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

1. A De-coupled Sliding Mode Controller and Observer for Satellite Attitude Control Ronald Fenton

2. Outline • Introduction • Spacecraft Dynamics • Sliding Mode Control Design • Sliding Mode Observer Dynamics • Conclusion

3. 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.

4. 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)

5. Sliding ModeDesign • 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.

6. Sliding Manifold

7. Spacecraft Dynamics and Kinematics • Rotational motion for a general rigid spacecraft acting under the influence of outside torques is given by the following equation.

8. Sliding Mode ControllerProblem Formulation: • 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

9. Sliding Mode ControllerStability 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.

10. Sliding Mode ControllerControl 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

11. Sliding Mode ControllerControl 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.

12. Sliding Mode ControllerControl 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.

13. Sliding 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.

14. Sliding 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.

15. 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 • (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

16. Example • 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

17. Figure 1. Sliding Mode Controller and Observer Implementation

18. Figure 2. Quaternion Profiles. Figure 3. Quaternion Error Norm.

19. Figure 4. Quaternion Observer Error Norm Figure 5. Angular Velocity Observer Error Norm

20. 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.

21. 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. 328-342.