3 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
4 Single Loop Control Load Noise The controller should preferably be able to follow the reference r, reject load changes l and noise disturbances n, but these requirements are in conflict with each other. We would like to transfer PID tuning methods to the fuzzy controller in order to have a tuning method.
5 Rule Base With 4 Rules1. If error is Neg and change in error is Neg then control is NB3. If error is Neg and change in error is Pos then control is Zero7. If error is Pos and change in error is Neg then control is Zero9. If error is Pos and change in error is Pos then control is PBThe four rules can handle many cases, and they are sufficient for a linear controller.
6 Textbook PID Controllers Continuous versionDiscrete versionIncremental, discrete version
7 Fuzzy P controller f Rule base u GU U GE E e Gain on error Gain on controlfRule baseuGUUGEEeProvided that the rule base acts like the identity functionBy comparison with the P controller equation
8 FP Rule Base 1. If E(n) is Pos then u(n) is 100 2. If E(n) is Neg then u(n) is -100With a proper choice of membership functions the controller will act like a linear P controller
9 Fuzzy PD Controller e GE GCE f Rule base E CE u GU U de/dt Provided that the rule base acts like a summationNow we know what the gains do
10 FPD Rule Base 1. If E(n) is Neg and CE(n) is Neg then u(n) is -200 3. If E(n) is Neg and CE(n) is Pos then u(n) is 07. If E(n) is Pos and CE(n) is Neg then u(n) is 09. If E(n) is Pos and CE(n) is Pos then u(n) is 200With a proper choice of membership functions the controller will act like a linear PD controller. Four rules are sufficient.
11 Fuzzy PD+I Controller CE e GE f PD rules GCE + GU E GIE IE u U de/dt It is better that the integral action bypasses the rule base. It saves rules.
12 Fuzzy Incremental Controller The output is a change to the previous stateeGEGCEfRule baseECEGCU1/sUCUcude/dtThis is an integrator. It could be a valve position, for instance.The increment. It is a change to the sum of all previous signals.
13 Fuzzy - PID Gain Relation ControllerKp1/TiTdFPGE×GUFIncGCE×GCUGE/GCEFPDGCE/GEFPD+IGIE/GEIt tells what each fuzzy gain does to the proportional gain, the derivative gain, and the integral gain. Conversely, given values for Kp, Ti and Td we can find one or more sets of values for the fuzzy gains. Very important table.
14 TuningProcess gainIf we increase Kp too much, the system might oscillate or even become unstableIf we increase Kp, we suppress load changes.If we increase Kp, the response will be more sensitive to noise.
15 Ziegler-Nichols Tuning Increase Kp until oscillation, Kp = KuRead period Tu at this settingUse Z-N table for approximate controller gains
16 Ziegler-Nichols (freq. method) ControllerKpTiTdP0.5KuPI0.45KuTu/1.2PID0.6KuTu/2Tu/8Given values for Ku and Tu, the table provides the gains in the three controller cases. Easy, but often the result is a poorly damped system.
17 Z-N oscillation of 1/(1+s)3 The ultimate gain Ku = 8, and the ultimate period is Tu = 15/4 s
18 PID control of 1/(1+s)3 Response to a reference step Response to a load step
19 Fuzzy FPD+I control of 1/(1+s)3 The response is the same as for PID controlTrajectory on the control surface, which is a planeThe membership functions are linear
20 Hand-Tuning Set Td = 1/Ti = 0 Tune Kp to satisfactory response, ignore any final value offsetIncrease Kp, adjust Td to dampen overshootAdjust 1/Ti to remove final value offsetRepeat from step 3 until Kp large as possible
21 Quick reference to controllers AdvantageDisadvantageFPSimpleMaybe too simpleFPDLess overshootNoise sensitive, derivative kickFIncRemoves steady state error, smooths control signalSlowFPD+IAll in oneWindup, derivative kick
22 Scaling e GE GCE f Rule base E CE u GU U α 1/α de/dt The linear controller is invariant towards scaling. In the nonlinear controller we can use it to avoid saturation in the input universes.
23 Summary Design crisp PID Replace it with linear fuzzy Make it nonlinearFine-tune it