Automated Beam Steering Using Model Reference Control and Optimal Control

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Automated Beam Steering Using Model Reference
Control and Optimal Control

• Christopher K. AllenLos Alamos National
  Laboratory

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Outline

• Overview
• Motivation
• Basic Approach
• Beam Dynamics Model
• State Estimation
• Steering
• Summary and Conclusion

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1. Overview

• Motivation
• Estimate the beam state, including momentum, at
  specific beamline locations
• Control beam trajectory throughout entire
  beamline, not just at BPM locations
• Automate the procedure for on-line operation
• Estimate misalignments in beamline

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1. Overview (cont.)

• Approach State Estimation
• Use many BPM measurements along beamline, and
  model predictions, to reconstruct entire beam
  state z(s) at locations (s0,s1,s2,).

• Recursively improve state estimate using multiple
  measurements (in time) k1,2, with differing
  corrector strengths

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1. Overview (cont.)

• Approach Beam Steering
• Minimize a quadratic functional J of beam state
  z(s) throughout beamline

• Compute the beam trajectory z(s) between BPM
  locations according to a transfer-matrix model
  ?(us) of the beamline

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1. Issues

• We do not know z(s)
• Have data only at BPM locations s0,s1,s2,
• BPMs provide only position coordinates (x,y,z)
• Noise and misalignments
• State Estimator to reconstruct momentum
  coordinates
• Multistage Network Model to compute z(s)
• Optimal Control to minimize J

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2. Beam Dynamics Model

• Beam state z(s) at axial position s is a point in
  phase space.

• Phase space parameterized by homogeneous
  coordinates in ?6?1

• State z contains position (x,y,z) and momentum
  (x,y,z) coordinates

Homogenous coordinates are used because
translation, rotation and scaling in phase space
can all be performed by matrix multiplication.
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2. Dynamics (cont.)

• Beamline is divided into N stages containing at
  least one element
• Entrance of stage n is at s ? sn so that
• zn ? z(sn) x(sn) x(sn) y(sn) y(sn) z(sn)
  z(sn) 1T
• The action of stage n is represented by a
  transfer matrix ?n(un)
• zn1 ?n(un)zn
• where un is the control vector of steering
  magnet strengths

• For example, a steering magnet can be model with
  a matrix of the form

where ?x, ?x, ?y, are the translations
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2. Dynamics (cont.)

• zn beam states at stage entrances
• un control vectors to stages (e.g., steering
  magnet strengths)
• hn measurement vectors at stages (e.g., BPM
  outputs)
• ?n state transfer matrices for each stage
• ?n observation matrix for each stage

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2. Dynamics (cont.)

• Dynamics with noise nn and 1st order
  misalignments ?n

• nn white noise Wiener process
• ?n generator matrix for misalignment
  (translation rotation)

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3. State Estimation

• Basic Idea - use all the measurements hn to
  compute a (model-based) least-squares estimate
  for the state vectors zn
• Let h ? (h0 h1hN-1)T and u ? (u0 u1uN-1)T be
  the vector of measurements and controls, resp.,
  for the entire network
• We build an equation for each stage n of the form

h measurement vectorHn measurement
matrix (model dependent)zn state vector at
stage n?n vector of measurement unknowns
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3. State Estimation (cont.)

• Measurement matrix Hn(u) represents the response
  between all the measurements h, all the controls
  u, and the state vector zn to stage n.
• For example, H0(u) appears as

• The remaining Hn(u) may be computed via the
  recursion relation

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3. State Estimation (cont.)

• The least-squares estimate to the state at stage
  n is given by

LnW weighted left psuedo-inverse of Hn HnT
measurement matrix transposeWn weighting
matrix for stage nh measurement vector
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3. State Estimation (cont.)

• Simulation Results
• Includes BPM noise and random misalignments
• Beamline composed of 4 stages
• Each stage is compose of 2 FODO periods

Parameters Ldrift 14.88 cm Lquad 6.10
cm kquad 90 deg
stage n schematic
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3. State Estimation (cont)
Simulation Results Estimating state vector zn
at each stage
Parameters xi 3 mm xi 0 mrad ?align 100
?m?noise 100 ?m
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3. State Estimation (cont.)
Simulation Results Estimating state vector zn
at each stage
Parameters xi 3 mm xi 0 mrad ?align 250
?m?noise 100 ?m
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3. State Estimation (cont.)
Simulation Results Estimating state vector zn
at each stage
Parameters xi 3 mm xi 0 mrad ?align 500
?m?noise 100 ?m
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3. State Estimation Current Work

• Recursive State Estimation
• Since the state z0 does not vary with network
  controls u it is possible to use new measurements
  to refine the estimate for z0
• Letting k index each additional measurement h(k)

• Misalignment Parameter Estimation
• The effectiveness of the above technique seems to
  be limited by the ability to estimate the
  misalignments ?n of the network

• Online estimation of both the states zn and
  misalignments ?n is nontrivial

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4. Distributed Steering Algorithm

• Basic Idea - rather than minimizing position
  errors at discrete BPM locations, we minimize a
  functional J of the beam state z(s) throughout
  the beamline
• The steering objective is thus defined by the
  functional J
• Functional J is decomposed into N terms Jn, one
  for each stage n
• The sub-functional for each stage n is has the
  form

where Qn symmetric, positive semi-definite
matrix z(s) ?n(uns)zn
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4. Steering (cont.)

• Example Objective Functional A Drift Space
• Consider only the x-plane so that the state
  vector is xn (xn xn 1) and x(s) is

• The cost functional Jd for a drift of length ld
  is then

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4. Steering (cont.)

• Terminal Cost
• Any constraints on the final state zN are
  enforced with a terminal cost functional ?(zN)
• Letting zf represent are the desired final state
  then

where P ? ?7?7 is another symmetric, positive
semi-definite tuning matrix
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4. Steering (cont.)

• Total Steering Objective
• The total steering objective is described by the
  total cost functional J

• Steering Problem Statement
• Our steering problem is described mathematically
  as

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4. Steering (cont.)

• Remarks
• We must find that control set u0,u1,,uN-1 that
  solves the above constrained optimization
  problem
• By selecting the matrices P and Q, and their
  relative magnitudes, we can stipulate different
  performance objectives for our beam steering
  algorithm.
• By applying optimal control theory we develop an
  efficient solution algorithm which is
  unconstrained

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4. Steering (cont.)

• Optimal Control
• Introducing the set of costate variable p0,,pN
  define the Hamiltonian Hn

• The necessary conditions for an optimal solution
  are given by

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4. Steering (cont.)

• Direct solution of the necessary conditions is
  nontrivial
• Two-point boundary value problem
• However, the gradient ?J/?un has a convenient
  expression

• where the zn and pn satisfy the propagation
  equations of the necessary conditions

• The gradients ?J/?un can be used for an
  unconstrained search algorithm to minimize J
• Simple and efficient algorithm for computing the
  optimal un

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4. Steering (cont.)

• Steering Algorithm

• given error tolerance ?
• while J gt ?
• forward propagate the zn
• backward propagate the pn
• compute the ?J/?un
• update the control vectors un
• compute J

• Unconstrained search in the controls un only
• More efficient and accurate than numerically
  computing ?J/?un
• Any standard gradient search technique may be
  applied

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4. Steering (cont.)

• Simulation Results
• Assume full state knowledge at each state
• state estimator not implemented
• Four cases
• Two cases with terminal constraints only (Q 0)
• Two cases with both distributed and terminal
  constraints
• Gradient search
• Polak-Ribiere variant of conjugate-gradients for
  search direction
• Armijos rule for search length
• Bisection method for optimal un

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4. Steering (cont.)

• Simulation Results cases 1 and 2

Parameters xi 3 mm xf 0 mmxi 0
mrad xf 0 mrad
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4. Steering (cont.)

• Simulation Results cases 3 and 4

Parameters xi 3 mm xf 0 mmxi 0
mrad xf 0 mrad
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5. Summary and Conclusions

• State Estimation
• Use several measurements to construct single
  state
• Misalignments are limiting factor
• BPM noise averages out
• Least-squares technique accurate for
  misalignments less than 250 ?m
• Misalignment parameter estimation under
  investigation

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5. Summary and Conclusion

• Steering Algorithm
• Considers beam behavior between measurement
  locations
• Supports variety of steering objectives via
  tuning parameters
• Dual of response matrix approach
• back-propagate steering errors back to actuator
  rather than actuator response to beam positions
• Requires a model reference
• For definition of cost functionals
• For full beam state at BPM locations (state
  estimator)

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5. Current and Future Work

• Simulate combined state estimation and steering
  algorithm
• Develop method for improving recursive state
  estimation
• Some aspect of misalignment parameter estimation?
• Apply similar techniques for beam shaping