Satellite Formation Control Using Continuous Adaptive Sliding Mode

Satellite Formation Control Using Continuous Adaptive Sliding Mode Controller

Hancheol CHO, Gaëtan KERSCHEN Marie Curie COFUND Postdoctoral Fellow Space Structures and Systems Laboratory (S3L) Department of Aerospace and Mechanical Engineering University of Liège, Belgium

2016 KOSEAbe Biannual Conference, KIC, Brussels, April 30

Contents

1. Objective & motivation of this presentation

2. SFF Model and System Equations of Motion

3. Design of Global Adaptive SMC Laws

4. Simulation results & conclusions 2

Objective & Motivation  Objectives – Development of a continuous SMC (sliding mode controller) for satellite formation control in the presence of unknown mass of the satellite and external disturbances (gravitational perturbations and atmospheric drag) – A simple adaptive law for automatic gain tuning (no exact modeling of the dynamic system and no exact information about the uncertainties) – A new design of the sliding manifold to improve the transient response (global SMC)

 Innovations – Continuous control (no chattering) – Very simple adaptive law for control gain by minimizing numerical complexity (only one parameter to be determined) – High robustness is guaranteed during the transient period (by removing the reaching phase) 3

Satellite Formation Flying Model  Leader-Follower Formation Flying System x rL     T mq  m   y    m  0   2mRR        0  z 

 x  rL   y   mRR  T    z 

 x  rL  m   y  3/2    2 2 2 x  r  y  z L  z   

 x  rL  u x   d x   y   u    d     y  y  z   u z   d z 

m : mass of the follower (unknown) q  x

y

T

z  : position of the follower (in LVLH frame)

rL : distance from the center of the Earth to the leader

 : gravitational parameter of the Earth R : rotation matrix (ECI  LVLH), i.e.,  x  rL  X   y   R Y       z   Z 

rL Leader

q

Follwer

ui : components of the control input vector (i  x, y, z ) d i : disturbance force on the follower (unknown) 4

Desired Formation Trajectories  Projected Circular Formation xd 

0 sin  nt    , yd  0 cos  nt    , zd   0 sin  nt    2

u?

q  qd 5

Sliding Mode Control (SMC) e

 Design Process of standard SMC 1.

2.

Selection of a sliding surface • When s = 0, e → 0 (e = q – qd) Ex) s  e   e    0 

e

Development of a control law that forces system trajectories to converge to the sliding surface and remain on it Ex) V  1 s 2 

2 V  ss  s  e   e   s   x   xd   e   s  u     0



Issues 1. 2.

0.5

sgn(s)

Choose u    sgn  s  , where     V        s  sgn  s         s  0

1

0 -0.5 -1 -1

-0.5

0

0.5

1

s

Chattering How to determine the control gain  ? 6

Sliding Mode Control (SMC) 1. Selection of a Sliding Surface → Global SMC • faster response, less overshoot in sliding phase (i.e., when s = 0)

1.2 1

e=q-q

d

0.8 0.6 0.4 0.2 0 -0.2 0

0.1

0.2

0.3

0.4

0.5

Time

7

Sliding Mode Control (SMC) 1. Selection of a Sliding Surface → Global SMC • faster response, less overshoot in sliding phase (i.e., when s = 0) • removal of reaching phase (i.e., s(0) = 0)





si  t   ei  t   ei  0  exp   it    i  i exp   ki ei2  t    1  ei  t   ei  0  exp    i t    i  x, y , z    where  i  0,  i  0, ki  0, i  i  0

 Check

1.2



1

slow

d

0.8



i  i exp  ki ei  t    1

: time constant (decreasing)

→ less overshoot, smaller se ling me

e=q-q

• When si = 0, ei → 0 • si(0) = 0 1 • i  2

0.6

fast

0.4 0.2 0 -0.2 0

0.1

0.2

0.3

0.4

0.5

Time

8

Sliding Mode Control (SMC) 2. Controller Design with Gain Adaptation x rL      T mq  m   y    m  0   2mRR        0  z 

 x  rL   x  rL  m   y   mRR  T  y   3/2      2 2 2 x  r  y  z L  z   z    1  q  f  t , q, q    u  d   m, d : unknown  m

 Input saturation:

 x  rL  u x   d x   y   u    d     y  y  z   u z   d z 

ui  U , i  x , y , z

1

1

 Tracking error: ei  qi  qd ,i   fi  m di  qd ,i   m ui 



→ si  t   ei  t    i ei  0  exp   it   i  i  exp  ki ei2  t    1 ei  t   i ei  0  exp   it  2ki i ei ei exp   ki ei2  t    ei  t   ei  0  exp    i t     i t  

1 ui  t  m

 i , m :

unknown 

9

Sliding Mode Control (SMC) 2. Controller Design with Gain Adaptation Vi 

m 2 si  Vi  msi si  si  m i  ui  2

 Standard SMC:

ui    sgn  s  , where m i  

 Instead, we use:

 si  0  chattering   ui    s, where m i     si  si    no chattering 

 How to determine the gain  ?  → The control law ui     s itself can be used as an es mator of the uncertainty, leading to a gain adaptation law.

10

Estimation of the Uncertainty Let u  

K 

K s 

Controller

Plant

s 

Plant

 s 2

 u s 

K  2

Controller

2  u s s   / 2 

K  Kˆ

Controller

Plant

s 

 Kˆ

Kˆ  u s s ˆ   /K





11

Estimation of the Uncertainty Kˆ  u   s  s   : ˆ  Kˆ

s(t) ˆ

 Unknown real gain (uncertainty bounds):

t Measured

ˆ ˆ = K 

Tried

Desired

 max(|u|) is not sensitive to the selected gain.

Kˆ Kˆ  max  u   max  s       const. ˆ   K

12

Gain Adaptation

ˆ ˆ = K 

K(t) K ( k 1) 

s(k )

K ( k )  K0

  u (k )  K0

 K0  0

U+K0 K

1   m  s    u  m  

K0 t This dynamic gain does not affect max(|u|). 13

Gain Adaptation  Main Result: For the SFF model described by q  f  t , q, q  

1 u  d , m

 m, d :

unknown 

if the sliding surface is selected as si  t   ei  t   ei  0  exp   i t     i  i  exp   ki ei2  t    1   ei  t   ei  0  exp   i t   ,   the control law is defined as ui  t   

K t  si  t  , 

and the gain adaptation law is designed as





  



K ( k 1)  max ui( k )  K 0 , max ui( k ) : max u x( k ) , u (yk ) , u z( k )



then the system tracking error ei  t  in each axis will converge to the region si  t    in a finite time and remain there thereafter. 

Proof: ui  

u s K si  i  i  1  K 

Rigorous proof can be found in the paper. 14

Gain Adaptation s  0

 Generalization to HOSMC: u t   

K t 



s t 

1st-order 

s0

K t  u  t      t  ,  2nd-order   where   t   s  t    s  t 

1/2

s  s  0

sgn  s  (Levant 1993* )

  t    , s t   

* Levant, A., “Sliding order and sliding accuracy in sliding mode control,” International Journal of Control, 58, 1993, pp. 1247–1263.

15

Simulation Results  Desired relative configuration: projected circular formation ( 0  1.0 km,   0 )  Disturbances: gravitational perturbations up to order and degree 10, atmospheric drag (NRLMSISE-00, January 1st, 2016 at 00:00:00 UTC) T T  Initial errors: e  0   1.0 1.0 1.0 (km), e  0    0 0 0  (m/s)  Control parameters: i 

1 1 , i  , i  0.03, i  0.026, ki  0.1, U  0.05, K 0  0.5U ,   5  10 4 , 0.01P 0.01P

 The control thrusts are saturated by U / m  5  103 N/kg.  Orbital and system parameters are

16

52 e 6rm km ,i210 ,3 /s M , rad C , ,,kg km 3.986 L878 S m D ref 

Simulation Results

17

52 e 6rm km ,i210 ,3 /s M , rad C , ,,kg km 3.986 L878 S m D ref 

Simulation Results

18

Conclusions & Future Work •

Conclusions

1. A simple adaptive-gain continuous SMC is proposed. (estimator + controller) 2. Only |u| in the previous time step is required for on-line tuning. 3. The dynamic gain does not affect max(|u|). 4. The only parameter to be determined: K0 5. Transient response can be greatly improved.

•

Future Work

1. Not sensitive to K0, but more sensitive to  i , i , i , i , and ki 2. Real test of rendezvous of two satellites (QARMAN CubeSat with another CubeSat of QB50 mission, in the end of this year)

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