I Delay-Dependent H Control of Linear Systems with input Delay Using Composite State-Derivative Feedback

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1Abstract— H∞ control problem for input-delayed systems is considered in this paper. A composite state-derivative control law is used, in which, a composition of the state variables and their derivatives appear in the control law. Thus, 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 control law parameters. This condition provides new challenging issues which has its own merits in theoretical research as well as application aspects. New delay-dependent sufficient condition for the existence of H_∞ controller in terms of matrix inequalities is derived in the present paper. The resulting H∞ controller guarantees asymptotic stability of the closed-loop system as well as a guaranteed limited H∞ norm smaller than a prescribed level. Numerical examples are presented to illustrate the effectiveness of the proposed methods.

I. INTRODUCTION

n various engineering systems such as chemical processes, rolling mills and nuclear reactors, time-delays are frequently a source of instability. Hence, many researchers have paid great attention to the stabilization and 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 [1]. 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 [2]. H∞ output feedback control of neutral systems has also been the centre of attention in some research such as [3] and [4]. Moreover, an H∞ output feedback controller has been designed in terms of three LMIs using bounded real lemma in [3], for a neutral system with multiple delays.

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 analysis and 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 these 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 a H∞ state-feedback controller for an input-delay active suspension system [5].

1 Advanced Robotics and Automated Systems (ARAS), Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, P.O. Box 1631-1355, Tehran, Iran (email: [email protected]).

Since the ride comfort is an 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 [6] and [7]. Abdelaziz and válašek [7] 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. [6] 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 composite state-derivative feedback in presence of small uncertain delays in the feedback loop was presented in [8]. 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 composite 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( −h_i) and

) (t h_i

x& − 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.

This paper is organized as follows. Problem formulation is introduced in Section 2, and in Section 3, an H∞ controller is designed 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 methods in some case studies, and real application.

Finally, the concluding remarks are given in Section 5.

Delay-Dependent H

∞

Control of Linear Systems with input Delay Using Composite State-Derivative Feedback

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

I

Proceedings of the 4th International Symposium on Communications,

Control and Signal Processing, ISCCSP 2010, Limassol, Cyprus, 3-5 March 2010

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II. PROBLEM FORMULATION

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

( ) ( ) ( ) ( )

( ) ( )

¹

( )

²

( )

x t Ax t Bu t Ew t z t Cx t D u 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 is the constant time-delay of the system and is assumed to satisfy 0^{< ≤}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 composite 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 BK x t Ew t z t Cx t D K x t D K x 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 two well known lemmas which will be used further in the main result of the paper.

Lemma 1 [9]: Assume a(.)∈ℜ^na, b(.)∈ℜ^nb and N∈ℜ^na×nb are defined on the interval Ω, then for any matrices X∈ℜ^na^×^nb, Y∈ℜ^na^×^nb and Z∈ℜ^na^×^nb, the following holds:

( ) ( ) ( )

( )

2 ( ) ,

T T

a X Y N a

a Nb d d

b Y N Z b

α α

α α α α α α

Ω Ω

⎡ ⎤ ⎡ − ⎤⎡ ⎤

⎢ ⎥ ⎢ ⎥⎢ ⎥

−

∫

≤

∫

⎢⎢⎣ ⎥⎥⎦ ⎢⎣ − ⎥⎦⎢⎢⎣ ⎥⎥⎦ Where ⎥>0

⎦

⎢ ⎤

⎣

⎡ Z Y

Y X

T

Remark 1: The above inequality can be extended to the similar inequalities with multiple integrals.

Lemma 2 [10]: (Schur Complement) The LMI ( ) ( )

( ) ( ) 0

T

Q y S y S y R y

⎡ ⎤

⎢ ⎥<

⎣ ⎦ is equivalent to , 0 ) ( ) ( ) ( ) ( , 0 )

(y < Q y −S y R y ⁻¹S y ^T <

R

Lemma 3 [11]: For a prescribed matrix, ^M ^A B

=⎡ ⎤⎢ ⎥

⎣ ⎦ or

[ ]

M= A B we have the following inequality:

( ) ( )

{ }

( )

{

( ) ( )

}

max σ A ,σ B ≤σ M ≤ 2 max σ A ,σ B III. H∞ CONTROL DESIGN

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 scalar τ >0, the closed-loop system (6) is asymptotically stable and ║T_zw║_∞<γ for any constant time- delay τ satisfying 0^{< ≤}τ τ , if there exist positive definite symmetric matrices L, T, H1, H2, F1, F2 ∈ℜⁿ^×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∞ composite state-derivative feedback control law is given by u=VL x WL x⁻¹ + ⁻¹&.

1 1

1

1 1

T 0

M N

N LF L⁻

⎡ ⎤

⎢ ⎥ >

⎢ ⎥

⎣ ⎦ (8)

and,

2 2

1

2 2

2 0

T 2

M N

N LF L τ

τ ⁻

⎡ ⎤

⎢ ⎥>

⎣ ⎦ (9)

in which,

( ) ( )

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 4: 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 we have ( ) ( ) ( )^,

d s w s sw s

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣ ⎦ then the transfer function from d to z can be expressed by

( )

(

( ) ( )

)

^T

(

( ) ( )

)

^T ^T

Tzd s = ⎢⎡ z s w s z s sw s ⎤⎥

⎣ ⎦

_____________________________________________________________________________________________________

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

( ) ( )

1 1 2 2 2

1 2 1

1

1 2 1 2 1

1 2

2

2 2

2

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

T 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 D V D BV BV ABV ABV D V

LH L N D W D BW BW BV ABW BV ABW D W

LH L BW BW

D D I E E E A E A

I E E

τ τ τ

τ τ

τ τ τ

τ

γ τ τ

γ τ

−

Ω − − − +

−

− − + +

−

− 1 1

2

0

0 0 0 0

* * * * * * * 0 0 0

* * * * * * * * 0 0

* * * * * * * * * 0

* * * * * * * * * *

F H

F

H

I τ

τ

⎡ ⎤

⎢ ⎥

⎢ ⎥<

⎢ ⎥

⎢ − ⎥

⎢ ⎥

−

⎢ ⎥

⎢ − ⎥

⎢ ⎥

⎢ − ⎥

⎣ ⎦

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The above equality can be rewritten as ( ) ^T( ) ¹ ^T( )^T

zd zw zw

T s T s T s

s

⎡ ⎤

⎢ ⎥

=⎢⎣ ⎥⎦

By the Lemma 3, the following inequality holds as ( )

( )

max , 1 1

zw

zw zw

zw

T s

T T

s T s

s

∞

⎡ ⎤

⎧ ⎫ ⎢ ⎥

⎪ ⎪

⎪ ⎪≤ ⎢ ⎥

⎨ ⎬ ⎢ ⎥

⎪ ⎪

⎩ ⎭ ⎢⎣ ⎥⎦

By the above inequality, it can be easily concluded that ( )

1 ( )

zw

T s sT s

γ

∞

⎡ ⎤

⎢ ⎥

⎢ ⎥ <

⎢ ⎥

⎣ ⎦

guarantees Tzw _∞^<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 T_zw _∞^<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 (10)

( ) ( )

0

2 ^t ^T 1 ,

V t x Z x d d

τ β α α α β

− +

=

∫ ∫

^& ^& ⁽¹¹⁾

( ) ( ) ( ) ( )

3 1

t T t T

t t

V x Qx d x R x d

τ α α α τ α α α

− −

=

∫

+

∫

^& ^& ⁽¹²⁾

( ) ( )

( ) ( ) ( )

1 0 4

1/ 2 ,

t T

t

t T

t

V x Z x d d d

x R x d

β

τ τ η

τ

τ α α α η β

α α α

−

− − +

−

= ′ + ′

∫ ∫ ∫

∫

&& &&

&& && (13)

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 τ BK x t τ Ew t

=

= + − + − +

& &

&

It is possible to write

( ) ( ) ^t ( )

x t x t t x d

τ ₋τ α α

− = −

∫

^& ⁽¹⁴⁾

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

( ) ¹ ( ) ( ) ⁰ ^t ( )

x t x t x t t ^βx d d

τ τ

τ τ⁻ τ ⁺ α α β

− −

⎡ ⎤

− = ⎢ − − − ⎥

⎢ ⎥

⎣

∫ ∫

⎦

& && (15)

Therefore,

( )

( ) ( )

( ) ( ) ( )

1

1 1 2

1

2 1

1 0 2

2 ( )

2 2

T

T T t

t

T t T

t

V x P A BK BK x t

x PBK x 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 Lemma 2 and Remark 1, the following upper bound for V&1 is obtained:

( ) ( )

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

( )

( ) ( )

( )

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

( ) ( ) ( ) ( )

1

1 1 1 1

2 2

1 1

1 0

1

{ / 2 }

2

T T T T

T T T

T T T T

t T t T

t t

V x A P PA X X Y Y Y Y x

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

x t Y x t x t Y x t x t PEw t

x Z x d d x Z x d

β

τ τ τ

τ τ

τ τ τ τ

τ α α α β α α α

−

− −

− +

− − −

′ ′ ′

≤ + + + + + + +

+ − − + − −

′ ′

+ − − + − −

′ ′

− − − − +

+

∫ ∫

′ +

∫

&

& &

&& && & &

(16) where

1 1

T 0 X Y Y Z

⎡ ⎤

⎢ ⎥ >

⎢ ⎥

⎣ ⎦ (17)

and,

T 0 X Y

Y Z

′ ′

⎡ ⎤

⎢ ′ ′⎥>

⎣ ⎦ (18)

Also, the time derivative of V4 can be represented as follows:

( ) ( ) ( ) ( ) ( )

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

1 0

4 / 2

1 / 2 1 / 2

t T T

t

T T

V x Z x d d x t Z x t

x t R x t x t R x t

β

τ τ

τ α α α β τ

τ τ

− +

− − ′ ′

= − +

′ ′

+ − − −

&

∫ ∫

&& && && &&

(19) It can be shown that the time derivative of V2 and V3 are

( )( ) ( ) ( ) ( )

2 1 1

T t T

V x t Z x t t x Z x d

τ ₋τ α α α

= −

∫

& & & & & (20)

( ) ( ) ( ) ( )

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

= − − −

+ − − −

&

& & & & (21)

Therefore we have ( ) ⁴

1 i i

V t V

=

∑

& & (22)

We consider (16), (19) ~ (22) and defineτ⁻¹Y^{′ =}Y₂. By the assumption of Y′=Y′^T<0and then adding and subtracting the terms x tT( )( ) ( )τY x t2 and

( )( ) (2 )

T T

x t& ⁻τ τY x t& ⁻τ in (22), an upper bound for V& is

obtained as

( ) ( ( ) )

( )( ) ( ) ( ) ( )

( ) ( )

( )

⁽ ⁾

(

^{( )} ⁽ ⁾

)

^{( )} ^{( )}

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

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

( ) ( ) ( )

( )

1 1 1 2

1 1 2

2

2 2

1 2 1

{ 1/ 2 2 }

2 2

2 / 2

1/ 2 1/ 2

.

T T T

T T

T T T T

T T T

T T

T

V t x A P PA X X Y Y Y x

x t PBK Y x t x t PBK x t

x t x t Y x t x t x t PEw t x t Y x t x t Y x t x t Z x t

x t R x t x t R x t Ax BK x t BK x t Ew t Z Ax

τ

τ τ

τ τ τ

τ τ

τ τ τ

≤ + + + ′ + + +

+ − − + −

+ + − − + − +

− − − − + ′

′ ′

+ − − −

+ + − + − +

+

&

& &

&& &&

&& && && &&

&

( ) ( ) ( )

( ) ^{( ) ( )}

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

( ) ( ) ( ) ( )

( ) ^{( ) ( )}

( )( ) ( ) ( ) ( )( ) ⁽ ⁾

1 2

1 1

1 2 1

2 2

2

T

T T T

T

T T T

BK x t BK x t Ew t x t Qx t x t Qx t x t R x t x t R x t

Ax t BK x t BK x t Ew t R Ew t x t Y x t x t t Y x t

τ τ

τ τ τ τ

τ τ

τ τ τ τ

− + − + +

− − − + − − −

+ + − + − +

+ + − −

&

& & & &

&

& &

(23)

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:

(4)

( ) ₀

(

^T ² ^T

)

Jzd w ⁼

∫

^∞ z z⁻γ d d dτ ⁽²⁴⁾

where ( ) ( ) ( )^,

d t w t w t

⎡ ⎤

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎣& ⎦

Therefore the performance index (24) can be rewritten as

( ) ₀

(

^T ² ^T ² ^T

)

Jzd w =

∫

^∞ z z−γ w w−γ w w d& & τ 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 1 2 1 1 2

2 1 1 2 2 1 2

2 2

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 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 K x t x t K D D w x t K D D K x t x t K D D w w D D w w w w w V t d

τ τ

τ τ τ

γ γ τ

≤ ∞ + − + −

+ + − −

+ − − + −

+ − − +

∫

^&

&

& & &

&

& &

(25) Considering (23) and 0^{< ≤}τ τ a new upper bound for (25) is obtained as

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

{

² ²

}

0

T T T T

Jzd≤

∫

^∞ ζ Πζ+x t τY x t +x t^& −τ τY x t^& −τ dτ ₍₂₆₎

with defined

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

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

ζ=⎡⎣ −τ & −τ && −τ & ⎤⎦

and Π= ⎢ ⎥⎡ ⎤⎣ ⎦Σij where Σij=Σ^Tij and i,j= 1,2,…6.

in witch

( ) ( )

( )

11 1 1 1 2

1 2

12 1 1 2 1 1 2 1 1 1

13 2 2 1 2 2 1 2 1 2

14 2 2 15 1 2 2

16 2

22 1 1 1

/ 2 2

T T

T T T

T T T T

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 BK ABK C D K

A A BK PE A E A A AE C D

A A E

Q BK BK

τ τ

τ Σ

ϒ ϒ

Σ ϒ ϒ

Σ ϒ Σ ϒ ϒ

Σ ϒ

= + + + + + ′ + −

+ + + +

= − − + + +

= − + + + +

= = + + +

=

= − + ( )

( ) ( ) ( )

1 2 1 1 1 1 1

23 1 1 2 1 1 1 2 1 2 1 2

24 1 2 2 25 1 1 1 2 1 1 2

T T T

T T T T T

T T T T T T

ABK ABK K D D K

BK BK K D D K BK A BK ABK

BK A BK BK E ABK AE K D D

ϒ

Σ ϒ ϒ

Σ ϒ Σ ϒ ϒ

+ +

= + + +

= = + +

( )

( ) ( )

( )

( ) ( )

26 1 2

33 1 2 2 1 2 2 1 1 2

1 2 2 1 2

34 1 2 2 2

35 2 1 2 1 2 2 1 2

36 1 2 2 44 2 2 2

45 2 2 46 2 2

/ 2

T

T T T

T

T T T T

T T

ABK E

R Y BK BK K D D K BK ABK BK ABK BK ABK BK

BK E ABK BK AE K D D

BK ABK E R BK BK

BK AE BK E

τ

Σ ϒ

ϒ

Σ ϒ

Σ ϒ ϒ

Σ ϒ Σ ϒ

=

=− − + +

+ + +

= +

= + + +

= + =− ′ +

= =

2

55 1 2 2 2

T T T T

E E E A AE D D γ I

Σ = ϒ + ϒ + −

2

56 2 66 2

T T T

E A E E E γ I

Σ = ϒ Σ = ϒ −

where ϒ₁⁼R₁⁺τZ₁ and ϒ2⁼R^′/ 2⁺(τ/ 2)Z^′.

Considering the constraintY₂=Y₂^T<0, ifζ^TΠζ<0, then the negative semi definiteness of Jzd in (26) is guaranteed for any constant time-delay τ satisfying 0^{< ≤}τ τ . Hence, when

assuming w t w t( ) ( ),& ∈L2[0 ∞)and Π<0 then implies that Jzd<0 and therefore ^T_zd _∞^<γ. This condition is the H∞

performance to guarantee the tracking performance. By Lemma 4, the inequality Tzd _∞^<γ guarantees Tzw _∞^<γ to be satisfied. Using Schur complement, the condition Π<0 is equivalent to the following matrix inequality:

11 12

12 22

T 0

Ξ Ξ

⎡ ⎤

⎢Ξ Ξ ⎥<

⎢ ⎥

⎣ ⎦ (27)

with LMI (17) and

2 2

X Y 0 Y Z

τ τ

⎡ ′ ⎤

⎢ ⎥ >

⎢ ′⎥

⎣ ⎦ (28)

where

1 1 1 2 2 2 2

1 1 2

1 2 2 1 2

11

2 2 2

2

0 0

* 0 0 0

* * 0 0

* * * / 2 0 0

* * * * 0

* * * * *

T

T T

T

PBK Y Y PBK Y PE C D

Q K D D

R Y K D D

R

D D I I τ

τ

γ γ

⎡Ω − − − + ⎤

⎢ ⎥

⎢ − ⎥

⎢ − − ⎥

Ξ =⎢ ⎥

− ′

⎢ ⎥

⎢ − ⎥

⎢ ⎥

⎢ − ⎥

⎣ ⎦

[ ]

12 τ 1 1Z 1 1R τ 2Z′/ 2 2R′/ 2 3

Ξ = Δ Δ Δ Δ Δ

where

(

( )

)

( )

1 A P PA Y Y^T 1 1^T τ X1 1 / 2 X′ 2 τ Y2 Q

Ω = + + + + + + − +

[ ] [ ]

[ ]

1 1 2 3 1 1 1 2

2 1 1 2 2

0 0^T 0 0 0^T

T

A BK BK E C D K D K

A A ABK BK ABK BK AE E

Δ = Δ =

Δ = +

and

( )

22 diag τZ1 R1 τZ′/ 2 R′/ 2 I

Ξ = − − − − −

Denote P⁻¹,Z1⁻¹,R1⁻¹as L, F1, and H1 respectively, by performing a congruence transformation to (27) by diag (

1 1

, , , , , , 2 , 2

L L L L F H Z′⁻ R′⁻ ) together with introducing the change of variables M₁=LX₁L, M₂=L(X′^{/2)L, N}1=LY₁L, N₂=LY₂L, F₁=Z₁^-1, T=LQL, F₂=2Z′^-1, H₁=R₁^-1, H₂=2R′^-1, V=K₁L, W=K₂L, the matrix inequality (7) is derived. Furthermore, pre and post multiplying the LMI (28) by diag (L L, ) and its transpose and defining the same change of variables, the matrix inequality (8) is provided.

Similarly, by performing a congruence transformation to (28) by diag (L L, ), we can obtain

2 2

T 0

LX L LY L LY L LZ L

τ τ

⎡ ′ ⎤

⎢ ⎥ >

⎢ ′ ⎥

⎣ ⎦

Using Schur complement, we have ( 2 )

( )

¹

(

2^T

)

0 LX L^′ ⁻τLY L LZ L^′ ⁻ τLY L ^>

Substituting M2=L X

(

′/ 2 ,

)

L N₂=LY L₂ and F2=2Z′⁻¹, the following matrix inequality is derived.

( )

(

¹

) (

¹

)

2 2 2 2

2M −τN 2LF L⁻ ⁻ τN^T >0

(29) On the other hand we have

(5)

( )

(

¹

)

¹⁽ ⁾ ⁽ ⁾

(

¹

)

¹⁽ ⁾

2 2 2 2 2 2 2 2

2M −τN 2LF L⁻ ⁻ τN ^T≥2M −τN 2LF L⁻ ⁻ τN ^T(30) Therefore,