MOP043

Using Lag Compensator in Orbit Feedback
Igor Pinayev, Brookhaven National Laboratory, Upton, NY, USA
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 createsproblems with systems stability due to the additional phase shifts caused by the trims (power supplies, eddy currents in vacuum chambers, etc.) and filtering of beam position data.
Conventionally employed systems have 20 dB/decade slope near the unity gain providing 90 degrees phase shift which is sufficient for stability. Utilizing one or more 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 bemodified.
Rejection of the disturbance in the linear system feedback is proportional to the gainof the open loop. If the open loop gain is described by a transfer function G(s) thesuppression of the disturbance is given by formula:1
𝑃 𝑠 =1+𝐺 𝑠
Increase of the slope to 40 dB/decade by adding additional pole into the signal chain
increases the phase shift to 180 degrees and makes system unstable. Increase of theimplemented gain is limited by the requirement of the system stability. The phase shift
caused by different elements of orbit feedback (trims with power supplies, eddy currents
in vacuum chamber, filters implemented in the beam position monitors, latency in thesignal propagation chain) can turn negative feedback into positive one and make system
unstable. Typically, the transfer function of the orbit feedback system is described byfew poles. The lowest frequency pole is user defined and might be at zero frequency if
integrator is utilized. The second pole is defined by hardware and is the main limiting
factor in the system bandwidth. Therefore, at high frequencies the loop gain is inversely
proportional to the frequency
𝐺 𝑓

The transfer function of lag compensator is
𝑇𝑠 + 1𝐷 𝑠 =𝛼,𝛼𝑇𝑠 + 1𝛼>1where a is the ratio between zero/pole breakpoint frequencies and T is time constant. At
DC the lag compensator provides additional gain a > 1. But it also creates negative
phase shift. For a=10 phase shift is almost 60 degrees and can create a long ringing
response or even cause instability. This drawback can be overcome by using morecompensators with different time constants. This will increase the slope of the Bodediagram above 20 dB/decade but keep phase margin acceptable as it can be seen below.

𝑓!
≅𝑓

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

Figure on the left shows
implementation of stages lag
compensator the single input – single
output (SISO) system with integrator
in error amplifier and single pole at
800 Hz. Three stages providing 20
dB gain at DC were used. Figure on
the right shows noise suppression
and phase of the residuals.

While the stability of the system does not changemuch with introduction of the lag compensator thetransient response changes noticeably. Figure showsthat the overshoot increases from 2% to 16 % forsystem with lag compensator and longer tail appears.