LMIs in Control/Applications/Mixed H2 and Hinf Satellite Attitude Control

The System

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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.

The Data

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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

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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

  1. The closed-loop eigenvalues are located in  ,
    • λ(A+BK)  
  2. That the H2 and Hinf performance conditions below are satisfied with a small   and  :
    •  
    •  

The LMI: Mixed H2/Hinf Feedback Control

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min  

s.t.

  •  
  • trace(Z) <  
  • AX + B1W + (AX + B1W)T + BBT < 0
  •  
  •  


Gives a set of solutions to  , and W, Z and X > 0, where   is equal to  .

Conclusion

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Once the solutions are calculated, the state feedback gain matrix can be constructed as K = WX-1, and   =  

Implementation

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This LMI can be translated into MATLAB code that uses YALMIP and an LMI solver of choice such as MOSEK or CPLEX.

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