Linear functional state bounding for positive singular systems with unbounded delay and disturbances varying within a bounded set

In this paper, the linear functional state bounding problem, which is considered in [25], is extended to the singular system with unbounded delay. Firstly, some conditions are presented to guaranteed positivity, regularity, impulse-free, and the component-wise bound for the state vector of the singular system without disturbance. Then, based on the results obtained and by using state transformations, the smallest component-wise ultimate bound of the state vector of the singular system with bounded disturbances is derived. By using the new technique, some sufﬁcient conditions were proposed given in terms of the linear programming/Hurwit matrix/spectral abscissa for linear functional state bounding problems of the singular system with unbounded delay. Finally, a numerical example is given to illustrate the obtained results.


INTRODUCTION
Singular systems (also called implicit systems, differentialalgebraic equations) have an essential position in system control theory and becoming increasingly important in various technical areas such as biological systems, economic systems, power systems, aircraft control systems etc. (see, [1][2][3] and the references therein). Moreover, disturbances often occur and are not eliminated in practical engineering systems for many reasons, such as measurement errors, external noises, linear approximation. In general, it is challenging to get asymptotic stability for the dynamic systems in which noise occurs. The objective of the state bounding problem is to find an ultimate bound, which is a set such that the state vector converges within it when the time tends to infinity [4] or the time tends to prespecified time [5,6]. Recently the problem of finding the smallest ultimate bound set for perturbed systems has been increasingly concerned with research and has become an essential issue in practical system control theory (see, [5,[7][8][9]).
To solve this problem, two commonly used methods. The first method is based on the Lyapunov way combined with linear matrix inequality, and the second for the positive system is based on the Hurwitz/Schur matrices combined with the solution comparison method [5,6,[10][11][12]. The first method is widely used for classes of linear systems whose matrices are Note that researching the singular system is much more complicated than the standard system because one needs to consider not only stability but also regularity and causality (discretetime systems) or non-impulsiveness (continuous-time systems), and due to the singularity of the derivative matrix and the positive restriction on variables, much of the developed theory for positive descriptor systems is still not up to a quantitative level [13,14,16,17]. Recently, the problem of state bounding for positive regular systems with time-varying delays and bounded disturbances has been considered in [6,10,15]. However, it is very difficult to extend the results of this work to positive singular systems because in order to deal with positive singular systems it requires more new techniques. Although there have been many results on the state bounding for linear systems or nonlinear systems with time-delay, little attention has been paid to this problem for positive singular systems with time-varying bounded/unbounded delay.
Very recently, in [16] studied the state bounding for positive singular systems problem, but the results only apply to discretetime systems with delay bounded. It is worth noting that in all of the papers cited above, the delays are assumed to be bounded, while time delays in many real-life dynamical systems such as neural networks systems are often unbounded. Therefore, many researchers have paid much attention to problem of stability of systems with unbounded delays and have obtained many results [18][19][20][21][22]. Recently, the linear functional state bounding problem for the positive system has been very interested in research and obtained many exciting results [23][24][25]. Besides, the approach used in [23][24][25] seems challenging to apply to the singular system with unbounded delay. Moreover, to the best of the authors' knowledge, there are no results for the linear functional state bounding problem of the singular system with unbounded delay. Note that, if we find component-wise state bounds of the system then manifest, we also obtain linear functional bound of the system. However, the results obtained by such a method do not give an optimal linear functional bound. Note that the problem of finding state bounds in previous studies, the disturbance vector is bounded (or bounded component-wise) by known constant limits [10,15,16,[26][27][28]. In this paper, we consider a more general case where the disturbance vector is assumed to vary within a known bounded set. Therefore, we can study the state bounding problem for broader classes of perturbed dynamical systems. Inspired by the work in [25], in this study, we consider the linear functional state bounding problem of the singular system with unbounded delay. Different from the techniques used in [25], by introducing new techniques and Lemma, we proposed some sufficient conditions given in terms of the linear programming/Hurwit matrix/spectral abscissa for linear functional state bounding problems of the singular system with unbounded delay. The main results of the paper are summarised as follows: i. We first derive some sufficient conditions to guarantee the considered singular system is regular, impulse-free, and positive. Then, we obtain component-wise bound for the singular system without disturbances with unbounded delay. Based on this result, we also derive sufficient conditions to guaranteed asymptotic stability for the singular system with unbounded delay. ii. We present a method for establishing the smallest component-wise ultimate bound of the state vector of the singular system with bounded disturbances. iii. Finally, based on the results obtained, by some new technique, we derive linear functional state bounding for positive singular systems with unbounded delay and disturbances varying within a bounded set.
The remaining of this paper is organised as follows. In Section 2, we provide the problem statement and preliminaries. The main results are given in Section 3. A numerical example is given in Section 4. A conclusion is presented in Section 5.

PROBLEM FORMULATION AND PRELIMINARIES
Notation: ℝ n + (ℝ n 0,+ ) denotes the set of all positive (nonnegative) vectors in ℝ n ; ℝ p×q denotes the space of all real p × q matrices. I m is the m-dimensional identity matrix. For x = (x 1 , x 2 , … , x k ) ∈ R k , x ⪰ 0(≻ 0) means that x i ≥ 0(> 0) for all i = 1, … , k. A matrix B ∈ ℝ n×n is Metzler if all its off diagonal elements are non-negative. C ([−h, 0], ℝ n ) denotes the set of all ℝ n − valued continuous functions on [−h, 0]; ‖ ‖ ∞ denotes the infinity norm of (⋅) ⌈a⌉ is the smallest integer greater than or equal to real number a.
[a] denotes the largest integer less than or equal to real number a. For Q ∈ ℝ p×p , we denote s(Q) = max{Re( ) : Consider the singular system with unbounded time-variable delay have the form below where (t ) = col (x(t ), y(t )) ∈ ℝ r × ℝ n−r is the state vector. The matrix E ∈ ℝ n×n is singular and rank(E ) = r < n. A 0 , A d ∈ ℝ n×n , F ∈ ℝ n×m ; (⋅) ∈ S ⊂ ℝ m 0,+ is the unknown disturbance, but suppose that it varies within a closed and bounded set S . Further, we assume that := max ∈S F can be determined. In this paper, we assume that delay r (⋅) ∈ ℝ + satisfies the following assumption: Assumption 1: There exist T > 0, 0 < < 1 such that From condition (2), we derive that t − r (t ) ≥ (1 − )t > 0, ∀t ≥ T . Hence, the initial condition of system (1) is given by r (t ). The initial func- Let us denote by (t, , ) the state trajectory with the initial function = ( 1 , 2 ) of system (1). Given an E matrix with rank(E ) = r < n, , we always have two X, Y matrices so that there is the representation XEY = , , where A, M ∈ ℝ r×r , B, N ∈ ℝ r×(n−r ) , C, P ∈ ℝ (n−r )×r , D, Q ∈ ℝ (n−r )×(n−r ) , F 1 ∈ ℝ r×m , F 2 ∈ ℝ (n−r )×m .
Consider the linear function vector of the state system of the following form: where R ∈ ℝ n×d 0,+ is a given matrix. In this paper, we find the smallest component-wise upper bound of the L(t ).
, then the singular system (1) is impulse free.
Lemma 1 [29]. Let D ∈ ℝ n×n be a Metzler matrix. Then the following conditions are equivalent.
iii. D is Hurwitz matrix. iv. For any ∈ ℝ n 0,+ , then T D has at least one strictly negative entry.
We see that, if A d ⪰ 0, A 0 is a Metzler matrix, then A 0 + A d is also a Metzler matrix. Based on Lemma 1, we get the following lemma: Lemma 2. Let A 0 be a matrix Metzler matrix and A d ⪰ 0. Then, the following conditions are equivalent: ii. s(D + Q) < 0 and iii. s(A + M ) < 0 and Since, (C + P ) 1 ⪰ 0, it follows that (D + Q) 2 ≺ 0. Using Lemma 1 we have s(D + Q) < 0, and −(D + Q) −1 ⪰ 0. Combine this with the inequality (5) we get We use the condition (i) and inequality (6) we obtain It is easy to check that: This implies that for sufficiently small . Setting We obtain Pre -multiplying both sides of equation with the matrix (D + Q) we get Therefore, (i) holds.
(i ) ⇒ (iii ). We have obtained from (i) the following: (B + N ) 2 ⪰ 0, and Using inequality (7) with the condition (i), we derive the following: This implies that This implies for sufficiently small . Setting Then, we obtain Pre -multiplying both sides of equation Therefore, (i) holds.
(i ) ⇔ (iv). Using Lemma 1, we have (i) and (iv) are equivalent. □ iii. There exists ∈ (0, 1) such that Proof. (i) It follows from A 0 is Metzler matrix and A d ⪰ 0 we obtain that A, D are Metzler matrices and B, C, M, N, P, Q are nonnegative matrices. It is easy to see that A 0 + A d is also Metzler matrix. Combining this with Lemma 1 and which is equivalent to It follows from (C + P ) ⪰ 0, and (14) that By Q ⪰ 0 and (15) we get D ≺ 0 and from D Metzler matrix, using Lemma 1, it can be deduced that det(D) ≠ 0 and −D −1 ⪰ 0. Note that, the matrix A is Metzler, Then, the proof of (i) is similar to [22]. Therefore, we omit it here.
(ii) Based on the linearity and positivity of the system (8) (9). The condition (10) can be proved by similar arguments.

MAIN RESULTS
From now on, we always assume that A 0 is a Metzler matrix,

State bounding for positive singular systems with unbounded time-variable delay
Firstly, we consider the systems as follows We get the following result for the system (18).

Theorem 1.
Assume that A 0 , A d satisfy one of the conditions in Lemma 2. Then system (18) is regular, impulse-free, positive, and ∃ ∈ (0, 1), ∈ ℝ n + and a sequence 0 = T 0 < T 1 < T 2 < ⋯ < T n < ⋯ < +∞ such that Proof. By A 0 + A d is Metzler and Hurwitz matrix, it follows from Lemma 1, ∃ ∈ ℝ n + such that: Let us denotes: = ( 0 , 0 ), where 0 ∈ ℝ r + , 0 ∈ ℝ n−r + . From (20), we obtain which is equivalent to Using the second inequality of (21), combined with the condition (C + P ) 0 ⪰ 0 we derive (D + Q) 0 ≺ 0. Use this together with Lemma 1 with the note that D + Q is a Metzler matrix we have D + Q is a Hurwitz matrix. This together with D is a Metzler matrix, Q ⪰ 0 we get D is also Hurwitz matrix and −D −1 ⪰ 0 by Lemma 1. This implies that the system (18) is regular and impulse-free. Besides, the system (18) can rewrite in vector form as follows ) .
(22) Applying Lemma 3 to the system (22), we derive the system (22) is a positive system, which implies the system (18) is also a positive system.
The following corollary of Theorem 1 gives a sufficient condition for the stability of the system (18) with any time-variable delay function satisfying (2). ( 1 , … , n ) ∈ ℝ n + . For any initial function (⋅) satisfying ‖ ‖ ∞ < , we obtain (s) < ‖ ‖ ∞ , s ∈ [−r 0 , 0). Since system (18) is linear and using (29), (30), we get Remark 2. We know that, for a standard positive system of the is bounded or unbounded, the necessary and sufficient condition to ensure the asymptotic stability of the system is A 0 + A d is the Hurwitz matrix (see in [30,31]). For the singular system (18) with unbounded delay considered in this paper, as Corollary 1 shows that, when the E matrix is of the form E = ( , the condition to guarantee that the system is asymptotic stability is also the A 0 + A d is the Hurwitz matrix. Note that this condition is independent of the matrix E. Now, we investigate the state bounding problem for the singular system (1). To do that, we first consider the following system: { The following lemma provides a relationship between the state trajectory of the system (1) and the state trajectory of the system (52).
(ii ) By the same method as in the proof of part (i ) and Lemma 3, we get that (ii ). □ The following theorem provides a condition sufficient to ensure that system (1) is regular, impulse-free, and the existence of an ultimate component-wise bound for the system.
(ii) For t → ∞, it follows from (54), we get (55). Then, is a component-wise ultimate bound of system (1). We now show that lim t →∞ (t, 0 , ) = , where 0 (s) = 0, s ∈ [−r 0 , 0]. Using coordinate transformation v(t ) = − (t ). (67) This together with (52), imply that and where 0 (s) = , s ∈ [−r 0 , 0]. It follows from Lemma 3 that v(t, 0 ) ⪰ 0, t ≥ 0. Applying Theorem 1 to system (68), implies ∃ ∈ (0, 1) and a sequence 0 = T 0 < T 1 < T 2 < ⋯ < T n < ⋯ < +∞ such that Combining (69) and (70) , ∀t ≥ 0. This implies that lim t →∞ (t, 0 , ) = . Then we have is the smallest component-wise ultimate bound of system (1). □ Remark 3. In [10], the author considered the problem of state bounding for positive coupled differential-difference equations with bounded disturbances of the forṁ It is not hard to see that this result is equivalent to Theorem 2 of this paper. However, to obtain this result, the authors of the work [10] have to assume that the system satisfies regular condition and impulse-free condition. In contrast, Theorem 2 of this paper can be applied to a general time-delay system which does not need to be satisfied regular condition and impulse-free condition.

Linear functional state bounding
Let us denote: where A 0 j the jth column vector of matrix A 0 , j = 1, … , n.

Lemma 5. Let A 0 ∈ ℝ n×n be a Metzler and Hurwitz matrix. For
Proof.
i) We denote A 0 = (a i j ) n×n , 1 ≤ i, j ≤ n, there always exist * = max 1≤i≤n |a ii | such that a ii + * .1 ≥ 0, for all 1 ≤ i ≤ n. Moreover, since A 0 is a Metzler matrix, a i j ≥ 0 for all i ≠ j . So there always exists * ∈ ℝ satisfying ii) We set A T 0 q = ( a 1 a 2 … a n ) T , I = {1, 2, … , n}, I − = {i : a i < 0, i ∈ I}, I + = {i : a i ≥ 0, i ∈ I}. Since matrix A 0 is Metzler and Hurwitz. According to part (iv) of Lemma 1, we have the set I − is nonempty. It is easy to see that if ∈ ℝ satisfies A T 0 q + q ⪰ 0 then > 0. Indeed, the opposite assumes that ≤ 0, since the set I − is nonempty, there exists an index j ∈ I − such that a j < 0, so a j + q j < 0. This contradicts the condition A T 0 q + q ⪰ 0. Note that, for all i ∈ I + , then a i + q i ≥ 0 for all ∈ ℝ + . Therefore, we get min Ξ = min{ ∈ ℝ : a j + q j ≥ 0, j ∈ I} = min{ ∈ ℝ + : a j + q j ≥ 0, j ∈ I + ∪ I − } = min{ ∈ ℝ + : a j + q j ≥ 0, j ∈ I − }. (72) as the jth row vector of matrix A T 0 . Then we get Because A T 0 is a Metzler matrix, we have c j j < 0, and c jk ≥ 0, k ≠ j . From this, we infer that for all j ∈ I − , a j < 0 deduces that q j > 0. Combining this with (72) we get the following result: Note that max iii) Based on Remark 7 in [25], we obtain iii). □

Theorem 3. Assume that A 0 , A d satisfy one of the conditions in Lemma 2. Then
is the smallest linear functional bound of (4).
Proof. Using (4) and Lemma 5 implies thaṫ Combining this with (1) we obtain where . We have We can immediately deduce that A 0 + A d is Hurwitz and Metzler matrix. Applying Theorem 2 to the system (76), we get the following estimation of x(t ): Setting So we get that: By some simple calculation, we get ] .
(80) Combining (78), (80) and W = R T A 0 − UR T , and  = It is note that max ( f + g) ≤ max( f ) + max(g), then we have R ⪯ R T . Combining this with (81) implies that L is smallest if and only if □ Remark 4. Recently, the linear functional state bounding problem for positive systeṁ Trajectories of (t ) with (t ) = 0, t ≥ 0 was studied in [25]. It is obvious that the system (1) is more general than the system (83), when the matrix E = I , applying Theorem 3, we obtain the smallest linear functional bound of (83) is Then, Theorem 9 in [25] is a particular case of Theorem 3. We know that studying the singular system is much more complicated than the standard system because one needs to consider not only stability but also regularity and causality (discrete-time systems) or non-impulsiveness (continuous-time systems). In this paper, we provide sufficient conditions to ensure that the singular system is regular, causal, positive and finding a linear functional bound for a class of positive singular systems with unbounded delay.  So by using the formula (74) in Theorem 3, we show that the smallest linear functional bound of the system:

CONCLUSIONS
We have considered the problem of linear functional state bounding for positive singular systems with unbounded delay and disturbances varying within a bounded set. (i) we have given a condition for the existence of component-wise bounds and a condition for the asymptotic stability of the singular systems without disturbances, (ii) we have provided a sufficient condition for the existence of component-wise ultimate bounds of the singular systems with bounded disturbances, (iii) we proposed some sufficient conditions given in terms of the linear programming/Hurwit matrix/spectral abscissa for linear functional state bounding problems of the singular system with unbounded delay. A numerical example is given to illustrate the effectiveness of the proposed results. Besides, extending the results of this paper to solve the controller design problem for the singular positive systems would be an interesting problem for future research. Furthermore, future works can apply the techniques used in this paper to study positive observer synthesis for positive singular systems with unbounded delay and interval parameter uncertainties.