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Non-fragile H∞ filtering for nonlinear discrete-time delay systems with randomly occurring gain variations
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Non-fragile H1 filtering for nonlinear discrete-time delay systems with randomly occurring gain variations$ Yajuan Liu a,b, Ju H. Park a,n, Bao-Zhu Guo b,c a
Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China c School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 February 2016 Received in revised form 2 April 2016 Accepted 8 April 2016 This paper was recommended for publication by Dr. Jeff Pieper
In this paper,the problem of H1 filtering for a class of nonlinear discrete-time delay systems is investigated. The time delay is assumed to be belonging to a given interval, and the designed filter includes additive gain variations which are supposed to be random and satisfy the Bernoulli distribution. By the augmented Lyapunov functional approach, a sufficient condition is developed to ensure that the filtering error system is asymptotically mean-square stable with a prescribed H1 performance. In addition, an improved result of H1 filtering for linear system is also derived. The filter parameters are obtained by solving a set of linear matrix inequalities. For nonlinear systems, the applicability of the developed filtering result is confirmed by a longitudinal flight system, and an additional example for linear system is presented to demonstrate the less conservativeness of the proposed design method. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear discrete-time systems Interval time-delay Non-fragile filter Randomly occurring gain variations
1. Introduction Over the past few decades, filtering problem has been widely addressed due to its capability of applications in various fields such as military, astronautics, and signal processing [1,2]. There are two major approaches to be used to deal with the filtering problem: the Kalman filtering approach and the H1 filtering approach. The Kalman filtering requires the external disturbance to be white Gaussian noise which is not always satisfied in many engineering applications. As an effective alternative, the H1 filtering does not require the external noise to be white Gaussian. Compared with classical Kalman filtering, the H1 filtering takes advantage of external noise signal to be in any form with energy bounded, and importantly, the exact noise statistics is not required to be known. Moreover, the H1 filtering has been shown to be robust against unmodeled dynamics [3]. On the other hand, since time delay is unavoidable in various engineering systems, it remains the source of instability and poor performance [4–7]. Hence, the H1 filtering for time delayed systems has attracted considerable attention and a wealth of important results are available in the literature like those reported in [8–13]. ☆ This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10005201). n Corresponding author. Tel.: þ 82 53 8102491. E-mail address: [email protected] (J.H. Park).
The main objective of H1 filtering is to design a suitable filter such that the mapping from exogenous input to estimation error is minimized or is no larger than a prescribed level in terms of the H1 norm. It has been shown that the filter gains can be obtained by solving a set of linear matrix inequalities (LMIs). In [3], a sufficient condition for H1 filter design is derived by employing a new finite sum inequality. More recently, some less conservative results are obtained by using the model transformation approach [9,11]. However, it is noted that these results with analytical tools, although complete in some extent, neglect time delays in estimation of difference upper bound of the constructed Lyapunov functional. On the other hand, the H1 filters developed in [3–13] are only concerned with linear systems. Since the nonlinearity is ubiquitous in many practical systems and the nonlinearity makes system analysis more difficult [14–18], it is significant to consider the H1 filtering problem for nonlinear systems [19–24]. Moreover, all the aforementioned H1 filtering problems in [3–13,19–24] are based on an implicit assumption that the filters can be implemented precisely. This is, however, not the case in practical situations, Actually, the inaccuracy or uncertainty is usually inevitable in filter implementation. Such uncertainty can be attributed to unexpected errors during the implementation such as actuator degradation, roundoff errors in numerical computation, aging of components, and requirement for parameters’ re-adjustment during the process. As a result, the designed filter is expected to be insensitive or non-fragile [25–28]. Moreover, the filter gain variations may be
http://dx.doi.org/10.1016/j.isatra.2016.04.009 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
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2
subject to random changes because of environment circumstance in the process of filter implementation among networks. In this case, the gain variations may be presented in a probabilistic way with certain types and intensity. It is therefore worth taking randomly occurring gain variations into account in filter design [29,30]. To the best of our knowledge, the problem of non-fragile H1 filtering for nonlinear discrete-time systems with randomly occurring gain variations has not been well addressed. Motivated by the aforementioned discussions, the aim of this paper is to investigate the H1 filtering for a class of discrete nonlinear systems with interval time varying delays and randomly occurring gain variations. Inspired by [14,21], the sector nonlinearity in this work is more general than usual Lipschitz condition. In terms of LMI technique, a sufficient condition for nonfragile H1 filtering of the resulting closed-loop nonlinear system is developed. Based on this condition, an improved filtering design for linear systems is proposed. Two numerical examples are presented to demonstrate the validity and less conservativeness of the proposed method. It should be pointed out that the main contributions of this paper are: (a) we consider the H1 filtering for nonlinear systems instead linear ones in the existing literature; (b) we take randomly occurring gain variations into account, where the gain variations appear in a random way based on a certain kind of probabilistic law; (c) for linear systems that are free of randomly occurring gain variations, the corresponding H1 filtering result proposed in this paper is, based on Lyapunov method and zero inequality approach, less conservative than that in the existing literature. The rest of the paper is organized as follows. The problem formulation is presented in Section 2. In Section 3, the main results on H1 filter analysis and design method are presented. Two numerical examples are given to illustrate the effectiveness of the proposed scheme in Section 4, and conclusions are given in Section 5. Notations: Throughout the paper, Rn denotes the n-dimensional Euclidean space. Rnm is the set of all n m real matrices. For a real matrix X, X 4 0 and X o 0 mean that X is a positive/negative definite symmetric matrix, respectively. I is an identity matrix with appropriate dimension. n denotes the symmetric part of given matrix. diagf⋯g denotes the diagonal matrix. J J refers to the induced matrix 2-norm. L2 means the space of square integral R 1=2 1 . vector functions on ½0; 1Þ with norm J J 2 0 J J 2 dt
2. Problem statement Consider a class of nonlinear discrete time-delay system as follows: xðk þ 1Þ ¼ AxðkÞ þ Ad xðk dðkÞÞ þ Ff ðxðkÞÞ þ Bw wðkÞ; yðkÞ ¼ CxðkÞ þ C d xðk dðkÞÞ þ H 1 hðxðkÞÞ þ H 2 hðxðk dðkÞÞ þ Dw wðkÞ; zðkÞ ¼ LxðkÞ þLd xðk dðkÞÞ þ Gw wðkÞ;
ð1Þ
where xðkÞ A Rn and yðkÞ A Rm are the state vector and measured output, respectively. wðkÞ A Rq is a disturbance input belonging to L2 ½0; 1Þ, zðkÞ A Rp is the signal to be estimated, A, Ad, F, Bw, C, Cd, H1, H2, Dw, L, Ld, and Gw are known constant matrices with appropriate dimensions. Assumption 1. The time delay d(k) is assumed to be time-varying and satisfies d1 rdðkÞ r d2 ;
where d1 4 0 and d2 4 0 are constants representing the lower and upper bounds of the delay, respectively. Assumption 2. f ðÞ and hðÞ are sector-bounded functions and satisfy f ð0Þ ¼ hð0Þ ¼ 0. Assumption 3. The nonlinear vector-valued functions are assumed to satisfy ½f ðxÞ f ðyÞ M 1 ðx yÞ ½f ðxÞ f ðyÞ M 2 ðx yÞ r 0;
8 x; y A Rn ;
ð2Þ
½hðxÞ hðyÞ N 1 ðx yÞ ½hðxÞ hðyÞ N 2 ðx yÞ r 0;
8 x; y A Rm ;
ð3Þ
where M1, M2, N1, and N2 are known real constant matrices, and M 2 M 1 and N 2 N1 are positive definite matrices, respectively. Remark 1. The nonlinear functions in Eqs. (2) and (3) are quite general than those with Lipschitz conditions. Our main objective is to estimate the signals z(k). To this purpose, we design the following non-fragile full-order filter: xf ðk þ 1Þ ¼ ðAf þ αðkÞΔAf ðkÞÞxf ðkÞ þ ðBf þ βðkÞΔBf ðkÞÞyðkÞ zf ðkÞ ¼ C f xf ðkÞ þ Df yðkÞ; xf ð0Þ ¼ 0;
ð4Þ
where xf ðkÞ A Rn is the state of filter and zf ðkÞ A Rm is the estimated output, the matrices Af, Bf, Cf, and Df are appropriately dimensioned filter gains to be determined, and αðkÞ and β ðkÞ are mutually independent Bernoulli-distributed white sequences. Assumption 4. A natural assumption on αðkÞ and βðkÞ is made as follows: ProbfαðkÞ ¼ 1g ¼ EfαðkÞg ¼ α ; ProbfαðkÞ ¼ 0g ¼ 1 α ; Probfβ ðkÞ ¼ 1g ¼ Efβ ðkÞg ¼ β ; Probfβ ðkÞ ¼ 0g ¼ β : Assumption 5. The uncertain perturbation matrices ΔAf ðkÞ and Δ Bf ðkÞ are defined as follows:
ΔAf ðkÞ ¼ Ma Δ1 ðkÞN a ; ΔBf ðkÞ ¼ Mb Δ2 ðkÞN b ; where M a ; M b ; N a and Nb are known constant matrices with appropriate dimensions, and Δ1 ðkÞ and Δ2 ðkÞ are unknown matrix functions satisfying
ΔTr ðkÞΔr ðkÞ r I;
r ¼ 1; 2:
Remark 2. In real networked systems, the additive gain variations are inevitable due to unexpected errors during the filter implementation, e.g., round off errors in numerical computation and programming errors [25,28]. Moreover, the filter parameters may subject to random changes which satisfy the Bernoulli distribution [29,30]. Such description can compensate the parameter variations of the filter in a random fashion and has not been investigated for nonlinear discrete-time systems. Denote " # xðkÞ ζ ðkÞ ¼ x ðkÞ; f eðkÞ ¼ zðkÞ zf ðkÞ: Then, we have the following error system ^ þ β BðkÞ ^ ^ þ β^ ðkÞBðkÞÞ ^ ζ ðk þ 1Þ ¼ ðA þ α AðkÞ ζ ðkÞ þ α^ ðkÞAðkÞ ^ ^ ^ þ ðA d þ β A d ðkÞ þ β ðkÞA d ðkÞÞζ ðk dðkÞÞ þ F f ðxðkÞÞ þ ðH 1 þ β H^ 1 ðkÞ þ β^ ðkÞH^ 1 ðkÞÞhðxðkÞÞ þ ðH 2 þ β H^ 2 ðkÞ þ β^ ðkÞH^ 2 ðkÞÞhðxðk dðkÞÞÞ þ ðB w þ β B^ w ðkÞ þ β^ ðkÞB^ w ðkÞÞwðkÞ;
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eðkÞ ¼ C ζ ðkÞ þ C d ζ ðk dðkÞÞ þD 1 hðxðkÞÞ þ D 2 hðxðk dðkÞÞÞ þ D w wðkÞ; ð5Þ where "
A
0
"
#
0
0
#
^ ; ; AðkÞ ¼ Bf C Af 0 ΔAf ðkÞ " # " Ad 0 0 Ad ¼ ; A^ d ðkÞ ¼ ΔBf ðkÞC d Bf C d 0 " # " # 0 0 ^ H1 ¼ ; H 1 ðkÞ ¼ Bf H 1 ΔBf ðkÞH1 ; " # " # 0 Bw ^ H 2 ðkÞ ¼ ΔBf ðkÞH2 ; B w ¼ Bf Dw ; h h i C ¼ L Df C C f ; C d ¼ Ld Df C d A¼
D 1 ¼ Df H 1 ;
α^ ðkÞ ¼ αðkÞ α ;
D 2 ¼ Df H 2 ;
" ^ BðkÞ ¼ #
0
0
;
m2 ðkÞ : N -Rn , the following inequality holds
1
η1 ðkÞ "
#
r
ΔBf ðkÞC 0 ;
F ; 0 0 " # 0 H2 ¼ ; Bf H 2 " # 0 ^ B w ðkÞ ¼ ΔBf ðkÞDw ; i 0 ;
0
mT1 ðkÞMm1 ðkÞ
m1 ðkÞ
"
M
S
n
M
mT2 ðkÞMm2 ðkÞ #
m1 ðkÞ
:
m2 ðkÞ
3. Main results For the sake of simplicity for matrix representation, we define e^ i A Rð19n þ qÞn to be block entry matrix. For example, e^ 2 ¼ ½0 I 0…0 T . We collect some scalars, vectors, and matrices as |ffl{zffl} 18
h
η ðkÞ ¼ ζ ðkÞ ζ T ðk d1 Þ ζ T ðk dðkÞÞ ζ T ðk d2 ÞΔζ T ðkÞ T
D w ¼ Gw Df Dw ;
Definition 2. For a given positive scalar γ, the filtering error system (5) is said to be asymptotically stable in mean square with a guaranteed H1 performance γ if it is asymptotically stable and the filtering error e(k) satisfies 1 1 n o n o X X E J eðkÞ J 2 r γ 2 E J wðkÞ J 2
T
k X 1 d1 þ 1 i ¼ k d
k-1
ð7Þ
k¼0
Δζ ðkÞ ¼ ζ ðk þ 1Þ ζ ðkÞ; e1 ¼ ½e^ 1 e^ 2 ; e2 ¼ ½e^ 3 e^ 4 ; e3 ¼ ½e^ 5 e^ 6 ; e4 ¼ ½e^ 7 e^ 8 ; e5 ¼ ½e^ 9 e^ 10 ; e6 ¼ ½e^ 11 e^ 12 ; e7 ¼ ½e^ 13 e^ 14 ; e8 ¼ ½e^ 15 e^ 16 ; e9 ¼ e^ 17 ; e10 ¼ e^ 18 ; e11 ¼ e^ 19 ; e12 ¼ e^ 20 ; " Ma ¼
K ¼ ½I 0;
U1 ¼
where yðiÞ ¼ xðiÞ xði 1Þ and 0 2 X xðiÞ: h þ1 i ¼ h
Remark 3. When Lemma 1 is employed to time-varying delay, the 1 term hh þ might be difficult to be dealt with. As pointed out in [31], 1 1 Lemma 2 is provided to make the term hh þ disappear from the 1 inequality. Lemma 2 (Seuret et al. [31]). For a given symmetric positive definite matrix Z A Snþ and any sequence of discrete-time variable x in ½ h; 0⋂Z-Rn , where h Z 1, the following inequality holds: " #T # " 0 X Θ0 Z 0 1 Θ0 T y ðiÞZyðiÞ r ; Θ1 h Θ1 0 3Z i ¼ hþ1
Θ1 are defined in Lemma 1.
S Lemma 3 (Park et al. [32]). For any matrix ½M Z0; scalars η1 ðkÞ n M 4 0; η2 ðkÞ 4 0 satisfying η1 ðkÞ þ η2 ðkÞ ¼ 1; vector functions m1 ðkÞ and
0
#
"
; N a ¼ 0 Na ; Ma
0 ; N bcd ¼ N b C d 0 ;
# 0 Mb ¼ ; Mb
U T1 U 2 þU T2 U 1 U T þ U T2 ; U2 ¼ 1 ; 2 2 T T T T V V 2 þ V 2V 1 V þV2 ; U2 ¼ 1 ; V1 ¼ 1 2 2 T T T W W2 þW2W1 W þ W T2 ; W2 ¼ 1 ; W1 ¼ 1 2 2 Σ 1 ¼ ½e1 þe5 ðd1 þ1Þe6 e2 ðdðkÞ d1 þ 1Þe7 þðd2 dðkÞ þ 1Þe8 e3 e4 ; Σ 2 ¼ ½e1 ðd1 þ1Þe6 e1 ðdðkÞ d1 þ 1Þe7 þ ðd2 dðkÞ þ 1Þe8 e3 e2 ;
Σ 3 ¼ ½e1 e2 e1 þ e2 2e6 ; Σ 4 ¼ ½e3 e4 e3 þ e4 2e8 e2 e3 e2 þ e3 2e7 ; 2
GA ¼ 4
Θ0 ¼ xð0Þ xð hÞ;
ζ T ðiÞ
2
Now, the filtering problem can be stated as follows: Filtering problem: Design a filter of the form (4) such that the filtering error system (5) is asymptotically stable in the meansquare sense and achieves a prescribed H1 performance level.
i ¼ hþ1
1
kX d1 1 ζ T ðiÞ dðkÞ d1 þ 1 i ¼ k dðkÞ
i T T T f ðxðkÞÞ h ðxðkÞÞ h ðxðk dðkÞÞÞ wT ðkÞ ;
N bc ¼ N b C
Lemma 1 (Seuret et al. [31]). For a given symmetric positive definite matrix Z A Snþ and any sequence of discrete-time variable x in ½ h; 0⋂Z-Rn , where h Z 1, the following inequality holds: #" " #T " # 0 X Z 0 Θ0 1 Θ0 T y ðiÞZyðiÞ r ; 1 Z 0 3 hh þ Θ1 h Θ1 1
ζ T ðiÞ
kX dðkÞ 1 d2 dðkÞ þ 1 i ¼ k d
for all nonzero wðÞ A L2 ½0; 1Þ subject to the zero initial condition.
where yðiÞ; Θ0 and
1
η2 ðkÞ
F¼
Definition 1. The filtering error system (5) with wðkÞ ¼ 0 is said to be asymptotically stable in the mean-square sense, if for any initial condition, there holds n o ð6Þ lim E J ζ ðkÞ J 2 ¼ 0:
Θ1 ¼ xð0Þ þ xð hÞ
#T
m2 ðkÞ
β^ ðkÞ ¼ βðkÞ β :
k¼0
3
G1 A þ B^ f C GT A þ B^ C f
2
" GF ¼
G1 F
#
3 A^ f 5; A^ f 2
GH 1 ¼ 4
; GT2 F 2 3 G1 Bw þ B^ f Dw 5; GB w ¼ 4 T G2 Bw þ B^ f Dw
2
3 G1 Ad þ B^ f C d 0 5; GT2 Ad þ B^ f C d 0 2 3 B^ f H 2 4 5; GH 2 ¼ ^ B H2
GAd ¼ 4 3 B^ f H 1 5; B^ H 1 f
" R2 ¼
f
R2
0
0
3R2
#
Φ ¼ ½GA G 0 GA d 0 G 0 0 0 0 GH 1 GH 2 GB w ; Ψ ¼ ½C 0 C d 0 0 0 0 0 0 D 1 D 2 D w ; ϵa ¼ ϵ2 þ ϵ3 þ ϵ4 þ ϵ5 þ ϵ6 ; ϵb ¼ ϵðϵ8 þ ϵ8 þ ϵ10 þ ϵ11 þ ϵ12 Þ; 1 2 þ e1 Q 1 eT1 e2 Q 1 eT2
1 2
Ω ¼ ðΣ 1 Σ 2 ÞPðΣ 1 þ Σ 2 ÞT þ ðΣ 1 þ Σ 2 ÞPðΣ 1 Σ 2 ÞT "
þ e1 Q 2 eT1 e4 Q 2 eT4 þ d1 e5 R1 eT5 Σ 3 2
R1
0
0
3λðd1 ÞR1
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#
Σ T3
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4
" 2
þ ðd2 d1 Þ " þ ½e1 e9
e5 R2 eT5
Σ4
X
n
3R 2 #
K T U 1K
K T U 2
n
I
" þ ½e1 e10 " þ ½e3 e11
#
R2
KT V 1K
KT V 2
n
I
Σ T4
where yðjÞ ¼ ζ ðjÞ ζ ðj 1Þ. We compute the difference of V(k) along the error system (5) with wðkÞ ¼ 0. Notice that 2 3 2 3 ζ ðkÞ þ Δζ ðkÞ ζ ðk þ1Þ 6 7 6 7 k k X X 6 7 6 7 6 7 6 7 ζ ðiÞ ζ ðk d Þ ζ ðiÞ 1 6 7 6 7 6 i ¼ k d1 þ 1 7¼6 7; i ¼ k d1 6 7 6 7 6 kX 7 6 7 kX dðkÞ d1 kX d1 6 7 6 7 4 ζ ðiÞ 5 4 ζ ðiÞ þ ζ ðiÞ ζ ðk dðkÞÞ ζ ðk d2 Þ 5
#
KT W 2
n
I
yT ðjÞR2 yðjÞ;
i ¼ d2 þ 1 j ¼ k þ i
½e1 e9 T
KT W 1K
k X
d1 X
V 5 ðkÞ ¼ ðd2 d1 Þ
½e1 e10 T # ½e3 e11 T
þ ½e1 þ ϵe5 Φ þ Φ ½e1 þ ϵe5 T T
T
i ¼ k d2 þ 1
T
þ e1 ½α ðϵ1 1 þ ϵϵ7 1 ÞN a N a þ β ðϵ2 1 þ ϵϵ8 1 ÞN bc N bc eT1
i ¼ k dðkÞ
i ¼ k d2
ð11Þ
T
þ e3 β ðϵ3 1 þ ϵϵ9 1 ÞN bcd N bcd eT3 1 1 þ e10 β ðϵ4 1 þ ϵϵ10 ÞH T1 N Tb N b H 1 eT10 þe11 β ðϵ5 1 þ ϵϵ11 ÞH T2 N Tb Nb H 2 eT11 1 þ e12 β ðϵ6 1 þ ϵϵ12 ÞDTw N Tb N b Dw eT12 γ 2 e12 eT12 :
Theorem 1. For given constants d2 4 d1 4 0, γ 4 0, ϵ 40, and ϵi 4 0 ði ¼ 1; 2; …; 12Þ, and matrices U1, U2, V1, V2, W1, and W2, there
exists an admissible non-fragile H1 filter (4) such that the filtering error system (5) is asymptotically stable in mean square for wðkÞ ¼ 0 and the filtering error e(k) satisfies (7) under the zero initial condition for any nonzero wðkÞ A L2 ½0; 1Þ, if there exist matrices P 4 0, Q 1 4 0, ^ , and G ¼ ½GT1 G2 satisfying Q 2 4 0, R1 4 0, R2 4 0, X , G, A^ f , B^ f , C^ f , D f G2 G2 the following LMIs 2 3 Ω Ψ T e1 αϵ1 GM a e1 ϵa β GM b e5 αϵϵ7 GM a e5 ϵb β GM b 6 7 6 n I 7 0 0 0 0 6 7 6n 7 n ϵ α 0 0 0 1 6 7 6 7 o 0; 6n 7 n n ϵ β 0 0 a 6 7 6n 7 n n n ϵϵ α 0 7 4 5 n n n n n ϵb β
ð8Þ "
R2
X
n
R2
# Z 0:
ð9Þ
Moreover, if the LMIs admit feasible solutions, the matrices of the admissible H1 filter in the form (4) can be obtained by Af ¼ G2 1 A^ f ;
Bf ¼ G2 1 B^ f ;
C f ¼ C^ f ;
^ : Df ¼ D f
Proof. Construct the following Lyapunov functional: VðkÞ ¼ V 1 ðkÞ þ V 2 ðkÞ þ V 3 ðkÞ þ V 4 ðkÞ þV 5 ðkÞ
and 2
ζ ðkÞ
3T 2
ζ ðkÞ
V 2 ðkÞ ¼
kX 1
i ¼ k d2
ζ ðkÞQ 1 ζ ðkÞ; T
i ¼ k d1
V 3 ðkÞ ¼
kX 1
ζ T ðkÞQ 2 ζ ðkÞ;
i ¼ k d2
V 4 ðkÞ ¼ d1
0 X
k X
i ¼ d1 þ 1 j ¼ k þ i
yT ðjÞR1 yðjÞ;
7 7 7 7 7: i ¼ k d1 7 7 kX d1 7 ζ ðiÞ ζ ðk dðkÞÞ ζ ðk d1 Þ 5 k X
ζ ðiÞ ζ ðkÞ
i ¼ k dðkÞ
i ¼ k d2
i ¼ k d2
3
ζ ðkÞ
ð12Þ The differences of V i ðkÞ ði ¼ 1; 2; …; 5Þ are given by n o T T T E ΔV 1 ðkÞ ¼ E ξ ðkÞðΣ 1 P Σ 1 Σ 2 P Σ 2 ÞξðkÞ n o T ¼ E ξ ðkÞðΣ 1 þ Σ 2 ÞPðΣ 1 Σ 2 ÞT ξðkÞ ;
ð13Þ
n o T T E ΔV 2 ðkÞ ¼ E ζ ðkÞQ 1 ζ ðkÞ ζ ðk d1 ÞQ 1 ζ ðk d1 Þ ;
ð14Þ
n o T T E ΔV 3 ðkÞ ¼ E ζ ðkÞQ 2 ζ ðkÞ ζ ðk d2 ÞQ 2 ζ ðk d2 Þ ;
ð15Þ
8 9 k < = X 2 T E ΔV 4 ðkÞ ¼ E d1 y ðk þ 1ÞR1 yðk þ1Þ d1 yT ðiÞR1 yðiÞ : ; i ¼ k d1 þ 1 8 9 k < = X T 2 yT ðiÞR1 yðiÞ ; ¼ E d1 Δζ ðkÞR1 Δζ ðkÞ d1 : ; i ¼ k d1 þ 1 n E ΔV 5 ðkÞ ¼ E ðd2 d1 Þ2 yT ðk þ 1ÞR2 yðkþ 1Þ ) kX d1 yT ðiÞR2 yðiÞ ðd2 d1 Þ i ¼ k d2 þ 1
8 < T ¼ E ðd2 d1 Þ2 Δζ ðkÞR2 Δζ ðkÞ ðd2 d1 Þ :
ð10Þ
kX d1 i ¼ k d2 þ 1
9 = yT ðiÞR2 yðiÞ : ;
ð16Þ
3
6 kX 7 6 kX 7 6 1 7 6 1 7 6 6 ζ ðiÞ 7 ζ ðiÞ 7 6 7 6 7 7 P 6 i ¼ k d1 7; V 1 ðkÞ ¼ 6 6 i ¼ k d1 7 6 7 6 k X 7 6 k X 7 6 d1 1 7 6 d1 1 7 4 5 4 ζ ðiÞ ζ ðiÞ 5 i ¼ k d2
2
6 kX 7 6 6 1 7 6 6 6 ζ ðiÞ 7 6 7 6 6 i ¼ k d1 7¼6 6 7 6 6 k X 7 6 kX 6 d1 1 7 6 dðkÞ 4 ζ ðiÞ 5 4 ζ ðiÞ þ
with 2
3
ζ ðkÞ
By Lemma 1, k X
d1
i ¼ k d1
"
η1 ðkÞ y ðiÞR1 yðiÞ r η2 ðkÞ þ1 T
#T "
R1 0
0 3λðd1 ÞR1
#"
#
η1 ðkÞ ; η2 ðkÞ ð17Þ
η1 ðkÞ ¼ ζ ðkÞ ζ ðk d1 Þ, η2 ðkÞ ¼ ζ ðkÞ þ ζ ðk d1 Þ d1 2þ 1 d1 þ 1 ζ ðiÞ, λ ðd Þ ¼ 1ðd ¼ 1Þ, and λ ðd 1 1 1 Þ ¼ d1 1ðd1 41Þ. i ¼ k d1
where Pk
By Lemmas 2 and 3, one can obtain
ðd2 d1 Þ
kX d1
yT ðiÞR2 yðiÞ
i ¼ k d2 þ 1
¼ ðd2 d1 Þ
kX dðkÞ i ¼ k d2 þ 1
yT ðiÞR2 yðiÞ ðd2 d1 Þ
kX d1 i ¼ k dðkÞ þ 1
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yT ðiÞR2 yðiÞ
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o T T þ ε9 1 ζ ðk dðkÞÞN bcd N bcd ζ ðk dðkÞÞ ; n T T T Ef2Δζ ðkÞGH^ 1 ðkÞhðxðkÞÞg r E Δζ ðkÞε10 GM b M b GΔζ ðkÞ o 1 T h ðxðkÞÞH T1 NTb N b H 1 hðxðkÞÞ ; þ ε10
" #T " # #" γ 1 ðkÞ R2 0 d2 d1 γ 1 ðkÞ r γ 2 ðkÞ 0 3R2 d2 dðkÞ γ 2 ðkÞ " #T " # #" γ 3 ðkÞ R2 0 d2 d1 γ 3 ðkÞ γ 4 ðkÞ 0 3R2 dðkÞ d1 γ 4 ðkÞ " # R2 X r γ T ðkÞ γ ðkÞ; n R2
ð18Þ
In view of Assumptions 2 and 3, for known constant matrices U 1 , U 2 , V 1 , V 2 , W 1 , and W 2 , it readily follows that #" " #T " # ζ ðkÞ ζ ðkÞ K T U 1K KT U 2 Z 0; ð19Þ f ðxðkÞÞ f ðxðkÞÞ n I
ζ ðkÞ
#T "
KT V 1K
KT V 2
n
I
hðxðkÞÞ "
ζ ðk dðkÞÞ
#T "
#"
ζ ðkÞ
K T W 1K
KT W 2
n
I
hðxðk dðkÞÞÞ
Z 0;
#"
hðxðk dðkÞÞÞ
^ ^ ^ þ β^ ðkÞBðkÞÞÞ þ α^ ðkÞAðkÞ þ β BðkÞ ζ ðkÞ ^ ^ ^ þ ðA þ β A ðkÞ þ β ðkÞA ðkÞÞζ ðk dðkÞÞ þ F f ðxðkÞÞ d
d
d
For positive scalars εi ði ¼ 1; 2; …; 12Þ, we have the following easily obtained inequalities: ^ ζ ðkÞg r Efζ T ðkÞðε1 GM a M T G þ ε 1 N T N a Þζ ðkÞg; Ef2ζ ðkÞGAðkÞ 1 a a T
^ ζ ðkÞg rEfζ ðkÞðε2 GM b M G þ ε 1 N N bc Þζ ðkÞg; Ef2ζ ðkÞGBðkÞ 2 b bc n T T T Ef2ζ ðkÞGA^ d ðkÞζ ðk dðkÞÞg rE ζ ðkÞε3 GM b M b Gζ ðkÞ o T T þ ε3 1 ζ ðk dðkÞÞN bcd N bcd ζ ðk dðkÞÞ ; T
T
T
T
Ef2ζ ðkÞGH^ 1 ðkÞhðxðkÞÞg rEfζ ðkÞε4 GM b M b Gζ ðkÞ T
T
T
þ ε4 1 h ðxðkÞÞH T1 N Tb N b H 1 hðxðkÞÞg; n T T T Ef2ζ ðkÞGH^ 2 ðkÞhðxðk dðkÞÞÞg r E ζ ðkÞε5 GM b M b Gζ ðkÞ o T þ ε5 1 h ðxðk dðkÞÞÞH T2 N Tb Nb H 2 hðxðk dðkÞÞÞ ;
T
T
1 T þ ε12 w ðkÞDTw NTb N b Dw wðkÞg:
ð23Þ
Here, we establish the H1 performance for filtering error system (5). If the difference of V(k) is negative, then zðkÞ-0 as k-1. Next, by assuming zero initial conditions for the filtering error system, the performance index is 1 n o X J¼ E eðkÞT eðkÞ γ 2 wðkÞT wðkÞ k¼0
1 n o X E feðkÞT eðkÞ γ 2 wðkÞT wðkÞ þ ΔVðkÞg þ EVð0Þ EVð1Þ: ¼ k¼0
ð22Þ
From (5), for any matrix G, n ^ 2E ½ζ ðkÞ þ ϵΔζ ðkÞG½ð Δζ ðkÞ þ ðA I þ α AðkÞ
þ ðH 1 þ β H^ 1 ðkÞ þ β^ ðkÞH^ 1 ðkÞÞhðxðkÞÞ þ ðH 2 þ β H^ 2 ðkÞ þ β^ ðkÞH^ 2 ðkÞÞhðxðk dðkÞÞÞ o þ ðB w þ β B^ w ðkÞ þ β^ ðkÞB^ w ðkÞÞwðkÞ ¼ 0:
T
and by a simple matrix calculation, it is straightforward to verify that " # " # G1 A þ G2 Bf C G2 Af G1 Ad þ G2 Bf C d 0 GA ¼ ; ; GAd ¼ T T G2 A þ G2 Bf C G2 Af G2 Ad þ G2 Bf C d 0 " # " # " # G2 B f H 1 G2 B f H 2 G1 F ; GH 1 ¼ ; GH 2 ¼ ; GF ¼ G2 B f H 1 G2 B f H 2 GT2 F " # G1 Bw þ G2 Bf Dw : GB w ¼ GT2 Bw þ G2 Bf Dw
# Z 0:
Ef2Δζ ðkÞGB^ w ðkÞwðkÞg r EfΔζ ðkÞε12 GM b M b GΔζ ðkÞ
ð21Þ
ð20Þ
ζ ðk dðkÞÞ
T
T
n o If E eðkÞT eðkÞ γ 2 wðkÞT wðkÞ þ ΔVðkÞ o0, then VðkÞ-0 as k-1. Taking G as " # G 1 G2 ; G¼ T G2 G2
#
hðxðkÞÞ
Ef2Δζ ðkÞGH^ 2 ðkÞhðxðk dðkÞÞÞg rEfΔζ ðkÞε11 GM b M b GΔζ ðkÞ o 1 T þ ε11 h ðxðk dðkÞÞÞH T2 N Tb N b H 2 hðxðk dðkÞÞÞ ; T
where γ ðkÞ ¼ ½γ T1 ðkÞ γ T2 ðkÞ γ T3 ðkÞ γ T4 ðkÞT , γ 1 ðkÞ ¼ ζ ðk dðkÞÞ ζ ðk d2 Þ, Pk dðkÞ 2 γ 2 ðkÞ ¼ ζ ðk dðkÞÞ þ ζ ðk d2 Þ d2 dðkÞ ζ ðiÞ, þ1 i ¼ k d2 γ 3 ðkÞ ¼ ζ ðk d1 Þ ζ ðk dðkÞÞ, P γ 4 ¼ ζ ðk d1 Þ þ ζ ðk dðkÞÞ dðkÞ 2d1 þ 1 ki ¼ dk1 dðkÞ ζ ðiÞ, and " # R2 0 R2 ¼ : 0 3R2
"
5
Define a set of variables as follows: A^ f ¼ G2 Af ;
B^ f ¼ G2 Bf ;
C^ f ¼ C f ;
^ f ¼ Df : D
Then, combining with Eqs. (13)–(23), one can obtain n o E ΔV ðkÞ þ eðkÞT eðkÞ γ 2 wðkÞT wðkÞ n T T T r E ξ ðkÞðΩ þ Ψ Ψ þ αε1 e1 GM a M a GeT1
o T T T þ βεa e1 GM b M b GeT1 þ αεε7 e5 GM a M a GeT5 þ βεb e5 GM b M b GeT5 ÞξðkÞ : ð24Þ
T
Ef2ζ ðkÞGB^ w ðkÞwðkÞg r Efζ ðkÞε T
þε
T 6 GM b M b G
T
ζ ðkÞ
T T 1 T 6 w ðkÞDw N b N b Dw wðkÞg;
^ ζ ðkÞg r EfΔζ ðkÞε7 GM a M GΔζ ðkÞ Ef2Δζ ðkÞGAðkÞ a T
T
T
T
þ ε7 1 ζ ðkÞN a N a ζ ðkÞg; T
^ ζ ðkÞg rEfΔζ ðkÞε8 GM M GΔζ ðkÞ Ef2Δζ ðkÞGBðkÞ b b T
T 1 T ðkÞN bc N bc 8
T
T
ζ ðkÞg; þε ζ n T T T Ef2Δζ ðkÞGA^ d ðkÞζ ðk dðkÞÞg r E Δζ ðkÞε9 GM b M b GΔζ ðkÞ
T
By Schur complement, Ω þ Ψ Ψ þ αε1 e1 GM a M a GeT1 þ βεa e1 G T
T M b M b GeT1 þ
T T 7 e5 GM a M a Ge5 þ
T T b e5 GM b M b Ge5 o 0
αεε βε is equivalent to (8), which means J o 0 for any nonzero wðkÞ A L2 ; i.e., the filtering error system has a guaranteed γ level of disturbance attenuation. This completes the proof of the theorem. □
If F ¼ H 1 ¼ H 2 ¼ 0, the system (1) is reduced to the following linear system: xðk þ 1Þ ¼ AxðkÞ þ Ad xðk dðkÞÞ þ Bw wðkÞ; yðkÞ ¼ CxðkÞ þ C d xðk dðkÞÞ þ Dw wðkÞ; zðkÞ ¼ LxðkÞ þ Ld xðk dðkÞÞ þ Gw wðkÞ:
ð25Þ
And by taking Δ1 ðkÞ ¼ Δ2 ðkÞ ¼ 0, the filter of the form (4) is
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6
Af ¼ G2 1 A^ f ;
reduced to xf ðk þ 1Þ ¼ Af xf ðkÞ þ Bf yðkÞ; zf ðkÞ ¼ C f xf ðkÞ þDf yðkÞ;
xf ð0Þ ¼ 0:
ð26Þ
In these cases, the corresponding error system is written as
ζ ðk þ 1Þ ¼ A ζ ðkÞ þ A d ζ ðk dðkÞÞ þB w wðkÞ; eðkÞ ¼ C ζ ðkÞ þ C d ζ ðk dðkÞÞ þD w wðkÞ;
ð27Þ
T T T T T η~ T ðkÞ ¼ 6 4ζ ðkÞ ζ ðk d1 Þ ζ ðk dðkÞÞ ζ ðk d2 Þ Δζ ðkÞ
e~ 1 ¼ ½e 1 e 2 ; e~ 2 ¼ ½e 3 e 4 ; e~ 3 ¼ ½e 5 e 6 ; e~ 4 ¼ ½e 7 e 8 ; e~ 5 ¼ ½e 9 e 10 ; e~ 6 ¼ ½e 11 e 12 ; e~ 7 ¼ ½e 13 e 14 ; e~ 8 ¼ ½e 15 e 16 ; e~ 9 ¼ e 17
þ ðd2 dðkÞ þ 1Þe~ 8 e~ 3 e~ 4 ; ~ Σ 2 ¼ ½e~ 1 ðd1 þ 1Þe~ 6 e~ 1 ðdðkÞ d1 þ1Þe~ 7 þ ðd2 dðkÞ þ 1Þe~ 8 e~ 3 e~ 2 ;
Σ~ 3 ¼ ½e~ 1 e~ 2 e~ 1 þ e~ 2 2e~ 6 ; Σ~ 4 ¼ ½e~ 3 e~ 4 e~ 3 þ e~ 4 2e~ 8 e~ 2 e~ 3 e~ 2 þ e~ 3 2e~ 7 ;
In this section, two examples are given to show the effectiveness of the proposed method on the design of the robust H1 filter.
1 2
" þ ðd2 d1 Þ
2~
e 5 R2 e~ T5
Σ~
4
R2
X
n
3R 2
3
R1 0
#
Example 1. Consider the following simplified longitudinal flight system [19]:
# 0 T Σ~ 3 3λðd1 ÞR1
x1 ðk þ1Þ ¼ 0:9944x1 ðkÞ 0:1203x2 ðkÞ 0:4302x3 ðkÞ;
T Σ~
R2 n
# X Z 0: R2
ð32Þ
ð28Þ
Corollary 1. For given constants d2 4 d1 4 0; γ 4 0 and ε 4 0, there exists an admissible non-fragile H1 filter (26) such that the filtering error system (27) is asymptotically stable in mean square for wðkÞ ¼ 0 and the filtering error e(k) satisfies (7) under the zero initial condition for any nonzero wðkÞ A L2 ½0; 1Þ, if there exist matrices iP 4 0, Q 1 4 0, h Q 2 4 0, R1 40, R2 4 0, X , A^ f , B^ f , C^ f , D^ f , and G ¼ GGT1 GG22 satisfying the 2 following LMIs " # ~ Ψ~ T Ω o 0; ð29Þ n I "
ð31Þ
x2 ðk þ2Þ ¼ 0:0017x1 ðkÞ þ 0:9902x2 ðkÞ ð0:0747 þ 0:01rðkÞÞx3 ðkÞ;
4
~ þΦ ~ T ½e~ þ εe~ T γ 2 e~ e~ T þ ½e~ 1 þ εe~ 5 Φ 5 1 9 9
When the optimal γ is obtained from the above optimization problem, the designed filters can guarantee that the filtering error system is exponentially stable and achieves a prescribed H1 performance level γ. It should be pointed out that it can easily minimize the disturbance attenuation level γ according to Theorem 1 (Corollary 1) by solving a convex optimization problem [33].
4. Numerical examples
~ ¼ ðΣ~ Σ~ ÞPðΣ~ þ Σ~ ÞT þ ðΣ~ þ Σ~ ÞPðΣ~ Σ~ ÞT Ω 1 2 1 2 1 2 1 2 "
Remark 5. Different from the method proposed in [9,11], neither the free-weighting matrices nor the transformation approach is applied to estimate the upper bound of the cross term that appears in the difference of the constructed Lyapunov functional. In order to obtain the less conservative results of the designed filter, Lemma 1 is used to estimate the upper bound of d1 Pk T i ¼ k d1 þ 1 y ðiÞR1 yðiÞ instead of using Jensen's inequality. And Lemmas 2 and 3 are employed to tackle with the term P d1 yT ðiÞR2 yðiÞ. ðd2 d1 Þ ki ¼ k d2 þ 1
min γ 2 , subject to the LMIs (8) and (9) ((29) and (30)).
~ ¼ ½GA G 0 GA 0 G 0 0 0 GB w ; Φ d Ψ~ ¼ ½C 0 C d 0 0 0 0 0 D w ; Σ~ 1 ¼ ½e~ 1 þ e~ 5 ðd1 þ 1Þe~ 6 e~ 2 ðdðkÞ d1 þ 1Þe~ 7
Σ~
Remark 4. Different from the Lyapunov functional constructed in [9,11], the augmented vector, which includes the terms such as Pk d 1 1 Pk 1 ζ ðiÞ, is included in (10). By these terms, i ¼ k d1 ζ ðiÞ; i ¼ k d2
Remark 7. In Theorem 1 (Corollary 1), the filter parameters can be obtained by solving the following optimization problem such that the H1 disturbance attenuation level is minimized:
2
2 þ e~ 1 Q 2 e~ T1 e~ 4 Q 2 e~ T4 þ d1 e~ 5 R1 e~ T5
^ : Df ¼ D f
Remark 6. In order to convert a nonlinear matrix inequality into a linear matrix one, the facts GP 1 G r P G GT are used in [9,22,23,27,28], and some additional positive condition must be satisfied in [11]. Here, the zero equality (22) is used to avoid such problem, which can give much flexibility in solving LMIs.
kX d1 k X 1 1 ζ T ðiÞ ζ T ðiÞ d1 þ 1 i ¼ k d dðkÞ d1 þ1 i ¼ k dðkÞ 1 3 kX dðkÞ 1 T T ζ ðiÞ w ðkÞ5; d2 dðkÞ þ1 i ¼ k d
þ e~ 1 Q 1 e~ T1 e~ 2 Q 1 e~ T2
C f ¼ C^ f ;
more past history of ζ ðkÞ can be used, which may lead to less conservative results.
where A; A d ; B w ; C ; C d and D w are defined in (5). Now, we focus on the H1 performance analysis for system (25). By Theorem 1, we have the following Corollary 1 which says that the error system (27) is asymptotically mean-square stable with H1 performance γ. To do this, we define e i A Rð16n þ qÞn to be the block entry matrix. For example, e 2 ¼ ½0 I 0 … 0 T . And some of |fflfflffl{zfflfflffl}15 scalar, vectors and matrices are defined as 2
1 2
Bf ¼ G2 1 B^ f ;
ð30Þ
Moreover, if the above LMIs admit feasible solutions, the matrices of the admissible H1 filter in the form (26) can be obtained by
x3 ðk þ1Þ ¼ 0:8187x2 ðkÞ þ 0:1wðkÞ;
ð33Þ
where r(k) is unknown but satisfies j rðkÞj r 1. The measurement and estimated signals are supposed to be yðkÞ ¼ 0:2x1 ðkÞ þ 0:1x2 ðkÞ þ ð0:1 þ 0:01rðkÞÞx3 ðkÞ þ 0:1x1 ðk dðkÞÞ þ ð0:1 þ 0:01rðkÞÞx2 ðk dðkÞÞ þ 0:02rðkÞx2 ðk dðkÞÞ þ 0:04rðkÞx3 ðkÞ þ 0:1wðkÞ; zðkÞ ¼ 0:1x2 ðkÞ þ 0:2x3 ðkÞ þ 0:1x1 ðk dðkÞÞ þ 0:2x2 ðk dðkÞÞ:
ð34Þ ð35Þ
It is easy to see that the system described by Eqs. (31)–(35) satisfies Eqs. (2) and (3) and has the form (1) with 2 3 0:9944 0:1203 0:4302 6 7 0:9902 0:0747 5; Ad ¼ 0; F ¼ 0; A ¼ 4 0:0017 0
0:8187
0
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2
0
3
0.5 Response of the filtering error e(k)
6 7 B w ¼ 4 0 5; 0:1
C ¼ 0:2 0:1 0 ; C d ¼ 0:1 0:1 0 ;
H 1 ¼ H 2 ¼ 0:2 0:2 0:2 ; Dw ¼ 0:1;
L ¼ 0 0:1 0:2 ; Ld ¼ 0 0:1 0 ; Gw ¼ 0: The parameters of nonlinear functions are given as 2 3 2 3 0 0 0 0 0 0 6 7 6 7 U 1 ¼ 4 0 0 0:01 5; U 2 ¼ 4 0 0 0:01 5; 0
0
0 6 V1 ¼ 4 0
0 0
2
0
0
3 0:01 0 7 5;
0
e(k)
0.4 0.3 0.2 0.1 0 −0.1 −0.2
0:04 2 3 2 0 0 0:01 0 60 0 7 0 5; W 1 ¼ 6 V2 ¼ 4 40 0 0 0:04 0 2 3 0 0 0 6 7 W 2 ¼ 4 0 0:01 0 5: 0 0:02 0 0
0
7
0
20
40
60
80
100 120 140 160 180 200 t
0
0 0:01 0:02
0
3
Fig. 2. Estimated error of e(k) in Example 1.
07 5; 0
Table 1 Comparison of H1 performance γ.
The uncertain parameters of the filter (4) are taken as follows: 2 3 0:1
6 7 M a ¼ M b ¼ 4 0:2 5; N a ¼ 0:1 0:1 0:1 ;
d(k)
d1 ¼ 1 d2 ¼ 4
d1 ¼ 1 d2 ¼ 5
d1 ¼ 2 d2 ¼ 5
d1 ¼ 2 d2 ¼ 6
[9] [11] Corollary 1
5.0782 4.9431 4.2786
6.4910 6.1604 4.8368
5.5151 5.3551 4.4473
7.0533 6.7581 4.9473
0:3
2
Let d1 ¼ 1; d2 ¼ 5; α ¼ 0:2; β ¼ 0:5; ε ¼ εi ¼ 1 ði ¼ 1; 2; …; 12Þ. The minimum H1 performance is γ ¼ 0:1583 by Theorem 1 and the corresponding filter parameters are 2 3 2 3 0:3807 0:1105 0:1159 0:5876 6 7 6 7 0:9015 0:1339 5; Bf ¼ 4 0:0561 5; Af ¼ 4 0:0741 0:0357 0:1104 0:4147 0:0225
C f ¼ 0:0089 0:1961 0:0206 ; Df ¼ 0:2786: For initial condition xð0Þ ¼ ½ 0:5 0 0:5T and wðkÞ ¼ sin ðkÞe 10k , the response of z(k) and zf(k), and the error e(k) are shown in Figs. 1 and 2, respectively. The simulations show that the desired system performance can be achieved by using the proposed modeling method and designed filter parameters.
0.6
z(k) zf(k)
Resonse of the Outputs
0.4
Response of the Outputs
Δ1 ðkÞ ¼ Δ2 ðkÞ ¼ sin ðkÞ:
N b ¼ 0:2;
z(k) zf(k)
1.5 1 0.5 0 −0.5 −1 0
50
100
150
t Fig. 3. Responses of z(t) and zf(k) in Example 2.
Example 2. Consider the system (25) with the following parameters: 0:2 0 0:1 0:85 0:1 ; Bw ¼ ; A¼ ; Ad ¼ 0:2 0:1 0:4 0:1 0:7
C ¼ 0:2 2:5 ; C d ¼ 0:5 0:5 ; Dw ¼ 1;
L ¼ 0 2:2 ; Ld ¼ 1:5 0:4 ; Gw ¼ 0:1:
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2 −1.4
0
20
40
60
80
100 120 140 160 180 200 t
Fig. 1. Responses of z(t) and zf(k) in Example 1.
The allowable H1 performance γ obtained by various methods are depicted in Table 1. When d1 ¼ 2; d2 ¼ 6 and ε ¼ 2, one can obtain the minimal H1 performance level γ ¼ 4:9473 by Corollary 1, which is much smaller than 7.0533 and 6.7581. It means that the derived result in this paper is less conservative than that proposed in [9,11]. Solving the LMIs in (29) and (30), the corresponding
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8
Response of the filtering error e(k)
0.5
e(k)
0 −0.5 −1 −1.5 −2 −2.5
0
50
100
150
t Fig. 4. Estimated error of e(k) in Example 2.
parameters of the filter gains are given by 1:2663 2:1020 0:5634 Af ¼ ; Bf ¼ ; 0:2140 0:2016 0:3286
C f ¼ 1:1348 4:8925 ; Df ¼ 1:0936: For initial condition xð0Þ ¼ ½0:5 0:5T and wðkÞ ¼ sin ðkÞe 10k , the response of z(k) and zf(k), and the error e(k) are shown in Figs. 3 and 4, respectively. From these simulation results, it is known that the disturbance is effectively attenuated by designed H1 filter for linear discrete-time system (25).
5. Conclusions The delay-dependent non-fragile H1 filtering for nonlinear discrete-time delay systems has been investigated in this paper. By introducing a zero equality and combining with the Lyapunov functional method, a sufficient condition has been developed to ensure that the filtering error system is asymptotically meansquare stable with a prescribed H1 performance. Furthermore, an improved H1 filter for linear system has been proposed. It should be noted that we do not employ the model transformation method unlike existing works. The numerical simulations illustrate the effectiveness of the proposed approach. In fact, the phenomenon of randomly occurring gain variations considered in this paper can be employed to deal with the stabilization for stochastic systems [4,5], synchronization of dynamical networks [15], tracking control for multi-agent systems [16], which could be our future work.
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[5] Li J-N, Li L-S. Mean-square exponential stability for stochastic discrete-time recurrent neural networks with mixed time delays. Neuroocmputing 2015;151:790–7. [6] Shi K, Zhu H, Zhong S, Zeng Y, Zhang Y, Wang W. Stability analysis of neutral type neural networks with mixed time-varying delays using triple-integral and delay-partitioning methods. ISA Trans 2015;58:85–95. [7] Kang W, Zhong S, Shi K, Cheng J. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Trans 2016;60:67– 73. [8] Zhang J, Xia Y, Shi P. Parameter-dependent robust H1 filtering for uncertain discrete-time systems. Automatica 2009;45(2):560–5. [9] He Y, Liu GP, Rees D, Wu M. H1 filtering for discrete-time systems with timevarying delay. Signal Process 2009;89(3):275–82. [10] Wu ZG, Park Ju H, Su H, Song B, Chu J. Reliable H1 filtering for discrete-time singular systems with randomly occurring delays and sensor failures. IET Control Theory Appl 2012;6(14):2308–17. [11] You J, Gao H, Basin MV. Further improved results on H1 filtering for discrete time-delay systems. Signal Process 2013;93:1845–52. [12] Zhang X-M, Han Q-L. Event-based H1 filtering for sampled-data systems. Automatica 2015;51:55–69. [13] Chang X-H, Park Ju H, Tang Z. New approach to H1 filtering for discrete-time systems wtih polytopic uncertainties. Signal Process 2015;113:147–58. [14] Wang Z, Ho DWC, Liu Y, Liu X. Robust H1 control for a class of nonlinear discrete-time delay stochastic systems with missing measurements. Automatica 2009;45:684–91. [15] Tang Y, Gao H, Kurths J. Distributed robust synchronization of dynamical networks with stochastic coupling. IEEE Trans Circuits Syst I, Reg Papers 2014;61:1508–19. [16] Tang Y, Xing X, Karimi HR, Kocarev L, Kurths J. Tracking control of networked multi-agent systems under new characterizations of impulses and its applications in robotic systems. IEEE Trans Ind Electron 2016:1299–307. [17] Li H, Wang J, Lam HK, Zhou Q, Du H. Adaptive sliding mode control for interval type-2 fuzzy systems. IEEE Trans Syst Man Cybern Syst http://dx.doi.org/10. 1109/TSMC.2016.2531676, in press. [18] Li H, Chen Z, Wu L, Lam HK, Du H. Event-Triggered Fault Detection of Nonlinear Networked Systems. IEEE Trans Cybern http://dx.doi.org/10.1109/TCYB. 2016.2536750, in press. [19] Xu S. Robust H1 filtering for a class of discrete-time uncertain nonlinear systems with state delay. IEEE Trans Circuits Syst I-Fundam Theory Appl 2002;50:1853–9. [20] Gao H, Lam J, Wang C. Induced L2 and generalized H2 filtering for systems with repeated scalar nonlinearities. IEEE Trans Signal Process 2005;53 (11):4215–26. [21] Yang R, Shi P, Liu G-P. Filtering for discrete-time networked nonlinear systems with mixed random delays and packet dropouts. IEEE Trans Autom Control 2011;56(November (11)):2655–60. [22] Zhong Q, Bai J, Wen B, Li S, Zhong F. Finite-time boundedness filtering for discrete-time Markovian jump system subject to partly unknown transition probabilities. ISA Trans 2014;53(4):1107–18. [23] Ding Y, Liu H, Cheng J. H1 filtering for a class of discrete-time singular Markovian jump systems with time-varying delays. ISA Trans 2014;53(4):1054– 60. [24] Zhang Z, Zhang Z, Zhang H, Shi P, Karimi HR. Finite-time H1 filtering for T–F fuzzy systems with time-varying delay and norm-bounded uncertainties. IEEE Trans Fuzzy Syst 2015;23(December (6)):2427–34. [25] Chang X-H, Yang G-H. Nonfragile H1 filter design for T-F fuzzy systems in standard form. IEEE Trans Ind Electron 2014;61(July (7)):3448–58. [26] Wu Y-Q, Su H, Lu R, Wu Z-G, Shu Z. Passivity-based non-fragile control for Markovian-jump systems with aperiodic sampling. Syst Control Lett 2015;84 (August):35–43. [27] Zhang D, Cai WJ, Xie LH, Wang QG. Nonfragile distributed filtering for T–F fuzzy systems in sensor networks. IEEE Trans Fuzzy Syst 2015;23(October (5)):1883–90. [28] Zhang D, Shi P, Zhang WA, Yu L. Non-fragile distributed filtering for fuzzy systems with multiplicative gain variation. Signal Process 2016;121:102–10. [29] Zhang S, Wang Z, Ding D, Dong H, Alsaadi FE, Hayat T. Non-fragile H1 fuzzy filtering with randomly occurring gain variations and channel fadings, IEEE Trans Fuzzy Syst http://dx.doi.org/10.1109/TFUZZ.2015.2446509, in press. [30] Shen H, Wu Z-G, Park Ju H. Reliable mixed passive and H1 filtering for semiMarkov jump systems with randomly occurring uncertainties and sensor failures. Int J Robust Nonlinear Control 2015;25:3231–5. [31] Seuret A, Gouaisbaut F, Fridman E. Stability of discrete-time systems with time-varying delay via a novel summation inequality. IEEE Trans Autom Control 2015;60(October (10)):2740–5. [32] Park PG, Ko JW, Jeong C. Reciprocally convex approach to stability of systems with time-varying delays. Automatica 2011;47(1):235–8. [33] Boyd S, Ghaoui El L, Feron E, Balakrishnan V. Linear Matrix inequalities in system and control theory. Philadelphia: SIAM; 1994.
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Non-fragile H1 filtering for nonlinear discrete-time delay systems with randomly occurring gain variations$ Yajuan Liu a,b, Ju H. Park a,n, Bao-Zhu Guo b,c a
Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China c School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 February 2016 Received in revised form 2 April 2016 Accepted 8 April 2016 This paper was recommended for publication by Dr. Jeff Pieper
In this paper,the problem of H1 filtering for a class of nonlinear discrete-time delay systems is investigated. The time delay is assumed to be belonging to a given interval, and the designed filter includes additive gain variations which are supposed to be random and satisfy the Bernoulli distribution. By the augmented Lyapunov functional approach, a sufficient condition is developed to ensure that the filtering error system is asymptotically mean-square stable with a prescribed H1 performance. In addition, an improved result of H1 filtering for linear system is also derived. The filter parameters are obtained by solving a set of linear matrix inequalities. For nonlinear systems, the applicability of the developed filtering result is confirmed by a longitudinal flight system, and an additional example for linear system is presented to demonstrate the less conservativeness of the proposed design method. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear discrete-time systems Interval time-delay Non-fragile filter Randomly occurring gain variations
1. Introduction Over the past few decades, filtering problem has been widely addressed due to its capability of applications in various fields such as military, astronautics, and signal processing [1,2]. There are two major approaches to be used to deal with the filtering problem: the Kalman filtering approach and the H1 filtering approach. The Kalman filtering requires the external disturbance to be white Gaussian noise which is not always satisfied in many engineering applications. As an effective alternative, the H1 filtering does not require the external noise to be white Gaussian. Compared with classical Kalman filtering, the H1 filtering takes advantage of external noise signal to be in any form with energy bounded, and importantly, the exact noise statistics is not required to be known. Moreover, the H1 filtering has been shown to be robust against unmodeled dynamics [3]. On the other hand, since time delay is unavoidable in various engineering systems, it remains the source of instability and poor performance [4–7]. Hence, the H1 filtering for time delayed systems has attracted considerable attention and a wealth of important results are available in the literature like those reported in [8–13]. ☆ This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10005201). n Corresponding author. Tel.: þ 82 53 8102491. E-mail address: [email protected] (J.H. Park).
The main objective of H1 filtering is to design a suitable filter such that the mapping from exogenous input to estimation error is minimized or is no larger than a prescribed level in terms of the H1 norm. It has been shown that the filter gains can be obtained by solving a set of linear matrix inequalities (LMIs). In [3], a sufficient condition for H1 filter design is derived by employing a new finite sum inequality. More recently, some less conservative results are obtained by using the model transformation approach [9,11]. However, it is noted that these results with analytical tools, although complete in some extent, neglect time delays in estimation of difference upper bound of the constructed Lyapunov functional. On the other hand, the H1 filters developed in [3–13] are only concerned with linear systems. Since the nonlinearity is ubiquitous in many practical systems and the nonlinearity makes system analysis more difficult [14–18], it is significant to consider the H1 filtering problem for nonlinear systems [19–24]. Moreover, all the aforementioned H1 filtering problems in [3–13,19–24] are based on an implicit assumption that the filters can be implemented precisely. This is, however, not the case in practical situations, Actually, the inaccuracy or uncertainty is usually inevitable in filter implementation. Such uncertainty can be attributed to unexpected errors during the implementation such as actuator degradation, roundoff errors in numerical computation, aging of components, and requirement for parameters’ re-adjustment during the process. As a result, the designed filter is expected to be insensitive or non-fragile [25–28]. Moreover, the filter gain variations may be
http://dx.doi.org/10.1016/j.isatra.2016.04.009 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
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2
subject to random changes because of environment circumstance in the process of filter implementation among networks. In this case, the gain variations may be presented in a probabilistic way with certain types and intensity. It is therefore worth taking randomly occurring gain variations into account in filter design [29,30]. To the best of our knowledge, the problem of non-fragile H1 filtering for nonlinear discrete-time systems with randomly occurring gain variations has not been well addressed. Motivated by the aforementioned discussions, the aim of this paper is to investigate the H1 filtering for a class of discrete nonlinear systems with interval time varying delays and randomly occurring gain variations. Inspired by [14,21], the sector nonlinearity in this work is more general than usual Lipschitz condition. In terms of LMI technique, a sufficient condition for nonfragile H1 filtering of the resulting closed-loop nonlinear system is developed. Based on this condition, an improved filtering design for linear systems is proposed. Two numerical examples are presented to demonstrate the validity and less conservativeness of the proposed method. It should be pointed out that the main contributions of this paper are: (a) we consider the H1 filtering for nonlinear systems instead linear ones in the existing literature; (b) we take randomly occurring gain variations into account, where the gain variations appear in a random way based on a certain kind of probabilistic law; (c) for linear systems that are free of randomly occurring gain variations, the corresponding H1 filtering result proposed in this paper is, based on Lyapunov method and zero inequality approach, less conservative than that in the existing literature. The rest of the paper is organized as follows. The problem formulation is presented in Section 2. In Section 3, the main results on H1 filter analysis and design method are presented. Two numerical examples are given to illustrate the effectiveness of the proposed scheme in Section 4, and conclusions are given in Section 5. Notations: Throughout the paper, Rn denotes the n-dimensional Euclidean space. Rnm is the set of all n m real matrices. For a real matrix X, X 4 0 and X o 0 mean that X is a positive/negative definite symmetric matrix, respectively. I is an identity matrix with appropriate dimension. n denotes the symmetric part of given matrix. diagf⋯g denotes the diagonal matrix. J J refers to the induced matrix 2-norm. L2 means the space of square integral R 1=2 1 . vector functions on ½0; 1Þ with norm J J 2 0 J J 2 dt
2. Problem statement Consider a class of nonlinear discrete time-delay system as follows: xðk þ 1Þ ¼ AxðkÞ þ Ad xðk dðkÞÞ þ Ff ðxðkÞÞ þ Bw wðkÞ; yðkÞ ¼ CxðkÞ þ C d xðk dðkÞÞ þ H 1 hðxðkÞÞ þ H 2 hðxðk dðkÞÞ þ Dw wðkÞ; zðkÞ ¼ LxðkÞ þLd xðk dðkÞÞ þ Gw wðkÞ;
ð1Þ
where xðkÞ A Rn and yðkÞ A Rm are the state vector and measured output, respectively. wðkÞ A Rq is a disturbance input belonging to L2 ½0; 1Þ, zðkÞ A Rp is the signal to be estimated, A, Ad, F, Bw, C, Cd, H1, H2, Dw, L, Ld, and Gw are known constant matrices with appropriate dimensions. Assumption 1. The time delay d(k) is assumed to be time-varying and satisfies d1 rdðkÞ r d2 ;
where d1 4 0 and d2 4 0 are constants representing the lower and upper bounds of the delay, respectively. Assumption 2. f ðÞ and hðÞ are sector-bounded functions and satisfy f ð0Þ ¼ hð0Þ ¼ 0. Assumption 3. The nonlinear vector-valued functions are assumed to satisfy ½f ðxÞ f ðyÞ M 1 ðx yÞ ½f ðxÞ f ðyÞ M 2 ðx yÞ r 0;
8 x; y A Rn ;
ð2Þ
½hðxÞ hðyÞ N 1 ðx yÞ ½hðxÞ hðyÞ N 2 ðx yÞ r 0;
8 x; y A Rm ;
ð3Þ
where M1, M2, N1, and N2 are known real constant matrices, and M 2 M 1 and N 2 N1 are positive definite matrices, respectively. Remark 1. The nonlinear functions in Eqs. (2) and (3) are quite general than those with Lipschitz conditions. Our main objective is to estimate the signals z(k). To this purpose, we design the following non-fragile full-order filter: xf ðk þ 1Þ ¼ ðAf þ αðkÞΔAf ðkÞÞxf ðkÞ þ ðBf þ βðkÞΔBf ðkÞÞyðkÞ zf ðkÞ ¼ C f xf ðkÞ þ Df yðkÞ; xf ð0Þ ¼ 0;
ð4Þ
where xf ðkÞ A Rn is the state of filter and zf ðkÞ A Rm is the estimated output, the matrices Af, Bf, Cf, and Df are appropriately dimensioned filter gains to be determined, and αðkÞ and β ðkÞ are mutually independent Bernoulli-distributed white sequences. Assumption 4. A natural assumption on αðkÞ and βðkÞ is made as follows: ProbfαðkÞ ¼ 1g ¼ EfαðkÞg ¼ α ; ProbfαðkÞ ¼ 0g ¼ 1 α ; Probfβ ðkÞ ¼ 1g ¼ Efβ ðkÞg ¼ β ; Probfβ ðkÞ ¼ 0g ¼ β : Assumption 5. The uncertain perturbation matrices ΔAf ðkÞ and Δ Bf ðkÞ are defined as follows:
ΔAf ðkÞ ¼ Ma Δ1 ðkÞN a ; ΔBf ðkÞ ¼ Mb Δ2 ðkÞN b ; where M a ; M b ; N a and Nb are known constant matrices with appropriate dimensions, and Δ1 ðkÞ and Δ2 ðkÞ are unknown matrix functions satisfying
ΔTr ðkÞΔr ðkÞ r I;
r ¼ 1; 2:
Remark 2. In real networked systems, the additive gain variations are inevitable due to unexpected errors during the filter implementation, e.g., round off errors in numerical computation and programming errors [25,28]. Moreover, the filter parameters may subject to random changes which satisfy the Bernoulli distribution [29,30]. Such description can compensate the parameter variations of the filter in a random fashion and has not been investigated for nonlinear discrete-time systems. Denote " # xðkÞ ζ ðkÞ ¼ x ðkÞ; f eðkÞ ¼ zðkÞ zf ðkÞ: Then, we have the following error system ^ þ β BðkÞ ^ ^ þ β^ ðkÞBðkÞÞ ^ ζ ðk þ 1Þ ¼ ðA þ α AðkÞ ζ ðkÞ þ α^ ðkÞAðkÞ ^ ^ ^ þ ðA d þ β A d ðkÞ þ β ðkÞA d ðkÞÞζ ðk dðkÞÞ þ F f ðxðkÞÞ þ ðH 1 þ β H^ 1 ðkÞ þ β^ ðkÞH^ 1 ðkÞÞhðxðkÞÞ þ ðH 2 þ β H^ 2 ðkÞ þ β^ ðkÞH^ 2 ðkÞÞhðxðk dðkÞÞÞ þ ðB w þ β B^ w ðkÞ þ β^ ðkÞB^ w ðkÞÞwðkÞ;
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eðkÞ ¼ C ζ ðkÞ þ C d ζ ðk dðkÞÞ þD 1 hðxðkÞÞ þ D 2 hðxðk dðkÞÞÞ þ D w wðkÞ; ð5Þ where "
A
0
"
#
0
0
#
^ ; ; AðkÞ ¼ Bf C Af 0 ΔAf ðkÞ " # " Ad 0 0 Ad ¼ ; A^ d ðkÞ ¼ ΔBf ðkÞC d Bf C d 0 " # " # 0 0 ^ H1 ¼ ; H 1 ðkÞ ¼ Bf H 1 ΔBf ðkÞH1 ; " # " # 0 Bw ^ H 2 ðkÞ ¼ ΔBf ðkÞH2 ; B w ¼ Bf Dw ; h h i C ¼ L Df C C f ; C d ¼ Ld Df C d A¼
D 1 ¼ Df H 1 ;
α^ ðkÞ ¼ αðkÞ α ;
D 2 ¼ Df H 2 ;
" ^ BðkÞ ¼ #
0
0
;
m2 ðkÞ : N -Rn , the following inequality holds
1
η1 ðkÞ "
#
r
ΔBf ðkÞC 0 ;
F ; 0 0 " # 0 H2 ¼ ; Bf H 2 " # 0 ^ B w ðkÞ ¼ ΔBf ðkÞDw ; i 0 ;
0
mT1 ðkÞMm1 ðkÞ
m1 ðkÞ
"
M
S
n
M
mT2 ðkÞMm2 ðkÞ #
m1 ðkÞ
:
m2 ðkÞ
3. Main results For the sake of simplicity for matrix representation, we define e^ i A Rð19n þ qÞn to be block entry matrix. For example, e^ 2 ¼ ½0 I 0…0 T . We collect some scalars, vectors, and matrices as |ffl{zffl} 18
h
η ðkÞ ¼ ζ ðkÞ ζ T ðk d1 Þ ζ T ðk dðkÞÞ ζ T ðk d2 ÞΔζ T ðkÞ T
D w ¼ Gw Df Dw ;
Definition 2. For a given positive scalar γ, the filtering error system (5) is said to be asymptotically stable in mean square with a guaranteed H1 performance γ if it is asymptotically stable and the filtering error e(k) satisfies 1 1 n o n o X X E J eðkÞ J 2 r γ 2 E J wðkÞ J 2
T
k X 1 d1 þ 1 i ¼ k d
k-1
ð7Þ
k¼0
Δζ ðkÞ ¼ ζ ðk þ 1Þ ζ ðkÞ; e1 ¼ ½e^ 1 e^ 2 ; e2 ¼ ½e^ 3 e^ 4 ; e3 ¼ ½e^ 5 e^ 6 ; e4 ¼ ½e^ 7 e^ 8 ; e5 ¼ ½e^ 9 e^ 10 ; e6 ¼ ½e^ 11 e^ 12 ; e7 ¼ ½e^ 13 e^ 14 ; e8 ¼ ½e^ 15 e^ 16 ; e9 ¼ e^ 17 ; e10 ¼ e^ 18 ; e11 ¼ e^ 19 ; e12 ¼ e^ 20 ; " Ma ¼
K ¼ ½I 0;
U1 ¼
where yðiÞ ¼ xðiÞ xði 1Þ and 0 2 X xðiÞ: h þ1 i ¼ h
Remark 3. When Lemma 1 is employed to time-varying delay, the 1 term hh þ might be difficult to be dealt with. As pointed out in [31], 1 1 Lemma 2 is provided to make the term hh þ disappear from the 1 inequality. Lemma 2 (Seuret et al. [31]). For a given symmetric positive definite matrix Z A Snþ and any sequence of discrete-time variable x in ½ h; 0⋂Z-Rn , where h Z 1, the following inequality holds: " #T # " 0 X Θ0 Z 0 1 Θ0 T y ðiÞZyðiÞ r ; Θ1 h Θ1 0 3Z i ¼ hþ1
Θ1 are defined in Lemma 1.
S Lemma 3 (Park et al. [32]). For any matrix ½M Z0; scalars η1 ðkÞ n M 4 0; η2 ðkÞ 4 0 satisfying η1 ðkÞ þ η2 ðkÞ ¼ 1; vector functions m1 ðkÞ and
0
#
"
; N a ¼ 0 Na ; Ma
0 ; N bcd ¼ N b C d 0 ;
# 0 Mb ¼ ; Mb
U T1 U 2 þU T2 U 1 U T þ U T2 ; U2 ¼ 1 ; 2 2 T T T T V V 2 þ V 2V 1 V þV2 ; U2 ¼ 1 ; V1 ¼ 1 2 2 T T T W W2 þW2W1 W þ W T2 ; W2 ¼ 1 ; W1 ¼ 1 2 2 Σ 1 ¼ ½e1 þe5 ðd1 þ1Þe6 e2 ðdðkÞ d1 þ 1Þe7 þðd2 dðkÞ þ 1Þe8 e3 e4 ; Σ 2 ¼ ½e1 ðd1 þ1Þe6 e1 ðdðkÞ d1 þ 1Þe7 þ ðd2 dðkÞ þ 1Þe8 e3 e2 ;
Σ 3 ¼ ½e1 e2 e1 þ e2 2e6 ; Σ 4 ¼ ½e3 e4 e3 þ e4 2e8 e2 e3 e2 þ e3 2e7 ; 2
GA ¼ 4
Θ0 ¼ xð0Þ xð hÞ;
ζ T ðiÞ
2
Now, the filtering problem can be stated as follows: Filtering problem: Design a filter of the form (4) such that the filtering error system (5) is asymptotically stable in the meansquare sense and achieves a prescribed H1 performance level.
i ¼ hþ1
1
kX d1 1 ζ T ðiÞ dðkÞ d1 þ 1 i ¼ k dðkÞ
i T T T f ðxðkÞÞ h ðxðkÞÞ h ðxðk dðkÞÞÞ wT ðkÞ ;
N bc ¼ N b C
Lemma 1 (Seuret et al. [31]). For a given symmetric positive definite matrix Z A Snþ and any sequence of discrete-time variable x in ½ h; 0⋂Z-Rn , where h Z 1, the following inequality holds: #" " #T " # 0 X Z 0 Θ0 1 Θ0 T y ðiÞZyðiÞ r ; 1 Z 0 3 hh þ Θ1 h Θ1 1
ζ T ðiÞ
kX dðkÞ 1 d2 dðkÞ þ 1 i ¼ k d
for all nonzero wðÞ A L2 ½0; 1Þ subject to the zero initial condition.
where yðiÞ; Θ0 and
1
η2 ðkÞ
F¼
Definition 1. The filtering error system (5) with wðkÞ ¼ 0 is said to be asymptotically stable in the mean-square sense, if for any initial condition, there holds n o ð6Þ lim E J ζ ðkÞ J 2 ¼ 0:
Θ1 ¼ xð0Þ þ xð hÞ
#T
m2 ðkÞ
β^ ðkÞ ¼ βðkÞ β :
k¼0
3
G1 A þ B^ f C GT A þ B^ C f
2
" GF ¼
G1 F
#
3 A^ f 5; A^ f 2
GH 1 ¼ 4
; GT2 F 2 3 G1 Bw þ B^ f Dw 5; GB w ¼ 4 T G2 Bw þ B^ f Dw
2
3 G1 Ad þ B^ f C d 0 5; GT2 Ad þ B^ f C d 0 2 3 B^ f H 2 4 5; GH 2 ¼ ^ B H2
GAd ¼ 4 3 B^ f H 1 5; B^ H 1 f
" R2 ¼
f
R2
0
0
3R2
#
Φ ¼ ½GA G 0 GA d 0 G 0 0 0 0 GH 1 GH 2 GB w ; Ψ ¼ ½C 0 C d 0 0 0 0 0 0 D 1 D 2 D w ; ϵa ¼ ϵ2 þ ϵ3 þ ϵ4 þ ϵ5 þ ϵ6 ; ϵb ¼ ϵðϵ8 þ ϵ8 þ ϵ10 þ ϵ11 þ ϵ12 Þ; 1 2 þ e1 Q 1 eT1 e2 Q 1 eT2
1 2
Ω ¼ ðΣ 1 Σ 2 ÞPðΣ 1 þ Σ 2 ÞT þ ðΣ 1 þ Σ 2 ÞPðΣ 1 Σ 2 ÞT "
þ e1 Q 2 eT1 e4 Q 2 eT4 þ d1 e5 R1 eT5 Σ 3 2
R1
0
0
3λðd1 ÞR1
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#
Σ T3
Y. Liu et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
" 2
þ ðd2 d1 Þ " þ ½e1 e9
e5 R2 eT5
Σ4
X
n
3R 2 #
K T U 1K
K T U 2
n
I
" þ ½e1 e10 " þ ½e3 e11
#
R2
KT V 1K
KT V 2
n
I
Σ T4
where yðjÞ ¼ ζ ðjÞ ζ ðj 1Þ. We compute the difference of V(k) along the error system (5) with wðkÞ ¼ 0. Notice that 2 3 2 3 ζ ðkÞ þ Δζ ðkÞ ζ ðk þ1Þ 6 7 6 7 k k X X 6 7 6 7 6 7 6 7 ζ ðiÞ ζ ðk d Þ ζ ðiÞ 1 6 7 6 7 6 i ¼ k d1 þ 1 7¼6 7; i ¼ k d1 6 7 6 7 6 kX 7 6 7 kX dðkÞ d1 kX d1 6 7 6 7 4 ζ ðiÞ 5 4 ζ ðiÞ þ ζ ðiÞ ζ ðk dðkÞÞ ζ ðk d2 Þ 5
#
KT W 2
n
I
yT ðjÞR2 yðjÞ;
i ¼ d2 þ 1 j ¼ k þ i
½e1 e9 T
KT W 1K
k X
d1 X
V 5 ðkÞ ¼ ðd2 d1 Þ
½e1 e10 T # ½e3 e11 T
þ ½e1 þ ϵe5 Φ þ Φ ½e1 þ ϵe5 T T
T
i ¼ k d2 þ 1
T
þ e1 ½α ðϵ1 1 þ ϵϵ7 1 ÞN a N a þ β ðϵ2 1 þ ϵϵ8 1 ÞN bc N bc eT1
i ¼ k dðkÞ
i ¼ k d2
ð11Þ
T
þ e3 β ðϵ3 1 þ ϵϵ9 1 ÞN bcd N bcd eT3 1 1 þ e10 β ðϵ4 1 þ ϵϵ10 ÞH T1 N Tb N b H 1 eT10 þe11 β ðϵ5 1 þ ϵϵ11 ÞH T2 N Tb Nb H 2 eT11 1 þ e12 β ðϵ6 1 þ ϵϵ12 ÞDTw N Tb N b Dw eT12 γ 2 e12 eT12 :
Theorem 1. For given constants d2 4 d1 4 0, γ 4 0, ϵ 40, and ϵi 4 0 ði ¼ 1; 2; …; 12Þ, and matrices U1, U2, V1, V2, W1, and W2, there
exists an admissible non-fragile H1 filter (4) such that the filtering error system (5) is asymptotically stable in mean square for wðkÞ ¼ 0 and the filtering error e(k) satisfies (7) under the zero initial condition for any nonzero wðkÞ A L2 ½0; 1Þ, if there exist matrices P 4 0, Q 1 4 0, ^ , and G ¼ ½GT1 G2 satisfying Q 2 4 0, R1 4 0, R2 4 0, X , G, A^ f , B^ f , C^ f , D f G2 G2 the following LMIs 2 3 Ω Ψ T e1 αϵ1 GM a e1 ϵa β GM b e5 αϵϵ7 GM a e5 ϵb β GM b 6 7 6 n I 7 0 0 0 0 6 7 6n 7 n ϵ α 0 0 0 1 6 7 6 7 o 0; 6n 7 n n ϵ β 0 0 a 6 7 6n 7 n n n ϵϵ α 0 7 4 5 n n n n n ϵb β
ð8Þ "
R2
X
n
R2
# Z 0:
ð9Þ
Moreover, if the LMIs admit feasible solutions, the matrices of the admissible H1 filter in the form (4) can be obtained by Af ¼ G2 1 A^ f ;
Bf ¼ G2 1 B^ f ;
C f ¼ C^ f ;
^ : Df ¼ D f
Proof. Construct the following Lyapunov functional: VðkÞ ¼ V 1 ðkÞ þ V 2 ðkÞ þ V 3 ðkÞ þ V 4 ðkÞ þV 5 ðkÞ
and 2
ζ ðkÞ
3T 2
ζ ðkÞ
V 2 ðkÞ ¼
kX 1
i ¼ k d2
ζ ðkÞQ 1 ζ ðkÞ; T
i ¼ k d1
V 3 ðkÞ ¼
kX 1
ζ T ðkÞQ 2 ζ ðkÞ;
i ¼ k d2
V 4 ðkÞ ¼ d1
0 X
k X
i ¼ d1 þ 1 j ¼ k þ i
yT ðjÞR1 yðjÞ;
7 7 7 7 7: i ¼ k d1 7 7 kX d1 7 ζ ðiÞ ζ ðk dðkÞÞ ζ ðk d1 Þ 5 k X
ζ ðiÞ ζ ðkÞ
i ¼ k dðkÞ
i ¼ k d2
i ¼ k d2
3
ζ ðkÞ
ð12Þ The differences of V i ðkÞ ði ¼ 1; 2; …; 5Þ are given by n o T T T E ΔV 1 ðkÞ ¼ E ξ ðkÞðΣ 1 P Σ 1 Σ 2 P Σ 2 ÞξðkÞ n o T ¼ E ξ ðkÞðΣ 1 þ Σ 2 ÞPðΣ 1 Σ 2 ÞT ξðkÞ ;
ð13Þ
n o T T E ΔV 2 ðkÞ ¼ E ζ ðkÞQ 1 ζ ðkÞ ζ ðk d1 ÞQ 1 ζ ðk d1 Þ ;
ð14Þ
n o T T E ΔV 3 ðkÞ ¼ E ζ ðkÞQ 2 ζ ðkÞ ζ ðk d2 ÞQ 2 ζ ðk d2 Þ ;
ð15Þ
8 9 k < = X 2 T E ΔV 4 ðkÞ ¼ E d1 y ðk þ 1ÞR1 yðk þ1Þ d1 yT ðiÞR1 yðiÞ : ; i ¼ k d1 þ 1 8 9 k < = X T 2 yT ðiÞR1 yðiÞ ; ¼ E d1 Δζ ðkÞR1 Δζ ðkÞ d1 : ; i ¼ k d1 þ 1 n E ΔV 5 ðkÞ ¼ E ðd2 d1 Þ2 yT ðk þ 1ÞR2 yðkþ 1Þ ) kX d1 yT ðiÞR2 yðiÞ ðd2 d1 Þ i ¼ k d2 þ 1
8 < T ¼ E ðd2 d1 Þ2 Δζ ðkÞR2 Δζ ðkÞ ðd2 d1 Þ :
ð10Þ
kX d1 i ¼ k d2 þ 1
9 = yT ðiÞR2 yðiÞ : ;
ð16Þ
3
6 kX 7 6 kX 7 6 1 7 6 1 7 6 6 ζ ðiÞ 7 ζ ðiÞ 7 6 7 6 7 7 P 6 i ¼ k d1 7; V 1 ðkÞ ¼ 6 6 i ¼ k d1 7 6 7 6 k X 7 6 k X 7 6 d1 1 7 6 d1 1 7 4 5 4 ζ ðiÞ ζ ðiÞ 5 i ¼ k d2
2
6 kX 7 6 6 1 7 6 6 6 ζ ðiÞ 7 6 7 6 6 i ¼ k d1 7¼6 6 7 6 6 k X 7 6 kX 6 d1 1 7 6 dðkÞ 4 ζ ðiÞ 5 4 ζ ðiÞ þ
with 2
3
ζ ðkÞ
By Lemma 1, k X
d1
i ¼ k d1
"
η1 ðkÞ y ðiÞR1 yðiÞ r η2 ðkÞ þ1 T
#T "
R1 0
0 3λðd1 ÞR1
#"
#
η1 ðkÞ ; η2 ðkÞ ð17Þ
η1 ðkÞ ¼ ζ ðkÞ ζ ðk d1 Þ, η2 ðkÞ ¼ ζ ðkÞ þ ζ ðk d1 Þ d1 2þ 1 d1 þ 1 ζ ðiÞ, λ ðd Þ ¼ 1ðd ¼ 1Þ, and λ ðd 1 1 1 Þ ¼ d1 1ðd1 41Þ. i ¼ k d1
where Pk
By Lemmas 2 and 3, one can obtain
ðd2 d1 Þ
kX d1
yT ðiÞR2 yðiÞ
i ¼ k d2 þ 1
¼ ðd2 d1 Þ
kX dðkÞ i ¼ k d2 þ 1
yT ðiÞR2 yðiÞ ðd2 d1 Þ
kX d1 i ¼ k dðkÞ þ 1
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yT ðiÞR2 yðiÞ
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o T T þ ε9 1 ζ ðk dðkÞÞN bcd N bcd ζ ðk dðkÞÞ ; n T T T Ef2Δζ ðkÞGH^ 1 ðkÞhðxðkÞÞg r E Δζ ðkÞε10 GM b M b GΔζ ðkÞ o 1 T h ðxðkÞÞH T1 NTb N b H 1 hðxðkÞÞ ; þ ε10
" #T " # #" γ 1 ðkÞ R2 0 d2 d1 γ 1 ðkÞ r γ 2 ðkÞ 0 3R2 d2 dðkÞ γ 2 ðkÞ " #T " # #" γ 3 ðkÞ R2 0 d2 d1 γ 3 ðkÞ γ 4 ðkÞ 0 3R2 dðkÞ d1 γ 4 ðkÞ " # R2 X r γ T ðkÞ γ ðkÞ; n R2
ð18Þ
In view of Assumptions 2 and 3, for known constant matrices U 1 , U 2 , V 1 , V 2 , W 1 , and W 2 , it readily follows that #" " #T " # ζ ðkÞ ζ ðkÞ K T U 1K KT U 2 Z 0; ð19Þ f ðxðkÞÞ f ðxðkÞÞ n I
ζ ðkÞ
#T "
KT V 1K
KT V 2
n
I
hðxðkÞÞ "
ζ ðk dðkÞÞ
#T "
#"
ζ ðkÞ
K T W 1K
KT W 2
n
I
hðxðk dðkÞÞÞ
Z 0;
#"
hðxðk dðkÞÞÞ
^ ^ ^ þ β^ ðkÞBðkÞÞÞ þ α^ ðkÞAðkÞ þ β BðkÞ ζ ðkÞ ^ ^ ^ þ ðA þ β A ðkÞ þ β ðkÞA ðkÞÞζ ðk dðkÞÞ þ F f ðxðkÞÞ d
d
d
For positive scalars εi ði ¼ 1; 2; …; 12Þ, we have the following easily obtained inequalities: ^ ζ ðkÞg r Efζ T ðkÞðε1 GM a M T G þ ε 1 N T N a Þζ ðkÞg; Ef2ζ ðkÞGAðkÞ 1 a a T
^ ζ ðkÞg rEfζ ðkÞðε2 GM b M G þ ε 1 N N bc Þζ ðkÞg; Ef2ζ ðkÞGBðkÞ 2 b bc n T T T Ef2ζ ðkÞGA^ d ðkÞζ ðk dðkÞÞg rE ζ ðkÞε3 GM b M b Gζ ðkÞ o T T þ ε3 1 ζ ðk dðkÞÞN bcd N bcd ζ ðk dðkÞÞ ; T
T
T
T
Ef2ζ ðkÞGH^ 1 ðkÞhðxðkÞÞg rEfζ ðkÞε4 GM b M b Gζ ðkÞ T
T
T
þ ε4 1 h ðxðkÞÞH T1 N Tb N b H 1 hðxðkÞÞg; n T T T Ef2ζ ðkÞGH^ 2 ðkÞhðxðk dðkÞÞÞg r E ζ ðkÞε5 GM b M b Gζ ðkÞ o T þ ε5 1 h ðxðk dðkÞÞÞH T2 N Tb Nb H 2 hðxðk dðkÞÞÞ ;
T
T
1 T þ ε12 w ðkÞDTw NTb N b Dw wðkÞg:
ð23Þ
Here, we establish the H1 performance for filtering error system (5). If the difference of V(k) is negative, then zðkÞ-0 as k-1. Next, by assuming zero initial conditions for the filtering error system, the performance index is 1 n o X J¼ E eðkÞT eðkÞ γ 2 wðkÞT wðkÞ k¼0
1 n o X E feðkÞT eðkÞ γ 2 wðkÞT wðkÞ þ ΔVðkÞg þ EVð0Þ EVð1Þ: ¼ k¼0
ð22Þ
From (5), for any matrix G, n ^ 2E ½ζ ðkÞ þ ϵΔζ ðkÞG½ð Δζ ðkÞ þ ðA I þ α AðkÞ
þ ðH 1 þ β H^ 1 ðkÞ þ β^ ðkÞH^ 1 ðkÞÞhðxðkÞÞ þ ðH 2 þ β H^ 2 ðkÞ þ β^ ðkÞH^ 2 ðkÞÞhðxðk dðkÞÞÞ o þ ðB w þ β B^ w ðkÞ þ β^ ðkÞB^ w ðkÞÞwðkÞ ¼ 0:
T
and by a simple matrix calculation, it is straightforward to verify that " # " # G1 A þ G2 Bf C G2 Af G1 Ad þ G2 Bf C d 0 GA ¼ ; ; GAd ¼ T T G2 A þ G2 Bf C G2 Af G2 Ad þ G2 Bf C d 0 " # " # " # G2 B f H 1 G2 B f H 2 G1 F ; GH 1 ¼ ; GH 2 ¼ ; GF ¼ G2 B f H 1 G2 B f H 2 GT2 F " # G1 Bw þ G2 Bf Dw : GB w ¼ GT2 Bw þ G2 Bf Dw
# Z 0:
Ef2Δζ ðkÞGB^ w ðkÞwðkÞg r EfΔζ ðkÞε12 GM b M b GΔζ ðkÞ
ð21Þ
ð20Þ
ζ ðk dðkÞÞ
T
T
n o If E eðkÞT eðkÞ γ 2 wðkÞT wðkÞ þ ΔVðkÞ o0, then VðkÞ-0 as k-1. Taking G as " # G 1 G2 ; G¼ T G2 G2
#
hðxðkÞÞ
Ef2Δζ ðkÞGH^ 2 ðkÞhðxðk dðkÞÞÞg rEfΔζ ðkÞε11 GM b M b GΔζ ðkÞ o 1 T þ ε11 h ðxðk dðkÞÞÞH T2 N Tb N b H 2 hðxðk dðkÞÞÞ ; T
where γ ðkÞ ¼ ½γ T1 ðkÞ γ T2 ðkÞ γ T3 ðkÞ γ T4 ðkÞT , γ 1 ðkÞ ¼ ζ ðk dðkÞÞ ζ ðk d2 Þ, Pk dðkÞ 2 γ 2 ðkÞ ¼ ζ ðk dðkÞÞ þ ζ ðk d2 Þ d2 dðkÞ ζ ðiÞ, þ1 i ¼ k d2 γ 3 ðkÞ ¼ ζ ðk d1 Þ ζ ðk dðkÞÞ, P γ 4 ¼ ζ ðk d1 Þ þ ζ ðk dðkÞÞ dðkÞ 2d1 þ 1 ki ¼ dk1 dðkÞ ζ ðiÞ, and " # R2 0 R2 ¼ : 0 3R2
"
5
Define a set of variables as follows: A^ f ¼ G2 Af ;
B^ f ¼ G2 Bf ;
C^ f ¼ C f ;
^ f ¼ Df : D
Then, combining with Eqs. (13)–(23), one can obtain n o E ΔV ðkÞ þ eðkÞT eðkÞ γ 2 wðkÞT wðkÞ n T T T r E ξ ðkÞðΩ þ Ψ Ψ þ αε1 e1 GM a M a GeT1
o T T T þ βεa e1 GM b M b GeT1 þ αεε7 e5 GM a M a GeT5 þ βεb e5 GM b M b GeT5 ÞξðkÞ : ð24Þ
T
Ef2ζ ðkÞGB^ w ðkÞwðkÞg r Efζ ðkÞε T
þε
T 6 GM b M b G
T
ζ ðkÞ
T T 1 T 6 w ðkÞDw N b N b Dw wðkÞg;
^ ζ ðkÞg r EfΔζ ðkÞε7 GM a M GΔζ ðkÞ Ef2Δζ ðkÞGAðkÞ a T
T
T
T
þ ε7 1 ζ ðkÞN a N a ζ ðkÞg; T
^ ζ ðkÞg rEfΔζ ðkÞε8 GM M GΔζ ðkÞ Ef2Δζ ðkÞGBðkÞ b b T
T 1 T ðkÞN bc N bc 8
T
T
ζ ðkÞg; þε ζ n T T T Ef2Δζ ðkÞGA^ d ðkÞζ ðk dðkÞÞg r E Δζ ðkÞε9 GM b M b GΔζ ðkÞ
T
By Schur complement, Ω þ Ψ Ψ þ αε1 e1 GM a M a GeT1 þ βεa e1 G T
T M b M b GeT1 þ
T T 7 e5 GM a M a Ge5 þ
T T b e5 GM b M b Ge5 o 0
αεε βε is equivalent to (8), which means J o 0 for any nonzero wðkÞ A L2 ; i.e., the filtering error system has a guaranteed γ level of disturbance attenuation. This completes the proof of the theorem. □
If F ¼ H 1 ¼ H 2 ¼ 0, the system (1) is reduced to the following linear system: xðk þ 1Þ ¼ AxðkÞ þ Ad xðk dðkÞÞ þ Bw wðkÞ; yðkÞ ¼ CxðkÞ þ C d xðk dðkÞÞ þ Dw wðkÞ; zðkÞ ¼ LxðkÞ þ Ld xðk dðkÞÞ þ Gw wðkÞ:
ð25Þ
And by taking Δ1 ðkÞ ¼ Δ2 ðkÞ ¼ 0, the filter of the form (4) is
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6
Af ¼ G2 1 A^ f ;
reduced to xf ðk þ 1Þ ¼ Af xf ðkÞ þ Bf yðkÞ; zf ðkÞ ¼ C f xf ðkÞ þDf yðkÞ;
xf ð0Þ ¼ 0:
ð26Þ
In these cases, the corresponding error system is written as
ζ ðk þ 1Þ ¼ A ζ ðkÞ þ A d ζ ðk dðkÞÞ þB w wðkÞ; eðkÞ ¼ C ζ ðkÞ þ C d ζ ðk dðkÞÞ þD w wðkÞ;
ð27Þ
T T T T T η~ T ðkÞ ¼ 6 4ζ ðkÞ ζ ðk d1 Þ ζ ðk dðkÞÞ ζ ðk d2 Þ Δζ ðkÞ
e~ 1 ¼ ½e 1 e 2 ; e~ 2 ¼ ½e 3 e 4 ; e~ 3 ¼ ½e 5 e 6 ; e~ 4 ¼ ½e 7 e 8 ; e~ 5 ¼ ½e 9 e 10 ; e~ 6 ¼ ½e 11 e 12 ; e~ 7 ¼ ½e 13 e 14 ; e~ 8 ¼ ½e 15 e 16 ; e~ 9 ¼ e 17
þ ðd2 dðkÞ þ 1Þe~ 8 e~ 3 e~ 4 ; ~ Σ 2 ¼ ½e~ 1 ðd1 þ 1Þe~ 6 e~ 1 ðdðkÞ d1 þ1Þe~ 7 þ ðd2 dðkÞ þ 1Þe~ 8 e~ 3 e~ 2 ;
Σ~ 3 ¼ ½e~ 1 e~ 2 e~ 1 þ e~ 2 2e~ 6 ; Σ~ 4 ¼ ½e~ 3 e~ 4 e~ 3 þ e~ 4 2e~ 8 e~ 2 e~ 3 e~ 2 þ e~ 3 2e~ 7 ;
In this section, two examples are given to show the effectiveness of the proposed method on the design of the robust H1 filter.
1 2
" þ ðd2 d1 Þ
2~
e 5 R2 e~ T5
Σ~
4
R2
X
n
3R 2
3
R1 0
#
Example 1. Consider the following simplified longitudinal flight system [19]:
# 0 T Σ~ 3 3λðd1 ÞR1
x1 ðk þ1Þ ¼ 0:9944x1 ðkÞ 0:1203x2 ðkÞ 0:4302x3 ðkÞ;
T Σ~
R2 n
# X Z 0: R2
ð32Þ
ð28Þ
Corollary 1. For given constants d2 4 d1 4 0; γ 4 0 and ε 4 0, there exists an admissible non-fragile H1 filter (26) such that the filtering error system (27) is asymptotically stable in mean square for wðkÞ ¼ 0 and the filtering error e(k) satisfies (7) under the zero initial condition for any nonzero wðkÞ A L2 ½0; 1Þ, if there exist matrices iP 4 0, Q 1 4 0, h Q 2 4 0, R1 40, R2 4 0, X , A^ f , B^ f , C^ f , D^ f , and G ¼ GGT1 GG22 satisfying the 2 following LMIs " # ~ Ψ~ T Ω o 0; ð29Þ n I "
ð31Þ
x2 ðk þ2Þ ¼ 0:0017x1 ðkÞ þ 0:9902x2 ðkÞ ð0:0747 þ 0:01rðkÞÞx3 ðkÞ;
4
~ þΦ ~ T ½e~ þ εe~ T γ 2 e~ e~ T þ ½e~ 1 þ εe~ 5 Φ 5 1 9 9
When the optimal γ is obtained from the above optimization problem, the designed filters can guarantee that the filtering error system is exponentially stable and achieves a prescribed H1 performance level γ. It should be pointed out that it can easily minimize the disturbance attenuation level γ according to Theorem 1 (Corollary 1) by solving a convex optimization problem [33].
4. Numerical examples
~ ¼ ðΣ~ Σ~ ÞPðΣ~ þ Σ~ ÞT þ ðΣ~ þ Σ~ ÞPðΣ~ Σ~ ÞT Ω 1 2 1 2 1 2 1 2 "
Remark 5. Different from the method proposed in [9,11], neither the free-weighting matrices nor the transformation approach is applied to estimate the upper bound of the cross term that appears in the difference of the constructed Lyapunov functional. In order to obtain the less conservative results of the designed filter, Lemma 1 is used to estimate the upper bound of d1 Pk T i ¼ k d1 þ 1 y ðiÞR1 yðiÞ instead of using Jensen's inequality. And Lemmas 2 and 3 are employed to tackle with the term P d1 yT ðiÞR2 yðiÞ. ðd2 d1 Þ ki ¼ k d2 þ 1
min γ 2 , subject to the LMIs (8) and (9) ((29) and (30)).
~ ¼ ½GA G 0 GA 0 G 0 0 0 GB w ; Φ d Ψ~ ¼ ½C 0 C d 0 0 0 0 0 D w ; Σ~ 1 ¼ ½e~ 1 þ e~ 5 ðd1 þ 1Þe~ 6 e~ 2 ðdðkÞ d1 þ 1Þe~ 7
Σ~
Remark 4. Different from the Lyapunov functional constructed in [9,11], the augmented vector, which includes the terms such as Pk d 1 1 Pk 1 ζ ðiÞ, is included in (10). By these terms, i ¼ k d1 ζ ðiÞ; i ¼ k d2
Remark 7. In Theorem 1 (Corollary 1), the filter parameters can be obtained by solving the following optimization problem such that the H1 disturbance attenuation level is minimized:
2
2 þ e~ 1 Q 2 e~ T1 e~ 4 Q 2 e~ T4 þ d1 e~ 5 R1 e~ T5
^ : Df ¼ D f
Remark 6. In order to convert a nonlinear matrix inequality into a linear matrix one, the facts GP 1 G r P G GT are used in [9,22,23,27,28], and some additional positive condition must be satisfied in [11]. Here, the zero equality (22) is used to avoid such problem, which can give much flexibility in solving LMIs.
kX d1 k X 1 1 ζ T ðiÞ ζ T ðiÞ d1 þ 1 i ¼ k d dðkÞ d1 þ1 i ¼ k dðkÞ 1 3 kX dðkÞ 1 T T ζ ðiÞ w ðkÞ5; d2 dðkÞ þ1 i ¼ k d
þ e~ 1 Q 1 e~ T1 e~ 2 Q 1 e~ T2
C f ¼ C^ f ;
more past history of ζ ðkÞ can be used, which may lead to less conservative results.
where A; A d ; B w ; C ; C d and D w are defined in (5). Now, we focus on the H1 performance analysis for system (25). By Theorem 1, we have the following Corollary 1 which says that the error system (27) is asymptotically mean-square stable with H1 performance γ. To do this, we define e i A Rð16n þ qÞn to be the block entry matrix. For example, e 2 ¼ ½0 I 0 … 0 T . And some of |fflfflffl{zfflfflffl}15 scalar, vectors and matrices are defined as 2
1 2
Bf ¼ G2 1 B^ f ;
ð30Þ
Moreover, if the above LMIs admit feasible solutions, the matrices of the admissible H1 filter in the form (26) can be obtained by
x3 ðk þ1Þ ¼ 0:8187x2 ðkÞ þ 0:1wðkÞ;
ð33Þ
where r(k) is unknown but satisfies j rðkÞj r 1. The measurement and estimated signals are supposed to be yðkÞ ¼ 0:2x1 ðkÞ þ 0:1x2 ðkÞ þ ð0:1 þ 0:01rðkÞÞx3 ðkÞ þ 0:1x1 ðk dðkÞÞ þ ð0:1 þ 0:01rðkÞÞx2 ðk dðkÞÞ þ 0:02rðkÞx2 ðk dðkÞÞ þ 0:04rðkÞx3 ðkÞ þ 0:1wðkÞ; zðkÞ ¼ 0:1x2 ðkÞ þ 0:2x3 ðkÞ þ 0:1x1 ðk dðkÞÞ þ 0:2x2 ðk dðkÞÞ:
ð34Þ ð35Þ
It is easy to see that the system described by Eqs. (31)–(35) satisfies Eqs. (2) and (3) and has the form (1) with 2 3 0:9944 0:1203 0:4302 6 7 0:9902 0:0747 5; Ad ¼ 0; F ¼ 0; A ¼ 4 0:0017 0
0:8187
0
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2
0
3
0.5 Response of the filtering error e(k)
6 7 B w ¼ 4 0 5; 0:1
C ¼ 0:2 0:1 0 ; C d ¼ 0:1 0:1 0 ;
H 1 ¼ H 2 ¼ 0:2 0:2 0:2 ; Dw ¼ 0:1;
L ¼ 0 0:1 0:2 ; Ld ¼ 0 0:1 0 ; Gw ¼ 0: The parameters of nonlinear functions are given as 2 3 2 3 0 0 0 0 0 0 6 7 6 7 U 1 ¼ 4 0 0 0:01 5; U 2 ¼ 4 0 0 0:01 5; 0
0
0 6 V1 ¼ 4 0
0 0
2
0
0
3 0:01 0 7 5;
0
e(k)
0.4 0.3 0.2 0.1 0 −0.1 −0.2
0:04 2 3 2 0 0 0:01 0 60 0 7 0 5; W 1 ¼ 6 V2 ¼ 4 40 0 0 0:04 0 2 3 0 0 0 6 7 W 2 ¼ 4 0 0:01 0 5: 0 0:02 0 0
0
7
0
20
40
60
80
100 120 140 160 180 200 t
0
0 0:01 0:02
0
3
Fig. 2. Estimated error of e(k) in Example 1.
07 5; 0
Table 1 Comparison of H1 performance γ.
The uncertain parameters of the filter (4) are taken as follows: 2 3 0:1
6 7 M a ¼ M b ¼ 4 0:2 5; N a ¼ 0:1 0:1 0:1 ;
d(k)
d1 ¼ 1 d2 ¼ 4
d1 ¼ 1 d2 ¼ 5
d1 ¼ 2 d2 ¼ 5
d1 ¼ 2 d2 ¼ 6
[9] [11] Corollary 1
5.0782 4.9431 4.2786
6.4910 6.1604 4.8368
5.5151 5.3551 4.4473
7.0533 6.7581 4.9473
0:3
2
Let d1 ¼ 1; d2 ¼ 5; α ¼ 0:2; β ¼ 0:5; ε ¼ εi ¼ 1 ði ¼ 1; 2; …; 12Þ. The minimum H1 performance is γ ¼ 0:1583 by Theorem 1 and the corresponding filter parameters are 2 3 2 3 0:3807 0:1105 0:1159 0:5876 6 7 6 7 0:9015 0:1339 5; Bf ¼ 4 0:0561 5; Af ¼ 4 0:0741 0:0357 0:1104 0:4147 0:0225
C f ¼ 0:0089 0:1961 0:0206 ; Df ¼ 0:2786: For initial condition xð0Þ ¼ ½ 0:5 0 0:5T and wðkÞ ¼ sin ðkÞe 10k , the response of z(k) and zf(k), and the error e(k) are shown in Figs. 1 and 2, respectively. The simulations show that the desired system performance can be achieved by using the proposed modeling method and designed filter parameters.
0.6
z(k) zf(k)
Resonse of the Outputs
0.4
Response of the Outputs
Δ1 ðkÞ ¼ Δ2 ðkÞ ¼ sin ðkÞ:
N b ¼ 0:2;
z(k) zf(k)
1.5 1 0.5 0 −0.5 −1 0
50
100
150
t Fig. 3. Responses of z(t) and zf(k) in Example 2.
Example 2. Consider the system (25) with the following parameters: 0:2 0 0:1 0:85 0:1 ; Bw ¼ ; A¼ ; Ad ¼ 0:2 0:1 0:4 0:1 0:7
C ¼ 0:2 2:5 ; C d ¼ 0:5 0:5 ; Dw ¼ 1;
L ¼ 0 2:2 ; Ld ¼ 1:5 0:4 ; Gw ¼ 0:1:
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2 −1.4
0
20
40
60
80
100 120 140 160 180 200 t
Fig. 1. Responses of z(t) and zf(k) in Example 1.
The allowable H1 performance γ obtained by various methods are depicted in Table 1. When d1 ¼ 2; d2 ¼ 6 and ε ¼ 2, one can obtain the minimal H1 performance level γ ¼ 4:9473 by Corollary 1, which is much smaller than 7.0533 and 6.7581. It means that the derived result in this paper is less conservative than that proposed in [9,11]. Solving the LMIs in (29) and (30), the corresponding
Please cite this article as: Liu Y, et al. Non-fragile :ISATransactionsð2016Þ; http : ==dx:doi:org=10:1016=j:isatra:2016:04:009i
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8
Response of the filtering error e(k)
0.5
e(k)
0 −0.5 −1 −1.5 −2 −2.5
0
50
100
150
t Fig. 4. Estimated error of e(k) in Example 2.
parameters of the filter gains are given by 1:2663 2:1020 0:5634 Af ¼ ; Bf ¼ ; 0:2140 0:2016 0:3286
C f ¼ 1:1348 4:8925 ; Df ¼ 1:0936: For initial condition xð0Þ ¼ ½0:5 0:5T and wðkÞ ¼ sin ðkÞe 10k , the response of z(k) and zf(k), and the error e(k) are shown in Figs. 3 and 4, respectively. From these simulation results, it is known that the disturbance is effectively attenuated by designed H1 filter for linear discrete-time system (25).
5. Conclusions The delay-dependent non-fragile H1 filtering for nonlinear discrete-time delay systems has been investigated in this paper. By introducing a zero equality and combining with the Lyapunov functional method, a sufficient condition has been developed to ensure that the filtering error system is asymptotically meansquare stable with a prescribed H1 performance. Furthermore, an improved H1 filter for linear system has been proposed. It should be noted that we do not employ the model transformation method unlike existing works. The numerical simulations illustrate the effectiveness of the proposed approach. In fact, the phenomenon of randomly occurring gain variations considered in this paper can be employed to deal with the stabilization for stochastic systems [4,5], synchronization of dynamical networks [15], tracking control for multi-agent systems [16], which could be our future work.
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Please cite this article as: Liu Y, et al. Non-fragile :ISATransactionsð2016Þ; http : ==dx:doi:org=10:1016=j:isatra:2016:04:009i