Dissipativity analysis for fixed-point interfered digital filters

Signal Processing ] (]]]]) ]]]–]]]

1

Contents lists available at ScienceDirect

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Signal Processing

5

journal homepage: www.elsevier.com/locate/sigpro

7 9 11

Dissipativity analysis for fixed-point interfered digital filters

13 Q1

15 Q4 Q3

Choon Ki Ahn a, Peng Shi b a b

School of Electrical Engineering, Korea University, Anam-Dong 5-Ga, Sungbuk-Gu, Seoul 136-701, South Korea School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia

17 19

a r t i c l e i n f o

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Article history: Received 13 July 2014 Received in revised form 17 September 2014 Accepted 24 October 2014

23 25 27 29

Keywords: (Q, S, R)-α-dissipativity Unified framework Digital filter Finite word length effect

abstract This paper is concerned with establishing a new criterion for the (Q, S, R)-α-dissipativity of fixed-point interfered state-space digital filters with saturation overflow arithmetic. The objective of this paper is to present the H 1 performance, passivity, and mixed H 1 =passivity criteria in a unified framework. By tuning the weight matrices, the proposed criterion reduces to the H 1 performance, passivity, and mixed H 1 =passivity criteria. Improved criteria are also proposed for reducing the conservatism of the proposed criterion. These criteria are expressed with linear matrix inequalities (LMIs). A numerical example shows the effectiveness of the proposed results. & 2014 Published by Elsevier B.V.

31

63

33

1. Introduction

35

The digital filter is a very important building block in many engineering areas, such as electrical and electronic engineering. The implementation of a digital filter with fixed-point arithmetic in digital hardware is associated with nonlinearities, such as quantization and saturation, because of finite word length effects and a constraint imposed on the maximum bound of signals. These nonlinearities can generate undesired effects, such as zero-input limit cycles, in digital filters. Hence, the analysis and design of digital filters under these nonlinearities are extremely important. So far, much research has focused on the analysis of the stability of digital filters under nonlinearities [1–11]. The hardware implementation of a large-scale digital filter usually requires its division into several small-scale digital filters. In this situation, interferences between these smallscale filters always exist, resulting in poor performance or final destruction [12,13]. Thus, analysis of the effects of interferences in digital filters is an important research subject. Recently, Ahn tackled this issue and established some new stability criteria for one-dimensional and two-dimensional digital filters in [14–18] and [18–23], respectively. The dissipativity concept [24,25], which originated from electrical networks, gives an important framework for synthesis and analysis of several control and signal

37 39 41 43 45 47 49 51 53 55 57 59

processing systems using input–output descriptions with energy-based considerations. The input–output description leads to a modular approach to the synthesis and analysis of signal processing systems (for example, digital filters). One of the important properties of dissipative systems is that the total energy stored in the dissipative system decreases through time. It turns out that the dissipativity concept is a very helpful guide for the design of state estimation filters and output feedback controllers [26]. Dissipativity is regarded as a generalization of some well-known performance indices, such as H 1 performance, passivity, and mixed H 1 =passivity. Thus, dissipativity provides a unified framework to cover H 1 performance, passivity, and mixed H 1 =passivity [27–31]. For this reason, many researchers have used the dissipativity approach to create new controller and observer design methods for several nonlinear systems [26,32–35]. Here, an interesting question arises: Is it possible to obtain a dissipativity criterion for fixed-point state-space interfered digital filters? This paper answers this question in the positive. To the best of the authors’ knowledge, the current literature contains no papers on the dissipativity of fixedpoint state-space interfered digital filters using saturation overflow arithmetic. This paper establishes a new (Q, S, R)-α-dissipativity criterion for fixed-point state-space interfered digital filters

http://dx.doi.org/10.1016/j.sigpro.2014.10.029 0165-1684/& 2014 Published by Elsevier B.V.

61 Please cite this article as: C.K. Ahn, P. Shi, Dissipativity analysis for fixed-point interfered digital filters, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.10.029i

65 67 69 71 73 75 77 79 81 83 85 87 89 91

C.K. Ahn, P. Shi / Signal Processing ] (]]]]) ]]]–]]]

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1 3 5 7 9 11 13 15 17 19

with saturation overflow arithmetic. The purpose of this paper is to provide a unified filter stability analysis approach for fixed-point digital filters with interferences. The proposed criterion covers the H 1 performance, passivity, and mixed H 1 =passivity criteria as special cases by tuning the weight matrices. With the introduction of slack matrices and diagonally dominant matrices, improved (Q, S, R)α-dissipativity criteria for fixed-point state-space digital filters are also proposed to reduce the conservatism of the proposed criterion. These criteria are described with linear matrix inequalities (LMIs) [36,37] and thus are computationally attractive. This paper is organized as follows. In Section 2, we present an LMI-based criterion for the (Q, S, R)-α-dissipativity of fixed-point state-space interfered digital filters. In Section 3, improved (Q, S, R)-α-dissipativity criteria are proposed using slack matrices and diagonally dominant matrices. In Section 4, we investigate some special cases of the proposed criterion. In Section 5, a numerical example is given, and finally, conclusions are presented in Section 6.

P A Rnn , a positive definite diagonal matrix M A Rnn , and a positive scalar δ such that Φ o0, where 2

Φ¼6 4

δAT A AT QA P

⋆ P  δI  2 M

MA

δA ST A QA

3

⋆

P þM

⋆ P þ ðδ þ αÞI R  Q  S ST

23 25 27

Consider the following form of digital filter:

69 71

31

T

75

T

f ðyðrÞÞf ðyðrÞÞ ¼ f ðAxðrÞ þ wðrÞÞf ðAxðrÞ þwðrÞÞ T

r½AxðrÞ þwðrÞ ½AxðrÞ þwðrÞ

33 35 37 39 41

þ wT ðrÞwðrÞ;

ð7Þ

from which we have

81 ð8Þ

83

Consider the following function: VðxðrÞÞ ¼ x ðrÞPxðrÞ. Using (8), its time difference (ΔVðxðrÞÞ 9Vðxðr þ 1ÞÞ  VðxðrÞÞ) satisfies

85

which are confined to the sector ½0; 1, i.e., f i ð0Þ ¼ 0; 0 r

45

In this paper, given a constant α Z0 and constant matrices Q A Rnn , S A Rnn , and RA Rnn with Q and R symmetric, we obtain a new criterion such that the digital filter (1) and (2) satisfies

49

T

51

T

i ¼ 1; 2; …; n:

T

ð4Þ

T

∑ y ðrÞQyðrÞ þ 2 ∑ y ðrÞSwðrÞ þ ∑ w ðrÞRwðrÞ Z α ∑ w ðrÞwðrÞ

r¼0

T

r¼0

T

r¼0

T

T

r¼0

ð5Þ 53 55 57 59 61

T

89

T

þwT ðrÞPwðrÞ xT ðrÞPxðrÞ þ 2f ðyðrÞÞM½AxðrÞ T

þwðrÞ  f ðyðrÞÞ  2f ðyðrÞÞM½yðrÞ f ðyðrÞÞ

91

þ δ½x ðrÞA AxðrÞ þ w ðrÞAxðrÞ þ x ðrÞA wðrÞ T

T

87

T

T

T

T

93

T

þwT ðrÞwðrÞ  f ðyðrÞÞf ðyðrÞÞ ¼ Ω ðrÞΦΩðrÞ þ yT ðrÞQyðrÞ þ 2yT ðrÞSwðrÞ T

95

þwT ðrÞ½R  αIwðrÞ 2f ðyðrÞÞM  ½yðrÞ  f ðyðrÞÞ; T

ð9Þ where

2

xðrÞ

97 99

3

7 ΩðrÞ ¼ 6 4 f ðyðrÞÞ 5

ð10Þ

wðrÞ

101 103

T

43

47

T

ð2Þ

where xðrÞ ¼ ½x1 ðrÞx2 ðrÞ⋯xn ðrÞT A Rn is the state vector, yðrÞ ¼ ½y1 ðrÞy2 ðrÞ⋯yn ðrÞT A Rn is the output vector, wðrÞ ¼ ½w1 ðrÞw2 ðrÞ⋯wn ðrÞT A Rn is the external interference, and A A Rnn is the coefficient matrix. Here, we consider the following saturation nonlinearities: 8 if yi ðrÞ 4 1; > <1 f i ðyi ðrÞÞ ¼ yi ðrÞ if  1 r yi ðrÞ r1; i ¼ 1; 2; …; n; ð3Þ > : 1 if y ðrÞ o 1; i

79

δ½xT ðrÞAT AxðrÞ þ wT ðrÞAxðrÞ þ xT ðrÞAT wðrÞ þwT ðrÞwðrÞ f T ðyðrÞÞf ðyðrÞÞ Z 0:

ð1Þ

f i ðyi ðrÞÞ r 1; yi ðrÞ

77

¼ xT ðrÞAT AxðrÞ þwT ðrÞAxðrÞ þxT ðrÞAT wðrÞ

ΔVðxðrÞÞ r f ðyðrÞÞPf ðyðrÞÞ þ f ðyðrÞÞPwðrÞ þw ðrÞPf ðyðrÞÞ

yðrÞ ¼ AxðrÞ þwðrÞ;

73

Proof. Condition (4) implies

¼ ½f 1 ðy1 ðrÞÞf 2 ðy2 ðrÞÞ⋯f n ðyn ðrÞÞT þ ½w1 ðrÞw2 ðrÞ⋯wn ðrÞT ;

29

67

and ⋆ denotes an entry that can be deduced from the symmetry of the matrix. Then, the digital filter (1) and (2) is (Q, S, R)-α-dissipative with the performance bound α.

T

xðr þ 1Þ ¼ f ðyðrÞÞ þ wðrÞ

65

ð6Þ

21 2. (Q, S, R)-α-dissipativity criterion for fixed-point digital filters

7 5

63

under the zero initial condition, where T 40. The digital filter is said to be (Q, S, R)-α-dissipative with the performance bound α if condition (5) is satisfied. The following theorem gives a new (Q, S, R)-α-dissipativity criterion for state-space fixed-point digital filters. Theorem 1. Given a constant α Z 0 and constant matrices Q A Rnn , S A Rnn , and R A Rnn with Q and R symmetric, assume that there exist a symmetric positive definite matrix

and the term  2f ðyðrÞÞM½yðrÞ f ðyðrÞÞ is not positive considering (3). Thus, if the LMI (6) is satisfied, we have

105

ΔVðxðrÞÞ o yT ðrÞQyðrÞ þ 2yT ðrÞSwðrÞ þ wT ðrÞ½R αIwðrÞ:

107 ð11Þ 109

Summation of both sides of (11) from 0 to T 1 gives T

T

T

r¼0

r¼0

r¼0

111

∑ yT ðrÞQyðrÞ þ 2 ∑ yT ðrÞSwðrÞ þ ∑ wT ðrÞ½R αIwðrÞ T

4 ∑ ΔVðxðrÞÞ ¼ VðxðT þ 1ÞÞ Vðxð0ÞÞ≧0 r¼0

ð12Þ

under the zero initial condition, which guarantees (5). This completes the proof. □ Remark 1. Given matrices Q, S, and R, the optimal dissipativity performance bound αn can be obtained by maximizing αsubject to (6), P 4 0, M 4 0, and δ 40. Remark 2. Let Q o0. When wðrÞ ¼ 0, we have ΔVðxðrÞÞ oyT ðrÞQyðrÞ r0 from (11) under the condition Φ o 0. This

Please cite this article as: C.K. Ahn, P. Shi, Dissipativity analysis for fixed-point interfered digital filters, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.10.029i

113 115 117 119 121 123

C.K. Ahn, P. Shi / Signal Processing ] (]]]]) ]]]–]]]

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guarantees limr-1 xðrÞ ¼ 0 from the Lyapunov stability theory.

3

Remark 3. If the (Q, S, R)-α-dissipativity criterion for fixed-point digital filters is established, this criterion can be widely used in testing the robustness performances (such as the H 1 performance, passivity performance, and mixed H 1 =passivity performance, of digital filters, including low-pass filters and high-pass filters) against external interference in a unified framework when they are implemented using digital hardware and/or software. For this reason, the establishment of the (Q, S, R)-α-dissipativity criterion for fixed-point digital filters is both important and desirable from a practical viewpoint.

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

Remark 4. In this paper, we propose new and the first results on the (Q, S, R)-α-dissipativity of fixed-point digital filters. The results proposed in this paper open new possibilities for the application of the (Q, S, R)-α-dissipativity approach to many digital filters, including timedelayed and shift-varying digital filters. Establishing these new criteria would make for interesting future work. Remark 5. Most existing work on the stability of fixed-point digital filters investigates the asymptotic stability of digital filters, where no external interference in digital filters is assumed [1–11]. However, since external interference between digital filters inevitably exists in their implementation in hardware or software, using the existing stability conditions, it is not possible to test the robustness performances for digital filters with external interference. In order to tackle this problem, conditions for the robustness performances of digital filters have been proposed in [14–18,38] independently. This paper proposes some conditions for the H 1 performance, passivity performance, and mixed H 1 =passivity performance of digital filters in a unified framework based on the (Q, S, R)-α-dissipativity. Remark 6. In recent years, some filter/controller design approaches, such as dissipativity-based sliding mode controller design [39], fuzzy distributed filter design [40], induced l2 fuzzy filter design [41], and passivity-based sliding mode controller design [42], have been proposed. These results consider state estimator or controller design problems for given linear or nonlinear systems. In [43], a finite impulse response (FIR) filter design method was proposed for obtaining the equiripple characteristic with changing notch frequencies. However, in this paper, we analyze the robustness performances of fixed-point digital filters (such as the low-pass filters and high-pass filters) based on the (Q, S, R)-α-dissipativity. For this reason, this paper deals with a different problem from [39–43].

3

P A Rnn , a positive definite diagonal matrix M A Rnn , matrices H i A Rnn (i ¼ 1; 2), and a positive scalar δ such that ~ o 0, where Φ 2

6 6 ~ ¼6 Φ 6 6 4

δAT A AT QA P

3

⋆ P  δI  2M

⋆ ⋆

⋆ ⋆

 H T1

H2

δA  ST A  QA þ HT1

P þM

 H2  HT2 H T2

~ ð4; 4Þ Φ

MA þ H T1

⋆

7 7 7 7; 7 5

63 65 67 69

ð13Þ

71

ð14Þ

73

Then, the digital filter (1) and (2) is (Q, S, R)-α-dissipative with the performance bound α.

75

~ ð4; 4Þ ¼ P þðδ þ αÞI  R Q S  ST : Φ

77

Proof. For any matrices H1 and H2, it is clear that    2 xT ðrÞH 1 þ xT ðr þ1ÞH2 f ðyðrÞÞ þ wðrÞ xðr þ 1Þ ¼ 0

ð15Þ

from (1). Substituting this into ΔVðxðrÞÞ yields ~ ðrÞΦ ~ ðrÞ þ yT ðrÞQyðrÞ þ 2yT ðrÞSwðrÞ ~Ω ΔVðxðrÞÞ r Ω T þwT ðrÞ½R  αIwðrÞ  2f ðyðrÞÞM  ½yðrÞ  f ðyðrÞÞ; T

ð16Þ where

2

3 xðrÞ 6 7 f ðyðrÞÞ 7 ~ ðrÞ ¼ 6 6 7 Ω 6 xðr þ 1Þ 7: 4 5 wðrÞ

ð17Þ

Theorem 3. Given a constant α Z 0 and constant matrices Q A Rnn , S A Rnn , and R A Rnn with Q and R symmetric, assume that there exist symmetric positive definite matrices P A Rnn , G ¼ ½g st  A Rnn , and matrices H i A Rnn (i ¼ 1; 2), b o0, where L ¼ ½lst  A Rnn such that Φ 2

AT GA  AT QA  P 6 6 LT A þ H T1 b ¼6 Φ 6 6  H T1 4 GA ST A  QA þ H T1

3

⋆ P  G L  LT

⋆ ⋆

⋆ ⋆

H2

 H2  HT2 H T2

n

In this section, first, we study an improved (Q, S, R)-αdissipativity criterion, which reduces the potential conservatism of the criterion (13) in Theorem 1 by using two slack matrices.

lss 4 0;

Theorem 2. Given a constant α Z 0 and constant matrices Q A Rnn , S A Rnn , and R A Rnn with Q and R symmetric, assume that there exist a symmetric positive definite matrix

14

t ¼ 1;t a s

jg st j;

85

89

b ð4; 4Þ Φ

P þL

s ¼ 1; 2; …; n;

93 95 97 99 101 103 105

7 7 7 7; 7 5

109

ð18Þ

111

⋆

b ð4; 4Þ ¼ P þ αI þ G  R Q S ST ; Φ ∑

83

91

In Theorem 2, δ is a positive scalar, and the diagonal structure is imposed on the matrix M. For this reason, the LMI condition in Theorem 2 may be conservative. In the following theorem, we present a less conservative condition based on diagonally dominant matrices.

g ss Z

81

87

Using the proof in Theorem 1, it is easy to show that (5) is satisfied under (13). This completes the proof. □

3. Improved criteria

79

107

113 ð19Þ

115 117

s ¼ 1; 2; …; n;

lst ¼ lss χ st ; n

∑

t ¼ 1;t a s

ð20Þ

s; t ¼ 1; 2; …; n ðs atÞ;

ð21Þ

jχ st j;

ð22Þ

119 121

s ¼ 1; 2; …; n:

Please cite this article as: C.K. Ahn, P. Shi, Dissipativity analysis for fixed-point interfered digital filters, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.10.029i

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C.K. Ahn, P. Shi / Signal Processing ] (]]]]) ]]]–]]]

4

1 3 5

Then, the digital filter (1) and (2) is (Q, S, R)-α-dissipative with the performance bound α. Proof. By Lemma 1 in [4,8], the saturation nonlinearity in (3) satisfies T

7

δA þA

T

f ðyðrÞÞGf ðyðrÞÞ r ½AxðrÞ þ wðrÞ G½AxðrÞ þwðrÞ;

15 17

2

xT ðrÞAT GAxðrÞ þ wT ðrÞGAxðrÞ þ xT ðrÞAT GwðrÞ þ wT ðrÞGwðrÞ

ð24Þ

Ξ ðrÞ ¼ yT ðrÞLf ðyðrÞÞ f T ðyðrÞÞLT yðrÞ þ f T ðyðrÞÞ½L þLT f ðyðrÞÞ: ΔVðxðrÞÞ o y ðrÞQyðrÞ þ 2y ðrÞSwðrÞ þ w ðrÞ½R  αIwðrÞ: T

T

T

ð25Þ

31 33 35

41 43

Remark 7. In Theorem 1, a diagonal matrix M and a scalar δ are introduced, which leads to the potential conservatism of the LMI condition in (6). We reduced the conservatism by employing the zero matrix equality (15) with the free matrix variables H1 and H2 in Theorem 2, which will be verified in Section 5. The potential conservatism of the condition (13) in Theorem 2 was further improved based on the diagonally dominant matrices G and L in Theorem 3. 4. Special cases This section examines several special cases of interest, which each appear for specific choices of the parameters Q, S, and R.

45

4.1. H 1 performance criteria

47

Let Q ¼ I, S¼0, and R ¼ ðα2 þ αÞI. Substitute these values into (11) and obtain

49

ΔVðxðrÞÞ o  yT ðrÞyðrÞ þ α2 wT ðrÞwðrÞ:

51

Summation of both sides of this relation from 0 to 1 yields 1

53

 ∑ yT ðrÞyðrÞ þ α2 ∑ wT ðrÞwðrÞ r¼0

r¼0

1

4 ∑ ΔVðxðrÞÞ

57

¼ Vðxð1ÞÞ Vðxð0ÞÞ Z 0

61

⋆ ⋆

H2

 H2  H T2 HT2

⋆

P þL

P þ G α2 I þ I

7 7 7 7o0 7 5

 ∑ yT ðrÞyðrÞ þ γ 2 ∑ wT ðrÞwðrÞ 4 Vðxð1ÞÞ Vðxð0ÞÞ r¼0 T

r¼0 2 T

r¼0

ð27Þ

ð30Þ

from (27). This implies  1  1 ∑ yT ðrÞyðrÞ o γ 2 ∑ wT ðrÞwðrÞ þ xT ð0Þxð0Þ :

ð31Þ

1

1

r¼0

r¼0

∑ yT ðrÞyðrÞ o α2 ∑ wT ðrÞwðrÞ:

ð28Þ

71 73 75 77 79

r¼0

83 85 87 89

4.2. Passivity criteria 91

Let Q¼0, S ¼I, and R ¼ 2αI. Then, we have T

T

r¼0

r¼0

2 ∑ yT ðrÞwðrÞ Z  α ∑ wT ðrÞwðrÞ

ð32Þ

93 95

from (5). This relation means that the digital filter (1) and (2) is passive from w(r) to y(r). In this case, we obtain the following passivity criterion: 2 T 3 δA A  P ⋆ ⋆ 6 7 ⋆ P  δI 2M ð33Þ 4 MA 5 o0: δA  A P þM P þ ðδ  αÞI  2I In addition, the improved criterion can be obtained as 2 3 AT GA  P ⋆ ⋆ ⋆ 6 T 7 6 L A þ H T1 7 P  G  L  LT ⋆ ⋆ 6 7 6 7 o0: T T 6 7 H2  H2  H2 ⋆  H1 4 5 P þL HT2 P þ G  αI  2I GA  A þ H T1 ð34Þ

97 99 101 103 105 107 109 111

4.3. Mixed H 1 =passivity criteria

113

Let μ A ½0; 1 be a weighting parameter that represents the trade-off between passivity and H 1 performances. By choosing Q ¼  μI, S ¼ ð1  μÞI, and R ¼ ½ðα2  αÞμ þ 2αI, we obtain the following mixed H 1 =passivity criterion:

115

2 6 6 4

under the zero initial state condition, which implies

69

81

Z  x ð0ÞPxð0Þ 4  γ x ð0Þxð0Þ

r¼0

67

1

ð26Þ

1

55

59

⋆ ⋆

Using the proof in Theorem 1, we can show that (5) is satisfied. This completes the proof. □

37 39

⋆ P  G  L  LT

1

23

29

3

from Theorem 3. The zero initial state condition can be removed by adding the condition P o γ 2 I. Under this condition, we have

ð23Þ

T

b o0 implies Ξ ðrÞ r 0 in view of (3), (20)–(22). Thus, Φ

27

ð29Þ

2

~ ðrÞΦ ~ ðrÞ þ yT ðrÞQyðrÞ þ 2yT ðrÞSwðrÞ bΩ ΔVðxðrÞÞ r Ω T þ w ðrÞ½R  αIwðrÞ þ Ξ ðrÞ;

21

25

7 5o0

P þ δI  α I þI

P þM

63 65

Using the arguments in Theorems 1, 2, and [8], ΔVðxðrÞÞ satisfies

T

 f ðyðrÞÞGf ðyðrÞÞ Z 0:

where 19

3

AT GA þ AT A  P 6 6 LT Aþ HT1 6 6 6  HT1 4 GAþ Aþ HT1

11 13

performance

from Theorem 1. A less conservative criterion for the H 1 performance is given by

which leads to 9

In this case, we have the following H 1 criterion [14,15]: 2 T δA A þ AT A P ⋆ ⋆ 6 ⋆ MA P  δI 2M 4

δAT A þ μAT A  P MA

δA A þ 2μA

⋆

⋆

P  δI  2M P þM

⋆ P þ ½δ  α  2 þ ð3 þ α  α2 ÞμI

117

3

119

7 7 o0: 5

121

ð35Þ

123

Please cite this article as: C.K. Ahn, P. Shi, Dissipativity analysis for fixed-point interfered digital filters, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.10.029i

C.K. Ahn, P. Shi / Signal Processing ] (]]]]) ]]]–]]]

3 5 7

Then, we can also obtain a less conservative the mixed H 1 =passivity as 2 T ⋆ ⋆ A GAþ μAT A P 6 T T T 6 A þ H P  G  L  L ⋆ L 1 6 6 6 H T1 H2  H 2  H T2 4 GA  Aþ 2μA þH T1 P þL H T2

criterion for ⋆

3

7 ⋆7 7 7 o 0; ⋆7 5

Θ

ð36Þ

9

where Θ ¼ P þ G þ ½  α 2 þð3 þ α  α ÞμI, from Theorem 3. 2

11 5. Numerical example 13 15 17 19 21 23 25

Consider a second-order filter (1) and (2) with   0:4  0:6 A¼ : 0:15 0:4

ð37Þ

Let Q ¼  λI, S ¼ ξI, and R ¼ ζ I. First, we set λ ¼ 1, ξ ¼ 3, and ζ ¼ 4. The criterion (6) in Theorem 1 is not feasible. Thus, we cannot check the dissipativity of the digital filter in this example using Theorem 1. However, if we use criterion (13) in Theorem 2, we can guarantee the dissipativity of the digital filter and obtain the optimal dissipativity performance bound αn ¼ 0:9986. Then, the following feasible solution is obtained as    1:5975 0 0:7652  0:1370 ; δ ¼ 0:0013; ; M¼ P¼ 0 1:5975 0:1370 1:3969      0:1653 0:0581 0:8286 0:1295 H1 ¼ ; H2 ¼ : 0:2329 0:1616 0:1381 0:2012

coefficient variations and/or time delays. In addition, the dissipativity analysis of interfered shift-varying digital filters would be an interesting research topic. Finally, the experimental demonstration of the proposed results with

5

33 35 37 39 41 43

Thus, it is verified that criterion (13) in Theorem 2 improves the conservatism of the criterion (6) in Theorem 1. Next, we investigate the variations of the optimal dissipativity performance bound αn with respect to λ, ξ, and ζ. Fig. 1 shows the plot of αn versus λ with fixed ξ ¼3 and ζ ¼4. Fig. 2 shows the plot of αn versus ξ with fixed λ ¼ 1 and ζ ¼ 4. Fig. 3 shows the plot of αn versus ζ with fixed λ ¼1 and ξ ¼3. From these results, we can find that the optimal dissipativity performance bound αn decreases monotonically with the increase of λ. However, the optimal dissipativity performance bound αn increases monotonically with the increase of ξ and ζ. Thus, in order to obtain a large optimal dissipativity performance bound, we have to select a small value of λ and large values of ξ and ζ.

45 6. Conclusion 47 49 51 53 55 57 59 61

This paper has established a new (Q, S, R)-α-dissipativity criterion for fixed-point state-space interfered digital filters with saturation overflow arithmetic. By adjusting the weight matrices in the proposed criterion, we obtained several stability criteria, such as H 1 performance, passivity, and mixed H 1 =passivity criteria for fixed-point state-space interfered digital filters in a unified framework. In addition, improved (Q, S, R)-α-dissipativity criteria were presented to reduce the conservatism of the obtained criterion using slack matrices and diagonally dominant matrices. These criteria were presented in the form of LMI. The effectiveness of the proposed approach was shown using a numerical example. In the future, we will consider extending the results proposed in this paper to digital filters with random

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Fig. 1. The optimal dissipativity performance bound (αn ) versus λ.

87 6 The optimal dissipativity performance bound (α*)

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The optimal dissipativity performance bound (α*)

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Please cite this article as: C.K. Ahn, P. Shi, Dissipativity analysis for fixed-point interfered digital filters, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.10.029i

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C.K. Ahn, P. Shi / Signal Processing ] (]]]]) ]]]–]]]

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field-programmable gate array (FPGA) or very large-scale integration (VLSI) implementation would be interesting future works.

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Please cite this article as: C.K. Ahn, P. Shi, Dissipativity analysis for fixed-point interfered digital filters, Signal Processing (2014), http://dx.doi.org/10.1016/j.sigpro.2014.10.029i