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Robust adaptive variable structure control

Yoni Habuba

Amichay Israel

adviser :Mark Moulin

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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.

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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.

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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.

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Equation set

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

( )xJ J sat u z

. 1( )

2x I

. 1

2o

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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 s k

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Controller's presentation

1. Equivalent control-based sliding mode controller2. Sliding mode control under control input saturation

using an approximate sign function3. Sliding mode control under control input saturation

using an accurate sign function4. Combination between the sliding surface and propo

rtional controller 5. proportional 6. Adaptive variable structure controller

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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

–

–

.

~

( )vs mu s u

return

1( )

2x x

equ J kJ I

Comparison between Controller 2 and Controller 3

Sliding mode control under control input saturation using an approximate sign function

0 200 400 600 800 1000 1200-30

-20

-10

0

10

20

30

40

50

60

0 100 200 300 400 500 600 700 800 900-30

-20

-10

0

10

20

30

40

50

60

~

( ) mu s u

Controller 2 Controller 3

NOTE: controller 3 has a chattering problem therefore we will use controller 2 only from now on return

W1-blue W2-red

W3-green

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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

–

–

.( )pru s K

return

1( )

2x x

equ J kJ I

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Controller 5

2

31 2 3 4 5 6

2

3

, , , , ,iu k k k k k k

return

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.

2

3

2

3

0

0

0

0

0

0

1o

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Controller 6

s k

3.

1

[sgn( ) ( ) sgn( )]i i ii

k abs smuk

• 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

~

( ) mu s u

return

.

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Simulation

The initial condition for all controllers:– W(0)=[29 29 29]– e(0) =[0.4 0.2 0.4 ]– eo(0)=0.8

The inertia matrix is

20 0 0.9

0 17 0

0.9 0 15NJ

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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.

W1 - sin source

Yellow - Controller1Magenta –Controller2Cyan – Controller4Red – Controller5Blue – Controller6

e1 - sin source

Yellow - Controller1Magenta –Controller2Cyan – Controller4Red – Controller5Blue – Controller6

U1 - sin source

Yellow - Controller1Magenta –Controller2Cyan – Controller4Red – Controller5Blue – Controller6

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Controller’s effectiveness to different parameters

We have cheeked the parameters in the presence of the following disturbances:

1. Square wave 20pp, 40pp, 60pp, f=0.5 Hz2. 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.

ss

20pp 40pp 60pp

Max value 58 71 75

εss 0.002 0.005 0.001

Max value i.c i.c Not converged

εss 0.003 0.3 Not converged

Max value i.c i.c i.c

εss 0.25 0.5 0.9

Max value i.c i.c i.c

εss 0.6 1.2 2

Max value i.c i.c Not converged

εss 0.3 0.3 Not converged

1245

6

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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)

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Checking Tsettle and Umax

Umax Tsettle(sec)

Controller 1 1200 0.65

Controller 2 70 2

Controller 4 300 0.1

Controller 5 120 1.4

Controller 6 70 1.2

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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

1 max(0)

Controller 1 Controller 2 Controller 4 Controller 5 Controller 6

rad/sec 50 30 Not limited 100 301 max(0)

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Discussion about the internal parameters of controllers 2,6

We have discussed the following parameters regarded to controllers 2 and 6 :

1. Discussion about limitation of k in controller 2

2. Discussion about γ and k(0) in controller 6

We will present the discussion about γ in controller 6

ε1

K(t)

Ω1

Ω1

ε1

K(t)

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Conclusion

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

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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.