1. Robust adaptive variable structure control
Yoni Habuba
Amichay Israel
adviser :
Mark Moulin

2. introduction
Future spacecraft will be expected to achieve highly accurate pointing, fast slewing, and other fast maneuvers from large initial conditions and in the presence of large environmental disturbances, measurement noise, large uncertainties, subsystem , component failures and control input saturation.

3. Introduction (Cont.)
In this project we propose globally stable control algorithms for robust stabilization of spacecraft in the presence of controls input saturation, parametric uncertainty, and external disturbances.

4. Introduction (Cont.) We will compare between 6 control algorithms.
One of the controllers is a simple proportional, the other - 5 of the 6 control algorithms are based on variable structure control design and have the following properties: fast and accurate response in the presence of bounded external disturbances and parametric uncertainty
explicit accounting for control input saturation
Computational simplicity and straightforward tuning.

5. Equation set The purpose is to stabilize Ω ,ε and ε0
J denotes inertia matrix
U denotes the control torques
Ω denotes the inertial angular velocity
ε and ε0 denote the Euler parameters
The purpose is to stabilize Ω ,ε and ε0

6. Sliding surface
In the context of spacecraft control, the control-based sliding mode control design is based on the use of the following sliding surface:
where k > 0 is a scalar

7. Controller's presentation
Equivalent control-based sliding mode controller
Sliding mode control under control input saturation using an approximate sign function
Sliding mode control under control input saturation using an accurate sign function
Combination between the sliding surface and proportional controller
proportional
Adaptive variable structure controller

8. Controller 1
Equivalent control-based sliding mode controller
u = ueq + uvs
Where ueq denotes the equivalent control component and is chosen to ensure that s(t) = 0 for all time .
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9. Comparison between Controller 2 and Controller 3 Sliding mode control under control input saturation using an approximate sign function
Controller 2
Controller 3
NOTE: controller 3 has a chattering problem therefore we will use controller 2 only from now on
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10. Controller 4
Combination between the sliding surface and proportional controller
u=ueq +upr
Where ueq denotes the equivalent control component and is chosen to ensure that s(t) = 0 for all time .
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11. Controller 5 proportional Controller law is
For getting the gain vector we made linearization of the system around the point:
After the linearization we determined the poles we used Ackerman's method for getting the gain vector.
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12. Controller 6 . Adaptive variable structure controller
Controller 6 is the similar to Controller 2 i.e.
but the different is that k in the equation
is time depend i.e.
while γ >0 denote the adaptive gain .
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13. Simulation The initial condition for all controllers:
W(0)=[ ]
e(0) =[ ]
eo(0)=0.8
The inertia matrix is

14. Disturbances rejection
We have checked the response of the difference controllers to a certain disturbances:
Square wave, 30pp, f=0.5 Hz.
Sinus wave, 30pp, f=0.5 Hz.
Triangle wave, 30pp, f=0.5 Hz.
We will present the sinus disturbance.

15. W1 - sin source Yellow - Controller1 Magenta –Controller2
Cyan – Controller4
Red – Controller5
Blue – Controller6

16. e1 - sin source Yellow - Controller1 Magenta –Controller2
Cyan – Controller4
Red – Controller5
Blue – Controller6

17. U1 - sin source Yellow - Controller1 Magenta –Controller2
Cyan – Controller4
Red – Controller5
Blue – Controller6

18. Controller’s effectiveness to different parameters
We have cheeked the parameters in the presence of the following disturbances:
Square wave 20pp, 40pp, 60pp, f=0.5 Hz
Inside disturbance 2pp f=0.5 Hz .
We will present the parameters for disturbance 1 .
We have checked the following parameters:
Max value
-steady error state.
Tsettle
Umax
Robustness to changing the initial conditions.

19. 20pp
40pp
60pp
Max value 58 71 75
εss
0.002
0.005
0.001
i.c
Not converged
0.003
0.3
0.25
0.5
0.9
0.6
1.2 2 1 2 4 5 6

20. Conclusions larger disturbance causes larger εss
The controllers 2, 6 which have only a non linear controller do not converge while the disturbance amplitude is 60 because it is ~Umax=70.
another conclusion that can’t be seen from the table and graphs but we have checked it by simulation The controllers are more effective for larger frequencies (~ 100 Hz)

21. Checking Tsettle and Umax
Tsettle(sec)
Controller 1
1200
0.65
Controller 2 70 2
Controller 4
300
0.1
Controller 5
120
1.4
Controller 6
1.2

22. Robust to changing the initial conditions
We have noticed that the controllers have a problem to settle the system while the initial conditions are too large
We have checked the for the different controllers
Controller 1
Controller 2
Controller 4
Controller 5
Controller 6
rad/sec 50 30
Not limited
100

23. Discussion about the internal parameters of controllers 2,6
We have discussed the following parameters regarded to controllers 2 and 6 :
Discussion about limitation of k in controller 2
Discussion about γ and k(0) in controller 6
We will present the discussion about γ in controller 6

24. Ω1 ε1
K(t)

25. Ω1 ε1
K(t)

26. Conclusion It can be seen immediately that for γ =0.01
e(t) - don’t converge to zero.
k(t) - converge to zero

27. Explanation This controller only guarantees that k(t)Xe(t) ,but not
necessarily e(t) ,will converge to zero. If k(t) converge
to zero faster than e(t) ,then e(t) may converge to some nonzero constant value.
To ensure that e(t) will converge to zero, one need to
keep k(t) from converging to zero. This can be
achieved by using a sufficiently small γ such that k(t)
changes slowly and that and hence will not deviate too
much from its initial value.