LMIs in Control/Applications/Mixed H2 and Hinf 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:

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

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

• 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.
• 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:

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

$\begin{array}{c}A=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ \frac{-4{\omega }_0^2{I}_{yz}}{I_x}& 0& 0& 0& 0& \frac{-{\omega }_0{I}_{yzx}}{I_x}\\ 0& \frac{-3{\omega }_0^2{I}_{xz}}{I_y}& 0& 0& 0& 0\\ 0& 0& \frac{-{\omega }_0^2{I}_{yx}}{I_z}& \frac{{\omega }_0{I}_{yzx}}{I_x}& 0& 0\end{array}\right]\\ \\ B_1=B_2=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ \frac{1}{I_x}& 0& 0\\ 0& \frac{1}{I_y}& 0\\ 0& 0& \frac{1}{I_z}\end{array}\right]\\ \\ C_1=10^{-3}\times \left[\begin{array}{ccccc}-4{\omega }_0^2{I}_{yz}& 0& 0& 0& -{\omega }_0{I}_{yxz}\\ 0& -3{\omega }_0^2{I}_{xz}& 0& 0& 0& 0\\ 0& 0& -{\omega }_0^2{I}_{yx}& {\omega }_0{I}_{yxz}& 0& 0\end{array}\right]\\ \\ C_2=\left[\begin{array}{cc}{I}_{3x3}& 0_{3x3}\end{array}\right]\\ \\ D_1=10^{-3}\times L_1,D_2=10^{-3}\times L_2\end{array}$

• 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 = [q q']^T , M = diag(I_x, I_y, I_z), 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 $𝔻=\{s|s\in ℂ,L+sM+\overline{s}{M}^T<0\}$, 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 $𝔻$,
   • λ(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/Hinf Feedback Control

min $c_{\mathrm{\infty }}{\gamma }_{\mathrm{\infty }}+c_2\rho$

s.t.

• $\left[\begin{array}{cc}-Z& C_{2}X\\ X{C}_2^T& -X\end{array}\right]<0$

• trace(Z) < $\rho$

• AX + B₁W + (AX + B₁W)^T + BB^T < 0

• $L\otimes X+M\otimes (AX+B_{1}W)+M^T\otimes (AX+B_{1}W{)}^T<0$
• $\left[\begin{array}{ccc}(AX+B_{1}W{)}^T+AX+B_{1}W& B_1& (C_{1}X+D_{2}W{)}^T\\ B& -{\gamma }_{\mathrm{\infty }}I& D_1^T\\ (C_{1}X+D_{2}W)& D_1& -{\gamma }_{\mathrm{\infty }}I\end{array}\right]<0$

Gives a set of solutions to ${\gamma }_{\mathrm{\infty }},\rho$, 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 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