536
M. Lamont et al.
G(s) = G 0 (s) · e
−λs G NL (s) .
(9.38)
Using the same pseudo-inverse G i
0 (s) as for the above example and inserting
Eq. (9.28) into (9.23) yields a controller parameterisation D NL (s) including a
classic Smith-Predictor and anti-windup paths, discussed in more detail in [6, 9].
Inserting Eq. (9.28) including the delay and non-linearities into Eq. (9.24) yields
the following closed loop transfer function:
T (s) = F Q (s) · e
−λs G NL (s) .
(9.39)
Similar to the linear case discussed above, the closed loop is essentially defined by
the function F Q (s) that within limits can be chosen arbitrarily based on the required
disturbance rejection and robustness during possibly different operational scenarios
(gain-scheduling). Further information and a review on Youla’s parameterisation
can be found in [6, 10].
9.4.2 Inter-Loop Dependencies
Above described individual parameter control complexity is dwarfed by the challenge of operating parallel feedback loops on for example orbit, tune, chromaticity, coupling, radial position and transverse bunch-by-bunch motion. Even in a
fully optimised scheme, some cross-talk is inevitable: the momentum modulation
required to measure chromaticity induces tune and radial offsets that are seen by the
tune, orbit and radial position feedbacks; transverse feedback, by design, minimises
the very same beam oscillations required to measure the tune. If not addressed at an
early design stage, a naïve one-by-one implementation of these feedback loops can
lead to serious interferences, coupling and instabilities.
There are various classic de-coupling strategies such as: diagonalisation, e.g
decoupling of horizontal and vertical planes; suppression of known cross-terms,
i.e. allowing certain variations which are required for measurements; dead-bands to
limit the operational ranges of one feedback in favour of another; time-scheduling
between feedback actions, such as alternating tune measurements with transverse
feedback operation; choosing different bandwidths for each loop.
An improved de-coupling strategy is to derive the dependent variable from the
compensated feedback actuator control signal. In this case the tune feedback is
operated at the maximum desired bandwidth, fully compensating radial modulation
induced tune changes. Chromaticity is in turn derived and corrected from the
amplitude of the actuator signal required to stabilise the tunes. Due to the finite
bandwidth and gain of the feedback, the actuator signal does not typically contain
the full modulation. An accurate chromaticity estimate needs to account for this
and should be complemented by the demodulation of the residual tune frequency
oscillation remaining on the beam. The required dispersion orbit variation and
M. Lamont et al.
G(s) = G 0 (s) · e
−λs G NL (s) .
(9.38)
Using the same pseudo-inverse G i
0 (s) as for the above example and inserting
Eq. (9.28) into (9.23) yields a controller parameterisation D NL (s) including a
classic Smith-Predictor and anti-windup paths, discussed in more detail in [6, 9].
Inserting Eq. (9.28) including the delay and non-linearities into Eq. (9.24) yields
the following closed loop transfer function:
T (s) = F Q (s) · e
−λs G NL (s) .
(9.39)
Similar to the linear case discussed above, the closed loop is essentially defined by
the function F Q (s) that within limits can be chosen arbitrarily based on the required
disturbance rejection and robustness during possibly different operational scenarios
(gain-scheduling). Further information and a review on Youla’s parameterisation
can be found in [6, 10].
9.4.2 Inter-Loop Dependencies
Above described individual parameter control complexity is dwarfed by the challenge of operating parallel feedback loops on for example orbit, tune, chromaticity, coupling, radial position and transverse bunch-by-bunch motion. Even in a
fully optimised scheme, some cross-talk is inevitable: the momentum modulation
required to measure chromaticity induces tune and radial offsets that are seen by the
tune, orbit and radial position feedbacks; transverse feedback, by design, minimises
the very same beam oscillations required to measure the tune. If not addressed at an
early design stage, a naïve one-by-one implementation of these feedback loops can
lead to serious interferences, coupling and instabilities.
There are various classic de-coupling strategies such as: diagonalisation, e.g
decoupling of horizontal and vertical planes; suppression of known cross-terms,
i.e. allowing certain variations which are required for measurements; dead-bands to
limit the operational ranges of one feedback in favour of another; time-scheduling
between feedback actions, such as alternating tune measurements with transverse
feedback operation; choosing different bandwidths for each loop.
An improved de-coupling strategy is to derive the dependent variable from the
compensated feedback actuator control signal. In this case the tune feedback is
operated at the maximum desired bandwidth, fully compensating radial modulation
induced tune changes. Chromaticity is in turn derived and corrected from the
amplitude of the actuator signal required to stabilise the tunes. Due to the finite
bandwidth and gain of the feedback, the actuator signal does not typically contain
the full modulation. An accurate chromaticity estimate needs to account for this
and should be complemented by the demodulation of the residual tune frequency
oscillation remaining on the beam. The required dispersion orbit variation and