Ma and Li Journal of Inequalities and Applications (2018) 2018:293 https://doi.org/10.1186/s13660-018-1886-5

R E S E A R C H Open Access

Stability criteria on delay-dependentrobust stability for uncertain neutralstochastic nonlinear systems with time-delayTengyu Ma1,2 and Longsuo Li1*

*Correspondence:lilongsuo6892@126.com1Department of Mathematics,Harbin Institute of Technology,Harbin, ChinaFull list of author information isavailable at the end of the article

AbstractThis work mainly studies the robust stability analysis and design of a controller foruncertain neutral stochastic nonlinear systems with time-delay. Using a modifiedLyapunov–Krasovskii functional and the free-weighting matrices technique, weestablish some new delay-dependent criteria in terms of linear matrix inequality (LMI).The innovative point of this work is that we generalize the robust stability analysis ofnonlinear stochastic time-delay systems to the uncertain neutral stochastic systems.Due to the added derivative term of time-delay, the proposed scheme can be appliedmore widely. Finally, numerical examples are provided to validate the derived results.

Keywords: Uncertain neutral stochastic systems; One-sided Lipschitz condition;Lyapunov–Krasovskii functional; Linear matrix inequality (LMI)

1 IntroductionTime-delay systems are widely used to model concrete systems in engineering sciences,such as biology, chemistry, mechanics [1–3]. Time-delay systems, with the rate of currentstate affected by past state, are negative for the analysis and design of control systemssince they may be responsible for performance degradation and instability [4, 5]. There aremany valuable results about stability conditions for time-delay systems [6–9]. Generally,the delay-dependent stability condition is less conservative than the delay-independentone. Thus, pursuing the delay-dependent stability condition motivates the present study[10–14].

Over the past years, the Brownian motion phenomenon has been common in biol-ogy, economics, and engineering applications. Considerable attention has been devoted tostochastic systems governed by Itô stochastic differential equations, where the noises aredescribed by Brownian motion [15, 16]. A large number of works that focused on stochas-tic time-delay systems have been published; see, for example, [17–20]. Wang et al. in [19]considered the problems of non-fragile robust stochastic stabilization and robust H∞ con-trol for uncertain stochastic nonlinear time-delay systems. Both the robust stability anal-ysis and non-fragile robust control for a class of uncertain stochastic nonlinear time-delaysystems that satisfy a one-sided Lipschitz condition were investigated in [17]. Based onthe stochastic Lyapunov–Krasovskii stability approach, the problem of stochastic stabilityanalysis was investigated for H∞ control of uncertain stochastic Markovian jump systems

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Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 2 of 19

(SMJSs) with mixed time-varying delays [18]; meanwhile, some delay-dependent sufficientconditions on the stochastic stability and γ -disturbance attenuation were presented.

The neutral stochastic systems can effectively model a class of physical dynamical sys-tems, since the mathematical models of them include the time-delays of state and itsderivative. These models have received considerable attention recently [21–25]. In fact,neutral stochastic systems are applied widely in automatic control, aircraft stabilization,lossless transmission lines, and system of turbojet engine [26–28]. Both the stability anal-ysis and synthesis of neutral stochastic systems have been extensively studied [29–36].By using the generalized integral inequality and the nonnegative local martingale conver-gence theorem, the authors in [30] investigated the exponential stability and the almostsure exponential stability of neutral stochastic delay systems (NSDSs) with Markovianswitching. The authors in [33] constructed a new sliding surface functional and consid-ered the H∞ sliding mode control (SMC) for uncertain neutral stochastic systems withMarkovian jumping parameters and time-varying delays. Using a delayed output-feedbackcontrol method, Karimi et al. in [36] designed a controller, which guarantees H∞ synchro-nization of the second-order neutral master and slave systems.

On the other hand, a useful one-sided Lipschitz condition was developed in [37]. Sincethe nonlinear part satisfying this condition can make positive contributions to the stabilityof systems, we can easily solve the observer design problem for nonlinear systems [17, 38–40]. Inspired by the above works, we investigate the robust stability of neutral stochastictime-delay nonlinear systems with one-sided Lipschitz condition.

In this paper, we propose a class of uncertain neutral stochastic nonlinear systems.Since the systems have a derivative term of time-delay of state, they can be used in lotsof fields. We investigate the nonlinear function with both one-sided Lipschitz conditionand a quadratic inner-bounded condition. Firstly, a delay-dependent sufficient conditionis proposed by constructing an appropriate Lyapunov–Krasovskii functional based on thefree-weighting matrices method. Secondly, we construct a memory-less non-fragile state-feedback controller to guarantee asymptotical stability of the closed-loop systems. Finally,we present some numerical examples to illustrate the advantages and effectiveness of ourresults and find that the proposed method is less conservative.

The organization of this paper is given as follows. In the next section, we recall somenotations, lemmas, and definitions of stochastic differential equations. In Sect. 3, themain problems are formulated. In Sect. 4, we give two delay-dependent sufficient con-ditions for uncertain neutral stochastic nonlinear time-delay systems. In Sect. 5, we de-sign a memory-less non-fragile state-feedback controller to guarantee that the closed-loopsystems are asymptotically stable, and in Sect. 6, we present two numerical examples todemonstrate the validity of the mentioned method. The last section contains a conclusion.

2 Notations and preliminariesIn this section, we introduce some basic concepts, properties, and notations. These basicfacts can be found in any introductory book on stochastic differential equations; see, forexample, [41–43].

Throughout this paper, let (Ω ,F ,P) be a complete probability space with a filtration{Ft}t≥0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and isright continuous). B(t) is a one-dimensional Brownian motion defined on the probabilityspace adapted to the filtration. Rn and R

m×n denote the n-dimensional Euclidean space

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 3 of 19

and the set of all m×n real matrices, respectively. ‖·‖2 stands for the usual L2[0,∞) norm.The inner product of vectors x and y in R

n is denoted by < x, y > or xT y. Let C2,1(Rn ×R+;R+) denote the family of all real-valued functions V (x(t), t) defined on R

n × R+ suchthat they are continuously twice differentiable in x and once in t. C([–τ , 0];Rn) denotesthe space of all continuous Rn-valued functions ϕ defined on [–τ , 0] with a norm ‖ϕ‖ =sup–τ≤θ≤0|ϕ(θ )|. The notation P > 0 means that P is real symmetric and positive definite;the asterisk “∗” denotes a matrix that can be inferred by symmetry and the superscript“T” represents the transpose of a matrix or a vector. In a matrix, (i, j) denotes an (i, j)-block element of the matrix. The notation E{·} represents the mathematical expectationoperator. I denotes the identity matrix of compatible dimension.

Definition 2.1 ([17]) The nonlinear function f (x, y) is said to be one-sided Lipschitz ifthere exist α1,α2 ∈R such that

⟨f (x, y), x

⟩ ≤ α1xT x + α2yT y (1)

for ∀x, y ∈Rn, where constants α1 and α2 are positive, zero, or even negative, and they are

called one-sided Lipschitz constants for f (x, y) with respect to x and y.

Definition 2.2 ([17]) The nonlinear function f (x, y) is called quadratic inner-bounded inthe region C if, for any x, y ∈ C, there exist constants β1, β2, and γ such that

f (x, y)T f (x, y) ≤ β1xT x + β2yT y + γ⟨x, f (x, y)

⟩. (2)

Lemma 2.1 (Schur complement [43]) For a given symmetric matrix

S =

[S11 S12

ST12 S22

]

,

the following conditions are equivalent:(1) S < 0,(2) S11 < 0, S22 – ST

12S–111 S12 < 0,

(3) S22 < 0, S11 – S12S–122 ST

12 < 0.

Lemma 2.2 ([44]) Let E ∈Rn, G ∈R

n, and ε > 0. Then we have

ET G + GT E ≤ εGT G + ε–1ET E.

Lemma 2.3 (S-procedure [45]) Denote the set Z = {z}, and let F (z), y1(z), y1(z), . . . , yk(z)be some functions or functionals. Further define the domain D as follows:

D ={

z ∈Z : y1(z) ≥ 0, y2(z) ≥ 0, . . . , yk(z) ≥ 0}

,

and the two following conditions:(1) F (z) > 0,∀z ∈D,

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 4 of 19

(2) ∃ ε1 ≥ 0, ε2 ≥ 0, . . . , εk ≥ 0 such that

S(ε, z) = F (z) –k∑

i=1

εiyi(z) > 0, ∀z ∈Z .

Then (2) implies (1). The procedure of replacing (1) by (2) is called the S-procedure.

3 Problem formulationConsider the following uncertain neutral stochastic time-delay system described in Itô’sform:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

d[x(t) – Cx(t – τ (t))]

= [(A + A(t))x(t) + (Aτ + Aτ (t))x(t – τ (t))

+ f (x(t), x(t – τ (t)) + Uu(t)] dt

+ [(H + H(t))x(t) + (Hτ + Hτ (t))x(t – τ (t))] dB(t),

x(t) = φ(t), t ∈ [–τ , 0],

(3)

where x(t) ∈Rn is the state vector, u(t) ∈R

p is the control input, and τ (t) is the unknowntime-varying delay satisfying 0 ≤ τ (t) < τ and τ (t) ≤ d with real constants τ , d. f (x(t), x(t –τ (t))) ∈R

n is a nonlinear function with respect to the state x(t) and the delayed state x(t –τ (t)), f (0, 0) = 0. φ(t) ∈ C([–τ , 0];Rn) is a continuous vector-valued initial function, andB(t) is a one-dimensional Brownian motion satisfying

E{

dB(t)}

= 0, E{

dB(t)}2 = dt,

in which A, Aτ , C, H , Hτ ∈Rn×n, and U ∈R

n×p are known real constant matrices of appro-priate dimensions. Moreover, A(t), Aτ (t), H(t), and Hτ (t) are unknown matricesrepresenting time-varying parameter uncertainties and are assumed to be of the form

[A(t) Aτ (t)H(t) Hτ (t)

]

=

[E1

E2

]

F(t)[G1 G2

], (4)

where E1, E2, G1, andG2 are known real constant matrices and F(t) is an unknown time-varying matrix function satisfying

FT (t)F(t) ≤ I, ∀t ∈R+. (5)

The parameter uncertainties A(t), Aτ (t), H(t), and Hτ (t) are said to be admissibleif both (4) and (5) hold [17].

4 Robust stability analysisLet

h(t) = F(t)[G1x(t) + G2x

(t – τ (t)

)],

h1(t) = Ax(t) + Aτ x(t – τ (t)

)+ f

(x(t), x

(t – τ (t)

))+ E1h(t) + Uu(t).

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System (3) can be rewritten as follows:⎧⎨

⎩d[x(t) – Cx(t – τ (t))] = h1(t) dt + [Hx(t) + Hτ x(t – τ (t)) + E2h(t)] dB(t),

h(t)T h(t) ≤ [G1x(t) + G2x(t – τ (t))]T [G1x(t) + G2x(t – τ (t))].(6)

We will consider the problem of robust stability for time-varying delay system (6) withu(t) = 0.

Theorem 4.1 Consider the neutral stochastic time-delay system (6) with u(t) = 0. The non-linear function f (x(t), x(t –τ (t))) satisfies (1) and (2). For given scalars τ and d, if there existmatrices P > 0, W1 > 0, W2 > 0, R > 0, Mi > 0 (i = 1, . . . , 5), and Nj (j = 1, . . . , 4) of appropri-ate dimensions and positive scalars ε1, ε2, ε3 satisfying the following LMIs:

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ω11 Ω12 0 NT2 0 Ω16 Ω17 AT R M1

∗ Ω22 NT3 Ω24 NT

4 –CT P Ω27 ATτ R M2

∗ ∗ Ω33 Ω34 Ω35 0 0 0 M3

∗ ∗ ∗ Ω44 Ω45 0 0 0 M4

∗ ∗ ∗ ∗ Ω55 0 0 0 M5

∗ ∗ ∗ ∗ ∗ –ε3I 0 R 0∗ ∗ ∗ ∗ ∗ ∗ Ω77 ET

1 R 0∗ ∗ ∗ ∗ ∗ ∗ ∗ –τ–1R 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ –M

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (7)

where

Ω11 = PA + AT P + HT PH + W1 + W2 + ε1GT1 G1 + ε2α1I + ε3β1I,

Ω12 = PAτ – AT PC + HT PHτ + NT1 + ε1GT

1 G2,

Ω16 = P –12ε2I +

12ε3γ I, Ω17 = PE1 + HT PE2,

Ω22 = –CT PAτ – ATτ PC + HT

τ PHτ – (1 – d)W1

– N1(C + I) – (C + I)T NT1 + ε1GT

2 G2 + ε2α2I + ε3β2I,

Ω24 = N1C – CT NT2 – NT

2 , Ω27 = –CT PE1 + HTτ PE2

Ω33 = –W2 – N3 – NT3 , Ω34 = –N3C, Ω35 = N3C – NT

4 ,

Ω44 = N2C + CT NT2 , Ω45 = –CT NT

4 , Ω55 = N4C + CT NT4 ,

Ω77 = ET2 PE2 – ε1I, M = diag

{τ–1M1, τ–1M2, τ–1M3, τ–1M4, τ–1M5

},

M1 = [M1, 0, 0, 0, 0], M2 = [0, M2, 0, 0, 0], M3 = [0, 0, M3, 0, 0],

M4 = [0, 0, 0, M4, 0], M5 = [0, 0, 0, 0, M5],

Λ =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

–M1 0 0 0 0 0∗ –M2 0 0 0 0∗ ∗ –M3 0 0 –N3

∗ ∗ ∗ –M4 0 0∗ ∗ ∗ ∗ –M5 –N4

∗ ∗ ∗ ∗ ∗ –R

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (8)

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Π =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

–M1 0 0 0 0 0∗ –M2 0 0 0 –N1

∗ ∗ –M3 0 0 0∗ ∗ ∗ –M4 0 –N2

∗ ∗ ∗ ∗ –M5 0∗ ∗ ∗ ∗ ∗ –R

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (9)

then the null solution of the stochastic time-delay system (6) is asymptotically stable in themean square.

Proof Choose the following Lyapunov–Krasovskii functional:

V(x(t), t

)=

4∑

i=1

Vi, (10)

where

V1 =[x(t) – Cx

(t – τ (t)

)]T P[x(t) – Cx

(t – τ (t)

)], V2 =

∫ t

t–τ (t)xT (s)W1x(s) ds,

V3 =∫ t

t–τ

xT (s)W2x(s) ds, V4 =∫ 0

–τ

∫ t

t+θ

hT1 (s)Rh1(s) ds dθ .

Using Itô’s formula [41], we obtain the stochastic differential of V (x(t), t) as follows:

dV(x(t), t

)

= LV(x(t), t

)dt + 2

[x(t) – Cx

(t – τ (t)

)]T P[Hx(t) + Hτ x

(t – τ (t)

)+ E2h(t)

]dB(t),

={

2[x(t) – Cx

(t – τ (t)

)]T P[Ax(t) + Aτ x

(t – τ (t)

)+ f

(x(t), x

(t – τ (t)

))+ E1h(t)

]

+[Hx(t) + Hτ x

(t – τ (t)

)+ E2h(t)

]T P[Hx(t) + Hτ x

(t – τ (t)

)+ E2h(t)

]

+ xT (t)W1x(t) – (1 – τ (t)xT(t – τ (t)

)W1x

(t – τ (t)

)

+ xT (t)W2x(t) – xT (t – τ )W2x(t – τ )

+ τhT1 (t)Rh1(t) –

∫ t

t–τ

hT1 (s)Rh1(s)ds

}dt

+{

2[x(t) – Cx

(t – τ (t)

)]T P[Hx(t) + Hτ x

(t – τ (t)

)+ E2h(t)

]}dB(t). (11)

Taking the expectation of both sides of (11), we have

E dV(x(t), t

)= ELV

(x(t), t

)dt. (12)

Set

xτ

(t – τ (t)

) ≡ x(t – τ (t) – τ

(t – τ (t)

)),

xτ (t – τ ) ≡ x(t – τ – τ (t – τ )

).

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 7 of 19

Using the Newton–Leibniz formula and the free-weighting matrices technique, we canderive the following equations:

2[xT(

t – τ (t))N1 + xT

τ

(t – τ (t)

)N2

][x(t) – Cx

(t – τ (t)

)– x

(t – τ (t)

)+ Cxτ

(t – τ (t)

)

–∫ t

t–τ (t)h1(s) ds –

∫ t

t–τ (t)

(Hx(s) + Hτ x

(s – τ (s)

)+ E2h(s)

)dB(s)

]= 0, (13)

2[xT (t – τ )N3 + xT

τ (t – τ )N4][

x(t – τ (t)

)– Cxτ

(t – τ (t)

)– x(t – τ ) + Cxτ (t – τ )

–∫ t–τ (t)

t–τ

h1(s) ds –∫ t–τ (t)

t–τ

(Hx(s) + Hτ x

(s – τ (s)

)+ E2h(s)

)dB(s)

]= 0, (14)

where Nj (j = 1, . . . , 4) are arbitrary matrices with appropriate dimensions. Using the prop-erties of the stochastic integral [41], we have

E{[

xT(t – τ (t)

)N1 + xT

τ

(t – τ (t)

)N2

] ∫ t

t–τ (t)

(Hx(s) + Hτ x

(s – τ (s)

)+ E2h(s)

)dB(s)

}= 0,

E{[

xT (t – τ )N3 + xτ (t – τ )NT4] ∫ t–τ (t)

t–τ

(Hx(s) + Hτ x

(s – τ (s)

)+ E2h(s)

)dB(s)

}= 0.

Adding the left-hand sides of (13) and (14) onto LV (x(t), t), (12) is transformed to

E dV(x(t), t

)= ELV

(x(t), t

)dt, (15)

where

LV(x(t), t

)

= LV(x(t), t

)+ 2

[xT(

t – τ (t))N1 + xT

τ

(t – τ (t)

)N2

][x(t)

– Cx(t – τ (t)

)– x

(t – τ (t)

)+ Cxτ

(t – τ (t)

)–

∫ t

t–τ (t)h1(s) ds

]

+ 2[xT (t – τ )N3 + xT

τ (t – τ )N4]

×[

x(t – τ (t)

)– Cxτ

(t – τ (t)

)– x(t – τ ) + Cxτ (t – τ ) –

∫ t–τ (t)

t–τ

h1(s) ds]

. (16)

For τ (t) ≤ d, we have

LV(x(t), t

)

= LV(x(t), t

)+ 2

[xT(

t – τ (t))N1 + xT

τ

(t – τ (t)

)N2

][x(t)

– Cx(t – τ (t)

)– x

(t – τ (t)

)+ Cxτ

(t – τ (t)

)–

∫ t

t–τ (t)z(s) ds

]

+ 2[xT (t – τ )N3 + xT

τ (t – τ )N4]

×[

x(t – τ (t)

)– Cxτ

(t – τ (t)

)– x(t – τ ) + Cxτ (t – τ ) –

∫ t–τ (t)

t–τ

h1(s) ds]

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 8 of 19

≤{

2[x(t) – Cx

(t – τ (t)

)]T P[Ax(t) + Aτ x

(t – τ (t)

)+ f

(x(t), x

(t – τ (t)

))

+ E1h(t)]

+[Hx(t) + Hτ x

(t – τ (t)

)+ E2h(t)

]T P

× [Hx(t) + Hτ x

(t – τ (t)

)+ E2h(t)

]+ xT (t)W1x(t)

– (1 – d)xT(t – τ (t)

)W1x

(t – τ (t)

)+ xT (t)W2x(t)

– xT (t – τ )W2x(t – τ ) + τhT1 (t)Rh1(t) –

∫ t–τ (t)

t–τ

hT1 (s)Rh1(s) ds

–∫ t

t–τ (t)hT

1 (s)Rh1(s) ds}

+ 2[xT(

t – τ (t))N1 + xT

τ

(t – τ (t)

)N2

]

×[

x(t) – Cx(t – τ (t)

)– x

(t – τ (t)

)+ Cxτ

(t – τ (t)

)–

∫ t

t–τ (t)h1(s) ds

]

+ 2[xT (t – τ )N3 + xT

τ (t – τ )N4]

×[

x(t – τ (t)

)– Cxτ

(t – τ (t)

)– x(t – τ ) + Cxτ (t – τ ) –

∫ t–τ (t)

t–τ

h1(s) ds]

. (17)

On the other hand, by using the one-sided Lipschitz (1) and the quadratically inner-bounded condition (2), we obtain the following inequalities:

α1xT (t)x(t) + α2xT(t – τ (t)

)x(t – τ (t)

)– xT (t)f

(x(t), x

(t – τ (t)

)) ≥ 0, (18)

β1xT (t)x(t) + β2xT(t – τ (t)

)x(t – τ (t)

)– f

(x(t), x

(t – τ (t)

))T f(x(t), x

(t – τ (t)

))

+ γ xT (t)f(x(t), x

(t – τ (t)

)) ≥ 0. (19)

Using the S-procedure in (17), we obtain that LV (x(t), t) < 0 is satisfied if there existpositive scalars ε1, ε2, ε3 satisfying

LV(x(t), t

)+ ε1

[G1x(t) + G2x

(t – τ (t)

)]T[G1x(t) + G2x

(t – τ (t)

)]– ε1h(t)T h(t)

+ ε2α1xT (t)x(t) + ε2α2xT(t – τ (t)

)x(t – τ (t)

)– ε2xT (t)f

(x(t), x

(t – τ (t)

))

+ ε3β1xT (t)x(t) + ε3β2xT(t – τ (t)

)x(t – τ (t)

)– ε3f

(x(t), x

(t – τ (t)

))T

× f(x(t), x

(t – τ (t)

))+ ε3γ xT (t)f

(x(t), x

(t – τ (t)

))< 0. (20)

Moreover, the following formula holds for any positive definite matrix M1(M2, . . . , M5)of appropriate dimensions:

τxT (t)M1x(t) –∫ t

t–τ

xT (t)M1x(t) ds = 0. (21)

We can decompose the integration interval [t –τ , t] into two subintervals that are [t –τ , t –τ (t)] and [t – τ (t), t], and let

ξT (t) =[xT (t)xT(

t – τ (t))xT (t – τ )xT

τ

(t – τ (t)

)xT

τ (t – τ )f T(x(t), x

(t – τ (t)

))hT (t)

],

ηT (t, s) =[xT (t)xT(

t – τ (t))xT (t – τ )xT

τ

(t – τ (t)

)xT

τ (t – τ )hT1 (s)

].

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 9 of 19

Combining the above formula (21) and rearranging (20), we have the following inequality:

ξT (t)Ωξ (t) + τhT1 (t)Rh1(t)

+∫ t–τ (t)

t–τ

ηT (t, s)Λη(t, s) ds +∫ t

t–τ (t)ηT (t, s)Πη(t, s) ds < 0, (22)

where

Ω =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ω11 Ω12 0 NT2 0 Ω16 PE1 + HT PE2

∗ Ω22 NT3 Ω24 NT

4 –CT P –CT PE1 + HTτ PE2

∗ ∗ Ω33 Ω34 Ω35 0 0∗ ∗ ∗ Ω44 Ω45 0 0∗ ∗ ∗ ∗ Ω55 0 0∗ ∗ ∗ ∗ ∗ –ε3I 0∗ ∗ ∗ ∗ ∗ ∗ ET

2 PE2 – ε1I

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (23)

Λ =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

–M1 0 0 0 0 0∗ –M2 0 0 0 0∗ ∗ –M3 0 0 –N3

∗ ∗ ∗ –M4 0 0∗ ∗ ∗ ∗ –M5 –N4

∗ ∗ ∗ ∗ ∗ –R

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

Π =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

–M1 0 0 0 0 0∗ –M2 0 0 0 –N1

∗ ∗ –M3 0 0 0∗ ∗ ∗ –M4 0 –N2

∗ ∗ ∗ ∗ –M5 0∗ ∗ ∗ ∗ ∗ –R

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0.

Ω11 = PA + AT P + HT PH + W1 + W2 + ε1GT1 G1 + τM1 + ε2α1I + ε3β1I,

Ω12 = PAτ – AT PC + HT PHτ + NT1 + ε1GT

1 G2,

Ω16 = P –12ε2I +

12ε3γ I,

Ω22 = –CT PAτ – ATτ PC + HT

τ PHτ – (1 – d)W1 – N1(C + I) – (C + I)T NT1

+ ε1GT2 G2 + τM2 + ε2α2I + ε3β2I,

Ω24 = N1C – CT NT2 – NT

2 ,

Ω33 = –W2 + τM3 – N3 – NT3 ,

Ω34 = –N3C,

Ω35 = N3C – NT4 ,

Ω44 = N2C + CT NT2 + τM4,

Ω45 = –CT NT4 ,

Ω55 = N4C + CT NT4 + τM5.

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 10 of 19

Utilizing the Schur complement Lemma 2.1, (23) is equivalent to the following LMI:

Ω =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ω11 Ω12 0 NT2 0 Ω16 Ω17 AT M1

∗ Ω22 NT3 Ω24 NT

4 –CT P Ω27 ATτ M2

∗ ∗ Ω33 Ω34 Ω35 0 0 0 M3

∗ ∗ ∗ Ω44 Ω45 0 0 0 M4

∗ ∗ ∗ ∗ Ω55 0 0 0 M5

∗ ∗ ∗ ∗ ∗ –ε3I 0 I 0∗ ∗ ∗ ∗ ∗ ∗ Ω77 ET

1 0∗ ∗ ∗ ∗ ∗ ∗ ∗ –τ–1R–1 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ –M

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (24)

Ω11 = PA + AT P + HT PH + W1 + W2 + ε1GT1 G1 + ε2α1I + ε3β1I,

Ω12 = PAτ – AT PC + HT PHτ + NT1 + ε1GT

1 G2,

Ω16 = P –12ε2I +

12ε3γ I, Ω17 = PE1 + HT PE2,

Ω22 = –CT PAτ – ATτ PC + HT

τ PHτ – (1 – d)W1

– N1(C + I) – (C + I)T NT1 + ε1GT

2 G2 + ε2α2I + ε3β2I,

Ω24 = N1C – CT NT2 – NT

2 , Ω27 = –CT PE1 + HTτ PE2,

Ω33 = –W2 – N3 – NT3 Ω34 = –N3C, Ω35 = N3C – NT

4 ,

Ω44 = N2C + CT NT2 , Ω45 = –CT NT

4 , Ω55 = N4C + CT NT4 ,

Ω77 = ET2 PE2 – ε1I, M = diag

{τ–1M1, τ–1M2, τ–1M3, τ–1M4, τ–1M5

},

M1 = [M1, 0, 0, 0, 0], M2 = [0, M2, 0, 0, 0], M3 = [0, 0, M3, 0, 0],

M4 = [0, 0, 0, M4, 0], M5 = [0, 0, 0, 0, M5].

Pre-and post-multiplying (24) by diag{I, I, I, I, I, I, I, R, I}, we obtain LMI (7). Combiningwith Λ < 0 and Π < 0, we find that ELV (ξ (t), t) < 0, i.e., it guarantees the asymptotic sta-bility of system (6) in the mean square. �

If the uncertain parameters A(t), Aτ (t), H(t), and Hτ (t) in system (6) are equalto zero, the system is simplified to the following deterministic stochastic system:

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

d[x(t) – Cx(t – τ (t))]

= [Ax(t) + Aτ x(t – τ (t)) + f (x(t), x(t – τ (t)) + Uu(t)] dt

+ [Hx(t) + Hτ x(t – τ (t))] dB(t),

x(t) = φ(t), t ∈ [–τ , 0].

(25)

The following conclusion of the robust asymptotic stability is obtained by Theorem 4.1for the deterministic stochastic system (25).

Corollary 4.1 Consider the stochastic time-delay system (25). The nonlinear functionf (x(t), x(t – τ (t))) satisfies (1) and (2). For given scalars τ and d, if there exist matrices

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 11 of 19

P > 0, W1 > 0, W2 > 0, R > 0, Mi > 0 (i = 1, . . . , 5), and matrices Nj (j = 1, . . . , 4) of appropri-ate dimensions and positive scalars ε1, ε2, ε3 satisfying the following LMIs:

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ω11 Ω12 0 NT2 0 Ω16 AT R M1

∗ Ω22 NT3 Ω24 NT

4 –CT P ATτ R M2

∗ ∗ Ω33 –N3C Ω35 0 0 M3

∗ ∗ ∗ Ω44 –CT NT4 0 0 M4

∗ ∗ ∗ ∗ Ω55 0 0 M5

∗ ∗ ∗ ∗ ∗ –ε3I R 0∗ ∗ ∗ ∗ ∗ ∗ –τ–1R 0∗ ∗ ∗ ∗ ∗ ∗ ∗ –M

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (26)

where

Ω11 = PA + AT P + HT PH + W1 + W2 + ε2α1I + ε3β1I,

Ω12 = PAτ – AT PC + HT PHτ + NT1 , Ω16 = P –

12ε2I +

12ε3γ I,

Ω22 = –CT PAτ – ATτ PC + HT

τ PHτ – (1 – d)W1

– N1(C + I) – (C + I)T NT1 + ε2α2I + ε3β2I,

Ω24 = N1C – CT NT2 – NT

2 , Ω33 = –W2 – N3 – NT3 ,

Ω35 = N3C – NT4 , Ω44 = N2C + CT NT

2 , Ω55 = N4C + CT NT4 ,

M = diag{τ–1M1, τ–1M2, τ–1M3, τ–1M4, τ–1M5

},

M1 = [M1, 0, 0, 0, 0], M2 = [0, M2, 0, 0, 0], M3 = [0, 0, M3, 0, 0],

M4 = [0, 0, 0, M4, 0], M5 = [0, 0, 0, 0, M5],

Λ =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

–M1 0 0 0 0 0∗ –M2 0 0 0 0∗ ∗ –M3 0 0 –N3

∗ ∗ ∗ –M4 0 0∗ ∗ ∗ ∗ –M5 –N4

∗ ∗ ∗ ∗ ∗ –R

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (27)

Π =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

–M1 0 0 0 0 0∗ –M2 0 0 0 –N1

∗ ∗ –M3 0 0 0∗ ∗ ∗ –M4 0 –N2

∗ ∗ ∗ ∗ –M5 0∗ ∗ ∗ ∗ ∗ –R

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (28)

the null solution of the stochastic time-delay system (25) is asymptotically stable in themean square.

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 12 of 19

5 Non-fragile robust state feedback controller designIn this section, we consider the design of a non-fragile state-feedback controller:

u(t) = K(t)x(t) =(K + K(t)

)x(t), (29)

guaranteeing the robust stability for the closed-loop system (3). K is the controller gain,and K(t) represents the gain perturbations with the following assumption:

K(t) = E3F(t)G3, (30)

where E3 and G3 are given real constant matrices with appropriate dimensions.

Theorem 5.1 Consider the stochastic time-delay system (6). The nonlinear functionf (x(t), x(t – τ (t))) satisfies (1) and (2). For given scalars τ and d, if there exist matricesP > 0, W1 > 0, W2 > 0, R > 0, Mi > 0 (i = 1, . . . , 5), and matrices Nj (j = 1, . . . , 4) of appropri-ate dimensions and positive scalars ε1, ε2, ε3, ε4, and σ satisfying the following LMIs:

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ω11 Ω12 0 NT2 0 Ω16 Ω17 AT R M1 J1 0

∗ Ω22 NT3 Ω24 NT

4 –CT P Ω27 ATτ R M2 0 0

∗ ∗ Ω33 Ω34 Ω35 0 0 0 M3 0 0∗ ∗ ∗ Ω44 Ω45 0 0 0 M4 0 0∗ ∗ ∗ ∗ Ω55 0 0 0 M5 0 0∗ ∗ ∗ ∗ ∗ –ε3I 0 R 0 0 0∗ ∗ ∗ ∗ ∗ ∗ Ω77 ET

1 R 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ –τ–1R 0 0 L1

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ –M 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ –J 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ –L

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (31)

Λ =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

–M1 0 0 0 0 0∗ –M2 0 0 0 0∗ ∗ –M3 0 0 –N3

∗ ∗ ∗ –M4 0 0∗ ∗ ∗ ∗ –M5 –N4

∗ ∗ ∗ ∗ ∗ –R

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (32)

Π =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

–M1 0 0 0 0 0∗ –M2 0 0 0 –N1

∗ ∗ –M3 0 0 0∗ ∗ ∗ –M4 0 –N2

∗ ∗ ∗ ∗ –M5 0∗ ∗ ∗ ∗ ∗ –R

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (33)

where

Ω11 = PA + AT P + HT PH + W1 + W2 + ε1GT1 G1 + ε2α1I + ε3β1I + 2ε4GT

3 G3,

Ω12 = PAτ – AT PC + HT PHτ + NT1 + ε1GT

1 G2,

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 13 of 19

Ω16 = P –12ε2I +

12ε3γ I,

Ω17 = PE1 + HT PE2,

Ω22 = –CT PAτ – ATτ PC + HT

τ PHτ – (1 – d)W1 – N1C – CT NT1 – N1

– NT1 + ε1GT

2 G2 + ε2α2I + ε3β2I,

Ω24 = N1C – CT NT2 – NT

2 ,

Ω27 = –CT PE1 + HTτ PE2, Ω33 = –W2 – NT

3 – N3, Ω34 = –N3C,

Ω35 = N3C – NT4 , Ω44 = N2C + CT NT

2 , Ω45 = –CT NT4 ,

Ω55 = N4C + CT NT4 , Ω77 = ET

2 PE2 – ε1I,

M = diag{τ–1M1, τ–1M2, τ–1M3, τ–1M4, τ–1M5

},

M1 = [M1, 0, 0, 0, 0], M2 = [0, M2, 0, 0, 0], M3 = [0, 0, M3, 0, 0],

M4 = [0, 0, 0, M4, 0], M5 = [0, 0, 0, 0, M5], J = diag

{12σ I, ε4I,σ I

},

J1 = [PU , PUE3, PU], L = diag{ε4I,σ I}, L1 = [RUE3, RU],

the closed-loop systems are asymptotically stable in the mean square with the non-fragilestate-feedback controller K = σ –1BT P.

Proof By using the controller K = σ –1UT P and letting

h1(t) =(A + U

(K + K(t)

))x(t) + Aτ x

(t – τ (t)

)+ f

(x(t), x

(t – τ (t)

))+ E1h(t),

system (6) can be rewritten as follows:

⎧⎪⎪⎨

⎪⎪⎩

d[x(t) – Cx(t – τ (t))]

= h1(t) dt + [Hx(t) + Hτ x(t – τ (t)) + E2h(t)] dB(t),

h(t)T h(t) ≤ [G1x(t) + G2x(t – τ (t))]T [G1x(t) + G2x(t – τ (t))].

(34)

Similar to Theorem 4.1, ELV (ξ (t), t) < 0 is guaranteed by the following matrix inequalityΩ < 0:

Ω =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ω11 Ω12 0 NT2 0 Ω16 Ω17 Ω18 M1

∗ Ω22 NT3 Ω24 NT

4 CT P Ω27 ATτ M2

∗ ∗ Ω33 –N3C Ω35 0 0 0 M3

∗ ∗ ∗ Ω44 –CT NT4 0 0 0 M4

∗ ∗ ∗ ∗ Ω55 0 0 0 M5

∗ ∗ ∗ ∗ ∗ –ε3I 0 I 0∗ ∗ ∗ ∗ ∗ ∗ Ω77 ET

1 0∗ ∗ ∗ ∗ ∗ ∗ ∗ –τ–1R–1 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ –M

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 14 of 19

where

Ω11 = P(A + UK) + (A + UK)T P + HT PH + W1 + W2 + ε1GT1 G1

+ ε2α1I + ε3β1I + PUE3F(t)G3 + GT3 FT (t)ET

3 UT P,

Ω12 = PAτ – AT PC + HT PHτ + NT1 + ε1GT

1 G2,

Ω16 = P –12ε2I +

12ε3γ I,

Ω17 = PE1 + HT PE2,

Ω18 = AT R + KT UT R + G3FT (t)ET3 UT R,

Ω22 = –CT PAτ – ATτ PC + HT

τ PHτ – (1 – d)W1 – N1C – CT NT1 – N1 – NT

1

+ ε1GT2 G2 + ε2α2I + ε3β2I,

Ω24 = N1C – CT NT2 – NT

2 ,

Ω27 = –CT PE1 + HTτ PE2, Ω33 = –W2 – NT

3 – N3, Ω35 = N3C – NT4 ,

Ω44 = N2C + CT NT2 , Ω55 = N4C + CT NT

4 , Ω77 = ET2 PE2 – ε1I

M = diag{τ–1M1, τ–1M2, τ–1M3, τ–1M4, τ–1M5

},

M1 = [M1, 0, 0, 0, 0], M2 = [0, M2, 0, 0, 0], M3 = [0, 0, M3, 0, 0],

M4 = [0, 0, 0, M4, 0], M5 = [0, 0, 0, 0, M5].

According to condition (30), we have

Ω = Ω + EFG + GT FT ET + K R + RT KT ,

where

Ω =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ω11 Ω12 0 NT2 0 Ω16 Ω17 AT R M1

∗ Ω22 NT3 Ω24 NT

4 –CT P Ω27 ATτ M2

∗ ∗ Ω33 –N3C N3C – NT4 0 0 0 M3

∗ ∗ ∗ Ω44 Ω45 0 0 0 M4

∗ ∗ ∗ ∗ Ω55 0 0 0 M5

∗ ∗ ∗ ∗ ∗ –ε3I 0 I 0∗ ∗ ∗ ∗ ∗ ∗ Ω77 ET

1 0∗ ∗ ∗ ∗ ∗ ∗ ∗ –τ–1R–1 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ –M

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where

Ω11 = P(A + UK) + (A + UK)T P + HT PH + W1 + W2 + ε1GT1 G1 + ε2α1I + ε3β1I,

Ω12 = PAτ – AT PC + HT PHτ + NT1 + ε1GT

1 G2,

Ω16 = P –12ε2I +

12ε3γ I, Ω17 = PE1 + HT PE2,

Ω22 = –CT PAτ – ATτ PC + HT

τ PHτ – (1 – d)W1 – N1C – CT NT1 – N1 – NT

1

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 15 of 19

+ ε1GT2 G2 + ε2α2I + ε3β2I,

Ω24 = N1C – CT NT2 – NT

2 ,

Ω27 = –CT PE1 + HTτ PE2,

Ω33 = –W2 – NT3 – N3,

Ω44 = N2C + CT NT2 ,

Ω45 = –CT NT4 ,

Ω55 = N4C + CT NT4 ,

Ω77 = ET2 PE2 – ε1I,

M = diag{τ–1M1, τ–1M2, τ–1M3, τ–1M4, τ–1M5

}, M1 = [M1, 0, 0, 0, 0],

M2 = [0, M2, 0, 0, 0], M3 = [0, 0, M3, 0, 0],

M4 = [0, 0, 0, M4, 0], M5 = [0, 0, 0, 0, M5].

Other notations are defined by (28), and E = {(PUE3)1,1, (RUE3)8,2} denotes a block matrixwith appropriate dimensions whose all nonzero blocks are the (1, 1)-block PUE3, the (8, 2)-block RUE3, and all other blocks are zero matrices. Similarly, G = {(G3)1,1, (G3)2,1}, K ={(KT )1,1}, R = {(UT R)1,8}. According to Lemma 2.2, for any scalars ε4 > 0 and σ > 0, wehave

Ω < Ω + ε–14 EET + ε4GT G + σ KKT + σ –1RT R. (35)

Let K = σ –1UT P, using the Schur Lemma 2.1, we arrive at

Ω + ε–14 EET + ε4GT G + σ KKT + σ –1RT R < 0

if LMI (31) is satisfied. �

6 Two illustrative numerical examplesIn order to illustrate the flexibility and reduced conservativeness of the proposed results,we present two numerical examples in this section.

Example 6.1 [17] Consider the neutral stochastic time-delay nonlinear system (25) with

A =

[–1.2 0.1–0.1 –1

]

, Aτ =

[–0.6 0.7–1 –0.8

]

,

and the following nonlinear function that satisfies the one-sided Lipschitz (1) and thequadratically innerbounded conditions (2):

f(x(t), x

(t – τ (t)

))=

[0.1

sin2(x2–2)x1(t–τ (t))0.1

sin1(x1–2)x2(t–τ (t))

]

,

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 16 of 19

with α1 = 0.5, α2 = 0.005, γ = 5,β1 = –2.5, β2 = –0.015 (see [17]). We consider the coeffi-cient matrix C as follows:

C =

[–0.1 0.09

–0.02 –1

]

.

By Corollary 4.1, we can obtain the allowable upper bound of the time-delay τ = 3.0345 forthe above systems, which is greater than the previous result (the allowable upper boundτ = 2.2487 in [17]). Comparing the allowable values of the systems, we see that our resultare less conservative. Since our system considers neutral stochastic systems with deriva-tives of state time-delay, our results may have more extensive applications.

Example 6.2 Consider the uncertain neutral stochastic nonlinear time-delay system (3)with

A =

[–2.0 0.00.0 –0.9

]

, Aτ =

[–1.0 0.0–1.0 –1.0

]

,

C =

[0.05 00.1 0.05

]

, B =

[12

]

,

H =

[–0.3 0.10.1 0.2

]

, Hτ =

[–0.2 0.20.3 –0.2

]

,

E1 =

[0.2 00 0.2

]

, E2 =

[0.2 00 0.2

]

, E3 =[0.2 0.2

],

G1 =

[0.2 00 0.2

]

, G2 =

[0.2 00 0.2

]

, G3 =

[0.2 00 0.2

]

.

Selecting the same nonlinear function f (x(t), x(t – τ (t))) as Example 6.1 and lettingd = 1.1, we solve LMIs (31), (32), and (33) to obtain the allowable bound τ = 3.2106. Hence,for any time-delay τ satisfying 0 < τ ≤ 3.2106, there exists a non-fragile state-feedbackcontroller such that the closed-loop systems are asymptotically stable in the mean square.For this example, if we choose the time-delay as τ = 2, according to Theorem 5.1, we canobtain a set of solutions as follows:

P =

[4.4021 –0.9004

–0.9004 0.8275

]

,

W1 =

[1.6119 –0.0127

–0.0127 2.6697

]

, W2 =

[28.4346 –0.5875–0.5875 28.2063

]

,

R =

[1.8078 –0.6499

–0.6499 0.8340

]

,

N1 =

[–0.0561 –0.1719–0.1719 0.0351

]

, N2 =

[–0.0367 –0.0039–0.0039 –0.0085

]

,

N3 =

[–0.0114 –0.1333–0.1333 0.0285

]

, N4 =

[–0.0190 –0.0003–0.0003 –0.0034

]

,

Ma and Li Journal of Inequalities and Applications (2018) 2018:293 Page 17 of 19

M1 =

[9.9084 –0.2109

–0.2109 10.0265

]

, M2 =

[0.4494 0.01970.0197 0.8863

]

,

M3 =

[10.1539 –0.1770–0.1770 10.2906

]

, M4 =

[0.0014 0.00040.0004 0.0003

]

,

M5 = 1.0 × 10–3

[0.4711 0.08750.0875 0.0850

]

,

ε1 = 8.9466, ε2 = 1.0763 × 103, ε3 = 255.2072, ε4 = 29.2135,

σ = 38.2553,

and the desired state-feedback matrix K = σ –1BT P = [0.0680 0.0197].

7 ConclusionsIn this work, the robust stability has been investigated for neutral stochastic time-delaysystems with disturbance, uncertainties, and one-sided Lipschitz nonlinearity. The para-metric uncertainties are assumed to be time-varying and norm bounded. Firstly, the al-lowable upper bound of time-delay has been obtained, which is a less conservative result.Secondly, based on Lyapunov stability, a novel non-fragile state-feedback controller hasbeen designed to guarantee the robust stability of the closed-loop systems. Finally, twonumerical examples have been given to illustrate the effectiveness of the proposed controlscheme.

AcknowledgementsThe authors would like to express their deep gratitude to the referee for his/her careful reading and invaluable comments.

FundingMa’s work was supported by the Fundamental Research Funds in Heilongjiang Provincial Universities (Nos. 135209246and 135209259). In the first funder (No. 135209246), the first author was the first participant and undertook thetheoretical proof of the work; in the second funder (No. 135209259), the first author was the fourth participant who hasundertaken the numerical simulation of this work. Ma’s work was also supported by the Science and TechnologyPlanning Project Fund of Qiqihar (No. GYGG-201718). Li’s work was supported by the National Natural Science Foundationof China (No. 61873071).

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsTM carried out the robust stability analysis and synthesis in this work. LL participated in the design and coordination andhelped to analyze the manuscript. All authors read and approved the final manuscript.

Author details1Department of Mathematics, Harbin Institute of Technology, Harbin, China. 2Department of Mathematics, QiqiharUniversity, Qiqihar, China.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 29 May 2018 Accepted: 18 October 2018

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