Reachable Set Estimation for Uncertain Markovian Jump Systems with Time-Varying Delay and Disturbances

Zhou, Shouwei; Gu, Jiangliu; Shen, Changchun; Jiang, Min

doi:https://doi.org/10.1155/2021/6695585

Journal of Mathematics

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Abstract Introduction Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6695585 | https://doi.org/10.1155/2021/6695585

Reachable Set Estimation for Uncertain Markovian Jump Systems with Time-Varying Delay and Disturbances

Shouwei Zhou,¹Jiangliu Gu,¹Changchun Shen,¹and Min Jiang¹

Academic Editor: Efthymios G. Tsionas

Received04 Nov 2020

Revised27 Dec 2020

Accepted04 Jan 2021

Published08 Feb 2021

Abstract

In this paper, we are concerned with the problem of reachable set estimation for uncertain Markovian jump systems with time-varying delays and disturbances. The main consideration is to find a proper method to obtain the no-ellipsoidal bound of the reachable set of Markovian jump system as small as possible. Based on an augmented Lyapunov–Krasovskii functional, by dividing the time-varying delay into two nonuniform subintervals, more general delay-dependent stability criteria for the existence of a desired ellipsoid are derived. An optimized integral inequality which is based on distinguished Wirtinger integral inequality and reciprocally convex combination inequality is used to deal with the integral terms. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.

1. Introduction

The reachable set for a dynamic system with disturbances is a set that bounds the state trajectories starting from the origin by inputs with peak value. In practice and engineering applications, many dynamical systems may cause abrupt variations in their structure due to stochastic failures or repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and so on. Markovian jump systems, modeled by a set of subsystems with transitions among the models determined by a Markov chain taking values in a finite set, have appealed to a lot of researchers in the control community. In the past few decades, the Markovian jump systems have been extensively studied (see [1–3] and the references therein).

For the bound of reachable sets for linear systems without any time delay, we can find a well-known result which has been formulated in terms of linear matrix inequality (LMI) [4], and it is widely used to design control systems that have saturating actuators [5, 6]. However, time-delay phenomenon is frequently encountered in many practical systems, such as biological systems, chemical systems, hydraulic systems, and electrical networks. In recent years, the problem of reachable set estimation for time-delay systems has attracted much attention. Then, an increasing number of researchers have devoted their efforts to the problem of reachable set estimation [7–11]. In [7], a delay-dependent condition for an ellipsoid bounding the set of reachable states was presented by using the Lyapunov–Razumikhin function and the S-procedure. Five nonconvex scalar parameters have to be treated as tuning parameters to find the smallest possible ellipsoid. Based on a relaxed Lyapunov–Krasovskii function, the delay-dependent and delay-rate-dependent conditions for the existence of a desired ellipsoid are obtained [8]. Chen and Zhong [10] studied the reachable set of neutral systems with perturbations and uncertainties via novel inequality. In [11], the time-varying delay is split into two nonuniform subintervals based on the Lyapunov–Krasovskii functional, and using the well-known Wirtinger integral inequality and reciprocating convex combination inequality, the RSE boundary of the time-delay system is obtained.

However, the Markov jump system is different from the general time-delay system. It is a random system with multiple modes. The jump transfer between the various modes of the system is determined by a set of Markov chains. In practical application, the equation of state of the system often has some randomness. On the one hand, the stability and stability of Markov jump system have been widely studied [12–16]. On the other hand, there are few studies on the reachable set estimation problem of the Markov jump system (see [17–20] and the references therein). Therefore, this paper studies the reachable set estimation of uncertain Markov jump system.

Besides, in real life, parameter uncertainty is inevitable in the mathematical modeling due to the failure or maintenance of parts, external perturbations, parameter fluctuations, data errors and the change of connection mode of subsystems, etc. Therefore, inspired by the issues discussed above, the problem of reachable set estimation for uncertain Markov jump systems with time-varying delays and disturbance is investigated in this study. The interval is divided by time-varying delay split based on Lyapunov–Krasovskii functional, and the partial integral term of derivative of Lyapunov function is optimized by using Wirtinger-based inequality and reciprocating convex matrix inequality. Finally, numerical examples are given to illustrate the validity of the results.

Notations: the notations used throughout the paper are fairly standard. The superscript “” stands for matrix transposition; denotes the n-dimensional Euclidean space; the notation means that P is a positive definite matrix; and represent identity matrix and zero matrix with dimension n, respectively; and . In symmetric block matrices, we use an asterisk (∗) to represent a term which is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Statement

Consider the following Markov jump systems with uncertainties:where is the state vector and is the disturbance which satisfies

The discrete time-varying delay satisfieswhere , are constants and , . is a Markovian process taking values on the probability space in a finite state with generator given bywhere , , , for is the transition probability from mode at time to mode at time , . , , and are known constant matrices of the Markovian process. For notational simplicity, when , the matrix will be represented by , and the other symbols are similarly defined.

Since the state transition probability of the Markovian jump process considered in this paper is partially known, the transition probability matrix of Markovian jumping process is defined aswhere ? represents the unknown transition rate. For notational clarity, , and the set with

Moreover, if , it is further described as , where is a non-negation integer with and , represent the known element of the row and column in the state transition probability matrix .

Besides, , , and are the parametric uncertainties in system (1), which are assumed to be in the following form:where is an unknown real and possibly time-varying matrix with Lebesgue measurable elements satisfyingand , , , and are known real constant matrices which characterize how the uncertainty enters the nominal matrices , , and .

Before proceeding further, system (1) can be written aswith the constraint: .

We further have

For the sake of brevity, is used to represent the solution of the system under initial conditions , and satisfies the initial condition . And its weak infinitesimal generator, acting on function , is defined in [21].

A reachable set that bounds the state of system (1) is defined by

Based on the ideas proposed in [4], this reachable set estimation problem can be transformed into the problem of finding an ellipsoid to bound the . We will bound by an ellipsoid of the form

Before proceeding further, we will state well-known lemmas.

Lemma 1 (see [4]). Let and ; ifthen we have for .

Lemma 2 (see [22]). For any positive definite matrix , scalar , and vector function such that the integrations concerned are well defined,

Lemma 3 (see [23, 24]). For given a matrix , the following inequality holds for all continuously differentiable functions in :where

Lemma 4 (reciprocally convex combination inequality [25]). For all vectors , the functionwhere and are matrices with appropriate dimensions. Then, the following inequality holds:if there exists a matrix X such that .

Lemma 5 (Schur complement [26]). Given constant symmetric matrices , where and , then holds if and only if

3. Main Results

In this section, some delay-dependent criteria for the existence of ellipsoid bounding the states of system (1) will be obtained. The notations for some matrices are defined in Appendixes.

Theorem 1. Consider the uncertain Markov jump system (1) with constraints (2) and (3); if there exist real matrices , , , , , and , , , any matrices , with appropriate dimension such that , , , scalar satisfying the following matrix inequality:where , , , , , , and , are defined in Appendix A. Then, the reachable sets of system (1) with (2) and (3) are bounded by boundaries , which is defined in (12).

Proof. We choose the following Lyapunov–Krasovskii functional candidate as follows:wherewithTaking derivative of along the trajectories of system (1), we can obtain the following:whereTaking into account the situation that the information of transition probabilities are not accessible completely, due to , the following equations hold for arbitrary appropriate matrices are satisfiedHence,Based on Lemma 2, can be rewritten asSo, according to Lemma 3, we haveWhen , based on the Lemmas 3 and 4, we havewhereUsing Lemma 3, we further haveThus, can be acquired asMeanwhile, for any matrices and with appropriate dimension, the following equation is true:Combining equations (25)–(37), we can obtainwhere , are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar such thatfor all . By using Lemma 5, the matrix inequalities (21), (23), and (24) imply (41).
Similarly, when or , inequality (35) can be reduced to the following two inequalities, respectively. When , we obtain the following inequality:Combining Equations (25)–(39) and (42), we can obtainwhere , are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar such thatfor all . By using Lemma 5, the matrix inequalities (21), (23), and (24) imply (44).
When , we getCombining Equations (25)–(39) and (45), we can obtainwhere , are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar such thatfor all . By using Lemma 5, the matrix inequalities (21), (23), and (24) imply (47).
It should be noted that , , according to inequalities (21), (23), and (24), which implies that .
When , the last two integral terms of are handled in the following way based on Lemmas 3 and 4:Using the same method to deal with the following integral inequality, we further havewhereThen, can be rewritten asBased on Equations (25)–(34), (39), and (51), we havewhere , are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar such thatfor all . By using Lemma 5, the matrix inequalities (22)–(24) imply (53).
When , inequality (49) can simplify the following equation:Based on Equations (25)–(34), (39), and (54), we havewhere , are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar such thatfor all . By using Lemma 5, the matrix inequalities (22)–(24) imply (56).
It should be noted that , according to inequalities (22)–(24), which implies that .
In conclusion, is true on the basis of equations (21)–(24). Since , there is a positive matrix such thatWe further have the inequality when . Hence, is true by using Lemma 1. This completes the proof.

Remark 1. We choose an augmented Lyapunov–Krasovskii functional to establish more general delay-dependent conditions, where is a fifth-order matrix and is a five-dimensional column vector. Therefore, the reachable set estimation criteria can utilize more information on state variables via using these augmented variables in the Lyapunov–Krasovskii functional.

Remark 2. Parameter is used to divide the time-delay interval into two subintervals in this paper. is the time-varying delay satisfying . Generally, the authors divide the time-delay interval into two or more equal subintervals in previous studies to get less conservative stability criteria. Different from them, the time-delay interval in our study is partitioned into and by introducing an adjustable parameter , where ; when , the two subintervals are equal as those in the literatures. Moreover, the integral interval is decomposed in the same way to estimate the bounds of integral terms more exactly.

Remark 3. An optimized integral inequality is provided to deal with the integral term and . Recently, the reciprocally convex combination approach [18, 25, 27] has been widely used to deduce the results. According to the applications shown in these literatures, it is easy to see that the results based on this method are less conservative than the existing ones. Therefore, we adopt the distinguished Wirtinger integral inequality [23] together with the reciprocally convex combination inequality to handle these two integral terms to get more general reachable set estimation criteria.

Corollary 1. Consider the uncertain Markov jump system (1)–(4) with all elements completely known in transition rate matrix (5); if there exist real matrices , , , , , and , , , any matrices , with appropriate dimension such that , , a scalar such that the equations (21)–(22) hold, which all elements , , , , , , and are defined in Appendix B.

Corollary 2. Consider the uncertain Markov jump system (1)–(4) with all elements completely unknown in transition rate matrix (5); if there exist real matrices , , , , , , , , , any matrices , with appropriate dimension such that , , , scalar such that the equations (21) and (22) hold, where , , , , , , and are defined in Appendix C.
Next, we consider the following Markov jump system with disturbancesTheorem 2 is a result for the ellipsoidal bound of a reachable set for a Markov jump time-delayed system (58) having constraints in equations (2) and (3).

Theorem 2. Consider the Markov jump system (56) with all elements partly known in transition rate matrix (5); if there exist real matrices , , , , , and , , , any matrices , , with appropriate dimension such that , , , satisfying the following matrix inequalities:where all the elements are defined in Appendix D.
Then, the reachable sets of system (58) having the constraints (2) and (3) are bounded by a boundary defined in equation (13).

Proof . of Theorem 2 Following a similar line as in the proof of Theorem 1, one can simply obtain this theorem. This completes our proof.

Corollary 3. Consider the Markov jump system (58) with all elements completely known in transition rate matrix (5); if there exist real matrices , , , , and , , , any matrices , , with appropriate dimension such that , , , satisfying the following matrix inequalities:where all elements are defined in Appendix E.

Corollary 4. Consider the Markov jump system (58) with all elements completely unknown in transition rate matrix (5); if there exist real matrices , , , , and , , , any matrices , , with appropriate dimension such that , , , satisfying the following matrix inequalities:where all elements are defined in Appendix F.

Remark 4. Since this is an estimation problem, our aim is to find an ellipsoid as small as possible to bound the reachable sets of system (1).We use a simple approximation as that in [7]. That is, maximize subject to , which can be transformed into the following optimization problem:

4. Numerical Examples

In this section, two examples are used to demonstrate the effectiveness and correctness of the main results derived above.

Example 1. Consider the following uncertain time-delayed system (58) which has been studied in [9, 11]:The transition rate matrix is considered as in the following three cases. Case 1. The transition rate matrix is completely known, which is considered as . Case 2. The transition rate matrix is partly known, which is considered as . Case 3. The transition rate matrix is completely unknown, which is considered as .By solving the optimization problem (59)–(62), the allowable minimum values of obtained by different methods for and are listed in Table 1. According the Table 1, it is inescapably clear that our results decrease the size of the ellipsoid significantly.
Furthermore, for the parameters listed above, let , , and we can get the following feasible solutions by Theorem 2 in this paper. In this example, when the transition probability matrix is completely unknown (at this point, the state trajectory is general switching system, not the system studied in this paper), it is a general switching system, such as [9, 11]. But when the is partly unknown or completely known, it is the Markov jump system. And there are two elliptical boundaries in the picture. So, we choose the smaller one as the reachable set boundary. Due to the limitation of the length of this paper, we just show some of them here, and the reachable set is .
By using Theorem 2 and solving the problem (59)–(62) in Case 2, we can obtainBy using Corollary 3 and solving the problems (63) and (64) in Case 1, we can obtainBy using Corollary 4 and solving the problems (65) and (66) in Case 3, we can obtainFigure 1 is the plot of the state trajectory of system (56). Figures 2–4 are the plots of the ellipsoidal sets defined in equation (13), which are obtained in Theorem 2 and Corollaries 3 and 4 when , respectively.

Example 2. Consider the uncertain with time-varying delays and disturbances:whereBy using Corollary 1 and solving the optimization problem (72) in Case 1, we can get that the minimization of is 0.0785 (the short half axis length of the ellipsoid is 0.2802) when and the corresponding feasible matrices are given as . The reachable sets of system (72) in Case 1 are bounded by a intersection of two ellipsoids: , which is depicted in Figure 5.
By using Theorem 1 and solving the optimization problem (72) in Case 2, we can get that the minimization of is 0.0524 (the short half axis length of the ellipsoid is 0.2288) when and the corresponding feasible matrices are given as . The reachable sets of system (72) in Case 1 are bounded by a intersection of two ellipsoids: , which is depicted in Figure 6.
By using Corollary 2 and solving the optimization problem (72) in Case 3, we can get that the minimization of is 0.0598 (the short half axis length of the ellipsoid is 0.2445) when and the corresponding feasible matrices are given as . The reachable sets of system (72) in Case 1 are bounded by a intersection of two ellipsoids: , which is depicted in Figure 7.
Figure 8 is the plot of the time responses of state variable for system (72). Figures 5–7 are the plots of the ellipsoidal sets defined in equation (13), which are obtained in Theorem 1 and Corollaries 1 and 2 when , respectively.

5. Conclusion

The reachable set bounding for uncertain Markov jump systems with time-varying delays and disturbances has been investigated in our study. We partition the time-varying delay into two nonuniform subintervals and consider the delay in these two cases separately. Furthermore, some new reachable set estimation conditions are derived in terms of linear matrix inequalities by constructing a novel augmented Lyapunov–Krasovskii functional, combining with optimized integral inequality which is based on distinguished Wirtinger integral inequality and reciprocally convex combination inequality. Finally, the feasibility and the comparisons with recent results obtained in the latest literatures are shown through numerical examples.

Appendix

A. The Representation of in Theorem 1

wherewherewith

The other elements in , , , , , , , , , , , , , , and are equal to zero.

B. The Representation of in Corollary 1

where

The other elements are the same as those defined in Appendix A.

C. The Representation of in Corollary 2

where

The other elements are the same as those defined in Appendix A.

D. The Representation of in Theorem 2

where

The other elements are defined in Appendix A.

E. The Representation of in Corollary 3

where

The elements and other are the same as those defined in Appendix D.

F. The Representation of in Corollary 4

where

The other elements are the same as those defined in Appendix D.

Data Availability

As a research paper, this paper mainly studies the theory of dynamic properties of time-delay differential system, and some numerical simulations were carried out by MATLAB. The data and program used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Science and Technology Foundation of Guizhou Province (nos. LKM[2013]21, J[2015]2074, and J[2016]1074), Doctoral Fund Project of Guizhou Minzu University (no. 006), and Project for Young Talents Growth of Guizhou Provincial Department of Education (no. Ky[2017]133).

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Copyright © 2021 Shouwei Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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