Two-Dimensional State-Space Digital Filters with
Minimum Frequency-Weighted L2-Sensitivity
under L2-Scaling Constraints
T. Hinamoto, T. Oumi, O. I. Omoifo
W.-S. Lu
Graduate School of Engineering
Hiroshima University, Japan
Electrical & Computer Engineering
University of Victoria, Canada
1
Outline
•
Early and Recent Work
•
System Model
•
Sensitivity Measure and Scaling Constraints
•
Problem Formulation
•
A Solution Method
•
Experimental Results
2
Early and Recent Work
• Kawamata, Lin, and Higuchi, 1987. (L2/L1 mixed sensitivity)
• Hinamoto, Hamanaka, and Maekawa, 1990. (L2/L1 mixed sensitivity)
• Hinamoto, Takao, and Muneyasu, 1992. (L2/L1 mixed sensitivity)
• Hinamoto and Takao, 1992. (L2/L1 mixed sensitivity)
• Hinamoto, Zempo, Nishino, and Lu 1999. (L2/L1 mixed sensitivity)
• Li, 1997 and 1998. (L2 sensitivity)
• Hinamoto, Yokoyama, Inoue, Zeng, and Lu, 2002. (L2 sensitivity)
• Hinamoto and Sugie, 2002. (L2 sensitivity)
Some of the above work have considered frequency weighted
sensitivity, but they do not impose scaling constraints on the
design variables. This paper presents a study concerning a
frequency weighted L2 measure subject to L2 scaling constraints.
3
System Model
• We consider stable and locally controllable and observable
2-D state-space digital filters that are modeled by Roesser’s
local state space model as
⎡ x h (i + 1, j ) ⎤ ⎡ A1 A2 ⎤ ⎡ x h (i, j ) ⎤ ⎡ b1 ⎤
⎢ v
⎥ = ⎢A A ⎥⎢ v
⎥ + ⎢ b ⎥ u (i , j )
3
4 ⎦ ⎣ x (i , j ) ⎦
2⎦
⎣N
⎣ x (i, j + 1) ⎦ ⎣
b
A
⎡ x h (i , j ) ⎤
y (i, j ) = [ c1 c2 ] ⎢ v
+ d u (i , j )
⎥
⎣ x (i , j ) ⎦
c
• Transfer function:
H ( z1 , z2 ) = c ( Z − A) b, Z = z1 I m ⊕ z2 I n
−1
4
Sensitivity Measure and Scaling Constraints
• Frequency-weighted L2-sensitivity
∂H ( z1 , z2 )
∂H ( z1 , z2 )
∂H ( z1 , z2 )
+ WB ( z1 , z2 )
+ WC ( z1 , z2 )
∂A
∂b
∂c
2
2
2
S = WA ( z1 , z2 )
2
• Evaluation of S
S = WA ( z1 , z2 )G ( z1 , z2 ) F ( z1 , z2 )
T
T
2
2
2
2
2
2
+ WB ( z1 , z2 )G ( z1 , z2 ) + WC ( z1 , z2 ) F ( z1 , z2 )
T
where
F ( z1 , z2 ) = ( Z − A) −1 b
G ( z1 , z2 ) = c( Z − A) −1
and the squared L2-norm Y ( z1 , z2 ) 2 can be computed as
2
5
2
2
trace of a certain matrix:
⎡ 1
Y ( z1 , z2 ) 2 = trace ⎢
2
(2
)
π
j
⎣
2
dz1dz2 ⎤
v∫ Γ1 v∫ Γ2 Y ( z1 , z2 )Y ( z1 , z2 ) z1z2 ⎥⎦
∗
which leads to
• An alternative expression of S:
S = trace [ M A ] + trace [WB ] + trace [ K C ]
• L2 signal scaling constraints:
( K1 )i ,i = 1, ( K 4 )k ,k = 1
where
⎡ K1
K =⎢
⎣ K3
K2 ⎤
1
2
=
=
(
,
)
X
z
z
1 2 2
K 4 ⎥⎦
(2π j ) 2
6
v∫ v∫
Γ1
Γ2
F ( z1 , z2 ) F ∗ ( z1 , z2 )
dz1dz2
z1 z2
Problem Formulation
• Minimization of the frequency-weighted L2-sensitivity
subject to L2 scaling constraints is achieved by using an
optimized state-space coordinate transformation
⎡ x h (i, j ) ⎤ ⎡T1 xˆ h (i, j ) ⎤ ⎡T1 0 ⎤ ⎡ xˆ h (i, j ) ⎤
⎥=⎢
⎢ v
⎥=⎢ v
⎥
⎥ ⎢ ˆv
0
T
ˆ
(
,
)
T
x
i
j
(
,
)
(
,
)
x
i
j
x
i
j
4⎦⎣
⎣
⎦ ⎣ 4
⎦
⎦ ⎣
T
• The transfer function is invariant under a state-space
transformation T, but system realization {A, b, c, d} is
changed to { Aˆ , bˆ, cˆ, dˆ} with
Aˆ = T −1 AT , bˆ = T −1b, cˆ = cT , dˆ = d
• Sensitivity measure S under transformation T is changed
7
accordingly to
S ( P ) = trace [ M A ( P ) P ] + trace [WB P ] + trace ⎡⎣ K C P −1 ⎤⎦
where P = TTT and
M A ( P) =
1
(2π j ) 2
v∫ v∫
Γ1
Γ2
Y ( z1 , z2 ) P −1Y ∗ ( z1 , z2 )
dz1dz2
z1 z2
with Y ( z1 , z2 ) = WA ( z1 , z2 )G T ( z1 , z2 ) F T ( z1 , z2 ) .
• And here is the point: one can select a state space
transformation T to minimize the sensitivity S ( P) subject to
L2 scaling constraints:
minimize
S ( P)
T
T , P =TT
subject to: (T1−1K1T1−T ) = 1,
i ,i
8
(T
−1
4
K 4T4−T )
k ,k
=1
A Solution Method
• The solution method proposed here eliminates the L2
scaling conditions to convert the problem at hand into an
unconstrained problem which is then solved using a
quasi-Newton method.
• Let
Tˆ1 = T1T K1−1/ 2 , Tˆ4 = T4T K 4−1/ 2
then constraints
(T
−1
1
K1T1−T ) = 1,
(T
K 4T4− T )
)
(
)
−1
4
i ,i
k ,k
=1
become
(
Tˆ1−T Tˆ1−1
i ,i
= 1,
9
Tˆ4−T Tˆ4−1
k ,k
=1
which are automatically satisfied if we set
⎡ t11
−1
ˆ
T1 = ⎢
⎣ t11
t12
t12
"
⎡ t41
t1m ⎤
−1
ˆ
⎥ , T4 = ⎢
t1m ⎦
⎣ t41
t42
t42
"
t4 n ⎤
⎥
t4 n ⎦
• The L2-sensitivity in terms of Tˆ1 and Tˆ4 are given by
S ( x ) = trace ⎡⎣Tˆ M A ( Pˆ )Tˆ T ⎤⎦ + trace ⎡⎣Tˆ Wˆ BTˆ T ⎤⎦ + trace ⎡⎣Tˆ − T Kˆ C Tˆ −1 ⎤⎦
where Pˆ = Tˆ T Tˆ and
x = ⎡⎣t11T t12T " t1Tm
t
T
41
t
T
41
T T
4n
" t ⎤⎦
• Minimizing S(x) is an unconstrained problem that can be
carried out using an efficient iterative algorithm such as a
quasi-Newton algorithm as follows:
10
(1) Start with an initial point x0 corresponding to
Tˆ1 = I m , Tˆ4 = I n . Set k = 0 and S0 = I.
(2) Update xk to xk +1 = xk + α k d k where
d k = − Sk ∇S ( xk ), α k = arg min S ( xk + α d k )
α
(3) Update Sk to
⎛ γ kT Sk γ k
Sk +1 = Sk + ⎜ 1 + T
γ k δk
⎝
⎞ δ k δ kT δ k γ kT Sk + Sk γ k δ kT
⎟ γ Tδ −
T
γ
k δk
⎠ k k
δ k = xk +1 − xk , γ k = ∇S ( xk +1 ) − ∇S ( xk )
(4) If S ( xk +1 ) − S ( xk ) < ε , terminate the iteration, otherwise
set k := k + 1 and repeat from step (2).
11
Experimental Results
• An Example: Consider a stable recursive digital filter
realization ( Ao , bo , c o , d ) 4,4 where
o
⎡
A
Ao = ⎢ 1o
⎣ A1
A2o ⎤ o ⎡ b1o ⎤ o
o
⎡
,
b
,
c
c
=
=
⎥
⎢ o⎥
⎣ 1
A2o ⎦
b
⎣ 2⎦
c2o ⎤⎦ with
0
0.481228
0
0
⎡
⎤
⎢
⎥
0
0
0.510378
0
⎥
A1o = ⎢
0
0
0
0.525287 ⎥
⎢
⎢ −0.031857 0.298663 −0.808282 1.044600 ⎥
⎣
⎦
⎡ −0.226080
⎢ −0.843550
A2o = ⎢
⎢ −1.260339
⎢ −1.121498
⎣
−0.000933⎤
1.610400 −0.309366 0.065898 ⎥
⎥
2.005100 −0.453220 0.203118 ⎥
1.636435 −0.590516 0.562890 ⎥⎦
0.776837
0.024693
12
b1o = b2o = [ 0 0 0 0.198473]
T
c1o = [ −0.567054 0.231913 0.197016 0.239932]
c2o = [ 0.464344 0.441837 −0.061100 0.105505]
d = 0.009430
• Frequency-weighted functions were z-transforms of
wA(i,j) = wB(i,j) = wC(i,j) = 0.256322 ⋅ e
−0.103203⎡( i − 4)2 + ( j − 4)2 ⎤
⎣
⎦
for (0,0) ≤ (i, j ) ≤ (20,20) and zero elsewhere.
• The frequency weighted L2-sensitivity of the filter was
found to be J 0 (Tˆ0 ) ≈ 8.60 × 103 (with Tˆ0 = I 4 ⊕ I 4 ).
• With an initial Tˆ = I 4 ⊕ I 4 and tolerance ε = 10−8 , it took the
algorithm 54 iterations to converge to a solution.
13
9000
8000
o
k
J (x )
7000
6000
5000
4000
0
10
20
30
k
14
40
50
• The optimized Tˆ was found to be
⎡ 3.056671 −2.673365 0.575882 −0.429287 ⎤
⎢ −0.331629 2.142411 −0.401503 −0.192081⎥
⎥
Tˆ opt = ⎢
⎢ −2.530651 0.932586 0.553002 −0.136935 ⎥
⎢
⎥
1.754363
−
0.312582
0.624509
0.515370
⎣
⎦
⎡ 1.307170 −0.419919 0.045538 −0.194118⎤
⎢ 0.762443 0.830435 −0.297531 0.062104 ⎥
⎥
⊕⎢
⎢ −0.405202 0.189220 0.976564 −0.250656 ⎥
⎢
⎥
1.071478
−
0.069804
0.315533
0.828727
⎣
⎦
• The minimized frequency weighted L2-sensitivity was
found to be J 0 (Tˆ opt ) ≈ 4.67 ×103 .
15