Event-triggered non-fragile control for linear systems with actuator saturation and disturbances

Information Sciences 429 (2018) 1–11

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Event-triggered non-fragile control for linear systems with actuator saturation and disturbances Dan Liu a, Guang-Hong Yang b,∗ a b

College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, 110819, PR China State Key Laboratory of Synthetical Automation of Process Industries, Northeastern University, Shenyang, Liaoning, 110819, PR China

a r t i c l e

i n f o

Article history: Received 18 January 2017 Revised 30 October 2017 Accepted 1 November 2017 Available online 2 November 2017 Keywords: Event-triggered control (ETC) Non-fragile control Linear systems Linear matrix inequalities (LMIs)

a b s t r a c t This paper investigates the problem of event-triggered non-fragile dynamic output feedback controller design for linear systems with actuator saturation and disturbances. The controller to be designed is supposed to include additive gain variations. By using Lyapunov stability theory and adding slack matrix variables, new sufficient conditions are derived to design the event-triggered parameters and the controller gains. Compared with the existing non-fragile dynamic output feedback controller design methods, the structural restriction on the Lyapunov matrix is relaxed. The effectiveness of the proposed method is demonstrated by two examples. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Networked control systems (NCSs) have received substantial attention due to their advantages, such as flexibility, maintainability and easy installation [12,28,31]. Nevertheless, the communication bandwidth in NCSs is inevitably constrained. Facing the situation, a so-called event-triggered control (ETC) has been proposed to save the communication resources. In comparison to the conventional time-driven control approach, in which the measurement information is sent with a constant sampling period, the ETC approach can conserve communication resources, and guarantee desired performance. Recently, the ETC problem has attracted increasing attention and several valuable results have been developed [7–9,13,21,25,36,39–41]. In [38], the dynamic output feedback controller (DOFC) is designed for ETC systems. In [29] and [30], the event-triggered filters and controllers are designed to detect faults. In [5] and [14], the event-triggered H∞ control problems for continuous and discrete stochastic systems are considered, respectively. Owing to physical and safety constraints, one of the common control problems is actuator saturation. If the physical constraint on input is not considered in the design of controllers, then the saturation nonlinearity can result in performance deterioration or even instability for systems, which may further lead to catastrophic accidents. Hence, it is of great significance to study dynamic systems with actuator saturation, and abundant results have been obtained [1,3,6,15,26]. Furthermore, the ETC problems for linear systems with actuator saturation have also attracted considerable interest [20,22,23,32,37]. In the above mentioned results, an implicit assumption is that the controller or filter can be implemented exactly. However, in practice, controllers or filters do have a certain degree of inaccuracies. Such inaccuracies may be caused by a variety of factors, including the aging of the components, roundoff errors in numerical computation, etc. Therefore, how to design an insensitive controller or filter with respect to some variations in its gains is a significant issue, and this has re∗

Corresponding author. E-mail addresses: [email protected] (D. Liu), [email protected] (G.-H. Yang).

https://doi.org/10.1016/j.ins.2017.11.003 0020-0255/© 2017 Elsevier Inc. All rights reserved.

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D. Liu, G.-H. Yang / Information Sciences 429 (2018) 1–11

Fig. 1. The event-triggered control loop.

ceived considerable interest [4,10,24,33,34]. In [4,34], the non-fragile filter design problems have been investigated for linear continuous-time systems. The non-fragile state feedback control problems and the output feedback control problems are discussed in [18] and [35], respectively. In [35], both the ETC and the non-fragile control problem is taken into account. Nevertheless, the control method in [35] is unapplicable when the states are not available. On the other hand, the actuator saturation is a common phenomenon. Hence, it is attractive and challenging to design an event-triggered non-fragile DOFC for linear systems with actuator saturation, but there is no result about it. Inspired by these points, this paper investigates the problem of event-triggered non-fragile DOFC design for linear systems with actuator saturation and disturbances. The main contributions are summarized as follows: 1) A discrete eventtriggered mechanism is adopted to save communication resources. Under this mechanism, the resulting closed-loop system is represented as a saturated linear system with time delay. By using Lyapunov theory and adding some slack variables, novel controller design conditions are derived in terms of linear matrix inequalities (LMIs). 2) Compared with [18], the proposed controller design conditions in this paper remove the strict restriction on the Lyapunov matrix by introducing the slack variables. 3) In contrast to the results without considering the impact of controller gain variations, the proposed non-fragile control approach can ensure better steady-state performance when the controller is not implemented exactly. Notation: For simplicity, the identity matrix with appropriate dimensions is abbreviated as I. The zero matrix with appropriate dimensions is denoted by 0. sgn( · ) denotes function. diag  the signum   {} means a block-diagonal matrix. He(X) X ZT X ∗ T denotes X + X . Sometimes, the symmetric matrices is written as . Z Y Y Z 2. Problem statement and preliminaries The structure diagram considered in this paper is shown as Fig. 1, which consists of the networks, the physical plant, the sampler, the event generator, the zero-order holder (ZOH), the controller and the actuator. In addition, the clock synchronization is adopted to keep the both samplers synchronized. 2.1. System description Consider the system



x˙ (t ) = Ax(t ) + Bσ (u(t )) + Gω (t ) y(t ) = C1 x(t ) z(t ) = C2 x(t ) + Lσ (u(t ))

(1)

where x(t ) ∈ Rn is the system state; y(t ) ∈ Rr is the measured output; z(t ) ∈ R p is the regulated output; u(t ) ∈ Rm is the control input; ω (t ) ∈ Rq is the external disturbance belonging to L2 [0, ∞ ); A, C1 , C2 , G, B and L are known matrices of appropriate dimensions, and assume that B is of full column rank (the same assumption is also used in [2,16,19]). σ ( · ) represents the standard saturation function, and it is defined as follows:

σ ( u ) = [ σ ( u 1 ) σ ( u 2 ) · · · σ ( u m )] T , among which σ (ui ) = sgn(ui )min{|ui |, 1} for i = 1, 2, · · · , m. Here, the notation σ is slightly abused to denote the vector function and the scalar valued function. 2.2. Event-triggered generator and non-fragile DOFC In order to reduce the communication load, an event generator is constructed between the sensor and the controller, which is used to determine whether the current sampled-data (y(tk h + ih ), tk h + ih ) will be sent to the controller by the

D. Liu, G.-H. Yang / Information Sciences 429 (2018) 1–11

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following event-triggered mechanism:

[y(tk h + ih ) − y(tk h )]T [y(tk h + ih ) − y(tk h )] ≤ ε 2 yT (tk h )y(tk h )

(2)

where h > 0 is the sampling interval, 0 < ε < 1 and  > 0 are event-triggered parameters. If the condition (2) is violated, then the current sampled-data will be sent to the controller; otherwise, it is discarded directly. Similar to [38], the interval [tk h, tk+1 h ) can be written as

[tk h, tk+1 h ) =

q 

Ti

i=1

where Ti = [tk h + (i − 1 )h, tk+1 h ), and q = tk+1 − tk . Define two functions η(t) and e(t) in the interval [tk h, tk+1 h ) as follows

⎧ ⎪ ⎨t − tk h,

t − tk h − h, η (t ) = ⎪· · · ⎩ t − tk h − (q − 1 )h,

t ∈ T1 t ∈ T2 , ··· t ∈ Tq

⎧ ⎪ ⎨y(tk h ) − y(tk h ), y(tk h ) − y(tk h + h ), e(t ) = ⎪ ⎩· · · y(tk h ) − y(tk h + (q − 1 )h ),

t ∈ T1 t ∈ T2 . ··· t ∈ Tq

It is clear that η(t) satisfies 0 ≤ η(t) < h. For the system (1), a non-fragile DOFC of the following form is considered on the interval [tk h, tk+1 h )



x˙ d (t ) = (Ad + Ad )xd (t ) + (Bd + Bd )y(tk h ) u˜ (t ) = (Kd + Kd )xd (t )

(3)

where xd (t ) ∈ Rn is the controller state; Ad , Bd and Kd are controller gain matrices to be designed; Ad = [δai j ]n×n , Bd = [δbi j ]n×r and Kd = [δki j ]m×n represent additive gain variations, where δ aij , δ bij and δ kij satisfy the following bound conditions

|δai j | ≤ δa , i, j = 1, 2, · · · , n, |δbi j | ≤ δb , i = 1, 2, · · · , n; j = 1, 2, · · · , r, |δki j | ≤ δk , i = 1, 2, · · · , m; j = 1, 2, · · · , n.

(4)

Remark 1. The interval-type uncertainty model can be found in [4,17,34], which has been widely applied to describe the finite word length (FWL) effects. From Fig. 1, it is easy to see that

u(t ) = u˜ (tk h + ih ) = (Kd + Kd )xd (t − η (t )), t ∈ [tk h, tk+1 h ). Let ξ (t ) =

[xT (t )

xTd (t )]T .

(5)

Combining with (1)-(5), the closed-loop system is obtained as follows

˙ ξ (t ) = Aξ (t ) + B1 ξ (t − η (t )) + B2 σ (Kξ (t − η (t ))) + B3 e(t ) + G ω (t ) y(t ) = C1 E1 ξ (t ) z(t ) = C2 E1 ξ (t ) + Lσ (Kξ (t − η (t )))

where t ∈ [tk h, tk+1 h ) and



A A= 0



G = GT





0 0 , B1 = A d + A d (Bd + Bd )C1

T



0 , K= 0









0 B , B2 = 0 0

Kd + Kd , E1 = I







0 , B3 = B d + B d

(6)

 ,

0 .

For the system (6), an initial condition of the state ξ (t) on [−h, 0] is supplemented as

ξ (t ) = ξ0 , t ∈ [−h, 0] where ξ 0 is a constant function on [−h, 0]. Furthermore, the event-triggered mechanism (2) is translated into

eT (t )e(t ) ≤ ε 2 (e(t ) + C1 E1 ξ (t − η (t )))T (e(t ) + C1 E1 ξ (t − η (t ))).

(7)

2.3. Preliminaries In this subsection, some mathematical tools are introduced, which will be used later. For a given w0 , Ww0 is defined by Ww0 = {ω (t ) : 0 <

∞ 0

ωT (t )ω (t )dt ≤ w0 }.

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For a given positive number ρ and a positive-definite matrix P ∈ R2n×2n , the set E (P, ρ ) is defined as

E (P, ρ ) = {ξ ∈ R2n : ξ T P ξ ≤ ρ}. For a matrix F ∈ Rm×n , the set L(F ) is defined as

L ( F ) = {ξ ∈ R 2n : |F i ξ | ≤ 1 } where F = [0 F ] and Fi is the ith row of F (i = 1, 2, · · · , m ). The set D contains all m × m diagonal matrices of which diagonal elements are either 0 or 1. Each element in D is denoted as Di , namely D = {Di : i = 1, 2, · · · , 2m }. Denote D− = I − Di . It is easy to see that D− ∈ D. i i Lemma 1 ([27]). For a matrix  and a symmetric matrix , the following statements are equivalent: 1. There exists a matrix P > 0 such that + P  + T P < 0. 2. For a given positive constant α , there exist matrices P > 0 and Y such that



− 2α P P + Y ( + α I )

P + ( + α I )T Y T −Y − Y T



< 0.

(8)

Lemma 2 ([1]). Given K, F ∈ Rm×2n , if ||F ξ ||∞ ≤ 1 for ξ ∈ R2n , then

σ (Kξ ) ∈ co{Di Kξ + D−i F ξ : i = 1, 2, · · · , 2m } where “co” denotes the convex hull. 2.4. Problem to be addressed This paper aims at designing an event-triggered mechanism and a non-fragile DOFC such that 1. The state trajectories of the system (6) that start from the region E (P + hQ, 1 ) will remain in the region E (P, w0 + 1 ) for ω (t ) ∈ Ww0 . 2. The L2 gain from the disturbance ω(t) to the regulated output z(t) is less than or equal to γ under zero initial conditions for ω (t ) ∈ Ww0 . 3. Stability analysis In the following theorem, novel sufficient conditions are presented to solve the event-triggered non-fragile control problem under the effect of the actuator saturation. Theorem 1. Given positive constants h, γ and λ1 , if there exist a scalar 0 < ε < 1, and matrices  > 0, R > 0, Q > 0, P > 0, F and S of appropriate dimensions such that E (P, 1 + w0 ) ⊂ L(F ) and



R ST

⎡



S R

T11 ⎢ T21 ⎢ −S ⎢ ⎢ λ1 PA ⎣ ( P G )T ( P B3 )T

≥ 0, ∗ T22 R+S T42 0 ε 2 C1 E1

(9) ∗ ∗ −Q − R 0 0 0

∗ ∗ ∗ h2 R − 2λ1 P λ1 (PG )T λ1 (PB3 )T

∗ ∗ ∗ ∗ −I 0

∗ ∗ ∗ ∗ ∗ − + ε 2 

⎤ ⎥ ⎥ ⎥ < 0. ⎥ ⎦

(10)

where

T11 = P A + AT P + Q − R +

1

γ2

(C2 E1 )T C2 E1 ,

T21 = (P B1 )T + (P B2 (Di K + D− F ))T + i

1

γ2

T22 = −2R − S − ST + ε 2 (C1 E1 )T C1 E1 +

(Di K + D−i F )T LT C2 E1 + R + S,

1

γ2

(Di K + D−i F )T LT L(Di K + D−i F ),

T42 = λ1 P B1 + λ1 P B2 (Di K + D− F ), ( i = 1, 2, · · · , 2m ) i then (1) the state trajectories of the closed-loop system (6) that start from the region E (P + hQ, 1 ) will remain in the region E (P, w0 + 1 ) for ω (t ) ∈ Ww0 .

D. Liu, G.-H. Yang / Information Sciences 429 (2018) 1–11

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(2) the L2 gain from the disturbance ω(t) to the regulated output z(t) is less than or equal to γ under zero initial conditions for ω (t ) ∈ Ww0 . Proof. 1) A Lyapunov–Krasovskii functional candidate is chosen as follows:

V (t ) = ξ T (t )P ξ (t ) +

t

t−h

ξ T ( s ) Q ξ ( s )d s + h

0 −h

t

t+θ

ξ˙ T (s )Rξ˙ (s )dsdθ .

Taking the time derivative along with the closed-loop system (6) yields

V˙ (t ) = 2ξ T (t )P ξ˙ (t ) + ξ T (t )Q ξ (t ) − ξ T (t − h )Q ξ (t ) + h2 ξ˙ T (t )Rξ˙ (t ) − h

t

t−h

ξ˙ T (s )Rξ˙ (s )ds.

(11)

Since (9), it follows

−h

t

t−h

ξ˙ T (s )Rξ˙ (s )ds ≤ − (ξ (t − η (t )) − ξ (t − h ))T R(ξ (t − η (t )) − ξ (t − h )) − (ξ (t ) − ξ (t − η (t )))T R(ξ (t ) − ξ (t − η (t ))) + 2(ξ (t − η (t )) − ξ (t − h ))T S(ξ (t ) − ξ (t − η (t ))).

(12)

Substituting (12) into (11) and taking the event-triggered mechanism (7) into consideration, it yields

V˙ (t ) ≤2ξ T (t )P (Aξ (t ) + B1 ξ (t − η (t )) + B2 σ (Kξ (t − η (t ))) + B3 e(t ) + G ω (t )) + h2 ξ˙ T (t )Rξ˙ (t ) + 2λ1 ξ˙ T (t )P (Aξ (t ) + B1 ξ (t − η (t )) + B2 σ (Kξ (t − η (t ))) + B3 e(t ) + G ω (t ) − ξ˙ (t )) − eT (t )e(t ) + ε 2 (e(t ) + C1 E1 ξ (t − η (t )))T (e(t ) + C1 E1 (t − η (t ))) + ξ T (t )Q ξ (t ) + ξ T (t − h )Q ξ (t − h ) − (ξ (t − η (t )) − ξ (t − h ))T R(ξ (t − η (t )) − ξ (t − h )) − (ξ (t ) − ξ (t − η (t )))T R(ξ (t ) − ξ (t − η (t ))) + 2(ξ (t − η (t )) − ξ (t − h ))T S(ξ (t ) − ξ (t − η (t ))).

(13)

Since E (P, 1 + w0 ) ⊂ L(F ), by Lemma 2, σ (Kξ (t − η (t ))) ∈ co{(Di K + D− F )ξ (t − η (t )) : i = 1, 2, · · · , 2m } for ξ ∈ E (P, 1 + i w0 ). Then, it yields

V˙ (t ) + ≤

1

γ2

zT (t )z(t ) − ωT (t )ω (t )

max

i=1,2,··· ,2m

2ξ T (t )P (Aξ (t ) + B1 ξ (t − η (t )) + B2 (Di K + D− F )ξ (t − η (t )) + B3 e(t ) + G ω (t )) i

+ 2λ1 ξ˙ T (t )P (Aξ (t ) + B1 ξ (t − η (t )) + B2 (Di K + D− F )ξ (t − η (t )) + B3 e(t ) + G ω (t ) − ξ˙ (t )) i + h2 ξ˙ T (t )Rξ˙ (t ) − eT (t )e(t ) + ε 2 (e(t ) + C1 E1 ξ (t − η (t )))T (e(t ) + C1 E1 (t − η (t ))) − ωT (t )ω (t ) − (ξ (t − η (t )) − ξ (t − h ))T R(ξ (t − η (t )) − ξ (t − h )) − (ξ (t ) − ξ (t − η (t )))T R(ξ (t ) − ξ (t − η (t ))) + γ −2 (C2 E1 ξ (t ) + L(Di K + D− F )ξ (t − η (t )))T (C2 E1 ξ (t ) + L(Di K + D− F )ξ (t − η (t ))) i i + 2(ξ (t − η (t )) − ξ (t − h ))T S(ξ (t ) − ξ (t − η (t ))) + ξ T (t )Q ξ (t ) + ξ T (t − h )Q ξ (t − h ). By (10), it ensures

V˙ (t ) +

1

γ2

zT (t )z(t ) − ωT (t )ω (t ) < 0.

(14)

Thus,

V˙ (t ) ≤ ωT (t )ω (t ).

(15)

Integrating both sides of (15) from 0 to t results in

V (t ) ≤ V (0 ) +

t 0

ω T ( s ) ω ( s )d s ≤ 1 + w 0 .

Therefore, the state trajectories of the system (6) starting from the region E (P, 1 ) will remain in the region E (P, w0 + 1 ) for ω (t ) ∈ Ww0 . ∞ ∞ 2) Integrate both sides of (14) from 0 to ∞, then 0 zT (t )z(t )dt ≤ γ 2 0 ωT (t )ω (t )dt under zero initial conditions.  Remark 2. Theorem 1 shows that the asymptotic stability cannot be achieved in the presence of disturbances in the system. Nevertheless, all states will eventually enter a region. It is common that the derived results guarantee the bounded stability, such as [10,11]. Remark 3. In the absence of disturbances, it is clear to see that V˙ < 0 for x ∈ E (P + hQ, 1 )\{0}. Consequently, the set E (P + hQ, 1 ) is included in the domain of attraction of the origin.

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4. Event-triggered controller design In Theorem 1, the controller gain matrices Ad , Bd and Kd are coupled with the Lyapunov matrix P, and therefore they cannot be directly computed. To obtain convex controller design conditions, the matrix B is assumed to be of full column  (BT B )−1 BT rank. Based on this assumption, let T = , where each row of matrix  is mutually independent and orthogonal 



to the columns of B. It is clear that T B = I 0

T

.

Theorem 2. For given positive scalars h, γ , α , λ1 , λ2 , β 1 , β 2 and a matrix Y0 , if there exist ε¯ > 0,  > 0, P > 0, R > 0, S, F, Kd , W1 , W2 , W3 , Y111 , Y112 , Y12 , Y13 , Y15 , Y16 , Y22 , Y23 , Y25 , Y26 , Y32 , Y33 , Y35 , Y36 , Y42 , Y43 , Y45 , Y46 , Y52 , Y53 , Y55 , Y56 , Y62 , Y63 , Y65 , Y66 of appropriate dimensions such that (9) and



P Fi

11 21 31

∗

22 0

 11 =

33 0

ψ112

0

(16)



∗ ∗

ψ111



ψ111

≥ 0,

(1 + w0 )−1 I



where



FiT

Q − R − 2α P = R+S −S

< 0 f or

δai j ∈ {−δa , δa }, δbi j ∈ {−δb , δb }, δki j ∈ {−δk , δk }.





ψ2112 ψ2122

ψ2111 , 21 = ψ2121

∗ −2R − S − ST − 2α I R+S

 , ∗ ∗ −Q − R − 2α I

 ,

ψ112 =diag{h2 R − 2αλ1 P, −I − 2α I, − − 2α I},    P + α 0 + 1 αY12 + 2 + 3 αY13 0 0 I + αY22 αY23 ψ2111 = , ψ2112 = 0 0 αY32 I + αY33 0   λ2 1 αY42 + λ2 2 + λ2 3 αY43 αY52 αY53 ψ2121 = 0 , 0 αY62 I + αY63   λ1 P + (α − 1 )λ2 0 αY45 + λ2 4 αY46 + λ2 5 0 I + αY55 αY56 ψ2122 = , 0 αY65 I + αY66  C 0 0 L(Di (Kd + Kd ) + D− F) 0 0 0 0 0 i 31 = 2 0

0

C1

0

33 =diag{−γ I,  −2ε¯ I}, ⎡ −He(0 ) ∗ −He(Y22 ) ⎢ −Y12T T ⎢ −Y T −Y32 − Y23 13 22 =⎢ ⎢ −Y42 0 ⎣ T T

(17)

0

0

0

0

0

αY15 + 4 αY25 αY35

αY16 + 5 αY26 αY36

 ,



0 , I

2

−Y15 T −Y16

 [Y0 Y111 ]T 0 = β1 [Y0 Y111 ]T

−Y52 − Y25 T −Y62 − Y26 Y112 β2Y112



∗ ∗ −He(Y33 ) −Y43 T −Y53 − Y35 T −Y63 − Y36



, 1 =

∗ ∗ ∗

−He(λ2 0 ) T −Y45 T −Y46

[Y0 Y111 ]T A β1 [Y0 Y111 ]T A

∗ ∗ ∗ ∗ −He(Y55 ) T −Y65 − Y56

W1 + Y112 Ad β2W1 + β2Y112 Ad

⎤

∗ ∗ ⎥ ⎥ ∗ ⎥, ⎥ ∗ ⎦ ∗ −He(Y66 )



,

    0 Y0 (Di (Kd + Kd ) + D− F) W2C1 + Y112 Bd C1 0 i 2 = , 3 = , − β2W2C1 + β2Y112 Bd C1 0 0 β1Y0 (Di (Kd + Kd ) + Di F )     [Y0 Y111 ]G W2 + Y112 Bd 4 = , 5 = , β1 [Y0 Y111 ]G β2W2 + β2Y112 Bd then (1) the state trajectories of the system (6) that start from the region E (P + hQ, 1 ) will remain in the region E (P, w0 + 1 ) for ω (t ) ∈ Ww0 .

D. Liu, G.-H. Yang / Information Sciences 429 (2018) 1–11

7

(2) the L2 gain from the disturbance ω(t) to the regulated output z(t) is less than or equal to γ under zero initial conditions for ω (t ) ∈ Ww0 . −1 −1 Moreover, the controller gains are given by Ad = Y112 W1 , Bd = Y112 W2 and Kd .

Proof. Rewrite the inequality (10) as

 + P  + T P < 0 where

(18)

⎡

Q − R + γ −2 (C2 E1 )T C2 E1 1 − T T ⎢R + S + γ 2 (Di K + Di F ) L C2 E1 ⎢ −S  =⎢ ⎢ 0 ⎣ 0 0

⎡

B 1 + B 2 ( Di K + D− F) i 0 0 B 1 + B 2 ( Di K + D− F) i 0 0

A ⎢0 ⎢0  =⎢ ⎢A ⎣ 0 0

∗

∗ ∗ −Q − R 0 0 0

φ1 R+S 0 0 ε 2 C1 E1

0 0 0 0 0 0

0 0 0 −I 0 0

G 0 0 G 0 0

⎤

∗ ∗ ∗ h2 R 0 0

∗ ∗ ∗ ∗ −I 0

⎤

∗ ∗ ⎥ ⎥ ∗ ⎥, ⎥ ∗ ⎦ ∗ 2 − + ε 

B3 0⎥ 0⎥ ⎥, B3 ⎥ ⎦ 0 0

P =diag{P, I, I, λ1 P, I, I},

φ1 = − 2R −S−ST+ ε 2 (C1 E1 )T C1 E1 +

1

γ2

(Di K + D−i F )T LT L(Di K + D−i F ).

By Lemma 1, (18) is equivalent to the following inequality



 − 2α P P + Y ( + α I )



∗ −Y − Y T

<0

(19)

where α is a given constant, and Y is an introduced slack matrix. Define

⎡

Y11 ⎢0 ⎢0 Y =⎢ ⎢0 ⎣ 0 0 with

 Y11 =

Y12 Y22 Y32 Y42 Y52 Y62

Y13 Y23 Y33 Y43 Y53 Y63

[Y0 Y111 ]T β1 [Y0 Y111 ]T

0 0 0 λ2Y11 0 0

Y112 β2Y112

Y15 Y25 Y35 Y45 Y55 Y65

⎤

Y16 Y26 ⎥ Y36 ⎥ ⎥ Y46 ⎥ ⎦ Y56 Y66

(20)

 ,

Substitute ε¯ = ε −1 , W1 = Y112 Ad , W2 = Y112 Bd and (20) to (19), and apply Schur complement lemma to (19). Then



11 21 31

∗

22 0

∗ ∗

33



<0

(21)

for δ aij , δ bij and δ kij satisfying (4). Obviously, (21) is equivalent to (17). Additionally, the constraint E (P, 1 + w0 ) ⊂ L(F ) is equivalent to (16). The inequalities (16) and (17) can ensure that the conditions in Theorem 1 are satisfied. This completes the proof.  Remark 4. The convex controller design conditions are given in Theorem 2, where the variables Y112 , W1 , W2 , Kd , ε¯ and −1 −1  can be obtained by solving a set of LMIs (9), (16) and (17). In other words, Ad = Y112 W1 , Bd = Y112 W2 and ε = ε¯ −1 are derived. Hence, the controller gain matrices and the event-triggered parameters are co-designed by Theorem 2.

Remark 5. In [18], the Lyapunov matrix P is partitioned as P = [(1 − α1 α2 )X−1 α2 I; α2 I − α2 α−1 X], which cannot still be 1 used for solving the problem in this paper. To obtain convex conditions, it is necessary to set X = α3 I. If let Yi j = 0 (i = j ), Y22 = Y33 = Y55 = Y66 = I, Y112 = α2 α−1 I, [Y0 Y111 ]T = (1 − α1 α2 )α−1 α−1 I, β1 = (1 − α1 α2 )−1 α2 α3 and β2 = −α3 α−1 , then a 3 1 set of solutions of Theorem 2 is obtained. It means that sufficient conditions for Theorem 2 can be obtained by applying the method in [18]. Hence, the convex controller design method in Theorem 2 relaxes the extra constraints on the Lyapunov matrix P by introducing some slack matrices.

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When the event-triggered mechanism and the actuator saturation are not considered, the following corollary provides a sufficient condition to design the non-fragile H∞ controller. Corollary 1. Given constants α > 0, γ > 0, β 1 , β 2 , and matrix Y0 , the system (1) is asymptotically stable satisfying a prescribed H∞ performance, if there exist matrices P > 0, Kd , W1 , W2 , Y111 , Y112 , Y12 and Y22 such that

⎡

−2α P 0 ⎢ ⎢P + αY11 + 1 + 2 + 1 ⎣ 0

∗ − ( 1 + 2α )I αY12 + Y11 G I + αY22 0

2

for where

 1 =

∗ ∗ ∗ T −Y22 −Y22 0

∗ ∗ ∗ ∗

⎤

⎥ ⎥ <0 ⎦

(22)

−γ 2 I

δai j ∈ {−δa , δa }, δbi j ∈ {−δb , δb }, δki j ∈ {−δk , δk } 

Y0 (Kd + Kd ) β1Y0 (Kd + Kd )

0 0

∗ ∗ T −Y11 −Y11 T −Y12 0

, 2 = [C2

LKd + LKd ].

−1 −1 The controller gains are given by Ad = Y112 W1 , Bd = Y112 W2 and Kd .

5. Examples To demonstrate that the proposed non-fragile controller design method is effective, the following examples are presented in this section. Example 1. Consider the plant (1) with



A=



0.1 0



 









1 , C2 = 0.5

C1 = 1



0.6 1 0.1 , B= , G= , −0.1 1 0.1 0.1 , L = 0.

By solving inequalities (9), (16) and (17) for δa = δb = 0, δk = 0.2, γ = 0.095 and h = 0.01, the controller gains are computed as follows



Ad =

−4.1952 −1.7714



 −1.2763 , Bd = −1.7982 −1.9553

T



−0.8429 , Kd = 2.0681



0.5819 ,

and the event-triggered parameters are ε = 0.2,  = 49.2237. Applying the standard controller design method, the controller gain matrices are given by



−2.9693 Ad = −1.2106



 −1.4468 , Bd = −1.2987 −1.2911

T



−0.4027 , Kd = 0.2020



−0.3809 ,

and the event-triggered parameters are ε = 0.2,  = 48.8071. In the simulation, the initial states are taken as x(0 ) = [0.1 0.15]T and xd (0 ) = [0 0]T , and the disturbance is described as



ω (t ) =

0.2sin(2t ), 0,

0 ≤ t ≤ 20 . t > 40

Additionally, the controller gain variations are assumed to be Kd = [−0.2 − 0.2cos(t )], Ad = 0 and Bd = 0. The state trajectories under non-fragile controller and standard controller are shown in Fig. 2 and Fig. 3, respectively. Fig. 2 shows that the controller obtained by Theorem 2 is insensitive for a small perturbation in Kd , and the system is still stable. Fig. 3 shows that a small perturbation in Kd leads to instability. In addition, the inter-event intervals obtained by Theorem 2 with or without controller gain variations are given in Fig. 4. Example 2. Consider the plant (1) with



A=



0 −1

C1 = 1



 





1 2 0.1 , B= , G= , 1 4 0.1





2 , C2 = 1



1 , L = 0.

Assume that the gain variations in the controller (3) are given as follows

Ad = 0, Kd = 0, Bd = [δb11 δb21 ]T

D. Liu, G.-H. Yang / Information Sciences 429 (2018) 1–11

9

Fig. 2. The state trajectories under the non-fragile controller with δk = 0 (left) and δk = 0.2 (right).

Fig. 3. The state trajectories under the standard controller with δk = 0 (left) and δk = 0.2 (right).

Fig. 4. The inter event-intervals under the non-fragile controller with δk = 0 (left) and δk = 0.2 (right).

where |δ b11 | ≤ 0.01 and |δ b21 | ≤ 0.01. Taking h = 0.01 and γ = 0.9, then inequality conditions (9), (16) and (17) in Theorem 2 have a set of feasible solutions



−4.1786 Ad = −3.5736



 −1.1749 , Bd = −0.7300 −2.3360

T





−0.5872 , Kd = 2.3031

0.6575 .

However, there is no feasible solution for the methods in [18] and [38]. In the following, a comparison between Corollary 1 and Theorem 3 in [18] will be given. For γ = 0.37, the controller gain matrices obtained by Theorem 3 in [18] are given by



−1.7147 Ad = 10 × −3.7216 3









T

−1.6963 16.2993 1.3774 , Bd = , Kd = 103 × −3.7421 16.3041 −9.1805

.

Applying Corollary 1 in this paper, the controller gain matrices are computed as follows



−8.6347 Ad = −5.7987



 −2.8192 , Bd = −1.7564 −4.4766

T



−2.2496 , Kd = 1.2047



0.3947 .

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It is clear that the controller gains obtained by Corollary 1 are much smaller than [18] for the same disturbance attenuation level γ . 6. Conclusion The problem of event-triggered non-fragile controller design for linear systems with actuator saturation and disturbances has been investigated in this paper. By using Lyapunov theory and adding slack matrix variables, the LMI-based sufficient conditions are given to design event-triggered parameters and controller gains. The resulting design conditions can achieve the prescribed control objectives. The simulation examples have illustrated the superiority of the proposed method. Acknowledgment This work was supported in part by the Funds of the National Natural Science Foundation of China (Grant Nos. 61621004 and 61420106016), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2013ZCX01). References [1] Y.Y. Cao, Z.L. Lin, T.S. 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