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Nonlinear Fuzzy PID Control Phase plane analysis Standard surfaces Performance

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Phase Plane

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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]

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Closed Loop (1/s 2 )

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Example: 1/s 2

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Example: Stopping a Car Open loop Closed loop

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Phase Plane

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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

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Surfaces: Linear and Saturation Linear Saturation

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Surfaces: Deadzone and Quantizer Deadzone Quantizer

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Example: FPD Control of 1/s 2

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Example: FPD+I Control of 1/s 2

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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

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Limit Cycle

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Input Universe Saturation

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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

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Bode Plot: Linear FPD

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Bode Plot: Nonlinear FPD

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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

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

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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

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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

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

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Summary Phase plane analysis Standard surfaces Performance

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Lecture 9: PID Controller.

Lecture 9: PID Controller.

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