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Filtering of Interval Type-2 Fuzzy Systems With Intermittent Measurements Hongyi Li, Chengwei Wu, Ligang Wu, Senior Member, IEEE, Hak-Keung Lam, Senior Member, IEEE, and Yabin Gao
Abstract—In this paper, the problem of fuzzy filter design is investigated for a class of nonlinear networked systems on the basis of the interval type-2 (IT2) fuzzy set theory. In the design process, two vital factors, intermittent data packet dropouts and quantization, are taken into consideration. The parameter uncertainties are handled effectively by the IT2 membership functions determined by lower and upper membership functions and relative weighting functions. A novel fuzzy filter is designed to guarantee the error system to be stochastically stable with H∞ performance. Moreover, the filter does not need to share the same membership functions and number of fuzzy rules as those of the plant. Finally, illustrative examples are provided to illustrate the effectiveness of the method proposed in this paper. Index Terms—Data packet dropouts, filter design, interval type-2 (IT2) fuzzy systems, nonlinear networked system.
I. I NTRODUCTION N MODERN industrial systems, the physical plants, sensors, filters, and actuators are usually located in different geographical places, which require the signal to be transmitted from one place to another. The traditional communication via point-to-point cables cannot satisfy the requirement. Thus, the network media is introduced into the system to connect these components, which gives rise to the networked systems [1], [2]. Networked systems have brought obvious advantages, such as low cost, simple installation and maintenance, and high reliability [3]. Antsaklis and Baillieul [4] have highlighted that
I
Manuscript received September 26, 2014; revised December 18, 2014 and February 27, 2015; accepted March 3, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61203002, Grant 61174126, Grant 61222301, and Grant 61333012, in part by the Program for New Century Excellent Talents in University under Grant NCET-13-0696, in part by the Chinese National Post-Doctoral Science Foundation under Grant 2014M551111, in part by the Post-Doctoral Science Foundation of Northeastern University, in part by the Key Laboratory of Integrated Automation for the Process Industry (Northeast University), and in part by the Fundamental Research Funds for the Central Universities under Grant HIT.BRETIV.201303. This paper was recommended by Associate Editor P. Shi. H. Li is with the College of Engineering, Bohai University, Jinzhou 121013, China (e-mail: [email protected]). C. Wu and Y. Gao are with the College of Information Science and Technology, Bohai University, Jinzhou 121013, China (e-mail: [email protected]; [email protected]). L. Wu is with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]). H.-K. Lam is with the Department of Informatics, King’s College London, London WC2R 2LC, U.K (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2413134
increasing attention has been received in modeling, analysis, and synthesis of networked systems. Recently, some stability and stabilization results concerning networked systems were reported in [5]–[11] and the filtering problem was studied in [12]–[15]. For unavailable system states, the filter and observer play a vital role in estimating them. Zhang et al. [16] proposed a filtering design method to estimate state variables. In [17]–[20], the observer was used to handle the unmeasurable systems states with different systems requirements. However, nonlinearities often exist in the complex plants [21]–[23], which makes it difficult to design them in nonlinear systems. The Takagi–Sugeno (T–S) fuzzy-model-based approach [24] can cope with nonlinearities existing in the application systems, which “blends” every local linear system through “IF-THEN” rules [25]–[33]. In networked environment, considerable filter design results with the T–S fuzzy model have been reported. The H∞ filtering problem for a new class of discrete-time nonlinear networked systems with mixed random delays and packet dropouts was investigated in [34]. Dong et al. [25] studied the distributed finite-horizon filtering problem for a class of time-varying systems over lossy sensor networks. The problem of H∞ filtering for a class of nonlinear discrete-time systems with measurement quantization and packet dropouts was investigated in [16]. In [35], the filter design problem for fault detection with missing measurement was researched. Zhang and Wang [14] proposed a novel approach for the filtering problem of T–S fuzzy systems in a network environment. Both delays of the premise variables and measurements were considered. However, it is worth noticing that the aforementioned results concerning networked systems were achieved through the T–S fuzzy model-based approach on the basis of the type-1 fuzzy set theory, which can perfectly deal with nonlinearities. There also exist parameter uncertainties in the application systems, which cannot be handled by the type-1 fuzzy set theory effectively. And the accuracy of system modeling will be degraded if parameter uncertainties are not fully taken into account. Mendel et al. [36] proposed an interval type-2 (IT2) fuzzy model to model the physical plants with parameter uncertainties. In view of its merits, researchers have been inspired to investigate it and many results concerning IT2 fuzzy systems were reported in [37]–[40]. Lam and Seneviratne [41] provided an IT2 T–S fuzzy model approach to model the plant subject to parameter uncertainties and achieved the stability condition. Additionally, it has been proved that the IT2 T–S fuzzy model performs better than the type-1 one when the
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parameter uncertainty exists in the system. In [42] and [43], the designed controller did not need to share the same premise variables, membership functions, and fuzzy rules as those of the plant, which enhance the flexibility of controller design. The filter design problem was studied in [44], in which the authors obtained the desired condition to guarantee the error system to be stochastically stable as well as the guaranteed H∞ performance. However, the results on IT2 T–S fuzzy model are related to traditional control systems and few ones are concerned with nonlinear networked systems. Therefore, it is challenging to investigate the filter design problem for a class of nonlinear networked systems subject to parameter uncertainties on the basis of IT2 T–S fuzzy model. This paper focuses on the problem of filter design for the nonlinear networked systems with parameter uncertainties in the framework of IT2 T–S fuzzy model. In the process of modeling the physical plant, the logarithmic quantizer and Bernoulli stochastic process are used to handle data quantization and intermittent data missing, respectively. The main contributions of this paper are summarized as follows. 1) The system considered in this paper is modeled by the IT2 T–S fuzzy model, in which lower and upper membership functions with weighting functions are utilized to capture the parameter uncertainty. 2) The footprint of uncertainty (FOU) [42], [43] is considered in order to take into account more information of uncertainties. 3) The filter design is independent of the premise variables, membership functions, and number of fuzzy rules of the system in order to enhance the flexibility of the filter design, that is, the filter does not need to share the same ones as those of the system. Furthermore, the filter to be designed guarantees the resulting error system preserves the stochastic stability as well as the prescribed H∞ performance. Finally, illustrative examples are provided to illustrate the effectiveness of the method proposed in this paper. The remainder of this paper is organized as follows. Section II describes the considered problem. The main results are presented in Section III. Illustrative examples are provided to verify the usefulness of the proposed method in Section IV. Finally, Section V concludes this paper. A. Notation The notation utilized in this paper is quite standard. The superscript “T” and “−1” denote matrix transposition and matrix inverse, respectively. The identity matrix and zero matrix with compatible dimensions are represented by I and 0, respectively. The notation P > 0 (≥ 0) suggests that P is positive definite (semi-definite) with the real-symmetric structure. The notation A indicates the norm of matrix A defined by A = tr(AT A). l2 [0, ∞) means the space of square-integrable vector functions over [0, ∞); Rn describes the n-dimensional Euclidean space; and . 2 shows the usual l2 [0, ∞) norm. In complex matrix, the symbol (∗) is utilized to represent a symmetric term, and we utilize diag{. . .} to describe the matrix with block diagonal
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Fig. 1.
Structure of networked systems with the quantizer.
structure. Furthermore, E{x|y} and E{x} signify expectation of x conditional on y and expectation of x, respectively. Prob(·) and λmin (A) represent the occurrence probability of the event “·” and the minimum eigenvalue of the matrix A, respectively. Matrices in this paper without dimensions explicitly stated, we assume they have compatible dimensions. II. P ROBLEM F ORMULATION In this section, the discrete-time nonlinear networked system presented in Fig. 1 is considered. As shown in Fig. 1, data packet dropouts and quantization are taken into consideration. In the framework of the IT2 T–S fuzzy model, a r-rule fuzzy model is given as follows. Plant Rule i: IF f1 (x(k)) is Mi1 , and f2 (x(k)) is Mi2 and , . . . , and fθ (x(k)) is Miθ , THEN x(k + 1) = Ai x(k) + Bi w(k) y(k) = Ci x(k) + Di w(k) z(k) = Li x(k)
(1)
where Mij presents the fuzzy set and f (x(k)) = f1 (x(k)), f2 (x(k)), . . . , fθ (x(k)) stands for the premise variable, in which θ is the number of the fuzzy sets. x(k) ∈ Rnx is the state; y(k) ∈ Rny is the measured output; z(k) ∈ Rnz is the controlled output; and w(k) ∈ Rnw is the disturbance input which belongs to l2 [0, ∞). Ai , Bi , Ci , Di , and Li are system matrices with appropriate dimensions. i ∈ 1, 2, . . . , r, the scalar r is the number of IF-THEN rules of the system. The following interval sets present the firing strength of the ith rule: Wi (x(k)) = mi (x(k)), mi (x(k)) where mi (x(k)) =
θ
uMip ( fp (x(k))) ≥ 0
p=1
mi (x(k)) =
θ
uMip ( fp (x(k))) ≥ 0
p=1
uMip ( fp (x(k))) ≥ uMip ( fp (x(k))) ≥ 0 mi (x(k)) ≥ mi (x(k)) ≥ 0
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uMip ( fp (x(k))), uMip ( fp (x(k))), mi (x(k)), and mi (x(k)) denote the lower membership function, upper membership function, lower grade of membership, and upper grade of membership, respectively. The inferred dynamics of the fuzzy system (1) is as follows: x(k + 1) =
r
y(k) = z(k) =
i=1 r
Then, the overall discrete-time T–S fuzzy filter is as follows: xf (k + 1) = zf (k) =
mi (x(k))[Ai x(k) + Bi w(k)]
s j=1 s
ωj (x(k))[Afj xf (k) + Bfj yf (k)] ωj (x(k))Lfj xf (k)
(5)
j=1
i=1 r
3
where mi (x(k))[Ci x(k) + Di w(k)] mi (x(k))Li x(k)
ωj (x(k)) =
ω˜ j (x(k)) = bj (x(k))ωj (x(k)) + bj (x(k))ωj (x(k)) 0 ≤ bj (x(k)) ≤ 1, 0 ≤ bj (x(k)) ≤ 1
i=1 s
mi (x(k)) = ai (x(k))mi (x(k)) + ai (x(k))mi (x(k)) mi (x(k)) ≥ 0 0 ≤ ai (x(k)) ≤ 1, 0 ≤ ai (x(k)) ≤ 1 r mi (x(k)) 1=
(3)
1 = ai (x(k)) + ai (x(k)) with ai (x(k)) and ai (x(k)) being nonlinear weighting functions and mi (x(k)) regarded as the grades of membership. In this paper, the IT2 filter does not share the same membership functions and number of fuzzy rules with the system for enhancing the flexibility of the filter design. The details of an s-rule fuzzy filter are as follows. Filter Rule j: IF g1 (x(k)) is Nj1 , and g2 (x(k)) is Nj2 and , . . . , and g¯ (x(k)) is Nj¯ , THEN
where ωi (x(k)) =
uNjq gq (x(k)) ≥ 0
q=1
ωi (x(k)) =
¯
(6)
bi (x(k)) and bj (x(k)) are nonlinear functions and ωj (x(k)) is regarded as the grades of membership. For brevity, mi mi (x(k)) and ωj ωj (x(k)). Remark 1: Compared with the existing filter design results, the main advantages of filter (4) designed in this paper lie in two sides. One is that the filter is designed in the framework of the IT2 T–S fuzzy model, which can deal with uncertainties. The other is that the membership functions and the number of fuzzy rules are not the same as those of the system model, which, to some degree, can reduce the implementation and computation complexity. Remark 2: It can be seen that the expression for ωj (x(k)) is not the same as that for mi (x(k)). In order to enhance the flexibility of the filter design and obtain less conservative results, this design strategy is adopted.
(4)
where xf (k) ∈ Rnx , zf (k) ∈ Rnz , and Afj , Bfj , Lfj are filter matrices to be determined; s is the number of rules of the filter. The following interval sets describe the firing strength of the jth rule: j (x(k)) = ωj (x(k)), ωj (x(k))
¯
ωj (x(k)) = 1, bj (x(k)) + bj (x(k)) = 1
j=1
i=1
xf (k + 1) = Afj xf (k) + Bfj yf (k) zf (k) = Lfj xf (k)
, ωj (x(k)) ≥ 0
ω˜ j (x(k))
j=1
(2)
where
ω˜ j (x(k))
s
uNjq gq (x(k)) ≥ 0
q=1
uNjq gq (x(k)) ≥ uNjq gq (x(k)) ≥ 0 ωi (x(k)) ≥ ωi (x(k)) ≥ 0 uNjq (gp (x(k))), uNjq (gp (x(k))), ωi (x(k)), and ωi (x(k)) denote lower membership function, upper membership function, lower grade of membership, and upper grade of membership, respectively.
A. Measurement Quantization The quantizer used in the system is as in [45] and the model is denoted as
˜ 2 (y2 (k)) . . . U ˜ ny yny (k) T ˜ ˜ 1 (y1 (k)) U y¯ (k) = U(y(k)) = U where y¯ (k) ∈ Rny is the quantized signal transmitted to the ˜ filter via the network. The quantizer U(y(k)) is the logarithmic ˜ c (·) (1 ≤ c ≤ ny ) type, whose quantization levels for each U are described by
(c) (c) (c) qc = ±χl , χl = κcl χ0 , l = 0, ±1, ±2, . . . ∪ {0} (c)
0 < κc < 1, χ0 > 0. The quantizer maps the whole segments to the quantization level via corresponding each of the quantization level to a ˜ c (·) as segment. Define the logarithmic quantizer U ⎧ (c) (c) (c) 1 1 ⎨ χl , 1+εc χl < yc (k) ≤ 1−εc χl ˜ Uc (yc (k)) = 0, yc (k) = 0 ⎩ ˜ c (−yc (k)), yc (k) < 0 −U with εc = (1 − κc /1 + κc ).
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According to the sector-bound method provided in [46], the quantization errors can be handled as
¯ y¯ (k) = I + (k) y(k) (7) ¯ c (yc (k)) < εc , ¯ ny (k)},
¯ ¯ 1 (k), . . . ,
where (k) = diag{
and c = 1, 2, . . . , ny .
FOU into ς +1 sub-FOUs. For l = 1, 2, . . . , ς +1, expressions of lower and upper membership functions in the lth sub-FOU are rewritten as follows: hijl (x(k)) =
As shown in Fig. 1, the network is used to exchange information in the system such that the intermittent data missing phenomenon arises. The quantized measurement y¯ (k) is hardly transmitted to the filter perfectly [e.g., y¯ (k) = yf (k)]. Therefore, the following model is utilized in this paper in order to describe the stochastic measurement missing: yf (k) = e(k)¯y(k)
(8)
in which e(k) obeys the Bernoulli process and describes the unreliable communication link between the quantizer and filter. Assume e(k) as follows: Prob{e(k) = 1} = E{e(k)} = e¯ Prob{e(k) = 0} = 1 − e¯ .
mi =
i=1
s
ωj =
j=1
q 2 2
υτ iτ zl (xτ (k))ϑ
(11)
...
nx 2
υτ iτ zl (xτ (k))ϑ
inx =1 τ =1
0 ≤ hijl (x(k)) ≤ hijl (x(k)) ≤ 1 ϑ ≤ϑ
(12)
where ϑ and ϑ are constant scalars to be determined respectively; and represent ϑ iji1 i2 ...inx zl and ϑ iji1 i2 ...inx zl , 0 ≤ υτ is zl (xτ (k)) ≤ 1 with the property υτ 1zl (xτ (k)) + υτ 2zl (xτ (k)) = 1 for τ, s = 1, 2, . . . , nx ; l = 1, 2, . . . , ς + 1; x(k) ∈ Hz ; and υτ is zl (xτ (k)) = 0 if otherwise. iτ = 1, 2; q Thus, z=1 2i1 =1 2i2 =1 . . . 2inx =1 nτ x=1 υτ iτ zl (xτ (k)) = 1 for all l, which is used to analyze the system stability. For brevity, hijl hijl (x(k)) and hijl hijl (x(k)). Then, system (10) can be rewritten as follows: x¯ (k + 1) =
r s
nx 2 inx =1 τ =1
z=1 i1 =1 i2 =1
From (3) and (6), we know r
...
z=1 i1 =1 i2 =1
hijl (x(k)) =
B. Communication Links
q 2 2
r s
hij (x(k)) A˜ ij x¯ (k) + B˜ ij w(k)
i=1 j=1
mi ωj = 1.
(9)
z¯(k) =
i=1 j=1
r s
hij (x(k))L¯ ij x¯ (k)
(13)
i=1 j=1
According to (2), (5), (7), and (8), the filtering error system is represented as x¯ (k + 1) = z¯(k) =
r s i=1 j=1 r s
mi ωj A˜ ij x¯ (k) + B˜ ij w(k)
where hij (x(k)) = mi ωj =
ς+1
ρijl (x(k)) ζ ijl (x(k))hijl
l=1
mi ωj L¯ ij x¯ (k)
+ ζ ijl (x(k))hijl
(10)
i=1 j=1
where
1=
r s
hij (x(k))
i=1 j=1
A˜ ij = A1ij + e˜ (k)A2ij , B˜ ij = B1ij + e˜ (k)B2ij 0 Ai A1ij = ¯ Afj e¯ Bfj (I + (k))C i 0 0 A2ij = ¯ 0 Bfj (I + (k))C i Bi B1ij = ¯ e¯ Bfj (I + (k))D i 0 B2ij = ¯ Bfj (I + (k))D i x(k) L¯ ij = [Li −Lfj ], x¯ (k) = xf (k) z¯(k) = z(k) − zf (k), e˜ (k) = e(k) − e¯ . It is obvious that E{˜e(k)} = 0, E{˜e(k)˜e(k)} = e¯ (1 − e¯ ). To analyze the system (10), we refer to [42]. The state space of interest H is divided into q connected sub-state spaces q Hz (z = 1, 2, . . . , q) with H = ∪z=1 Hz . Then, we divide the
0 ≤ ζ ijl (x(k)) ≤ ζ ijl (x(k)) ≤ 1 1, hijl (x(k)) ∈ the sub-FOU l ρijl (x(k)) = 0, else where ζ ijl (x(k)) and ζ ijl (x(k)) are two functions and unneces-
sary to be known. ζ ijl (x(k)) + ζ ijl (x(k)) = 1 for i, j, and l. The following definition is used to obtain the main result in this paper. For brevity, hij hij (x(k)), ζ ijl ζ ijl (x(k)),
ζ ijl ζ ijl (x(k)), and ρijl ρijl (x(k)). Definition 1 [35]: The filtering error system in (13) is stochastically stable in the mean square when w(k) ≡ 0 for any initial condition x¯ (0) if there is a finite W > 0 such that ∞ 2 |¯x(k)| |x¯ (0) < x¯ (0)T W x¯ (0). E k=0
The purpose of this paper is to design a filter on the basis of IT2 framework such that the following conditions are satisfied simultaneously. 1) The error system in (13) is stochastically stable.
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2) Under zero initial condition, with a given positive scalar γ , the error output z¯(k) satisfies ⎫ ⎧ ∞ ⎬ ⎨ E |¯z(k)|2 ≤ γ w2 . ⎭ ⎩
5
where ˜ ij = A˜ ij B˜ ij ˇ ijl A¯ T P¯ B¯ ij Q ij Qijl = . ∗ B¯ Tij P¯ B¯ ij
k=0
Remark 3: The parameter uncertainties can lead to the state-variable uncertainties, which results in a less accurate model when we model the physical plants. They are fully taken into consideration in the framework of IT2 T–S fuzzy model, in which the membership function is an interval rather than a specific one. The membership functions are determined by lower and upper functions with weighting functions, which makes the parameter uncertainties concluded in the membership functions. III. M AIN R ESULTS In this section, for given filter gain matrices Afj , Bfj , Lfj ( j = 1, 2, . . . , s), the sufficient condition is provided which ensures error system (13) is stochastically stable with a predefined H∞ performance. Theorem 1: With the FOU and state space divided into ς +1 sub-FOUs and q connected sub-state spaces, for the given filter gain matrices Afj , Bfj , Lfj (j = 1, 2, . . . , s), and positive scalar γ , the system in (13) is stochastically stable and satisfies a given H∞ performance, if there exist symmetric matrices P > 0, Wijl > 0 (i = 1, 2, . . . , r, j = 1, 2, . . . , s, l = 1, 2, . . . , ς + 1) and M with appropriate dimensions such that the following inequalities hold: r s ˇ − M < 0 (14)
To achieve the sufficient condition, the following slack matrices are introduced: ⎫ ⎧ r s ς+1 ⎬ ⎨
ρijl 1 − ζ¯ijl hijl + ζ¯ijl hijl − 1 M = 0 (17) ⎭ ⎩ i=1 j=1 l=1
−
(15)
!
ρijl 1 − ζ¯ijl hijl − hijl Wijl ≥ 0. (18)
i=1 j=1 l=1
Substituting (17) and (18) into (16), it can be found that ⎧ r s ς+1 ⎨ ! ρijl hijl Qijl − hijl − hijl Wijl
V(k) ≤ χ T (k) ⎩ i=1 j=1 l=1 ⎫ ⎬ + hijl M − M χ (k) ⎭ − χ T (k)
r s ς+1
ρijl ζ¯ijl hijl − hijl
!
i=1 j=1 l=1
× Qijl − Wijl + M χ (k).
(19)
When w(k) = 0, it can be seen from (14) and (15) that < 0, where
i=1 j=1
¯ ijl − Wijl + M < 0 Q
r s ς+1
=
r s ς+1 i=1 j=1 l=1
where ˇ = ˇ ijl = Q ¯ ijl = Q A¯ ij = B¯ ij =
! ˇ ijl − hijl − hijl W1ijl + hijl M1 − M1 × ρijl hijl Q
¯ ijl − ϑ − ϑ Wijl + ϑM ϑQ A¯ Tij P¯ A¯ ij − P ˇ ijl + L¯ T L¯ ij Q A¯ Tij P¯ B¯ ij ij ∗ −γ 2 I + B¯ Tij P¯ B¯ ij T AT1ij fAT2ij , f = e¯ (1 − e¯ ) T BT1ij fBT2ij , P¯ = diag{P, P}.
−
ˇ ijl − W1ijl + M1 × Q where
V(k) = x¯ T (k)P¯x(k) where P > 0 is the matrix to be determined. Let χ T (k) = [¯xT (k) wT (k)]. Then, the difference is computed as
V(k) = E{V(k ⎧ + 1)|χ (k)} − V(k) ⎫ r s ⎨ ⎬ ˜ ij χ (k) ˜ Tij P ≤ E χ T (k) hij ⎩ ⎭ i=1 j=1
T
i=1 j=1
ρijl ζ¯ijl hijl − hijl
i=1 j=1 l=1
Proof: Consider the following Lyapunov function for system (13):
− x¯ (k)P¯x(k) r s T = χ (k) hij Qijl χ (k)
r s ς+1
(16)
Wijl =
W1ijl ∗
!
!
W2ijl M1 , M= W3ijl ∗
M2 . M3
Then, the following inequality holds: " T # E x¯ (k + 1)P¯x(k + 1) − x¯ T (k)P¯x(k) ≤ −λmin (−)¯xT (k)¯x(k). Summing the mathematical expectation for both sides of the inequality from k = 0, 1, . . . , d with any d ≥ 1, one can obtain # " E x¯ T (d + 1)P¯x(d + 1) − x¯ T (0)P¯x(0) d 2 |¯x(k)| ≤ −λmin (−)E k=0
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which yields d 2 |¯x(k)| ≤ (λmin (−))−1 x¯ T (0)P¯x(0) E k=0
with the initial " condition x¯#(0). When d = 1, 2, . . . , ∞, considering E x¯ T (∞)P¯x(∞) ≥ 0, we have d 2 |¯x(k)| ≤ x¯ T (0)W x¯ (0) E k=0
where W (λmin (−))−1 P, which means W > 0. From Definition 1, the error system in (13) is stochastically stable. Next, consider the H∞ performance of the error system in (13). Under the zero initial condition, the H∞ performance index is J E{V(k + 1)|χ (k)} − V(k) " # + E z¯T (k)¯z(k)|χ (k) − γ 2 wT (k)w(k). Then, we obtain " # J = E z¯T (k)¯z(k) − γ 2 wT (k)w(k) + V(k) ⎧ r s ς+1 ⎨ ! ¯ ijl − hijl − hijl Wijl ρijl hijl Q ≤ χ T (k) ⎩ i=1 j=1 l=1 ⎫ ⎬ + hijl M − M χ (k) ⎭ − χ (k) T
r s ς+1
ρijl ζ¯ijl hijl − hijl
where
!
i=1 j=1 l=1
¯ ijl − Wijl + M χ (k). × Q
(20)
Substituting (11) and (12) into (20), we have J ≤ χ T (k)
r s ς+1
ρijl
i=1 j=1 l=1 q 2 2
...
z=1 i1 =1 i2 =1
nx 2
ˇ (k) υτ iτ zl (xτ (k))χ
inx =1 τ =1
− χ T (k)Mχ (k) − χ T (k)
r s ς+1
ρijl ζ¯ijl
i=1 j=1 l=1
×
q 2 2 z=1 i1 =1 i2 =1
...
nx 2
υτ iτ zl (xτ (k)) ϑ − ϑ
inx =1 τ =1
¯ ijl − Wijl + M χ (k). × Q
the guaranteed H∞ performance. However, due to the exis¯ tence of quantization effect (k), it has difficulty on checking the effectiveness of Theorem 1. Then, the quantization effect can be eliminated. Next, we will show the new sufficient condition. Theorem 2: For the given filter gain matrices Afj , Bfj , and ˜ j (·) and positive scalar γ , the system in Lfj , the quantizer U (13) is stochastically stable and satisfies a given H∞ performance, if there exists a scalar ε > 0 and symmetric matrices ˜ 2ijl , W ˜ 3ijl > 0 (i = 1, 2, . . . , r, ˜ 1ijl > 0, W P > 0, W ˜ 1 –M ˜ 3 with approj = 1, 2, . . . , s, l = 1, 2, . . . , ς + 1), and M priate dimensions such that the following inequalities hold: ⎤ ⎡ ¯1 ¯ 3 ε ¯T 2 ⎣ ∗ (22) −εI 0 ⎦
1
Filtering of Interval Type-2 Fuzzy Systems With Intermittent Measurements Hongyi Li, Chengwei Wu, Ligang Wu, Senior Member, IEEE, Hak-Keung Lam, Senior Member, IEEE, and Yabin Gao
Abstract—In this paper, the problem of fuzzy filter design is investigated for a class of nonlinear networked systems on the basis of the interval type-2 (IT2) fuzzy set theory. In the design process, two vital factors, intermittent data packet dropouts and quantization, are taken into consideration. The parameter uncertainties are handled effectively by the IT2 membership functions determined by lower and upper membership functions and relative weighting functions. A novel fuzzy filter is designed to guarantee the error system to be stochastically stable with H∞ performance. Moreover, the filter does not need to share the same membership functions and number of fuzzy rules as those of the plant. Finally, illustrative examples are provided to illustrate the effectiveness of the method proposed in this paper. Index Terms—Data packet dropouts, filter design, interval type-2 (IT2) fuzzy systems, nonlinear networked system.
I. I NTRODUCTION N MODERN industrial systems, the physical plants, sensors, filters, and actuators are usually located in different geographical places, which require the signal to be transmitted from one place to another. The traditional communication via point-to-point cables cannot satisfy the requirement. Thus, the network media is introduced into the system to connect these components, which gives rise to the networked systems [1], [2]. Networked systems have brought obvious advantages, such as low cost, simple installation and maintenance, and high reliability [3]. Antsaklis and Baillieul [4] have highlighted that
I
Manuscript received September 26, 2014; revised December 18, 2014 and February 27, 2015; accepted March 3, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61203002, Grant 61174126, Grant 61222301, and Grant 61333012, in part by the Program for New Century Excellent Talents in University under Grant NCET-13-0696, in part by the Chinese National Post-Doctoral Science Foundation under Grant 2014M551111, in part by the Post-Doctoral Science Foundation of Northeastern University, in part by the Key Laboratory of Integrated Automation for the Process Industry (Northeast University), and in part by the Fundamental Research Funds for the Central Universities under Grant HIT.BRETIV.201303. This paper was recommended by Associate Editor P. Shi. H. Li is with the College of Engineering, Bohai University, Jinzhou 121013, China (e-mail: [email protected]). C. Wu and Y. Gao are with the College of Information Science and Technology, Bohai University, Jinzhou 121013, China (e-mail: [email protected]; [email protected]). L. Wu is with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]). H.-K. Lam is with the Department of Informatics, King’s College London, London WC2R 2LC, U.K (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2413134
increasing attention has been received in modeling, analysis, and synthesis of networked systems. Recently, some stability and stabilization results concerning networked systems were reported in [5]–[11] and the filtering problem was studied in [12]–[15]. For unavailable system states, the filter and observer play a vital role in estimating them. Zhang et al. [16] proposed a filtering design method to estimate state variables. In [17]–[20], the observer was used to handle the unmeasurable systems states with different systems requirements. However, nonlinearities often exist in the complex plants [21]–[23], which makes it difficult to design them in nonlinear systems. The Takagi–Sugeno (T–S) fuzzy-model-based approach [24] can cope with nonlinearities existing in the application systems, which “blends” every local linear system through “IF-THEN” rules [25]–[33]. In networked environment, considerable filter design results with the T–S fuzzy model have been reported. The H∞ filtering problem for a new class of discrete-time nonlinear networked systems with mixed random delays and packet dropouts was investigated in [34]. Dong et al. [25] studied the distributed finite-horizon filtering problem for a class of time-varying systems over lossy sensor networks. The problem of H∞ filtering for a class of nonlinear discrete-time systems with measurement quantization and packet dropouts was investigated in [16]. In [35], the filter design problem for fault detection with missing measurement was researched. Zhang and Wang [14] proposed a novel approach for the filtering problem of T–S fuzzy systems in a network environment. Both delays of the premise variables and measurements were considered. However, it is worth noticing that the aforementioned results concerning networked systems were achieved through the T–S fuzzy model-based approach on the basis of the type-1 fuzzy set theory, which can perfectly deal with nonlinearities. There also exist parameter uncertainties in the application systems, which cannot be handled by the type-1 fuzzy set theory effectively. And the accuracy of system modeling will be degraded if parameter uncertainties are not fully taken into account. Mendel et al. [36] proposed an interval type-2 (IT2) fuzzy model to model the physical plants with parameter uncertainties. In view of its merits, researchers have been inspired to investigate it and many results concerning IT2 fuzzy systems were reported in [37]–[40]. Lam and Seneviratne [41] provided an IT2 T–S fuzzy model approach to model the plant subject to parameter uncertainties and achieved the stability condition. Additionally, it has been proved that the IT2 T–S fuzzy model performs better than the type-1 one when the
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parameter uncertainty exists in the system. In [42] and [43], the designed controller did not need to share the same premise variables, membership functions, and fuzzy rules as those of the plant, which enhance the flexibility of controller design. The filter design problem was studied in [44], in which the authors obtained the desired condition to guarantee the error system to be stochastically stable as well as the guaranteed H∞ performance. However, the results on IT2 T–S fuzzy model are related to traditional control systems and few ones are concerned with nonlinear networked systems. Therefore, it is challenging to investigate the filter design problem for a class of nonlinear networked systems subject to parameter uncertainties on the basis of IT2 T–S fuzzy model. This paper focuses on the problem of filter design for the nonlinear networked systems with parameter uncertainties in the framework of IT2 T–S fuzzy model. In the process of modeling the physical plant, the logarithmic quantizer and Bernoulli stochastic process are used to handle data quantization and intermittent data missing, respectively. The main contributions of this paper are summarized as follows. 1) The system considered in this paper is modeled by the IT2 T–S fuzzy model, in which lower and upper membership functions with weighting functions are utilized to capture the parameter uncertainty. 2) The footprint of uncertainty (FOU) [42], [43] is considered in order to take into account more information of uncertainties. 3) The filter design is independent of the premise variables, membership functions, and number of fuzzy rules of the system in order to enhance the flexibility of the filter design, that is, the filter does not need to share the same ones as those of the system. Furthermore, the filter to be designed guarantees the resulting error system preserves the stochastic stability as well as the prescribed H∞ performance. Finally, illustrative examples are provided to illustrate the effectiveness of the method proposed in this paper. The remainder of this paper is organized as follows. Section II describes the considered problem. The main results are presented in Section III. Illustrative examples are provided to verify the usefulness of the proposed method in Section IV. Finally, Section V concludes this paper. A. Notation The notation utilized in this paper is quite standard. The superscript “T” and “−1” denote matrix transposition and matrix inverse, respectively. The identity matrix and zero matrix with compatible dimensions are represented by I and 0, respectively. The notation P > 0 (≥ 0) suggests that P is positive definite (semi-definite) with the real-symmetric structure. The notation A indicates the norm of matrix A defined by A = tr(AT A). l2 [0, ∞) means the space of square-integrable vector functions over [0, ∞); Rn describes the n-dimensional Euclidean space; and . 2 shows the usual l2 [0, ∞) norm. In complex matrix, the symbol (∗) is utilized to represent a symmetric term, and we utilize diag{. . .} to describe the matrix with block diagonal
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Fig. 1.
Structure of networked systems with the quantizer.
structure. Furthermore, E{x|y} and E{x} signify expectation of x conditional on y and expectation of x, respectively. Prob(·) and λmin (A) represent the occurrence probability of the event “·” and the minimum eigenvalue of the matrix A, respectively. Matrices in this paper without dimensions explicitly stated, we assume they have compatible dimensions. II. P ROBLEM F ORMULATION In this section, the discrete-time nonlinear networked system presented in Fig. 1 is considered. As shown in Fig. 1, data packet dropouts and quantization are taken into consideration. In the framework of the IT2 T–S fuzzy model, a r-rule fuzzy model is given as follows. Plant Rule i: IF f1 (x(k)) is Mi1 , and f2 (x(k)) is Mi2 and , . . . , and fθ (x(k)) is Miθ , THEN x(k + 1) = Ai x(k) + Bi w(k) y(k) = Ci x(k) + Di w(k) z(k) = Li x(k)
(1)
where Mij presents the fuzzy set and f (x(k)) = f1 (x(k)), f2 (x(k)), . . . , fθ (x(k)) stands for the premise variable, in which θ is the number of the fuzzy sets. x(k) ∈ Rnx is the state; y(k) ∈ Rny is the measured output; z(k) ∈ Rnz is the controlled output; and w(k) ∈ Rnw is the disturbance input which belongs to l2 [0, ∞). Ai , Bi , Ci , Di , and Li are system matrices with appropriate dimensions. i ∈ 1, 2, . . . , r, the scalar r is the number of IF-THEN rules of the system. The following interval sets present the firing strength of the ith rule: Wi (x(k)) = mi (x(k)), mi (x(k)) where mi (x(k)) =
θ
uMip ( fp (x(k))) ≥ 0
p=1
mi (x(k)) =
θ
uMip ( fp (x(k))) ≥ 0
p=1
uMip ( fp (x(k))) ≥ uMip ( fp (x(k))) ≥ 0 mi (x(k)) ≥ mi (x(k)) ≥ 0
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uMip ( fp (x(k))), uMip ( fp (x(k))), mi (x(k)), and mi (x(k)) denote the lower membership function, upper membership function, lower grade of membership, and upper grade of membership, respectively. The inferred dynamics of the fuzzy system (1) is as follows: x(k + 1) =
r
y(k) = z(k) =
i=1 r
Then, the overall discrete-time T–S fuzzy filter is as follows: xf (k + 1) = zf (k) =
mi (x(k))[Ai x(k) + Bi w(k)]
s j=1 s
ωj (x(k))[Afj xf (k) + Bfj yf (k)] ωj (x(k))Lfj xf (k)
(5)
j=1
i=1 r
3
where mi (x(k))[Ci x(k) + Di w(k)] mi (x(k))Li x(k)
ωj (x(k)) =
ω˜ j (x(k)) = bj (x(k))ωj (x(k)) + bj (x(k))ωj (x(k)) 0 ≤ bj (x(k)) ≤ 1, 0 ≤ bj (x(k)) ≤ 1
i=1 s
mi (x(k)) = ai (x(k))mi (x(k)) + ai (x(k))mi (x(k)) mi (x(k)) ≥ 0 0 ≤ ai (x(k)) ≤ 1, 0 ≤ ai (x(k)) ≤ 1 r mi (x(k)) 1=
(3)
1 = ai (x(k)) + ai (x(k)) with ai (x(k)) and ai (x(k)) being nonlinear weighting functions and mi (x(k)) regarded as the grades of membership. In this paper, the IT2 filter does not share the same membership functions and number of fuzzy rules with the system for enhancing the flexibility of the filter design. The details of an s-rule fuzzy filter are as follows. Filter Rule j: IF g1 (x(k)) is Nj1 , and g2 (x(k)) is Nj2 and , . . . , and g¯ (x(k)) is Nj¯ , THEN
where ωi (x(k)) =
uNjq gq (x(k)) ≥ 0
q=1
ωi (x(k)) =
¯
(6)
bi (x(k)) and bj (x(k)) are nonlinear functions and ωj (x(k)) is regarded as the grades of membership. For brevity, mi mi (x(k)) and ωj ωj (x(k)). Remark 1: Compared with the existing filter design results, the main advantages of filter (4) designed in this paper lie in two sides. One is that the filter is designed in the framework of the IT2 T–S fuzzy model, which can deal with uncertainties. The other is that the membership functions and the number of fuzzy rules are not the same as those of the system model, which, to some degree, can reduce the implementation and computation complexity. Remark 2: It can be seen that the expression for ωj (x(k)) is not the same as that for mi (x(k)). In order to enhance the flexibility of the filter design and obtain less conservative results, this design strategy is adopted.
(4)
where xf (k) ∈ Rnx , zf (k) ∈ Rnz , and Afj , Bfj , Lfj are filter matrices to be determined; s is the number of rules of the filter. The following interval sets describe the firing strength of the jth rule: j (x(k)) = ωj (x(k)), ωj (x(k))
¯
ωj (x(k)) = 1, bj (x(k)) + bj (x(k)) = 1
j=1
i=1
xf (k + 1) = Afj xf (k) + Bfj yf (k) zf (k) = Lfj xf (k)
, ωj (x(k)) ≥ 0
ω˜ j (x(k))
j=1
(2)
where
ω˜ j (x(k))
s
uNjq gq (x(k)) ≥ 0
q=1
uNjq gq (x(k)) ≥ uNjq gq (x(k)) ≥ 0 ωi (x(k)) ≥ ωi (x(k)) ≥ 0 uNjq (gp (x(k))), uNjq (gp (x(k))), ωi (x(k)), and ωi (x(k)) denote lower membership function, upper membership function, lower grade of membership, and upper grade of membership, respectively.
A. Measurement Quantization The quantizer used in the system is as in [45] and the model is denoted as
˜ 2 (y2 (k)) . . . U ˜ ny yny (k) T ˜ ˜ 1 (y1 (k)) U y¯ (k) = U(y(k)) = U where y¯ (k) ∈ Rny is the quantized signal transmitted to the ˜ filter via the network. The quantizer U(y(k)) is the logarithmic ˜ c (·) (1 ≤ c ≤ ny ) type, whose quantization levels for each U are described by
(c) (c) (c) qc = ±χl , χl = κcl χ0 , l = 0, ±1, ±2, . . . ∪ {0} (c)
0 < κc < 1, χ0 > 0. The quantizer maps the whole segments to the quantization level via corresponding each of the quantization level to a ˜ c (·) as segment. Define the logarithmic quantizer U ⎧ (c) (c) (c) 1 1 ⎨ χl , 1+εc χl < yc (k) ≤ 1−εc χl ˜ Uc (yc (k)) = 0, yc (k) = 0 ⎩ ˜ c (−yc (k)), yc (k) < 0 −U with εc = (1 − κc /1 + κc ).
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According to the sector-bound method provided in [46], the quantization errors can be handled as
¯ y¯ (k) = I + (k) y(k) (7) ¯ c (yc (k)) < εc , ¯ ny (k)},
¯ ¯ 1 (k), . . . ,
where (k) = diag{
and c = 1, 2, . . . , ny .
FOU into ς +1 sub-FOUs. For l = 1, 2, . . . , ς +1, expressions of lower and upper membership functions in the lth sub-FOU are rewritten as follows: hijl (x(k)) =
As shown in Fig. 1, the network is used to exchange information in the system such that the intermittent data missing phenomenon arises. The quantized measurement y¯ (k) is hardly transmitted to the filter perfectly [e.g., y¯ (k) = yf (k)]. Therefore, the following model is utilized in this paper in order to describe the stochastic measurement missing: yf (k) = e(k)¯y(k)
(8)
in which e(k) obeys the Bernoulli process and describes the unreliable communication link between the quantizer and filter. Assume e(k) as follows: Prob{e(k) = 1} = E{e(k)} = e¯ Prob{e(k) = 0} = 1 − e¯ .
mi =
i=1
s
ωj =
j=1
q 2 2
υτ iτ zl (xτ (k))ϑ
(11)
...
nx 2
υτ iτ zl (xτ (k))ϑ
inx =1 τ =1
0 ≤ hijl (x(k)) ≤ hijl (x(k)) ≤ 1 ϑ ≤ϑ
(12)
where ϑ and ϑ are constant scalars to be determined respectively; and represent ϑ iji1 i2 ...inx zl and ϑ iji1 i2 ...inx zl , 0 ≤ υτ is zl (xτ (k)) ≤ 1 with the property υτ 1zl (xτ (k)) + υτ 2zl (xτ (k)) = 1 for τ, s = 1, 2, . . . , nx ; l = 1, 2, . . . , ς + 1; x(k) ∈ Hz ; and υτ is zl (xτ (k)) = 0 if otherwise. iτ = 1, 2; q Thus, z=1 2i1 =1 2i2 =1 . . . 2inx =1 nτ x=1 υτ iτ zl (xτ (k)) = 1 for all l, which is used to analyze the system stability. For brevity, hijl hijl (x(k)) and hijl hijl (x(k)). Then, system (10) can be rewritten as follows: x¯ (k + 1) =
r s
nx 2 inx =1 τ =1
z=1 i1 =1 i2 =1
From (3) and (6), we know r
...
z=1 i1 =1 i2 =1
hijl (x(k)) =
B. Communication Links
q 2 2
r s
hij (x(k)) A˜ ij x¯ (k) + B˜ ij w(k)
i=1 j=1
mi ωj = 1.
(9)
z¯(k) =
i=1 j=1
r s
hij (x(k))L¯ ij x¯ (k)
(13)
i=1 j=1
According to (2), (5), (7), and (8), the filtering error system is represented as x¯ (k + 1) = z¯(k) =
r s i=1 j=1 r s
mi ωj A˜ ij x¯ (k) + B˜ ij w(k)
where hij (x(k)) = mi ωj =
ς+1
ρijl (x(k)) ζ ijl (x(k))hijl
l=1
mi ωj L¯ ij x¯ (k)
+ ζ ijl (x(k))hijl
(10)
i=1 j=1
where
1=
r s
hij (x(k))
i=1 j=1
A˜ ij = A1ij + e˜ (k)A2ij , B˜ ij = B1ij + e˜ (k)B2ij 0 Ai A1ij = ¯ Afj e¯ Bfj (I + (k))C i 0 0 A2ij = ¯ 0 Bfj (I + (k))C i Bi B1ij = ¯ e¯ Bfj (I + (k))D i 0 B2ij = ¯ Bfj (I + (k))D i x(k) L¯ ij = [Li −Lfj ], x¯ (k) = xf (k) z¯(k) = z(k) − zf (k), e˜ (k) = e(k) − e¯ . It is obvious that E{˜e(k)} = 0, E{˜e(k)˜e(k)} = e¯ (1 − e¯ ). To analyze the system (10), we refer to [42]. The state space of interest H is divided into q connected sub-state spaces q Hz (z = 1, 2, . . . , q) with H = ∪z=1 Hz . Then, we divide the
0 ≤ ζ ijl (x(k)) ≤ ζ ijl (x(k)) ≤ 1 1, hijl (x(k)) ∈ the sub-FOU l ρijl (x(k)) = 0, else where ζ ijl (x(k)) and ζ ijl (x(k)) are two functions and unneces-
sary to be known. ζ ijl (x(k)) + ζ ijl (x(k)) = 1 for i, j, and l. The following definition is used to obtain the main result in this paper. For brevity, hij hij (x(k)), ζ ijl ζ ijl (x(k)),
ζ ijl ζ ijl (x(k)), and ρijl ρijl (x(k)). Definition 1 [35]: The filtering error system in (13) is stochastically stable in the mean square when w(k) ≡ 0 for any initial condition x¯ (0) if there is a finite W > 0 such that ∞ 2 |¯x(k)| |x¯ (0) < x¯ (0)T W x¯ (0). E k=0
The purpose of this paper is to design a filter on the basis of IT2 framework such that the following conditions are satisfied simultaneously. 1) The error system in (13) is stochastically stable.
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2) Under zero initial condition, with a given positive scalar γ , the error output z¯(k) satisfies ⎫ ⎧ ∞ ⎬ ⎨ E |¯z(k)|2 ≤ γ w2 . ⎭ ⎩
5
where ˜ ij = A˜ ij B˜ ij ˇ ijl A¯ T P¯ B¯ ij Q ij Qijl = . ∗ B¯ Tij P¯ B¯ ij
k=0
Remark 3: The parameter uncertainties can lead to the state-variable uncertainties, which results in a less accurate model when we model the physical plants. They are fully taken into consideration in the framework of IT2 T–S fuzzy model, in which the membership function is an interval rather than a specific one. The membership functions are determined by lower and upper functions with weighting functions, which makes the parameter uncertainties concluded in the membership functions. III. M AIN R ESULTS In this section, for given filter gain matrices Afj , Bfj , Lfj ( j = 1, 2, . . . , s), the sufficient condition is provided which ensures error system (13) is stochastically stable with a predefined H∞ performance. Theorem 1: With the FOU and state space divided into ς +1 sub-FOUs and q connected sub-state spaces, for the given filter gain matrices Afj , Bfj , Lfj (j = 1, 2, . . . , s), and positive scalar γ , the system in (13) is stochastically stable and satisfies a given H∞ performance, if there exist symmetric matrices P > 0, Wijl > 0 (i = 1, 2, . . . , r, j = 1, 2, . . . , s, l = 1, 2, . . . , ς + 1) and M with appropriate dimensions such that the following inequalities hold: r s ˇ − M < 0 (14)
To achieve the sufficient condition, the following slack matrices are introduced: ⎫ ⎧ r s ς+1 ⎬ ⎨
ρijl 1 − ζ¯ijl hijl + ζ¯ijl hijl − 1 M = 0 (17) ⎭ ⎩ i=1 j=1 l=1
−
(15)
!
ρijl 1 − ζ¯ijl hijl − hijl Wijl ≥ 0. (18)
i=1 j=1 l=1
Substituting (17) and (18) into (16), it can be found that ⎧ r s ς+1 ⎨ ! ρijl hijl Qijl − hijl − hijl Wijl
V(k) ≤ χ T (k) ⎩ i=1 j=1 l=1 ⎫ ⎬ + hijl M − M χ (k) ⎭ − χ T (k)
r s ς+1
ρijl ζ¯ijl hijl − hijl
!
i=1 j=1 l=1
× Qijl − Wijl + M χ (k).
(19)
When w(k) = 0, it can be seen from (14) and (15) that < 0, where
i=1 j=1
¯ ijl − Wijl + M < 0 Q
r s ς+1
=
r s ς+1 i=1 j=1 l=1
where ˇ = ˇ ijl = Q ¯ ijl = Q A¯ ij = B¯ ij =
! ˇ ijl − hijl − hijl W1ijl + hijl M1 − M1 × ρijl hijl Q
¯ ijl − ϑ − ϑ Wijl + ϑM ϑQ A¯ Tij P¯ A¯ ij − P ˇ ijl + L¯ T L¯ ij Q A¯ Tij P¯ B¯ ij ij ∗ −γ 2 I + B¯ Tij P¯ B¯ ij T AT1ij fAT2ij , f = e¯ (1 − e¯ ) T BT1ij fBT2ij , P¯ = diag{P, P}.
−
ˇ ijl − W1ijl + M1 × Q where
V(k) = x¯ T (k)P¯x(k) where P > 0 is the matrix to be determined. Let χ T (k) = [¯xT (k) wT (k)]. Then, the difference is computed as
V(k) = E{V(k ⎧ + 1)|χ (k)} − V(k) ⎫ r s ⎨ ⎬ ˜ ij χ (k) ˜ Tij P ≤ E χ T (k) hij ⎩ ⎭ i=1 j=1
T
i=1 j=1
ρijl ζ¯ijl hijl − hijl
i=1 j=1 l=1
Proof: Consider the following Lyapunov function for system (13):
− x¯ (k)P¯x(k) r s T = χ (k) hij Qijl χ (k)
r s ς+1
(16)
Wijl =
W1ijl ∗
!
!
W2ijl M1 , M= W3ijl ∗
M2 . M3
Then, the following inequality holds: " T # E x¯ (k + 1)P¯x(k + 1) − x¯ T (k)P¯x(k) ≤ −λmin (−)¯xT (k)¯x(k). Summing the mathematical expectation for both sides of the inequality from k = 0, 1, . . . , d with any d ≥ 1, one can obtain # " E x¯ T (d + 1)P¯x(d + 1) − x¯ T (0)P¯x(0) d 2 |¯x(k)| ≤ −λmin (−)E k=0
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which yields d 2 |¯x(k)| ≤ (λmin (−))−1 x¯ T (0)P¯x(0) E k=0
with the initial " condition x¯#(0). When d = 1, 2, . . . , ∞, considering E x¯ T (∞)P¯x(∞) ≥ 0, we have d 2 |¯x(k)| ≤ x¯ T (0)W x¯ (0) E k=0
where W (λmin (−))−1 P, which means W > 0. From Definition 1, the error system in (13) is stochastically stable. Next, consider the H∞ performance of the error system in (13). Under the zero initial condition, the H∞ performance index is J E{V(k + 1)|χ (k)} − V(k) " # + E z¯T (k)¯z(k)|χ (k) − γ 2 wT (k)w(k). Then, we obtain " # J = E z¯T (k)¯z(k) − γ 2 wT (k)w(k) + V(k) ⎧ r s ς+1 ⎨ ! ¯ ijl − hijl − hijl Wijl ρijl hijl Q ≤ χ T (k) ⎩ i=1 j=1 l=1 ⎫ ⎬ + hijl M − M χ (k) ⎭ − χ (k) T
r s ς+1
ρijl ζ¯ijl hijl − hijl
where
!
i=1 j=1 l=1
¯ ijl − Wijl + M χ (k). × Q
(20)
Substituting (11) and (12) into (20), we have J ≤ χ T (k)
r s ς+1
ρijl
i=1 j=1 l=1 q 2 2
...
z=1 i1 =1 i2 =1
nx 2
ˇ (k) υτ iτ zl (xτ (k))χ
inx =1 τ =1
− χ T (k)Mχ (k) − χ T (k)
r s ς+1
ρijl ζ¯ijl
i=1 j=1 l=1
×
q 2 2 z=1 i1 =1 i2 =1
...
nx 2
υτ iτ zl (xτ (k)) ϑ − ϑ
inx =1 τ =1
¯ ijl − Wijl + M χ (k). × Q
the guaranteed H∞ performance. However, due to the exis¯ tence of quantization effect (k), it has difficulty on checking the effectiveness of Theorem 1. Then, the quantization effect can be eliminated. Next, we will show the new sufficient condition. Theorem 2: For the given filter gain matrices Afj , Bfj , and ˜ j (·) and positive scalar γ , the system in Lfj , the quantizer U (13) is stochastically stable and satisfies a given H∞ performance, if there exists a scalar ε > 0 and symmetric matrices ˜ 2ijl , W ˜ 3ijl > 0 (i = 1, 2, . . . , r, ˜ 1ijl > 0, W P > 0, W ˜ 1 –M ˜ 3 with approj = 1, 2, . . . , s, l = 1, 2, . . . , ς + 1), and M priate dimensions such that the following inequalities hold: ⎤ ⎡ ¯1 ¯ 3 ε ¯T 2 ⎣ ∗ (22) −εI 0 ⎦
