Outline

2. Outline • Introduction • Modeling • Simulation • Implementation • Demo • Conclusion

3. Outline • Introduction • Modeling • Simulation • Implementation • Demo • Conclusion

4. Introduction • Rotating arm and inverted pendulum. • Rotating arm is actuated by a DC motor. • The angular disturbance will be sensed by the potentiometer.

5. Introduction • The system is controlled by a PID control circuit. • Two equilibrium points existed. • Use a cut-off device to protect the system.

6. Outline • Introduction • Modeling Find the transfer function of input voltage and the angle of inversed pendulum. • Equation of motion. • Linearization • Laplace transform • Transfer function • Simulationment • Implementation • Demo • Conclusion

7. Modeling -Equation of motion • Step 1 : Find the equation of motion by Lagrange equation

8. Modeling -Equation of motion

9. Modeling -Linearization • Step 2 : Linearization • To do the linearization, we have to find the equilibrium points first. • Find the position where the extreme value of the potential energy exist.

10. Modeling -Linearization • In this case, we set the equilibrium point at θ=0° • Expand the nonlinear terms in Taylor series.

11. System modeling -Linearization • If the angle of disturbance is 5°, the max. error between linear and nonlinear model is 0.046°, less then 1%.

12. System modeling -Laplace transform • Step 3 : Laplace transform of the motion equations

13. System modeling -Transfer function • Step 4 : Find the transfer function of a DC motor • According to Kirchhoff’s voltage law (KVL) Where is the voltage of coil is the induced voltage of the motor is the torque generate by motor Equivalent circuit of a DC motor

14. System modeling -Transfer function • Step 5 : Transfer function of the system

15. Modeling -Transfer function • Set the values we need • Assume the values we need but we don’t know Ref. : Stephen J. Chapman “Electric Machinery Fundamentals” Chap. 9 McGraw. Hill

16. Modeling -Transfer function • Transfer function.

17. Modeling -Transfer function • Unit step command test

18. Modeling -Transfer function • Command unit step and disturbance is zero to check transfer function.

19. Modeling –Routh-Hurwitz Stability • Using Routh-Hurwitz stability to find the stable range of the gain of PID or PD controller.

20. Modeling -Reference • S. Awtar, N. king, T. Allen, I. Bang, M, Hagan, D.Skidmore, K. Craig, “Inverted pendulum systems: rotary and arm-driven- a mechatronic system design case study.” Mechatronic 12 (2002) • Y. Yavin, “Control of a Rotary Inverted Pendulum.” Applied Mathematics Letters 12 (1999)

21. Outline • Introduction • Modeling • Simulation • Open loop • PD controller • PI controller • PID controller • Implementation • Demo • Conclusion

22. Simulation • Use SimMechanics to build a nonlinear system model

23. Simulation • Use Simulink to build a nonlinear system model

24. Simulation • Use Simulink to build a linear system model

25. 。Simulation–open loop (angular V)

26. Simulation -PD controller

27. Simulation -PD controller

28. Simulation-PD controller • Response simulation.(PD controller) • Absolute error between the simulation of SimMechanics and Simulink.

29. Simulation -PI controller

30. Simulation-PI controller

31. Simulation -PI controller • Response simulation.(PI controller) • Absolute error between the simulation of SimMechanics and Simulink.

32. Simulation-PID controller

33. Simulation-PID controller

34. Simulation-PID controller • Response simulation.(PID controller) • Absolute error between the simulation of SimMechanics and Simulink.

35. Outline • System introduction • System modeling • Simulation • Implementation • Inversed pendulum • Control circuit • Demo • Conclusion

36. Implementation • System block diagram

37. Implementation -Inversed pendulum • The length and mass of pendulum:32 cm and 28.41g • The length and mass of rotating arm: 10 cm and 46 g • Gear ratio: 5

38. Implementation -Control circuit • Circuit block diagram

39. Implementation -Control circuit • Circuit board Cut-off circuit Powersupply I Limit switch Signal light On/Off Motor Power amplifier PID controller Sensor Powersupply II

40. Implementation -Potentiometer • Use a variable resistor as a potentiometer. Inverted pendulum Potentiometer

41. Implementation - Potentiometer • How does it work?

42. Implementation -PID controller • Use 17741 operational amplifier • Modes switch • Elements shiftable PID controller

43. Implementation -PID controller

44. Implementation -Cut-off circuit, signal light 500 ohm resistances 7404 NOT Resistance with signal light 7408 AND NPN transistor Relay 5V 2 Form C Contact 7408 7404

45. Implementation-Cut-off circuit, signal light

46. Implementation -Power amplifier NPN TIP41 Diode NPN TIP107

47. Implementation • Why we use two power supply? • The DC motor turns on, the voltage of power supply drops. Output： DC power supply +15V port normal Input： triangular ±200mV;2Hz The DC motor use the power from +15V port

48. Outline • Introduction • Modeling • Simulation • Implementation • Demo • Conclusion

49. Demo -PD controller • Steady state error exist

50. Demo -PID controller • Steady state error is zero