Nonlinear Fuzzy PID Control Phase plane analysis Standard surfaces Performance

Phase Plane

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]

Closed Loop (1/s 2 )

Example: 1/s 2

Example: Stopping a Car Open loop Closed loop

Phase Plane

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 CE E

Surfaces: Linear and Saturation Linear Saturation

Surfaces: Deadzone and Quantizer Deadzone Quantizer

Example: FPD Control of 1/s 2

Example: FPD+I Control of 1/s 2

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

Limit Cycle

Input Universe Saturation

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

Bode Plot: Linear FPD

Bode Plot: Nonlinear FPD

Nyquist: Nonlinear FPD+I of 1/(s+1) Kp = 4.8, Ti = 15/8, Td = 15/32 quantizer saturation deadzone linear

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

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

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

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

Summary Phase plane analysis Standard surfaces Performance