LMIs in Control/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:

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

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:

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

${\begin{aligned}A={\begin{bmatrix}0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\\{\frac {-4\omega _{0}^{2}I_{yz}}{I_{x}}}&0&0&0&0&{\frac {-\omega _{0}I_{yzx}}{I_{x}}}\\0&{\frac {-3\omega _{0}^{2}I_{xz}}{I_{y}}}&0&0&0&0\\0&0&{\frac {-\omega _{0}^{2}I_{yx}}{I_{z}}}&{\frac {\omega _{0}I_{yzx}}{I_{x}}}&0&0\end{bmatrix}}\\\\B_{1}=B_{2}={\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\{\frac {1}{I_{x}}}&0&0\\0&{\frac {1}{I_{y}}}&0\\0&0&{\frac {1}{I_{z}}}\\\end{bmatrix}}\\\\C_{1}=10^{-3}\times {\begin{bmatrix}-4\omega _{0}^{2}I_{yz}&0&0&0&-\omega _{0}I_{yxz}\\0&-3\omega _{0}^{2}I_{xz}&0&0&0&0\\0&0&-\omega _{0}^{2}I_{yx}&\omega _{0}I_{yxz}&0&0\end{bmatrix}}\\\\C_{2}={\begin{bmatrix}I_{3x3}&0_{3x3}\end{bmatrix}}\\\\D_{1}=10^{-3}\times L_{1},D_{2}=10^{-3}\times L_{2}\end{aligned}}$

• I_ab = I_a - I_b, I_abc = I_a - I_b - I_c
• $q={\begin{bmatrix}\phi \\\theta \\\psi \end{bmatrix}}$, 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 $\mathbb {D} =\{s|s\in \mathbb {C} ,L+sM+{\bar {s}}M^{T}<0\}$, 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 $\mathbb {D}$,
   • λ(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/Hinf Feedback Control

min $c_{\infty }\gamma _{\infty }+c_{2}\rho$

s.t.

• ${\begin{bmatrix}-Z&C_{2}X\\XC_{2}^{T}&-X\end{bmatrix}}<0$

• trace(Z) < $\rho$

• AX + B₁W + (AX + B₁W)^T + BB^T < 0

• $L\otimes X+M\otimes (AX+B_{1}W)+M^{T}\otimes (AX+B_{1}W)^{T}<0$
• ${\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{1}&(C_{1}X+D_{2}W)^{T}\\B&-\gamma _{\infty }I&D_{1}^{T}\\(C_{1}X+D_{2}W)&D_{1}&-\gamma _{\infty }I\end{bmatrix}}<0$

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