H∞ Control of Uncertain Neutral Systems with Time-Varying Delays

Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008

Robust Mixed H 2 / H ∞ Control of Uncertain Neutral Systems with Time-Varying Delays Hamid Reza Karimi * Ningsu Luo **

* Center for Industrial Mathematics, University of Bremen, D-28359 Bremen, Germany. e-mail: [email protected] ** Institute of Informatics and Applications, University of Girona, Girona, Spain. e-mail: [email protected] Abstract: This paper considers the problem of robust mixed H 2 / H ∞ delayed state feedback control for a class of uncertain neutral systems with time-varying discrete and distributed delays. Based on the Lyapunov-Krasovskii functional theory, new required sufficient conditions are established in terms of delay-range-dependent linear matrix inequalities (LMIs) for the stability and stabilization of the considered system using some free matrices. The desired robust mixed H 2 / H ∞ delayed control is derived based on a convex optimization method such that the resulting closed-loop system is asymptotically stable and satisfies H 2 performance with a guaranteed cost and a prescribed level of H ∞ performance, simultaneously. Finally, a numerical example is given to illustrate the effectiveness of our approach. 1. INTRODUCTION Delay systems represent a class of infinite-dimensional systems largely used to describe propagation and transport phenomena or population dynamics. Neutral delay systems constitute a more general class than those of the retarded type. Stability of these systems proves to be a more complex issue because the system involves the derivative of the delayed state. Especially, in the past few decades increased attention has been devoted to the problem of robust delayindependent stability or delay-dependent stability and stabilization via different approaches for linear neutral systems with delayed state and/or input and parameter uncertainties (see, Han, 2004; He et. al., 2007; Han and Yu, 2004; Lam et. al., 2005). Among the past results on neutral delay systems, the LMI approach is an efficient method to solve many control problems such as stability analysis and stabilization (Fridman, 2001; Chen and Zheng, 2007) and H ∞ control problems (Chen, 2005; Fridman and Shaked, 2003; Gao and Wang, 2003; Xu et. al., 2001; Chen, 2006; Xu et. al., 2002). It is also worth citing that some appreciable works have been performed to design a guaranteed-cost (observer-based) control for the neutral system performance representation (Karimi, 2008; Chen et. al., 2006; Lien, 2005; Park, 2003; Xu et. al., 2003). To the best of our knowledge, a robust mixed H 2 / H ∞ delayed state feedback control for uncertain neutral systems with time-varying discrete and distributed delays has not been fully investigated in the past and remains to be important and challenging. This paper develops an efficient approach for robust mixed H 2 / H ∞ delayed state feedback control problem of uncertain neutral systems with discrete and distributed time-varying delays. The main merit of the proposed method is the fact that it provides a convex problem such the control gain can be found from the LMI formulations. New required sufficient 978-3-902661-00-5/08/$20.00 © 2008 IFAC

conditions are established in terms of delay-range-dependent LMIs combined with the Lyapunov-Krasovskii method for the existence of the desired robust mixed H 2 / H ∞ control such that the resulting closed-loop system is asymptotically stable and satisfies both, H 2 performance with a guaranteed cost and a prescribed level of H ∞ performance. A numerical example is given to illustrate the use of our results. 2. PROBLEM DESCRIPTION Consider a class of neutral systems with discrete and distributed delays and norm-bounded time-varying uncertainties represented by x (t ) − A2 x (t − d (t )) = ( A + ∆ A(t )) x(t ) + ( A1 + ∆ A1 (t )) x (t − h (t )) + ( A3 + ∆ A2 (t ))

t

∫ x( s) ds + ( B + ∆ B (t )) u (t ) + ( B1 + ∆ B1(t )) w(t ),

t −τ ( t )

t ∈ [− max {h 2 , d 2 , τ 2 }, 0]

x (t ) = φ (t ),

z (t ) = (C + ∆ C (t )) x (t ) + ( D + ∆ D(t )) u (t ),

(1a-c) where x(t ) ∈ ℜ n , u(t ) ∈ ℜ m , w(t ) ∈ Ls2 [0, ∞) and z (t ) ∈ ℜ z are state, input, disturbance and controlled output, respectively. The time-varying function φ (t ) is continuous vector valued initial function and the time-varying delays h(t ), d (t ) and τ (t ) are functions satisfying, respectively, h1 ≤ h (t ) ≤ h2 , h(t ) ≤ h3 , (2a) 0 ≤ d (t ) ≤ d 1 , 0 ≤ τ (t ) ≤ τ 1 ,

d (t ) ≤ d 2 < 1, τ(t ) ≤ τ 2 < 1.

(2b) (2c)

Moreover, ∆ A(t ), ∆ A1 (t ), ∆ A2 (t ), ∆ A3 (t ), ∆ B(t ), ∆ B1 (t ), ∆ C (t ), and ∆ D (t ) are bounded uncertainties and defined as follows:

4899

[∆A(t )

∆A1 (t ) ∆A2 (t ) ∆B(t ) ∆B1 (t ) ] = H 1 ∆(t )[E

[∆C (t )

E1

E2

∆D(t ) ] = H 2 ∆ (t )[E 5

E3

E4 ]

E6 ]

(3)

10.3182/20080706-5-KR-1001.0544

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

where the uncertain matrix ∆(t ) satisfies ∆T (t ) ∆(t ) ≤ I . Definition 1: The H 2 and H ∞ performance measures of the system (1) are defined, respectively, as ∞

J 2 = ∫ χ T ( x(t )) S 1 χ ( x (t )) + u T (t ) S 2 u (t ) dt ,

(4a)

0

∞

J ∞ = ∫ z T (t ) z (t ) − γ 2 wT (t ) w(t ) dt ,

V5 (t ) =

V7 (t ) =

A2 x(t − d (t ))

In this paper, the authors’ attention will be focused on the design of the following robust mixed H 2 / H ∞ delayed state feedback control law, u (t ) = K χ ( x(t )) (6) where the matrix K of the appropriate dimension is to be determined such that for any delays satisfying (2) the resulting closed-loop system is asymptotically stable and J 2 ≤ J 0 , where the constant scalar J 0 is an upper bound of the H 2 performance measure which satisfies an H ∞ norm bound γ . It can be easily seen that the resulting closed-loop system (1) and (6) is of the following form, x (t ) = ( A + ∆ A(t ) + ( B + ∆ B (t )) K ) χ (t ) + 1−1d 2 ( A + ∆ A(t )) × A2 x (t − d (t )) + ( A1 + ∆ A1 (t )) x(t − h(t )) + ( A3 + ∆ A2 (t ))

− h2 t +θ

t − d (t )

t

t

t

T T ∫ x( s) R7 x( s) ds , V8 (t ) = ∫ ∫ x( s) R8 x( s) ds dβ . t −τ (t ) β

t − d (t )

+ ( A1 + ∆ A1 (t )) x(t − h(t ) + + ( A3 + ∆ A2 (t ))

(5)

and S1 > 0 , S 2 > 0 and the positive scalar γ are given. Assumption 1: The full state variable x(t ) is available for measurement.

t

V1 (t ) ≤ 2 χ ( x(t ))T P {( A + ∆ A(t ) + ( B + ∆ B (t )) K ) χ ( x(t ))

(4b)

where the operator χ ( x(t )) in (4a) is defined by χ ( x (t )) = x (t ) −

t

T T ∫ ∫ x (s) R5 x (s) ds dθ , V6 (t ) = ∫ x( s) R6 x( s) ds ,

Differentiating V1 (t ) in t we obtain

0

1 1− d 2

− h1

1 1− d 2

( A + ∆ A(t )) A2 x(t − d (t ))

t

∫ x( s) ds + ( B1 + ∆ B1 (t )) w(t )

t −τ (t )

Differentiating other terms in (8) give ( h12 := h2 − h1 ) 2

V2 (t ) = ∑ x(t ) T Ri x (t ) − x(t − hi ) T Ri x(t − hi )

V3 (t ) ≤ xT (t ) R3 x(t ) − (1 − h3 ) xT (t − h(t )) R3 x (t − h(t )) t

t −h (t )

t −h (t )

t − h2

V4 (t ) = h2 x (t )T R4 x (t ) − ∫ x (s)T R4 x (s) ds − ∫

V5 (t ) = h12 x (t )T R5 x (t ) −

x (s)T R4 x (s ) ds

t −h( t )

t − h1

t − h2

t −h( t )

∫

x(s) T R5 x(s) ds −

∫

(11) (12)

x(s)T R5 x (s) ds (13)

V6 (t ) ≤ xT (t ) R6 x (t ) − (1 − d 2 ) xT (t − d (t )) R6 x(t − d (t ))

(14)

V7 (t ) ≤ x T (t ) R7 x (t ) − (1 − d 2 ) xT (t − d (t )) R7 x (t − d (t ))

(15)

V8 (t ) ≤ τ1 x(t )T R8 x (t ) − (1 − τ 2 )

t

T ∫ x( s ) R8 x( s) ds

(16)

t −τ (t )

Moreover, from the Leibniz-Newton formula, the following equations hold for {N i }i4=1 with appropriate dimensions: 2( χ T (t ) N 1 + x T (t − h (t )) N 2 )( x(t ) − x(t − h (t )) −

(7)

∫ x(s) ds + A2 x (t − d (t )) + (B1 + ∆ B1 (t )) w(t ),

t

∫ x (s) ds ) = 0

(17)

t −h(t )

2( χ T (t ) N 3 + x T (t − h(t )) N 4 )( x(t − h1 ) − x(t − h(t )) −

t −τ ( t )

t − h1

∫ x(s) ds) = 0 (18)

t −h(t )

Lemma 1: (Wang et. al., 1992) Given matrices Y = Y T , D , E and F of appropriate dimensions with F T F ≤ I , then the matrix inequality Y + sym ( D F E ) < 0 , the operator sym ( A) represents A + A T , holds for all F if and only if there exists a scalar ε > 0 such that Y + ε D D T + ε −1 E T E < 0 3. ROBUST CONTROL SYNTHESIS In this section, both the asymptotic stability and mixed H 2 / H ∞ performance of the interconnection of plant and the control are investigated such sufficient stability conditions are derived for the existence of the control (6) combined with the Lyapunov method in terms of LMIs. In the literature, extensions of the quadratic Lyapunov functions to the quadratic Lyapunov-Krasovskii functionals have been proposed for time-delayed systems (Park, 1999). Now, we choose a Lyapunov functional candidate for the uncertain neutral system (1) as 8

V (t ) = ∑ Vi (t ) ,

(8)

i =1

where t

2

V1(t ) = χ ( x(t )) P χ ( x (t )) , V2 (t ) = ∑ ∫ x ( s ) T Ri x( s ) ds T

i =1 t − hi

V3 ( t ) =

(10)

i =1

t

×

(9)

t

0

t

T T ∫ x(s ) R3 x(s ) ds , V4 (t ) = ∫ ∫ x (s) R4 x( s) ds dθ ,

t − h (t )

− h2 t +θ

Now, to establish the H ∞ performance measure for the system (1), assume zero initial condition, then we have V (t ) t =0 = 0 . Consider the index J ∞ in (4b), then along the solution of (1) for any nonzero w(t ) there holds ∞

J ∞ ≤ ∫ z T (t ) z (t ) − γ 2 wT (t ) w(t ) + V (t ) dt

(19)

0

Substituting (1c), (7) and from (8)-(16) and adding the left sides of equations (17) and (18) into (19), we obtain ∞

J ∞ ≤ ∫ ϑ T (t ) Σϑ (t ) dt

(20)

0

where ϑ (t ) := col {χ ( x(t )), x(t − h(t )), x(t − h1 ), x(t − h2 ), x(t − d (t )), x (t − d (t )),

t

∫ x(s) ds, w(t )} is an augmented state vector and the t −τ ( t )

matrix Σ is given by Σ = Π + A T (h2 R4 + h12 R5 + R7 ) A + h2 M 1 R4−1 M 1T + h12 M 2 R5−1 M 2T

(21) where 0 Π11 Π12 N3  ∗ Π 0 N4 22   ∗ 0 ∗ − R1  R2 ∗ ∗ ∗ − Π=  ∗ ∗ ∗ ∗  ∗ ∗ ∗  ∗  ∗ ∗ ∗ ∗  ∗ ∗ ∗  ∗

4900

Π15 1 1−d2

0

N2 A2 0

0 0

0

0

Π55

0

∗ ∗

− (1− d2 )R7 ∗

∗

∗

P( A3 + ∆A2 (t )) P(B1 + ∆B1(t))  0 0   0 0  0 0   0 0  0 0   0 − (1− τ 2 )R8  ∗ −γ 2I 

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

with M 1 = col {N1 , N 2 ,0} , M 2 = col {N 3 , N 4 ,0} and

ˆ ˆ Π Π  11 ˆ 12  ∗ Π 22  ∗ ∗   ∗ ∗ ˆ = Π 1  ∗  ∗  ∗ ∗  ∗  ∗  ∗ ∗ 

A = [ A + ∆A(t ) + ( B + ∆B (t )) K , A1 + ∆A1 (t ) , 0, 0, 1−1d ( A + ∆A(t )) A2 , 2

A2 , A3 + ∆A2 (t ), B1 + ∆B1 (t )]

Π11 = sym {P (( A + ∆A(t )) + ( B + ∆B (t )) K ) + N1} + K T ( B + ∆B (t ))T ( R3 + h2 R4 ) ( B + ∆B (t )) K + ((C + ∆C (t )) + ( D + ∆D (t )) K )T ((C + ∆C (t )) 3

+ ( D + ∆D (t )) K ) + ∑ Ri + R6 + τ 1 R8 ,

1 1− d 2

i =1

Π 12 = P ( A1 + ∆ A1 (t )) − N 1 + N − N 3 , T 2

Π15 =

3

(∑ Ri + R6 + τ1R8 ) A2 +

1 1− d 2

1 1− d 2

i =1

+ 1−1d ((C + ∆C (t )) + ( D + ∆D (t )) K )T (C + ∆C (t )) A2 2

Π 22 = − (1 − h3 ) R 3 − sym{N 4 + N 2 } ,

h2 M 1

0

0 0

0 0

∗ ∗

− R7−1 ∗

0 − h2 R 4

∗

∗

∗

−

h12 R5−1

h12 M 2   0  0  <0. 0  0  − h12 R5 

∗

∗

∗

∗

∗

∗ 0

A3 R8

0 0

0 0

0

0

0

0 − (1 − d 2 ) R7

0 0

∗

∗

− (1 − τ 2 ) R8

∗

∗

∗

i =1

B1   0  0   0   0  0   0  − γ 2 I 

ˆ = [ω , 0, 0, 0, ω , 0, 0, 0]T , Π 2 1 2

where the symbol ∗ denotes the elements below the main diagonal of a symmetric block matrix. Thus, if the inequality Σ < 0 holds, the inequality J ∞ < 0 is satisfied. The inequality Σ < 0 yields (by Schur complement) AT

− R2 ∗

− (1 − d 2 ) R6 ∗

3

h12 A T

0

∗ ∗

R3 N 2 A2 R6 0

Π 55 = −(1 − d 2 )R6 + (1 − d 2 ) −2 A2T ((C + ∆C(t))T (C + ∆C(t )) + ∑ Ri + R6 + τ 1 R8 ) A2

Π h2 A T  −1  ∗ − h2 R 4 ∗ ∗  ∗ ∗ ∗ ∗   ∗ ∗

− R1

( I + XN1 ) A2 R6

1 1− d 2

( P + N1 ) A2

0 0

X N3 R1 R3 N 4 R1

ˆ = diag {− R , − h R , − I , − R , − R , − R ,− R ,−τ R } , Π 3 3 2 4 1 2 3 6 1 8

[

Ψ2 = ϕ

with

Ψ1 = diag {ψ 1 ,ψ 2 ,ψ 3 } , 0 ϕ

T 1

h2ϕ 2T

T 2

ϕ 3T

0 h2ϕ 4T

h12 ϕ 4T

ϕ 4T

]

T

0 0 ,

ˆ = sym { ( AX + BKX + XN X } , Π 1 11

ˆ = A R + X (− N + N T − N ) R , Π 1 3 1 2 3 3 12

(22)

ˆ = − (1 − h ) R − R (sym{N + N }) R , Π 3 3 3 4 2 3 22

~ A = [ AX + BKX , A1 R3 , 0, 0, 1−1d AA2 R6 , A2 R7 , A3 R8 , B1 ] , 2

M 1 = col { XN 1 R 4 , R3 N 2 R 4 ,0} , M 2 = col { XN 3 R5 , R3 N 4 R5 ,0} ,

ξ = diag { X , R3 , R1 , R2 , R6 , R7 , R8 , I , I , I , I , R4 , R5 }

Let

X := P

−1

and Ri :=

Ri−1

where

for i = 1,", 8 . Pre-multiplying ξ and

post-multiplying ξ T to the matrix inequality (22) yield ˆ Π 1  ∗ ∗  ∗  ∗ ∗   ∗

ˆ Π 2 ˆ Π

~ h2 A T

~ h12 A T

~ AT

h2 M 1

0

0

0

0

∗

− h2 R4

0

0

0

∗ ∗

∗ ∗

− h12 R5 ∗

0 − R7

0 0

∗

∗

∗

∗

− h2 R4

∗

∗

∗

∗

∗

3

+ sym (Ψ1 ( I 21 ⊗ ∆(t )) Ψ2 ) < 0

h12 M 2   0  0   0   0  0   − h12 R5 

where ˆ Π  1 ∗ ∗  ˆ =∗ Π  ∗ ∗   ∗

Ψ   0 <0 − ε1I   T 2

2

2

2

2

2

2

ψ 3 := diag {H 1 , H 1 , H 1 ,0,0} ,

[

ϕ 1 = EX + E 3 KX

(23)

ϕ 2 = [E 3 KX

[

E1 R3

0 0 0 0 E 2 R8

0] , ϕ 3 = E5 X + E6 KX

[

ϕ 4 = EX + E 3 KX

E1 R3

0 0

1 1− d 2

0 0 0 EA2 R 6

1 1− d 2

E4

]

0,

E5 A2 R6

0 E 2 R8

E4

] 0]

0 ,

On the other hand, by applying the same LyapunovKrasovskii functional candidate (8) for the uncertain neutral system (1), under w(t ) ≡ 0 , for the index J 2 in (4a) we get ∞

J 2 ≤ ∫ χ T ( x (t )) S1 χ ( x (t )) + χ T ( x (t )) K T S 2 K χ ( x(t )) + V (t ) dt 0

(24)

∞

(25)

≤ ∫ ϑˆ T (t ) Σˆ ϑˆ (t ) dt 0

ˆ Π 2 ˆ Π

~ h2 A T 0

~ h12 A T 0

~ AT 0

h2 M 1 0

∗ ∗

− h2 R4 ∗

0 − h12 R5

0 0

0 0

∗ ∗

∗ ∗

∗ ∗

− R7 ∗

0 − h2 R4

∗

∗

∗

∗

∗

3

τ

ω2 := col{0, 1−1d CA2R6 , 1−1d A2 R6 , 1−1d A2R6 , 1−1d A2R6 , 1−1d A2 R6 , 1−d1 A2R6}

ψ 1 := col {H 1 ,0} , ψ 2 := col {H 1 , H 1 , H 2 ,0} ,

and by Lemma 1 and applying Schur complement, the following inequality is obtained for any scalar ε 1 > 0 , ˆ ε Ψ Π 1 1   ∗ − ε1I ∗ ∗ 

ω1 := col {BKX , h2 BKX , (C + DK ) X , X , X , X , X ,τ 1 X } ,

h12 M 2   0  0   0   0  0   − h12 R5 

where ϑˆ (t ) := col {χ ( x(t )), x(t − h(t )), x (t − h1 ), x(t − h2 ), x(t − d (t )), x (t − d (t )),

t

ˆ ∫ x( s ) ds} and the matrix Σ is given by

t −τ ( t )

~ ~ ~ Σˆ = Π + A T ( h2 R 4 + h12 R5 + R7 ) A + h2 M 1 R 4−1 M 1T + h12 M 2 R5−1 M 2T

(26) where

4901

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

~ Π 11   ∗  ∗ ~  Π= ∗  ∗   ∗   ∗

Π 12

N3

~ Π 15

0

Π 22 ∗

N4 − R1

0 0

∗ ∗

∗ ∗

− R2 ∗

1 1−d 2

P ( A3 + ∆A2 (t ))   0   0  0   0   0  − (1 − τ 2 ) R8 

0

N 2 A2 0

0 0

0 ~ Π 55

0 0

∗

∗

∗

∗

− (1 − d 2 ) R 7

∗

∗

∗

∗

∗

ˆ Π  11  ∗  ∗  ~ Π1 =  ∗   ∗  ∗   ∗

with

~ A = [ A + ∆A(t ) + ( B + ∆B (t )) K , A1 + ∆A1 (t ) , 0, 0, 1−1d ( A + ∆A(t )) A2 ,

ˆ Π 12 ˆ Π

22

∗ ∗ ∗ ∗ ∗ 0 0 0 − R2 ∗ ∗ ∗

2

A2 , A3 + ∆A2 (t )] ~ Π 11 = sym {P (( A + ∆A(t )) + ( B + ∆B(t )) K ) + N 1 } + K T ( B + ∆B (t )) T 3

× ( R3 + h2 R4 )( B + ∆B(t )) K + K T S 2 K + S1 + ∑ Ri + R6 + τ 1 R8 , i =1

3 ~ Π 15 = 1−1d 2 (∑ Ri + R6 + τ 1 R8 ) A2 + 1−1d 2 ( P + N1 ) A2 ,

1 1− d 2

3

Π 55 = −(1 − d 2 ) R6 + ∑ Ri + R6 + τ 1 R8 ) A2 .

Therefore, the condition Σˆ < 0 in (25) implies ∞

(27)

0

with

≤ − ∫ χ ( x(t )) S 1 χ ( x (t )) + χ ( x (t )) K S 2 K χ ( x (t )) dt T

0

G

By Theorem 1.6 of the reference Kolmanovskii and Myshkis (1992), we conclude that the system (1) with w(t ) ≡ 0 is asymptotically stabilizabe by (6). Now, by considering the asymptotically stability of the system (1) by (6) the H 2 performance measure for the system is established as ∞

∫χ

T

i =1 − hi

+

0

0

∫ ∫ φ( s )

T

0

R 4 φ( s ) ds dθ +

0

∫ φ ( s)

T

∫ ∫ φ( s )

T

0

∫ φ( s)

− d ( 0)

T

R 5 φ( s ) ds dθ

R 7 φ(s ) ds +

0

Similar to the case of H ∞

2

1

[

0

∫ ∫ φ (s)

T

R 8 φ ( s ) ds d β

~ Π 1  ∗  I ∗ Π= ∗  ∗ ∗   ∗

~ Ψ2T   0 <0 − ε2I 

(29)

~ Π2 ~ Π3 ∗

I h2 AT

I h12 A T

I AT

I h2 M 1

0 − h2 R 4

0 0

0 0

0 0

∗ ∗

∗ ∗

− h12 R5 ∗

0 − R7

0 0

∗ ∗

∗ ∗

∗ ∗

∗ ∗

− h2 R4 ∗

2

2

2

1

1

3

1

1

E1R3 0 0 0 0 E2 R8 ~ ϕ 2 = [E3 KX 0] ,

E1R3

0 0

1 1− d 2

EA2 R6

]

0 ,

0 E2 R8

ˆ ε Ψ Π Ψ2T  1 1   0 <0  ∗ − ε1I ∗ ∗ − ε1I    I IT ~ Π ε 2 Ψ  Ψ2 1   0 <0  ∗ − ε 2I ∗ ∗ − ε 2I   

applying some matrix manipulations to the inequality Σˆ < 0 we obtain for any scalar ε 2 > 0 ,

where

]

4

0

].

and positive definite matrices X and {Ri }8i =1 , satisfying the following LMIs,

(28) performance measure, after

I ~ Π ε 2 Ψ 1  ∗ − ε2I ∗ ∗ 

4

ϕ~1 = EX + E3 KX

−τ ( 0 ) β

− d ( 0)

12

Theorem 1: Consider the system (1)-(3) and let γ > 0 be a given scalar. If there exists scalars {ε i }3i=1 > 0 , a matrix Y ,

− h2 θ

R 6 φ ( s ) ds +

4

τ

[

R 3 φ ( s ) ds

− h1 0

2

2

ϕ~4 = EX + E3 KX

−h(0)

− h2 θ

+

T

2

I M 1 = col { XN 1 R 4 , R3 N 2 R 4 ,0} , I M 2 = col { XN 3 R5 , R3 N 4 R5 ,0} , ψ~1 := col {H1 ,0} , ψ~ := col {H , H ,0} , ψ~ := diag {H , H , H ,0} ,

J 0 = (φ (0) − 1−1d 2 A2 φ (− d (0))) T P (φ (0) − 1−1d 2 A2 φ ( − d (0)))

∫ φ ( s)

2

2

0

0

2

ω 2 := col {0, 1−1d A2 R6 , 1−1d A2 R6 , 1−1d A2 R6 , 1−1d A2 R6 , 1− d1 A2 R6 } ,

where 2

1

I A = [ AX + BKX , A1 R3 , 0, 0, 1−1d AA2 R6 , A2 R 7 , A3 R8 ] , 2 G ω1 := col { BKX , h2 BKX , S1 X , S 2 KX , X , X , X , X , τ 1 X } ,

( x (t )) S1 χ ( x (t )) + χ T ( x (t )) K T S 2 K χ ( x (t )) dt ≤ V (0) = J 0

+ ∑ ∫ φ (s ) T R i φ ( s) ds +

[

2

V (t ) − V ( 0) ∫ V (t ) dt = lim t→∞ T

 A3 R 8  0   0  0   0   0  − (1 − τ 2 ) R 8 

0 0 0 0 0 − (1 − d 2 ) R 7 ∗

~ Π 3 = diag {− R3 , − h2 R4 , − S1 , − S 2 ,− R1 , − R2 , − R3 ,− R6 ,−τ 1 R8 } , ~ Ψ1 = diag {ψ~1 ,ψ~2 ,ψ~3 } , ~ T Ψ = ϕ~ T 0 ϕ~ T h ϕ~ T 0 h ϕ~ T h ϕ~ T ϕ~ T 0 ,

i =1

T

( I + XN 1 ) A 2 R 6 0 0 0 − (1 − d 2 ) R 6 ∗ ∗

G G ~ T Π 2 = [ω 1 , 0, 0, 0, ω 2 , 0, 0, 0,0 ]

i =1

∞

X N 3 R1 R 3 N 4 R1 − R1 ∗ ∗ ∗ ∗

where

I h12 M 2   0  0   0   0  0   − h12 R5 

4902

 Π1  ∗ ∗  ˆ Π=∗  ∗ ∗   ∗

∗

G h2 AT 0 − h2 R4

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

Π2 ˆ Π

3

G h12 AT 0 0

G AT 0 0

(30a)

(30b)

~ h2 M 1 0 0

− h12 R5 0 0 ∗ − R7 0 ∗ ∗ − h2 R4 ∗ ∗ ∗ ~ Π ξb  Π1 =  1 , 2  ∗ − γ I 

~ h12 M 2 0 0

     0 ,  0  0   − h12 R5 

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

 sym { ( A + I ) X + BY } ( A1 − 2 I ) R3 + ε 3 X  ∗ − 2ε 3 R3   ∗ ∗  ~ Π1 =  ∗ ∗  ∗ ∗   ∗ ∗  ∗ ∗ 

−

0 0 − (1 − d 2 ) R6

0 0 0 0 0

∗ ∗

− (1 − d 2 ) R7 ∗

2 1− d 2 1 1− d 2

A2 R6

R

ε 3 A2 R6

Proof: Omitted due to space constraints. ■

0 0 0 − R2 ∗ ∗ ∗

1 1− h3 2

R1 − R1 ∗ ∗ ∗ ∗

4. SIMULATION RESULTS Consider the system (1) with the following matrices 0   −1  0.01 −0.04 0 0.1 A=   ; A1 =   ; A2 =  ; 0.2 − 1.2 0.02 0.01  0 0.1 0.1 0   2 0  0 0  A3 =   ; B =   ; B1 =   ; C =  ;  0 0.1 1  1  0 1  0  1 0   0.5  D =   ; S1 =   ; S 2 = 1 ; φ (t ) =  ; 1 0 1      − 0.3

         0  − (1 − τ 2 ) R8  A3 R8 0 0 0 0

Π 2 = [ω 1 , 0, 0, 0, ω 2 , 0, 0, 0] ,

1  0  H 1 =   ; H 2 =   ; E = [1 0.1] ; E1 = [0.5 0.2] ; 0   1  E2 = [0.2 0.3] ; E3 = 0.3 ; E4 = 0 ; E5 = [0.1 0] ; E6 = 0.1 .

T

[

Ψ2 = ϕ1T

0 ϕ 2T

~ Π 1  ∗  ∗ I  Π=∗  ∗ ∗   ∗

h2ϕ 2T

I Π2 ~ Π3

H h2 AT

∗ ∗

ϕ 3T

0 h2ϕ 4T

h12ϕ 4T

H h12 AT

H AT

I h2 M 1

0

0

0

0

− h2 R4

0

0

0

∗

− h12 R5

0

0

∗

∗

∗

− R7

0

∗

∗

∗

∗

− h2 R4

∗

∗

∗

∗

∗

I I Ψ2 = ϕ1T

[

I I G T Π 2 = [ω 1 , 0, 0, 0, ω 2 , 0, 0, 0,0 ] , I I I I 0 ϕ 2T h2ϕ 2T 0 h2ϕ 4T h12ϕ 4T

0 0 ,

I h12 M 2   0  0   0 ,  0  0   − h12 R5 

IT

ϕ4

]

T

ϕ 4T

u (t ) = [- 0.5835 - 0.0008] x(t ) + [0 0.0835] x (t − 0.3 sin(t ) 2 ) .

]

T

0 ,

ξb := col {B1 ,0} ,

G A = [ AX + BY , A1 R3 , 0, 0, 1−1d AA2 R6 , A2 R7 , A3 R8 , B1 ] , 2 ~ ~ 1− h M 1 = col {R4 , ε 3 R4 ,0} , M 2 = col {R5 ,− 2 3 R5 ,0} ,

ω 1 := col {BY , h2 BY , C X + DY , X , X , X , X , τ 1 X } , I

ω 1 := col {BY , h2 BY , S1 X , S 2 Y , X , X , X , X , τ 1 X } ,

[

ϕ1 = EX + E3Y

[

E1R3

0 0 0 0 E2 R8

ϕ 2 = [E3Y 0] , ϕ 3 = E5 X + E6 Y 0 0 0

[

1 1− d 2

E4

]

0 ,

] 0] ,

0 ,

E 5 A2 R 6

ϕ 4 = EX + E3Y

E1 R 3 0 0 1−1d EA2 R 6 0 E 2 R8 E 4 2 H A = [ AX + BY , A1 R3 , 0, 0, 1−1d AA2 R 6 , A2 R7 , A3 R8 ] , 2 I I ϕ1 = EX + E3Y E1R3 0 0 0 0 E2 R8 0 , ϕ 2 = [E3Y 0] , I ϕ 4 = EX + E3Y E1R3 0 0 1−1d EA2 R6 0 E2 R8 0

[

]

[

2

]

then, the robust mixed H 2 / H ∞ delayed state feedback control gain in (6) is given by K = Y X −1 and an upper bound of the H 2 performance measure is obtained by J 0 = (φ (0) − 1−1d 2 A2 φ ( −d (0)))T X −1 (φ (0) − 1−1d 2 A2 φ (− d (0))) 2

i =1 − hi

+

0

0

+ ∑ ∫ φ ( s ) T Ri−1 φ ( s ) ds + 0 0

∫ ∫ φ( s)

T

∫ φ (s)

0

∫ φ (s)

T

R4−1 φ( s ) ds dθ +

0

0

− h1 0

∫ ∫ φ(s)

T

R5−1 φ( s ) ds dθ

− h2 θ

R6−1 φ ( s ) ds +

−d (0)

+

R3−1 φ ( s ) ds

∫ ∫ φ (s) −τ ( 0 ) β

0

∫ φ(s)

−d (0) T

(32) For simulation purpose, we simply choose a unit step in the time interval [1, 2] as the disturbance, ∆(t ) = sin(t ) as the norm-bounded uncertainty and select time delays as h(t ) = 0.1 + 0.7 sin(t ) 2 , d (t ) = 0.3 sin(t ) 2 and τ (t ) = 0.3 . The simulation results are shown in Figures 1 and 2. Responses of two states, i.e., x1 (t ), x 2 (t ) , of the closed-loop system are depicted in Figure 1 and compared with the corresponding state trajectories in the open-loop system under the initial condition x(0) = [0.5 − 0.3]T . It is seen from Figure 1 that the closed-loop system is asymptotically stable. The corresponding control signal (32) is also shown in Figure 2. 5. CONCLUSION The problem of robust mixed H 2 / H ∞ delayed state feedback control was proposed for a class of uncertain neutral systems with time-varying discrete and distributed delays. Based on the Lyapunov-Krasovskii functional theory, new required sufficient conditions were established in terms of delayrange-dependent LMIs for the stability and stabilization of the considered system with considering a mixed H 2 / H ∞ performance measure.

−h (0)

− h2 θ

+

T

Using Theorem 1 and solving LMIs (30a, b) with parameters h1 = 0.1, h2 = 0.8, h3 = 0.7, d1 = d 2 = 0.3, τ 1 = 0.3, and τ 2 = 0 , the corresponding suboptimal H 2 performance measure of the resulting closed-loop system is given by J 0 = 1.5539 and the minimum value of the parameter γ in optimal H ∞ performance measure is obtained as 0.578 . Hence, according to Theorem 1, a robust mixed H 2 / H ∞ delayed state feedback control law is given by

R8−1 φ ( s ) ds dβ

T

R7−1 φ( s ) ds

(31)

ACKNOWLEDGMENT This work has been partially funded by the European Union (European Regional Development Fund) and the Commission of Science and Technology of Spain (CICYT) through the coordinated research projects DPI-2005-08668-C03 and by the Government of Catalonia (Spain) through SGR00296.

4903

17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008

REFERENCES Chen, J.D., ‘LMI-based robust H ∞ control of uncertain neutral systems with state and input Delays.’ J. Optimiz. Theory and Appl., 126:553-570, 2005. Chen J.D., ‘LMI approach to robust delay-dependent mixed H 2 / H ∞ controller of uncertain neutral systems with discrete and distributed time-varying delays’ J. Optimiz. Theory and Appl., 131(3):383-403, 2006. Chen, B., Lam, J., and Xu, S., ‘Memory state feedback guaranteed cost control for neutral systems’ Int. J. Innovative Computing, Information and Control, 2(2): 293-303, 2006. Chen, W.H., and Zheng W.X., ‘Delay-dependent robust stabilization for uncertain neutral systems with distributed delays’ Automatica, 43:95-104, 2007. Fridman, E., ‘New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems.’ Systems & Control Letters, 43:309-319, 2001. Fridman, E., and Shaked, U., ‘Delay-dependent stability and H ∞ control: constant and time-varying delays.’ Int. J. Control, 76:48-60, 2003. Gao, H., and Wang, C., ‘Comments and further results on ‘A descriptor system approach to H ∞ control of linear time-delay systems.’ IEEE Trans. Automatic control, 48:520-525, 2003. Han, Q. L., ‘A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays.’ Automatica, 40:1791-1796, 2004. Han, C.L. and Yu, L., ‘Robust stability of linear neutral systems with nonlinear parameter perturbations.’ IEE Proc. Control Theory Appl., 151(5):539-546, 2004. He Y., Wang Q.G., Lin C. and Wu M., ‘Delay-rangedependent stability for systems with time-varying delay’ Automatica, 43:371-376, 2007. Karimi H.R., ‘Observer-based mixed H 2 / H ∞ control design for linear systems with time-varying delays: An LMI approach’ Int. J. Control, Automation, and Systems, 6(1):1-14, February 2008. Kolmanovskii V.B. and Myshkis A., ‘Applied theory of functional differential equations’ Kluwer Academic Publishers, Dordrecht. Netherlands, 1992. Lam, J., Gao, H., and Wang, C., ‘ H ∞ model reduction of linear systems with distributed delay.’ IEE Proc. Control Theory Appl., 152(6):662-674, 2005. Lien, C.H., ‘Guaranteed cost observer-based controls for a class of uncertain neutral time-delay systems.’ J. Optimiz. Theory and Appl., 126(1):137-156, 2005. Park, J.H. ‘Guaranteed cost stabilization of neutral differential systems with parametric uncertainty’ J. Comp. and Applied math., 151, 371-382, 2003. Park P., ‘A delay-dependent stability criterion for systems with uncertain time-invariant delays.’ IEEE Trans. Automatic Control, 44:876-877, 1999. Wang Y., Xie L. and de Souza C.E., ‘Robust control for a class of uncertain nonlinear systems’ System Control Letters, 19:139-149, 1992. Xu, S., Lam, J., and Yang, C., ‘ H ∞ and positive-real control for linear neutral delay systems.’ IEEE Transactions on Automatic control, 46: 1321-1326, 2001.

Xu, S., Lam, J., and Yang, C., ‘Robust H ∞ control for uncertain linear neutral delay systems.’ Optimal Control Applications and Methods, 23, 113-123, 2002. Xu S., J. Lam, C. Yang, and E.I. Verriest, ‘An LMI approach to guaranteed cost control for uncertain linear neutral delay systems’ Int. J. Robust and Nonlinear Control, 13:35-53, 2003.

First state 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1

0

5

10 15 Time (sec)

20

25

20

25

(a) Second state 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

0

5

10 15 Time (sec)

(b) Fig 1. Response of the states x1 (t ) and x2 (t ) : closed-loop system (solid line) and b) open-loop system (dashed line).

Control signal 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35

0

5

10 Time (sec)

Fig. 2. Control law for system.

4904

15