Analysis of feedback control of transverse beam oscillation by sampled-data theory

Nuclear Instruments and Methods in Physics Research 222 (1984) 411-419 North-Holland, Amsterdam

411

ANALYSIS OF FEEDBACK C O N T R O L OF T R A N S V E R S E BEAM O S C I L L A T I O N BY SAMPLED-DATA THEORY Atsushi O G A T A National Laboratory for High Energy Physics, Oho - machL Tsukuba - gun, lbaraki- ken, 305, Japan

Received 8 July 1983 and in revised form 21 November 1983

Feedback control of coherent transverse oscillation of a beam in a storage ring is studied by the sampled-data control theory. The betatron oscillation equation combined with the beam optics relation is z-transformed to find the stable loop gain region. The response in the time domain is simulated by inverse-z-transform. It is shown that the apparent shift of the betatron oscillation phase can improve the performance of the feedback.

1. Introduction A transverse beam feedback control technique has been used successfully in several circular accelerators and storage rings to stabilize coherent beam oscillation [1-5]. A few papers report subjects such as mechanical structure of a sensor (monitor in another popular word) and an actuator (kicker, damper or deflector) [6], electronics [5], etc. Few studies have, however, been made from the viewpoint of control engineering except Pelligrin's [2], which derived the transfer function of the feedback system, constructed the block diagram, and studied the root locus and frequency response. The present paper aims at extending his effort. The results obtained in this paper must be useful in the design of the feedback system, prediction of its performance, and parameter adjustment at its operation. The treatment in this paper has two major features. First, the basic equations include transformation of betatron oscillation from the kicker point to the monitor point. Second, the z-transform method based on the sampled-data theory is introduced. The usual Laplace transform methods needs an approximation that the system is continuous, where the kicker magnetic field is applied to a beam continuously both in time and in space. The present method does not need such a rough approximation. In the next section we start from the differential equation describing the betatron oscillation, combine it to the beam optics relation, construct a block diagram and perform ihe z-transform to derive the time response of the feedback system. We will find that the feedback gain value has a limit which is hidden by the continuous model. In section 3 we give further specific considerations to a popular case where the kicker and the monitor are closely positioned. The performance may be found unsatisfactory in some cases, so we explore a method to improve it in section 4. Sections 5 and 6 contain discussion and conclusion, respectively.

2. Basic relations Fig. 1 shows the essential components of the feedback system. The beam displacement at the kicker given by the solution of the following equation at t = n T , where T = 2 ~ r R / c is the revolution time and n is an integer:

Xk(t ) is

dexk/dt

z + 2ax k + o~x k = eC3Bk(t)/E.

0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(1)

A. Ogata / AnaO:sisof feedback control

412 s:s, /~ ,

~-----~

o/:q~ Q~rrl

......

JL / /

\\ \\~

j¢~.., ~k_e_r_. . . . . .

"-.~2__k "" ......

"

s:o, qJ=O /~k , Clk

Fig. 1. Essential components of the transverse feedback system.

Here, a represents the oscillation, which has positive or negative sign in the case of amplitude decreasing or increasing, respectively. ~0a = 2~r~,/Tis the angular frequency of the betatron oscillation, where ~ is the tune or betatron number. Hereafter we consider the case of a = 0, but it is easy to adapt the present method to a general case of a 4: 0. The magnetic field of the kicker Bk(t ) is effective only at the moment t = nT when the beam passes the kicker position. We factor Bk(t ) into two terms:

Bk(t)=B(t)P(t ), where

B(t)

(2)

is the feedback input, while

P(t)

represents a series of impulses whose area equals

P(t)= IT ~ 8(t+nT). 2~rR ,=-~

lT/2qrR,

i.e., (3)

l is the length of the kicker and R is the mean radius of the accelerator ring, so l/(2#R)equals the duty factor of the pulse of the magnetic field as seen by the beam. It is readily shown that the relation describing the transformation of the betatron oscillation from the kicker position to the monitor position is

Xm[(n+A)T]

=

~k

~m (COSq)+ a k s i n d p ) x k ( n r ) + l ~ m ~ sinq~ c

dXk ( nT ) dt

(4)

where

a, = - (1/2 )( dflk/ dS ), and a=

s 2err '

(5)

with s the distance along the beam orbit from the kicker to the monitor, and

=

fo

ds

B(s)"

(6)

Next, we Laplace-transform the above relations. Let us write the Laplace operator p. Eqs. (1) and (4) then become

(p2 +co~)Xk =e¢3 - B( t )P( t )] , __E_L[

(7)

413

A. Ogata / Analysis of feedback control

Xm

(8)

e

We give the feedback input with feedback gain J as B((n + k )T)=

-Jxm((n-l-A)T)

(9)

under the assumption that the input is fed ( k - A + 1) revolutions after the beam displacement is monitored. We regard our system as a sampled-data control system. Following the custom, we hereafter replace e pr by z, and call the Laplace transform of the sampled data the z-transform, writing Z[f(t)] or Z[F(p)]. Fig. 2 shows the block diagram of the feedback loop. Samplers are represented by switches which make contact instantly and periodically. The sampler at the monitor makes contact T s after the one at the kicker does. The block (1 - e pT)/p is a holder, which acts transparent in this case. The assumption of a = 0 means that the stationary oscillation is started by any external disturbance. We assume a unit impulse with the dimension of a displacement as the disturbance in this paper, which is fed as shown in fig. 2. We can readily derive the z-transform of the response of x k(nT) to the external disturbance Z[Xd(nT)]:

Z[x (nr)]

8

Z[xd(.r)]

A '

B=Z A=I+Z[Je

{~-~)pr.l-e-Pr]p

×Z EwB p 2 + 4

e

/V~-£k ( c o s ~ + a k s i n , )

q}

.

(lO)

The explicit z-transform of each factor on the right-hand side is obtained from tables in a textbook [9]: ~o~

_

Z p2 + ¢o~

z sin ~0BT (11)

Z 2 -- 2z cos ~0aT+ 1 '

Z[je-(a-x)t,r.l-e-pt]_

J

p-

ZtE~o B e -arp

(121

z-i- , ,

(COS*+ak s i n ~ ) + P

-elC2Eo)BZm[~{~'m

~m~

sin,

(C°Sf+aksinf)+P/~m/~f~--mm/Bffk s i n * ) ] m=t--A

elc2 b{z

s i n [ ( 1 - A)coBT ] + sin(&oaT)}

Ecop a

¢'J9/3~ m ~ c

+a{z

cos[(1 - zl)~oCT]- cos(A~oBT)}

z 2 - 2z cos ~0~T+ 1 sin,,

b=

~m ~

(cos~p+a ksin,),

0
(13)

A, Ogata/ Analysisoffeedbackcontrol

414

xd

EwB i +~~-L5 ~- - ~ e.~c z

-

xo +"(

[-711

eE~cco;..... ~°13 1 p z-K~Bz

= Xk

T

Fig. 2. Block diagram of the transverse feedback system.

where

denotes the modified z-transform of F ( p ) . Eq. (10) thus reduces to

Z m[F(p)]

Z[xk(.r)]

B

Z[Xd(nT) ]

A'

B=

z sin 2~'v z 2 - 2z cos 2~rv + 1 '

A=l+--

G

1

z~ ~ z 2-2zcos2rrv+l

× [b { z sin[2~'(1 - a ) v ] + sin(2rrkv) } + a { z cos[2~'(1 - a ) ] - cos(2~'kv}}], (14)

O
G = elc2J/Eo~a is the normalized gain which has no dimension. The relation ~aaT= 2~ru is introduced in t h e course of modification. The characteristic equation to describe the system stability is derived from the denominator of the fight-hand side of eq. (14):

z k - ' ( z 2 - 2z cos 2~rv + 1 ) + G [ b { z sin(2~r(1 - A)v) + sin(2rrkv) } +a{zcos(2~(1-a)v)-cos(2raav)}]

=0,

O
(15)

The case A = 1 will be considered in the following section in detail. The values of G and k should be designed as follows. Given the kicker and the monitor positions, we calculate q) according to eq. (6). We then apply a stability test to eq. (15) to know the stable area in the k - G plane. Selecting a reasonable k - G pair in the plane, we inverse-transform eq. (14) to simulate the behavior of x k in the time domain. If the result is unsatisfactory, we can try another k - G pair, or, if possbile, another kicker-monitor position pair. The specific procedure of the stability test and the response simulation will be found in the next section.

3. Special case In this section we consider the case of A = 1 and k -- 1. The c o n d i t i o n A = 1 specifies the case where the m o n i t o r and the kicker are closely positioned on the ring, and k = 1 specifies that the monitored value is

A. Ogata / Analysis of feedback control

415

5.0[G

~ \\

/ \\

Xd

L

O0

Xo

+09_2

/

//

I0

Xk

- 5 0

Fig. 3. Block diagram of the case k = 1 and A = 1 (i.e., ~ = 2~rv).

-

/

"

Fig. 4. Stable region of the system of fig. 3 in the v-G plane.

fed back to the kicker at the next revolution. The feedback control system of the T R I S T A N accumulation ring is designed on these conditions. In this case the relations /~m = ~ k and ~ = 2~rv hold. Modifying eq. (4) with the help of v = c/B, we obtain the following natural relation:

Xm((n + 1 ) T ) = Xk((n + 1 ) T ) ,

(18)

or xm(nT)=Xk(nT ). The block diagram of this case is shown in fig. 3. The z-transform of the displacement is given by Z [ x k (t)]

z sin 2~rv

Z[Xd(t)]

z 2 - 2z cos 2 e v + ( G sin 2~rv + 1)

(19)

The system characteristic equation is z 2 - 2zcos2~rv + Gsin2~rv + 1 = 0.

(20)

Let us test the stability of the system. A sampled-data system is stable if all the roots of the system, characteristic equation lie within the unit circle Izl = 1. The bilinear transformation

(21)

z=(w+ l)/(w-1), m a p s the unit circle into the region Re (w)~< 0. In this m a n n e r eq. (21) is converted to w2{2(1 - cos 2 v v ) + G sin 2~rv} - 2 w G sin 2~rv + 2(1 + cos 27rv) + G sin 2~rv = 0.

(22)

Applying the Hurewitz test, we find the stability conditions: B 0 = 2(1 - cos 2~ru) + G sin 2Try > 0, B 1 = - 2 G sin 27ru > 0,

(23)

B 2 = 2(1 + cos 2~ru) + G sin 2~rv > 0. The boundaries B 0 = 0 and B 2 = 0 are reduced to G = 2 tan~ru and G = - 2 c o t ~ru, respectively. The stability region in the v-G plane is depicted in fig. 4. Though the abscissa of fig. 4 covers only the region [0,1], enough information is included because the stability region is periodic. Fig. 4 tells two facts: 1) The polarity of the gain has to be negative for v ~ (0, 0.5) and positive fgr u ~ (0.5, 1). 2) The gain for the stability limit depends on v. Its absolute value is m a x i m u m at v = 0.25 and 0.75, while feedback is impossible at v = 0 and 0.5. We now examine the time response of the system. If x d is a unit impulse at t = 0, we have Z[xa(t)] = 1, so the right-hand side of eq. (19) directly gives the impulse response. We first examine the case v = 0.25, giving G various values. Because 2~rv = 1/2, we obtain

Z[Xk(t)]

2

z2+G+I

"

(24)

A. Ogata / Analysis of feedback control

416

T h e value of X k ( n T ) c a n be obtained as the coefficient of z " F o r example, if G = 0 (i.e., no feedback) in eq. (24), we have Z[x~(t)]

=O.z°+

1.z

'+o.

z 2-

].:

'+o.~-

in the power series expansion of

4+ 1._5+

2 [ A k ( / )].

. . . .

i.e.,

(Xk (0),

Xk(T), Xk(2t), Xk(3T)

....

) = (0, l, 0,-- 1, 0, 1, 0,-- ] . . . .

).

T h e stability region of this is G ~ ( - 2, 0) from fig. 4. Fig. 5 shows the impulse response for the case G = 0, -0.5, - 1 , - 1 . 5 , - 2 , obtained by this procedure. In general the time response is described by the three equations: Xk(2nT ) = 0, Xk(T ) = 1, Xk((2n -- 1)T) = - (G + 1)Xk((2n -- 3)T). In the case G = - 1 we find the deadbeat response, where the response settles within finite time. This p h e n o m e n o n is known to be characteristic to the s a m p l e d - d a t a system in the field of process control [9]. The characteristic equation reudces to z = 0 in this case. The impulse response for any given v value is given by z sin 2~n,

_

Z[Xk(t)]

,o-

TI TTIT! T~ (o)

(25)

zZ-2zcos2~ru+Gsin2~rl,+ 1 I.O

9

G = -20

-IO I.o ]

-,o4

o.o _T;:,:o:.:_.......................... .......... _.,,_.,_,,

-to ]

(b) G = -I.5

~L/li~~

(o)z~=oo5

oo/,- .Tllrr,. . . . . . . . . . . . . . . . . . . . . . 1[?

(b)

Z/: O,I

-IO

::rl ................................

-IO.

_T . . . . . . . . . . . . . . . l

T oo Jl/S' .... " ......... 51 ................. "

oli 0,0

,.o.

(d)

G : -0.5

_I,,,]I.F-~,I.....

- i .o

oo]l_l:_I__[[ o

'

°lT

'It]

t .......

(~)

z,: 0.2

I0

_,oi[:-1-l-[-[-l--[-[-[[-[[[- 1 (e)

G = 0

Fig. 5. Time response of x k to an impulsive x d for the case of k = 1, A = 1, v = 0.25 and various gain values. The interval o n the abscissa is equal to the revolution time. Fig. 6. Time response of x k to an impulsive x d for the case of k = 1 , A = 1 , G = - t a n ( ¢ r v ) and various i, values. The interval on the abscissa is equal to the revolution time.

A. Ogata / Analysis of feedback control

417

The power series expansion gives the time response as follows:

xk(0)

=

O,

x k ( T ) = sin 2~rv,

x k ( n T ) = 2xk((n-- 1)T) cos 2~rv- x k ( ( n - 2 ) T ) ( G sin 2~v + 1).

(26)

Fig. 6 shows time responses for v = 0.05, 0.1, 0.15, 0.2, 0.25, where the gain G is set at the half of the critical value: i.e., G = tan 2Try. 4. Performance

improvement

The system in fig. 3 requires an accelerator operator to adapt the gain G to the tune v. In spite of this effort, the feedback is not very effective when the fraction of v takes a value far from 0.25 or 0.75. One established method to solve this problem is to shift the apparent phase of the betatron oscillation [3,4]. In this section we consider how this method improves the performance. The monitored signal is a pulse series, so the following procedure is necessary to accomplish the phase shift. We first filter out a high component of the revolution frequency from the pulse series and shift the phase of that component. Sampling it at the interval T, we are then able to have the same result as sampling the phase-shifted betatron oscillation signal itself. The block diagram directly corresponding to the actual circuitry is given in fig. 7. Note that the b l o c k w~/(p2+ 6oi) in fig. 3 multiplies the input by sin ~0#t, and that what we want to multiply is sin(% + 0), with 0 the phase shift. Expanding sin(%t + 0) = sin ¢oBtcos 0 + cos % t sin 0 and applying the Laplace transform, we find that the addition of the block (cos 0 + p / % sin 0) accomplishes the phase shift. We thus obtain fig. 8, which is mathematically equivalent to fig. 7. Note that fig. 8 is quite similar to fig. 2; i.e., the phase shift has a similar effect to change the relative position of the kicker-monitor pair. The following discussion will be based on the diagram of fig. 8. The response of x k to x d is given by z sin % T Z[Xk ]

z 2 - 2z COS % T + 1

Z[xd]

1 + G z sin % T cos O + z ( z - cos % r ) sin O Z

Z 2 -- 2Z COS ¢0~ -Jr- 1

z sin 2Try (27)

z= + z ( G sin O - 2 cos 2~rv) + G sin(2~rv- O) + 1 The characteristic equation becomes z 2 + z ( G sin 0 - 2 cos 2~rv) + G sin(2zrv - 0) + 1 = 0.

(28) xd

x ° ~

°--~ pZ~-ws2

•

Xk

x°

p2 +w~2

s0+-~BsiOn P

Fig. 7. Block diagram of the system to improve feedback performance by the phase-shift method. Fig. 8. Block diagram mathematically equivalent to the one shown in fig. 7.

Xk

418

A. Ogata / Analysis of feedback control

Let us consider the case where the following two conditions are satisfied. Gsin0-2cos2~rv=0,

Gsin(2~rv-0)+l=0.

(29)

In this case, the characteristic equation reduces to z 2= 0, and the response reduces to Z[Xk]/Z[xd] = sin 2~rv/z, which is the deadbeat response. The time response becomes the one shown in fig. 5c, except for its amplitude. We can rewrite eq. (29) as cot 0 G=

1 2 cos 2~rv(cos 2~rv - sin 2~rv) '

2 cos 2~rv sin0 '

(sin0~:0).

(30)

These are the conditions of phase shift and gain to obtain the deadbeat response for the case of sin 0 :/: 0. The case of sin 0 = 0 has already been treated in the previous section. 5. Discussion

It is interesting to compare the present result to the one obtained based on the continuous model ignoring the sampling. Following the method given in refs. [3] and [7], we start from the following equations instead of eqs. (7)-(9):

( p2 + 60~)X k

=

ec3g/E,

(31)

(33)

n = - J x m.

We can readily combine them to obtain the characteristic equation of the system: f l m ~ sinep)+[to~+ec3j~ tim ( c o s ~ + a k s i n t h ) ] = O . P2+2P ( Jec2 2E T

(34)

It gives the damping time as 2E -

(35)

Jec2fl~mfl~ sin q~ "

It should be noted that eq. (32) and, consequently, eq, (34) ornits the time delay from eq. (8), which is the time necessary for a beam to travel from the monitor to the kicker. Eq. (32) certainly gives the phase delay between two positions. However, the model on which it is based assumes that the b o r n exists all around the ring, as if wave-transmissible fluid filled the whole ring. Eq. (32) gives the phase difference the fluid has at the two points, and, at the same moment. The dynamics of eq. (34) thus holds for the time range where the time delay is smaller than l / t % To clarify the limit of the continuous model, let us again consider the case described in section 3, where the kicker and the monitor share the same position and the monitored value is fed back to the kicker at the next revolution. We cannot derive eq. (18), i.e., x m = x k, from eq. (32), so we are forced to insert relations = 2cry and ~8m = flk into eq. (34) to have

p2 +po~aGsin 2vrv + e~k C2

G=--~ J.

[

co: ] • + ---~--k (COS 2~rv + a k sin 2~rv) = 0,

(36)

,4. Ogata / Analysis of feedback control

419

The stability condition requires the coefficient of the second term to be positive: G sin 2~ru > 0.

(37)

It is equivalent to G>0 G<0

for 0 < u < 0.5, for 0 . 5 < ~ < 1 .

(38)

The stability region suggested above is quite opposite to that of fig. 4. This is because eq. (34) does not take account of the phase shift caused by the time delay. What if the time delay is correctly taken into account? It is the description of sections 2 and 3 that answers the question. We have introduced several assumptions in previous sections. First of all the model is linear, which means small beam displacement and unlimited kicker power. The latter problem is discussed in ref. [8]. Secondly, we have limited our considerations to the case a = 0. The present result is, however, applicable if the absolute value of a is much smaller than ~ . Adaptation of the described treatment to a general case a :g 0 is not difficult. Thirdly, zero-pulse width is assumed as the kicker and the monitor signals for the z-transform to be applicable. It implies that the geometrical length of the kicker and the monitor is neglected. In an actual kicker, traveling-wave-type electrodes are often used which have a long stripline shape. An example is the one to be used in the TRISTAN AR which is 2 m long [5]. Though the length is still small compared with the ring circumference, a more exact solution can be obtained by a technique called the p-transform [9]. In section 4 we have pursued the possibility of the deadbeat response. Two comments are left to be mentioned. Firstly, the ideal deadbeat is attainable only when the assumed disturbance is realized. In our case, it is true only for the impulsive disturbance; the situation is different for a ramp or step disturbance, or for any real disturbance encountered in accelerator operation. Secondly, to attain the deadbeat response, the feedback system has to be adapted to the change of tune number; an operator has to adjust the gain and the phase shift occasionally. A feedback system synchronous with a tune measurement setup is otherwise needed. The z-transform method here introduced will possibly be applied to any other problems where the beam is actuated on certain points of the ring; i.e., acceleration, scraping, etc.

6. Conclusion We have applied the z-transform method to the betatron oscillation equation and the beam optics equation in studying the transverse feedback control problem. Specific consideration is made of the case where the kicker and the monitor are closely postioned. We have found that the loop gain and its polarity must be adapted to the fraction of the tune number Au. In spite of this adaptation, the feedback has little effect around the region Av = 0 or 0.5. Adjustment of both the gain and the betatron phase is a good method to cope with this difficulty.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

C.H. Pruett, R.A. Otte and F.E. Mills, Proc. 5th Int. Conf. on High energy accelerators (1965) p. 343. J.-L. Pellegrin, IEEE Trans. Nucl. Sci. NS-22 (1975) 1500. K. Wille, ibid. NS-26 (1979) 3281. R. Bossart, L. Burnod, J. Gareyte, B. de Rees and V. Rossi, ibid. NS-26 (1979) 3284. C.W. Olson, J.M. Paterson, J.-L. Pellegrin and J.R. Rees, ibid. NS-28 (1981) 2296. J.-L. Pellegrin, T. leiri and Z. Mizumachi, KEK 82-10 (1982) TRISTAN (A). J.-L. Pellegrin and J.R. Rees, PEP-Note-315 (1979). K. Wille, Internal Report DESY PEP-79/01 (1979). For example, E.I. Jury, Sampled-data control system (Robert E. Krieger, Huntington, NY, 1958).