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Verification of the Design of the Beam-based Controller. Jürgen Pfingstner 2. June 2009. Content. Analysis of the contoller with standard control engineering techniques Uncertanty studies of the response matix and the according control performance. - PowerPoint PPT Presentation
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CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Verification of the Design of the Beam-based Controller
Jürgen Pfingstner
2. June 2009
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Content
1. Analysis of the contoller with standard control engineering techniques
2. Uncertanty studies of the response matix and the according control performance
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Analysis of the contoller with standard control engineering techniques
• Daniel developed a controller, with common sense and feeling for the system
• I tried to verify this intuitive design with, more abstract and standardized methods:
• Standard nomenclature
• z – transformation
• Time-discrete transfer functions
• Pole-zero plots
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
The model of the accelerator
1.) Perfect aligned beam line
BPMiQPi
Laser-straight beam
2.) One misaligned QP
yixi
Betatron- oscillation
(given by the beta function of the
lattice)
yjxj(=0)
a.) 2 times xi -> 2 times amplitude -> 2 times yj b.) xi and xj are independent
Linear system without ‘memory’
2
1
2221
1211
2
1
x
x
rr
rr
y
y
Rxy
y … vector of BPM readingsx … vector of the QP displacementsR … response matrix
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Mathematical model of the controlled system
(R*)-1+ + z-1 R +
z-1
+
vi
ni
yiri xixi+1
uiui+1 G(z)
C(z)
-
ui, ui+1 … controller state variablesxi, xi+1 … plant state variables
(QP position)
C(z) … ControllerG(z) … Plant
ri … set value (0) … BPM measurementsyi … real beam position vi … ground motionni … BPM noise
iy
iy
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
System elements(SISO analogon!)
• I controller
z
RzG )(
1)()( 1*
z
zRzC
Be aware about the mathematical not correct writing of the TF (matrix instead of scalar)
• simple
• allpass
• non minimum phase
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Stability and Performance
• Stability • necessary attenuation at high frequencies
• all poles at zero
• Performance of the interesting transfer functions
•
•
•
))(1(
11
)(1
)(
)(
)(:)(
1*
RRzz
zR
zL
zG
zv
zyzV
))(1(
1
)(1
1
)(
)(:)(
1*
RRz
z
zLzn
zyzN
))(1(
1)(
)(1
)(
)(
)(:)(
1*1*
RRzRR
zL
zL
zr
zyzR
1
1)()()(:)( 1*
zRRzGzCzL
z
1
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Important transfer functions
)(zV)()( zNzR
(ground motion behavior)(set point following and
measurement noise)
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Conclusions• Controller is:
• very stable and robust (all poles at zero)
• integrating behavior (errors will die out)
• good general performance
• simple (in most cases a good sign for robustness)
• measurement noise has a strong influence on the output
Further work• H∞ optimal control design
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Uncertanty studies of the response matix and the according control performance
Controller is robust, but is it robust enough?
yes no
• Plan A:• Use methods from robust control to adjust the controller to the properties of the uncertainties (e.g. pole shift)
• Plan B:• Use adaptive control techniques to estimate R first and than control accordingly
done
• Answer to that question by simulations in PLACET
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Tests in PLACET
• Script in PLACET where the following disturbances can be switched on and off:
• Initial energy Einit
• Energy spread ΔE• QP gradient jitter and systematic errors• Acceleration gradient and phase jitter• BPM noise and failures • Corrector errors• Ground motion
• Additional PLACET function PhaseAdvance • 2 Test series (Robustness according to machine drift):
• Robustness regarding to machine imperfection with perfect controller model
• Robustness regarding to controller model imperfections with perfect machine
• Analysis of the controller performance and the resulting R
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Test procedure
1.) Misalign the QP at the begin of the simulation to create an emittance growth at the end of the CLIC main linac
2.) Observe the feedback action in respect to the resulting emittance over time
3.) Change certain accelerator parameter and repeat 1 and 2.
=> The test focus on the stability and convergence speed more than on the steady-state emittance growth and growth rate. (see metric)
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Performance metric
0
max
st
t
dtt
m4.0
0 0max
0)(:
)( initialEm
]20[ mspulsestime ][GeVEinitial
Normalized emittance ][)( radnmt
Allowed range for iswhere
initialE05.0)( max mEm initial
maxm
… Emittance produced by the DR and RTML … Emittance at the end of the main linacTheoretical minimum for
0)(t
01.002.05.0min m
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
ResultsQuantity Acceptable values Nominal values
Einit 8.6 – 9.4 GeV 9.0 GeV ( ± 1 %)
σE 0.0 – 9.0 % 2.0 % (± <1%)
QP gradient jitter 0.0 - 0.25 % < 0.1%
QP gradient error (syst.) 0.0 – 0.25 %
Acceleration gradient variation
0.0 – 0.3 ‰ < 0.3 ‰
Corrector jitter 0.0 – 8.0 nm (std) < 10 nm (std)
BPM noise 0.0 – 500 nm (std) < 50 nm (std)
BPM error distributed *2 110 of 2010 (5%)
BPM block errors *2 6 of 2010
Ground motion 0.68 x 10-3nm/min ( 5d)*1 … Values are according to the multi-pulse projected emittance*2 … Failed BPMs that deliver zeros are worse then the one giving random values.
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Conclusions
• Controller is very robust in respect to stability
• It is sufficient in respect to disturbances
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Thank you for your attention!
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
z – Transformation • Method to solve recursive equations• Equivalent to the Laplace – transformation for
time-discrete, linear systems • Allows frequency domain analysis
i
i
zigzG
0
][)(dTiez
z – transfer function
G(z)
Discrete Bode plot][ig
Discrete
System
z
1
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Measures for the performance
• Goal: Find properties of Rdist that correspond with the controller performance
i jijnorm
i jij
norm
nomdist
r
m
m
RRM
,
normabs
Rdist … disturbed matrixRnom … nominal matrixabsnorm … absolut matrix norm
normev
)(max1
)(
)( 1*
ii
norm evev
LeigEV
RREL
evnorm … eigenvalue normL … matrix that determines the poles of the control loop
CERN, BE-ABP-CC3
Jürgen Pfingstner Verification of the Design of the Beam-based Controller
Information from absnorm and evnorm
evnormabsnorm
Controller works well for:• absnorm = 0.0 – 0.4
Controller works well for:• evnorm = 0.5 – 1.0