Active Packet Dropout-Based Networked Control ... - IEEE Xplore

2012 American Control Conference Fairmont Queen Elizabeth, Montréal, Canada June 27-June 29, 2012

Active Packet Dropouts-Based Networked Control System Performance Optimization Yu-Long Wang, Qing-Long Han∗ and Xian-Ming Zhang Abstract— This paper is concerned with active packet dropouts-based anti-disturbance performance optimization for a continuous-time networked control system (NCS) under consideration of time-varying network-induced delays. By proposing the active packet dropouts method, a new model for an NCS is established. Based on the established model, stabilizing controller design criteria are derived to optimize the anti-disturbance performance of the considered NCS. When transferring nonlinear matrix inequalities into linear matrix inequalities, a parameter searching algorithm and new bounding inequalities are proposed to introduce less conservatism. The mutually exclusive distribution characteristic of the interval time-varying delays is made full use to deal with integral inequalities for products of vectors. A numerical example is given to illustrate the effectiveness of the proposed active packet dropouts-based anti-disturbance performance optimization.

I. INTRODUCTION Networked control systems are digital control systems in which the sensor, the controller and the actuator reside in physically different nodes communicating over shared communication networks. The sharing of networks simplifies the cabling and increases overall system reliability. However, the insertion of communication networks into control systems will inevitably lead to network-induced delays and packet dropouts. Then, it is significant to study the effects of network-induced delays and packet dropouts on NCSs. There have been some nice results available in the existing literature dealing with NCSs [1]–[10]. The problem of predictionbased NCSs design has also been paid much attention [11]– [13]. The non-uniform distribution characteristic of networkinduced delays was employed in [14], [15] to study the stabilization of NCSs. In the last decade there are some developments in the design of observer-based control schemes [16], [17]. Notice that the negative effect of network-induced delays and packet dropouts have been paid much attention in the above mentioned literature. It should be pointed This work was supported in part by the Australian Research Council Discovery Projects under Grant DP1096780 and Grant DP0986376; the Research Advancement Awards Scheme Program (January 2010 - December 2012), and the RDI Merit Grant Scheme Project under Grant RDIM1109 (January 2011 - December 2011) at Central Queensland University, Australia. The research work of Y.-L. Wang was also partially supported by the National Science Foundation of China under Grant No. 61004025. * Corresponding author, Tel: +61 7 4930 9270; Fax: +61 7 4930 9729, E-mail: [email protected] Y.-L. Wang, Q.-L. Han and X.-M. Zhang are with the Centre for Intelligent and Networked Systems, and School of Information and Communication Technology, Central Queensland University, Rockhampton QLD 4702, Australia. Y.-L. Wang is also with the School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China.

[email protected]; [email protected] 978-1-4577-1096-4/12/$26.00 ©2012 AACC

out that introducing a time delay purposely may lead to some positive effect [18], [19]. By utilizing the delayed force sensor measurements, Robinett et al. [18] presented a force feedback controller for damping initial and residual oscillations of a slewing flexible arm. Zhang et al. [19] investigated the positive effects of a small time delay on an offshore steel jacket structure. Generally speaking, for NCSs under consideration of network-induced delays and packet dropouts, the longer the network-induced delays and the larger the number of consecutive packet dropouts, the worse the system performance. However, this is not always the truth. Take the offshore steel jacket structure subject to a wave-induced force [19] for example, if one drops some control input packets purposely such that the latest available control inputs counteract the disturbance-induced oscillation, the overall oscillation of the system may be reduced. For NCSs, dropping some control input packets purposely may improve the anti-disturbance performance of the systems. However, such a problem has not been paid full attention in the literature, which motivates the present study. The bounding inequalities that −Mi−1 ≤ P −1 Mi P −1 − 2P −1 and −W −1 ≤ α2 X −1 W X −1 − 2αX −1 , where Mi , P , X, W are symmetric positive definite matrices and α is a given scalar, were presented in [3] and [20], respectively, to transfer nonlinear matrix inequalities into linear matrix inequalities (LMIs), which are solvable by using Matlab LMI control toolbox. Obviously, if α = 1, −W −1 ≤ α2 X −1 W X −1 − 2αX −1 will be reduced to −W −1 ≤ X −1 W X −1 − 2X −1 . It should be pointed out that the bounding inequality that −W −1 ≤ α2 X −1 W X −1 −2αX −1 is derived from (X − αW )(αW )−1 (X − αW ) ≥ 0. If the eigenvalues of (X − αW )(αW )−1 (X − αW ) are far larger than zero, such bounding inequalities will introduce much conservatism. This paper will propose less conservative bounding inequalities −Ri−1 ≤ µ2 P −1 Ri P −1 − (2µ + ai )P −1 to transfer nonlinear matrix inequalities into linear matrix inequalities. Furthermore, a new parameter searching algorithm for choosing appropriate µ and ai will be given to reduce the conservatism of the corresponding bounding inequalities in [3], [20], [21]. For t ∈ [ik h + τk , ik+1 h + τk+1 ), where ik h and τk denote the sampling instant and the length of the networkinduced delays, respectively, the interval time-varying delay τ (t) = t − ik h was introduced in [4] to study stabilization of an NCS. Suppose that τ (t) ∈ [τm , σ1 ) and introduce a scalar τ¯ = (τm + σ1 )/2, then one can conclude that at any instant t, where t ∈ [ik h + τk , ik+1 h + τk+1 ), τ (t) ∈ [τm , τ¯) or τ (t) ∈ [¯ τ , σ1 ). On the other hand, for

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the specific instant t, τ (t) ∈ [τm , τ¯) and τ (t) ∈ [¯ τ , σ1 ) can not occur simultaneously, which is named as mutually exclusive distribution in this paper. If the mutually exclusive distribution characteristic of the interval time-varying delays is adopted to deal with integral inequalities for products of vectors, less conservative results will be obtained. However, such a mutually exclusive distribution characteristic has not been taken into full consideration in the literature. This paper is concerned with active packet dropouts-based anti-disturbance performance optimization for a continuoustime NCS under consideration of time-varying networkinduced delays. The active packet dropouts method is proposed, and new models for the active packet dropoutsbased NCS are established. Based on the established models, stabilizing controller design criteria are given to optimize the anti-disturbance performance of the considered NCS. When transferring nonlinear matrix inequalities into linear matrix inequalities, less conservative bounding inequalities are proposed. The mutually exclusive distribution characteristic of the interval time-varying delays is made full use to deal with integral inequalities for products of vectors. Notation. Rn denotes n-dimensional Euclidean space. I and 0 represent an identity matrix and a zero matrix with appropriate dimensions, respectively. E stands for the expectation operation. ∗ denotes the entries of a matrix implied by symmetry. Matrices, if not explicitly stated, are assumed to have appropriate dimensions.

Fig. 1 depicts the signal transmission for an NCS under consideration of network-induced delays, where tk , tk+1 , · · · (k = 0, 1, 2, · · · ) denote the sampling instants, τk denotes the time from the instant tk when the sensor samples data from the plant to the instant when the actuator transmits data to the plant. Suppose that τm ≤ τk ≤ τM . Then, for t ∈ [tk + τk , tk+1 + τk+1 ), the control input used by the plant is described as u(t) = Kx(tk )

(2)

where K is a state feedback controller gain which will be designed in this paper, k = 0, 1, 2, · · · . Define τ (t) = t − tk , then τ (t) ∈ [τk , h + τk+1 ). Considering that τm ≤ τk ≤ τM , one has τ (t) ∈ [τm , h + τM ). Then, the system (1) is converted into ( x(t) ˙ = Ax(t) + B1 Kx(t − τ (t)) + B2 ω(t) (3) z(t) = Cx(t) + DKx(t − τ (t)) where t ∈ [tk + τk , tk+1 + τk+1 ). In this paper, we will propose the active packet dropoutsbased method to improve the anti-disturbance performance of the considered NCS.

II. ACTIVE PACKET DROPOUTS-BASED NCS MODELLING Consider a linear time-invariant networked control system described by  ˙ = Ax(t) + B1 u(t) + B2 ω(t)   x(t) z(t) = Cx(t) + Du(t) (1)   x(t0 ) = x0

where x(t) ∈ Rn , u(t) ∈ Rm , z(t) ∈ Rr , and ω(t) ∈ Rq are the state vector, control input vector, controlled output, and disturbance, respectively; ω(t) is assumed to belong to L2 [t0 , ∞); x0 ∈ Rn denotes the initial condition; A, B1 , B2 , C, and D are known constant matrices of appropriate dimensions. Throughout this paper, we assume that network-induced delays will occur in both sensor-to-controller and controllerto-actuator channels; the sensor is time-driven, while the controller and actuator are event-driven.

Fig. 1.

Signal transmission for an NCS without packet dropouts

Fig. 2.

Responses of z(t) under active packet dropouts

Fig. 2 depicts the responses of the controlled output z(t), where the solid lines denote the responses without considering packet dropouts. If one drops some control input packets purposely and transfer the active packet dropoutsbased control inputs to the plant, the additive response curves, which are described by the dashed lines, will be obtained. Then, the overall oscillations of z(t) are the lump sum of the solid lines and the dashed lines. As one can see in Fig. 2 (a), since the solid line is counteracted partially by the dashed line, the overall oscillation of z(t) will be reduced. An ideal case is given in Fig. 2 (b), in which figure the solid line is counteracted totally by the dashed line, then the overall oscillation of z(t) is reduced greatly. If one drops some control input packets purposely, the signal transmission depicted in Fig. 1 will be converted into Fig. 3, where h is the length of the sampling period, τk is the length of the network-induced delays, and the dashed lines

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Fig. 3.

In this paper, we will take the mutually exclusive distribution characteristic of the interval time-varying delays τ (t), d1 (t) and d2 (t) into full consideration to deal with integral inequalities for products of vectors. For this purpose, define a scalar ρ  1, τ (t) ∈ [τm , τ¯) ρ= (8) 0, τ (t) ∈ [¯ τ , σ1 )

Signal transmission for an NCS under packet dropouts

denote that the corresponding control inputs are dropped. Then, for t ∈ [ik h + τk , ik+1 h + τk+1 ), the control input used by the plant is described as u(t) = Lx(ik h)

(4)

where L is a state feedback controller gain which will be designed in this paper, k = 0, 1, 2, · · · . Define δ as the upper bound of the number of consecutive packet dropouts, d1 (t) = t − ik h. Then d1 (t) ∈ [τk , (ik+1 − ik )h + τk+1 ). From τm ≤ τk ≤ τM , one has d1 (t) ∈ [τm , η), where η = (δ + 1)h + τM . Then, the system (1) is converted into ( x(t) ˙ = Ax(t) + B1 Lx(t − d1 (t)) + B2 ω(t) (5) z(t) = Cx(t) + DLx(t − d1 (t)) where t ∈ [ik h + τk , ik+1 h + τk+1 ). In fact, for the signal transmission mode depicted in Fig. 1, one can use the newly received control inputs and the active packet dropouts-based control inputs to construct new control inputs. That is, for t ∈ [tk + τk , tk+1 + τk+1 ), the control input used by the plant is described as u(t) = Kx(tk ) + Lx(tk − dk h)

(6)

where K and L are the controller gains, dk denotes the number of packet dropouts and dk = 1, 2, · · · , δ. Define τ (t) = t − tk , d2 (t) = t − (tk − dk h). Then τ (t) ∈ [τm , h + τM ) and d2 (t) ∈ [τk + dk h, h + τk+1 + dk h). Considering that τm ≤ τk ≤ τM , one has d2 (t) ∈ [τm + h, η), where η is the same as the one defined above. Then, the system (1) is converted into  ˙ = Ax(t) + B1 Kx(t − τ (t))   x(t)  

Similarly, one can define scalars λ1 and λ2 , where  1, d1 (t) ∈ [τm , d¯1 ) λ1 = 0, d1 (t) ∈ [d¯1 , η)  1, d2 (t) ∈ [σ2 , d¯2 ) λ2 = 0, d2 (t) ∈ [d¯2 , η)

(9) (10)

By proposing the active packet dropouts method, we are devoted to improving the anti-disturbance performance of the considered NCS. III. ACTIVE PACKET DROPOUTS-BASED STABILIZING CONTROLLER DESIGN

This section is devoted to stabilizing controller design for the active packet dropouts-based NCSs (5) and (7). Theorem 1. For given positive scalars h, δ, τm , τM , µ, γ, scalars ai ≥ 0 (i = 1, 2, 3), and under the control law (4) with L = N T W −1 , the closed-loop system (5) is asymptotically stable with an H∞ norm bound γ if there e 1, Q e2 , Q e3 , exist symmetric positive definite matrices W , Q e1 , R e2 , R e3 , and a matrix N , such that the following linear R matrix inequality holds for every feasible value of λ1 " # e 12 e 11 Π Π (11) e 22 < 0 ∗ Π where

+ B1 Lx(t − d2 (t)) + B2 ω(t) (7) z(t) = Cx(t) + DKx(t − τ (t)) + DLx(t − d2 (t))

where t ∈ [tk + τk , tk+1 + τk+1 ). Define σ1 = h + τM , σ2 = h + τm . Then, one has τ (t) ∈ [τm , σ1 ), d1 (t) ∈ [τm , η), d2 (t) ∈ [σ2 , η). To make full use of the mutually exclusive distribution characteristic of the interval time-varying delays τ (t), d1 (t) and d2 (t), one can define τ¯ = (τm + σ1 )/2, d¯1 = (τm + η)/2, d¯2 = (σ2 + η)/2. Remark 1. Notice that in (4) and (6), the numbers of dropped control input packets are ik+1 −ik −1 and dk , respectively. To improve the anti-disturbance performance of the closed-loop systems, one should select ik+1 − ik − 1 and dk according to the following selection criterion: the selection of ik+1 −ik −1 and dk should ensure that the additive response curves in Fig. 2 will counteract the oscillation, which is induced by the external disturbances, of the controlled output z(t). 1239

e 11 Π



    =   

e 11 Ω ∗ ∗ ∗ ∗ ∗

1 e τm R1

e 22 Ω ∗ ∗ ∗ ∗

e 13 Ω e 23 Ω e Ω33 ∗ ∗ ∗

0 e Ω24 e 34 Ω e 44 Ω ∗ ∗

0 0 e 35 Ω e 45 Ω e 55 Ω ∗

B2 0 0 0 0 −γI

        

e 11 =AW + W AT + Q e1 − 1 R e1 , Ω e 13 = B1 N T Ω τm 1 e1 − e 22 =Q e2 − Q e1 − 1 R e2 R Ω τm d¯1 − τm e 23 = λ1 R e2 , Ω e 24 = 1 − λ1 R e2 Ω ¯ d1 − τm d¯1 − τm e3 e 33 = − 2λ1 R e2 − 2(1 − λ1 ) R Ω η − d¯1 d¯1 − τm e3 e 34 = λ1 R e2 + 1 − λ1 R Ω ¯ η − d¯1 d1 − τm e 35 = 1 − λ1 R e3 Ω η − d¯1 1 e 44 =Q e3 − Q e2 − e2 − 1 R e3 Ω R d¯1 − τm η − d¯1 e 45 = λ1 R e3 , Ω e 55 = −Q e3 − 1 R e3 Ω ¯ η − d1 η − d¯1

e 12 Π e 22 Π



 W AT W AT W C T 0 0 0   T T N B1 N B1 N D T   0 0 0   0 0 0  B2T B2T 0 −1 2 e =diag{τm [µ R1 − (2µ + a1 )W ], (d¯1 − τm )−1 W AT  0   N B1T =  0   0 B2T

e2 − (2µ + a2 )W ], (η − d¯1 )−1 [µ2 R e3 × [µ2 R − (2µ + a3 )W ], − γI}

Proof: The proof is omitted here due to page limitation.  The bounding inequalities −Ri−1 ≤ µ2 P −1 Ri P −1 − (2µ + ai )P −1 are introduced in Theorem 1. The following algorithm describes the method of choosing appropriate ai and µ. Algorithm 1. Step 1. For given positive scalars h, δ, τm , τM , choose the initial value µ0 > 1, final value µult > 0 and an appropriate step length µdec > 0 for µ, where µult < µ0 . Define γopt , µopt and ai,opt as the optimal values of γ, µ and ai (i = 1, 2, 3), respectively. Choose a large enough value for γopt , and set µopt = µ0 , ai,opt = 0. Set appropriate step lengths ςi > 0 for ai . Step 2. Choose large enough initial values ai > 0 such that the LMI in (11) is feasible for every feasible value of λ1 . Step 3. For a given positive scalar γ, solve the LMI presented in (11), and compare −Ri−1 with µ2 P −1 Ri P −1 − (2µ + ai )P −1 . Step 4. If −Ri−1 > µ2 P −1 Ri P −1 − (2µ + ai )P −1 , set ai = ai − ςi and go to step 5; otherwise, go to step 6. Step 5. If ai > 0, go to step 3; otherwise, go to step 8. Step 6. Compare the H∞ norm bound γ with γopt . If γ < γopt , set γopt = γ, µopt = µ, ai,opt = ai , go to step 7; otherwise, go to step 7 directly. Step 7. Set µ = µ−µdec . If µ ≥ µult , go to step 2; otherwise, output γopt , µopt and ai,opt , stop. Step 8. Set the scalars ai = 0. For a given positive scalar γ, solve the LMI presented in (11). Compare the H∞ norm bound γ with γopt . If γ < γopt , set γopt = γ, µopt = µ, ai,opt = ai , go to step 7; otherwise, go to step 7 directly. If packet dropouts are not considered, similar to the proof of Theorem 1, we can derive the following result. Corollary 1. For given positive scalars h, τm , τM , µ, γ, scalars ai ≥ 0 (i = 1, 2, 3), and under the control law (2) with K = N T W −1 , the closed-loop system described by (3) is asymptotically stable with an H∞ norm bound γ if there e 1, Q e2 , Q e3 , exist symmetric positive definite matrices W , Q e e e R1 , R2 , R3 , and a matrix N , such that the following LMI holds for every feasible value of ρ   e 12 ˆ 11 Π Π (12) ˆ 22 < 0 ∗ Π

ˆ 11 is derived from Π e 11 in (11) by replacing λ1 , d¯1 where Π e 12 is the same as the and η with ρ, τ¯ and σ1 , respectively. Π ˆ 22 is derived from Π e 22 in (11) by replacing one in (11). Π d¯1 and η with τ¯ and σ1 , respectively.

In what follows, we will study the problem of stabilizing controller design for the closed-loop NCS (7), and the mutually exclusive distribution characteristic of the interval timevarying delays τ (t) and d2 (t) is taken into full consideration. Theorem 2. For given positive scalars h, δ, τm , τM , µ, γ, scalars bj ≥ 0 (j = 1, 2, 3, 4), and under the control law (6) with K = N1T W −1 , L = N2T W −1 , the closed-loop system described by (7) is asymptotically stable with an H∞ norm bound γ if there exist symmetric positive definite matrices e1, Q e2 , Q e3 , Q e 4, R e1 , R e2 , R e3 , R e4 , and matrices N1 , N2 , W, Q such that the following LMI holds for every feasible value of ρ and λ2   ¯ 12 ¯ 11 Π Π (13) ¯ 22 < 0 ∗ Π where

¯ 11 Π

 Ω ¯ 11  ∗  ∗   ∗ =  ∗  ∗  ∗ ∗

¯ 12 Ω ¯ 22 Ω ∗ ∗ ∗ ∗ ∗ ∗

¯ 13 Ω ¯ 23 Ω ¯ 33 Ω ∗ ∗ ∗ ∗ ∗

0 0 ¯ 34 Ω ¯ 44 Ω ∗ ∗ ∗ ∗

¯ 15 Ω 0 0 0 ¯ 55 Ω ∗ ∗ ∗

¯ 16 Ω 0 0 0 ¯ 56 Ω ¯ 66 Ω ∗ ∗

0 0 0 0 0 ¯ 67 Ω ¯ 77 Ω ∗

B2 0 0 0 0 0 0 −γI

        

¯ 11 =AW + W AT + Q e1 − 1 R e1 + Q e3 − 1 R e3 Ω τm σ2 e1 , Ω ¯ 13 = B1 N T , Ω ¯ 15 = 1 R e3 , Ω ¯ 16 = B1 N T ¯ 12 = 1 R Ω 1 2 τm σ2 1 ρ 1 − ρ ¯ 22 =Q e2 − Q e1 − e1 − ( e2 Ω R + )R τm τ¯ − τm σ1 − τm 1−ρ e ¯ 23 =( ρ + )R2 Ω τ¯ − τm σ1 − τm 1 1−ρ e ¯ 33 = − ( ρ + + )R2 Ω τ¯ − τm σ1 − τm σ1 − τ¯ 1−ρ e ρ ¯ 34 =( + )R2 Ω σ1 − τm σ1 − τ¯ 1−ρ e ρ ¯ 44 = − Q e2 − ( + )R2 Ω σ1 − τm σ1 − τ¯ ¯ 55 =Q e4 − Q e3 − 1 R e4 e3 − ( λ2 + 1 − λ2 )R Ω σ2 η − σ2 d¯2 − σ2 ¯ 56 =( λ2 + 1 − λ2 )R e4 Ω η − σ2 d¯2 − σ2 e4 ¯ 66 = − ( λ2 + 1 + 1 − λ2 )R Ω η − σ2 d¯2 − σ2 η − d¯2 e4 ¯ 67 =( λ2 + 1 − λ2 )R Ω η − σ2 η − d¯2 ¯ 77 = − Q e 4 − ( λ2 + 1 − λ2 )R e4 Ω η − σ2 η − d¯2   Ψ1 Ψ1 Ψ1 Ψ1 W C T  0 0 0 0 0     Ψ2 Ψ2 Ψ2 Ψ2 N1 D T     0 0 0 0 0  ¯   Π12 =  0 0 0 0   0   Ψ3 Ψ3 Ψ3 Ψ3 N2 D T     0 0 0 0 0  B2T B2T B2T B2T 0

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TABLE I P EAK VALUES AND OSCILLATION SCOPES OF z

0.8 ω(t) 0.6 0.4

Negative peak values φ1 Positive peak values φ2 Oscillation scopes |φ1 | + |φ2 |

Fig. 8 -0.2347 0.0360 0.2707

Fig. 9 -0.1332 0 0.1332

Fig. 10 -0.1884 0 0.1884

0.2 0 −0.2 −0.4 −0.6 −0.8

¯ 22 =diag{τ −1 [µ2 R e1 − (2µ + b1 )W ], (σ1 − τm )−1 Π m e2 − (2µ + b2 )W ], σ −1 [µ2 R e3 − (2µ + b3 )W ], .[µ2 R

0

Fig. 4.

5

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25

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Curve of disturbance input ω(t)

2

e4 − (2µ + b4 )W ], − γI} (η − σ2 )−1 [µ2 R

Ψ1 =W AT , Ψ2 = N1 B1T , Ψ3 = N2 B1T

0.8 τ(t) 0.7 0.6

Proof: The proof is omitted here due to page limitation. 

0.5

IV. SIMULATION RESULTS AND DISCUSSION

0.4 0.3

Example 1. To illustrate the effectiveness of the proposed active packet dropouts-based NCS anti-disturbance performance optimization methods, we consider the following open-loop unstable NCS       0 1 0.2 0.6 x(t) ˙ = x(t) + u(t) + ω(t) 0 −0.1 −0.5 −0.1 z(t) = [−0.2 0.3]x(t) − 0.6u(t) (14) For Theorem 1 and Corollary 1 in this paper, suppose that h = 0.4s, τm = 0.1s, τM = 0.3s, and suppose that δ = 4 for Theorem 1. For Theorem 2 in this paper, suppose that h, δ, τm , τM are the same as the ones presented above. For convenience of comparison, we suppose that µ = 1, a1 = a2 = a3 = b1 = b2 = b3 = b4 = 0. Solving the LMIs in Theorem  1 and Corollary 1, one can 0.1142 0.6773 and K = get the controller gains L =   0.5603 1.9107 , respectively. Solving the LMI in Theorem 2, one can get the controller  gains K = 0.2820 1.2065 and L = 0.0218 0.1362 , and the H∞ norm bound γ = 0.6013. Suppose that the initial state of the system is x0 = [−0.2 0.2]T , the disturbance input ω(t) is presented in Fig. 4, the interval time-varying delay τ (t) is given in Fig. 5. For the systems (5) and (7), suppose that ik+1 h − ik h = 2h and dk = 2, respectively. Then the interval time-varying delays d1 (t) and d2 (t) are given in Fig. 6 and Fig. 7, respectively. Then curves of plant state and controlled output corresponding to the closed-loop systems (3), (5) and (7) are pictured in Fig. 8, Fig. 9 and Fig. 10, respectively. As one can see, Fig. 9 and Fig. 10 will provide better H∞ performance than Fig. 8. Based on Fig. 8, Fig. 9 and Fig. 10, we will analyze the peak values and oscillation scopes of the controlled output z. Define φ1 as the negative peak value of z, and φ2 , which occurs later than φ1 , as the positive peak value of z. Define |φ1 | + |φ2 | as the oscillation scope of z, where |φ1 | and |φ2 | denote the absolute values of φ1 and φ2 , respectively. As one can see from Fig. 8, the negative peak value and positive peak value of z are -0.2347 and 0.0360, respectively. Then the oscillation scope is 0.2347+0.0360=0.2707. 1241

0.2 0.1 0

Fig. 5.

0

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Curve of interval time-varying delay τ (t)

1.4 d1(t) 1.2

1

0.8

0.6

0.4

0.2

0

Fig. 6.

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Curve of interval time-varying delay d1 (t)

1.8 d2(t) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Fig. 7.

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Curve of interval time-varying delay d2 (t)

0.4 x1 x2

0.3

z 0.2

0.1

0

−0.1

−0.2

−0.3

Fig. 8.

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Curves of plant state and controlled output for (3)

0.5 x1 x2

0.4

z 0.3

0.2

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0

−0.1

−0.2

Fig. 9.

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Curves of plant state and controlled output for (5) 0.5 x1 x2

0.4

z 0.3

0.2

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−0.1

−0.2

Fig. 10.

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Curves of plant state and controlled output for (7)

Similarly, one can get the oscillation scopes of z in Fig. 9 and Fig. 10. The peak values and oscillation scopes of z corresponding to different cases are given in Table I, which illustrates the effectiveness of the proposed active packet dropouts-based NCS anti-disturbance performance optimization methods. V. CONCLUSIONS The problem of active packet dropouts-based antidisturbance performance optimization for a continuous-time NCS under consideration of time-varying network-induced delays has been studied. The active packet dropouts method has been proposed and new models for the considered NCS have been established. The mutually exclusive distribution characteristic of the interval time-varying delays has been taken into full consideration to derive some stabilization criteria. When transferring nonlinear matrix inequalities into linear matrix inequalities, new bounding inequalities and a parameter searching algorithm have been proposed to introduce less conservatism. A numerical example has illustrated the effectiveness of the proposed active packet dropoutsbased anti-disturbance performance optimization methods. R EFERENCES [1] W. P. M. H. Heemels, A. R. Teel, N. van de Wouw, and D. Neˇsi´c, Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance, IEEE Transactions on Automatic Control, vol. 55, no. 8, 2010, pp. 1781-1796. [2] Y.-B. Zhao, J. Kim, and G.-P. Liu, Error bounded sensing for packetbased networked control systems, IEEE Transactions on Industrial Electronics, vol. 58, no. 5, 2011, pp. 1980-1989. [3] H. Gao and T. Chen, Network-based H∞ output tracking control, IEEE Transactions on Automatic Control, vol. 53, no. 3, 2008, pp. 655-667. [4] D. Yue, Q.-L. Han, and J. Lam, Network-based robust H∞ control of systems with uncertainty, Automatica, vol. 41, no. 6, 2005, pp. 999-1007.

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