9 Accelerator Operations
533
reduce the measurement noise—prior to closing the feedback loop. While this is
suitable for systems that are used for verification or monitoring only, it has certain
disadvantages when used in feedbacks e.g. through introducing additional sampling
delays. Combining the filtering of beam instruments and closed-loop responses
inside D(s) is not just equivalent to this from a noise point of view but also improves
the feedback response by minimising the total loop delay in these cases.
The discussed simple scalar controller are usually only used if the beam
parameter response is much faster than the required reference changes (e.g. feedback
that act on a cycle-to-cycle basis) but need to be extended for time dependent
components for the other cases. Classic feedback designs typically rely on the
discussion of denominator zeros in Eqs. (9.19) and (9.20) while keeping constraints
such as required bandwidth, minimisation of overshoot, limits on the maximum
possible excitation signal and robustness with respect to model and measurement
errors. For ideal processes, this yields adequate controller designs but often falls
short in providing a simple comprehensive method for estimating and modifying the
loop sensitivity (robustness) in the presence of process uncertainties, non-linearities
and noise.
Here, Youla’s affine parameterisation method for optimal controllers is briefly
introduced, which is based on the analytic process inversion, first introduced in [8].
For an open-loop stable process G(s), the nominal closed-loop transfer function is
stable if and only if Q(s) is an arbitrary stable proper transfer function and D(s)
parameterised as:
D(s) =
Q(s)
1 − Q(s)G(s)
.
(9.23)
The stability of the closed loop system follows immediately out of the above
definition if inserted into Eqs. (9.19)–(9.22). The sensitivity functions in the Q(s)
form are given as:
T (s) = Q(s)G(s) ,
(9.24)
S d (s) = 1 − Q(s)G(s) ,
(9.25)
S i (s) = (1 − Q(s)G(s))G(s) ,
(9.26)
S u (s) = Q(s) .
(9.27)
Assuming G(s) is stable, the only requirement for closed loop stability is for Q(s)
to be stable. The strength of this method is the explicit controller design with respect
to required closed loop performance, as visible in Eq. (9.24), and required stability
(Eqs. (9.25)–(9.27)). Equations (9.24) and (9.25) are complementary and illustrate
the intrinsic limiting trade-off of feedbacks that either have a good disturbance
rejection or are robust with respect to noise. The ultimate limit is thus defined
rather by the bandwidth and noise performance of the corrector circuits and beam
measurements than by the feedback loop design itself.
533
reduce the measurement noise—prior to closing the feedback loop. While this is
suitable for systems that are used for verification or monitoring only, it has certain
disadvantages when used in feedbacks e.g. through introducing additional sampling
delays. Combining the filtering of beam instruments and closed-loop responses
inside D(s) is not just equivalent to this from a noise point of view but also improves
the feedback response by minimising the total loop delay in these cases.
The discussed simple scalar controller are usually only used if the beam
parameter response is much faster than the required reference changes (e.g. feedback
that act on a cycle-to-cycle basis) but need to be extended for time dependent
components for the other cases. Classic feedback designs typically rely on the
discussion of denominator zeros in Eqs. (9.19) and (9.20) while keeping constraints
such as required bandwidth, minimisation of overshoot, limits on the maximum
possible excitation signal and robustness with respect to model and measurement
errors. For ideal processes, this yields adequate controller designs but often falls
short in providing a simple comprehensive method for estimating and modifying the
loop sensitivity (robustness) in the presence of process uncertainties, non-linearities
and noise.
Here, Youla’s affine parameterisation method for optimal controllers is briefly
introduced, which is based on the analytic process inversion, first introduced in [8].
For an open-loop stable process G(s), the nominal closed-loop transfer function is
stable if and only if Q(s) is an arbitrary stable proper transfer function and D(s)
parameterised as:
D(s) =
Q(s)
1 − Q(s)G(s)
.
(9.23)
The stability of the closed loop system follows immediately out of the above
definition if inserted into Eqs. (9.19)–(9.22). The sensitivity functions in the Q(s)
form are given as:
T (s) = Q(s)G(s) ,
(9.24)
S d (s) = 1 − Q(s)G(s) ,
(9.25)
S i (s) = (1 − Q(s)G(s))G(s) ,
(9.26)
S u (s) = Q(s) .
(9.27)
Assuming G(s) is stable, the only requirement for closed loop stability is for Q(s)
to be stable. The strength of this method is the explicit controller design with respect
to required closed loop performance, as visible in Eq. (9.24), and required stability
(Eqs. (9.25)–(9.27)). Equations (9.24) and (9.25) are complementary and illustrate
the intrinsic limiting trade-off of feedbacks that either have a good disturbance
rejection or are robust with respect to noise. The ultimate limit is thus defined
rather by the bandwidth and noise performance of the corrector circuits and beam
measurements than by the feedback loop design itself.