August,
August ,, 1981
1981
LIDS-P-1116
ROUNDOFF NOISE AND SCALING IN
THE DIGITAL IMPLEMENTATION OF CONTROL COMPENSATORS
by
Paul Moroney
LINKABIT Corporation
1L453
Roselle Street
92121
San Diego, California
Alan S. Willsky
Laboratory for Information and Decision Systems
Department of Electrical Engineering and
Computer Science
02139
Cambridge, Massachusetts
Paul K. Houpt
Laboratory for Information and Decision Systems
Department of Mechanical Engineering
Massachusetts Institute of Technology
02139
Cambridge, Massachusetts
ABSTRACT
Researchers in digital signal processing have examined at
length the effects of finite wordlength in the design of digital
The issues that have been considered apply to any
filters.
In particular, the design of digital control
digital system.
In this paper we will use,
systems must consider these issues.
adapt, and extend the ideas developed in digital signal
processing to the issue of roundoff noise in digital LQG
We will then examine
(linear-quadratic-Gaussian) compensators.
the roundoff noise effects for a particular LQG example and
several different implementation structures.
This work was performed in part at the MIT Laboratory for
Information and Decision Systems with support provided by NASA
Ames under grant NGL-22-009-124 and in part at the Charles
Stark Draper Laboratory.
1.
INTRODUCTION
In the design of digital filters it has
strated
that
been
amply
demon-
one must consider the effects of the finite preci-
sion inherent in the digital implementation.
This finite
preci-
leads to degradation due to quantization noise, coefficient
sion
inaccuracy, and limit cycle
the
been
effects
have
of a great deal of research in digital signal
subject
processing.
These
oscillations.
[1,2,3]
It is also important to
investigate
issues
these
in
the
application of digital processing to other fields - specifically,
to
discrete-time
designs
control
usually
have
systems.
been
implemented
that
could
effectively
past,
controller
large,
expensive,
the
on
However, the number of applica-
floating-point computer systems.
tions
In
use small-scale hardware control
systems that work in real time has greatly increased,
with
advent of the inexpensive microprocessor.
the
menting such compensators, we must
consider
the
especially
When imple-
problems
that
arise in dealing with the fixed-point arithmetic and finite wordlengths
of
small-scale
digital systems.
As these problems are
not addressed at all in the idealized mathematical design
proce-
dures that have been developed to date for control, a methodology
must
be established for treating the digital implementation of a
compensator design.
For
developed
this
methodology,
we
have
for
implementing
digital
turned
filters.
to
the
results
On one level, a
single-input
digital
single-output
filter.
However,
feedback loop around
require 'us
to
the
adapt
implementations.
ideas
digital
as
compensator
is
simply
a
we will show, the existence of a
digital
compensator
frequently
will
and extend the ideas developed for filter
These adaptations of digital signal
processing
will serve as the basis for techniques dealing with round-
off noise, scaling, coefficient rounding,
and
limit
cycles
in
digital feedback compensators.
Some work has already been done concerning controller implementation
[5-14].
However,
these
have been limited in scope,
typically treating only a few points, or only one
example.
Our
intention is quite different - a systematic extension and adaptation of digital signal processing ideas to controller implementation.
We have reported results concerning the coefficient quan-
tization issue in [15] and
the
issues
of
[16].
In this paper, we will
examine
scaling and roundoff quantization in fixed-point
controller implementations.
The basic idea behind the examination of roundoff
the
noise
same for digital filters and digital compensators.
mating the results of intermediate computations, or
values,
node
is
Approxisignal
with a finite number of bits will cause a degradation in
the system's performance as compared to the ideal.
Assuming that
a given quantitative performance measure is provided, we can measure the tradeoff in the number
Then,
assuming
of
bits
vs.
the
degradation.
that we specify an acceptable amount of degrada-
-2-
needed
meet this goal.
to
In fixed-point digital systems, this
can be done by bounding the effects
also
would
this
however,
of
quantization
additive white noise, thus allowing
analysis
com-
More
noise.
that the roundoff operation can be modeled as
assume
we
[8-11,17];
include limit cycle effects and thus
constitute very loose bounds on quantization
monly,
bits
can determine the minimum number of node variable
tion, we
[3].
techniques
the
linear
of
use
system
We will adopt the latter approach in
our investigation.
In order to bring out specific
results
concerning
control
system implementations, we will consider a specific class of con-
problems
trol
linear-quadratic-Gaussian (LQG) problems.
The
in
the
compensator resulting from the LQG framework is
sense
that
it minimizes a quadratic functional of the state and
This
control fluctuations.
detail
in
optimal
Section
2.
compensator
will
be
described
in
It is very convenient also to treat this
quadratic functional as the index of performance by which we measure the relative merit of different digital implementations of a
controller design.
performance
index
Thus any given implementation will produce
greater
than
the 'optimal' ideal (infinite-
This increase will reflect the degradation due
precision) value.
to some finite wordlength effect, such as
noise.
The
fact
roundoff
*the
output
quantization
that we use this performance metric, somewhat
different from but more appropriate than the
of
a
noise
used
in
-3-
digital
1 2 -norm
(variance)
filtering analysis, is
from
another reason out results differ
signal processing literature.
digital
in
reported
those
the
If the problem under con-
sideration had been a Kalman or Weiner filter,
a
then
suitable
performance measure would have been the trace of the error covariance
matrix.
Our results extend in a straightforward manner to
this case, and also to the output noise power
case,
measure.
In
any
the control problem, it is the presence of a feedback
for
loop through some controlled system which makes our
techniques novel.
for
structure
a
results
and
Specifically, as we will see, the concept of a
digital compensator, the notation developed by
Chan [18] for describing
the
operations
in
a
filter
digital
structure (including precedence), approaches to scaling, roundoff
noise
analysis techniques, and the minimum roundoff noise struc-
tures of Mullis and Roberts 119,20] and Hwang 121] all need to be
adapted for the control problem.
deal
only
with
Also, in this
single-output
single-input
systems,
control
multiple-
although our results can be extended to multiple-input
output systems
The
shall
we
paper
[16].
organization of this paper will be as follows.
In Sec-
tion 2 we will describe the LQG control problem and the resulting
ideal optimal compensator - it is
this
ideal
compensator
that
must be implemented with as little degradation as possible due to
finite precision effects.
somewhat
different
The notion of a compensator structure,
than a conventional filter structure, and an
adaptation of the Chan notation for
-4-
describing
such
structures
will
be presented in Section 3.
In Section 4 we will review the
digital signal processing techniques
for
scaling
structures to
satisfy the dynamic range constraints of fixed-point digital filters,
and
will
we
how
show
digital -control systems.
these ideas must be modified for
The scaling issue, of course,
is
cen-
tral to any meaningful measurement of roundoff quantization noise
effects.
noise
In Section 5 we will review the techniques for roundoff
analysis in digital filtering, and extend these to digital
controllers.
The minimum roundoff noise structures of Mullis and
Roberts [19,20] and Hwang [21] will be adapted for controllers in
Section 6, along with a treatment of the more
tion
techniques
introduced
troller implementation.
developed)
we
by
Chan,
optimiza-
general
also as adapted for con-
Finally, (using the techniques
we
have
will compare the roundoff quantization noise per-
formance of several scaled structures for implementing a specific
LQG controller example.
pensator
structure,
We will also show that a 'default'
quite
natural
com-
for the control designer to
implement, is in fact not a very desirable structure
to
select.
2.
LQG CONTROLLER DESIGN
In
section we will introduce the single-input single-
this
design - one that would only be possible with
infinite-precision
From this ideal we wish to select a finite-precision
arithmetic.
implementation
which
in
results
a
digital
filter
as little degradation (in the
directly
This is
performance index) as possible.
designing
that
compensator
of course only specify an ideal
will
procedure
This
results.
optimal
the
and
problem
output LQG control
using
a
analogous
to
bilinear transformation,
implementing
impulse invariance, or whatever technique, and then
the 'ideal' design in finite-precision.
Let us assume that we wish to design a digital discrete-time
compensator
for
a
continuous-time system (a 'plant'), and that
piecewise
the control signal will be
assume
that
constant.
We
will
also
the output of the plant is sampled at the rate 1/T.
The term linear-quadratic-Gaussian refers to the following design
problem:
given a linear discrete-time model of a continuous-time
that
system subject to disturbances
Gaussian
noises,
design
a
can
be
modeled
as
white
linear compensator that minimizes a
quadratic performance index.
Consider the following discrete-time model of a
continuous-
time plant:
x(k+l) = ux(k) + ru(k) + w (k)
(1)
y(k)
= Lx(k) + w 2 (k)
-6-
where x is the state n-vector, u and y are the control and output
variables,
is an n x n state transition matrix, r is an n x--l
4
input gain matrix, and L is a L x. nt
output
gain
matrix.
The
quantities w 1 and w 2 are the discrete Gaussian noises referred to
above.
These noises are zero mean, with covariance matrices el,
(n x n) and e 2
(1 x 1), respectively.
The performance index can
be written as follows(2)
ix I(k)Q x(k)+ 2x'(k) Mu(k) + Ru2(k)I
-
E
J = i+
k=-i
Thus we see that J reflects the weighted squared
of the states and of the control.
deviations
The weighting parameters Q, M,
and R can be specified by the designer.
The
determination
of a linear compensator that minimizes J
involves the solution of
plant
and
two
Ricatti
weighting parameters.
equations
involving
the
However, the resulting control
u(k) typically will depend on past values of the plant output
to
and
including y(k)
[22].
Unfortunately,
pensator is not directly feasible
for
up
the resulting com-
implementation,
since
a
certain amount of time must be allowed to compute u(k) from y(k),
y(k-l),
etc.
Yet
u(k) and y(k) refer to the control and plant
output at identical times.
Some delay must be accounted for
and
thus the design as described so far is unfeasible.
Fortunately,
Kwakernaak
and
Sivan
[23]
have presented a
design procedure that does account for this delay.
compensator is optimal in the sense that
that
it
The resulting
produces
the
u(k)
minimizes J, but based only on a linear function of y(k-l),
-7-
y(k-2)..., and not on y(k).
mented,
essentially
Such a
allowing
one
compensator
full
sample
cxCputattrr of-u(k) after the y(k-l) sample
however,
the
computation
time
available
long
before
it
is
be
imple-
period for the
generated.
If,
is much shorter than the sample
interval, this implies some inefficiency;
be
can
i
the output
used. as
a
u(k)
will
coatra..,
Thus
Kwakernaak and Sivan also include a method for skewing the sample
time of the plant output
sator.
up
with respect to the rest of the compen-
The compensator output u(k) will still depend
on
to and including y(k-l), but now y(k-1) is produced
calculation time before u(k)
inefficiency [16,23].
is
needed.
This
inputs
only one
eliminates
any
We will return to the implications of this
necessary calculation time in Section 3.
The
optimal compensator described above is of the following
form:
x(k+l) =
xP(k)
+ ru(k) + K(y(k) - Lx(k))
(3)
u(k+i) = -Gx(k+l)
where x is the estimated state vector, and the n x 1 Kalman
ter
matrix
K and 1 x n regulator matrix G result from the solu-
tion of two discrete-time algebraic Ricatti equations [23].
that in equations
inputs
y(k),
fil-
(3), the next control u(k+l)
y(k-l)....
depends
only
Note
on
Thus the computational delay has been.
allowed for in this formulation.
-8-
Now if we treat this compensator as a discrete linear system
and, examine its transfer function, we have -l
gY(=z)
-G(z-4+KL+rG)
1K
In a more conventional form, this can be written:
-1
U(z)
Y
Note
the
alz
n
+a2
Y+b lz
lack
-n
-2
+b2z2
of
2z)
(5)
+...+bnz n
a term a o in the numerator.
The presence of
such a term would reflect a dependence of the present
the
present
input.
Since
output
on
(5) represents a compensator that can
be implemented, the a o term must be zero.
This delay has an important implication in the way
at
structures
we
look
for implementing digital compensators, as we will
show in Section 3.
1 Note
that we have taken u to be the output of the digital network
and y to be the input. This may be contrary to the expectations
of many readers.
-9-
3.
ALGORITHMS AND STRUCTURES FOR DIGITAL COMPENSATORS
In the nomenclature of digital
term
structure
precision)
is
refers
to
signal
processing
the specific combination of
arithmetic operations by which a filter output
generated from intermediate values and the input.
a structure can be represented by a signal-flow
examine
a
simple
filter
structure
to
Specifically, let us examine
(1]
filter structure:
a
(n=4)
0o
u(k)
a
-b
- 4b
<
3
z
\/-l
.>
a4
Figure 3-1:
Direct Form II Filter Structure
-10-
sample
Let
us
(5).
(See Figure 3-1)
a
(finite-
Typically,
graph.
fourth-order
y(k)
the
see whether it will be
appropriate for representing the compensator of
form II
[IJ
direct
Note
the
presence
of
the
ao
term.
exactly represent an implementation,
has not been accounted for.
taken
to
represent
a
assumed
to
be
since
computational
delay
needed
present, and is ignored.
for
computations
iorates
when
However, in any
adequately
repre-
If series delay exists in the
compensator, and has not been accounted for, the
system may be unstable.
entire
control
Control system performance always deter-
extra delay is added to the loop.
Thus any treat-
ment of compensator structures must include specification of
calculation delays.
is
In most digital filter
control system, all delays that exist must be
in the structure notation.
ais
in digital signal processing;
applications, series delay is of no consequence.
sented
delay
However, such a signal-flow graph
structure
basically, the extra series
Such a structure cannot
all
This consideration basically led to the form
of equation (5).
Now,
tion (5).
let us take Figure 3-1 and set a° to zero, as in equa(See Figure 3-2)
>.
y(k)
1.T
...
1 a
z ..-
.
% -b4
__
u(k)
a4
Figure 3-2:
Direct Form II Filter Structure (a=0O)
an
accurate
implementation of equation (5).
The only
The signal-flow graph of Figure 3-2
representation
of
an
is
time available for computation is between
sample-skewing
for now).
still
not
sample
times
(ignore
Yet Figure 3-2 shows u(k) depending on
compensator state nodes (defined to be the outputs of delay
ments),
also
at
the same time k.
Time must be allowed for the
Thus u(k) cannot be in
multiplications al through a4 .
ele-
existence
until after the state node values are calculated.
A
structure
Figure 3-3.
for
implementing
equation (5) is depicted in
This can be derived from Figure
signal-flow graph manipulation.
-12-
3-2
by
elementary
aal
delaymustthe
precede
) u(k
node.
For
controller
'direct form II
Thus the
-1
uk)node is
<41
Direct Form II Compensator Structure
implementations theis
structure.
compensatorned.
state
will
One clear result
delay must precede the u(k) node.
The
be
design
procedure,
careful
to
include
a
unit
Thus the u(k) node is always a
that this organization of the com-5
unit
allowing
u(k)
ao.
Thus
to depend only on past y
values, results in a controller which can be
are
delays
ined as the
emerges
putations was only possible due to the zero value
our
a
design
-1ur
u(k)
procedure,
allowing
to depend only on past y
V
Figure 3-3:
z
all
of
implemented
if we
the actual delays inherent in the
structure.
Another difference between filter and controller
should
be
mentioned.
The
structure
structures
of Figure 3-3 has 5 unit
delays, while that of Figure 3-1 has only 4.
This
carries
to all types of filter and compensator structures [16].
over
In fact,
for an nth-order transfer function, a delay-canonic filter struc-
-13-
ture has n unit delays, while a delay-canonic compensator
ture
has
n+I.
This fact has an interesting consequence when we
TQok at certain filter structures.
composed
struc-
One example is the
structure
of cascaded direct form I second-order sections.
fourth-order transfer function, such a
filter
structure
For a
has
6
unit delaysr and is not canonic (a canonic structure would have 4
delays):
(See Figure 3-4)
u(k)
1t
-y(k)
z
Z
d2
Figure 3-4:
z -l
z-I
Z
i
d.
-b 2
-b_
Direct Form I Filter Structure
-14-
-
ld
However,
such
a compensator structure for a fourth-order system
(n=4) would have only 5 delays, which is
tors:
[16]
for
compensa-
(See Figure 3-5)
-u
(k)
4
Figure 3-5:
This
canonic
-uk
Direct Form I Compensator Structure
again brings out some of the differences between the filter
and compensator cases.
In addition to representing compensator structures with
the
signal-flow graph, we need a mathematical notation for describing
In order to accomplish this, we will adapt the fil-
a structure.
ter
developed
notation
structures.
by Chan [18] to the case of compensator
Chan's notation accounts for the specific multiplier
coefficients in the structure, and for the exact sequence, or precedence, to the computations and quantizations involved.
Using y
and u to represent a filter output and input respectively, and
the
filter states
v
(delay-element outputs), the Chan notation can
be written as:
-15-
v(k)
v(k+l)
_
vV
u,1(6)
1u
q-l
y(k)
(rounded)
Each
(k)
coefficient
in the filter structure occurs
once and only once as an entry in one of the
dence to the operations
is
by
shown
the
The
matrices.
the matrix entries are ones and zeros.
of
remainder
Yi
ordering
The precethe
of
The operations involved in computing the intermediate
matrices.
(non-state) nodes
r
are
v(k )
(k) = Y1
(k)
~lu(k)
1 =
A
forth.
The
=
then r2(k)
first,
completed
T2
r (k) next,
and
so
parameter q specifies the number of such precedence
levels.
above,
For representing compensator structures as discussed
several
changes
definition:
are
necessary.
First, u and y are reversed in
u is now the compensator output, and
y
input.
the
But more importantly, the u(k) node is now a state of the compensator.
Inclusion
of these changes produces the following modi-
fied state space notation:
v(k+l)
u(k+l)
v(k)
=T 1 - u(k)(7)
qq-1'
(k
Examples of the modified state space representation can be
in Section 7 and in [16].
-16-
found
Notationally, it is also useful to define T,
nite
precision product of iq,
to be the infi-
Yq_,-f...' 'Yt and to partition it
as follows:
TCO_
_X [,1 1 . 1 2 ]
= CT 11
1
-(8)
where T11 is (n+l) x (n+l) and T12 is (n+l) x 1.
We can summarize the main point presented in this section as
th
follows: an n h-order filter transfer function can be impleth
(states-).
However, the nt -order
mented with n unit delays
th
(for an n -order system
transfer function of a compensator
model)
requires
unit delays.
a compensator structure representation with nil
This altered form of a structure reflects the exact
consideration of the computation delays that must
digital implementation.
-17-
exist
in
any
4.
DYNAMIC RANGE CONSTRAINTS AND SCALING
Researchers in digital signal
length
node signals within a
digital
arithmetic operations.
flow
have
treated
at
for scaling to reduce the dynamic range of the
need
the
processing
structure
employing
fixed-point
[1,3,18-21,24] The tradeoff between over-
roundoff quantization noise must be resolved before we
and
can measure and compare the roundoff noise
performance
of
dif-
ferent structures.
One seems
Several methods of scaling digital filters exist.
particularly
stochastic
appropriate
for use with stochastic systems.
(12) scaling procedure
[19,20,21] deals with the probSpecifi-
ability of overflow at each node within the structure.
cally,
if
the
This
input is zero-mean Gaussian noise with a
filter
given variance, then scalers are selected such that the probabil-
ity of overflow at each node is identical to the
overflow at the input.
variance
at
each
This
node.
is
Chan
probability
accomplished by equalizing
[183
has
applied
12
of
the
scaling
to digital filters, using his state space notation.
Certainly
one
obvious
approach
to the scaling of digital
compensators would be to follow the method developed by Chan, but
with alterations to include the
modified
-18-
state
space
notation
in
introduced
in
[16].
Any
In fact, we have shown this in detail
Section 3.
treatment
of
that
problem
However,- there is one important
a compensator as a stand-alone filter ignores
This
the remainder of the feedback loop through the plant.
have
resulting
when designing LQG systems
that
the
stable
even
though
itself,
in-and-of
procedure
would
fail,
*or
at
compensator
'is
the overall closed-loop
system performs as desired and is stable. "In
scaling
can
Specifically, there is no guarantee
implications.
serious
arises..
such
a
case
the
least produce incorrect
results.
compensator node will
Therefore, since the variance of each
depend
on the closed loop,
the closed-loop
compensator must be accounted for.
performance of
Thus we have adapted the
the
12
scaling procedure for digital compensators as follows. Recall the
plant and compensator equations (1) *and
(7).
To
compute
the
state
and
system's closed-loop performance, we must combine the
compensator equations into a
single
augmented
state
space for
the overall closed-loop system:
x(k+l)
u(k+l)J
wl(k)
=Ax(k)
v(k+l) +
=Av(k)
u(k) L
1 2 w 2 (k)]
(9)
where
I
A = ....
12
L
On -1
.....
11
-19-
and
0n
represents an all-zero n x n matrix and Y11' Y 12 represent
the unscaled compensator as partitioned in (8).
Given
this complete form (9),
let us now follow in general
Scaling
the basic 12 scaling procedure as applied by Chan 1181.
will
to diagonal transformations of the Yi matrices,
correspond
sinTilarity
the
analogous to
in
transformation
linear
system
Let the scaled compensator have the modified state space
theory.
Then
parameters Yq',.'l
t i(Si
=
for i=q, ... ,1
)-1
(10)
where
S
all
S
and
and
all
are
Si
i
0
i00
We now describe a modification of
1diagonal.
diagonal.
We
now describe a modification of
Chan's procedure for determining the elements of the S i.
The (unscaled) state covariance matrix
written
Z
for
(9)
be
in [19-21]) assuming a unit-variance Gaussian noise
(as
input y(k), as the solution of a steady-state Lyapunov
where Z is the (2n+l) x (2n+l) covariance matrix.
(
x' u v'
equation:
(11)
Z = AZA' + C
Z = E
can
')
-20-
and
· ~T 12E021i2j
0
At this point we must break from the usual 12 scaling procesince the plant states, of course, cannot be scaled (nor do
dure
The usefulness of
we wish to scale them).
us-
an
expression
node variances.
the compensator
for
MYlI is that it
gives
Let us
partition Z as follows:
(12)
z=
Z22
[Z12
Since we wish to equalize the
where Zll is nxn.
2
with the variance of y, let us compute a.
node
variances
Since y = Lx,
(13)
a = LZ1 L'
y
11
(7),
From
we
see that the node covariances will depend on
the covariances of v, u, and y.
Combining (1) and
(12), we
pro-
duce:
E)u
Using
(v
(7)
UYj y'[
and
1
222
212;L '
LZ12
LZ11L'"
(14)
(14), the covariance matrix of the nodes r
can be written as:
E(
1 ri
Z22
Z 12L']
LLZ12
LZ11L
=
-21-
1
i
(15)
In general, for the ith intermediate nodes,
Z2 2
E(r i r!)
= i
i
Z12,'L
LZ
LL,
'1
(16)
With this information, we can again apply the
by
Chan
methods
[18] and others to compute a set of matrices Ki
(Note the normalization by a
used
... Kq:
of (13).)
y
2O
LZ
El
X0
Yi2-l~
(17)
12
The diagonal elements of the K matrices represent the gains
the variance of y to the specific intermediate node variance.
this
definition,
Kq
must
then
from
By
represent the gains from the y
variance to the variances of the compensator state nodes v and u,
and should thus equal Z22 /ay.
[16,18].
As
in
This can
be
shown
to
be
true
[18], the scaling matrices Si1 shown in (10) are
computed as:
[Si]
(K
[Ki
)l/2
i=l,.-.q
(18)
for all j
Thus the scaled system K matrices will have unit
all
node
ance of y.
diagonals;
variances will be equal to each other and to the variThis ensures the
desired
equal
overflow
probabil-
ities.
Two
other points of difference between filter and compensa-
tor scaling arise.
Firstly, the issue of
-22-
A/D
and
D/A
scaling
must
be
addressed.
Note
the
that
In order to keep
actually scales the output node u.
loop
system
transfer
above scaling
procedure
the
closed
function unchanged due to this scaling, a
gain factor of p" must be included in the
D/A
conversion
of
u.
From (10),
(19)
[sq-l 2
n+l,n+l
beyond this point, how do we set the A/D and D/A scale
=
But
factors in general?
probability of overflow.
will
ability
procedure
The scaling
equalized
the
node
However, the actual value of that prob-
be determined by the way we employ the A/D conver-
ter, in other words, how we set its scale factor.
This
must
be
decided by the expected level of transients in the system (due to
the
control
action)
and
by
the
closed-loop RMS fluctuations
(variance) of y. This issue will be less complex for digital filters, where there is no external closed loop.
Again, to keep our
ideal closed-loop response unchanged, the D/A scale
counteract
kda
any A/D gain.
factor
The D/A scale factor could be written:
P/ka
(20)
Whatever is selected for kad' the compensator scaling
meter
matrices
must
Si
para-
do not change; the compensator and converter
scalings are independent.
Remember that in
comparing
different
compensator structures for a given ideal compensator design, only
the
scaling of the compensator itself will be important.
-23-
The final point of difference between filter and compensator
scaling
LQG configurations, as described in
the
will
have
set
have
the
to
to some non-zero output, and also to minimize
system
[23]
regulators
Such set-point
fluctuations around that value.
same parameter values as described in Section 2,
independent of the set point.
plant
2,
Thus the control action will try
tion of the control system.
will
Section
- in other words, reference inputs for the regulator por-
points
drive
Most of the
the nature of the control system.
concerns
However,
any
offset
DC
in
the
output y (the compensator input) certainly will affect the
probability of overflow at
internal
the
and
input
compensator
at
The 12
compensator nodes, and thus affect the scaling.
scaling procedure assumed a Gaussian zero-mean input to the
pensator.
This
all
com-
procedure would unfortunately not be valid with
DC inputs.
described
in
Sivan [23], where ur is the reference input.
If
Figure 4-1 presents the set-point LQG
Kwakernaak
and
we wish to drive the output y to yr,
H1
(l)Yr,
then
system
ur
must
be
set
to
where Hc(z) is the closed-loop transfer function from
ur to y:
Hc(z) = L(zI -
~ +
rG)
r
(21)
Unfortunately, this compensator has a
steady-state
value
of
y
is
scaling is not possible.
-24-
non-zero.
DC
input
Thus,
since
the
as we said, 12
IU
~i~~
Uri
'
plant
".
y
---
-[L-----
i
I
I
I
I
i
I
+
Kalman
I --I
I
CompensatorI
Figure 4-1:
Set-Point Compensator Configuration
However, we will show that we
system
can
set-point
Let us define i, n, and y
Figure 4-1 in another way.
of
the
describe
to be the deviations of the states, input, and output
steady-state DC values xo, uo, and yo.
their
and Y = y-y..
u-u
Thus
i
Using equation (1), the following
away
=
x-x0 ,
from
n'
relation-
ship must hold between these steady-state values:
=
x
y=
o
Ox
+
ru
(22)
Lx%
o
Now,
;
I
we can design an LQG compensator for the system devia-
tions, using a model for the deviations:
e(k+l) =
i(k) + r n(k) + wl(k)
(23)
x(k)
= LE
(k) + w 2 (k)
0--·
00' --
--
-
-
---
-25- --
-
-
This will, of course, produce the same parameters as the LQG
design with equation
system,
Now, if we take the resulting
(1).
and substitute for n and y, we produce Figure 4-2.
YO
uo
-
--
- --
-
-
--
- -
-
--_
-- -
- - -_
-
u
? I+ /
(-,')I
actual
Y
+
plant
n
I
I
I
- -
I
I
..
-
(Z,i;,y) plant
Compensator (3)
designed for
(th/,y) system
Figure 4-2:
Alternate LQG Set-Point Configuration
-26-
Thus
it
is possible to use an alternate LQG set-point con-
figur.ation where the compensator input Y has an average value- of.
thereby allowing us to apply stochastic
zero,
(-12)
scaling.
The
disadvantage to this alternate configuration is the necessity
two reference inputs which maintain the precise relation-
having
typically in the presence of
ship (22),
plant
(at least one pole at the origin), which is
series integration
very common occurence in control systems.
is
integrator
before
added
an
an actuator (part of the plant) to
r ))-ly 0 .
the
if
plant
has
figuration of Figure 4-2 need
YO=yr,
and
However,
since
DC
the
not
two.
u0 is
forced
to
zero.
gain
inputs
u
In
other
any series integration, the LQG conhave
only
one
reference
input,
Note that the configuration of Figure 4-1
does not change when the plant has integrator poles;
pensator
ef-
infinite if there are any open-loop integrator poles
in the plant (poles at z=l),
words,
the
To see
integrator pole on the configuration of Figure 4-2,
let us write uO as (L(I-L(I-f) Fris
a
In fact, frequently an
provide desensitivity to constant disturbances.
of
uncer-
parameter
disadvantage will vanish whenever the plant has a
This
tainty.
fect
of
both
com-
and y will still have DC components, and the
system as a whole still requires the reference
input
ur.
From
this point on, the Figure 4-2 configuration is assumed so that 12
scaling can be applied.
-27-
5.
ROUNDOFF NOISE ANALYSIS
In
this section we will develop a method for evaluating the
effects of roundoff quantization noise in
As
stated
in
the
digital
introduction, we will
adapt a method used in
evaluating digital filter roundoff effects [18-21].
digital
filtering
applications,
length of the internal registers
compensators.
As
in
most
we will assume a uniform wordof the
compensator.
We
will
also not consider the effect of the input A/D converter quantization on system performance, since this will be independent of the
structure chosen.
Typically, many less bits are required for the
converters, as compared to the internal registers.
In
most
filtering applications, the filter output variance
due to roundoff quantization is taken to be the measure
formance
for the structure [3,18-21,25].
each
We can
repre-
roundoff operation on some value x as x plus additive
white noise, uniformly distributed between
quantization
step
size.
+
where
A
is
independent [3].
the
Furthermore, in a structure with many
roundoff operations, all such additive noises are assumed
tion
per-
The analysis of round-
off effects is based on the following assumption.
sent
of
to
be
Given this statistical description of quantiza-
effects, one can easily apply linear system analysis tech-
niques to compute the output node variance due to roundoff.
In examining the effects of roundoff
tors,
we
on
digital
compensa-
can make the same assumptions concerning the quantizer
-28-
However, as with scaling, we cannot
model.
procedures
blindly
the
follow
Again, the performance of the com-
used for filters.
pensator will depend on the entire closed-loop system, and not on
the compensator alone.
compen-
Furthermore, the variance of the
output node is not the best measure of performance for the
sator
control system.
original
metric
Instead, it is much more reasonable to
use
the
which figured in the ideal compensator design -
the quantity J shown in (2).
Curry
Some work has been done previously along elated lines.
[7] has considered the second moment of the system
output
error
due to rounding for a specific sampled-data control system with a
direct
form
II
compensator structure.
Knowles and Edwards
also used the additive white noise model for generating
a
[6]
bound
on the quantization noise effects of direct form II, cascade, and
parallel
compensator structures.
Sripad [12] has considered the
increase in the performance index J due to
additive
white
model,
noise
but
did
scaling issue or the general concepts of
and representations.
be
more
general
roundoff,
not
using
the
address either the
compensator
structures
The results we present in this section will
since
we can consider any type of compensator
structure, using the modified state space notation, and since
we
have accounted for the necessary scaling operation.
The following roundoff analysis procedure results if we consider
J
to
be
the performance measure, and if we consider the
closed-loop nature of the
control
-29-
system.
Let
us
model
the
roundoff errors after each compensator multiplication as additive
A/D
structure-independent
(The
noise..
will
contribution
be
The scaled, augmented system of plant and compensator,
ignored.)
sources,
roundoff
including all the internal
can
written:
be
(see (9) and (10))
(k
ii' W
<
(k+l)
(k+l) I
(k+l)
q
e ( =
++(k
q
A(k)+(k)+.
...
q
)+ 1
2
k da k2 (
d2
(24)
L
The
(D 0
-
where. A =
l12
Lk
irkda
11
'
ad
will refer to scaled quantities, Si(k) will represent
tilde
the noise vector due to the product quantizations associated with
the precedence level matrix Ti, and q will be the number of
cedence levels.
be
Since this is a linear system, superposition can
The noises wl(k) and w 2(k) are present even in the
applied.
idealized infinite precision system - they are the
that
pre-
the ideal design value of J.
produce
uncertainties
Thus if we treat the
Ei(k) noises alone, our analysis will yield the inease
to internal roundoff quantization.
Thus
our
system
in J due
model
for
roundoff analysis is:
x(k)
x(k+l)
(k+l)
A
(k)
s (k+q1
a(k)
q
+
L
-30-
...
=2 q-30-
()
this model, the resulting (scaled) state covariance matrix
Given
equation:
can be computed as in Section 4, by solving a Lyapunov
Z
AZA
+
o
(26)
where
q q--L ~ q +
i-={Aq+jqAq_l
The
A.
matrices
q q-1 q-2q2 i4l~
Tq"'-+ ''+
Wq.
'Y 2 A 1N2'qj
are diagonal matrices whose (j,j)th entry
th
row
equals the number of non-integer coefficients in the jth
Ti,
of
that is, the number of roundoff error sources associated with
the
jth
component of r i.
This expression assumes that roundoff
occurs after every non-trivial product.
ble
produce
to
double-precision
precision addition.
It would also be
and
products,
do
possi-
a double-
A single roundoff quantization would then be
needed to generate the new node value, in which case all the nonzero entries of A i would be ones. This method requires more hardware, but results in a reduced roundoff noise effect.
To relate the state covariance
index
J
in
matrix
to
the
performance
(2 ), we must rewrite J as described in [26].
yields the following expression in terms of the unscaled
ties:
-31-
This
quanti-
J = trace(Qxx') + 2 trace(M ux') + trace(R uu')
where
On
n
On
On
0O
M'
O
R
T-
Now
off noise,
x(k)
that
directly.
(27)
TZ
= trace
us apply this to the increase (dJ) in J due round-
let
and then relate dJ to the scaled covariance Z.
is
scaled
not
and
The only scaled quantity
computation is u.
However, u =
does
v(k)
kda
not
Note
figure into dJ
enter into the
1
u, or u = pkad u. Thus, subthat
does
stituting into (27), and considering dJ;
(28)
dJ = trace TZ
where
Q
n
On
' n _daM'
Thus
our
analysis
kaM
kdaM
n
da
0
procedure
roundoff involves solving
for
(26) for
(28).
-32-
the
Z,
increase
and
dJ
evaluating
due
to
equation
6.
MINIMUM ROUNDOFF NOISE STRUCTURES
Mullis
and
Roberts
[19,20]
and Hwang [211 have developed
techniques for computing structures with minimum
effects.
The
most
to
for
(q=l) second-order sections, each of which
minimum
roundoff noise.
will have about 4n coefficients, where n
is
which
overall
be composed of a parallel or series combination of
one-precence level
optimized
noise
of these structures is the block
In this filter structure, one assumes the
optimal form.
structure
practical
roundoff
is
The resulting structure
is
the
filter
order,
about twice the number for a direct form II cascade or
parallel structure.
Certainly we could treat a compensator as a stand-alone filabove-mentioned
ter and
follow
the
minimum
roundoff noise structure.
procedure
to
generate
a
However, as explained in Sec-
tions 4 and 5, in order to obtain an accurate picture of roundoff
effects we must consider the closed-loop
system.
Thus
certain
changes will be required in the procedure.
For
filters,
the
involves the computation of a
procedure
block transformation matrix, which when applied to
the
original
unscaled filter structure, generates a new structure with minimum
roundoff noise.
K1
and
W 1.
The keys to this transformation are the matrices
K 1 is related to the filter scaling procedure dis-
cussed in Section 4,
reflects
the
gain
and
(in
W1
is
a
"noise
gain"
matrix.
It
variance) from each noise source to the
-33-
output node.
to
node
Recall that Ki reflects the
Rober-ts, Hwang technique to digital
minimization
if
we
for
technique
input
compensator
noise
roundoff
can compute K 1 and W 1 matrices that account
for the closed-loop control system.
the
the
Thus, we can apply the Mullis and
internal node.
each
from
gain
finding
in
K1
We
have
already
specified
the section on scaling
(see
In this section, we will develop an expression for W 1 .
(17.)).
Since W 1 represents the gain from roundoff noise sources
filters), we need a matrix which reflects
(for
variance
output
to
the gain from these sources to the performance index increase dJ.
The following will be adapted from Mullis and Roberts, and Hwang.
Let us rewrite (26) in terms of the unscaled
eters T11 and T12:
= TAT -T-1
1
compensator
param-
(As in [173 this assumes q=l.)
AT
+
A
(29)
_
where T includes the scaling matrix S1 ;
T =
n
S1
I
is the n x n identity matrix, and
I
A
-
A__
-
before.
Ir
n____ _
11
12
as
0
(We have assumed kad is 1; kad would at any rate not
affect the optimal structure.)
the
of
definition
just equals T
1
ZT
1.
Z
By manipulating
and
using
and T, we can recognize that the matrix Z
Substituting in (29), we have
ZAZA'
+Z = AZA' +
(10)
(30)
(30)
°
-12-34-
The expression for the
increase
in
performance
index
due
to
roundoff noi.se for the scaled system can also be written in terms.
of the unscaled covariance matrix Z:
(See (27).)
= trace {TT- ZTI}
dJ = trace {TZ}
= trace {T1TTZ}
= tracetTZl
Using
an
(31)
adjoint Lyapunov equation (see Appendix 1), equa-
tions (30) and (31) can be replaced by (32) and (33):
dJ
trace
12
=
12
tL
{
AlSl}
ts
W
(32)
alJ
where
W = A'WA +
(33)
T
The trace expression in (32) can be simplified if we
define
W 1 to be the lower right-hand (n+l) x (n+l) portion of W:
A2
dJ = -
W1
is
trace
(34)
W1
(-W1
{
12
the matrix needed to apply the Mullis and Roberts, Hwang
techniques for
generating
Using
W1
K1
and
as
the
optimal
presented
transformation
above and in Section 4, we can
follow the remainder of their technique to generate
minimum roundoff noise compensator structure.
be
quite
different
matrix.
a
one-level
It may, of course,
from that resulting from a treatment of the
compensator as a stand-alone filter.
-35-
Conceptually,
extended
to
the
multiple
described
technique
above
could
be
However, the iterative structure
levels.
optimization procedure developed by Chan [18] for filters is
far
more useful for minimizing roundoff noise for general structures.
In Chan's method, an initial structure is subjected to continuous
transformations,
each
of
reduces some overall objective
which
function, given constraints on certain coefficient
filters,
an
equation
similar
Almost any of the coef-
ficients of the initial structure can be held
process.
fixed
during
the
Thus we can use this method to trade off
an increase in the number of
The
For
to (34) (but without the closed-
loop) is used as the objective function.
optimization
values.
coefficients
versus
performance.
extension of this very useful procedure to low-noise compen-
sator design is presented in [16].
-36-
7.
AN LQG EXAMPLE
A specific sixth-order LQG example was
could
examine
so
that
f16].
This
example
adapted from the longitudinal control system design done for
the F8 digital fly-by-wire
and
parameters
plant
continuous-time
fighter
[27].
The
performance
index
parameters
0
-15.77
are given
below:
Continuous Time System Parameters:
5.7x10
-0.6696
A
=
we
roundoff noise performance of several struc-
the
tures using the techniques developed above
was
chosen
4
0
-0.01357
1
-1.2x10
-9.01
-14.11
4
-32.2
-1.214
0
0
-0.433
0
-0.1394
0
1
0
0
0
0
0
0
0
0
0
-12
12
O
0
0
0
B=
CO
0
0
C =
[1
0.003091
0
0
31.28
0
1l
1
3.592
-37-
0]
0
Continuous-Time Performance Index Parameters
r=6.637
=
Q
0
0
2.6554x10-7
0
2.686x10-3
0
0
0
3.085x10
Lo0
R =
0O
0
2.686x10
3
27.174
0
-4
3.085x10 -4
0
0
3.121
0
0
7
0O
0O
0
0
0
0
0.3585
0
O
0
0
27.174
3.121
0O
0
4j
5.252
Continuous-Time Noise Covariances
-1=
=
diag[O
0
0
0
10 - 6
10
-
6
0.0018441
This continuous-time system was discretized at a sample rate
of 10 HZ and the optimal regulator and
The
double-precision parameters 0,
K - can be found in
r,
Kalman
L, Q,
filter
M,
R,
designed.
e 1 , e,
G, and
[16].
Before discussing the different structures tested,
it
will
be helpful to mention the A/D noise contribution for this example
(independent of structure).
in J
due
similar
to
.to
this
that
If we allow a five per cent increase
single. noise. source,. then
Outlined
....
in
-38-
Section
5
.a
procedure
requires
a 4.98
bit
A/D
the
next
be
fix
wordlength.
largest
Including a.sign
integer
value, our
selecting
actual wordlength would
for
bears out the need
As will be shown, this
bits.
and
bit,
shorter A/D wordlengths as compared to internal wordlengths.
compen-
Five types of structures for implementing .the ideal
sator transfer funtion (4) were examined.
direct
filter-ing-based
form
These were the digital
II structure for compensators (see
form
Section 3), several cascade structures consisting of direct
II
or
direct
form
I
sections, several parallel
second-order
structures composed of direct form II sections,
roundoff
minimum
noise
structure,
a
and a 'default' compensator
structure which we will call the simple structure.
cases,
In
alter
generally
4.
five
all
before
will indicate the initial design coefficients
we
implementing the 12 scaling procedure of Section
This
will
the initial values, and frequently create a few
extra scaling coefficients.
the
block-optimal
In any case where a unity
entry
in
unscaled structure has become a multiplier coefficient (non-
unity, non-power of two) when scaled, we have indicated this with
an asterisk.
The first structure we will examine is the
compensator
structure.
the
output
form II structure come
function
(35).
II
Figure 3-3 presents a signal-flow graph
for the 4th-order case (n=4).
preceding
direct form
Note the
presence
of
the
delay
The 12 coefficients of the direct
node.
directly
from
the
unfactored
transfer
Its modified state space representation (two
precedence levels) is shown in (36).
-39-
H(z)
alz
+a 2 z -2+a3 z
1l'~
2
+blz-l+b2z
+a 4 z
+az-D+a6 .z -6
(35)-
3
4
5
6z
+b3z3
+b4z-4+b5z-5+b6z-6
1
0
0
0
0
0
o
1
o
o
o
0]
ol
0
0
0
1
0
O
0
0
0
0 1
0
0
0
0
a6
a5
a4
a3
0
0
01
a2 a
(36)
0
1
0
0
0
0
0
0
O
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
O
0
0
0
1
0
0
0
O
O
0'
0
1
0
0
-b 2
-b 1
0
1*
O
sb
-b
5
-b6
When this structure is
-b 3
12 scaled, the
unity
value
marked
with an asterisk will become a 13 th non-unity coefficient.
The second type of structure, the cascade, derives its coefficients
from
a
multiplicative
second-order sections.
factorization
into
The factored transfer function,
3
series
assuming
direct form II sections, is shown in (37).
H(z)
(dz-l+d2z 2)(l+d 3 z 1+d4 z 2 ) (l+d5 zl1 d6 z 2 )
2
3
4
5
6
'+
-1
(l+clz l+c2 z -2)(l+c 3 z-1-+c4 z -2 )(l+c5z -1+c6z -22)
-40-
(
(37)
a structure will have 12 coefficients, 4 precedence levels,
Such
and will require 3 additional scaling multipliers when
1 2 -scaled
(see 38)).
7r
4
1
0
0
0
0
0
1
-O
0
0
0
0
0
o
6
0
1
0
0
0
0
0
1
0
0
0
0
O
O
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
d6
d5
1*
(38)
1
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
O
0
0
0
0
0
1
0
1
0.0
0
0o
0
0
1
0
0
0
0
0
0
0
1
0
01
O
O
0
0.
0
0
0
0
1
0
0
0
0
0
0
d4
d
3
7!1=
0
-c4 -C3
1d2
-c 6 -c5
0
5 d2
1 -0
0
0
0
0
0
0-
O
.1
0
0
0
0
0
O
0
0
1
0
0
0
0
°
0
°
0
°
0
0
1 1 =0
00
00
0
0
0
0
0
1
0 0
-c 2 -C 1
0
0
0
0
0
-41-
,3
-4
1*
1 0
0
0
1
O:
0
0
d
0
Details
are available in [16].
tures can be formed
ferently,
or
by
Note that several cascade struc-
simply by grouping the poles and zeraos
ordering
the sections differently, or even by
implementing the sections differently [16].
ters,
the
typical
digital compensatot
second-order
poles together.
using
direct
ordering.
presence
of
more
we
Unlike digital
than
further complicates
section,
dit-
the
fil-
one real pole in a
cnoices.
In
each
will also have to pair different real
Two cascade structures will be considered,
both
form II sections, but with a different pairing and
In addition, a direct form I structure will be tested.
Its modified state space is given in [16].
The third type of structure, the parallel form,
to
corresponds
a partial-fraction expansion of (35), which allows the use of
parallel first and/or second-order sections.
parallel
direct
form
II
sections,
before
of
5
4 first-order and 1 second
order, the expanded transfer function
cients
For the case
(also
having
12
coeffi-
scaling) is shown in (39), and its modified state
space is given in (40):
H(z)
=
e1
+e2
ez1
ez-+e 2 z2
+ e3z 1
l+c2z-l+c2-2
+dz-1
e zea-1
e4z
+dz-1
-42-
-42-
-1
e 5z
e z-1
e 6z
6z
l+d
(39)
0
0
0
0
0o
0
1
0
0
0
0
0
0
1
0
0
0
O
O
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
e2
el
e3
e4
e5
e6
i
2
=
(.40)
0
1
0 10
0
0
0
0O
-C1
0
0
0
0
0
1*
0
0
-C 3
0
0
0
0
1*
0
0
0
-c
0
0
0
1*
0
0
0
0
-C 5
0
0
1*
,0
0
0
0
0
0
1*
-C
2
-
6
To scale this structure, 5 additional scalers (one per
tion)
are
required.
Of
course, there are many other parallel
forms, depending on how one groups the poles and zeros,
one
implements
each
section
[16].
We
and
The
two
how
will also examine two
parallel structures having only second-order direct form II
tions.
sec-
sec-
structures differ only in the pairing of the 4
real poles of the compensator.
The fourth type of structure
tested
is
a
parallel
block
optimal minimum roundoff noise structure, as described in Section
-43-
6.
This structure will have 25 coefficients, many more than the
previous 3 types.
Its modified state space is given in (41), and
precedence
shows three second-order sections and only one
level
[16].
fl
f2
0
0
0
0
0
f3
0
0
415
0
0
f
f
0
fl
f1
f24
0
f25
f
f24
f25
f2 6
(41)
0
f
if
19
The
last
0
0 0
if
if
20
21
0
ff 1 6
f18
22 f23
structure
filtering literature.
we
f26
will consider has no analog in the
structure
This
would result if
we tried to directly implement the compensator equations shown in
(3).
It is important to analyze this type of 'simple'
because
a
naive
structure
implementation might employ it more or less by
default, as it is the natural result of the ideal design.
the equations shown in (3), we must
implementable
form.
(Note
first
rewrite
them
in
an
that u(k+l) cannot be computed from
x(k+l) since some computation delay must be allowed for the
tiplication by G.)
Taking
Substituting for £(k+l), we get:
-44-
mul-
x(k+l) = Ix(k) + ru(k) + K(y(k)
- Lx(k))
(42)
u(k+l) = -Gbx(k)
These
ture.
- Gru(k)
equations
- GK(y(k) - Li(k))
do represent a feasible compensator struc-
Its modified state space representation would have 3
pre-
cedence levels:
=
= |
3h2IW
-:]_0
T,
! r lK
]---
The
The
important
!(43)
isa ------
0
-0
!°!%
_ -L
disadvantage
of
this
type
of structure can be
For a sixth-order example, there will
easily seen from (43).
be
up to 60 coefficients.
The
actual scaled coefficient values for these 9 structures
Table 7-1 summarizes the
are presented in [16].
results
for
these structures.
roundoff
noise
As mentioned previously, the A/D
noise contribution is the same for
all
and
structures
is
not
included.
structure
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
direct form II
parallel direct form II
parallel direct form II
parallel direct form II
block optimal parallel
cascade direct form II
cascade direct form II
cascade. direct form I
simple, default structure
Table 7-1:
The
levels,
'levels'
levels
N
2
2
2
2
1
4
4
3
3
13
17
15
15
25
15
15
14
50
wordlength
dpa
spa
19.65
8.05
10.18
14.74
7.88
15.69
10.51
15.52
9.01
18.25
7.45
9.39
13.94
7.06
14.68
9.47
14.36
7.54
Roundoff Noise Results
column
and the 'N' column
lists
lists
-45-
the
the
number
number
of
of
precedence
coefficents
including
The roundoff noise results
in the structure.
scalers
are presented in terms of the number of signal
bits (wordlength)
that are required to hold the increase in J due to product roundnoise to 5% of the ideal value.
off
Two wordlengths
include the sign bit.
are
presented
The left-hand column (larger) corresponds to
structure.
of
Again, these numbers do not
roundoff
for
the case
after every nontrivial multiplication, and then the
use of single-precision adders, while the right-hand column
responds
each
cor-
to the case of double-precision adders and quantization
after addition.
From Table 7-1, we can see that the different pole
with parallel structures (c) and (d) produced results
associated
that differed by 4.5 bits.
poles
pairings
Placing the near-unit magnitude
real
of the compensator in different sections was significantly
Of the two similar cascades
superior.
two
poles
(f) and (g), the one
with
in different sections required 5.2 fewer
these
same
bits.
As with filters, the pairing/ordering issue is clearly not
a trivial question.
sections,
(h),
Also note that the cascade of direct form
which uses the same pairing/ordering as (g), has
nearly identical performance, but 1 less precedence level
less
I
coefficient.
Unlike
digital
filter
and
1
applicationsi it is
worth considering for feedback compensator implementation.
Structure (b), the combination of
parallel
sections,
first-
and
second-order
with its 17 coefficients, outperformed every
-46-
other structure except the block optimal.
coefficients
of
the
block
Even so, the
extra
8
optimal structure with second-order
sections only gained 0.2 bits of performance over this structure.
Thus, when evaluating different structures, it
know
the
important
to
block optimal result (for various pairings) so that we
can judge whether a suboptimal structure like
enough.
(b)
is
effective
In this case it clearly is.
As
expected
from
direct form II has very
important
it
cients.
the
literature
on digital filters, the
poor noise performance.
It is also very
to note that even though the simple structure (i) per-
formed fairly well
(b)),
is
has
(but still 1
bit
worse
(comparatively)
far
too many multiplier coeffi-
than
the
structure
Were such a structure to be built, this surfeit of coef-
ficients would
require
either a slower sampling rate, or a more
expensive compensator, than (b) or (e), which would outperform it.
The second wordlength column in table
7-1
shows
the
gain
possible when using double precision adders and fewer quantizers.
Depending
on the structure tested, a savings of from 0.6 to 1.47
bits was realized.
tify
the
Whether this small savings is enough to
higher-precision
jus-
adders will depend on the particular
application.
-47-
8.
SUMMARY
We.have shown that the implementation of
compensa-
digital
can benefit greatly from the concepts developed for dealing
tors
in
effects
digital
and
structures
with different computational
finite
wordlength
However, due to the nature of the
filters.
Thug
control problem, new issues arise that must bae cdrTdered.
we
had. to adapt and extend these digital signal processing
have
concepts and techniques.
unaccounted
First, the importance of
system
that
required
structure.
included
we
rethink
to
shown
be
When dealing with the LQG
ideal
the
proce-
design
resulting
compensator
have its output node as a state, that is, the
output of a unit delay element.
Thus the
used
structures
com-
for filters must be modified when applied to compensators.
The concept of scaling for fixed-point digital
affect
technique
response.
compensators
Since the entire closed-loop system
also had to be reconsidered.
will
control
in our model so that the closed-loop system performs as
dure. Any structure for implementing
monly
a
Any computational delay that will be present must
control problem, this led to a specific
be
in
notion of a compensator
the
close to the ideal design as possible.
can
delay
the internal compensator node variance, any scaling
must
take
into
account
the
overall
closed-loop
That is, it is inappropriate to treat the compensator
as a stand-alone filter.
The
12
-48-
scaling
technique
was
thus
for
adapted
LQG
In addition, the form of a set-
compensators.
important
in
that would be effective.
An
alternate set-point configuration that would allow the use of
12
point regulator control system
the
determining
type
of
was
scaling
shown
to
be
scaling -was proposed.
scalirrg has
Once
beer accomplished, we can begin to analyze
performance.
the effects of quantization noise on control system
As
with
scaling,
alone filter.
the
include
the compensator cannot be treated as a stand-
Thus we adapted the filter analysis techniques
to
plant and feedback loop, and to use the natural LQG
performance index as our figure of merit rather than output noise
variance due to roundoff.
Minimum
roundoff
noise
structures,
such as those developed for filters by Mullis and Roberts, Hwang,
and
also Chan, were treated next.
These structural design tech-
niques also needed modification to include
the
effects
of
the
closed-loop on system performance.
Finally, several example structures for a single LQG compenwere
sator
scaled and their roundoff performance compared.
pairing and ordering issues involved with
parallel
and
cascade
structures were shown to be even more complex for compensa-
type
tors, due to the number of real poles that are common in
system
compensators.
LQG
compensator.
control
It was also shown that the default type of
structure for LQG controllers is a poor choice of
the
The
structure
for
Its extremely large number of multiplier
coefficients would impose unnecessary speed limitations
-49-
or
com-
plexity
embedded.
on the compensator and the control system in which it is
In fact, the result points out the need for
ing the issues we have dealt with in this paper.
-50-
consider-
APPENDIX 1
If we take the trace of the product of two matrices to
an inner
product on the space of matrices, and
be
i to be a matrix
operator, then:
(7r(X)
trace
where
U)
= trace
(X
ir is the adjoint operator of
operator
XT
can be derived from
trace ((X-AXA')U)
(A.1)
(U))
r. For
T(X)=X-AXA', the
(A.1):
= trace
(XU)-trace(AXA'U)
(XU)-trace(XA'UA)
trace
=
X
(A.2)
= trace(X(U-A'UA))
Thus
XT
(U)=U-A'UA.
As used in section 6, the Lyapunov equation (30)
trace (31) were replaced by the equivalent equations
Relating this to the derivation above:
X=
Z
U=W
r(X)
A12
120
AlS
* (U) = T
-51-
2
and
(32) and
the
(33).
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-54-