Experimental Results on the Design for the
APS PID Global Orbit Control System
Jeffrey A. Kirchman and Youngsoo Chung
Argonne National Laboratory
James P. Bobis
Northern Illinois University
Abstract The Advanced Photon Source third generation synchrotron light source needs a stabilized particle beam
position to produce high brightness and low emittance radiation. Global orbit correction control is introduced and isutilized to satisfy the demanding needs of the accelerator. This paper presents the experimental results fordetermining an effective and optimal controller to meet the global orbit correction requirements. These requirements
include frequency/time domain demands consisting of vibrational noise attenuation, limiting of controller gains forstability and improving the system time response. Experiments were conducted with a digital signal processor
implementing various PID sets to make comparisons between simulations and experiments. Measurements at these
PID sets supported the results of software simulation.

I INTRODUCTION
The 7-GeV Advanced Photon Source (APS) at Argonne National Laboratory (ANL) utilizes state of the art
technology to produce the brightest beam of high-energy x-rays available for research purposes. The APS storage
ring requires stabilization of the particle beam position to achieve the low emittance and high brightness radiation
critical to a third generation light source. To achieve this, global and local beam orbit correction feedback is
employed [1]. The global feedback control system consists of a PID controller in a digital communication systemthat collects beam position monitor (BPM) data in digital form, transmits it to a digital signal processor (DSP) andthen forwards the resulting power supply correction currents calculated by the DSP to the relevant correctors. This
paper presents the experiments that were conducted with a DSP implementing various PID sets to make comparisons
between simulations [2] and experiments.

II REQUIREMENTS
The block diagram of the APS closed loop feedback system is shown in Figure 1. The low pass filter(LPF) is designed to provide system stability and to reduce feedback from the noise source. The controller is the
PID. The compensation filter (CF) is a derived transfer function that attempts to neutralize the eddy current effects ofthe magnet and vacuum chamber.

Fig. 1. Simplified block diagram of the APS diagnostic closed loop feedback system

The controller has four main constraints on its design. These are both time-domain and frequency-domain
stipulations that should be achieved to allow the closed loop feedback to perform its task. These requirements are:
1. The noise transfer function attenuation should be no less than -12 dB at 20 Hz,
2. The maximum noise transfer function should be less than 2 dB,
3. The system should operate near critical damping to improve time response,
4. The controller gains should be limited to prevent power supply saturation and clipping, which would leadto stability concerns.
The results of the procedure to define a parameter region that satisfies the four main constraints defined by
the three gain factors of the PID are shown in Figure 2 [2] where the digital implementation of the PID transfer
function is [3]Gc (Z) = Kp +KiT/Z-1 +Kd (Z-1)/TZ
and the sampling period is 0.25 ms [2].
The acceptable parameter graph shown in Figure 2 indicates a theoretical practical range of PID parameters
to achieve the necessary time and system responses [2]. Various complications can arise in practice that cannot be
predicted or compensated. There are numerous areas where these problems can arise; for example the compensation
filter designed to cancel the effect of the vacuum chamber, magnet and beam position monitor may perform
adequately but not cancel perfectly over the frequency spectrum desired. This in turn would adjust the acceptable
ranges of values shown in Figure 2. Other obstacles include temperature, eddy currents, a power supply transfer
function that is not truly unity and numerous others which have not been factored into the model. To do so would
either be impossible to calculate or would complicate the model to a point of complexity that would make it
unusable. Thus simulations may not be entirely accurate, and experiments need to be done to prove the legitimacyof the claims made.

Fig. 2. Acceptable range of parameters for Ki and Kp that satisfy all conditions listed: 20 HZ noise
attenuation, maximum noise allowed; system time response and system stability

III EXPERIMENTAL RESULTS
Experiments were conducted using the block diagram shown in Figure 3, the blocks performing thefunctions of closed loop control. This was implemented using the test setup shown in Figure 4. The DSP
completes several tasks in the system. It reduces the bandwidth of the feedback with a low pass filter. This isnecessary because the conversion of the analog to digital signals requires the reduced feedback. The low pass filter isanti-aliasing to limit and attenuate the high frequency components. The DSP performs the control portion of theclosed loop scheme with a digital PID. It uses the backward rectangular integration digital PID to accomplish this.
The final duty of the processor is to compensate for the magnet/vacuum chamber effect in the system.

Fig. 3. Block diagram of the actual closed loop system of the global orbit control
Positioned after the DSP is a digital to analog converter on the output. This signal is transmitted to theanalog power supply, which for lab experimentation was a Kepco model BOP 20-20M 400 W device. This is abipolar operational power supply/amplifier capable of being driven by voltage or current and can source or sink 20 A
with a voltage capability of 20 V.

Fig. 4. DSP global orbit feedback test setup

The magnetic field is measured by a Hall effect probe which for purposes of experimentation performs thefunction of beam position monitor. The beam position is directly affected by the corrected magnet's field, so
adequate position simulation is achieved using a probe, together with a gaussmeter, as feedback transducers. This
combination measures the magnetic field intensity that reaches the probe through the aluminum alloy vacuum
chamber. Eddy currents are formed in this chamber and cause a distortion in the field. This distortion is not static
but is dynamically changing, causing difficulty in predicting its transfer function contribution.
The first task performed was the determination of the transfer function for the compensation filter which isin essence the inverse of the magnet/vacuum chamber transfer function. A system analyzer, Hewlett Packard model
3563A, was used to determine this. The analyzer can calculate a digital or analog transfer function for a network bysourcing a test voltage into it and measuring the output. The user may choose the number of poles and zeros in thefit so that a trade-off may be made between fit quality and the complexity and delay required by the filter. The morepoles and zeros added the more calculations are needed by the processor, and in general a compromise solution mustbe accepted.
The first fit contains four poles and three zeros and is depicted in Table 1. The fit is a very good one, andby adding additional poles and/or zeros the response is not significantly improved. The nature of the magnet would
lead one to believe that the magnet/vacuum chamber frequency response would have low pass tendencies, and the dataand curve fit support this. Since the number of poles is greater than the number of zeros, the net effect is a 20
dB/decade rolloff per unmatched pole for high frequencies.
A different fit contains four poles and two zeros, as is shown in Table 2. This fit does not appear to be sogood as the previous one, but the inverted filter does have less gain than in the first case. This is an advantage
because the more gain produced by the DSP, the larger the outgoing signal therefore requiring a higher voltage fromthe power supply, a disadvantage.
The four pole and two zero case becomes a two pole and four zero function when inverted and thus has
much gain. Most of this gain appears at high frequencies where there is significant attenuation, but some is placed
low enough in the spectrum that it forces the power supply to limit its output. Hence the four pole and three zero
curve fit appears to be the better of the two and is the one used for the remainder of this work. Other curve-fit
pole/zero combinations were examined, but little was gained by increasing the filter order and much was lost when
the order was decreased.
Figure 5 shows the frequency response of both the measured magnet/vacuum chamber and the four pole andtwo zero digital filter. To cancel the effects of this low pass response of the magnet/vacuum chamber, the digitalmodel filter is inverted to force the product of the two transfer functions toward unity. The ideal case would be unity
over the entire frequency spectrum, but since this is extremely difficult frequencies under approximately 1kHz weretargeted. That is, more effort went into the design of the compensation filter below 1kHz so that the bestcompensation would be accomplished in this region. It is for this reason that the number of poles or zeros that are
nondominating at high frequency are not critical to the compensation filter design. That design was low order and its
performance was adequate with a minimum amount of time delay.

Fig. 5. Frequency response of the digital filter and the actual magent and vacuum chamber
The DSP was programmed with numerous sets of PID values to measure and record the responses. Aftertrying sets of control parameters that ranged from very low to very high gain, it was observed that any set that had

even a modest gain caused the power supply to saturate. One method to alleviate such saturation is the addition of asingle pole, low-pass analog filter that helps remove high frequency components which are actually amplified in thecontroller due to the high-pass compensation filter. For it to truly be a compensation filter, its transfer function had
to be the inverse of that for the low pass magnet/vacuum chamber. To cancel the magnet/vacuum chamber transfer
function, the compensation filter had then to amplify high frequencies, and approached infinite gain as frequency
increased. It could not realistically do this, although it may possess an extremely high amplification in thisparticular range. This is the justification for the analog reconstruction filter (ARF), which prevents the power
supply from limiting due to its inability to switch at that rate.

IV PID INFLUENCE ON ATTENUATION OF NOISE TRANSFER FUNCTION
The first constraint considered is the noise transfer function attenuation, which is a means to measure theability of the control system to reduce the effects of feedback noise. It is the goal of the control system to have thelargest attenuation possible at 20 Hz, below which most vibrational noise is predicted to be present. Figure 6
indicates the PID values used for the experiment.

Fig. 6. Points selected to measure system response in the frequency and time domain
The first group of points to be examined are the Ki=0.25 with Kd=0 and Kp ranging from 2.5 to 4.0. This
allows analysis of Kp and its impact on noise feedback. Each group of PID sets is subjected to three ARF cutoff
frequencies. Figure 7 displays the frequency response of the noise transfer function with Kp=2.5, Ki=0.25, and Kd=0.
The three traces shown are taken using this PID with different filter cutoffs. The high frequency noise in the system
is readily apparent using a 1000 Hz ARF. The lower frequency range under 1 Hz seems erratic apparently due to the
DSP's compensation filter amplifying high frequency noise, and the ARF's cutoff not being low enough to adjust it.
The 20 Hz attenuation does not appear to be shaped by the filter. Table 3 shows the frequency results of a series ofexperiments for ranges of Kp, Ki, and Kd. From this table, all of the first grouping is seen as approximately -18
dB. The second group deviates from a barely unacceptable -11 dB to an adequate -14 dB, depending upon Kp .

Figure 7
Since the proportional and integrative constants are most important, they have received the principal
attention. It should be noted that the best attenuation occurred with the last point (Kp=2, Ki=0.3, Kd=0) and theworst with one of the first (Kp=2.5, Ki=0.05, Kp=0). It appears from the table that the integration constant has themost influence on noise control. It is the integration portion of the PID that gives the closed loop control a pole atDC, so this should be reflected in the value chosen for Ki. It should also be noted that a higher gain assigned to Kp
seems to diminish noise. As the gain of G is increased, the effective noise transfer function is decreased. Since thegain of G is directly proportional to the controller proportional constant, any increase in controller gain will reducenoise output.

.
V PID INFLUENCE ON SYSTEM TIME RESPONSE
In a subsequent experiment a step function was applied to the system and by saving the data points stored in
the signal analyzer the output response was measured and recorded. The step response was taken using the points of
Figure 6 and, as is shown in Figure 7b, the effect of the ARF is apparent. The added pole increases the overshootand influences the rise time. Because it was necessary to reduce the input signal for the 1000 Hz ARF, it is difficult
to compare that case with the other two. It does appear that the 1000 Hz condition has a smaller overshoot than theothers, with the 200, 400 and 1000 Hz ARF cases having 60%, 40% and 25% overshoot respectively.
Examination of Figures 8 and 9 reveals more about the system time response. The value of Ki has been
reduced from 0.25 to 0.05 and this has changed the step response significantly. When the proportional constant Kp
is 2.5 the system response is slightly underdamped with minimal overshoot. However as Kp is increased theovershoot and settling time escalate. This indicates that decreasing the integrator gain Ki while increasing the
proportional gain does not diminish the overshoot.

What happens if derivative control is utilized to help offset integration control is indicated in Figures 10 and 11.
Using values of Kp =3, Ki =0.1 and Kd either 0, no control, or 1, one can see that without derivative control the 200
Hz ARF overshoot is approximately 25%, but with it the overshoot decreases to 20%, not a huge benefit but anadvance nonetheless. With the 400 Hz ARF a more pronounced change in that response appears, being converted
from underdamped to critically damped.

Table 1 Filter coefficients of the first compensation filter design.
a0 = 9.41294 x 10-4
a1 = 1.30784 x 10-2
a2 = -8.52438 x 10-4
a3 =-1.11792 x 10-2

b1 = -1.87602
b2= 7.05058 x 10-1
b3 = 3.46876 x 10-1
b4 =-1.74941 x 10-1

Σakz-kH(z) = ––––––––––
1 + Σbkz-k
Table 2 Filter coefficients of the second compensation filter design.
a0= 1.36339 x 10-2
a1 = -7.6617 x 10-4
a2 = -1.091 x 10-2

b1 =-2.87463
b2 = 7.16485 x 10-1
b3 = 3.25489 x 10-1
b4=-1.66391 x 10-1

Σakz-kH(z) = ––––––––––
1 + Σbkz-k

Table 3 Noise transfer function measurements including maximum gain, frequency of
maximum gain, 20 Hz attenuation, and phase margin.
Kp

Ki

Kd

ARF

sweep

Gpk

fpk

G@20Hz

P.M.

1

0.25

0

3

0.25

0

98
119
119
108
130

-18
-18
-19
-18
-18

114
99
89
109
93

0.25

0

9

143

-19

104

1

0.05

0

4
4
3

125
172
202

-11
-11
-11

60
51
45

3

0.05

0

4

0.05

0

3

0.1

0

3

0.1

1

5

0.125

2

0.4

0
0
1
0
0
1

200
200
200
200
200
200
200
200
200
500
500
500
500
500
500
500
500
500
200
200
200
200
200
200
200
200
200
200
200
200

10
6
5
9
6

4

200
400
1000
200
400
1000
200
400
1000
200
400
1000
200
400
1 000
200
400
1000
200
400
1000
200
400
1000
200
200
200
200
200
200

7
6
6
6
6
7
6
6
6
7
7
7
8
8
8

147
202
202
124
124
113
124
124
119
173
173
165
81
81
81

-14
-14
-14
-13
-13
-14
-13
-14
-13
-16
-16
-16
-22
-22
-22

75
64
59
75
63
61
70
59
64
88
87
88
152
149
148

The most important questions are: Did the simulation tests support the data taken for the actual time
responses? Does the optimal line live up to its name? If the 1 kHz ARF cases are examined and compared the
optimal line seems to hold true. However it is important to realize that the 1 kHz ARF case is closest to simulation
conditions where there actually was no analog reconstruction filter. The 1 kHz cutoff puts the pole far enough awayfrom the rest of the response that its influence is not great.
The points 4, 7,8 and 9 of Figure 6, nearest the optimal line, display little or no overshoot with the
exception of point 9. This point uses a 200 Hz ARF, which has adversely influenced the response to reflect theadded pole. Since the other two points on this line approach critical damping, there is no other reason for point 9 todo so. These other points also have excellent rise times and settling times, while if one deviates from this optimalline the effects can be readily seen.

VI CONCLUSION
The APS third generation synchrotron light source requires a global orbit control system that reduces noise
effects and stabilizes the beam. This was accomplished through a high speed feedback loop established completelyaround the accelerator ring. This encompassing control system requires a specific kind of control that balances its
needs without compromising any of these demands too heavily.
The need exists for a controller to achieve noise reduction, but other consequences must be considered aswell. Such notable system attributes as time response, stability, power supply bandwidth and cost are all concernsof closed loop control and it was necessary to find a control algorithm that met or exceeded the design criteria. The
PID controller proved to be the best in this situation, and design centered on the choice of proportional, derivativeand integral gains and sampling time
Experimentation has justified the type of controller design, where optimal control was achieved using threedimensional plots of the performance in question. These plots were constructed with two of the design parameters as
variables; they can also be organized into tabular format to facilitate comparison viewing. The performancemeasures were rated in terms of noise filtering ability, system response time or system stability. This paper has
described methods to assess system performance of both time and frequency domain functions.

Performance of noise rejection was measured by plotting the noise transfer function value at a criticalfrequency for different values of Kp, Ki, and Kd where the values that met or exceeded the criterion for handling
injected noise were plotted. The same procedure holds true for determination of maximum gain of the noise transfer
function.
The work performed in this paper examined the controller’s design effects on the performance of the systemwith one “BPM” and one corrector. A better experiment could be performed using all 40 of the BPMs and correctors.
This would be the final proof of the efficiency and capability of the controller design. This experiment would need
to be performed on an existing accelerator where a PID loop could be programmed into a global orbit feedback
system.

REFERENCES
1. D.Bulfone et al., “Design Considerations for the ELETTRA Beam Position Feedback Systems,” European
Particle Accelerator Conference, 1990, PP 895-897.
2. J. A. Kirchman and J.P. Bobis, “Preliminary Design of the APS PID Global Orbit Control System,” IEEE
International Conference on Industrial Electronics, Control and Instrumentation, Maui, Hawaii, Nov. 1993.
3. Benjamin C. Kuo, Automatic Control Systems, Prentice-Hall, Englewood Cliffs, NJ 1991.