Finite-Time Stabilization of Uncertain Switched Positive Linear Systems with Time-Varying Delays
This paper is concerned with finite-time stabilization (FTS) analysis for a class of uncertain switched positive linear systems with time-varying delays. First, a new definition of finite-time boundedness (FTB) is introduced for switched positive system. This definition can simplify FTS analysis. Taking interval and polytopic uncertainties into account, a robust state feedback controller is built such that the switched positive linear system is finite-time bounded. Finally, an example is employed to illustrate the validities of obtained results.
As a special kind of hybrid dynamical systems, the switched system consists of finite subsystems and a switching law. The switching law orchestrates the switches between subsystems . The switched positive system means that its output and state are nonnegative whenever the initial condition and input are nonnegative . The switched positive systems are widely applied in practice, such as formation flying , communication system , viral mutation , and vehicle control .
In control theory, the stability and stabilization are basic problems for switched system. Many valuable results on these problems have been obtained, such as stability [7–12] and stabilization [13–15]. Most of the existing results focus on Lyapunov asymptotic stability (LAS). As we all know, LAS is defined over infinite-time interval. However, due to large state amplitude in transient process, some asymptotic stable systems may be useless . Thus, FTB and FTS are introduced to analyze the transient performance of system. FTB implies that system state remains within a prescribed bound over a fixed interval. FTS means that designing a stabilization controller guarantees that the system is finite-time bounded. The main concern of FTB is the behavior of system state over a fixed finite time interval. It should be pointed out that there is not necessary relation between FTB and LAS. In practice, the finite-time bounded systems are employed in many fields, such as networked control systems  and network congestion control .
Since switched system contains several dynamic subsystems and a switching law, it is difficult to analyze FTB and FTS. Recently, these problems have been extensively studied. Based on the average dwell-time approach, Lin et al. investigated FTB for switched systems with fixed delays . However, the time delay is always time-varying in practice. Then the application of obtained results is limited. Taking time-varying delay into account, M. Xiang and Z. Xiang analyzed FTB and FTS for switched linear systems . In the above-mentioned literatures, the switching instants must be scheduled in advance. Therefore the corresponding switching law is time-dependent. However, the switching instants are determined by system state in some switched system. In this case, the obtained approaches may be useless. In , a state-dependent switching law was designed such that the switched linear system was finite-time bounded. In order to simplify the mathematical calculation, Zhong and Chen adopted a new method to analyze the FTB . In fact, the switches between subsystems produce great effect on system state. Thus, for the problems of FTB and FTS, it is significant to investigate this issue. In , Xiang and Xiao investigated the switching behavior’s impacts on system state. Furthermore, a state feedback controller was built such that the system was finite-time bounded. In practice, the researchers sometimes care about not only whether the system could be stabilized but also whether the stabilization time meets the requirements. In , some criteria were proposed on designing an optimal switching law. Naveed et al. investigated the problem of finite-time H-infinity state estimation for switched systems under asynchronous switching , and the developed method is useful for FTB and FTS. There are some other latest literatures [26–29] related to FTB and FTS, and the references are cited therein.
About the mentioned literatures, we need to pay attention to the following facts. First, switched positive system is an important kind of switched system. However, there are few works on its FTB and FTS. Second, most of the obtained results are based on a common assumption that the norm of external disturbance is bounded over infinite time interval. But the external disturbance usually does not satisfy this strict assumption. In this case, the obtained results may be unsuitable for application. Thus, it is very significant to reduce the restriction on external disturbance. Besides, the definition of FTB contains a product term of gain matrix and state vector; therefore complex mathematical calculations are unavoidable. Finally, the state of switched positive system is nonnegative. Then, whether a more concise definition of FTB could be proposed for switched positive system to simplify the analysis of FTS is a problem that naturally arises.
Motivated by the above considerations, we investigate FTS for uncertain switched positive linear systems. The main object of this paper is to design a robust state feedback controller such that the system is finite-time bounded. The main contributions and novelties of this paper are summarized as follows: for switched positive system, a much more concise definition of FTB is introduced to simplify mathematical calculation; a state feedback controller is built for the uncertain switched positive linear system with time-varying delay; different from most of other results, the disturbance is only required to be bounded in this paper; taking the interval and polytopic uncertainties into account, the obtained controller is robust to uncertainties.
The remainder of the paper is organized as follows. In Section 2, some necessary definitions and lemmas are introduced. In Section 3, a robust stabilization controller is designed for uncertain switched positive linear systems with time-varying delays. A numerical example is given in Section 4. The paper is concluded by Section 5.
Notation. denotes the -dimensional real (positive) vector space; represents Metzler matrices whose off-diagonal entries are nonnegative; (, ) means that all elements of matrix are positive (nonnegative, negative); let , where is positive integer.
2. Preliminaries and Problem Formulation
Consider a switched linear system described as where represents the system state; is the switching law which is a piecewise continuous function with ; the switched system is composed of subsystems; if , the th subsystem is activated; , , , and are known constant matrices with appropriate dimension; and represent uncertainties satisfying ; represents the time-varying delay; is the control input; denotes the external disturbance; is the continuous vector-valued initial function over ; and is a positive constant.
The state feedback controller is given as follows: where is the gain matrix to be determined.
The main task is to choose appropriate such that system (1) is finite-time bounded.
Assume that system (1) satisfies the following assumptions.
Assumption 1 (see ). The external disturbance satisfies
Remark 2. In most of the existing literatures, the external disturbance should satisfy Obviously, condition (6) is much stricter than (5). In Assumption 1, the external disturbance is only required to be bounded. In addition, Assumption 1 conforms much more to the practical situation.
Assumption 3. The time-varying delay satisfies
Assumption 4. The state trajectory is continuous everywhere. Furthermore, the switching number of is finite over fixed finite-time interval.
Next, Lemma 5 is introduced for subsequent analysis.
Definition 6 is proposed as follows.
Definition 6 (see ). Forgiven positive constants , , , and a switching signal , if then system (3) is said to be finite-time bounded with respect to , where ; is the th element of vector . In this paper, we specify .
Remark 7. According to the definition of FTB in most of other literatures, system (3) should satisfy Note the differences between (8) and (9). First, compared with (9), the gain matrix is removed from (8). This improvement brings in much convenience for subsequence process. Some complex matrix transformations could be avoided. Besides, if we choose the definition (9), it is not convenient to adopt multiply copositive Lyapunov-Krasovskii function to analyze FTB due to the term of . Therefore, is replaced by which can simplify subsequent mathematical analysis.
Remark 8. The essence of FTB lies in that the system state must remain within the prescribed bound over fixed interval. Since both (8) and (9) can guarantee it, the two different definitions are consistent in essence.
Definition 9. For , let denote the switching number of over . If holds for and an integer , then is called an average dwell-time .
3. Main Result
Assume that the uncertainties in system (3) are interval or polytopic uncertainties. First, system (3) with interval uncertainties is considered. In this case, we build a robust state feedback controller such that system (3) is finite-time bounded.
Theorem 10. Assume that , for . , , , and are known matrices. For given positive constants , , , , , and , if there exist vectors , , , and such that
and the average dwell-time satisfies
then under the controller with gain matrix satisfying
system (3) is finite-time bounded with respect to under the switching law (14).
, , , ; denotes the th row of matrix ; , , , and represent the th element of vectors , , , and , respectively; is the th row and pth column element of matrix ; , and denotes all eigenvalues of matrix ; , , , and for , . In addition, ; .
Remark 11. Due to the existence of interval uncertainties, system matrices and are unknown matrices. However, their value ranges usually are known in practice. In fact, and denote the upper bounds of and , respectively; and are the lower bounds of and , respectively.
Proof. Since (13) holds, . Noting that , then . Besides, , , , and . By Lemma 5, system (3) is a switched positive system.
Construct multiply copositive Lyapunov-Krasovskii function for system (3) as follows: where
Taking the derivatives of , , and with respect to along the trajectory of system (3) on , we have
From (18), it follows that
It is derived from (11) that
Applying (20) to (19), we get
Since , (21) is transformed into
Let stand for the instant of the th switching and denote the instant just before . Integrating from to on both sides of (22), then
Noting (12) and the continuity of , then
It is obtained via (24) that
Assume that the switching number of over is . We get
According to the definition of ,
Substituting (27) into (26) yields
It is derived from (28) and (29) that
On the other hand, (31) could be derived from (14):
According to Definition 6, the system (3) is finite-time bounded. Thus, the designed controller is the desired controller. This completes the proof of Theorem 10.
Remark 12. Note (11) and (15). If the in (11) is replaced by , then the inequality (11) is not LMIS. In this case, we could not solve (11) via MATLAB. Thus, we should analyze (11) directly. Note that and . , and are known; , and are unknown. We find that and do not contain any product term between two unknown variables. Therefore inequality (11) is LMIS. and could have been obtained by solving (11). Then substitute them into (15). In this way, is obtained finally.
Remark 13. Compared with other literatures, such as [19, 21, 22], the derivation process of Theorem 10 in this paper is much more concise. The main reason lies in that the definition on FTB in this paper is much more concise compared with other literatures, such as [19, 21, 22]. Thus complex matrix translations are eliminated. Besides, the description on FTB is also very concise in [20, see Definition 5]. However, the physical meaning of Definition 6 in this paper is much clearer.
Next, the attention is focused on designing a state feedback controller such that the system (3) with polytopic uncertainties is finite-time bounded.
Theorem 14. Assume that and ; and denote the convex hull of the vertices and , respectively. and are known matrices. For given positive constants , and , if there exist vectors , , , and such that
and the average dwell-time satisfies
then under the controller with gain matrix satisfying
system (3) is finite-time bounded with respect to under the switching law (35).
, , ; denotes the th row of matrix ; , , and represent the th element of vectors , , and , respectively; is the th row and th column element of matrix ; let ; denotes all eigenvalues of matrix ; , , , and for , ; In addition, ; .
Proof. By the property of convex hull, , , and for . Furthermore, , and ; therefore which implies that . Besides, , , and . By Lemma 5, system (3) is a switched positive system.
Construct multiply copositive Lyapunov-Krasovskii functions which have the same structure of (16). Taking the derivatives of , , and with respect to along the trajectory of system (3) on , we get
It is derived from (32) that
It follows that
Note that and ; then
Similarly, we get
Applying (40) and (41) to (37) yields
The remainder of the proof is similar to (23)–(31) and omitted. Therefore the system (3) is finite-time bounded under the designed controller. This completes the proof of Theorem 14.
Remark 15. Inequalities (11), (13), (32), and (34) are LMI. Once inequality (11) or (32) is resolved, the values of and are obtained. Substituting and into , is obtained. Furthermore, if inequality (13) or (34) is satisfied, then is the desired gain matrix. By this way, the desired controller is built such that system (3) is finite-time bounded.
4. Numerical Example
A numerical example is given to illustrate the validities of the obtained results in this section. Consider the following system with interval uncertainties: where
Then, it follows that
Noting that , we obtain
Consequently, and meet the requirements of the design. Then the controller is obtained.
On the other hand, from (14), we get
Finally, the simulation of system (43) with the designed controller is performed. The simulation results are shown in Figures 1–3. The switching law of system (43) is shown in Figure 1. Obviously, the switching times of system (43) is 7 over . Thus, and meets the requirement of (14). In Figure 2, we could find that every element of falls in the range . Therefore, every element of system state is bounded which implies that the system state is bounded over . In other words, system (43) is FTB. The same conclusion could be reached via Figure 3. From Figure 3, it is easy to find over . Thus, the state of system (43) remains within the prescribed bound over . Therefore, under the designed controller, system (43) is indeed bounded over .
In addition, though system (43) is finite-time bounded, its state trajectory is an oscillating curve over . Thus, system (43) is not Lyapunov stable over . It is illustrated that FTB is different from LAS. There is not a necessary relation between them.
This paper focuses on the FTS analysis for a class of uncertain switched positive linear systems. Based on multiply copositive Lyapunov-Krasovskii functions and average dwell-time method, a robust state feedback controller is built such that the uncertain system is finite-time bounded. The innovation points lie in that a concise definition of FTB is proposed for switched positive system. Since the new definition is adopted, the subsequent mathematical derivation is greatly simplified. Finally, a numerical example illustrates the validities of the obtained results and the difference between LAS and FTB.
However, multiply copositive Lyapunov-Krasovskii functions in this paper are traditional. In the literature [32, 33], barrier Lyapunov functions and P-times differentiable unbounded functions are presented, respectively. In future work, we will try to utilize the two new kinds of functions to deal with the FTS problem for switched positive system and make a comparative analysis with the traditional functions. In addition, interval and polytopic uncertainties could not cover all kinds of the uncertainties in practice. Therefore, we will analyze the problem of FTS for switched positive linear system with arbitrary uncertainties. As mentioned in Section 1, the switching behavior is an important factor for the problems of FTB and FTS. For FTB and FTS, it is one of the critical issues to analyze asynchronous switching , slow switching, and fast switching impacts on the system state. In this paper, the issue has not been discussed in detail. Therefore we will try to investigate this issue for switched positive system in further study.
Conflict of Interests
The authors (Tianjian Yu, Yanke Zhong, Tefang Chen, and Chunyang Chen) declare that there is not any conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under Grant no. 61273158.
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