Design of Multi-Array Controllers for Electron Beam Stabilisation on ...
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
Design of Multi-Array Controllers for Electron Beam Stabilisation on Synchrotrons Sandira Gayadeen1 , Stephen R. Duncan1 and William P. Heath2 Abstract— This paper considers the design of electron beam stabilisation control systems for synchrotrons that simultaneously determine the inputs to multiple arrays of actuators for regulating variations in the electron beam position. An approach based on determining the interaction of the controllable subspaces is used. This enables the control problem to be decomposed into a series of single-input, single-output; multiple-input, single output and multiple-input, multipleoutput problems. An Internal Model Control structure is used to design the controllers for each case and mid-ranging control is proposed for the case where the control directions align. Results from a simulation study using machine data from the Booster synchrotron of the UK’s national synchrotron facility, Diamond Light Source are presented.
I. INTRODUCTION At synchrotron radiation facilities, electrons generated by an electron gun are accelerated until they reach the nominal energy (≈ 3 GeV for medium energy machines and ≈ 8 GeV for high energy machines), before being transferred into a storage ring. Storage rings consist of straight sections connected through a series of bends to form a closed ring. For example, the storage ring at Diamond Light Source, Oxfordshire, UK has 24 sections which create a ring of 561.6 m in circumference [1]. Large bending magnets (dipoles) are used to curve the electron beam between adjacent straight sections, and as the electron beam passes through a bending magnet, it emits a wide fan of synchrotron radiation that is channeled into a photon beamline. This synchrotron radiation spans the electromagnetic spectrum from infrared through visible and ultra violet light to X-rays. The storage rings of modern synchrotron radiation facilities, such as Diamond, also use arrays of dipole magnets called Insertion Devices that cause the electron beam to oscillate along their length and so provide a source of synchrotron radiation that can be tuned by varying the magnetic field. Although the position of the electron beam is maintained by the magnetic fields within the storage ring, the electron beam is subjected to disturbances from environmental effects that are coupled through the girders supporting the magnets. Several external disturbances cause electron beam perturbations, ranging from long-term disturbances, such as changes in air temperature during the day, to re-injections of the electron beam, which are typically at frequencies lower *S.G. was supported by Diamond Light Source 1 S. Gayadeen and S.R. Duncan are with Department of Engineering Science, University of Oxford, Parks Road, OX1 3PJ, Oxford, UK.
(s.gayadeen,s.duncan)@eng.ox.ax.uk 2 W.P. Heath is with the Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, Manchester, M13 9PL, UK. [email protected]
978-1-4799-0176-0/$31.00 ©2013 AACC
than 0.01 Hz. Most disturbances arise from ground vibrations and residual disturbances after feedforward correction of Insertion Device field changes and lie within the range 0.01 Hz to 100 Hz [2]. Above 100 Hz, the effects of vibration arising from cooling water pumps and girder cooling water flow resonances appear in the beam spectra [3]. Modern synchrotron radiation facilities are built for producing high brilliance photon beams, which is achieved by reducing the emittance in both planes. The vertical beam size defines the emittance, which leads to requirements on the electron beam stability [2]. In order to achieve optimum performance, electron beam stability is a crucial parameter for third generation light sources. In particular, sub-micron stability is now a common requirement for the vertical position of the beam. At Diamond, the electron beam must be controlled to within 10% of the beam size, which corresponds to an RMS variation less than 12.3 µm in the horizontal plane and 0.6 µm in the vertical plane [4]. To achieve such performance, electron beam stabilisation feedback systems are used. Some synchrotron machines use a single set of corrector magnets for control system actuation, while others use two sets of correctors, strong and fast. Historically correctors with strong magnetic fields were used for slow correction but the use of these correctors located over high conductivity vacuum chambers produce eddy currents that limit the feedback bandwidth to a few Hertz [2]. Therefore a set of fast correctors is required to compensate for high frequency disturbances. The control problem at these facilities is the design of two global feedback systems that work together. The common approach has been to design the two feedback controllers separately, but when the controllers share a common frequency range, they fight against each other, resulting in the faster actuators saturating. Several approaches are used to deal with this problem, one of which is the use of a frequency deadband to separate the fast and slow systems so that the two systems are independent [5]. However, with this approach, the beam spectrum components within the deadband are not corrected. This is undesirable as there are components of the disturbance that cannot be left without correction in this range, such as transient orbit distortions from Insertion Devices, which change the magnetic field as part of an experiment. Another approach is to use the fast feedback system alone with an active frequency range down to DC. This approach solves the uncorrected perturbations at low frequencies, but in some cases this presents a limitation due to the relative weakness of fast correctors. Because the fast correctors are designed to
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regulate small perturbations at high speeds and therefore have limited amplitude ranges, the actuators tend to saturate [2]. In some machines, this is compensated for by using the slow correctors to periodically remove the DC components of the fast correctors so that the fast actuators do not saturate even after long term drift of the closed orbit [6]. Other solutions seek to use the slow system to predict the different orbit at the next iteration and transfer it as a new reference to the fast system. As a consequence, the fast system will not see the perturbation created by the slow correction and will not try to compensate for it [5],[7]. However this approach has caused instabilities on systems where the fast correctors are different from the slow ones [2]. Some machines combine all these approaches. In this paper, the beam stabilisation systems are considered to be analogous to cross-directional processes that are common in process industries such as paper machines, plastic film extrusion and metal rolling [8],[9],[10],[11],[12]. The term cross-directional describes systems where it is required to control variations of a measured variable in a profile that is orthogonal to the direction of propagation of the variable. A cross-directional control design approach is appropriate for synchrotron applications because there is considerable interaction between the spatial responses of each of the magnets, and because this spatial response is decoupled from the dynamic response of the actuators. Similar to single-array cross-directional control, several control designs based on large-scale optimisation have been used to handle multi-array cross-directional problems which uses long horizon model predictive control (MPC) [13]. The size of the control problem in industrial cross-directional systems is usually overcome by reducing the model size or using a fast quadratic programming algorithm [14]. However, because of the large dimensions of synchrotron feedback systems with sample rates of 10 kHz, robust constrained MPC is computationally demanding and will extend the time delay, hence reducing closed loop bandwidth. In [15],[16] a method of exploiting the knowledge of the spatial responses of the arrays of actuators is used to determine the interaction between the controllable subspaces in order to decompose the problem into a series of single-input, single-output (SISO), multiple-input, singleoutput (MISO) and multiple-input, multiple-output (MIMO) problems. For example, for spatial modes where the control directions are orthogonal, the arrays are decoupled and SISO control design can be used. However where the control directions align, the choice of control direction is not straightforward. In this paper, the approach in [16] is extended to consider the case where the controllable subspaces of the arrays overlap and mid-ranging control is proposed as a useful solution. The term mid-ranging control is used to describe a control problem with two actuators but one measured variable where an additional specification is used to return one of the actuators to a setpoint. In some cases of two-input, single-output problems, it may be possible to manipulate one input at a time, but in this application it is desirable
to simultaneously manipulate both inputs as the strong correctors are bandwidth limited. An important characteristic of mid-ranging applications is that input constraints are often encountered [17] which is the case with most fast corrector magnets. There are several approaches to designing mid-ranging controllers [17], [18] but in [19] an Internal Model Control (IMC) structure is used which is adopted in this paper. An IMC approach is appropriate for synchrotron feedback systems which are subject to significant delays and is consistent with previous designs of single array controllers for this application in [3], [20], [21]. More recently in [22] a dual-mode MPC approach based on sum-of-norms regularisation is used to develop a smart allocation policy to use the least number of actuators. In the approach of [22], actuators are set to zero for extensive periods. For the current application such an approach is inappropriate, as it is desirable to maximise the use of actuators since this ensures that the maximum DC correction is achieved. Framing the control problem as an optimal control problem may not take advantage of the different actuator characteristics such as dynamics and gain whereas the IMC approach to the multi-array problem, including the use mid-ranging control, has the benefit of not only reducing computation over MPC in general but allowing explicit design trade-offs. The paper is structured as follows. In Section II the multiarray controller structure for the Diamond Light Source Booster synchrotron is presented. In Section III, a design for decoupling the control problem is presented. The stability of the system is considered in Section IV. In Section V the Booster synchrotron controller design and results are presented and conclusions are drawn in Section VI. II. C ONTROLLER S TRUCTURE For two actuator arrays with different dynamics, the nominal plant model for the system at Diamond can be described by, y(k) = z−d1 g1 (z−1 )B1 u1 (k) + z−d2 g2 (z−1 )B2 u2 (k) + d(k) (1) where y(k) ∈ Rm represents the signals measured at M sensors at times {t = kTs : k ∈ Z+ } with Ts being the sample interval so that y(k) = y(kTs ). The process disturbance is represented by d(k) ∈ Rm and u1 (k) ∈ Rr1 and u2 (k) ∈ Rr2 represent the inputs applied to the two distinct arrays of actuators. It is assumed that the all the actuators of a given array have the same dynamics. The discrete transfer function representing the actuator and process dynamics are represented by z−d1 g1 (z−1 ) and z−d2 g2 (z−1 ) where (without loss of generality) g1 (1) = 1 and g2 (1) = 1 i.e. the dc gains of the dynamics can be taken as unity. The static response for each array is represented by B1 and B2 where r1 = rank(B1 ) and r2 = rank(B2 ) and r1 ≥ r2 . Therefore B1 and B2 can be expressed in terms of reduced singular value decompositions so that,
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B1 = Φ1 Σ1 ΨT1
and
B2 = Φ2 Σ2 ΨT2
(2)
𝚽𝟏 𝑻
𝚺𝟏 −𝟏 +
𝑸𝟏 (𝑧 −1 )
𝚿𝟏
𝚿𝟏 𝑻
𝒈𝟏𝒊 (𝑧 −1 )
𝚺𝟏
𝚽𝟏
𝑑(𝑘)
Array 1 𝑟(𝑘)
𝒈𝟏𝒊 (𝑧 −1 )
𝐗𝐇
−
+
𝑦(𝑘)
+
𝚺𝟐 −𝟏
𝚽𝟐 𝑻
+
𝐗𝑻
𝑸𝟐 (𝑧 −1 )
+
𝚿𝟐
𝚿𝟐 𝑻
𝒈𝟐𝒊 (𝑧 −1 )
𝚺𝟐
𝚽𝟐
Array 2 𝒈𝟐𝒊 (𝑧 −1 )
Fig. 1.
Internal Model Control structure for multi-array problem
where Φ1 ∈ Rm×r1 , Σ1 ∈ Rr1 ×r1 , Ψ1 ∈ Rr1 ×r1 , Φ2 ∈ Rm×r2 , Σ2 ∈ Rr2 ×r2 , Ψ2 ∈ Rr2 ×r2 . Even though B1 and B2 are independent, there may be some overlap between the controllable subspaces. The relationship between the subspaces of B1 and B2 can be found by comparing the subspaces of Φ1 and Φ2 which are orthogonal and can be determined from algorithm 12.4.3 in [23] where ΦT1 Φ2 = C
(3)
and the singular value decomposition of C is given by S T C =Y Z 0
(4)
where Y ∈ Rr1 ×r1 , Z ∈ Rr2 ×r2 and S ∈ Rr2 ×r2 such that S = diag{cos(θ1 ), . . . , cos(θr2 } where θi is the angle between the principal vectors spanning Φ1 and Φ2 which span the controllable subspaces of B1 and B2 . Preserving this structure for a single array of actuators in [10], [20], the closed loop for a multi-array system is shown in Fig. 1 so that the closed loop response can be described by, y(k) = g1 (z−1 )B1 Q˜ 1 (z−1 ) + g2 (z−1 )B2 Q˜ 2 (z−1 ) r(k)+ . I − g1 (z−1 )B1 Q˜ 1 (z−1 ) − g2 (z−1 )B2 Q˜ 2 (z−1 ) d(k) (5) For the current regulator problem r(k) is set to zero. From (5), the closed loop response from d(k) to y(k) can be written as, g1 (z−1 )Σ1 ΨT1 Q˜ 1 (z−1 ) y(k) = I − [Φ1 Φ2 ] d(k) (6) g2 (z−1 )Σ2 ΨT2 Q˜ 2 (z−1 ) where −1 Q˜ 1 (z−1 ) = Ψ1 Σ−1 1 Q1 (z )F1
(7)
and −1 Q˜ 2 (z−1 ) = Ψ2 Σ−1 2 Q2 (z )F2
(z−1 )
(z−1 )
(8) ∈ Rr1 ×m
for some diagonal Q1 and Q2 and for F1 and F2 ∈ Rr2 ×m . Therefore (6) can be expressed as g1 (z−1 )Q1 (z−1 )F1 y(k) = I − [Φ1 Φ2 ] d(k). g2 (z−1 )Q2 (z−1 )F2
If the diagonal transfer function matrices Q1 (z−1 ) and Q2 (z−1 ) are chosen to have unity dc gain, then for steady state T yss = I − [Φ1 Φ2 ] 1 dss (10) T2 Similar to [20], pre-multiplying both sides by [Φ1 Φ2 ]T , projects the response into “modal space” so that I ΦT1 Φ2 H11 H12 y˜ss = I − T d˜ss (11) H21 H22 Φ2 Φ1 I and for some decoupling matrix, H such that H11 ∈ Rr1 ×r1 , H12 ∈ Rr1 ×r2 , H21 ∈ Rr2 ×r1 and H22 ∈ Rr2 ×r2 and where T T Φ1 Φ1 ˜ y˜ss = y and dss = d (12) ΦT2 ss ΦT2 ss with
T F1 Φ = H 1T . F2 Φ2
(13)
III. MULTI-ARRAY DESIGN HEURISTIC From (11), let M=
I ΦT2 Φ1
ΦT1 Φ2 I
where from (3) and (4), M can be written as I 0 S M = X 0 I 0 X T S 0 I where X is a diagonal the decoupling matrix H = M −1 which gives, D H¯ = 0 0
(14)
(15)
matrix X = diag{Y, Z}. In principle, H can then be designed by setting ¯ T where H¯ ∈ Rm×m is H = X HX 0 0 I 0 −S I 0 0 I 0 (16) 0 D −S 0 I
and D ∈ Rr1 ×r2 is a diagonal matrix with elements
(9) 1205
[D]i j =
1 . 1 − cos2 (θi )
(17)
When cos(θi ) = 0, the directions Φ1 (:, i) and Φ2 (:, i) are orthogonal so that the system is considered as a decoupled two-input, two-output system i.e. two SISO structures. The matrix H¯ gives this automatically where the corresponding elements of H¯ 11 and H¯ 22 are set to 1 and H¯ 12 and H¯ 21 set to 0. Similarity, when 0 < cos(θi ) < 1, H¯ automatically splits the effort between the two actuators. However there are some special cases to be considered when the choice of control direction is no longer straightforward: 1) When cos(θi ) = 1, the directions Φ1 (:, i) and Φ2 (:, i) align, giving a two-input, single-output structure. In this case mid-ranging control is appropriate due to the actuator characteristics. However difficulties arise with the use of this formulation for the decoupling matrix, so instead set corresponding elements of H¯ 11 and H¯ 22 to 1 and the corresponding elements of H¯ 12 and H¯ 21 to 0 to control in both directions and use mid-ranging control. 2) When cos(θi ) is close to 0, then the choice to control in a single direction is appropriate. 3) When cos(θi ) is close to 1, then the choice is to use both directions for control and use mid-ranging as when cos(θi ) = 1. IV. S TABILITY OF THE MULTI - ARRAY DESIGN Suppose that the true plant behaviour is given by −1 y(k) =z−d1 g1 (z−1 )B1 + ∆Φ 1 (z ) u1 (k)+ −1 z−d2 g2 (z−1 ) B2 + ∆Φ 2 (z )u2 (k) + d(k)
(18)
−1 Φ −1 where ∆Φ 1 (z ) and ∆2 (z ) are additive uncertainties in the response of the two arrays. Then the controller is robustly stable provided
Q˜ 1 (z−1 ) Φ −1 Φ −1
(19)
Q˜ 2 (z−1 ) ∆1 (z ) ∆2 (z ) < 1 ∞
From (7) and (8), T Q˜ 1 (z−1 ) Ψ1 Σ−1 Q1 (z−1 ) 0 Φ1 1 = −1 ) H ΦT Q˜ 2 (z−1 ) 0 Ψ2 Σ−1 Q (z 2 2 2 (20) so the uncertainty can be expressed as Φ −1 l −1 −1 ∆ (z ) ∆Φ 2 (z ) = Φ1 Φ2 W (z ). 1 −1 T (21) 0 ∆11 (z ) ∆12 (z−1 ) r −1 Ψ1 W (z ) 0 ΨT2 ∆21 (z−1 ) ∆22 (z−1 ) where W l (z−1 ) = diag{W1l (z−1 ),W2l (z−1 )} and W r (z−1 ) = diag{W1r (z−1 ),W2r (z−1 )} with ∆11 (z−1 ) ∈ Rr1×r1 , ∆12 (z−1 ) ∈ Rr1×r2 , ∆21 (z−1 ) ∈ Rr2×r1 and ∆22 (z−1 ) ∈ Rr2×r2 representing the plant uncertainty lying within the subspace that is controllable by the combination of the two actuator arrays. The weights Wl (z−1 ) and Wr (z−1 ) are known matrices that characterize the uncertainty in the modal space [16]. Using these decompositions, the robust stability condition becomes
−1
Σ Q˜ 1
0 l ∆11 ∆12 r
1
˜ 1 HMW ∆21 ∆22 W < 1 (22) 0 Σ−1 Q 2 ∞
TABLE I VALUES OF PARAMETERS USED IN DESIGN Parameter a τd
Array 1 2π×200 rad.s−1 400 µs
Array 2 2π×1000 rad.s−1 700 µs
so for any uncertainty satisfying
∆11 ∆12
∆21 ∆22 < 1 ∞
(23)
the closed loop system will be robustly stable provided that
−1
r Σ Q˜ 1
0 l 1
W (24) HMW < 1. −1 ˜
0 Σ 2 Q1 ∞ V. B OOSTER SYNCHROTRON CASE STUDY The Booster synchrotron at Diamond is used to ramp the energy of the electrons to the nominal 3 GeV before transferring the electrons into the storage ring. The Booster synchrotron, which has a circumference of 158 m, is a substantial accelerator in its own right that has the same hardware as the storage ring for beam position detection and control of the corrector magnets. Closed loop beam orbit can therefore be applied on the Booster during the acceleration ramp or by running the Booster as an electron storage ring at 100 MeV. The benefit of having closed orbit correction on the Booster is the improvement of beam position stability over long periods of time and hence operational efficiency. The feedback controller uses 22 horizontal and 22 vertical corrector magnets in total which correct the electron beam position when subjected to disturbances. The disturbances on the Booster synchrotron are mainly between 10 to 30 Hz with a peak around 50 Hz, which are associated with one of the resonant modes of the girders, so a key requirement of the control design is to regulate the horizontal and vertical position of the electron beam in the presence of disturbances in the range 1 to 100 Hz, which includes the first two harmonics of the girder disturbance. The process models for the vertical plane of each array are given by g(1,2) (z−1 ) = z−d(1,2)
b0(1,2) + b1(1,2) z−1 1 − a1(1,2) z−1
(25)
where d(1,2) is the smallest integer satisfying d(1,2) Ts > τd and a1(1,2) = e−a(1,2) Ts b0(1,2) = 1 − e
0 −a(1,2) (Ts −τ(1,2) )
0 −a(1,2) (Ts −τ(1,2) )
b1(1,2) = e
(26)
− e−a(1,2) Ts
when τd(1,2) is the delay in the system, a(1,2) is the bandwidth 0 of the actuator response (in rad.s−1 ) and τ(1,2) = τd(1,2) − (d(1,2) − 1)Ts [20]. The sample rate for the feedback system is Ts = 100 µs. The parameters in (25) for each array are given in Table I, so that g2 (z−1 ) has faster dynamics than g1 (z−1 ). The Booster has M = 22 sensors in each plane and r1 = 17 and r2 = 5, so that B1 ∈ R22×17 and B2 ∈ R22×5 . The controllable subspaces of B1 and B2 are determined from (3)
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TABLE II A NGLES BETWEEN CONTROLLABLE SUBSPACES OF B1 AND B2
cos(θi )
i=1 0.99
i=2 0.98
i=3 0.91
i=4 0.59
i=5 0
and (4) and Table II shows the cosine of the angles between the first 5 columns of Φ1 and Φ2 . A. Two-input, single-output controller design From Table II, the angle between Φ1 (:, i) and Φ2 (:, i) for i = 1 to 3 are close to 1, so the system can be approximated as a two-input, single output structure and mid-ranging control is used. The mid-ranging problem within the IMC structure in [19] and [24] is used so that the closed loop dynamics are described by (5). However by setting H¯ 11 and H¯ 22 to 1 and H¯ 12 and H¯ 21 set to 0, the closed loop dynamics for the twoinput, one-output problem can be written as can be written as, y(k) = g1 (z−1 )q1 (z−1 ) + g2 (z−1 )q2 (z−1 ) r(k)+ I − g1 (z−1 )q1 (z−1 ) − g2 (z−1 )q2 (z−1 ) d(k)+ (27) g1 (z−1 )p1 (z−1 ) + g2 (z−1 )p2 (z−1 ) ur (k) where the pre-filters p1 (z−1 ) and p2 (z−1 ) are designed to obtain a desired response from the reference signal on the faster actuator, ur (k) to y(k) as in [19]. The dynamics of g2 (z−1 ) are faster than that of g1 (z−1 ), so the following is defined, T f (z−1 ) = g1 (z−1 )q1 (z−1 ) + g2 (z−1 )q2 (z−1 ) Ts (z−1 ) = g1 (z−1 )q1 (z−1 )
(28)
and for steady state behaviour in mid-ranging control, it is required that g1 (1)q1 (1)+g2 (1)q2 (1) = 1. If it is desired that one of the actuators (typically the faster one) is zero in steady state then q2 (1) = 0. The slow closed loop transfer function Ts (z−1 ) is selected so that Ts (z−1 ) = Ts− (z−1 )T˜s (z−1 ) where Ts− (z−1 ) includes the non-minimum phase components (including delays) of both g1 (z−1 ) and g2 (z−1 ). Then for i = 1 to 3, q1i (z−1 ) = Ts (z−1 )g1 (z−1 ). Likewise T f (z−1 ) is selected so that T f (z−1 ) = T f− (z−1 )T˜ f (z−1 ) where T f− (z−1 ) includes the non-minimum phase components (including delays) of g2 (z−1 ) only. Then for i = 1 to 3, q2i (z−1 ) = [T f (z−1 ) − Ts (z−1 )]/g2 (z−1 ). B. Single-input, single-output or decoupled two-input, twooutput controller design For i = 4, one control direction can be chosen or the effort split between the both directions using H. For i = 5, the directions are decoupled so both directions are used. For i = 6 to 17 a single SISO structure is used where array 1 actuators are used. In the IMC structure, the pseudo plant inverse dynamics can be selected as 1 − λ1i 1 − a11 i z−1 1 − λ1i z−1 b01 i + b11 i 1 − λ2i 1 − a12 i z−1 q2i (z−1 ) = 1 − λ2i z−1 b02 i + b12 i q1i (z−1 ) =
i = 4, . . . , 17 (29) i = 4, 5
Fig. 2. Vertical beam displacement y(k) for a step change in disturbance d(k) with no mid-ranging (dashed, red line) and with mid-ranging (solid, blue line) for i = 1. The inputs u1 (k) and u2 (k) and references r(k) = 0 and ur (k) = 0 (dot-dashed, black line). are shown for each case.
with λ(1,2)i = e−ζi Ts where ζi is a tuning parameter that can be different for each mode [20]. C. Simulation Results In Fig. 2 the performance of the mid-ranging controllers to a step change in the disturbance is compared to using two SISO controllers for i = 1. The mid-ranging controllers are able to return the fast input to its reference while not overshooting its steady state value. In Fig. 3 real machine data is used to simulate the performance of the orbit feedback controller by considering the integrated disturbance over frequency. For this simulation mid-ranging controllers are used for i = 1 to 3. For frequencies below 100 Hz the controller is able to suppress the beam motion to within 10 % beam size specification. Suppressing the beam motion in the lower frequencies results in an amplification at higher frequencies (above 500 Hz), but a major component of disturbances at these frequencies is considered to be sensor noise from the sensors and not as a result of beam motion. For the Booster synchrotron the main source of uncertainty is in the spatial response resulting from slow drifts of the acceleration gradient [25]. The response matrix measured off the machine was compared with the ideal response matrix to −1 Φ −1 determine ∆Φ 1 (z ) and ∆2 (z ). Using the formulation in (21), it is found for this design that
−1
Σ Q˜ 1
0 l ∆11 ∆12
1 HMW W r −1 ˜
= 0.13 (30) ∆21 ∆22 0 Σ 2 Q1 ∞ indicating that the closed loop system is guaranteed stable in the presence of spatial uncertainties. VI. CONCLUSIONS This paper has considered the design of an electron beam stabilisation control system for synchrotrons where multiple arrays of actuators are used to regulate the electron beam position in synchrotron machines. In this paper a method to determine the interaction of the actuator arrays is proposed. By considering the angles between the overlapping
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Fig. 3. Uncontrolled integrated beam motion at first vertical sensor (dashed, red line) and controlled integrated beam motion for the multi-array design using mid-ranging controllers (solid, blue line)
controllable subspaces of the actuator arrays, an appropriate choice of control direction can be made. The design of the decoupling matrix presented in this paper automatically selects the appropriate control direction for most cases. For the case where the controllable subspaces align, the control problem becomes a two-input, one-output structure for which mid-ranging control is appropriate. Rather than choosing one control direction and thereby not using an actuator, the mid-ranging approach splits the effort between the two actuators, making use of the different actuator characteristics. Using all actuators is advantageous because the maximum DC correction can be achieved. The mid-ranging structure also allows a reference to be placed on the faster input to avoid actuator saturation. In this paper the Internal Model Control structure used for single array controllers is extended for the multi-array approach which is also appropriate for mid-ranging control. This approach to multi-array problems on synchrotrons for electron beam stabilisation improves on current approaches which design independent controllers without taking into consideration their interaction. VII. ACKNOWLEDGMENTS The authors would like to thank Diamond Light Source for supporting this research and for contributing to the experimental work carried out on the Booster. The authors would also like to thank Nicolas Hubert at Soleil Synchrotron for insightful discussions on multi-array control for synchrotrons. R EFERENCES [1] R. Bartolini, “The commissioning of the Diamond Storage Ring,” in 22nd Particle Accelerator Conf., Albuquerque, New Mexico, 2007. [2] N. Hubert, L. Cassinari, J.-C. Denard, A. Nadji, and L. Nadolski, “Global orbit feedback systems down to DC using fast and slow correctors,” in 9th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators, Basel, Switzerland, 2009. [3] A. Napier, S. Gayadeen, and S. R. Duncan, “Fast orbit beam stabilisation for a synchrotron,” in IEEE Int. Conf. on Control Applicat., vol. 47, Denver, CO, 2011, pp. 1770–1775.
[4] J. Rowland, M. G. Abbott, J. A. Dobbing, M. T. Heron, I. Martin, G. Rehm, and I. Uzun, “Status of the Diamond fast orbit feedback system,” in Int. Conf. on Accelerator and Large Experimental Physics Control Systems, Knoxville, Tennessee, 2007. [5] C. Schwartz and L. Emery, “Compensating the frequency deadband of the APS real-time and DC transverse orbit correction systems,” in Proc. of Particle Accelerator Conf., 2001, vol. 2, 2001, pp. 1234 –1236. [6] L. Yu, S. Krinsky, O. Singh, and F. Willeke, “The performance of a fast closed orbit feedback system with combined fast and slow correctors,” in 11th European Particle Accelerator Conf., Genoa, Italy, 2008. [7] C. Steier, E. Domning, T. Scarvie, and E. Williams, “Operational experience integrating slow and fast orbit feedbacks at the ALS,” in 9th European Particle Accelerator Conf., Lucerne, Switzerland, 2004. [8] G. Goodwin, B. Carny, and W. Edwards, “Analysis of thermal camber control in rolling mills,” in Proc. of the IFAC World Congr., Tallinn, USSR, 1990, pp. 160–164. [9] S. R. Duncan, “The design of robust cross-directional control systems for paper making,” in Proc. of the American Control Conf. 1995, vol. 3, Seattle, WA, 1995, pp. 1800 –1805. [10] W. P. Heath, “Orthogonal functions for cross-directional control of web forming processes,” Automatica, vol. 32, no. 2, pp. 183–198, 1996. [11] S. R. Duncan and G. F. Bryant, “The spatial bandwidth of crossdirectional control systems for web processes,” Automatica, vol. 33, no. 2, pp. 139–153, 1997. [12] P. E. Wellstead, M. B. Zarrop, and S. R. Duncan, “Signal processing and control paradigms for industrial web and sheet manufacturing,” Intl. J. Adaptive Control and Signal Processing, vol. 14, no. 1, pp. 51–76, 2000. [13] J. Backstr¨om, C. Gheorghe, G. Stewart, and R. Vyse, “Constrained model predictive control for cross-directional multi-array processes,” Pulp and Paper Canada, vol. 102, pp. T128–T131, 2001. [14] J. Fan, G. E. Stewart, and G. A. Dumont, “Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional processes,” Automatica, vol. 40, no. 11, pp. 1891–1903, 2004. [15] S. R. Duncan, W. P. Heath, and A. Taylor, “A new approach to multiarray cross-directional control,” in Proc. PAPTAC Control Systems 2008 Conf., Vancouver, BC, Canada, June 2008, pp. 69–74. [16] S. R. Duncan and W. P. Heath, “The robustness of multi-array crossdirectional control systems,” in Proc. Control Systems 2010 Conf. 2010, Stockholm, Sweden, September 2010, pp. 180–185. [17] B. Allison and S. Ogawa, “Design and tuning of valve position controllers with industrial applications,” Transactions of the Institute of Measurement and Control, vol. 25, pp. 3–16, 2003. [18] M. A. Henson, B. A. Ogunnaike, and J. Schwaber, “Habituating control strategies for process control,” AIChE Journal, vol. 41, pp. 604–618, 1995. [19] S. Gayadeen and W. P. Heath, “An Internal Model Control approach to mid-ranging control.” ADCHEM 2009, IFAC International Symposium on Advanced Control of Chemical Processes, Koc¸ University, Istanbul, July 12-15, 2009. [20] S. R. Duncan, “The design of a fast orbit beam stabilisation system for the Diamond synchrotron,” Dept. of Eng. Sci., University of Oxford, Tech. Rep. 2296/07, 2007. [21] S. Gayadeen, S. R. Duncan, C. Christou, M. T. Heron, and J. Rowland, “Diamond Light Source Booster fast orbit feedback system,” in ICALEPCS 2011, Grenoble, France, October 2011. [22] M. Gallieri and J. Maciejowski, “`asso MPC: Smart regulation of overactuated systems,” in Proc. of American Control Conference (ACC), 2012, june 2011, pp. 1217 –1222. [23] G. H. Golub and C. F. Van Loan, Matrix computations (3rd ed.). Baltimore, MD, USA: Johns Hopkins University Press, 1996. [24] W. P. Heath and S. Gayadeen, “Simple robustness measures for control of MISO and SIMO plants,” in 18th IFAC World Congress, vol. 18, Milano, Italy, Aug 2011, pp. 8176 –8181. [25] S. Gayadeen and S. R. Duncan, “Uncertainty modeling and robust stability analysis of a synchrotron electron beam stabilisation control system,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on, 2012, pp. 931–936.
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Design of Multi-Array Controllers for Electron Beam Stabilisation on Synchrotrons Sandira Gayadeen1 , Stephen R. Duncan1 and William P. Heath2 Abstract— This paper considers the design of electron beam stabilisation control systems for synchrotrons that simultaneously determine the inputs to multiple arrays of actuators for regulating variations in the electron beam position. An approach based on determining the interaction of the controllable subspaces is used. This enables the control problem to be decomposed into a series of single-input, single-output; multiple-input, single output and multiple-input, multipleoutput problems. An Internal Model Control structure is used to design the controllers for each case and mid-ranging control is proposed for the case where the control directions align. Results from a simulation study using machine data from the Booster synchrotron of the UK’s national synchrotron facility, Diamond Light Source are presented.
I. INTRODUCTION At synchrotron radiation facilities, electrons generated by an electron gun are accelerated until they reach the nominal energy (≈ 3 GeV for medium energy machines and ≈ 8 GeV for high energy machines), before being transferred into a storage ring. Storage rings consist of straight sections connected through a series of bends to form a closed ring. For example, the storage ring at Diamond Light Source, Oxfordshire, UK has 24 sections which create a ring of 561.6 m in circumference [1]. Large bending magnets (dipoles) are used to curve the electron beam between adjacent straight sections, and as the electron beam passes through a bending magnet, it emits a wide fan of synchrotron radiation that is channeled into a photon beamline. This synchrotron radiation spans the electromagnetic spectrum from infrared through visible and ultra violet light to X-rays. The storage rings of modern synchrotron radiation facilities, such as Diamond, also use arrays of dipole magnets called Insertion Devices that cause the electron beam to oscillate along their length and so provide a source of synchrotron radiation that can be tuned by varying the magnetic field. Although the position of the electron beam is maintained by the magnetic fields within the storage ring, the electron beam is subjected to disturbances from environmental effects that are coupled through the girders supporting the magnets. Several external disturbances cause electron beam perturbations, ranging from long-term disturbances, such as changes in air temperature during the day, to re-injections of the electron beam, which are typically at frequencies lower *S.G. was supported by Diamond Light Source 1 S. Gayadeen and S.R. Duncan are with Department of Engineering Science, University of Oxford, Parks Road, OX1 3PJ, Oxford, UK.
(s.gayadeen,s.duncan)@eng.ox.ax.uk 2 W.P. Heath is with the Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, Manchester, M13 9PL, UK. [email protected]
978-1-4799-0176-0/$31.00 ©2013 AACC
than 0.01 Hz. Most disturbances arise from ground vibrations and residual disturbances after feedforward correction of Insertion Device field changes and lie within the range 0.01 Hz to 100 Hz [2]. Above 100 Hz, the effects of vibration arising from cooling water pumps and girder cooling water flow resonances appear in the beam spectra [3]. Modern synchrotron radiation facilities are built for producing high brilliance photon beams, which is achieved by reducing the emittance in both planes. The vertical beam size defines the emittance, which leads to requirements on the electron beam stability [2]. In order to achieve optimum performance, electron beam stability is a crucial parameter for third generation light sources. In particular, sub-micron stability is now a common requirement for the vertical position of the beam. At Diamond, the electron beam must be controlled to within 10% of the beam size, which corresponds to an RMS variation less than 12.3 µm in the horizontal plane and 0.6 µm in the vertical plane [4]. To achieve such performance, electron beam stabilisation feedback systems are used. Some synchrotron machines use a single set of corrector magnets for control system actuation, while others use two sets of correctors, strong and fast. Historically correctors with strong magnetic fields were used for slow correction but the use of these correctors located over high conductivity vacuum chambers produce eddy currents that limit the feedback bandwidth to a few Hertz [2]. Therefore a set of fast correctors is required to compensate for high frequency disturbances. The control problem at these facilities is the design of two global feedback systems that work together. The common approach has been to design the two feedback controllers separately, but when the controllers share a common frequency range, they fight against each other, resulting in the faster actuators saturating. Several approaches are used to deal with this problem, one of which is the use of a frequency deadband to separate the fast and slow systems so that the two systems are independent [5]. However, with this approach, the beam spectrum components within the deadband are not corrected. This is undesirable as there are components of the disturbance that cannot be left without correction in this range, such as transient orbit distortions from Insertion Devices, which change the magnetic field as part of an experiment. Another approach is to use the fast feedback system alone with an active frequency range down to DC. This approach solves the uncorrected perturbations at low frequencies, but in some cases this presents a limitation due to the relative weakness of fast correctors. Because the fast correctors are designed to
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regulate small perturbations at high speeds and therefore have limited amplitude ranges, the actuators tend to saturate [2]. In some machines, this is compensated for by using the slow correctors to periodically remove the DC components of the fast correctors so that the fast actuators do not saturate even after long term drift of the closed orbit [6]. Other solutions seek to use the slow system to predict the different orbit at the next iteration and transfer it as a new reference to the fast system. As a consequence, the fast system will not see the perturbation created by the slow correction and will not try to compensate for it [5],[7]. However this approach has caused instabilities on systems where the fast correctors are different from the slow ones [2]. Some machines combine all these approaches. In this paper, the beam stabilisation systems are considered to be analogous to cross-directional processes that are common in process industries such as paper machines, plastic film extrusion and metal rolling [8],[9],[10],[11],[12]. The term cross-directional describes systems where it is required to control variations of a measured variable in a profile that is orthogonal to the direction of propagation of the variable. A cross-directional control design approach is appropriate for synchrotron applications because there is considerable interaction between the spatial responses of each of the magnets, and because this spatial response is decoupled from the dynamic response of the actuators. Similar to single-array cross-directional control, several control designs based on large-scale optimisation have been used to handle multi-array cross-directional problems which uses long horizon model predictive control (MPC) [13]. The size of the control problem in industrial cross-directional systems is usually overcome by reducing the model size or using a fast quadratic programming algorithm [14]. However, because of the large dimensions of synchrotron feedback systems with sample rates of 10 kHz, robust constrained MPC is computationally demanding and will extend the time delay, hence reducing closed loop bandwidth. In [15],[16] a method of exploiting the knowledge of the spatial responses of the arrays of actuators is used to determine the interaction between the controllable subspaces in order to decompose the problem into a series of single-input, single-output (SISO), multiple-input, singleoutput (MISO) and multiple-input, multiple-output (MIMO) problems. For example, for spatial modes where the control directions are orthogonal, the arrays are decoupled and SISO control design can be used. However where the control directions align, the choice of control direction is not straightforward. In this paper, the approach in [16] is extended to consider the case where the controllable subspaces of the arrays overlap and mid-ranging control is proposed as a useful solution. The term mid-ranging control is used to describe a control problem with two actuators but one measured variable where an additional specification is used to return one of the actuators to a setpoint. In some cases of two-input, single-output problems, it may be possible to manipulate one input at a time, but in this application it is desirable
to simultaneously manipulate both inputs as the strong correctors are bandwidth limited. An important characteristic of mid-ranging applications is that input constraints are often encountered [17] which is the case with most fast corrector magnets. There are several approaches to designing mid-ranging controllers [17], [18] but in [19] an Internal Model Control (IMC) structure is used which is adopted in this paper. An IMC approach is appropriate for synchrotron feedback systems which are subject to significant delays and is consistent with previous designs of single array controllers for this application in [3], [20], [21]. More recently in [22] a dual-mode MPC approach based on sum-of-norms regularisation is used to develop a smart allocation policy to use the least number of actuators. In the approach of [22], actuators are set to zero for extensive periods. For the current application such an approach is inappropriate, as it is desirable to maximise the use of actuators since this ensures that the maximum DC correction is achieved. Framing the control problem as an optimal control problem may not take advantage of the different actuator characteristics such as dynamics and gain whereas the IMC approach to the multi-array problem, including the use mid-ranging control, has the benefit of not only reducing computation over MPC in general but allowing explicit design trade-offs. The paper is structured as follows. In Section II the multiarray controller structure for the Diamond Light Source Booster synchrotron is presented. In Section III, a design for decoupling the control problem is presented. The stability of the system is considered in Section IV. In Section V the Booster synchrotron controller design and results are presented and conclusions are drawn in Section VI. II. C ONTROLLER S TRUCTURE For two actuator arrays with different dynamics, the nominal plant model for the system at Diamond can be described by, y(k) = z−d1 g1 (z−1 )B1 u1 (k) + z−d2 g2 (z−1 )B2 u2 (k) + d(k) (1) where y(k) ∈ Rm represents the signals measured at M sensors at times {t = kTs : k ∈ Z+ } with Ts being the sample interval so that y(k) = y(kTs ). The process disturbance is represented by d(k) ∈ Rm and u1 (k) ∈ Rr1 and u2 (k) ∈ Rr2 represent the inputs applied to the two distinct arrays of actuators. It is assumed that the all the actuators of a given array have the same dynamics. The discrete transfer function representing the actuator and process dynamics are represented by z−d1 g1 (z−1 ) and z−d2 g2 (z−1 ) where (without loss of generality) g1 (1) = 1 and g2 (1) = 1 i.e. the dc gains of the dynamics can be taken as unity. The static response for each array is represented by B1 and B2 where r1 = rank(B1 ) and r2 = rank(B2 ) and r1 ≥ r2 . Therefore B1 and B2 can be expressed in terms of reduced singular value decompositions so that,
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B1 = Φ1 Σ1 ΨT1
and
B2 = Φ2 Σ2 ΨT2
(2)
𝚽𝟏 𝑻
𝚺𝟏 −𝟏 +
𝑸𝟏 (𝑧 −1 )
𝚿𝟏
𝚿𝟏 𝑻
𝒈𝟏𝒊 (𝑧 −1 )
𝚺𝟏
𝚽𝟏
𝑑(𝑘)
Array 1 𝑟(𝑘)
𝒈𝟏𝒊 (𝑧 −1 )
𝐗𝐇
−
+
𝑦(𝑘)
+
𝚺𝟐 −𝟏
𝚽𝟐 𝑻
+
𝐗𝑻
𝑸𝟐 (𝑧 −1 )
+
𝚿𝟐
𝚿𝟐 𝑻
𝒈𝟐𝒊 (𝑧 −1 )
𝚺𝟐
𝚽𝟐
Array 2 𝒈𝟐𝒊 (𝑧 −1 )
Fig. 1.
Internal Model Control structure for multi-array problem
where Φ1 ∈ Rm×r1 , Σ1 ∈ Rr1 ×r1 , Ψ1 ∈ Rr1 ×r1 , Φ2 ∈ Rm×r2 , Σ2 ∈ Rr2 ×r2 , Ψ2 ∈ Rr2 ×r2 . Even though B1 and B2 are independent, there may be some overlap between the controllable subspaces. The relationship between the subspaces of B1 and B2 can be found by comparing the subspaces of Φ1 and Φ2 which are orthogonal and can be determined from algorithm 12.4.3 in [23] where ΦT1 Φ2 = C
(3)
and the singular value decomposition of C is given by S T C =Y Z 0
(4)
where Y ∈ Rr1 ×r1 , Z ∈ Rr2 ×r2 and S ∈ Rr2 ×r2 such that S = diag{cos(θ1 ), . . . , cos(θr2 } where θi is the angle between the principal vectors spanning Φ1 and Φ2 which span the controllable subspaces of B1 and B2 . Preserving this structure for a single array of actuators in [10], [20], the closed loop for a multi-array system is shown in Fig. 1 so that the closed loop response can be described by, y(k) = g1 (z−1 )B1 Q˜ 1 (z−1 ) + g2 (z−1 )B2 Q˜ 2 (z−1 ) r(k)+ . I − g1 (z−1 )B1 Q˜ 1 (z−1 ) − g2 (z−1 )B2 Q˜ 2 (z−1 ) d(k) (5) For the current regulator problem r(k) is set to zero. From (5), the closed loop response from d(k) to y(k) can be written as, g1 (z−1 )Σ1 ΨT1 Q˜ 1 (z−1 ) y(k) = I − [Φ1 Φ2 ] d(k) (6) g2 (z−1 )Σ2 ΨT2 Q˜ 2 (z−1 ) where −1 Q˜ 1 (z−1 ) = Ψ1 Σ−1 1 Q1 (z )F1
(7)
and −1 Q˜ 2 (z−1 ) = Ψ2 Σ−1 2 Q2 (z )F2
(z−1 )
(z−1 )
(8) ∈ Rr1 ×m
for some diagonal Q1 and Q2 and for F1 and F2 ∈ Rr2 ×m . Therefore (6) can be expressed as g1 (z−1 )Q1 (z−1 )F1 y(k) = I − [Φ1 Φ2 ] d(k). g2 (z−1 )Q2 (z−1 )F2
If the diagonal transfer function matrices Q1 (z−1 ) and Q2 (z−1 ) are chosen to have unity dc gain, then for steady state T yss = I − [Φ1 Φ2 ] 1 dss (10) T2 Similar to [20], pre-multiplying both sides by [Φ1 Φ2 ]T , projects the response into “modal space” so that I ΦT1 Φ2 H11 H12 y˜ss = I − T d˜ss (11) H21 H22 Φ2 Φ1 I and for some decoupling matrix, H such that H11 ∈ Rr1 ×r1 , H12 ∈ Rr1 ×r2 , H21 ∈ Rr2 ×r1 and H22 ∈ Rr2 ×r2 and where T T Φ1 Φ1 ˜ y˜ss = y and dss = d (12) ΦT2 ss ΦT2 ss with
T F1 Φ = H 1T . F2 Φ2
(13)
III. MULTI-ARRAY DESIGN HEURISTIC From (11), let M=
I ΦT2 Φ1
ΦT1 Φ2 I
where from (3) and (4), M can be written as I 0 S M = X 0 I 0 X T S 0 I where X is a diagonal the decoupling matrix H = M −1 which gives, D H¯ = 0 0
(14)
(15)
matrix X = diag{Y, Z}. In principle, H can then be designed by setting ¯ T where H¯ ∈ Rm×m is H = X HX 0 0 I 0 −S I 0 0 I 0 (16) 0 D −S 0 I
and D ∈ Rr1 ×r2 is a diagonal matrix with elements
(9) 1205
[D]i j =
1 . 1 − cos2 (θi )
(17)
When cos(θi ) = 0, the directions Φ1 (:, i) and Φ2 (:, i) are orthogonal so that the system is considered as a decoupled two-input, two-output system i.e. two SISO structures. The matrix H¯ gives this automatically where the corresponding elements of H¯ 11 and H¯ 22 are set to 1 and H¯ 12 and H¯ 21 set to 0. Similarity, when 0 < cos(θi ) < 1, H¯ automatically splits the effort between the two actuators. However there are some special cases to be considered when the choice of control direction is no longer straightforward: 1) When cos(θi ) = 1, the directions Φ1 (:, i) and Φ2 (:, i) align, giving a two-input, single-output structure. In this case mid-ranging control is appropriate due to the actuator characteristics. However difficulties arise with the use of this formulation for the decoupling matrix, so instead set corresponding elements of H¯ 11 and H¯ 22 to 1 and the corresponding elements of H¯ 12 and H¯ 21 to 0 to control in both directions and use mid-ranging control. 2) When cos(θi ) is close to 0, then the choice to control in a single direction is appropriate. 3) When cos(θi ) is close to 1, then the choice is to use both directions for control and use mid-ranging as when cos(θi ) = 1. IV. S TABILITY OF THE MULTI - ARRAY DESIGN Suppose that the true plant behaviour is given by −1 y(k) =z−d1 g1 (z−1 )B1 + ∆Φ 1 (z ) u1 (k)+ −1 z−d2 g2 (z−1 ) B2 + ∆Φ 2 (z )u2 (k) + d(k)
(18)
−1 Φ −1 where ∆Φ 1 (z ) and ∆2 (z ) are additive uncertainties in the response of the two arrays. Then the controller is robustly stable provided
Q˜ 1 (z−1 ) Φ −1 Φ −1
(19)
Q˜ 2 (z−1 ) ∆1 (z ) ∆2 (z ) < 1 ∞
From (7) and (8), T Q˜ 1 (z−1 ) Ψ1 Σ−1 Q1 (z−1 ) 0 Φ1 1 = −1 ) H ΦT Q˜ 2 (z−1 ) 0 Ψ2 Σ−1 Q (z 2 2 2 (20) so the uncertainty can be expressed as Φ −1 l −1 −1 ∆ (z ) ∆Φ 2 (z ) = Φ1 Φ2 W (z ). 1 −1 T (21) 0 ∆11 (z ) ∆12 (z−1 ) r −1 Ψ1 W (z ) 0 ΨT2 ∆21 (z−1 ) ∆22 (z−1 ) where W l (z−1 ) = diag{W1l (z−1 ),W2l (z−1 )} and W r (z−1 ) = diag{W1r (z−1 ),W2r (z−1 )} with ∆11 (z−1 ) ∈ Rr1×r1 , ∆12 (z−1 ) ∈ Rr1×r2 , ∆21 (z−1 ) ∈ Rr2×r1 and ∆22 (z−1 ) ∈ Rr2×r2 representing the plant uncertainty lying within the subspace that is controllable by the combination of the two actuator arrays. The weights Wl (z−1 ) and Wr (z−1 ) are known matrices that characterize the uncertainty in the modal space [16]. Using these decompositions, the robust stability condition becomes
−1
Σ Q˜ 1
0 l ∆11 ∆12 r
1
˜ 1 HMW ∆21 ∆22 W < 1 (22) 0 Σ−1 Q 2 ∞
TABLE I VALUES OF PARAMETERS USED IN DESIGN Parameter a τd
Array 1 2π×200 rad.s−1 400 µs
Array 2 2π×1000 rad.s−1 700 µs
so for any uncertainty satisfying
∆11 ∆12
∆21 ∆22 < 1 ∞
(23)
the closed loop system will be robustly stable provided that
−1
r Σ Q˜ 1
0 l 1
W (24) HMW < 1. −1 ˜
0 Σ 2 Q1 ∞ V. B OOSTER SYNCHROTRON CASE STUDY The Booster synchrotron at Diamond is used to ramp the energy of the electrons to the nominal 3 GeV before transferring the electrons into the storage ring. The Booster synchrotron, which has a circumference of 158 m, is a substantial accelerator in its own right that has the same hardware as the storage ring for beam position detection and control of the corrector magnets. Closed loop beam orbit can therefore be applied on the Booster during the acceleration ramp or by running the Booster as an electron storage ring at 100 MeV. The benefit of having closed orbit correction on the Booster is the improvement of beam position stability over long periods of time and hence operational efficiency. The feedback controller uses 22 horizontal and 22 vertical corrector magnets in total which correct the electron beam position when subjected to disturbances. The disturbances on the Booster synchrotron are mainly between 10 to 30 Hz with a peak around 50 Hz, which are associated with one of the resonant modes of the girders, so a key requirement of the control design is to regulate the horizontal and vertical position of the electron beam in the presence of disturbances in the range 1 to 100 Hz, which includes the first two harmonics of the girder disturbance. The process models for the vertical plane of each array are given by g(1,2) (z−1 ) = z−d(1,2)
b0(1,2) + b1(1,2) z−1 1 − a1(1,2) z−1
(25)
where d(1,2) is the smallest integer satisfying d(1,2) Ts > τd and a1(1,2) = e−a(1,2) Ts b0(1,2) = 1 − e
0 −a(1,2) (Ts −τ(1,2) )
0 −a(1,2) (Ts −τ(1,2) )
b1(1,2) = e
(26)
− e−a(1,2) Ts
when τd(1,2) is the delay in the system, a(1,2) is the bandwidth 0 of the actuator response (in rad.s−1 ) and τ(1,2) = τd(1,2) − (d(1,2) − 1)Ts [20]. The sample rate for the feedback system is Ts = 100 µs. The parameters in (25) for each array are given in Table I, so that g2 (z−1 ) has faster dynamics than g1 (z−1 ). The Booster has M = 22 sensors in each plane and r1 = 17 and r2 = 5, so that B1 ∈ R22×17 and B2 ∈ R22×5 . The controllable subspaces of B1 and B2 are determined from (3)
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TABLE II A NGLES BETWEEN CONTROLLABLE SUBSPACES OF B1 AND B2
cos(θi )
i=1 0.99
i=2 0.98
i=3 0.91
i=4 0.59
i=5 0
and (4) and Table II shows the cosine of the angles between the first 5 columns of Φ1 and Φ2 . A. Two-input, single-output controller design From Table II, the angle between Φ1 (:, i) and Φ2 (:, i) for i = 1 to 3 are close to 1, so the system can be approximated as a two-input, single output structure and mid-ranging control is used. The mid-ranging problem within the IMC structure in [19] and [24] is used so that the closed loop dynamics are described by (5). However by setting H¯ 11 and H¯ 22 to 1 and H¯ 12 and H¯ 21 set to 0, the closed loop dynamics for the twoinput, one-output problem can be written as can be written as, y(k) = g1 (z−1 )q1 (z−1 ) + g2 (z−1 )q2 (z−1 ) r(k)+ I − g1 (z−1 )q1 (z−1 ) − g2 (z−1 )q2 (z−1 ) d(k)+ (27) g1 (z−1 )p1 (z−1 ) + g2 (z−1 )p2 (z−1 ) ur (k) where the pre-filters p1 (z−1 ) and p2 (z−1 ) are designed to obtain a desired response from the reference signal on the faster actuator, ur (k) to y(k) as in [19]. The dynamics of g2 (z−1 ) are faster than that of g1 (z−1 ), so the following is defined, T f (z−1 ) = g1 (z−1 )q1 (z−1 ) + g2 (z−1 )q2 (z−1 ) Ts (z−1 ) = g1 (z−1 )q1 (z−1 )
(28)
and for steady state behaviour in mid-ranging control, it is required that g1 (1)q1 (1)+g2 (1)q2 (1) = 1. If it is desired that one of the actuators (typically the faster one) is zero in steady state then q2 (1) = 0. The slow closed loop transfer function Ts (z−1 ) is selected so that Ts (z−1 ) = Ts− (z−1 )T˜s (z−1 ) where Ts− (z−1 ) includes the non-minimum phase components (including delays) of both g1 (z−1 ) and g2 (z−1 ). Then for i = 1 to 3, q1i (z−1 ) = Ts (z−1 )g1 (z−1 ). Likewise T f (z−1 ) is selected so that T f (z−1 ) = T f− (z−1 )T˜ f (z−1 ) where T f− (z−1 ) includes the non-minimum phase components (including delays) of g2 (z−1 ) only. Then for i = 1 to 3, q2i (z−1 ) = [T f (z−1 ) − Ts (z−1 )]/g2 (z−1 ). B. Single-input, single-output or decoupled two-input, twooutput controller design For i = 4, one control direction can be chosen or the effort split between the both directions using H. For i = 5, the directions are decoupled so both directions are used. For i = 6 to 17 a single SISO structure is used where array 1 actuators are used. In the IMC structure, the pseudo plant inverse dynamics can be selected as 1 − λ1i 1 − a11 i z−1 1 − λ1i z−1 b01 i + b11 i 1 − λ2i 1 − a12 i z−1 q2i (z−1 ) = 1 − λ2i z−1 b02 i + b12 i q1i (z−1 ) =
i = 4, . . . , 17 (29) i = 4, 5
Fig. 2. Vertical beam displacement y(k) for a step change in disturbance d(k) with no mid-ranging (dashed, red line) and with mid-ranging (solid, blue line) for i = 1. The inputs u1 (k) and u2 (k) and references r(k) = 0 and ur (k) = 0 (dot-dashed, black line). are shown for each case.
with λ(1,2)i = e−ζi Ts where ζi is a tuning parameter that can be different for each mode [20]. C. Simulation Results In Fig. 2 the performance of the mid-ranging controllers to a step change in the disturbance is compared to using two SISO controllers for i = 1. The mid-ranging controllers are able to return the fast input to its reference while not overshooting its steady state value. In Fig. 3 real machine data is used to simulate the performance of the orbit feedback controller by considering the integrated disturbance over frequency. For this simulation mid-ranging controllers are used for i = 1 to 3. For frequencies below 100 Hz the controller is able to suppress the beam motion to within 10 % beam size specification. Suppressing the beam motion in the lower frequencies results in an amplification at higher frequencies (above 500 Hz), but a major component of disturbances at these frequencies is considered to be sensor noise from the sensors and not as a result of beam motion. For the Booster synchrotron the main source of uncertainty is in the spatial response resulting from slow drifts of the acceleration gradient [25]. The response matrix measured off the machine was compared with the ideal response matrix to −1 Φ −1 determine ∆Φ 1 (z ) and ∆2 (z ). Using the formulation in (21), it is found for this design that
−1
Σ Q˜ 1
0 l ∆11 ∆12
1 HMW W r −1 ˜
= 0.13 (30) ∆21 ∆22 0 Σ 2 Q1 ∞ indicating that the closed loop system is guaranteed stable in the presence of spatial uncertainties. VI. CONCLUSIONS This paper has considered the design of an electron beam stabilisation control system for synchrotrons where multiple arrays of actuators are used to regulate the electron beam position in synchrotron machines. In this paper a method to determine the interaction of the actuator arrays is proposed. By considering the angles between the overlapping
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Fig. 3. Uncontrolled integrated beam motion at first vertical sensor (dashed, red line) and controlled integrated beam motion for the multi-array design using mid-ranging controllers (solid, blue line)
controllable subspaces of the actuator arrays, an appropriate choice of control direction can be made. The design of the decoupling matrix presented in this paper automatically selects the appropriate control direction for most cases. For the case where the controllable subspaces align, the control problem becomes a two-input, one-output structure for which mid-ranging control is appropriate. Rather than choosing one control direction and thereby not using an actuator, the mid-ranging approach splits the effort between the two actuators, making use of the different actuator characteristics. Using all actuators is advantageous because the maximum DC correction can be achieved. The mid-ranging structure also allows a reference to be placed on the faster input to avoid actuator saturation. In this paper the Internal Model Control structure used for single array controllers is extended for the multi-array approach which is also appropriate for mid-ranging control. This approach to multi-array problems on synchrotrons for electron beam stabilisation improves on current approaches which design independent controllers without taking into consideration their interaction. VII. ACKNOWLEDGMENTS The authors would like to thank Diamond Light Source for supporting this research and for contributing to the experimental work carried out on the Booster. The authors would also like to thank Nicolas Hubert at Soleil Synchrotron for insightful discussions on multi-array control for synchrotrons. R EFERENCES [1] R. Bartolini, “The commissioning of the Diamond Storage Ring,” in 22nd Particle Accelerator Conf., Albuquerque, New Mexico, 2007. [2] N. Hubert, L. Cassinari, J.-C. Denard, A. Nadji, and L. Nadolski, “Global orbit feedback systems down to DC using fast and slow correctors,” in 9th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators, Basel, Switzerland, 2009. [3] A. Napier, S. Gayadeen, and S. R. Duncan, “Fast orbit beam stabilisation for a synchrotron,” in IEEE Int. Conf. on Control Applicat., vol. 47, Denver, CO, 2011, pp. 1770–1775.
[4] J. Rowland, M. G. Abbott, J. A. Dobbing, M. T. Heron, I. Martin, G. Rehm, and I. Uzun, “Status of the Diamond fast orbit feedback system,” in Int. Conf. on Accelerator and Large Experimental Physics Control Systems, Knoxville, Tennessee, 2007. [5] C. Schwartz and L. Emery, “Compensating the frequency deadband of the APS real-time and DC transverse orbit correction systems,” in Proc. of Particle Accelerator Conf., 2001, vol. 2, 2001, pp. 1234 –1236. [6] L. Yu, S. Krinsky, O. Singh, and F. Willeke, “The performance of a fast closed orbit feedback system with combined fast and slow correctors,” in 11th European Particle Accelerator Conf., Genoa, Italy, 2008. [7] C. Steier, E. Domning, T. Scarvie, and E. Williams, “Operational experience integrating slow and fast orbit feedbacks at the ALS,” in 9th European Particle Accelerator Conf., Lucerne, Switzerland, 2004. [8] G. Goodwin, B. Carny, and W. Edwards, “Analysis of thermal camber control in rolling mills,” in Proc. of the IFAC World Congr., Tallinn, USSR, 1990, pp. 160–164. [9] S. R. Duncan, “The design of robust cross-directional control systems for paper making,” in Proc. of the American Control Conf. 1995, vol. 3, Seattle, WA, 1995, pp. 1800 –1805. [10] W. P. Heath, “Orthogonal functions for cross-directional control of web forming processes,” Automatica, vol. 32, no. 2, pp. 183–198, 1996. [11] S. R. Duncan and G. F. Bryant, “The spatial bandwidth of crossdirectional control systems for web processes,” Automatica, vol. 33, no. 2, pp. 139–153, 1997. [12] P. E. Wellstead, M. B. Zarrop, and S. R. Duncan, “Signal processing and control paradigms for industrial web and sheet manufacturing,” Intl. J. Adaptive Control and Signal Processing, vol. 14, no. 1, pp. 51–76, 2000. [13] J. Backstr¨om, C. Gheorghe, G. Stewart, and R. Vyse, “Constrained model predictive control for cross-directional multi-array processes,” Pulp and Paper Canada, vol. 102, pp. T128–T131, 2001. [14] J. Fan, G. E. Stewart, and G. A. Dumont, “Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional processes,” Automatica, vol. 40, no. 11, pp. 1891–1903, 2004. [15] S. R. Duncan, W. P. Heath, and A. Taylor, “A new approach to multiarray cross-directional control,” in Proc. PAPTAC Control Systems 2008 Conf., Vancouver, BC, Canada, June 2008, pp. 69–74. [16] S. R. Duncan and W. P. Heath, “The robustness of multi-array crossdirectional control systems,” in Proc. Control Systems 2010 Conf. 2010, Stockholm, Sweden, September 2010, pp. 180–185. [17] B. Allison and S. Ogawa, “Design and tuning of valve position controllers with industrial applications,” Transactions of the Institute of Measurement and Control, vol. 25, pp. 3–16, 2003. [18] M. A. Henson, B. A. Ogunnaike, and J. Schwaber, “Habituating control strategies for process control,” AIChE Journal, vol. 41, pp. 604–618, 1995. [19] S. Gayadeen and W. P. Heath, “An Internal Model Control approach to mid-ranging control.” ADCHEM 2009, IFAC International Symposium on Advanced Control of Chemical Processes, Koc¸ University, Istanbul, July 12-15, 2009. [20] S. R. Duncan, “The design of a fast orbit beam stabilisation system for the Diamond synchrotron,” Dept. of Eng. Sci., University of Oxford, Tech. Rep. 2296/07, 2007. [21] S. Gayadeen, S. R. Duncan, C. Christou, M. T. Heron, and J. Rowland, “Diamond Light Source Booster fast orbit feedback system,” in ICALEPCS 2011, Grenoble, France, October 2011. [22] M. Gallieri and J. Maciejowski, “`asso MPC: Smart regulation of overactuated systems,” in Proc. of American Control Conference (ACC), 2012, june 2011, pp. 1217 –1222. [23] G. H. Golub and C. F. Van Loan, Matrix computations (3rd ed.). Baltimore, MD, USA: Johns Hopkins University Press, 1996. [24] W. P. Heath and S. Gayadeen, “Simple robustness measures for control of MISO and SIMO plants,” in 18th IFAC World Congress, vol. 18, Milano, Italy, Aug 2011, pp. 8176 –8181. [25] S. Gayadeen and S. R. Duncan, “Uncertainty modeling and robust stability analysis of a synchrotron electron beam stabilisation control system,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on, 2012, pp. 931–936.
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