Jan Jantzen jj@inference.dk www.inference.dk 2013 Fuzzy PID Control Jan Jantzen jj@inference.dk www.inference.dk 2013

Summary Reduce design choices Tuning

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

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.

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 The four rules can handle many cases, and they are sufficient for a linear controller.

Textbook PID Controllers Continuous version Discrete version Incremental, discrete version

Fuzzy P controller f Rule base u GU U GE E e Gain on error Gain on control f Rule base u GU U GE E e Provided that the rule base acts like the identity function By comparison with the P controller equation

FP Rule Base 1. If E(n) is Pos then u(n) is 100 2. If E(n) is Neg then u(n) is -100 With a proper choice of membership functions the controller will act like a linear P controller

Fuzzy PD Controller e GE GCE f Rule base E CE u GU U de/dt Provided that the rule base acts like a summation Now we know what the gains do

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 0 7. If E(n) is Pos and CE(n) is Neg then u(n) is 0 9. If E(n) is Pos and CE(n) is Pos then u(n) is 200 With a proper choice of membership functions the controller will act like a linear PD controller. Four rules are sufficient.

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.

Fuzzy Incremental Controller The output is a change to the previous state e GE GCE f Rule base E CE GCU 1/s U CU cu de/dt This is an integrator. It could be a valve position, for instance. The increment. It is a change to the sum of all previous signals.

Fuzzy - PID Gain Relation Controller Kp 1/Ti Td FP GE×GU FInc GCE×GCU GE/GCE FPD GCE/GE FPD+I GIE/GE It 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.

Tuning Process gain If we increase Kp too much, the system might oscillate or even become unstable If we increase Kp, we suppress load changes. If we increase Kp, the response will be more sensitive to noise.

Ziegler-Nichols Tuning Increase Kp until oscillation, Kp = Ku Read period Tu at this setting Use Z-N table for approximate controller gains

Ziegler-Nichols (freq. method) Controller Kp Ti Td P 0.5Ku PI 0.45Ku Tu/1.2 PID 0.6Ku Tu/2 Tu/8 Given 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.

Z-N oscillation of 1/(1+s)3 The ultimate gain Ku = 8, and the ultimate period is Tu = 15/4 s

PID control of 1/(1+s)3 Response to a reference step Response to a load step

Fuzzy FPD+I control of 1/(1+s)3 The response is the same as for PID control Trajectory on the control surface, which is a plane The membership functions are linear

Hand-Tuning Set Td = 1/Ti = 0 Tune Kp to satisfactory response, ignore any final value offset Increase Kp, adjust Td to dampen overshoot Adjust 1/Ti to remove final value offset Repeat from step 3 until Kp large as possible

Quick reference to controllers Advantage Disadvantage FP Simple Maybe too simple FPD Less overshoot Noise sensitive, derivative kick FInc Removes steady state error, smooths control signal Slow FPD+I All in one Windup, derivative kick

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.

Summary Design crisp PID Replace it with linear fuzzy Make it nonlinear Fine-tune it

Advanced Section

Nyquist 1/(s+1)3 with PID -2 2 -1 1 Kp = 4.8, Ti = 15/8, Td = 15/32

Tuning Map 1/(s+1)3 -2 2 000 a) 001 b) 010 c) 011 d) 100 e) 101 f) 110 2 000 a) 001 b) 010 c) 011 d) 100 e) 101 f) 110 g) 111 h)