Overview of recent advances in stability of linear systems with timevarying delays
Abstract
This study provides an overview and indepth analysis of recent advances in stability of linear systems with timevarying delays. First, recent developments of a delay convex analysis approach, a reciprocally convex approach and the construction of Lyapunov–Krasovskii functionals are reviewed insightfully. Second, indepth analysis of the Bessel–Legendre inequality and some affine integral inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a timevarying delay, where information on the bounds of the timevarying delay and its derivative is totally known, partly known and completely unknown, respectively. Third, a number of stability criteria are developed for the above three cases of the timevarying delay by employing canonical Bessel–Legendre inequalities, together with augmented Lyapunov–Krasovskii functionals. It is shown through numerical examples that these stability criteria outperform some existing results. Finally, several challenging issues are pointed out to direct the near future research.
1 Introduction
A timedelay system is also called a system with aftereffect or deadtime [1]. A defining feature of a timedelay system is that its future evolution is related not only to the current state but also to the past state of the system. Timedelay systems are a particular class of infinite dimensional systems, which have complicated dynamic properties compared with delayfree systems. A large number of practical systems encountered in areas, such as engineering, physics, biology, operation research and economics, can be modelled as timedelay systems [2, 3]. Therefore, timedelay systems have attracted continuous interest of researchers in a wide range of fields in natural and social sciences, see, e.g. [4–9].
Recalling some existing results, there are two types of approaches to delaydependent stability of linear timedelay systems: frequency domain approach and time domain approach. Frequency domain approachbased stability criteria have been long in existence [1, 9]. For some recent developments in the frequency domain, we mention an integral quadratic constraint framework [21–23], which describes the stability of a system in the frequency domain in terms of an integral constraint on the Fourier transform of the input/output signals [24]. In the time domain approach, the direct Lyapunov method is a powerful tool for studying stability of linear timedelay systems [1, 9]. Specifically, there are complete Lyapunov functional methods and simple Lyapunov–Krasovskii functional methods for estimating the maximum admissible delay upper bound that the system can tolerate and still maintain stability. Complete Lyapunov functional methods can provide necessary and sufficient conditions on stability of linear systems with a constant timedelay [25–28]. Simple LyapunovKrasovskii functional methods only provide sufficient conditions on stability of linear timedelay systems. Compared with stability criteria based on complete Lyapunov functional methods, although stability criteria based on simple LyapunovKrasovskii functional methods are more conservative, they can be applied easily to control synthesis and filter design of linear timedelay systems [29].

Model transformation approach. The model transformation approach employs the Leibniz–Newton formula to transform the system (1) to a system such that a crossterm is introduced in the derivative of the Lyapunov–Krasovskii functional. Then using the basic inequality (or called Young's inequality)
(4)or the improved basic inequality [30], foror the general basic inequality [31]: if(5)the crossterm can be bounded by , which exactly ‘offsets ’ the quadratic integral term in the derivative of the LyapunovKrasovskii functional, where is a proper quadratic function on and . As a result, a delaydependent stability criterion can be derived in terms of linear matrix inequalities. There are a number of model transformation approaches proposed in the literature, namely, ‘firstorder transformation’ [32], ‘parameterised firstorder transformation’ [33], ‘secondorder transformation’ [34], ‘neutral transformation’ [35] and ‘descriptor model transformation’ [36, 37]. As pointed out in [38, 39], under the firstorder transformation, or the parameterised firstorder transformation, or the secondorder transformation, the transformed system is not equivalent to the original one due to the fact that additional eigenvalues are introduced into the transformed system. Under the neutral transformation, although no explicit additional eigenvalue is introduced, some additional eigenvalue constraints for the stability of an appropriate operator should be satisfied [39]. The descriptor model transformation delivers some larger delay upper bounds since the transformed system is equivalent to the original one. 
Freeweighting matrix approach. Compared with model transformation approaches, a freeweighting matrix approach can provide an easier way to deal with the quadratic integral term . By introducing some proper zerovalued terms as [40, 41]
where and are called freeweighting matrices, the derivative of the Lyapunov–Krasovskii functional can be expressed as , where and is a certain matrix. Then a delaydependent stability criterion is obtained. It is clear that the model transformation and the bounding of crossterms are obviated by the freeweighting matrix approach. Moreover, simulation results have shown that freeweighting matrix approaches can produce larger delay upper bounds than model transformation approaches. 
Integral inequality approach. An integral inequality approach directly provides an upper bound for the quadratic integral term [42–44]. By using the Leibniz–Newton formula, an integral inequality for is proposed, which reads as
(6)where and is a free matrix.
In this paper, we provide an overview and indepth analysis of integral inequality approaches to stability of linear systems with timevarying delays. First, an insightful overview is made on convex delay analysis approaches, reciprocally convex approaches and the construction of Lyapunov–Krasovskii functionals. Second, indepth analysis of Bessel–Legendre inequalities and some affine integral inequalities is made, and recent stability results based on these inequalities are reviewed. Specifically, the refined allowable delay sets are discussed with insightful understanding. Third, we develop a number of stability criteria based on a canonical Bessel–Legendre inequality recently reported, taking three cases of timevarying delay into account. Simulation results show that the canonical Bessel–Legendre inequality plus an augmented Lyapunov–Krasovskii functional indeed can produce a larger delay upper bound than some existing methods. Finally, some challenging issues are proposed for the near future research.
The remaining part of the paper is organised as follows. Section 2 gives an overview of recent advances in convex and reciprocally convex delay analysis approaches, as well as the construction of Lyapunov–Krasovskii functionals. Recent integral inequalities and their applications to stability of linear systems with timevarying delay are reviewed in Section 3. A canonical Bessel–Legendre inequality and its affine version, together with a proper augmented Lyapunov–Krasovskii functional is developed to derive some stability criteria for three cases of timevarying delays. Section 5 concludes this paper and proposes some challenging problems to be solved in the future research.
Notation: The notations in this paper are standard. . and denote a blockdiagonal matrix and a blockcolumn vector, respectively. stands for a polytope generated by two vertices and . represents the set of polynomials of degree less than N, where N is a positive integer; and the notation refers to binomial coefficients given by . A symmetric term in a symmetric matrix is symbolised by a ‘ ’.
2 Recent advances in convex analysis approaches, reciprocally convex approaches and the construction of Lyapunov–Krasovskii functionals
2.1 Convex delay analysis approach
2.2 Reciprocally convex delay analysis approach
In the sequel, it will be shown that the inequality (23) is a special case of the inequality obtained from the following lemma (see [74, Lemma 1]).
Lemma 1.Let and be real column vectors with dimensions of and , respectively. For given real positive symmetric matrices and , the following inequality holds for any scalar and matrix satisfying
2.3 Construction of Lyapunov–Krasovskii functionals
2.3.1 Augmented Lyapunov–Krasovskii functionals
The purpose of the augmentation of a Lyapunov–Krasovskii functional is to help provide a tighter estimate on its derivative by introducing some new matrix variables as well as some new staterelated vectors. It is true that the estimate of the derivative of a Lyapunov–Krasovskii functional depends mainly on the treatment with some integral terms. However, such an estimate sometimes is not enough for a less conservative stability criterion. In both [72, 80], it has been proven that the Wirtingerbased inequality can produce a tighter estimate on the derivative of a Lyapunov–Krasovskii functional than Jensen integral inequality, but both the obtained stability criteria are of the same conservatism if the Lyapunov–Krasovskii functional is not augmented. Recent research [81, 82] shows that using an augmented Lyapunov–Krasovskii functional plus the N order Bessel–Legendre inequality indeed can yield nice stability criteria of less conservatism.
2.3.2 Lyapunov–Krasovskii functionals with multipleintegral terms
2.3.3 Lyapunov–Krasovskii functionals for linear systems with interval timevarying delays
It should be mentioned that, if both lower and upper bounds of are known to be constants, a novel Lyapunov–Krasovskii functional is introduced in [61], where the Lyapunov matrix P is chosen as a convex combination on , where and are two positivedefinite symmetric matrices. This idea is also applicable to the case of .
2.3.4 Lyapunov–Krasovskii functionals based on a delayfractioning approach
3 Recent developments of integral inequality approaches to stability of linear systems with timevarying delays
In this section, we focus on the recent developments of integral inequality approaches to stability of the system (1). To begin with, we first give an overview of integral inequalities developed recently.
3.1 Recent integral inequalities
Employing the Wirtinger inequality provides a larger lower bound than the wellused Jensen integral inequality for a nonnegative integral term [47]. Soon after, by introducing a proper auxiliary function, an auxiliaryfunctionbased integral inequality is reported in [53]. Both of them are given in the following.
Lemma 2.For any constant matrix , two scalars a and b with , and a vector function such that the integrations below are well defined, the following inequalities hold:

Wirtingerbased integral inequality:
(32) 
Auxiliaryfunctionbased integral inequality:
(33)
where
Clearly, the auxiliaryfunctionbased integral inequality (33) is an improvement over the Wirtingerbased integral inequality (32). A natural inspiration from (33) is to extend the inequality to a general form, which is completed by introducing the Legendre polynomials, leading to the canonical Bessel–Legendre inequality [81, 82].
Lemma 3.Under the assumption in Lemma 2, the following inequality holds:
Since in (35) depends on the Legendre polynomial, the inequality (35) is not convenient for use. In [82], a useful form of the canonical integral inequality is developed for stability analysis of timedelay systems, which is given in the following lemma.
Lemma 4.For an integer , a real symmetric matrix , two scalars a and b with , and a vectorvalued differentiable function such that the integrations below are well defined, then
Notice that is a k integral of the vector . Thus, the integral inequality (40) discloses an explicit relationship between and the vectors , and the multiple integrals .
The affine versions of (32) and (33) can be found in [67, 97] or [98]. As pointed out in [97] and [58], the affine version and its corresponding integral inequality provide an equivalent lower bound for the related integral term. It should be mentioned that those affine integral inequalities can be regarded as special cases of (46). For example, Lemma 1 in [98] is a special case of (46) with .
3.2 Recent developments on stability of the system (1) using recent integral inequalities
Although the canonical BesselLegendre inequality in Lemma 3 provides a lower bound for the integral term as tight as possible if , in the recent years, most researchers' interest is focused on its special cases such as [47, 99, 100] and [52, 53, 64, 101]. It is proven in [80] that a tighter bound of the integral term in the derivative of the Lyapunov–Krasovskii functional should not be responsible for deriving a less conservative stability criterion. Therefore, although the integral inequality (33) provides a tighter bound than (32), it is not a trivial thing to derive a less conservative stability criterion using the inequality (33). The main difficulty is that the vectors and in (34) are not easily handled in the stability analysis of the system (1). It is shown from [72] that the vectors and should occur in the derivative of the Lyapunov–Krasovskii functional so that a less conservative stability criterion for the system (1) can be obtained using the integral inequality (33). The recent development on this issue is briefly summarised as follows.

Case 1: is differentiable and satisfies
(47) 
Case 2: is differentiable and satisfies
(48) 
Case 3: is continuous and satisfies
(49)where , and are real constants.
In what follows, we consider the three cases.
3.2.1 Case 1
Since information on the upper and lower bounds of the timevarying delay and its timederivative is available, in order to formulate some less conservative stability criteria, an augmented Lyapunov–Krassovskii functional is introduced in [69] on the basis of the Lyapunov–Krassovskii functional in (28), where the first term is augmented with , where , , , , . Then taking the timederivative of the term yields some vectors similar to and induced by the auxiliaryfunctionbased integral inequality (33). Combining with the extended reciprocally convex inequality (18), some nice results are derived therein.
3.2.2 Case 2
3.2.3 Case 3
The timevarying delay is only known to be continuous (possibly not differentiable), which implies that information on the derivative of the timevarying delay is unavailable. Thus, the above Lyapunov–Krasovskii functionals in Cases 1 and 2 can be no longer used to produce the vectors similar to and in its timederivative. In this case, an augmented Lyapunov–Krasovskii functional is constructed in [52], where the quadratic in (28) is augmented by with , . By dividing the integral interval into two parts as and , those vectors induced from the integral inequality (35) appear in the derivative of the Lyapunov–Krasovskii functional, for details see [52, Theorem 1]. It should be pointed that stability of the system (1) with unknown information on the derivative of is also investigated in [53] using the auxiliaryfunctionbased integral inequality (33), but those vectors induced from (33) do not exist in the derivative of the chosen Lyapunov–Krasovskii functional. Thus, one can claim that [53, Theorem 1] is of the same conservatism as that using the Wirtingerbased integral inequality (32) instead of (33).
From the above analysis, one can see that, it is still challenging to investigate the stability for the system (1) with timevarying delay based on the recent integral inequalities. When both the timevarying delay and its derivative are bounded from above and from below, most existing stability criteria are based on the Wirtingerbased or the auxiliaryfunctionbased integral inequalities or the secondorder Bessel–Legendre inequality. In the other cases where the information on the derivative of the timevarying delay is partly known or completely unknown, relatively few results on stability of the system (1) are obtained, even using the secondorder Bessel–Legendre inequality.
3.3 Refinement of allowable delay sets
On the one hand, the above analysis just keeps an eye on the two vertices and in while no attention is paid to the other vertices and . In some situation, the vertices and can also not be reached. Let us still consider the above delay function in (55). Set . Then , leading to . Similarly, if setting , then we have that . In a word, these two vertices and for this delay function can never be reached.
It is a good idea to refine the allowable delay set such that less conservative stability criteria can be obtained. However, based on the above analysis, one cannot claim that the set in (57) is a refinement of . A set is called a refinement of only if , where denotes the real domain of . Therefore, the refinement of aims at seeking a possible ‘minimum’ polytope within to cover the real domain . How to do it depends on the delay function itself. For the delay given in (55), one can build a polygon (such as the octagon with green dashed lines in Fig. 3) as small as possible to cover the ellipsoid, while for other different it may be not the case. It should be pointed out that the vertices of a refinement delay set are not necessarily reached by when a stability criterion is established.
4 Stability criteria based on the canonical Bessel–Legendre inequalities (40) and (46)
In this section, we develop some stability criteria using the canonical Bessel–Legendre inequalities (40) and (46), in order to show the effectiveness of canonical Bessel–Legendre inequalities, and confirm some claims made in the previous sections as well.
4.1 Stability criteria for case 1
The first augmented term in (59) is motivated from Lemma 4 such that the vectors in (60) induced from the integral inequality (40) appear in the derivative of the Lyapunov–Krasovskii functional . The second and the third augmented terms are taken from [62]. It should be mentioned that the Lyapunov–Krasovskii functional in (59) is different from the one in (51), which is dependent on the Legendre polynomials.
4.1.1 N dependent stability criteria
Proposition 1.For constants , and a positive integer N, the system (1) subject to is asymptotically stable if there exist real matrices , , and and real matrices and with appropriate dimensions such that, for
Proof.First, we introduce a vector as , , , , , , . It is easy to verify that and , where and are defined in Proposition 1. Taking the derivative of in (59) along with the trajectory of the system (1) yields
Instead of the integral inequality (40), one can also use its affine version (46) to derive another N dependent stability criterion by slightly modifying the Lyapunov–Krasovskii functional (59), where the term is replaced with . The result is stated in the following proposition.
Proposition 2.For constants , and an integer , the system (1) subject to is asymptotically stable if there exist real matrices , , and and real matrices and with appropriate dimensions such that, for
Remark 1.Propositions 1 and 2 deliver two N dependent stability criteria for the system (1) subject to (47), thanks to the canonical integral inequality (40). The number of required decision variables can be calculated as for Proposition 1 and for Proposition 2. Moreover, the positive definiteness of the matrices and can be relaxed if one follows the line in [76] or [81].
4.1.2 Hierarchy of LMI stability criteria
In [81], it is proven that the stability criterion in terms of LMIs forms a hierarchy. In the following, it is shown that such a hierarchical characteristic is also hidden in the LMIs of Propositions 1 and 2. Based on Proposition 1, one has
Proposition 3.For the system (1) subject to (47), one has that
Proof.Without loss generality, suppose that is not empty. From the definition of , there exist real matrices , , and and real matrices and with appropriate dimensions such that and , which are equivalent to
Similar to Proposition 3, one can prove that the LMIs in Proposition 2 also form a hierarchy. For given scalars and , we denote by the admissible maximum upper bound of the timevarying delay using Proposition 1 or 2. Then from the hierarchy feature, one can draw a conclusion that .
4.2 Stability criteria for case 2
Proposition 4.For constants and , the system (1) subject to (48) is asymptotically stable if there exist real matrices , , and and real matrices and with appropriate dimensions such that
Proof.Taking the timederivative of in (74) yields
Similar to Proposition 2, if using the affine integral inequality (46) instead of (40), we have the following result.
Proposition 5.For constants and , the system (1) subject to (48) is asymptotically stable if there exist real matrices , , and and real matrices and with appropriate dimensions such that
Remark 2.Propositions 4 and 5 provide two stability criteria for the system (1) subject to (48). Compared with [64, Theorem 1], the main difference lies in that Propositions 4 and 5 are derived based on such a condition as for , where is a linear matrixvalued function on , leading to a necessary and sufficient condition and such that for . This linear matrixvalued function contributes to the introduction of the vectors and in (81). However, in the proof of [64, Theorem 1], is a quadratic function on . Thus, applying the quadratic convex approach in (13) only gives a sufficient condition such that for .
Remark 3.The purpose of introducing the vectors and in (81) is to absorb such that is linear on ; Otherwise, will be a triple matrixvalued polynomial function on , which is difficult in deriving a stability criterion for the system (1) in Case 2. The number of decision variables required is for Proposition 4 and for Proposition 5.
4.3 Stability criteria for case 3
Proposition 6.For a constant , the system (1) subject to (49) is asymptotically stable if there exist real matrices , , and and real matrices and with appropriate dimensions such that one of the following two statements is true.
Proof.The proof can be completed by the following the proof of Proposition 4.□
Remark 4.In Case 3, Proposition 6 presents two stability criteria for the system (1). By using the secondorder Bessel–Legendre inequality, a stability criterion for the system (1) with (49) is also reported in [52, Theorem 1]. The main difference between them lies in the chosen Lyapunov–Krasovskii functional. In Proposition 6, an augmented vector is introduced in in (88), but not in [52, Theorem 1]. As a result, taking the derivative of the augmented term yields
Remark 5.The number of decision variables required in Proposition 6 is for the condition (i) and for the condition (ii), which are smaller than in [52, Theorem 2].
Remark 6.It should be pointed out that, the proposed results in this section can be easily extended to a linear system with an interval timevarying delay provided that one modifies the chosen Lyapunov–Krasovskii functionals by taking the lower bound into account. Because of their similarities, those results are omitted in the paper.
4.4 Illustrative examples
In this section, we compared the above stability criteria with some existing ones through two numerical examples.
Example 1.Consider the system (1), where
Example 1 is well used to calculate the admissible maximum upper bound (AMUB) for the timevarying delay . In order to make a comparison, we consider two cases of .
Case 1: satisfies (47) with . We compared the stability criteria with some existing ones obtained for defined in (54). For different values of , Table 1 lists the obtained AMUBs of by Seuret and Gouaisbaut [102, Theorem 1], Zhang et al. [72, Proposition 1], Zhang et al. [69, Theorem 2], Lee et al. [100, Theorem 1], Zeng et al. [67, Corollary 1], Seuret and Gouaisbaut [81, Theorem 8 with ], the IQC approach [21], the quadratic separation approach [104] and Propositions 1 and 2 with in this paper. From Table 1, one can see that
Method  0  0.1  0.5  0.8 

[21]  6.117  4.714  2.280  1.608 
[104]  6.117  4.794  2.682  1.957 
[69]  6.165  4.714  2.608  2.375 
[67]  6.059  4.788  3.055  2.615 
[102]  6.0593  4.71  2.48  2.30 
[100]  6.0593  4.8313  3.1487  2.7135 
[72]  6.168  4.910  3.233  2.789 
[81]  6.1725  5.01  3.19  2.70 
Proposition 1 ()  6.0593  4.8344  3.1422  2.7131 
Proposition 1 ()  6.1689  4.9192  3.1978  2.7656 
Proposition 1 ()  6.1725  4.9203  3.2164  2.7875 
Proposition 1 ()  6.1725  4.9246  3.2230  2.7900 
Proposition 2 ()  6.0593  4.8377  3.1521  2.7278 
Proposition 2 ()  6.1689  4.9217  3.2211  2.7920 
Proposition 2 ()  6.1725  4.9239  3.2405  2.8159 
Proposition 2 ()  6.1725  4.9297  3.2527  2.8230 
 Results in this line are obtained from Theorem 8 with in [81].
 • Propositions 1 and 2 with obtain a larger upper bound than the criteria in [67, 69, 72, 100, 102], the IQC approach [21] and the quadratic separation approach [104]. Even for and , Propositions 1 and 2 outperform [69, Theorem 2], [100, Theorem 1], [67, Corollary 1], the IQC approach [21] and the quadratic separation approach [104].
 • For , Seuret and Gouaisbaut [81, Theorem 8 with ] give a larger delay upper bound than Propositions 1 and 2 with due to that the positive definiteness of the matrices and are relaxed. However, for and , Propositions 1 and 2 offer better results that [81, Theorem 8].
 • For the same N, Proposition 2 delivers a larger upper bound than Proposition 2 at the cost of higher computation burden, which means that a stability criterion using the affine integral inequality (46) can derive a larger upper bound than that using the integral inequality (40) and the improved reciprocally convex inequality (21).
Case 2: The timevarying delay satisfies (48). In order to show the effectiveness of Propositions 4 and 5, the AMUBs of are listed in Table 2 for different values of . From this table, one can see that (i) Propositions 4 and 5 indeed can derive some larger upper bounds of than [64, Theorem 1] and [79, Theorem 1], while Propositions 4 and 5 require more decision variables than [79, Theorem 1] () and [64, Theorem 1] (); and (ii) The affine integral inequality (46) can result in a larger upper bound than the integral inequality (40) plus the improved reciprocally convex inequality (21).
Example 2.Consider the system (1) subject to (49), where
This example is taken to illustrate the validity of Proposition 6.
For comparison, we calculate the upper bound of such that the system remains stable. Applying [47, Theorem 7], [53, Theorem 1], [52, Theorem 2] and Proposition 6, the obtained results and the required number of decision variables are listed in Table 3, from which one can see that Proposition 6 outperforms those methods in [47, 52, 53]. Moreover, it is clear that using the affine integral inequality (46) can yield a larger upper bound than that using the integral inequality (40) though more decision variables are required.
In summary, through two wellused numerical examples, it is shown that, the obtained stability criteria in this paper are more effective than some existing ones in deriving a larger upper bound for a linear system with a timevarying delay.
As a counterpart of integral inequalities, finitesum inequalities for stability analysis of discretetime systems with timevarying delays also have gained much attention. A large number of finitesum inequalities and stability criteria have been reported in the published literature, see, [105–114]. Since discretetime systems with timevarying delays are not the focus of the paper, stability criteria based on finitesum inequalities developed recently are not mentioned in the paper.
5 Conclusion and some challenging issues
An overview and indepth analysis of recent advances in stability analysis of timedelay systems has been provided, including recent developments of integral inequalities, convex delay analysis approaches, reciprocally convex approaches and augmented Lyapunov–Krasovskii functionals. Then, some existing stability conditions have been reviewed by taking into consideration three cases of timevarying delay, where information on the upper and lower bounds of the delayderivative are totally known, partly known and completely unknown. Furthermore, a number of stability criteria have been developed by employing the recent canonical Bessel–Legendre integral inequalities and an augmented Lyapunov–Krasovskii functional. When information on the lower and upper bounds of both and is known, the obtained stability criteria have been proven to be hierarchical.
 • If the positive integer N approaches to infinity, the canonical N order Bessel–Legendre inequality can provide an accurate estimate on the integral term. Thus, using such an integral inequality, it is possible to derive a necessary and sufficient condition on stability for linear systems with timevarying delays, which is interesting but challenging. Moreover, extending the canonical N order Bessel–Legendre inequality to multidimensional systems like 2D systems with timevarying delays is also an interesting topic [115, 116].
 • For the system (1) subject to (48) or (49), no N dependent stability criteria are derived using the integral inequality (40) due to the vectors in which the scalar appears in the form of its inverse. Applying the integral inequality (40) to the system (1) possibly yields such a stability condition as , where are real matrices irrespective of the timevarying delay . How to obtain a necessary and sufficient feasible condition such that for is a significant problem.
 • In the proof of Proposition 4, four extra vectors and are introduced to absorb such that the obtained stability condition is dependent linearly on the timevarying delay . If not doing so, will be a triple matrixvalued polynomial function on . As a result, a necessary and sufficient condition, i.e. and , can be derived such that for . How to extend this technique to a general case is an interesting issue.
 • The integral inequality (40) is established based on a sequence of orthogonal polynomials. Is it possible to formulate some integral inequality based on a sequence of nonorthogonal polynomials such that the scalars disappear or appears linearly? Answering this question is beneficial for deriving less conservative stability criteria for linear systems with timevarying delay, which is significant and challenging.
 • Simulation in this paper shows that the canonical BesselLegendre inequality approach can yield some nice results on stability. However, how to apply it to deal with control problems of a number of practical systems, such as networked control systems [117, 118], eventtriggered control systems [119–121], vibration control systems [122–124], formation control systems [125] and multiagent systems [126–129], deserves much effort of researchers.
6 Acknowledgment
This work are supported in part by the Australian Research Council Discovery Project under Grant DP160103567.