Takagi–Sugeno Fuzzy-Model-Based Fault Detection for Networked

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Takagi–Sugeno Fuzzy-Model-Based Fault Detection for Networked Control Systems with Markov Delays Ying Zheng, Huajing Fang, and Hua O. Wang Abstract—A Takagi–Sugeno (T–S) model is employed to represent a networked control system (NCS) with different network-induced delays. Comparing with existing NCS modeling methods, this approach does not require the knowledge of exact values of network-induced delays. Instead, it addresses situations involving all possible network-induced delays. Moreover, this approach also handles data-packet loss. As an application of the T–S-based modeling method, a parity-equation approach and a fuzzy-observer-based approach for fault detection of an NCS were developed. An example of a two-link inverted pendulum is used to illustrate the utility and viability of the proposed approaches. Index Terms—Fault detection, fuzzy observer, Markov transfer matrix, networked control system (NCS), parity equation, Takagi–Sugeno (T–S) fuzzy model.

I. I NTRODUCTION A networked control system (NCS) refers to a control system whose feedback loop is closed through some network channels. Modeling, analysis, and design of NCSs have received increasing attention in recent years. In an NCS, sensor and/or controller data are transmitted through network channels. NCSs can be applied to a wide variety of engineering systems including manufacturing plants, aircrafts, automobiles, etc. In this correspondence, we consider NCSs as depicted in Fig. 1. Here, an NCS consists of a plant, sensors, actuators, and a controller, as in a typical control system. However, in an NCS, the sensor data packets reach the controller, and controller data packets arrive at the actuators via network channels. In such a setting, the network load and the limited communication bandwidth can cause network-induced delays. As shown in Fig. 1, such delays include the sensor–controller delay τSC and the controller–actuator delay τCA . The overall network-induced delay, which is also the transfer delay of data packets, can be computed by τ = τSC + τCA . Due to many uncertain factors, the network-induced delays are generally considered random. Therefore, an NCS is generally a time-varying system. A number of NCS modeling approaches have been developed in literature. For instance, Halevi and Ray [1], [2] constructed a timevarying discrete augmented NCS model. Zhang et al. [3], [4] presented a simplified discrete model for an NCS in which the controller and the actuators are event driven. Walsh et al. [5], [6] provided a continuoustime NCS model. In this correspondence, an approach based on a Takagi–Sugeno (T–S) fuzzy model is employed to represent a timevarying NCS with an event-driven controller and clock-driven actuators. As shown in Fig. 2, the actuator accepts the data packet that is delayed by several periods at each sampling instant. Thus, the networkinduced delay for this class of NCS is discrete and equals to integer

Manuscript received September 1, 2004; revised November 2, 2004, March 29, 2005, and August 10, 2005. This work was supported in part by the Chinese National Natural Science Foundation under Grant 60274014 and in part by the Hubei Natural Science Foundation under Grant 2005ABA252. This paper was recommended by Associate Editor W. J. Wang. Y. Zheng and H. Fang are with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]; [email protected]). H. O. Wang is with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China and also with the Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCB.2005.861879

sampling periods. A model named T–S model is proposed by Takagi and Sugeno to represent nonlinear systems [7]. In recent years, there has been a rapidly growing popularity of the applications of T–S fuzzy models [8]. The main feature of a T–S fuzzy model is to express the local dynamics of each fuzzy rule by a simple linear-system model. The overall fuzzy model of a system is achieved by fuzzy “blending” of the local models with membership functions [8]. In this correspondence, a T–S model is used to handle the time-varying aspect of an NCS. Comparing with existing NCS modeling methods, our modeling approach not only incorporates all possible network-induced delays but can also account for data-packet loss. As a result, the proposed modeling approach does not require the knowledge of exact values of the networkinduced delays. As an application of the T–S-based modeling method, the problem of fault detection for NCS is considered. A great deal of effort has been devoted to model-based fault detection approaches [9]–[11]. Among them, the parity-equation approach and the observer-based approach are of interest in this correspondence. In these methods the generation of the residual r(t) is a crucial issue [12]. In general, a fault is detected by comparing the residual with a threshold (generally zero). Based on the T–S-based NCS model, a parity-equation approach and a fuzzy-observer-based approach (FOA) for NCS fault detection are investigated in this correspondence. To begin with, a parity equation or a fuzzy observer for each local model is constructed. Then, the global residual is derived to decouple the disturbance and to detect the faults. The global residual is the fuzzy “blending” of the local ones. A number of papers have been published on fault detection of control systems using T–S model [13], [14]. But to our knowledge, none considered time-delay systems. This correspondence is organized as follows. In Section II, a T–Sbased model for an NCS is presented. Results on the computation of Markov transfer matrix are included. In Section III, based on a new modeling paradigm, the problem of fault detection for an NCS is addressed using a parity relation approach and an FOA. A detailed study of an example of a two-link inverted pendulum is provided in Section IV. Concluding remarks are collected in Section V.

II. T–S F UZZY M ODEL FOR NCS A. Local Model In this correspondence, we assume that the sensors and the actuators are clock-driven whereas the controller is event-driven, and the data packets reach the controller and the actuators by their original transmitting sequence if they are not lost. Some protocols satisfy the latter assumption, such as TCP/IP at the application level and IEEE 802.5 for token ring. Thus, in each sample period h, the actuator may receive several control data packets; whereas it only accepts the latest one as the plant input (as shown in Fig. 2). Assume that the transfer delay of the data packet, which is accepted by the actuator at the instant kh, is τk (∈ N ) periods, i.e., at the instant kh the actuators accept the sensor data packet that is sent out at the instant (k − i)h. Assume max(τk ) = n. Therefore, τk+1 is only affected by τk and is irrelevant to τ1 , . . . , and τk−1 . That is, {τ1 , τ2 , . . . , τk , . . .} construct a Markov chain [15]. Assume that the continuous state-space model of the plant dynamics is x(t) ˙ = Ax(t) + Bu(t) + Bd d(t) + Bf f (t) y(t) = Cx(t) + Df (t)

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Fig. 1.

925

Structure of NCS.

where µi (k) is the membership function. Hersh and Caramazza [16] and Stallings [17] give a point of view that regards the membership function as a probability function. Given a population of individuals U and a fuzzy event, each individual u ∈ U is asked whether it can call an event F or not. The likelihood P ( F  |u) is then obtained and represents the proportion of individuals that answered yes to the above question [18]. So it is obvious that µF (u) = P ( F  |u).

Fig. 2.

Time diagram for data packets.

where x(t) is the state vector; u(t) is the control vector, which is constant over sampling period h; y(t) is the plant output vector; d(t) is the disturbance (or the unknown input) vector; and f (t) is the fault vector, which includes the actuator faults and the sensor faults. Matrices A, B, Bd , Bf , C, D are of compatible dimensions. Discretizing (1), we can obtain the model of NCS at the instant kh as follows:




ˆ ˆ ˆd d(k) + B ˆf f (k) x(k + 1) = Ax(k) + Bu(k − τk ) + B ˆ ˆ y(k) = Cx(k) + Df (k)

(2)

ˆ B, ˆ B ˆ ,B ˆ , C, ˆ D ˆ are of compatible dimensions and where matrices A,  h As d f h  Ah ˆ ˆ ˆf = h eAs Bf ds, ˆ A = e , B = 0 e Bds, Bd = 0 eAs Bd ds, B 0 ˆ = C, and D ˆ = D. C B. Global Model The global model is the fuzzy fusion of the local ones. IF–THEN rules provide the relationship between the local models and the NCS global model. As τk has n different values, the number of fuzzy rules is also n. So the nonlinear system is regarded as the blending of n local linear models. Rule i(i = 1, 2, . . . , n − 1, n): IF τk is i, THEN the NCS model is




ˆ ˆ ˆd d(k) + B ˆf f (k) x(k+1) = Ax(k)+ Bu(k − i)+ B ˆ ˆ y(k) = Cx(k) + Df (k).

In this correspondence, U is regarded as the n values of τk and F is regarded as the ith local NCS model. According to this viewpoint, the membership µi (k) represents the probability of τk = i, i.e., µi (k) = P (τk = i). It satisfies

 n

i=1

∀i = 1, 2, . . . , n.



(a)

C. Model Probability µi (k) As described above, {τ1 , τ2 , . . . , τk , . . .} is a Markov chain. So P (τk+1 |τ1 , . . . , τk ) = P (τk+1 |τk ). Assume pi (i = 0, 1, . . . , n) denotes the probability of data packet with transfer delay i and pl denotes the probability of the data-packet loss. As shown in Fig. 2, it must take a certain time for a data packet to transfer between the sensor and the actuator. Thus, the delay τk cannot equal to 0, i.e., p0 = 0. Set P (τk+1 = j|τk = i) = pnj|i , i, j ∈ {1, 2, . . . , n}. We can n conclude that i=1 pi + pl = 1, j=1 pj|i = 1. The Markov state probability distribution is µ(k) = [µ1 (k) µ2 (k) · · · µn (k)], where µi (k) = P (τk = i). Thus, µ(k + 1) = µ(k) × T ; where the Markov transfer matrix T = [pj|i ] i, j ∈ {1, 2, . . . , n} and i, and j are the row and line numbers, respectively. If the initial probability distribution µ(0) is known, then µ(k) can be achieved according to pj|i . Theorem 1 extends the results in [15] to obtain pj|i . Theorem 1: If max(τk ) = n, then pj|i = 0

(3) p(i+1)|i =

(b)

(4)

(6)

0 ≤ µi (k) ≤ 1

n 

∀i + 2 ≤ j ≤ n pk + pl

k=i+1 n



∀i ≤ n − 1

pk + pl

k=i

ˆd d(k) + B ˆf f (k) +B

 n     ˆ ˆ (k)  + Df µi (k) Cx(k)  y(k) =

µi (k) = 1

i=1

Therefore, the global model of NCS can be concluded as follows:

 n    ˆ ˆ  + Bu(k − i) x(k + 1) = µi (k) Ax(k)    i=1

(5)

pj|i =

pi + pl

n 

n 

×

pk + pl

k=i

∀j ≤ i ≤ n.

pi−1 + pl

n 

k=i−1

pk + pl

× ··· ×

pk + pl

k=j n



k=j−1

pk + pl

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Proof: As shown in Fig. 2, τk = i means that, at the instant kh, the actuators accept the sensor data packet that is sent out at the instant (k − i)h. Since the data packets are transferred according to their original transmitting sequence; if the next sensor data packet that is sent out at the instant (k − i + 1)h is not lost, its transfer delay must be larger than i − 1 periods. It means that the next data packet may be delayed by i, i + 1, · · · , or n periods or lost if τk = i. The probabilities are pi , pi+1 , . . . , pn , and pl , respectively. Set τk+1 = j. Based on the probability formula P (A|B) = P (AB)/P (A), we will discuss pj|i according to the transfer delay of the sensor data packet that is sent out between the instant (k − i)h and (k − j + 1)h as follows. 1) If τk+1 = j > i + 1, it is obvious that the data packets cannot arrive at the actuators by their original transmitting sequence. So when ∀i + 1 < j ≤ n, pj|i = 0. 2) If τk+1 = j = i + 1; then at the instant (k + 1)h, actuators will still accept the sensor data packet that is sent out at the instant (k − i)h. Therefore, if the next sensor data packet that is sent out at the instant (k − i + 1)h is not lost, its transfer delay must be larger than i periods. Consequently

According to Theorem 1, |T | < 1, we conclude that {µ(1), µ(2), . . .} is convergent. Therefore, µ(∞) = µ(∞) × T where µ(∞) = [µ1 (∞) µ2 (∞) · · · µn (∞)] = limk→∞ µ(k). III. F AULT -D ETECTION A PPROACH FOR NCS A. Parity Relation Approach The parity equation is relevant to the control vector u and the plant output y. First, we construct the parity equation on local models. z-transforming (3), we obtain ˆ −i u(z) + (zI − A) ˆd d(z) ˆ −1 B ˆ −1 Bz x(z) = (zI − A) −1 ˆ ˆ + (zI − A) Bf f (z) ˆ ˆ (z) y(z) = Cx(z) + Df ˆ −i u(z) + C(zI ˆd d(z) ˆ ˆ −1 B ˆ ˆ −1 Bz − A) = C(zI − A)  −1 ˆ ˆ B ˆf + D ˆ f (z). − A) + C(zI

 ˆ ˆ  x(k) = Ax(k − 1) + Bu(k − i − 1)

n 

=

pk

k=i

pk +pl

ˆ −i u(z) − C ˆd d(z) ˆA ˆ−1 zx(z) − C ˆA ˆ−1 Bz ˆ Aˆ−1 B y(z) = C ˆf − D)f ˆ (z). ˆ Aˆ−1 B −(C

pk +pl

pi

n 

pk



pk + pl ∀i ≤ n − 1.

pk + pl

ˆd d(z) = B ˆ Aˆ−1 zx(z) − B ˆ Aˆ−1 Bz ˆ −i u(z) ˜d C ˜d C B −1 ˆf − D)f ˆ Aˆ B ˆ (z) − B ˜d y(z) ˜d (C −B

k=i

3) If τk+1 = j ≤ i, then at the instant (k + 1)h, the actuators will accept the sensor data packet that is sent out at the instant (k − j + 1)h. So if the sensor data packets that are sent out at the instant (k − i + 1)h, (k − i + 2)h, . . . , (k − j)h, and (k − j + 1)h are not lost; their transfer delay must be i, i − 1, . . . j + 1, and j periods, respectively. Furthermore, if the sensor data packet sent out at the instant (k − j + 2)h is not lost, its transfer delay must be larger than j periods. Thus pj|i =

P (τk = i, τk+1 = j) P (τk = i) pi

n 

=

pk

×  n

k=i

n 

pi−1 +pl

pi +pl pk +pl

×  n

k=i

pk +pl

× ··· × 

pk +pl

k=j−1

pi

n

pi + pl

 n

×

pk + pl

k=i

pi−1 + pl

 n

k=i−1

pk + pl

(11)

ˆd )+ . ˜d = B ˆd (C ˆ Aˆ−1 B where B Combining (8) and (11), we get





ˆ Aˆ−1 + I)zI − Aˆ x(z) = −B ˆ Aˆ−1 ) ˜d C ˜d y(z) + (I − B ˜d C (−B  ˆf − B ˆf − D) ˆ −i u(z) + B ˜d (C ˆ Aˆ−1 B ˆ f (z). (12) × Bz

ˆ Aˆ−1 + I)zI − A. ˜ = (−B ˜d C ˆ Combining (12) into (8), we Set A obtain ˜d )y(z) − C ˆ Aˆ−1 )B ˆ ˜−1 ˜d C ˆi z −i u(z) ˆA ˜−1 B (I + C  −1  A (I − B−1 ˜ ˆ ˆ ˆ ˆ f (z). ˆ ˜ ˆ ˆ = CA Bf − Bd (C A Bf − D) + D

(13)

Thus, the local residual is defined as follows. It is only relevant to actuator and sensor faults. Rule i(i = 1, 2, . . . , n − 1, n): IF τk is i, THEN



n 

× ··· ×

(10)

˜d )y(z) ˆ A˜−1 B ri (z) = Qi (z) (I + C −1 ˆ Aˆ−1 )B ˆ ˜ ˜d C ˆi z −i u(z) − C A (I − B

pk

k=i

=

pk +pl

k=j n

k=i−1



(9)

Assume NCS local models (3) satisfy the necessary condition of ˆd ) [19]. Since A ˆ = eAT ˆB ˆd ) = rank(B disturbance decoupling rank(C −1 ˆ ˆ ˆ is full rank, the pseudoinverse of C A Bd exists. Thus, the following equation can be concluded from (10):

k=i

k=i+1 n

ˆf f (k − 1) (a) ˆd d(k − 1) + B +B ˆ ˆ (k − 1) (b). y(k − 1) = Cx(k − 1) + Df

Combining (9a) and (9b) and z-transforming the result, we achieve

k=i

n 

=



k=i+1

×  n

pi

(8)

The following equation can also be inferred from (3):

pj|i = p(i+1)|i P (τk = i, τk+1 = i + 1) = P (τk = i) n 

(7)

pk + pl

k=j n



= Grf f (z)

pk + pl

k=j−1

∀j ≤ i ≤ n.


(14)

ˆF − D)) ˆ A˜−1 (B ˆf − B ˜d (C ˆ Aˆ−1 B ˆ + D]. ˆ The where Grf = Qi (z)[C coefficient Qi (z) can add design freedom to achieve robust control, fault isolation, etc. It can be designed by the optimization method, such as H∞ optimization [20]. In addition, a directional residual vector can be used here to achieve fault isolation.

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The global residual is a fuzzy combination of local ones. It is constructed as the weighted sum of local residuals as follows: r(z) =

n



˜d )y(z) ˆA ˜−1 B µi (k)Qi (z) (I − C

i=1



ˆA ˆ−1 )B ˆ A˜−1 (I − B ˜d C ˆi z −i u(z) . (15) −C Thus, the faults can be diagnosed using a simple logic |r(k)|

∼

= 0 Normal > 0 Faulty.

(16) Fig. 3. Two-link inverted pendulum.



B. Fuzzy-Observer-Based Approach A fuzzy observer can be designed for fault detection based on ˆ C) ˆ is observable, which T–S model [21]. Assume that the system (A, means all local models are observable, and each local fuzzy observer is constructed according to its NCS local model. All local observers combine to form a global one. It is associated with its corresponding fuzzy rules as follows. Rule i(i = 1, 2, . . . , n − 1, n): IF τk is i, THEN the NCS fuzzy observer is




ˆ ˆ x ˆ(k + 1) = Ax(k) + Bu(k − i) + Li (y − yˆ) ˆx yˆ(k) = C ˆ(k)

(18)

e(k+1) = x(k + 1) − x ˆ(k + 1)





ˆ ˆ ˆ ˆ ˆ µi (k) (A−L i C)e(k)+ Bd d(k)+(BF −Li D)f (k) .

i=1

(19) The stability condition of NCS fuzzy observer (18) is given in the following corollary. Corollary 1 [22]: The NCS fuzzy observer (18) is asymptotically stable if there exists a common positive definite matrix P such that ˆ T P + P (A ˆ < 0, ˆ − Li C) (Aˆ − Li C)

i = 1, 2, . . . , n.

ˆd = 0. A B

0  9.8 x˙ =  0 −9.8 y=

1 0

0 0

0 1







0 x 0

(21)



1.005 0.01 1.0005  0.0980 x(k + 1) =  −0.0005 0 −0.0981 −0.0005





−0.0005 0 −0.0981 −0.0005  x(k) 1.0015 0.01  0.2942 1.0015



−0.01 0.03  −0.0099 0.0299  + u(k − i) −0.01 0.03  −0.0101 0.0303










−0.1 0  0.1  0 + d(k) +  0.1  0 0 0

ˆ ˆ (z) = QCe(z) + QDf (20)

n ˆ −1 B ˆd and Grf = ˆ − Aˆ + i=1 µi (k)Li C) where Grd = QC(zI  n −1 ˆ ˆ ˆ ˆ ˆ ˆ QD + QC i=1 µi (k)(zI − A + Li C) (Bf − Li D). To design a disturbance decoupled residual, we need to satisfy Grd = 0 and Grf = 0. The following corollary is designed to achieve Grd = 0.



1 0 0 0 0 0 −9.8 0   1 −2  x+ u 0 0 1 0 0  0 29.4 0 −2 5

where all point masses m1 = m2 = 1 kg, all links have length l1 = l2 = 1 m, and u1 , u2 denote torques about the respective pivots, the system output yi = θi , i = 1, 2. Assume that h = 0.01 s and n = max(τk ) = 3. After discretizing (21), we obtain a T–S-based fuzzy model according to (2). Rule i(i = 1, 2, 3): IF τk is i, THEN the NCS model is

r(z) = Q [y(z) − yˆ(z)]



ˆd = 0 HB



Thus, the residual can be constructed as follows:

= Grd d(z) + Grf f (z)

2)

A two-link inverted pendulum (as shown in Fig. 3) is considered here. It has the following dynamics [23]:



Combining (4) and (18), we obtain the error dynamics as follows:

n

HA = 0


Equation (16) can also be adopted here to realize fault detection. Moreover, fault isolation can be achieved with a structured residual set designed by a dedicated fuzzy-observer scheme.

i=1

=

ˆd = 0 HB

IV. E XAMPLES

 n   ˆx(k) + Bu(k ˆ  − i) + Li (y − yˆ) µi (k) Aˆ  xˆ(k + 1) = i=1

1)

(17)

where Li is the observer gain matrix. The overall fuzzy observer is the weighted sum of the individual local ones

n   ˆx ˆx  yˆ(k) = µi (k)C ˆ(k) = C ˆ(k).

ˆ ˆ and A = A ˆ − n µi (k)Li C. Corollary 2 [9]: Set H = QC i=1 Sufficient conditions for satisfying the disturbance decoupling requirement Grd = 0 are either of the following:



y(k) =

1 0

0 0

0 1





0 0 0 0

1 0 0 0



0 0 f (k) 1 0



0 1 0 0 0 x(k) + f (k) 0 0 1 0 0

where f (k) comprises of two sensor faults and two actuator faults, i.e., f (k) = [fs1 (k) fs2 (k) fa1 (k) fa2 (k)]T ; and the unknown disturbance d(k) represents a random sequence here.

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Fig. 4. Residual generated by the parity relation approach. (a) r1 (solid line) and (b) r2 (dashed line).

Fig. 5. Residual generated by the FOA. (a) r1 (solid line) and (b) r2 (dashed line).

Assume that max(τk ) = 3, pl = 0.05, and µ(0) = [0.3 0.4 0.3]. According to Theorem 2, the Markov transfer matrix is

where r1 (k) is only relevant to fs1 (k), and r2 (k) is only relevant to fs2 (k). Using MATLAB, we depict the simulation curves of r1 (k) and r2 (k) in Fig. 4(a) and (b), respectively. When there are no faults in NCS, the residual is shown in Fig. 4(a). Both r1 and r2 fluctuate near the zero region. When two step faults with amplitude 10 are added—one is added to sensor 1 at 0.4 s and the other is added to sensor 2 at 0.6 s, respectively—the residual will change at those times [see Fig. 4(b)].



T =

0.35 0.225 0.225



0.65 0 0.4179 0.3571 . 0.4179 0.3571

After 50 sampled periods, µ(50) = [0.2571 0.4776 0.2653]. Since µ(50) ≈ µ(50) × T , we can conclude that µ(50) ≈ µ(∞) by (7). The two fault-detection approaches are given below.

B. Fuzzy-Observer-Based Approach A. Parity Relation Approach





0.5002 1.0050 0.5002 2.0100 . Accord0.4998 −1.0050 0.4998 −2.0100 ing to (15), the residual r(k) becomes Choose Q =



r(k) =

 =

r1 (k) r2 (k)



0.0403 −0.0203

L1 = L2 = L3



=





1 0 −0.9053 −0.4942 y(k + 1) + y(k) 0 1 −0.0992 −0.5068



+



A fuzzy observer can be constructed by (18). According to Corollary 1 and Corollary 2, Li (i = 1, 2, 3) and Q can be chosen as follows:



−0.1210 (µ1 (k)u(k − 1) + µ2 (k)u(k − 2) −0.0610 + µ3 (k)u(k − 3))

 Q=

−0.0688 0.0193



−136.6286 0.0113 1.8424 −2.6286 −0.1686 −976.4336

T

1 1 . 1 1

The simulation curves of r1 (k) and r2 (k) are depicted in Fig. 5(a) and (b), respectively. When there are no faults in NCS, the residual is shown in Fig. 5(a). Both r1 and r2 fluctuate near the zero region. When two step faults with amplitude 10 are added—one is added to sensor 1

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at 0.4 s and the other is added to sensor 2 at 0.6 s, respectively—the residual is shown in Fig. 5(b). V. C ONCLUSION In this correspondence, a new NCS modeling approach—a T–Sbased fuzzy model—is presented. Comparing with existing NCS modeling methods, our method has the following advantages. First, the method does not require the knowledge of the exact values of network-induced delays. Second, it addresses situations involving all possible network-induced delays. Third, it handles the data-packet loss. Within the new NCS modeling framework, we have obtained formulas for computing the Markov transfer matrix for the associated NCS problem. On the basis of this modeling method, the parity relation approach and the FOA are presented to detect the sensor/actuator faults. As discussed in Section III, it is easy to construct the parity equation. But it is hard, sometimes even impossible, to design the equation coefficients. When using the FOA, one or several observers must be constructed. Thus, it is more complicated to construct the observers than the parity equation. However, the design of the observer coefficients is relatively easier. Thus, we will choose a different approach according to a different situation. Much work should be done to apply this T–S-based model to more complex NCSs. For example, a nonlinear plant will add the complexity of the modeling process; an event-driven actuator will make τk not always an integer; and data packets that are not accepted by its original transmitting sequence will not have the Markov characteristic. Those are the directions of the future research work. R EFERENCES [1] Y. Halevi and A. Ray, “Integrated communication and control systems: Part I—Analysis,” J. Dyn. Syst. Meas. Control, vol. 110, no. 4, pp. 367–373, Dec. 1988. [2] A. Ray and Y. Halevi, “Integrated communication and control system: Part II—Design considerations,” J. Dyn. Syst. Meas. Control, vol. 110, no. 4, pp. 374–381, Dec. 1988. [3] W. Zhang, M. S. Branicky, and S. M. Philips, “Stability of networked control system,” IEEE Control Syst. Mag., vol. 21, no. 1, pp. 84–99, Feb. 2001. [4] M. S. Branicky, S. M. Phillips, and W. Zhang, “Stability of networked control systems: Explicit analysis of delay,” in Proc. Amer. Control Conf., Chicago, IL, 2000, pp. 2352–2357.

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