SLACPUB6503
June
1994
(A)

A
Formal
Approach
to
the
Design
of
Multibunch
Feedback
Systems:
LQG
Controllers
H. Hindi,
J. Fox,
S. Prabhakar,
L. Sapozhnikov,
G. Oxoby,
I. Linscott,
and
D. Teytelman
Stanford
Linear
Accelerator
Center,
Stanford
University,
Stanford,
CA 94309
Abstr~t
We formulate
the multibunch
feedback
problem
~
a stan
dard
controlsystems
design
problem
and
solve
it
using
Linear
Quadratic
Gaussian
(LQG)
regulator
theory.
Use
of a specific
optimality
criterion
allows
quantitative
eval
uation
of different
controllers
and
leads
to
the
design
of
opt imal
LQG
cent rollers.
Computer
simulations
are used
to show
that,
as compared
to the existing
Finite
Impulse
Response
(FIR)
control,
LQG
control
can provide
the same
closedloop
damping
for
less
peak
power,
thus
making
more
effective
use of limited
kicker
power.
furthermore,
 LQG
cotitrol
enables
us to use more
power
to provide
bet
ter damping
wit bout
the
problem
of
driving
instabilities
with
higher
loop
gains.
The
code
for the LQG
filters de
scribed
h=
been written
for the Quick
prototype
installed
at ALS.
1
INTRODUCTION
The problem of designing the controller (filtering alg~
rithm) for the longitudinal feedback system is bmically a
regulator pm~le~.
The regulator problem has been stud
ied extensively in control theory, and there are many tech
niques available to solve it.
This paper will compare the
performances of the existing FIR filterbased control tech
nique to that of the LQG %lterbased technique.
Oneof
the distinguishing features of LQG design is the use of a
specificoptimality crfierion. This is in centrast to the FIR
based technique, which is based on an adhoc discretetime
approximation
of a differentiator
[1,2].
Intheprocessof dampingsynchrotronoscillations,mea
surementsof the beamphasearetakenat discretetimes
andthefeedbackcorrectionsignalsareappliedat discrete
times. Therefore,our analysiswill be carriedout using
discretetimecontrolformalism.We will also usestate
spacenotationin our descriptionof the plantand con
troller[3,4].Givena systemdescribedby stat+spacema
trices
{A, B, C, D},
the transfer
function
is obtained
by
H(z)
=
C(zI–
A)lB+D
.
(1)
2
LINEAR
QUADRATIC
GAUSSIAN
~GULATOR
THEORY

Figure 1 sh~ws a block diagram of the LQG regulator prob
lem. A precise statement oft he problem
follows.
Given a
*Work supported
by Department
of Energy contract
DW
AC0376SFO0515 .
w+
P(z)
Cx
{A, B, C, D}
Figure 1. Block diagram of the LQG regulator
problem.
linear model of the plant P,
described
by state matrices
{A, B, C, D}
(D=
O),
z(k +
1)
=
Ax(k)
+ Bu(k)
+ w(k)
y(k)
=
Cz(k)
+ v(k)
.
(2)
where
z
is the
plant
state
(energy
and
phase),
u is the
cent rol input
(kicker
signal),
y is the measured
regulated
output
(phase),
and w and v are process
and measurement
noises,
respectively,
both
of which
are assumed
to be un
correlated
white
noises
with
covariance
matrices
W
and
V.