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

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

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

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

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

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

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

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

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

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

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

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

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

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 + SM 0/2 BPM hn zn+1 zn un stage n schematic Sept 30, 2004 C.K. Allen

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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