#### Abstract

The problem of stability analysis for a class of neutral systems with mixed time-varying neutral, discrete and distributed delays and nonlinear parameter perturbations is addressed. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent. The conditions are presented in terms of linear matrix inequalities (LMIs) and can be efficiently solved using convex programming techniques. Two numerical examples are given to illustrate the efficiency of the proposed method.

#### 1. Introduction

Delay (or memory) systems represent a class of infinite-dimensional systems largely used to describe propagation and transport phenomena or population dynamics [1–3]. Delay differential systems are assuming an increasingly important role in many disciplines like economic, mathematics, science, and engineering. For instance, in economic systems, delays appear in a natural way since decisions and effects are separated by some time interval. The presence of a delay in a system may be the result of some essential simplification of the corresponding process model. The problem of delay effects on the stability of systems including delays in the state, and/or input is a problem of recurring interest since the delay presence may induce complex behaviors (oscillation, instability, bad performances) for the schemes [1, 2]. Some improved methods pertaining to the problems of determining robust stability criteria and robust control design of uncertain time-delay systems have been reported; see, for example, [4, 5] and the references cited therein. When dealing with time-varying delays and the reduction of the level of design conservatism, one has to select appropriate Lyapunov-Krasovskii functional (LKF) with moderate number of terms [6].

Neutral delay systems constitute a more general class than those of the retarded type. Stability of these systems proves to be a more complex issue because the system involves the derivative of the delayed state. Especially in the past few decades, increased attention has been devoted to the problem of robust delay-independent stability or delay-dependent stability and stabilization via different approaches (e.g., model transformation techniques [2, 7–9], the improved bounding techniques [10, 11], and the properly chosen Lyapunov-Krasovskii functionals [12, 13]) for a number of different neutral systems with delayed state and/or input, parameter uncertainties, and nonlinear perturbations (see, e.g., [14–25] and the references therein).

Among the existing results on neutral delay systems, the linear matrix inequality (LMI) approach is an efficient method to solve many problems such as stability analysis, stabilization [9, 15, 26, 27], control problems [28–30], filter designs [31, 32], and guaranteed-cost (observer-based) control [33–39]. Besides, for neutral systems with mixed neutral and discrete delays, most of the aforementioned methods can only provide neutral-delay-independent and discrete-delay-dependent results. Furthermore, the subject of the robust stability and feedback stabilization of continuous- and discrete-time systems (within the framework LMI) under additive perturbations which are nonlinear functions in time and state of the systems are investigated in [40, 41], respectively.

In the recent literature on neutral systems, He et al. in [42] proposed a new approach to analyze the stability of the systems with mixed delays by incorporating some free-weighting matrices, and the less conservative criteria, which were both discrete-delay-dependent and neutral-delay-dependent, were obtained without considering the model transformations. However, some of the free matrices did not serve to reduce the conservatism of the results that were obtained. Moreover, in [9, 20], the authors studied the problem of the robust stability of neutral systems with nonlinear parameter perturbations and mixed time-varying neutral and discrete delays and presented neutral-delay-independent stability criteria, that cannot be directly applied to the systems with different time-varying neutral, discrete, and distributed delays. Furthermore, from the published results, it appears that general results pertaining to neutral systems with mixed time-varying neutral, discrete, and distributed delays and nonlinear parameter perturbations are few and restricted; see [9, 10, 18, 20, 42] where most of the efforts were virtually neutral-delay-range-independent or were not centered on distributed delays.

In this paper, we develop new stability criteria for the stability analysis of the neutral systems with nonlinear parameter perturbations based on a descriptor model transformation. The dynamical system under consideration consists of time-varying neutral, discrete, and distributed delays without any restriction on upper bounds of derivatives of time-varying delays. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent. The conditions are presented in terms of LMIs and can be easily solved by existing convex optimization techniques. Two numerical examples are given to demonstrate the less conservatism of the proposed results over some existence results in the literature.

*Notations*. The superscript stands for matrix transposition; denotes the *n*-dimensional Euclidean space; is the set of all real *m* by *n* matrices. refers to the Euclidean vector norm or the
induced matrix 2-norm. and represent, respectively, a column vector and a
block diagonal matrix, and the operator represents . and denote, respectively, the smallest and largest
eigenvalue of the square matrix . The notation means that is real symmetric and positive definite; the
symbol denotes the elements below the main diagonal
of a symmetric block matrix.

#### 2. Problem Description

Consider a class of linear neutral systems with different time-varying neutral, discrete, and distributed delays and nonlinear parameter perturbations represented by where , and is the state vector. The time-varying vector valued initial function is a continuously differentiable functional, and the time-varying delays , , and are functions satisfying, respectively, The time-varying vector-valued functions () are continuous and satisfy , and the Lipschitz conditions, that is, for all and for all such that are some known matrices.

*Remark 2.1. *In this case, is called an interval-like or
range-like time-varying delay [14]. It is also noted that this kind of
time-delay describes the real situation in many practical engineering systems. For example, in the field of networked control systems, the network
transmission induced delays (either from the sensor to the controller or from
the controller to the plant) can be assumed to satisfy (2.2a) without loss of generality [43, 44].

Throughout the paper, the following assumptions are needed to enable the application of Lyapunov’s method for the stability of neutral systems [1]:

(A1)let the difference operator given by be delay-independently stable with respect to all delays. A sufficient condition for (A1) is that(A2)all the eigenvalues of the matrix are inside the unit circle.

Before ending this section, we recall the following lemmas, which will be used in the proof of our main results.

Lemma 2.2 (see [9]). * For any arbitrary column
vectors , any matrix , and positive-definite matrix , the following inequality holds: *

Lemma 2.3 (see [45]). * Given matrices , , and of appropriate
dimensions with , then the following matrix inequality holds: ** for all if and only
if there exists a scalar such that *

#### 3. Main Results

In this section, new delay-range-dependent sufficient conditions for the asymptotic stability of the neutral system (2.1) are presented. By utilizing the Leibniz-Newton formula, the following two zero equations hold: then, we can represent the system (2.1) as with and where the matrices will be chosen in the following theorem.

Theorem 3.1. * Under (A1), for given scalars , the neutral system (2.1) is asymptotically stable, if there exist some scalars , matrices , and positive-definite matrices , such that the following LMI is feasible: ** with ** where (3.4). *

*Proof*
Firstly, we represent () in an
equivalent descriptor model form as Define the Lyapunov-Krasovskii functional where with , , and , where .

On the other
hand, noting that . According to [34], using the Cauchy-Schwarz
inequality and after
some manipulations, we obtain where with Differentiating along the system trajectory becomes where

Using Lemma 2.2 for and , we obtain Differentiating the second Lyapunov term in (3.6) gives and the time
derivative of the third term of in (3.6) is and, similarly, and also the
time derivative of the fifth and sixth terms of in (3.6) are, respectively, For nonlinear functions , we have Moreover, from the Leibniz-Newton formula and the system (2.1), the following equations
hold for any matrices with appropriate dimensions: where Using the
obtained derivative terms (3.10)–(3.17) and adding the right- and the left-hand
sides of (3.18) and (3.19) into , the following result is obtained: where , and the matrix is given by with Now, if holds, then which means that the neutral system (2.1) is asymptotically stable. By applying the Schur
complement, the matrix inequity results in with where .

Following [34, 35], we choose , , where is a tuning scalar parameter (which may be
restrictive). Let Premultiplying and postmultiplying to the matrix inequality (3.24) and considering , , and () to eliminate the nonlinearities in the matrix inequality, we obtain the LMI
(3.3). Moreover, from (2.1) and the fact that is square integrable on , it follows that . Under (A1), the later implies that . Therefore, by [1, Theorem 1.6], we conclude that the neutral
system (2.1) is asymptotically stable.

*Remark 3.2. *The
results given in Theorem 3.1 are derived for system (2.1) with time-varying delays , , and satisfying (2.2a), (2.2b), and (2.2c), where the derivatives
of the time-varying delays are available. However, in many situations, the
information on the derivative of time-varying delays is unknown *a prior*. In such circumstances, the corresponding delay-rate-independent stability
analysis results for time-delays only satisfying can be easily obtained by
setting in Theorem 3.1.

*Remark 3.3. * The reduced conservatism of Theorem 3.1
benefits from the construction of the new Lyapunov-Krasovskii
functional in (3.6), utilizing Leibniz-Newton formula, using a free-weighting matrix technique, and no
bounding technique is needed to estimate the inner product of the involved
crossing terms (see, e.g., [12, 20]). It can be easily seen that
results of this paper are quite different from most existing results in the recent
literature in the following
perspectives.
(a) Theoretically stability analysis of neutral
systems with different time-varying neutral, discrete, and distributed delays is
much more complicated, especially, for the case where the delays are
time-varying and different.
(b) In this paper, the derived sufficient
conditions are convex, neutral-delay-dependent, discrete-delay-range-dependent, and
distributed-delay-dependent, which
make the treatment in the present paper more general with less conservative in
compare to most existing results in the literature which are independent of the
neutral or distributed delays; see for instance [21, 22, 38].

#### 4. Uncertainty Characterization

In this section, we will discuss the uncertainty characterization for the linear neutral system (2.1) with different time-varying neutral, discrete, and distributed delays and nonlinear parameter perturbations.

##### 4.1. Polytopic Uncertainty

The first class of uncertainty frequently encountered in practice is the polytopic uncertainty [2]. In this case, the matrices of the system (2.1) are not exactly known, except that they are within a compact set denoting We assume that for some scalars satisfying where the vertices of the polytope are described by In order to take into account the polytopic uncertainty in the system (2.1), we derive the following result from applying the same transformation that was used in deriving Theorem 3.1.

Theorem 4.1. * Under
(A1), for given scalars , if the uncertainty set is polytopic with vertices , , then the system described by (2.1)), (2.2a), (2.2b), (2.2c), and (4.2)–(4.4) is asymptotically stable if
there exist some scalars , matrices , and positive-definite matrices such that LMI (3.3) is satisfied for all *

*Proof. * It follows
directly from the proof of Theorem 3.1 and using properties of (4.2)–(4.4).

##### 4.2. Norm-Bounded Uncertainty

There are also other uncertainties that cannot be reasonably modeled by a polytopic uncertainty set with a number of vertices. In such a case, it is assumed that the deviation of the system parameters of an uncertain system from their nominal values is norm bounded [2]. In our case, consider the time-varying structured uncertain neutral system where the time-varying structured uncertainties are said to be admissible if the following form holds: where are constant matrices with appropriate dimensions, is an unknown, real, and possibly time-varying matrix with Lebesgue measurable elements, and its Euclidean norm satisfies In this section, we modify (A1)-(A2) in order to enable the application of Lyapunov’s method for the stability of the time-varying structured uncertain neutral system (4.6) as follows:

()let the difference operator given by be delay-independently stable with respect to all delays. A sufficient condition for () is that ()all the eigenvalues of the matrix are inside the unit circle, that is, .

Theorem 4.2. *
Under
, for given scalars , the neutral system (4.6) with admissible uncertainties (4.7)) and (4.8) is
robustly asymptotically stable if there
exist some scalars , , matrices , and positive-definite matrices , such that the following LMI is feasible: ** where , , and with *

*Proof. * If the matrices in (3.3) are replaced with , , , , , , and , respectively, then (3.3) with the admissible uncertainties (4.7) is equivalent to
the following condition: By Lemma 2.3, a necessary and sufficient condition for (4.11) is that there
exist some such that Applying
Schur complements, we find that (4.12) is equivalent to (4.9).

*Remark 4.3. * It
is noted that
our approach is
different from that in the reference [20] in several perspectives.
(a) The
system structure in [20] considers a system with different time-varying neutral
and discrete delays and in compare to our case do not center on time-varying distributed delays, that is, the
results in [20] cannot
be directly applied to the systems with different time-varying neutral, discrete, and distributed delays.
(b) Their system only considers the case that
the range of the time-varying delay is from 0 to an upper bound in compare to our
case that the time-varying discrete delay () lies in a range, in which the lower bound is not 0.
(c) The derived neutral-delay-range-independent
conditions and using the inequality bounding technique [11, Lemma 1] employed
for some cross terms encountered in their analysis conditions may produce
conservative results in comparison with the present paper.

#### 5. Numerical Results

In this section, two examples are provided to illustrate the effectiveness of the results obtained in the previous sections.

*Example 5.1. *
Consider the neutral system (2.1) with
the following matrices adopted from [9]: and the nonlinear parameter perturbations and , where .

*Case 1. *Assume that , , and . Applying the criteria in [10, 18, 21, 22] and Theorem 3.1 in this paper, the maximum value of for stability of system under consideration is
listed in Table 1. Furthermore, in the case of and by the criteria in [20], the nominal system is
asymptotically stable for any satisfying . It is easy to see that the stability criterion in this paper gives a much less
conservative result than that in [10, 20–22], and also the maximum
value of decreases as increases. Moreover, unlike the results obtained in [10, 18, 21, 22] our
approach can also
consider the case and handle fast time-varying delays
completely.

*Case 2. *Assume that , , , and . We
also calculate the maximum delay bounds that guarantee the asymptotic stability
of the system under consideration by Theorem 3.1 in this paper and developed
methods in [9, 20] for different values of the parameter , as listed in Table 2. It can be seen from the table that decreases as increases, and the
stability condition in Theorem 3.1 of this paper is less conservative than that
in [9, 20].

*Example 5.2 (see [46]). *Consider the system (4.6) with the following matrices: Recently, Kwon and Park in [46]
derived the stability bound of with as with the parameters above for the system (4.6). However, by applying Theorem 4.2 to the system under consideration, one can see
that our criterion is feasible for . And for the condition , for fixed or , the upper bounds of delays and are shown in Tables 3 and 4, respectively. From Tables 3 and 4, it can be seen that Theorem 4.2 provides a condition
for guaranteeing stability with respect to given delay bounds and .

#### 6. Conclusion

The problem of stability analysis has been presented in this paper for a class of neutral systems with different time-varying neutral, discrete, and distributed delays and nonlinear parameter perturbations using an appropriate Lyapunov-Krasovskii functional. By combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new feasibility conditions, which are neutral-delay-dependent, discrete-delay-range-dependent, and distributed-delay-dependent, have been developed to ensure that the considered system is asymptotically stable. The conditions were presented in terms of linear matrix inequalities (LMIs) and solved by existing convex optimization techniques. Two numerical examples were given to demonstrate the less conservatism of the proposed results over some existence results in the literature.

#### Acknowledgments

This work has been partially funded by the European Union (European Regional Development Fund) and the Ministry of Science and Innovation (Spain) through the coordinated research project DPI2008-06699-C02-01. H. R. Karimi is grateful to the grant of Juan de la Cierva program of the Ministry of Science and Innovation (Spain), and M. Zapateiro is grateful to the FI grant of the Department for Innovation, University and Enterprise of the Government of Catalonia (Spain). The authors would also like to thank the associate editor and anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper.