Stability Analysis for Neutral Delay Markovian Jump Systems with Nonlinear Perturbations and Partially Unknown Transition Rates

The problem of exponential stability for the uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations is investigated in this paper. This study starts from the corresponding nominal systems with known and partially unknown transition rates, respectively. By constructing a novel augmented Lyapunov functional which contains tripleintegral terms and fully utilizes the bound of the delay, the delay-range-dependent and rate-dependent exponential stability criteria are developed by the Lyapunov theory, reciprocally convex lemma, and free weighting matrices. Then, the results about nominal systems are extended to the uncertain case. Finally, numerical examples are given to demonstrate the effectiveness of the proposed methods.


Introduction
Neutral time-delay systems have been the focus of the research community, which are often encountered in such practical situations as distributed networks, population ecology, processes including steam or heat exchanges [1], and robots in contact with rigid environments [2], and so forth.The existence of time delay may cause the instability of the systems, thus making the stability analysis of timedelay systems an interesting topic.Existing results can be roughly classified into two categories, delay-independent criteria and delay-dependent criteria, where the latter is generally regarded as less conservative.In addition, it should be pointed out that the stability of neutral time-delay systems is more difficult to tackle since the derivative of the delayed state is involved.The situation is similar as singular systems [3,4], whose stability problem is more complicated than that for regular systems because more factors need to be considered.In the past decades, considerable attention has been devoted to the robust delay-independent stability and delay-dependent stability of linear neutral systems, which are mainly obtained based on the Lyapunov-Krasovskii (L-K) method [5][6][7][8].Furthermore, when nonlinear perturbations or parameter uncertainties appear in neutral systems, some results on stability analysis have been also presented [9][10][11][12][13][14]. Various techniques have been proposed in these papers, for example, model transformation techniques, the improved bounding techniques, and matrix decomposition approaches.In particular, He et al. [14] propose a new method for dealing with time-delay systems, which employs free weighting matrices to express the relationships between the terms in the Newton-Leibniz formula and has brought novel results.However, many complex systems with uncertainties and neutral types as well as time-varying or state-dependent delays are still inviting further investigation.
It is noted that many practical dynamics such as solar thermal central receivers, robotic manipulator systems, aircraft control systems, and economic systems, experience abrupt changes in their structures, caused by phenomena such as component failures or repairs, changes in subsystem interconnections, and sudden environmental changes.This class of systems can be described as Markovian jump systems (MJSs) where the abrupt variation in the structures and parameters can be naturally represented by the jumps in MJSs.Since its first introduction by Krasovskii and Lidskii in 1961, MJSs have received much attention, Advances in Mathematical Physics and considerable progress has been made; see, for example, [15][16][17][18][19][20][21][22] and references therein for more details.In view of these results, although related research has made good achievement, we have to admit that the transition probabilities in the jumping process determine the system behavior to a large extent.However, the likelihood of obtaining such available knowledge is actually questionable, and the cost is probably expensive.Thus, it is significant and necessary, from control perspectives, to further study more general jump systems with partly unknown transition probabilities.Recently, many results on the Markovian jump systems with partly unknown transition probabilities are obtained [23][24][25][26][27].Most of these improved results just require some free matrices or the knowledge of the known elements in transition rate matrix, such as the bounds or structures of uncertainties, and some else of the unknown elements do not need to be considered.It is a great progress on the analysis of Markovian jump systems.However, few of these results are concerned with neutral delay systems.It is urgent and significant to consider the problem of delay-dependent exponential stability for Markovian jumping neutral systems with partially unknown transition rates.Besides, to the best of the authors' knowledge, the neutral Markovian jump systems have not been fully investigated, and it is very challenging, especially when nonlinear perturbations exist.These facts thus motivate our study.
In this paper, the exponential stability problem of neutral Markovian jump systems with mixed interval time-varying delay, nonlinear perturbations, and partially unknown transition rates is investigated.A new augmented Lyapunov functional containing triple-integral terms is constructed by dividing the delay interval into two subintervals, and then the delay-range-dependent and rate-dependent exponential stability criteria are obtained by reciprocally convex lemma and free weighting matrices.Moreover, in contrast with the recent research on uncertain transition rates, our proposed concept of the partly unknown transition rates does not require any knowledge of the unknown elements, such as the bounds or structures of uncertainties.On the basis of the obtained results about nominal systems, we further extend the criteria to the uncertain case.All the obtained results are presented in terms of LMIs that can be solved numerically.
The main contributions of this paper can be summarized as follows: (1) the bound of the delay is fully utilized in this paper; that is, improved bounding technique is used to reduce the conservativeness.(2) The constructed Lyapunov functional contains some triple-integral terms which is very effective in the reduction of conservativeness and has not appeared in the context of neutral Markovian jump systems with nonlinear perturbations before.(3) The reciprocally convex lemma is used to derive the delay-range-dependent stability conditions, which can reduce the conservativeness of the investigated systems.(4) The proposed results are applicable to the uncertain transition rates and expressed in a new representation, which are proved to be less conservative than some existing ones.
The remainder of the paper is organized as follows.Section 2 presents the problem and preliminaries.Section 3 gives the main results, which are then verified by numerical examples in Section 4. Section 5 concludes the paper.
Notation.The following notations are used throughout the paper.R  denotes the  dimensional Euclidean space, and R × is the set of all  ×  matrices. <  ( > ), where  and  are both symmetric matrices, means that  −  is negative (positive) definite. is the identity matrix with proper dimensions.For a symmetric block matrix, we use * to denote the terms introduced by symmetry.E is defined to be the expectation operator with respect to the probability measure.‖V‖ is the Euclidean norm of vector V, ‖V‖ = (V  V) 1/2 , while ‖‖ is spectral norm of matrix , ‖‖ = [ max (  )] 1/2 . max(min) () is the eigenvalue of matrix  with maximum (minimum) real part.Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

Problem Statement and Preliminaries
Given a probability space {Ω, F, P} where Ω is the sample space, F is the algebra of events, and P is the probability measure defined on F. {  ,  ≥ 0} is a homogeneous, finitestate Markovian process with right continuous trajectories taking values in a finite set  = {1, 2, 3, . . ., }, with the mode transition probability matrix being where Δ > 0, lim Δ → 0 ((Δ)/Δ) = 0, and   ≥ 0 (,  ∈ ,  ̸ = ) is the transition rate from mode  to , and for any state or mode  ∈ , it satisfies The following uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations over the space {Ω, F, P} are considered: where () ∈ R  is the system state and () is time-varying neutral delay which satisfies 0 ≤ () ≤ , τ () ≤   .The time-varying retarded delay () is such that where  1 < where  ≥ 0,  ≥ 0, and  ≥ 0 are given constants, for simplicity,  1 ≜ where   ,   , and   are known mode-dependent constant matrices with appropriate dimensions and   () is an unknown and time-varying matrix satisfying Particularly, the following nominal systems can be obtained for   () = 0: Before proceeding with the main results, we present the following definitions, assumptions, and lemmas.Assumption 1. System matrices   , (∀ ∈ ) are Hurwitz, and all the eigenvalues have negative real parts for each mode.  , (∀ ∈ ) is full rank in row.
Assumption 2. The Markov process is irreducible, and the system mode   is available at time .Definition 3 (see [28]).Define operator is uniformly asymptotically stable.In this paper, that is, ‖  ‖ +  < 1.

Main Results
This section will first state the exponential stability analysis for (9) with known and partly unknown transition rates, respectively.Then, the uncertain systems described by (3) are considered.With creative Lyapunov functional and novel matrix inequalities analysis, delay-range-dependent and ratedependent exponential stability conditions are presented.
On the other hand, for () ∈ [  ,  2 ], the integral terms The above equations: are disposed and estimated by Lemma 8.

Advances in Mathematical Physics
Remark 11.It is noted that the integral intervals in (61) are enlarged as follows: Equation ( 61) can be obtained by letting  be defined as previously.
Remark 12.In Theorem 10, the factors may be enlarged as (exp { 2 } − exp { 1 })/.This will lead conservative results due to the fact that () cannot achieve  1 and  2 at the same time.While we apply Lemma 7 to these terms, the method by using reciprocally convex lemma [34] can achieve less conservative results.Moreover, for  > 0, the factor ( ḋ () − 1) exp {−()} that appeared in the derivative of Lyapunov functional may be directly enlarged as  exp {− 1 } − exp {− 2 }.In this paper, we enlarge it as (, ) to reduce the conservativeness of the obtained criteria.In the literature [32,36,37], this factor is enlarged as ( − 1) exp {− 2 }, which only holds for  < 1.
Remark 13.The proposed Lyapunov functional (28) contains some triple-integral terms, which has not been used in any of the existing literatures in the same context before.Compared with the existing ones, [33] has shown that such a Lyapunov functional type is very effective in the reduction of conservatism.Besides, the information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms such as ∫ Remark 14.It should be also mentioned that the result obtained in Theorem 10 is delay-range-dependent and decay rate-dependent stability condition for (9), which is less conservative than the previous ones and will be verified in Section 4.Although the large number of introduced free weighting matrices may increase the complexity of computation, utilizing the technique of free weighting matrices would reduce the conservativeness.In addition, the given results can be extended to more general systems with neutral delay ().That is,  1 ≤ () ≤  2 .The results can be obtained by using the similar methods.
and the remaining notations are the same as Theorem 10.
In Theorem 10, it is assumed that  ̸ = 0.For  = 0, by L'Hospital's rule, the following asymptotic stability criterion can be obtained.

Exponential Stability for the Nominal Systems with Partially Unknown Transition Rates.
In this subsection, we take into account the situation that the information of transition rates is not accessible completely and propose the following conditions to guarantee the exponential stability of system (9) with partially unknown transition rates.Since the transition rates of the Markov chain are partially unknown, then some elements in matrix Π = [  ] × are inaccessible.For instance, the system (3) with five operation modes, the jump rates matrix Π may be viewed as where ?represents the unknown element.For notation clarity, we denote that S  = S   ⋃ S   , for all  ∈  and If S   ̸ = 0, it is further described as where    , ( = 1, 2, . . ., ) represent the th known element of the set S   in the th row of the transition rate matrix Π. Specially, when S   = 0, S  = S   , the full information about Π is obtained, and it becomes the case of Section 3.1.
Consider the system (9) with partially unknown transition rates, and the following corollaries are given, for unknown information on the delay derivative and  = 0, respectively.
Remark 20.Corollaries 16 and 19 provide new delay-rangedependent asymptotic stability conditions for the systems described by ( 9) with known and partially unknown transition rates.Though reciprocally convex lemma has not helped here, the results are still less conservative than some previous ones because of some triple-integral terms, which will be verified in Section 4.

Extension to the
Considering (ii) of ( 89) and combining the uncertainties condition (7), we have By the definition of A, we obtain According to (8), by Lemmas 9 and 6, with (92), we obtain (ii) of (86).Following the same procedure, (90) is considered and (iv) of (87) can be obtained.Finally, following the proof of Theorem 10, the systems described by (3) are exponentially stable with a decay rate /2.This completes the proof. where and other notations are the same as Theorem 17.
The proof of Theorem 22 is omitted here because it is identical with the proof of Theorem 21.In addition, considering the system (3) with fully or partly known transition rates, the corollaries for unknown information on the delay derivative and  = 0 are also omitted here because they are uniform results as the nominal case.

Numerical Examples
In this section, numerical examples are provided with known and partially unknown transition rates, respectively, which demonstrate that the proposed theoretical results in this paper are effectiveness.

Systems with Known Transition
Rates.In this subsection, numerical systems with full information on transition rates are given to show the effectiveness of ours.
Consider the previous system again but with parameter uncertainties as follows: where and the uncertain matrices Δ  () and Δ  () satisfy For given , by Theorem 21, the maximum exponential decay rate ε, which satisfies the LMIs in ( 22), ( 86) and (87), can be calculated by solving a quasi-convex optimization problem.The results are presented in Table 2, where the method of [40] is no longer applicable.
From Table 2, we know that the maximum exponential decay rate ε = 1.0419 in this paper by setting  = 0.3, while the maximum exponential decay rate ε = 0.6255 for [39] and ε = 1.0108 for [41].The results are also given by setting  = 0.5,  = 0.7,  = 0.9,  = 1.1,  = 1.3, and  = 1.5, and it is found that the maximum exponential decay rate in this paper is larger than those in [3,22,26].So, it can be seen that the delay-range-dependent and rate-dependent exponential stability conditions in Theorem 21 in this paper are less conservative than previous results in [39,41].

Systems with Partially Unknown Transition
Rates.In this subsection, numerical systems with partly unknown transition rates are given to show the effectiveness of the approaches presented in this paper.( Following the same procedure in Example 1, we solve the LMIs ( 22), (75), and (93) in Theorem 22 and obtain the feasible solutions to guarantee the exponential stability of the uncertain neutral delay Markovian jump system with nonlinear perturbations and partially unknown transition rates.
Example 5. Consider the nominal system (9) with four operation modes  = {1, 2, 3, 4},  =  =  = 0, and the following parameters: The partially unknown transition rate matrix is considered as the following two cases: For given  = 0 and () = (), we have  1 = 0,  2 = , and  =   .Set   =  2 /2 and employ Corollary 19, and the maximum upper bound of the time delay  2 , which satisfies LMIs ( 22), (75), and (84), can be calculated by solving a quasi-convex optimization problem.This neutral Markovian jump system with partially unknown transition rates was also considered in reference [43].The results on the maximum upper bound of  2 are compared in Tables 6 and 7.
From Tables 6 and 7, we consider the previous system with Π 1 , Π 2 and obtain the maximum upper bound of delay  2 = 0.4763,  2 = 12.4382, respectively, in this paper by setting  = 0.18, while the maximum upper bound of delay  2 = 0.4489,  2 = 12.0750, respectively for [43].The results are also given by setting  = 0.19,  = 0.20,  = 0.21,  = 0.22,  = 0.23,  = 0.24, and  = 0.25, and it is found that the maximum upper bound of delay in this paper is larger than that of [43].So, it can be seen that our proposed method is less conservative than the result in [43].Besides, we know that the maximum upper bound of delay to guarantee stability is dependent on transition rates knowledge.

Conclusions
This paper addresses the exponential stability for neutral Markovian jumping systems with interval time-varying delays, nonlinear perturbations, and partially unknown transition rates.According to the Lyapunov theory, a novel augmented Lyapunov-Krasovskii functional, which contains some triple-integral terms and sufficiently takes advantage of the delay bound, is constructed by dividing the time-varying delay interval into two subintervals.Then, less conservative

Figure 1 :
Figure 1: The mode switching of Example 1.
Neutral time-varying delay

Table 1 :
Maximum exponential decay rate ε with different parameter .

Table 2 :
Maximum exponential decay rate ε with different parameter .

Table 3 :
Maximum upper bound of  2 with different  and parameter  = 0.

Table 4 :
Maximum upper bound of  2 with different  and parameter  = 0.5.

Table 5 :
Maximum upper bound of  2 with different  and parameter  = 0.9.

Table 7 :
Maximum upper bound of  2 with Π 2 and different parameter .-dependent and rate-dependent exponential stability criteria are obtained by the reciprocally convex lemma and free weighting matrices, both of which can be used to reduce the conservativeness.Furthermore, these theoretical results are successfully verified through some numerical examples.