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

2. Example: a nonlinear valve Valve opening between 0 and 1 Nonlinear flow through valve Three steps up on the reference. The response gets worse and worse. The third response is marginally stable and the valve saturates in the upper limit (fully open). 2

3. Standard nonlinearities Nonlinear systems can be mathematically unpredictable. Instead we simulate the behaviour using a number of standard blocks that model nonlinear components. The simulation will be approximate when we cannot solve the equations, but it is often good enough. 3

4. Standard rule base 1.If error is Neg and change in error is Neg then control is NB 2.If error is Neg and change in error is Zero then control is NM 3.If error is Neg and change in error is Pos then control is Zero 4.If error is Zero and change in error is Neg then control is NM 5.If error is Zero and change in error is Zero then control is Zero 6.If error is Zero and change in error is Pos then control is PM 7.If error is Pos and change in error is Neg then control is Zero 8.If error is Pos and change in error is Zero then control is PM 9.If error is Pos and change in error is Pos then control is PB 4 With two inputs and three fuzzy terms (Neg, Zero, Pos) for each, we can build nine rules that cover the whole state space. However, just four rules may be sufficient in many cases: rules 1, 3, 7, and 9. These avoid rules with Zero, and in that case Neg and Pos must overlap each other in order to account for mid-range values.

5. Phase plot 5 Trajectory of the response on the control surface Phase plane 4 3 2 1 We have chosen one point in each quadrant Step response

6. Linear and saturation surface 6 A plane. The values are between -200 and 200. The 'saturation' surface. The four corners are in the same positions as the linear surface. The surfaces are built from four rules using these membership functions. We have constructed four standard control surfaces. The choice of shapes was inspired by the standard nonlinearities presented earlier. We can thus study a variety of standard behaviours.

7. Dead zone and quantizer surface 7 There is a dead zone in the middle with almost zero control signal. The 'quantizer' surface. The four corners are in the same positions in all four standard surfaces. The quantizer surface requires three input sets and nine rules.

8. Saturation and limit cycle 8 An example of saturation in the limit of the universe. The trajectory touches the edge. An example of a limit cycle. The trajectory cycles round and round for ever. These are two typical phenomena in nonlinear systems.

9. Example: unstable frictionless vehicle 9 1.Design a crisp PD controller 2.Replace it with a linear fuzzy controller 3.Make it nonlinear 4.(Fine-tune it) Linear equation of motion: Newton's 2. law. Force Mass Acceleration

10. Linear FPD 10 Same performance as the PD controller. The PD controller was hand- tuned.

11. Saturation surface FPD 11 The saturation surface provides tighter control around the centre of the surface. The response is more damped.

12. FPD with squared memberships 12 Making the surface steeper improves damping even more. In this case, at least. They are the previous ones squared. The control signal looks highly nonlinear now. A very good response.

13. Example: nonlinear valve compensator Three steps up on the reference. The response is considerably better than without the compensator. 13 1.If u is Low then x = 4.62u 2.If u is High then x = 0.46u + 0.543

14. Example: motor actuator with limits The control signal saturates, and the integrator in PI winds up. The response is poor. 14 Motor Actuator

15. Example: motor actuator with limits The response is considerably better with the FInc controller. It has an integrator at the end of its signal path, and it is relatively easy to limit it to the same limits as the actuator. It avoids windup. 15

16. Autopilot example: mass load A train car is standing on a curved track. At t = 10 the car is loaded with 5 times its own mass. It is taken off again at t = 30. A PI controller attempts to keep the train car in place (we do not consider using the brakes for the sake of the example). 16

17. Autopilot example: mass load The PID response has a large dip, and it creeps in towards the setpoint (x = 5). 17 The FInc response has a smaller dip, a smaller flare, and it avoids creeping. This membership function is squared using Very in the rule base.

18. Summary The linear fuzzy performs as a PID. We introduced three nonlinear surfaces in order to standardize. Fuzzy can cope with some nonlinearities that PID cannot. Human beings can read the rule base. 18