Improved Stability Criteria for Time-Varying Delay System Using Second and First Order Polynomials

This article concerns the problem of stability analysis of systems with time-varying delay. Recent developments in this direction involves approximation of a second order polynomial function of time-delay. This article proposes a new Lyapunov-Krasovskii Functional that does not introduce the second-order polynomial and thereby avoid the approximation involved in obtaining the stability criterion. Two stability criterion are presented, one introduces the second-order polynomial and the other one does not. A comparison using numerical examples shows that the avoidance of second-order polynomial formulation leads to improved results.


I. INTRODUCTION
Consider a Time-delay system as: where x(t) ∈ R n is the state vector; A, A τ ∈ R n×n are the constant system matrices with continuously differentiable initial condition. The time-varying delay τ t is otherwise represented as τ (t) that satisfies the following properties: whereh, µ 0 and µ 1 are constants. The problem of stability analysis for time-delay system (1) has been widely investigated over the years and consistent improvement have been attained using the Lyapunov-Krasovskii (LK) approach [1]- [4]. Performance of such stability criteria is assessed by the maximum permissible upper bound (MPUB) of τ t .
The key steps involved in obtaining improved results are by constructing appropriate LK Functional (LKF) involving augmentation of several states including state integrals and delayed states, and to find precise bound of the integral function in the derivative of LKF. It was shown in [5]- [8] that LKF The associate editor coordinating the review of this manuscript and approving it for publication was Jun Shen . must include the states involved in the integral function such that interaction among various states are established in the resulting criterion. Therefore, construction of LKFs depends on the integral function to be used in the derivation of the stability criterion. On the other hand, regarding integral inequalities, Jensen and Wirtinger inequalities [5], [9] have been widely used to obtain bound of the integral function. Due to this reason, stability criteria are obtained as a first-order polynomials f (τ t ) = a 1 τ t + a 0 , where a i (i = 0, 1) ∈ R and independent of τ t . To make f (τ t ) < 0 for all τ t ∈ [0,h], two inequalities are to be satisfied as f (0) < 0 and f (h) < 0.
Recent developments involve the use of (i) double integral states to construct the LKF and (ii) higher order inequalities, such as auxiliary-polynomial based inequality [10] and Bessel Legendre function based inequality (BLBI) [11] to obtain bound of the integral function. The stability criteria are then obtained in quadratic form as: f (τ t ) = a 2 τ 2 t + a 1 τ t + a 0 with a i (i = 0, 1, 2) ∈ R. In [12], the above is approximated as f (0) < 0 and f (h) < 0 to obtain f (τ t ) < 0, but it applies when a 2 ≥ 0. The requirement a 2 ≥ 0 has been relaxed in [13] by introducing an additional condition −h 2 a 2 + f (h) < 0. Similar works have been reported in [14]- [17], where some new set of conditions for negativity of a quadratic function VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ have been developed. In [15] and [16], tuning parameters are introduced that lead to solving inequalities that are dependent on a tuning parameter to obtain less conservative results. In [14], the entire delay interval [0,h] is divided uniformly in multiple subintervals and for each of the sub-intervals, two end points are considered. It has been illustrates that less conservative results can be obtained by increasing the number of subintervals. Recently, two lemmas have been introduced in [17] to ensure the negativity of quadratic polynomial. The first inequality uses the property of cross-point between two tangent lines of the end points whereas the other inequality exploits the condition of finite interval. It should be noted that such quadratic inequalities also introduce conservativeness, since one has to approximate the polynomial. Then question arises ''Can the appearance of quadratic form of τ t be avoided while still using higher-order integral inequalities?''. An investigation on the appearance of τ 2 t dependent terms in the derivative of LKFs reveals that it is due to the combination of product of delay interval and interval-normalized states. To this end, a further inspection leads to the fact that these terms can be avoided by treating integral and its intervalnormalized states as separate individual states. Moreover, not only this, out of the several possible LKFs, one requires to choose appropriate LKF that leads to less conservative result. This motivates the present work, which incorporates the following.
(i) An augmented-type delay-product LKF has been constructed by including single and double integral states along with their interval-normalized forms to incorporate more τ 2 t dependent terms in the resulting stability criterion.
(ii) Based on this LKF and negative-determination lemma (NDL) introduced in [13] to obtain negativedefiniteness of the resulting criterion, a stability criteria is derived leading to quadratic function based LMI conditions.
(iii) Another new stability criterion is derived by avoiding quadratic function based LMI conditions by treating integral states and their normalized forms as individual states in the derivative of the LKF.
Finally, three examples are considered to illustrate the less conservativeness of the proposed stability criterion that does not involve the NDL. It is demonstrated that not using the NDL leads to considerably less conservative results.
Notations:-In this article, 0 and I represent the zero and identity matrices respectively. For any square matrix N, we denoted Sym{N } = N + N T . Also, n-dimensional Euclidean space and set of all n × m real matrices are represented by R and R n×m , respectively.

II. USEFUL LEMMAS
In this section, the following Lemmas are recalled that will be used to derive the main result. These are useful to deal with inversely weighted positive convexity parameters, integral function and second order polynomial conditions. Lemma 1 [18]: For a real scalar α ∈ (0, 1), symmetric matrices R 1 ≥ 0, R 2 ≥ 0 and any matrices U 1 and U 2 , the following inequality holds: The above is known as the Reciprocity Lemma and it is used to obtain a bound of quadratic terms involving reciprocal parameters. Next, we introduce the second-order BLBI, which supersede the inequalities proposed in [10] and [19], respectively.
Lemma 2 [11]: For any constant matrix R ≥ 0, the following inequality holds for all continuously differentiable where The next Lemma is known as the NDL and it guarantees the negative definiteness of a quadratic function in the interval [0,h] irrespective of its concave or convex nature.
Lemma 3 [13]: The given quadratic function z(u) = a 2 u 2 + a 1 u + a 0 , where a 0 , a 1 , a 2 ∈ R, satisfy z(u) < 0 for all u ∈ [0, ] if the following three inequalities hold: (a) z(0) < 0 (b) z(h) < 0 (c) −h 2 a 2 + z(0) < 0 To simplify matrix and vector representations, the following notations are subsequently used: VOLUME 8, 2020 Note that the vectors w i , (i = 1, 2) contain the integral states for the delay intervals [t −τ t , t] and [t −h, t −τ t ]. Their interval normalized forms are obtained by multiplying reciprocal of the delay interval with the integral states, for example, 1 τ t w 1 (t) and 1 h−τ t w 2 (t). In this article, both the integral and their interval normalized states are utilized to define the LKF.

III. MAIN RESULTS
In this section, two stability criteria are derived for system (1). One leads to a criterion involving the second-order polynomial and the other one does not. The construction of LKF is discussed next.

A. LYAPUNOV-KRASOVSKII FUNCTIONAL
In [20], a new type of LKF has been introduced in which single integral states have been used as pivot elements of augmented vectors to construct delay-coefficient based quadratic terms. The time-derivative of this LKF introduces terms involving both the delay and its derivative, so that these terms contribute to reduce conservativeness in the stability criterion. By extending this idea, in [21] and [22], new delayproduct based LKF have been constructed by using double integral states. On the basis of these works, a new LKF is constructed by involving both the integral states and its interval-normalized forms. Further, appropriate zero equalities are used to exploit the time-dependency of the states. The following LKF is used in this work. where and ψ i , i = 0, 1, 2, 3 are defined in section II. P, P 1 , P 2 , Q 1 , Q 2 , R 1 and R 2 are positive definite matrices. Remark 1: In the functional V 0 (t), additional integral states w 1 (t) and w 2 (t) are introduced in the augmented vectors of delay-coefficient based quadratic terms corresponding to the delay intervals [t − τ t , t] and [t −h, t − τ t ], such that more τ t andτ t dependent terms appear in the derived conditions. Note that these terms have not been used in LKFs defined earlier in [23]- [25]. Also, using these states, delaydependent zero-equalities are formulated to exploit the timerelation of the delayed states.
Remark 2: A new form of single integral functionals have been proposed in [20], in which the cross terms

been introduced. By extending this idea, V 1 (t) incorporates crossterms s t−τ tẋ (s)ds,ẋ(s), x(s) in one interval whereas x(s) in the other one to avoid the inclusion of x(t −h) in the delay coefficient based terms and in the Lyapunov-matrix related
term. This also avoids the appearance of additional state in the form of derivative of x(t −h) in the derivative of the LKF. By doing so the number of LMI variables and maximum order of LMI decreases which in turn reduces the computational burden. Also similar concept has been addressed in [26] in the construction of new LKF, by not including the information of delayed states in the augmented vectors. Hence, the delayed state derivative terms can be eliminated from the augmented vectors of derivative of LKF to decrease the dimension of stability criteria in LMI form. In addition, the work in [27] demonstrated the advantage of using V 2 (t) that leads to delay-derivative dependent integral terms.

B. ZERO-EQUALITIES
Zero-equalities are often used to exploit the time relation of the states [28], [29]. In [30], the integral states and its interval-normalized forms are utilized to construct new zeroequalities. The following two zero-equalities with slack variable matrices N i , i = 1, 2, 3, 4, of appropriate dimensions are used in this work.
whereÊ 1 = I 0 ,Ê 2 = 0 I and ζ (t) is a column vector to be appropriately chosen. Note that the above zero-equalities (6) and (7) are quadratic forms of interval-normalized states and involves zero-functions of both w 1 (t) and w 2 (t) in the sense that the bracketed terms are zero by virtue of their time-dependency. These inequalities will be augmented in the derivative of the quadratic LKF so that the time relation among the integrals can be exploited. The slack variables introduce some freedom in the interplay of the constraints imposed on other variables arising from the LKF.

C. STABILITY ANALYSIS
This section presents two stability criteria for system (1) incorporating LKF (5). The first method leads to terms involving τ 2 t and thereby leads to a criterion as a second-order polynomial function of τ t . Whereas the second one involves only a first-order polynomial that does not require further approximation for a polytopic representation. The first result is presented next.
In Theorem 1,V 0 (t) yields τ 2 t terms because the integral states w 1 (t) and w 2 (t) are considered as product of delayinterval and their interval-normalized form. Hence, the stability criterion is in the form of quadratic function of τ t , and this enforces using Lemma 3. However, Lemma 3 has inherent conservatism that make the stability result conservative.
To deal with this issue, an improved criteria is proposed in the next theorem by treating w 1 (t) and w 2 (t) and their interval-normalized forms 1 τ t w 1 (t) and 1 h−τ t w 2 (t) are considered as separate individual states. This leads to no τ 2 t term in the derivative of the LKF and thereby not introducing conservativeness invited by the use of Lemma 3. Note that, this step invokes more decision variables in the resulting criterion, which is a trade-off with the reduction in conservativeness.
TheΛ related quadratic term in the above consists of the following vectors: Proof 2: Similar to Theorem 1, consider the LKF (5). Next, by taking the derivative of V 0 (t) and by treating the integral and its normalized version as individual states to reformulate second order polynomial into first order, one obtainṡ V 0 (t) = ξ T 2 (t) 0 (τ t ,τ t )ξ 2 (t) (32) where 0 (τ t ,τ t ) is defined in (29) and In the zero equalities of (6) and (7), the integral states w 1 (t) and w 2(t) and their normalized version are considered as separate individual states then one obtains Next, by including (33) and (34) in the derivatives of V 1 (t), we haveV where 1 (τ t ,τ t ) is defined in (10). Similarly, the derivative of V 2 (t) can be expressed as: where By bounding the integral function I(t) using Lemma 1 and 2, and invoking (34), one obtainṡ where 2 (τ t ,τ t ) is defined in (31) and
Remark 3: In Theorem 2, the integral and their intervalnormalized forms are treated as separate individual states, to avoid the use of Lemma 3. Therefore, in Theorem 2, additional states w 1 (t) and w 2 (t) are considered as compared to Theorem 1. Two zero equalities are also modified to take these states into account. In doing so, the maximum order of the LMI criterion and number of LMI variables are increased. This results in computational burden and convergence time. In general, the potential to yield better result using Theorem 2 in comparison to Theorem 1 is at the price of more computational burden and convergence time. This yet again provides the trade-off between complexity and conservativeness.

IV. NUMERICAL EXAMPLES
In order to demonstrate the less conservativeness of Theorem 2, following three examples are considered and comparisons are constructed in terms of MPUB ofh and number of LMI variables (NLVs).

A. SYSTEM PARAMETERS
Consider system (1) with the following three sets of system matrices: (1) Example 1: (2) Example 2: (3) Example 3:  Tables 1, 2 and 3 for example 1, 2 and 3, respectively. It can be seen that the MPUBs obtained using Theorem 2 is larger than Theorem 1. This shows that use of Lemma 3 due to the involvement of τ 2 t terms introduces considerably conservativeness. This can be avoided if the states are appropriately augmented.
For a comparison among the proposed results and the existing ones in literature, one can observe that Theorem 1 in example 1 is more conservative than all the approaches except Theorem 1 and 2 of [14] and [27], respectively. However, Theorem 2 yields better results as compared to all other approaches consistently. In case of example 2, Theorem 1 mostly provides conservative result except Proposition of [30] and Theorem 2(N = 2) of [35]. Further, for example 3, Theorem 1 yields better results among all that are listed in table 3 except Theorem 1 of [27].
(b) Complexity computation: The computational complexity depends on the maximum order of LMIs (MOL) involved in the stability criteria and also on the number of scalar LMI variables (NLV) used. Due to the use of extra states and zero equalities in the stability analysis of Theorem 2, the MOL is increased by four as compared to Theorem 1. Also the NLVs used in Theorem 2 are larger than that of Theorem 1. So, the computational burden and convergence time is more in Theorem 2 as compared to Theorem 1. Further, for comparison with the existing methods, the NLVs are listed in all the Tables. One can observe that the NLVs required in proposed Theorem 2 are less as compared to the other approaches except in Theorem 1 (C3) of [14], Theorem 2(C1) of [27]    and Theorem 1 of [32]. Similarly, for example 2, the NLVs used in proposed Theorem 2 are more in comparison to the other methods listed in Table 2 except Theorem 3 of [31] and Theorem 2 of [16]. Further solving in example 3 requires large number of LMI variables in comparison to all the criterion listed in Table 3. Therefore, on the basis of the above comparisons, it can observed that Theorem 2 requires more convergence time and computational burden than the some of the existing approaches.
For verification of proposed results, we consider the following parameters.   Figs. 1-3. In all the cases, the systems are seen to be asymptotically stable that corroborates the obtained results.

V. CONCLUSION
Two new criteria in the form of second and first-order polynomial functions of τ t have been proposed. Both the criteria are obtained using same LKF for the stability analysis of systems with time-varying delay. Using comparative studies, it is observed that the conservativeness in the stability criterion due to use of NDL is considerable and it can be removed by avoiding appearence of the quadratic term of τ t . However, zero-equalities are used with additional states which leads to more number of LMI variables and complexities as well. . He is currently involved with several research projects on renewable energy and grid tied microgrid system with the Oregon Institute of Technology (Oregon Tech), where he has been an Assistant Professor with the Department of Electrical Engineering and Renewable Energy, since 2015. He has been working in the area of distributed power systems and renewable energy integration for last 10 years. He has published a number of research papers and posters in this field. He and his dedicated research team is looking forward to explore methods to make the electric power systems more sustainable, cost-effective, and secure through extensive research and analysis on energy storage, microgrid systems, and renewable energy sources. His research interests include the modeling, analysis, design, and control of power electronic devices, energy storage systems, renewable energy sources, the integration of distributed generation systems, microgrid and smart grid applications, and robotics and advanced control systems. He is also serving as an Associate Editor of IEEE ACCESS.