System identification for the main linac of CLIC

CERN, BE-ABP (Accelerators and Beam Physics group)

Jürgen Pfingstner System Identification for the main linac as CLIC

System identification for the main linac of CLIC

Jürgen Pfingstner

8th of Dezember 2009



Content

1. What is system identification and what is it good for?

2. Basic system identification

3. Problems with the basic scheme and possible solutions

4. Summery

ControllerAccelerator

R

System

identification

Controller

design

Param.

riui+1

yi

Ri & gmi

Adaptive Controller (STR) System



What is system identification and what is it good for?



What is system identification ?

Real-world system R(t)

Estimation algorithm

y(t)u(t)

... Input data

… Output data

… real-world system (e.g. accelerator)

… estimated system

• Goal:

Fit the model system in some sense to the real

system,

using u(t) and y(t)

• Ingredients

• Model assumption

• Estimation algorithm

• System excitation

Excitation



Application 1: adaptive controller

ControllerAccelerator

R

System

identification

Controller

design

Param.

riui+1

yi

Ri & gmi

Adaptive Controller (STR) System

• 3 adaptive control schema [1]:

- Model-Reference Adapt. Sys. (MRAS)

- Self-tuning Regulators (STR)

- Dual Control

STR

• The main linac of CLIC changes over time due to drifts.

• The assumed system behavior by the controller is not the correct

one anymore.

• Controller performance degrades over time, e.g.:

• If the assumed response matrix R differs more than 40%

from the real response matrix, the controller starts to

perform very poorly [2]

• Idea: Tackle problem of system changes by an online system

identification in an adaptive controller



Application 2: system diagnostics

Idea 1: LOCO-like tool

• LOCO is a tool from J. Safranek [3]

• It estimates important system parameters out of the

response matrix

• Used in rings and maybe could be adapted to linacs

R

• beta function

• dispersion

• sensor and

actuator scaling

Idea 2: Eigenvalue pattern

• Observe

• If , all eigenvalues are at 0

• If the

systems

are different:

• Meth. from pattern recognition could detect system

changes



Basic system identification



The matrix R• N’s Columns correspond to the measured

beam motion in the linac, created by the N’s QP

• Motion is characterized by phase advance , beta function and Landau damping

• R is ‘nearly’ triangular and the elements close to the diagonal are most important.

-1

-1

-1

…

0



RLS algorithm and derivative [1]

• can e.g. be formalized as

• Offline solution to this Least Square problem by pseudo inverse (Gauss):

... Estimated parameter

… Input data

… Output data

• LS calculation can be modified for recursive

calculation (RLS):

• is a forgetting factor for time varying systems

• Derivatives (easier to calculate)

- Projection algorithm (PA)

- Stochastic approximation (SA)

- Least Mean Square (LMS)



Computational effort

• Normally the computational effort for RLS is very high.

• For most general form of linac problem size:

- Matrix inversion (1005x1005)

- Storage of matrix P (1 TByte)

• Therefore often just simplifications as PA, SA and LMS are used.

• For the linac system and

have a simple diagonal form.

• The computational effort can be reduced strongly

- Matrix inversion becomes scalar inversion

- P (few kByte)

- Parallelization is possible

• Full RLS can be calculated easily



First simulation results

• Identification of one line of R and the gm-vector d • Simulation data from PLACET [4]

• ΔT = 5s

• λ = 0.85

• R changes according to

last slide

• Ground motion as by A.

Seri (model A [5])



Problems with the basic scheme and possible solutions



Problems with the basic approach

Problem 2: Nature of changes

• No systematic in system change• Adding up of many indep. changes• Occurs after long excitation

Problem 1: Excitation

• Particles with different energies move

differently

• If beam is excited, these different movements

lead to filamentation in the phase space

(Landau Damping)

• This increases the emittance

=> Excitation cannot be arbitrary



Semi-analytic identification scheme

Δx’1 Δ x’2 Δ x’3

Excitation Strategy:

• Necessary excitation can not be arbitrary, due to

emittance increase

• Strategy: beam is just excited over short distance and

caught again.

• Beam Bump with min. 3 kickers is necessary

Practical system identification:

• Just parts of R can be identified

• Rest has to be interpolated

- Transient landau damping model

- Algorithm to calculate phase advance from BPM/R data

-1

-1-1

…

0



Allover identification algorithm

Accelerator R(t)

Local

RLS 1

y(t1)

u(t1), u(t2), u(t3), …

Excitation

unit

Phase reconstr.

Local

RLS 2Phase reconstr.

y(t2)

y(t3).

.

.

.

.

..

.

.

Amplitude

Model

Phase

information

Amplitude

information

Estimated

matrix



Model of the transient Landau DampingResult: (Kick at 390 and

6350m)• Approach [6] to describe Landau damping is

adopted to lattice of the CLIC main linac

• Envelope extracted by peak detection

algorithm

• Limitation: Works just for time independent

energy distribution

• Not the case at injection into linac => fit to data

• l2 error: 4 to 7 %



Phase reconstruction from BPM data

Principle:

• Use one part of a column of R, estimated by RLS

• Detect zero-crossings• Between two zero-crossings

-> 180°• Estimate first and last point • Low pass-filtering of data

• Noise can add “spurious” zero-crossings !!!

-> more robust by deleting too close zeros

Results:

S/N ratio l1 error l2 error

∞ (no noise) 1.47° 2.07°

8 2.46° 3.26°

4 3.56° 4.78°

2 6.89° 8.58°



Summery

• System identification is a tool to get online information about a system

• It can be used for adaptive control or system diagnostics

• Ingredients

• Proper Excitation

• Model assumption

• Estimation Algorithm

• An approach to overcome the problem with excitation was presented

• Amplitude Model

• Phase reconstruction

• excitation scheme

• combination of the elements

• The idea looks feasible, but it has to be seen if the additive inaccuracies of the scheme cause to big errors to be useful.



References

[1] K. J. Åström and B. Wittenmark. Adaptive Control. Dover Publications, Inc., 2008. ISBN: 0-486-46278-1.

[2] J. Pfingstner, W. http://indico.cern.ch/conferenceDisplay.py?confId=54934. Beam-based feedback for the main linac, CLIC Stabilisation Meeting 5, 30 th March 2009.

[3] J. Safranek. Experimental determination of storage ring optics using

orbit response measuremets. Nucl Inst. and Meth. A, 388:27–36, 1997.

[4] E. T. dAmico, G. Guignard, N. Leros, and D. Schulte. Simulation Package based on PLACET. In Proceedings of the 2001 Particle Accelerator Converence (PAC01), volume 1, pages 3033–3035, 2001.

[5] Andrey Sery and Olivier Napoly. Influence of ground motion on the time evolution of beams in linear colliders. Phys. Rev. E, 53:5323, 1996..

[6] Alexander W. Chao. Physics of Collective Beam Instabilities in High Energy Accelerators. John Wiley & Sons, Inc., 1993. ISBN: 0-471-55184-8.



Thank you for your attention!



The model of the main linac

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

Þ Linear system without ‘memory’

2

1

2221

1211

2

1

x

x

rr

rr

y

y

Rxy

y … vector of BPM readings

x … vector of the QP displacements

R … response matrix



• Noise/Drift generation

• Parameter of noise (for similar emittance growth; ΔT = 5s):

- BPM noise: white noise (k = 5x10-8)

- RF disturbance: 1/f2 drift (k = 7x10-4) + white noise (k = 1.5x10-2)

- QP gradient errors: 1/f2 drift (k = 4x10-6) + white noise (k = 3x10-4)

- Ground motion: According to Model A of A. Sery [8]

• RF drift much more visible in parameter changes than QP errors

Modeling of the system change

Random number z-1

+z-1

+k

white noise 1/f

noise 1/f2

noisewhite and gaussian



Forgetting factor λ

d10: λ to big (overreacting)

d500: λ fits

d1000: λ is to small

=> Different positions in the linac should use a different λ (work)

System identification for the main linac of CLIC

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