1. Nonlinear Fuzzy PID Control Phase plane analysis Standard surfaces Performance

2. Phase Plane

3. Equilibrium Points x1 x2 Stable node x1 Time [s]x1 x2 Unstable node x1 Time [s] x1 x2 Stable focus x1 Time [s]x1 x2 Unstable focus x1 Time [s] x1 x2 Center point x1 Time [s]x1 x2 Saddle point x1 Time [s]

4. Closed Loop (1/s 2 )

5. Example: 1/s 2

6. Example: Stopping a Car Open loop Closed loop

7. Phase Plane

8. Rule Base With 4 Rules 1. If error is Neg and change in error is Neg then control is NB 3. If error is Neg and change in error is Pos then control is Zero 7. If error is Pos and change in error is Neg then control is Zero 9. If error is Pos and change in error is Pos then control is PB -1000100 -100 -50 0 50 100 CE E 1 3 7 9

9. Surfaces: Linear and Saturation Linear Saturation

10. Surfaces: Deadzone and Quantizer Deadzone Quantizer

11. Example: FPD Control of 1/s 2

12. Example: FPD+I Control of 1/s 2

13. Hand-Tuning 1.Adjust GE (or GCE) to exploit universe 2.Set GIE = GCE = 0; tune GU 3.Increase GU, then increase GCE 4.Increase GIE to remove final offset 5.Repeat from 3) until GU is large as possible

14. Limit Cycle

15. Input Universe Saturation

16. Design Procedure * Build and tune a conventional PID controller first. Replace it with an equivalent linear fuzzy controller. Make the fuzzy controller nonlinear. Fine-tune the fuzzy controller. *) Relevant whenever PID control is possible, or already implemented

17. Bode Plot: Linear FPD

18. Bode Plot: Nonlinear FPD

19. Nyquist: Nonlinear FPD+I of 1/(s+1) 3 -202 0 1 2 Kp = 4.8, Ti = 15/8, Td = 15/32 quantizer saturation deadzone linear

20. Nyquist: Nonlinear FPD+I of 1/s 2 -202 0 1 2 Kp = 0.5, Ki = 0, Td = 1 quantizer saturation deadzone linear

21. Nyquist: Nonlinear FPD+I of e -2s /(s+1) -202 0 1 2 Kp = 4.8, 1/Ti = 1, Td = 0.46875 quantizer saturation deadzone linear

22. Nyquist: Nonlinear FPD+I of 25/(s+1)(s 2 +25) -202 0 1 2 Kp = -0.25, 1/Ti = -1, Td = 0 quantizer saturation deadzone linear

23. Fuzzy + PID Configurations ProcessPID Fuzzy ProcessPID Fuzzy ProcessPID Fuzzy ProcessPID (a)(b) (c)(d)

24. Summary Phase plane analysis Standard surfaces Performance