1. Flat Phase PID Controllers - 1 Presented at Control 2008 © Dr Richard Mitchell 2008 FLAT PHASE PID CONTROLLERS Dr Richard Mitchell Cybernetics Intelligence Research Group Cybernetics, School of Systems Engineering University of Reading, UK R.J.Mitchell@reading.ac.uk

2. Flat Phase PID Controllers - 2 Presented at Control 2008 © Dr Richard Mitchell 2008 Overview Flat Phase controllers have constant (flat) phase shift round the feedback loop around the key design frequency – this sets the Phase Margin. As such, if the gain of the plant changes, the Phase Margin remains (almost) constant Hence %overshoot to a step input is largely unchanged. This is termed an iso-robustness property. Flat Phase PID controllers have been designed using Bode Integrals and by setting the phase at two frequencies. This paper describes a novel three point controller. The different strategies are compared.

3. Flat Phase PID Controllers - 3 Presented at Control 2008 © Dr Richard Mitchell 2008 Flat Phase and Bode’s Ideal Loop TF Bode (1948) Ideal Loop Transfer Function Nyquist locus: straight line through origin at angle Φ from –ve real axis Barbosa et al (2005): design PID controller so closed loop TF is as close as poss to having given loop TF L(jω) Chen and Moore (2005): design PID controller using Bode Integrals, at ω c locus is constant : follows above line then Mitchell (2006): design PID controller so locus passes through two points on this line, around ω c  -1,j0 Re Im

4. Flat Phase PID Controllers - 4 Presented at Control 2008 © Dr Richard Mitchell 2008 On Chen & Moore – Bode Integrals Phase flat at ω c where gain = cos(Φ) not unity. No way of knowing ‘width’ of flat phase. Complicated formulae Mitchell - Two Point PID Controller Specify two freqs near ω c where phase set at -π + Φ. Better defines the width. Not as flat, better re %o/s ? How set two freqs?

5. Flat Phase PID Controllers - 5 Presented at Control 2008 © Dr Richard Mitchell 2008 Improvement : Two Point Controller Aim : system ‘isorobust’ when plant gain changes. So specify key design freq ω c, desired Φ for normal plant AND amount system can tolerate plant gain change Then to allow gain factor gfac, that is the gain to change between G*gfac and G/gfac, phase should be –π + Φ at But phase not that flat between ω 2 and ω 1. So … Use formulae (Mitchell 2006) to calculate parameters:

6. Flat Phase PID Controllers - 6 Presented at Control 2008 © Dr Richard Mitchell 2008 Three Point Controller Here design PID controller so that phase is –π + Φ at three frequencies, ω 1, ω c and ω 2. Need to have PID controller with extra parameter: Paper shows how matrices can be used to form cubic eqn for c, in terms of p, q and r (related to Plant phase, ω,..): T i = a/b – c = q/pT d = 1 / (a – bc) – c = 1/qc – c N = T d / c K p = K * (a – bc) = K * qc

7. Flat Phase PID Controllers - 7 Presented at Control 2008 © Dr Richard Mitchell 2008 Experiments The Three Flat Phase Controllers were tested on different plants (eg those in Chen & Moore, Astrom etc) The C&M and 2 pt controllers were designed. For comparison with 3 pt, versions of C&M and 2pt were included where the T d term is filtered with N = 10 : problem as designs not cope with N term.

8. Flat Phase PID Controllers - 8 Presented at Control 2008 © Dr Richard Mitchell 2008 On the Results See paper for actual controller parameters. Results show how flat the phase is, and how iso-robust re time to peak, %overshoot and settling time

9. Flat Phase PID Controllers - 9 Presented at Control 2008 © Dr Richard Mitchell 2008 Plant 1 – Variation of phase ωC FP1 C FPN1 C 2P1 C 2PN1 C 3PA1 C 3PB1 0.351.021.090.5350.6890.2060.16 0.360.710.7820.270.440.0990.056 0.370.450.530.080.260.035-0.002 0.380.250.31-0.050.080.004-0.024 0.390.100.14-0.13-0.02-0.005-0.020 0.40-0.00-0.16-0.100 0.41-0.06-0.12-0.14 0.0070.026 0.42-0.09-0.21-0.08-0.160.0060.046 0.43-0.09-0.30.01-0.17-0.020.052 0.44-0.07-0.380.12-0.17-0.060.034 0.45-0.04-0.450.26-0.17-0.14-0.016 3 Point better than Flat Phase better than 2 Point

10. Flat Phase PID Controllers - 10 Presented at Control 2008 © Dr Richard Mitchell 2008 Plant 1 – Iso-Robustness GainC(s)T pk %osT set C(s)T pk %osT set 0.8C FP1 11.0416.526.81C FPN1 10.9416.026.41 1.09.84117.631.129.35615.830.25 1.28.23417.229.256.16521.228.42 0.8C 2P1 10.9917.433.13C 2PN1 11.0016.726.62 1.09.83618.331.599.80116.231.02 1.28.75417.429.626.09620.928.99 0.8C 3PA1 11.9716.434.73C 3PB1 11.9116.634.91 1.07.19915.833.037.21615.433.00 1.26.23525.430.966.20024.830.89 2 Point controllers best re o/s and Tpk – long Tset

11. Flat Phase PID Controllers - 11 Presented at Control 2008 © Dr Richard Mitchell 2008 Step Responses : Gain *0.8, 1, 1.2 05101520 0 0.5 1 1.5 C FP1 05101520 0 0.5 1 1.5 C FPN1 05101520 0 0.5 1 1.5 C 2P1 05101520 0 0.5 1 1.5 C 2PN1 0 0 5101520 0.5 1 1.5 C 3PA1 0 5 10 0 0.5 1 1.5 t 051520 C 3PB1 C FP1 and C 2P1 good re constant o/s; long settling time

12. Flat Phase PID Controllers - 12 Presented at Control 2008 © Dr Richard Mitchell 2008 Plant 2 – Nyquist Plots -2-1.5-0.500.5 -2 0 1 dotted line for C FP1 dashed line for C FPB2 solid line for C 2P2 Others like that for C 2P2 Clearly Two and Three Point Controllers much better than C&M as regards flat phase

13. Flat Phase PID Controllers - 13 Presented at Control 2008 © Dr Richard Mitchell 2008 Plant 2 – Iso-Robustness GainC(s)T pk %osT set C(s)T pk %osT set 0.8C FP2 0.28018.11.02C FPN2 0.27818.11.01 1.00.24119.90.890.23820.00.88 1.20.21321.40.790.20921.60.79 0.8C 2P2 0.27326.50.68C 2PN2 0.27026.40.54 1.00.23526.50.800.23126.60.46 1.20.20726.50.760.20126.70.74 0.8C 3PA2 0.20826.50.75C 3PB2 0.20826.50.76 1.00.17826.40.670.17826.40.67 1.20.15626.30.610.15626.30.61 Clearly 2 and 3 point controllers much better than C&M

14. Flat Phase PID Controllers - 14 Presented at Control 2008 © Dr Richard Mitchell 2008 In fact can change gain by more GainC(s)T pk %osT set C(s)T pk %osT set 0.25C 2P2 0.60425.91.80C 3PB2 0.45826.31.32 0.50.37526.21.010.28526.50.72 1.00.23126.60.460.17826.40.67 1.50.17227.00.660.13326.30.53 20.13827.50.550.10726.40.43 30.10029.50.410.07827.40.17 40.08032.60.320.06329.40.23

15. Flat Phase PID Controllers - 15 Presented at Control 2008 © Dr Richard Mitchell 2008 Conclusions and Further Work For Plant with repeated poles, the two point controller is marginally better as regards iso-robustness For Plant with distributed poles, three point controller best … much better than the Chen&Moore method Further tests needed on other plants Re The arbitrary value of N = 10 when filtering the derivative term – it would be better if could include this formally in designs for 2point and C&M controllers The flat phase approach is interesting, though of concern is the long settling time that occurs. Perhaps worth noting alternative Bode Integral design which gives better settling time (Karimi et al 2002).