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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 3, MAY 2013

Communication-Delay-Distribution-Dependent Decentralized Control for Large-Scale Systems With IP-Based Communication Networks Chen Peng, Qing-Long Han, and Dong Yue

Abstract— This paper investigates the decentralized control for a large-scale system with an IP-based communication network. The decentralized controller design specifically takes the nonuniform communication-delay-distribution characteristic of the IP-based communication network into account. First, a networked decentralized control modeling for the large-scale system is proposed. The two main points are: 1) a decentralized controller is designed, which does not depend on the full-order state of the system and 2) the nonuniform communicationdelay-distribution characteristic of the IP-based communication network is fully considered in the controller design. Second, if the probability distribution of communication delay is known a priori in the design process, sufficient stability and stabilization conditions for the networked large-scale system are derived in terms of linear matrix inequalities. It shows that, the solvability of the design depends not only on the probability distribution of the communication delay but also on the network topology. Finally, the design method is applied to two pendulums coupled by a spring and a quadruple-tank process. Simulation results demonstrate the effectiveness of the proposed method. Index Terms— Decentralized control, large-scale systems, linear matrix inequalities, networked control systems (NCSs), probability distribution.

I. I NTRODUCTION

L

ARGE-SCALE systems have become more and more important in an ever-increasingly interconnected technological society, which has many successful applications, for example, electrical power systems, computer communication systems, and economic systems [1]–[3]. Networked

Manuscript received August 21, 2011; accepted April 12, 2012. Manuscript received in final form April 18, 2012. Date of publication May 15, 2012; date of current version April 17, 2013. This work was supported in part by the Australian Research Council Discovery Project under Grant DP1096780, the Research Advancement Awards Scheme Program (January 2010– December 2012) at Central Queensland University, Australia, the National Science Foundation of China, under Grant 61074024 and Grant 61074025, and the Natural Science Foundation of Jiangsu Province of China, under Grant BK2010543. Recommended by Associate Editor Z. Wang. C. Peng is with the Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton QLD 4702, Australia, and also with the School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China (e-mail: [email protected]). Q.-L. Han is with the Centre for Intelligent and Networked Systems and the School of Information and Communication Technology, Central Queensland University, Rockhampton QLD 4702, Australia (e-mail: [email protected]). D. Yue is with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, 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/TCST.2012.2196573

control systems (NCSs) use serial communication channels to exchange information and control signals between spatially distributed system components and have several advantages, such as cost effectiveness, reduced weight and power requirements, simple installation, and maintenance [4]. However, due to the bandwidth limitation and other characteristics of shared communication networks, NCSs are quite different from traditional point-to-point control systems, which has motivated a lot of interesting research, thereby leading to a significant number of publications, to name a few, see [2], [5]–[8] and references therein. Notice that there are two limitations in some existing researches. First, the system under consideration generally is a linear system with single packet transmission [4] and [9]. However, if the sensors are geographical distribution, e.g., for a large-scale system, the nodes are located in different places, then it will be very difficult to lump all of the sampled signals into a single packet to transmit over the identical communication channel. Therefore, the multipacket transmission scheme is more appropriate to be employed in large-scale systems. Second, assume that the controller depends on all of the system’s state and/or output. These assumptions cannot be directly generalized to large-scale control systems because the multipacket transmission scheme in a large-scale system cannot ensure all the packets arrive at the controller at the same time. Therefore, although some theoretical results including those in [9]–[13] are important and have some significant technical merits, there are still major difficulties in their applications to large-scale control systems with multipacket transmissions [6]. To stabilize a large-scale control system in traditional point-to-point connections, generally speaking, there are three types of control strategies in the literature, that is, centralized, quasi-decentralized, and decentralized control strategies. The centralized control requires that all measurements must be collected and sent to a central unit for processing simultaneously [2]. Under a decentralized control strategy, a set of local controllers are connected via a communication channel to each distributed sub-system, and there is no signal transfer between different local controllers [14]. When a decentralized control cannot provide the required performance and to avoid the complexity associated with the traditional centralized control under network environments, a quasi-decentralized control scheme may be used, where the quasi-decentralized control refers to a situation in which most signals used for control are collected and processed locally, while some signals still need to be

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PENG et al.: COMMUNICATION-DELAY-DISTRIBUTION-DEPENDENT DECENTRALIZED CONTROL

transferred between local plants and/or controllers [2]. However, considering to the spatially distributed system components, the multipacket transmission scheme must be employed due to the fact that the multipacket transmission scheme cannot guarantee that all of the transmitted information arrive at their destinations at the same time. So, it is still a problem to the centralized and quasi-decentralized control strategies since they depend on all/part of the coupled measurements. Therefore, the decentralized control strategy will be adopted in this paper. Many different network types have been promoted for using in NCSs. For example, the Ethernet with carrier sense multiple access with collision detection, token-passing bus (e.g., ControlNet), and controller area network bus (e.g., DeviceNet) [15] and [16]. Those based on internet protocol (IP) networks are generally employed to control wide area plant automation, since an IP networking technology is convenient and low-cost for communication over a large geographical area. However, the statistical features of IP-based communication networks have not been fully explored in current research [17]–[19]. It is evident that a better control performance may be achieved if the statistical features of network behaviors are fully utilized in the analysis and design of NCSs [20]. Subsequently, how to effectively utilize the nonuniform information of communication delay in the analysis and synthesis of large-scale systems with multipackets transmission scheme is still an open problem. This motivates the present study. The purpose of this paper is to develop a new communication-delay-distribution-dependent decentralized control framework for a class of large-scale systems controlled over an IP-based communication network. Our main idea is to transfer the modeling of a large-scale system to a group of linear systems with the decoupled structural and coupled nonlinear uncertainties, while the structural and nonlinear uncertainties are bounded by the practical operating-range conditions, and the multipacket transmission and synchronous sampling are unified in a decentralized control framework in network environments. Compared with the centralized control scheme in the literature, the decentralized control scheme is independent of all of other subsystems’ information. Two communication parameters-dependent stability and stabilization criteria for system analysis and synthesis are derived in a linear matrix inequalities (LMI) framework, by constructing a novel Lyapunov–Krasovskii functional and making full use of the probability distribution information of the IP communication network. Moreover, the designed decentralized controller depends not only on the statistic distribution characteristic of the IP-based communication network but also on the NCS’s configuration. If there are communication networks among the sensors and controllers and the controllers and the actuators, the identical controller gains may be employed to make the control scheme feasibility. If there is only a communication network between the sensors and the controllers, the different controller gains may be employed in different subintervals to reduce the conservativeness. The main contributions of this paper are as follows.

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1) A decentralized control framework for a large-scale system controlled over an IP communication network is proposed. Compared with the centralized control scheme adopted in the literature, the requirement of signalpacket transmission among the distributed subsystems is no longer required. 2) The analysis and synthesis criteria for the system under consideration not only capture the nonuniform distribution characteristic of an IP communication network but also depend on the topology of the communication network. The decentralized control scheme with the identical or different controller gains is decided by whether or not there is a communication between the controllers and actuators. The rest of this paper is organized as follows. Section II presents a decentralized control model for a large-scale system to incorporate the specific characteristics of the IP-based communication delays. Section III investigates the stability and stabilization for the system modeled in Section II. Two numerical examples are given in Section IV. Finally, Section V concludes this paper. Notation: Throughout this paper, N stands for set of positive integers. Rn denotes the n-dimensional Euclidean space, Rn×m is the set of n × m real matrices, and I is the identity matrix of appropriate dimensions. For X ∈ Rn×n , the notation X > 0 (respectively, X ≥ 0) means that the matrix X is a real symmetric positive definite (respectively, positive semi-definite). For an arbitrary matrix B and two symmetric matrices A and C A ∗ B C denotes a symmetric matrix, where ∗ denotes the entries implied by symmetry. II. S YSTEM AND P ROBLEM D ESCRIPTIONS A. Large-Scale System Modeling Consider a large-scale system [21], [22] S : x(t) ˙ = Ax(t) + Bu(t) + g(t, x(t)) + f (t, x(t), u(t))

(1)

which is an interconnection of N subsystems Si : x˙i (t) = Ai x i (t) + Bi u i (t) + gi (t, x i (t)) i = 1, 2, . . . , N Rn i

(2)

is the state and u i (t) ∈ is the input where x i (t) ∈ of (2), and Ai and Bi are constant matrices of appropriate dimensions, which constitute stabilizable pairs (Ai , Bi ). In (1), x(t) = (x 1T (t), x 2T (t), . . . , x NT (t))T ∈ Rn is the state, u(t) = (u 1T (t), u 2T (t), . . . , u TN (t))T ∈ Rm is the input of the interN N connected system (1), where n = i=1 n i , m = i=1 m i , g : R × Rn → Rn and f : R × Rn × Rm → Rn express nonlinear functions. The system matrices in (1) are defined as A = diag{A1 , A2 , . . . , A N }, B = diag{B1 , B2 , . . . , B N }. In (2), the gi (t, x i (t)) represents the structured uncertainty and is independent of other subsystem’s information. Similar to [21] and [22], we assume that gi (t, x i (t)) satisfies gi (t, x i (t))2 ≤ G i x i (t)2

Rm i

∀ (t, x i (t)) ∈ R × Rni

(3)

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Fig. 1.

Typical NCS setup for decentralized control.

where gi : R × Rni → Rni is a vector component of g = (g1T , g2T , . . . , g TN )T and G i ∈ Rni ×ni . For the decoupled nonlinear function g(t, x(t)), from (3), one can get g(t, x(t))2 ≤ Gx(t)2

∀ (t, x(t)) ∈ R × Rn

(4)

where G = diag{G 1 , G 2 , . . . , G N }. For the coupled interconnected function f (t, x(t), u(t)), we assume that f (t, x(t), u(t))2 ≤ H1 x(t)2 + H2u(t)2

(5)

where H1 ∈ Rn×n , H2 ∈ Rm×m , for all (t, x(t), u(t)) ∈ R × Rn × R m . It is assumed that g(t, x(t)) and f (t, x(t), u(t)) are sufficiently smooth so that the solution of (1) exists and is unique for all initial conditions and all fixed inputs u, g(t, 0) = 0, f (t, 0, 0) = 0, and x = 0 is assumed to be the unique equilibrium of S when u = 0. For convenience, in the sequel, we use g and f to denote g(t, x(t)) and f (t, x(t), u(t)), respectively. Throughout this paper, we assume that (1) is controlled through an IP-based network and the system state is available for feedback control, and that the quality of service (QoS) provided by the network is not perfect. Therefore, there are communication delays, packet dropouts, and disorder packets in the communications. B. Networked Decentralized Control for a Large-Scale System A networked control structure for a large-scale system (1) is shown in Fig. 1, where solid lines represent physical links and broken lines for signal flows, and the decentralized controller is adopted for the large-scale system where there are multisensing sources and multiactuator nodes. In Fig. 1, each labeled “Sensor”, such as “Sensor 1”, represents a sensing source of a group of sensors, which can send several measured signals from one location in one data packet. In contrast to a typical NCS structure with the centralized controller in [22], the decentralized controller is employed in Fig. 1. Moreover, if there is only one sensing source and only one actuator node, this becomes one sensor and one actuator structure with single packet transmission in [9]–[12]. Under the proposed framework, we need the following assumptions. Assumption 1: Multipacket transmission is used for the system under consideration. That is, for different sensor nodes

located in different places, the measurements taken there are sent via different packets. Assumption 2: Each data packet is time stamped and with the identifier, where the role of identifier is used to identify the sequence number of the subsystem. The sensors are timetriggered, i.e., the system states are sampled periodically at a constant period h. Moreover, the sampling instances tk h are synchronized among the sensors in different subsystems, where tk (k = 1, 2, 3, . . .) are some integers such that {t1 , t2 , t3 , . . .} ⊂ {0, 1, 2, 3, . . .}. Assumption 3: The decentralized controller is eventtriggered. It calculates the control signal as soon as it receives a packet from its uplink. Upon the completion of the calculation, the control packet is transmitted to its downlink. Moreover, there is a buffer among the subsystems to choose and store the largest communication delay ηtk in different subsystems, i.e., ηtk = max νtik (i = 1, 2, . . . , N), where νtik is the communication delay in subsystem i at sampling instant tk h. Communication delays in every subsystems are chosen as ηtk in the modeling. Assumption 4: The actuators are event-triggered with a serial of logic zeroth-order-hold (ZOH). The function of a logic ZOH is to select and to store the latest control signal based on the time-stamps of received packets. Based on the above assumptions, the following decentralized networked state feedback controller is designed for subsystem Si , i = 1, 2, . . . , N: u i (t + ) = ki x i (tk h), t ∈ {tk h + ηtk , k = 1, 2, . . .}

(6)

where u i (t + )=limtˆ→t +0 u i (tˆ), ki is controller gain to be determined. Define τ (t) = t − tk h, t ∈ = [tk h + ηtk , tk+1 h + ηtk+1 ).

(7)

From Assumption 2, it is known that each data packet is time-stamped. Then the controller can choose the latest timestamped packet as the input. In this sense, the condition of tk+1 h + ηtk+1 > tk h + ηtk is guaranteed in (7). Although, the subsystems use the different channels of the same network to transmit the sampled signal, it is reasonable to assume that there are the identical statistic characteristics of the packets transferred in these communication channels. Also, it is known that to a specified communication network, communication delay and the number of consecutive packet losses are bounded [23]. From (7), we have τ1 ≤ ηtk ≤ τ (t) < (tk+1 − tk )h + ηtk+1 ≤ τ3

(8)

where τ1 = inf k {ηtk }, τ3 = supk [(tk+1 − tk )h + ηtk+1 ]. Since τ3 is dependent on the communication delay and packet dropout, this enables that all kinds of nonideal network QoS can be incorporated in an integrated maximum allowable equivalent delay bound τ3 [24]. From (6) and (7), the control output in the logic ZOH can be represented by u i (t + ) = ki x i (t − τ (t))

t ∈ , i = 1, 2, . . . , N.

(9)


From (9), one can see that the decentralized controller for (1) can be written as u(t + ) = K x(t − τ (t))

t ∈

(10)

where K is of the compatible block-diagonal structure, that is, K = diag{k1, k2 , . . . , k N }, x(t − τ (t)) means col{x 1(t − τ (t)), . . . , x N (t − τ (t))}. Based on (1) and (10), the dynamics of the system is x(t) ˙ = Ax(t) + B K x(t − τ (t)) + g + f,

t ∈ .

(11)

Remark 1: In Fig. 1, the signal transfers from the sensors to the controllers and from the controllers to the actuators are not perfect. For example, there are communication delay, possible data packet dropout and/or out-of-order. However, because of the time stamp, the logic ZOH guarantees that tk+1 > tk . In other words, the out-of-order packets are discarded in the actuators. Moreover, notice that in (8) the communication delay and the delay due to the packet dropouts are included into the overall τ (t). Therefore, the data dropout and communication delay have been included in (10). C. Delay-Distribution-Dependent Modeling It is known from (8) that the overall equivalent delay τ (t) is an interval time-varying delay and has its lower and upper bounds. Furthermore, it is known that communication delays in the IP-based NCS are of nonuniform distribution [18] and a multifractal nature [19], [20]. From (7), it is seen that the τ (t) inherits the similar property from the communication delay. That is, the probability of τ (t) ∈ [τ1 , τ2 ) does not equal to the probability of τ (t) ∈ [τ2 , τ3 ), where τ2 ∈ [τ1 , τ3 ) is a value in a statistical sense to distinguish that the delay belongs to the large or small intervals with the different probability. To fully use this statistic characteristic of the IP-based networks and assume there is a network between the sensors and controllers, the following delay-distribution-dependent feedback control law (12) is therefore adopted to replace the general form of the control law given in (10) u(t) = δ(t)K 1 x(t − τ1 (t)) + (1 − δ(t))K 2 x(t − τ2 (t)) t ∈

(12)

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From (11) and (12), we have the following closed-loop system model: x(t) ˙ = Ax(t) + δ(t)B K 1 x(t − τ1 (t)) +(1 − δ(t))B K 2 x(t − τ2 (t)) + g + f, t ∈ .

(15)

Remark 2: Notice that (15) makes use of information about probability distribution of an IP communication network. Therefore, less conservative results may be expected than those without considering this characteristic. This will be demonstrated by a numerical example in Section IV. Remark 3: If there is a network between the controllers and the actuators, since the communication delays between the controller and the actuators are unknown to the controller, the τ (t) in (13) is unknown to the controller. As a result, it is difficult for the controller to determine how to choose K 1 and K 2 based on the unmeasurable τ (t). In this case, (12) may be redesigned as u(t) = δ(t)K x(t − τ1 (t)) + (1 − δ(t))K x(t − τ2 (t)) t ∈ .

(16)

However, since the same controller gains are used in (16), the less conservative results may be obtained based on (12) than those based on (16), this will be shown in Section IV. The lemma below is an improved version of Jensen’s integral inequality [25]. It gives a tighter bound to deal with integral terms. The proof of Lemma 1 is similar to those in [26], due to the limited space, it is omitted here. Lemma 1: For any constant matrices Ri ∈ Rn×n and Ui ∈ n×n R Ri ∗ i = ≥0 Ui R i scalars 0 ≤ τ1 ≤ τ (t) ≤ τ3 , and vector function x˙ : [−τi+1 , −τi ] → Rn , i = 1, 2, such that the following integrations are well defined, then for τi ≤ τ (t) ≤ τi+1 , we have t −τi ξ T (t)iT i+1 i ξ(t) − x˙ T (v)Ri+1 x(v)dv ˙ ≤− (17) τi+1 − τi t −τi+1 where ξ(t) = col[x(t), x(t − τ1 ), x(t − τ1 (t)), x(t − τ2 ),

where τ1 (t) = δ(t)τ (t), δ(t) = 1,

if τ (t) ∈ [τ1 , τ2 )

τ2 (t) = (1 − δ(t))τ (t), δ(t) = 0,

if τ (t) ∈ [τ2 , τ3 ).

x(t − τ2 (t)), x(t − τ3 ), g, f ] 0 1 −1 0 0 0 0 0 1 = 0 0 1 −1 0 0 0 0 0 0 0 1 −1 0 0 0 2 = . 0 0 0 0 1 −1 0 0

(13)

The τ1 (t) and τ2 (t) in (12) are important in the analysis and synthesis of the networked closed-loop system (11). τ1 (t) means that the distribution of τ (t) is within a lower-end range of [τ1 , τ2 ), while τ2 (t) means that the distribution of τ (t) is within a higher-end range of [τ2 , τ3 ). Similar to [19] and [20], it is also assumed that δ(t) in (12) is a Bernoulli-distributed sequence with Prob{δ(t) = 1} = E{δ(t)} := δ¯ (14) Prob{δ(t) = 0} = 1 − E{δ(t)} := 1 − δ¯ where 0 ≤ δ¯ ≤ 1 is a constant and is determined by the chosen network and the NCSs configuration.

III. M AIN R ESULTS In this section, we develop an approach for stability analysis and controller synthesis of (1). Now, we first state and establish the following stability analysis result. The proof can be seen in the appendix. Theorem 1: For some given constants 0 < τ1 < τ2 < τ3 , 0 < δ¯ < 1 and matrices K 1 and K 2 , (15) is asymptotically stable in the mean square if there exist real matrices P > 0, Q i > 0, Ri > 0 (i = 1, 2, 3), scalars εi > 0 (i = 1, 2),

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and matrices U j ( j = 2, 3) with appropriate dimensions such that ⎡ ⎤

11 ∗ ∗ ∗ ⎢ 21 22 ∗ ∗ ⎥ ⎢ ⎥ (18) ⎣ 31 0 33 ∗ ⎦ < 0

41 0 0 44 Rj ∗ j = ≥ 0, j = 2, 3 (19) Uj Rj where

⎡

11 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ 21 22 ∗ ∗ ∗ ∗ ⎢ 31 32 33 ∗ ∗ ∗ ∗ ∗ ⎢ ⎢ 0 42 32 44 ∗ ∗ ∗ ∗

11 = ⎢ ∗ ∗ ⎢ 51 0 0 54 55 ∗ ⎢ 0 0 0 64 54 66 ∗ ∗ ⎣ P 0 0 0 0 0 −ε1 I ∗ P 0 0 0 0 0 0 −ε2 I

21 =

31 = I1 = I2 =

22 =

33 =

41 =

44 =

I1T

T I2

I1T I2T

T I1T T I2T

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

with 11

3 τ1 Q i − R i = P A + AT P + , = 1 − δ¯ H2 K 2 τ1

21

3 3 Ri Ri R2 + R3 = , 22 = − − − Q1 τ1 τ1 τ2 − τ1

i=1

31 =

¯ 1T δK

T

B P, 32 =

i=1 3 i=2

33

R i − Ui τ2 − τ1

3 UiT + Ui − 2Ri U2 + U3 = , 42 = τ2 − τ1 τ2 − τ1 i=2

K T BT P R2 + R3 R3 − , 51 = 2 ¯ −1 τ2 − τ1 τ3 − τ2 (1 − δ) T U + U3 − 2R3 R 3 − U3 = , 55 = 3 τ3 − τ2 τ3 − τ2 U3 R3 = , 66 = −Q 3 − . τ3 − τ2 τ3 − τ2

44 = −Q 2 − 54 64

where

˜ 11

¯ K2 0 I I A 0 δ¯ B K 1 0 (1 − δ)B

A 0 B K 1 0 −B K 2 0 0 0 R1−1 R2−1 R3−1 −diag , , τ1 τ2 τ3

22 ¯δ(1 − δ) ¯ ⎤ ⎡ G 0 0 0 0 000 ⎣ H1 0 0 0 0 0 0⎦ √ 0 0 0 δ¯ H2 K 1 0 0 0 0 diag −ε1−1 I, −ε2−1 I, −ε2−1 I

i=1

Theorem 2: For some given constants 0 < τ1 < τ2 < τ3 , 0 < δ¯ < 1, (15) is asymptotically stable in the mean square with feedback gains K i = Yi X −1 (i = 1, 2) if there exist real matrices X > 0, Q˜ i > 0, R˜ i > 0 (i = 1, 2, 3), scalars ε˜ i > 0 (i = 1, 2), and matrices U˜ j ( j = 2, 3) with appropriate dimensions such that ⎡ ⎤

˜ 11 ∗ ∗ ∗ ⎢ ˜ 21 ˜ 22 ∗ ∗ ⎥ ⎢ ⎥ (20) ⎣ ˜ 31 0 ˜ 33 ∗ ⎦ < 0

˜ 41 0 0 ˜ 44 R˜ j ∗ ˜ j = ˜ ˜ ≥ 0, j = 2, 3 (21) Uj Rj

Based on Theorem 1, we are in a position to design the state feedback controller for the closed-loop system (15). When designing the networked decentralized feedback controller, the control objective is to guarantee the asymptotic stability of (15) in the mean square.

⎡ ˜ 11 ⎢ ˜ 21 ⎢ ˜ ⎢ 31 ⎢ 0 =⎢ ⎢ ˜ 51 ⎢ ⎢ 0 ⎣ ε˜ I

∗ ∗ ∗ ∗ ∗ ˜ 22 ∗ ∗ ∗ ∗ ˜ 32 ˜ 33 ∗ ∗ ∗ ˜ 32 ˜ 44 ∗ ˜ 42 ∗ ˜ 55 ∗ ˜ 54 0 0 ˜ 54 ˜ 66 ˜ 64 0 0 0 0 0 0 0 1 ε˜ 2 I 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ −I 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

T T

˜ 21 = I˜ 1T I˜1T I˜1T , ˜ 31 = I˜2T I˜2T I˜ 2T X R1−1 X X R2−1 X X R3−1 X , ,

˜ 22 = −diag τ1 τ2 τ3

¯ D Y2 , 0, ε˜ 1 I, ε˜ 2 I I˜ 1 = A D X, 0„δ¯ B D Y1 , 0, (1 − δ)B

˜ 22 I˜ 2 = [0, 0, BY1 , 0, −BY2 , 0, 0, 0], ˜ 33 = ¯δ(1 − δ) ¯

˜ 41 =

GX 0 0 0 0 000 H1 X 0 √ 0 0 0 000 ˜ 000 0 0 δ¯ H2 Y1 0

˜ =

˜ 44 = diag{−˜ε1 I, −˜ε2 I, −˜ε2 I },

1 − δ¯ H2 Y2

with ˜ 11 = AX + X A T +

3 3 ˜ τ1 Q˜ i − Ri Ri ˜ 21 = , τ1 τ1 i=1

˜ 22 = −

3 i=1

˜ 32

i=1

R˜ i R˜ 2 + R˜ 3 ¯ 1T B T ˜ 31 = δY − − Q˜ 1 , τ1 τ2 − τ1

3 ˜ 3 ˜T Ui + U˜ i − 2 R˜ i Ri − U˜ i ˜ 33 = = , τ2 − τ1 τ2 − τ1 i=2

i=2

Y2T B T U˜ + U˜ 3 R˜ − U˜ 3 ˜ 54 = 3 ˜ 51 = ˜ 42 = 2 , , ¯ −1 τ2 − τ1 τ3 − τ2 (1 − δ) R˜ + R˜ 3 R˜ 3 U˜ 3 ˜ 44 = − Q˜ 2 − 2 ˜ 64 = − , τ2 − τ1 τ3 − τ2 τ3 − τ2 ˜3 U˜ 3T + U˜ 3 − 2 R˜ 3 R ˜ 66 = − Q˜ 3 − ˜ 55 = , . τ3 − τ2 τ3 − τ2 Proof: Pre- and post-multiply both sides of (20) with diag(X, X, X, X, X, X, ε1−1 , ε2−1 , I , I , I , I , I , ε1−1 , ε2−1 , ε2−1 ) and its transpose, and (21) with diag(X, X) and its transpose, respectively. Define ε˜ i = εi−1 (i = 1, 2), X = P −1 , X Ri X T = R˜ i , X Q i X T = Q˜ i (i = 1, 2, 3) and Yi = K i X T


TABLE I U PPER D ELAY B OUNDS τ2 AND C ONTROLLER F EEDBACK G AINS FOR τ1 = 0.01 s AND δ(t) = 1 Method

τ2

Yang et al. [22]

0.06

Theorem 2

0.13

K −8.9190 −2.4270 −0.0034 −0.0004 −0.0018 0.0003 −4.5521 −1.2410 −7.9824 −2.0262 0 0 0 0 −4.0850 −1.0361

(i = 1, 2). Then using Schur complement, we have (20). This completes the proof. Remark 4: Notice that Theorem 2 cannot be directly implemented by the standard numerical software due to nonlinear terms X R˜ i−1 X. However, since R˜ i > 0 and X > 0, the conditions of ( R˜ i − X) R˜ i−1 ( R˜ i − X) > 0 are true and are equivalent to −X R˜ i−1 X < R˜ i − 2X. Then using R˜ i − 2X to replace −X R˜ i−1 X in Theorem 2, the original nonlinear matrix inequalities are transferred to LMIs, and the M ATLAB LMI Toolbox may be used to obtain the desired results.

825

l2 = 0.8 m, m 1 = 1 kg, m 2 = 0.8 kg, L = 1.2 m, κ = 0.04 N/m, we obtain the state space model (1) with the nonlinear functions as described in (4) and (5) with the following parameters: ⎤ ⎡ ⎤ ⎡ 0 1 0 0 0 ⎥ ⎢ 13.7 0 0 0 ⎥ ⎢ ⎥, g = ⎢ 13.7(sin θ1 − θ1 ) ⎥ A= ⎢ ⎦ ⎣ 0 0 0 1⎦ ⎣ 0 17.2(sin θ2 − θ2 ) 0 0 17.2 0 ⎡ ⎤ ⎤ ⎡ 0 0 0 ⎢3 0 ⎥ ⎢ 3F cos(θ1 − φ) ⎥ ⎥. ⎥ ⎢ B =⎢ ⎣ 0 0 ⎦, f = ⎣ ⎦ 0 0 5.9 −4.68F cos(θ2 − φ) Note that the nonlinear functions in each subsystem Si satisfy |g1 (sin θi − θi )| ≤ 0.45 |θi | , |θi | ≤ π/6. Then we have

2 3 g1 (sin θ1 − θ1 )2 2l1 3 +( g1 )2 (sin θ2 − θ2 )2 2l2 3 ∗ 0.45 2 2 3 ∗ 0.45 2 2 ≤( ) θ1 + ( ) θ2 2l1 2l2 = Gx(t)2 (24)

g T (t, x(t))g(t, x(t)) ≤

IV. E XAMPLES In this section, we introduce two examples to show the effectiveness of the proposed method. One is two pendulums coupled by a spring (TPCS) [21] and [22], which is a classical test bed for the study of decentralized control of large unstable nonlinear systems, another is a quadruple-tank process [27], [28]. Here, both of the systems are to be extended to include the communication network in the transmission. Example 1: TPCS, where each pendulum is treated as a subsystem and they are coupled by a spring between them [21] and [22]. The dynamic equations for the TPCS are given as [m 1 (l1 )2 /3]θ¨1 [m 1 (l2 )2 /3]θ¨2

= π1 + m 1 g1 (l1 /2) sin θ1 + l1 F cos(θ1 − φ) = π2 + m 2 g1 (l2 /2) sin θ2 − l2 F cos(θ2 − φ) (22) where g1 = 9.8 m/s2 is the constant of gravity, θi is the angular displacement of pendulum i (i = 1, 2), πi is the torque input generated by the actuator for pendulum i (i = 1, 2), F is the spring force, ls is the spring length, φ is the angle of the spring to the earth, li is the length of pendulum i , m i is the mass of pendulum, L is the distance of two pendulums, κ is the spring constant, and F = κ(ls − [L 2 + (l2 − l1 )2 ]1/2 ) ls = (L + l2 sin θ2 − l1 sin θ1 )2 + (l2 cos θ2 − l1 cos θ1 )2 l1 cos θ1 − l2 cos θ2 φ = tan−1 . L + l2 sin θ2 − l1 sin θ1 The mass of each pendulum is uniformly distributed. The length of the spring is chosen so that F = 0 when θ1 = θ2 = 0, which implies that ( θ1 θ˙1 θ2 θ˙2 )T = 0 is an equilibrium of the system if πi = 0. For simplicity, we assume that the mass of the spring is zero. Define x(t) = [θ1 , θ˙1 , θ2 , θ˙2 ]T , u(t) = [π1 (t), π2 (t)]T . Based on the above description, with the given l1 = 1 m,

(23)

where G = diag{0.675, 0, 0.844, 0}. One can draw a 3-D figure of |F| against |θ1 | and |θ2 |, and obtain |F| ≤ κ(1.49 |θ1 | + 0.18 |θ2 |). (25) From (25), it is clear F 2 ≤ κ 2 (2.738θ12 + 0.55θ22 ). This leads to

(26)

2 3 f (t, x(t)) f (t, x(t)) = F cos2 (θ1 − φ) m 1l 1 2 3 + F cos2 (θ2 − φ) m 2l2 2 2 3 3 ≤ F + F m 1l 1 m 2l2 T T ≤ x (t)H1 H1 x(t) (27) T

where H1 = diag{0.368, 0, 0.165, 0}. First, we consider a simple case, that is, all delays are assumed within the range of [τ1 , τ2 ) (δ(t) = 1). With the given lower delay bound τ1 = 0.01 s , Table I gives the results of the two designs: one is based on the method given in Yang et al. [22] and the other is based on Theorem 2 of this paper. From Table I one can see that the method given in this paper is much less conservative than that in [22]. Compared with the centralized controller in [22], it is clear that the designed controller in this paper is a decentralized controller since it only depends on its local subsystem state. Moreover, define the performance T T T index J = 0 [x (s)x(s) + u (s)u(s)]ds. For a given simulation time T = 5 s, it is obtained J equals 141.3 and 150.3

826


TABLE II U PPER D ELAY B OUNDS τ3 AND C ONTROLLER F EEDBACK G AINS FOR THE D IFFERENT VALUES OF τ2 AND G IVEN δ¯ = 0.7 τ2

τ3

0.05

0.31

0.07

0.28

0.09

0.25

K1 −10.0265 −2.9244 0 0 0 0 −5.2126 −1.1309 −9.5750 −2.8007 0 0 0 0 −4.8707 −1.4231 −9.2688 −2.6708 0 0 0 0 −4.7368 −1.3679

K2 −5.2126 −1.1309 0 0 0 0 −2.6784 −0.5780 −5.6966 −1.2015 0 0 0 0 −2.9157 −0.6166 −5.9633 −1.2461 0 0 0 0 −3.0561 −0.6360

Position in Radian

Pendulum 1 Pendulum 2

U PPER D ELAY B OUNDS τ3 AND C ONTROLLER F EEDBACK G AINS FOR D IFFERENT VALUES OF τ2 AND G IVEN δ¯ = 0.7

0.2

0

τ2

τ3

0.05

0.25

0.07

0.24

−0.2

0.09

Fig. 2.

TABLE III

0.4

−0.4 0

1

2 3 Time (Second)

4

0.21

K −3.9481 −1.0017 0

0

−7.7418 −1.9654 0

0

−7.6746 −1.9528 0

0

0

0

−3.8056 −0.9465 0

0

−3.9481 −1.0017 0

0

−3.8979 −0.9919

5

System’s response with 0.01 s ≤τ (t) < 0.31 s.

for the centralized control in [22] and the decentralized control in this paper, respectively. Furthermore, for a larger given spring constant κ = 0.1, we have τ2 = 0.11 s, J = 140.7, and 164.4 in [22] and in this paper, respectively. Compared with the cases of different spring constants, one can see the larger spring constant, the less allowable τ2 . Moreover, from the above results, it is observed that if all of the system’s states are available at the same time, then smaller J may be expected in centralized control than that in decentralized control. Second, considering that the communication delay is nonuniformly distributed. With the given δ¯ = 0.7 and τ1 = 0.01 s, the results of the designs based on the method in this paper are listed in Table II. Comparing the results in Tables I and II, one can see that the values of the upper bound τ3 in Table II are larger than τ2 in Table I. This shows that if taking the communication characteristic into account, then less conservative results may be obtained. For a given nonzero initial condition x(t) = [−0.4, 0, 0.4, 0]T , Fig. 2 depicts the time responses of the system under consideration with 0.01 s ≤ τ (t) < 0.31 s and Prob{τ (t) ∈ [0.01, 0.05)} = 0.7, Prob{τ (t) ∈ [0.05, 0.31)} = 0.3. Third, when there exists a communication network between the controller and the actuator, as mentioned in Remark 3, only the same controller gain can be used in (12). In this case, Table III also lists the obtained controller gain and upper delay bounds τ3 with the given τ1 = 0.01 s and the different τ2 . Compared with the τ3 listed in Tables II and III, it is clear the less conservative results can be obtained based on the dis-

tributed controller gain (12) than those based on the identical controller gain in (16). Fig. 3 depicts the time responses of the system under consideration with 0.01 s ≤ τ (t) < 0.24 s and Prob{τ (t) ∈ [0.01, 0.07)} = 0.7, Prob{τ (t) ∈ [0.07, 0.24)} = 0.3. As same as the above case, it is observed that the designed decentralized controller with the same controller gain still works well with the smaller adjustment time. As a by-product, according to the definition τ3 in (8), if τ3 is known, for the stability the restrictions on the sampling period h, the number of packet lost (tk+1 − tk ) − 1, and the communication delay ηtk+1 can be obtained. For example, for the case of τ3 = 0.13 s in Table II, if h = 0.01 s and the numbers of consecutive packet loss are zero, one, two, and three, then the designed decentralized controller can stabilize the system as long as the upper bounds of the communication delay are less than 0.12, 0.11, 0.10, and 0.09 s, respectively. If h = 0.03 s and the numbers of consecutive packet loss are the same as before, then the designed decentralized controller can stabilize the system as long as the upper bounds of the communication delay are less than 0.10, 0.07, 0.04, and 0.01 s, respectively. Example 2: The quadruple-tank process consists of four interconnected water tanks and two pumps, the schematic diagram of the quadruple-tank process from [27] is shown in Fig. 4. The target is to control the level in the lower two tanks with two pumps through a communication network. The process inputs are v 1 and v 2 (input voltages to the pumps) and the outputs are y1 and y2 (voltages from level measurement devices). The nonlinear model equations are


827

0.4

Position in Radian

Pendulum 1 Pendulum 2 0.2

0

−0.2

−0.4 0

Fig. 3.

1

2 3 Time (Second)

4

5

System response with 0.01 s ≤ τ (t) < 0.24 s. Fig. 4.

given as follows: ⎧ dh 1 a1 √ a3 √ ⎪ ⎪ dt = A1 2gh 1 + A1 2gh 3 + ⎪ ⎪ ⎪ dh 2 a2 √ a4 √ ⎪ ⎪ ⎪ dt = A2 2gh 2 + A2 2gh 4 + ⎨ √ dh 3 1 )k1 = Aa33 2gh 3 + (1−η v1 dt A3 ⎪ ⎪ dh √ ⎪ (1−η )k a4 2 2 4 ⎪ ⎪ ⎪ dt = A4 2gh 4 + A4 v 2 ⎪ ⎪ ⎩ y = k h , i = 1, 2 i

Based on (30) and (31) with a good approximation [28], we have

η1 k1 A1 v 1 η2 k2 A2 v 2

(28)

x˙1 = 0.0073x 12 − 0.017x 1 − 0.0047x 22 + 0.049x 2 +0.0951u 1 x˙2 = 0.0047x 22 − 0.049x 2 + 0.0314u 1 x˙3 = 0.0374x 4 + 0.0837u 2 − 0.00052x 32 − 0.011x 3

c i

A2 = A4 = 32 are cross secwhere A1 = A3 = 28 tions of the tanks, a1 = a3 = 0.071 cm2 , a2 = a4 = 0.057 cm2 are cross sections of the outlet holes, g = 981 cm/s2 , kc = 0.5 V/cm. Suppose the operating-range of the system is cm2 ,

Schematic diagram of the quadruple-tank process [27].

cm2

0 ≤ h i ≤ 20, i = 1, 2, 0 ≤ h i ≤ 6, i = 3, 4.

(29)

For the minimum phase-case, give the operating point h 1 = 12.4 cm, h 2 = 12.7 cm, h 3 = 1.8 cm, h 1 = 1.4 cm v 1 = v 2 = 3 V, η1 = 0.8, η2 = 0.3, k1 = 3.33 cm3 /Vs k2 = 3.35 cm3 /V s. Introducing x 1 = h 1 − 12.4, x 2 = h 3 − 1.8, x 3 = h 2 − 12.7, x 4 = h 4 − 1.4, u i = v i − 3, i = 1, 2, (28) can be evolved as x˙1 = 0.1123 x 2 + 1.8 − 0.1123 x 1 + 12.4 +0.0951(u 1 + 3)

x˙2 = 0.0314(u 1 + 3) − 0.1123 x 2 + 1.8 x˙3 = 0.0789 x 4 + 1.4 − 0.0789 x 3 + 12.7 +0.0837(u 2 + 3)

x˙4 = 0.0208(u 2 + 3) − 0.1123 x 1 + 1.4.

(30)

From (29), the operating-range of the transformed system is −12.4 ≤ x 1 ≤ 7.6, −1.8 ≤ x 2 ≤ 4.2 −12.7 ≤ x 3 ≤ 7.3, −1.4 ≤ x 4 ≤ 4.6.

(31)

−0.0035x 42 x˙ 4 = 0.0035x 42 − 0.0374x 4 + 0.0208u 2 y1 = 0.5x 1 , y2 = 0.5x 3.

(32)

From the (31) and (32), we have (1) with the following parameters: ⎡ ⎤ −0.017 0.0492 0 0 ⎢ 0 ⎥ −0.0492 0 0 ⎥ A= ⎢ ⎣ 0 0 −0.011 0.0374 ⎦ 0 0 0 −0.0374 ⎤ ⎡ 0.0951 0 ⎢ 0 0 0.0314 0 ⎥ ⎥ ⎢ , H2 = B =⎣ 0.0208 0 0 0.0837 ⎦ 0 0 ⎡ ⎤ 0.011 0 0 0 ⎢ 0 0.0036 0 0 ⎥ ⎥ , H1 = 0. G=⎢ (33) ⎣ 0 0 0.011 0 ⎦ 0 0 0 0.004 Assume there is a communication network between the controllers and the actuators, and given τ1 = 0.01 s and τ2 = 10 s, τ3 = 50 s with Prob{τ (t) ∈ [0.01, 10)} = 0.7. We obtain the following distributed controller gain: −0.0696 −0.1365 0 0 K = . (34) 0 0 −0.1208 −0.1769 With the given initial condition x(t) = [−1, −1, 2, 2]T , Fig. 5 depicts the time responses of the system’s output. Since there is a communication network between the controllers and the actuators, this is out of the scope of the proposed local decentralized PI control scheme in [28]. Moreover, from (32),

828


1.5

to demonstrate the effectiveness of the theoretical results presented.

y (t) 1

y (t) 2

1

Output

A PPENDIX 0.5

Proof: Construct a Lyapunov–Krasovskill functional candidate as 3 t V (x t ) = x T (t)Px(t) + x T (s)Q i x(s)ds

0

−0.5

−1 0

Fig. 5.

100

200 300 Time (Second)

400

+

500

−τi

i=1

i=1 t t +s

t −τi

x˙ T (v)Ri x(v)dvds ˙

(37)

where P > 0, Q i > 0, Ri > 0 (i = 1, 2, 3) are to be determined, and τ0 = 0. It follows from (14) that ¯ − δ). ¯ Then, the ¯ = 0, E{(δ(t) − δ) ¯ 2 } = δ(1 E{δ(t) − δ} mathematical expectation of the generator LV (x t ) for the evolution of V (x t ) along the solutions of (15) is given by

Response to nonzero initial condition (Example 2).

B and G in (33) can also be written as ⎡ ⎤ 0.0951 0 ⎢ 0 0.0314 ⎥ ⎥ B=⎢ ⎣ 0 0.0837 ⎦, H1 = H2 = 0. 0.0208 0

3 0

(35)

E{LV (x t )} = 2x T (t)Pϕ(t) +

Then, we obtain the following controller gain with the same initial parameters as the above case: −0.068139 −0.14545 0.014992 0.027113 K = . (36) 0.012094 0.044672 −0.10936 −0.16792 Compared with the decentralized controller based on (34), one can see that the controller based on (36) is a centralized controller since it depends on all of the system’s states. From the given simulation time T = 500 s, it is obtained that J equals 92.3 and 85.7 for the controllers with the gains (34) and (36), respectively. Although in this case, the centralized control outperforms the decentralized control performance under the network environment, the latter is more practical to a class of large-scale system controlled over a communication network.

x T (t)Q i x(t) − x T (t − τi )Q i x(t − τi )

i=1

+

3

E x˙ (t)τi Ri x(t) ˙ − T

t

"

T

x˙ (v)Ri x(v)dv ˙ .

t −τi

i=1

(38) From (15), we have 3

E x˙ T (t)τi Ri x(t) ˙

i=1

=

3 T

¯ ¯ E ϕ + (δ(t) − δ)ψ τi Ri ϕ + (δ(t) − δ)ψ i=1

=

3 !

¯ − δ)ψ ¯ T τi R i ψ ϕ T τi Ri ϕ + δ(1

" (39)

i=1

V. C ONCLUSION The delay-distribution-dependent decentralized control for a class of large-scale systems controlled over a communication network has been studied, where the communication delay was caused by an IP-based communication network and has a probabilistic interval distribution characteristic. By coupling the decentralized networked control modeling and the delay-distribution characteristic of IP-based communication networks in a unified framework, two delay-distributiondependent stability and stabilization conditions have been achieved. The designed decentralized controller depends not only on the statistic distribution characteristic of the IP-based communication network but also on the NCS’s configuration. To the case of there is only a communication network between the sensors and the controllers, the different controller gains have been employed in different subintervals to reduce the conservativeness. All these have been applied to a system of TPCS and a practical quadruple-tank process

3 !

where ϕ A D x(t) + δ¯ B D K 1 x(t − τ1 (t)) ¯ D K 2 x(t − τ2 (t)) + g + f +(1 − δ)B ψ B D K 1 x(t − τ1 (t)) − B D K 2 x(t − τ2 (t)). The integral items in (38) can be written as 3 i=1

t t −τi

x˙ T (v)Ri x(v)dv ˙ =

t −τ2

t −τ3

+ +

x˙ T (v)R3 x(v)dv ˙

t −τ1

t −τ2

x˙ T (v)(R2 + R3 )x(v)dv ˙

3 t i=1

t −τ1

x˙ T (v)Ri x(v)dv. ˙ (40)


Applying Lemma 1 and Jensen’s inequality [25] to deal with the integral terms in (40), for Ri > 0, i = 1, 2, 3, one can get −

3 i=1

t t −τ1

≤− −

x˙ T (v)Ri x(v)dv ˙

3

x(t) − x(t − τ1 )

i=1 t −τ 1 t −τ2

≤ −ξ T 1T −

(41)

x˙ T (v)(R2 + R3 )x(v)dv ˙

2 + 3 1 ξ τ2 − τ1

t −τ2

t −τ3

T Ri x(t) − x(t − τ1 ) τ1

x˙ T (v)R3 x(v)dv ˙ ≤ −ξ T 2T

(42) 3 2 ξ (43) τ3 − τ2

where i+1 and i (i = 1, 2) are defined in Lemma 1, and ξ denotes ξ(t) for simplicity. Considering (37)–(43) together, for any εi > 0 (i = 1, 2), we have T −1 T −1

22 21 − 31

33 31 ]ξ + E{LV (x t )) < E ξ T [ 11 − 21 (44) where i j (i, j = 1, 2) and ξ are defined in Theorem 1 and Lemma 1, respectively, ε1 g(t, x(t))2 + ε2 f (t, x(t), u(t))2 . For (4) and (5), it is readily derived that ≤ ε1 Gx(t)2 + ε2 H1 x(t)2 + ε2 H2 u(t)2 .

(45)

Using Schur complements, (18) and (45) guarantee that E{LV (x t )} < 0 in (44). This means E{LV (x t )} < −ρ x(t)2 for a sufficiently small ρ > 0, and ensures the asymptotic stability of system (15) in the mean square. This completes the proof. R EFERENCES [1] L. Bakule, “Decentralized control: An overview,” Annu. Rev. Control, vol. 32, no. 1, pp. 87–98, 2008. [2] T.-C. Yang, “Networked control system: A brief survey,” IEE Proc. Control Theory Appl., vol. 153, no. 4, pp. 403–412, Jul. 2006. [3] C. Ocampo-Martínez, S. Bovo, and V. Puig, “Partitioning approach oriented to the decentralised predictive control of large-scale systems,” J. Process Control, vol. 21, no. 5, pp. 775–786, 2011. [4] W. Zhang, M. S. Branicky, and S. M. Phillips, “Stability of networked control systems,” IEEE Control Syst. Mag., vol. 21, no. 1, pp. 84–99, Feb. 2001. [5] J. Hespanha, P. Naghshtabrizi, and Y. Xu, “A survey of recent results in networked control systems,” Proc. IEEE, vol. 95, no. 1, pp. 138–162, Jan. 2007. [6] J. Baillieul and P. Antsaklis, “Control and communication challenges in networked real-time systems,” Proc. IEEE, vol. 95, no. 1, pp. 9–28, Jan. 2007. [7] S. Dai, H. Lin, and S. Ge, “Scheduling-and-control codesign for a collection of networked control systems with uncertain delays,” IEEE Trans. Control Syst. Technol., vol. 18, no. 1, pp. 66–78, Jan. 2010. [8] X. Wang and M. Lemmon, “Event-triggering in distributed networked control systems,” IEEE Trans. Autom. Control, vol. 56, no. 3, pp. 586– 601, Mar. 2011. [9] D. Yue, Q.-L. Han, and C. Peng, “State feedback controller design of networked control systems,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 51, no. 11, pp. 640–644, Nov. 2004.

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[10] F. Yang, Z. Wang, Y. Hung, and M. Gani, “H∞ control for networked systems with random communication delays,” IEEE Trans. Autom. Control, vol. 51, no. 3, pp. 511–518, Mar. 2006. [11] H. Gao, X. Meng, and T. Chen, “Stabilization of networked control systems with a new delay characterization,” IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2142–2148, Oct. 2008. [12] X. Jiang, Q.-L. Han, S. Liu, and A. Xue, “A new H∞ stabilization criterion for networked control systems,” IEEE Trans. Autom. Control, vol. 53, no. 4, pp. 1025–1032, May 2008. [13] Y. Zhao, G. Liu, and D. Rees, “Packet-based deadband control for internet-based networked control systems,” IEEE Trans. Control Syst. Technol., vol. 18, no. 5, pp. 1057–1067, Sep. 2010. [14] M. Vaccarini, S. Longhi, and M. Katebi, “Unconstrained networked decentralized model predictive control,” J. Process Control, vol. 19, no. 2, pp. 328–339, 2009. [15] F. Lian, J. Moyne, and D. Tilbury, “Performance evaluation of control networks: Ethernet, controlnet, and devicenet,” IEEE Control Syst. Mag., vol. 21, no. 1, pp. 66–83, Feb. 2001. [16] Y.-C. Tian, X. Jiang, D. Levy, and A. Agrawala, “Local adjustment and global adaptation of control periods for QoC management of control systems,” IEEE Trans. Control Syst. Technol., vol. 20, no. 3, pp. 846– 854, May 2012. [17] Y.-C. Tian, Z.-G. Yu, and C. Fidge, “Multifractal nature of network induced time delay in networked control systems,” Phys. Lett. A, vol. 361, nos. 1–2, pp. 103–107, Jan. 2007. [18] Y. Tipsuwan and M. Y. Chow, “Gain scheduling middleware: A methodology to enable existing controllers for networked control and teleoperation-part I: Network control,” IEEE Trans. Ind. Electron., vol. 51, no. 6, pp. 1218–1227, Dec. 2004. [19] C. Peng, D. Yue, E. G. Tian, and Z. Gu, “A delay distribution based stability analysis and synthesis approach for networked control systems,” J. Franklin Inst., vol. 346, no. 4, pp. 349–365, May 2009. [20] C. Peng and T. C. Yang, “Communication-delay-distribution-dependent networked control for a class of T-S fuzzy system,” IEEE Trans. Fuzzy Syst., vol. 18, no. 2, pp. 326–335, Apr. 2010. [21] M. Ikeda and D. D. Šiljak, Robust Stabilisation of Nonlinear Systems via Linear State Feedback, vol. 51, C. T. Leondes, Ed. New York: Academic, 1992. [22] T. C. Yang, C. Peng, D. Yue, and M. R. Fei, “New study of controller design for networked control systems,” IET Control Theory Appl., vol. 4, no. 7, pp. 1109–1121, Jul. 2010. [23] J. L. Xiong and J. Lam, “Stabilization of networked control systems with a logic ZOH,” IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 358–363, Feb. 2009. [24] C. Peng, Y.-C. Tian, and M. O. Tadé, “State feedback controller design of networked control systems with interval time-varying delay and nonlinearity,” Int. J. Robust Nonlinear Control, vol. 18, no. 12, pp. 1285– 1301, Aug. 2008. [25] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems. Basle, Switzerland: Birkhäuser, 2003. [26] P. Park, J. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, pp. 235–238, Jan. 2011. [27] K. Johansson, “The quadruple-tank process: A multivariable laboratory process with an adjustable zero,” IEEE Trans. Control Syst. Technol., vol. 8, no. 3, pp. 456–465, May 2000. [28] B. Labibi, H. Marquez, and T. Chen, “Decentralized robust output feedback control for control affine nonlinear interconnected systems,” J. Process Control, vol. 19, no. 5, pp. 865–878, 2009. Chen Peng was born in Jiangsu, China, in 1972. He received the M.Sc. and Ph.D. degrees from the Chinese University of Mining Technology, Beijing, China, in 1999 and 2002, respectively. He was a Post-Doctoral Research Fellow in applied math with Nanjing Normal University, Nanjing, China, from 2002 to 2004. From November 2004 to January 2005, he was a Research Associate with Hong Kong University, Pokfulam, Hong Kong. From July 2006 to August 2007, he was a Visiting Scholar with the Queensland University of Technology, Brisbane, Australia. From July 2010 to August 2012, he was a Visiting Professor with Central Queensland University, Rockhampton, Australia. He is currently a Full Professor with Nanjing Normal University. His current research interests include analysis and synthesis of networked control systems, distributed control systems, and fuzzy control systems.

830


Qing-Long Han received the B.Sc. degree in mathematics from Shandong Normal University, Jinan, China, in 1983, and the M.Eng. and Ph.D. degrees in information science (electrical engineering) from the East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively. He was a Post-Doctoral Research Fellow with LAII-ESIP, Université de Poitiers, Poitiers, France, from September 1997 to December 1998. From January 1999 to August 2001, he was a Research Assistant Professor with the Department of Mechanical and Industrial Engineering, Southern Illinois University, Edwardsville. In September 2001, he joined Central Queensland University, Rockhampton, Australia, where he is currently a Professor with the School of Information and Communication Technology, Associate Dean with Research and Innovation, Faculty of Arts, Business, Informatics and Education, and Director with the Centre for Intelligent and Networked Systems. In March 2010, he was a Chang Jiang (Yangtze River) Scholar Chair Professor by the Ministry of Education, Beijing, China. In October 2011, he was appointed a 100 Talents Program Chair Professor by Shanxi Province, China. He was a Visiting Professor with LAII-ESIP, Université de Poitiers, a Chair Professor with Hangzhou Dianzi University, Hangzhou, China, as well as a Guest Professor with the Huazhong University of Science and Technology, Wuhan, China, East China University of Science and Technology, Shanghai, and Nanjing Normal

University, Nanjing, China. His current research interests include time-delay systems, robust controls, networked control systems, neural networks, and complex dynamical systems.

Dong Yue received the Ph.D. degree from the South China University of Technology, Guangzhou, China, in 1995. He is currently a Changjiang Professor with the Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, China. He has published more than 100 papers in international journals, domestic journals, and international conferences. His current research interests include analysis and synthesis of networked control systems, multiagent systems, interconnected systems and its application to power systems, and internet of things. Dr. Yue is currently an Associate Editor of the IEEE Control Systems Society Conference Editorial Board, the International Journal of Systems Science, the Journal of Mathematical Control Science and Applications, and the International Journal of Systems, Control and Communications.