OPTIMAL SPARSE CONTROLLER STRUCTURE WITH MINIMUM
ROUNDOFF NOISE GAIN
Jinxin Hao, Teck Chew Wee, Lucas S. Karatzas and Yew Fai Lee
School of Engineering, Temasek Polytechnic, 529757, Singapore
Keywords:
Roundoff noise gain, Sparse controller structure, Optimization, Direct-form II transposed (DFIIt) structure.
Abstract:
This paper investigates the roundoff noise effect in the digital controller on the closed-loop output for a
discrete-time feedback control system. Based on a polynomial parametrization approach, a sparse controller
structure is derived. The performance of the proposed structure is analyzed by deriving the corresponding ex-
pression of closed-loop roundoff noise gain and the problem of finding optimized sparse structures is solved.
A numerical example is presented to illustrate the design procedure and the performance of the proposed
structure compared with those of some existing well-known structures.
1 INTRODUCTION
Finite word length (FWL) effects have been a well
studied field in the design of digital filers for more
than three decades (Mullis and Roberts, 1976),
(Hwang, 1977), (Roberts and Mullis, 1987), (Gevers
and Li, 1993). However, they have received less at-
tention in the area of digital control. Nowadays, many
researchers have recognized the importance of the nu-
merical problems caused by FWL effects in digital
controller implementation. The optimal FWL con-
troller structure design (Fialho and Georgiou, 1994),
(Li, 1998), (Wu et al., 2001), (Yu and Ko, 2003) has
been considered as one of the most effective methods
to minimize the effects of FWL errors on the perfor-
mance of closed-loop control systems. The basic idea
behind this approach is that for a given digital con-
troller, there exist different structures which have dif-
ferent numerical properties, and the optimal structure
problem is to identify those structures that optimize a
certain FWL performance criterion.
Generally speaking, there are two types of FWL
errors in the digital controller. The first one is the per-
turbation of the controller parameters implemented
with FWL, and the second one is the rounding er-
rors that occur in arithmetic operations, which are
usually measured with the so-called roundoff noise
gain. The effects of roundoff noise have been well
studied in digital signal processing, particularly in
digital filter implementation (Wong and Ng, 2000),
(Wong and Ng, 2001). However, it was not un-
til the late 1980s that the problem of optimal con-
troller realizations minimizing the roundoffnoise gain
was addressed. The roundoff noise gain was de-
rived for a control system with a state-estimate feed-
back controller and the corresponding optimal real-
ization problem was solved in (Li and Gevers, 1990),
while the roundoff error effect on the linear quadratic
regulation (LQG) performance was investigated in
(Williamson and Kadiman, 1989) and the optimal so-
lution was obtained by Liu et al (Liu et al., 1992). The
problem of finding the optimum roundoff noise struc-
tures of digital controllers in a sampled-data system
has been investigated in (Li et al., 2002).
It has been noted that the optimal controller real-
izations obtained with the above design methods are
usually fully parametrized, which increase the com-
plexity for real-time implementations. From a prac-
tical point of view, it is desired that the actually im-
plemented controller have a nice performance against
the FWL effects as well as a sparse structure that
possesses many trivial parameters
1
which produce no
FWL errors. As far as we know, a few results have
been published on the sparseness issue for the con-
troller structure design (Li, 1998), (Wu et al., 2003),
however, it is noted that in these approaches, sophisti-
cated numerical algorithms were utilized and the po-
sitions of trivial parameters were not predictable. In
(Hao et al., 2006), we proposed two sparse structures
1
By trivial parameters we mean those that are 0 and ±1,
other parameters are, therefore, referred to as nontrivial pa-
rameters.
13
Hao J., Wee T., Karatzas L. and Lee Y.
OPTIMAL SPARSE CONTROLLER STRUCTURE WITH MINIMUM ROUNDOFF NOISE GAIN.
DOI: 10.5220/0002170900130019
In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2009), page
ISBN: 978-989-674-001-6
Copyright
c
 2009 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

for digital controllers, which have some degrees of
freedom that can be used to enhance the closed-loop
stability robustness against the FWL effects.
In this paper, a new sparse controller structure is
derived by adopting the polynomial parametrization
approach in (Hao et al., 2006) and using the l
2
-scaling
scheme. This structure can be considered as a l
2
-
scaled generalized DFIIt (direct-form II transposed)
structure. The expression of the roundoff noise gain
is derived for a closed-loop feedback control system,
in which the digital controller is implemented with the
proposed structure. The problem of finding optimized
sparse structures is solved by minimizing the corre-
sponding closed-loop roundoff noise gain. A numeri-
cal example is given to illustrate the design procedure,
which shows that the proposed structure beats the tra-
ditional DFIIt structures greatly in terms of roundoff
noise performance, and furthermore, outperforms the
fully parametrized optimal realization (Li et al., 2002)
in terms of both roundoff noise gain and computation
efficiency.
2 A SPARSE CONTROLLER
STRUCTURE
Consider a discrete-time feedback control system de-
picted in Fig. 1, where P
d
(z) is the discrete-time plant
and C
d
(z) is a well-designed digital controller. The
controller can be represented by its transfer function
which is parametrized with {ξ
k
,ζ
k
} in the shift oper-
ator z:
C
d
(z) =
∑
K
k=0
ζ
k
z
K−k
z
K
+
∑
K
k=1
ξ
k
z
K−k
. (1)
This controller can be implemented with many differ-
ent structures, such as the direct forms or the follow-
ing state-space equations:

x(n+ 1) = Ax(n) + Bu(n)
y(n) = Cx(n) + du(n)
(2)
where x(n) ∈ R
K×1
is the state variable vector and
u(n), y(n) are the input and output of the controller
C
d
(z), respectively, while r(n) is the input signal of
the closed-loop system. R , (A, B,C,d) is called a
realization of C
d
(z) with A ∈ R
K×K
,B ∈ R
K×1
,C ∈
R
1×K
and d ∈ R , satisfying
C
d
(z) = d +C(zI − A)
−1
B.
Denote S
C
as the set of all the realizations: S
C
,
{(A,B,C,d) : C
d
(z) = d +C(zI − A)
−1
B}. Let R
0
,
(A
0
,B
0
,C
0
,d) ∈ S
C
be an initial realization. It can be
shown that S
C
is characterized by
A = T
−1
A
0
T, B = T
−1
B
0
, C = C
0
T (3)
where T ∈ R
K×K
is any nonsingular matrix. Such a
matrix T is usually called a similarity transformation.
Once an initial realization R
0
is given, different con-
troller realizations correspond to different similarity
transformations T.
-
r(n)


∑
-
C
d
(z)
y(n)
u(n)
6
P
d
(z)
- -

Figure 1: A discrete-time feedback control system.
2.1 A Generalized DFIIt Structure
Based on the approach in (Hao et al., 2006), we define
ρ
k
(z) ,
z− γ
k
∆
k
, k = 1,2,...,K, (4)
where {γ
k
} and {∆
k
> 0} are two sets of constants to
be discussed later. Let
p
k
(z) ,
K
∏
m=k+1
ρ
m
(z), ∀k ∈ {0,1,···,K − 1},
p
K
(z) , 1. (5)
It can be shown that (1) can be rewritten as
C
d
(z) =
β
0
p
0
(z) + β
1
p
1
(z) + ... + β
K
p
K
(z)
p
0
(z) + α
1
p
1
(z) + ... + α
K
p
K
(z)
, (6)
where
¯
α , [1 α
1
··· α
K
]
T
= κ
¯
T
−T
p
[1 ξ
1
··· ξ
K
]
T
¯
β , [β
0
β
1
··· β
K
]
T
= κ
¯
T
−T
p
[ζ
0
ζ
1
··· ζ
K
]
T
with κ=
∏
K
k=1
∆
−1
k
such that
¯
α(1) = 1 and
¯
T
p
an upper
triangular matrix whose kth row is formed with the
coefficients of p
k−1
(z) defined above. Equation (6)
implies that the controller transfer function C
d
(z) is
reparametrized with {α
k
} and {β
k
} in the new set of
polynomial operators {p
k
(z)}.
It follows from (5) and (6) that the output of the
controller can be computed with the following equa-
tions
y(n) = β
0
u(n) + w
1
(n)
w
k
(n) = ρ
−1
k
[β
k
u(n) − α
k
y(n) + w
k+1
(n)]
w
K
(n) = ρ
−1
K
[β
K
u(n) − α
K
y(n)] (7)
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
14

where w
k
(n) is the output of ρ
−1
k
(z) and can be com-
puted with the structure depicted in Fig. 2. Fig. 3
shows the corresponding structure to (7). For conve-
nience, a structure defined by Fig.s 2 and 3 is called a
generalized DFIIt structure, denoted as ρDFIIt. This
structure possesses {α
k
, β
k
, ∆
k
} and a set of free pa-
rameters {γ
k
}. For a given digital controller C
d
(z),
there exists a class of such structures, depending on
the space within which {γ
k
} take values. Clearly,
when γ
k
= 0, ∆
k
= 1, ∀k, Fig. 3 is the conventional
direct-form II transposed (DFIIt) structure.
direct-form II transposed (DFIIt) structure.
- l
+
-
z
−1
-
∆
k
- w
k
(n)
x
k
(n)

γ
k
6
1
Figure 2: A realization of ρ
−1
k
(z) defined in (4).
u(n)
β
K
?
?
- -
ρ
−1
K
f
+
−α
K
6
6
β
K−1
?
?
- -
ρ
−1
K−1
f
+
−α
K−1
6
6
β
K−2
?
?
···
-
···
f
+
−α
K−2
6
6
···
-
β
2
?
?
--
ρ
−1
2
f
+
−α
2
6
6
β
1
?
?
--
ρ
−1
1
f
+
−α
1
6
6
β
0
?
?
f
+
?
y(n)
Figure 3: Block diagram of the ρDFIIt structure.
with
sion. Then
Figure 3: Block diagram of the ρDFIIt structure.
With {x
k
(n)} indicated in Fig. 2 as the state vari-
ables and x(n) denoting the state vector, one can
find the equivalent state-space realization, denoted as
(A
ρ
,B
ρ
,C
ρ
,β
0
), of the proposed ρDFIIt structure:
C
d
(z) = β
0
+C
ρ
(zI − A
ρ
)
−1
B
ρ
(8)
with B
ρ
=
¯
V
β
− β
0
¯
V
α
, where
¯
V
x
, [x
1
··· x
k
··· x
K
]
T
for x = α,β, C
ρ
= [∆
1
0 · · · 0 0], and
A
ρ
,






a
11
∆
2
0 ··· 0 0
a
21
γ
2
∆
3
··· 0 0
.
.
.
a
(K−1)1
0 0 ··· γ
K−1
∆
K
a
K1
0 0 ··· 0 γ
K






with a
11
= γ
1
− ∆
1
α
1
and a
k1
= −∆
1
α
k
, k ∈
{2,3, · · · , K}.
2.2 Scaling Scheme
It is well known that in an implementation system, all
the signals should be sustained within a certain dy-
namic range in order to avoid overflow. Under the as-
sumption that the input r(n) and the output u(n) of the
closed-loop system are properly pre-scaled, the only
signals which may have overflow are the elements of
the controller state vector x(n), which, therefore, have
to be scaled.
There exist different scaling schemes for prevent-
ing variables from overflow. The popularly used ones
are the l
2
- and l
∞
-scalings. In what follows, we will
concentrate on the l
2
-scaling scheme. The l
2
-scaling
means that each element of the controller state vec-
tor x(n) should have a unit variance when the input
r(n) is a white noise with a unit variance. This can be
achieved if
¯
K (l,l) = 1, l = N + 1,N + 2,...,N + K (9)
where
¯
K is the controllability Gramian of the closed-
loop system of order N + K. Assuming that P
d
(z) is
strictly proper and has a realization (A
z
,B
z
,C
z
,0), let
(A
cl
,B
cl
,C
cl
,0) be the closed-loop realization, where
A
cl
=

A
z
+ dB
z
C
z
B
z
C
BC
z
A

B
cl
=

B
z
0

C
cl
= [C
z
0] (10)
with 0 denoting the zero vector of appropriate dimen-
sion. Then
¯
K is given by
¯
K =
+∞
∑
k=0
A
k
cl
B
cl
B
T
cl
(A
T
cl
)
k
(11)
satisfying
¯
K = A
cl
¯
K A
T
cl
+ B
cl
B
T
cl
.
Let (A
cl
,B
cl
,C
cl
) and (A
0
cl
,B
0
cl
,C
0
cl
) be two real-
izations of the closed-loop system with A
cl
,B
cl
and
C
cl
defined in (10), corresponding to the two digi-
tal controller realizations R , (A,B,C, d) and R
0
,
(A
0
,B
0
,C
0
,d) which are related with (3), respectively.
It can be shown that
A
cl
=

I 0
0 T

−1
A
0
cl

I 0
0 T

B
cl
=

I 0
0 T

−1
B
0
cl
C
cl
= C
0
cl

I 0
0 T

. (12)
It then follows from (12) that
¯
K =

I 0
0 T

−1
¯
K
0

I 0
0 T

−T
where
¯
K
0
is the closed-loop controllability Gramian
corresponding to R
0
. Let
¯
K ,

K
11
K
12
K
21
K

,
¯
K
0
,

K
0
11
K
0
12
K
0
21
K
0

(13)
OPTIMAL SPARSE CONTROLLER STRUCTURE WITH MINIMUM ROUNDOFF NOISE GAIN
15

have the same partition as

I 0
0 T

, then
K = T
−1
K
0
T
−T
(14)
where K
0
is a positive-definite matrix independent of
T.
It is easy to see from above equations that the l
2
-
scaling constraint (9) can be satisfied if the diagonal
elements of K are all equal to one, that is
K (k,k) = 1,∀k. (15)
When the ρDFIIt structure is used to implement a
digital controller, it has to be l
2
-scaled in order to pre-
vent the signals in the controller from overflow, which
can be achieved by choosing {∆
k
} properly. It is in-
teresting to note that
p
k
(z) = [
K
∏
l=k+1
∆
−1
l
] ¯p
k
(z), ∀k (16)
where all ¯p
k
(z) are obtained using (5) with ∆
k
= 1, ∀k.
Let (A
0
ρ
,B
0
ρ
,C
0
ρ
,β
0
) be the equivalent state-space
realization corresponding to ∆
k
= 1, ∀k. With (16), it
can be shown that
A
ρ
= T
sc
A
0
ρ
T
−1
sc
, B
ρ
= T
sc
B
0
ρ
, C
ρ
= C
0
ρ
T
−1
sc
where T
−1
sc
is a diagonal scaling similarity transfor-
mation, and
T
sc
= diag(d
1
,d
2
,···,d
K
), d
k
=
k
∏
l=1
∆
−1
l
, ∀k.
Denote
¯
K
ρ
and
¯
K
0
ρ
as the closed-loop controlla-
bility Gramians, corresponding to the controller real-
izations (A
ρ
,B
ρ
,C
ρ
,β
0
) and (A
0
ρ
,B
0
ρ
,C
0
ρ
,β
0
), respec-
tively. Let K
ρ
be the sub-matrix of
¯
K
ρ
with the
partition defined in (13), then (14) becomes K
ρ
=
T
sc
K
0
ρ
T
T
sc
with K
0
ρ
the corresponding sub-matrix of
¯
K
0
ρ
. It is easy to see that the l
2
-scaling can be
achieved if K
ρ
(k,k) = 1,∀k, or equivalently,
d
2
k
K
0
ρ
(k,k) = 1, k = 1,2,...,K
which leads to
∆
1
=
q
K
0
ρ
(1,1), ∆
k
=
s
K
0
ρ
(k,k)
K
0
ρ
(k− 1,k− 1)
, (17)
k = 2,3,...,K.
In the sequel, all the structures under discussion,
including the ρDFIIt structure, are assumed to have
been l
2
-scaled. Here we should note that the l
2
-scaled
ρDFIIt structure to be analyzed in this paper is differ-
ent from the structure in (Hao et al., 2006) where {∆
k
}
are free parameters used for maximizing the stability
robustness measure.
3 PERFORMANCE ANALYSIS
AND OPTIMIZED STRUCTURE
In this section, we will analyze the performance of
the ρDFIIt structure in terms of closed-loop round-
off noise gain. The problem of finding the optimized
structure will then be formulated and solved.
One notes that for a given digital controller C
d
(z),
there exists a class of l
2
-scaled ρDFIIt structures,
which are determined by a space, denoted as S
γ
, from
which the free parameters {γ
k
} take values. It is easy
to see that {γ
k
} are the parameters to be implemented
directly in the structure. Since we are confined to
fixed-point implementation for which the FWL ef-
fects are more serious, it is desired that γ
k
be abso-
lutely not bigger than one and of FWL format. For a
fixed-point implementation of B
p
bits, define
S
FWL
, {−1,1} ∪ {±
B
p
∑
l=1
b
l
2
−l
, b
l
= 0, 1, ∀l} (18)
which is a discrete space, containing 2
B
p
+1
+ 1 ele-
ments. Therefore, one can choose S
γ
⊂ S
FWL
, which
means that all γ
k
are of exact B
γ
-bit format with B
γ
≤
B
p
.
3.1 Closed-loop Roundoff Noise Gain
In practice, a designed digital controller has to be im-
plemented with finite precision and a rounding oper-
ation has to be applied if less-than-double precision
fixed-point arithmetic is utilized. Assuming round-
ing occurs after multiplication (RAM), a variable, say
x, computed with a multiplication, has to be replaced
by its quantized version, denoted as Q[x], in the ideal
computation model. The difference Q[x] − x is the
corresponding roundoff noise, which is usually mod-
elled as a white noise sequence and statistically inde-
pendent of those produced by other sources.
Let µ be a parameter in a controller structure and
Q[µs(n)] the quantized version of the product µs(n).
The roundoff noise due to the parameter µ can be de-
fined as
ψ(µ)ε
µ
(n) , Q[µs(n)] − µs(n)
where ψ(µ) = 1 if µ is nontrivial, otherwise, ψ(µ) = 0.
In fact, the function ψ(µ) is used for indicating the
fact that µ produces no roundoff noise when it is triv-
ial. Denote ∆u(n) as the corresponding output de-
viation of the closed-loop system to ψ(µ)ε
µ
(n) and
F(z) as the transfer function between ψ(µ)ε
µ
(n) and
∆u(n). It is well known (see, e.g., (Gevers and Li,
1993)) that ∆u(n) is a stationary process and the vari-
ance E[(∆u(n))
2
] = ψ(µ)kF(z)k
2
2
E[ε
2
µ
(n)]. Then the
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
16

roundoff noise gain for µ is defined as
G
µ
,
E[(∆u(n))
2
]
E[ε
2
µ
(n)]
= ψ(µ)kF(z)k
2
2
(19)
where ||.||
2
is the L
2
-norm:
k F(z) k
2
,
(
1
2π
Z
2π
0
l
∑
i=1
m
∑
k=1
| f
ik
(e
jω
)|
2
dω
)
1/2
=

tr[
1
j2π
I
|z|=1
F(z)F
H
(z)z
−1
dz]

1/2
(20)
with F(z) = { f
ik
(z)} ∈ R
l×m
, and H , tr[.] denoting
the conjugate-transpose and trace operators, respec-
tively. Let F(z) = D+ L(zI −Φ)
−1
J. It can be shown
that
kF(z)k
2
2
= tr[DD
T
+ LW
c
L
T
] = tr[D
T
D+ J
T
W
o
J]
(21)
where W
c
,W
o
are the controllability and observability
Gramians of the realization (Φ,J, L, D), respectively.
Consider a digital controller implemented with a
ρDFIIt structure. We note that the parameters in a
ρDFIIt structure are {α
k
},{β
k
}, {∆
k
}, and {γ
k
}. It
follows from (6) that
y(n) = β
0
u(n) +
K
∑
l=1
[
p
l
(z)
p
0
(z)
β
l
u(n) −
p
l
(z)
p
0
(z)
α
l
y(n)].
(22)
Let us first look at the effect of roundoff noise
ψ(β
0
)ε
β
0
(n) due to β
0
on the closed-loop output. Let
u
∗
(n) and y
∗
(n) be the corresponding output of the
closed-loop system and the controller, respectively.
Clearly, they obey (22) with β
0
u
∗
(n) replaced by
β
0
u
∗
(n)+ψ(β
0
)ε
β
0
(n). Denote ∆y(n) , y
∗
(n)−y(n).
Then one can show that
∆y(n) = [β
0
∆u(n) + ψ(β
0
)ε
β
0
(n)]
+
K
∑
l=1
p
l
(z)
p
0
(z)
β
l
∆u(n) −
K
∑
l=1
p
l
(z)
p
0
(z)
α
l
∆y(n) (23)
where ∆u(n) , u
∗
(n) − u(n), satisfying
∆u(n) = P
d
(z)∆y(n). (24)
Let H
cl
(z) be the transfer function of the closed-
loop system, which is given by
H
cl
(z) =
P
d
(z)
1− P
d
(z)C
d
(z)
where P
d
(z) is the transfer function of plant and C
d
(z)
the polynomial parametrized controller transfer func-
tion given by (6). It is easy to see that
H
cl
(z) = D
cl
+C
cl
(zI − A
cl
)
−1
B
cl
(25)
with (A
cl
,B
cl
,C
cl
,D
cl
) the realization of closed-loop
system. It then follows from (23) and (24) that
∆u(n) = S
0
(z)ψ(β
0
)ε
β
0
(n)
where S
0
(z) is the transfer function between
ψ(β
0
)ε
β
0
(n) and ∆u(n), which is given by
S
0
(z) = H
cl
(z)V
0
(z)
with
V
0
(z) ,
p
0
(z)
p
0
(z) +
∑
K
l=1
α
l
p
l
(z)
.
Comparing V
0
(z) with (6), it follows from (8) that
V
0
(z) = [β
0
+C
ρ
(zI − A
ρ
)
−1
B
ρ
]
|
β
0
=1,
¯
V
β
=0
= 1−C
ρ
(zI − A
ρ
)
−1
¯
V
α
.
One observes that S
0
(z) is of the form S
0
(z) =
[D
2
+ C
2
(zI
2
− A
2
)
−1
B
2
][D
1
+ C
1
(zI
1
− A
1
)
−1
B
1
],
where A
1
= A
ρ
,B
1
= −
¯
V
α
,C
1
= C
ρ
,D
1
= 1, A
2
=
A
cl
,B
2
= B
cl
,C
2
= C
cl
,D
2
= D
cl
, and I
k
,k = 1,2 de-
notes the identity matrix of a proper dimension. It is
easy to verify that
S
0
(z) ,
˜
D+
˜
C(z
˜
I −
˜
A)
−1
˜
B
where
˜
D = D
2
D
1
,
˜
C = [D
2
C
1
C
2
]
˜
I =

I
1
0
0 I
2

˜
A =

A
1
0
B
2
C
1
A
2

,
˜
B =

B
1
B
2
D
1

.
According to (19) and (21), the roundoff noise gain
due to parameter β
0
is given by
G
β
0
= ψ(β
0
)||S
0
(z)||
2
2
= ψ(β
0
)tr(
˜
D
T
˜
D+
˜
B
T
˜
W
˜
B)
, ψ(β
0
)G
0
where
˜
W is the observability Gramian of the realiza-
tion (
˜
A,
˜
B,
˜
C,
˜
D).
Using the same procedure, one can analyze the
roundoff noise gain due to the parameter β
k
. Let
ψ(β
k
)ε
β
k
(n) be the corresponding roundoff noise.
It can be shown that the transfer function from
ψ(β
k
)ε
β
k
(n) to ∆u(n), denoted as S
k
(z), is
S
k
(z) = H
cl
(z)V
k
(z)
with H
cl
(z) given by (25) and
V
k
(z) =
p
k
(z)
p
0
(z) +
∑
K
l=1
α
l
p
l
(z)
= C
ρ
(zI − A
ρ
)
−1
e
k
for k = 1,2, · · · , K, where e
k
is the kth elementary vec-
tor whose elements are all zero except the kth one
which is 1. Therefore,
G
β
k
= ψ(β
k
)||S
k
(z)||
2
2
, ψ(β
k
)G
k
, ∀k
OPTIMAL SPARSE CONTROLLER STRUCTURE WITH MINIMUM ROUNDOFF NOISE GAIN
17

with
G
k
= tr(
˜
D
T
k
˜
D
k
+
˜
B
T
k
˜
W
k
˜
B
k
)
where (
˜
A
k
,
˜
B
k
,
˜
C
k
,
˜
D
k
) is the realization of S
k
(z) and
˜
W
k
is the corresponding observability Gramian.
Comparing the positions of α
k
,γ
k
and ∆
k+1
with
that of β
k
in Fig. 3, one can see easily that
G
α
k
= ψ(α
k
)G
k
, G
γ
k
= ψ(γ
k
)G
k
, G
∆
k
= ψ(∆
k
)G
k−1
for k = 1,2,··· , K.
Therefore, the total closed-loop roundoff noise
gain of the ρDFIIt structure is
G
ρ
,
K
∑
k=1
[G
α
k
+ G
γ
k
+ G
∆
k
] +
K
∑
k=0
G
β
k
,
K
∑
k=0
υ
k
G
k
(26)
where the coefficients υ
k
can be specified easily with
the expressions, obtained above, of roundoff noise
gain for all the parameters.
3.2 Structure Optimization
For a given digital controllerC
d
(z) and any given free
parameters {γ
k
}, one can obtain the l
2
-scaled ρDFIIt
structure with the procedure presented in Section II.
The roundoff noise gain G
ρ
can then be evaluated
with (26). Since different sets of {γ
k
} yield different
ρDFIIt structures and hence lead to different roundoff
noise gain G
ρ
, an interesting problem is to minimize
G
ρ
with respect to these free parameters, which leads
to the following optimal ρDFIIt structure problem:
min
γ
k
∈S
γ
G
ρ
. (27)
It seems impossible to obtain analytical solutions
to the problem (27) due to the high nonlinearity of G
ρ
in {γ
k
}. However, noting that S
γ
is of finite number
of elements, the problem can be well solved using the
exhaustive searching method.
4 A DESIGN EXAMPLE
In this section, we illustrate our design procedure
and the performance of the proposed structure with
a numerical example, in which S
γ
= {±1,±(2
−1
+
2
−2
),±2
−1
,±2
−2
,0}. The elements in the set S
γ
are
of exact 3-bit fixed-point format (including one bit for
the sign). Using more bits or floating-point formats
will lead to a further improved performance, which
can also confirm the effectiveness of our design pro-
cedure.
Consider a discrete-time control sys-
tem, where the digital plant P
d
(z) =
10
−1
×
0.0181z
4
+0.0033z
3
−0.1628z
2
+0.0111z+0.0163
z
5
−3.7174z
4
+5.7458z
3
−4.6673z
2
+2.0336z−0.3953
Table 1: Comparison of Different Structures.
zDFIIt δDFIIt R
f
ρDFIIt
G 1.5191 × 10
4
7.1763 4.9919 1.0085
N
p
19 19 49 24
and controller C
d
(z) = 0.0577 +
0.2258z
5
−0.6588z
4
+0.8195z
3
−0.5320z
2
+0.1814z−0.0234
z
6
−3.6172z
5
+5.9513z
4
−5.6335z
3
+3.2509z
2
−1.0895z+0.1690
.
The corresponding poles of the closed-
loop system are {0.4523 ± j0.5315,0.4837 ±
j0.4556, 0.6055 ± j0.4108, 0.7814 ±
j0.3099, 0.8886 ± j0.3326,0.9113}.
Applying exhaustive searching to (27), one gets
the optimal ρDFIIt structure, denoted as ρDFIIt, for
which γ
1
= 1, γ
5
= 0.5, γ
k
= 0.75, k ∈ {2,3, 4,6}.
For comparison, an optimal fully parametrized state-
space realization, denoted by R
f
, is obtained using
the procedure in (Li et al., 2002). zDFIIt and δDFIIt
are the traditional DFIIt structures in the shift- and δ-
operators, corresponding to γ
k
= 0, ∀k and γ
k
= 1, ∀k,
respectively.
The comparative results of different structures are
presented in Table I, where G is the roundoff noise
gain and N
p
is the number of nontrivial parameters in
each structure.
From this example, one can see that zDFIIt yields
a very large roundoffnoise gain, though it has only 19
parameters to implement, while δDFIIt has a much
better performance. The fully parametrized optimal
realization R
f
yields a further better performance,
however, all the 49 parameters in R
f
are nontrivial.
It is interesting to see that ρDFIIt beats R
f
in terms
of the roundoff noise performance. Moreover, ρDFIIt
is very sparse and has only 24 nontrivial parameters,
which is less than half of those in R
f
.
5 CONCLUSIONS
In this paper, we have addressed the optimal con-
troller structure problem in a discrete-time control
system with roundoff noise consideration. Our ma-
jor contribution is twofold. Firstly, a sparse controller
structure, which is a l
2
-scaled generalized DFIIt struc-
ture, has been derived. Secondly, the performance
of the proposed structure has been analyzed by de-
riving the corresponding expression of closed-loop
roundoff noise gain and the problem of finding op-
timized sparse structures has been solved. Finally,
a numerical example has been given, which shows
that the proposed structure can achieve much bet-
ter performance than some well-known structures and
particularly, outperforms the traditional optimal fully
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
18

parametrized realization greatly in terms of reducing
roundoff noise and implementation complexity. This
optimal controller design strategy with high precision
arithmetic can be utilized to develop suitable con-
trol systems for robotic platforms performing com-
plex movements, where efficiency, accuracy and fast
speed are essential.
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