Int. J. Mach. Learn. & Cyber. DOI 10.1007/s13042-017-0689-1

ORIGINAL ARTICLE

Delay-dependent robust absolute stability of uncertain Lurie singular systems with neutral type and time-varying delays Yuechao Ma1 · Pingjing Yang1 · Qingling Zhang2 

Received: 13 March 2016 / Accepted: 29 April 2017 © Springer-Verlag Berlin Heidelberg 2017

Abstract  This paper is concerned with the robust absolute stability analysis for Lurie control of the neutral singular systems. By using Lyapunov–Krasovskii function method, delay dependent criteria for absolute stability of system are derived in terms of Linear Matrix Inequality. The difference between this paper and other existing results is that the lower bounds and upper bounds of discrete-delay are considered, which will obtain some less conservative stability analysis results. Finally, some numerical examples are presented to illustrate the effectiveness of the method. Keywords  Lurie singular systems · Neutral type · Absolute stability · Delay-dependent · Linear matrix inequalities

1 Introduction A great number of nonlinear systems in engineering, such as neural networks, vehicle suspension systems, automatic control etc, can be modeled as Lurie control systems whose nonlinear element satisfies certain sector constraints [1–7]. Moreover, the notion of absolute stability was introduced by Lurie [1] for the plant of automatic pilot in the 1940s, is one of the oldest open problems in the theory of stability. Since that, absolute stability analysis for the Lurie systems has been extensively studied in [2–7]. For example, Gao * Pingjing Yang [email protected] 1

College of science, Yanshan University, Qinhuangdao 066004, Hebei, People’s Republic of China

2

College of Science, Northeastern University, Shenyang 110004, Liaoning, People’s Republic of China



et al. [3] considered the absolute stability analysis for timedelay Lurie control systems with nonlinearity located in a finite sector. On the other hand, the time delay is frequently viewed as a source of instability and encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems, networked control systems etc [8–14]. Many researchers have investigated the absolute stability of Lurie control systems with time-delays and derived some stability criteria [9, 10, 12, 13], but most of the existing criteria are delay-independent. It limits the applicable ranges of those results. Generally speaking, the conservativeness of delay-dependent criteria is smaller than the delay-independent. To avoid this, we will give delaydependent absolute stability criteria. On the other hand, singular control systems have been extensively studied in the past years due to the fact that singular systems better describe physical systems than regular ones. A large number of results based on the theory of the regular systems have been extended to the area of singular systems [15–24]. Recently, the stability of Lurie singular systems has been considered [16–20]. Yang [18, 19] discussed the problem of the stability of the Lure singular system, but he did not consider the uncertain parameter and time delays. Lu [16] studied the robust control for Lurie singular systems with time-delays and parameter uncertainties. By using the free-weighting matrices approach, delaydependent stability criteria of stability of Lurie singular systems with time-varying delay are derived in terms of Linear Matrix Inequality in [17]. Recently, the stability analysis of neutral differential systems, which have delays in both their state and the derivatives of their state, has been widely investigated by many researchers [25–29]. The problem of absolute stability and robust stability of neutral systems with sector-bounded nonlinearity and mixed time-varying delays is solved in [26, 27] discussed

13

Vol.:(0123456789)



Int. J. Mach. Learn. & Cyber.

the absolute stability for neutral Lurie system by dividing the discrete delay interval into multiple segments. Delay-dependent robust stability of uncertain mixed neutral and Lurie systems with interval time-varying delays and sector bounded nonlinearity have been proposed [28]. But Yin et al. [28] did not consider their time-derivative which contains uncertainties [29] studied the problems of robust delay-dependent stability criteria for time-varying delayed Lurie systems of neutral type. However, it is just apply to nonsingular systems. It should be pointed out that when the absolute stability problem of neutral singular systems is investigated, the regularity and absence of impulses (for continuous systems) and causality (for discrete systems) are required to be considered simultaneously [20, 21]. Hence, the absolute stability problem for singular systems neutral type is much more complicated than that for state-space ones. Recently, Li studied the stability of the neutral type descriptor system with mixed delays, and presented some absolute stability criteria. But the existing criteria are all delay-independent which does not include the information on delay in [30]. In [31] the authors studied the problem of the Lurie singular systems of neutral type and presented the stability criteria without considering parameter uncertainties. Hence, it is not only theoretical but also practical importance to study the problem of the robust absolute stability analysis for Lurie control of the neutral singular systems. To the best of our knowledge, the robust absolute stability problem for uncertain Lurie singular systems of neutral type remains open, which motivates this paper. The problem of robust absolute stability analysis for Lurie singular systems of neutral type with parameter uncertainties and time-varying delays in a finite sector is discussed in this paper. In this paper, by employing Laypunov–Krasovskii function, LMI method, a sufficient condition for the absolute stability of the system is provided. The lower bounds and upper bounds of the discrete-delay make the result of this paper less conservative. Finally, some numerical examples are presented to illustrate the effectiveness of the method.

2 Description of the problem and main results

d(t) and h(t) are time-varying bounded delays satisfying ̇ ≤ u, h(t) ̇ ≤ v. The matrix d1 ≤ d(t) ≤ d2 , 0 ≤ h(t) ≤ h, d(t) E ∈ Rn×n may be singular, and assume that rank(E) = r ≤ n. A, B, E0 , G and C are known real constant matrices with appropriate dimensions. The nonlinear function f (𝜎(t)) has of the form: [ ( ) ( ) ( )]T f (𝜎(t)) = f1 𝜎1 (t) , f2 𝜎2 (t) , … , fm 𝜎m (t) where the output vector 𝜎(t) = (𝜎1 (t), 𝜎2 (t), … , 𝜎m (t)) and fi (⋅) satisfies. [ ] { ( ) } fi (⋅) ∈ Ki 0, ki = fi (⋅)|fi (0) = 0, 0 < 𝜎i fi 𝜎i ≤ ki 𝜎i2 , 𝜎i ≠ 0 (2) where K = diag{k1 , k2 , … km }, 0 < ki < ∞, (i = 1, 2, … , m). Moreover, ΔA, ΔB, ΔG, ΔE0 are unknown matrices representing norm-bounded parameter uncertainties, which are assumed to be of the following form: [ ] [ ] ΔA, ΔB, ΔG, ΔE0 = LF(t) Na , Nb , Nc , Nd . (3) where L, Na , Nb , Nc , Nd are known real constant matrices with appropriate dimensions, and F(t) ∈ ℝq×k is an unknown real and possibly time-varying matrix satisfying:

F T (t)F(t) ≤ I, ∀t (4) I is a unit matrix with appropriate dimensions. The nominal unforced Lurie singular system of neutral type (1) can be written as: ⎧ Ex(t) ̇ = Ax(t) + Bx(t − d(t)) + Gx( ̇ t − h(t)) + E0 f (𝜎(t)) ⎪ 𝜎(t) = Cx(t) ⎨ ⎪ x(t) = 𝜑(t), t ∈ �− max �h, d �, 0�. 2 ⎩ (5) Definition 1  The pair (E, A) is said to be regular, impulse free, if there exists a nonsingular matrix Psuch that PT E = ET P and PT A + AT P < 0. Definition 2 [25]

(1)

1. The neutral type singular system (5) is said to be regular, impulse free, if the pair (E, A) is regular and impulse free. 2. The neutral type singular system (5) is said to be asymptotically stable if for any 𝜀 > 0, there exists a scalar 𝛿(𝜀) > 0 such that for any compatible initial conditions 𝜙(t) satisfying sup− max (h,d)≤t≤0 ‖𝜙(t)‖ ≤ 𝛿(𝜀), the solution x(t) of the system (5) satisfies ‖x(t)‖ ≤ 𝜀 for t ≥ 0. Furthermore, limt→∞ x(t) = 0.

where x(t) ∈ Rn is the state vector, 𝜑(t) [is a given { continu} ously differentiable function specified on − max h, d2 , 0].

Definition 3 [11] The uncertain Lurie singular system of neutral type (1) is said to be robustly absolute stable if its zero solution is asymptotically stable for any nonlinearity function f (⋅) satisfying the condition (2).

Consider the following Lurie neutral singular system with time-varying delays and uncertainties:

⎧ Ex(t) ̇ = (A + ΔA) x(t) + (B + ΔB) x(t − d(t)) ⎪ ⎪ + (G + ΔG) x( ̇ t − h(t)) � � ⎪ + E0 + ΔE0 f (𝜎(t)) ⎨ ⎪ 𝜎(t) = Cx(t) ⎪ � � � � ⎪ x(t) = 𝜑(t), t ∈ − max h, d2 , 0 . ⎩

13

Int. J. Mach. Learn. & Cyber.

The aim of this paper is to establish absolute stability criteria for the uncertain Lurie singular system of neutral type (1), which will enlarge application field than the existing results. Throughout this paper we will always assume that all the eigenvalues of G are inside the unit circle, which guarantees that the zero solution of the homȯ − 𝜏(t)) = 0 is geneous “difference” equation x(t) − Gx(t uniformly asymptotically stable. Lemma 1  (Jessens inequality [14]) For any constant symmetric matrix M > 0, scalar d > 0, d2 > d1 > 0, d12 = d2 − d1, vector function 𝜑:[0, d] → Rn such that the integrations concerned are well defined, the following inequality holds:

𝜑(s)ds, 𝜑T (s)M𝜑(s)ds ≤ − 𝜑T (s)dsM �0 �0 �0 ) ( 2 −d1 t d2 − d12 − 𝜑T (s)dsd𝜃M𝜑(s) �−d2 �t+𝜃 2 d

d

d

−d

≤−

−d1

t

�−d2 �t+𝜃

𝜑T (s)dsd𝜃M

−d1

t

[

�t−h(t)

ẋ T (s)ET MEx(s)ds ̇ ≤ ]

Theorem  1  Given positive scalars 0 ≤ d1 ≤ d2 , 𝛽 > 0, h > 0, 0 ≤ u < 1, 0 ≤ v < 1, the system (5) is absolute stability, if there exist positive definite symmetric matrices P, Q1 , Q2 , Q3 , Q4 , R1 , R2 , S1 , S2 , S3 , S4 , Z1 , Z2 such that the following LMI holds:

PT E = ET P ≥ 0, ⎡ Ω11 ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢ ∗ Ω=⎢ ⎢∗ ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢∗ ⎣

�−d2 �t+𝜃

𝜑(s)dsd𝜃.

PT G 0 Ω33 ∗ ∗ ∗ ∗ ∗ ∗ ∗

Ω14 0 0 Ω44 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 Ω55 ∗ ∗ ∗ ∗ ∗

0 Ω26 0 0 0 Ω66 ∗ ∗ ∗ ∗

0 Ω27 0 0 0 0 Ω77 ∗ ∗ ∗

Ω18 0 0 0 0 0 0 Ω88 ∗ ∗

Ω19 0 0 0 0 0 0 0 Ω99 ∗

AT W ⎤ BT W ⎥⎥ GT W ⎥ E0T W ⎥ ⎥ 0 ⎥ < 0, 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 −W ⎥⎦

(7)

where

Ω11 = PT A + AT P + Q1 − (1 − u)ET Q3 E + hQ4 2 S2 − d12 ET S3 E + R1 + R2 + d12 S1 + d12

T

x(t) x(t − h(t))

Ω12 Ω22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

(6)

t

Remark 1  For any constant symmetric matrix M > 0, a ̇ 0] → Rnsuch that scalar h > 0 and vector function x(⋅):[−h, the following integration is[well-defined, then it holds that ] [ ] −h(t)

3 Main results

2 T E Z2 E, Ω12 − ET S4 E − d12 ET Z1 E − d12

−ET ME ET ME ET ME −ET ME

x(t) . The following lemma is used to deal with the x(t − h(t))

time-varying structured uncertainties in the system (1). Lemma 2  Given matrices U , W and symmetric matrix M , we have M + UF(t)W + W T F T (t)U T< 0 for any F T (t)F(t) ≤ I , if and only if there exists a scalar 𝜀 > 0, such that

M + 𝜀UU T + 𝜀−1 W T W < 0. Lemma 3 [23] Given matrices X , Y , Z with appropriate dimensions, and Y is symmetric positive definite. Then the following inequality holds:

−Z T YZ ≤ X T Z+Z T X+X T Y −1 X.

Lemma 4 [15] Consider the function 𝜑:R+ → R, if 𝜑̇ is bounded on [0, ∞), that is, there exists a scalar, such that ̇ ≤ 𝛼 for all t ∈ [0, ∞), that is uniformly continuous on |𝜑(t)| [0, ∞). Lemma 5  (Barbarlat’s Lemma [15]) Consider the function 𝜑:R+ → R, if 𝜑̇ is uniformly continuous and t ∫0 ‖𝜑(s)‖ds < ∞. Then, lim 𝜑(t) = 0.

= PT B + (1 − u)ET Q3 E + d12 ET S3 E, Ω14 = PT E0 + 𝛽CT K T , Ω18 Ω22

= d1 ET Z1 E, Ω19 = d12 ET Z2 E, ) ( = −(1 − u)(Q1 + ET Q3 E) − d12 + d2 ET S3 E − 2ET S4 E, Ω26 = ET S4 E,

Ω27 = d2 ET S3 E + ET S4 E, Ω33 ( ) = −(1 − v) ET + E + 2m + m2 Q2 , Ω44 = −2𝛽I, Ω55 = −h(1 − v)Q4 , Ω66 = −R1 − ET S4 E, Ω77 = −d2 ET S3 E − ET S4 E − R2 , Ω88 = −S1 − ET Z1 E, Ω99 = −S2 − ET Z2 E, W = Q2 + d22 Q3 + d12 d2 2 S3 2 + d12 S4 +

d14 4

Z1 +

2 d12 (d22 − d12 )

4

Z2 < 0.

Proof  Firstly, we proof Lurie singular system of neutral type Eq. (5) is regular and impulse free. From Eq. (7), we obtain that

t→∞

13



Int. J. Mach. Learn. & Cyber.

Ω11 = PT A + AT P + Q1 − (1 − u)ET Q3 E + hQ4 + R1 + R2 + −

2 S2 − d12 ET S3 E + d12 2 T − d12 E Z2 E < 0.

d12 S1

d12 ET Z1 E

T

− E S4 E

Since rank(E) ≤ r, there exist two invertible matrices M and N ∈ Rn×n such that

] ] [ ] [ Ir 0 A1 A2 P1 P2 , MAN = , M −T PN = 0 0 P3 P4 A3 A4 [ ] X1 X2 , X = Q3 , S3 , Z1 , Z2 M −T XM = X3 X4 [

MEN =

Pre- and post-multiply Ω11 and Eq.  (6) by N T and N , respectively, from Eq. (6) we can get P2 = 0. So it is easy to obtain that [ ] ∗∗ <0 ∗ A4 P4 + PT4 AT4 where “∗” denote insignificant terms. So we can deduce A4 P4 + PT4 AT4 < 0. According to Definition 1 the pair (E, A) is regular and impulse free combining with Eq. (6). Thus, the system (5) is regular and impulse free. Next, we will show the stability of the singular system (5). Dai [15] presented that if the pair (E, A) is regular and impulse free, there exits another two invertible matrices J1 and J2 ∈ Rn×n such that ( ) ( ) Ir 0 Ā r 0 Ē = J1 EJ2 = , Ā = J1 AJ2 = (8) 0 0 0 In−r where Ir ∈ Rr×r , In−r ∈ R(n−r)×(n−r) are identity matrices, Ar ∈ Rr×r. According to Eq. (8), let ( ) ( ) ̄ ) ( B̄ B̄ ̄ G ̄ , Ē 0 = J1 E0 = E01 , B̄ = J1 BJ2 = ̄ 11 ̄ 12 B̄ = B, Ē 02 B21 B22

( ) P̄ P̄ P̄ = J1−T PJ2 = ̄ 11 ̄ 12 , C̄ = J2−1 CJ2 , K̄ = KJ2 , P21 P22 ( ) −T −1 ̄ X = J1 XJ1 X = Q3 , Z1 , Z2 , S3 , S4 , ( ) Ῡ = J T ΥJ2 Υ = Qi , Q4 , Ri , Si ;i = 1, 2 2

̄ 11 ⎡Ω ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢ ̄ = ⎢∗ Ω ⎢∗ ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢∗ ⎣

̄ 12 Ω ̄ Ω22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

̄ 14 Ω 0 0 ̄ 44 Ω ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 ̄ 55 Ω ∗ ∗ ∗ ∗ ∗

0 ̄ 26 Ω 0 0 0 ̄ 66 Ω ∗ ∗ ∗ ∗

0 ̄ 27 Ω 0 0 0 0 ̄ 77 Ω ∗ ∗ ∗

̄ 18 Ω 0 0 0 0 0 0 ̄ 88 Ω ∗ ∗

̄ 19 Ω 0 0 0 0 0 0 0 ̄ 99 Ω ∗

̄ ⎤ Ā T W ̄ ⎥ ̄BT W ⎥ T ̄ ⎥ ̄ G W ̄ ⎥ Ē 0T W ⎥ 0 ⎥<0 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ̄ ⎥⎦ −W

(11)

where ̄ 11 = P̄ T Ā + Ā T P̄ + Q ̄ 1 − (1 − u)Ē T Q ̄ 3 Ē + hQ ̄ 4 + R̄ 1 + R̄ 2 Ω 2 ̄ 2 ̄T̄ ̄ + d12 S̄ 1 + d12 S2 − d12 Ē T S̄ 3 Ē − d12 Ē T Z̄ 1 Ē − d12 E Z2 E, ̄ 14 ̄ 12 = P̄ T B̄ + (1 − u)Ē T Q ̄ 3 Ē + d12 Ē T S̄ 3 E, ̄ Ω Ω ̄ 18 = d1 Ē T Z̄ 1 E, ̄ = P̄ T Ē 0 + 𝛽 C̄ T K̄ T , Ω

̄ 22 = −(1 − u)Q ̄ 19 = d12 Ē T Z̄ 2 E, ̄ 1 − (1 − u)Ē T Q ̄ 3 Ē ̄ Ω Ω ( ) T T ̄ − d12 + d2 Ē S̄ 3 Ē − 2Ē S̄ 4 E, T̄ ̄ ̄ T̄ ̄ T̄ ̄ ̄ ̄ ̄ ̄ ̄ Ω26 = E S4 E, Ω27 = d2 E S3 E + E S4 E, Ω33 ( ) ̄2 , = −(1 − v) Ē + Ē T + 2m + m2 Q ̄ 44 = −2𝛽I, Ω ̄ 66 = −R̄ 1 − Ē T S̄ 4 E, ̄ 55 = −h(1 − v)Q ̄ 4, Ω ̄ Ω ̄ 88 ̄ 77 = −d2 Ē T S̄ 3 Ē − Ē T S̄ 4 Ē − R̄ 2 , Ω Ω T ̄ 99 = −S̄ 2 − Ē T Z̄ 2 E, ̄ Ω ̄ = −S̄ 1 − Ē Z̄ 1 E, ̄ =Q ̄2 + W

̄3 d22 Q

+

d12 d22 S̄ 3

+

2 ̄ d12 S4

( Now let y(t) = J2 −1 x(t) =

+

d14 4

Z̄ 1 +

( 2 ) 2 d2 − d12 d12 4

Z̄ 2 .

) y1 (t) , where y1 (t) ∈ Rr , y2 (t)

y2 (t) ∈ Rn−r. Plugging Eqs.  (8) and (9) into the system (5), we get ̄ y( ̄ ̄ − d(t)) + Ē 0 f (𝜂(t)) Ē y(t) ̇ −G ̇ t − h(t)) = Ay(t) + By(t

(12) ̄ where 𝜂(t) = Cy(t) . The system (12) can be decomposed as ̄ 11 ẏ 1 (t − h(t)) ẏ 1 (t) = Ā r y1 (t) + B̄ 11 y1 (t − d(t)) + B̄ 12 y2 (t − d(t)) + G ̄ 12 ẏ 2 (t − h(t)) + E01 f (𝜂(t)) +G

(9)

Pre-multiplying and post-multiplying the left and the right side of (6) by J2 and J2 T , we have

(10) P̄ T Ē = Ē T P̄ ≥ 0 Pre-multiplying and post-multiplying left and { T T the T T T T diag J , J , J , I, J , J2 , J2 , the right side of Eq.  (7) by 2 2 2 2 } T T −T J2 , J2 , J1 and it transpose, we have

(13) ̄ 21 ẏ 1 (t − h(t)) 0 = y2 (t) + B̄ 21 y1 (t − d(t)) + B̄ 22 y2 (t − d(t)) + G ̄ ̄ + G22 ẏ 2 (t − h(t)) + E02 f (𝜂(t)). (14) Next, we shall show the stability of the system (5). Choose a Lyapunov functional candidate to be

V(y(t)) =

7 ∑ l=1

13

̄ P̄ T G 0 ̄ 33 Ω ∗ ∗ ∗ ∗ ∗ ∗ ∗

Vl (y(t))

Int. J. Mach. Learn. & Cyber.

where

( ) 2 ̄ S2 y(t) − d1 V̇ 5 (y(t)) = yT (t) d12 S̄ 1 + d12 ̄T̄

T

V1 (y(t)) = y (t)E Py(t) ∫t−d(t)

̄ 1 y(s)ds + yT (s)Q

0

V3 (y(t)) = d2

t

̄ 3 Ē y(s)dsd𝛽 ̇ +h ẏ T (s)Ē T Q

t

V4 (y(t)) =

yT (s)R̄ 1 y(s)ds + 0

̄ 4 y(s)ds yT (s)Q

yT (s)S̄ 2 y(s)ds ∫t−d2 ( ) 2 ̄ ̇ S4 Ē y(t) V̇ 6 (y(t)) = ẏ T (t)Ē T d12 d22 S̄ 3 + d12 t

− d12 d2

y (s)S̄ 1 y(s)dsd𝛽 t

∫−d2 ∫t+𝛽 0

V6 (y(t)) = d12 d2

yT (s)R̄ 2 y(s)ds,

T

−d1

yT (s)S̄ 2 y(s)dsd𝛽

∫−d2 ∫t+𝛽

̇ ẏ T (s)Ē T S̄ 4 Ē y(s)ds ∫t−d2 ( 4 ) ) ( 2 d2 − d2 d1 d12 2 1 T T Z̄ + Z̄ 2 Ē y(t) ̇ V̇ 7 (y(t)) = ẏ (t)Ē 4 1 4

d12

0

+ d12

∫−d2 ∫𝜃 ∫t+𝛽

0

̇ ẏ T (s)Ē T S̄ 4 Ē y(s)dsd𝛽

d12

−d1

0

̄T̄

̇ ẏ (s)E Z1 Ē y(s)dsd𝛽d𝜃 T

t

̇ ẏ T (s)Ē T Z̄ 2 Ē y(s)dsd𝛽d𝜃

0

t

2 ∫−d1 ∫t+𝛽

2 − d12

t

2 ∫−d1 ∫𝜃 ∫t+𝛽 −d1

−

t

∫−d2 ∫t+𝛽

̇ ẏ T (s)Ē T S̄ 3 Ē y(s)ds

t−d1

̇ ẏ T (s)Ē T S̄ 3 Ē y(s)dsd𝛽

+ d12

∫t−d2

− d12

t

−d1

V7 (y(t)) =

∫t−h(t)

t

∫−d1 ∫t+𝛽

+ d12

∫t−d2

yT (s)S̄ 1 y(s)ds

− d12

t

∫t−d1

V5 (y(t)) = d1

∫t−h(t)

̄ 2 Ē y(s)ds ̇ ẏ T (s)Ē T Q

t

∫−d2 ∫t+𝛽

∫t−d1

t−d1

t

t

V2 (y(t)) =

t

̇ ẏ T (s)Ē T Z̄ 1 Ē y(s)dsd𝛽 t

∫−d2 ∫t+𝛽

̇ ẏ T (s)Ē T Z̄ 2 Ē y(s)dsd𝛽

By Lemma 1 and Remark 1, the following equation is true: t

The time-derivative of V(y(t)) along the trajectory of system (5) is given by

− d12 d2

�t−d2

t−d(t)

̇ = −d12 d2 ẏ T (s)Ē T S̄ 3 Ē y(s)ds

�t−d2

̇ ẏ T (s)Ē T S̄ 3 Ē yds T

⎤ ⎡ y(t) ⎥ ⎢ ẏ T (s)Ē T S̄ 3 Ē y(s)ds ̇ ≤ ⎢ y(t − d(t)) ⎥ − d12 d2 �t−d(t) ⎢ y�t − d � ⎥ 2 ⎦ ⎣ t

̇ V(y(t)) =

7 ∑

V̇ l (y(t))

l=1

where V̇ 1 (y(t)) = 2yT (t)Ē T P̄ y(t) ̇ ) ( ̄ − d(t)) ≤ yT (t) P̄ T Ā + Ā T P̄ y(t) + 2yT (t)P̄ T By(t ̄ y(t + 2yT (t)P̄ T G ̇ − h(t)) + 2yT (t)P̄ T Ē 0 f (𝜂(t)) [ ] T ̄ + 2𝛽f (𝜂(t)) K𝜂(t) − f (𝜂(t))

̄ 1 y(t) − (1 − u)yT (t − d(t))Q ̄ 1 y(t − d(t)) V̇ 2 (y(t)) ≤ yT (t)Q T T ̄ ̄ T ̄ ̄ 2 Ē y(t ̇ − (1 − v)ẏ (t − h(t))Ē T Q ̇ − h(t)) + ẏ (t)E Q2 Ey(t)

( ) ̄ 3 Ē y(t) V̇ 3 (y(t)) ≤ ẏ T (t)Ē T d22 Q ̇ − d2 (1 − u) t

�t−d(t)

̄ 4 y(t) ̄ 3 Ē y(s)ds ̇ + hyT (t)Q ẏ T (s)Ē T Q

̄ 4 y(t − h(t)) − h(1 − v)yT (t − h(t))Q ) ( ) ( ) ( ̇V4 (y(t)) = yT (t) R̄ 1 + R̄ 2 y(t) − yT t − d1 R̄ 1 y t − d1 ( ) ( ) − yT t − d2 R̄ 2 y t − d2

⎡ −d12 Ē T S̄ 3 Ē ⎢ ⎢∗ ⎢∗ ⎣

d12 Ē T S̄ 3 Ē � � − d12 + d2 Ē T S̄ 3 Ē ∗

⎤⎡ y(t) ⎤ 0 ⎥⎢ ⎥ T̄ ̄ ̄ d2 E S3 E ⎥⎢ y(t − d(t)) ⎥ � � −d2 Ē T S̄ 3 Ē ⎥⎦⎢⎣ y t − d2 ⎥⎦

Similarly, by Lemma 1 and Remark 1, we can shrink V̇ 3 (y(t)), V̇ 5 (y(t)), V̇ 6 (y(t)), V̇ 7 (y(t)). By Lemma 3, there exists a appropriate matrix, positive scalar m > 0, such that

̄ 2 Ē y(t ̇ − h(t)) −(1 − v)ẏ T (t − h(t))Ē T Q ( ) T T ̄ ̄ 2 y(t ≤ −(1 − v)ẏ (t − h(t)) E + Ē + 2mI + m2 Q ̇ − h(t)) Lurie singular system of neutral type (5) is equivalent to [ ] ̄ Ē 0 0 0 0 0 0 , Ē y(t) ̇ =[ Γ𝜍(t), Γ = Ā B̄ G 𝜍(t) = yT (t) yT (t − d(t)) ẏ T (t − h(t)) f T (𝜂(t)) yT (t − h(t)) ] ( ) ( ) t t−d yT t − d1 yT t − d2 ∫t−d yT (s)ds ∫t−d 1 yT (s)ds . 2

13



Int. J. Mach. Learn. & Cyber.

From the above discussion, we can get ( ) ̇ ̄ 𝜍(t) V(y(t)) ≤ 𝜍 T (t) Φ + ΓT WΓ where Φ is the first nine rows and nine columns of Eq. (11) ̄ . By Schur complement we can obtain inequality matrix Ω ̇ x(t)) < 0. Then (11). Therefore, if Ω < 0, in other words V(̄ we have

T �2 ̄ 𝜆1 � �y1 � − V(y(0)) ≤ y1 (t)P11 y1 (t) − V(y(0)) T T̄ ̄ ≤ y (t)E Py(t) − V(y(0)) ≤ V(y(t)) − V(y(0))

=

�0

t

̇ V(y(s))ds ≤ −𝜆2

≤ −𝜆2

�0

Theorem  2  Given positive scalars 0 ≤ d1 ≤ d2 , h > 0, 0 ≤ u < 1, 0 ≤ v < 1, the system (1) is absolute stability, if there exist positive definite symmetric matrices P, Q1 , Q2 , Q3 , Q4 , R1 , R2 , S1 , S2 , S3 , S4 , Z1 , Z2 such that the following LMI holds:

PT E = ET P ≥ 0

t

‖y(s)‖2 ds

t

�y (s)�2 ds < 0. �0 � 1 �

(15)

where

( ) 𝜆1 = 𝜆min ET P > 0, 𝜆2 = −𝜆max (Ξ) > 0. Taking into account (15), we can deduce that t‖ ‖2 ‖2 the above 𝜆1 ‖ ‖y1 (t)‖ + 𝜆2 ∫0 ‖y1 (s)‖ ds ≤ V(y(0)). From 2 ‖ ‖ discussion, we have0 < ‖y1 (t)‖ ≤ 𝜆1 V(y(0)), t 1 ‖2 0 < ∫0 ‖ ‖y1 (s)‖ ds ≤ 𝜆2 V(y(0)). t‖ ‖ ‖2 Thus ‖ ‖y1 (t)‖ and ∫0 ‖y1 (s)‖ ds are bounded. Moreover, taking into Eq. (2), we can deduce that 1

�2 �Ē f (𝜂(t))�2 ≤ �Ē K̄ Cy(t) ̄ � � 02 � � 02 √ � � � � � � � ̄ ⇒ �E20 f (𝜂(t))� ≤ 𝜆3 �y1 (t)� � + �y2 (t)� .

(16)

[

] T ̄ where 𝜆3 = 𝜆max C̄ T K̄ T Ē 02 E02 K̄ C̄ . Taking into account Eqs. (14) and (16), we have ‖x̄ (t)‖ − ‖B̄ y (t − d(t))‖ − ‖B̄ y (t − d(t))‖ ‖ 2 ‖ ‖ 21 1 ‖ ‖ 22 2 ‖ ̄ 21 ẏ 1 (t − h(t))‖ + ‖G ̄ 22 ẏ 2 (t − h(t))‖ + ‖Ē 02 f (𝜂(t))‖ G ≤‖ ‖ ‖ ‖ ‖ ‖ ‖

(17) ‖ is bounded. Using So it can deduce that ‖ y (t) ‖ 2 ‖ same method, considering Eqs. (13) and (16), we have ‖ that ‖ ‖ẏ 1 (t)‖ is bounded, and using Lemma 5, we get limt→∞ y1 (t) = 0. Using same method, it follows from Eq. (15) that t‖ 2 ‖ ‖2 ∫0 ‖y2 (s)‖ ‖ ds and ‖ẏ 2 (t)‖ are bounded. Therefore limt→∞ y2 (t) = 0. Then, according to the Definition 2 and 3, the system (5) is robustly absolute stable, this completes the proof.

Remark 2  The conservatism of stability conditions is a topic of research. Constructing multi-Lyapunov functions is an efficient approach to reduce the conservatism. And introducing those free weighting matrices P, Q1 , Q2 , Q3 , Q4 , R1 , R2 , S1 , S2 , S3 , S4 , Z1 , Z2 play an important role in reducing the conservatism, which can be seen from Example 1 and 2.

13

According to Theorem 1 and Lemma 2, it can be generalized to its structure uncertain the Lurie neutral singular system, we have the following theorem:

𝜃2T ⎡ Ω 𝜃1 ⎤ ⎢ ∗ −𝜀0 I 0 ⎥ < 0, ⎢ −1 ⎥ ∗ ∗ −𝜀 I ⎣ ⎦ 0

(18)

(19)

where

[ ] 𝜃2 = Na Nb Nc Nd 01×6 . Proof  Firstly, uncertain Lurie singular system of neutral type (1) is regular and impulse free. Then replacing A, B, G, E0 in (7) with A + ΔA, B + ΔB, G + ΔG, E0 + ΔE0, then (7) can be written

Ω + 𝜃1 F𝜃2 + 𝜃2T F T 𝜃1T < 0.

(20)

By Lemma 2, a sufficient condition guaranteeing Eq. (20) for Eq. (1) is that there exists a positive number 𝜀0 > 0 such that

Ω + 𝜀−1 𝜃 𝜃 T + 𝜀0 𝜃2T 𝜃2 < 0. 0 1 1

(21)

Applying the Schur complement shows Eq. (21) is equivalent to Eq. (19). The proof is completed. Remark 3 When E = I, G + ΔG = 0, d(t) = d2 = d, d1 = 0, the system (1) reduces to the uncertain Lurie control system in [3]: � � ⎧ x(t) ̇ = (A + ΔA) x(t) + (B + ΔB) x(t − d) + E0 + ΔE0 f (𝜎(t)) ⎪ ⎨ 𝜎(t) = Cx(t) ⎪ x(t) = 𝜑(t), t ∈ [−d, 0] ⎩

(22) Based on the method of Theorem 2, it is easy to obtain the following results for the uncertain Lurie control system (22). Corollary 1  Given a positive scalar d the system (22) is absolute stability, if there exist positive definite symmetric

Int. J. Mach. Learn. & Cyber.

matrices P, Q1 , Q3 , R2 , S2 , S3 , S4 , Z2 , such that the following LMI holds:

Λ11 = PT A + AT P + Q1 − (1 − u)Q3 + hQ4 + R1 2 2 + R2 + d12 S1 + d12 S2 − d12 S3 − d12 Z1 − d12 Z2 ,

Λ12 = PT B + (1 − u)Q3 + d12 S3 , Λ14 = PT E0 + 𝛽CK, T ⎡ X11 X12 P E0

⎢∗ ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢∗ ⎢ ⎣∗

X22 ∗ ∗ ∗ ∗ ∗

+

𝛽CT K T

0 −2𝛽I ∗ ∗ ∗ ∗

dZ2 0 0 X44 ∗ ∗ ∗

AT W

PT L

BT W 0 0 −W ∗ ∗

0 0 0 XT L −𝜀I ∗

T

Na ⎤ Nb T ⎥ Nd T ⎥ ⎥ 0 ⎥ < 0, ⎥ 0 ⎥ 0 ⎥ −𝜀−1 I ⎦

where

X11 = PT A + AT P + Q1 − Q3 + R2 + d2 S2 − dS3 − d2 Z2 , X12 = PT B + Q3 + dS3 , X44 = −dS2 − Z2 , X22 = −Q1 − Q3 − 2dS3 − 2S4 , W=d3 S3 + d2 S4 +

d4 Z. 4 2

E = I, ΔA = 0, ΔB = 0, ΔG = 0, Remark 4 When ΔE0 = 0, the system (1) reduces to neutral Lurie control system with mixed time-varying delays in [7]: ⎧ x(t) ̇ − Gx( ̇ t − h(t)) = Ax(t) + Bx(t − d(t)) + E0 f (𝜎(t)), ⎪ ⎨ 𝜎(t) = Cx(t), ⎪ x(t) = 𝜑(t), t ∈ �− max �h, d �, 0�. 2 ⎩ (23) Based on the method of Theorem 1, it is easy to obtain the following results for neutral Lurie control system. Corollary 2  Given positive scalars 0 ≤ d1 ≤ d2 , h > 0, 0 ≤ u < 1, 0 ≤ v < 1, the system (23) is absolute stability, if there exist positive definite symmetric matrices P, Q1 , Q2 , Q3 , Q4 , R1 , R2 , S1 , S2 , S3 , S4 , Z1 , Z2 such that the following LMI holds:

⎡ Λ11 ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢∗ ⎣ where

Λ12 Λ22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

PT G 0 Λ33 ∗ ∗ ∗ ∗ ∗ ∗ ∗

Λ14 0 0 Λ44 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 Λ55 ∗ ∗ ∗ ∗ ∗

0 Λ26 0 0 0 Λ66 ∗ ∗ ∗ ∗

0 Λ27 0 0 0 0 Λ77 ∗ ∗ ∗

Λ18 0 0 0 0 0 0 Λ88 ∗ ∗

Λ19 0 0 0 0 0 0 0 Λ99 ∗

AT W ⎤ BT W ⎥⎥ GT W ⎥ E0T W ⎥ ⎥ 0 ⎥ < 0, 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 −W ⎥⎦

Λ18 = d1 Z1 , Λ19 = d12 Z̄ 2 ,

( ) Λ22 = −(1 − u)Q1 − (1 − u)Q3 − d12 + d1 S3 − 2S4 , Λ26 = S4 , Λ27 = d2 S3 + S4 , Λ33 = −(1 − v)Q2 , Λ44 = −2𝛽I, Λ55 = −h(1 − v)Q4 , Λ66 = −R1 − S4 , Λ77 = −d2 S3 − S4 − R2 , Λ88 = −S1 − Z1 , Λ99 = −S2 − Z2 , ( ) d2 d2 − d12 d4 2 Z2 . S4 + 1 Z1 + 12 2 W = Q2 + d22 Q3 + d12 d2 2 S3 + d12 4 4

G + ΔG = 0, B + ΔB = 0, ΔA = Remark 5 When ΔE0 = 0, the system (1) reduces to the following Lurie descriptor system in [18]: { Ex(t) ̇ = Ax(t) + E0 f (𝜎(t)), (24) 𝜎(t) = Cx(t). Corollary 3  Given positive a scalar 𝛽, the system (24) is absolute stability, if there exists positive definite symmetric matrix P such that the following LMI holds:

PT E = ET P ≥ 0, [

PT A + AT P PT E0 + 𝛽CK ∗ −2𝛽I

] < 0.

4 Numerical example In this section, we will give some simple numerical examples to show the effectiveness of the proposed conditions. Example 1  Consider the following system: when E = I, ΔA = ΔB = 0, ΔG = ΔE0 = 0 the system can lead to the neutral Lurie system as in Qiu [4]: [

] [ ] [ ] −0.9 0.2 −1.1 −0.2 −0.2 0 , B= , G= , 0.1 −0.9 −0.1 −1.1 0.2 −0.1 [ ] [ ] −0.2 0.1 0.3 −0.2 , C= . E0 = −0.45 −0.3 0.3 0.1 A=

Let the neutral delay h(t) be time-varying with derivative v = 0.1. Table 1 shows the maximum values of d2 = h which guarantee stability of this system by applying Theorem 1 in this work. Example 1 shows that the delaydependent stability condition in this note gives better results.

13



Int. J. Mach. Learn. & Cyber.

Table 1  Comparisons of maximum allowed delay d1 = 0, d2 = h with v = 0.1 (Example 1)

𝜇

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Qiu [4] Theorem 1 (f ∈ K[0, 0.5])

1.14 1.71

0.89 1.60

0.81 1.49

0.79 1.39

0.78 1.29

0.76 1.20

0.75 1.10

0.75 1.00

0.74 0.90

Example 2  Consider the following neutral Lurie system with same parameters given in Liu [7]: ( ) x(t) ̇ − Cx(t ̇ − 𝜏(t)) = (A + ΔA(t))x(t) + A1 + ΔA1 (t) x(t − h(t)) 𝜎(t) = DT x(t), t ≥ 0.

+ (B + ΔB(t))f (𝜎(t)),

where [ A=

] [ ] [ ] [ ] −2 0.5 1 0.4 10 11 , B= , E0 = , C= , 0 −1 0.4 −1 01 00

[

] [ ] 2𝛾1 0 10 L= , Na = Nb = 0, Nd = , 0 2𝛾2 01 [ ] { } 1 cos t × sgn x1 (t − d2 ) 0 2 { } F(t) = 1 sin t × sgn x1 (t − d2 ) 0 2 Let u = 0.01, 𝜈 = 0.01, 𝛽 = 0.04, 𝜀 = 0.1 and the lower bound delay be d1 = 0, we can obtain the maximum allowable delay bound d2 = h is listed in Table 2. From the Table 2, we can see that the method in this paper produces better result than [7, 31, 32]. Moreover, in order to illustrate the stability, the simulation result is given for d2 = h = 1.83 in Fig. 1. It is easy to see that the state trajectories approach to zero asymptotically stable. Example 3  Consider a system is first given in [33], which is represented by and

Fig. 1  State trajectories with [ ]T x(0) = 1.0 −1.0 (Example 2)

d2 = h = 1.832

current, v and are vR voltages. The characteristic of the current controlled resistor represented by vR = g(iR ) satisfies g ≤ g� (iR ) ≤ ḡ , where g] ≥ 0. Let U = 1, V = 1, g = 1, ḡ = 2 [ − −T − and denote x = v iR and f (iR ) = g(iR ) − giR. Then the − system can be described in the form of system (5), with [ E=

] [ ] [ ] [ ]T 10 −1 1 0 0 ,A= , E0 = ,C= ,K=1 00 −1 −1 1 1

According to the Corollary 3, by using LMI toolbox in MATLAB, we have [ ] 1.2191 −0.2816 P= −0.2816 1.1580

U v̇ = −Vv + iR 0 = −v − g(iR ) The system describe a nonlinear RC circuit shown in Fig.  2, where V is assumed to be time-varying and g(iR ) is only sector constrained, U is a linear capacitance, iR is a

Table 2  Comparison of the maximum allowable delay bound d2 = h (d1 = 0) (Example 2) Park [31]

Yin et al. [32]

Liu [7]

Theorem 2 in the paper

0583

0.672

1.7553

1.832

13

and

Fig. 2  State trajectories with x(0) = [1.0;0.5] (Example 3)

Int. J. Mach. Learn. & Cyber.

[

] [ ] 0.0045 −0.0004 0.0382 −0.0030 Q4 = , S2 = , −0.0004 0.0001 −0.0030 0.0006 ] [ 0.0118 0.0004 S3 = 0.0004 0.0029 ] ] [ [ 0.0391 0.0013 0.0118 −0.0008 , R1 = , S4 = 0.0013 0.0015 −0.0008 0.0001 ] [ 0.0120 −0.0008 R2 = −0.0008 0.0001 [ ] 2.8699 0.0933 Z1 = 1.0e + 005 ∗ , 0.0933 1.2719 [ ] 0.0473 0.0017 Z2 = 0.0017 0.0234

Fig. 3  A nonlinear RC circuit (Example 3) Table 3  Maximal sector bounds (Example 4) max(𝛿)

k1 = k2 = 𝛿

k1 = 0.5k2 = 𝛿

Theorem 2 in this paper

0.298

0.439

For simulation, let the nonlinearity be f (𝜎(t)) = 0.5(x2 + sin(x2 )). Fig.  3 shows the state responses of the nonlinear descriptor system. Example 4  Consider the system (1) described by [

]

[

]

[

]

[

−2 0.2 −0.5 0.2 2 0 2 0 ,B = ,G = , E0 = 0.1 −0.9 0.1 −0.2 0 2 1 2 ] ] ] [ [ [ 0.01 0 0.01 0.01 0.01 0 Na = , Nb = Nc = 0 0.01 0 0.01 0 0.01 ] [ ] [ [ ] ] [ k1 0 1 0 0.01 0 1 0 ,K = ,E = Nd = ,L = 0 k2 0 0 0.01 0.01 0 1

]

A=

when k1 = 0.5k2 = 𝛿 lead to [ ] [ ] 0.0245 −0.0080 0.0243 −0.0087 P= , Q1 = , −0.0080 0.0072 −0.0087 0.0037

[

] 0.4543 −0.0097 Q2 = 1.0e − 003 ∗ , −0.0097 0.7077 [ ] 0.0029 −0.0001 , Q3 = −0.0001 0.0028 [

] [ ] 0.0048 −0.0005 785.7067 −11.2515 , S1 = , −0.0005 0.0001 −11.2515 1.9722

[

] [ ] 0.0438 −0.0048 0.0087 −0.0002 , S3 = , −0.0048 0.0013 −0.0002 0.0056

[

] [ ] 0.0469 −0.0013 0.0109 −0.0011 , R1 = , −0.0013 0.0029 −0.0011 0.0002

Q4 =

S2 =

S4 =

v = 0.01, u = 0.3, m = 0.8, d2 = 0.5, d1 = 0.01, 𝛽 = 0.04, h = 2, 𝜀0 = 0.1

In this example, we will consider two cases: k1 = k2 = 𝛿 and k1 = 0.5k2 = 𝛿. Using Theorem 2 to calculate the maximal sector bounds max (𝛿) by choosing a big, the maximal bounds are summarized in Table 3. And using LMI toolbox in MATLAB, when k1 = k2 = 𝛿 lead to.

[

] [ ] 0.0238 −0.0055 0.0193 −0.0064 , Q1 = , −0.0055 0.0060 −0.0064 0.0027 ] [ 0.0029 0.0001 Q3 = 0.0001 0.0014 ] [ 0.3535 0.0126 , Q2 = 1.0e − 003 ∗ 0.0126 0.3590 ] [ 1.5654 −0.0078 S1 = 1.0e + 003 ∗ −0.0078 0.0009 P=

[

] 0.0110 −0.0011 , −0.0011 0.0002 [ ] 5.6145 −0.0245 Z1 = 1.0e + 004 ∗ , −0.0245 4.3399 R2 =

[

] 0.0608 −0.0013 Z2 = . −0.0013 0.0455 From the example we can obtain feasible solution by applying the Theorem 2 for above the system in this paper. We consider the regularity and absence of impulses on the base of the neutral Lurie system. So our system is more extensive.

13



5 Conclusions In this paper, the problem of delay-dependent robust absolute stability for uncertain Lurie singular systems of neutral type with time-varying delays is concerned. By introducing a new integral inequality, a sufficient condition is obtained which guarantees that the Lurie singular systems of neutral type with time delay is regular, impulse free and robust absolute stability. At last the numerical examples presented clearly endorses that proposed method could obtain less conservative results compared to existing ones in the same frame work. Moreover, H∞ performance is a hot spot in the robust control. To this point, further study would focus on robust H∞ control for uncertain Lurie singular systems of neutral type with time-varying delays. Acknowledgements  This paper was supported by the National Natural Science Foundation of China, under Grant Number 61273004 and Natural Science Foundation of Hebei Province, under Number F2014203085. The authors would like to thank the editor and anonymous reviewers for their many helpful comments and suggestions to improve the quality of this paper.

References 1. Lurie AI (1957) Some nonlinear problem in the theory of automatic control. H. M. Stationary Office, London 2. Basar T (1961) Absolute stability of nonlinear systems of automatic control. Autom Remote Control 22(22):961–979 3. Gao JF, Su HY, Ji XF (2008) New delay-dependent absolute stability criteria for Lurie control systems. Acta Autom Sin 34:1275–1280 4. Qiu F, Cui BT, Ji Y (2010) Delay-dividing approach for absolute stability of Lurie control system with mixed delays. Nonlinear analysis. Real World Appl 11:3110–3120 5. Liao F, Yu X, Deng J (2017) Absolute stability of time-varying delay Lurie indirect control systems with unbounded coefficients. Adv Diff Equ 2017(1):38 6. Mukhija P, Kar IN et  al (2014) Robust absolute stability criteria for uncertain Lurie system with interval time-varying delay. J Dyn Syst Measur Control 136(4):041020 7. Liu PL (2016) Robust absolute stability criteria for uncertain Lurie interval time-varying delay systems of neutral type. ISA Trans 60:2–11 8. Xie L, De Souza CE (1992) Robust control for linear systems with norm-bounded time-varying uncertainty. IEEE Trans Autom Control 37:1188–1191 9. Wang D, Liao FC (2003) Absolute stability of Lurie direct control systems with time-varying coefficients and multiple nonlinearities. Appl Math Comput 219:4465–4473 10. Yang CY, Zhang QL, Zhou LN (2007) Lur’e Lyapunov functions and absolute stability criteria for Lur’e systems with multiple nonlinearities. Robust Linear Control 17:829–841 11. Wang Y, Zhang X, Zhang X (2016) Neutral-delay-range-dependent absolute stability criteria for neutral-type Lur’e systems with time-varying delays. J Franklin Inst 353(18):5025–5039 12. Mukhija P, Kar IN et  al (2012) Delay-distribution-dependent robust stability analysis of uncertain Lurie systems with timevarying delay. Autom Remote Control 38(7):1100–1106

13

Int. J. Mach. Learn. & Cyber. 13. Mukhija P, Kar IN et  al (2014) Absolute stability analysis of neutral type Lurie system with unbounded distributed delay and interval time-varying probabilistic delay. Int J Model Identif Control 21(1):72–81 14. Wang X, She K, Zhong S et al (2016) New result on synchronization of complex dynamical networks with time-varying coupling delay and sampled-data control. Neurocomputing 214:508–515 15. Dai L (1989) Singular control systems. Springer, Berlin 16. Lu RQ, Su HY, Chu J (2003) Robust control for a class of uncertain Lur’e singular systems with time-delays. IEEE Conf Decis Control 6:5585–5590 17. Lu RQ, Wang JH, Xue AK, Su HY, Chu J (2007) Robust filtering for a class of uncertain Lur’e time-delay singular systems. Acta Autom Sin 33:292–296 18. Yang CY, Zhang QL, Zhou LN (2009) Strongly absolute stability problem of Lur’e descriptor systems: Popov criteria. Int J Robust Nonlinear Control 19:786–806 19. Yang CY, Zhang QL, Sun J, Chai TY (2011) Lur’e Lyapunov function and absolute stability criterion for Lur’e singularly perturbed systems. IEEE. Trans Autom Control 56:2666–2671 20. Zhou PF, Wang YX, Wang QB, Chen J, Duan DP (2013) Stability and passivity analysis for Lur’e singular systems with markovian switching. Int J Autom Comput 40:79–84 21. Wang HJ, Xue AK, Lu RQ (2009) Absolute stability criteria for a class of nonlinear singular systems with time delay. Nonlinear Anal 70:621–630 22. Ma YC, Yang PJ, Yan YF et  al (2017) Robust observer-based passive control for uncertain singular time-delay systems subject to actuator saturation. ISA Trans 67:9–18 23. Ma SP, Zhang CH (2012) H ­ ∞ Control for discrete-time singular Markov jump systems subject to actuator saturation. J Frankl Inst 349:1880–1902 24. Sing HV (2004) Robust stability of cell neural networks with delay. Linear matrix inequality approach. IEEE Proc Control Theory Appl 151:125–129 25. Ma YC, Yang PJ, Zhang QL (2016) Robust control for uncertain singular discrete T–S fuzzy time-delay systems with actuator saturation. J Franklin Inst 353(13):3290–3311 26. Wang Y, Zhang X, He Y (2012) Improved delay-dependent robust stability criteria for a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays and sector-bounded nonlinearity. Nonlinear Anal Real World Appl 13(5):2188–2194 27. Gao JL, Gao Q (2012) Stability research of delay neutral Lurie systems. International conference on measurement. Inf Control 2:857–860 28 Yin C, Zhong SM, Chen WF (2010) On delay-dependent robust stability of a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays. J Franklin Inst 347(9):1623–1642 29 Liu YJ, Lee SM et  al (2015) Robust delay-dependent stability criteria for time-varying delayed Lur’e systems of neutral type. Circuits Syst Signal Process 34(5):1481–1497 30 Liu PL (2013) A delay decomposition approach to stability analysis of neutral systems with time-varying delay. Appl Math Model 37:5013–5026 31 Liu PL (2015) Delayed-decomposition approach for absolute stability of neutral-type Lurie control systems with time-varying delays. J Dyn Syst Meas Control 137(8):1002–1001 32 Park JH (2005) Novel robust stability criterion for a class of neutral systems with mixed delays and nonlinear perturbations. Appl Math Comput 161(2):413–421 33 Vladimir BB (1986) Partial stability of motion of semi-state systems. Int J Control 44:1383–1394