Abstract

An observer-based finite-time - control law is devised for a class of positive Markov jump systems in a complex environment. The complex environment parameters include bounded uncertainties, unknown nonlinearities, and external disturbances. The objective is to devise an appropriate observer-based control law that makes the corresponding augment error dynamic Markov jump systems be positive and finite-time stabilizable and satisfy the given - disturbance attenuation index. A sufficient condition is initially established on the existence of the observer-based finite-time controller by using proper stochastic Lyapunov-Krasovskii functional. The design criteria are presented by means of linear matrix inequalities. Finally, the feasibility and validity of the main results can be illustrated through a numerical example.

1. Introduction

As a special kind of hybrid systems, Markov jump systems (MJSs) consist of two kinds of hybrid dynamic forms. One form is characterized by a discrete state and continuous-time Markov process, called mode; the other form is described by state space equations in each mode, called state. This kind of MJSs was firstly proposed by Krasovskii and Lidskii [1] in 1960s. Due to the stochastic Markov process, it is always considered as a special stochastic system. Moreover, MJSs tend to describe systems in which structures are subjected to abrupt stochastic variations. Such variations usually come from sudden failure of connection between system components, abrupt environmental changes, or changes in the operating point of nonlinear dynamics. MJSs are widely used in many applications, for example, economic systems [2], power electronic system [3], communication systems [4], and circuit network systems [5]. Due to the wide applications, the research of MJSs has been paid much attention in the past decades. For more details about this issue, we can refer to [615].

On another research front, we notice that in some dynamic systems there always exist nonnegative characteristics, such as the number of animals, absolute temperature, density of matter, and the concentration of chemical reactions [1618]. To describe these characteristics, we usually use a positive system to illustrate the nonnegative (positive, strictly) dynamic behavior of state variables. Positive system is a special system; the states and the outputs are both nonnegative (positive, strictly) for any nonnegative (positive, strictly) initial conditions. More recently, the research of positive system has become a heated topic and many publications about this issue have been developed; see, for example, [1923] and the references therein. However, the main result in above references only considered that the states of the systems can be measurable case. It should be noted that the states always cannot be measurable in some practical application systems [2426]. Up to present, the observer-based finite-time control problem of positive Markov jump systems (PMJSs) with some complex environment parameters has not been intensively studied, saying nothing of the simultaneous presence of uncertainties [2732], unknown nonlinearities [21, 3337], and external disturbances [3840]. These motivate our research on observer-based finite-time controller design problem for MJSs.

This paper analyzes the problems of observer-based finite-time - control for a class of PMJSs in a complex environment. Compared with existing results related to Markovian jump systems, the main contributions and difficulties are addressed as follows.

(i) The main contributions of this paper mainly consist of three aspects. Firstly, we aim to analyze the stabilizable problem of a class of PMJSs with some complex environment parameters. Secondly, we try to design an appropriate observer-based finite-time - control law to ensure that closed-loop Markov jump systems be finite-time stabilizable when the states of the system cannot be measured. Thirdly, we attempt to give a sufficient condition to guarantee the positiveness of the augment error dynamic MJSs and we essay to illustrate the validity of the designed method through a numerical example.

(ii) Different from the existing results in [3, 6, 12, 13], the main difficulties of the paper are how to design an appropriate control law for PMJSs in a complex environment with nonmeasurable states such that the corresponding augment error dynamic Markov jump systems be positive and finite-time stabilizable and satisfy the given - disturbance attenuation index. It is necessary to point out that the main results in [3, 6, 12, 13] only considering the states of the systems can be measurable case.

In this paper, all matrices are assumed with proper dimensions and all notations are quite standard. The implication of the symbols is given in Table 1.

2. Preliminaries

2.1. System Description

For a probability space composed of sample space , algebra of events , and the probability measure which is defined on , assume that the stochastic process is a continuous-time discrete-state Markov stochastic process in a finite set over the probability space . The transition probability matrix is defined aswhere and ; is the transition probability rate from mode at time to mode at time and satisfies .

In this paper, we investigate a class of PMJSs in complex environments including uncertainties, unknown nonlinearities, and unknown external disturbances. The PMJSs in the probability space are described bywhere denote the state, denote the measured output, denote the controlled output, denote the unknown external disturbance, denote the controlled input, and denote the unknown nonlinearities, which indicate the nonlinear disturbances related to the state. and are initial mode and initial state, respectively. is a mode-dependent Metzler matrix; , , , , , , , , and are mode-dependent positive matrices. For convenience, we use , , , , , , , , , , and to denote the relevant parameter matrices with . Moreover, the time-varying uncertain matrices satisfywhere is a mode-dependent Lebesgue norm measurable function and satisfies ; , , , , and are known mode-dependent matrices.

Remark 1. In this paper, the uncertain matrices , , , and in (3) can be considered as admissible conditions. In actual applications, it is usually impossible directly to get the accurate mathematical model of realistic dynamics because of some complex environment including unknown nonlinearities, environmental noises, and time-varying parameters [10, 27]. Thus, the uncertain dynamics existing in PMJSs (2) reflect the inexactness in mathematical modeling of such Markov jump systems. Moreover, the mode-dependent Lebesgue norm measurable function is selected as a full row rank matrix and it also can be considered as state-dependent; that is, if . For more results of this issue, we refer readers to [3032].

When the states of PMJSs (2) cannot be available, we can construct the following state observer and feedback control law: where denote the estimated state; denote the observer output; denote the estimation output. and are observer and control law gains to be devised, respectively. The state estimated error and the controlled output error of are defined by and , respectively. Therefore, we have the following MJSs by (2) and (4):

Letting , the closed-loop augment error dynamic MJSs can be represented aswhere

2.2. Main Definitions, Lemmas, and Assumptions

The following main definitions, lemmas, and assumptions are important for analyzing and giving the main results of the paper.

Definition 2. For given constants and , the augment error dynamic MJSs (6) are finite-time stabilizable (FTS) with regard to , if there exist constants and positive-definite matrix , such that

Definition 3. For given constants and , is said to be the finite-time - observer-based state feedback control law of MJSs (6) under the zero initial condition, if the augment error dynamic MJSs (6) are FTS with regard to and satisfywhere ,,, and.

Definition 4. The weak infinitesimal operator of stochastic Lyapunov-Krasovskii functional (SLKF) is defined as

Definition 5. MJSs (2) are positive, if, for the initial conditions and and the controlled input , the relevant trajectories of satisfy and .

Lemma 6 (see [41]). is said to be a Metzler matrix, if there exists a mode-dependent constant satisfying , where is a real square matrix.

Lemma 7 (see [41]). MJSs (2) are said to be positive if and only if is a Metzler matrix and ,,,,, and.

Lemma 8 (see [30]). Suppose that and are mode-dependent real matrices and is a mode-dependent Lebesgue norm measurable function and satisfies . There exists a mode-dependent constant , satisfying

Lemma 9 (see [30]). Suppose that and are mode-dependent real matrices. There exists a positive-definite matrix and a mode-dependent constant , satisfying

Assumption 10. The mode-dependent nonlinear function satisfies the following Lipschitz condition:where is a real matrix with proper dimension.

Assumption 11. For given constant , is energy-bounded and satisfies

Remark 12. Assumption 10 guarantees that we can use linearization method to study the nonlinear systems by means of linear matrix inequalities [3, 5, 11]. In the design of observer-based finite-time - control law, Assumption 11 is given to assume that the unknown external disturbance is to be an arbitrary deterministic signal of bounded energy [20, 24, 27].

3. Main Results

3.1. FTS Analysis

In this subsection, the FTS analysis for the augment error dynamic MJSs (6) will be considered. Based on the SLKF approach and LMIs techniques, a sufficient condition of FTS will be given in Theorem 13.

Theorem 13. For given constants ,,, and, the augment error dynamic MJSs (6) are FTS with regard to , where and , if there exists a set of mode-dependent positive-definite symmetric matrix , positive-definite symmetric matrix , matrix , a set of mode-dependent constant , and constant , such thatwhere , , , and .

Proof. See Appendix A.

3.2. Finite-Time - Disturbance Attenuation Index Analysis

Theorem 13 gives a sufficient condition of FTS for augment error dynamic MJSs (6). Recalling Definition 3, we will give the following Theorem 14 to analyze the observer-based finite-time - control law design.

Theorem 14. For given constants ,,, and, the augment error dynamic MJSs (6) are FTS with regard to , where and satisfy the given - disturbance rejection disturbance attenuation index (10), if there exists a set of mode-dependent positive-definite symmetric matrix , positive-definite symmetric matrix , matrix , a set of mode-dependent constant , and constant , such that inequalities (15)-(16) and the following relation hold:

Proof. See Appendix B.

3.3. Positiveness and Observer-Based Control Law Gain Solution

Recalling Definition 5, following sufficient condition will be given to ensure the positiveness of the augment error dynamic MJSs (6) in Theorem 15. It should be noted that there exist some time-varying uncertain matrices in Theorem 13. Therefore, it is difficult to obtain the observer gain and the control law gain from matrix inequalities (15)–(17). It is necessary for us to convert the nonlinear matrix inequalities (15)–(17) into the solvable inequalities, which can be directly solved by Matlab LMI toolbox.

Theorem 15. For given constants ,,, and, there exists a finite-time - observer-based control law with and and the augment error dynamic MJSs (6) are positive and FTS and satisfy the given - disturbance rejection disturbance attenuation index (9) with regard to , where , if there exists a set of mode-dependent positive-definite symmetric matrix , positive-definite symmetric matrix , a set of mode-dependent matrices and , a set of mode-dependent constants ,,,,,, and, and constants ,,, and, such thatwhere

Proof. See Appendix C.

Corollary 16. The sufficient conditions to design the stochastic finite-time - observer-based control law for a class of PMJSs in complex environments have been presented in Theorems 1315. Considering that the coupling inequalities (18)–(25) are related to , , , , , , , , , , , , and , we have the optimization algorithm by setting as an optimization variable value:

Remark 17. It should be pointed out that the optimization algorithm (27) is given to solve the unknown matrices and parameters through Matlab LMI toolbox. Recalling inequality (20), it is known that all of the parameters in are linear and also can be solved through Matlab LMI toolbox by setting as an unknown parameter. Considering - disturbance attenuation index in Definition 3, we select as an optimization variable value in optimization algorithm (27).

Remark 18. For PMJSs (2) in complex environments without uncertainties in probability space , we have the following dynamic systems:The state observer and feedback control law for PMJSs (28) can be designed asLetting , we can obtain the following closed-loop augment error dynamic MJSs: where

The main results in Theorem 15 will reduce to the following Corollary 19.

Corollary 19. For given constants , , , and , there exists a finite-time - observer-based control law with and and the augment error dynamic MJSs (29) are positive and FTS and satisfy the given - disturbance rejection disturbance attenuation index (10) with regard to , where and , if there exists a set of mode-dependent positive-definite symmetric matrices , a set of mode-dependent matrix , matrix , a set of mode-dependent constants and , and constants , , , and , such that inequalities (20)–(25) and the following relations hold:where , , and .

4. A Numerical Example

Consider a class of PMJSs with two modes described as follows.

Mode 1.

Mode 2.

The values of the relevant parameters are given as , , , , , and . The mode of PMJSs (2) is converted according to the following Markov chain conversion rate matrix: and we select the unknown nonlinear function as .

Solving LMIs (18)–(25) in Theorem 15, we can get the observer and the control law gain as , , , and with .

The jumping modes, the response of the real states, the estimated state , the estimated error , the evolution , and the controlled output error are shown in Figures 16.

Figure 2 gives the real state trajectory in open case and it shows that the open-loop PMJSs are unstable. The responses of the real state and the observer state are depicted in Figure 3. Figure 4 shows the state estimated error and Figure 5 shows the state trajectory of the closed-loop MJSs. From Figure 5, we know that the designed observer-based control law can ensure that the closed-loop MJSs are FTS in the given finite-time interval. Obviously, it can be seen from Figure 6 that the controlled output error of the closed-loop MJSs is positive and FTS.

5. Conclusions

In this paper, we studied the observer-based finite-time - control law design problem of a class of PMJSs in a complex environment. Based on the designed SLKF methods and LMIs technique, sufficient conditions on the existence of the observer-based finite-time - control law are proposed and proven. The designed finite-time - control law makes the closed-loop augment error dynamic MJSs be positive and FTS and satisfy the given induced - disturbance attenuation index. A numerical example was delivered to demonstrate the contribution of the main results.

Appendix

A. Proof of Theorem 13

Proof. We select a SLKF candidate as . Recalling Definition 2 and along the trajectories of the augment error dynamic MJSs (6), the weak infinitesimal operator of can be written asAccording to Lemma 9 and Assumption 10, we know that there exists a set of mode-dependent constant and matrix with proper dimension such thatBy considering Schur complement lemma and substituting inequality (A.2) into equality (A.1), we can getwhereConsidering Definition 2, we introduce the following inequality:Thus, inequality (A.5) can be obtained by inequality (15).
Multiplying inequality (A.5) by and integrating inequality (A.5) from to , we can obtainIf we define , , and , we have the following inequality by and :Considering thatwe havewhich is equivalent toTherefore, , can be ensured through inequality (16); that is, the augment error dynamic MJSs (6) are FTS with regard to . This completes the proof.

B. Proof of Theorem 14

Proof. We select the same SLKF as Theorem 13. From inequality (A.6), we haveUnder the zero initial condition, inequality (B.1) can be rewritten:From inequality (17), it yields . Recalling Definition 3, we haveThus, we have by (B.2)-(B.3). Recalling Definition 3, we know that the finite-time - disturbance rejection disturbance attenuation index (10) can be guaranteed by , . This completes the proof.

C. Proof of Theorem 15

Proof. Considering the uncertainties in inequality (6), we substitute , , , , and into inequalities (15) and (17) and define ; it yieldswhereFrom inequalities (C.1)-(C.2), we know that , , and are nonlinear. Let and get , , and . Recalling equality (3), and can be rewritten aswhere , , , , , , , and .
According to Lemma 8, it follows from inequalities (C.1)-(C.2) thatUse to pre- and postmultiply matrix (C.6) and use to pre- and postmultiply matrix (C.7). Considering , , , and and using Schur complement lemma, we can obtain inequalities (18)-(19). Noting that and , inequality (16) can be guaranteed by inequalities (20)-(21).
Next, we prove the positiveness of the augment error dynamic MJSs (6). From inequalities (22)-(23), we know that and are Metzler matrices. Since , is also a Metzler matrix. From (24)-(25), we know that and are positive; that is, and are positive matrices. Considering that is a positive matrix, is a Metzler matrix and and are positive matrices. Recalling Lemma 6, we know that the augment error dynamic MJSs (6) are positive. This completes the proof.

Data Availability

The data 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 work was supported in part by the National Natural Science Foundation of China under Grants 61673001 and 61203051, the Foundation for Distinguished Young Scholars of Anhui Province under Grant 1608085J05, the Key Support Program for University Outstanding Youth Talent of Anhui Province under Grant gxydZD2017001, and the open fund for Discipline Construction, Institute of Physical Science and Information Technology, Anhui University.