Control
 * Infelizmente LMI for Satellite Attitude Control