Outline

2. Outline • Introduction • Reaction Wheels • Modelling • Control System • Real Time Issues • Questions • Conclusions

3. The Plant • Pendulum • Reaction Wheel • Motor • Encoders • TI Digital Signal Processor • PWM Motor Driver

4. Reaction Wheel • Wheel acceleration by torque from motor • Torque on motor from wheel inertia • Torque is transferred to the whole pendulum

5. Applications of Reaction Wheels • Satellite adjustment • Motorcycle mid-jumpcorrection

6. Model Derivation • Three states • Model derived by laws of physics and measurements

7. Validating the Model

8. Hybrid Automaton • Two discrete states • Swinging State • Balancing State

9. Swing Up Controller • Bang-bang energy control • Energy of pendulum • Wtotal = Wpotential + Wkinetic • Reference value is the potential energy at the upright position • The pendulum will reach the catch angle with the right amount of speed

10. Continuous time Plant h Continuous time Controller Discretized Plant h Discrete Controller Two Approaches of Controller Design • Design in Continuous Time • Design in Discrete Time

11. Process Controller State Feedback State observer Design in Continuous Time • Design of a State Feedback Controller • Investigate PD controller:

12. Analysis of the Root Locus • Root locus : closed-loop pole trajectories as a function of

13. A Stable Closed-loop System necessity of a feedback on

14. hold sample State Feedback State observer Sampling of the controller • Discrete transformation of the derivatives in using backward difference • Filtering of the velocities and  First order low pass filter

15. Higher overshoot in reality (Nonlinearities such as dry friction) High rising time (>1.5s) Open loop plant has 3 poles : 8.82, -8,72, 7.64 2 turns around -1  stable closed loop Performance of the PD-controller

16. LQ-controller • How to choose for optimal results? Computed from the continuous plant state matrices With , and gives optimal solution

17. Performance of the LQ-controller • No overshoot. • Phase margin 60 degrees

18. Performance of the LQ-controller Demo of the continuous LQ…

19. Discrete Plant State observer Design in Discrete Time • Plant is sampled with a zero-order hold approximation. • LQ controller derived with the discrete plant state matrices : • with • Gives optimum solution for any sampling period h :

20. Performance of the Discrete LQ-controller Demo of continuous and discrete LQ…

21. Deadbeat Control • Use a state feedback • The strategy: drive the state into the origin in at most 3 steps • Possible if • Cayley-Hamilton theorem states that if the desired closed loop poles are put at the origin,

22. Performance of the Deadbeat Controller Demo of the Deadbeat Controller…

23. Embedded behaviour • CPU time 2%  no pb with deadlines not met • Sampling frequency VS control performance • Maximum sampling period h=150 ms according to rising time of motor

24. Conclusion • Energy controller for swinging up the pendulum gives good results. • Continuous LQ works fine with high sampling frequency • For lower sampling frequencies, discrete design of controller needed. • Deadbeat controller does not work because of voltage limitations

25. Questions?