Design the Finite-time L∞ Controller of a Class of

Nonlinear Switched time-delay Systems

Qilong Ai, Chengcheng Ren Shuping He*

School of Electrical Engineering and Automation, Anhui Key Laboratory of Intelligent Computing & Signal
University, Processing, Ministry of Education,
Hefei, China School of Electrical Engineering and Automation, Anhui
e-mail addresses: 279550868@qq.com (Q.L.Ai) University, Hefei, China
e-mail:chengcheng_ren@ahu.edu.cn (C.C.Ren) e-mail addresses: shuping.he@ahu.edu.cn

Abstract—This paper investigated the problem of finite-time controllable systems is studied. Moreover, the author
L’ controller design for time-delay nonlinear switched systems. concerned with finite-time boundedness for switched delay
Our purpose is to construct appropriate conditions to guarantee systems in [7]. For more research results of this subjects, the
the switched systems are not only finite-time bounded, but also readers can refer to the literature [8]-[10] and related reference.
have the L’ disturbance rejection performance. Based on the In our life, most of the control systems are nonlinear
multiple Lyapunov functions and average dwell-time method, the systems, therefore nonlinear systems control have more
closed-loop systems have a specified bounded. According to practical meaning to study. In [11], the author research for a
Lipschitz conditions, the nonlinear functions have been piecewise-affine approximation of nonlinear systems. In [12],
effectively solved. By using linear matrix inequalities technology ,
the author discussed about property benefits analysis for
the L’ performance have been analysis and described as an
optimization problem. In the end, a simulation example is used to
nonlinear systems. The issue of finite-time control of
proved the feasibility of the presented methods. nonlinear systems is studied in [13]. Furthermore, the issue of
L’ control problem has regained focus in analysis and
Keywords—nonlinear time-delay switched systems; finite-time; synthesis near the recent years. In [14], based on the fuzzy
L’ controller; average dwell-time; multiple Lyapunov functions. control theory, the author considered L’-gain of nonlinear
dynamic systems with persistent bounded disturbance. In [15],
I. INTRODUCTION the author studied for L’-gain analysis for positive time-
Switched system is one of the significant model of the delayed systems. However, the finite-time L’ controller design
hybrid systems, it is now a mainstream research direction at for a category of time-delayed nonlinear switched systems
the international level. Switched systems consist of a series have not been studied. This has motivated us to study about
subsystems and switching signals described the relationship this topic.
among the subsystems. In recent years, most of the literature In this paper, we discussed about the issue of finite-time
pay more attention to the stability of switched systems, some L’ controller for a category of time-delayed nonlinear
of them research for this topic with average dwell-time switched systems. Our purpose is to design a suitable L’
method [1], others study for the Lyapunov asymptotic stability controller and proper switching signal, which can ensure the
[2-4]. In [5], the author concerned the problem with linear close-loop system is finite-time bounded(FTB), and satisfies
matrix inequalities. the L’ disturbance rejection performance. To achieve this
On the other hand, based on the stability research, The purpose, we construct appropriate multiple Lyapunov
analysis of finite-time problems is gaining more and more functions , utilize the average dwell-time(ADT) approach and
attention. In practice, we tend to achieve the desired effect in a linear matrix inequalities (LMIs) technology. Moreover, the
short time rather than infinite bounded. Comparing to the finite-time L’ controller problem is constructed as an
classical Lyapunov stability, it would be more efficiency and optimization problem. Finally, a simulation example
apply to more fields, thus finite-time problems have more demonstrates the effectiveness of the above methods.
research value. In [6], the issue of finite-time stability of The main contributions of this paper are that the finite-
time L’ controller is constructed for nonlinear time-delay
switched systems. Different from [14], [15], we have
This work was supported in part by the Foundation for researched for the transient performance in the limited time
Distinguished Young Scholars of Anhui Province under Grant No interval. And the system model is based on the nonlinear time-
1608085J05, and the Key Support Program of University delay switched system, which can make the results more
Outstanding Youth Talent of Anhui Province under Grant No general.
gxydZD201701.
Corresponding author to Shuping He at: School of Electrical II. SYSTEM FORMULATION
Engineering and Automation, Anhui University, Hefei 230601, PR Considering the following nonlinear switched system:
China. Tel./fax: +86 55163861413. E-mail addresses:
shuping.he@ahu.edu.cn.

978-1-5386-3524-7/17/$31.00 ©2017 IEEE 189

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 Tf
­ x (t ) = Aσ ( t ) x(t ) + Adσ ( t ) x(t − τ ) + Bσ ( t )u (t )
° ³
0
= Τ (t ) =(t )dt ≤ h . (6)
° + Gσ ( t ) fσ ( t ) ( x(t )) + H σ ( t ) =(t ) (1) Definition 3. The controller (4) is called as a finite-time
® L’ controller with disturbance rejection performance γ > 0 for
° z (t ) = Cσ ( t ) x(t )
° x(t ) = ϕ (t ), t ∈ [t − τ , t ] switched system (1), if the switched system (5) is FTB under
¯ 0 0 the define of Definition 2, and satisfies the following
where x(t ) ∈ ℜ n is the state , x (t − τ ) ∈ ℜ n is the time-delay condition:
state, u (t ) ∈ ℜ m denotes the control input, z (t ) ∈ ℜ p denotes x(t ) ∞ ≤ γ =(t ) ∞ (7)
the controlled output, =(t ) ∈ ℜ q is the disturbance input, = sup x Τ (t ) x(t ) , =(t ) = sup = Τ (t )=(t ) .
where x(t ) ∞ ∞
fσ ( t ) ( x(t )) is the nonlinear function, ϕ (t ) denotes the initial t∈[ 0 ,T f ] t∈[ 0 ,T f ]

The purpose of this paper is to design a finite-time L’

continuous function. Aσ (t ) , Adσ (t ) , Bσ (t ) , Gσ (t ) , H σ (t ) , Cσ (t ) controller which under definition 2 and definition 3 for the
are constant matrices with appropriate dimension. σ (t ) is the nonlinear switched system (1). Meanwhile, the solving
switching law which define on the interval Ι = {1,2," , M } , method of finite-time L∞ state feedback controller gains and
M represents the number of subsystems. The switching law the conditions of controller are given by linear matrix
has the switching sequences as {(t 0 , σ (t0 )), (t1 , σ (t1 )),! , (t k , σ (t k ))}, inequalities .
when t ∈ [t k , t k +1 ) the σ (t k ) subsystem is activated. To III. MAIN RESULTS
facilitate, Aσ ( t ) , Adσ ( t ) , Bσ ( t ) , Gσ ( t ) , H σ ( t ) , Cσ ( t ) are indicated Theorem 1. For given constants T f > 0 , c1 > 0 , α > 0 ,
as Ai , Adi , Bi , Gi , H i , Ci respective with σ (t ) = i . μ ≥ 1 , ε i > 0 , the switched system (5) is FTB as for
Definition 1. For ∀T f ≥ t ≥ 0 , define Nσ (t , T f ) as the (
c1 c2 T f R h σ , let )
Pi = R −1/ 2 Pi R −1/ 2 ,
switching number of σ (t ) over (t , T f ) . As if τ a > 0 , N 0 ≥ 0 , Qi = R −1/ 2
Qi R −1 / 2
. Where 0 < c1 < c2 , R > 0 , if there exists
switched number Nσ (t , T f ) satisfied the following inequality positive-definite symmetric matrices Pi ∈ ℜ n×n , Qi ∈ ℜ n×n and
Nσ (t , T f ) ≤ N 0 + (T f − t ) / τ a , we called τ a as average dwell positive constants η , c2 , such that the following inequalities
time (ADT). hold:
Assumption 1. For ∀i ∈ N , the function of nonlinear ªΛ i Pi Aˆ di Pi H i º
section f i ( x(t )) satisfies global Lipschitz condition as: « »
Ω1 = « ∗ − Qi 0 » < 0, (8)
f i ( x(t )) ≤ U i x(t ) (2) «∗ »
¬ ∗ − η I ¼
where U i are known Lipschitz constant matrices.
Pi ≤ μPj , Qi ≤ μQ j , ∀i, j ∈ Ι , i ≠ j , (9)
Lemma 1. For the proper dimension matrices X , Y,
positive definite matrix T and positive constant ρ , the (λ1 + τλ3 )c1 + ηh < λ2 c2e
−αT f
, (10)
following matrix inequality is established:
∗ T f ln μ
X ΤY + Y Τ X ≤ ρX ΤTX + ρ −1Y ΤT −1Y . (3) τa >τa = , (11)
For system (1), we construct the following controller: ln(λ2 c2 ) − ln[(λ1 + τλ3 )c1 + ηh] − αT f
Λ = Aˆ iΤ Pi + Pi Aˆ i + Qi − αPi + ε iU i U i + ε i Pi Gi Gi Pi ,
Τ −1 Τ
u (t ) = K1,σ ( t ) x(t ) + K 2,σ ( t ) x(t − τ ) (4) where
where K1,σ ( t ) , K 2,σ ( t ) are the state feedback controller gains λ1 = max (λmax ( Pi )) , λ2 = min (λmin ( Pi )) , λ3 = max (λmax (Qi )) ,
i∈Ι i∈Ι i∈Ι
require to be designed. λ4 = min (λmin (Qi )) .
Substituting controller (4) into system (1), the closed-loop i∈Ι
switched system is obtained as follows: Proof. Consider the following Lyapunov function as:
t
­ x (t ) = Aˆσ ( t ) x(t ) + Aˆ dσ ( t ) x(t − τ ) + Gσ ( t ) fσ ( t ) ( x(t )) + H σ ( t ) =(t ) Vi ( x(t )) = V1,i ( x(t )) + V2,i ( x(t)) = xΤ (t)Pi x(t) + ³ xΤ (s)Qi x(s)ds .(12)
°° t −τ

® z (t ) = Cσ ( t ) x(t ) (5) Along the track of the system (5), according to Assumption
° 1 and Lemma 1 and solve the derivative of V [x(t )] , we can get:
°¯ x (t ) = ϕ (t ), t ∈ [t 0 − τ , t 0 ]
where Aσ ( t ) = Aσ ( t ) + Bσ ( t ) K1,σ ( t ) , Aˆ dσ ( t ) = Adσ ( t ) + Bσ ( t ) K 2,σ ( t ) .
ˆ
Definition 2. For given positive constants c1 , h , T f ,
positive definite matrix R , the switched system (5) is FTB as
for (c1 c2 T f R h σ ) , if there exist a positive constant
c2 > c1 , such that the following condition holds for all state
x(t ) and disturbance input =(t ) :
sup −τ ≤θ ≤ 0 x Τ (θ ) Rx(θ ) ≤ c1 Ÿ x Τ (t ) Rx(t ) < c2 , ∀t ∈ [0, T f ] .

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 Vi ( x (t )) = x Τ (t )( Aˆ i Pi + Pi Aˆ i + Qi ) x (t )
T Furthermore, with Pi = R −1/ 2 Pi R −1/ 2 , Qi = R −1/ 2Qi R −1/ 2 ,
+ x Τ (t − τ ) Aˆ Τ P x (t ) + x Τ (t ) P Aˆ x (t − τ ) according to the define of λ1 , λ2 , λ3 , we can get:
di i i di
Τ Vσ (t ) ( x(t )) ≥ x Τ (t ) Pσ (t ) x(t )
+ f i Τ ( x (t )) Gi Pi x (t ) + x Τ (t ) Pi Gi f i ( x (t ))
Τ
+ = Τ (t ) H i Pi x (t ) + x Τ (t ) Pi H i = (t ) ≥ λmin ( Pσ (t ) ) x Τ (t ) Rx(t ) , (23)
− x Τ ( t − τ ) Qi x ( t − τ ) . (13) ≥ λ2 x (t ) Rx(t )
Τ

0
ˆ ˆ Vσ ( 0) ( x(0)) = x Τ (0) Pσ (0 ) x(t ) + ³ x Τ ( s )Qσ ( 0 ) x( s) ds
Τ Τ
Τ
≤ x (t )( Ai Pi + Pi Ai + Qi + ε iU i U i
−τ
+ ε P G G P ) x (t ) + x Τ (t − τ ) Aˆ Τ P x (t )
−1 Τ
i i i i i di i ≤ (λmax ( Pσ (t ) ) + τλmax (Qσ (t ) )) sup {x Τ ( s ) Rx( s )} .
+ x (t ) Pi Aˆ di x (t − τ ) + = (t ) H i Pi x (t )
Τ Τ Τ −τ ≤ s ≤ 0

≤ (λ1 + τλ3 )c1

+ x Τ (t ) Pi H i = (t ) − x Τ (t − τ )Qi x (t − τ ) (24)
Define the following formula:
Above inequalities are equal to:
J1i = Vi ( x(t )) − αVi ( x(t )) − η= Τ (t )=(t ) . (14)
1
If inequality (8) is established, we can get: x Τ (t ) Rx(t ) ≤ Vσ (t ) ( x(t ))
t λ2
J1i + α ³ x Τ ( s )Qi x( s) ds = ξ Τ Ω1ξ < 0 , (15)
1
t −τ
≤ e [(λ1 + τλ3 )c1 + ηh] .
αT f
μ
T f /τ a

[
where ξ = x Τ (t ) x Τ (t − τ ) = Τ (t ) . ]
Τ
λ2
(25)

Considering α > 0 and Qi > 0 , the inequality (15) means (λ + τλ3 )c1 + ηh αT T /τ
= 1 e μ f f a

J1i < 0 . Thus, we can obtain: λ2

Vi ( x(t )) < αVi ( x(t )) + η= Τ (t )=(t ) . (16) (1) If μ = 1 , according to condition (10), inequality (25) is
equivalent to:
When t ∈ [tk , t k +1 ] , integrate the above inequality on t k to
(λ + τλ3 )c1 + ηh αT f
t , we have: x Τ (t ) Rx(t ) ≤ 1 e < c2 . (26)
t d t λ2
³tk dt e Vi ( x(t ))dt < η ³tk e = (s)=(s)ds .
−αt −αs Τ
(17) (2) If μ > 1 , on the base of inequality (26), we can
Above inequality is equal to: recognize:
t ln(λ2c2 ) − ln[(λ1 + τλ3 )c1 + ηh] − αT f > 0 , (27)
Vσ ( t ) ( x(t )) < eα ( t −t k )Vσ ( t k ) ( x(tk )) + η ³ eα ( t − s ) = Τ ( s) =( s) ds . (18)
t k according to inequality (11), we have:
Assuming that at the switching time t k , σ (t k ) = i , T f ln(λ2 c2 ) − ln[(λ1 + τλ3 )c1 + ηh ] − αT f
< . (28)
σ (t k − ) = j , consider that x(t k ) = x(tk − ) , if satisfied τa ln μ
inequalities (9), we can get: Substitute inequality (28) into (25), we can get:
−
Vσ (tk ) ( x(t k )) ≤ μVσ (t − ) ( x(t k )) . (19) x Τ (t ) Rx(t ) < c2 . (29)
k
This completes the proof.
According to the inequalities (18) to (19), we have:
t
Considering of the performance index function (7),
Vσ ( t ) ( x(t )) < μeα ( t −tk )Vσ ( t − ) ( x(t k )) + η ³ eα ( t − s ) = Τ ( s) =( s) ds .(20)
−
Theorem 2 can be obtained as follows.
k t
Theorem 2. For given constants T f > 0 , c1 > 0 , α > 0 ,
k

For ∀t ∈ [0, T f ] , consider that N is switched number of

μ ≥ 1 , ε i > 0 , nonlinear time-delay switched system (5) is
switching signal σ (t ) , which means Nσ ( t ) (0, t ) ≤ N ,
FTB and satisfies the given L’ disturbance rejection
N ≤ T f / τ a . Based on the condition μ ≥ 1 and above methods, performance (7) with respect to c1 c2 T f R h γ σ , ( )
we can get:
where 0 < c1 < c2 , Ri > 0 , if there exists positive definite
t1
Vσ (t ) ( x(t )) < eαt μ Nσ ( 0,t )Vσ ( 0) (0) + μ Nσ ( 0,t )η ³ eα (t − s ) = Τ ( s )=( s )ds + symmetric matrices Pi ∈ ℜ n×n , Qi ∈ ℜ n×n and positive
0
t2 t 2αT f
μ Nσ ( 0,t ) −1η ³ eα ( t − s ) = Τ ( s) =( s )ds + " + η ³ eα ( t − s ) = Τ ( s)=( s) ds η (e − 1) η
t1 tk constants γ = , , c2 , such that the following
t 2αλ 2
= eαt μ Nσ ( 0,t )Vσ ( 0) (0) + η ³ μ Nσ ( s ,t ) eα (t − s ) = Τ ( s )=( s )ds .
0 inequalities hold:
ª∏ i Pi Aˆ di Pi H i º
t
μ Nη ³ = Τ ( s )=( s )ds
αT f αT f
≤e μ NVσ ( 0 ) (0) + e
0 « »
≤e
αT f
μ N (Vσ ( 0 ) (0) + ηh) « ∗ − Qi 0 » < 0, (30)
(21) « ∗ ∗ − ηI »¼
¬
Based on Definition 1, we have: Pi ≤ μPj , Qi ≤ μQ j , ∀i, j ∈ Ι , i ≠ j ,
αT T /τ
(31)
Vσ ( t ) ( x(t )) ≤ e f μ f a (Vσ ( 0 ) (0) + ηh) . (22) −αT f
(λ1 + τλ3 )c1 + ηh < λ2 c2e , (32)

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 ­° T f ln μ ln μ ½° sup x(t ) 2αT f
τ a > τ a ∗ = max ® , ¾, t∈[ 0 ,T f ] η (e − 1)
°̄ ln(λ2 c2 ) − ln[(λ1 + τλ3 )c1 + ηh] − αT f α °¿ ≤ . (43)
sup =(t ) 2αλ2
(33) t∈[ 0 ,T f ]

∏ i = AˆiΤ Pi + Pi Aˆi + Qi − αPi + ε iU i U i + ε i Pi Gi Gi Pi , This completes the proof.

Τ −1 Τ
where
Remark 1. Theorem 2 proposes a new method to solve the
λ1 = max (λmax ( Pi )) , λ2 = min (λmin ( Pi )) , λ3 = max (λmax (Qi )) , finite-time L’ disturbance rejection performance, making full
i∈Ι i∈Ι i∈Ι

λ4 = min (λmin (Qi )) . use of switching numbers Nσ (t , T f ) through the proof of

i∈Ι
Proof. Consider the following Lyapunov function as: Theorem 2, which can gets a less conservative results.
Vi ( x(t )) = V1,i ( x(t )) + V2,i ( x(t )) Theorem 3. For given constants T f > 0 , α > 0 , c1 > 0 ,
t . (34) μ ≥ 1 , ε i > 0 , the switched system (5) is about
= x Τ (t ) Pi x(t ) + ³ x Τ ( s )Qi x( s )ds ( )
c1 c2 T f h R γ σ FTB, where c2 > c1 , R > 0 .
t −τ
Along the track of the system (5), the derivative of And satisfies finite short time L’ disturbance rejection
V [x(t )] is given as follows: 2αT f
performance (7) of norm bounded γ =
− 1) , if there η (e
Vi ( x(t )) ≤ x Τ (t )( Aˆ i Pi + Pi Aˆ i + Qi + ε iU i U i
Τ Τ
2α
+ ε i Pi Gi Gi Pi ) x(t ) + x Τ (t − τ ) Aˆ diΤ Pi x(t ) exist scalars c2 > 0 , η > 0 , σ 1 > 0 , σ 2 > 0 , positive definite
−1 Τ

. (35)
+ x Τ (t ) Pi Aˆ di x(t − τ ) + = Τ (t ) H i Pi x(t )
Τ symmetric matrix X i ∈ ℜ n×n , Qi ∈ ℜ n×n , M i ∈ ℜ n×n and

+ x Τ (t ) Pi H i =(t ) − x Τ (t − τ )Qi x(t − τ ) matrix Yi ∈ ℜ m×n , such that the following inequalities hold:
Based on inequality matrix (30), we have: ªΨi Adi + Bi K 2,i Hi X iU iΤ Gi º
Vi ( x(t )) < αVi ( x(t )) + η= Τ (t )=(t ) . (36) « »
«∗ − Qi 0 0 0 »
For t ∈ [tk , t k +1 ] , integrate the above inequality on t k to t , «∗ ∗ − ηI 0 0 » <0, (44)
we have: « »
t «∗ ∗ ∗ − ε i−1I 0 »
Vσ (t ) ( x(t )) < eα (t −t k )Vσ (t k ) ( x(tk )) + η ³ eα (t − s ) = Τ ( s) =( s) ds . (37) «∗ ∗ ∗ ∗ − ε i I »¼
t k ¬
According to the condition (31), we have: σ 1 R −1 < X i < R −1 , (45)
−
Vσ (tk ) ( x(t k )) ≤ μVσ (t − ) ( x(t k )) .
(38) 0 < Qi < σ 2 R , (46)
k

As the same as the proof of Theorem 1, through the ªc1τσ 2 + ηh − c2 e c1 º

−αT f

iterative operation we can get: « » < 0, (47)

t1 ¬« c1 − σ 1 ¼»
Vσ ( t ) ( x(t )) < eαt μ Nσ ( 0,t )Vσ ( 0 ) (0) + μ Nσ ( 0,t )η ³ eα ( t − s ) = Τ ( s)=( s )ds
0 X j ≤ μX i , Qi ≤ μQ j , ∀i, j ∈ Ι , i ≠ j (48)
t2 t
+ μ Nσ ( 0,t ) −1η ³ eα ( t − s ) = Τ ( s)=( s )ds + " + η ³ eα ( t − s ) = Τ ( s )=( s )ds (39) the switching signal satisfies the following inequality:
t t
­° T f ln μ ln μ ½°
1 k

< e μ Vσ ( 0 ) (0) + η ³ μ
αt N
t N σ ( s ,t ) α ( t − s )
e Τ
= ( s )=( s )ds τ a > τ a ∗ = max ® , ¾,
0 °̄ ln(c2 ) − ln[(1 / σ 1 + τσ 2 )c1 + ηh] − αT f α °¿
Since 0 ≤ Nσ ( s, t ) ≤ (t − s ) / τ a ≤ α (t − s ) / ln μ , we can get (49)
1 ≤ μ N σ ( s , t ) ≤ eα ( t − s ) . Τ Τ Τ
where Ψi = Ai X i + X i Ai + BiYi + Yi Bi − αX i + M i .
With the zero initial conditions, we have: −1
t
In addition, controller gain can be designed as: K1,i = Yi X i .
Vσ (t ) ( x(t )) < η ³ e 2α ( t − s ) Τ
= ( s )=( s )ds Proof . Substituting Aˆ = A + B K , Aˆ = A + B K
0
t
. (40) i i i 1,i di di i 2 ,i

< η ³ e 2αs = Τ (t − s )=(t − s )ds into (30), we can get:

0
ªΦ Pi ( Adi + Bi K 2,i ) Pi H i º
For convenience, we assume min (λmin ( Pi )) ≤ min (λmin ( Pi )) . «∗
i∈Ι i∈Ι
« − Qi 0 »» < 0 , (50)
2
Considering Vσ (t ) ( x(t )) ≥ min (λmin ( Pi )) x(t ) , (41) «¬ ∗ ∗ − ηI »¼
i∈Ι
we have the following inequalities: where
t
η ³ e 2αs = Τ (t − s )=(t − s )ds ≥ min (λmin ( Pi )) x(t )
2 Φ = ( AiΤ + K1Τ,i BiT ) Pi + Pi ( Ai + Bi K1,i ) + Qi − αPi
0 i∈Ι . (51)
2 + ε iU iΤU i + ε i−1PiGiGiΤ Pi
η sup =(t ) . (42) −1
t∈[ 0 ,T f ] 2αT f 2 Then pre- and post-matrix (50) with diag{Pi , I , I } , and
(e − 1) ≥ min (λmin ( Pi )) sup x(t )
2α i∈Ι t∈[ 0 ,T f ] considering X i = Pi −1 , Yi = K1,i X i , M i = X i Qi X i , we have:
Based on Definition 2, we have:

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 ªΦ 1 Adi + Bi K 2,i Hi º ª0.5 0º ª0 0 º
«∗ matrix as: U1 = « » , U2 = « » . Then based on the
« − Qi 0 »» < 0 , (52) ¬ 0 0¼ ¬0 0.2¼
method of Remark, using the LMI toolbox to solve these
¬« ∗ ∗ − ηI ¼»
problems.
Where According to the above methods and conditions, we found
Τ Τ Τ
Φ1 = Ai X i + X i Ai + BiYi + Yi Bi − αX i the LMI constrains are feasible, and the feasible solutions are
Τ
, as follow:
+ M i + ε i−1Gi Gi + ε i X iU iΤU i X i
u1 (t ) = [− 35.8459 − 23.8133]x(t )
then according to Schur complement, we can get inequality
matrix (44). + [− 0.3784 − 0.5800]x(t − τ )
Finally, define X i = R1/ 2 X i R1/ 2 , Qi = R −1/ 2Qi R −1/ 2 , and set u2 (t ) = [− 27.0288 − 18.0498]x(t )
σ 1 ≤ min (λmin ( X i )) , max(λmax ( X i )) < 1 , max (λmax (Qi )) ≤ σ 2 ,
i∈Ι i∈Ι i∈Ι
+ [− 0.3333 − 0.0514]x(t − τ )
*
and consider max (λmax ( X i )) = 1/ min (λmin ( Pi )) . the ADT of switching signal is τ a = 0.6156 , so we choose
i∈Ι i∈Ι
The inequality (32) can be ensured by the following τ a = 0.62 , and we can obtain γ = 0.8186 , c2 = 1.9698 .
Considering the initial state condition x0 = [0.6 0.4] , we
Τ
inequality:
1 −αT Τ
c1 + τσ 2 c1 + ηh < c2e f . (53) can get x0 Rx0 = 0.52 < c1 . The result of simulink in Fig.1-4.
σ1
Applying the Schur complement, LMIs (45)-(47) from
inequality (53) can be obtained. The proof is completed.
Remark 2. Theorem 3 gives the sufficient conditions for the
existence of finite-time L’ controller of the nonlinear time-
delay switched system (1). Considering the coupling
inequality (44)-(48) are limited to X i , Yi , Qi , c1 , c2 , T f , α ,
h , ε i , σ 1 , σ 2 and η , we can get the following optimization
problem by setting η as an optimization variable:
min η
X i ,Yi ,Qi ,α , h ,T f ,ε i ,σ 1 ,σ 2 ,c1 , c2
. (54)
s.t. inequaliti es˄44˅˄ - 48˅
Fig.1 The state curve of open-loop system
IV. NUMERAL EXAMPLE
Consider a category of nonlinear time-delay switched
systems with two subsystems are described as:
ª2 0 º ª − 0.2 0.3º ª 0.5º
A1 = « » , Ad 1 = « » , B1 = « » ,
¬0 − 3¼ ¬ 0.5 0.4¼ ¬0.7 ¼
ª− 0.3 − 0.1º ª0.5º
C1 = [1 0.5], G1 = « » , H1 = «0.9 » ,
¬ 0 . 1 0 . 3 ¼ ¬ ¼
ª− 3 4 º ª0.2 0.4 º ª 0 .6 º
A2 = « » , Ad 2 = « » , B2 = « » ,
¬ 4 − 3¼ ¬0.3 − 0.2¼ ¬ 0 .9 ¼
ª 0.12 0.23 º ª 0.3º
C 2 = [0.5 0.1] , G2 = « » , H2 = « » . Fig.2 The state curve of closed-loop system
¬− 0.23 − 0.12 ¼ ¬0.4¼
Define the initial constants as : α = 0.01 , ε 1 = 3 , ε 2 = 2 ,
c1 = 0.6 , τ = 0.1 , T f = 5 , μ = 1.002 , R = I ,
and choose disturbance function as: = (t ) = (0.5sin (5t ))e −0.01t ,
5
integrate = Τ (t ) =(t ) on 0 to T f , we can get: ³ = Τ (t )= (t ) dt = 0.5977 ,
0
so we choose h = 1 .
ª0.2 sin( x1 (t ))º
Considering the nonlinear function as: f1 ( x (t )) = « »,
¬ 0 ¼
ª 0 º
f 2 ( x(t )) = « » , and consider the Lipschitz constant
¬0.1sin( x2 (t ))¼ Fig.3 The trajectory of switching signal

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