Abstract
In this paper, the problem of input-output finite-time control of positive switched nonlinear systems with time-varying and distributed delays is investigated. Nonlinear functions considered in this paper are located in a sector field. Firstly, the proof of the positivity of switched positive nonlinear systems with time-varying and distributed delays is given, and the concept of input-output finite-time stability ( IO-FTS) is firstly introduced. Then, by constructing multiple co-positive-type nonlinear Lyapunov functions and using the average dwell time (ADT) approach, a state feedback controller is designed and sufficient conditions are derived to guarantee the corresponding closed-loop system is IO-FTS. Such conditions can be easily solved by linear programming. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.
1. Introduction
In the past decades, positive switched systems have been paid much attention due to its broad applications in many areas such as communication systems [1], biological systems [2], and systems theory [3–6]. Many meaningful results have been presented in the literatures [7–11], and the references therein. Most of related results are mainly concerned with positive switched linear systems. In practice, many systems are nonlinear, such as physical systems, chemical systems, and transport systems. Although the study of dynamic behavior of positive switched nonlinear systems is not an easy work, because it is difficult to define the positivity of a nonlinear system, and few effective control techniques with respect to positive switched nonlinear systems are proposed. There are already some available results about positive switched nonlinear systems [12–16].
However, the literatures mentioned above mainly focus on asymptotic stability, but in many practical applications, one is more interested in the behavior of the systems within a finite-time interval. Reference [17] firstly defined the concept of finite-time stability (FTS) for linear deterministic systems. For positive switched nonlinear systems, only [18] considered the guaranteed cost finite-time control problem. Recently, the definition of input-output finite-time stability (IO-FTS) is proposed in [19], which is fully consistent with the definition of FTS. IO-FTS means that given a class of norm bounded input signals over a specified time interval , the outputs of the system do not exceed an assigned threshold during such time intervals, and it does not necessarily require the inputs and outputs to belong to the same class. It is noteworthy that the concept of IO-FTS is different from that of input-output stability, the later deals with the behavior of a system within an infinite time interval. As we know, only a few results considered the IO-FTS problem of positive switched linear systems [20–22]. Reference [20] extended IO-FTS to positive switched linear systems with state delays, but the time-varying delay and distributed delay were not considered. References [21, 22] discussed the input-output finite-time control problem of positive switched systems and uncertain positive impulsive switched systems, respectively. However, for positive switched nonlinear systems, the IO-FTS problem has not been studied.
Moreover, multiple time delays have great effect on the systems, such as oscillation and instability. On the other hand, compared with performance, gain performance is more suitable to describe the performances of the systems with positive properties [23–25]. When taking multiple time delays and gain performance into account, the problem of choosing an appropriate nonlinear copositive Lyapunov function and analyzing the IO-FTS of the positive switched nonlinear systems will be more complex and challenging, which motivates our study.
In this paper, we are interested in the problem of IO-FTS for a class of positive switched nonlinear systems with time-varying and distributed delays via ADT. The proposed system model is more general such that the systems dealt with in [6, 13, 17, 21–23] can be regarded as special forms. The main contributions of this paper are (1) the proof of the positivity of positive switched nonlinear systems with time-varying and distributed delays is given, (2) integrating performance index into IO-FTS, the concept of IO-FTS is for the first time extended to positive switched nonlinear systems with time-varying and distributed delays, and (3) a state feedback controller is designed and sufficient conditions for IO-FTS of the corresponding closed-loop system are given. Such conditions can be easily solved by linear programming. The rest of this paper is organized as follows. In Section 2, problem statements and necessary lemmas are given. Main results are given in Section 3. A numerical example is provided in Section 4. Section 5 is the conclusion.
Notations. The representation means that , which is applicable to a vector. means that . The symbols R, , and denote the set of real numbers, the space of the vectors of n-tuples of real numbers, the space of matrices with real numbers, respectively. is the n-dimensional nonnegative (positive) vector space. denotes the l-dimensional vector . denotes the transpose of matrix A. Let , 1-norm is defined by . denotes the space of absolute integrable vector-valued functions on the interval ; that is, if holds. denotes the space of the uniformly bounded vector-valued functions on the interval ; that is, if holds. Matrices are assumed to have compatible dimensions for calculating if their dimensions are not explicitly stated.
2. Preliminaries and Problem Statements
Consider a class of positive switched nonlinear systems with time-varying and distributed delays:where is the system state, and represent the control input and output, respectively. , . The function is the system switching signal, and it is continuous from the right everywhere for a switching sequence, denotes the number of subsystems. is the initial time, denotes the kth switching instant. , , , , , , and are constant matrices with appropriate dimensions. The differentiable function represents time-varying delay, which satisfies , is the distributed delays, where and h are known positive constants. is the initial condition on , . is the exogenous disturbance and is defined aswith a known scalar .
Next, we will present some definitions and lemmas for the positive switched nonlinear systems (1).
Definition 1. System (1) is said to be positive if for any switching signals , any initial conditions , , , the corresponding trajectory satisfies and for all .
Definition 2 (see [20]). A is called a Metzler matrix if the off-diagonal entries of matrix A are nonnegative.
Definition 3 (see [20]). For any , let denote the switching number of , over the interval , for given and , if the inequalityholds, then the positive constant is called an ADT, and is called a chattering bound. Generally speaking, is chosen in this paper.
Assumption 1. The nonlinear functions and lie in sector fields satisfyingfor and , where , and , .
Remark 1. The system model (1) is a more general form. If , then system (1) is turned into the system in [13]. If (it means , ), then system (1) is transformed into the systems in [21, 22]. Moreover, if and , then system (1) will be turned into the systems in [6, 10, 17, 23].
Remark 2. Assumption 1 is a necessary assumption, because it takes full advantage of the characteristics of nonnegative states of positive switched nonlinear systems.
Lemma 1 (see [25]). Let be a Metzler matrix, then there exists a vector such that .
Lemma 2. Under Assumption 1. System (1) is positive if and only if , , are Metzler matrices and , , , , , and .
Proof (Necessity). Firstly, let on and . Then, for some . By definition 1, for every , which means that for each . In the same way, we can get .
Secondly, we prove that , are Metzler matrices. Let and . Suppose there exists an element . From the system (1), we have
We can see that is possible if , , and takes a small value enough. Thus, , which brings a contradiction with the positivity of the system (1). So, , , are Metzler matrices.
Thirdly, we prove by the same way. Suppose there exists an element , then we have
It is easy to get that is possible if takes a small value enough. Furthermore, , which brings a contradiction with the positivity of the system (1). So, .
Finally, suppose there exists an element , similar to the above process, we have
In the same way, it is possible that if takes a small value enough. It shows that system (1) is not positive. Based on these four points addressed above, the necessity is obtained.
2.1. Sufficiency
Let . To prove that for all , it is sufficient to ensure that the vector does not point towards the outside of whenever is on the boundary of . This is equivalent to verifying that the components of the vector corresponding to the zero components of are nonnegative. Denoting by the set of indices of such components, that is, for . Then, for some , we havewhere , , , , are the ith row jth column element of , , , , , respectively. From the condition (4), it follows from that for . According to system (1), for , , , , . So, we have for and . This means for . From (5), we have for . Combining this with , it yields .
That reveals the system (1) is positive under any switching signals if and only if are Metzler matrices, , , , , and , .
From the above, we can conclude the system (1) is positive.
Remark 3. Lemma 2 proves the positivity of a new class of switched nonlinear systems, which plays a key role in our latter work. From Assumption 1, if and for , then Lemma 2 still holds. That is to say, the system (1) is positive under any switching signals if and only if it consists of a family of positive nonlinear subsystems.
Next, we will give the definitions of IO-FTS and IO-FTS for the positive switched nonlinear system (1).
Definition 4. (IO-FTS) Consider zero initial condition (), for a given time constant , disturbances signals defined by (2) and a vector , the system (1) is said to be IO-FTS with respect to (δ, , ε, ), ifIf the disturbance satisfies , where is defined asthen, we give Definition 5.
Definition 5. (IO-FTS) Consider zero initial condition (), for a given time constant , disturbances signals , and a vector . System (1) is said to be IO-FTS with respect to (δ, , ε, ), if
Remark 4. In Definition 4 and 5, two classes of exogenous disturbances are norm bounded integrable signals and the uniformly bounded signals , respectively. Because of the similarity of the proof process, we only focus on the former in this paper.
Definition 6. ( IO-FTS) Under the zero initial condition (), for a given time constant , the system (1) is said to be IO-FTS with respect to (δ, , ε, ), if the following conditions are satisfied: (1) system (3) is IO-FTS with respect to (δ, , ε, ); (2) the output satisfieswhere , , and satisfies (2).
Remark 5. In Definition 6, -gain performance index provides a more useful description for the disturbance attenuation performance of positive switched systems. Integrating -gain performance index into IO-FTS, the concept of IO-FTS is more suitable to describe the system behavior during a specified time interval.
The aim of this paper is to design a state feedback controller and find a class of switching signals for positive switched nonlinear system (1) such that the corresponding closed-loop system is IO-FTS.
3. Main Results
3.1. IO-FTS Analysis
Consider the system (1) with , the system is described by
In this subsection, we concern with the IO-FTS analysis of positive switched nonlinear system (14). The following theorem gives some sufficient conditions of IO-FTS for system (14) via the ADT technique.
Theorem 1. Consider the system (14), for given constants , γ, , , and a vectors , if there exist positive vectors , , , , such that the following inequalities hold:where , , , , satisfies (20), then the system (14) is IO-FTS for switching signal with the ADT scheme:
Proof. Construct the multiple co-positive-type nonlinear Lyapunov functional for the system (14):the form of each can be given bywhere , , and , .
Along the trajectory of system (14), we haveConsider (4) and , can be obtained, then (25) is transformed intoNoting that , we haveAccording to (27), (26) can be rewritten asFrom (15)–(17) and (21), we getIntegrating both sides of (29) during the period for leads toFrom (20) and (24), we getDenote , as the switching instants over the interval . From (2), (30), and (31), we haveFrom (24), (32) and the zero initial condition , we haveFrom (3), (33) and (34), one hasAccording to (22) and (35), we can easily obtainBy (5) and (19), we getFrom (36) and (37), we obtainThus, the proof is completed.
3.2. IO-FTS Performance Analysis
In this section, we will consider the problem of IO-FTS for system (14).
Theorem 2. Consider the system (14), for given constants , , , η, and a vector , if there exist positive vectors , , , , such that (16), (17), (19), (20), and the following inequalities hold,where , , , , satisfies (20), then the system (14) is IO-FTS for any switching signal with the ADT scheme
Proof. (39) can be easily derived from (15). Let in Theorem 1, from (16), (17), (19), (20), (39), (40), and (41), we can obtain that the system (14) is IO-FTS with respect to (δ, , ε, ).
Choosing the multiple co-positive-type Lyapunov functional (24). Similar to the proof process of Theorem 1, from (16), (17), and (39), we have
Denoting and integrating both sides of (43) from to t for , it gives rise to
Similar to the proof process of (32), for any , we can obtain
Considering the zero initial condition, we have
Multiplying both sides of (46) by leads to
Noting that , , and , we obtain that ; that is, . Then (47) can be turned into
Let , then multiplying both sides of (48) by yields
Denoting , (49) can be rewritten as
Thus, the proof is completed.
3.3. IO-FTS Controller Design
In this subsection, we consider the design of IO-FTS controller. Considering system (1), under the controller , the corresponding closed-loop system is given by
By Lemma 2, to guarantee the positivity of system (51), should be Metzler matrices.
Theorem 3. Consider the system (51), for given constants , , , and a vector , if there exist positive vectors , , , , , such that (16), (17), (19), (20), (40)–(42), and the following inequalities hold:where , , , and ; then, under the ADT scheme (42), the resulting closed-loop system (51) is IO-FTS with respect to (δ, , ε, ), where satisfies (20).
Proof. Replacing in (39) with , then letting , similar to Theorem 2, we can get (53); then, the resulting closed-loop system (51) is IO-FTS with ADT scheme (42).
The proof is completed.
Next, an algorithm is presented to obtain the feedback gain matrices , .
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Remark 6. There is not a systemic method to adjust these parameters; the selection is generally by experience.
4. Numerical Examples
Example 1. Consider system (1) with the parameters as follows:where , ; then, we get , , . Choosing , , , , , , , , , . Solving the inequalities in Theorem 3 by linear programming, we getBy , we obtainIt is easy to verify that are Metzler matrices. Then, from (42), we can obtain . Choosing . The simulation results are shown in Figures 1–3, where the . Figure 1 shows the state trajectories of the closed-loop system (1). The switching signal is depicted in Figure 2. Figure 3 plots the evolution of of system (1). From the simulation results, we know , which implies the effectiveness of our proposed method.
Example 2. A practical price dynamics model is considered as an uncertain positive switched system with time-varying delays in [3]. In an economic system, affected by the national macroeconomic regulation and control policy, the relation between supply and demand always presents nonlinear behavior. So, it is more suitable to describe the price dynamic model by positive switched nonlinear systems [18]. Consider the parameters as follows:where , ; then, we get , , . Choosing , , , , , , , , . Solving the inequalities in Theorem 3 by linear programming, we getBy , we obtainIt is easy to verify that are Metzler matrices. Then, from (42), we can obtain . Choosing . The simulation results are shown in Figures 4–6, where the . Figure 4 shows the state trajectories of the closed-loop system (1). The switching signal is depicted in Figure 5. Figure 6 plots the evolution of of system (1). From the simulation results, we know , which implies the effectiveness of our proposed method.
5. Conclusions
This paper has investigated the problem of IO-FTS for positive switched nonlinear systems with time-varying and distributed delays. Firstly, the concept of IO-FTS of positive switched nonlinear systems is firstly proposed. Then, by constructing nonlinear Lyapunov–Krasovskii functions, a state feedback controller is designed. Based on the ADT approach, some sufficient conditions are obtained to guarantee that the closed-loop system is IO-FTS.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful for the supports of the National Natural Science Foundation of China under grants U1404610 and U1704157 and Young Key Teachers Plan of Henan Province (2016GGJS-056).