Finite-time asynchronous filtering for switched linear systems with an event-triggered mechanism

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Journal of the Franklin Institute 356 (2019) 5503–5520 www.elsevier.com/locate/jfranklin

Finite-time asynchronous filtering for switched linear systems with an event-triggered mechanism Liu Yang, Chaoxu Guan∗, Zhongyang Fei The Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150000, China Received 15 October 2018; received in revised form 29 January 2019; accepted 8 March 2019 Available online 11 May 2019

Abstract This paper investigates the event-triggered finite-time H∞ filtering for a class of continuous-time switched linear systems. Considering that the system may switch within an inter-event interval, the asynchronous problem is taken into account for the system and filter modes. By adopting the average dwell time (ADT) technique and multiple Lyapunov functions, new conditions are obtained to guarantee that the filtering error system is finite-time bounded with a prescribed disturbance attenuation performance. Further, the finite-time H∞ filter together with event-triggered mechanism is co-designed for the switched linear systems. Finally, a numerical example is provided to demonstrate the effectiveness of the method proposed in this paper. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Switched systems, as an important kind of hybrid systems, have attracted much attention over the past decades since their widespread applications in chaos generators, networked control systems, and robot control systems [1–3]. Switched systems are usually composed of a finite number of subsystems and a switching signal that administrates the switching behaviors among subsystems. Owing to its theoretical value and practical applications, lots of results have been reported on various issues of switched systems [4–8]. As a typical switching ∗

Corresponding author. E-mail addresses: [email protected] (L. Yang), [email protected] (C. Guan), zhongyang.fei@hit. edu.cn (Z. Fei). https://doi.org/10.1016/j.jfranklin.2019.03.019 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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signal in switched systems, average dwell time (ADT) switching means that the number of switches in a finite interval is bounded and the average time between consecutive switching is not less than a constant [9]. Recently, considerable attention has been paid to the research on switched systems with ADT switching [10–12]. For example, the asynchronous H∞ control was investigated for a class of discrete-time switched systems by using Lyapunov functional method and ADT switching technique [13]. Zhao and Hill introduced the concept of general Lyapunov-like functions and presented a necessary and sufficient condition for the stability of switched nonlinear systems [14]. On the other hand, with the rapid growth of digital computing, the problem of sampleddata systems has been excessively studied over the past decades [16–18]. Usually, the system information is sampled with a fixed sampling period for controller or filter updating, which is the so-called periodic sampling or time-triggered sampling mechanism. Though this method is simple and easy to be implemented, it may lead to unnecessary data transmission, which is undesired in practical engineering, especially for some networked control systems with limited communication bandwidth. To overcome such problem, event-triggered mechanism was first proposed by Hendricks et al. in the 1990s [19]. Then, much effort has been successively made for the research on various issues of event-triggered control systems since the wide applications in offshore platforms, neural networks, decentralised control and etc. [15,20–22]. As for linear systems, Heemels et al. developed the periodic event-triggered control (PETC) scheme and applied it to both static state-feedback and dynamical output-based control design for both centralized and decentralized (periodic) event-triggering systems [23]. Gu et al. proposed an adaptive event-triggered scheme for a class of networked nonlinear interconnected systems, which achieved a balance between the control performance and utilization of the network resource [24]. Gu et al. also designed an adaptive event-triggered scheme to the decentralized filtering for a class of networked nonlinear interconnected systems [25]. At present, the research on switched systems is mainly focused on Lyapunov asymptotic stability, while in practical applications, one may pay more attention to the dynamic behavior of a system in the fixed time interval. Therefore, it is very important to study the control problem of switched systems in the fixed time interval, which gives rise to the concept of finite-time stability and boundedness. The finite-time filter design was first discussed in [26]. Then many results have been proposed for the related issues in the finite-time domain [27–29]. To mention a few, the finite-time control was considered for linear systems subject to time-varying parametric uncertainties and exogenous disturbances, and a sufficient condition was obtained for finite-time stability via a state feedback controller [30]. Qi et al. developed a systematic control method to study the finite-time boundedness and stabilization of switched linear systems using event-triggered controllers [31]. The global finite-time stabilization was addressed for switched nonlinear systems under arbitrary switchings by non-Lipschitz continuous state feedback controllers [32]. However, to the best of our knowledge, rare result can be found on the finite-time H∞ filtering for switched linear systems, especially with the event-triggered control mechanism, which motivates the current study. In this paper, we investigate the finite-time H∞ filtering for switched linear systems with asynchronous phenomenon. The main contributions can be summarized as follows. A novel framework of output-based control for filtering error system with asynchronous switching and external disturbance is established. An event-triggered mechanism is adopted for the switched linear systems, which can avoid some unnecessary data transmission. Based on the ADT

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switching technique and multiple Lyapunov functional method, new criteria are obtained to guarantee that the filtering error system is finite-time bounded and satisfies a weighted H∞ disturbance attenuation performance. Moreover, the finite-time H∞ filter is designed for the switched linear systems. The remainder of this paper is organized as follows. In Section II, the problem formulation and some preliminaries are given. Section III establishes the sufficient conditions for the finite-time boundedness and finite-time H∞ performance of the event-triggered switched linear system. Then, numerical simulations are provided to illustrate the effectiveness and validity of the proposed method in Section IV. Finally, Section V concludes this paper. Notation: The notations used in this paper are quite standard. Rn stands for the n dimensional Euclidean space. N+ represents the set of all positive integers and  ·  refers to the Euclidean vector norm. PT and P−1 denote the transposition and the inverse of matrix P, respectively. P > 0( < 0) means that P is positive (negative) define. λmin (P) and λmax (P) represent the minimum and maximum eigenvalues of matrix P, respectively. ∗ denotes the symmetric elements in a symmetric matrix, I and 0 are the appropriately dimensioned identity matrix and zero matrix. diag{. . .} refers to a block-diagonal matrix. L2 [0, ∞) is the space of square integrable infinite sequence. 2. Problem formulation and preliminaries Consider the following continuous-time switched linear system: ⎧ ⎪ ⎨x˙(t ) = Aσ (t ) x(t ) + Bσ (t ) ω(t ), y(t ) = Cσ (t ) x(t ) + Dσ (t ) ν(t ), ⎪ ⎩z(t ) = L x(t ), σ (t )

(1)

where x(t ) ∈ Rnx is the state vector, z(t ) ∈ Rnz denotes the signal to be estimated and y(t ) ∈ Rny is the controlled output. ω(t ) ∈ Rnw is the exogenous disturbance and ν(t ) ∈ Rnv denotes the measurement noise, which belong to L2 [0, ∞). σ (t ) : [0, ∞ ) → S = {1, 2, . . . , p} is the switching signal which is a piecewise constant function depending on t, and p is the number of subsystems. Let {tlσ , l = 1, 2, . . .} be a given time sequence satisfying t1σ < t2σ < · · · < σ tlσ < tl+1 < · · · , where tlσ is the switching instant. Assumption 1. [17] There exists a number τ d > 0 such that any two switches are separated σ by at least τ d , i.e., tl+1 − tlσ ≥ τd for any l > 0. Define the event-triggered time instants as t0 < t1 <  < tk < , where limk→∞ tk = ∞. We assume that the length of variable inter-event interval hk = tk − tk−1 is bounded by H, which satisfies 0 < hk ≤ H ≤ τd , ∀k ∈ N,

(2)

where H denotes the maximum length of inter-event intervals. For given positive definite matrices  and , consider the switched system (1) with the event-triggered communication represented by

tk+1 = min{tk + H, tk+1 },

(3)

where

tk+1 = min{t , eTy (t )ey (t ) ≥ yT (tk )y(tk )}. t>tk

(4)

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ey (t ) = y(t ) − y(tk ) is output error for t ∈ [tk , tk+1 ) and ,  are event-triggered matrices to be determined. For simplicity, we call [tk , tk+1 ) an inter-event interval. Obviously, for any t ∈ [tk , tk+1 ), we have eTy (t )ey (t ) ≤ yT (tk )y(tk ).

(5)

Furthermore, the inequality (5) can be expressed by eTy (t )ey (t ) ≤ (y(t ) − ey (t ))T (y(t ) − ey (t )), t ∈ [tk , tk+1 ).

(6)

Based on the above event-triggered mechanism, the H∞ filter system is formulated as follows:  x˙ f (t ) = A f σ (tk ) x f (t ) + B f σ (tk ) y(tk ), (7) z f (t ) = C f σ (tk ) x f (t ) + D f σ (tk ) y(tk ), where xf (t) denotes the filter state, zf (t) is the estimator of z(t). A f σ (tk ) , B f σ (tk ) , C f σ (tk ) , D f σ (tk ) are filter parameters to be designed.  T  T Define e(t ) = z(t ) − z f (t ), η(t ) = x T (t ) x Tf (t ) , and ω˜ (t ) = ωT (t ) ν T (t ) , δ(t ) = (σ (t ), σ (tk )), we can obtain the following filtering error system: η(t ˙ ) = A˜ δ(t ) η(t ) + B˜ 1σ (tk ) ey (t ) + B˜ ωδ(t ) ω˜ (t ), (8) e(t ) = C˜δ(t ) η(t ) + D˜ 1σ (tk ) ey (t ) + D˜ ωδ(t ) ω˜ (t ), where A˜ δ(t ) =



Aσ (t ) B f σ (tk )Cσ (t )

B˜ 1σ (tk ) =



0 A f σ (tk )

 ,



B ˜ , Bωδ(t ) = σ (t ) −B f σ (tk ) 0 0

0 B f σ (tk ) Dσ (t )

 ,

C˜δ(t ) = [Lσ (t ) − D f σ (tk )Cσ (t ) , −C f σ (tk ) ],  D˜ 1σ (tk ) = D f σ (tk ) , D˜ ωδ(t ) = 0

 −D f σ (tk ) Dσ (t ) .

Assumption 2. For a given interval Tf and a positive constant d, the exogenous disturbance ω˜ (t ) satisfies the following condition: Tf ω˜ T (t )ω˜ (t )d t ≤ d , d > 0. (9) 0

Now we present some definitions for latter development. Definition 1. [33] For any T > t > 0 and any switching signal σ (t), Nσ (t, T) defines the switching number of σ (t) in the interval (t, T). We said that σ (t) has an average dwell time τ α if there exist positive numbers N0 and τ α such that Nσ (t, T ) ≤ N0 +

T −t , ∀0 < t < T , τα

where N0 is called the chatter bound of σ (t).

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Definition 2. [34] Given a positive definite matrix R, positive constants c1 , c2 , Tf with c1 < c2 and a switching signal σ (t), the filtering error system (8) is said to be finite-time bounded (FTB) with respect to (c1 , c2 , Tf , R, d, σ (t)), if ηT (t0 )Rη(t0 ) ≤ c1 ⇒ ηT (t )Rη(t ) ≤ c2 , ∀t ∈ [0, T f ], Tf ∀ω˜ (t ) : ω˜ T (t )ω˜ (t )d t ≤ d . 0

Definition 3. [34] Given a positive definite matrix R, two positive constants c2 and Tf and a switching signal σ (t), the filtering error system (8) under zero initial condition is said to be FTB with a prescribed finite-time non-weighted H∞ attenuation performance index γ > 0, if the system is finite-time bounded and the following inequality holds: Tf Tf eT (t )e(t )dt < γ 2 ω˜ T (t )ω˜ (t )dt, 0

0

where ω˜ (t ) satisfies the Assumption 2. 3. Main results In this section, we will present the sufficient conditions for the finite-time boundedness and finite-time H∞ performance of the filtering error system (8). Theorem 1. For given positive scalars γ , α > 1, ρ 1 , ρ 2 , μ ≥ 1, if there exist positive definite matrices Pii , Pij , , , such that the following matrix inequalities hold for any i, j ∈ S, i = j, ⎤ ⎡ Pi j A˜ i j + A˜ Ti j Pi j − αPi j Pi j B˜ 1 j Pi j B˜ ωi j C¯iT  ⎢ ∗ − 0 − ⎥ ⎢ ⎥ (10) ⎢ ⎥ < 0, ⎣ D¯ iT  ⎦ ∗ ∗ −γ 2 I ∗ ∗ ∗ − ⎡

Pj j A˜ j j + A˜ Tj j Pj j − αPj j ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗

Pj j B˜ 1 j − ∗ ∗

Pj j B˜ ω j j 0 −γ 2 I ∗

⎤ C¯ Tj  − ⎥ ⎥ ⎥ < 0, D¯ Tj  ⎦ −

Pii ≤ μPi j , Pi j ≤ Pj j ,   ρ2 c1 ρ1 c2 2 + γ d eε < , λ1 λ2

(11)

(12) (13)

with the ADT switching signal σ (t) satisfying τ ≥ τα∗ =

ln ( ρλ1 c2 2 )

ln μT f , − ln ( ρλ2 c1 1 + γ 2 d ) − ε

(14)

then the filtering error system (8) is finite-time bounded with respect to (c1 , c2 , Tf , R, d, σ (t)), where ρ 1 I ≤ Pδ(t) , ρ 2 I ≥ Pδ(t) , λ1 = λmin (R), λ2 = λmax (R), ε = N0 ln μ + αT f + αH . Proof. Choose the following Lyapunov function: Vδ(t ) (t ) = ηT (t )Pδ(t ) η(t ).

(15)

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When t ∈ [tk , tk+1 ), taking the derivation of Vδ(t) (t) along the trajectories of system (8) yields V˙δ(t ) (t ) = 2ηT (t )Pδ(t ) η(t ˙ ) T = 2η (t )Pδ(t ) [A˜ δ(t ) η(t ) + B˜ 1σ (tk ) ey (t ) + B˜ ωδ(t ) ω˜ (t )]. Case 1). Suppose that there is one switching in [tk , tk+1 ). When t ∈ [tlσ , tk+1 ), according to the prescribed event-triggered inequality (8), we have V˙i j (t ) − αVi j (t ) − γ 2 ω˜ T (t )ω˜ (t ) ≤ 2ηT (t )Pi j η(t ˙ ) − αηT (t )Pi j η(t ) − γ 2 ω˜ T (t )ω˜ (t ) −eTy (t )ey (t ) + (y(t ) − ey (t ))T (y(t ) − ey (t )) = 2ηT (t )Pi j (A˜ i j η(t ) + B˜ 1 j ey (t ) + B˜ ωi j ω˜ (t )) − αηT (t )Pi j η(t ) + (C¯i η(t ) + D¯ i ω˜ (t ) − ey (t ))T (C¯i η(t ) + D¯ i ω˜ (t ) − ey (t )) − γ 2 ω˜ T (t )ω˜ (t ) = ξ T (t )i j ξ (t ),

(16)

where  T     ξ (t ) = x˜T (t ) eTy (t ) ω˜ T (t ) , C¯i = Ci 0 , D¯ i = 0 Di ,   1i = C¯i −I D¯ i , ⎡ ⎤ Pi j A˜ i j + A˜ Ti j Pi j − αPi j Pi j B˜ 1 j Pi j B˜ ωi j ⎢ ⎥ i j = ⎣ ∗ − 0 ⎦ + T1i 1i . ∗ ∗ −γ 2 I

(17)

Similarly, when t ∈ [tk , tlσ ), we can obtain V˙ j j (t ) − αV j j (t ) − γ 2 ω˜ T (t )ω˜ (t ) ≤ 2ηT (t )Pj j η(t ˙ ) − αηT (t )Pj j η(t ) − γ 2 ω˜ T (t )ω˜ (t ) −eTy (t )ey (t ) + (y(t ) − ey (t ))T (y(t ) − ey (t )) = 2ηT (t )Pj j (A˜ j j η(t ) + B˜ 1 j ey (t ) + B˜ ω j j ω˜ (t )) −αηT (t )Pj j η(t ) + (C¯ j η(t ) + D¯ j ω˜ (t ) − ey (t ))T (C¯ j η(t ) + D¯ j ω˜ (t ) − ey (t )) − γ 2 ω˜ T (t )ω˜ (t ) = ξ T (t ) j j ξ (t ), where j j

⎡

Pj j A˜ j j + A˜ Tj j Pj j − αPj j ⎢ =⎣ ∗ ∗

(18) Pj j B˜ 1 j − ∗

⎤ Pj j B˜ ω j j ⎥ 0 ⎦ + T1 j 1 j . −γ 2 I

(19)

Case 2). There is no switching in [tk , tk+1 ). By setting tlσ → tk+1 , we can obtain the similar result as the interval [tk , tlσ ) in Case 1. Therefore, V˙δ(t ) (t ) − αVδ(t ) (t ) − γ 2 ω˜ T (t )ω˜ (t ) ≤ ξ T (t )δ(t ) ξ (t ),

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where δ(t ) = (i, j) in Eq. (17) and δ(t ) = ( j, j) in Eq. (19), respectively. Based on Schur complement, inequality δ(t) < 0 can be guaranteed by Eqs. (10) and (11) for any t ∈ [tk , tk+1 ). Consequently, the following inequality holds: V˙δ(t ) (t ) − αVδ(t ) (t ) − γ 2 ω˜ T (t )ω˜ (t ) < 0.

(20)

Denote F(t ) = γ 2 ω˜ T (t )ω˜ (t ). For any t ∈ [tk , tk+1 ), it holds that tlσ σ σ V j j (tlσ )
(21)

tk

On the other hand, from Eqs. (12) and (15), Vi j (tlσ ) ≤ V j j (tlσ ) σ

= eα(tl



−tk )

tlσ

V j j (tk ) +

σ

eα(tl

−s)

F (s)ds,

(22)

tk

From Eqs. (21) and (22), we have tk+1 α(tk+1 −tlσ ) σ Vi j (tk+1 ) ≤ e Vi j (tl ) + eα(tk+1 −s) F (s)ds tlσ

σ

σ

= eα(tk+1 −tl ) (eα(tl



−tk )

tlσ

V j j (tk ) +

σ

eα(tl

−s)

F (s)ds) +

tk

σ

≤ eα(tk+1 −tk )V j j (tk ) + eα(tk+1 −tl ≤e



α(tk+1 −tk )

V j j (tk ) +

)



tlσ

σ

eα(tl

−s)

F (s)ds +

tk tlσ

e

α(tk+1 −s)

F (s)ds +

tk

= eα(tk+1 −tk )V j j (tk ) +



tk+1

tk+1

tlσ

tk+1

tlσ tk+1

tlσ

eα((tk+1 −s) F (s)ds eα(tk+1 −s) F (s)ds

eα(tk+1 −s) F (s)ds

eα(tk+1 −s) F (s)ds.

(23)

tk

Moreover, Eqs. (12) and (23) imply Vii (tk+1 ) ≤ μVi j (tk+1 ) ≤ μeα(tk+1 −tk )V j j (tk ) + μ



tk+1

eα(tk+1 −s) F (s)ds

tk

≤ μe

α(tk+1 −tk )

V j j (tk ) + μe

≤ μeα(tk+1 −tk ) (V j j (tk ) +

α(tk+1 −tk )





tk+1

F (s)ds

tk tk+1

F (s)ds).

tk

Therefore, for any t ∈ [tk , tk+1 ), we can obtain t Vδ(t ) (t ) ≤ μeα(t−tk ) (Vδ(tk ) (tk ) + F (s)ds) tk

 ≤ μeα(t−tk ) μeα(tk −tk−1 ) Vδ(tk−1 ) (tk−1 ) +

tk

tk−1

F (s)ds) + tk

t

 F (s)ds

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≤ μNσ (t0 ,t ) eα(t−t0 )Vδ(t0 ) (t0 ) + μNσ (t0 ,t ) eα(t−t0 ) + · · · + μ2 eα(t−tk−1 )





t1

t0 tk

F (s)ds



F (s)ds + μeα(t−tk )

tk−1

Nσ (t0 ,t ) α(t−s+s−t0 )

Vδ(t0 ) (t0 ) + μ

e

F (s)ds

tk

Nσ (t0 ,t ) α(t−s+s−t0 )

=μ

t

+ · · · + μ2 eα(t−s+s−tk−1 )

t0



tk

t1

F (s)ds

e

F (s)ds + μeα(t−s+s−tk )

tk−1



t

F (s)ds.

tk

Since the inter-event interval cannot be larger than H, t1 Nσ (t0 ,t ) α(t−t0 ) Nσ (t0 ,t ) αH Vδ(t ) (t ) ≤ μ e Vδ(t0 ) (t0 ) + μ e eα(t−s) F (s)ds + · · · + μ2 eαH



t0 tk

eα(t−s) F (s)ds + μeαH

tk−1

≤ μNσ (t0 ,t ) eα(t−t0 )Vδ(t0 ) (t0 ) + eαH



≤μ

e

t

eα(t−s) F (s)ds

μNσ (s,t ) eα(t−s) F (s)ds

Nσ (t0 ,T f ) αH +αT f

Vδ(t0 ) (t0 ) + μ

t

tk

t0 Nσ (t0 ,T f ) αT f





e

Tf

F (s)ds

t0

≤ μNσ (t0 ,Tf ) eαTf +αH (Vδ(t0 ) (t0 ) + γ 2 d ).

(24)

In addition, it yields from Eq. (15) that λmax Pδ(t ) T η (t0 )Rη(t0 ) λmin (R) ρ2 c1 ≤ . λ1

Vδ(t0 ) (t0 ) ≤

(25)

On the other hand, λmin Pδ(t ) T η (t )Rη(t ) λmax (R) ρ1 ≥ ηT (t )Rη(t ). λ2

Vδ(t ) (t ) ≥

(26)

Then, it follows from Eqs. (24)–(26) that λ2Vδ(t ) (t ) ρ1   λ2 Nσ (t0 ,Tf ) αTf +αH ρ2 c1 ≤ μ e + γ 2d ρ1 λ1   ρ2 c1 λ2 = eNσ (t0 ,Tf )lnμ eαH +αTf + γ 2d . ρ1 λ1

ηT (t )Rη(t ) ≤

By combining the inequality (13) and the switching constraint (14), we have ηT (t)Rη(t) ≤ c2 , which means the filtering error system (8) is FTB with respect to (c1 , c2 , Tf , R, d, σ (t)). The proof is completed.

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Based on the result in Theorem 1, the H∞ performance analysis is proposed in the following theorem: Theorem 2. For given positive scalars γ˜ , α > 1, ρ 1 , ρ 2 , μ ≥ 1, if there exist positive definite matrices Pii , Pij , , , such that the following matrix inequalities hold for any i, j ∈ S, i = j: ⎤ ⎡ ˜ C¯iT  Pi j Ai j + A˜ Ti j Pi j − αPi j Pi j B˜ 1 j Pi j B˜ ωi j C˜iTj ⎢ ∗ − 0 D˜ 1T j − ⎥ ⎢ ⎥ ⎢ ⎥ 2 T (27) ⎢ ∗ ∗ −γ I D˜ ωi j D¯ iT  ⎥ < 0, ⎢ ⎥ ⎣ ⎦ ∗ ∗ ∗ −I 0 ∗ ⎡

Pj j A˜ j j + A˜ Tj j Pj j − αPj j ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

∗

∗

Pj j B˜ 1 j − ∗ ∗ ∗

∗

Pj j B˜ ω j j 0 −γ 2 I ∗ ∗

− ⎤ C¯ Tj  − ⎥ ⎥ ⎥ D¯ Tj  ⎥ < 0, ⎥ 0 ⎦

C˜ Tj j D˜ 1T j D˜ ωT j j −I ∗

(28)

−

Pii ≤ μPi j , Pi j ≤ Pj j ,

(29)

ρ1 c2 /λ2 , γ 2d

(30)

eε ≤

the filtering error system (8) is FTB with respect to (0, c2 , Tf , R, d, σ (t)) and has an H∞ performance index γ˜ for any ADT switching signals satisfying  ln μT f ln μ ∗ τ ≥ τα = max , , (31) ln ( ρλ1 c2 2 ) − ln (γ 2 d ) − ε α where γ˜ = γ eN0 ln μ/2+αTf . Proof. By performing the similar procedure as Theorem 1, we can easily obtain that conditions (27) and (28) yield V˙δ(t ) (t ) − αVδ(t ) (t ) − γ 2 ω˜ T (t )ω˜ (t ) + eT (t )e(t ) < 0. Since (t ) = γ 2 ω˜ T (t )ω˜ (t ) − eT (t )e(t ) and tk+1 Vii (tk+1 ) ≤μeα(tk+1 −tk )V j j (tk ) + μ eα(tk+1 −s) (s)ds, tk

we get Vδ(t ) (t ) ≤ μeα(t−tk )Vδ(tk ) (tk ) + μ



t

eα(t−s) (s)ds

tk

≤ μeα(t−tk ) (μeα(tk −tk−1 )Vδ(tk−1 ) (tk−1 ) + μ



tk

tk−1

eα(tk −s) (s)ds) + μ

tk

t

eα(t−s) (s)ds

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= μ2 eα(t−tk−1 )Vδ(tk−1 ) (tk−1 ) + μ2 ≤ μNσ (t0 ,t ) eαt Vσ (t0 ) (t0 ) +



tk

eα(t−s) (s)ds + μ

tk−1



t



t

eα(t−s) (s)ds

tk

μNδ (s,t ) eα(t−s) (s)ds.

t0

Then Vδ(t ) (t ) ≤ μeα(t−tk )Vδ(tk ) (tk ) + μ



t

eα(t−s) (s)ds

tk

≤ μeα(t−tk ) (μeα(tk −tk−1 )Vδ(tk−1 ) (tk−1 ) + μ = μ2 eα(t−tk−1 )Vδ(tk−1 ) (tk−1 ) + μ2 ≤ μNσ (t0 ,t ) eαt Vδ(t0 ) (t0 ) +







tk

eα(tk −s) (s)ds) + μ

tk−1 tk

eα(t−s) (s)ds + μ

tk−1 t





t

eα(t−s) (s)ds

tk t

eα(t−s) (s)ds

tk

μNσ (s,t ) eα(t−s) (s)ds.

t0

When ω˜ (t ) = 0, under zero initial condition, we have t μNσ (s,t ) eα(t−s) (s)ds ≥ 0. t0

Equivalently, t μNσ (s,t ) eα(t−s) eT (s)e(s)ds t0



≤

t

γ 2 μNσ (s,t ) eα(t−s) ω˜ T (s)ω˜ (s)ds.

t0

According to the ADT constraint (31), Nσ (s, t ) ln μ ≤ N0 ln μ + T (s, t )α. Due to μ ≥ 1, it holds that t t eT (s)e(s)ds ≤ μNσ (s,t ) eα(t−s) eT (s)e(s)ds. t0

t0

Let t = T f , t0 = 0 and we obtain Tf Tf eT (s)e(s)ds ≤ eN0 ln μ+αTf eα(Tf −s) γ 2 ω˜ T (s)ω˜ (s)ds 0 0 Tf N0 ln μ+2αT f ≤e γ 2 ω˜ T (s)ω˜ (s)ds 0 Tf = γ˜ 2 ω˜ T (s)ω˜ (s)ds. 0

According to Definition 2, the conclusion is derived. Remark 1: This work focuses on achieving the finite-time boundedness and H∞ performance of filtering error system and meanwhile saving limited network resource. To deal with

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the limited bandwidth, an event-triggered mechanism is adopted to determine the data sending, which effectively reduces the data transmission compared with periodic sampling. Now we are in the position to design the event-triggered H∞ filter on the basis of Theorem 2. Assume Pi j = Pj j , for any i, j ∈ p, i = j, then decompose Pij as follows:



−1  Nj Y12 j Mj X12 j −1 , Pi j = . Pi j = T X12 X22 j Y12T j Y22 j j Define

−1 Nj T1 j = Y12T j



I I , T2 j = 0 0

 Mj . T X12 j

It follows from Pi j Pi−1 j = I that Pi j T1 j = T2 j , T1Tj Pi j T1 j = T1Tj T2 j =

−1 Nj I

 I > 0, Mj

T which implies that Y12 j X12 j , Y12j and X12j are nonsingular matrices.

Theorem 3. For given positive scalars γ˜ , α > 1, ρ 1 , ρ 2 , μ ≥ 1, if there exist positive definite matrices Mj , Nj , , , a nonsingular matrix X12j ,and matrix X22j , AFj , BFj , CFj , DFj , such that the following linear matrix inequalities hold for any i, j ∈ S, i = j, ⎡ ⎤ 11 12 0 N j Bi 0 16 CiT  ⎢ ∗ 22 BF j M j Bi 25 26 CiT  ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ∗ − 0 0 DFT j − ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ −γ 2 I 0 0 0 ⎥ (32) ⎢ ⎥ < 0, ⎢ ∗ 2 T ⎥ ∗ ∗ ∗ −γ I  D  56 ⎢ ⎥ i ⎢ ⎥ ∗ ∗ ∗ ∗ −I 0 ⎦ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ − ⎡

11 ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗

12 22 ∗ ∗ ∗ ∗ ∗



  Mj X12i −μ T X12 j X22i

Mi T X12i

0 BF j − ∗ ∗ ∗ ∗

NjB j MjBj 0 −γ 2 I ∗ ∗ ∗

0 25 0 0 −γ 2 I ∗ ∗

 X12 j ≤ 0, X22 j

16 26 DFT j 0 56 −I ∗

⎤ C Tj  C Tj  ⎥ ⎥ ⎥ − ⎥ ⎥ 0 ⎥ ⎥ < 0, T ⎥ D j ⎥ ⎥ 0 ⎦ −

(33)

(34)

N j − M j < 0,

(35)

ρ1 c2 /λ2 , γ 2d

(36)

eε ≤

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where 11 = ATi N j + N j Ai − αN j , 12 = ATi M j + ATF j + N j Ai − αN j + CiT BFT j , 16 = LiT − CiT DFT j − CFT j , 22 = ATi M j + M j Ai + CiT BFT j + BF j Ci − αM j , 25 = BF j Di , 26 = LiT − CiT DFT j , 56 = −DiT DFT j , 11 = ATj N j + N j A j − αN j , 12 = ATj M j + ATF j + N j A j − αN j + C Tj BFT j , 16 = L Tj − C Tj DFT j − CFT j , 22 = ATj M j + M j A j + C Tj BFT j + BF j C j − αM j , 25 = BF j D j , 26 = L Tj − C Tj DFT j , 56 = −DTj DFT j , the filtering error system (8) is FTB with respect to (c1 , c2 , Tf , R, d, σ (t)) and has an H∞ performance index γ˜ for any ADT switching signals satisfying τ ≥ τα∗ = max{

ln μT f ln μ }. , ln ( ρλ1 c2 2 ) − ln (γ 2 d ) − ε α

(37)

Moreover, the filter parameters can be computed as follows: −1 −1 −1 A f j = X12 j AF j (N j − M j ) X12 j , B f j = X12 j BF j ,

C f j = CF j (N j − M j )−1 X12 j , D f j = DF j . Proof. Pre-multiplying and post-multiplying both sides of (27) and (28) with diag{T1Tj , I , I , I , I } and diag{T1j , I, I, I, I}, and then pre-multiplying and post-multiplying both sides of the resulted inequalities with diag{Nj , I, I, I, I, I, I}, we can obtain ⎡ ⎤ 11 12 0 N j Bi 0 16 CiT  ⎢ ∗ 22 BF j M j Bi 25 26 CiT  ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ∗ − 0 0 DFT j − ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ −γ 2 I 0 0 0 ⎥ (38) ⎢ ⎥ < 0, ⎢ ∗ 2 T ⎥ ∗ ∗ ∗ −γ I 56 Di  ⎥ ⎢ ⎢ ⎥ ∗ ∗ ∗ ∗ −I 0 ⎦ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −

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⎡

11 ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗

12 22 ∗ ∗ ∗ ∗ ∗

0 BF j − ∗ ∗ ∗ ∗

NjB j MjBj 0 −γ 2 I ∗ ∗ ∗

0 25 0 0 −γ 2 I ∗ ∗

16 26 DFT j 0 56 −I ∗

⎤ C Tj  C Tj  ⎥ ⎥ ⎥ − ⎥ ⎥ 0 ⎥ ⎥ < 0, T ⎥ D j ⎥ ⎥ 0 ⎦ −

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(39)

where 11 = ATi N j + N j Ai − αN j , T T T T 12 = ATi M j + S Tj Y12 j ATf j X12 j + N j Ai − αN j + Ci B f j X12 j ,

16 = LiT − CiT DTf j − S Tj Y12 j C Tf j , T 22 = ATi M j + M j Ai + CiT BTf j X12 j + X12 j B f j Ci − αM j ,

25 = X12 j B f j Di , 26 = LiT − CiT DTf j , 56 = −DiT DTf j , 11 = ATj N j + N j A j − αN j , T T T T 12 = ATj M j + S Tj Y12 j ATf j X12 j + N j A j − αN j + C j B f j X12 j ,

16 = L Tj − C Tj DTf j − S Tj Y12 j C Tf j , T 22 = ATj M j + M j A j + C Tj BTf j X12 j + X12 j B f j C j − αM j ,

25 = X12 j B f j D j , 26 = L Tj − C Tj DTf j , 56 = −DTj DTf j . By denoting AF j = X12 j A f j Y12T j S j , BF j = X12 j B f j , CF j = C f j Y12T j S j , DF j = D f j , it is easy to find the equivalence of Eqs. (32) and (38), Eq. (33) and Eq. (39), respectively. Hence, it can be concluded from Theorem 2 that the finite-time boundedness and H∞ disturbance attenuation performance are ensured based on the choice of the above filter gains, which thus completes the proof. Remark 2: Notice that some parameters need to be predetermined to solve the inequalities in the proposed theorems, such as α, μ, ρ 1 , ρ 2 . α is related to the increasing rate of Lyapunov function during the inter-event intervals. μ ≥ 1 is the coefficient of the Lyapunov function at switching instants, which limits the energy increasing while the switching happens. ρ 1 and ρ 2 limit the range of matrix P. If the switched system is FTB and α, μ, ρ 1 and ρ 2 are all chosen for the performance requirements of the filtering error system, then one could use Theorem 3 to find a possible filter.

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4. Numerical example In this section, we will use a technical example to show the effectiveness and superiority of the method developed in this paper. Example 1. Consider a switched system borrowed from [35] with the following two system modes:



 −0.8 −0.2 0.1 , B1 = , A1 = 1 −0.6 −0.1



 0.2 −0.5 0.1 , B2 = , A2 = −0.5 −0.4 −0.1     C1 = −0.1 0.2 , D1 = 0.1, L1 = −0.1 0.1 ,     C2 = 0.1 0.2 , D2 = −0.2, L2 = 0.1 0.1 . Suppose that μ = 1.1, α = 1.1, γ = 0.4. By solving the inequalities in Theorem 3, we obtain the following filtering gains:



 27.49 −24.427 0.0038 , Bf1 = , Af1 = −2.666 −24.105 0.0060   C f 1 = 0.3900 −0.8403 , D f 1 = 0.0454,



 −3.8195 −0.7096 −0.1796 Af2 = , Bf2 = , 1.9512 −1.4749 −0.1902   C f 2 = 0.5662 −0.4404 , D f 2 = 0.0252. The corresponding event-triggered parameters are calculated as follows:  = 0.9590,  = 0.3762. Let c2 = 10, T f = 8, h = 0.05s, N0 = 0, R = I2×2 , ρ1 = 1, ρ2 = 1.2, d = 2, and the maximum event-triggered sampling interval is chosen to be H = 0.4. According to Eq. (14), we get the admissible switching signal σ (t) to ensure the finite-time boundedness and H∞ perfor0.762 762 mance with τ ≥ τα∗ = 9.210+1 = 0. = 0.6871. Suppose that the external disturbance .139−9.24 1.109 −0.01t −0.01t are ω(t ) = 0.5e , and ν(t ) = −0.2e , respectively. The state trajectories of system output z(t) and filter system output zf(t) are drawn in Fig. 1, respectively. Fig. 2 shows system’s switching signal σ (t) and the event-triggered switching signal σ (tk ). Fig. 3 displays the evolution of event triggering. Here we would like to compare the amounts of sending measurements between the proposed method and periodic sampling in [17]. If the system performance index is required to be γ = 0.4, based on periodic sampled-data, 160 data packets need to be sent within the time interval [0,8], and the event-triggered method can effectively reduce the amount of packets to 30 according to the result in Fig. 3. It is almost a reduction of eighty percent. Then we focus on some certain intervals, such as [3.16, 3.2], [5.88, 6.1] and et al, from which one can clearly see the asynchronous phenomenon existing in the system and filter modes. It can be seen from Fig. 4 that ηT (t)Rη(t) 10, ∀t ∈ [0, 8], i.e., the filter error system (8) is finite bounded. Hence, the effectiveness of the proposed method is demonstrated.

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0.08 z(t) z (t)

0.06

f

0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 0

1

2

3

4

5

6

7

8

Fig. 1. System output z(t) and filter system output z f (t ).

System mode

2

1

Filter mode

2

1 0.86 0.9 0

1

3.16 3.2 4.32 4.35 2

3

4

5.88 5

6.1 7.18 7.2 6

7

time(s)

Fig. 2. The signals of system mode and filter mode.

8

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Release instants

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

1

2

3

4

5

6

7

8

t(s)

Fig. 3. Evolution of event triggering based on event-triggered scheme (3). 0.3 T

(t)R (t)

0.25

0.2

0.15

0.1

0.05

0

-0.05 0

1

2

3

4

5

6

7

8

Time t Fig. 4. The trajectory of ηT (t)Rη(t) of the filtering error system.

5. Conclusions This paper aims to investigate the finite-time H∞ filtering for a class of switched linear systems under networked control framework. The event-triggered control mechanism is adopted and the asynchronism between system and filter is taken into consideration. By utiliz-

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ing multiple Lyapunov functions and ADT switching property, sufficient criteria are proposed to guarantee the filtering error system is finite-time bounded with a disturbance attenuation performance. Meanwhile, the asynchronous filter as well as the event-triggered matrices is co-designed for the underlying system. In the end, an illustrative example is introduced to demonstrate the effectiveness of the obtain results. The method proposed in this paper can also be extended to the system with MDADT switching signals, which is a more general switching approach, and will be our future development. Acknowledgment This research is partially supported by the National Science Foundation of China(61873310), Heilongjiang Science Foundation (F2018015) the Fundamental Research Funds for the Central Universities and the 111 Project(No. B16014). References [1] Y. Liu, X. Li, R. Cheung, S. Chan, H. Wong, in: High-speed discrete Gaussian sampler with heterodyne chaotic laser inputs, 65, 2018, pp. 794–798. [2] S. Sharifi, M. Monfared, Series and tapped switched-coupled-inductors impedance networks, IEEE Trans. Ind. Electron. 65 (12) (2018) 9498–9508. [3] I. Hiskens, Stability of hybrid system limit cycles: application to the compass gait biped robot, in: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, USA, 2001. [4] J. Hespanha, A. Morse, Switching between stabilizing controller, Automatica 38 (11) (2002) 1905–1917. [5] Z. Li, H. Gao, H.R. Karimi, Stability analysis and H∞ controller synthesis of discrete-time switched systems with time delay, Syst. Control Lett. 66 (2014) 85–93. [6] X. Zhao, X. Zheng, B. Niu, L. Liu, Adaptive tracking control for a class of uncertain switched nonlinear systems, Automatica 52 (2014) 185–191. [7] Z. Li, H. Gao, R. Agarwal, O. Kaynak, H∞ control of switched delayed systems with average dwell time, Int. J. Control 86 (12) (2013) 2146–2158. [8] C. Wang, Z. Zou, Z. Qi, Z. Ding, Predictor-based extended-state-observer design for consensus of MASs with delays and disturbance, IEEE Trans. Cybern. (2018), doi:10.1109/TCYB.2018.2799798. [9] D. Liberzon, A. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag. 19 (5) (1999) 59–70. [10] J. Xiong, J. Lam, H. Gao, W. Daniel, On robust stabilization of Markovian jump systems with uncertain switching probabilities, Automatica 41 (5) (2005) 897–903. [11] L. Zhang, H. Gao, Asynchronously switched control of switched linear systems with average dwell time, Automatica 46 (5) (2010) 953–958. [12] H. Liu, Y. Shen, X. Zhao, Delay-dependent observer-based H∞ finite-time control for switched systems with time-varying delay, Nonlinear Anal.: Hybrid Syst. 6 (2012) 885–898. [13] L. Zhang, P. Shi, Stability, l2 -gain and asynchronous H∞ control of discrete-time switched systems with average dwell time, IEEE Trans. Autom. Control 54 (9) (2009) 2192–2199. [14] J. Zhao, D. Hill, On stability, l2 -gain and H∞ control for switched systems, Automatica 44 (5) (2008) 1220–1232. [15] Z. Fei, C. Guan, H. Gao, Exponential synchronization of networked chaotic delayed neural network by a hybrid event trigger scheme, IEEE Trans. Neural Netw. Learn. Syst. 29 (6) (2018) 2558–2567. [16] C. Briat, Stability analysis and stabilization of stochastic linear impulsive, switched and sampled-data systems under dwell-time constraints, Automatica 74 (2016) 279–287. [17] J. Lian, C. Li, B. Xia, Sampled-data control of switched linear systems with application to an F-18 aircraft, IEEE Trans. Ind. Electron. 64 (2) (2017) 1332–1340. [18] C. Guan, D. Sun, Z. Fei, C. Ren, Synchronization for switched neural networks via variable sampled-data control method, Neurocomputing 311 (2018) 325–332. [19] E. Hendricks, M. Jensen, A. Chevalier, T. Vesterholm, Problems in event based engine control, in: Proceedings of the 1994 American Control Conference, Baltimore, MD, 1994. [20] B. Zhang, Q. Han, X. Zhang, Event-triggered H∞ reliable control for offshore structures in network environments, J. Sound Vib. 368 (2016) 1–21.

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