a de-coupled sliding mode controller and observer for satellite attitude control ronald fenton

A De-coupled Sliding Mode Controller and A De-coupled Sliding Mode Controller and Observer for Satellite Attitude ControlObserver for Satellite Attitude Control

Ronald Fenton

OutlineOutline

Introduction Spacecraft Dynamics Sliding Mode Control Design Sliding Mode Observer Dynamics Conclusion

IntroductionIntroduction

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 ControlSliding 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 ModeSliding Mode DesignDesign

Define your sliding manifold in terms of the tracking error.

Select a Lyapunov candidate function dependent on the sliding manifold and calculate the derivative of 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.

0)( eq

)()( VV

ieqi

iii

uu

satuumax

max )(

Sliding ManifoldSliding Manifold

Spacecraft Dynamics and KinematicsSpacecraft Dynamics and Kinematics

Rotational motion for a general rigid spacecraft acting under the influence of outside torques is given by the following equation.

DC TTΩJωωJ

0

0

0

12

13

23

)(2

1QMq

qq T2

14

TIqQM x 334)(

0

0

0

12

13

23

qq

qq

qq

T

Sliding Mode ControllerSliding Mode ControllerProblem Formulation:Problem Formulation:

24 1;1 qq

To avoid the singularity in M(Q)-1 that occurs at q4 =0 the workspace is restricted by the following:

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

0)()(lim tqtqd

Sliding Mode ControllerSliding Mode ControllerStability AnalysisStability 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.

Now choose a Lyapunov candidate function to provide σ (q) with asymptotic stability.

eCd KqTMJMJqq 114 2

1

2

1

2

1

1

2

1 JMV T

1T

2

1 JMV

1111

41

2

1

2

1

2

1MJMJTqKJMqJMJqJMV Ced

T

ee Kqqq )(

)(2

1)()()( qMtqtqtqq dde

Sliding Mode ControllerSliding Mode ControllerControl Law Design 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

When the substitution is made, the derivative of V shows the existence of a de-coupled sliding mode controller that is asymptotically stable

)(signTT EQC

11114

1 2222 MJMJqKJMqJMJqJMT edEQ

n

iisignV

1

)(

Sliding Mode ControllerSliding Mode ControllerControl Law DesignControl 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.

ieqC

iCCi

uT

satTTmax

max )(

Sliding Mode ControllerSliding Mode ControllerControl Law DesignControl 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.

i

ii

i

isat

1

1

Sliding Mode ObserverSliding Mode Observer

Nonlinear Observer Dynamics (Drakunov)

Once again two sliding manifolds were given in terms of the observer estimate errors to prove the convergence of the observers above.

)ˆ(ˆ 1 qqsignLq

)ˆ(1 qqsignLzz )ˆ

2

1(ˆ 2 zMsignL

zMe

qqqe

2

1

0ˆ

Sliding Mode ObserverSliding Mode Observer

• Now choose three Lyapunov candidate function to provide the previous sliding manifolds with asymptotic stability.

• 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.

eTe

T

eTee

wV

QeMeeV

qqqV

2

1)(

2

1)(

2

1)(

1

)()(

ˆ2

1ˆ

2

1)(

)()(

2

111

1

esignLV

eMzMMMeeV

qqqsignqLqV

ee

T

eeTee

Sliding Mode ObserverSliding 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

– (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

i

i zMqqsignMLz

MMMMML

1

11111

2

)ˆ(ˆ

2

1max2)(

ExampleExample

Spacecraft Parameters, Initial Conditions, Disturbance Torques, and Desired Trajectories

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

2

45.027.03.

027.48.02.

03.02.49.

kgmJ

TQ 7071;.5;.5;.0;0

Nm

t

t

t

Td

)sin(5.

)sin(5.

)sin(5.

)5.12

cos(5.

)5.12

cos(5.

)5.12

cos(5.

t

t

t

qd

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

ConclusionsConclusions

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.

ReferencesReferences

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. 328-342.