978-1-4244-8092-0/10/$26.00 ©2010 IEEE1006 18th Mediterranean Conference on Control & AutomationCongress Palace Hotel, Marrakech, MoroccoJune 23-25, 2010

(1)

1Abstract— This paper considers H∞ control problem for input-delayed systems for time-varying delays. A proportional- derivative state feedback control law is used in this paper. By this means, the resulting closed-loop system turns into a specific time-delay system of neutral type. The significant specification of this neutral system is that its delayed term coefficients depend on the controller parameters. This condition provides new challenging issues in theoretical research as well as providing new applications. In the present paper, new delay-dependent sufficient condition is derived for the existence of H∞ controller in terms of matrix inequalities, in presence of varying time- delays. The resulting H∞ controller guarantees asymptotic stability of the closed-loop system as well as a guaranteed limited system induced norm smaller than a prescribed level.

Numerical examples are presented to illustrate the effectiveness of the proposed method.

I. INTRODUCTION

ime-delay phenomena appear in many systems and processes, such as chemical and thermal processes [1], population dynamic model [2], rolling mill [3] and systems with long transmission line [4]. In many systems, time-delay is a source of instability. Hence, many researchers have paid great attention to the control of time-delay systems of retarded or neutral type.

Referring to H∞ control of neutral systems, robust H∞ state feedback control of uncertain neutral system has been considered in [5]. An optimization problem has been formulated with linear matrix inequality constraints to obtain an H∞ state feedback controller. Observer-based H∞ state feedback control for a class of uncertain neutral systems is another topic which Lien has considered in [6]. H∞ output feedback control of neutral systems has also been the centre of attention in some research such as [7] and [8]. Moreover, Xu et al. have used bounded real lemma to design an H∞ state feedback and positive real control for a linear neutral delay system [9].

A general representation of a neutral system is shown as

( ) ( ) ( )

( )

^{( )}

1 ⁱ 1 ^j

n k

h i d j

i j

x t Ax t A x t h A x t d Bu t

= =

= +∑ − +∑ − +

& & (1)

To the best knowledge of the authors, in all developed theories for neutral systems, no synthesis has been derived when both x(t−h_i) and x&(t−h_i)s’ coefficients are dependent on the controller’s parameter, whereas this condition has its own merits in practical application as well as leading to a new challenging theoretical problem. The importance of the above condition is observed in the control systems such as active vibration suppression. Du and Zhang proposed an H∞ state-feedback controller for an input-delay active suspension system [10]. Since the ride comfort is an

1Authors are with Advanced Robotics and Automated Systems (ARAS), Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, P.O. Box 16315-1355, Tehran, Iran (email:

[email protected]).

important objective which is related to the body acceleration sensed by the passenger, an acceleration feedback can effectively improve this performance objective in active suspension system or generally, in active vibration suppression systems. Some researchers have paid considerable attention to this idea such as [11] and [12].

Abdelaziz and válašek [12] proposed a formula similar to Ackermann for solving the pole-placement problem for non- delay linear single-input/single-output systems and multi- input/multi-output systems using state-derivative feedback.

Assoncao et.al. [11] used this idea to design a stabilizing state-derivative controller for a delay-free system which bounds the output peak as well as the state-derivative feedback. Moreover, an analysis for the stability of a system controlled by proportional-derivative state feedback in presence of small uncertain delays in the feedback loop was presented in [17]. To the best of our knowledge, no synthesis of state-derivative feedback has been presented for input- delay systems in the literature, whereas, as described earlier, it could be of great significance in practice. To benefit the advantages of the state feedback as well as acceleration feedback or generally, state derivative feedback, we employ proportional-derivative state feedback control law as it is shown by the following equation:

x K x K

u= 1 + 2& (2)

Assume the general representation of linear input-delayed systems as follows:

( ) ( ) ( ) ( )

1 n

i i

i

x t Ax t B u t h Ew t

=

= +∑ − +

& (3)

Applying the control law (2) to the input time-delay system (3) leads to a time-delay closed system of neutral type which is represented as follows:

( ) ( ) 1 ( ) 2 ( ) ( )

1 1

n n

i i i i

i i

x t Ax t B K x t h B K x t h Ew t

= =

= +∑ − +∑ − +

& & (4)

As it is seen in the Equation (4), both x(t-hi) and x&(t−h_i) s’ coefficients are functions of the control law parameters.

Therefore, finding K1 and K2 introduces a new challenging problem theoretically, whereas the choice of K1 and K2 can be very effective in obtaining desired performance in such applications. The main purpose of this paper is to elaborate this problem in detail and to design H∞-based controller for the closed-loop system in presence of varying time-delay.

This paper is organized as follows. Problem formulation is introduced in Section 2, and in Section 3, an H∞ controller is designed for the time-varying delay case. This is accomplished in terms of some matrix inequalities for the closed-loop time-delay system of neutral type. Illustrative examples are provided in section 4 to show the effectiveness of the proposed method in some case studies, and real application. Finally, the concluding remarks are given in section 5.

Delay-Dependent H

_∞

Control of Linear Systems with Time-Varying Delays Using Proportional-Derivative State Feedback

A. Shariati, H. D Taghirad and B. Labibi

T

18th Mediterranean Conference on Control & Automation Congress Palace Hotel, Marrakech, Morocco

June 23-25, 2010

(2)

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

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

( ) ( )

2

1 2 2 2

2

1 2 1

2 2

1 2 1 2 1

2 2

3 4

0 0 0 0 0 0

* 1 0 0 0 0 0 0 0

* * 0 0 0 0

* * * 0 0 0 0 0 0 0 0 0

* * * * 0 0 0 0 0 0 0 0 0 0

* * * * * 0 0 0 0 0 0 0 0 0

* * * *

T T T T T T T T

T T T T T T

d

T T T T T T T T

T T

BV N N BW N E LC D LA LA LA A LA A LC

T DV D BV BV ABV ABV DV

N DW D BW BW ABW ABW BV BV DW

BW BW

τ τ τ

τ τ τ τ

τ

Ω − − − +

− −

−Θ −

−Θ

( )

2 2

1 1

2 2

2 2 1

2 1

2

* * 0 0 0 0

* * * * * * * 0 0 0 0 0

* * * * * * * * 0 0 0 0 0 0

* * * * * * * * * 0 0 0 0 0

* * * * * * * * * * 0 0 0 0

* * * * * * * * * * * 0 0 0

* * * * * * * * * * * * 0 0

* * * * * * * * * * * * * 0

* * * * * * * * * * * * * *

T T T T T T T

T T

D D I E E E A E A

I E E

F H

F

H

F H

I

γ τ τ

γ τ

τ

τ αβ β

−

⎡⎢

⎢⎢

⎢ −

⎢⎢ −

⎢ −

⎢⎢ −

⎢ −

−

⎣

0

⎤⎥

⎥⎥

<

⎥⎥

⎢ ⎥⎥

⎢ ⎥

⎢ ⎥⎦

(7) II. PROBLEM FORMULATION

In this paper, we consider the following time-delay system with input delay:

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

¹

( ( ) )

²

( )

x t Ax t Bu t t Ew t z t Cx t Du t t D w t

τ τ

= + − +

&

(5)

where x is the state, w∈ℜ^p is the disturbance input of system that belongs to L2[0,∞), τ(t) is the time-varying delay of the system and is assumed to satisfy 0<τ( )t ≤τ and

( )t _d 1/ 4

τ& ^{< <}τ , u∈ℜ^m is the system input and z∈ℜ^q is the

controlled system output. The matrices A∈ℜ^n¥n, B∈ℜ^n¥m, E∈ℜ^n¥p, C∈ℜ^q¥n, D1∈ℜ^q¥m, D2∈ℜ^q¥p are assumed to be known. In this paper we assume that all the state variables are measured. Considering the control law (2), the state space equations of the closed-loop system is given by

( ) ( )

(

( )

) (

^{( )}

)

^{( )}

( ) ( )

(

( )

) (

( )

)

( )

[ ]

1 2

1 1 1 2 2

(0 ) ( ) ,0

x t Ax t BK x t t BK x t t Ew t z t Cx t D K x t t D K x t t D w t x t

τ τ

θ φ θ θ τ

= + − + − +

+ = ∀ ∈ −

& &

& (6)

Therefore, the resulting closed-loop system (6) is a time- delay system of neutral type which both coefficients of

( )

x t−τ and (x t& −τ)depending on the controller parameters.

Here, we state the following lemma which will be used further in the main result of the paper.

Lemma 1 [14]: For a prescribed matrix, M ^A B

=⎡ ⎤⎢ ⎥

⎣ ⎦ or

[ ]

M= A B we have the following inequality:

( ) ( )

{ }

^{( )}

{

^{( ) ( )}

}

maxσ A,σ B ≤σ M ≤ 2 max σ A,σ B

III. H∞CONTROL DESIGN WITH TIME-VARYING DELAY

In many developed theories, conventional state feedback controller has been used for obtaining stability as well as performance objectives of the closed-loop system. In spite of the effectiveness of state feedback controller in many applications, it is not suitable for the cases that we need to have an acceleration feedback or generally derivative of the state in the feedback. On the other hand, H∞ control is an effective method which guarantees asymptotic stability as well as performance objectives. This is why H control for

time-delay systems has been among the most challenging topics in recent years. All the aforementioned facts motivate us to elaborate on the following Theorem and one Lemma which are stated in this section.

Theorem 1: Given scalars ,τ τ_d>0, the closed-loop system (6) is asymptotically stable and ║Tzw║∞<γ, if there exist positive definite symmetric matrices L, T, H1, H2, F1, F2

∈ℜ^n¥n, negative definite symmetric matrix N2 and matrices M1, M2, N1, ∈ℜ^n¥n, V, W ∈ℜ^m¥n satisfying matrix inequalities (7) ~ (9). Moreover, H∞ proportional-derivative state feedback control law is given by u=VL x WL x⁻¹ + ⁻¹&.

1 1

1

1 1

/ 2 0

T

M N

N LF L⁻

⎡ ⎤

⎢ ⎥>

⎢ ⎥

⎣ ⎦ (8)

( )

2 2

1 1

2 2

2 0

T 2

d

M N

N LF L

τ

τ τ τ ⁻ ⁻

⎡ ⎤

⎢ ⎥>

+

⎢ ⎥

⎣ ⎦ (9)

and,

( )^T ₀

L LT BV

BV I

⎡ ⎤

>

⎢ ⎥

⎣ ⎦ (10)

in which Θ1=(1−τ_d)LH L1⁻¹ , Θ2=(1−τ_d)(2+τ_d)⁻¹LH L2⁻¹ ,

( ) ( ) ( )( )¹

( )

²

3 1 2τ_d F1/ 2 ,τ 4 1 4τ_d 2 τ_d F2/ 2τ ,β 1 τ_d

Θ = − Θ = − + ⁻ = +

( ) ( )

1 1 1 2 2 2

T T

LA AL N N τ M M τ N T

Ω = + + + + + + − +

First, let us prove following useful lemma which will be applied in the proof of Theorem 1.

Lemma 2: Consider the neutral system (6) and assume d(t)=[^{w t}^T

( )

^{w t}^&^T

( )

^]^T^{. If ║T}^zd^║^∞^<γ^, then the inequality

║Tzw║∞<γ is satisfied.

Proof: Since z(s)=Tzw(s)w(s) and

( ) ( ) ( )

^,

d s w s sw s

⎡ ⎤

⎢ ⎥

=⎢ ⎥

⎢ ⎥

⎣ ⎦

then we have ( ) zw( ) ¹ zw( ) ( )

z s mT s mT s d s s

⎡ − ⎤

⎢ ⎥

=⎢⎣ ⎥⎦ .

where m is a real scalar value. Therefore Tzd(s) can be written as

( ) ( ) ¹ ( )

zd zw zw

T s mT s mT s

s

⎡ − ⎤

⎢ ⎥

=⎢⎣ ⎥⎦

By the Lemma 1 the following inequality holds as

(3)

1 ( )

max _zw , m _zw _zd

mT T T s

∞ s ∞

∞

⎧ ⎫

⎪ − ⎪

⎪ ⎪≤

⎨ ⎬

⎪ ⎪

⎩ ⎭

Setting m=1, it can be easily concluded that if

zd( )

T s _∞<γ , then T_zw _∞^<γ. This completes the proof. ■ Corollary 1: Consider the neutral system (6) and two following performance indices:

( )

(

²

)

^{( )} ²

1 0 ^T ^T , 2 0 ( ^T ^T )

J w ⁼

∫

^∞ z z⁻γ w w dτ J w ⁼

∫

^∞ z z⁻γ d d dτ Where d(t)=[^{w t}^T

( )

^{w t}^&^T

( )

^]^T. Since the inequalities J1<0 and J2<0 corresponds to H∞ constraints Tzw _∞^<γ and

Tzd _∞^<γ respectively, then for the inequality J1<0 to be satisfied, it suffices to show that the condition J2<0 is satisfied.

Proof of Theorem 1: In this case a Lyapunov-Krasovskii functional candidate for the system (6) has the form

1 2 3 4

V= +V V + +V V Where

( ) ( )

1

V=x t Px tT (11)

( ) ( )

( ) 0

2 2 ^t ^T 1 ,

t t

V x Z x d d

τ β& α &α α β

− +

=

∫ ∫

⁽¹²⁾

( ) ( )

( ) ( ) ( )

3 ( ) 1

t t

T T

t t t t

V x Qx d x R x d

τ α α α τ & α &α α

− −

=

∫

+

∫

⁽¹³⁾

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) 0 4

0 0

1 / 2

1 / 2 ,

t T

t t t

t T

t t t

t T

t t

V x Z x d d d

x Z x d d d

x R x d

β

τ τ η

τ

α α α η β α α α η β

α α α

&& &&

− − +

−

= ′ + ′ + ′

∫ ∫ ∫

∫

(14)

where P=P^T>0, Q=Q^T>0, R1=R1T>0, R′=R′^T>0, Z1=Z1T>0 and Z′=Z′^T>0. Differentiating V1 with respect to t gives us

( ) ( ) ( ) ( )

(

( )

)

(

( )

)

( )

1 1

2

2 2 {

}

T T

V x t Px t x t P Ax t BK x t t BK x t t Ew t

τ τ

= = + −

+ − +

& &

&

It is possible to write

(

( )

)

( ) ^t_{( )} ( )

t t

x t t x t x d

τ ₋τ α α

− = −

∫

^& ⁽¹⁵⁾

We introduce the following relation for the delayed derivative of the state:

( ) ( ) ( )

(

( )

)

( )

( ) ( )

1 0 t

t t t

x t t x t x t t ^β x d d

τ τ

τ τ ⁻ ⎡ τ ₋ ₋⁺ α α β⎤

− = ⎢ − − − ⎥

⎢ ⎥

⎣

∫ ∫

⎦

& && (16)

Therefore,

( )

(

( )

)

( )

( ) ( ) ( ) ( )

( ) 1

1 1 2

1

2 1

1 0 2

2 ( )

2 2

T

T T t

t t

T t T

t t t

V x P A BK BK x t

x PBK x t t x PBK x d

x PBK x d d x t PEw t

τ β

τ τ

τ

τ τ α α

τ α α β

−

− +

− −

= + +

− − −

− +

∫

∫ ∫

&

&&

Applying an extension of the proposed inequality in [13], the following upper bound for V&1 is obtained:

( )

^{( )}

(

^{( )}

)

( ) ( ) ( )

( ) ( )

( )

( ) ( )

1 1

1 1 2 2

1 1 1

1 1

0 2

1

2

2 2

2

T T

t T T

t t T

t T t t t T

V x P A BK BK x t x PBK x t t

X Y PBK

x t x t

x t PEw t d

x Y PBK Z x

X Y PBK

x t x t

x Y PBK Z x d d

τ

β

τ τ

τ τ τ

α α α

τ α β

α τ α

− −

−

− +

− −

≤ + + − − +

⎡ − ⎤

⎡ ⎤ ⎢ ⎥⎡ ⎤

⎢ ⎥ ⎢ ⎥

+ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ − ⎥⎥⎦⎢⎢⎣ ⎥⎥⎦

⎡ ′ ′− ⎤

⎡ ⎤ ⎢ ⎥⎡ ⎤

⎢ ⎥ ⎢ ⎥

+ ⎢⎢⎣ ⎥⎥⎦ ⎢⎢⎣ ′− ′ ⎥⎥⎦⎢⎢⎣ ⎥⎥⎦

∫

∫ ∫

&

& &

&& &&

( )

^{( )(} ⁾

(

^{( )}

)

( )( )

(

( )

)

( )

(

( )

)

( ) ( ) _{( )} ( ) ( )

( ) ( )

1 1 1

1

1 1

1 2

1 0

{ / 2

} 2

2 2

2

T T T

T T

T t T

t t

t T

t t t

x A P PA X X Y Y

Y Y x x t PBK Y x t t x t PBK Y x t t x t Y x t t

x t PEw t x Z x d

x Z x d d

τ β

τ τ

τ

τ τ

τ τ τ

α α α

α α α β

−

− +

− −

= + + + ′ + +

′ ′

+ + + − −

′ ′

+ − − − −

+ +

+ ′

∫

∫ ∫

&

& &

&& &&

(17)

where

1 1

T 0

X Y

Y Z

⎡ ⎤

⎢ ⎥>

⎢ ⎥

⎣ ⎦ (18)

and,

T 0

X Y

Y τZ

′ ′

⎡ ⎤

⎢ ′ ′⎥>

⎣ ⎦ (19)

Also, the time derivative of V2 can be represented as follows:

( ) ( ) ( ) _{( )} ( ) ( )

( ) ( )

( )

2 1 1

1

2 2

2

T t T

t t

t T

t t

V t x t Z x t x Z x d

x Z x d

τ τ

τ α α α

& & & & &

& & &

−

= −

+

∫

( ) ( ) ( ) _{( )} ( ) ( )

( ) _{( )} ( ) ( )

1 1

1

2 1 2

T t T

t t

t T

t t

t x t Z x t x Z x d

x Z x d

τ

τ α α α

−

= −

− −

∫

& & & &

& & &

Using Lemma 1 in [18], we have

( ) ( ) ( ) ( )

( )

(

( ) ( ) ( ) ( )

)

( ) ( )

( ) (

( ) ^{( )}

)

2 1 2

1 1 2

1

. 2

1 2

T

t T

t

t T t

t t t t

V Ax t BK x t BK x t Ew t Z Ax t BK x t BK x t Ew t

x Z x d

x d Z x d

τ

τ τ

τ τ τ

α α α

τ α α α α

τ

−

− −

≤ + − + − +

+ − + − +

−

⎛ − ⎞⎟

−⎜⎜⎜⎝ ⎟⎟⎟⎠

∫

∫ ∫

& &

&

& &

& & &

(20)

It can be shown that the time derivative of V3 and V4 are

( ) ( ) ( ) ( ) ( )

3

1 1

T T

V x t Qx t x t Qx t

x t R x t x t R x t

τ τ τ

= − − − −

+ − − − −

& &

& & & & & (21)

( )

^{( )}

(

^{( )}

) (

^{( )}

)

( ) ( )

( ) ( ) ( )

( ) ( )

( ) _{( )} _{( )} ( ) ( )

( ) ( ) ( )

( )

2 4

0

2

/ 2 1

2

1/ 2 3 / 2

/ 2

/ 2 1

T T

t T

t t t

t T

t t t

t T

t t t

t T

t t

T

V x t Z R x t x t t R x t t

x Z x d d

x t Z R x t

β

τ τ

τ β

τ τ τ τ

α α α β

τ α α α β

τ

&

& && && && &&

&& &&

& && &&

&& &&

+

− −

− +

⎛ − ⎟⎞

′ ′ ⎜ ′

= + −⎜⎜⎜⎝ ⎟⎟⎟⎠ − −

− ′

− ′ + ′ + ′

′ ′ −

≤ + −

∫ ∫

(

( )

) (

^{( )}

)

2

xT t t R x t t

τ^& && τ && τ

⎛ ⎞⎟

⎜ ⎟ − ′ −

⎜ ⎟⎟

⎜⎜⎝ ⎠

(4)

( ) ( )

( )

(

( )

) (

^{( )}

)

( ) ( )

( ) 0

0

2

0

0 2

1 4 2

/ 2 1

2

1 4 2

t T

t t t

t T

t t t

T T

t T

t t t

t T

t t t

x Z x d d

x t Z R x t x t t R x t t

x Z x d d

x d d

β

τ τ

β

τ τ

α α α β

τ α α α β

τ τ τ τ

α α α β

τ α α β

τ

&& &&

& && &&

&

&& && && &&

&& &&

& &&

+

− −

+

− −

− ′

⎛ − ⎞⎟

⎜ ′

−⎜⎜⎜⎝ ⎟⎟⎟⎠

⎛ − ⎟⎞

′ ′ ⎜ ′

≤ + −⎜⎜⎜⎝ ⎟⎟⎟⎠ − −

− ′

⎛ − ⎞⎛⎟

−⎜⎜⎜⎜⎝ ⎟⎟⎟⎠⎝

∫ ∫

Z ⁰_{( )}t t^t _{( )}t x^{( )}d d

τ τ ^&&α α β

− −

⎞ ⎛ ⎞

⎟ ⎟

⎜ ⎟ ′⎜ ⎟

⎜ ⎟⎟ ⎜ ⎟⎟

⎜ ⎠ ⎜⎝

∫ ∫

⎠

(22) Therefore we have

( ) ⁴

1 i i

V t V

=

∑

& & (23)

We consider (17), (20) ~ (23) and defineτ⁻¹Y^′⁼Y2. By the assumption of Y′=Y′^T<0and then adding and subtracting the terms x t^T( )( ) ( )τY x t₂ and ( )( ) (2 )

T T

x t& ⁻τ τY x t& ⁻τ in

(23), an upper bound for V& is obtained as

( ) ( )

( )

^{( )(} ⁾

(

^{( )}

)

4

1 1 1

1

2 2 1 1

{ / 2

} 2

T T T

i i

T T

V t V x A P PA X X Y Y

Y Y x x t PBK Y x t t τ

τ

& &

=

= ≤ + + + ′ + +

+ + + − −

∑

( )

(

( )

) (

( )

(

( )

) )

⁽ ⁾

( )

(

( )

)

( )

^{( )}

(

^{( )}

) (

^{( )}

)

^{( )}

( )( ) ( ) ( )

( ) ( )

( )

( ) ( ) ( )

(

( )

) (

( )

)

( ) ( )

( )

(

( )

) (

^{( )}

)

^{( )}

2 2

1 1

2 .

2 1 2 1 1

T T T

t t

T d T

t t t t

T T T

d

T T

d

x t PBK x t t x t x t t Y x t x t t x t Y x t t x t t Y x t

x t Z x t x d Z x d

x t Qx t x t t Qx t t x t R x t x t t R x t t x t

τ τ

τ τ τ

τ τ α α α α

τ

τ τ τ

τ τ τ τ

& &

&

& & & &

& &

& & &&

− −

+ − + + − −

+ − − − − −

⎛ − ⎞⎛⎟ ⎞ ⎛⎟ ⎞⎟

⎜ ⎜ ⎜

+ −⎜⎜⎝ ⎟⎟⎟⎠⎜⎜⎝ ⎟⎟⎠ ⎝⎜⎜ ⎟⎟⎠

+ − − − − +

− − − − +

∫ ∫

( )

^{( )}

(

( )

) (

^{( )}

)

( ) ( )

2

0 0

2

/ 2 1

2 1 4

2

T d

t t

d T

t t t t t t

Z R x t x t t R x t t

x d d Z x d d

τ τ τ τ

τ τ τ

τ α α β α α β

τ

&&

&& &&

− − − −

′+ ′

⎛ − ⎟⎞

⎜ ′

−⎜⎜⎝ ⎟⎟⎟⎠ − −

⎛ − ⎞⎛⎟ ⎟⎞ ⎛ ⎞⎟

⎜ ⎜ ′⎜

−⎜⎜⎝ ⎟⎟⎟⎠⎜⎜⎝

∫ ∫

⎟⎟⎠ ⎝⎜⎜

∫ ∫

⎟⎟⎠

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

( ) ( )

( )

( ) ( ) ( )

( )

2 2

T T T T

T T T

x t Y x t x t Y x t x t t Y x t x t t Y x t

τ τ τ τ ζ ζ

τ τ τ τ

Π

& &

+ + − − = +

+ − − (24)

Assume zero initial condition, i.e. φ(t)=0, ∀t∈[-τ,0] we have V(q(t))|t=0=0. For a prescribedγ >0, consider the following performance index J2 in Corollary 1. Therefore, J2

can be rewritten as

( ) ₀

(

^T ² ^T ² ^T

)

Jzd w =

∫

^∞ z z−γ w w−γ w w d& & τ⁽²⁵⁾ Since V(t)|t=0=0 and V(t)|t→∞≥0, we obtain

( )

(

( )

)

^{( )} ^{( )}

(

( )

)

2 2

0 0

2 2

0

T T T

zd t t

T T T

J w z z w w w w V t d V t V t

z z w w w w V t d

γ γ τ

∞

= →∞

∞

= − − + + −

≤ − − +

∫

&

& &

&

& &

Hence the following inequality is obtained:

( )

{

( ) ⁽ ⁾

( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 1 2

0

2 1 1 1 1 1 1 2

1 1 1 2 2 1 1 2

2 2

2

T T T T T T

zd

T T T T T T T T

T T T T T T

J w x C Cx x C D K x t x C D K x t x C D w x t K D D K x t x t K D D w x t K D D K x t x t K D D K x t

τ τ

τ τ τ

τ τ τ τ

&

& & &

≤ ∞ + − + −

+ + − − + −

+ − − + − −

∫

( ) 2 1 2 2 2 ² ² ( )

}

2x t&^T τ K D D w^T ^T w D D w^T ^T γw w^T γw w V t d& &^T ^& τ

+ − + − − +

(26) Considering (24), 0<τ( )t ≤τ and ( )τ&t ^{< <}τ_d 1/ 4 a new upper bound for (26) is obtained as

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

{

² ²

}

0 zd

T T T T

J

x t t Y x t x t t Y x t d

ζ Πζ τ τ τ τ τ

∞

≤

+ + − −

∫

^& ^&

with defined

( ) ( ) ( ) ( ) 1 2 ( ) ( )

x t x t x t x t w t w t

ζ⁼^⎡_⎣ ⁻τ & ⁻τ && ⁻τ ζ ζ & ^⎤_⎦

where ( )

( ) ( )

( ) ( ) 0

1 ^t , 2 ^t

t tx d t t tx d d

τ τ τ

ζ ⁼

∫

₋ ^& α α ζ ⁼

∫ ∫

₋ ₋ ^&&α α β

and Π=⎡ ⎤⎢ ⎥⎣ ⎦Σij where Σij=Σ^Tij and i,j= 1,2,…8.

in which,

( ) ( )

11 1 1 1 2

1 2

12 1 1 2 1 1 2 1 1 1

13 2 2 1 2 2 2 1 2

/ 2 2

T T

T T T

T T T T

A P PA Y Y X X Y

A A A A AA Q C C

PBK Y Y A BK A A ABK C D K PBK Y A BK A A ABK C D K

τ τ

τ Σ

ϒ ϒ

Σ ϒ ϒ

= + + + + + ′ + −

+ + + +

= − − + + +

= − + + +

( ) ( ) ( )

( ) ( )

( )

14 17 1 2 2

18 2

22 1 1 1 1 2 1 1 1 1 1

23 1 1 2 1 2 2 1 1 1 2

24 27 1 1 1 2 1 1 2

28 1 2

0,

1

0,

T T T T

T T

T T T T

d

T T T T

T

PE A E A A AE C D A A E

Q BK BK ABK ABK K D D K

BK BK ABK ABK K D D K