Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays

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Research Article

Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays Pin-Lin Liu n Department of Automation Engineering Institute of Mechatronoptic Systems, Chienkuo Technology University, 1 Chien-Shous N. Load, Changhua, Taiwan, ROC

art ic l e i nf o

a b s t r a c t

Article history: Received 9 July 2015 Received in revised form 18 October 2015 Accepted 3 November 2015 This paper was recommended for publication by Jeff Pieper

In this paper, we consider the problem of delay-interval-dependent robust stability and stabilization of a class of linear uncertain neutral-type systems with time-varying delay. By constructing a candidate Lyapunov–Krasovskii functional (LKF), that takes into account the delay-range information appropriately, less conservative robust stability criteria are proposed in terms of linear matrix inequalities (LMIs) to compute the maximum allowable upper bounds (MAUB) for the delay-interval within which the uncertain neutraltype system under consideration remains asymptotically stable. The verifiable stabilizability conditions and memoryless state feedback control design are stated. Finally, numerical examples are also designated to demonstrate the effectiveness and reduced conservatism of the developed results. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Neutral systems Integral inequality approach (IIA) Interval time-varying delay Delayed decomposition approach (DDA) Linear matrix inequality (LMI) Maximum allowable upper bounds (MAUB)

1. Introduction Dynamical systems with delay are abundant in nature. One reason is that nature is full of transparent delays. Another reason is that time-delay systems are often used to model a large class of engineering systems, where propagation and transmission of information or material are involved. Since time-delay which exists in many applications is a natural phenomenon and often causes instability and poor performance of systems, there have been considerable efforts to the stability analysis of time-delay systems during the last decade [1,3,12] and references therein. As far as we know, time-varying delay is more important and common in real engineering processes and has more complex impacts on system dynamics than constant time delay. Actually, the constant time-delay is a special case of time-varying delay. Therefore, some approaches have been proposed to research the systems with time-varying delays [2,5,9–18,20,21,23–25,26,28,29]. The stabilization problem via state feedback controls of linear control systems represents a significant challenge. Despite the efforts over the last decades, there still remain many important problems in this area to be solved analytically or numerically, especially for time-varying control systems [13]. Neutral-type system contains time delay in the state and its derivative. That makes it more complicated than a system with a delay in only the state. Neutral n

Tel.: þ 886 4 7111190; fax: þ 886 4 7121953.

delays occur not only in physical systems, but also in control systems, where they are sometimes artificially added to boost the performance. Neutral delay systems often appear in the study of automatic control, steam or water pipes, heat exchangers, population dynamics, lossless transmission lines and vibrating masses attached to an elastic bar [2–8,10,12,17,19–23,25,26,31]. On the other hand, there are often uncertainties due to errors in system modeling and changes in operating conditions. Since systems often have uncertainties, there is keen interest in neutral-type systems either with just uncertainties or with both uncertainties and a delay. Many sufficient conditions for the robust absolute stability of neutral-type systems with uncertainties have been derived by the different techniques mentioned [4,5,7,8,12,17,19–23,26]. Recently, stability of systems with interval time varying delays has been an important topic which describes that the propagated speed of signals is finite and uncertain in systems, such as networked control systems, some recent surveys can be found in [2,8,9,14,15,16,18,24,29] and references cited therein. More recently, the stability analysis of interval time-varying delay system has been a focused topic of theoretical and practical importance [2,8,9,24,28,29]. In most of these works, the delay-range-dependent of time-varying delay is assumed to go from zero to an upper bound. In practical, time delay may vary in a range for which the lower bound is not restricted to be zero: such time delay is called interval time-varying delay. A typical example of dynamic systems with interval time-varying delays is networked control systems [15]. However, in certain time-delay neutral-type systems [9,28], the delay-range- dependent may have

http://dx.doi.org/10.1016/j.isatra.2015.11.004 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

2

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non-zero lower bound, and such neutral-type systems are referred to as interval time-varying delay systems [2,8,24,29]. In [8], by employing a new Lyapunov–Krasovskii functional (LKF), a delayinterval-dependent stability criterion for time-varying parametric perturbations is derived. However, in the delay-dependent stability analysis of [29], while dealing with the time-derivative of the Lyapunov–Krasovskii functional (LKF), a few useful terms are neglected. In order to further improve the performance of delay-intervaldependent stability criteria, free-weighting method was proposed in [7], where neither system transformation nor bounding techniques on some cross terms were involved, thus avoiding the conservatism induced by model transformation. Note that the free weighting matrix method makes stability criteria complicated, therefore the criteria still leave some room for improvement in accuracy as well as complexity due to the method used. In addition, a memoryless state feedback controller with input constraints yields less conservative sufficient conditions in terms of linear matrix inequalities (LMIs) so that it allows wider feasible region of numerical optimization. The author in [13] has presented novel stability and stabilization criteria for uncertain systems with time-varying delay by developing a delayed decomposition approach. In [13], they consider the timevarying delay hðtÞ satisfying 0 r hðtÞ r h and then divide the time delay ½t  h; t into ½t  h; t  αh and ½t  αh; h with 0o α o 1: A new Lyapunov–Krasovskii functional (LKF) which includes the information of the time-varying delay is constructed in [16], to exact delayrange-dependent stability measure for uncertain systems with interval time-varying delay based on an integral inequality approach (IIA). However, uncertainties of the neutral-type system were not considered [16]. It should be pointing that, dealing with stabilization problem for uncertain neutral-type systems with time-varying delays is still an open problem, few results have been given in the literature dealing with the issue for uncertain neutral-type systems with delayinterval-dependent robust stability and stabilization criteria. Furthermore, the existing results still remain some conservatism, which inspires us for this study. In [16], in order to reduce the conservatism of the delaydependent stability criteria, the delay range ½h1 ; h2  was divided into two equally spaced subintervals: ½h1 ; δ and ½δ; h2 ; where δ ¼ h2 þ h1 2 : Very recently, Ramakrishnan and Ray [24] presented some delay-interval-dependent robust stability criteria, which are less conservative than the previous results. In this work, when estimating the upper bound of the derivative of the LKF, the term R t h R t  hðtÞ  t  h21 x_ T ðsÞRx_ ðsÞds was divided into  t  h2 x_ T ðsÞRx_ ðsÞds and R t  h1 T x_ ðsÞRx_ ðsÞds rather than directly estimation, which reduces  t  hðtÞ considerable conservatism. However, in the previous works, the variation interval of the time delay ½h1 ; h2  is directly considered when constructing the Lyapunov–Krasovskii functional (LKF), which may neglect some information on the variation interval of the time delay according to the delay decomposition method proposed in [16]. However, according to the idea of optimally dividing the delay interval proposed in Zhang and Liu [30], in many cases, segmentations with an unequal width can lead to better results than those with equal width. This offers the motivation that how to derive less conservative delay-interval-dependent robust stability criteria. This paper focuses on the delay-interval-dependent robust stability and stabilization criteria for neutral-type systems with time-varying delays. Contributions of this paper can be summarized as follows aspects. Fist, the present paper is an extended version of our earlier works [11–17] to design a controller which will ensure delay-intervaldependent robust stability and stabilization for a class of linear uncertain neutral-type systems with time-varying delay by using the integral inequality approach (IIA) and delayed decomposition approach (DDA). Second, by dividing the whole delay interval into two segmentations with an unequal width and checking the variation of the Lyapunov–Krasovskii functional (LKF) for each subinterval of delay,

much less conservative delay-dependent robust stability and stabilization criteria are derived. Third, all the conditions are presented in terms of linear matrix inequalities (LMIs) can be easily calculated by using Matlab LMI control toolbox. Finally, some numerical simulations are presented to illustrate the effectiveness and the preponderance of the proposed design methods. 1.1. Notations Throughout this paper, the superscripts ‘  1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively; Rnn denotes an n-dimensional Euclidean space; Rmn is the set of all m  n real matrices; P 4 0 means that matrix P is symmetric positive definite; For real symmetric matrices X and Y; the notation X Z Y (respectively, X 4 Y) means that the matrix X  Y is positive semi-definite (respectively, positive definite); I is an appropriately dimensional identity matrix;X ij denotes the element in row i and column j of matrix X; The notation * always denotes the symmetric block in one symmetric matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

2. Problem formulation Consider the neutral-type systems with time-varying delays x_ ðtÞ  C x_ ðt  τðtÞÞ ¼ A0 xðtÞ þ A1 xðt  hðtÞÞ þ BuðtÞ

ð1aÞ

xðsÞ ¼ ϕ ðsÞ;

ð1bÞ

8 t A ½  max fh2 ; τg; 0;

with xðtÞ A Rn as state vector of the system, A0 ; A1 ; B; C A Rnn constant matrices. The initial condition function, ϕ ðsÞ A L2 ½  max fh2 ; τg; 0; is a given continuous vector valued function. The delays, hðtÞ and τðtÞ are time-varying continuous function that satisfies 0 o h1 r hðtÞ rh2 ;

τðtÞ r τ;

_ r h ; τ_ ðtÞ r τ o 1 hðtÞ d d

8t Z0

ð2Þ

where h1 ; h2 ; τ; hd ; and τd are constants. For system (1), we will design a memoryless state-feedback controller uðtÞ ¼ KxðtÞ

ð3Þ

where K A Rmn is a constant matrix to be designed later. Plugging the controller expression (3) in (1) we get the following closed-loop dynamics: x_ ðtÞ  C x_ ðt  τðtÞÞ ¼ AxðtÞ þ A1 xðt  hðtÞÞ

ð4Þ

where A ¼ A0 BK: In this paper, our goal is to design a state feedback controller guaranteeing that the closed-loop is stable using some appropriate weighting matrices to reduce the conservatism. In order to make the following clearly, we let δ ¼ h2  h1 . To obtain the main results, the lemmas are indispensable in deriving the proposed stability criteria. This result will require the lemmas in the Appendix A.

3. Stability analysis This section constructs a new Lyapunov–Krasovskii functional (LKF) that contains information of the lower bound of delay h1 and upper bound h2 . Theorem 1 presents delay-interval-dependent result in terms of LMIs and expresses relationships between the terms of the Leibniz–Newton formula and integral inequality approach. Theorem 1. In the case h1 r hðtÞ rh1 þ ρδ; for the nominal unforced neutral time-varying system (4) subject to (2) is asymptotically stable

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

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for given scalars ρð0 o ρ o1Þ; τ; τd ; and hd ; if there exist symmetry positive-definite matrices T

T

P ¼ P 4 0; W ¼ W 40; Q i ¼

Q Ti

Z 0; Rj ¼ RTj

Z 0ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4Þ; 2 3 2 3 Y 11 Y 12 Y 13 X 11 X 12 X 13 6 7 6 X 22 X 23 5 Z 0; Y ¼ 4  Y 22 Y 23 7 X¼4 5 Z 0;   X 33   Y 33 2 3 2 3 Z 11 Z 12 Z 13 U 11 U 12 U 13 6 6 Z 22 Z 23 7 U 22 U 23 7 Z ¼4 5 Z 0; and U ¼ 4  5Z0   Z 33   U 33 such that the following LMIs hold: 2 Ω11 Ω12 Ω13 Ω14 0 0 6 Ω22 0 Ω24 Ω25 0 6  6 6   Ω33 0 0 0 6 6   Ω44 0 0 6  Ω¼6 6  Ω55 Ω56    6 6 6  Ω66     6 6       4      

3

7 7 7 o0 7 7 7 0 7 7 0 7 5 0 0

0

Ω77

6 Z ¼4 

Z 12 Z 22 

Z 13

3

2

U 11

6 Z 23 7 5 Z 0; and U ¼ 4  Z 33 

U 12 U 22 

0

Ω 26 0 0 0

Ω 66

U 13

3

U 23 7 5 Z0 U 33 3

Ω17 Ω18 0 Ω28 7 7 7 0 Ω38 7 7 0 0 0



Ω77





7 7 7o0 7 7 7 0 7 7 0 7 5 0 0

ð6aÞ

Ω88

and

ð5aÞ

Ω88



Z 11

such that the following LMIs hold: 2 Ω11 Ω12 Ω13 Ω14 0 6 Ω 22 0 0 Ω 25 6  6 6   Ω33 0 0 6 6   Ω 44 Ω 45 6  Ω¼6 6  Ω 55    6 6 6      6 6      4     

Ω17 Ω18 0 Ω28 7 7 7 0 Ω38 7 7 0 0

2

3

R1  X 33 Z0

ð6bÞ

R2  Y 33 Z 0

ð6cÞ

R3  Z 33 Z 0

ð6dÞ

ð1  τd ÞR4  U 33 Z 0

ð6eÞ

where

and R1  X 33 Z 0

ð5bÞ

R2  Y 33 Z 0

ð5cÞ

R3  Z 33 Z 0

ð5dÞ

ð1  τd ÞR4  U 33 Z0

ð5eÞ

where

Ω11 ¼ AT0 P þPA0 þ Q 1 þ Q 4 þ Q 5 þ h1 X 11 þ X 13 þ X T13 þ τU 11 þ U 13 þU T13 ; Ω12 ¼ PA1 ; Ω13 ¼ PC; Ω14 ¼ h1 X 11 þX 13 þ X T13 ; Ω17 ¼ τU 12  U 13 þ U T23 ; Ω18 ¼ AT0 ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 ; Ω22 ¼  ð1  hd ÞQ 4 þ ρδY 11 þ Y 13 þ Y T13 þ ρδY 22  Y 23  Y T23 ; Ω24 ¼ ρδY T12  Y T13 þY 23 ; Ω25 ¼ ρδY 12  Y 13 þY T23 ; Ω28 ¼ AT1 ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 ; Ω33 ¼  ð1  τd ÞW; Ω38 ¼ C T ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 ; Ω44 ¼ Q 2 Q 1 þ h1 X 22  X 23  X T23 þ ρδY 11 þ Y 13 þY T13 ; Ω55 ¼ Q 3 Q 2 þ ρδY 22  Y 23  Y T23 þ ð1  ρÞδZ 11 þ Z 13 þ Z T13 ; Ω56 ¼ ð1  ρÞδZ 12  Z 13 þ Z T23 ; Ω66 ¼  Q 3 þ ð1  ρÞδZ 22  Z 23 Z T23 ; Ω77 ¼  ð1  τd ÞQ 5 þ τU 22 U 23  U T23 ; Ω88 ¼  ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 :

Ω 22 ¼ ð1 hd ÞQ 4 þ ð1  ρÞδZ 11 þ Z 13 þ Z T13 þ ð1  ρÞδZ 22  Z 23  Z T23 ; Ω 25 ¼ ð1  ρÞδZ T12  Z T13 þ Z 23 ; Ω 26 ¼ ð1  ρÞδZ 12  Z 13 þ Z T23 ; Ω 44 ¼ Q 2  Q 1 þ h1 X 22 X 23  X T23 þ ρδY 11 þ Y 13 þ Y T13 ; Ω 45 ¼ ρδY 12  Y 13 þ Y T23 ; Ω 55 ¼ Q 3  Q 2 þ ρδY 22  Y 23  Y T23 þ ð1  ρÞδZ 11 þ Z 13 þZ T13 ; Ω 66 ¼  Q 3 þð1  ρÞδZ 22  Z 23  Z T23 : and Ωij ; ði; j ¼ 1; 2; 3; :::; 8; i o j r8Þ are defined in (5a). Proof. See the Appendix C.

4. Robust stability analysis When the system contains uncertainty, it can be described by the following neutral-type uncertain time-varying delay systems _  τðtÞÞ ¼ ðA0 þ ΔA0 ðtÞÞxðtÞ þ ðA1 þ ΔA1 ðtÞÞxðt  hðtÞÞ _  ðC þ ΔCðtÞÞxðt xðtÞ þ ðB þ ΔBðtÞÞuðtÞ;

h

The uncertainties are assumed to be the form i  ΔA0 ðtÞ ΔA1 ðtÞ ΔBðtÞ ΔCðtÞ ¼ DFðtÞ E0 E1 Eb

ð7Þ

Ec



ð8Þ

where D; E0 ; E1 ; Eb ; and Ec are constant matrices with appropriate dimensions, and FðtÞ is an unknown, real, and possibly timevarying matrix with Lebesgue-measurable elements satisfying F T ðtÞFðtÞ r I;

8 t:

ð9Þ

Proof. See the Appendix B.

Now, extending Theorem 1 to system (7) with time-varying structured uncertainties yields Theorem 3.

Theorem 2. In the case h1 þ ρδ r hðtÞ rh2 ; for the nominal unforced neutral time-varying system (4) subject to (2) is asymptotically stable for given scalars ρð0 o ρ o1Þ; τ; τd ; and hd ; if there exist symmetry positive-definite matrices

Theorem 3. In the case h1 r hðtÞ r h1 þ ρδ; for the unforced neutral uncertain time-varying delay system (7) subject to (2) and (8) is asymptotically stable for given scalars ρð0 o ρ o 1Þ; τ; τd ; and hd ; if there exist

P ¼ P T 4 0; W ¼ W T 40; Q i ¼ Q Ti Z 0; Rj ¼ RTj

P ¼ P T 4 0; W ¼ W T 4 0;

Z 0ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4Þ; 2 3 2 Y 11 X 11 X 12 X 13 6 7 X 22 X 23 5 Z 0; Y ¼ 6 X¼4 4   X 33 

Y 12 Y 22 

Y 13

3

Y 23 7 5 Z 0; Y 33

Q i ¼ Q Ti Z 0; Rj ¼ RTj Z0ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4Þ; ε 4 0; X 2 3 X 11 X 12 X 13 6 X 22 X 23 7 ¼4 5 Z 0;   X 33

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

2

Y 11

6 Y ¼4  2

Y 12

Y 13

3

2

Z 11

Y 22 

6 Y 23 7 5 Z0; Z ¼ 4  Y 33 

U 12

U 13

U 11

6 U ¼4 

Z 12

Z 13

3

such that the following LMIs hold:

Z 23 7 5 Z 0; and Z 33

Z 22 

2

3

U 22

U 23 7 5Z0



U 33

such that the following LMIs hold: 2 6 6 6 6 6 6 6 6 6 Ωu ¼ 6 6 6 6 6 6 6 6 6 6 4

Ω11 Ω12 Ω13 Ω14 0 0 Ω17 Ω18 Ω19  Ω22 Ω23 Ω24 Ω25 0 0 Ω28 0   Ω33 0 0 0 0 Ω38 0    Ω44 0 0 0 0 0 Ω55 Ω56 0 0 0     Ω66 0 0 0      Ω77 0 0       Ω88 Ω89        Ω99         

















Ω110 Ω210 Ω310 0 0 0 0 0 0

Ω1010

3 7 7 7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 5

ð10aÞ

6 6 6 6 6 6 6 6 6 Ωu ¼ 6 6 6 6 6 6 6 6 6 6 4

Ω11 Ω12 Ω13 Ω14 0 0 Ω17 Ω18 Ω19  Ω22 0 0 Ω 25 Ω 25 0 Ω28 0   Ω33 0 0 0 0 Ω38 0    Ω 44 Ω 45 0 0 0 0 Ω 55 0 0 0 0     Ω 66 0 0 0            Ω77 0 0 Ω88 Ω89        Ω99         

















3

Ω110 Ω210 Ω310 0 0 0 0 0 0

Ω1010

7 7 7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 5

ð11aÞ and R1  X 33 Z0

ð11bÞ

R2  Y 33 Z 0

ð11cÞ

R3  Z 33 Z 0

ð11dÞ

ð1  τd ÞR4  U 33 Z 0

ð11eÞ

where and

Ω 22 ¼ ð1 hd ÞQ 4 þ ð1  ρÞδZ 11 þ Z 13 þZ T13 þ ð1  ρÞδZ 22  Z 23 Z T23 ;

R1  X 33 Z 0

ð10bÞ

R2  Y 33 Z 0

ð10cÞ

R3  Z 33 Z0

ð10dÞ

ð1  τd ÞR4  U 33 Z0

ð10eÞ

where

Ω19 ¼ εPD; Ω89 ¼ εDT ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 ; Ω99 ¼ Ω1010 ¼  εI; Ω110 ¼

ET0 ;

Ω210 ¼ ET1 ; Ω310 ¼ ETc ; and Ωij ; ði; j ¼ 1; 2; :::; 8; i o jr 8Þ are defined in (5a). Proof. See the Appendix D. Theorem 4. In the case h1 þ ρδ r hðtÞ rh2 ; for the unforced neutral uncertain time-varying delay system (7) subject to (2) and (8) is asymptotically stable for given scalars ρð0 o ρ o 1Þ; τ; τd ; and hd ; if there exist P ¼ P T 4 0; W ¼ W T 40; Q i ¼ Q Ti Z 0; Rj ¼ RTj Z 0ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4Þ; ε 4 0; 2

X 11

6 X ¼4  2

Y 11 6 Y ¼4  2

U 11 6 U ¼4 

X 12

X 13

3

X 22

X 23 7 5 Z 0;



X 33 3

2



Y 33

U 12

U 13

U 22 

 3

U 23 7 5Z0 U 33

Proof. The proof can be completed in a similar formulation to Theorem 3.

5. State feedback control for time-varying delay systems In the section, we will focus on the controller synthesis for the neutral-type time-varying delay system with interval delays (4) and the objective is to construct a memoryless state-feedback control law (3).The results of the previous section can be applied to analyze the closed-loop system. The following Theorem 5 gives an LMI- based computational procedure to determine a memoryless state-feedback controller for the case h1 r hðtÞ rh1 þ ρδ: Then we have the following result. Theorem 5. In the case h1 rhðtÞ r h1 þ ρδ; for the nominal neutral time-varying system (4) subject to (2) under the control uðtÞ ¼ L 1 P xðtÞ is asymptotically stabilizable for given scalars ρð0 o ρ o 1Þ; τ; τd ;and hd ; if there exist symmetry positive-definite matrices T

Z 11 6 Y 23 7 5 Z0; Z ¼ 4  Y 13

and Ω ij ; ði; j ¼ 1; 2; 3; :::; 8; i oj r 8Þ are defined in (5a) and (6a).

T

T

T

P ¼ P 4 0; W ¼ W 4 0; Q i ¼ Q i Z 0; S j ¼ S j Z0ði ¼ 1; 2; 3; 4; 5;

Y 22

Y 12

Ω 25 ¼ ð1  ρÞδZ T12  Z T13 þ Z 23 ; Ω 26 ¼ ð1  ρÞδZ 12  Z 13 þZ T23 ; Ω 44 ¼ Q 2  Q 1 þ h1 X 22  X 23  X T23 þ ρδY 11 þ Y 13 þ Y T13 ; Ω 45 ¼ ρδY 12  Y 13 þ Y T23 ; Ω 55 ¼ Q 3  Q 2 þ ρδY 22  Y 23  Y T23 þ ð1  ρÞδZ 11 þ Z 13 þ Z T13 ; Ωij Ω 66 ¼  Q 3 þ ð1  ρÞδZ 22 Z 23  Z T23 ; Ω19 ¼ εPD; Ω89 ¼ εDT ½h1 R1 þ ρδR2 þð1  ρÞδR3 þ τR4 ; Ω99 ¼ Ω1010 ¼  εI; Ω110 ¼ ET0 ; Ω210 ¼ ET1 ; Ω310 ¼ ETc ;

Z 12 Z 22 

Z 13

3

Z 23 7 5 Z 0; and Z 33

j ¼ 1; 2; 3; 4Þ; 2 3 2 3 X 11 X 12 X 13 Y 11 Y 12 Y 13 6 T 7 6 T 7 7 6 7 X ¼6 4 X 12 X 22 X 23 5 Z0; Y ¼ 4 Y 12 Y 22 Y 23 5 Z 0; T T T T X 13 X 23 X 33 Y 13 Y 23 Y 33 2 3 2 3 Z 11 Z 12 Z 13 U 11 U 12 U 13 6 T 7 6 T 7 7 6 7 Z ¼6 4 Z 12 Z 22 Z 23 5 Z 0 U ¼ 4 U 12 U 22 U 23 5 Z 0 T T T T Z 13 Z 23 Z 33 U 13 U 23 U 33

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

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and a matrix L with appropriate dimension such that the following LMIs hold: 2 6 6 6 6 6 6 6 6 6 6 Ψ ¼6 6 6 6 6 6 6 6 6 6 6 4

Ψ 11 Ψ 12 Ψ 13 Ψ 14 0 0 Ψ 17 Ψ 18 Ψ 19  Ψ 22 0 Ψ 24 Ψ 25 0 0 Ψ 28 Ψ 29   Ψ 33 0 0 0 0 Ψ 38 Ω39    Ψ 44 0 0 0 0 0 Ψ 55 Ψ 56 0 0 0     Ψ 66 0 0 0      Ψ 77 0 0       Ψ 88 0        Ψ 99        

Ψ 110 Ψ 210 Ψ 310 0

0

0

0

0

0

0

0

0

0

0

0 0



















Ψ 1010





















3

Ψ 111 Ψ 211 Ψ 311

Ψ 1111

7 7 7 7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 7 5

2

3 2 3 X 11 X 12 X 13 Y 11 Y 12 Y 13 6 T 7 6 T 7 7 6 7 X ¼6 4 X 12 X 22 X 23 5 Z0; Y ¼ 4 Y 12 Y 22 Y 23 5 Z 0; T T T T X 13 X 23 X 33 Y 13 Y 23 Y 33 2 3 2 3 Z 11 Z 12 Z 13 U 11 U 12 U 13 6 T 7 6 T 7 7 6 7 Z ¼6 4 Z 12 Z 22 Z 23 5 Z 0 U ¼ 4 U 12 U 22 U 23 5 Z0 T T T T Z 13 Z 23 Z 33 U 13 U 23 U 33 such that the following LMIs hold: 2

ð12aÞ and

Ψ

P  X 33 Z0

ð12bÞ

P  Y 33 Z 0

ð12cÞ

P  Z 33 Z 0

ð12dÞ

ð1  τd ÞP  U 33 Z 0

ð12eÞ

where

Ψ 11 ¼ P AT0 LT BT þA0 P þBL þ Q 1 þ Q 4 þ Q 5 þ h1 X 11 þ X 13 T

T

þX 13 þ τU 11 þ U 13 þ U 13 ; T

Ψ 12 ¼ P A1 ; Ψ 13 ¼ P C; Ψ 14 ¼ h1 X 12 X 13 þ X 23 ; Ψ Ψ Ψ Ψ

T 17 ¼ U 12  U 13 þ U 23 ; T T T ðP AT0  LT BT Þ; 18 ¼ h1 ðP A0 L B Þ; 19 ¼ T T T 110 ¼ ð1  Þ ðP A0  L B Þ; T T T 111 ¼ ðP A0 L B Þ; 22 ¼  ð1  hd ÞQ 4 þ

τ

Ψ

ρδ

Ψ

 

 

 

 

 

 

 

 

Ψ 111 Ψ 211 Ψ 311

0

0

0 0

0 0

0

0

0

0

0

0

Ψ 1010

3

0

Ψ 1111



7 7 7 7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 7 5

ð13aÞ and P  X 33 Z 0

ð13bÞ

P  Y 33 Z 0

ð13cÞ

P  Z 33 Z 0

ð13dÞ

ð1  τd ÞP  U 33 Z 0

ð13cÞ

T

T

T

T

Ψ 25 ¼ ð1  ρÞδZ 12 Z 13 þ Z 23 ; Ψ 26 ¼ ð1  ρÞδZ 12  Z 13 þ Z 23 ;

T

T

 

Ψ 110 Ψ 210 Ψ 310

T

þY 13 þ Y 13 þ ρδY 22  Y 23  Y 23 ; T

Ψ 11 Ψ 12 Ψ 13 Ψ 14 0 0 Ψ 17 Ψ 18 Ψ 19  Ψ 22 0 0 Ψ 25 Ψ 26 0 Ψ 28 Ψ 29   Ψ 33 0 0 0 0 Ψ 38 Ω39    Ψ 44 Ψ 45 0 0 0 0 Ψ 55 0 0 0 0     Ψ 66 0 0 0      Ψ 77 0 0       Ψ 88 0        Ψ 99        

Ψ 22 ¼  ð1  hd ÞQ 4 þ ð1  ρÞδZ 11 þ Z 13 þ Z 13 þ ð1  ρÞδZ 22  Z 23  Z 23 ;

ρδY 11

T

6 6 6 6 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 6 4

where

ρδ

τ

5

T

T

Ψ 44 ¼ Q 2  Q 1 þ h1 X 22  X 23  X 23 þ ρδY 11 þ Y 13 þ Y 13 ;

T

Ψ 24 ¼ ρδY 12  Y 13 þ Y 23 ; Ψ 25 ¼ ρδY 12  Y 13 þ Y 23 ; Ψ 28 ¼ h1 P AT1 ; Ψ 29 ¼ ρδP AT1 ;

Ψ 45 ¼ ρδ

T Y 12 Y 13 þ Y 23 ;

T

Ψ 55 ¼ Q 3 Q 2 þ ρδY 22 Y 23  Y 23 T

Ψ 210 ¼ ð1  ρÞδP AT1 ; Ψ 211 ¼ τP AT1 ; Ψ 33 ¼  ð1  τd ÞW ; Ψ 38 ¼ h1 P C T ; Ψ 39 ¼ ρδP C ; T

T

Ψ 310 ¼ ð1  ρÞδP C T ; Ψ 311 ¼ τC T P ; Ψ 44 ¼ Q 2  Q 1 þ h1 X 22  X 23  X 23 T

þ ρδY 11 þ Y 13 þ Y 13 ; T

þ ð1  ρÞδZ 11 þ Z 13 þ Z 13 ; T

Ψ 66 ¼  Q 3 þ ð1  ρÞδZ 22  Z 23  Z 23 : and Ψ ij ; ði; j ¼ 1; 2; 3; :::; 11; i oj r 11Þ are defined in (12a). Proof. The proof can be completed in a similar formulation to Theorem 5.

T

Ψ 55 ¼ Q 3  Q 2 þ ρδY 22  Y 23  Y 23 þ ð1  ρÞδZ 11 þ Z 13 þ Z 13 ; T

Ψ 56 ¼ ð1  ρÞδZ 12  Z 13 þ Z 23 ; T

Ψ 66 ¼ Q 3 þ ð1  ρÞδZ 22 Z 23  Z 23 ; Ψ 77 ¼  ð1  τd ÞQ 5 T

þ τU 22 U 23  U 23 ; Ψ 88 ¼  h1 S1 ;

Ψ 99 ¼  ρδS2 ; Ψ 1010 ¼  ð1  ρÞδS3 ; Ψ 1111 ¼  τS4 : Proof. See the Appendix E. Now, extending Theorem 2 to system (4) for h1 þ ρδ r hðtÞ r h2 ; yields the following theorem. Theorem 6. In the case h1 þ ρδ r hðtÞ r h2 ; for the nominal neutral time-varying system (4) subject to (2) under the control uðtÞ ¼  L 1 P xðtÞ is asymptotically stabilizable for given scalars ρð0 o ρ o1Þ; τ; τd ;and hd ;if there exist symmetry positive-definite matrices T

T

T

P ¼ P 40; W ¼ W 4 0; Q i ¼ Q i T

Z 0; S j ¼ S j Z 0ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4Þ;

6. State feedback control for uncertain time-varying delay systems Now, extending Theorems 5and 6 to neutral-type uncertain time-varying delay systems (13) yield the following Theorems. Theorem 7. In the case h1 r hðtÞ r h1 þ ρδ; for the unforced neutral uncertain time-varying delay system (7) subject to (2) and (8) under 1 the control uðtÞ ¼  LP xðtÞ is robustly stable for all admissible uncertainties, for given scalars ρð0 o ρ o 1Þ; τ, τd ; and hd ; if there exist T

T

T

T

P ¼ P 4 0; W ¼ W 4 0; Q i ¼ Q i Z 0; S j ¼ S j Z 0ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4Þ; 2 3 2 X 11 X 12 X 13 Y 11 6 T 7 6 T 7 6 ε 4 0; X ¼ 6 4 X 12 X 22 X 23 5 Z0; Y ¼ 4 Y 12 T T T X 13 X 23 X 33 Y 13

Y 12 Y 22 T Y 23

Y 13

3

7 Y 23 7 5 Z 0; Y 33

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

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6

2

Z 11 6 T 6 Z ¼ 4 Z 12 T Z 13

Z 12 Z 22 T

Z 23

2 U 11 7 6 T 6 Z 23 7 5 Z 0 and U ¼ 4 U 12 T Z 33 U 13 Z 13

3

U 12 U 22 T

U 23

U 13

3

Theorem 8. In the case h1 þ ρδ r hðtÞ r h2 ; for the unforced neutral uncertain time-varying delay system (7) subject to (2) and (8) under 1 the control uðtÞ ¼  LP xðtÞ is robustly stable for all admissible

7 U 23 7 5Z0 U 33

such that the following LMIs hold: 2

Ψ 11 Ψ 12 Ψ 13 Ψ 14 6 Ψ 22 0 Ψ 24 6 6 6  Ψ 0 33 6 6   Ψ 44 6

6 6 6 6 6 6 Ψu ¼ 6 6 6 6 6 6 6 6 6 6 6 6 6 4 

0

0

Ψ 17 Ψ 18 Ψ 19 Ψ 110 0 Ψ 28 Ψ 29 Ψ 210 0 Ψ 38 Ψ 39 Ψ 310

0

0

0

0

0

Ψ 25 0

0

Ψ 111 Ψ 211 Ψ 311

Ψ 112 0

Ψ 113 Ψ 213 Ψ 313

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

























Ψ 55 Ψ 56 0 0 Ψ 66 0 0  Ψ 77 0   Ψ 88   

 

 

 

 

 

 

 















Ψ 99 0 0 Ψ 1010 0 Ψ 1111  





















Ψ 812 Ψ 912 Ψ 1012 Ψ 1112 Ψ 1212

























and P  X 33 Z0

ð14bÞ

P  Y 33 Z 0

ð14cÞ

P  Z 33 Z 0

ð14dÞ

ð1  τd ÞP  U 33 Z 0

ð14eÞ

Ψ 11 Ψ 12 Ψ 13 Ψ 14 6 Ψ 22 0 0 6 6 6  Ψ 0 33 6 6   Ψ 44 6

6 6 6 6 6 6 Ψu ¼ 6 6 6 6 6 6 6 6 6 6 6 6 6 4 

0

Ψ 25 Ψ 26 0

0

T

T

X 11

6 T j ¼ 1; 2; 3; 4Þ; ε 4 0; X ¼ 6 4 X 12 T X 13

2

0

Ψ 1313

2

Y 12 Y 22 T

Y 23

Z 11

Z 12

6 T Z ¼6 4 Z 12 T Z 13

Proof. See the Appendix F.

2

0

T

are defined in (12a).

Now, extending Theorem 6 to system (13) h1 þ ρδ r hðtÞ r h2 ; yields the following theorem.

0

ð14aÞ

T

P ¼ P 4 0; W ¼ W 4 0; Q i ¼ Q i Z 0; R j ¼ R j Z 0ði ¼ 1; 2; 3; 4; 5;

Y 11 6 T 6 Y ¼ 4 Y 12 T Y 13

Ψ 112 ¼ εD; Ψ 812 ¼ εh1 DT ; Ψ 912 ¼ ερδDT ; Ψ 1012 ¼ εð1  ρÞδDT ; Ψ 1112 ¼ ετDT ; Ψ 1212 ¼ Ψ 1313 ¼  εI and Ψ ij ; ði; j ¼ 1; 2; :::; 11; i o j r11Þ

0 0

7 7 7 7 7 7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 7 7 7 7 5

uncertainties, for given scalars ρð0 o ρ o1Þ; τ; τd ; and hd ; if there exist

2

where

3

Z 22 T Z 23

Y 13

X 12

X 13

7 X 23 7 5 Z 0;

X 22 T X 23

X 33

3

7 Y 23 7 5 Z 0; Y 33 Z 13

3

2

U 11

6 T 7 6 Z 23 7 5 Z 0 and U ¼ 4 U 12 T Z 33 U 13

Ψ 17 Ψ 18 Ψ 19 Ψ 110 0 Ψ 28 Ψ 29 Ψ 210 0 Ψ 38 Ψ 39 Ψ 310

T U 23

Ψ 112 0

Ψ 113 Ψ 213 Ψ 313

0





















































Ψ 88 0 0 0 Ψ 99 0 0  Ψ 1010 0   Ψ 1111   





















Ψ 812 Ψ 912 Ψ 1012 Ψ 1112 Ψ 1212























 

U 22

Ψ 111 Ψ 211 Ψ 311



 

U 12

U 13

3

7 U 23 7 5 Z0 U 33

such that the following LMIs hold:

Ψ 45 0 0 Ψ 55 0 0 Ψ 66 0  Ψ 77  

 

3

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0 0 0 0 0 0

Ψ 1313

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 7 7 7 7 5

ð15aÞ

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

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7

and

Time delay 1.86 sec

P  X 33 Z0

1

ð15bÞ

x1 x2

0.8

P  Y 33 Z 0

ð15cÞ

P  Z 33 Z 0

ð15dÞ

0.4

ð1  τd ÞP  U 33 Z 0

ð15eÞ

0.2

Ψ 112 ¼ εD; Ψ 812 ¼ εh1 D ; Ψ 912 ¼ ερδD ; T

T

Ψ 1012 ¼ εð1  ρÞδDT ; Ψ 1112 ¼ ετDT ; Ψ 1212 ¼ Ψ 1313 ¼  εI; Ψ ij and Ψ ij ði; j ¼ 1; 2; :::; 11; i oj r 11Þ

Output x1,x2

where

0.6

0 -0.2 -0.4 -0.6 -0.8

are defined in (12a) and (13a).

-1

Proof. The proof can be completed in a similar formulation to Theorem 7.

0

2

4

6

8

10

12

14

16

18

20

Time(sec) Fig. 1. The simulation of the Example 1 forh2 ¼ τ ¼ 1.86 s.



7. Illustrative examples In this section, five examples will be given to show the less conservatism of the proposed stability results. Example 1 is corresponding to the nominal neural-type systems with time-varying delays and Examples 2–4 are corresponding to the ones with uncertainty. Example 5 is used to demonstrate the effectiveness of the control design. Example 1. Consider a class of neural-type systems with timevarying delays as follows: x_ ðtÞ  C x_ ðt  τðtÞÞ ¼ A0 xðtÞ þ A1 xðt  hðtÞÞ ð16Þ where  1 A0 ¼ 0

  0:5 ; A1 ¼ 0:5 1 1

  0:4 ;C ¼  0:5 0 0

0 0:4

 :

Find the neutral delay τðtÞ r τ and delay-interval 0 r h1 r hðtÞ r h2 guaranteeing system (16) to be asymptotically stable. Solution The purpose is to compute the upper bound of neutral delay τðtÞ r τ and delay-interval 0 r h1 r hðtÞ r h2 as the neutraltype system with time-varying delays is asymptotically stable when the derivative of the time-varying delay hd and τd are unknown. For the case of h1 ¼ 0:5; h2 ¼ τ is found to be 0.912 by using the method in [29], 1.071 by [8], and 1.676 by [2]. According to Theorem 1 in this paper, however, it is found that the system is robustly asymptotically stable for h2 ¼ τ ¼ 1:9643, which shows that Theorem 1 yields less conservative stability test than previous related results for this example. Let h1 ¼ 0:5; h2 ¼ τ ¼ 1:9643 by using LMI Toolbox in MATLAB (with accuracy 0.01), then solutions of LMI given in (5) are found to be     2:6724  5:4018 7:1929  16:9456 P¼ ; ; Q1 ¼  5:4018 15:3648  16:9456 46:6429   0:6490  0:0584 ; Q2 ¼ 0:0584 0:0063     0:0005  0:0011 0:0004  0:0009 ; Q5 ¼ ; Q3 ¼ 0:0011 0:0030  0:0009 0:0023 Table 1 Maximum allowable upper bound (MAUB) h2 ¼ τ whenhd and τd are unknown and h1 ¼ 0 (Example 1). [25]

[26]

[29]

[31]

[8]

[2]

Ours

1.0574

1.0606

1.0606

1.0852

1.0864

1.2223

1.8611

R1 ¼

4:5973

 5:8435

 5:8435

8:4615

2 6 R2 ¼ 4

4:2583

 5:5516

 ; 3

2

7 6 5; R3 ¼ 4

0:0010

 0:0022

3 7 5;

 5:5516 11:7280 0:0022 0:0058 3 0:0011 0:0138 6 7 R4 ¼ 4 5;  0:0138 0:2000 2 3 2 3 2:4354  3:0731 12:7693  9:8669 6 7 6 7 W ¼4 5; X 11 ¼ 4 5;  3:0731 10:2261  9:8669 10:3459 2 3 2 3  15:3507 16:2826  7:6289 9:4575 6 7 6 7 X 12 ¼ 4 5; X 13 ¼ 4 5; 11:5863  14:5759 5:9133  8:5616 2 3 2 3 19:1420  21:0596 9:3265  11:9997 6 7 6 7 X 22 ¼ 4 5; X 23 ¼ 4 5;  21:0596 26:5349  10:1428 14:9119 2 3 2 3 4:5970  5:8428 7:7839  10:0036 6 7 6 7 X 33 ¼ 4 5; Y 11 ¼ 4 5;  5:8428 8:4598  10:0036 21:1131 2 3 2 3  7:7695 9:9630  5:7569 7:4693 6 7 6 7 Y 12 ¼ 4 5; Y 13 ¼ 4 5; 9:9763 21:0360 7:4343 15:7340 2 3 2 3 7:7553 9:9361 5:7461 7:4489 6 7 6 7 Y 22 ¼ 4 5; Y 23 ¼ 4 5;  9:9361 20:9602  7:4041 15:6765 2 3 2 3 4:2581 5:5512 0:0018  0:0046 6 7 6 7 Y 33 ¼ 4 5; Z 11 ¼ 4 5;  5:5512 11:7268  0:0046 0:0120 2 3 2 3  0:0018 0:0046  0:0009 0:0022 6 7 6 7 Z 12 ¼ 4 5; Z 13 ¼ 4 5; 0:0046  0:0119 0:0022  0:0058 2 3 2 3 0:0021  0:0053 0:0009  0:0022 6 7 6 7 Z 22 ¼ 4 5; Z 23 ¼ 4 5;  0:0053 0:0140  0:0022 0:0058 2 3 2 3 0:0007  0:0016 3:9127  0:0037 6 7 6 7 Z 33 ¼ 4 5; U 11 ¼ 4 5;  0:0016 0:0043 0:0037 3:9537 2

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

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Table 2 Maximum allowable upper bound (MAUB) h2 ¼ τ for h1 ¼ 0; τd ¼ 0; hd ¼ 0:1 with different values of cðρ ¼ 0:5Þ (Example 2). jc j

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

[31] [4] [20] [29] [2] [22] [21] [23] [10] [16] Theorem 3

0.92 0.97 1.072 1.166 1.172 1.2723 1.3209 1.3418 1.3519 2.2165 2.2403

0.73 0.78 0.9208 0.962 0.966 1.0360 1.0777 1.0841 1.1127 1.7187 2.2305

0.55 0.60 0.7516 0.778 0.781 0.8302 0.8701 0.8585 0.8972 1.3022 2.2292

0.41 0.45 0.5991 0.616 0.618 0.6512 0.6827 0.6664 0.7055 0.9579 2.1979

0.29 0.31 0.4625 0.472 0.474 0.4958 0.5123 0.6664 0.5366 0.6764 2.1970

0.19 0.19 0.3402 0.346 0.347 0.3607 0.3596 0.3649 0.3882 0.4488 2.1461

0.11 0.10 0.2310 0.235 0.235 0.2409 0.2419 0.2438 0.2568 0.2660 2.1019

0.04 0.02 0.1227 0.130 0.130 0.1312 0.1450 0.1320 0.1355 0.1479 2. 0809

Table 3 Maximum allowable upper bound (MAUB) h2 ¼ τfor τd ¼ 0:1; h1 ¼ 0; c ¼ 0:1 with different values of hd ðρ ¼ 0:5Þ (Example 2). hd

0.1

0.3

0.5

0.7

0.9

Z1

[4] [20] [29] [8] [2] [21] [10] [23] [16] Theorem 3

0.77 0.9109 0.962 0.966 0.968 1.0657 1.1004 1.1026 1.7187 2.2208

0.68 0.8537 0.907 0.920 0.957 0.9426 1.0266 1.0086 1.6135 2.1004

0.57 0.7980 0.850 0.887 0.958 0.8732 0.9858 0.9751 1.5120 1.9170

0.42 0.7472 0.789 0.866 0.958 0.85558 0.9709 0.9751 1.4247 1.7158

0.17 0.7199 0.714 0.853 0.958 0.8943 0.9708 0.9751 1.3901 1.6882

– 0.7216 0.658 0.852 0.958 0.9376 0.9708 0.9751 1.3899 1.6824

Table 4 Maximum allowable upper bound (MAUB) h2 ¼ τ for τd ¼ 0:1; h1 ¼ 0:5; c ¼ 0:1; with different values of hd (Example 2). hd

0.1

0.3

0.5

0.7

0.9

hd is unknown

[29] [5] [8] [2] Theorem 3

0.73 0.962 0.966 1.431 1.9120

0.57 0.907 0.922 1.323 1.6994

0.41 0.850 0.895 1.214 1.5671

0.24 0.793 0.894 1.105 1.4317

0.07 0.793 0.894 1.089 1.4315

– 0.793 0.894 1.089 1.4315

Table 5 Maximum allowable upper bound (MAUB) h2 with different values of h1 when hd is unknown and τd ¼ 0 (Example 3). h1

0

0.5

1.0

100

1000

[29] Theorem 3

1.8842 5.7072

2.3032 5.9942

2.8032 6.5735

101.8032 105.3353

1001.8032 1004.9783

Table 6 Maximum allowable upper bound (MAUB) h2 with different values of h1 when hd is unknown and τd ¼ 0:6 (Example 3). h1

0

1.0

10

1000

[29] Theorem 3

1.6904 5.7001

2.6904 6.4296

11.6904 15.2940

1001.6904 1004.8651

2 6 U 12 ¼ 4

 0:0002 0:0026

2 6 U 22 ¼ 4

0:0004  0:0042

0:0026

3

2

7 6 5; U 13 ¼ 4

 0:0376  0:0042 0:0584

0:0000  0:0001

3

2

7 6 5; U 23 ¼ 4

0:0004  0:0052

 0:0001

3 7 5;

0:0018  0:0052 0:0757

3 7 5;

2 6 U 33 ¼ 4

0:0008  0:0104

0:0104

3 7 5:

0:1502

For the case of h1 ¼ 0; we give the comparison of our results obtained by applying Theorem 1 with other ones, listed in Table 1. It is clear that the proposed method in Example 1 is more general than the ones in [2,8,25,26,29,31]. In order to verify the stability condition, the simulation result is given for h2 ¼ τ ¼ 1:86 in Fig. 1. One can see that the state trajectories approach to zero asymptotically stable. Example 2. Consider the linear neutral delay-differential systems with parametrical perturbations as follows: ð17Þ x_ ðtÞ  C x_ ðt  τðtÞÞ ¼ ðA0 þ ΔA0 ðtÞÞxðtÞ þ ðA1 þ ΔA1 Þxðt  hðtÞÞ;       2 0 1 0 c 0 where A0 ¼ ; A1 ¼ ;C ¼ ; 0rco 0 1 1  1 0 c 1 and ΔAðtÞ and ΔBðtÞ are of the form of (18) with D ¼ I; E0 ¼ dia gf1:6; 0:05g; E1 ¼ diagf0:1; 0:3g: Find the neutral delay τðtÞ r τ and delay-interval 0 r h1 r hðtÞ r h2 within which the uncertain neutral system (17) under consideration remains asymptotically stable. Solution For hd ¼ 0:1 and τd ¼ 0; the maximum allowable bound h2 ¼ τ provided by the proposed robust stability criterion for different values of c is listed in Table 2 along with the existing results. It can be seen from Table 2 that our results are less conservative than the existing ones [2,4,10,16,20,2,22,23,29,31]. Similarly, the allowable upper bound (MAUB) h2 ¼ τ for c¼ 0.1 and τd ¼ 0:1 which the uncertain neutral-type system with timevarying delay is robustly stable are obtained for different values of the derivative of the time-varying delay hd and when hd is unknown. The results for the case of h1 ¼ 0 and h1 ¼ 0:5 are listed in Tables 3 and 4, respectively. For this example, it shown from Tables 3 and 4 that, comparing results with that of [2,4,8,10,16,20, 21,23,29] and [2,5,8,29], respectively. From the Tables 3 and 4, they are clear that the proposed robust stability criterion is less conservative than the recently reported results Example 3. Consider the linear neutral delay-differential systems with parametrical perturbations as follows: _  C xðt _  τðtÞÞ ¼ ðA0 þ ΔA0 ðtÞÞxðtÞ þ ðA1 þ ΔA1 Þxðt  hðtÞÞ; ð18Þ xðtÞ where  2 A0 ¼ 0

  0 ; A1 ¼ 0:4 2 0

  0:4 0:1 ;C ¼ 0 0

 0 ; 0:1

and ΔA0 ðtÞ; ΔA1 ðtÞ; and ΔCðtÞ are of the form of (18) with D ¼ I; E0 ¼ diagf0:1; 0:1g; E1 ¼ Ec ¼ diagf0:05; 0:05g; Eb ¼ diagf0:1; 0:3g: Find the neutral delay τðtÞ r τ and delay-interval 0 r h1 r hðtÞ r h2 within which the uncertain neutral system (18) under consideration remains asymptotically stable.

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

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Table 7 Comparison of MAUB using different methods when hd ¼ 0 ðρ ¼ 0:5Þ (Example 4) jc j

0

0.05

0.1

0.15

0.20

0.25

0.30

0.35

[3] [7] [27] [12] [19] [6] [18] [17] Theorem 3

1.77 2.39 2.3970 2.4009 2.44 3.13 3.45 4.0005 5.6574

1.63 2.05 2.1536 2.1358 2.17 2.98 3.21 3.7329 4.6958

1.48 1.75 1.9373 1.8982 1.93 2.83 3.02 3.5140 4.3631

1.33 1.49 1.7440 1.6848 1.72 2.66 2.88 3.0015 4.2139

1.16 1.27 1.5702 1.4924 1.52 2.49 2.62 2.5589 4.0151

0.98 1.08 1.4127 1.3183 1.35 2.31 2.54 2.1750 3.9838

0.79 0.91 1.2691 1.1604 1.19 2.12 2.51 1.8407 3.3631

0.59 0.76 1.1373 1.0167 1.104 1.93 2.23 1.5492 3.2952

Example 4. Consider the following uncertain neutral system with time-varying delays as follows: _ C xðt _  τðtÞÞ ¼ ðA0 þ ΔA0 ðtÞÞxðtÞ þ ðA1 þ ΔA1 Þxðt hðtÞÞ; t 4 0 xðtÞ

Time delay 2.2975sec 1 x1 x2

0.8

ð19Þ

0.6

where

Output x1,x2

0.4

 A0 ¼

0.2 0

  c ;C ¼ 1 0 0

0 c

 ; jcj o 1;

ΔA0 ðtÞ and ΔA1 ðtÞ are unknown matrices satisfying D ¼ I; E0 ¼ E1 ¼ diagf0:2; 0:2g: Find the neutral delay τðtÞ r τ and delayinterval 0 r h1 r hðtÞ r h2 within which the uncertain neutral system (19) under consideration remains asymptotically stable. Solution Table 7 gives out the related comparative results, from which one can see that Theorem 3 obtained in this paper are less conservative than those established in [3,6,7,12,17,18,19,27].

-0.2 -0.4 -0.6 -0.8 -1

  0 1 ; A1 ¼  0:9 1

2 0

0

5

10

15

20

25

30

35

40

45

50

Time(sec) Fig. 2. Trajectories xðtÞ of Example 5 when control uðtÞ deactivated.

Example 5. Consider the following time-varying delay neutraltype system as follows: ð20Þ x_ ðtÞ  C x_ ðt  τðtÞÞ ¼ A0 xðtÞ þA1 xðt hðtÞÞ þ BuðtÞ where

Time delay 2.2975sec 1



x1 x2

0.8

A0 ¼

0 0

       0:10 2  0:5 0 ; A1 ¼ ;B¼ ;C ¼ : 00:1 1 0 1 1 0

0.6

It is found that the nominal system (20) is unstable and it is intended to stabilize the controlled system and find the range of delay time hðtÞ by using memoryless state feedback controller to guarantee that the above system is asymptotically stable. Solution Letτ ¼ h2 ; τd ¼ hd ¼ ρ ¼ 0:5;and h1 ¼ 0:1solving the quasi-convex optimization problem (20), the maximum upper bound, h2 ; for which the system is stabilized by the corresponding

Output x1,x2

0.4 0.2 0 -0.2

1

-0.4 -0.6 -0.8 -1

0

5

10

15

20

25

30

35

40

45

50

Time(sec)

Fig. 3. Trajectories xðtÞ of Example 5 when control uðtÞ activated.

Solution The delay bounds which the uncertain system (18) is robustly stable are obtained for different values of the derivative of the time-varying delay h1 r hðtÞ r h2 and when hd is unknown. The results for the case of τd ¼ 0 and τd ¼ 0:6 are listed in Tables 5 and 6, respectively. Hence, for this example, the criteria proposed here significantly improve the estimate of the stability limit compared for the result of [29].

state feedback gain K ¼ LP ¼ ½2:39321:8786 and h2 r2:2975: Fig. 2 shows open-loop response, and it becomes obvious that, the system is unstable in the absence of any control. Fig. 3 shows closed-loop system response and clearly demonstrates that, under the influence of the proposed controller (3), the system (20) is asymptotically stable for 0 r hðtÞ rh2 r2:2975: Remark 1. The major contributions and novelties of our paper are as follows: (1) the delay interval ½h1 ; h2  is divided into two subintervals with an unequal width as ½h1 ; h1 þ ρδ and ½h1 þ ρδ; h2 ; 0 o ρ o 1;which is different from the existing methods; (2) the R th R t h term  t  h21 x_ T ðsÞRx_ ðsÞds is divided into  t  h11 ρδ x_ T ðsÞRx_ ðsÞds R t  h1  ρδ T x_ ðsÞRx_ ðsÞds; and is respectively calculated in two and  t  h2 subintervals, where different integral inequality matrix variables are fully used in each subinterval; (3) much less conservative delay-interval-dependent robust stability criteria are derived compared with existing methods; (4) moreover, for this paper,

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

segmentations with a suitable unequal width lead to better results than those with tow equal length segmentations ρ ¼ 0:5: Remark 2. Some comparisons have been made with the same examples that appear in many recent papers. Our results show them less conservative than those reports.

þxT ðt  hðtÞÞ½hX T12  X T13 þ X 23 xðtÞ þ xT ðt  hðtÞÞ½hX 22 X 23  X T23 xðt  hðtÞÞ

ðA3Þ

Lemma 2. [1]. The following matrix inequality " # Q ðxÞ SðxÞ o0 T S ðxÞ RðxÞ

ðA4Þ

8. Conclusions

where Q ðxÞ ¼ Q T ðxÞ; RðxÞ ¼ RT ðxÞ and SðxÞ depend on affine on x; is equivalent to

In this paper, the delay-interval-dependent robust stability and stabilization criteria for uncertain neutral-type system with timevarying delays have been considered. By discretising the delay interval into two segmentations with an unequal width, improvement delay-interval-dependent sufficient conditions for the robust stability of uncertain neutral-type systems are proposed in terms of linear matrix inequalities (LMIs) by employing a Lyapunov–Krasovskii functional (LKF) and integral inequality approach (IIA). These conditions reduce the conservativeness in computing the maximum allowable upper bounds (MAUB) in many cases. Some benchmark examples are given to illustrate that the proposed stability criteria are less conservative than some existing results. Further possible research directions would include the extension of the main results to the neural networks with interval time-varying delays. Furthermore, we are interested to extend this work to the proportionalintegral-derivative (PID)-type state feedback controller for congestion control of transmission control protocol networks. The results will appear soon in further.

RðxÞ o0

ðA5Þ

Q ðxÞ o0

ðA6Þ

Lemma 1. [11-17]. For any positive semi-definite matrices

X¼

X 11

X 12

6 XT 4 12 X T13

X 13

3

X 23 7 5 Z 0;

X 22 X T23

ðA1Þ



t t  hðtÞ

Z r

x_ T ðsÞX 33 x_ ðsÞds 2



t

t  hðtÞ

x ðtÞ T

x ðt  hðtÞÞ T

X 11 6 T x_ ðsÞ 4 X 12 X T13 T

Lemma 3. [1]. Given matrices Q ¼ Q T ; D;andE;of appropriate dimensions then Q þDFðtÞE þ ET F T ðtÞDT o0

for all FðtÞ satisfying F ðtÞFðtÞ r I; if and only if there exists some

ε 4 0 such that

Q þ ε  1 DDT þ εET E o0

X 12

X 13

32

3

xðtÞ 7 6 X 23 7 54 xðt  hðtÞÞ 5ds x_ ðsÞ 0

X 22 X T23

Proof. The New-Leibniz formula and integral inequality approach are used to derive the lemma; they are stated below: Z t x_ T ðsÞX 33 x_ ðsÞds  Z r



t

t  hðtÞ

xT ðtÞ

xT ðt  hðtÞÞ

X 11 6 T x_ T ðsÞ 4 X 12 X T13

32

X 12 X 22 X T23

t  hðtÞ

Z

þ xT ðt  hðtÞÞhX 22 xðt  hðtÞÞ þ xT ðt  hðtÞÞX 23 þ

t t  hðtÞ

¼x

T

x_ T ðsÞdsX T13 xðtÞ þ

Z

t

t  hðtÞ

3

X 13 xðtÞ 6 7 X 23 7 54 xðt  hðtÞÞ 5ds _ xðsÞ 0

r xT ðtÞhX 11 xðtÞ þxT ðtÞhX 12 xðt  hðtÞÞ Z t x_ ðsÞdsþ xT ðt  hðtÞÞhX T12 xðtÞ þ xT ðtÞX 13

Z

ðA9Þ

Appendix B. Proof of Theorem 1

Case 1. When h1 r hðtÞ rh1 þ ρδ

Z þ þ

Z Z þ Z þ

t  h1  ρδ

t  h1

t  h1  ρδ

t  h2 Z t

þ

2

ðA8Þ

T

ðA2Þ

t  hðtÞ

ðA7Þ

The proposed stability analysis is based on the following LKF candidate Z t Z t  h1 VðtÞ ¼ xT ðtÞPxðtÞ þ xT ðsÞQ 1 xðsÞdsþ xT ðsÞQ 2 xðsÞds

X 33

the following integral inequality holds Z

Q ðxÞ SðxÞR  1 ðxÞST ðxÞ o 0

Proof For facilitate searching the optimal division of the delay interval ½h1 ; h2 : Based on the idea of delayed-decomposition approach, we divide the interval delay ½h1 ; h2  into two subintervals ½h1 ; h1 þ ρδ and ½h1 þ ρδ; h2 ; 0 o ρ o 1; δ ¼ h2  h1 ; if we can proof that Theorem 1 holds for the two subintervals, then Theorem 1 is true.

Appendix A

2

and

Z

t  hðtÞ t t  τðtÞ 0  h1

Z

xT ðsÞQ 4 xðsÞds þ

t t  τðtÞ

xT ðsÞQ 5 xðsÞds

x_ T ðsÞW x_ ðsÞds t tþθ

 h1  ρδ  h2

xT ðsÞQ 3 xðsÞds

x_ T ðsÞR1 x_ ðsÞdsdθ þ Z

t t þθ

Z

Z

 h1

t

 h1  ρδ

t þθ

0

t

x_ T ðsÞR3 x_ ðsÞdsdθ þ

Z

 τðtÞ

Z

x_ T ðsÞR2 x_ ðsÞdsdθ

t þθ

x_ T ðsÞR4 x_ ðsÞdsdθ ðB1Þ

The derivatives of this Lyapunov–Krasovskii functional (B1) with respect to time along the solution of the unforced system (4) are given by: _ ¼ xT ðtÞðAT P þ PA0 ÞxðtÞ þ xT ðtÞPA1 xðt  hðtÞÞ þ xT ðtÞPC x_ ðt  τðtÞÞ VðtÞ 0

t

t  hðtÞ

x_ ðsÞds

x_ T ðsÞdsX T23 xðt  hðtÞÞ

ðtÞ½hX 11 þ X 13 þX T13 xðtÞ þ xT ðtÞ½hX 12  X 13 þ X T23 xðt  hðtÞÞ

þ xT ðt  hðtÞÞAT1 PxðtÞ þ x_ T ðt  τðtÞÞC T PxðtÞ þ xT ðtÞQ 1 xðtÞ  xT ðt h1 ÞQ 1 xðt  h1 Þ þ xT ðt  h1 ÞQ 2 xðt  h1 Þ  xT ðt  h1  ρδÞQ 2 xðt  h1  ρδÞ þ xT ðt  h1  ρδÞQ 3 xðt  h1  ρδÞ  xT ðt  h2 ÞQ 3 xðt  h2 Þ T _ þ xT ðtÞQ 4 xðtÞ  xT ðt hðtÞÞð1  hðtÞÞQ 4 xðt  hðtÞÞ þx ðtÞQ 5 xðtÞ

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Z

 xT ðt  τðtÞÞð1  τ_ ðtÞÞQ 5 xðt  τðtÞÞ þ x_ T ðtÞW x_ ðtÞ



 xT ðt  τðtÞÞð1  τ_ ðtÞÞWxðt  τðtÞÞ Z t x_ T ðsÞR1 x_ ðsÞds þ x_ T ðtÞρδR2 x_ ðtÞ þ x_ T ðtÞh1 R1 x_ ðtÞ 

t  h2

t  h1

t  h1  ρδ

x_ T ðsÞR2 x_ ðsÞds

þ x_ T ðtÞð1  ρÞδR3 x_ ðtÞ 

Z

t  h1  ρδ t  h2

þ x_ T ðtÞτðtÞR4 x_ ðtÞ  ð1  τ_ ðtÞÞ

Z

x_ T ðsÞR3 x_ ðsÞds

t t  τ ðtÞ

x_ T ðsÞR4 x_ ðsÞds

τ

ρδ

þ xT ðt  hðtÞÞ½ρδY 12  Y 13 þY T23 xðt  h1  ρδÞ þ xT ðt  h1  ρδÞ½ρδY T12 Y T13 þY 23 xðt  hðtÞÞ

t  h1  ρδ

Z 

þ xT ðt  h1  ρδÞ½ρδY 22 Y 23 Y T23 xðt  h1  ρδÞ

t  h1

t

t  τðtÞ

x_ ðsÞR2 x_ ðsÞds

t  h1  ρδ

t  h2

 x_ ðsÞR3 x_ ðsÞds T

R t  h1

t  hðtÞ

x_ T ðsÞY 33 x_ ðsÞds

þ xT ðt  h1 Þ½ρδY 12  Y 13 þ Y T23 xðt  hðtÞÞ

x_ T ðsÞð1  τd ÞR4 x_ ðsÞds

þ xT ðt  hðtÞÞ½ρδY T12  Y T13 þY 23 xðt  h1 Þ þ xT ðt  hðtÞÞ½ρδY 22  Y 23 Y T23 xðt  hðtÞÞ

þ x ðtÞPC xðt  τðtÞÞ



þ xT ðt  hðtÞÞAT1 PxðtÞ þ x_ T ðt  τðtÞÞC T PxðtÞ

R t  h1  ρδ t  h2

r xT ðt  h1  ρδÞ½ð1  ρÞδZ 11 þZ 13 þ Z T13 xðt h1  ρδÞ

þ xT ðt  h1  ρδÞðQ 3  Q 2 Þxðt  h1  ρδÞ  xT ðt  h2 ÞQ 3 xðt  h2 Þ

þ xT ðt  h1  ρδÞ½ð1  ρÞδZ 12 Z 13 þ Z T23 xðt  h2 Þ

 xT ðt  hðtÞÞð1  hd ÞQ 4 xðt  hðtÞÞ

þ xT ðt  h2 Þ½ð1  ρÞδZ T12  Z T13 þ Z 23 xðt  h1  ρδÞ þ xT ðt  h2 Þ½ð1  ρÞδZ 22  Z 23  Z T23 xðt  h2 Þ

 xT ðt  τðtÞÞð1  τd ÞQ 5 xðt  τðtÞÞ  x ðt  τ_ ðtÞÞð1  τd ÞWxðt  τ_ ðtÞÞ þ x_ ðtÞ½W þ h1 R1 þ ρδR2 þð1  ρÞδR3 þ τR4 x_ ðtÞ Z t  h1 Z t x_ T ðsÞðR1  X 33 Þx_ ðsÞds x_ T ðsÞðR2  Y 33 Þx_ ðsÞds  T

T

t  h1  ρδ

t  h1



t  h2 t

Z 

t  τðtÞ

Z 

Z

t  h2

x_ T ðsÞZ 33 x_ ðsÞds 

Z

t t  h1

x_ T ðsÞX 33 x_ ðsÞds

 

t  h2 Z t

Z

t  τðtÞ t

¼

t t  τðtÞ

x_ T ðsÞU 33 x_ ðsÞds

t  h1

Z 

t  h1

t  hðtÞ

þ ρδR2 þ ð1  ρÞδR3 þ τR4 

½A0 xðtÞ þ A1 xðt  hðtÞÞ þ xT ðtÞPC x_ ðt  τðtÞ ¼ xT ðtÞAT0 ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 A0 xðtÞ þ xT ðtÞAT0 ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 A1 xðt  hðtÞÞ þ xT ðtÞAT0 ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 C x_ ðt  τðtÞ þ xT ðt  hðtÞÞAT1 ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 A1 xðt hðtÞÞ

x_ ðsÞU 33 x_ ðsÞds T

x_ T ðsÞY 33 x_ ðsÞds

ðB2Þ

þ xT ðt  hðtÞÞAT1 ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 A0 xðtÞ

x_ ðsÞZ 33 x_ ðsÞds T

x_ T ðsÞX 33 x_ ðsÞds

With the operator for the term x_ ðtÞ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδ R3 þ τR4 x_ ðtÞ as follows: x_ T ðtÞ½W þh1 R1 þ ρδR2 þð1  ρÞδR3 þ τR4 x_ ðtÞ ¼ ½A0 xðtÞ þA1 xðt hðtÞÞ þ xT ðtÞPC x_ ðt  τðtÞT  ½W þh1 R1

t  h1  ρδ

t  h1  ρδ

¼ xT ðtÞ½τU 11 þ U 13 þU T13 xðtÞ þ xT ðtÞ½τU 12  U 13 þ U T23 xðt  τðtÞÞ

T

Alternatively, the following equations are true: Z t  h1 Rt x_ T ðsÞY 33 x_ ðsÞds  t  h1 x_ T ðsÞX 33 x_ ðsÞds  Z

x_ T ðsÞU 33 x_ ðsÞds

ðB8Þ Z

x_ ðsÞY 33 x_ ðsÞds

t  h1  ρδ



x_ ðsÞðR3  Z 33 Þx_ ðsÞds

T

t  h1  ρδ

t  τ ðtÞ

ðB7Þ

þ xT ðt  τ ðtÞÞ½τU T12  U T13 þ U 23 xðtÞ þ xT ðt  τðtÞÞ½τ U 22 U 23  U T23 xðt  τ ðtÞÞ

T

x_ T ðsÞðð1  τd ÞR4  U 33 Þx_ ðsÞds 

t  h1



and Rt

ðB6Þ

x_ T ðsÞZ 33 x_ ðsÞds

þ xT ðt  h1 ÞðQ 2 Q 1 Þxðt  h1 Þ

t  h1  ρδ

ðB5Þ

r xT ðt  h1 Þ½ρδY 11 þ Y 13 þ Y T13 xðt  h1 Þ

¼ xT ðtÞðAT0 P þ PA0 þ Q 1 þ Q 4 þ Q 5 ÞxðtÞ þ xT ðtÞPA1 xðt  hðtÞÞ T _

Z

ðB4Þ

1

 xT ðt  τ_ ðtÞÞð1  τd ÞWxðt  τ_ ðtÞÞ þ x_ T ðtÞ½W þ h1 R1 þ ρδR2 Z t þ ð1  ρÞδR3 þ τR4 x_ ðtÞ  x_ T ðsÞR1 x_ ðsÞds 

ðB3Þ

r xT ðt  hðtÞÞ½ρδY 11 þ Y 13 þ Y T13 xðt  hðtÞÞ

 x ðt  hðtÞÞð1  hd ÞQ 4 xðt  hðtÞÞ

Z

x_ T ðsÞU 33 x_ ðsÞds

Similarly, we obtain R t  hðtÞ  t  h  ρδ x_ T ðsÞY 33 x_ ðsÞds

 xT ðt  τðtÞÞð1  τd ÞQ 5 xðt  τðtÞÞ

T

t  τ ðtÞ

þxT ðt  h1 Þ½h1 X 22  X 23  X T23 xðt  h1 Þ

T

t  h1

t

þxT ðt  h1 Þ½h1 X T12  X T13 þ X 23 xðtÞ

τ

Z

Z

¼ xT ðtÞ½h1 X 11 þ X 13 þ X T13 xðtÞ þxT ðtÞ½h1 X 12  X 13 þ X T23 xðt  h1 Þ

r x ðtÞðAT0 P þ PA0 þQ 1 þ Q 4 þ Q 5 ÞxðtÞ þ xT ðtÞPA1 xðt  hðtÞÞ þxT ðtÞPC x_ ðt  ðtÞÞ þ xT ðt  hðtÞÞAT1 PxðtÞ þ x_ T ðt  ðtÞÞC T PxðtÞ þ xT ðt  h1 ÞðQ 2 Q 1 Þxðt  h1 Þ þ xT ðt  h1  ÞðQ 3  Q 2 Þxðt  h1  Þ  xT ðt  h2 ÞQ 3 xðt  h2 Þ T

ρδ

x_ T ðsÞZ 33 x_ ðsÞds 

By utilizing Lemma 1 [12–17] and the Leibniz–Newton formula, we have Rt  t  h1 x_ T ðsÞX 33 x_ ðsÞds 2 32 3 X 11 X 12 X 13 xðtÞ Z t   T 6 7 6 7 r xT ðtÞ xT ðt  h1 Þ x_ T ðsÞ 4 X 12 X 22 X 23 54 xðt  h1 Þ 5ds t  h1 T T x_ ðsÞ X 13 X 23 0

t  h1

Z 

t  h1  ρδ

11

þ xT ðt  hðtÞÞAT1 ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 C x_ ðt  τðtÞ Z

t  hðtÞ

t  h1  ρδ

x_ T ðsÞY 33 x_ ðsÞds

þ xT ðt  τðtÞÞC T ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 A0 xðtÞ þ x_ T ðt  τðtÞÞC T ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 A1 xðt hðtÞÞ þ x_ T ðt  τðtÞÞC T ½W þ h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 C x_ ðt  τðtÞÞ ðB9Þ

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

12

Z

Substituting (B3)–(B9) into (B2), we obtain Rt T _ VðtÞ r ξ ðtÞΞξðtÞ  t  h1 x_ T ðsÞðR1  X 33 Þx_ ðsÞds Z t  h1 x_ T ðsÞðR2 Y 33 Þx_ ðsÞds  Z

Z 

t  h1  ρδ

t  h2 Z t t  τðtÞ

t t  h1

x_ T ðsÞX 33 x_ ðsÞds 

t  hðtÞ

t  h2

t  h1  ρδ

 

¼

Z

t  h1 t  h1  ρδ

x_ T ðsÞZ 33 x_ ðsÞds 

Z

x_ T ðsÞY 33 x_ ðsÞds

t  h1  ρδ t  hðtÞ

x_ T ðsÞZ 33 x_ ðsÞds

ðC1Þ

The proof can be completed in a similar formulation and (C1) to Theorem 1.

x_ T ðsÞðR3  Z 33 Þx_ ðsÞds

x_ T ðsÞðð1  τd ÞR4  U 33 Þx_ ðsÞds

ðB10Þ

Appendix D. Proof of Theorem 3

where 

Proof. Replacing A0 ; A1 ; and C in (5a) with A0 þ DFðtÞE0 ;A1 þ DFðt ÞE1 ; and C þ DFðtÞEc ; respectively, we find that Lemma 3 [1] for the unforced neutral uncertain time-varying delay system is equivalent to the following condition:

ξT ðtÞ ¼ xT ðtÞ xT ðt  hðtÞÞ x_ T ðt  τðtÞÞxT ðt h1 ÞxT  ðt  h1  ρδÞxT ðt  h2 ÞxT ðt  τðtÞÞ and

2

Ξ 11 Ξ 12 Ξ 13 Ξ 22 Ξ 23 Ξ T23 Ξ 33 Ξ T24 0 Ξ T25 0

Ξ 14 Ξ 24 Ξ 25 Ξ 44

0

0

0

0

0

0

6 ΞT 6 12 6 T 6Ξ 6 13 6 T Ξ¼6 6 Ξ 14 6 6 0 6 6 6 0 4

Ξ T17

0

0

0

0

0

0

0

0

0

Ξ 17

Ξ þ Γ d FðtÞΓ e þ Γ Te FðtÞΓ Td o 0

0 7 7 7 0 7 7 7 0 7 7 7 0 7 7 7 0 7 5

Ξ 55 Ξ 56 Ξ T56 Ξ 66 0

3

where h Γ d ¼ PD 0

τ

U 11 þU 13 þ U T13

þ AT0 ½h1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 A0 ;

Ξ 23 Ξ 24 Ξ 33 Ξ 44 Ξ 55 Ξ 56 Ξ 66 Ξ 77

ρδ ρδ Ξ ρδ

ρδ ρδ

τ τ

¼ PA1 þ AT0 ½h1 R1 þ R2 þ ð1  Þ R3 þ R4 A1 ; ¼ PC þ AT0 ½h1 R1 þ R2 þ ð1  Þ R3 þ R4 C; ¼ h1 X 11 þ X 13 þ X T13 ; 17 ¼ U 12  U 13 þU T23 ; ¼  ð1  hd ÞQ 4 þ Y 11 þ Y 13 þ Y T13 þ Y 22  Y 23  Y T23 þ AT1 ½h1 R1 þ R2 þ ð1  Þ R3 þ R4 A1 ; ¼ AT1 ½h1 R1 þ R2 þ ð1  Þ R3 þ R4 C; ¼ Y T12  Y T13 þ Y 23 ; 25 ¼ Y 12  Y 13 þ Y T23 ; ¼  ð1  d ÞW þ C T ½h1 R1 þ R2 þ ð1  Þ R3 þ R4 C; ; ¼ Q 2  Q 1 þ X 22  X 23  X T23 þ Y 11 þ Y 13 þ Y T13 ; ¼ Q 3  Q 2 þ Y 22  Y 23  Y T23 þ ð1  Þ Z 11 þZ 13 þ Z T13 ; ¼ ð1  Þ Z 12  Z 13 þ Z T23 ; ¼  Q 3 þ ð1  Þ Z 22  Z 23  Z T23 ; ¼  ð1  d ÞQ 5 þ U 22 U 23  U T23 :

ρδ ρδ

ρδ

τ

ρδ ρδ

τ

ρδ

τ

ρδ

ρδ τ

Since R1  X 33 Z0;R2  Y 33 Z 0; R3  Z 33 Z 0; and ð1  τd ÞR4  U 33 Z 0then the last four terms in (B10) is less than or equal to 0. Furthermore, using the Schur complement, of Lemma 2 [1] with some effort we show that (B10) guarantees of V_ ðtÞis negative, if _ VðtÞis negative, then Ξ is negative. From this, we can conclude that, for all delay hðtÞand τðtÞsatisfying (2).Therefore, the nominal unforced neutral system (4) with time-varying delay is asymptotically stable if linear matrix inequality (5a) is true.

Appendix C. Proof of Theorem 2 Proof. Case 2: When h1 þ ρδ r hðtÞ r h2 Alternatively, the following equations are true: Z t  h1 Rt x_ T ðsÞY 33 x_ ðsÞds  t  h1 x_ T ðsÞX 33 x_ ðsÞds 

t  h1  ρδ t  h2

t  h1  ρδ

x_ ðsÞZ 33 x_ ðsÞds T

ðh1 R1 þ ρδR2 þ ð1  ρÞδR3 þ τR4 ÞD

i



By the , a sufficient condition guaranteeing (5a) for the unforced neutral uncertain time-varying delay system (7) is that there exists a positive number ε 4 0 such that

Ξ þ ε  1 Γ Td Γ d þ εΓ Te Γ e o 0

ðD2Þ

Applying the Schur complement of Lemma 2 [1] shows that (D2) is equivalent to (10a). This completes the proof.

Appendix E. Proof of Theorem 5 Proof. In view of Theorem 1, to prove the asymptotic stability of the closed-loop system with control law uðtÞ ¼  KxðtÞ, it suffices to show that there exist P ¼ P T 4 0; W ¼ W T 40; Q i ¼ Q Ti Z 0; Rj ¼ RTj Z 0ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4Þ; 2 3 2 3 X 11 X 12 X 13 Y 11 Y 12 Y 13 6 7 6 X 22 X 23 5 Z 0; Y ¼ 4  Y 22 Y 23 7 X¼4 5 Z0;   X 33   Y 33 2 3 2 3 U 11 U 12 U 13 Z 11 Z 12 Z 13 6 6 Z 22 Z 23 7 U 22 U 23 7 Z¼4  5 Z0; and U ¼ 4  5Z0   Z 33   U 33 such that (5a) remains valid with A0 replaced by A0  BK: By Schur complement of Lemma 2 [1], Ω o 0 is equivalent to 2

Z

0

ρδ

ρδ τ ρδ τ Ξ ρδ ρδ ρδ

ρδ τ

0



T T 11 ¼ A0 P þ PA0 þ Q 1 þ Q 4 þQ 5 þ h1 X 11 þ X 13 þX 13 þ

Ξ 12 Ξ 13 Ξ 14 Ξ 22

0

Γ e ¼ E0 E1 Ec 0 0 0 0 0 :

where

Ξ

0

and

Ξ 77

0

0

ðD1Þ

6 6 6 6 6 6 6 6 6 6 Ωk ¼ 6 6 6 6 6 6 6 6 6 6 6 4

Ω11 Ω12 Ω13 Ω14 0 0 Ω17 Ω18 Ω19  Ω22 0 Ω24 Ω25 0 0 Ω28 Ω29   Ω33 0 0 0 0 Ω38 Ω39    Ω44 0 0 0 0 0 Ω55 Ω56 0 0 0     Ω66 0 0 0      Ω77 0 0       Ω88 0        Ω99        

Ω110 Ω210 Ω310

Ω111 Ω211 Ω311

0

0

0

0

0

0

0

0

0

0

0

0



















Ω1010





















0

Ω1111

3 7 7 7 7 7 7 7 7 7 7 7 7o0 7 7 7 7 7 7 7 7 7 5

ðE1Þ

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

h

where

Ω11 ¼ ðA0 BKÞT P þ PðA0 BKÞ þ Q 1 þ Q 4 þ Q 5 þ h1 X 11 þ X 13 þ X T13 þ τU 11 þU 13 þ U T13 ; Ω12 ¼ PA1 ; Ω13 ¼ PC; Ω14 ¼ h1 X 12  X 13 þ X T23 ; Ω17 ¼ τU 12 U 13 þ U T23 ; Ω18 ¼ h1 ðA0  BKÞT R1 ; Ω19 ¼ ρδðA0  BKÞT R2 ; Ω110 ¼ ð1  ρÞδðA0  BKÞT R3 ; Ω111 ¼ τAT0 R4 ; Ω22 ¼  ð1  hd ÞQ 4 þ ρδY 11 þ Y 13 þ Y T13 þ ρδY 22  Y 23  Y T23 ; Ω24 ¼ ρδY T12  Y T13 þY 23 ;

R3 1

P

1

13

" # i R3  Z 33

h 1 ¼ P  Z 33 ; R4

P 1

P 1

" # i ð1  τd ÞR4 P  1 ¼ ð1  τd ÞP  U 33 :  U 33

leads to (12a)–(12e). This ends the proof.

Appendix F. Proof of Theorem 7

Ω25 ¼ ρδY 12  Y 13 þ Y T23 ; Ω28 ¼ h1 AT1 R1 ; Ω29 ¼ ρδAT1 R2 ; Ω210 ¼ ð1  ρÞδAT1 R3 ; Ω211 ¼ τAT1 R4 ; Ω33 ¼  ð1  τd ÞW; Ω38 ¼ h1 C T R1 ; Ω39 ¼ ρδC T R2 ; Ω310 ¼ ð1  ρÞδC T R3 ; Ω311 ¼ τC T R4 ; Ω44 ¼ Q 2  Q 1 þ h1 X 22  X 23  X T23 þ ρδY 11 þ Y 13 þ Y T13 ;

Proof. Replacing A0  BK; A1 ;and C in (E1) with A0  BK þ DFðtÞE0  DFðtÞEb K; A1 þ DFðtÞE1 ; and C þ DFðtÞEc ; respectively, we find that Lemma 3 [1] for the unforced neutral uncertain time-varying delay system is equivalent to the following condition: T

T

Ωk þ Γ d FðtÞΓ e þ Γ e FðtÞΓ d o 0

Ω55 ¼ Q 3 Q 2 þ ρδY 22  Y 23  Y T23 þ ð1  ρÞδZ 11 þ Z 13 þ Z T13 ; Ω56 ¼ ð1  ρÞδZ 12  Z 13 þ Z T23 ; Ω66 ¼  Q 3 þ ð1  ρÞδZ 22  Z 23 Z T23 ; Ω77 ¼  ð1  τd ÞQ 5 þ τU 22  U 23 U T23 ; Ω88 ¼  h1 R1 ; Ω99 ¼  ρδR2 ; Ω1010 ¼  ð1  ρÞδR3 ; Ω1111 ¼  τR4 :

ðF1Þ

where h

Γ d ¼ εPD 0 0 0 0 0 0 εh1 R1 D ερδR2 D εð1  ρÞδR3 D ετR4 D and  Γ e ¼ E0  Eb K

E1

Ec

0

0

0

0

0

0

0

i

 0 :

Pre- and post-multiplying both sides of (E1) by n o diag P  1 ; P  1 ; P  1 ; P  1 ; P  1 ; P  1 ; P  1 ; R1 1 ; R2 1 ; R3 1 ; R4 1

By the , a sufficient condition guaranteeing (E1) for the neutral uncertain time-varying delay system (7) is that there exists a positive number ε 4 0 such that

and letting

Ωk þ ε  1 Γ d Γ d þ εΓ e Γ e o 0

T

P ¼ P  1 ; L ¼ KP ; P  1 WP  1 ¼ W ; P  1 Q i P  1 ¼ Q i ;

2

Ω11 Ω12 Ω13 Ω14 6 Ω22 0 Ω24 6 6 6  Ω 0 33 6 6   Ω44 6



P

 X 33

h 1 ¼ P  X 33 ; R2

0

0

Ω25 0

0

Ω111 Ω211 Ω311

Ω112 0 0

Ω113 Ω213 Ω313

0

0

0

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0

0

0

0



 

 

 







Ω55 Ω56 0 0 Ω66 0 0  Ω77 0   Ω88   











































Ω99 0 0 Ω1010 0 Ω1111  





















Ω812 Ω912 Ω1012 Ω1112 Ω1212

























0 0 0 0

Ω1313

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 o0 7 7 7 7 7 7 7 7 7 7 7 7 5

ðF3Þ

where

Ω112 ¼ εPD; Ω812 ¼ εh1 R1 D; Ω912 ¼ ερδR2 D; Ω1012 ¼ εð1  ρÞδR3 D; Ω1112 ¼ ετR4 D; Ω113 ¼ E0  Eb K; Ω213 ¼ E1 ; Ω313 ¼ Ec ; and Ωij ; ði; j ¼ 1; 2; 3; :::; 11; io j r 11Þ are defined in (E1).

and R1 1

0

0

0



Si ¼ Ri 1 ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4; m; n ¼ 1; 2; 3Þ;

h

0

Ω17 Ω18 Ω19 Ω110 0 Ω28 Ω29 Ω210 0 Ω38 Ω39 Ω310

0



P  1 Z mn P  1 ¼ Z mn ; P  1 U mn P  1 ¼ U mn ;

" # i R1 1

ðF2Þ

Applying the Schur complement of Lemma 2 [1] shows that (F2) is equivalent to

P  1 X mn P  1 ¼ X mn ; P  1 Y mn P  1 ¼ Y mn ;

6 6 6 6 6 6 Ωk ¼ 6 6 6 6 6 6 6 6 6 6 6 6 6 4

T

P 1

P 1

i

"

R2 Y 33

Pre- n and post- multiplying both sides of (F3) o by diag P  1 ; P  1 ; P  1 ; P  1 ; P  1 ; P  1 ; P  1 ; R1 1 ; R2 1 ; R3 1 ; R4 1 ; I; I

# P  1 ¼ P Y 33 ;

P ¼ P  1 ; L ¼ KP ; P  1 WP  1 ¼ W ; P  1 Q i P  1 ¼ and letting Q i ;P  1 X mn P  1 ¼ X mn ;

Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i

P.-L. Liu / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

14

P  1 Y mn P  1 ¼ Y mn ;P  1 Z mn P  1 ¼ Z mn ; P  1 U mn P  1 ¼ U mn ;Si ¼ Ri 1 ði ¼ 1; 2; 3; 4; 5; j ¼ 1; 2; 3; 4; m; n ¼ 1; 2; 3Þ; and " # h i R1 R1 1 P  1 P 1  X 33 " # h i R2 1 P 1 ¼ P X 33 ; R2 P  1 ¼ P  Y 33 ;  Y 33 h

R3 1

P 1

" # i R3

¼ P Z 33 ;

 Z 33 h

R4 1

P 1

P

1

" # i ð1  τd ÞR4 P  1 ¼ ð1  τd ÞP  U 33 :  U 33

leads to (14a)–(14e). This ends the proof.

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Please cite this article as: Liu P-L. Improved results on delay-interval-dependent robust stability criteria for uncertain neutral-type systems with time-varying delays. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.004i