Abstract
This paper is concerned with the problem of finitetime control for a class of Markovian jump system with modedependent timevarying delay. By using the new augmented multiple Lyapunov function with more general decomposition approach, a novel sufficient condition for finitetime bounded with an performance index is derived. Based on the derived condition, the reliable control problem is solved, and an explicit expression of the desired controller is also given, the system trajectory stays within a prescribed bound during a specified time interval. Finally, numerical examples are given to demonstrate that the proposed approach is more effective than some existing ones.
Keywords:
finitetime stability; timevarying delay; Markov jump system; control; delay partition approach1 Introduction
Markovian jump systems were introduced by Krasovskii and Lidskii [1], which can be described by a set of systems with the transitions in a finite mode set. In the last few decades, there has been increasing interest in Markovian jump systems because this class of systems is appropriate to model many physical systems mainly those random failures, repairs and sudden environment disturbance [25]. Such class of systems is a special class of stochastic hybrid systems with finite operation modes, which may switch from one to another at different time. As a crucial factor, it is shown that such jumping can be determined by a Markovian chain [6]. For linear Markovian jumping systems, many important issues have been studied extensively such as stability, stabilization, control synthesis and filter design [612]. In finite operation modes, Markovian jump systems is a special class of stochastic systems that can switch from one to another at different time. It is worth pointing out that time delay is one of the instability sources for dynamical systems and is a common phenomenon in many industrial and engineering systems [1318]. Hence, it is not surprising that much effort has been made to investigate of Markovian jump systems with time delay during the last two decades [1923]. The exponential stabilization of Markovian jump systems with time delay was first studied in [19] where the decay rate was estimated by solving linear matrix inequalities [20]. However, in the aforementioned works, the networkinduced delays have been commonly assumed to be deterministic, which is fairly unrealistic since delays resulting from network transmissions are typically time varying.
Generally speaking, the delaydependent criterions are less conservative than delayindependent ones, especially when the time delay is small enough in Markovian jump systems. Thus, recent efforts were devoted to the delaydependent Markovian jump systems stability analysis by employing LyapunovKrasovskii functionals [2432]. However, most efforts have been given on how to construct an appropriate Lyapunov functional by dividing the delay interval into N equal length subintervals [22]. It should be pointed out that the delay decomposition method is not effective when the lower bound of timevarying delay is zero. Furthermore, although the decay rate can be computed, it is a fixed value that one cannot adjust to deduce if a larger decay rate is possible. Therefore, how to obtain the improved results without increasing the computational burden has greatly improved the current study.
Over the years, many research efforts have been devoted to the study of finitetime stability of systems. In finitetime interval, finitetime stability is investigated to address these transient performances of control systems. Recently, the concept of finitetime stability has been revisited in the light of linear matrix inequalities (LMIs) and Lyapunov function theory, some results are obtained to ensure that system is finitetime stable or finitetime bounded [3349]. It is noted that there are still some related issues to be solved, to the best of our knowledge, the finitetime control for a class of Markovian jump systems with timevarying delay has not been fully developed. The analysis method in the existing references seems still conservative to study Markovian jump system. There is room for further investigation.
The main contribution of this paper is as follows: Firstly, we present a new augmented Lyapunov functional by employing the more general decomposition of a delay interval for a class of Markovian jump systems with modedependent timevarying delay. Secondly, in order to reduce the possible conservativeness and computational burden, some slack matrices are introduced [18]. Several sufficient conditions are derived to guarantee the finitetime stability and boundedness of the resulting closedloop system. Last but not the least, it is shown that less conservative and more general results can be derived since the timevarying delays are divided into a more general decomposition. We find that finitetime stability is a concept independent from Lyapunov stability and can always be affected by switching behavior significantly, thus it deserves our investigation. The finitetime boundedness criteria can be tackled in the form of LMIs. Finally, numerical examples illustrate the effectiveness of the developed techniques.
Notations: Throughout this paper, we let (, , ) denote a symmetric positive definite matrix P (positive semidefinite, negative definite and negative semidefinite). For any symmetric matrix P, and denote the maximum and minimum eigenvalues of matrix P, respectively. denotes the ndimensional Euclidean space and refers to the set of all real matrices and . The identity matrix of order n is denoted as . ∗ represents the elements below the main diagonal of a symmetric matrix. The superscripts ⊺ and −1 stand for matrix transposition and matrix inverse, respectively.
2 Preliminaries
Given a probability space where Ω, F and P respectively represent the sample space, the algebra of events and the probability measure defined on F. In this paper, we consider the following Markov jump system over the space described by
where is the state vector of the system, is the controlled output, is the control input and , and are initial conditions of continuous state and the mode. is the disturbance input satisfying
Let the random form process be the Markov stochastic process taking values on a finite set with the transition rate matrix , , and the transition probabilities described as
where , , for , is the transition rate from mode i at time t to mode j at time and
for each mode , . denotes the mode dependent timevarying state delay in the system and satisfies the following condition:
where and are prescribed integers representing the lower and upper bounds of timevarying delay . Similarly, and are prescribed integers representing the lower and upper bounds of timevarying delay . , , , , , , and are known modedependent matrices with appropriate dimension functions of the random jumping process and represent the nominal systems for each . For notation simplicity, when the system operates in the ith mode (), , , , , , , and are denoted as , , , , , , and , respectively.
Remark 1 In this paper, the lower bound of is required in order to implement the proposed delay decomposition method. If , then corresponds to the constant delay.
Moreover, the transient process of a system can be clearly characterized if its decay rate is available. The objective of this study is to develop a new approach to designing a state feedback controller
via a novel Lyapunov functional such that the resulting closedloop system is finitetime stable, where is the controller gains to be designed.
In this paper, we split the delay interval into two segments: . Moreover, we further subdivide each interval into l, m equal length subsegments and , respectively, where
and l, m are given positive integers.
Remark 2 The delay intervals are divided subsegments dependent on t, thus the proposed delay decomposition method is more general than those in [1317,1924]. The conservatism will be reduced with the partitioning number l and m increase.
In order to more precisely describe the main objective, we introduce the following definitions and lemmas for the underlying system.
Definition 2.1System (1) is said to be finitetime bounded with respect toif condition (2) and the following inequality hold:
Definition 2.2[49]
Consideras the stochastic Lyapunov function of the resulting system (1), its weak infinitesimal operator is defined as
Definition 2.3Given a constant, for all admissiblesubject to condition (2), under zero initial conditions, if the closedloop Markovian jump system (1) is finitetime bounded and the control outputs satisfy condition (5) with attenuation,
then the controller system (1) is called the finitetime bounded with disturbance attenuation γ.
Remark 3 It should be pointed that the assumption of zero initial condition in system (1) is only for the purpose of technical simplification in the derivation, and it does not lose generality. In fact, if this assumption is lost, the same control result can still be got along the same lines, except adding extra manipulations in the derivation and extra terms in the control presentation. However, in real world applications, the initial condition of the underlying system is generally not zero.
Lemma 2.1[18]
Let () have positive values in an open subsetof. Then the reciprocally convex combination ofoversatisfies
subject to
Lemma 2.2For a given function, (), there exist four functions, , andsatisfyingand, respectively, such that, the following equation holds:
Lemma 2.3[50]
For matricesA, and, the following matrix inequality
holds if and only if there exists a matrixGof appropriate dimension such that
Lemma 2.4 ([14] Schur complement)
Given constant matricesX, Y, Z, whereand, thenif and only if
3 Finitetime performance analysis
We first consider the problem of stability analysis for system (1) with . The following results actually present the finitetime stability for the Markov jump system with timevarying delay.
Theorem 3.1System (1) is finitetime bounded with respect toif there exist matrices, , (), , , , , , scalars, , , , , (), , and, such that for all, , , the following inequalities hold:
where
Proof First, in order to cast our model into the framework of the Markov processes, we define a new process by
Now, we consider the following LyapunovKrasovskii functional:
where
Then, let the mode at time t be i, i.e., , we have
Moreover, denote
By using Lemma 2.1, it yields that
It follows from (9) and () that
Similarly, (10) implies
From (12)(16), we can eventually obtain
where
and
By Lemma 2.2, there exist functions , , and satisfying and , respectively. Using the Schur complement such that
where , , and are defined in Theorem 3.1. Substituting (18) into (17), then (18) can be rewritten as
Therefore, the following relation holds by condition (8) and (19):
Multiplying the aforementioned inequality by , we can get
By integrating the aforementioned inequality between 0 and t, it follows that
Denote , , , , , it yields that
On the other hand, it follows from (12) that
It can be derived from (21)(22) that
From (11) and (23), we have
Then the system is finitetime bounded with respect to . □
Remark 4 It should be mentioned that novel terms and are continuous at is included in the LyapunovKrasovskii functional (12), which plays an important role in reducing conservativeness of the derived result.
Remark 5 In this paper, and may have different upper bounds in various delay intervals satisfying (3) and (4), respectively. While in previous work such as [16,17], and are enlarged to and , respectively, which may lead to conservativeness inevitably. However, the case above can be taken fully into account by employing the LyapunovKrasovskii functional (12).
Remark 6 When dealing with term , the convex combination is not employed, Lemma 2.1 is used in this paper, then the freeweighting matricesdependent null add items are necessary to be introduced in our proof, which leads to the decrease in the number of LMIs and LMIs scalar decision variables.
Remark 7 The feature of this paper is the way to deal with the integral term. Many researchers have enlarged the derivative of the Lyapunov functional in order to deal with the integral term in mathematical operations. In this paper, we transform different integral intervals with the same integral length into an integral interval. It is worth pointing out that in the proof of the theorem no extra inequality is introduced. We propose a novel delaydependent sufficient criterion, which ensures that the Markovian jump system with timevarying delays is finitetime stable.
Remark 8 One can clearly see from the proof of Theorem 3.1 that neither freeweighting matrices nor model transformation has been employed to deal with the sum terms, and none of useful items are ignored, resulting in better results with the less number of LMIs scalar decision variables, which deduces some conservatism in some sense.
By using the novel Lyapunov functionals with the more general decomposition of delay interval, a state feedback controller (3) can be designed such that the resulting closedloop system is finitetime bounded with performance. When , the closedloop system is expressed by
where
Theorem 3.2System (25) is finitetime bounded with respect toif there exist matrices, , (), , , , , scalars, , , , , , (), , and, such that for all, , , the following inequalities hold:
where
Proof We now consider the performance of system (25). Select the same LyapunovKrasovskii functional as Theorem 3.1 and the Schur complement, it yields that
It follows from (26) that
Multiplying the aforementioned inequality by , one has
Under zero initial condition and , by integrating the aforementioned inequality between 0 and T, we can get
Then it yields
Thus it is concluded by Definition 2.3 that system (25) is finitetime bounded with an performance γ. The proof is completed. □
Remark 9 From the proof process of Theorem 3.1 and Theorem 3.2, it is easy to see that neither bounding technique for cross terms nor model transformation is involved. In other words, the obtained result is expected to be less conservative.
Remark 10 Lyapunov asymptotic stability and finitetime stability of a class of systems are independent concepts. Lyapunov asymptotically stable system may not be finitetime stable. Moreover, finitetime stable system may also not be Lyapunov asymptotically stable. There exist some results on Lyapunov stability, while finitetime stability also needs our full investigation, which was neglected by most previous work.
4 Finitetime control
Theorem 4.1System (25) is finitetime bounded with respect toif there exist matrices, , (), , , , , scalars, , , , , (), , and, such that for all, , , the following inequalities hold:
where
Moreover, the state feedback gain matrices can be designed as
Proof Consider Theorem 3.2 and the overall closedloop Markov jump system (25). Pre and postmultiplying inequality (26) by blockdiagonal matrix and its transpose, respectively. Letting
It can be easily obtained that
where
From Lemma 2.3, for any , and , one can obtain , , . Then (33) is equivalent to (39). Therefore, if (33) holds, system (1) is finitetime bounded with a prescribed performance index γ. The proof is completed. □
Remark 11 By solving the Markovian jumping system with finitetime observerbased controller, the modedependent positivedefinite weighting matrices in inequalities (33)(37) should be known first. For convenience, we always choose the initial value for .
Remark 12 In many actual applications, the minimum value of is of interest. In Theorem 3.2, with a fixed λ, can be obtained through the following optimization procedure:
In Theorem 4.1, as for finitetime stability and boundedness, once the state bound is not ascertained, the minimum value is of interest. With a fixed λ, and define , , then the following optimization problem can be formulated to get the minimum value
5 Illustrative example
Example 1 Consider the Markovian jump system (1) with two operation modes and the following data:
and the transition probability matrix is
Under different levels of the upper bound and λ, Table 1 and Table 2 list the results of the maximum allowable upper bound, the decay rate λ for different time delays and maximum values of derived from various methods including the one proposed in this paper, respectively. One can see from Table 1 and Table 2 that the same results are obtained in [21,22]. It is clear from Table 1 and Table 2 that the performance achieved by our method is much better than those by [21,22]. Therefore, our results not only are less conservative, but also require the less number of scalar decision variables.
Table 1. Comparison of the upper bounds of the decay rate for different delays
Table 2. Comparison of the allowable values of time delayfor different decay rates
Example 2 Consider a twomode Markovian jump system (1) with
In addition, the transition rate matrix is given by
Then we choose , , , , through Theorem 4.1, it yields that . Moreover, we also can obtain the following controller gains:
It confirms the effectiveness of Theorem 4.1 for the state feedback controller design to finitetime Markovian jump systems with timevarying delay.
6 Conclusions
In this paper, we have examined the problems of finitetime control for a class of Markovian jump systems with modedependent timevarying delay. Based on a novel approach, a sufficient condition is derived such that the closedloop Markovian jump system is finitetime bounded and satisfies the prescribed level of disturbance attenuation in a finite time interval. The controller and observer gains can be solved directly by using the existing LMIs optimization techniques. Finally, numerical examples are also given to illustrate the effectiveness of the proposed design approach.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have equal contributions.
Acknowledgements
The authors would like to thank the associate editor and anonymous reviewers for their detailed comments and suggestions. This work was supported by the National Basic Research Program of China (2010CB732501).
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