A De-coupled Sliding Mode Controller and
Observer for Satellite Attitude Control
Ronald Fenton
Outline





Introduction
Spacecraft Dynamics
Sliding Mode Control Design
Sliding Mode Observer Dynamics
Conclusion
Introduction




Develop a de-coupled sliding mode controller and observer
for attitude tracking maneuvers in terms of the quaternion.
Show that the controller sliding manifold guarantees
globally stable asymptotic convergence to the desired time
dependent quaternion.
Show the tracking error responds as a linear homogeneous
vector differential equation with constant coefficients and
desired eigenvalue placement.
Design a full order sliding mode observer to avoid
quaternion differentiation noise and the need for angular
velocity measurement.
Sliding Mode Control

Provides continuous control of linear and nonlinear
systems with a discontinuous controller.
 The sliding mode control laws primarily uses either the
sign function or the sat function in the control law.
 By guaranteeing that the sliding manifold reaches zero
asymptotically and in a finite time, the controller design is
also able to stabilize the equilibrium point of the original
system
 Most importantly, the sliding mode controller has the
ability to deal with parameter variations in the original
nonlinear system (i.e. Robustness)
Sliding Mode Design

Define your sliding manifold in terms of the tracking error.
 (qe )  0

Select a Lyapunov candidate function dependent on the sliding manifold and
calculate the derivative of V.
V ( ) 
V ( )
 Choose a control law u = ueq + ρsign(σ) where ueq cancels out all system
dynamics in the derivative of V showing proving that the derivative of V is
less than zero at all times, and the sliding manifold will asymptotically
converge to the sliding manifold σ =0 in a finite time
 In sliding mode control, there is a problem with chattering because of the
imperfections in switching devices and delays. In order to minimize chattering
the sign can be replaced by the saturation function.

ui  uimax sat ( i )

max
ui  uieq  
Sliding Manifold
Spacecraft Dynamics and Kinematics

Rotational motion for a general rigid spacecraft acting
under the influence of outside torques is given by the
following equation.
Jω  ΩJω  TC  TD
q 
1
M (Q)
2
1
q 4    T q
2
M (Q)  q4 I 3 x3  T
 0
   3
 2
 3
 0
T   q3
 q2
 q3
0
1
0
q1
2 
 1 
0 
q2 
 q1 
0 
Sliding Mode Controller
Problem Formulation:

To avoid the singularity in M(Q)-1 that occurs at q4 =0 the
workspace is restricted by the following:
q    1; q4  1   2

The overall task of the sliding mode controller is to track a
desired quaternion such that the limit of the norm of the
difference between the desired and actual quaternion was
equal to zero
lim qd (t )  q(t )  0
Sliding Mode Controller
Stability Analysis

A suitable sliding manifold had to be chosen such that the discontinuous
control guaranteed that the surface σ (q) =0 was reached in finite time and is
maintained thereafter.
 (q)  qe  Kqe
1
qe  qd (t )  q(t )  qd (t )  M (q)
2
1
1
1
1
  qd  q 4  MJ   MJ 1TC  Kqe
2
2
2

Now choose a Lyapunov candidate function to provide σ (q) with asymptotic
stability.
1
V   T JM 1
2
1
V   T JM 1
2
1
1
1


V   T  JM 1q 4  J  JM 1qd  JM 1 Kqe  TC  JM 1  JM 1 
2
2
 2

Sliding Mode Controller
Control Law Design

Choose the proper control torque to cancel out all the terms
in the derivative of V such that it is always less than zero
TC  TEQ  sign ( )
TEQ   JM 1q4  J  2 JM 1qd  2 JM 1Kqe  2 JM 1  2 JM 1

When the substitution is made, the derivative of V shows
the existence of a de-coupled sliding mode controller that
is asymptotically stable
n
V   sign ( )      i
i 1
Sliding Mode Controller
Control Law Design

Because Ueq is costly for implementation and an inherent
chattering problem with with the sign function exists, a
discontinuous control law was implemented satisfying all
requirements for stability with the following discontinuous
control law.
i
)


TCi  TCmax sat (
TCmax  uieq
Sliding Mode Controller
Control Law Design


To help mediate the chattering problem that occurs with the sign
function the saturation function was used.
As epsilon approaches zero, the saturation function becomes the sign
function.
 1 i 
  i   i
sat    
i 


  
 1  i 





Sliding Mode Observer

Nonlinear Observer Dynamics (Drakunov)
qˆ   L1sign (qˆ  q)
z   z  L1sign (qˆ  q)
1
2
ˆ   L2 sign ( Mˆ  z )

Once again two sliding manifolds were given in terms of
the observer estimate errors to prove the convergence of
the observers above.
qe  qˆ  q  0
e
1
M̂  z
2
Sliding Mode Observer
•
Now choose three Lyapunov candidate function to provide the previous sliding
manifolds with asymptotic stability.
1 T
qe qe
2
1
V (e)  eT M 1Qe
2
1
V ( w)  eT e
2
V (qe ) 
•
Find the derivate of V, to in order to prove that the derivative of V was less
than zero for two positive definite functions L1 and L2.
V (qe )   L1qeT sign (qe )  qe q
1
1

V (e)  eT  M 1M ˆ  ˆ  M 1 z   M 1e
2
2

V (e )  e  L2 sign (e)   
Sliding Mode Observer

Lyapunov Candidate Derivative of V Conditions:
– (1) qe = 0 in finite time if (L1)I > max|qi|
– (2) Substituting the angular velocity estimate equation into the
previous equation

1  1
L1M 1sign (qˆ  q)  1 
1 
1 z
( L2 )i  2 max M M  M M ˆ  M

 M z
2



i
– (3) we = 0 in finite time if (L2)I > max|wi|



If the following three conditions hold then the sliding
mode observer converges in finite time and is
asymptotically stable
Example

Spacecraft Parameters, Initial Conditions, Disturbance Torques, and
t 
Desired Trajectories

.5 cos(
)
 .49 .02  .03
J   .02 .48 .027  kgm2
 .03 .027 .45 
  0; Q  0;.5;.5;.7071T


.5 sin( t )
Td  .5 sin( t ) Nm
.5 sin( t )

12.5 

t 
qd   .5 cos(
)
12.5 

 .5 cos( t )

12.5 
For the sliding mode controller Uimax = 1 Nm for an ε = .0019 and
controller gains of K = 0.8.
For the observer (L1)i =50 and (L2)I = 1000 for initial conditions equal
to zero and ε =.02 for the quaternion observer and ε = 10 for the
angular velocity observer
Figure 1. Sliding Mode Controller and Observer Implementation
Figure 2. Quaternion Profiles.
Figure 3. Quaternion Error Norm.
Figure 4. Quaternion Observer Error Norm
Figure 5. Angular Velocity Observer Error Norm
Conclusions

The controller sliding manifold has several advantages:
– De-coupling the rigid body dynamics is provide through control
– The sliding manifold is suitable for both tracking and regulation without
modification and has a simpler implementation then previously designed
manifolds.
 The observer also has several advantages when implemented:
– It eliminates the need to measure angular velocity and the derivative of the
quaternion error.
– The observer combination provides smoother control and allows
robustness to parameter variations.
References


James H. McDuffie and Yuri B. Shtessel, A De-coupled Sliding Mode
Controller and Observer for Satellite Attitude Control, IEEE 29th
Symposium on System Theory, March 9-11, 1997 pg 92.
K. David Young, Vadim I. Utkin, and Umit Ozguner, A Control
Engineer’s Guide to Sliding Mode Control, IEEE Transactions on
Control Systems Technology, Vol. 7, No. 3, May 1999, pp. 328342.