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RST Digital ControlsPOPCA3 Desy Hamburg 20 to 23rd may 2012Fulvio BoattiniCERN TE\EPC

Bibliography “Digital Control Systems”: Ioan D. Landau; Gianluca Zito “Computer Controlled Systems. Theory and Design”: Karl J. Astrom; BjornWittenmark “Advanced PID Control”: Karl J. Astrom; Tore Hagglund; “Elementi di automatica”: Paolo BolzernPOPCA3 Desy Hamburg 20 to 23rd may 2012

SUMMARYSUMMARY RST Digital control: structure and calculation RST equivalent for PID controllers RS for regulation, T for tracking Systems with delays RST at work with POPS Vout Controller Imag Controller Bfield Controller ConclusionsPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST Digital control: structure and calculationPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST calculation: control structureR r0 r1 z 1 r2 z 2 . rn z nS s0 s1 z 1 s2 z 2 . sn z nT t0 t1 z 1 t 2 z 2 . t n z nA combination of FFW and FBK actionsthat can be tuned separatelyS u T r R yTSRHfb SHff REGULATIONyA S d A S B RA( z 1 ) S ( z 1 ) B ( z 1 ) R ( z 1 ) Pdes ( z 1 )TRACKINGPOPCA3 Desy Hamburg 20 to 23rd may 2012Pdes ( z 1 )T B (1)Pdes (1)B (1)yB T r A S B R

RST calculation: Diophantine EquationGetting the desired polynomial 2s2 2 s 2 2 100 0 .30.001949 z -1 0.001924 z -21 - 1.959 z -1 0.963 z -2Sample Ts 100usPdes 1 - 1.959 z -1 0.963 z -2Calculating R and S: Diophantine EquationA( z 1 ) S ( z 1 ) B ( z 1 ) R ( z 1 ) Pdes ( z 1 ) M x pMatrix form:nB dxT 1, s1 ,.,snS , r0 , r1 ,.,rnR POPCA3 Desy Hamburg 20 to 23rd may 20120b1b2bnB000 0 0 b1 b2 0 bnB . .0nA nB dp 1, p1 ,., pnP ,0,.,0 T 1 0 . 0 a0 1 a21 M a1 anAa2 0 0 . 0 anA nA

RST calculation: fixed polynomialsController TF is:R ( z 1 )Hfb S ( z 1 )Add integratorR ( z 1 )1 z 1 S * ( z 1 ) Add 2 zeros more on S A 1 z 1 1 1 z 1 2 z 2 S *yA S A S B Rd A S B RIntegrator active on step reference and step disturbance.Attenuation of a 300Hz disturbanceCalculating R and S: Diophantine EquationA( z 1 ) Hs ( z 1 ) S ( z 1 ) B ( z 1 ) Hr ( z 1 ) R ( z 1 ) Pdes ( z 1 )Hs ( z 1 ) fixed part of SHr ( z 1 ) fixed part of RPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST equivalent for PID controllersPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST equivalent of PID controller: continuous PID designConsider a II order system:B 2 2A s 2 s 2 2 100 0 .3Pole Placement forcontinuous PID1 PID Kp 1 s Td s Ti HclPIDc KpTiKd Kp TdKi PID Hsy HclPIDc.Den 1 PID Hsy 1 PID Hsy s 3 2 Kd 2 s 2 Kp 2 2 s Ki 2 s 0 0 s 2 2 0 0 s 0POPCA3 Desy Hamburg 20 to 23rd may 20122

RST equivalent of PID controller: continuous PID designConsider a II order system:B 2 2A s 2 s 2 2 100 0 .3 02Kp 1 2 0 0 2 1 03Ki 0 2 2 0 0 2 Kd 0 2 1s Td PIDf Kp 1 s Ti 1 s Td N POPCA3 Desy Hamburg 20 to 23rd may 2012KpTiKd Kp TdKi

RST equivalent of PID: s to z substitutionAll control actions on errorT ( z 1 ) R( z 1 )Proportional on error;Int deriv on outputT ( z 1 ) R(1)R( z 1 ) r0 r1 z 1 r2 z 2 1Choose R and S coeffs such that the 2 TF PIDd(z)are equalS ( z 1 ) s0 s1 z 1 s2 z 2 Ts 1 z 1 R r 0 r1 z 1 r 2 z 2N Td 1 z 1 PIDd Kp 1 1 1 Td N Ts N Ts 1 Td z S s0 s1 z 1 s2 z 21 z Ti 0 forward Euler 1 backward EulerPOPCA3 Desy Hamburg 20 to 23rd may 2012 0.5 Tustin

RST equivalent of PID: pole placement in zChoose desired poles 2s2 2 s 2 2 150 0 .8Sample Ts 100us0.001949 z -1 0.001924 z -21 - 1.959 z -1 0.963 z -2Pdes 1 - 1.959 z -1 0.963 z -2Choose fixed parts for R and SHs 1 z 1Hr 1Calculating R and S: Diophantine Equation 1 1* 1A( z ) Hs ( z ) S ( z ) B ( z 1 ) Hr ( z 1 ) R * ( z 1 ) Pdes ( z 1 )POPCA3 Desy Hamburg 20 to 23rd may 2012

RST equivalent of PID: pole placement in zThe 3 regulators behave verysimilarlyIncreasing KdManual Tuning with Ki, Kd andKp is still possiblePOPCA3 Desy Hamburg 20 to 23rd may 2012

RS for regulation, T for TrackingPOPCA3 Desy Hamburg 20 to 23rd may 2012

RS for regulation, T for tracking Pdes s 0 0 s 2 2 0 0 s 0REGULATIONDiophantine Equation : A Hs S * B R Pdes Paux 942r/s 0.9 31400r/s 0.004Hol 0 1.5 0 940r / s 0 0.9 1260r/s 1T B R 0.022626 z -1 (1 0.9875z -1 ) (1 - 1.929z -1 0.9322z -2 ) A S(1 - z -1) (1 - 0.6494z -1 ) (1 - 1.959z -1 0.963z -2)Closed Loop TF without THcl Pure delayHcl After the D. Eq solved we get the following open loop TF:Hs [1 1] integratorTRACKING2 1410r/s 1Pdes PauxB(1)T B0.50314 z -1 (1 0.9875z -1) A S B R1T polynomial compensate most ofthe system dynamicPOPCA3 Desy Hamburg 20 to 23rd may 2012B0.0019487 z -1 (1 0.9875z -1 ) A S B R (1 - 0.8819z -1 ) (1 - 0.8682z -1 ) (1 - 1.836z -1 0.844z -2)

Systems with delaysPOPCA3 Desy Hamburg 20 to 23rd may 2012

Systems with delaysII order system with pure delayB 2 e s 2A s 2 s 2 2 100r / s 0.3 3msSample Ts 1ms0.1692 z -4 0.149 z -51 - 1.368 z -1 0.6859 z -2Continuous timePID is muchslower than beforePOPCA3 Desy Hamburg 20 to 23rd may 2012

Systems with delaysPredictive controlsDiophantine EquationA( z 1 ) S ( z 1 ) z d B0 ( z 1 ) R ( z 1 ) A( z 1 ) Paux ( z 1 )Choose fixed parts for R and SHs 1 z 1Hr 1POPCA3 Desy Hamburg 20 to 23rd may 2012

RST at work with POPSPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST at work with POPSVout ControllerImag or Bfield controlAC/DC converter - AFEDC/DC converter - charger moduleDC/DC converter - flying modulePpk 60MWIpk 6kAVpk 10kV18KV ACScc 2 CF1DCCF2Lw2-DCRF1DC3 RF2CF11DCCF21DC4Crwb1DCCrwb2 -TW2TW1-DCDC DC1DCDC2Lw1CF12DCCF22- DCMAGNETSDC5POPCA3 Desy Hamburg 20 to 23rd may 2012 -DCDCDC6

RST at work with POPS: Vout ControlLfiltIm agIindC dm pVoutVinIcdmpIcfCfIII order output filterR dm pDecide desired dynamics 2s2 2 s 2 2 150 c 2 d Ts 1m s 0.1431 z -1 0.1043 z -2 0.851 - 1.142 z -1 0.3897 z -2Pdes 1 - 1.142 z -1 0.3897 z -2Eliminate process well dumped zerosPzeros z - 0.8053yB T r A S B RSolve Diophantine EquationA( z 1 ) S ( z 1 ) B( z 1 ) R( z 1 ) Pdes ( z 1 ) Pzeros ( z 1 )Calculate T to eliminate all dynamicsPdes ( z 1 )T (z ) B (1) 1POPCA3 Desy Hamburg 20 to 23rd may 2012HF zeroresponsible foroscillations

RST at work with POPS: Vout ControlWell not very nice performance .There must be something odd !!!Identification of output filter with a stepPut this back in the RSTcalculation sheetPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST at work with POPS: Vout ControlIn reality the response is a bit less nice but still very good.Performance to date (identified with initial stepresponse):Ref following: 130HzDisturbance rejection: 110HzPOPCA3 Desy Hamburg 20 to 23rd may 2012Ref following -----Dist rejection ------

RST at work with POPS: Imag ControlPS magnets deeply saturate:Magnet transfer function for Imag:I magVmag 1s Lmag RmagLmag 0.96HRmag 0.32 The RST controller was badly oscillating at the flattop because the gain of the system was changed26Gev without Sat compensationPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST at work with POPS: Imag Control26Gev with Sat compensationPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST at work with POPS: Bfield ControlTsampl 3ms.Ref following: 48HzDisturbance rejection: 27HzMagnet transfer function for Bfield:BmagVmag s K mag 1RmagLmagLmag 0.96HRmag 0.32 K magK mag 2.5Error 0.4GaussError 1GaussPOPCA3 Desy Hamburg 20 to 23rd may 2012

RST control: Conclusions RST structure can be used for “basic” PID controllers andconserve the possibility to manual tune the performances It has a 2 DOF structure so that Tracking and Regulationcan be tuned independently It include “naturally” the possibility to control systems withpure delays acting as a sort of predictor. When system to be controlled is complex, identification isnecessary to refine the performances (no manual tuning isavailable). A lot more . But time is over !POPCA3 Desy Hamburg 20 to 23rd may 2012

Thanks for the attentionQuestions?POPCA3 Desy Hamburg 20 to 23rd may 2012

Towards more complex systems(test it before !!!)POPCA3 Desy Hamburg 20 to 23rd may 2012

Unstable filter magnet delay150HzTs 1ms1.2487 (z 3.125) (z 0.2484)-------------------------------------z 3 (z-0.7261) (z 2 - 1.077z 0.8282)POPCA3 Desy Hamburg 20 to 23rd may 2012

Unstable filter magnet delayChoose Pdes as 2nd order system 100Hz well dumpedAux Poles for Robusteness loweredthe freq to about 50Hz @-3dB(not Optimized)POPCA3 Desy Hamburg 20 to 23rd may 2012

POPCA3 Desy Hamburg 20 to 23rd may 2012 SUMMARY RST Digital control: structure and calculation RST equivalent for PID controllers RS for regulation, T for tracking Systems with delays RST at work with POPS Vout Controller Imag Controller Bfield Controller Conclusions SUMMARY

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