1. Christopher K. Allen Los Alamos National Laboratory
Automated Beam Steering Using Model Reference Control and Optimal Control
Christopher K. Allen Los Alamos National Laboratory

2. Outline Overview Beam Dynamics Model State Estimation Steering
Motivation
Basic Approach
Beam Dynamics Model
State Estimation
Steering
Summary and Conclusion
Sept 30, 2004
C.K. Allen

3. 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*
Sept 30, 2004
C.K. Allen

4. 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) k=1,2,… with differing corrector strengths
Sept 30, 2004
C.K. Allen

5. 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 (u;s) of the beamline
Sept 30, 2004
C.K. Allen

6. 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
Sept 30, 2004
C.K. Allen

7. 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.
Sept 30, 2004
C.K. Allen

8. 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) 1]T
The action of stage n is represented by a transfer matrix n(un)
zn+1 = 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
Sept 30, 2004
C.K. Allen

9. 2. Dynamics (cont.) 0 1 N-1 2 h0 h1 h2 hN-1
multistage control network 0 1
N-1 2 z0 z1 z2 z3
zN-1 zN u0 u1 u2
uN-1 h0 h1 h2
hN-1
{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
Sept 30, 2004
C.K. Allen

10. 2. Dynamics (cont.) 0 1
N-1 2 z0 z1 z2 z3
zN-1 zN u0 u1 u2
uN-1 h0 h1 h2
hN-1
Dynamics with noise {nn} and 1st order misalignments {n}
{nn} – “white noise” Wiener process
{n} – “generator” matrix for misalignment (translation + rotation)
Sept 30, 2004
C.K. Allen

11. 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 h1…hN-1)T and u  (u0 u1…uN-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 vector Hn – “measurement” matrix (model dependent) zn – state vector at stage n n – vector of measurement unknowns
Sept 30, 2004
C.K. Allen

12. 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
Sept 30, 2004
C.K. Allen

13. 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 transpose Wn – weighting matrix for stage n h – measurement vector
Sept 30, 2004
C.K. Allen

14. 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 = cm Lquad = 6.10 cm
kquad = 90 deg + SM
0/2
BPM hn
zn+1 zn un
stage n schematic
Sept 30, 2004
C.K. Allen

15. 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
Sept 30, 2004
C.K. Allen

16. 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
Sept 30, 2004
C.K. Allen

17. 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
Sept 30, 2004
C.K. Allen

18. 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
Sept 30, 2004
C.K. Allen

19. 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(un;s)zn
Sept 30, 2004
C.K. Allen

20. 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
Sept 30, 2004
C.K. Allen

21. 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
Sept 30, 2004
C.K. Allen

22. 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
Sept 30, 2004
C.K. Allen

23. 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
Sept 30, 2004
C.K. Allen

24. 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
Sept 30, 2004
C.K. Allen

25. 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}
Sept 30, 2004
C.K. Allen

26. 4. Steering (cont.) Steering Algorithm
given error tolerance 
while J >  :
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
Sept 30, 2004
C.K. Allen

27. 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
Armijo’s rule for search length
Bisection method for optimal un
Sept 30, 2004
C.K. Allen

28. 4. Steering (cont.) Simulation Results – cases 1 and 2 Parameters
xi = 3 mm xf = 0 mm xi’ = 0 mrad xf’ = 0 mrad
Sept 30, 2004
C.K. Allen

29. 4. Steering (cont.) Simulation Results – cases 3 and 4 Parameters
xi = 3 mm xf = 0 mm xi’ = 0 mrad xf’ = 0 mrad
Sept 30, 2004
C.K. Allen

30. 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
Sept 30, 2004
C.K. Allen

31. 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)
Sept 30, 2004
C.K. Allen

32. 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
Sept 30, 2004
C.K. Allen

33. Abstract
We present a steering algorithm which, with the aid of a model, allows the user to specify beam behavior throughout a beamline, rather than just at specified beam position monitor (BPM) locations. The model is used primarily to compute the values of the beam phase vectors from BPM measurements, and to define cost functions that describe the steering objectives. The steering problem is formulated as constrained optimization problem; however, by applying optimal control theory we can reduce it to an unconstrained optimization whose dimension is the number of control signals.
Sept 30, 2004
C.K. Allen

34. 4. Steering (cont.) Remarks Functional Jn(zn,un) is quadratic in zn
There is a unique minimum zn* for each un
The matrix Q determines the form of the solution zn*
Chosen by the operator to specify objective
Can specify objectives on beam momentum
Functional Jn(zn,un) contains contributions of beam behavior throughout stage n
Avoid “over correcting”
Sept 30, 2004
C.K. Allen