Two-dimensional Digital Filters Described by ... - Semantic Scholar
Two-dimensional Digital Filters Described by Roesser Model with Interference Attenuation Prof. Choon Ki Ahn, IEEE Senior Member School of Electrical Engineering, Korea University 145, Anam-ro, Seongbuk-gu, Seoul, 136-701 Korea. Tel:+82-2-3290-4831, E-mail: [email protected]
Abstract
To the present time, stability criteria have been proposed for one-dimensional digital filters with external interference, but no stability criterion exists for cases where two-dimensional digital filters have external interference. In this paper, we propose a new criterion for the elimination of overflow oscillations in twodimensional digital filters described by Roesser model with saturation arithmetic and external interference. This criterion ensures asymptotic stability with a guaranteed H∞ performance. The proposed criterion is represented in terms of linear matrix inequality (LMI); thus, it is computationally efficient. Illustrative examples are given to demonstrate the effectiveness of the proposed criterion. Key Words: H∞ criterion, two-dimensional digital filters, finite wordlength effects, linear matrix inequality (LMI)
1
1
Introduction
Two-dimensional systems have several applications such as image processing, thermal processes, seismographic data processing, and gas absorption [1]. Thus, two-dimensional system design is an important and challenging task. When designing discrete systems using fixedpoint arithmetic, quantization and overflow nonlinearities inevitably occur. The presence of such nonlinearities causes the instability of the designed system [2, 3, 4]. The zero-input limit cycles, which are undesirable, may possibly occur because of such nonlinearities. The quantization and overflow nonlinearities may interact with each other. However, if the number of quantization steps is large, or in other words, the internal wordlength is long enough, quantization effects can be neglected when we investigate the effects of overflow [5, 6]. The stability of two-dimensional discrete systems described by the Roesser model [7] has been investigated extensively [6, 8, 9, 10, 11, 12, 13]. The stability of two-dimensional systems described by the Fornasini-Marchesini second model [14] has also received much attention [10, 15, 16, 17]. The result in [6] consists of a generalization and improvement over the criterion proposed in [8]. In [11], an alternative stability criterion was presented based on reduced sector characterization of saturation nonlinearity. An improvement and generalization of [11] was proposed in [18]. In [19], another criterion for the asymptotic stability of two-dimensional digital filters has been established. The criterion proposed in [19] provides improvement over previous stability criteria [8, 6, 12]. In [20], a modified version of [19] was presented. When we implement a high-order digital filter in a hardware component, we usually break down it into several filters before hardware implementation. Then, there are interferences between the filters. These cause malfunction, as well as ultimate destruction [21, 22]. However, most criteria for the stability of digital filters are only available under specific conditions. In unfavorable environments with external interferences, these criteria will be useless. Thus, it is desirable to have an alternative criterion that can overcome the shortcomings of existing criteria. In order to solve these shortcomings, Ahn recently proposed new stability criteria for digital filters with external interferences [23, 24, 25, 26, 27]. However, these results were restricted to one-dimensional digital filters. To the best of our knowledge, there is no result in the literature so far concerning the robust stability of two-dimensional digital filters with
2
saturation arithmetic and external interference; this remains an open and challenging topic. In this paper, a new criterion for the H∞ stability of two-dimensional digital filters described by Roesser model with saturation arithmetic and external interference is proposed. This criterion ensures that the effect of the external interference on the two-dimensional digital filter can be attenuated to an interference attenuation level. The criterion is represented by linear matrix inequality (LMI), which can be solved easily using existing convex optimization algorithms [28, 29]. The novelty of this paper is to establish a new and first criterion for the H∞ performance and asymptotic stability of two-dimensional digital filters with external interference. The present paper is organized as follows. In Section 2, a new LMI criterion for the H∞ stability of two-dimensional digital filters with saturation arithmetic and external interference is proposed. In Section 3, numerical examples are given, and finally, conclusions are presented in Section 4.
2
New Interference Attenuation Criterion
Consider the following two-dimensional digital filter: h h + 1, j) w (i, j) y (i, j) + =f y v (i, j) wv (i, j) xv (i, j + 1)
xh (i
=
f h (y h (i, j)) f v (y v (i, j))
+
wh (i, j) wv (i, j)
y h (i, j)
,
xh (i, j)
A11 A12 = , y v (i, j) A21 A22 xv (i, j)
h h f h (y h (i, j)) = [f1h (y1h (i, j)), · · · , fm (ym (i, j))]T ,
f v (y v (i, j)) = [f1v (y1v (i, j)), · · · , fnv (ynv (i, j))]T , h y h (i, j) = [y1h (i, j), · · · , ym (i, j)]T ,
y v (i, j) = [y1v (i, j), · · · , ynv (i, j)]T ,
3
(1)
(2)
where xh (i, j) ∈ Rm is the horizontal state vector, xv (i, j) ∈ Rn is the vertical state vector, wh (i, j) ∈ Rm is the horizontal external interference, wv (i, j) ∈ Rn is the vertical external interference, y h (i, j) is the horizontal output vector, y v (i, j) is the vertical output vector, f h (·) is the horizontal overflow nonlinearity, f v (·) is the vertical overflow nonlinearity, A11 ∈ Rm×m , A12 ∈ Rm×n , A21 ∈ Rn×m , and A22 ∈ Rn×n are state matrices. The overflow arithmetic to be considered presently is the saturation arithmetic given by 1, fkh (ykh (i, j)) = ykh (i, j), −1, 1, fkv (ykv (i, j)) = ykv (i, j), −1,
if ykh (i, j) > 1 if − 1 ≤ ykh (i, j) ≤ 1 ,
k = 1, 2, · · · , m,
(3)
k = 1, 2, · · · , n.
(4)
if ykh (i, j) < −1 if ykv (i, j) > 1 if − 1 ≤ ykv (i, j) ≤ 1 , if ykv (i, j) < −1
Given a level γ > 0, the purpose of this paper is to find a new LMI criterion such that the two-dimensional digital filter (1)-(2) with wh (i, j) = wv (i, j) = 0 is asymptotically stable and ∑∞ ∑∞
h T (i, j)P xh (i, j) 1 j=0 [x i=0 ∑∞ ∑∞ T h (i, j)wh (i, j) i=0 j=0 [w
+ xvT (i, j)P2 xv (i, j)] + wvT (i, j)wv (i, j)]
< γ2
(5)
under zero boundary conditions for all nonzero wh (i, j) and wv (i, j), where P1 and P2 are positive symmetric matrices. Parameter γ is called the H∞ norm bound or the interference attenuation level. In this case, the two-dimensional digital filter (1)-(2) is said to be asymptotically stable with a guaranteed H∞ performance γ. A new H∞ stability criterion for the two-dimensional digital filter (1)-(2) is given in the following theorem. Theorem 1. For a given level γ > 0, if there exist symmetric positive definite matrices P1 ,
4
P2 , Q, and R, positive diagonal matrices M1 and M2 , and positive scalars δ1 and δ2 such that
Φ1,1
Φ1,2
AT11 M1
AT21 M2
0
0
Φ Φ2,2 AT12 M1 AT22 M2 0 0 2,1 0 Q 0 M1 A11 M1 A12 Q − δ1 I − 2M1 M2 A21 M2 A22 0 R − δ2 I − 2M2 0 R 0 0 Q 0 Q − γ2I 0 0 0 0 R 0 R − γ2I
< 0,
(6)
where Φ1,1 = δ1 AT11 A11 + δ2 AT21 A21 + P1 − Q, Φ1,2 = δ1 AT11 A12 + δ2 AT21 A22 , Φ2,1 = δ1 AT12 A11 + δ2 AT22 A21 , Φ2,2 = δ1 AT12 A12 + δ2 AT22 A22 + P2 − R, then the two-dimensional digital filter (1)-(2) is asymptotically stable with a guaranteed H∞ performance γ. Proof. Consider the following two-dimensional quadratic Lyapunov function: V (xh (i, j), xv (i, j)) =
T
xh (i, j)
xh (i, j)
Q 0 . xv (i, j) 0 R xv (i, j)
Along the trajectory of (1)-(2), we have ∆V (xh (i, j), xv (i, j)) = V (xh (i + 1, j), xv (i, j + 1)) − V (xh (i, j), xv (i, j)) =
T xh (i
xh (i
+ 1, j) Q 0 + 1, j) xv (i, j + 1) 0 R xv (i, j + 1)
5
(7)
−
T
xh (i, j)
xh (i, j)
Q 0 xv (i, j) 0 R xv (i, j)
=
T f h (A11 xh (i, j)
+ A12
xv (i, j))
+
wh (i, j)
+
wh (i, j)
Q 0 f v (A21 xh (i, j) + A22 xv (i, j)) + wv (i, j) 0 R
×
f h (A
11
xh (i, j)
+ A12
xv (i, j))
f v (A21 xh (i, j) + A22 xv (i, j)) + wv (i, j)
−
T
xh (i, j)
xh (i, j)
Q 0 . xv (i, j) 0 R xv (i, j)
(8)
From (3) and (4), it is clear that T
f h (A11 xh (i, j) + A12 xv (i, j))f h (A11 xh (i, j) + A12 xv (i, j)) = ∥f h (A11 xh (i, j) + A12 xv (i, j))∥2 ≤ ∥A11 xh (i, j) + A12 xv (i, j)∥2 = [A11 xh (i, j) + A12 xv (i, j)]T [A11 xh (i, j) + A12 xv (i, j)],
(9)
f vT (A21 xh (i, j) + A22 xv (i, j))f v (A21 xh (i, j) + A22 xv (i, j)) = ∥f v (A21 xh (i, j) + A22 xv (i, j))∥2 ≤ ∥A21 xh (i, j) + A22 xv (i, j)∥2 = [A21 xh (i, j) + A22 xv (i, j)]T [A21 xh (i, j) + A22 xv (i, j)].
(10)
Then, for positive scalars δ1 and δ2 , we have { δ1 [A11 xh (i, j) + A12 xv (i, j)]T [A11 xh (i, j) + A12 xv (i, j)] } T − f h (A11 xh (i, j) + A12 xv (i, j))f h (A11 xh (i, j) + A12 xv (i, j)) ≥ 0, { δ2 [A21 xh (i, j) + A22 xv (i, j)]T [A21 xh (i, j) + A22 xv (i, j)]
6
(11)
} − f vT (A21 xh (i, j) + A22 xv (i, j))f v (A21 xh (i, j) + A22 xv (i, j)) ≥ 0.
(12)
Consider the following two quantities: T
2f h (A11 xh (i, j) + A12 xv (i, j))M1 [A11 xh (i, j) + A12 xv (i, j) − f h (A11 xh (i, j) + A12 xv (i, j))], 2f vT (A21 xh (i, j) + A22 xv (i, j))M2 [A21 xh (i, j) + A22 xv (i, j) − f v (A21 xh (i, j) + A22 xv (i, j))].
Using (11)-(12) and adding to and subtracting from (8) these two quantities, a new bound for ∆V (xh (i, j), xv (i, j)) can be obtained as ∆V (xh (i, j), xv (i, j)) ≤
T f h (A11 xh (i, j)
+ A12
xv (i, j))
+
wh (i, j)
+
wh (i, j)
+
wv (i, j)
Q 0 f v (A21 xh (i, j) + A22 xv (i, j)) + wv (i, j) 0 R
× −
f h (A11 xh (i, j) f v (A21 xh (i, j)
+ A12
xv (i, j))
+ A22
xv (i, j))
T
{ Q 0 h v T + δ 1 [A11 x (i, j) + A12 x (i, j)] xv (i, j) 0 R xv (i, j)
xh (i, j)
xh (i, j)
T
× [A11 xh (i, j) + A12 xv (i, j)] − f h (A11 xh (i, j) + A12 xv (i, j)) { × f (A11 x (i, j) + A12 x (i, j)) + δ2 [A21 xh (i, j) + A22 xv (i, j)]T h
h
}
v
× [A21 xh (i, j) + A22 xv (i, j)] − f vT (A21 xh (i, j) + A22 xv (i, j)) } T × f v (A21 xh (i, j) + A22 xv (i, j)) + 2f h (A11 xh (i, j) + A12 xv (i, j))M1 × [A11 xh (i, j) + A12 xv (i, j) − f h (A11 xh (i, j) + A12 xv (i, j))] + 2f vT (A21 xh (i, j) + A22 xv (i, j))M2 [A21 xh (i, j) + A22 xv (i, j) T
− f v (A21 xh (i, j) + A22 xv (i, j))] − 2f h (y h (i, j))M1 [y h (i, j) − f h (y h (i, j))] − 2f vT (y v (i, j))M2 [y v (i, j) − f v (y v (i, j))]
7
T xh (i, j)
xv (i, j) h f (A11 xh (i, j) + A12 xv (i, j)) = v f (A21 xh (i, j) + A22 xv (i, j)) wh (i, j) wv (i, j)
Φ1,1
AT11 M1
Φ1,2
AT21 M2
0
Φ Φ2,2 AT12 M1 AT22 M2 0 2,1 M A 0 Q 1 11 M1 A12 Q − δ1 I − 2M1 × M2 A21 M2 A22 0 R − δ2 I − 2M2 0 0 0 Q 0 Q − γ2I 0
0
0
R
0
0
0 0 R 0 R − γ2I
xh (i, j)
xv (i, j) h f (A11 xh (i, j) + A12 xv (i, j)) × v f (A21 xh (i, j) + A22 xv (i, j)) wh (i, j) wv (i, j)
− xh T (i, j)P1 xh (i, j)
T
− xvT (i, j)P2 xv (i, j) + γ 2 wh (i, j)wh (i, j) + γ 2 wvT (i, j)wv (i, j) + Ω(i, j),
(13)
where T
Ω(i, j) = −2f h (y h (i, j))M1 [y h (i, j) − f h (y h (i, j))] − 2f vT (y v (i, j))M2 [y v (i, j) − f v (y v (i, j))].
(14)
Note that Ω(i, j) is nonpositive in view of (3) and (4). If the LMI (6) is satisfied, we have T
∆V (xh (i, j), xv (i, j)) < −xh (i, j)P1 xh (i, j) − xvT (i, j)P2 xv (i, j)
8
T
+γ 2 wh (i, j)wh (i, j) + γ 2 wvT (i, j)wv (i, j).
(15)
Summation of both sides of (15) from i, j = 0 to i, j = ∞ gives ∞ ∑ ∞ ∑
∆V (xh (i, j), xv (i, j))
i=0 j=0
0 and p2 > 0, we have p1 ∑ p2 ∑
∆V (xh (i, j), xv (i, j))
i=0 j=0
=
p2 ∑
T
T
[xh (p1 + 1, j)Qxh (p1 + 1, j) − xh (0, j)Qxh (0, j)]
j=0 p1 ∑ + [xvT (i, p2 + 1)Rxv (i, p2 + 1) − xvT (i, 0)Rxv (i, 0)], i=0
which together with (16) implies that −
∞ ∑ ∞ ∑
T
xh (i, j)P1 xh (i, j) −
i=0 j=0
+γ
2
∞ ∑ ∞ ∑
∞ ∑ ∞ ∑
xvT (i, j)P2 xv (i, j)
i=0 j=0
w
hT
h
(i, j)w (i, j) + γ
i=0 j=0
2
∞ ∞ ∑ ∑
wvT (i, j)wv (i, j) > 0
(17)
i=0 j=0
under zero boundary conditions. Thus, we have the relation (5). When wh (i, j) = wv (i, j) = 0, we have T
∆V (xh (i, j), xv (i, j)) < −xh (i, j)P1 xh (i, j) − xvT (i, j)P2 xv (i, j) ≤ 0
9
(18)
from (15). This guarantees that lim
i,j→∞
xh (i, j) xv (i, j)
= 0.
(19)
This completes the proof. Let
A11 A12 P1 0 A= , , S = 0 P2 A21 A22
Q 0 M1 0 P = , M = , δ = δ1 = δ2 . 0 R 0 M2
(20)
Then, the criterion (6) becomes
δAT A
+S−P
AT M
0
MA
P − δI − 2M
P
0
P
P − γ2I
< 0,
(21)
which is the disturbance attenuation criterion for one-dimensional digital filters [23]. Remark 1. Criteria in the existing works [6, 8, 9, 10, 11, 12, 13] were derived assuming that there are no external interferences in two-dimensional digital filters. These existing criteria fail to check the interference attenuation of two-dimensional digital filters. In contrast to these existing results, in this paper, we consider two-dimensional digital filters with external interferences and propose a new LMI criterion (6) for interference attenuation and asymptotic stability of these digital filters. Based on Theorem 1, we present a method to obtain the optimal H∞ norm bound. Corollary 1. The optimal H∞ norm bound γ is obtained by solving the following semi-definite programming problem:
min γ 2 γ>0
10
(22)
subject to the LMI (6), P1 > 0, P2 > 0, Q > 0, R > 0, M1 > 0, M2 > 0, δ1 > 0, and δ2 > 0.
3 3.1
Numerical Examples Example 1
Consider the two-dimensional digital filter (1)-(2) with
0.01 A12 = , 0
1.2 0.55 A11 = , −0.09 0.11 [ A21 =
(23)
] 0.01 0
,
A22 = 0.01,
(24)
cos(5i) wh (i, j) = , 2 sin (2j)
wv (i, j) = cos(i + j).
(25)
For the design objective (5), let the H∞ performance be specified by γ = 0.3. A feasible solution to the LMI (6) is
0.0007 −0.0008 P1 = , −0.0008 0.0012
0.0028 0.0008 Q= , 0.0008 0.0100
R = 0.0661,
0 0.0062 M1 = , 0 0.0062 δ1 = 1.3878 × 10−4 ,
P2 = 0.0321,
M2 = 3.0307,
δ2 = 1.1073.
Criteria in the existing works [6, 8, 9, 10, 11, 12, 13] were derived assuming that there are no external interferences in two-dimensional digital filters. If we consider two-dimensional digital filters with external interferences, it is not possible to guarantee interference attenuation with the criteria in the existing works. Thus, each of the criteria in the existing works [6, 8, 9, 10, 11, 12, 13] fails in the example given by the two-dimensional digital filter (1)-(2) with the 11
parameters (23)-(25). However, it is clear that the proposed criterion (6) verifies the asymptotic stability result with a guaranteed H∞ performance in this example.
3.2
Example 2
Consider the two-dimensional digital filter (1)-(2) with the following parameters:
0.01 −0.5 A12 = , 0 0.35
−0.1 0.9 A11 = , 0.39 0.4
−0.15 0.1 A21 = , 0.3 −0.1 wh (i, j) =
n2 (i, j)
,
(26)
0.1 0.23 A22 = , 0.8 0.03
n1 (i, j)
wv (i, j) =
(27)
n3 (i, j) n4 (i, j)
,
(28)
where n1 (i, j), n2 (i, j), n3 (i, j), and n4 (i, j) are mutually independent white noises with mean 0 and variance 0.1. Let the disturbance attenuation level γ = 0.4. Then, we obtain the following feasible solution:
0.0063 −0.0028 P2 = , −0.0028 0.0039
0 0.0012 P1 = , 0 0.0012
0.0200 −0.0014 Q= , −0.0014 0.0305
0 0.0270 M1 = , 0 0.0270 δ1 = 0.0042,
0.0606 0.0049 R= , 0.0049 0.0320
0 0.0590 M2 = , 0 0.0590
δ2 = 0.0064.
The existing criteria [6, 8, 9, 10, 11, 12, 13] fail to check the interference attenuation of the two-dimensional digital filter (1)-(2) with the parameters (26)-(28) because they were derived assuming that there are no external interferences in two-dimensional digital filters. The bound-
12
ary conditions are given by
3.1 xh (0, j) = , −4
0 ≤ j ≤ 20,
−1.8 xv (i, 0) = , 5.5
0 ≤ i ≤ 30.
Figure 1 and Figure 2 shows the first and second state trajectories of xh (i, j), respectively. Figure 3 and Figure 4 shows the first and second state trajectories of xv (i, j), respectively. These figures demonstrate that xh (i, j) and xv (i, j) of the two-dimensional digital filter are bounded around the origin due to the H∞ performance (5). The first state trajecory of xh(i,j)
4 3 2 1 0 −1 −2 40 30
40 30
20 20
10 j
10 0
0
i
Figure 1: The first state trajectory of xh (i, j)
4
Conclusion
This paper has proposed a new criterion for the H∞ performance and asymptotic stability of two-dimensional digital filters described by Roesser model with saturation arithmetic and external interference. The proposed criterion ensured reduction in the effect of the external interference to an interference attenuation level. Thus, it solved the shortcomings of the existing stability criteria for two-dimensional digital filters. The criterion was represented by LMI, which can be easily checked using existing numerical algorithms. Via a numerical example, we demonstrated the usefulness of the presented criterion. 13
The second state trajecory of xh(i,j)
2 1 0 −1 −2 −3 −4 40 30
40 30
20 20
10
10 0
j
0
i
Figure 2: The second state trajectory of xh (i, j) The first state trajecory of xv(i,j)
1.5 1 0.5 0 −0.5 −1 −1.5 −2 40 30
40 30
20 20
10
10 0
j
0
i
Figure 3: The first state trajectory of xv (i, j) v
The second state trajecory of x (i,j)
6
4
2
0
−2 40 30
40 30
20 20
10 j
10 0
0
i
Figure 4: The second state trajectory of xv (i, j)
14
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l2 -l∞ stability criterion for fixed-point state-space digital filters
with saturation arithmetic.
International Journal of Electronics, in press (doi:
10.1080/00207217.2012.743083), 2012. [27] C. K. Ahn. IOSS criterion for the absence of limit cycles in digital filters employing saturation overflow arithmetic. Circuits, Systems & Signal Processing, in press (doi: 10.1007/s00034-012-9520-0), 2012. [28] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishinan. Linear matrix inequalities in systems and control theory. SIAM, Philadelphia, PA, 1994. [29] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali. LMI Control Toolbox. The Mathworks Inc., 1995.
17
Abstract
To the present time, stability criteria have been proposed for one-dimensional digital filters with external interference, but no stability criterion exists for cases where two-dimensional digital filters have external interference. In this paper, we propose a new criterion for the elimination of overflow oscillations in twodimensional digital filters described by Roesser model with saturation arithmetic and external interference. This criterion ensures asymptotic stability with a guaranteed H∞ performance. The proposed criterion is represented in terms of linear matrix inequality (LMI); thus, it is computationally efficient. Illustrative examples are given to demonstrate the effectiveness of the proposed criterion. Key Words: H∞ criterion, two-dimensional digital filters, finite wordlength effects, linear matrix inequality (LMI)
1
1
Introduction
Two-dimensional systems have several applications such as image processing, thermal processes, seismographic data processing, and gas absorption [1]. Thus, two-dimensional system design is an important and challenging task. When designing discrete systems using fixedpoint arithmetic, quantization and overflow nonlinearities inevitably occur. The presence of such nonlinearities causes the instability of the designed system [2, 3, 4]. The zero-input limit cycles, which are undesirable, may possibly occur because of such nonlinearities. The quantization and overflow nonlinearities may interact with each other. However, if the number of quantization steps is large, or in other words, the internal wordlength is long enough, quantization effects can be neglected when we investigate the effects of overflow [5, 6]. The stability of two-dimensional discrete systems described by the Roesser model [7] has been investigated extensively [6, 8, 9, 10, 11, 12, 13]. The stability of two-dimensional systems described by the Fornasini-Marchesini second model [14] has also received much attention [10, 15, 16, 17]. The result in [6] consists of a generalization and improvement over the criterion proposed in [8]. In [11], an alternative stability criterion was presented based on reduced sector characterization of saturation nonlinearity. An improvement and generalization of [11] was proposed in [18]. In [19], another criterion for the asymptotic stability of two-dimensional digital filters has been established. The criterion proposed in [19] provides improvement over previous stability criteria [8, 6, 12]. In [20], a modified version of [19] was presented. When we implement a high-order digital filter in a hardware component, we usually break down it into several filters before hardware implementation. Then, there are interferences between the filters. These cause malfunction, as well as ultimate destruction [21, 22]. However, most criteria for the stability of digital filters are only available under specific conditions. In unfavorable environments with external interferences, these criteria will be useless. Thus, it is desirable to have an alternative criterion that can overcome the shortcomings of existing criteria. In order to solve these shortcomings, Ahn recently proposed new stability criteria for digital filters with external interferences [23, 24, 25, 26, 27]. However, these results were restricted to one-dimensional digital filters. To the best of our knowledge, there is no result in the literature so far concerning the robust stability of two-dimensional digital filters with
2
saturation arithmetic and external interference; this remains an open and challenging topic. In this paper, a new criterion for the H∞ stability of two-dimensional digital filters described by Roesser model with saturation arithmetic and external interference is proposed. This criterion ensures that the effect of the external interference on the two-dimensional digital filter can be attenuated to an interference attenuation level. The criterion is represented by linear matrix inequality (LMI), which can be solved easily using existing convex optimization algorithms [28, 29]. The novelty of this paper is to establish a new and first criterion for the H∞ performance and asymptotic stability of two-dimensional digital filters with external interference. The present paper is organized as follows. In Section 2, a new LMI criterion for the H∞ stability of two-dimensional digital filters with saturation arithmetic and external interference is proposed. In Section 3, numerical examples are given, and finally, conclusions are presented in Section 4.
2
New Interference Attenuation Criterion
Consider the following two-dimensional digital filter: h h + 1, j) w (i, j) y (i, j) + =f y v (i, j) wv (i, j) xv (i, j + 1)
xh (i
=
f h (y h (i, j)) f v (y v (i, j))
+
wh (i, j) wv (i, j)
y h (i, j)
,
xh (i, j)
A11 A12 = , y v (i, j) A21 A22 xv (i, j)
h h f h (y h (i, j)) = [f1h (y1h (i, j)), · · · , fm (ym (i, j))]T ,
f v (y v (i, j)) = [f1v (y1v (i, j)), · · · , fnv (ynv (i, j))]T , h y h (i, j) = [y1h (i, j), · · · , ym (i, j)]T ,
y v (i, j) = [y1v (i, j), · · · , ynv (i, j)]T ,
3
(1)
(2)
where xh (i, j) ∈ Rm is the horizontal state vector, xv (i, j) ∈ Rn is the vertical state vector, wh (i, j) ∈ Rm is the horizontal external interference, wv (i, j) ∈ Rn is the vertical external interference, y h (i, j) is the horizontal output vector, y v (i, j) is the vertical output vector, f h (·) is the horizontal overflow nonlinearity, f v (·) is the vertical overflow nonlinearity, A11 ∈ Rm×m , A12 ∈ Rm×n , A21 ∈ Rn×m , and A22 ∈ Rn×n are state matrices. The overflow arithmetic to be considered presently is the saturation arithmetic given by 1, fkh (ykh (i, j)) = ykh (i, j), −1, 1, fkv (ykv (i, j)) = ykv (i, j), −1,
if ykh (i, j) > 1 if − 1 ≤ ykh (i, j) ≤ 1 ,
k = 1, 2, · · · , m,
(3)
k = 1, 2, · · · , n.
(4)
if ykh (i, j) < −1 if ykv (i, j) > 1 if − 1 ≤ ykv (i, j) ≤ 1 , if ykv (i, j) < −1
Given a level γ > 0, the purpose of this paper is to find a new LMI criterion such that the two-dimensional digital filter (1)-(2) with wh (i, j) = wv (i, j) = 0 is asymptotically stable and ∑∞ ∑∞
h T (i, j)P xh (i, j) 1 j=0 [x i=0 ∑∞ ∑∞ T h (i, j)wh (i, j) i=0 j=0 [w
+ xvT (i, j)P2 xv (i, j)] + wvT (i, j)wv (i, j)]
< γ2
(5)
under zero boundary conditions for all nonzero wh (i, j) and wv (i, j), where P1 and P2 are positive symmetric matrices. Parameter γ is called the H∞ norm bound or the interference attenuation level. In this case, the two-dimensional digital filter (1)-(2) is said to be asymptotically stable with a guaranteed H∞ performance γ. A new H∞ stability criterion for the two-dimensional digital filter (1)-(2) is given in the following theorem. Theorem 1. For a given level γ > 0, if there exist symmetric positive definite matrices P1 ,
4
P2 , Q, and R, positive diagonal matrices M1 and M2 , and positive scalars δ1 and δ2 such that
Φ1,1
Φ1,2
AT11 M1
AT21 M2
0
0
Φ Φ2,2 AT12 M1 AT22 M2 0 0 2,1 0 Q 0 M1 A11 M1 A12 Q − δ1 I − 2M1 M2 A21 M2 A22 0 R − δ2 I − 2M2 0 R 0 0 Q 0 Q − γ2I 0 0 0 0 R 0 R − γ2I
< 0,
(6)
where Φ1,1 = δ1 AT11 A11 + δ2 AT21 A21 + P1 − Q, Φ1,2 = δ1 AT11 A12 + δ2 AT21 A22 , Φ2,1 = δ1 AT12 A11 + δ2 AT22 A21 , Φ2,2 = δ1 AT12 A12 + δ2 AT22 A22 + P2 − R, then the two-dimensional digital filter (1)-(2) is asymptotically stable with a guaranteed H∞ performance γ. Proof. Consider the following two-dimensional quadratic Lyapunov function: V (xh (i, j), xv (i, j)) =
T
xh (i, j)
xh (i, j)
Q 0 . xv (i, j) 0 R xv (i, j)
Along the trajectory of (1)-(2), we have ∆V (xh (i, j), xv (i, j)) = V (xh (i + 1, j), xv (i, j + 1)) − V (xh (i, j), xv (i, j)) =
T xh (i
xh (i
+ 1, j) Q 0 + 1, j) xv (i, j + 1) 0 R xv (i, j + 1)
5
(7)
−
T
xh (i, j)
xh (i, j)
Q 0 xv (i, j) 0 R xv (i, j)
=
T f h (A11 xh (i, j)
+ A12
xv (i, j))
+
wh (i, j)
+
wh (i, j)
Q 0 f v (A21 xh (i, j) + A22 xv (i, j)) + wv (i, j) 0 R
×
f h (A
11
xh (i, j)
+ A12
xv (i, j))
f v (A21 xh (i, j) + A22 xv (i, j)) + wv (i, j)
−
T
xh (i, j)
xh (i, j)
Q 0 . xv (i, j) 0 R xv (i, j)
(8)
From (3) and (4), it is clear that T
f h (A11 xh (i, j) + A12 xv (i, j))f h (A11 xh (i, j) + A12 xv (i, j)) = ∥f h (A11 xh (i, j) + A12 xv (i, j))∥2 ≤ ∥A11 xh (i, j) + A12 xv (i, j)∥2 = [A11 xh (i, j) + A12 xv (i, j)]T [A11 xh (i, j) + A12 xv (i, j)],
(9)
f vT (A21 xh (i, j) + A22 xv (i, j))f v (A21 xh (i, j) + A22 xv (i, j)) = ∥f v (A21 xh (i, j) + A22 xv (i, j))∥2 ≤ ∥A21 xh (i, j) + A22 xv (i, j)∥2 = [A21 xh (i, j) + A22 xv (i, j)]T [A21 xh (i, j) + A22 xv (i, j)].
(10)
Then, for positive scalars δ1 and δ2 , we have { δ1 [A11 xh (i, j) + A12 xv (i, j)]T [A11 xh (i, j) + A12 xv (i, j)] } T − f h (A11 xh (i, j) + A12 xv (i, j))f h (A11 xh (i, j) + A12 xv (i, j)) ≥ 0, { δ2 [A21 xh (i, j) + A22 xv (i, j)]T [A21 xh (i, j) + A22 xv (i, j)]
6
(11)
} − f vT (A21 xh (i, j) + A22 xv (i, j))f v (A21 xh (i, j) + A22 xv (i, j)) ≥ 0.
(12)
Consider the following two quantities: T
2f h (A11 xh (i, j) + A12 xv (i, j))M1 [A11 xh (i, j) + A12 xv (i, j) − f h (A11 xh (i, j) + A12 xv (i, j))], 2f vT (A21 xh (i, j) + A22 xv (i, j))M2 [A21 xh (i, j) + A22 xv (i, j) − f v (A21 xh (i, j) + A22 xv (i, j))].
Using (11)-(12) and adding to and subtracting from (8) these two quantities, a new bound for ∆V (xh (i, j), xv (i, j)) can be obtained as ∆V (xh (i, j), xv (i, j)) ≤
T f h (A11 xh (i, j)
+ A12
xv (i, j))
+
wh (i, j)
+
wh (i, j)
+
wv (i, j)
Q 0 f v (A21 xh (i, j) + A22 xv (i, j)) + wv (i, j) 0 R
× −
f h (A11 xh (i, j) f v (A21 xh (i, j)
+ A12
xv (i, j))
+ A22
xv (i, j))
T
{ Q 0 h v T + δ 1 [A11 x (i, j) + A12 x (i, j)] xv (i, j) 0 R xv (i, j)
xh (i, j)
xh (i, j)
T
× [A11 xh (i, j) + A12 xv (i, j)] − f h (A11 xh (i, j) + A12 xv (i, j)) { × f (A11 x (i, j) + A12 x (i, j)) + δ2 [A21 xh (i, j) + A22 xv (i, j)]T h
h
}
v
× [A21 xh (i, j) + A22 xv (i, j)] − f vT (A21 xh (i, j) + A22 xv (i, j)) } T × f v (A21 xh (i, j) + A22 xv (i, j)) + 2f h (A11 xh (i, j) + A12 xv (i, j))M1 × [A11 xh (i, j) + A12 xv (i, j) − f h (A11 xh (i, j) + A12 xv (i, j))] + 2f vT (A21 xh (i, j) + A22 xv (i, j))M2 [A21 xh (i, j) + A22 xv (i, j) T
− f v (A21 xh (i, j) + A22 xv (i, j))] − 2f h (y h (i, j))M1 [y h (i, j) − f h (y h (i, j))] − 2f vT (y v (i, j))M2 [y v (i, j) − f v (y v (i, j))]
7
T xh (i, j)
xv (i, j) h f (A11 xh (i, j) + A12 xv (i, j)) = v f (A21 xh (i, j) + A22 xv (i, j)) wh (i, j) wv (i, j)
Φ1,1
AT11 M1
Φ1,2
AT21 M2
0
Φ Φ2,2 AT12 M1 AT22 M2 0 2,1 M A 0 Q 1 11 M1 A12 Q − δ1 I − 2M1 × M2 A21 M2 A22 0 R − δ2 I − 2M2 0 0 0 Q 0 Q − γ2I 0
0
0
R
0
0
0 0 R 0 R − γ2I
xh (i, j)
xv (i, j) h f (A11 xh (i, j) + A12 xv (i, j)) × v f (A21 xh (i, j) + A22 xv (i, j)) wh (i, j) wv (i, j)
− xh T (i, j)P1 xh (i, j)
T
− xvT (i, j)P2 xv (i, j) + γ 2 wh (i, j)wh (i, j) + γ 2 wvT (i, j)wv (i, j) + Ω(i, j),
(13)
where T
Ω(i, j) = −2f h (y h (i, j))M1 [y h (i, j) − f h (y h (i, j))] − 2f vT (y v (i, j))M2 [y v (i, j) − f v (y v (i, j))].
(14)
Note that Ω(i, j) is nonpositive in view of (3) and (4). If the LMI (6) is satisfied, we have T
∆V (xh (i, j), xv (i, j)) < −xh (i, j)P1 xh (i, j) − xvT (i, j)P2 xv (i, j)
8
T
+γ 2 wh (i, j)wh (i, j) + γ 2 wvT (i, j)wv (i, j).
(15)
Summation of both sides of (15) from i, j = 0 to i, j = ∞ gives ∞ ∑ ∞ ∑
∆V (xh (i, j), xv (i, j))
i=0 j=0
0 and p2 > 0, we have p1 ∑ p2 ∑
∆V (xh (i, j), xv (i, j))
i=0 j=0
=
p2 ∑
T
T
[xh (p1 + 1, j)Qxh (p1 + 1, j) − xh (0, j)Qxh (0, j)]
j=0 p1 ∑ + [xvT (i, p2 + 1)Rxv (i, p2 + 1) − xvT (i, 0)Rxv (i, 0)], i=0
which together with (16) implies that −
∞ ∑ ∞ ∑
T
xh (i, j)P1 xh (i, j) −
i=0 j=0
+γ
2
∞ ∑ ∞ ∑
∞ ∑ ∞ ∑
xvT (i, j)P2 xv (i, j)
i=0 j=0
w
hT
h
(i, j)w (i, j) + γ
i=0 j=0
2
∞ ∞ ∑ ∑
wvT (i, j)wv (i, j) > 0
(17)
i=0 j=0
under zero boundary conditions. Thus, we have the relation (5). When wh (i, j) = wv (i, j) = 0, we have T
∆V (xh (i, j), xv (i, j)) < −xh (i, j)P1 xh (i, j) − xvT (i, j)P2 xv (i, j) ≤ 0
9
(18)
from (15). This guarantees that lim
i,j→∞
xh (i, j) xv (i, j)
= 0.
(19)
This completes the proof. Let
A11 A12 P1 0 A= , , S = 0 P2 A21 A22
Q 0 M1 0 P = , M = , δ = δ1 = δ2 . 0 R 0 M2
(20)
Then, the criterion (6) becomes
δAT A
+S−P
AT M
0
MA
P − δI − 2M
P
0
P
P − γ2I
< 0,
(21)
which is the disturbance attenuation criterion for one-dimensional digital filters [23]. Remark 1. Criteria in the existing works [6, 8, 9, 10, 11, 12, 13] were derived assuming that there are no external interferences in two-dimensional digital filters. These existing criteria fail to check the interference attenuation of two-dimensional digital filters. In contrast to these existing results, in this paper, we consider two-dimensional digital filters with external interferences and propose a new LMI criterion (6) for interference attenuation and asymptotic stability of these digital filters. Based on Theorem 1, we present a method to obtain the optimal H∞ norm bound. Corollary 1. The optimal H∞ norm bound γ is obtained by solving the following semi-definite programming problem:
min γ 2 γ>0
10
(22)
subject to the LMI (6), P1 > 0, P2 > 0, Q > 0, R > 0, M1 > 0, M2 > 0, δ1 > 0, and δ2 > 0.
3 3.1
Numerical Examples Example 1
Consider the two-dimensional digital filter (1)-(2) with
0.01 A12 = , 0
1.2 0.55 A11 = , −0.09 0.11 [ A21 =
(23)
] 0.01 0
,
A22 = 0.01,
(24)
cos(5i) wh (i, j) = , 2 sin (2j)
wv (i, j) = cos(i + j).
(25)
For the design objective (5), let the H∞ performance be specified by γ = 0.3. A feasible solution to the LMI (6) is
0.0007 −0.0008 P1 = , −0.0008 0.0012
0.0028 0.0008 Q= , 0.0008 0.0100
R = 0.0661,
0 0.0062 M1 = , 0 0.0062 δ1 = 1.3878 × 10−4 ,
P2 = 0.0321,
M2 = 3.0307,
δ2 = 1.1073.
Criteria in the existing works [6, 8, 9, 10, 11, 12, 13] were derived assuming that there are no external interferences in two-dimensional digital filters. If we consider two-dimensional digital filters with external interferences, it is not possible to guarantee interference attenuation with the criteria in the existing works. Thus, each of the criteria in the existing works [6, 8, 9, 10, 11, 12, 13] fails in the example given by the two-dimensional digital filter (1)-(2) with the 11
parameters (23)-(25). However, it is clear that the proposed criterion (6) verifies the asymptotic stability result with a guaranteed H∞ performance in this example.
3.2
Example 2
Consider the two-dimensional digital filter (1)-(2) with the following parameters:
0.01 −0.5 A12 = , 0 0.35
−0.1 0.9 A11 = , 0.39 0.4
−0.15 0.1 A21 = , 0.3 −0.1 wh (i, j) =
n2 (i, j)
,
(26)
0.1 0.23 A22 = , 0.8 0.03
n1 (i, j)
wv (i, j) =
(27)
n3 (i, j) n4 (i, j)
,
(28)
where n1 (i, j), n2 (i, j), n3 (i, j), and n4 (i, j) are mutually independent white noises with mean 0 and variance 0.1. Let the disturbance attenuation level γ = 0.4. Then, we obtain the following feasible solution:
0.0063 −0.0028 P2 = , −0.0028 0.0039
0 0.0012 P1 = , 0 0.0012
0.0200 −0.0014 Q= , −0.0014 0.0305
0 0.0270 M1 = , 0 0.0270 δ1 = 0.0042,
0.0606 0.0049 R= , 0.0049 0.0320
0 0.0590 M2 = , 0 0.0590
δ2 = 0.0064.
The existing criteria [6, 8, 9, 10, 11, 12, 13] fail to check the interference attenuation of the two-dimensional digital filter (1)-(2) with the parameters (26)-(28) because they were derived assuming that there are no external interferences in two-dimensional digital filters. The bound-
12
ary conditions are given by
3.1 xh (0, j) = , −4
0 ≤ j ≤ 20,
−1.8 xv (i, 0) = , 5.5
0 ≤ i ≤ 30.
Figure 1 and Figure 2 shows the first and second state trajectories of xh (i, j), respectively. Figure 3 and Figure 4 shows the first and second state trajectories of xv (i, j), respectively. These figures demonstrate that xh (i, j) and xv (i, j) of the two-dimensional digital filter are bounded around the origin due to the H∞ performance (5). The first state trajecory of xh(i,j)
4 3 2 1 0 −1 −2 40 30
40 30
20 20
10 j
10 0
0
i
Figure 1: The first state trajectory of xh (i, j)
4
Conclusion
This paper has proposed a new criterion for the H∞ performance and asymptotic stability of two-dimensional digital filters described by Roesser model with saturation arithmetic and external interference. The proposed criterion ensured reduction in the effect of the external interference to an interference attenuation level. Thus, it solved the shortcomings of the existing stability criteria for two-dimensional digital filters. The criterion was represented by LMI, which can be easily checked using existing numerical algorithms. Via a numerical example, we demonstrated the usefulness of the presented criterion. 13
The second state trajecory of xh(i,j)
2 1 0 −1 −2 −3 −4 40 30
40 30
20 20
10
10 0
j
0
i
Figure 2: The second state trajectory of xh (i, j) The first state trajecory of xv(i,j)
1.5 1 0.5 0 −0.5 −1 −1.5 −2 40 30
40 30
20 20
10
10 0
j
0
i
Figure 3: The first state trajectory of xv (i, j) v
The second state trajecory of x (i,j)
6
4
2
0
−2 40 30
40 30
20 20
10 j
10 0
0
i
Figure 4: The second state trajectory of xv (i, j)
14
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