Finite-time L1 control for positive switched linear systems with time-varying delay

Accepted Manuscript

Finite-time L 1 control for positive switched linear systems with time-varying

delay

Mei Xiang Zhengrong Xiang

PII S1007-5704(13)00158-5

DOI httpdxdoiorg101016jcnsns201304014

Reference CNSNS 2774

To appear in Communications in Nonlinear Science and Numer‐

ical Simulation

Received Date 27 September 2012

Revised Date 13 April 2013

Accepted Date 14 April 2013

Please cite this article as Xiang M Xiang Z Finite-time L 1 control for positive switched linear systems with

time-varying delay Communications in Nonlinear Science and Numerical Simulation (2013) doi httpdxdoiorg

101016jcnsns201304014

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1

Finite-time L1 control for positive switched linear

systems with time-varying delay


School of Automation Nanjing University of Science and Technology

Nanjing 210094 Peoplersquos Republic of China

corresponding author e-mail xiangzrmailnjusteducn

Tel 0086-13951012297 Fax 0086-25-84313809

Abstract This paper is concerned with the problem of finite-time L1 control for a class of

positive switched linear systems with time-varying delay Firstly by using the average dwell time

approach sufficient conditions which can guarantee the L1 finite-time boundedness of the

underlying system are given Then in virtue of the results obtained a state feedback controller is

designed to ensure that the resulting closed-loop system is finite-time bounded with L1-gain

performance All the obtained results are formulated in terms of linear matrix inequalities (LMIs)

which can be solved conveniently Finally an example is given to illustrate the efficiency of the

proposed method

Keywords positive systems switched systems time-varying delay finite-time boundedness

L1-gain performance average dwell time

1 Introduction

2

Positive systems are dynamic systems with state variables and outputs constrained to be

positive (or at least nonnegative) at all times whenever the initial condition and input are

nonnegative The applications of such systems can be found in various areas for instance

biomedicine [1-2] ecology [3] industrial engineering [4] TCP-like Internet congestion control

[5-6] and so on Recently positive switched systems which consist of a family of positive

subsystems and a switching signal governing the switching among them have also been

highlighted by many researchers due to their broad applications in communication systems [7]

formation flying [8] and systems theory [9-13]

Time delays arise quite naturally in many dynamical systems and are frequently a source of

instability and poor performance Therefore considerable attention has been devoted to the study

of different issues related to time-delay systems and many results on these systems have been

presented in the literature [14-23] and the references therein

The stability problem has been a subject of considerable research and a major concern in the

area of positive switched systems [22-27] And up to now most of the existing literature related to

the stability of positive switched systems focuses on Lyapunov stability which is defined over an

infinite time interval However in practice one not only is interested in system stability (usually

in the sense of Lyapunov) but also concerns a bound of state trajectory over a fixed short time

[28] The finite-time stability is a different stability concept which admits that the state does not

exceed a certain bound during a fixed finite-time interval Some early results on finite-time

stability and stabilization can be found in [29-31] It should be pointed out that a finite-time stable

system may not be Lyapunov stable and a Lyapunov stable system may not be finite-time stable

since the transient of a system response may exceed the bound [33] Recently finite-time control

3

for switched linear systems with and without delays has been investigated in [32-36] For positive

switched linear systems the definition of finite-time stability has been given in [37] However to

the best of our knowledge there are no results available on finite-time stability and finite-time

boundedness of positive switched systems with time-varying delay which motivates our present

study

In this paper we are interested in investigating the problem of finite-time L1 control for a class

of positive switched linear systems with time-varying delay Compared with the existing works

the main contribution of this paper is threefold 1) Definitions of finite-time boundedness and 1L

finite-time boundedness are for the first time extended to positive switched linear systems with

time-varying delay 2) Sufficient conditions for the existence of 1L finite-time boundedness of

the underlying system are given 3) A state feedback controller is designed to guarantee that the

closed-loop system is 1L finite-time bounded

The paper is organized as follows In Section 2 problem statements and necessary lemmas are

given 1L finite-time boundedness analysis and controller design are developed in Section 3 A

numerical example is provided in Section 4 Finally Section 5 concludes this paper

Notations In this paper 0( 0)A means that all entries of matrix A are non-negative

(non-positive) 0( 0)A means that all entries of A are positive (negative) ( )A B A B

means that 0( 0)A B A B TA is the transpose of a matrix A R

is the set of all

positive real numbers nR

is the n-dimensional non-negative (positive) vector space nR is the

set of real vectors of n -dimension n kR

is the set of all real matrices of ( )n k -dimension

The notation 1

n

k

k

x x

where kx is the k th element of nx R

2 Problem Statements and Preliminaries

4

Consider the following positive switched linear systems with time-varying delay

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ( )) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t d t t t

t t t

x t A x t A x t d t G u t B w t

z t C x t D u t E w t

x

(1)

where ( ) nx t R ( ) mu t R and ( ) zz t R

denote the state control input and controlled

output respectively ( ) lw t R is the

disturbance input satisfying

0( ) 0

fT

w t dt d d (2)

( ) [0 )t 12 M M is the switching signal with M being the number of

subsystems pΑ dpΑ pG pB pC pD

and pE p M are constant matrices with

appropriate dimensions ( ) is the initial condition on [ 0]

0 0 0t is the initial

time and qt denotes the q th switching instant )(td

denotes the time-varying delay satisfying

)(0 td htd )( where and h are positive scalars

Next we will give the positive definition for the following switched system

( ) ( ) ( )

( ) ( )

( ) ( ) ( ( )) ( )

( ) ( ) ( )

( ) ( ) 0

t d t t

t t

x t A x t A x t d t B w t

z t C x t E w t

x

(3)

Definition 1 System (3) is said to be positive if for any initial conditions ( ) 0 0

( ) 0w t and any switching signals ( )t the corresponding trajectory 0)( tx

and 0)( tz

hold for all 0t

Definition 2 [38] A is called a Metzler matrix if the off-diagonal entries of the matrix A are

non-negative

The following lemma can be obtained from Lemma 3 in [39] and Proposition 1 in [22]

Lemma 1 System (3) is positive if and only if pA p M are Metzler matrices and

0 0 0 0dp p p pA B C E p M

5

Definition 3 [40] For any switching signals ( )t and any 012 TT let ( ) 1 2( )tN T T

denotes the number of switching of ( )t over the interval 1 2[ )T T For given 0aT and

0 0N if the inequality

2 1( ) 1 2 0( )t

a

T TN T T N

T

holds then the positive constant aT is called an average dwell time and 0N is called a

chattering bound As commonly used in the literature we choose 0 0N in this paper

Now we are in a position to give the definitions of finite-time stability finite-time boundedness

and finite-time 1L boundedness for the positive switched system (3)

Definition 4 (Finite-time stability) For a given time constant fT and two vectors 0

switched system (3) with ( ) 0w t is said to be finite-time stable with respect to

( ( ))fT t if 1)(sup0

txT ( ) 1Tx t [0 ]ft T If the above condition is

satisfied for any switching signals ( )t system (3) is said to be uniformly finite-time stable with

respect to ( )fT

Remark 1 As can be seen from Definition 4 the concept of finite-time stability is different from

the one of Lyapunov asymptotic stability A Lyapunov asymptotically stable switched system may

not be finite-time stable because its states may exceed the prescribed bounds during the interval

time

Definition 5 (Finite-time boundedness) For a given time constant fT and two vectors

0 positive switched system (3) is said to be finite-time bounded with respect to

( ( ))fT d t where ( )w t satisfies (2) if 1)(sup0

txT ( ) 1Tx t

[0 ]ft T

Definition 6 (Finite-time 1L boundedness) For a given time constant fT positive switched

6

system (3) is said to be 1L finite-time bounded with respect to ( ( ))fT d t if the

following conditions are satisfied

1) Positive switched system (3) is finite-time bounded with respect to ( ( ))fT d t

2) Under zero-initial condition ( ) 0 0 the output ( )z t satisfies

0 0( ) ( )

f fT Tte z t dt w t dt

where 0 0 and ( )w t satisfies (2)

The aim of this paper is to find a class of switching signals ( )t and determine a state

feedback controller ( )( ) ( )tu t K x t for positive switched system (1) such that the

corresponding closed-loop system is 1L finite-time bounded

3 Main Results

31 Finite-time stability and boundedness analysis

This section will focus on the problem of finite-time boundedness for positive switched system

(3)

Theorem 1 Consider system (3) for a given time constant fT and two vectors 0 if

there exist positive vectors pv p and p p M and positive constants p 1 2 3

and 4 such that the following inequalities hold

1 2 1 2 0p p p pn p p pnΨ diag ψ ψ ψ ψ ψ ψ (4)

1 2 3 4 p p pv (5)

22 3 4 1 fT

e e d e

(6)

where

prprprpp

T

prpr vva prp

T

dprpr hva )1(

m a x pp M

12 r n n

7

( )pr dpra a represents the r th column vector of the matrix ( )p dpA A and

1 2 T

p p p pnv v v v 1 2 T

p p p pn 1 2 T

p p p pn

then under the following average dwell time scheme

21 2 3 4

ln

ln lnf

fa a T

T μT T

e e e d

(7)

the system is finite-time bounded with respect to ( ( ))fT d t where

( )max ( )pp l M

12l l p is the th element of the vector 2

Tp pB

and 1μ satisfies

p q p q p qv v μ p q M (8)

Proof Choose the following piecewise co-positive type Lyapunov-Krasovskii functional for

system (3)

( )( ) ( ( )tV t V t x t

(9)

the form of each ( ( )pV t x t ( p M ) is given by

1 2 3( ( )) ( ( )) ( ( )) ( ( ))p p p pV t x t V t x t V t x t V t x t

where

1( ( )) ( ) T

p pV t x t x t v

( )

2( )

( ( )) ( ) pt t s T

p pt d t

V t x t e x s ds

0 ( )

3( ( )) ( ) pt t s T

p pt

V t x t e x s dsd

and n

p p pv R p M

For the sake of simplicity ( ( ))pV t x t is written as ( )pV t in this paper

Along the trajectory of system (3) we have

1( ) ( ) ( ) ( ( )) ( ) T T T T T T Tp p p p dp p p pV t x t v x t A v x t d t A v w t B v (10)

8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p

t t s d tT T Tp p p p p

t d t

t t s T T Tp p p p

t d t

V t e x s ds x t d t e x t d t

e x s ds x t h x t d t

(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p

t t s T T Tp p p p p

t

t tt s T T Tp p p p

t t d t

V t e x s dsd x t e x t d

e x s dsd x t x s ds

(12)

Combining (10)-(12) leads to

( ) ( ) ( )( )

( ( ))( (1 ) )

( )

T Tp p p p p p p p p

T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)

According to (5) and (13) we can easily obtain

2( ) ( ) ( ) ( ) T T T Tp p p p p pV t V t w t B v w t B (14)

Denoting 2T

p pB it follows from (14) that for 1[ )k kt t t

( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)

Let N be the switching number of ( )t over [0 )fT and denote 1 2 Nt t t as the

switching instants over the interval [0 )fT Then for [0 )ft T we obtain from (8) that

( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t

t tN t N t s T N t s T

tt

t s T

t

V t e V t e w s ds

e V t e w s ds

e V e w s ds e w s ds

e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)

Considering the definition of ( ) ( )tV t it yields that

( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)

Combining (16)-(18) we obtain

ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)

Substituting (7) into (19) we have

( ) 1Tx t

According to Definition 5 we can conclude that the positive switched system (3) is finite-time

bounded with respect to ( ( ))fT d t

The proof is completed

Remark 2 In the proof of Theorem 1 there is no requirement of negative definitiveness on

( ) ( )tV t which is different from the case of classical Lyapunov stability In addition when

1 in (7) one obtains 0aT which means that the switching signal can be arbitrary

When the exogenous noise signal ( ) 0w t the result on finite-time stability can be obtained

as follows

Corollary 1 Consider system (3) with ( ) 0w t for a given time constant fT and two vectors

0 if there exist positive vectors pv p and p p M and positive constants p

1 2 3 and 4 such that (4) (5) and the following inequality

22 3 4 1 fT

e e e

(20)

holds then under the following average dwell time scheme

21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)

the system is finite-time stable with respect to ( ( ))fT t where max pp M

and

10

1μ satisfies (8)

Remark 3 The general idea of finite-time stability concerns the boundedness of the state of a

system over a finite interval for given initial conditions The idea of finite-time bondedness on the

other hand concerns the behavior of the state in the presence of both given initial conditions and

external disturbances It is easy to see from Definitions 4 and 5 that finite-time stable can be

regarded as a special case by setting 0d

32 1L performance analysis

In this section we will consider the problem of 1L finite-time boundedness of positive

switched system (3)


there exist positive vectors pv p and p p M and positive constants p 1 2

3 and 4 such that (5) and the following inequalities hold

1 2 1 2 1 2 0p p p pn p p pn p p pnΨ diag ψ ψ ψ ψ ψ ψ ψ ψ ψ (22)

2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T

pr pr p p pr pr pr pra v v c (1 )T

pr dpr p pra v h

m a x pp M

T

pr pr p prb v e 12 r n n

pr dpr pr pr pra a b c e represents the r th column vector of the matrix p dp p p pA A B C E

and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)

the system is 1L finite-time bounded with respect to ( ( ))fT d t where 1μ

satisfies (8)

Proof (4) can be directly derived from (22) Setting in Theorem 1 we can obtain from

(5) (24) and (25) that system (3) is finite-time bounded with respect to ( ( ))fT d t

Choosing the piecewise co-positive type Lyapunov-Krasovskii functional (9) and following the

proof line of Theorem 1 we can get from the condition (22) that

( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)

Let ( ) ( ) ( )w s z s s then for 1[ )k kt t t (26) gives rise to

( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)

Following the proof line of (16) for any [0 )ft T we can obtain

( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds

Under the zero initial condition we have

( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t

t tN s t N s tt s t se z s ds e w s ds (28)

Multiplying both sides of (28) by ( ) (0 )tN t

leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t

t tN s N st s t se z s ds e w s ds

Noting that ( ) (0 )t

a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )

t tt s s t se e z s ds e w s ds (29)

Let ft T then multiplying both sides of (29) by fT

e

leads to

2

0 0( ) ( )

f fT Tse z s ds w s ds

Setting 2 according to Definition 6 we can conclude that the claim of the theorem is true

12


33 1L controller design

Consider system (1) under the controller ( )( ) ( )tu t K x t the corresponding closed-loop

system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t

x t A G K x t A x t d t B w t

z t C D K x t E w t

x

(30)

By Lemma 1 to guarantee the positivity of system (30) p p pA G K should be Metzler

matrices and 0p p pC D K p M



3 and 4 such that (5) (23) (24) and the following conditions are satisfied

p p pA G K are Metzler matrices 0p p pC D K (31)

1 2 1 2 1 2 0p p p pn p p pn p p pndiag (32)

where

T

pr pr p pr p pr pr pr pr pra v g v c f

(1 ) T

pr dpr p pra v h max pp M

T


T Tp p p pg K G v p p pF D K pr dpr pr pr pr pra a b c e f

represents the r th column vector

of matrix p dp p p p pA A B C E F prg represents the r th element of vector pg and

1 2 T


p p p pn 1 2 T

p p p pn

then under the average dwell time scheme (25) the resulting closed-loop system (30) is 1L

finite-time bounded with respect to ( ( ))fT d t where 1μ satisfies (8)

13

Proof Replacing pA

and pC in (22) with p p pA G K and p p pC D K respectively and

letting T T

p p p pg K G v and p p pF D K we can get (32)


We are now in a position to present an algorithm for constructing the state feedback controller

gain matrices pK p M

Algorithm

Step 1 Input the matrices pΑ dpΑ pG pB pC pD

and pE

Step 2 By adjusting the parameters p we can obtain the solutions pv p p p pg F

such that (5) (23) (24) and (31) hold

Step 3 By T T

p p p pg K G v we can compute pK and then p p pF D K is obtained If

0p pF F p p pA G K are Metzler matrices and 0p p pC D K then pK are

admissible Otherwise return to Step 2

4 Numerical Example

Consider system (1) with parameters as follows

1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T

and solving the inequalities in Theorem 3 we get

1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T

p p p pg K G v 12p we obtain

1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K

It is easy to verify that p p p pF F D K p p pA G K

are Metzler matrices and

0p p pC D K

From (8) and (25) we have 21028 and 45679aT Choosing 46aT

simulation results are shown in Figs 1-3 where (0) [012 01 01]Tx ( ) 0x

[ 0) and

05( ) 005 tw t e Fig1 depicts the switching signal The state trajectory of the

closed-loop system with the average dwell time 46aT is shown in Fig2 Fig 3 plots the

15

evolution of ( )Tx t it can be seen that the closed-loop system is 1L finite-time bounded with

respect to ( ( ))fT d t

0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e

Fig1 Switching signal

0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3

Fig2 State trajectory of the closed-loop system

16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε

Fig3 The evolution of ( )Tx t

5 Conclusions

Finite-time boundedness and L1 finite-time boundedness for a class of positive switched linear

systems have been investigated in this paper Some sufficient conditions have been provided for

the finite-time stability of positive switched linear systems and the L1 finite-time boundedness is

also studied Bases on the results obtained the state feedback controllers and a class of switching

signals with the average dwell time are designed to guarantee that the closed-loop system is

finite-time stable with L1-gain performance In our further work we will extend the proposed

method to discrete-time positive switched systems with time-varying delay

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant No

61273120

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19

[21] M Tang Y W Wang C Wen Improved delay-range-dependent stability criteria for linear

systems with interval time-varying delays IET Control Theory Appl 6(6) (2012) pp

868-873

[22] X Zhao L Zhang and P Shi Stability of a class of switched positive linear time-delay

systems Int J Robust Nonlinear Control 23(5) (2013) pp 578-589

[23] X Liu C Dang Stability analysis of positive switched linear systems with delays IEEE

Trans Autom Control 56(7) (2011) pp 1684-1690

[24] E Fornasini M Valcher Stability and stabilizability of special classes of discrete-time

positive switched systems in Proc Am Control Conf San Francisco USA (2011) pp

2619-2624

[25] L Gurvits R Shorten O Mason On the stability of switched positive liner systems IEEE


[26] F Knorn O Mason R Shorten On linear co-positive Lyapunov functions for sets of linear

positive systems Automatica 45(8) (2009) pp 1943-1947

[27] X Liu Stability analysis of switched positive systems a switched linear co-positive

Lyapunov function method IEEE Trans Circuits Syst II 56(5) (2009) pp 414-418

[28] X Lin H Du and S Li Finite-time boundedness and L2-gain analysis for switched delay

systems with norm-bounded disturbance Appl Math Comp 217(12) (2011) pp 5982-

5993

[29] L Weiss and E F Infante Finite-time stability under perturbing forces and on product spaces

IEEE Trans Autom Control 12(1) (1967) pp 54-59

[30] A N Michel and S H Wu Stability of discrete systems over a finite interval of time Int J

20

Control 9(6) (1969) pp 679-693

[31] P Dorato Short time stability in linear time-varying systems in Proc IRE Int Conv Record

New York (1961) pp 83-87

[32] Z Xiang Y Sun M S Mahmoud Robust finite-time Hinfin control for a class of uncertain

switched neutral systems Commun Nonlinear Sci Numer Simulat 17(2012) pp

1766-1778

[33] W Xiang and J Xiao Hinfin finite-time control for switched nonlinear discrete-time systems

with norm-bounded disturbance J Franklin Institute 348(2) (2010) pp 331-352

[34] H Du X Lin and S Li Finite-time stability and stabilization of switched linear systems

Joint 48th IEEE Conf Dec Control and 28th Chinese Control Conf Shanghai (2009) pp

1938-1943

[35] H Liu Y Shen Hinfin finite-time control for switched linear systems with time-varying delay

Intel Control Auto 2(3) (2011) pp 203-213

[36] Y Shen H Liu Finite-time stabilization of switched time-delay system via dynamic output

feedback control Mechanical Engin Tech AISC 125 (2012) pp 523-528

[37] G Chen and Y Yang Finite-time stabilization of switched positive linear systems Int J

Robust Nonlinear Control (2012) DOI 101002rnc2870

[38] L Hetel J Daafouz C Iung Stability analysis for discrete time switched systems with

temporary uncertain switching signal in Proc 46th IEEE Conf Dec Control New Orleans

LA (2007) pp 5623-5628

[39] P Li J Lam and Z Shu Hinfin positive filtering for positive linear discrete-time systems an

augmentation approach IEEE Trans Autom Control 55(10) (2010) pp 2337-2342

21

[40] M S Mahmoud and P Shi Asynchronous Hinfin filtering of discrete-time systems Signal

Process 92(10) (2012) pp 2356-2364

1



Mei XIANG Zhengrong XIANG



Corresponding author e-mail xiangzrmailnjusteducn

Highlights

1 Some sufficient conditions for the existence of finite-time boundedness for

positive switched linear systems with time-varying delay are proposed

2 1L performance analysis for positive switched linear systems is investigated

via the average dwell time approach

3 A state memory feedback controller is designed to stabilize the time-delay

positive switched system such that the corresponding closed-loop system is

1L finite-time bounded

1






corresponding author e-mail xiangzrmailnjusteducn

Tel 0086-13951012297 Fax 0086-25-84313809

Abstract This paper is concerned with the problem of finite-time L1 control for a class of

positive switched linear systems with time-varying delay Firstly by using the average dwell time

approach sufficient conditions which can guarantee the L1 finite-time boundedness of the

underlying system are given Then in virtue of the results obtained a state feedback controller is

designed to ensure that the resulting closed-loop system is finite-time bounded with L1-gain

performance All the obtained results are formulated in terms of linear matrix inequalities (LMIs)

which can be solved conveniently Finally an example is given to illustrate the efficiency of the

proposed method

Keywords positive systems switched systems time-varying delay finite-time boundedness

L1-gain performance average dwell time

1 Introduction

2























3





study














is the set of all





The notation 1

n

k

k

x x



4


( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ( )) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t d t t t

t t t



x

(1)





0( ) 0

fT

w t dt d d (2)










( ) ( ) ( )

( ) ( )

( ) ( ) ( ( )) ( )

( ) ( ) ( )

( ) ( ) 0

t d t t

t t


z t C x t E w t

x

(3)



and 0)( tz

hold for all 0t


non-negative




5




2 1( ) 1 2 0( )t

a

T TN T T N

T







( ( ))fT t if 1)(sup0



respect to ( )fT




time




txT ( ) 1Tx t

[0 ]ft T


6





0 0( ) ( )






3 Main Results



(3)





1 2 3 4 p p pv (5)

22 3 4 1 fT

e e d e

(6)

where

prprprpp

T

prpr vva prp

T

dprpr hva )1(

m a x pp M

12 r n n

7


1 2 T


p p p pn 1 2 T

p p p pn


21 2 3 4

ln

ln lnf

fa a T

T μT T

e e e d

(7)


( )max ( )pp l M


Tp pB

and 1μ satisfies



system (3)

( )( ) ( ( )tV t V t x t

(9)



where

1( ( )) ( ) T

p pV t x t x t v

( )

2( )

( ( )) ( ) pt t s T

p pt d t

V t x t e x s ds

0 ( )

3( ( )) ( ) pt t s T

p pt

V t x t e x s dsd

and n

p p pv R p M




8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p


t d t

t t s T T Tp p p p

t d t



(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p


t

t tt s T T Tp p p p

t t d t



(12)


( ) ( ) ( )( )

( ( ))( (1 ) )

( )


T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)



Denoting 2T


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)



( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t


tt

t s T

t

V t e V t e w s ds

e V t e w s ds


e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)


( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








2























3





study














is the set of all





The notation 1

n

k

k

x x



4


( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ( )) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t d t t t

t t t



x

(1)





0( ) 0

fT

w t dt d d (2)










( ) ( ) ( )

( ) ( )

( ) ( ) ( ( )) ( )

( ) ( ) ( )

( ) ( ) 0

t d t t

t t


z t C x t E w t

x

(3)



and 0)( tz

hold for all 0t


non-negative




5




2 1( ) 1 2 0( )t

a

T TN T T N

T







( ( ))fT t if 1)(sup0



respect to ( )fT




time




txT ( ) 1Tx t

[0 ]ft T


6





0 0( ) ( )






3 Main Results



(3)





1 2 3 4 p p pv (5)

22 3 4 1 fT

e e d e

(6)

where

prprprpp

T

prpr vva prp

T

dprpr hva )1(

m a x pp M

12 r n n

7


1 2 T


p p p pn 1 2 T

p p p pn


21 2 3 4

ln

ln lnf

fa a T

T μT T

e e e d

(7)


( )max ( )pp l M


Tp pB

and 1μ satisfies



system (3)

( )( ) ( ( )tV t V t x t

(9)



where

1( ( )) ( ) T

p pV t x t x t v

( )

2( )

( ( )) ( ) pt t s T

p pt d t

V t x t e x s ds

0 ( )

3( ( )) ( ) pt t s T

p pt

V t x t e x s dsd

and n

p p pv R p M




8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p


t d t

t t s T T Tp p p p

t d t



(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p


t

t tt s T T Tp p p p

t t d t



(12)


( ) ( ) ( )( )

( ( ))( (1 ) )

( )


T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)



Denoting 2T


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)



( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t


tt

t s T

t

V t e V t e w s ds

e V t e w s ds


e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)


( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








3





study














is the set of all





The notation 1

n

k

k

x x



4


( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ( )) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t d t t t

t t t



x

(1)





0( ) 0

fT

w t dt d d (2)










( ) ( ) ( )

( ) ( )

( ) ( ) ( ( )) ( )

( ) ( ) ( )

( ) ( ) 0

t d t t

t t


z t C x t E w t

x

(3)



and 0)( tz

hold for all 0t


non-negative




5




2 1( ) 1 2 0( )t

a

T TN T T N

T







( ( ))fT t if 1)(sup0



respect to ( )fT




time




txT ( ) 1Tx t

[0 ]ft T


6





0 0( ) ( )






3 Main Results



(3)





1 2 3 4 p p pv (5)

22 3 4 1 fT

e e d e

(6)

where

prprprpp

T

prpr vva prp

T

dprpr hva )1(

m a x pp M

12 r n n

7


1 2 T


p p p pn 1 2 T

p p p pn


21 2 3 4

ln

ln lnf

fa a T

T μT T

e e e d

(7)


( )max ( )pp l M


Tp pB

and 1μ satisfies



system (3)

( )( ) ( ( )tV t V t x t

(9)



where

1( ( )) ( ) T

p pV t x t x t v

( )

2( )

( ( )) ( ) pt t s T

p pt d t

V t x t e x s ds

0 ( )

3( ( )) ( ) pt t s T

p pt

V t x t e x s dsd

and n

p p pv R p M




8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p


t d t

t t s T T Tp p p p

t d t



(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p


t

t tt s T T Tp p p p

t t d t



(12)


( ) ( ) ( )( )

( ( ))( (1 ) )

( )


T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)



Denoting 2T


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)



( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t


tt

t s T

t

V t e V t e w s ds

e V t e w s ds


e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)


( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








4


( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ( )) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t d t t t

t t t



x

(1)





0( ) 0

fT

w t dt d d (2)










( ) ( ) ( )

( ) ( )

( ) ( ) ( ( )) ( )

( ) ( ) ( )

( ) ( ) 0

t d t t

t t


z t C x t E w t

x

(3)



and 0)( tz

hold for all 0t


non-negative




5




2 1( ) 1 2 0( )t

a

T TN T T N

T







( ( ))fT t if 1)(sup0



respect to ( )fT




time




txT ( ) 1Tx t

[0 ]ft T


6





0 0( ) ( )






3 Main Results



(3)





1 2 3 4 p p pv (5)

22 3 4 1 fT

e e d e

(6)

where

prprprpp

T

prpr vva prp

T

dprpr hva )1(

m a x pp M

12 r n n

7


1 2 T


p p p pn 1 2 T

p p p pn


21 2 3 4

ln

ln lnf

fa a T

T μT T

e e e d

(7)


( )max ( )pp l M


Tp pB

and 1μ satisfies



system (3)

( )( ) ( ( )tV t V t x t

(9)



where

1( ( )) ( ) T

p pV t x t x t v

( )

2( )

( ( )) ( ) pt t s T

p pt d t

V t x t e x s ds

0 ( )

3( ( )) ( ) pt t s T

p pt

V t x t e x s dsd

and n

p p pv R p M




8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p


t d t

t t s T T Tp p p p

t d t



(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p


t

t tt s T T Tp p p p

t t d t



(12)


( ) ( ) ( )( )

( ( ))( (1 ) )

( )


T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)



Denoting 2T


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)



( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t


tt

t s T

t

V t e V t e w s ds

e V t e w s ds


e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)


( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








5




2 1( ) 1 2 0( )t

a

T TN T T N

T







( ( ))fT t if 1)(sup0



respect to ( )fT




time




txT ( ) 1Tx t

[0 ]ft T


6





0 0( ) ( )






3 Main Results



(3)





1 2 3 4 p p pv (5)

22 3 4 1 fT

e e d e

(6)

where

prprprpp

T

prpr vva prp

T

dprpr hva )1(

m a x pp M

12 r n n

7


1 2 T


p p p pn 1 2 T

p p p pn


21 2 3 4

ln

ln lnf

fa a T

T μT T

e e e d

(7)


( )max ( )pp l M


Tp pB

and 1μ satisfies



system (3)

( )( ) ( ( )tV t V t x t

(9)



where

1( ( )) ( ) T

p pV t x t x t v

( )

2( )

( ( )) ( ) pt t s T

p pt d t

V t x t e x s ds

0 ( )

3( ( )) ( ) pt t s T

p pt

V t x t e x s dsd

and n

p p pv R p M




8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p


t d t

t t s T T Tp p p p

t d t



(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p


t

t tt s T T Tp p p p

t t d t



(12)


( ) ( ) ( )( )

( ( ))( (1 ) )

( )


T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)



Denoting 2T


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)



( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t


tt

t s T

t

V t e V t e w s ds

e V t e w s ds


e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)


( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








6





0 0( ) ( )






3 Main Results



(3)





1 2 3 4 p p pv (5)

22 3 4 1 fT

e e d e

(6)

where

prprprpp

T

prpr vva prp

T

dprpr hva )1(

m a x pp M

12 r n n

7


1 2 T


p p p pn 1 2 T

p p p pn


21 2 3 4

ln

ln lnf

fa a T

T μT T

e e e d

(7)


( )max ( )pp l M


Tp pB

and 1μ satisfies



system (3)

( )( ) ( ( )tV t V t x t

(9)



where

1( ( )) ( ) T

p pV t x t x t v

( )

2( )

( ( )) ( ) pt t s T

p pt d t

V t x t e x s ds

0 ( )

3( ( )) ( ) pt t s T

p pt

V t x t e x s dsd

and n

p p pv R p M




8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p


t d t

t t s T T Tp p p p

t d t



(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p


t

t tt s T T Tp p p p

t t d t



(12)


( ) ( ) ( )( )

( ( ))( (1 ) )

( )


T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)



Denoting 2T


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)



( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t


tt

t s T

t

V t e V t e w s ds

e V t e w s ds


e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)


( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








7


1 2 T


p p p pn 1 2 T

p p p pn


21 2 3 4

ln

ln lnf

fa a T

T μT T

e e e d

(7)


( )max ( )pp l M


Tp pB

and 1μ satisfies



system (3)

( )( ) ( ( )tV t V t x t

(9)



where

1( ( )) ( ) T

p pV t x t x t v

( )

2( )

( ( )) ( ) pt t s T

p pt d t

V t x t e x s ds

0 ( )

3( ( )) ( ) pt t s T

p pt

V t x t e x s dsd

and n

p p pv R p M




8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p


t d t

t t s T T Tp p p p

t d t



(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p


t

t tt s T T Tp p p p

t t d t



(12)


( ) ( ) ( )( )

( ( ))( (1 ) )

( )


T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)



Denoting 2T


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)



( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t


tt

t s T

t

V t e V t e w s ds

e V t e w s ds


e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)


( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








8

( ) ( )

2( )

( )

( )

( ) ( ) ( ) (1 ( )) ( ( ))

( ) ( ) (1 ) ( ( ))

p p

p


t d t

t t s T T Tp p p p

t d t



(11)

0 0( )

3-

0 ( )

- ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p

p


t

t tt s T T Tp p p p

t t d t



(12)


( ) ( ) ( )( )

( ( ))( (1 ) )

( )


T Tdp p p

T Tp p

V t V t x t A v v

x t d t A v h

w t B v

(13)



Denoting 2T


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s T

t t k pt

V t e V t e w s ds

(15)



( ) ( )

1 2

11

( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

( ) 1 ( )

(0) (0) ( )0

( )

(

( ) ( ) ( )

( ) ( )

(0) ( ) ( )

( )

t N tN N

N NN

N

NN N

tt t t s T

t t k tt

tt t t s T

N tt t


tt

t s T

t

V t e V t e w s ds

e V t e w s ds


e w s

( )

)

( ) ( )

(0) ( )0

(0) ( )0

(0)

(0) ( )

(0) ( )

(0)

NN

f t

f f

f

t

t

tT N s tN t s T

s

tT TN N T

s

TN

ds

e V e w s ds

e V e w s ds

e V d

(16)


( ) 1( ) ( ) T

tV t x t (17)

9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








9

2

(0) 2 3 40 0

2

2 3 40

2

2 3 4

(0) (0) sup ( ) sup ( )

sup ( )

T T T

T

V x e x e x

e e x

e e

(18)


ln

( )2

2 3 4

1

1( )

fa

TTTx t e e e d

(19)


( ) 1Tx t








as follows




22 3 4 1 fT

e e e

(20)


21 2 3 4

ln

ln( ) lnf

f

a a T

TT T

e e e

(21)


and

10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








10

1μ satisfies (8)








switched system (3)





2 Tprb (23)

22 3 4 1 fT

e e d e

(24)

where

T


pr dpr p pra v h

m a x pp M

T



and 1 2 T

p p p pnv v v v 1 2

T

p p p pn 1 2 T

p p p pn


11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








11

21 2 3 4

ln lnmax

ln lnf

fa a T

T μT T

e e e d

(25)


satisfies (8)





( ) ( ) ( ) ( ) 0p p pV t V t z t w t (26)


( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) t k tk k

kk

tt t t s

t t kt

V t e V t e s ds

(27)


( ) ( )(0 ) ( )( )

( ) (0)0

( ) (0) ( ) t ttN t N s tt t s

tV t e V e s ds


( ) ( )( ) ( )( ) ( )

0 0( ) ( ) t t



leads to

( ) ( )(0 ) (0 )( ) ( )

0 0( ) ( ) t t



a

sN s

T and

lnaT

we have

( ) ( )

0 0( ) ( )



e

leads to

2

0 0( ) ( )



12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








12




system is given by

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ( )) ( )

( ) ( ) ( ) ( )

( ) ( ) [ 0]

t t t d t t

t t t t


z t C D K x t E w t

x

(30)








where

T


(1 ) T


T





1 2 T


p p p pn 1 2 T

p p p pn



13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








13

Proof Replacing pA


letting T T





Algorithm


and pE



Step 3 By T T




4 Numerical Example


1 1 1

4 1 2 01 02 02 01

1 3 2 02 01 02 02

1 2 35 01 01 02 02

dA A B

1 1 1 1

03 02

04 01 03 02 02 06 05 03

05 02

G C D E

14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








14

2 2 2

2 2 2 2

5 2 2 02 02 01 02

3 5 2 02 01 01 01

1 2 3 01 02 01 02

02 01

03 01 01 03 02 02 03 02

01 02

dA A B

G C D E

Choosing 10fT 1 03 2 03 01 01h 2 4 25T

10

001d 001 001 001T


1

94939

151328

160609

v

2

115015

121141

160666

v

1

68860

61238

95759

2

73265

77388

46553

1

54118

47900

46451

2

26719

25244

22634

1

06343

05065

05126

g

2

02851

02590

02357

g

1

06343

05065

05126

F

2

02851

02590

02357

F

By T T


1

00325 00259 00263

00127 00101 00103K

2

00244 00222 00202

00181 00164 00149K



0p p pC D K



[ 0) and



15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








15



0 1 2 3 4 5 6 7 8 9 100

1

2

3

Time(s)

Syste

m m

od

e


0 1 2 3 4 5 6 7 8 9 1001

015

02

025

03

035

04

045

05

Time(s)

Sta

te r

esp

on

se

x1

x2

x3


16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








16

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

12x 10

-3

Time(s)

xT(t)ε


5 Conclusions








Acknowledgment


61273120

References


17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








17








pp 217-226







Syst (2003)








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








18








(2010) pp 173-185


54(7) (2009) pp 1596-1600





1041-1050







Control 23(2) (2013) pp 190-207

19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








19



868-873







2619-2624









5993




20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








20

Control 9(6) (1969) pp 679-693


New York (1961) pp 83-87



1766-1778





1938-1943









LA (2007) pp 5623-5628



21


Process 92(10) (2012) pp 2356-2364

1







Highlights








21


Process 92(10) (2012) pp 2356-2364

1







Highlights








1







Highlights








Documents

Finite-time L1 control for positive switched linear systems with time-varying delay