Using the same formulation of the problem as in the H2 and Hinf feedback applications (linked below) the system is of the form:
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:
- Iab = Ia - Ib, Iabc = Ia - Ib - Ic
, 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.
Data required for this problem include the moment of inertias and angular velocities of the system. Knowledge of expected disturbances d would be beneficial.
There are two requirements of this problem:
- Closed-loop poles are restricted to a desired LMI region
- Where
, 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
- The closed-loop eigenvalues are located in
,
- λ(A+BK)
![{\displaystyle \subset \mathbb {D} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/57816c0c1091b1fe410b32bed760eb8d7cf20b5a)
- That the H2 and Hinf performance conditions below are satisfied with a small
and
:
![{\displaystyle \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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef6d71f59ca69a660e9f5c3a5e7b5653a20600e)
![{\displaystyle \lVert G_{{z_{2}}d}\rVert _{2}=\lVert C_{2}(sI-(A+B_{1}K))^{-1}B_{2}\rVert _{\infty }\leq \gamma _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1daf94b25972d2a0fed73d44c669c350574a8f)
min
s.t.
![{\displaystyle {\begin{bmatrix}-Z&C_{2}X\\XC_{2}^{T}&-X\end{bmatrix}}<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efbbbefee827f2e787999a7a945638fecacd2755)
- trace(Z) <
![{\displaystyle \rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64)
- AX + B1W + (AX + B1W)T + BBT < 0
![{\displaystyle L\otimes X+M\otimes (AX+B_{1}W)+M^{T}\otimes (AX+B_{1}W)^{T}<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819cb94de5eebdbd878a8615fc583af356e8c154)
![{\displaystyle {\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26aa4119f237142d848ae00feb951b8e434a8c34)
Gives a set of solutions to
, and W, Z and X > 0, where
is equal to
.
Once the solutions are calculated, the state feedback gain matrix can be constructed as K = WX-1, and
=
This LMI can be translated into MATLAB code that uses YALMIP and an LMI solver of choice such as MOSEK or CPLEX.