User:Margav06/sandbox/Click here to continue/Applications of Linear systems/Mixed H2/Hinf LMI Satellite Attitude Control

The System
Using the same formulation of the problem as in the H2 and Hinf feedback applications (linked below) the system is of the form:

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

This formulation comes from the process described in Duan, page 371-373, steps 12.1 to 12.8.


 * Tc and Td are the flywheel torque and the disturbance torque respectively.
 * Ix, Iy, and Iz are the diagonalized inertias from the inertia matrix Ib.
 * ω0 = 7.292115 x 10-5 rad/s is the rotational angular velocity of the Earth, and θ, Φ, and ψ are the three Euler angles.

The LMI below utilizes the state space representation of the above system, which is described on the H2 and Hinf pages as well:

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

$$ \begin{align} A = \begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\               0 & 0 & 0 & 0 & 0 & 1 \\                \frac{-4\omega_0^2 I_{yz}}{I_x}  & 0 & 0 & 0 & 0 & \frac{-\omega_0 I_{yzx}}{I_x} \\ 0 & \frac{-3\omega_0^2 I_{xz}}{I_y} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{-\omega_0^2 I_{yx}}{I_z} & \frac{\omega_0 I_{yzx}}{I_x} & 0 & 0 \end{bmatrix}\\ \\ B_1 = B_2 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\                           0 & 0 & 0 \\                            \frac{1}{I_x} & 0 & 0 \\ 0 & \frac{1}{I_y} & 0 \\ 0 & 0 & \frac{1}{I_z} \\ \end{bmatrix}\\ \\ C_1 = 10^{-3} \times \begin{bmatrix} -4\omega_0^2I_{yz} & 0 & 0 & 0 & -\omega_0 I_{yxz}\\ 0 & -3\omega_0^2 I_{xz} & 0 & 0 & 0 & 0\\ 0 & 0 & -\omega_0^2 I_{yx} & \omega_0 I_{yxz} & 0 & 0 \end{bmatrix}\\ \\ C_2 = \begin{bmatrix} I_{3x3} & 0_{3x3} \end{bmatrix} \\ \\ D_1 = 10^{-3} \times L_1, D_2 = 10^{-3} \times L_2 \end{align}$$


 * Iab = Ia - Ib, Iabc = Ia - Ib - Ic
 * $$q = \begin{bmatrix} \phi \\ \theta \\ \psi \end{bmatrix}$$, x = [q  q']T,   M = diag(Ix, Iy, Iz),     zinf = 10-3 M q ''' ,       z2 = q

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

The Data
Data required for this problem include the moment of inertias and angular velocities of the system. Knowledge of expected disturbances d would be beneficial.

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) * λ(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/Hinf Feedback Control
min $$c_\infin \gamma_\infin + c_2\rho$$

s.t.


 * $$\begin{bmatrix} -Z & C_2X \\ XC_2^T & -X \end{bmatrix} < 0$$


 * trace(Z) < $$\rho$$


 * AX + B1W + (AX + B1W)T + BBT < 0


 * $$L \otimes X + M\otimes (AX+B_1W) + M^T \otimes (AX+B_1W)^T < 0$$
 * $$\begin{bmatrix} (AX+B_1W)^T +AX + B_1W & B_1 & (C_1X+D_2W)^T \\ B & -\gamma_\infin I & D_1^T \\ (C_1X+D_2W) & D_1 & -\gamma_\infin I \end{bmatrix} < 0$$

Gives a set of solutions to $$ \gamma_\infin, \rho$$, 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 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