Abstract
This paper investigates the event-triggered asynchronous filter of positive switched systems with state saturation using linear programming and multiple Lyapunov functions. First, a filter is constructed for continuous-time positive switched systems. Under the asynchronous switching law, an error system is proposed with respect to positive switched systems and their filters, where the state saturation term is described in a polytopic form by virtue of the saturation property. A novel event-triggering condition is addressed based on a 1-norm inequality. Under the event-triggering condition, the error system is transformed into an interval system with lower and upper bounds. By using multiple Lyapunov functions and linear programming, the positivity and stability of the error system are achieved by considering the corresponding properties of the lower and upper bounds, respectively. Then, the event-triggered l1-gain filter and nonfragile filter are also proposed for the systems with disturbances. Moreover, the presented filter framework is extended to the discrete-time case. Finally, two examples are given to verify the effectiveness of the proposed filters.
1. Introduction
There exists a class of positive switched systems whose states and outputs are always nonnegative, for which a switching rule is designed to specify the switching between subsystems. Positive switched systems have attracted much attention over past decades [1, 2]. They have extensive applications in the fields of biology systems and pharmacokinetics [3, 4]. In practice, there have been many systems that can be modeled as positive switched systems such as formation flying [5] and network employing TCP [6, 7]. In the literature [8], the stability of positive switched linear systems with average dwell time (ADT) was studied using multiple linear copositive Lyapunov functions (MLCLFs). A reverse time-dependent linear copositive Lyapunov function was constructed and a novel stability criterion of positive switched system was proposed in [9]. In [10], a matrix decomposition-based control approach was introduced for positive systems. It should be pointed out that linear Lyapunov functions are powerful for solving the control problems of positive switched systems.
In practical applications, saturation is a universal phenomenon owing to various restrictions of elements and unexpected environment factors. Zhao et al. [11] investigated the finite-time control of a class of Markovian jump delayed systems with input saturation. The literature [12] focused on the constraint control of positive Markovian jump systems with actuator saturation. The stabilization of switched linear systems subject to actuator saturation was solved in [13]. More information about input/actuator and sensor saturation can be seen in [14–16]. The literature mentioned above focus on the input/actuator saturation. However, as far as the authors’ knowledge, there are few results on state saturation. Indeed, the states of most of the practical systems are subject to constraints due to physical limitations. For example, the limited water storage capacity of pipes will lead to the saturation of the state in water systems, the limited bandwidth in network communication systems will bring the transmission restriction of the data packages, and the road bearing capacity in transportation systems has an upper limit. These states can be modeled via saturation. The state saturation will not only affect the stability of the systems but also cause other fault phenomena. Therefore, it is significant to explore the filter issue of positive systems with state saturation. Derong Liu and Michel [17] analyzed the stability of systems with partial state saturation nonlinearities using the Lyapunov function approach, and Kolev et al. [18] addressed the state saturation nonlinearities for discrete-time neural networks. Ji et al. [19] were concerned with the stability analysis of discrete-time linear systems with state saturation using a saturation-dependent Lyapunov functional. For positive systems, they also have some significant results on saturation [20–24]. Regrettably, these mentioned literature studies are concerned with input/actuator saturation, and few efforts are devoted to the state saturation issue of positive switched systems. In [25], the filter design of positive systems with state saturation was proposed using linear copositive Lyapunov function and linear programming. However, the filter design problem of positive systems with state saturation has not been completely solved. There are still many open issues such as the filter of hybrid positive systems with state saturation and the event-triggered filter of positive systems with state saturation.
In recent years, event-triggered strategy has attracted much attention owing to its advantages in relaxing on the traditional time-triggered strategy and guaranteeing the safe running of systems [26–29]. Event-triggered strategy has many advantages such as less computation burden, less sampling time, and lower energy requirement. This strategy has been applied to multiagent systems, networked control systems, etc. [30, 31]. The tool of linear matrix inequalities was employed for the event-triggered control of linear systems subject to actuator saturation [32]. In [33], an event-triggered control framework was introduced for nonlinear systems. An event-triggered filter for networked systems with signal transmission delay was designed by utilizing the Lyapunov–Krasovskii functional and linear matrix inequalities in [34]. In [35], the event-triggered fuzzy filter was applied for nonlinear time-varying systems. More results on the event-triggered filter can refer to [36, 37] and references therein. Up to now, the event-triggered filter of positive switched systems with state saturation is still open. Moreover, the synchronization switching is hard to be realized since it needs to take time to detect which subsystem is active. Therefore, the asynchronous switching is more important and practical than the synchronous switching. In [38], the problem of fault detection filter for continuous-time switched control systems under asynchronous switching was investigated, and the solution was provided in the form of a mixed filter approach. In [39], an asynchronous - filter for stochastic Markovian jump systems with randomly occurred sensor nonlinearities was proposed based on linear matrix inequalities. By applying the average dwell time technique and the piecewise Lyapunov–Krasovskii functional technique, sufficient conditions were obtained in [40] for designing an asynchronous finite-time filter of switched networked systems. In [41], positive -gain asynchronous filter of positive Markovian systems was designed. These existing results inspire us to investigate the event-triggered asynchronous filter of positive switched systems with state saturation. An asynchronous filter of positive switched systems with overlapped detection delay was designed in [42]. In [43], a class of clock-dependent Lyapunov function was constructed for positive switched systems to obtain less conservative asynchronous filter design approach. It is necessary to point out several points. First, the event-triggered strategy is still open to positive systems. In the event-triggered control of positive systems, it is not easy to transform the error term into the state term. Meanwhile, how to achieve the positivity of the filter error system is also complex. Second, the introduction of state saturation increases the complexity of the filter design. Positive systems with state saturation contain two classes of constraints: saturation and positivity. Each of them is difficult to be handled. Finally, the filter gain design is open to positive systems. Up to now, there are no tractable filter design framework on positive systems. The literature [41–43] proposed some design approaches to the filter of positive systems. They only considered the filter design for one class of positive systems. These presented approaches cannot be developed for the filter issues of other hybrid systems. Therefore, it is necessary to construct a unified filter design for positive systems. These points motivate us to present the work.
This paper investigates the event-triggered asynchronous filter of the positive switched system. First, a simple linear event-triggering condition is introduced for the systems. By using multiple copositive Lyapunov functions and the event-triggering condition, the error system between the original systems and the corresponding filters is transformed into an interval system. A polytope is used to deal with the state saturation term. By virtue of the matrix decomposition approach, the filter matrices are designed. An asynchronous switching law is also proposed. The contributions of this paper are as follows: (i) a filter is constructed for positive switched systems with state saturation, (ii) an event-triggered asynchronous switching design is given, and (iii) a linear programming-based approach is presented for designing the filter matrices. The rest of the paper is organized as follows: Section 2 gives the problem formulation and preliminaries, Section 3 addresses main results, Section 4 provides two illustrative examples, and Section 5 concludes the paper.
Notations: denote and as the sets of -dimensional real vectors (or non-negative) and -dimensional real matrices, respectively. The symbols and represent the sets of nonnegative and positive integers, respectively. For a vector , its 1-norm is defined by . The and norms of a vector are defined as and , respectively. The and spaces are denoted as and . Let and , and be an matrix whose elements are all one. Matrix is the identity matrix with appropriate dimension. For a vector , is the th component of and means that . Given a matrix , is the th row th column element of A. means that . Similarly, means that . A matrix is Metzler if all its off-diagonal elements are nonnegative. The symbol stands for the convex hull.
2. Problem Formulation
Consider a class of switched systems:where , , , and are the state, output, disturbance input belonging to an space, and output to be estimated, respectively. The system matrices have appropriate dimensions. The symbol denotes the derivative operator in the continuous-time context and the shift forward operator in the discrete-time context . The function is the switching signal taking values in a finite set and a switching sequence is given as . The function is the saturation function with , where . Assume that is Metzler matrix and , , , , in the continuous-time case and that in the discrete-time case. Assume that the time-derivative of disturbance exists and is bounded. In order to estimate the output , an asynchronous filter is designed as follows:where is the filter state, is the estimate output, is the switching signal of the filter, and represents the asynchronous time. The matrices , , , and are to be determined.
Next, we introduce some definitions and lemmas of positive switched systems to facilitate later development.
Definition 1. (see [1, 2]). A system is said to be positive if its states and outputs are non-negative for any non-negative initial conditions and any non-negative inputs.
Lemma 1 (see [1, 2]). A continuous-time system is positive if and only if is a Metzler matrix, and , , , , and . A discrete-time system is positive if and only if , , , , , and .
Lemma 2 (see [1, 2]). For a positive continuous-time system , the following conditions are equivalent:(i)The system matrix is Hurwitz(ii)There exists a vector such that
Lemma 3 (see [1, 2]). For a positive discrete-time system , the following conditions are equivalent:(i)The system matrix is Schur(ii)There exists a vector such that
Lemma 4 (see [1, 2]). Given vectors and , if , thenwhere is an diagonal matrix with elements either 1 or 0 and , . Then, .
A cone domain is defined as in the continuous-time state space and in the discrete-time state space, respectively. Let the matrix with be a cone attract domain matrix. A symmetric polyhedron is defined as in the continuous-time state space and in the discrete-time state space, respectively, where is the th row of the matrix and .
Definition 2. (see [44]). For a switching signal and , let denote the switching number of . If the condition holds, then is called average dwell time of and is the chatter bound.
Next, we introduce the event-triggering mechanism. Define the event-triggered error function:where is the output value of the event generator, , and is the event-triggering time instance. An event-triggering condition is established based on 1-norm:where is called event-triggering constant. Under the event-triggering condition, the asynchronous filter can be written asThe filter equation (6) is constructed based on the event-triggered mechanism, while the filter equation (2) is a time-triggered one. Replacing the output in the filter equation (2) by the output value of event generator , the filter equation (6) is obtained. The objective of this paper is to design the filter equation (6).
Denote , , and . Then, we haveAssume that . By Lemma 4, it derives thatwherewhere with and .
Definition 3. (see [11]). The system equation (8) is said to be (or ) gain stable if the following two conditions hold:(i)The system equation (8) with is asymptotically stable(ii)Under the zero initial condition, the relation holds for , where is the gain value, , and .
3. Main Results
In this section, we first consider the filter design of the continuous-time system with . Then, the filter of the discrete-time system is proposed.
3.1. Continuous-Time Case
Given a time interval , where the asynchronous time interval is and the synchronous time interval is . The th original subsystem is active in . The th filter is active in , and then the th filter is active in , where is the switching time instants and is the time lag between the subsystem and the filter and . When , the error system can be written aswhere
When , it follows thatwhere
Theorem 1. If there exist constants ; vectors , , , , , , , and ; and vectors , , , and such thathold for , then under the event-triggered asynchronous filter equation (6) withand the switching law satisfiesThe filter error system equation (8) is positive and asymptotically stable, where , , denotes the total time length of synchronous, and denotes the total time length of asynchronous of the switched systems. Moreover, all states starting from will remain inside the cone set , where for .
Proof. First, consider the positivity of the error system equation (8). For and , it gives by equation (5). Then, we deduce that . By equations (11) and (13), we havefor , whereFor , whereBy equations (15a) and (15e), we haveUsing equation (16) gives . Thus, and , which implies that . Similarly, we can also obtain . By equations (15b) and (15f), we get that and . It is easy to know that , , and are Metzler matrices. Due to and , by Lemma 1. Using recursive derivation gives , that is to say, the error system equation (8) is positive. Therefore, it holds thatNext, consider the stability of the system equation (8). Choose piecewise multiple Lyapunov functions:where and . By equation (23), we havefor , wherefor , whereThen,Moreover, we haveUsing equations (15i) and (15j) and , it yieldsBy equations (15c), (15d), (15g), and (15h), it holds that .
Thus,Together with equations (15j) and (15k), it yieldsFor , we getwhere . Then, we haveFrom equation (17a), we can getFurthermore, it follows that . By Definition 2 and equation (17b), it givesThen,where and are the minimal and maximal elements of . Therefore, the filter error system equation (8) is stable with equations (17a) and (17b).
Finally, we provide the invariance of the state. Given any initial conditions satisfying , that is, . By equation (38), . So, . Given , we have , where is the th row of . By , , and equation (15l), it follows that , which leads to . Then, .
Remark 1. Input/actuator saturation is frequently investigated since limited implementation ability of elements will lead to the saturation phenomenon. In [20–23], the input saturation of positive systems had also been explored, where copositive Lyapunov functions and linear programming were employed for coping with the control synthesis of positive systems. In practice, many quantities are subject to constraints such as the population of animals in a species, the volume of water storage, and the number of vehicles accessing to a circle road. These refer to state saturation, which is a new kind of saturation. Theorem 1 proposes a filter design for positive switched systems in terms of linear programming. The presented design in equation (16) is different from the design approaches in [20–23].
Remark 2. Positive systems have distinct research approaches from general systems. Existing filter design approaches [34–37] cannot be developed for positive systems. A direct development will bring some conservatism in describing the positivity condition, the computation, and so on. In [10], it was verified that a matrix decomposition approach is more suitable for dealing with the synthesis of positive systems. In Theorem 1, the filter gains are designed as equation (16) by following the approach in [10]. Under equation (16), the corresponding positivity and stability conditions can be solved via linear programming, which is more powerful for positive systems.
Remark 3. In [41–43], the asynchronous filter was designed for positive Markovian jump systems, positive switched systems, and positive systems, respectively. For different kind of positive systems, the filter design approaches are different. A question is whether a unified filter framework can be constructed. Such a framework is more significant for positive systems. In Theorem 1, the filter gains are designed as in equation (16) by following the matrix decomposition-based design in [10]. In [10], it had been pointed out that the matrix decomposition-based design approach can be easily developed for other syntheses of positive systems. The successful application in Theorem 1 implies that the filter framework is a unified one and can be applied for the related filter issues of positive systems. Based on this point, Theorem 1 can be applied for the design in [41–43].
It is clear that condition equation (15a)–(15l) cannot be directly solved in terms of linear programming when the parameters , and are unknown. How to choose these parameters such that equations (15a)–(15l) is feasible is key to the validity of Theorem 1. Considering this point, we give Algorithm 1 to transform equations (15a)–(15l) into a linear programming form. In this algorithm, the range of and should be selected as large as possible to guarantee the feasibility of equations (15a)–(15l) and the parameter should be chosen close to 1 to obtain a lower ADT.
Theorem 2. If there exist constants ; vectors , , , , , , , and ; and vectors , , , and such thathold for given , and any , then the filter error system equation (8) is positive and gain is stable under the event-triggered asynchronous filter equation (13) with equation (16), where , , and the switching law satisfyingwhere denotes the total time length of synchronous and denotes the total time length of asynchronous of the switched systems. Moreover, the system states starting from will remain inside the set .
Proof. Using a similar method to Theorem 1, we have