Circuits Syst Signal Process (2009) 28: 65–83 DOI 10.1007/s00034-008-9070-7

Robust Disturbance Attenuation with Stabilization for Uncertain Networked Control Systems Dan Huang · Sing Kiong Nguang

Received: 7 June 2007 / Revised: 1 February 2008 / Published online: 15 October 2008 © Birkhäuser Boston 2008

Abstract This paper investigates the problem of robust stabilization and attenuation for a class of uncertain networked control systems (NCSs) with random communication network-induced delays. Based on the Lyapunov–Razumikhin method, a controller is designed such that both robust stability and a prescribed disturbance attenuation performance for the closed-loop NCS are achieved, irrespective of the uncertainties and network-induced delays. The controller design technique is given in terms of the solvability of bilinear matrix inequalities. An iterative algorithm is proposed to change this non-convex problem into quasi-convex optimization problems, which can be solved effectively using available mathematical tools. The effectiveness of the proposed design methodology is verified by a numerical example. Keywords Networked control systems · Disturbance attenuation · Time delay · Linear matrix inequality 1 Introduction In recent years, due to the expansion of system physical setups and functionality, networked control systems (NCSs) have been introduced into the design of control systems. NCSs are a type of distributed control system where sensors, actuators, and controllers are interconnected by communication networks. An NCS can improve the efficiency, flexibility, and reliability of integrated applications, and reduce installation, reconfiguration, and maintenance time and costs. Due to their low cost, flexibility, and less wiring, the use of NCSs is rapidly increasing in industrial applications, D. Huang () · S.K. Nguang The Department of Electrical and Computer Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand e-mail: [email protected] S.K. Nguang e-mail: [email protected]

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including telecommunications, remote process control, altitude control of airplanes, and so on, and therefore considerable attention has been devoted to the problem of NCSs [2, 4, 5, 8, 15, 16, 18, 20–24]. The implementation of the NCS is closely associated with the advance in ASIC chip design and significant price drops in silicon. Today, sensors and actuators can be equipped with a network interface and thus can be independent nodes on a realtime control network. Current candidate networks for NCS implementations are DeviceNet, Ethernet, and FireWire, see [4, 24]. Network-induced delays and data transmission dropouts are the main issues raised in the research of NCSs, see [4, 8, 18, 24]. In the NCS, data is sent through the network in packets. Due to this network characteristic, any continuous-time signals from the plant are first sampled to be carried over the communication network. Chances are that those packets can be lost during transmission because of uncertainty and noise in the communication channels. They may also be lost at the destination when out-of-order delivery takes place. Furthermore, the network-induced delays are also a challenging problem in the control of NCSs that occurs while exchanging data among devices connected by the communication network. Depending on network characteristics, such as topologies, routing schemes, etc., these delays can be constant, time varying, or even random. They can degrade the performance of control systems and can even destabilize the system. The severity of the network-induced delays is aggravated when data packet dropouts occur during a network transmission. On the other hand, the problem of performance control with disturbance attenuation for time-delay systems has gathered much attention in recent years [3, 12, 14]. However, these results are mostly obtained for systems with state delays. Due to the characteristics of communication networks, network-induced time delays are input delays. So far, performance control with disturbance attenuation has not been addressed for systems with uncertain time-varying input delays. We attempt to solve this problem in this work. Note that for systems with time-varying input delays, it is difficult to analyze disturbance attenuation based on the gain characterization, because the state variation depends not only on the current but also on the history of the exterior disturbance input. Therefore, in this paper, a generalized disturbance attenuation will be introduced. This generalized disturbance attenuation reduces to the standard disturbance attenuation characterized by the L2 gain when the delay is zero. Based on the Lyapunov– Razumikhin method, a delay-dependent controller is obtained in the form of bilinear matrix inequalities. Note that the Lyapunov–Krasovskii approach [8, 23] can also be applied to time-delay systems. The methodology applied in this paper does not require the calculation of the time derivative of the Lyapunov–Krasovskii functionals. In our paper, the resulting delay-dependent controller guarantees both robust stability and a prescribed disturbance attenuation performance for the closed-loop NCS, irrespective of the uncertainties and network-induced effects, i.e., networked-induced delays and packet dropouts in both the sensor-controller and controller-actuator channels. An iterative algorithm is proposed to change this non-convex problem into quasi-convex optimization problems, which can be solved effectively using available mathematical tools.

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This paper is organized as follows. The problem formulation and preliminaries are given in Sect. 2. Section 3 gives the main results of the paper. An illustrating example is presented in Sect. 4, and the concluding remarks are given in Sect. 5.

2 Problem Formulation and Preliminaries Assume that the uncertain linear continuous state-space model of the plant dynamics is described by the following equations: ⎧ ˙ = (A + A)x(t) + (B1 + B1 )w(t) + (B2 + B2 )u(t), ⎪ ⎨ x(t) z(t) = (C1 + C1 )x(t) + (D1 + D1 )u(t), (2.1) ⎪ ⎩ y(t) = (C2 + C2 )x(t) + (D2 + D2 )w(t), where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input, w(t) ∈ Rp is the exogenous disturbance input and/or measurement noise, and y(t) ∈ Rl and z(t) ∈ Rs denote the measurement and regulated output respectively. Matrices A, B1 , B2 , C1 , C2 , D1 , and D2 are of appropriate dimensions. Matrices A, B1 , B2 , C1 , C2 , D1 , and D2 characterize the uncertainties in the system and satisfy the following assumption. Assumption 2.1 [ A

B1

B2 ] = H1 F (t)[ E1

E2

[ C1

D1 ] = H2 F (t)[ E1

E3 ],

[ C2

D2 ] = H3 F (t)[ E1

E2 ],

E3 ],

where H1 , H2 , H3 , E1 , E2 , and E3 are known real constant matrices of appropriate dimensions, and F (t) is an unknown matrix function with Lebesgue-measurable elements that satisfies F (t)T F (t) ≤ I , in which I is the identity matrix of appropriate dimension. In this paper, we consider an NCS for which the plant setup is depicted in Fig. 1. The plant outputs are sampled with periodic sampling interval hs and sent through the network at times khs , k ∈ N. In the absence of data dropouts, it can be noted that the measurement signals {y(khs ), k ∈ N} are received by the controller side at times khs + τks where τks is the delay that the measurement sent at khs experiences. A controller is constructed as follows:    ˙ˆ = Aˆ x(t)

x(t) ˆ + Bˆ yˆ khs , s ∀t ∈ khs + τks , (k + 1)hs + τk+1 , (2.2) ˆ u(t) = C x(t), ˆ s ) is equal to the last successfully received measurement signal, and mawhere y(kh ˆ ˆ ˆ trices A, B, Cˆ are the controller’s parameters.

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Fig. 1 An NCS with random delay, packet dropouts, and disturbance input

The controller sends control signal at times lha . These signals, assuming there are no dropouts, arrive at the plant side at time lha + τla where τla is the delay that the control signal sent at lha experiences. This leads to

  a . (2.3) u(t) = Cˆ xˆ lha , ∀t ∈ lha + τla , (l + 1)ha + τl+1 Defining τ (t) := t − khs , ρ(t) := t − lha ,

s , ∀t ∈ khs + τks , (k + 1)hs + τk+1

a a , ∀t ∈ lh + τla , (l + 1)ha + τl+1

(2.2) and (2.3) can be rewritten as    ˙ˆ = Aˆ x(t) ˆ t − τ (t) , x(t) ˆ + By   u(t) = Cˆ xˆ t − ρ(t) , and

(2.4) (2.5)

(2.6)

 s  τ (t) ∈ min τks , hs + max τk+1 ,

∀k ∈ N, τ˙ (t) = 1,

(2.7)

 a  ρ(t) ∈ min τla , ha + max τl+1 ,

∀l ∈ N, ρ(t) ˙ = 1.

(2.8)

k

k

l

l

Figure 2 shows τ (t) with respect to time where for all k, τks = τ s , and with constant sampling interval hs with T = khs + τ s . The derivative of τ (t) is almost always one, except at the sampling times, where τ (t) drops to τ s . Furthermore, data packet dropout can be viewed as a delay growing beyond the defined boundary in (2.7) and (2.8). Let us define ns and na as the number of consecutive dropouts in the sensor and actuator channels, respectively. Then   s   , ∀k ∈ N, τ˙ (t) = 1, (2.9) τ (t) ∈ min τks , ns + 1 hs + max τk+1 k

k

  a   ρ(t) ∈ min τla , na + 1 ha + max τl+1 , l

l

∀l ∈ N, ρ(t) ˙ = 1.

(2.10)

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Fig. 2 Evolution of τ (t) with respect to time without packet dropout

Fig. 3 Evolution of τ (t) with respect to time with packet dropout at khs

If the measurement packet sent at khs is lost, then τ (t) increases up to 2hs + τ s . We can see this scenario from Fig. 3. Therefore, with regard to (2.1) and (2.6), the closed-loop system can be written as 

      A + A 0 x(t) ˙ x(t) B1 + B1 = + w(t) ˙ˆ x(t) ˆ 0 x(t) 0 Aˆ     0 w t − τ (t) + ˆ B(D2 + D2 )    x(t − ρ(t)) 0 (B2 + B2 )Cˆ + x(t ˆ − ρ(t)) 0 0    0 0 x(t − τ (t)) . + ˆ ˆ − τ (t)) B(C2 + C2 ) 0 x(t

(2.11)

Define x(t) ˜ = [x T (t) xˆ T (t)]T , w(t − τ (t)) = v(t), and ω(t) = [w T (t) v T (t)]T . Hence (2.11) can be written in the following concise form:     ˙˜ = Ax(t) x(t) ˜ + Bω(t) + C1 x˜ t − ρ(t) + C2 x˜ t − τ (t)

(2.12)

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where

 A=  C1 =

 A + A 0 , 0 Aˆ

 B=

 0 (B2 + B2 )Cˆ , 0 0

 B1 + B1 0 ˆ 2 + D2 ) , 0 B(D   0 0 . C2 = ˆ B(C2 + C2 ) 0

Definition 2.1 The system (2.1) is said to be robust stabilizable with a disturbance attenuation level γ if its zero state response (x(φ) = 0, ω(φ) = 0, −χ ≤ φ ≤ 0) satisfies  Tf  Tf zT (t)z(t) dt ≤ γ 2 sup ωT (t + φ)ω(t + φ) dt (2.13) 0

0

−χ≤φ≤0

for any nonzero ω(t) ∈ L2 [0, Tf ] and Tf ≥ 0. Remark 2.1 From Definition 2.1, it is easy to find that once there is no time delay in the system, i.e., φ = 0, (2.13) reduces to L2 gain  Tf  Tf zT (t)z(t) dt ≤ γ 2 ωT (t)ω(t) dt. (2.14) 0

0

Hence, (2.14) can be regarded as a special case of (2.13). Before proceeding with our controller design, we recall the following matrix inequality lemmas which will be used throughout the proof. Lemma 2.1 [3] For constant matrices H and E, a symmetric matrix G, and scalar ε > 0, the following inequality holds: G + H F E + E T F T H H < 0, where F satisfies F T F ≤ I , if and only if for any ε > 0, G + εH H T + ε −1 E T E < 0. Lemma 2.2 [17] (Moon’s Inequality) For vectors a, b ∈ n , symmetric matrix P ∈ n×n > 0, and scalar ε > 0, we have 1 −2aT b ≤ aT P −1 a + εbT P b. ε In this paper, we assume u(t) = 0 before the first control signal reaches the plant. From here, we use (*) as an ellipsis for terms that are induced by symmetry in the symmetric block matrices. 3 Main Result In this paper, we assume that τks and τla are bounded. According to (2.7) and (2.8), there is no loss of generality to assume τ (t) ≤ τ ∗ and ρ(t) ≤ ρ ∗ . Let us denote the

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71

total maximal delay  = τ ∗ + ρ ∗ . The following theorem provides sufficient conditions for the existence of a dynamic output feedback controller for the system (2.12) that satisfies requirements for robust attenuation with stabilization. Theorem 3.1 Consider the system (2.12) satisfying Assumption 2.1. For given positive delay-free attenuation constant γdf , positive constants , ε1 , ε2 , ε3 , ε4 , ε5 , ε6 , ε7 , ε8 , and ε9 , if there exist symmetric matrices X = X T and Y = Y T , matrices F and L, and positive scalars β1 , β2 , such that the following inequalities hold:   Y I > 0, (3.1) I X ⎡ T ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

AY + Y A + B2 L + LT B2T + (β1 + 6β2 )Y

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(β1 + 6β2 )I

XA + AT X + F C2 + C2T F T + (β1 + 6β2 )X

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

B1T

B1T X

−γdf I

(∗)T

(∗)T

(∗)T

(∗)T

0

D2T F T

0

−γdf I

(∗)T

(∗)T

(∗)T

C1 Y + D 1 L

C1

0

0

−I

(∗)T

(∗)T

E1 Y + E3 L

0

E2

0

0

−ε1 I

(∗)T

ε1 H1T

0

0

0

0

0

−ε1 I

E1 Y

E1

E2

0

0

0

0

0

ε2 H1T X

0

0

0

0

0

0

E1

0

E2

0

0

0

0

ε3 H3T F T

0

0

0

0

0

E1 Y + E3 L

E1

0

0

0

0

0

0

ε4 H2T

0

0

0

(∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T −ε2 I 0 0 0 0 0

(∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T −ε2 I 0 0 0 0

0

(∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T −ε3 I 0 0 0

(∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T −ε3 I 0 0

0

(∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T −ε4 I 0

⎤

(∗)T (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ < 0, (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎦ −ε4 I

(3.2)

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⎡

−β1 Y (∗)T (∗)T (∗)T (∗)T ⎢ −β1 I −β1 X (∗)T (∗)T (∗)T ⎢ ⎢ Y AT −LT B T X − Y C T F T − A −Y (∗)T (∗)T 2 2 ⎢ ⎢ AT AT X −I −X (∗)T ⎢ T ⎣ ε5 H T ε5 H1 X 0 0 −ε5 I 1 0 0 E1 Y E1 0 ⎡ ⎤ −β2 Y (∗)T (∗)T (∗)T (∗)T (∗)T ⎢ −β2 I (∗)T (∗)T (∗)T (∗)T ⎥ 2X ⎢ T T −β ⎥ T T ⎢L B L B2 X −Y (∗)T (∗)T (∗)T ⎥ 2 ⎢ ⎥ < 0, ⎢ 0 0 −I −X (∗)T (∗)T ⎥ ⎢ ⎥ ⎣ ε6 H T ε6 H T X 0 0 −ε6 I (∗)T ⎦ 1 1 0 0 E3 L 0 0 −ε6 I ⎡ ⎤ −β2 Y (∗)T (∗)T (∗)T (∗)T (∗)T ⎢ −β2 I −β2 X (∗)T (∗)T (∗)T (∗)T ⎥ ⎢ ⎥ T T ⎢ 0 Y C2 F −Y (∗)T (∗)T (∗)T ⎥ ⎢ ⎥ < 0, ⎢ 0 0 −I −X (∗)T (∗)T ⎥ ⎢ ⎥ ⎣ 0 ε7 H3T F T 0 0 −ε7 I (∗)T ⎦ 0 0 E1 Y 0 0 −ε7 I ⎡ −Y (∗)T (∗)T (∗)T (∗)T (∗)T (∗)T ⎢ −I −X (∗)T (∗)T (∗)T (∗)T (∗)T ⎢ T T ⎢ B1 B1 X −I (∗)T (∗)T (∗)T (∗)T ⎢ ⎢ 0 D2T F T 0 −I (∗)T (∗)T (∗)T ⎢ ⎢ T T ⎢ ε8 H1 ε8 H1 X 0 0 −ε8 I (∗)T (∗)T ⎢ ⎢ 0 0 E2 0 0 −ε8 I (∗)T ⎢ ⎢ ⎣ 0 ε9 H3T F T 0 0 0 0 −ε9 I 0 0 0 E2 0 0 0

⎤ (∗)T (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ < 0, (∗)T ⎥ ⎥ (∗)T ⎦ −ε5 I

(3.3)

(3.4)

(3.5)

⎤ (∗)T (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ ⎥ < 0, (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ ⎥ T (∗) ⎦

(3.6)

−ε9 I

then (2.13) holds for all delays τ (t) and ρ(t) satisfying τ (t) + ρ(t) ≤ τ ∗ + ρ ∗ =  with γ 2 = γdf + . Furthermore, the controller is of the form (2.2) with −1

 −AT − XAY − F C2 Y − XB2 L Y −1 , Aˆ = Y −1 − X  −1 Bˆ = Y −1 − X F,

(3.8)

Cˆ = LY −1 .

(3.9)

Proof Note that for the system (2.12) at time t,  0   x˜ t − τ (t) = x(t) ˜ − x(t ˙ + θ ) dθ −τ (t)

 = x(t) ˜ −

0

−τ (t)



Ax(t ˜ + θ ) + Bω(t + θ )

   

+ C1 x˜ t − ρ(t) + θ + C2 x˜ t − τ (t) + θ dθ.

(3.7)

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73

Applying the same transformation to x(t ˜ − ρ(t)), the closed-loop system (2.12) can be rewritten as 

˙˜ = Dx(t) x(t) ˜ − C1

0

−ρ(t)

  Ax(t ˜ + σ ) + Bω(t + σ ) + C1 x˜ t − ρ(t) + σ

 

+ C2 x˜ t − τ (t) + σ dσ  0   Ax(t ˜ + θ ) + Bω(t + θ ) + C1 x˜ t − ρ(t) + θ − C2 −τ (t)

 

+ C2 x˜ t − τ (t) + θ dθ + Bω(t)

(3.10)

where D = A + C1 + C2 for simplification of the notation. Select a Lyapunov function candidate as   ˜ V x(t) ˜ = x˜ T (t)P x(t).

(3.11)

2 2     ˜  ≤ V x(t) ˜  ˜ ≤ α2 x(t) α1 x(t)

(3.12)

It follows that

where α1 = λmin (P ) and α2 = λmax (P ). Differentiating V (x(t)) ˜ along the closed-loop system (2.12) and using (3.10) yields   T ˙˜ ˜ + x˜ T (t)P x(t) V˙ x(t) ˜ = x˙˜ (t)P x(t)   ˜ + x˜ T (t)P Bω(t) + ωT (t)B T P x(t) ˜ = x˜ T (t) DT P + P D x(t)  0 ˜ + θ ) + Bω(t + θ ) x˜ T (t)P C2 Ax(t −2 −τ (t)

   

+ C1 x˜ t − ρ(t) + θ + C2 x˜ t − τ (t) + θ dθ  0 ˜ + σ ) + Bω(t + σ ) x˜ T (t)P C1 Ax(t −2 −ρ(t)

   

+ C1 x˜ t − ρ(t) + σ + C2 x˜ t − τ (t) + σ dσ   ˜ + x˜ T (t)P Bω(t) + ωT (t)B T P x(t) ˜ ≤ x˜ T (t) DT P + P D x(t)  1 T x˜ (t)P C2 AP −1 AT C2T P x(t) ˜ + β1 x˜ T (t + θ )P x(t ˜ + θ) + τ (t) β1 1 T x˜ (t)P C2 C1 P −1 C1T C2T P x(t) ˜ β2     + β2 x˜ T t − τ (t) + θ P x˜ t − τ (t) + θ +

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Circuits Syst Signal Process (2009) 28: 65–83

1 T x˜ (t)P C2 C2 P −1 C2T C2T P x(t) ˜ β2     + β2 x˜ T t − ρ(t) + θ P x˜ t − ρ(t) + θ

+

+ x˜

T

(t)P C2 BB T C2T P x(t) ˜ + ωT (t

 + θ )ω(t + θ )

 1 T + ρ(t) x˜ (t)P C1 AP −1 AT C1T P x(t) ˜ + β1 x˜ T (t + σ )P x(t ˜ + σ) β1 1 T x˜ (t)P C1 C1 P −1 C1T C1T P x(t) ˜ β2     + β2 x˜ T t − τ (t) + σ P x˜ t − τ (t) + σ +

1 T x˜ (t)P C1 C2 P −1 C2T C1T P x(t) ˜ β2     + β2 x˜ T t − ρ(t) + σ P x˜ t − ρ(t) + σ

+

+ x˜

T

(t)P C1 BB T C1T P x(t) ˜ + ωT (t

 + σ )ω(t + σ ) .

(3.13)

Following the Razumikhin theorem [11], we assume that for any δ > 1, the following inequality holds:     V x(ξ ˜ ) < δV x(t) ˜ ,

t − 2χ ≤ ξ ≤ t

(3.14)

where χ = max(τ (t), ρ(t)). Suppose that AP −1 AT < β1 P −1 ,

(3.15)

C1 P −1 C1T < β2 P −1 ,

(3.16)

C2 P −1 C2T < β2 P −1 ,

(3.17)

BB T < P −1 .

(3.18)

Furthermore, by adding and subtracting −zT (t)z(t) + γdf ωT (t)ω(t) to and from (3.13), we can get   ˜ + x˜ T (t)P Bω(t) + ωT (t)B T P x(t) ˜ V˙ (x(t)) ˜ ≤ x˜ T (t) DT P + P D x(t)  

− γdf ωT (t)ω(t) + τ (t) + ρ(t) 4β2 + (β1 + 2β2 )δ x˜ T (t)P x(t) ˜   + τ (t) + ρ(t) sup ωT (t + φ)ω(t + φ) −χ≤φ≤0

˜ − zT (t)z(t) + γdf ωT (t)ω(t) + x˜ T (t)E T E x(t)

Circuits Syst Signal Process (2009) 28: 65–83

75

     = x˜eT (t)M τ (t) + ρ(t) , δ x˜e + τ (t) + ρ(t)

sup

−χ≤φ≤0

ωT (t + φ)ω(t + φ)

− zT (t)z(t) + γdf ωT (t)ω(t) ˆ 1 + D1 )], and M(·, ·) is given where x˜e (t) = [x˜ T (t) ωT (t)]T , E = [C1 + C1 C(D by    M τ (t) + ρ(t) , δ  DT P + P D + (τ (t) + ρ(t))[4β2 + (β1 + 2β2 )δ]P + E T E = BT P

PB



−γdf I

.

In this paper the time delays are assumed to be bounded, hence τ (t) + ρ(t) can also be assumed to be bounded, that is, τ (t) + ρ(t) ≤ , where  is the constant given in the theorem. Using this fact, we learn that    M τ (t) + ρ(t) , δ ≤ M(, δ). Hence, if (3.2) holds, it can be shown later that M(, δ) < 0 for δ = 1. Then we get   V˙ x(t) ˜ <

ωT (t + φ)ω(t + φ) − zT (t)z(t) + γdf ωT (t)ω(t)

sup

−χ≤φ≤0

< ( + γdf ) < γ2

sup

−χ≤φ≤0

sup

−χ≤φ≤0

ωT (t + φ)ω(t + φ) − zT (t)z(t)

ωT (t + φ)ω(t + φ) − zT (t)z(t).

Integrating both sides yields 

Tf

  V˙ x(t) ˜ dt <

0



Tf 

γ2

0

    ˜ < V x(T ˜ f ) − V x(0)



Tf 

sup

 ωT (t + φ)ω(t + φ) − zT (t)z(t) dt,

sup

 ωT (t + φ)ω(t + φ) − zT (t)z(t) dt.

−χ≤φ≤0

γ2

−χ≤φ≤0

0

Using the fact that x(0) ˜ = 0 and V (x(T ˜ f )) ≥ 0 for all Tf = 0, we have 

Tf 0

 zT (t)z(t) dt ≤ γ 2

Tf 

sup

0

−χ≤φ≤0

 ωT (t + φ)ω(t + φ) dt.

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Circuits Syst Signal Process (2009) 28: 65–83

This satisfies the conditions set in Definition 2.1 and we can say the system (2.1) has a disturbance attenuation level γ . Hereafter, we will show that (3.2) guarantees M(, 1) < 0. Applying Schur’s complement to M(, 1) < 0, we can have ⎡

DT P + P D + (β1 + 6β2 )P ⎣ BT P E Multiplying (3.19) to the left by





(∗)T −γdf I 0

and to the right by ⎡

JT =⎣ 0 0

⎤ (∗)T 0 ⎦ < 0. −I

T

(3.19)

where

⎤ 0 0⎦, I

0 I 0

using Assumption 2.1 and Schur’s complement, and applying the controllers defined as in (3.7)–(3.9) yields  ⎡  AY + Y AT + B L 2 (∗)T (∗)T (∗)T ⎢ + LT B2T + (β1 + 6β2 )Y ⎢   ⎢ XA + AT X + F C2 ⎢ (∗)T (β1 + 6β2 )I (∗)T ⎢ + C2T F T + (β1 + 6β2 )X ⎢ ⎢ ⎢ B1T B1T X −γdf I (∗)T ⎢ ⎣ 0 D2T F T 0 −γdf I

⎡

C1 Y + D1 L

0

C1

⎤

H1 ⎢ 0 ⎥ ⎢ ⎥ ⎥ +⎢ ⎢ 0 ⎥ F (t)[ E1 Y + E3 L 0 E2 ⎣ 0 ⎦ 0

0 0] ⎡

+ [ E1 Y + E3 L 0 ⎡

E2

⎤ 0 ⎢ XH1 ⎥ ⎢ ⎥ ⎥ +⎢ ⎢ 0 ⎥ F (t)[ E1 Y ⎣ 0 ⎦ 0

⎤T H1 ⎢ 0 ⎥ ⎢ ⎥ ⎥ 0 0 ]T F T (t) ⎢ ⎢ 0 ⎥ ⎣ 0 ⎦ 0

E1

E2

0 0]

0

⎤ (∗)T

⎥ ⎥ ⎥ ⎥ (∗)T ⎥ ⎥ ⎥ T (∗) ⎥ ⎥ (∗)T ⎦ −I

Circuits Syst Signal Process (2009) 28: 65–83

77

⎡

+ [ E1 Y

E1

E2

⎡

⎤ 0 ⎢ F H3 ⎥ ⎢ ⎥ ⎥ +⎢ ⎢ 0 ⎥ F (t)[ 0 ⎣ 0 ⎦ 0

⎤T 0 ⎢ XH1 ⎥ ⎢ ⎥ ⎥ 0 0 ]T F T (t) ⎢ ⎢ 0 ⎥ ⎣ 0 ⎦ 0

0

E1

E2

0]

⎡

+ [0

0

E1

E2

⎡

⎤T 0 ⎢ F H3 ⎥ ⎢ ⎥ ⎥ 0 ]T F T (t) ⎢ ⎢ 0 ⎥ ⎣ 0 ⎦ 0

⎤ 0 ⎢ 0 ⎥ ⎢ ⎥ ⎥ +⎢ ⎢ 0 ⎥ F (t)[ E1 Y + E2 L E1 ⎣ 0 ⎦ H2

0 0 0] ⎡

+ [ E1 Y + E2 L E1

0

0

⎤T 0 ⎢ 0 ⎥ ⎢ ⎥ T T ⎥ 0 ] F (t) ⎢ ⎢ 0 ⎥ ⎣ 0 ⎦ H2

< 0.

(3.20)

Using Lemma 2.1, it is easy to see that (3.2) guarantees the existence of (3.20), which infers M(, 1) < 0. Using the continuity property of the eigenvalues of M(·, ·) with respect to δ, there exists a sufficiently small  > 0 such that M(, 1 + ) < 0. Hence, there exists a δ > 1 such that M(, δ) < 0 still holds. Next, it will be shown that (3.3)–(3.6) are derived from (3.15)–(3.18). First, the inequality (3.15) can be rewritten as follows by applying Schur’s complement: 

−β1 P −1 AT

A −P

 < 0.

(3.21)



−1 Using the partition P = −1X Y −X and Assumption 2.1, multiplying (3.21) to −1 Y −X X−Y



J TP 0

and to the right by P0J J0 where J = YY 0I , and using the the left by T 0

J

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controllers defined as in (3.7)–(3.9) yields ⎡

−β1 Y

⎢ −β I 1 ⎢ ⎢ ⎣ Y AT AT

⎡

(∗)T

−β1 X

(∗)T

−A − LT B2T X − Y C2T F T

−Y

(∗)T ⎥ ⎥ ⎥ (∗)T ⎦

AT X

−I

−X

⎤

H1 ⎢ XH1 ⎥ ⎥ +⎢ ⎣ 0 ⎦ F (t)[ 0 0

(∗)T

⎤

(∗)T

E1 ]

0 E1 Y ⎡

+ [0

0 E1 Y

⎤T H1 ⎢ XH1 ⎥ ⎥ E1 ]T F T (t) ⎢ ⎣ 0 ⎦ 0

< 0.

(3.22)

Applying Lemma 2.1 and Schur’s complement, it is not hard to see that (3.3) guarantees (3.22). Equations (3.4)–(3.6) can be derived from (3.16)–(3.18) using the same procedure. Besides, P > 0 is equivalent to  J TP J =

Y I

 I > 0. X

We therefore have the inequality condition (3.1). This completes the proof.

(3.23)



Remark 3.1 When there is no time delay in the system, i.e.,  = 0, the result in Theorem 3.1 reduces to be the same as (38) and (39) in [6] after some algebraic manipulations by using Schur’s complement and neglecting the Markovian terms. Note that terms β1 X and β1 Y in (3.2)–(3.6) are not convex constraints, which are difficult to solve. Some linear matrix inequality (LMI)-based algorithms have recently been proposed to solve such problems, to name a few, the min-max algorithm [9], XYcentering algorithm [13], and cone complementarity linearization (CCL) algorithm [10]. According to the LMI characteristics of the results given in Theorem 3.1, in this paper, we propose the following iterative algorithm to change this non-convex feasibility problem into quasi-convex optimization problems [19]. • Step 1: Find X, Y, F , and L such that the following LMIs hold: 

Y I

 I > 0, X

Circuits Syst Signal Process (2009) 28: 65–83

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

AY + Y AT

79

(∗)T

+ B2 L + LT B2T

XA + AT X

0

+ F C2 + C2T F T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

−γdf I

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

B1T

B1T X

0

D2T F T

0

−γdf I

(∗)T

C1 Y + D 1 L

C1

0

0

−I

(∗)T

(∗)T

E1 Y + E3 L ε1 H1T

0

E2

0

0

−ε1 I

(∗)T

0

0

0

0

0

−ε1 I

E1 Y

E1

E2

0

0

0

0

0

ε2 H1T X

0

0

0

0

0

0

E1

0

E2

0

0

0

0

ε3 H3T F T

0

0

0

0

0

E1 Y + E3 L

E1

0

0

0

0

0

0

0

0

0

ε4 H2T

0

0

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

(∗)T

−ε2 I

(∗)T

(∗)T

(∗)T

(∗)T

0

−ε2 I

(∗)T

(∗)T

(∗)T

0

0

−ε3 I

(∗)T

(∗)T

0

0

0

−ε3 I

(∗)T

0

0

0

0

−ε4 I

0

0

0

0

0

(∗)T

⎤

(∗)T ⎥ ⎥ ⎥ T (∗) ⎥ ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ < 0. ⎥ T (∗) ⎥ ⎥ (∗)T ⎥ ⎥ ⎥ (∗)T ⎥ ⎥ (∗)T ⎥ ⎥ ⎥ (∗)T ⎦

(3.24)

−ε4 I

• Step 2: For X and Y given in the previous step, find β1 , β2 , F , and L such that the following generalized eigenvalue problem (GEVP) has solutions by replacing  with τ : max

β1 , β2 , F, L

τ

s.t. (3.2)–(3.6) hold for X and Y fixed.

• Step 3: For β1 , β2 , F , and L given in the previous step, find X and Y such that the following GEVP has solutions: max τ X, Y

s.t. (3.1)–(3.6) hold for β1 , β2 , F , and L fixed.

80

Circuits Syst Signal Process (2009) 28: 65–83

• Step 4: Return to Step 2 until the convergence of τ is attained with a desired accuracy. • Step 5: For X, Y , F , and L found in previous steps, find minimal β1∗ , and β2∗ according to the following constraints: min β1

s.t.

(3.3) holds for X, Y , F , and L fixed,

min β2

s.t.

(3.4)–(3.5) hold for X, Y , F , and L fixed.

• Step 6: τ ∗ is defined as follows: τ∗ =

τ × (β1 + 6β2 ) . β1∗ + 6β2∗

• Step 7: If τ ∗ < , stop. Else, return to Step 2. Remark 3.2 In Step 1, the initial data is obtained by assuming that the system has no time delay. Note that Steps 2 and 3 are quasi-convex optimization problems [1]. Hence, these two steps guarantee the convergence of τ . In order to reduce the conservatism of upper bound τ , we have Steps 5 and 6. Note that Step 5 is a convex problem.

4 Numerical Example Consider the following example taken from [18], where the plant parameters are described as follows:       −1.7 3.8 0.1 5 A= , B1 = , B2 = , −1 1.8 0.1 2.01 C1 = [ 1 0 ],   0.01 H1 = , 0 E2 = −1,

C2 = [ 10.1 4.5 ], H2 = H3 = 0.01,

D1 = 0.1, E1 = [ 1

D2 = 0,

0 ],

E3 = −1.

In our simulation, we set  = 0.05. We assume the sampling period is 0.01 for both sensor and actuation channels, that is, ha = hs = 0.01, and ns = na = 0, which means no data packet dropout occurs in the communication channel. We also choose τks = τla = 0.01. From this, it is not hard to see that the longer the sampling period is or the more data packets are lost, the smaller the time delay the communication channel can tolerate. Delay-free attenuation constant γdf is set to be 1, while constants ε1 , ε2 , ε3 , ε4 , ε5 , ε6 , ε7 , ε8 , and ε9 are set be equivalent to 1. By applying Theorem 3.1 and the algorithm in the previous section, we get the following controller gains by the calculation of (3.7)–(3.9):     −15.3467 21.8452 0.2344 ˆ ˆ A= , B= , Cˆ = [ −2.453 4.2682 ]. −6.8852 7.0634 0.1995

Circuits Syst Signal Process (2009) 28: 65–83

81

Fig. 4 The ratio of the regulated output energy to the disturbance input noise

Fig. 5 The disturbance input signal

The ratio of the regulated output energy to the disturbance input noise is depicted in Fig. 4. In our simulation, we use a uniform distributed random disturbance input signal w(t) with maximum value 3, which is depicted in Fig. 5. It can be seen that the −4 level ratio tends √ to a constant value of about 1.19×10 , which means the attenuation √ −4 equals 1.19 × 10 ≈ 0.011, less than the prescribed level γ = 1 + 0.0453 ≈ 1.022. Therefore the designed controller meets the performance requirement.

5 Conclusion In this paper, a technique has been proposed for designing a delay-dependent dynamic output feedback controller with robust disturbance attenuation and stability for an uncertain NCS with random communication network-induced delays and data packet dropouts. The main contribution of this work is that both the sensor-to-controller and controller-to-actuator delays/dropouts have been taken into account. Furthermore,

82

Circuits Syst Signal Process (2009) 28: 65–83

these delays are regarded as input delays and are dealt with in the scope of disturbance attenuation. The Lyapunov–Razumikhin method has been employed to derive a controller for this class of systems, and sufficient conditions for the existence of such a controller for this class of NCSs are derived. We have also used a numerical example to demonstrate the effectiveness of this methodology. We mention that in [7] a quantization-effect-based approach is introduced which takes into consideration the way that information is exchanged in a network. Therefore, the quantization effect could be incorporated into the framework in future work.

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