International Journal of General Systems

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Finite-time event-triggered extended dissipative

control for discrete time switched linear systems

Hui Gao, Hongbin Zhang, Dianhao Zheng, Liangliang Zhang & Yang Li

To cite this article: Hui Gao, Hongbin Zhang, Dianhao Zheng, Liangliang Zhang & Yang Li (2019)
Finite-time event-triggered extended dissipative control for discrete time switched linear systems,
International Journal of General Systems, 48:5, 476-491, DOI: 10.1080/03081079.2019.1608983

To link to this article: https://doi.org/10.1080/03081079.2019.1608983

Published online: 16 May 2019.

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INTERNATIONAL JOURNAL OF GENERAL SYSTEMS
2019, VOL. 48, NO. 5, 476–491
https://doi.org/10.1080/03081079.2019.1608983

Finite-time event-triggered extended dissipative control for

discrete time switched linear systems
Hui Gaoa , Hongbin Zhanga , Dianhao Zhenga,b , Liangliang Zhanga and Yang Lia
a School of Information and Communication Engineering, University of Electronic Science and Technology of

China, Chengdu, Sichuan, China; b Faculty of Engineering and Information Technology, University of
Technology Sydney, Sydney, Australia

ABSTRACT ARTICLE HISTORY

This paper considers the problem of finite-time event-triggered Received 2 September 2018
extended dissipative control for a class of discrete time switched Accepted 19 February 2019
linear systems. The proposed system is modeled as a discrete time KEYWORDS
switched linear system with an event-triggered control scheme. Extended dissipative;
Under the event-triggered transmission schemes, we give some suf- finite-time; switched
ficient conditions to guarantee the finite-time extended dissipative systems; event-trigger
performance of the closed-loop switched system in terms of lin-
ear matrix inequalities. Furthermore, the state feedback controller
gains are proposed by solving a set of linear matrix inequalities.
Finally, a numerical example is given to show the effectiveness of the
proposed methods.

1. Introduction
Switched system belongs to a special class of hybrid system. Recently, switched systems
have drawn considerable attention (Liberzon and Morse 1999; Sun and Ge 2005; Ding
and Yang 2010; Xiao, Park, and Zhou 2018). Generally, stability are the main concerns
for switched systems. In conventional, most results about stability of switched systems are
focusing on Lyapunov asymptotic stability, which discussed the stability analysis over infi-
nite time interval (Daafouz, Riedinger, and Iung 2002; Lien et al. 2009; Liu et al. 2009;
Zhang and Shi 2009). However, the finite time system behavior is also important in some
practical situations, such as missile systems, power electronics and air traffic control. Actu-
ally, finite time stability of continuous-time switched systems has attracted much attention
(Lin, Du, and Li 2011; Liu and Shen 2012; Liu and Zhao 2014; Zong et al. 2015; Zong, Ren,
and Hou 2016; Xia et al. 2018; Gao et al. 2018a; Gao et al. 2018b). However, only few results
discussed the finite-time stability analysis of discrete-time switched systems.
On the other hand, a novel performance index named extended dissipative was firstly
introduced by Zhang in Zhang, Zheng, and Xu (2013), which unified the H∞ performance,
L2 − L∞ performance (Qi et al. 2019), Passivity performance and (Q, S, R)-dissipativity
performance together. The extended dissipative concept has been used in many systems
(Wei et al. 2013; Shen et al. 2017). All in all, the above reports are foucusing on continuous

CONTACT Hongbin Zhang zhanghb@uestc.edu.cn

© 2019 Informa UK Limited, trading as Taylor & Francis Group

 INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 477

time system cases, Shen et al. (2017) extend the extended dissipative concept to the discrete
time cases. To the best of our knowledge, the extended dissipative concept has not been
discussed in discrete-time switched linear systems yet, which motivates our current study.
The event-triggered scheme was proposed in some systems (Heemels and Donkers 2010;
Mazo, Anta, and Tabuada 2010; Donkers and Heemels 2012; Heemels, Donkers, and
Teel 2013; Zhang and Feng 2014; Ren, Zong, and Karimi 2019). The event-triggered scheme
can reduce the waste of network resources, and have its advantage in many areas. Therefore,
event-triggered control for switched systems is worth studying (Ren, Zong, and Li 2018).
However, to the best of our knowledge, there is few works on the event-triggered control
for discrete-time switched systems reported yet. Therefore, we are interested in designing
a suitable event-triggered scheme and a controller such that the discrete-time closed-loop
switched systems are finite-time stable.
This paper is organized as follows. In Section 2, Problem statements are provided. In
Section 3, the analysis and synthesis of finite-time boundness and finite-time extended
dissipative for discrete-time switched linear systems are given, state feedback controller
gains are proposed. In Section 4, we give simulation examples to show the efficiency of the
method. Section 5 gives the conclusion.
Notation: In this paper. M T denotes the transpose. X > 0 denote a positive definite
matrix. δij is the standard Kronecker delta function. λmin (P) denote the minimum eigen-
value of matrix P. Z+ = {1, 2 · · · } and I is the identity matrix.

2. Preliminaries and problem formulations

Consider system:

x(k + 1) = Aσ (k) x(k) + Bσ (k) u(k) + Cσ (k) w(k),

z(k) = Dσ (k) x(k), k ∈ Z+ . (1)

where x(k) ∈ Rn is the state vector, u(k) the control input, w(k) the exogenous disturbance,
z(k) ∈ Rn the output. The switching signal σ (k) : Z+ → M̃ = {1, 2 · · · l} is a piecewise
constant function, σ (k) = i means that the ith subsystem is activated. Ai , Bi , Ci , Di , i =
1, 2 · · · l are real matrices with appropriate dimensions.

Assumption 2.1 (Xia et al. 2018): The external disturbance satisfies


t
wT (k)w(k) ≤ d, ∀ t ∈ Z+ ,
k=0

where d is a positive number.

Assumption 2.2: Assume that σ (k) has a dwell time τd and σ (k) ∈ S[τa , N0 ].

Assumption 2.3: For ∀ μ ≥ 1, N0 ≥ 0, τa > 0, τd > 0, M > 0, η > 0, 0 < ρ < 1, and
α = ((1 + η)/(1 − ρ))N0 τd , β ∈ (1 − ρ, 1), we have

μ(2N0 +τd /τa +2M/τa ) (1 + η)2 (1 − ρ)−2 α 2 β 2M ≤ h,

h is a big positive number.

478 H. GAO ET AL.

Assumption 2.4: The state vector of the system satisfies


t
xT (k)x(k) ≤ H, ∀ t ∈ Z+ ,
k=0

H is a big positive number.

Definition 2.1 (Xia et al. 2018): For given positive constants c1 , c2 , M with c1 < c2 , and a
positive definite matrix R, if

xT (0)Rx(0) ≤ c1 ⇒ xT (k)Rx(k) ≤ c2 , ∀ k ∈ [1, M],

holds, then switched system (1) is finite-time bounded with respect to (c1 , c2 , R, M, σ ). If
we set w(k) ≡ 0, ∀ k ∈ [1, M], the inequality also holds, then system is finite-time stable.

Definition 2.2 (Xia et al. 2018): For any T2 > T1 ≥ 0, let Nσ (T1 , T2 ) denotes the switch-
ing number of σ (k) over (T1 , T2 ). If

T2 − T1
Nσ (T1 , T2 ) ≤ N0 +
τa
holds for τa > 0 and an integer N0 ≥ 0, then τa is called the average dwell-time and N0 is
the chatter bound.

Definition 2.3 (Xia et al. 2018): Given matrices ψ1 = ψ1T ≤ 0, ψ2 , ψ3 = ψ3T > 0 and
ψ4 = ψ4T ≥ 0 satisfying ( ψ1 + ψ2 )ψ4 = 0, if


M
J(k) ≥ sup zT (k)ψ4 z(k), ∀ M > 0, (2)
k=0 0≤k≤M

holds for any w(k) ∈ L2 [0, ∞), where J(k) = zT (k)ψ1 z(k) + 2zT (k)ψ2 w(k) + wT (k)ψ3 w
(k). Then system (1) is extended dissipative.

Remark 2.1: Through tuning weighting matrices, we have

(1) H∞ performance: ψ1 = −I, ψ2 = 0, ψ3 = γ 2 I, ψ4 = 0;

(2) L2 − L∞ performance: ψ1 = 0, ψ2 = 0, ψ3 = γ 2 I, ψ4 = I;
(3) Passivity performance: ψ1 = 0, ψ2 = I, ψ3 = γ I, ψ4 = 0;
(4) (Q, S, R)-dissipativity performance: ψ1 = Q, ψ2 = S, ψ3 = R − βI, ψ4 = 0.

The event-triggered scheme is considered in this paper.

ks+1 = min{ks+1 , ks + τd }, k0 = 0, (3)

where ks+1 = mink>ks {k | [x(k) − x(ks )]T σ (ks ) [x(k) − x(ks )] ≥ νx(ks )T σ (ks ) x(ks )}, the
positive scalar ν and the positive definite matrix σ (ks ) are mode dependent event-
triggered parameters.
 INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 479

We assume s = [ks , ks+1 ) be a holding interval of the event-triggered transmission

mechanism. We can deduce from (3) that ks+1 − ks ≤ τd .
The control input and switched signal can be given by

x̂(k) = x(ks ), σ̂ (k) = σ (ks ), k ∈ s .

The state feedback control law can be written as

u(k) = Kσ (ks ) x(ks ), k ∈ s . (4)

We assume τ1 (k) = k − ks and e(k) = x(k) − x(ks ), k ∈ s . then σ̂ (k) = σ (k − τ1 (k)),

and for ∀ k ∈ s , the closed-loop system of (1) and (4) can be obtained

x(k + 1) = Aσ (k) x(k) + B̃θ (k) (x(k) − e(k)) + Cσ (k) w(k),

z(k) = Dσ (k) x(k), k ∈ Z+ . (5)

where B̃θ (k) = Bσ (k) K σ̂ (k), τ1 (k) ∈ [0, τd ).

For ∀ k ∈ s , we can deduce from (3) that

eT (k) σ (ks ) e(k) < ν[x(k) − e(k)]T σ (ks ) [x(k) − e(k)]}.

Through merging the switching signal σ (k) with σ̂ (k), the augmented switching signal can
be obtained θ(k) = (σ (k), σ̂ (k)). Note that σ̂ (k) = σ (k − τ1 (k)), then σ̂ (k) can be viewed
as the delayed switching signal of σ (k). By τ1 (k) ∈ [0, τd ) and σ̂ (k) ∈ S[τa , N0 + τd /τa ] we
can derive the following Lemma.

Lemma 2.4 (Xiao, Park, and Zhou 2018): Considering the switching signal σ (k) ∈
S[τa , N0 ], then θ(k) ∈ S[τa /2, 2N0 + τd /τa ].

Lemma 2.5 (Xiao, Park, and Zhou 2018): We assume Ts (k0 , k) be the total synchronous
time in any interval [k0 , k) of switching signals σ (k) and σ̂ (k), and denote Tas (k0 , k) =
k − k0 − Ts (k0 , k) as their total asynchronous time in [k0 , k). Then, for some positive con-
stants 0 < ρ < 1, η > 0 and β ∈ (1 − ρ, 1), if (1 + η)τd (1 − ρ)τa −τd ≤ β τa , then (1 −
ρ)Ts (k0 ,k) (1 + η)Tas (k0 ,k) ≤ αβ k−k0 , where α = ((1 + η)/(1 − ρ))N0 τd .

Lemma 2.6 (Xia et al. 2018): Assume a,b be real matrices of appropriate dimensions, they
satisfy 2aT b ≤ aT a + bT b.

3. Main results
3.1. Finite-time boundedness analysis
Consider system:

x(k + 1) = Aσ (k) x(k) + B̃θ (k) (x(k) − e(k)) + Cσ (k) w(k),

z(k) = Dσ (k) x(k), k ∈ Z+ . (6)
480 H. GAO ET AL.

Theorem 3.1: For given positive scalars 0 < ρ < 1, η > 0, μ ≥ 1, ν > 0, if there exist
state feedback gain matrix Kj and positive definite matrices Pij , Qij , j with appropriate
dimensions, such that
⎡ 2 ⎤
−λij Pij + ν j −ν j 0 (Ai + B̃ij )T Pij
⎢ ∗ −(1 − ν) j 0 −B̃Tij Pij ⎥
⎢ ⎥ < 0, (7)
⎣ ∗ ∗ −Qij CiT Pij ⎦
∗ ∗ ∗ −Pij
μ−1 Pii ≤ Pij ≤ μPjj , Pii ≤ μPjj , ∀ i, j ∈ M̃, (8)

the average dwell-time satisfies

(τd + 2k) ln μ
τa ≥ τa∗ = , (9)
ln(λ1 c2 ) − ln(μ2N0 α 2 β 2k (λ2 c1 + (1 + η)2 (1 − ρ)−2 β −2k−2 λ3 d))
where

λij = δij (1 − ρ) + (1 − δij )(1 + η), λ1 = min(λmin (P̃θ (k) )), λ2 = max(λmax (P̃θ (k) )),
λ3 = max(λmax (Qθ (k) )), P̃θ (k) = R−1/2 Pθ (k) R−1/2 ,

then switched system (6) is finite-time bounded with respect to (c1 , c2 , M, R, d, σ ).

Proof: Consider the Lyapunov function

V(k) = Vθ (k) (k) = ξ T (k)Pθ (k) ξ(k). (10)

We assume k̄ as an arbitrary switching time of θ (k). We can deduce from (8) that

Vθ (k̄) (k̄) ≤ uVθ (k̄−1) (k̄). (11)

for ∀ k ∈ s , we have

Vθ (k) (k + 1) − λ2θ (k) Vθ (k) (k) − wT (k)Qθ (k) w(k)

= xT (k + 1)Pθ (k) x(k + 1) − λ2θ (k) xT (k)Pθ (k) x(k) − wT (k)Qθ (k) w(k)
= [Aσ (k) x(k) + B̃θ (k) (x(k) − e(k)) + Cσ (k) w(k)]T Pθ (k) [Aσ (k) x(k) + B̃θ (k) (x(k)
− e(k)) + Cσ (k) w(k)] − λ2θ (k) xT (k)Pθ (k) x(k) − wT (k)Qθ (k) w(k)
+ v[x(k) − e(k)]T σ̂ (k) [x(k) − e(k)] − e(k)
T
σ̂ (k) e(k) ≤ X T (k)θ (k) X(k), (12)

where
1 − ρ, k ∈ Ts (s );
λθ (k) = X(k) = [xT (k) eT (k) wT (k)]T ,
1 + η, k ∈ Tas (s ),

and
⎡ ⎤
−λ2θ (k) Pθ (k) 0 0
θ (k) =⎣ ∗ − σ̂ (k) 0 ⎦ + ÃTθ (k) Pθ (k) Ãθ (k) + vET j E,
∗ ∗ −Qθ (k)
 INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 481

Ãθ (k) = [Aσ (k) + B̃θ (k) − B̃θ (k) Cσ (k) ],

E = [I − I].

If there exists one switching in [ks , ks+1 ) for θ(k). And let σ (k̄) = i. When k ∈ [ks , k̄), we
have θ(k) = (j, j), θ (k) = jj ; and if k ∈ [k̄, ks+1 ), then θ (k) = (i, j), θ (k) = ij ;
On the other hand, if there is no switching in [ks , ks+1 ) for θ (k), then θ (k) = (j, j) in
k ∈ [ks , ks+1 ) and θ (k) = jj .
Considering (7), by Schur complement, we have θ (k) < 0.
Thus, Vθ (k) (k + 1) − λ2θ (k) Vθ (k) (k) − wT (k)Qθ (k) w(k) < 0 holds.
Denote k̃1 , . . . , k̃Nθ (0,k) as the switching instants of θ (k) in (0, k). We assume k̃1 > 0 and
k̃Nθ (0,k) < k. Then, from (11) and Vθ (k) (k + 1) − λ2θ (k) Vθ (k) (k) − wT (k)Qθ (k) w(k) < 0, we
have
Vθ (k) (k) ≤ λ2θ (k) Vθ (k) (k − 1) + wT (k − 1)Qθ (k) w(k − 1)
≤ λ4θ (k) Vθ (k) (k − 2) + λ2θ (k) wT (k − 2)Qθ (k) w(k − 2) + wT (k − 1)Qθ (k) w(k − 1)
≤ ···

2(k−k̃N (0,k) ) 
k−1
≤ λθ (k) θ Vθ (k) (k̃Nθ (0,k) ) + λ2(k−s−1)
θ (k) wT (s)Qθ (k) w(s)
s=k̃Nθ (0,k)

2(k−k̃Nθ (0,k) ) 
k−1
2(k−s−1) T
≤ λθ (k) μVθ (k̃ −1) (k̃Nθ (0,k) ) + λθ (k) w (s)Qθ (k) w(s)
Nθ (0,k)
s=k̃Nθ (0,k)

2(k−k̃Nθ (0,k) )
≤ μλθ (k) (λ2 V (k̃ − 1)
θ (k̃Nθ (0,k) −1) θ (k̃Nθ (0,k) −1) Nθ (0,k)

+ wT (k̃Nθ (0,k) − 1)Qθ (k̃ w(k̃Nθ (0,k) − 1))

Nθ (0,k) −1)


k−1
2(k−s−1) T
+ λθ (k) w (s)Qθ (k) w(s)
s=k̃Nθ (0,k)

2(k−k̃Nθ (0,k) )
= μλ (λ2 V (k̃ − 1)
θ (k̃Nθ (0,k) ) θ (k̃Nθ (0,k) −1) θ (k̃Nθ (0,k) −1) Nθ (0,k)

+ wT (k̃Nθ (0,k) − 1)Qθ (k̃ w(k̃Nθ (0,k) − 1))

Nθ (0,k) −1)


k−1
+ λ2(k−s−1) wT (s)Qθ (k̃ w(s)
θ (k̃Nθ (0,k) ) Nθ (0,k) )
s=k̃Nθ (0,k)

Through recursive calculation, we have

Vθ (k) (k) ≤ μNθ (0,k) (1 − ρ)2Ts (0,k) (1 + η)2Tas (0,k) Vθ (0) (0)

k−1
+ μNθ (s,k) λ2θ (k̄ (1 − ρ)2Ts (s,k−1) (1 + η)2Tas (s,k−1) λ−2 T
θ (s) w (s)Qθ (s) w(s)
Nθ (s,k) )
s=0
482 H. GAO ET AL.

where k̃0 = 0, k̃Nθ (0,k)+1 = k.k̄1 , . . . , k̄Nθ (s,k) denote switching times in (s, k).
By Lemma 2.4, we have Nθ (0, k) ≤ 2N0 + τd /τa + 2k/τa .
For all β ∈ [(1 − ρ)((1 + η)/(1 − ρ))τd /τa , 1), we have β ∈ (1 − ρ, 1), (1 + η)τd (1 −
ρ) a −τd ≤ β τa . It follows from the definition of λθ (k) we have 1 − ρ ≤ λθ (k̄N (s,k) ) ≤ 1 +
τ
θ
η, (1 + η)−1 ≤ λ−1 −1
θ (s) ≤ (1 − ρ) , by Lemma 2.5 we can deduce that

Vθ (k) (k) ≤ μNθ (0,k) (1 − ρ)2Ts (0,k) (1 + η)2Tas (0,k) Vθ (0) (0)

k−1
+ μNθ (s,k) λ2θ (k̄ (1 − ρ)2Ts (s,k−1) (1 + η)2Tas (s,k−1) λ−2 T
θ (s) w (s)Qθ (s) w(s)
Nθ (s,k) )
s=0

≤ μ(2N0 +τd /τa +2k/τa ) α 2 β 2k Vθ (0) (0) + μ(2N0 +τd /τa +2k/τa )

k−1
× (1 + η)2 (1 − ρ)−2 α 2 β 2(k−s−1) wT (s)Qθ (s) w(s)
s=0

≤ μ(τd +2k)/τa μ2N0 α 2 β 2k (Vθ (0) (0)


k−1
+ (1 + η)2 (1 − ρ)−2 β −2s−2 wT (s)Qθ (s) w(s))
s=0

≤ μ(τd +2k)/τa μ2N0 α 2 β 2k (xT (0)Pθ (0) x(0)


k−1
+ (1 + η)2 (1 − ρ)−2 β −2k−2 λ3 wT (s)w(s))
s=0
(τd +2k)/τa
≤μ μ 2N0 2 2k
α β (x (0)R T 1/2
(R−1/2 Pθ (0) R−1/2 )R1/2 x(0)
+ (1 + η)2 (1 − ρ)−2 β −2k−2 λ3 d)
≤ μ(τd +2k)/τa μ2N0 α 2 β 2k (λ2 xT (0)Rx(0) + (1 + η)2 (1 − ρ)−2 β −2k−2 λ3 d)
≤ μ(τd +2k)/τa μ2N0 α 2 β 2k (λ2 c1 + (1 + η)2 (1 − ρ)−2 β −2k−2 λ3 d) (13)

On the other hand, for ∀k ∈ s , we have

Vθ (k) (k) = xT (k)Pθ (k) x(k) = xT (k)R1/2 (R−1/2 Pθ (k) R−1/2 )R1/2 x(k) ≥ λ1 xT (k)Rx(k).
(14)
We can deduce from (13) and (14) that

μ(τd +2k)/τa μ2N0 α 2 β 2k (λ2 c1 + (1 + η)2 (1 − ρ)−2 β −2k−2 λ3 d)

xT (k)Rx(k) < .
λ1

Using (9), one obtains

xT (k)Rx(k) < c2 .

The proof is completed. 

 INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 483

Corollary 3.2: If we set w(k) = 0, for the given positive scalars 0 < ρ < 1, η > 0, μ ≥
1, ν > 0, if there exist state feedback gain matrix Kj and positive definite matrices Pij , j
with appropriate dimensions, such that
⎡ ⎤
−λ2ij Pij + ν j −ν j (Ai + B̃ij )T Pij
⎢ ⎥
⎣ ∗ −(1 − ν) j −B̃Tij Pij ⎦ < 0,
∗ ∗ −Pij
μ−1 Pii ≤ Pij ≤ μPjj , Pii ≤ μPjj , ∀ i, j ∈ M̃,

the average dwell-time satisfies

(τd + 2k) ln μ
τa ≥ τa∗ = ,
ln(λ1 c2 ) − ln(μ2N0 α 2 β 2k (λ2 c1 ))
where

λij = δij (1 − ρ) + (1 − δij )(1 + η), λ1 = min(λmin (P̃θ (k) )), λ2 = max(λmax (P̃θ (k) )),
P̃θ (k) = R−1/2 Pθ (k) R−1/2 ,

then switched system (6) is finite-time stable with respect to (c1 , c2 , M, R, d, σ ).

Proof: The proof is omitted. 

3.2. Finite time extended dissipative analysis

Theorem 3.3: For given matrices ψ1 , ψ2 , ψ3 , ψ4 and positive scalars 0 < ρ < 1, η >
0, μ ≥ 1, ν > 0, if there exist state feedback gain matrix Kj and positive definite matrices
Pij , j with appropriate dimensions, such that
⎡ ⎤
−λ2ij Pij + ν j − DTi ψ1 Di −ν j −DTi ψ2 (Ai + B̃ij )T Pij
⎢ ∗ −(1 − ν) 0 −B̃Tij Pij ⎥
⎢ j ⎥ < 0, (15)
⎣ ∗ ∗ −ψ3 CiT Pij ⎦
∗ ∗ ∗ −Pij
μ−1 Pii ≤ Pij ≤ μPjj , Pii ≤ μPjj , ∀ i, j ∈ M̃, (16)
1
Pθ (k) − DTi ψ4 Di > 0, ∀ i ∈ M̃, (17)
h
the average dwell-time satisfies
(τd + 2k − 2) ln μ
τa ≥ τa∗ = , (18)
ln(λ1 c2 ) − ln(μ (1 + η)2 (1 − ρ)−2 α 2 β 2(k−1) [λ
2N 0
4H + (λ5 + λ6 )d])
and

λ1 = min(λmin (P̃θ (k) )), λ2 = max(λmax (P̃θ (k) )), λ4 = max (λmax (DTi Di )),
∀ i∈M̃
−1/2
λ5 = λmax (ψ2T ψ2 ), λ6 = λmax (ψ3 ), P̃θ (k) = R Pθ (k) R−1/2 .
484 H. GAO ET AL.

Then the system is finite-time bounded and satisfies the extended dissipative performance.

Proof: Consider the Lyapunov function

V(k) = Vθ (k) (k) = ξ T (k)Pθ (k) ξ(k). (19)

Denote k̄ as an arbitrary switching time of θ(k). We can deduce from (16) that

Vθ (k̄) (k̄) ≤ uVθ (k̄−1) (k̄). (20)

for ∀ k ∈ s , we have

Vθ (k) (k + 1) − λ2θ (k) Vθ (k) (k) − J(k)

= xT (k + 1)Pθ (k) x(k + 1) − λ2θ (k) xT (k)Pθ (k) x(k) − zT (k)ψ1 z(k) − 2zT (k)ψ2 w(k)
− wT (k)ψ3 w(k) = [Aσ (k) x(k) + B̃θ (k) (x(k) − e(k)) + Cσ (k) w(k)]T Pθ (k)
[Aσ (k) x(k) + B̃θ (k) (x(k) − e(k)) + Cσ (k) w(k)] − λ2θ (k) xT (k)Pθ (k) x(k)
− xT (k)DTi ψ1 Di x(k) − 2xT (k)DTi ψ2 w(k) − wT (k)ψ3 w(k)
+ v[x(k) − e(k)]T σ̂ (k) [x(k) − e(k)] − e(k)
T
σ̂ (k) e(k) ≤ X T (k)θ (k) X(k), (21)

where

1 − ρ, k ∈ Ts (s );
λθ (k) = X(k) = [xT (k) eT (k) wT (k)]T ,
1 + η, k ∈ Tas (s ),

and
⎡ ⎤
−λ2θ (k) Pθ (k) − DTi ψ1 Di 0 −DTi ψ2
θ (k) =⎣ ∗ − σ̂ (k) 0 ⎦ + ÃTθ (k) Pθ (k) Ãθ (k) + vET j E,
∗ ∗ −ψ3
Ãθ (k) = [Aσ (k) + B̃θ (k) − B̃θ (k) Cσ (k) ],
E = [I − I].

Similar to the proof of Theorem 3.1, considering (15), by Schur complement, we obtain
θ (k) < 0.
Thus, Vθ (k) (k + 1) − λ2θ (k) Vθ (k) (k) − J(k) < 0 holds.
Denote k̃1 , . . . , k̃Nθ (0,k) as the switching instants of θ (k) in (0, k). We assume k̃1 > 0 and
k̃Nθ (0,k) < k. Then, from (16) and Vθ (k) (k + 1) − λ2θ (k) Vθ (k) (k) − J(k) < 0, similar to the
proof of Theorem 3.1, using the iterative method, we have

Vθ (k) (k) ≤ μNθ (0,k) (1 − ρ)2Ts (0,k) (1 + η)2Tas (0,k) Vθ (0) (0)

k−1
+ μNθ (s,k−1) λ2θ (k̄ (1 − ρ)2Ts (s,k−1) (1 + η)2Tas (s,k−1) λ−2
θ (s) J(s)
Nθ (s,k−1) )
s=0

where k̃0 = 0, k̃Nθ (0,k)+1 = k.k̄1 , . . . , k̄Nθ (s,k) denote switching times in(s, k).
 INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 485

By Lemma 2.6, we have Nθ (0, k) ≤ 2N0 + τd /τa + 2k/τa .

For all β ∈ [(1 − ρ)((1 + η)/(1 − ρ))τd /τa , 1), we have β ∈ (1 − ρ, 1), (1 + η)τd (1 −
ρ) a −τd ≤ β τa . It follows from the definition of λθ (k) we know 1 − ρ ≤ λθ (k̄N (s,k) ) ≤ 1 +
τ
θ
η, (1 + η)−1 ≤ λ−1 −1
θ (s) ≤ (1 − ρ) . By Lemma 2.5 and zero initial condition, we have


k−1
0 ≤ Vθ (k) (k) ≤ μNθ (s,k−1) λ2θ (k̄ (1 − ρ)2Ts (s,k−1) (1 + η)2Tas (s,k−1) λ−2
θ (s) J(s)
Nθ (s,k−1) )
s=0


k−1
≤ μ(2N0 +τd /τa +2(k−1)/τa ) (1 + η)2 (1 − ρ)−2 α 2 β 2(k−s−1) J(s)
s=0


k−1
≤ μ(2N0 +τd /τa +2(k−1)/τa ) (1 + η)2 (1 − ρ)−2 α 2 β 2(k−1) J(s)
s=0

Then we have

Vθ (k) (k)  k−1

0≤ ≤ J(s)
μ(2N0 +τd /τa +2(k−1)/τa ) (1 + η)2 (1 − ρ)−2 α 2 β 2(k−1) s=0

Setting k−1 = M, we have

Vθ (k) (k)  M
0 ≤ (2N +τ /τ +2M/τ ) ≤ J(s)
μ 0 d a a (1 + η)2 (1 − ρ)−2 α 2 β 2M
s=0

by Assumption 2.3, we have

Vθ (k) (k) 
M
0≤ ≤ J(s)
h
s=0

Considering

M
J(k) − sup zT (k)ψ4 z(k) ≥ 0, (22)
k=0 0≤k≤M

when ψ4 = 0, we can deduce that


M
J(k) > 0, (23)
k=0

when ψ4 > 0, by (17) we have


M
Vθ (k) (k) xT (k)Pθ (k) x(k)
J(k) > = > xT (k)DTi ψ4 Di x(k) = zT (k)ψ4 z(k), (24)
h h
k=0

so we obtain that

M
J(k) − sup zT (k)ψ4 z(k) ≥ 0. (25)
k=0 0≤k≤M
486 H. GAO ET AL.

Similar to the above proof, we have


k−1
(2N0 +τd /τa +2(k−1)/τa ) 2 −2 2 2(k−1)
Vθ (k) (k) ≤ μ (1 + η) (1 − ρ) α β J(s),
s=0

thus

k−1
(2N0 +τd /τa +2(k−1)/τa ) 2 −2 2 2(k−1)
Vθ (k) (k) ≤ μ (1 + η) (1 − ρ) α β (2zT (s)ψ2 w(s)
s=0
T
+ w (s)ψ3 w(s)), (26)

so we have
Vθ (k) (k) μ(2N0 +τd /τa +2(k−1)/τa ) (1 + η)2 (1 − ρ)−2 α 2 β 2(k−1)
xT (k)Rx(k) < <
λ1 λ1

k−1
(2zT (s)ψ2 w(s) + wT (s)ψ3 w(s)). (27)
s=0

Considering


k−1 
k−1
T T
(2z (s)ψ2 w(s) + w (s)ψ3 w(s)) = (2xT (s)DTi ψ2 w(s) + wT (s)ψ3 w(s)), (28)
s=0 s=0

by Lemma 2.6 we can deduce that

2xT (s)DTi ψ2 w(s) ≤ xT (s)DTi Di x(s) + wT (s)ψ2T ψ2 w(s), (29)

then
Vθ (k) (k) μ(2N0 +τd /τa +2(k−1)/τa ) (1 + η)2 (1 − ρ)−2 α 2 β 2(k−1)
xT (k)Rx(k) < <
λ1 λ1

k−1
(2zT (s)ψ2 w(s) + wT (s)ψ3 w(s))
s=0

μ(2N0 +τd /τa +2(k−1)/τa ) (1 + η)2 (1 − ρ)−2 α 2 β 2(k−1)

<
λ1

k−1
(xT (s)DTi Di x(s) + wT (s)ψ2T ψ2 w(s) + wT (s)ψ3 w(s))
s=0

μ(τd +2k−2)/τa μ2N0 (1 + η)2 (1 − ρ)−2 α 2 β 2(k−1)

< [λ4 H + (λ5 + λ6 )d]. (30)
λ1
It can be verified from (18) that

xT (k)Rx(k) ≤ c2 . (31)

Thus the proof is completed. 

 INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 487

Let Pij = Pj in Theorem 3.3, then pre-and post-multiply both sides of (15) with
diag{Pj−1 , Pj−1 , I, Pj−1 }. Denote Rj = Pj−1 , j = Pj−1 j Pj−1 , Yj = Kj Pj−1 , by using Schur
complement, we can obtain the following design method for the controller gains and
event-trigged parameters.

Theorem 3.4: For given matrices ψ1 , ψ2 , ψ3 , ψ4 and positive scalars 0 < ρ < 1, η >
0, μ ≥ 1, ν > 0, under the event-triggered scheme (3), if there exist positive definite matrices
Rj , j , and a matrix Yj such that
⎡ ⎤
−λ2ij Rj + νj −νj −Rj DTi ψ2 Rj ATi + YjT BTi Rj DTi
⎢ ∗ −(1 − ν)j −YjT BTi 0 ⎥
⎢ 0 ⎥
⎢ ⎥
⎢ ∗ ∗ −ψ3 CiT 0 ⎥ < 0,
⎢ ⎥
⎣ ∗ ∗ ∗ −Rj 0 ⎦
∗ ∗ ∗ ∗ ψ1−1
μ−1 Pii ≤ Pij ≤ μPjj , Pii ≤ μPjj , ∀ i, j ∈ M̃,

the average dwell-time satisfies

(τd + 2k) ln μ
τa ≥ τa∗ = ,
ln(λ1 c2 ) − ln(μ α β (λ2 c1 + (1 + η)2 (1 − ρ)−2 β −2k−2 λ3 d))
2N 0 2 2k

where λij = δij (1 − ρ) + (1 − δij )(1 + η). Then the system (6) is finite time bounded and
satisfies the extended dissipative performance. The controller gains and event-trigger param-
eters can be given by Kj = Yj R−1 −1 −1
j and j = Rj j Rj respectively.

Proof: The proof is omitted. 

4. Numerical example
In this section, we provide an example to show the efficiency of the proposed method.

Example: Consider system (6) with two subsystems and the parameters are given as
follows

0.1 0.2 0.1 0 0.2 0 0.3 0.1

A1 = , B1 = , C1 = , D1 = ,
0.1 0 0 0.2 0.1 0.2 0 0.3
0.2 0.3 0.1 0 0.1 0.4 0.2 0.1
A2 = , B2 = , C2 = , D2 = .
0 0.2 0.1 0.3 0 0.2 0 0.2

Table 1 listed the matrices in Remark 2.1:

In Theorem 3.4, we choose μ = 1.05, ρ = 0.01, η = 0.01, ν = 0.01, and through solv-
ing the LMIs, we obtain the optimized variables in Table 2, controller gains in Table 3 and
event-trigger parameters in Table 4.
488 H. GAO ET AL.

Table 1. Matrices for each case.

Analysis 1 2 3 4
L2 − L∞ performance 0 0 γ 2I I
H∞ performance −I 0 γ 2I 0
Passivity 0 I γ 0
Dissipativity −I I I−β ∗I 0

Table 2. Optimized variable for each case.

Subsystem L2 − L∞ performance H∞ performance Passivity Dissipativity
1 2 = 1 ∗ 10−10
γmin γmin
2 = 0.009 γmin = 0.097 βmax = 0.898
2 2 = 1 ∗ 10−9
γmin γmin
2 = 0.02 γmin = 0.112 βmax = 0.883

Table 3. Controller gain for each case.

Subsystem 1 2
−0.00258 −0.04455 −0.01720 −0.03347
L2 − L∞ performance K1 = K2 =
−0.04836 −0.00284 −0.03216 −0.06498
−1.4523 −0.4493 −1.7228 −0.4772
H∞ performance K1 = K2 =
−0.3406 −0.5907 −0.1084 −1.0734
2.7655 1.8743 1.0860 −0.7785
Passivity K1 = K2 =
1.9911 2.8399 −0.3248 1.7992
3.3610 1.5464 0.6763 −0.8089
Dissipativity K1 = K2 =
1.5044 2.7759 −0.2419 1.7129

Table 4. Event-trigger parameters.

Subsystem 1 2
3.2869 0.0009 4.0779 −0.0218
L2 − L∞ performance = 10−8 ∗ = 10−8 ∗
1
0.0009 3.2835 2
−0.0218 4.0658
168.3871 −105.8638 312.1159 −91.9407
H∞ performance = =
1
−105.8638 119.5161 2
−91.9407 269.6666
11.0695 −4.4444 20.9220 −17.5210
Passivity = =
1
−4.4444 8.9659 2
−17.5210 23.7134
13.6843 −6.5640 11.1850 −11.8119
Dissipativity = =
1
−6.5640 9.9360 2
−11.8119 17.6362

5. Conclusion
This paper investigated the problem of finite-time event-triggered extended dissipative
control for a class of discrete time switched linear systems. Under an event-triggered
scheme, we discussed the finite-time extended dissipative performance of the considered
systems. The design methods for state feedback controller gains are proposed. A simula-
tion example is given to show the effectiveness of the proposed method. In addition, in the
future, the concept of extended dissipative should be extended to more complex sysytems
such as multi-agent systems, Markovian jump systems, T-S fuzzy systems and so on, which
is more challenging.

Disclosure statement
No potential conflict of interest was reported by the authors.
 INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 489

Funding
The work was supported by the National Natural Science Foundation of China [Grants no.
61374117].

Notes on contributors
Gao Hui received the B.E degree from University of Jinan, Jinan, China, in 2015,
and the M.S. degree in system science from Liaocheng University, Liaocheng, in
2018. He is currently working towards his Ph.D. degree in circuits and systems
at the University of Electronic Science and Technology of China, Chengdu. His
current research interests include switched systems, robust control and event-
triggered control.

Zhang Hongbin received the B.Eng. degree in aerocraft design from Northwest-
ern Polytechnical University, Xian, China, in 1999, and the MEng and PhD
degrees in circuits and systems from the University of Electronic Science and
Technology of China, Chengdu, in 2002 and 2006, respectively. He has been
with the School of Electrical Engineering, University of Electronic Science and
Technology of China, since 2002, where he is currently a professor. From August
2008 to August 2010, he has served as a research fellow with the Department
of Manufacturing Engineering and Engineering Management, City University
of Hong Kong, Kowloon, Hong Kong. His current research interests include intelligent control,
autonomous cooperative control and integrated navigation.

Zheng Dianhao received the B.Eng. degree in electronic science and technol-
ogy from China University of Mining and Technology, Xuzhou, in 2009, and
the M.S. degree in circuits and systems from University of Electronic Science
and Technology of China, Chengdu, in 2012. He is currently working towards
two Ph.D. degrees in circuits and systems at the University of Electronic Sci-
ence and Technology of China and in engineering and information technology
at the University of Technology Sydney. He was an engineer from 2012 to 2014.
His research interests include cooperative control, multi-agent systems, and
switched systems.

Zhang Liangliang received the B.E. degree in the Zhejiang University City Col-
lege and M.S. degree in circuits and systems from University of Electronic
Science and Technology of China, Chengdu, Sichuan, in 2012 and 2014, respec-
tively. She is currently working towards her Ph.D. degree in circuits and systems
at the University of Electronic Science and Technology of China, Chengdu. Her
key research topics are fuzzy control, event-triggered H-infinity filtering and
switched systems.

Li Yang received the B.E. degree in electronic and information engineering and
M.S. degree in circuits and systems from University of Electronic Science and
Technology of China, Chengdu, in 2014 and 2017, respectively. He is currently
working towards his Ph.D. degree in circuits and systems at the University of
Electronic Science and Technology of China, Chengdu. His research interests
include fuzzy control and switched systems.
490 H. GAO ET AL.

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