Novel Equalities for Stability Analysis of Asynchronous Sampled-Data Systems

This paper is concerned with the stability analysis problems of asynchronous sampled-data systems with use of a input-delay approach, where a sampled-data system can be reformulated into a time-delay system having incremental delays. For these systems, this paper introduces a new looped-functional to utilize both integral states and their interval-normalized ones. Utilization of these two types of integral state variables in a construction of stability conditions has been effective in reducing the conservatism. Further, generalized equalities are proposed for the case utilizing these two types of integral states. The sampled-data system formulation consists of a sampled state, a system state and its derivative. Here, the sampled state between two consecutive sampling instants is a constant value and thus can be eliminated by an integration with Legendre polynomials of positive degrees. Consequently, the system formulation turns into the equalities consisting of state variables, sampled-state variables, integral state variables and their interval-normalized ones without additional slack matrices. Based on the proposed equalities, a large computational burden can be reduced while reducing the conservatism. The effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of asynchronous sampled-data systems.


I. INTRODUCTION
In the past decades, networked control systems (NCSs) have attracted considerable attention [1]- [5]. The NCSs consist of several distributed plants connected through digital communication networks and thus have several advantages such as reduced wiring, simple maintenance, and long distance control. However, because of limited network resources, a heavy temporary computational burden in a processor can corrupt the constancy of sampling periods. Additionally, packet losses in a wireless communication network and transmission delays cannot guarantee a stability of the systems. Such phenomena make the NCSs time-varying and asynchronous. It is thus an important issue to develop robust stability criteria for sampled-data systems with time delays and asynchronous samplings [6].
Since sampled-data control only requires the information of a system at the sampling instants, increasing sampling The associate editor coordinating the review of this manuscript and approving it for publication was Feiqi Deng .
interval reduces a computational burden of overall systems. It is thus an important topic to increase a sampling period while maintaining a system stability. Therefore, sampled-data systems with periodic/asynchronous samplings have been extensively researched in the literature [7]- [15]. In the case of periodic samplings, tools for stability analysis and controller synthesis problems have been well established [11]. However, in the case of asynchronous samplings, there still exist open problems. To deal with the stability analysis of asynchronous sampled-data systems, there have been three approaches according to modelings of a sampled-data system: impulsive models [7], [9], discrete-time systems [8], [10], input-delay systems [12], [13], [15], [16]. Among them, this paper focuses on the input-delay system approaches, where the original sampled-data system is reformulated into a time-delay system with an incremental delay. Therefore, the reformulated system can be analyzed by the Lyapunov-Krasovskii(L-K) theorem which has been widely used for the analysis of time-delay systems [17]- [20]. Recently, a framework based on a looped-functional was proposed for the 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/ analysis of sampled-data systems [21], [22]. Differently from the L-K functional satisfying positivity conditions, the looped-functional consists of a quadratic Lyapunov function and the differentiable functional satisfying the looping conditions. This additional functional is zero at the sampling instants, and thus the looped-functional approach can relax the positivity conditions on the functional. Commonly, a construction of the functional has played key roles in reducing the conservatism of stability criteria, which results in development of many functionals with use of various quadratic functions and integral quadratic functions [13], [15], [21]. Therefore, the time derivative of the looped-functional also contains integral quadratic functions which cannot be directly utilized as stability conditions expressed in terms of linear matrix inequalities (LMIs). To derive and relax LMIs guaranteeing the negativity of a derivative of the functional, integral inequalities [17], [20] and zero equalities [12], [13], [15], [18] have been proposed. The zero equalities can be obtained from the relation among system state variables such as a system state x(t), two consecutive sampled states r m x(r m+1 ) dr m+1 . . . dr 1 for real numbers t, t k , t k+1 , a non-negative integer m, and a positive integer k.
According to integrands in integral quadratic functions, upper bounds obtained from the integral inequalities consist of integral state variables or their interval-normalized versions. In the literature [15], [18], it was numerically shown that simultaneously utilizing these two types of integral state variables contributes to reducing the conservatism of stability criteria for time-delay systems. However, due to slack matrices of the zero equalities, which come from the relation between these two types of integral terms, a computational burden increases. Consequently, the common difficulties when utilizing the zero equalities come from computational burden. To make LMIs from the zero equalities, additional slack matrices are needed because of the convexity with respect to a length of an integration interval. Such observations motivate our work.
This paper aims at deriving sufficient conditions in terms of LMIs which guarantee that asynchronous sampled-data systems are stable with a sampling period as large as possible. To archive this goal, the main contributions of this paper can be summarized as follows.
• This paper refines a looped-functional suggested by [13] into a more general one to fully utilize integral state variables and their interval-normalized versions. By imposing certain structure on slack matrices, it can be shown that the looped-functional in [13] is a special case of the proposed functional.
• This paper proposes novel equalities obtained by fully utilizing both Legendre polynomials and a sampled-data system formulation. Between two consecutive sampling instants, a sampled state variable in the system formulation is a constant value, and thus it can be eliminated by an integration with Legendre polynomials of positive degrees. Consequently, the system formulation turns into the equalities consisting of state variables, sampledstate variables, integral state variables and their intervalnormalized versions without additional slack matrices. In the case utilizing the two types of integral state variables, large computational burden can be reduced while reducing the conservatism of stability criteria.
• Less conservative stability criteria for asynchronous sampled-data systems are developed in terms of LMIs by employing the proposed looped-functional and the equalities. Compared to the recent work [15], the proposed stability criteria have less number of variables while reducing the conservatism. The proposed stability criteria can deal with the cases of both periodic and asynchronous samplings. Three numerical examples of the stability analysis for sampled-data systems are given to show the effectiveness of the proposed methods in terms of maximum sampling intervals for given minimum sampling intervals.
Notations: X > 0 (X ≥ 0) means that X is a real symmetric positive definitive matrix (positive semidefinite).
N and R n denote the set of positive integer and the n-dimensional Euclidean space, respectively. Z ≥0 denotes the set of non-negative integer. R n×m is the set of all n×m real matrices. S n and S + n represent the set of symmetric matrices and the set of positive definite matrices of R n×n , respectively. The matrix I n represents the identity matrix in R n×n . The notation 0 n,m stands for the matrix in R n×m whose entries are zero and, when no confusion is possible, the subscript will be omitted. Define K, as the set of differentiable functions from an interval of the form [0, T ] to R n .

II. PRELIMINARIES
This section revisits several lemmas, a definition, and a system formulation to derive main results. Consider the following sampled-data system.
where x(t) ∈ R n is the system state, A, A d ∈ R n×n are system matrices, and t k is the sampling instant such that The sampling interval is defined as In [21], it was shown that integrating the differential equation (1) yields Under the periodic sampling h, the dynamics in (3) and (4) become Thus, under the periodic sampling, the system (1) is asymptotically stable if and only if (h) has all eigenvalues inside the unit circle. In the case of time-varying sampling interval h k , however, this method does not hold. Thus, an inputdelay approach with the following looped-functional has been widely utilized for the stability analysis of asynchronous sampled-data systems. Lemma 1 (Looped-Functional [21]): Let 0 < T 1 < T 2 be two positive scalars and V : R n → R + be a differentiable function for which there exist positive scalars µ 1 < µ 2 and p such that Then, the two following statements are equivalent.
1) The increment of the Lyapunov function is strictly negative for all k ∈ N and T k ∈ [T 1 , 2) There exists a continuous and differentiable functional and such that, for all k ∈ N, Moreover, if one of these two statements is satisfied, then the solutions of the system are asymptotically stable. Definition 1 (Legendre Polynomials [17]): For scalars a, b ∈ R, Legendre polynomials over the integral interval [a, b] can be defined as follows. where This polynomial function satisfies the following properties.
Integral inequalities have played essential roles to develop stability criteria for time-delay systems in terms of LMIs. The following lemma contains major two general integral inequalities.
Lemma 3 (Integral Inequalities [17], [20]): For scalars a, b ∈ R, let x(t) ∈ R n be a continuous function: Then, for integers m, k ∈ N, an arbitrary vector ζ (t) ∈ R kn , a positive definite matrix R ∈ S + n , and a matrix S ∈ R kn×(m+1)n , the following inequalities hold: 1) Orthogonal polynomials based integral inequality [20] − 2 where, for a non-negative integer i ∈ Z ≥0 , The relation between these two integral inequalities was shown in [20]. For the same integrand such asẋ T (r)Rẋ(r) or x T (r)Rx(r), the B-L inequality is a special case, which provides the tightest upper bound, of the orthogonal polynomials based integral inequality adopting ζ (t) = m (a, b) in (16). However, the B-L inequality inevitably provides reciprocally convex upper bounds with respect to the length (b − a) of the integral interval [a, b], which leads to non-convex stability conditions. On the other hand, the orthogonal polynomialsbased integral inequality provides the upper bound in terms of convex conditions with respect to (b−a). Thus, it might be less conservative in the case of time-varying (b − a) at a price of more computational burden for calculating the slack matrix S in (14). To reduce the conservatism and computational burden, this paper simultaneously utilize the two integral inequalities (14) and (15). Also, the two types of integral state variables (21) and (22) occurred in the upper bounds of the integral inequalities are also employed. VOLUME 8, 2020

III. STABILITY ANALYSIS OF ASYNCHRONOUS SAMPLED-DATA SYSTEMS
This section derives stability criteria for asynchronous sampled-data systems by utilizing a new looped-functional and new generalized equalities. First, we introduce a new looped-functional V (t) such that where Remark 1: To reduce the conservatism of stability criteria, this paper introduces a new looped-functional containing two types of integral quadratic functions, where integrands arė x T (r)R iẋ (r), (i = 1, 2) and x T (r)R j x(r), (j = 3, 4), respectively. The proposed functional is more general than that of [13], which can be shown as follows. For matrices Q 1 , Q 2 ∈ R 2n×2n ,X ∈ R n×n , and Z ∈ S 2n , if the matrices in the functional (23) are defined by the Lyapunov functional (23) reduces to the functional of [13]. Due to such generality, the stability criteria based on the proposed functional has the same or less conservatism than those of [13], which will be shown in numerical examples.
Remark 2: By simultaneously utilizing the two types of integral state variables (21) and (22), the B-L inequality in Lemma 3 can be modified as follow.
This modification provides the upper bound of integral quadratic functions containing the integrand x T (r)Rx(r) without additional slack matrices, the reciprocally convexity and the convexity. Second, we propose the following lemma. Lemma 4: Let x(r) ∈ R n be a state of the sampled-data system (1) t k+1 ), the following equality holds: where   (21) and l i j is defined in (10). The proof of Lemma 4 is provided in the appendix.
Remark 3: In the literature [12], [13], [15], [18], the following two types of zero equalities have been employed to fully utilize the relations among the state variables in a construction of the stability criteria. 1) In [15], [18], the relation between integral state variables U 0 (a, b), U 1 (a, b) and their interval-normalized versions I 0 (a, b), I 1 (a, b) yields 2) In [12], [13], [15], single and double integration of the system formulation (1) on the integral interval In the case utilizing the two types of integral state variables U i (a, b) and I i (a, b), (i = 1, 2), these zero equalities can be represented as follows.
The common difficulties when utilizing these zero equalities come from computational burden. Since the zero equalities Between two consecutive sampling instants, the sampled state variable x(t k ) in the system formulation (1) is a constant value, and thus it can be eliminated by an integration with Legendre polynomials of positive degrees. By fully utilizing Legendre polynomials (9) and the system formulation (1), generalized equalities are derived without the convexity. Consequently, an augmented integral state term I m (a, b) can be represented as combination of x(t), x(t k ), x(t k+1 ) and U m+1 (a, b) without any slack matrices and the convexity. Thus, slack matrices are only needed in the zero equaliteis obtained from the relation between the two types of integral state terms (31) and the single integration of the system formulation (32). Before deriving stability criteria for sampled-data systems, several vectors and matrices are defined as follows.
Here, ζ (t) is an arbitrary vector in Lemma 1. According to the integral interval [a, b], an arbitrary vector is chosen by defining arbitrary matrices γ 1 and γ 2 in (39). Based on the proposed looped-functional (23) and the definitions (36)-(39), we have the following theorem.
2) If ζ (t) = ξ (t), The proof of Theorem 1 is provided in the appendix. Defining the matrices γ 1 and γ 2 in (40)-(43) by γ 1 = γ 2 = I provides the most general upper bound among those obtained with choices of ζ (t). In this case, stability conditions might be less conservative than the others. However, since the dimensions of slack matrices S 1 and S 2 increase, a number of variables also increases. According to the choices of the matrices γ 1 and γ 2 , there exist trade-offs between a number of variables and the conservatism of stability criteria, which will be shown in numerical examples.

Remark 4:
It has been noted that increasing the degree m of the integral inequalities (14) and (15) contributes to reducing the conservatism of stability criteria at a price of more computational burden. With increasing the degree of the integral inequalities, the proposed equalities is effective in reducing a number of variables. For a non-negative integer i ∈ N, let us definē Then, according to availability of the proposed equalities, the augmented vector ξ (t) (36) and the dimension of the slack matrix M in Theorem 1 can be defined as follows. 1) First, if Lemma 4 is utilized, the interval-normalized integral state term x(t k ), x(t k+1 ) and U i (a, b), and thus there is no need to include I i−1 (a, b) into ξ (t). In this case, 2) Second, if the equalities (32)-(35) and those obtained from their multiple integration are utilized, In the construction of less conservative stability criteria, the dimension of the augmented vector ξ (t) also affects the sizes of the slack matrices S 1 , S 2 ∈ R p×(i+2)n . Thus, in the first and the second cases, the number of variables from the slack matrices S 1 , S 2 , and M is (6i 2 + 33i + 42)n 2 and (20i 2 + 59i + 42)n 2 , respectively. The difference between two number of variable is (14i 2 + 26i)n 2 , where i is the highest degree of the integral state terms. If the integral inequalities of the degree m in Lemma 3 is employed, the highest degree of the integral state term is m − 1. Consequently, compared to the previous approaches, Lemma 4 reduces the number of variables (14(m − 1) 2 + 26(m − 1))n 2 . Since this gap is an increasing function of m ∈ N, the proposed approach is more effective when the degree m increases.
Utilizing Lemma 3 of the degree m = 3, the following theorem is obtained.
2) If ζ (t) = ξ (t), The proof of Theorem 2 is provided in the appendix.

IV. NUMERICAL EXAMPLES
This section illustrates the effectiveness of the proposed approaches in terms of allowable maximum sampling intervals and numbers of variables. Example 1: Consider the system (1) with The conservatism of stability criteria for sampled-data systems has been checked in terms of an allowable maximum sampling interval h M for a given minimum sampling interval h m . A detailed comparison to other results of the literature are shown in Table 1 and Table 2, where the numbers of variables and the allowable maximum sampling intervals are listed, respectively. In the case of periodic sampling, an analytic bound obtained by the eigenvalue analysis (5) for this example is (0, 3.2715]. In the case of aperiodic sampling, it can be seen that the stability criteria in Theorem 1 and Theorem 2 have the less conservatism than those of [12]- [15]. In [15], the two types of zero equalities (32)-(31) were simultaneously utilized, which leads to less conservative stability criteria than those of [12]- [15] at a price of a more number of variables. Compared to the result of [15], Theorem 1 and Theorem 2 with (85) and (86) have the less number of variables while reducing the conservatism. Theorem 2 utilizes  higher inequalities than others, and thus number of variables increase with reduction of conservatism. However, utilizing the proposed methods, Theorem 2 has still less number of variables than that of [15]. Example 2: Consider the system (1) with Table 3 and Table 1 list the allowable maximum sampling intervals and numbers of variables, respectively. In the literature [14], [15], [24], it has been noted that this system is stable for any constant sampling time. In the case of aperiodic sampling, it can be seen that the stability criteria in Theorem 1 and Theorem 2 have the less conservatism than those of [12]- [15]. Example 3: Consider the system (1) with  [15]. In this example, Theorem 1 has the same conservatism as that of [15] whereas the number of variables is reduced.

V. CONCLUSION
In this paper, we have proposed novel equalities and a new looped-functional to develop the less conservative stability criteria for asynchronous sampled-data systems with reduction of the number of variables. In the construction of stability conditions, two types of integral state variables are fully utilized based on the proposed equalities and the functional.
Compared to the existing results from the literature, notable improvements have been obtained. Since the proposed methods mainly have dealt with the sampled-data systems, they cannot be directly utilized for the general time-delay systems. However, in the future work, the proposed methods can be applied to sampled-data controller synthesis problems of various systems such as fuzzy systems with sampled data and synchronization of chaotic Lur'e systems with time delays.

APPENDIX
This section provides proofs of Lemma 4, Theorem 1, and Theorem 2. Also, a method calculating number of variables listed in Table 1 is provided.
A. PROOF OF LEMMA 4 From the sampled-data system formulation (1), the following equality holds for a, b Based on Definition 1, Lemma 2, and Lemma 3, the following Due to the property (12), for integers i ≥ 1, combining (91)-(93) yields the following equality: where D m is a triangular matrix and its diagonal components are nonzeros. Thus, the matrix D m is nonsingular. It ends the proof.

B. PROOF OF THEOREM 1
The time-derivative of the functional (23) is obtained as followsV

C. PROOF OF THEOREM 2
Let us define ξ (t) and e i instead of (36) and (37), respectively. Interval-normalized integral state variables I 1 (t k , t) and I 1 (t, t k+1 ) are represented as follows.

D. NUMBER OF VARIABLES
Stability conditions in Theorem 1 and Theorem 2 consist of slack matrices which will be chosen by LMI solvers. As listed in Table 1, number of variables in slack matrices have been calculated to compare computational burden. For a symmetric matrix P ∈ S + n and a positive integer n ∈ N, a number of variables in P is n(n+1) 2 . For a matrix M ∈ R n×m and positive integers n, m ∈ N, a number of variables in M is nm. Thus, the numbers of variables in Theorem 1 and Theorem 2 can be summarized in Table 5. In Theorem 1, the positive integer q is defined by q = 3n for (60) and (61) 9n for (62).