Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 4969470, 10 pages
https://doi.org/10.1155/2019/4969470

Research Article

Novel Delay-Decomposing Approaches to Absolute Stability Criteria for Neutral-Type Lur’e Systems

Liang-Dong Guo, Sheng-Juan Huang, and Li-Bing Wu

School of Science, University of Science and Technology Liaoning, Anshan 114051, China

Correspondence should be addressed to Liang-Dong Guo; ldguo@ustl.edu.cn

Received 27 August 2019; Accepted 9 November 2019; Published 2 December 2019

Academic Editor: Mingshu Peng

Copyright © 2019 Liang-Dong Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The problem of absolute stability analysis for neutral-type Lur’e systems with time-varying delays is investigated. Novel delay-decomposing approaches are proposed to divide the variation interval of the delay into three unequal subintervals. Some new augment Lyapunov–Krasovskii functionals (LKFs) are defined on the obtained subintervals. The integral inequality method and the reciprocally convex technique are utilized to deal with the derivative of the LKFs. Several improved delay-dependent criteria are derived in terms of the linear matrix inequalities (LMIs). Compared with some previous criteria, the proposed ones give the results with less conservatism and lower numerical complexity. Two numerical examples are included to illustrate the effectiveness and the improvement of the proposed method.

1. Introduction

Over the last 30 years, time delay system has been one of the hottest research areas in control engineering for time delay often appears in many control systems either in the state, the control input, or the measurements [1, 2]. Since time delay frequently occurs in practical systems and is often the source of instability, there have been many results for stability of delayed systems [3–9]. Also stabilization [10–12], filtering [13], and adaptive control [14] of time-delay systems have received considerable attention. The neutral systems often appear in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar [15]. A considerable number of studies related to this topic have been reported (see, for example, [16–21], and the references therein).

Lur’e system is originally from a pilot robot [22], which is one of the important classes of nonlinear systems whose nonlinear element satisfies certain sector constraints. Many systems such as Chua’s circuits, Goodwin models, Swarm models, n-scroll attractors, and hyperchaotic attractors can be represented as Lur’e-type systems [23, 24]. In recent years, stability analysis of the neutral-type Lur’e system with time delays has attracted the attention of many researchers [25–35]. The main purpose of stability analysis is to calculate the maximum allowable delay bounds (MADBs), which is a key index for judging the conservatism of stability criteria, such that the Lur’e system maintains absolute stability for any time delay less than the MADBs by using the LKF method. In [25], absolute stability of Lur’e systems with sector-bounded nonlinearities and constant delay was discussed by using a Lur’e-Postnikov function. To avoid involving a considerable number of free-weighting matrices and leading to a computationally expensive stability criterion in [26], the general free weighting matrix method was proposed and some improved delay-range-dependent stability criteria were obtained [27, 28]. By using integral inequality and the Wirtinger integral inequality method instead of free weighting matrix one, some improved delay-dependent robust stability criteria were derived in [29, 30], respectively. Also, by eliminating nonlinearity and reducing the number of free-weighting matrices, Lur’e systems with interval time-varying delays were discussed in [31]. By discretizing the delay interval into two segmentations with an unequal width, some delay-dependent sufficient conditions were given for the robust stability of neutral-type Lur’e systems in [32, 33]. By employing an augmented LKF method, reciprocally convex combination approach, and convex combination technique, several robust absolute stability criteria for uncertain neutral-type Lur’e systems with time-varying delays were presented in [34]. Very recently, by constructing a LKF including both double-integral terms and triple-integral terms, using the piecewise analysis method, Wirtinger-based integral inequality, and the reciprocally convex combination technique, some new stability criteria were obtained in [35]. In fact, one of the term in LKFs in [35] is the special case of the delay-partitioning approach, where . The delay-partitioning method is an effective one to reduce a criterion’s conservativeness, which is widely used in the stability analysis for various systems (see details in [36–39]). However, when utilizing the delay-partitioning approach, the term is inevitably involved in LKFs, where and N is the delay-partitioning number. It is clear that the derived conditions become more complicated and the computational burden grows bigger when N increases. In addition, to relate to the Wirtinger-based integral inequality and deal with the derivative of triple-integral terms introduced in LKFs, it is ineluctable that the extra terms , , , and had to be introduced into the derivation process in [35], which leads to a sharp increase in the dimensions of the LMIs involved.

Motivated by this mentioned above, the aim of this work is to revisit the stability analysis for the neutral-type Lur’e system. In this study, novel delay-decomposing approaches are proposed firstly. Different from the delay-partitioning approaches or the delay-decomposing ones in [32, 33, 36–40], the interval of the state time delay is divided into three unequal subintervals , , and . In particular, to establish the relationship of the vectors such as , and , the novel terms and are introduced in LKFs in each of subintervals, which are more general than the ones in [28, 35], where and . It is worth mentioning that the merit of the proposed delay-decomposing method lies in that the dimensions of the LMIs involved are independent of the number of subinterval. In addition, to avoid introducing the extra vectors by Wirtinger-based integral inequality, the reciprocally convex combination method [41] and the integral inequality [42] are utilized to deal with the bounds of integral terms. Some novel LKFs related to the above inequalities are constructed on the obtained three subintervals. The presented stability criteria are given in terms of LMIs. Compared with the related literature, the conclusions of this paper have the advantages of less conservatism conservatism and the dimensions of the LMIs. Finally, two well-known numerical examples are given to demonstrate the effectiveness and less conservatism over the existing results.

Notation: in this paper, denotes n-dimensional Euclidean space and is the set of all real matrices. For symmetric matrices X and Y, the notation (respectively, ) means that the matrix is positive definite (respectively, nonnegative). denotes the block diagonal matrix. The subscript “T” denotes the transpose of the matrix. denotes the identity matrix.

2. Problem Statements and Preliminaries

Consider a class of Lur’e systems of neutral type with time-varying delays and sector-bound nonlinearities described as follows:

where are constant matrices with appropriate dimensions. is the state vector, and is the output vector. The time delay is a time-varying continuous function satisfying

where are the known constant scalars.

is the nonlinear function and is assumed to satisfy the finite sector restriction:

with known positive scalar or the infinite sector restriction:

The objective of this paper is to formulate the delay-dependent stability conditions of system (1). The following lemmas will play important roles in deriving the criteria.

Lemma 1 (see [41]). Let have positive values in an open subset D of . Then, the reciprocally convex combination of over D satisfies

subject to

Lemma 2 (see [42]). For any matrices with compatible dimensions, any continuous time vector function with compatible dimensions, and any scalar satisfying , the following integral inequality holds:

if and .

Remark 1. Let and , the integral inequality reduces the one in [43].

3. Main Result

In this section, some new delay-dependent stability criteria are proposed for system (1).

Note and (). It is easy to see that holds. Then, we divide the time interval into three subintervals , and . In the following, we will propose some criteria for the three subintervals.

Now, we give the stability criteria for system (1) with conditions (2) and (4) when as follows.

Theorem 1. For given scalars , , , , and , , system (1) with conditions (2) and (4) is absolutely stable if there exist symmetric positive definite matrices , symmetric positive definite matrices , symmetric semi-positive definite matrices , positive diagonal matrices , , any matrix , and matrices , such that LMIs (5) and (6) hold when :

where

Proof. For positive diagonal matrices and positive definite matrices , let us consider the following LKF candidates for the case :

where

The time derivative of along the trajectory of system (4) is given by

where

If holds, one can compute out the following according to Lemma 1:

where is given in (6).
Based on Lemma 2 (in which and ), when , the following inequality holds:

and using Lemma 2 again (in which and ), when , one can conclude

Under the assumption on nonlinear function (4), the following inequality holds

It is equivalent to

Rewrite the above as follows:

where

Then, combining (3)–(15) leads to

Therefore, if LMIs (5) and (6) hold, one has , which shows the absolute stability of system (1) subject to (2) and (4), when satisfies . This completes our proof.
For the case , we construct the LKFs as follows:

where

and are defined in (7).
Then, the derivative of can be obtained as

According to Lemma 1, if , the following inequality holds:

where is given in (20).
In addition, by using Lemma 2 (in which and ), if , one can obtain

The other procedure is straightforward from the proof of Theorem 1; we can cope with the situation of and have the following.

Theorem 2. For given scalars , system (1) with conditions (2) and (4) is absolutely stable if there exist symmetric positive definite matrices , symmetric positive definite matrices , symmetric semi-positive definite matrices , positive diagonal matrices , , any matrix , and matrices , such that LMIs (19) and (20) hold for the case :

where

and and K are defined in Theorem 1.
For the case , we construct the following LKF:

where

and are defined in (7).
Taking the derivative of yields

By using Lemma 1, if , one can obtain

where .
Handle the integral terms and in the same way as in (13) and (17), respectively. Following the same procedure, we can cope with the situation of and have the following.

Theorem 3. For given scalars , system (1) with conditions (2) and (4) is absolutely stable if there exist symmetric positive definite matrices , symmetric positive definite matrices , symmetric semi-positive definite matrices , positive diagonal matrices , , any matrices , and matrices , such that LMIs (23) and (24) hold for the case :

where

and , K, and are defined in Theorems 1 and 2, respectively.

Remark 2. Theorems 1–3 present the absolute stability criteria for the Lur’e system. Unlike the delay-partitioning approach or delay-decomposing one used in [28, 32, 33, 35, 40], the interval of the state time delay has been divided into three subintervals , , and . Obviously, the range of the subintervals , , and is , , and . Thus, to know more about the time-varying , one can select the value of the parameter α as close as possible to 0 for the case or case and as close as possible to 0.5 for the case , respectively. The following examples in the work will illustrate the point.

Remark 3. The terms and have been introduced in LKFs in the work instead of the terms in [28, 35] and in [36–39], where . And the relations between , and have been effectively expressed by the derivative of . One can easily see that or deduces into when . This is to say the established LKFs are more general than ones in [28, 35]. The idea is expected to reduce the conservatism of obtained criteria. On the other hand, it can effectively overcome the weakness of the computational burden growing bigger when delay-partitioning number increases in [36–39].

Remark 4. In [35], the Wirtinger-based integral inequality and double-integral inequality were used to deal with bounds of the derivative of double-integral and triple-integral terms in LKFs. Thus, the extra terms , , , and were unavoidably introduced in the derivation process, which leads the dimensions of the LMIs involved to increase sharply. A detailed comparison is given in Example 1.

Remark 5. If function of system (1) satisfies sector condition (3), for any , one has the following:

which is equivalent to

Thus, if function of system (1) satisfies sector condition (3), we have the following corollary.

Corollary 1. For given scalars , the system (1) with conditions (2) and (3) is absolutely stable if there exist symmetric positive definite matrices , , , with appropriate dimensions, positive diagonal matrices , , and matrices with appropriate dimensions and properties, such that LMIs (6) and (26) hold for the case , (3) and (28) hold for the case , and (20) and (28) hold for the case , respectively:

where , , are defined in Theorems 1–3.

4. Illustrative Example

In this section, we will use two well-known numerical examples to show the effectiveness and benefits of our results.

Example 1. Consider the following nominal neutral-type Lur’e system (1) subject to (2) and (4) with the parameters:

The purpose of this example is to compute MADBs of such that the neutral-type Lur’e system (1) remains stable for different and . For given , , and , the acceptable upper bounds of is 0.271, 2.277, and 2.652 when by using Corollary 1 () in our work, respectively. In order to make a comparison with some existing stability criteria, we calculate the MADBs and list them in Tables 1–3. Together with all derived MADBs listed in Tables 1–3, one can check that Corollary 1 can be superior over some present ones. The number of decision variables, maximal order of LMIs between our work, and the criteria in [29, 30, 33, 35] are listed in Table 4. It shows that our proposed method involves smaller decision variables or lower maximal order of LMIs than the relative ones. To confirm the obtained result, a simulation result is shown in Figure 1 when µ is unknown, .

Table 1: MADBs of for different , when for Example 1.

Method

[44]
2.6052
1.9538
1.6232
1.1659

[31]
2.6062
1.9730
1.6408
1.1854

[27]
2.6488
2.0056
1.6679
1.2049

[30]
2.8890
2.0678
1.7048
1.6059

[32]
2.9962
2.1316
1.7138
1.3204

[28]
2.9997
2.1012
1.7307
1.3359

[29]
3.0717
2.1171
1.7310
1.3367

[33]
3.1034
2.1579
1.7436
1.7088

[34]
3.1743
2.1789
1.7467
1.7153

[35]
3.4880
2.3787
1.8062
1.4625

Cor 1 ()
5.886
4.456
3.705
2.676

Cor 1 ()
3.784
2.865
2.385
2.176

Cor 1 ()
3.060
2.157
2.000
1.999

Cor 1 ()
3.729
3.729
3.729
3.729

Table 2: MADBs of for different , when for Example 1.

Method

[29]
2.5131
1.8146
1.4951
1.1168

[44]
2.1532
1.6986
1.3687
1.0066

[31]
2.1537
1.7041
1.3828
1.0095

[27]
2.1883
1.7322
1.4056
1.0262

[30]
2.3628
1.7752
1.5027
1.4364

[32]
2.4472
1.8289
1.4551
1.1058

[28]
2.4563
1.8009
1.4823
1.1157

[33]
2.5302
1.8461
1.5371
1.5208

[35]
2.8538
2.0236
1.5679
1.2425

Cor 1 ()
4.862
3.848
3.119
2.280

Cor 1 ()
3.126
2.474
2.101
1.922

Cor 1 ()
2.877
2.877
2.877
2.877

Cor 1 ()
2.504
1.847
1.738
1.737

Table 3: MADBs of for different , when for Example 1.

Method

[31]
0.1122
0.1088
0.1086
0.1086

[27]
0.1122
0.1089
0.1086
0.1086

[32]
0.1126
0.1102
0.1102
0.1102

[28]
0.1130
0.1105
0.1105
0.1105

[44]
0.1221
0.1087
0.1083
0.1083

[29]
0.1227
0.1197
0.1197
0.1197

[35]
0.1799
0.1697
0.1660
0.1658

[30]
0.2495
0.2366
0.2300
0.2285

[33]
0.2906
0.2745
0.2653
0.2621

Cor 1 ()
0.321
0.311
0.310
0.310

Cor 1 ()
0.281
0.272
0.271
0.271

Cor 1 ()
0.189
0.189
0.189
0.189

Cor 1 ()
0.113
0.113
0.113
0.113

Cor 1 ()
0.119
0.119
0.119
0.119

Table 4: Number of decision variables (NDV) and maximal order of LMIs (MOL) of different methods.

Method
NDV
MOL

[29] Theorem 1

[35] Theorem 10

[30] Theorem 1

[33] Theorem 1

Our Cor 1

Figure 1: State responses of the system considered in Example 1.

Example 2. Consider Chua’s circuit example discussed in [35]:

The nonlinear function . Let , and ; then, Chua’s circuit can be expressed as a Lur’e-type system with

The feedback nonlinear function belongs to .
Now, we calculate MADBs of . For different , the obtained results are given in Table 5. From this table, it is clear to see that Theorems 1 and 2 offer larger MADBs of than those methods in existing references.

Table 5: MADBs for different , when for Example 2.

Method

[24]
0.1622
0.1591
0.1566
0.1541
0.1527

[27]
0.1745
0.1698
0.1698
0.1698
0.1698

[29]
0.1747
0.1710
0.1703
0.1703
0.1703

[32]
0.1771
0.1721
0.1715
0.1715
0.1715

[35]
0.2790
0.2691
0.2553
0.2463
0.2463

Theorem 1 (α = 0.45)
0.361
0.359
0.359
0.359
0.359

Theorem 2 (α = 0.45)
0.295
0.295
0.295
0.295
0.295

Theorem 3 (α = 0.01)
0.163
0.163
0.163
0.163
0.163

5. Conclusions

This paper has investigated the absolute stability analysis for neutral-type Lur’e systems with time-varying delays. Based on the new delay-decomposition approaches in combination with the integral inequality and reciprocally convex technique, several improved stability criteria have been derived by constructing some appropriate LKFs on the subintervals. The merit of the obtained stability criteria lies in the significant less conservativeness and lower computational complexity than some existing ones. Finally, two examples have been given to demonstrate the effectiveness and less conservatism of the proposed method. In the future works, we will be dedicated to study the stability analysis for systems with infinite delays and devote to the study of output feedback, tracking control and filtering of the neutral-type Lur’e systems with time-varying delays based on the method proposed in this paper.

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61503058 and 61773013.

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