H Control for Nonlinear Systems with Interval

7/31/2019 H Control for Nonlinear Systems with Interval

1/7

Hi ngh ton quc v iu khin v T ng ho - VCCA-2011

VCCA-2011

H Control for Nonlinear Systems with Interval

Non-Differentiable Time-Varying Delays

N. T. Thanh* and V. N. Phat**

*Department of Mathematics,Hanoi University of Mining and Geology, Hanoi, Vietnam

e-Mail: [email protected]

**Institute of Mathematics, VAST,

18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam

e-Mail: [email protected]

AbstractIn this paper we develop a linear matrix inequality

(LMI) approach for studying H control problem for

a class of nonlinear system with interval time-varying

delay. The time delay is assumed to be a continuousfunction belonging to a given interval. By

constructing a set of improved Lyapunov-Krasovskii

functionals, a new delay-dependent sufficient

condition for the H control with exponential

stability of the system is established in terms of LMIs.

An application to H control of uncertain linear

systems with interval time-varying delay is also

given. Numerical examples are given to show the

effectiveness of the obtained results.

SymbolsSymbol Meaning

(.)x The state function

(.)u The control function

(.)z The observation output

(.) The intial function

(.) The uncertainty input

(.)h Time varying delay

Exponential index

Optimal level index

1. IntroductionThe H control of time-delay systems is of practical

and theoretical interest since the time delay is often

encountered in many industrial and engineering

processes [1]. The expository monographs [2, 3, 4]

present a lucid account of the early development in

the H control theory. A significant development in

the H control theory has recently been the

introduction of state-space methods [5]. This has ledto a rather transparent solution to the standard

problem of H control with the objective being to

find a feedback controller stabilizing a given system

that is subject to some normed suboptimal conditions

on perturbations/uncertainties.For the H control

problem, appropriate methods for linear time-delaysystems usually make use of the Lyapunov functional

approach, whereby the H conditions are obtained

via solving either matrix inequalities or algebraic

Riccati-type equations [6, 7]. More recently, a simple

and systematic procedure for constructing time-

varying Lyapunov functionals has been studied in [8,

9], but for H filtering. In [10-13], a modification of

standard the LMI-type exponential stability conditions

for linear time-delay systems is proposed, allowing to

compute two bounds that characterize theexponential nature of the solution in the case of

nominal as well as uncertain systems. Lyapunov

function method there was developed to H control

of linear systems with interval time-varying delays,

where the assumption on the derivative of the delay

function is either strictly bounded or by one is

removed, but the time-delay function is still assumed

to be differentiable. Paper [14] first time studies H

control of linear systems with interval non-

differentiable delays, but without the delay in

observation and the condition obtained for asymptotic

stability. To the best of our knowledge, there has been

no investigation on the H control of delayed

systems, where the time delay involved in state and

output is interval time-varying and non-diiferentiable .

In fact, this problem is difficult to solve; particularly,

when the time-varying delays are interval, non-

differentiable and the output is subjected to such time-varying delay functions. The time delay is assumed to

be any continuous function belonging to a given

interval, which means that the lower and upper

bounds for the time-varying delay are available, but

the delay function is bounded but not necessary to be

differentiable. This allows the time-delay to be a fasttime-varying function and the lower bound is not

restricted to being zero. It is clear that the application

493
mailto:[email protected]:[email protected]


2/7


VCCA-2011

of any memoryless feedback controller to such time-

delay systems would lead to closed-loop systems with

interval time-varying delays. The difficulties then

arise when one attempts to derive exponential

stabilizability conditions and to extract the controller's

parameters for these systems. Indeed, existing

Lyapunov-Krasovskii functionals and their associatedresults in [8, 11, 13, 14] cannot be applied to solve the

problem posed in this paper as they would either fail

to cope with the non-diffierentiability aspects of the

delays, or lead to very complex matrix inequality

conditions and any technique such as matrix

computation or transformation of variables fails to

extract the parameters of the memoryless feedbackcontrollers. This has motivated our research.

In this paper, we develop the results of [11, 13,14] for

the problem ofH control for linear systems with

time-varying delay. Compared to the existing results,

our result has its own advantages. First, the time delay

is assumed to be any continuous function belonging to

a given interval, which means that the lower and

upper bounds for the time-varying delay are available,

but the delay function is bounded but not necessary to

be differentiable. This allows the time-delay to be a

fast time-varying function and the lower bound is notrestricted to being zero. Second, both problems of

exponential stabilization and H control will be

treated simultaneously. For the former, the controller

is required to guarantee the global exponential

stability for the closed-loop system. As to the latter, a

prescribed performance in an H sense is also

required to be achieved for all admissible

uncertainties. By constructing a set of augmented

Lyapunov Krasovskii functionals, a new delay-

dependent condition for the robust H control is

established in terms of LMIs, that can be solved

numerically in an e_cient manner by using standard

computational algorithms [15]. The approach alows

us to apply to H control of uncertain linear

systems with interval non-differentiable time-varying

delay.

The paper is organized as follows. Section 2 presents

definitions and some well-known technical

propositions needed for the proof of the main results.

H controller design for exponential stability and an

application to uncertain systems with non-differentiable time-varying delay are presented in

Section 3. The paper ends with conclusions and cited

references.

2. PreliminariesConsider a nonlinear uncertain time-varying delay

system of the form

( ) ( ) ( ( )) ( )

( ) ,

( ) ( ) ( ( )) ( ) ,

( ) ( ), [ , 0],

x t Ax t Dx t h t Bu t

C t f

z t Ex t Gx t h t Fu t g

x t t t h

(2.1)

where ( ) nx t R is the state;

2(.) ([0, ], ),mu L s R 0,s is the control;

2(.) ([0, ], )rL R is the uncertain input;

( ) sz t R is the observation output. The initial

function1( ) ([ , 0], )nt C h R with the norm

0

sup ( ) , ( ) ,h t

t t

and the time delay

function ( )h t satisfies

1 20 ( ) , 0.h h t h t

Let ( ) : ( ( ))hx t x t h t , the nonlinear function

: ,n n m r nf R R R R R R

: n n m sg R R R R R satisfy the following

growth conditions :

, , , 0,a b c d ( , , , , ),ht x x u

( , , , , ) ,h hf t x x u a x b x c u d

and

1 1 1, , 0,a b c ( , , ),h

x x u

1 1 1( , , , ) .h h

g t x x u a x b x c u

Definition 1. Given 0. The zero solution of

system (1.1), where ( ) 0, ( ) 0,u t t is

exponentially stable if there is a positive number

0N such that every solution of the system

satisfies: ( , ) , 0.tx t N e t

Definition 2. Given 0, 0. The H control

problem for system (2.1) has a solution if there existsa memoryless state feedback controller ( ) ( )u t Yx t satisfying the following two requirements:

The zero solution of the closed-loop system:( ) [ ] ( ) ( ( ))

( , , , , 0),

( ) ( ), [ , 0],

h

x t A BY x t Dx t h t

f t x x u

x t t t h

is exponentially stable;

There is a number 0 0c such that

494


3/7


VCCA-2011

2

0

2 2

0

0

( )

sup ,

( )

z t dt

c t dt

where the supermum is taken over all1([ , 0], )nC h R and the non-zero uncertainty

2( ) ([0, ], ).rt L R In this case we say that

the feedback control ( ) ( )u t Yx t exponentiallystabilizes the system.

Proposition 1. Let ,P Q be matrices of appreciate

dimentions and Q is symmetric positivedefinite.

Then1(2 , ) ( , ) ( , ), .TPx y Qy y PQ P x x x

Proposition 2. (Shur complement lemma). Givenconstants matrices , , ,X Y Z where

1 0TX Z Y Z if and only if 0.T

X Z

Z Y

Proposition 3. For any symmetric positive definite

matrix ,n nM M scalar 0a vector function

:[0, ] na R such that the integrations concernedare well defined, the following inequality holds

0 0 0

( ) ( ) ( ) ( ) .

Ta a a

Ts ds M s ds a s M s ds

3. Main resultsIn this section for simplicity of expresstion, we

assume that [ , ] 0, .T TF E G F F I Let us denote

1 1 11 1, ,P P Q P QP

1 1 1 11 1, ,R P RP U P UP

212,

da b c

1 min 1( ),P 1 2 2

2 max 1 max 1 1 max 2 max 1

2

2 1 max 1

( ) ( ) ( ) ( )

( ) ( ),

P Q h R h R

h h U

1 2

111

2 2

6 4 22 2 2

4

242 2 ,

T T

h h T

c cT PA A P BB Q

P e R e R CC I

12 2 0.5T TT DP PA BB , 1213 0.5 ,h T TT e R PA BB 2214 0.5 ,h T TT e R PA BB

15 0.5 ,T TT P PA BB 2222 122 ,h T TT e U DP PD I CC

2223 ,

h T

T e U PD

2224 ,h TT e U PD 25 ,TT P PD 1 1 22 2 2

33

12,

h h h TT e Q e R e U I CC

2 2 22 2 244

12,

h h h TT e Q e R e U I CC

2 2 2

55 1 2 2 1( )

122 .T

T h R h R h h U

P I CC

max ( ),TK K K max ( ),a TL a aL L max ( ),b TL b bL L max ( ),c TL c cL L max ( ),d TL d dL L max ( ),e TL e eL L max ( ),g TL g gL L max ( ).f TL f fL L Theorem 1.

The H

control of system (2.1) has a solution if

there exist symmetric positive definite matrices

, , ,P Q R U such that the following LMI holds

11 12 13 14 15

22 23 24 25

33

44

55

*

* * 0

* * *

* * * *

* * * * *

* * * * *

* * * * *

* * * * *

T T T T T

T T T T

T P

T P

T

495


4/7


VCCA-2011

1

1

0 0

0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0.3

* 0 03

* * 06 4

* * *6 4

T

T

PE P

PG P

I

I

I

a a

I

b b

(3.1)

Moreover, stabilizing feedback control is1( ) 0.5 ( ),Tu t B P x t

and the solution of the systems satisfies

1

2

( ) , 0.tx t e t

Proof.

Consider the following Lyapunov-Krasovskii

functional for the closed loop system:6

1

( , ) ( , ),t i ti

V t x V t x

where

1 1( ) ( ),T

V x t P x t

1

2 ( )

2 1( ) ( ) ,

t

s t T

t h

V e x s Q x s ds

2

2 ( )

3 1( ) ( ) ,

t

s t T

t h

V e x s Q x s ds

1

0

2 ( )

4 1 1( ) ( ) ,

t

t T

h t s

V h e x R x d ds

2

02 ( )

5 2 1( ) ( ) ,t

t T

h t s

V h e x R x d ds

1

2

2 ( )

6 2 1 1( ) ( ) ( ) .

h t

t T

h t s

V h h e x U x d ds

It is easy to verify that2

1 ( ) ( , ), 0,tx t V t x t

and2

0 2

(0, ) .V x (3.2)

Taking the derivative of iV , 1, 2,3,i in t alongthe solution of the system (2.1), we obtain

1 1 1 1 1

1 1

2 ( ) ( ) 2 ( ) ( ) ( ) ( )

2 ( ) ( ) 2 ( ) .

T T h T T

T T

V x t P Ax t x t P ADx t x t PB P x t

x t PC t x t P f

12

2 1 1 1 1 2( ) 2 ( ) ( ) ( ) 2 .hT TV x t Q x t e x t h Q x t h V

22

3 1 2 1 2 3( ) ( ) ( ) ( ) 2 .hT TV x t Q x t e x t h Q x t h V

Applying Proposition 3 and the Newton-Leibniz

formula

( ) ( ) ( ), 1, 2,

i

t

i

t h

x s ds x t x t h i

we have

1

2

4 1 1 4

2

1 1 1

( ) ( ) 2

[ ( ) ( )] [ ( ) ( )].

T

h T

V h x t R x t V

e x t x t h R x t x t h

2

2

5 2 1 5

22 1 2

( ) ( ) 2

[ ( ) ( )] [ ( ) ( )].

T

h T

V h x t R x t V

e x t x t h R x t x t h

2

2

2

6 2 1 1 6

2

2 1 2

2

1 1 1

( ) ( ) ( ) 2

[ ( ) ( )] [ ( ) ( )]

[ ( ) ( )] [ ( ) ( )].

T

h h T h

h h T h

V h h x t U x t V

e x t h x t R x t h x t

e x t h x t R x t h x t

From the following identity relation

1 1 1 1

2 1 1

2 ( ) 2 ( ) 2 ( )

2 ( ) 2 ( )

( ) ( ) ( )0

( ) ( )

T h T T

T T

h

x t P x t P x t h P

x t h P x t P

x t Ax t Dx t

Bu t C t f

and the following inequalities

22 2 2 2

1 12 ,12

T hx P f a x b x c u P x

22 2 2 2

1 12 ,

12

T hx P f a x b x c u P x

2 22 2 2

1 12 ,12

hT h hx P f a x b x c u P x

22 2 2 2

1 1 1 12 ( ) ( ) ,

12

T hx t h P f a x b x c u P x t h

22 2 2 22 1 1 22 ( ) ( ) ,

12T hx t h P f a x b x c u P x t h

22 2 2 2

1 12 ( ) ( ) ,

12

T hx t P f a x b x c u P x t

2

1 1 1

122 ( ) ( ) ( ),

12

T T Tx t PC x t PCC P x t

2

1 1 1

122 ( ) ( ) ( ),

12

h T h T T hx t PC x t PCC P x t

2

1 1 1 1 1 1

122 ( ) ( ) ( ),

12

T T Tx t h PC x t h PCC Px t h

496


5/7


VCCA-2011

2

2 1 2 1 1 2

122 ( ) ( ) ( ),

12

T T Tx t h PC x t h PCC P x t h

2

1 1 1

122 ( ) ( ) ( ),

12

T T Tx t PC x t PCC P x t

setting 1( ) ( ),y t P x t we have2

2 1

1

2

1

2 ( ) ( )

4 2( ) 3 4 ( )

4

( ) 3 4 ( ),

T

T T T

h T T h

V V t M t

cy t PE EP a P BB y t

y t PG GP b P y t

(3.3)

where

1 2( ) ( ) ( ) ( ) ( ) ( ) ,T T h T T T T t y t y t y t h y t h y t

11 12 13 14 15

22 23 24 25

33

44

55

** * 0 ,

* * *

* * * *

M M M M M

M M M MM M P

M P

M

1 2

111

2 2

2

1

6 4 22 2 2

4

242 2

[6 4 ] 3 ,

T T

h h T

T

c cM PA A P BB Q

P e R e R CC I

a a P PE EP

12

2 0.5T TM DP PA BB ,

1213 0.5 ,h T TM e R PA BB 2214 0.5 ,h T TM e R PA BB 15 0.5 ,T TM P PA BB

22

22 2

123 ,

h T

T T

M e U DP PD I

PG GP CC

22

23 ,h T

M e U PD

2224 ,h TM e U PD 25 ,TM P PD 1 1 22 2 2

33

12,h h h T M e Q e R e U I CC

2 2 22 2 244

12,

h h h TM e Q e R e U I CC

2 2 2

55 1 2 2 1( )

122 .

T

M h R h R h h U

P I CC

Using the Schur complement lemma, the condition

(3.1) is equivalent to the condition 0M and fromthe inequality ( 3.3), it follows that

2 1

1

22

1

4 22 ( ) 3 4 ( )

4

( ) 3 4 ( ) .

T T T

h T T h

cV V y t PE EP a P BB y t

y t PG GP b P y t

(3.4)

Letting ( ) 0t , we finally obtain from (3.4) that

( , ) 2 ( , ) 0, 0.t tV t x V t x t (3.5)Differentiating inequality (2.5) from t to 0, we have

2

0( , ) (0, ) , 0.t

tV t x V x e t

Taking the condition (3.2) into account, we finally

obtain that

1

2

( ) , 0,tx t e t

which implies that the zero solution of the closed-loop

system is exponentially stable. To complete theproof of the theorem, it remain to show the optimal

level condition. For this, we consider the following

relaition

2 2 2 2

0 0 0

2 2

0

0

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

( ) ( ) ( ) (0, ).

s s s

s

z t t dt z t t V t dt V t dt

z t t V t dt V x

(3.6)

Conbining the condition (3.2) and the inequality

( , ) ( ) ( ),TtV t x y t Py t

we obtain from (2.4) that

2 1

1

22

1

4 2( ) 3 4 ( )

4

( ) 3 4 ( ) 2 ( ) ( ).

T T T

h T T h T

cV y t PE EP a P BB y t

y t PG GP b P y t y t Py t

(3.7)

Observe that the value of2

( )z t

2 2 1

1

2

1

4 2( ) ( ) 3 4 ( )

4

( ) 3 4 ( ).

T T T

h T T h

cz t y t PE EP a P BB y t

y t PG GP b P y t

(3.8)

Submitting the estimation of V and2

( )z t defined

by (3.7) and (3.8), respectively into (3.6), we obtain

2 2

0

0 0

2

0 2

( ) ( ) 2 ( ) ( ) (0, )

(0, ) .

s s

Tz t t dt y t Py t dt V x

V x

Letting s , and setting 20 0,c

we have

497


6/7


VCCA-2011

2

0

2 2

0

0

( )

sup ,

( )

z t dt

c t dt

for all non-zero 2( ) ([0, ], ),rt L R

1([ , 0], )nC h R . This completes the proof ofthe theorem.

In the sequel, we give an application to H control

of uncertain systems with interval time-varying

delay. Consider the following uncertain linear systemswith time-varying delay:

( ) [ ] ( ) [ ] ( ( )) [ ] ( )

[ ] ( ),

( ) [ ] ( ) [ ] ( ( )) [ ] ( ),

( ) ( ), [ , 0],

x t A A x t D D x t h t B B u t

C C t

z t E E x t G G x t h t F F u t

x t t t h

(3.9)

where the time varying uncertainties ,A ,D

,B ,C ,E ,G F satisfy[ ] ( )[ ],a d b c e g gA D B C E G F KH t L L L L L L L

where , , , , , , ,a d b c e g g

K L L L L L L L are known real

constant matrices of appreciate dimentions and

( )H t is an unknown matrix function with Lebesgue

measurable elements satifying( ) ( ) , 0.TH t H t I t (3.10)

To apply Theorem 1, let us denote

( ) ( ) ( ) ( ),hf Ax t Dx t Bu t C t

( ) ( ) ( ),hg Ex t Gx t Fu t

max ( ),T

K K K max ( ),aT

L a aL L

max ( ),bT

L b bL L max ( ),cT

L c cL L

max ( ),dT

L d dL L max ( ),eT

L e eL L

max ( ),gT

L g gL L max ( ).f

T

L f fL L

Observe that22 2 2

22 2

( ) 3 ( ) 3 ( ) 3 ( )

3 ( ) 3 ( ) 3 ( ) ,e g f

h

h

K L K L K L

g t Ex t Gx t Fu t

x t x t u t

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( ) .

a d

b e

h

h

K L K L

K L K L

f t Ax t Dx t Bu t C t

x t x t

u t t

Applying Theorem 1 with

1 3 ,eK La 1 3 ,gK Lb 1 3 ,fK Lc

,aK L

a ,dK L

b

,bK L

c ,eK L

d

we have

Corrolary 1.

The H control of system (3.9) has a solution if there

exist symmatrix positive definite mtrices

, , ,P Q R U such that the following LMI holds

11 12 13 14 15

22 23 24 25

33

44

55

*

* * 0

* * *

* * * *

* * * * *

* * * * *

* * * * *

* * * * *

T T T T T

T T T T

T P

T P

T

1

1

0 0

0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0.3

* 0 03

* * 06 4

* * *6 4

T

T

PE P

PG P

I

I

I

a a

I

b b

Moreover, stabilizing feedback control is

1( ) 0.5 ( ),T

u t B P x t

and the solution of the systems satisfies

1

2

( ) , 0.tx t e t

In the sequel, we illustrate the effectiness of the

proposed method which yields a computationally

solution to the H control in the contex of LMIs.

Example 1.

Consider the following nonlinear system with interval

time varying delay (2.1), where

498


7/7


VCCA-2011

1

3( ) 0.1 0.1sin ( ) [2 , 2 1 ],

( ) 0.1 ,

k N

h t t if t H k k

h t if t H

1.3 0.2,

0.1 1.4

A

0.3 0.2

,

0.1 0.5

D

0.01,

0.01B

0.03 0.01,

0.01 0.01C

0.006 0.006,

0.008 0.008E G

0.8,

0.6F

and

1 1 1 0.01.a b c d a b c It is worth

nothing that, the delay function ( )h t is

nondifferenible and the result obtained in [8, 11,

13,14] are not be applicable to this system. By usingLMI Toolbox in MATLAB [15], the LMI (3.1) is

feasible with

1 0.1,h 2 0.2,h 0.1, 0.4, and

1.5623 0.9473,

0.9473 2.3635P

1.0837 0.8515,

0.8515 1.6742Q

313.2418 0,0 313.2418R

313.2418 0.

0 313.2418U

Moreover, the solution ( , )x t of the system satisfies

( , ) 5.5147 .x t

4. ConclusionIn this paper, the problem of the exponential stability

and H control for nonlinear systems with time

varying delays and polytopic uncertainties has beenstudied. By introducing improved parameterdependent Lyapunov-Krasovskii functionals, new

delay-dependent conditions for theH and

exponential stability bave been established in terms of

linear matrix inequalities. A numerical example is

given showing the effeciveness of proposed results.

Acknowledgments.

This work was suppoeted by the National Foundation

for Sience and Technology Development, Vietnam.

References

[1] Francis BA., A Course in H Control Theory.Spinger, Berlin, 1987.

[2] Niculescu S., Gu K.,Advances in Time DelaySystems. Spinger, Berlin, 2004.

[3] Michiels W., Niculescu S., Stability andStabilization of Time Delay Systems.

Philadelphia, SIAM, 2007.[4] Petersen IR., Ugrinovskii VA., Savkin AV.,

Robust Control Design Using HMethods.

Spinger, London, 2000.[5] Ravi R., Nagpal KM., Khargonekar PP.,

H control of linear time varying systems: A

state space approach, SIAMJ. on Control and

Optimazation, 1991;29: 1394-1413.

[6] Ichikawa A., Product of nonnegative operatorsand infinite dimentional H Ricati equations.

Systems and Control Letters, 2000; 41:183-188.

[7] Phat VN., Nonlinear H optimal control inHilbert spaces via Ricati operator equations.Nonl. Funct. Anal. Appl., 2004; 9: 79-92.

[8] Fridman E., Shaked U., Delay dependentstability and H control: constant and time

varying delays, Int. J. Control, 2003; 76: 48-60.

[9] Kwon OM.,, Park JH, Robust H filtering foruncertain time delay systems: matrix inequality

approach. J. Optim. Theory Appl., 2006; 129:

309-324.

[10] Niculescu S, H memoryless control withstability constraint for time delay systems: an

LMI approach, IEEE Trans. Aut. Contr., 1998;

43: 739-743.

[11] Xu S., Lam J., Xie L., New results on delaydependent robust H control for systems with

time varyting delays, Autmatica, 2006; 42: 43-348.

[12] Phat VN., Memoryless H controller design forswitched nonlinear systems with mixed time

varying delays, Int. J. of Control, 209; 82: 1889-

1898.

[13] Phat VN., Ha QP., H control and exponentialstability for a class of nonlinear non

autonomuous systems with time varying delay,

J. Optim. Theory Appl., 2009; 142: 603-618.

[14] Jiang X., Han QL., OnH control for linearsystems with interval time varying delay,

Automatica, 205; 41: 2099-2106.

[15] Gahinet P., Nemirovskii A., Laub AJ., ChilaliM., LMI Control Tollbox for use with

MATLAB. The MathWorks, Inc., 1995.

499

H Control for Nonlinear Systems with Interval

Documents

Transcript of H Control for Nonlinear Systems with Interval