12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3

IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 4 3Igor Pinayev†, Brookhaven National Laboratory, Upton, NY, USA
Abstract

LAG COMPENASTOR

Growing demand on the beam orbit stability requires
higher loop gain within the operational bandwidth. Increasing the gain leads to the increase of the unity gain frequency and creates problems with systems stability due tothe additional phase shifts caused by the trims (power supplies, eddy currents in vacuum chambers, etc.) and filteringof beam position data. Conventionally employed systems
have 20 dB/decade slope near the unity gain providing 90°
phase shift which is sufficient for stability. Utilizing one ormore lag compensators allows to increase the gain at low
frequencies while keeping phase margin acceptable. The
paper provides more details on the proposed solution as
well as simulations of how the transients will be modified.

INTRODUCTION

Usage of lag compensation provides higher low-frequency gain [4]. The transfer function of lag compensatoris𝐷 𝑠 =𝛼, 𝛼>1(3)where  is the ratio between zero/pole breakpoint frequencies and T is time constant. At DC the lag compensator provides additional gain  > 1. But it also creates negative
phase shift. For =10 phase shift is almost 60° and can create a long ringing response or even cause instability. Thisdrawback can be overcome by using more compensatorswith different time constants. This will increase the slopeof the Bode diagram above 20 dB/decade but keep phasemargin acceptable as it can be seen in Fig. 1.

Rejection of the disturbance in the linear system feedback is proportional to the gain of the open loop. If the open
loop gain is described by a transfer function G(s) the suppression of the disturbance is given by formula:𝑃 𝑠 =(1)Increase of the slope to 40 dB/decade by adding additional pole into the signal chain increases the phase shift to180° and makes system unstable. Increase of the implemented gain is limited by the requirement of the systemstability. The phase shift caused by different elements oforbit feedback (trims with power supplies [1], eddy currents in vacuum chamber [1], filters implemented in thebeam position monitors [2], latency in the signal propagation chain [3]) can turn negative feedback into positive one
and make system unstable. Typically, the transfer functionof the orbit feedback system is described by few poles. The
lowest frequency pole is user defined and might be at zerofrequency if integrator is utilized. The second pole is defined by hardware and is the main limiting factor in thesystem bandwidth. Therefore, at high frequencies the loop
gain is inversely proportional to the frequency|𝐺 𝑓 | ≅(2)

where fc is crossover frequency where gain is unity. Since
the slope of the Bode plot is usually fixed (-20 dB/decadefor single pole), increase in suppression of the high-frequency disturbances requires increase of the crossover frequency.

___________________________________________

Figure 1: Bode plot of the open loop with two and threelag compensators. Total gain increase is the same but
maximal phase shift different for three-stage it is 40°
and for two-stage it is 50°.

IMPLEMENTATION IN ORBIT FEEDBACK
Figure 2 shows implementation of stages lag compensator the single input – single output (SISO) system with integrator in error amplifier and single pole at 800 Hz. Threestages providing 20 dB gain at DC were used. Figure 3shows noise suppression and phase of the residuals.

* Work supported by Brookhaven Science Associates, LLC under Contract No. DE-SC0012704 with the U.S. Department of Energy† pinayev@bn.gov

MOP04308 Feedback Systems and Beam Stability

123Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

USING LAG COMPENSATOR IN ORBIT FEEDBACK*Content from this work may be used under the terms of the CC-BY-4.0 licence (© 2023). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

12th Int. Beam Instrum. Conf.
ISBN: 9 7 8 - 3 - 9 5 4 5 0 - 2 3 6 - 3IBIC2023, Saskatoon, Canada
JACoW Publishing
ISSN: 2 6 7 3 - 5 3 5 0d o i : 1 0 . 1 8 4 2 9 / J A C o W - I B I C 2 0 2 3 - M O P 0 4 3

Figure 2: Example of the open loop gain with integrator
and a pole at 800 Hz. Additional gain at low frequencies
is about 20 dB, the phase margin is about 60° with andwithout compensator. Dots correspond to the digital system with 10 kHz sampling rate.

Figure 4: Transient response to the disturbance of thefeedback system.

CONCLUSION
Implementation of the lag compensator allows to increase the suppression of the low-frequency disturbances
without increase of the feedback bandwidth. Such solution
does not change system stability but affect slightly transient dynamics.

REFERENCES

Figure 3: Noise suppression plot for closed loop.
While the stability of the system does not change muchwith introduction of the lag compensator the transient response changes noticeably. Figure 4 shows that the overshoot increases from 2% to 16 % for system with lag compensator and longer tail appears.

[1] N. Hubert, L. Cassinari, J.-C. Denard, A. Nadji, L. S. N.
Nadolski, and D. Pedeau, “Design of a Fast Orbit Feedbackfor SOLEIL”, in Proc. DIPAC'05, Lyon, France, Jun. 2005,
paper POT019.
[2] E. Plouviez, “Optimizing the BPM processor filter for the fastorbit correction”, presented in 6th DEELS Workshop, ESRF,Jun. 2019.
[3] D. Bulfone, R. De Monte, V. Forchi, G. Gaio, and M. Lonza,“Design of a Fast Global Orbit Feedback System for theELETTRA Storage Ring”, in Proc. ICALEPCS'05, Geneva,
Switzerland, Oct. 2005, paper P3_011.
[4] G.F. Franklin, J.D. Powell, and A. Emami-Naeini, Feedback
Control of Dynamic Systems, Upper Saddle River, NJ, USA:
Pearson Prentice Hall, 2006.

MOP04312408 Feedback Systems and Beam Stability