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Robust Filtering for a Class of Networked Nonlinear Systems With Switching Communication Channels Lixian Zhang, Senior Member, IEEE, Xunyuan Yin, Zepeng Ning, and Dong Ye
Abstract—This paper is concerned with the problem of robust filter design for a class of discrete-time networked nonlinear systems. The Takagi–Sugeno fuzzy model is employed to represent the underlying nonlinear dynamics. A multi-channel communication scheme that involves a channel switching phenomenon described by a Markov chain is proposed for data transmission. Two typical communication imperfections, network-induced time-varying delays and packet dropouts are considered in each channel. The objective of this paper is to design an admissible filter such that the filter error system is stochastically stable and ensures a prescribed disturbance attenuation level bound. Based on the Lyapunov–Krasovskii functional method and matrix inequality techniques, sufficient conditions on the existence of the desired filter are obtained. A numerical example is provided to illustrate the effectiveness of the proposed design approach. Index Terms—Channel switching, multi-channel communication, network control systems, robust filter design, Takagi–Sugeno (T–S) fuzzy systems.
I. I NTRODUCTION VER the past several decades, study on networked control systems (NCSs) has gradually become an appealing research frontier due to many inherent advantages, such as low cost, low weight, flexible structures, simple implementation, and maintenance requirements. So far, we have witnessed extensive applications in modern industrial communities [1]. However, since sensors, controllers, and actuators in typical NCSs are connected via communication links which are not constantly reliable, network-based systems are frequently accompanied by information-transmission delays and intermittent packet dropouts [1], [2], which inevitably lead to deteriorated performance or even instability of the systems. As a result, it is necessary to take into account the negative influences brought by stochastic time delays as well as packet dropouts when performing analysis and design
O
Manuscript received August 30, 2015; revised December 11, 2015; accepted January 21, 2016. This paper was recommended by Associate Editor C. Hua. L. Zhang and Z. Ning are with the School of Astronautics, Harbin Institute of Technology, Harbin 150080, China (e-mail: [email protected]; [email protected]). X. Yin is with the School of Astronautics, Harbin Institute of Technology, Harbin 150080, China, and also with the Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB T6G2V4, Canada (e-mail: [email protected]). D. Ye is with the Research Centre of Satellite Technology, Harbin Institute of Technology, Harbin 150080, China (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.2016.2523811
for NCSs. Owing to the dedicated work from forward-looking researchers, fruitful results focusing on time-delay-induced effects on NCSs have been obtained. To name some representative studies, fault detection problems for NCSs considering uncertain time delays were investigated in [3] and [4], a model predictive control approach was proposed for constrained NCSs subject to packet dropouts [5]. Recently, different controller design methods for NCSs were reported in [6] and [7]. During the past few years, filtering problems have attracted great attention and have been extensively investigated for various systems within linear frameworks. For instance, an effective filter design was proposed in [8] for linear timedelay systems, and fault isolation filter design problems have been successfully addressed for linear stochastic systems in [9] and [10]. However, from a practical point of view, it is more significant to investigate filter design issues for nonlinear systems, since almost all the dynamic systems involve nonlinearities to some extent. In the literature, some representative results on filtering problems of various nonlinear systems were reported in [11]–[13], and the references therein. On the other hand, filtering problems for NCSs constitute another attractive research topic, to which much attention has been attached. As a result, fruitful results have been obtained for both linear and nonlinear network-based systems. To mention a few, an effective filtering strategy was proposed in [14] for networked systems involving time-varying transmission delays and packet dropouts. Taking into account certain communication constraints, a resilient filter design method has been developed for a class of NCSs [15] and was proved to be effective. For a class of linear systems with uncertainties, robust filtering problem has been successfully addressed in [16]. Filter design based on moving horizon estimation for largescale networked processes featured by nonlinearities has been developed in [17]. More detailed results with respect to filter design for NCSs can be referred to [18], and the references therein. In practice, reliability in terms of data transmission is a significant factor that affects the performance of NCSs [19]–[23], and it is especially true for mobile NCSs in high-performance automobiles [24], [25], aircrafts [26], space shuttles, etc. Recently, considerable efforts have been devoted to issues in terms of multi-channel communication schemes for their potential advantages in the improvement of communication reliability in modern industry. Multi-channel communication has been employed for networked control and state estimation issues in [27]–[29]. Under a multi-channel communication
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scheme, it is not necessary to utilize all the channels for communication simultaneously because communication resources are limited and should be properly arranged in NCSs. For a certain class of NCSs incorporating multiple communication channels, it is desirable to perform channel switching actions by exploiting an effective channel switching scheduler, since each channel is featured by fluctuating performance that is affected by sudden environmental changes or unavoidable system failures. To precisely describe the channel switching phenomenon caused by internal or external inducements, a proper hybrid model should be introduced for mathematical modeling. From a practical point of view, one effective approach is to use a Markov chain to depict such phenomenon. Interestingly, although there are some available results focusing on control synthesis for NCSs under a multi-channel environment, to the best of the authors’ knowledge, there are not satisfying results on filter design problems for NCSs subject to channel switching strategies, which is practical and important for complex systems that are operated under networked environments. Inspired by above considerations, we first propose a multichannel communication scheme subject to channel switching. Under this communication framework, we aim to carry out the H∞ filter design for a class of network-based nonlinear systems. To account for more general and practical situations, time-varying delays and intermittent packet dropouts will be considered in this paper. Specifically, the norm-bounded uncertainties involved in the original system is represented by a Takagi–Sugeno (T–S) fuzzy model [30], which has been extensively employed as approximations of the nonlinear systems for control and filtering problems [31]–[34]. A multichannel transmission strategy is proposed, then a more applicable data transmission model with consideration of communication delays and packet dropouts is established. Bernoulli random distribution is employed to characterize the data-missing phenomena occurring in communication channels. Based on the consideration that at each sampling time, every communication channel has different performance in terms of communication delays and packet dropout rate subject to environmental changes as well as other inducements, we propose a novel approach that allows switching actions among all communication channels to select an appropriate channel and improve communication reliability. Subsequently, the switching phenomenon is described by a discrete-time transition process that is governed by a Markov chain from a practical perspective. Based on a quadratic Lyapunov–Krasovskii functional combined with several effective linear matrix inequality techniques, sufficient conditions on stability of the obtained filtering error system and the existence of the desired filter are derived. The remainder of this paper is summarized as follows. In Section II, the T–S fuzzy model for a class of nonlinear systems is developed, and a channel switching strategy is proposed and modeled. Section III is devoted to introducing indispensable techniques and main results of this paper. In Section IV, a numerical example is given to illustrate the effectiveness of our proposed filter design technique. Finally, the conclusions are drawn in Section V.
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A. Notation The notation used throughout this paper is fairly standard. The superscripts “T” and “−1” stand for matrix transposition and inverse, respectively. Rn denotes the n-dimensional Euclidean space. I and 0 represent, respectively, an identity matrix and a zero matrix. In symmetric block matrices or complex matrix expressions, we adopt the symbol “∗” as an ellipsis for the terms that are introduced by symmetry and diag{...} stands for a block-diagonal matrix. · is used to refer to the Euclidean vector norm. ·2 stands for the usual l2 [0, ∞) norm. λmax (·) denotes the maximum eigenvalue of a symmetric matrix. In addition, E{x} and E{x|y} denote, respectively, the expectation of x and the expectation of x conditional on y. The notation P > 0 (P ≥ 0) implies that P is symmetric and positive (semi-positive) definite. II. P RELIMINARIES AND P ROBLEM F ORMULATION A. Physical Plant In this paper, we consider a class of networked nonlinear system that is expressed as follows: ⎧ ⎨ x(k + 1) = f (x(k), w(k)) y(k) = r(x(k), w(k)) (1) ⎩ z(k) = q(x(k)) where f (·), r(·), and q(·) are continuously vector-valued nonlinear functions. Then, the discrete-time nonlinear system (1) can be represented by the following T–S fuzzy dynamic model considering bounded uncertainties. Plant Rule i: IF θ1 (k) is Mi1 , and θ2 (k) is Mi2 , and . . . , and θp (k) is Mip , THEN ⎧ ⎨ x(k + 1) = Ai (k)x(k) + Bi (k)w(k) y(k) = Ci x(k) + Di w(k) ⎩ z(k) = Hi x(k) where i ∈ S := {1, 2, . . . , r}, with r representing the number of IF-THEN rules, and Mi1 is the fuzzy set; θ (k) := θ1 (k) θ2 (k) · · · θp (k) is the premise variable vector, and p is the number of these premise variables; x(k) ∈ Rn is the state vector, w(k) ∈ Rl is the disturbance input belonging to l2 [0, ∞), y(k) ∈ Rq is the measured output of the system based on the assumption that all sensors are in ideal conditions, and z(k) ∈ Rp is the objective output to be estimated; Ai (k) := Ai + Ai (k), Bi (k) := Bi + Bi (k). Ai , Bi , Ci , Di , and Hi are known constant matrices of compatible dimensions, while the real-valued matrices Ai (k) and Bi (k) represent the norm-bounded uncertainties of the system and are assumed to be with the following structure: [Ai (k)
Bi (k)] = GF(k)[E1i
E2i ]
(2)
where G, E1i , and E2i , for any i ∈ S, are assumed to be known constant matrices with appropriate dimensions, and F(k) is an unknown time-varying matrix function that satisfies F T (k)F(k) ≤ I.
(3)
Then the parameter uncertainty matrices Ai (k) and Bi (k) are said to be admissible if (2) and (3) hold simultaneously.
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Consider a pair of (x(k), w(k)), the nonlinear system that has been approximated by the T–S fuzzy model can be represented as ⎧ x(k + 1) = ri=1 hi (θ (k)) (Ai + GF(k)E1i )x(k) ⎪ ⎪ ⎨ + (Bi + GF(k)E 2i )w(k) (4) r y(k) = h (θ (k)) C x(k) + D w(k) ⎪ i i i ⎪ i=1 ⎩ z(k) = ri=1 hi (θ (k))Hi x(k) where the fuzzy basis functions are given by
μi (θ (k)) , μi (θ (k)) := Mij θj (k) hi (θ (k)) := r i=1 μi (θ (k)) p
j=1
with Mij (θj (k)) representing the grade of membership of θj (k) in Mij . In this paper, it is assumed that μi (θ (k)) ≥ 0, which implies ri=1 μi (θ (k)) ≥ 0. Due to the nature of the T–S fuzzy model, the normalized fuzzy weighting function satisfies 0 ≤ hi (θ (k)) ≤ 1, and
r
hi (θ (k)) = 1.
i=1
Remark 1: It should be emphasized that in (4), y(k) ∈ Rq does not represent the actual measured outputs of the system. Instead, y(k) is used to describe the measured outputs with hypothesis that all sensors are in ideal conditions. This assumption is a preliminary for latter analysis on communication behaviors. Remark 2: In many literatures with respect to different dynamic system models, especially the models of NCSs, the dynamic processes are hypothesized to be free of uncertainties, which cannot be neglected in many circumstances in practice. Therefore, in order to improve the adaptability of the system model, it is sensible to introduce norm-bounded uncertain matrices to precisely describe the parametric uncertainties. B. Communication Channels In a typical NCS, information of the system should be transmitted via a network-based medium, which is frequently unreliable to some extent. To be specific, information transmission between the sensors of the physical plant and the filter via communication links may experience nonignorable packet dropouts and time-varying delays, which should be properly considered in order to design a more applicable filter for the system. Since phenomena in terms of packet dropouts and timevarying delays are frequently encountered in many networkbased systems and may bring significantly negative effects, it is necessary to develop an effective approach to alleviate the negative influences caused by stochastic packet dropouts and time delays. Based on this perspective, a necessary assumption in terms of the communication structure and timevarying delays in each communication channel is ready to be proposed. Assumption 1: There are n communication channels available for the same information transmission task, and these n communication channels are mutually uncorrelated and have different characteristics in terms of time-delay bounds and packet dropout rates. The time-varying delays of the cth (c ∈ C) communication channel is represented by dc (k), which
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is constrained within a certain interval dc ≤ dc (k) ≤ d¯ c where dc and d¯ c are the lower bound and upper bound of the time-varying delays for the cth communication channel, respectively. Since it is known that there are q outputs in the original system, a series of stochastic variables αlc (k) ∈ R (l = 1, 2, . . . , q), which are introduced to account for the packet dropout occurrence probability when data is transmitted in the cth communication channel, abide by binary distributed random sequences and take the values of either 1 or 0. In addition, partial measurement failures of the sensors that constantly occur but have been extensively neglected are taken into account in the communication process. Therefore, we introduce a symbol λl (k), which is assumed to satisfy 0 < λl (k) ≤ λl max
(5)
where λl (k) is used to the accuracy rate of the measurement in the lth sensor corresponding to the lth output yl (k) as λl (k). Also, since the measurements at the end of the communication links are assumed to be intermittent, the signal received by the filter is no longer equivalent to the output y(k) of the system plant. In this paper, the packet dropout phenomenon can be described by the stochastic expression yf (k) = ϒc y(k − dc (k)) r = ϒc hi (θ (k))[Ci x(k − dc (k) + Di w(k − dc (k))] i=1
where yf (k) represents the measured output detected by the sensor of the filter, that is, the input signal of the filter; ϒ := diag{λ1 (k), λ2 (k), . . . , λq (k)}, c := diag{α1c (k), α2c (k), . . . , αqc (k)}. It is hypothesized that the premise variables of the system model are independent of the mentioned stochastic variables αlc (k) and λl (k) throughout this paper. In addition, the two separate sequences of stochastic variables αlc (k) and λl (k) are also supposed to be uncorrelated. Then, it is calculated that
ϒc = diag λ1 (k), λ2 (k), . . . , λq (k)
× diag α1c (k), α2c (k), . . . , αqc (k)
= diag λ1 (k)α1c (k), λ2 (k)α2c (k), . . . , λq (k)αqc (k)
= diag β1c (k), β2c (k), . . . , βqc (k) where βlc (k) := λl (k)αlc (k) is defined as the overall packet delivery ratio that takes a value at each sampling time according to certain probability density functions on the interval [0, λl max ]. Moreover, the mathematical expectation and variance of this ratio are denoted as E{βlc (k)} = β¯lc and Var{βlc (k)} = σlc2 , respectively. Then, the input of the filter, if employing cth communication channel, is written as yfc (k) = ϒc y(k − dc (k)) r = ϒc hi (θ (k)) Ci x(k − dc (k)) + Di w(k − dc (k)) i=1
=
r i=1
hi (θ (k))
q
βlc l Ci x(k − dc (k))
l=1
+ l Di w(k − dc (k))
(6)
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where l ∈ Rq×q is defined as ⎧ ⎫ ⎪ ⎪ ⎨ ⎬
l := diag 0, 0, . . . , 0, 1, 0, . . . , 0, 0 . ⎪ ⎩ ⎪ ⎭ l−1
(7)
q−l
For computational simplicity and analytical convenience, the input of the filter (6) is represented in the following form for the remaining part of this paper: yfc (k) = ϒc
r
hi (θ (k)) Ci x(k − dc (k)) + Di w(k)
i=1
=
r i=1
hi (θ (k))
q
βlc l Ci x(k − dc (k)) + l Di w(k) .
l=1
For analytical convenience, we define a matrix c := E{ϒc } = diag{β¯1c , β¯2c , . . . , β¯qc }, which will be used in the theoretical derivation. Remark 3: It is sensible to retain time delays in the system state term while ignoring the delays for the disturbance term. The reasons for this decision are twofold. 1) If system disturbance w(k) can be described or approximated by a typical function, the value at sampling instant k will be similar to the one at sampling instant k − dc (k) if time delay for each channel at every sampling instant is restricted within a certain small interval, which is not difficult to be realized in practice. 2) If the disturbance w(k) is a white noise sequence, then the expectation of w(k) equals the expectation of w(k) − dc (k). In the stability analysis and filter design, only the expectation of the disturbance is used. Therefore, (7) is a well-balanced and usable model. Remark 4: Practically, the performance of a sensor may gradually decrease due to the aging problem, which will lead to inaccurate measurements. Also, the sensor may also suffer from temporary measurement inaccuracy or partial measurement missing phenomenon because of random errors or sudden environmental changes. The concept of accuracy rate λl (k) is proposed to describe different accuracy proportions on the measurements. Therefore, βlc (k) can by no means simply take values of 1 or 0. Instead, βlc (k) will be valued on the interval [0, λl max ] and are closely dependent on different sensors for different outputs and different channels used for information transmission. Remark 5: It is worth mentioning that λl max denotes the upper bound of the accuracy rate of the measurement of each sensor, which is more general and sensible in terms of depicting sensor measurement errors compared with the missing probability index defined in the previous literatures. From a mathematical perspective, the upper bound may take any value subject to λl max ≥ 1. However, in a practical system, if λl max takes a comparatively large value, the lth sensor is suggested to be immediately replaced by a new one with satisfying performance. Therefore, assuming λl max is slightly greater than 1 is sensible and sufficient when applied in practical circumstances. At each sampling time when system measured outputs are required to be transmitted, the system is allowed to make a
selection among these communication channels according to a reasonably prescribed rule. Another assumption is made that real-time information subject to time delays dc (k) and packet dropout rate αlc (k) of each channel is accessible for the system plant, such that the channel selection rule can be constructed based upon these important factors. For example, the rule can be regulated to choose the channel with minimum time delay at sampling time k in order to guarantee the fastest transmission speed. Consequently, the switch between each channel is indirectly affected by the environmental disturbances, component failures or repairs as well as undetectable interconnections, which all may lead to the changes in accuracy rate λl (k), time delays dc (k) and packet dropout rate 1 − E{αlc (k)} at each sampling time. As a result, it is reasonable to model this unpredictable switching process by a discrete-time transition process governed by a Markov chain. rk is used to represent the index of channel that is used for communication at sampling instant k. Let {rk , k ≥ 0} be a Markov chain with a state space C. Denote the state transition matrix by P := [πab ]a,b∈C which means the transition probabilities of the Markov chain are dominated by Pr [ck+1 = b|ck = a] := πab , ∀a, b ∈ C where πab ≥ 0, ∀a, b ∈ C, and nb=1 πab = 1, ∀a ∈ C. Remark 6: Employing this multi-channel communication strategy will definitely bring a couple of advantages. First, it will substantially increase the reliability of the information transmission system. Second, the performance of an individual communication channel varies under different operating conditions. At each sampling time, the best communication performance may not be obtained by a fixed channel, thus the real-time optimal communication performance can only be guaranteed if switching among several channel candidates is allowed. Last but not least, a multi-channel scheme enables us to propose a back-up plan in case of an emergency situation, which is an important part of our future work. C. Fuzzy-Rule-Based Filter In this paper, our main objective is to obtain the estimation of the signal z(k) based on measured outputs of the given system. Therefore, the construction of the filter system constitutes an important focus of this paper. For the physical plant (5) to be investigated, a full-order fuzzy-rule-based filter is established with the structure given as follows. Filter Rule i: IF θ1 (k) is Mi1 , and θ2 (k) is Mi2 , and . . . , and θp (k) is Mip , THEN xf (k + 1) = Afi xf (k) + Bfi yfc (k) zf (k) = Cfi xf (k) where Afi , Bfi , and Cfi are filter matrices to be determined; xf (k) is the state vector of the filter; zf (k) represents the estimate of signal z(k); and yfc (k) is the input vector of the filter.
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Fig. 1. Fuzzy-rule-dependent filter error system under multi-channel communication.
Then, the overall full-order filter is given as follows: xf (k + 1) = rj=1 hj (θ (k)) Afj xf (k) + Bfj yfc (k) r zf (k) = j=1 hj (θ (k))Cfj xf (k).
(8)
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sure stability [35]. It is defined that all the mentioned criteria are second-order-based except the almost sure stability. It is demonstrated that the stability definition employed in this paper is extensively applicable and effective in most cases. In the rest of this paper, as a result, all the analysis to be conducted and sufficient conditions for our results to be presented will be based on the stochastic stability criterion. To end this section, the H∞ filter design problem to be addressed in this paper is formulated as follows. Considering a fuzzy model-based nonlinear system with uncertainties in (4) and a prescribed noise attenuation performance index γ > 0, design a full-order fuzzy-rule-based filter in the form of (8), such that the following two requirements can be simultaneously satisfied. 1) Stochastic Stability: The filter error system presented in (9) is stochastically stable in the sense of Definition 1. 2) H∞ Performance: Under zero initial condition, the following inequality for the filter error system holds:
e(k)E2 ≤ γ w(k)E2 In the rest of this paper, hi (θ (k)) will be denoted by hi , ∀i ∈ S ∞ for the sake of simplicity. Augment the plant and filter state as T := where e(k) E{ E 2 k=0 e (k)e(k)}. T If these two conditions are satisfied, the filter error system ξ(k) := xT (k) xfT (k) , and denote the filter error as e(k) := in (9) is said to be stochastically stable with a guaranteed H∞ z(k) − zf (k). Then, the filter error system, if using the cth disturbance attenuation level bound. communication channel at time k, is obtained in the following form: ⎧ r r III. M AIN R ESULTS ⎨ ξ(k + 1) = i=1 j=1 hi hj A¯ ij (k)ξ(k) (9) We first introduce the following lemmas that are indispens+ A¯ dij Gξ(k − dck (k)) + B¯ ij (k)w(k) ⎩ able for the derivation of our main results. zf (k) = ri=1 hi (θ (k))Cfi xf (k) Lemma 1 [36]: For any real matrices Qi , Qj , and K> 0 where j ∈ S), we define a set of coefficients hi that satisfy ri=1 (i, 0 0 Ai (k) hi = 1, then the following inequality holds: A¯ ij (k) := , A¯ dij := 0 Afj Bfj c Ci ⎤ r ⎡ r r (k) B i , C¯ ij := Hi −Cfj B¯ ij (k) := hi QTi K⎣ hj Qj ⎦ ≤ hi QTi KQi . Bfj c Di i=1 j=1 i=1 G := I 0 . Lemma 2 [37]: Assume that a ∈ Rna , b ∈ Rnb , and N ∈ The structure of the network-based filter system is presented Rna ×nb . Then for any matrices X ∈ Rna ×nb , Y ∈ Rna ×nb , and in Fig. 1. The filter error system mainly consists of an approX Y n ×n a b that satisfy ≥ 0, the following inequality priate channel switching scheduler, communication channels, Z ∈ R ∗ Z and corresponding filters for each of communication channel. holds: At each sampling time, a channel is picked up by the channel T X Y −N a a scheduler for information exchange, and the channel related ≥ −2aT N b. ∗ Z b b to this channel is used to give state estimates for the actual system state. Lemma 3 (S-Procedure): Let L = LT , and H, E, F be real Before further proceeding, it is necessary to introduce the matrices with appropriate dimensions. If F T F ≤ I is satisfied, following definition in terms of stochastic stability. then L + HFE + (HFE)T < 0 holds, if and only if there exists Definition 1: The filter error system (9) is said to be a positive scalar ε > 0, such that L + ε−1 HH T + εET E < 0 or stochastically stable, if w(k) ≡ 0, and for any initial condition equivalently the following inequality holds: ξ(0), there exists a finite W > 0, such that the following ⎡ ⎤ inequality holds: L H εET ∞ ⎣ ∗ −εI 0 ⎦ < 0. 2 T ∗ ∗ −εI ξ(k)2 ξ(0) < ξ (0)Wξ(0). E k=0
Remark 7: Generally, there are several available definitions in terms of system stability, e.g., mean square stochastic stability, mean exponential stability, stochastic stability, and almost
A. Stability Analysis Theorem 1: Consider the filter error system (9). Given a scalar γ > 0, if there exist matrices R > 0, X > 0, Z > 0,
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a > 0 (a ∈ C) and matrices Yij ⎡ Lij H ⎣ ∗ −εI ∗ ∗ X ∗ where
(i, j ∈ S), satisfying ⎤ εEiT 0 ⎦ b. If this occurs, it is assumed that bk=a f (k) = 0, where f (k) is any function with respect to the variable k. This assumption makes sense in this paper and is not contradictory with any mathematical rules. Remark 9: It is worth mentioning that although sufficient conditions in terms of stochastic stability are presented in
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Theorem 1, unknown matrices from the Lyapunov–Krasovskii functional are cross-coupled with system matrices to be calculated. Therefore, it is impossible to directly complete the filter design using linear matrix inequality (LMI) techniques based on Theorem 1. To combat this obstacle, further sufficient conditions are presented in the following theorem. Theorem 2: Consider the filter error system (9). Given a scalar γ > 0, if there exist matrices R > 0, X > 0, Z > 0, Yij , a > 0, φa (i, j ∈ S, a ∈ C), satisfying (11), and the following LMIs and matrices Yij : ⎤ ⎡ Wij N εEiT ⎣ ∗ (18) −εI 0 ⎦ 0, X1 > 0, X3 > 0, Z > 0, R > 0 (i, j ∈ S, a ∈ C), and a scalar ω, satisfying ⎡ ⎤ X1 X2 Y1,ij ⎣ ∗ X3 Y2,ij ⎦ < 0 ∗ ∗ Z ⎡ ⎤ T Oij N+ εE+i ⎣ ∗ (19) −εI 0 ⎦
1
Robust Filtering for a Class of Networked Nonlinear Systems With Switching Communication Channels Lixian Zhang, Senior Member, IEEE, Xunyuan Yin, Zepeng Ning, and Dong Ye
Abstract—This paper is concerned with the problem of robust filter design for a class of discrete-time networked nonlinear systems. The Takagi–Sugeno fuzzy model is employed to represent the underlying nonlinear dynamics. A multi-channel communication scheme that involves a channel switching phenomenon described by a Markov chain is proposed for data transmission. Two typical communication imperfections, network-induced time-varying delays and packet dropouts are considered in each channel. The objective of this paper is to design an admissible filter such that the filter error system is stochastically stable and ensures a prescribed disturbance attenuation level bound. Based on the Lyapunov–Krasovskii functional method and matrix inequality techniques, sufficient conditions on the existence of the desired filter are obtained. A numerical example is provided to illustrate the effectiveness of the proposed design approach. Index Terms—Channel switching, multi-channel communication, network control systems, robust filter design, Takagi–Sugeno (T–S) fuzzy systems.
I. I NTRODUCTION VER the past several decades, study on networked control systems (NCSs) has gradually become an appealing research frontier due to many inherent advantages, such as low cost, low weight, flexible structures, simple implementation, and maintenance requirements. So far, we have witnessed extensive applications in modern industrial communities [1]. However, since sensors, controllers, and actuators in typical NCSs are connected via communication links which are not constantly reliable, network-based systems are frequently accompanied by information-transmission delays and intermittent packet dropouts [1], [2], which inevitably lead to deteriorated performance or even instability of the systems. As a result, it is necessary to take into account the negative influences brought by stochastic time delays as well as packet dropouts when performing analysis and design
O
Manuscript received August 30, 2015; revised December 11, 2015; accepted January 21, 2016. This paper was recommended by Associate Editor C. Hua. L. Zhang and Z. Ning are with the School of Astronautics, Harbin Institute of Technology, Harbin 150080, China (e-mail: [email protected]; [email protected]). X. Yin is with the School of Astronautics, Harbin Institute of Technology, Harbin 150080, China, and also with the Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB T6G2V4, Canada (e-mail: [email protected]). D. Ye is with the Research Centre of Satellite Technology, Harbin Institute of Technology, Harbin 150080, China (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.2016.2523811
for NCSs. Owing to the dedicated work from forward-looking researchers, fruitful results focusing on time-delay-induced effects on NCSs have been obtained. To name some representative studies, fault detection problems for NCSs considering uncertain time delays were investigated in [3] and [4], a model predictive control approach was proposed for constrained NCSs subject to packet dropouts [5]. Recently, different controller design methods for NCSs were reported in [6] and [7]. During the past few years, filtering problems have attracted great attention and have been extensively investigated for various systems within linear frameworks. For instance, an effective filter design was proposed in [8] for linear timedelay systems, and fault isolation filter design problems have been successfully addressed for linear stochastic systems in [9] and [10]. However, from a practical point of view, it is more significant to investigate filter design issues for nonlinear systems, since almost all the dynamic systems involve nonlinearities to some extent. In the literature, some representative results on filtering problems of various nonlinear systems were reported in [11]–[13], and the references therein. On the other hand, filtering problems for NCSs constitute another attractive research topic, to which much attention has been attached. As a result, fruitful results have been obtained for both linear and nonlinear network-based systems. To mention a few, an effective filtering strategy was proposed in [14] for networked systems involving time-varying transmission delays and packet dropouts. Taking into account certain communication constraints, a resilient filter design method has been developed for a class of NCSs [15] and was proved to be effective. For a class of linear systems with uncertainties, robust filtering problem has been successfully addressed in [16]. Filter design based on moving horizon estimation for largescale networked processes featured by nonlinearities has been developed in [17]. More detailed results with respect to filter design for NCSs can be referred to [18], and the references therein. In practice, reliability in terms of data transmission is a significant factor that affects the performance of NCSs [19]–[23], and it is especially true for mobile NCSs in high-performance automobiles [24], [25], aircrafts [26], space shuttles, etc. Recently, considerable efforts have been devoted to issues in terms of multi-channel communication schemes for their potential advantages in the improvement of communication reliability in modern industry. Multi-channel communication has been employed for networked control and state estimation issues in [27]–[29]. Under a multi-channel communication
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scheme, it is not necessary to utilize all the channels for communication simultaneously because communication resources are limited and should be properly arranged in NCSs. For a certain class of NCSs incorporating multiple communication channels, it is desirable to perform channel switching actions by exploiting an effective channel switching scheduler, since each channel is featured by fluctuating performance that is affected by sudden environmental changes or unavoidable system failures. To precisely describe the channel switching phenomenon caused by internal or external inducements, a proper hybrid model should be introduced for mathematical modeling. From a practical point of view, one effective approach is to use a Markov chain to depict such phenomenon. Interestingly, although there are some available results focusing on control synthesis for NCSs under a multi-channel environment, to the best of the authors’ knowledge, there are not satisfying results on filter design problems for NCSs subject to channel switching strategies, which is practical and important for complex systems that are operated under networked environments. Inspired by above considerations, we first propose a multichannel communication scheme subject to channel switching. Under this communication framework, we aim to carry out the H∞ filter design for a class of network-based nonlinear systems. To account for more general and practical situations, time-varying delays and intermittent packet dropouts will be considered in this paper. Specifically, the norm-bounded uncertainties involved in the original system is represented by a Takagi–Sugeno (T–S) fuzzy model [30], which has been extensively employed as approximations of the nonlinear systems for control and filtering problems [31]–[34]. A multichannel transmission strategy is proposed, then a more applicable data transmission model with consideration of communication delays and packet dropouts is established. Bernoulli random distribution is employed to characterize the data-missing phenomena occurring in communication channels. Based on the consideration that at each sampling time, every communication channel has different performance in terms of communication delays and packet dropout rate subject to environmental changes as well as other inducements, we propose a novel approach that allows switching actions among all communication channels to select an appropriate channel and improve communication reliability. Subsequently, the switching phenomenon is described by a discrete-time transition process that is governed by a Markov chain from a practical perspective. Based on a quadratic Lyapunov–Krasovskii functional combined with several effective linear matrix inequality techniques, sufficient conditions on stability of the obtained filtering error system and the existence of the desired filter are derived. The remainder of this paper is summarized as follows. In Section II, the T–S fuzzy model for a class of nonlinear systems is developed, and a channel switching strategy is proposed and modeled. Section III is devoted to introducing indispensable techniques and main results of this paper. In Section IV, a numerical example is given to illustrate the effectiveness of our proposed filter design technique. Finally, the conclusions are drawn in Section V.
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A. Notation The notation used throughout this paper is fairly standard. The superscripts “T” and “−1” stand for matrix transposition and inverse, respectively. Rn denotes the n-dimensional Euclidean space. I and 0 represent, respectively, an identity matrix and a zero matrix. In symmetric block matrices or complex matrix expressions, we adopt the symbol “∗” as an ellipsis for the terms that are introduced by symmetry and diag{...} stands for a block-diagonal matrix. · is used to refer to the Euclidean vector norm. ·2 stands for the usual l2 [0, ∞) norm. λmax (·) denotes the maximum eigenvalue of a symmetric matrix. In addition, E{x} and E{x|y} denote, respectively, the expectation of x and the expectation of x conditional on y. The notation P > 0 (P ≥ 0) implies that P is symmetric and positive (semi-positive) definite. II. P RELIMINARIES AND P ROBLEM F ORMULATION A. Physical Plant In this paper, we consider a class of networked nonlinear system that is expressed as follows: ⎧ ⎨ x(k + 1) = f (x(k), w(k)) y(k) = r(x(k), w(k)) (1) ⎩ z(k) = q(x(k)) where f (·), r(·), and q(·) are continuously vector-valued nonlinear functions. Then, the discrete-time nonlinear system (1) can be represented by the following T–S fuzzy dynamic model considering bounded uncertainties. Plant Rule i: IF θ1 (k) is Mi1 , and θ2 (k) is Mi2 , and . . . , and θp (k) is Mip , THEN ⎧ ⎨ x(k + 1) = Ai (k)x(k) + Bi (k)w(k) y(k) = Ci x(k) + Di w(k) ⎩ z(k) = Hi x(k) where i ∈ S := {1, 2, . . . , r}, with r representing the number of IF-THEN rules, and Mi1 is the fuzzy set; θ (k) := θ1 (k) θ2 (k) · · · θp (k) is the premise variable vector, and p is the number of these premise variables; x(k) ∈ Rn is the state vector, w(k) ∈ Rl is the disturbance input belonging to l2 [0, ∞), y(k) ∈ Rq is the measured output of the system based on the assumption that all sensors are in ideal conditions, and z(k) ∈ Rp is the objective output to be estimated; Ai (k) := Ai + Ai (k), Bi (k) := Bi + Bi (k). Ai , Bi , Ci , Di , and Hi are known constant matrices of compatible dimensions, while the real-valued matrices Ai (k) and Bi (k) represent the norm-bounded uncertainties of the system and are assumed to be with the following structure: [Ai (k)
Bi (k)] = GF(k)[E1i
E2i ]
(2)
where G, E1i , and E2i , for any i ∈ S, are assumed to be known constant matrices with appropriate dimensions, and F(k) is an unknown time-varying matrix function that satisfies F T (k)F(k) ≤ I.
(3)
Then the parameter uncertainty matrices Ai (k) and Bi (k) are said to be admissible if (2) and (3) hold simultaneously.
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Consider a pair of (x(k), w(k)), the nonlinear system that has been approximated by the T–S fuzzy model can be represented as ⎧ x(k + 1) = ri=1 hi (θ (k)) (Ai + GF(k)E1i )x(k) ⎪ ⎪ ⎨ + (Bi + GF(k)E 2i )w(k) (4) r y(k) = h (θ (k)) C x(k) + D w(k) ⎪ i i i ⎪ i=1 ⎩ z(k) = ri=1 hi (θ (k))Hi x(k) where the fuzzy basis functions are given by
μi (θ (k)) , μi (θ (k)) := Mij θj (k) hi (θ (k)) := r i=1 μi (θ (k)) p
j=1
with Mij (θj (k)) representing the grade of membership of θj (k) in Mij . In this paper, it is assumed that μi (θ (k)) ≥ 0, which implies ri=1 μi (θ (k)) ≥ 0. Due to the nature of the T–S fuzzy model, the normalized fuzzy weighting function satisfies 0 ≤ hi (θ (k)) ≤ 1, and
r
hi (θ (k)) = 1.
i=1
Remark 1: It should be emphasized that in (4), y(k) ∈ Rq does not represent the actual measured outputs of the system. Instead, y(k) is used to describe the measured outputs with hypothesis that all sensors are in ideal conditions. This assumption is a preliminary for latter analysis on communication behaviors. Remark 2: In many literatures with respect to different dynamic system models, especially the models of NCSs, the dynamic processes are hypothesized to be free of uncertainties, which cannot be neglected in many circumstances in practice. Therefore, in order to improve the adaptability of the system model, it is sensible to introduce norm-bounded uncertain matrices to precisely describe the parametric uncertainties. B. Communication Channels In a typical NCS, information of the system should be transmitted via a network-based medium, which is frequently unreliable to some extent. To be specific, information transmission between the sensors of the physical plant and the filter via communication links may experience nonignorable packet dropouts and time-varying delays, which should be properly considered in order to design a more applicable filter for the system. Since phenomena in terms of packet dropouts and timevarying delays are frequently encountered in many networkbased systems and may bring significantly negative effects, it is necessary to develop an effective approach to alleviate the negative influences caused by stochastic packet dropouts and time delays. Based on this perspective, a necessary assumption in terms of the communication structure and timevarying delays in each communication channel is ready to be proposed. Assumption 1: There are n communication channels available for the same information transmission task, and these n communication channels are mutually uncorrelated and have different characteristics in terms of time-delay bounds and packet dropout rates. The time-varying delays of the cth (c ∈ C) communication channel is represented by dc (k), which
3
is constrained within a certain interval dc ≤ dc (k) ≤ d¯ c where dc and d¯ c are the lower bound and upper bound of the time-varying delays for the cth communication channel, respectively. Since it is known that there are q outputs in the original system, a series of stochastic variables αlc (k) ∈ R (l = 1, 2, . . . , q), which are introduced to account for the packet dropout occurrence probability when data is transmitted in the cth communication channel, abide by binary distributed random sequences and take the values of either 1 or 0. In addition, partial measurement failures of the sensors that constantly occur but have been extensively neglected are taken into account in the communication process. Therefore, we introduce a symbol λl (k), which is assumed to satisfy 0 < λl (k) ≤ λl max
(5)
where λl (k) is used to the accuracy rate of the measurement in the lth sensor corresponding to the lth output yl (k) as λl (k). Also, since the measurements at the end of the communication links are assumed to be intermittent, the signal received by the filter is no longer equivalent to the output y(k) of the system plant. In this paper, the packet dropout phenomenon can be described by the stochastic expression yf (k) = ϒc y(k − dc (k)) r = ϒc hi (θ (k))[Ci x(k − dc (k) + Di w(k − dc (k))] i=1
where yf (k) represents the measured output detected by the sensor of the filter, that is, the input signal of the filter; ϒ := diag{λ1 (k), λ2 (k), . . . , λq (k)}, c := diag{α1c (k), α2c (k), . . . , αqc (k)}. It is hypothesized that the premise variables of the system model are independent of the mentioned stochastic variables αlc (k) and λl (k) throughout this paper. In addition, the two separate sequences of stochastic variables αlc (k) and λl (k) are also supposed to be uncorrelated. Then, it is calculated that
ϒc = diag λ1 (k), λ2 (k), . . . , λq (k)
× diag α1c (k), α2c (k), . . . , αqc (k)
= diag λ1 (k)α1c (k), λ2 (k)α2c (k), . . . , λq (k)αqc (k)
= diag β1c (k), β2c (k), . . . , βqc (k) where βlc (k) := λl (k)αlc (k) is defined as the overall packet delivery ratio that takes a value at each sampling time according to certain probability density functions on the interval [0, λl max ]. Moreover, the mathematical expectation and variance of this ratio are denoted as E{βlc (k)} = β¯lc and Var{βlc (k)} = σlc2 , respectively. Then, the input of the filter, if employing cth communication channel, is written as yfc (k) = ϒc y(k − dc (k)) r = ϒc hi (θ (k)) Ci x(k − dc (k)) + Di w(k − dc (k)) i=1
=
r i=1
hi (θ (k))
q
βlc l Ci x(k − dc (k))
l=1
+ l Di w(k − dc (k))
(6)
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where l ∈ Rq×q is defined as ⎧ ⎫ ⎪ ⎪ ⎨ ⎬
l := diag 0, 0, . . . , 0, 1, 0, . . . , 0, 0 . ⎪ ⎩ ⎪ ⎭ l−1
(7)
q−l
For computational simplicity and analytical convenience, the input of the filter (6) is represented in the following form for the remaining part of this paper: yfc (k) = ϒc
r
hi (θ (k)) Ci x(k − dc (k)) + Di w(k)
i=1
=
r i=1
hi (θ (k))
q
βlc l Ci x(k − dc (k)) + l Di w(k) .
l=1
For analytical convenience, we define a matrix c := E{ϒc } = diag{β¯1c , β¯2c , . . . , β¯qc }, which will be used in the theoretical derivation. Remark 3: It is sensible to retain time delays in the system state term while ignoring the delays for the disturbance term. The reasons for this decision are twofold. 1) If system disturbance w(k) can be described or approximated by a typical function, the value at sampling instant k will be similar to the one at sampling instant k − dc (k) if time delay for each channel at every sampling instant is restricted within a certain small interval, which is not difficult to be realized in practice. 2) If the disturbance w(k) is a white noise sequence, then the expectation of w(k) equals the expectation of w(k) − dc (k). In the stability analysis and filter design, only the expectation of the disturbance is used. Therefore, (7) is a well-balanced and usable model. Remark 4: Practically, the performance of a sensor may gradually decrease due to the aging problem, which will lead to inaccurate measurements. Also, the sensor may also suffer from temporary measurement inaccuracy or partial measurement missing phenomenon because of random errors or sudden environmental changes. The concept of accuracy rate λl (k) is proposed to describe different accuracy proportions on the measurements. Therefore, βlc (k) can by no means simply take values of 1 or 0. Instead, βlc (k) will be valued on the interval [0, λl max ] and are closely dependent on different sensors for different outputs and different channels used for information transmission. Remark 5: It is worth mentioning that λl max denotes the upper bound of the accuracy rate of the measurement of each sensor, which is more general and sensible in terms of depicting sensor measurement errors compared with the missing probability index defined in the previous literatures. From a mathematical perspective, the upper bound may take any value subject to λl max ≥ 1. However, in a practical system, if λl max takes a comparatively large value, the lth sensor is suggested to be immediately replaced by a new one with satisfying performance. Therefore, assuming λl max is slightly greater than 1 is sensible and sufficient when applied in practical circumstances. At each sampling time when system measured outputs are required to be transmitted, the system is allowed to make a
selection among these communication channels according to a reasonably prescribed rule. Another assumption is made that real-time information subject to time delays dc (k) and packet dropout rate αlc (k) of each channel is accessible for the system plant, such that the channel selection rule can be constructed based upon these important factors. For example, the rule can be regulated to choose the channel with minimum time delay at sampling time k in order to guarantee the fastest transmission speed. Consequently, the switch between each channel is indirectly affected by the environmental disturbances, component failures or repairs as well as undetectable interconnections, which all may lead to the changes in accuracy rate λl (k), time delays dc (k) and packet dropout rate 1 − E{αlc (k)} at each sampling time. As a result, it is reasonable to model this unpredictable switching process by a discrete-time transition process governed by a Markov chain. rk is used to represent the index of channel that is used for communication at sampling instant k. Let {rk , k ≥ 0} be a Markov chain with a state space C. Denote the state transition matrix by P := [πab ]a,b∈C which means the transition probabilities of the Markov chain are dominated by Pr [ck+1 = b|ck = a] := πab , ∀a, b ∈ C where πab ≥ 0, ∀a, b ∈ C, and nb=1 πab = 1, ∀a ∈ C. Remark 6: Employing this multi-channel communication strategy will definitely bring a couple of advantages. First, it will substantially increase the reliability of the information transmission system. Second, the performance of an individual communication channel varies under different operating conditions. At each sampling time, the best communication performance may not be obtained by a fixed channel, thus the real-time optimal communication performance can only be guaranteed if switching among several channel candidates is allowed. Last but not least, a multi-channel scheme enables us to propose a back-up plan in case of an emergency situation, which is an important part of our future work. C. Fuzzy-Rule-Based Filter In this paper, our main objective is to obtain the estimation of the signal z(k) based on measured outputs of the given system. Therefore, the construction of the filter system constitutes an important focus of this paper. For the physical plant (5) to be investigated, a full-order fuzzy-rule-based filter is established with the structure given as follows. Filter Rule i: IF θ1 (k) is Mi1 , and θ2 (k) is Mi2 , and . . . , and θp (k) is Mip , THEN xf (k + 1) = Afi xf (k) + Bfi yfc (k) zf (k) = Cfi xf (k) where Afi , Bfi , and Cfi are filter matrices to be determined; xf (k) is the state vector of the filter; zf (k) represents the estimate of signal z(k); and yfc (k) is the input vector of the filter.
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Fig. 1. Fuzzy-rule-dependent filter error system under multi-channel communication.
Then, the overall full-order filter is given as follows: xf (k + 1) = rj=1 hj (θ (k)) Afj xf (k) + Bfj yfc (k) r zf (k) = j=1 hj (θ (k))Cfj xf (k).
(8)
5
sure stability [35]. It is defined that all the mentioned criteria are second-order-based except the almost sure stability. It is demonstrated that the stability definition employed in this paper is extensively applicable and effective in most cases. In the rest of this paper, as a result, all the analysis to be conducted and sufficient conditions for our results to be presented will be based on the stochastic stability criterion. To end this section, the H∞ filter design problem to be addressed in this paper is formulated as follows. Considering a fuzzy model-based nonlinear system with uncertainties in (4) and a prescribed noise attenuation performance index γ > 0, design a full-order fuzzy-rule-based filter in the form of (8), such that the following two requirements can be simultaneously satisfied. 1) Stochastic Stability: The filter error system presented in (9) is stochastically stable in the sense of Definition 1. 2) H∞ Performance: Under zero initial condition, the following inequality for the filter error system holds:
e(k)E2 ≤ γ w(k)E2 In the rest of this paper, hi (θ (k)) will be denoted by hi , ∀i ∈ S ∞ for the sake of simplicity. Augment the plant and filter state as T := where e(k) E{ E 2 k=0 e (k)e(k)}. T If these two conditions are satisfied, the filter error system ξ(k) := xT (k) xfT (k) , and denote the filter error as e(k) := in (9) is said to be stochastically stable with a guaranteed H∞ z(k) − zf (k). Then, the filter error system, if using the cth disturbance attenuation level bound. communication channel at time k, is obtained in the following form: ⎧ r r III. M AIN R ESULTS ⎨ ξ(k + 1) = i=1 j=1 hi hj A¯ ij (k)ξ(k) (9) We first introduce the following lemmas that are indispens+ A¯ dij Gξ(k − dck (k)) + B¯ ij (k)w(k) ⎩ able for the derivation of our main results. zf (k) = ri=1 hi (θ (k))Cfi xf (k) Lemma 1 [36]: For any real matrices Qi , Qj , and K> 0 where j ∈ S), we define a set of coefficients hi that satisfy ri=1 (i, 0 0 Ai (k) hi = 1, then the following inequality holds: A¯ ij (k) := , A¯ dij := 0 Afj Bfj c Ci ⎤ r ⎡ r r (k) B i , C¯ ij := Hi −Cfj B¯ ij (k) := hi QTi K⎣ hj Qj ⎦ ≤ hi QTi KQi . Bfj c Di i=1 j=1 i=1 G := I 0 . Lemma 2 [37]: Assume that a ∈ Rna , b ∈ Rnb , and N ∈ The structure of the network-based filter system is presented Rna ×nb . Then for any matrices X ∈ Rna ×nb , Y ∈ Rna ×nb , and in Fig. 1. The filter error system mainly consists of an approX Y n ×n a b that satisfy ≥ 0, the following inequality priate channel switching scheduler, communication channels, Z ∈ R ∗ Z and corresponding filters for each of communication channel. holds: At each sampling time, a channel is picked up by the channel T X Y −N a a scheduler for information exchange, and the channel related ≥ −2aT N b. ∗ Z b b to this channel is used to give state estimates for the actual system state. Lemma 3 (S-Procedure): Let L = LT , and H, E, F be real Before further proceeding, it is necessary to introduce the matrices with appropriate dimensions. If F T F ≤ I is satisfied, following definition in terms of stochastic stability. then L + HFE + (HFE)T < 0 holds, if and only if there exists Definition 1: The filter error system (9) is said to be a positive scalar ε > 0, such that L + ε−1 HH T + εET E < 0 or stochastically stable, if w(k) ≡ 0, and for any initial condition equivalently the following inequality holds: ξ(0), there exists a finite W > 0, such that the following ⎡ ⎤ inequality holds: L H εET ∞ ⎣ ∗ −εI 0 ⎦ < 0. 2 T ∗ ∗ −εI ξ(k)2 ξ(0) < ξ (0)Wξ(0). E k=0
Remark 7: Generally, there are several available definitions in terms of system stability, e.g., mean square stochastic stability, mean exponential stability, stochastic stability, and almost
A. Stability Analysis Theorem 1: Consider the filter error system (9). Given a scalar γ > 0, if there exist matrices R > 0, X > 0, Z > 0,
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a > 0 (a ∈ C) and matrices Yij ⎡ Lij H ⎣ ∗ −εI ∗ ∗ X ∗ where
(i, j ∈ S), satisfying ⎤ εEiT 0 ⎦ b. If this occurs, it is assumed that bk=a f (k) = 0, where f (k) is any function with respect to the variable k. This assumption makes sense in this paper and is not contradictory with any mathematical rules. Remark 9: It is worth mentioning that although sufficient conditions in terms of stochastic stability are presented in
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Theorem 1, unknown matrices from the Lyapunov–Krasovskii functional are cross-coupled with system matrices to be calculated. Therefore, it is impossible to directly complete the filter design using linear matrix inequality (LMI) techniques based on Theorem 1. To combat this obstacle, further sufficient conditions are presented in the following theorem. Theorem 2: Consider the filter error system (9). Given a scalar γ > 0, if there exist matrices R > 0, X > 0, Z > 0, Yij , a > 0, φa (i, j ∈ S, a ∈ C), satisfying (11), and the following LMIs and matrices Yij : ⎤ ⎡ Wij N εEiT ⎣ ∗ (18) −εI 0 ⎦ 0, X1 > 0, X3 > 0, Z > 0, R > 0 (i, j ∈ S, a ∈ C), and a scalar ω, satisfying ⎡ ⎤ X1 X2 Y1,ij ⎣ ∗ X3 Y2,ij ⎦ < 0 ∗ ∗ Z ⎡ ⎤ T Oij N+ εE+i ⎣ ∗ (19) −εI 0 ⎦
