Wolfgang Hofle Wolfgang.Hofle@cern.ch CERN CAS Darmtadt, October 9
Feedback i a mechanim that influence a ytem by looping back an output to the input a concept which i found in abundance in nature and eential to regulate the procee in any form of life Feedback ytem in engineering 788 Watt adopted an automated regulation mechanim uing a centrifugal fly ball governor to control a team engine, uing the angular peed to manipulate a valve for the team 868 Maxwell publihed an analyi of Watt centrifugal governor mathematical analyi of a feedback ytem 876 Vyhnegradkii independently analyzed tability of team engine governor tability
What i a ytem, what mean feedback? What are the purpoe of feedback ytem? recap the mathematical tool to analyze ytem behavior criteria for the tability of cloed loop feedback ytem tep in deigning feedback ytem 3
Sytem input time domain x (t) y(t) Sytem output Laplace domain X ( Y ( ( X ( called tranfer function characterized by a fixed rule determining an evolution in time of the output determinitic, output can be calculated from input and initial tate non linear ytem can be linearized in the vicinity of a working point Linear Time Invariant ytem LTI ytem 4
input Plant output time domain x (t) y(t) Laplace domain X ( Y ( ( X ( control theory tell u how to influence the output of a ytem referred to a plant plant can be: team engine, living cell, your home temperature, hip croing a river, particle beam trajectory 5
open loop imple input plant output but require precie knowledge of plant feed forward plant FF anticipate, require alternate mean of influencing output feedback plant feed back mean influencing the ytem output by acting back on the input FB new ytem! new propertie? 6
H. S. Black, working at Bell lab in the 9 on looking for a way to improve the linearity and bandwidth of vacuum tube amplifier tube amplifier FF Black patented a feed forward cheme in 98 tube contributing to feedback control theory at Bell lab: Nyquit, Bode amplifier FB patent for negative FB amplifier in 93 H. S. Black 7
model uncertaintie plant diturbance ytem input dynamic ytem plant ytem output actuator control ignal controller tracking error reference purpoe of feedback control: cancel plant imperfection precie tracking of reference parameter tabilize a potentially untable ytem reduce the effect of diturbance render output inenitive to model uncertaintie enor enor noie 8
X ( ytem input or plant Y ( ytem output F without feedback feedback Y ( ( X ( feedback output of cloed loop cloed loop tranfer function Y ( ( X ( ( F( Y ( CL ( Y ( ( X ( ( F( open loop tranfer function ( F( OL negative feedback ign convention or depending on ign convention 9
LTI ytem decribed by differential equation continuou time domain or difference equation dicrete time domain olving equation ubject to initial condition uing Laplace tranform, tranfer function F in domain complex frequency jfor continuou time domain ignal and ytem z tranform for dicrete time domain ignal and ytem Fourier tranform, continuou and dicrete decription in frequency domain by the repone to a inuoidal input concept of tranfer function F amplitude and phae, for purpoe of ytem characterization by meaurement 95 Ragazzini & Zadeh
...... t δ ( ) ( ) F( ) f ( t)e jt dt f ( t) co( t) F ) ( ) ( ) ( f ( t) F( )e j t d
function are zero for t j j...... zero t j pole complex frequency F ( f ( t) e t d t j f ( t) co( t) F( t f ( t) j j j F ( e t d
tep function delta function caling u(t), (t) f (at) a F a a t hift rule frequency e at F( a ) f ( t) nd hift rule time a f ( t a) e F( a differentiation t derivative f (t) F( f ( ) differentiation nd derivative integration t f (t) F( f ( ) f ( ) f ( ) d F( 3
Y Y Y 4 y y y t y t in e ) ( ) ( y y differential equation free ocillation initial condition for example (t) f ) ( ) ( f F Y f (t) ) ( ) ( ) ( f f F un damped ocillation for olving differential equation uing the Laplace tranform
y y y x( t) ( x( t) ( t) (t) Sytem input Sytem output x (t) y(t) X ( Y ( ( X ( The tranfer function decribe the repone of the ytem to a t ignal input j j pole x x x j j y e in t t pole in right half plane untable a mall diturbance will grow 5
d(t) plant diturbance x(t) Sytem input model uncertaintie dynamic ytem plant y(t) Sytem output actuator control ignal w(t) controller tracking error e(t) r(t) enor enor noie n(t) Feedback objective? identify plant what are input & output? 6
What do you want to achieve? damp longitudinal intability identify plant what are input & output? beam dynamic ytem plant beam: ynchrotron ocillation of the centroid of a bunch in a bucket How to ene output? enor phae pick up How to act on input? actuator cavity, longitudinal kicker What type of controller? controller Let try! 7
8 ) ( t x y y y differential equation ) ( K K F ) ( CL ) ( ) ( ) ( K K K example: beam bunched longitudinal intability plant: model of beam ynchrotron motion centroid chooe a feedback tranfer function educated gue velocity feedback feedback proportional to differentiated output beam untable without feedback
CL pole K K x j K / x x cloed loop tranfer function K pole of reulting tranfer function moved by feedback into left half plane for K/ amount of damping can be adjuted by the gain of the feedback K K K, j, j K / K / j, j, j note: feedback doe not influence the zero of the tranfer function pole in left half plane if K/ table, damped ocillation 9
Very often plant and feedback can be characterized or approximated by tranfer function which are rational function of. x (t) y(t) X ( Y ( ( X ( differential equation tranfer function impule repone tep repone P y( t) K x( t) ( K y( t) K ( t) y( t) K u( t) Delay PT y( t) x( t T ) y y x T T ( e y( t) ( t T) T ( T y( t) y( t) u( t T) t / T t / T T e y( t) T e
x (t) y(t) F X ( Y ( ( X ( differential equation tranfer function D y( t) K x( t) F( K PDT y KT y K x F( T T I y( t) K x( )d t F( K
tranfer function x (t) y(t) F X ( Y ( ( X ( P I DT F( K K K3 T T baic control, but offet K preciion, but over hoot K K 3 T T reduce overhoot
cont. feedback tranfer function cloed loop tranfer function F( CL pole: j T j T K e T Ke K T X ( Sytem input n x logk j T T j K K pole on imaginary axi T plant F feedback K Im OL K Y ( Sytem output OL if and CL Re OL Nyquit plot K T K locu of open loop tranfer function for j 3
cont. feedback tranfer function F( K T cloed loop tranfer function e T X ( Sytem input plant Y ( Sytem output CL F F CL Ke T T feedback pole: trancendental equation look at plot of open loop tranfer function OL T Ke T 4
Nyquit plot: open loop tranfer function for j OL T Ke T K 5 T T cloed loop CL OL d untable 5
Nyquit plot: open loop tranfer function for j OL T Ke T K 5 T T 4 cloed loop CL OL d table For tability the point, mut alway lie left of locu when followed 6
x x Bunch on turn n= x Bunch on turn n= area J Bunch on turn n= x area J x x Bunch on turn n= at pick-up at kicker 7
Kicker K q fractional tune pectrum n( q) nq ignal beam Pick-up Signal proceing gain g n n f / f rev PU each line i a harmonic ocillator 8
X ( model of beam harmonic ocillator Y ( Nyquit plot: F OL due to ign definition in meaurement the untable point to avoid i here in right half plane NWA open loop, inert network analyzer K F( / Q / Q F feedback choice of pick up/kicker location / / Q Q table kicker tranfer function 35.4 n 3 db @ 4.5 MHz two econd order all pae a corrector electronic extend table region to MHz open loop tranfer function meaurement by network analyzer F, SPS vertical TFB weep from 9.5 to 9.6 MHz Exact treatment for high gain require z tranform ampled ytem, once per turn 9
I ( t B ) plant diturbance I ( t) I (t) Sytem input power amplifier control ignal w(t) model uncertainty: cavity tune RF cavity impedance delay compenating network tracking error e(t) V r ( t) V acc ( t) Sytem output probe to meaure cavity field purpoe of above feedback control for cavity: precie etting of accelerating voltage reduce the effect of diturbance beam induced voltage Render the ytem inenitive to mall model uncertaintie reonant frequency of cavity, hunt impedance 3
feedback controller plant diturbance reference V I V I B B Z L V ref Z L L VZ ZL ZL Vref Z Z L L plant Z L Z R Z R actuator, gain ampère/volt Cavity feedback equivalent circuit CERN PS 3
V I B ZL ZL Vref Z Z L L Z L V I B V ref In the limiting cae of large feedback gain beam impedance only depend on feedback and amplifier gain! Beam impedance reduction by cavity feedback CERN PS hown on a relative cale 3
magnet current regulation direct RF feedback around accelerating cavity DC beam current tranformer feedback loop cloed around equipment orbit, tune, chromaticity feedback RF control loop beam phae and radial feedback loop feedback loop cloed around beam tranvere and longitudinal coupled bunch feedback Feed forward alo ued to control and reduce the effect of diturbance in accelerator: adaptive trajectory control in a tranfer line pule to pule RF cavity feed forward from meaured beam current, turn by turn Exiting pole of a ytem repone are not influenced by feed forward, only by feedback! 33
What i a ytem, what mean feedback? What are the purpoe of feedback ytem? mathematical tool to analyze ytem behavior criteria for the tability of cloed loop feedback ytem tep in deigning feedback ytem example, feed forward veru feedback 34