Event-Triggered Asynchronous Filter of Positive Switched Systems with State Saturation

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 l 1 -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.


Introduction
ere 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]. ey have extensive applications in the elds 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 ying [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 timedependent linear copositive Lyapunov function was constructed and a novel stability criterion of positive switched system was proposed in [9]. In [10], a matrix decompositionbased 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 nite-time H ∞ control of a class of Markovian jump delayed systems with input saturation. e literature [12] focused on the constraint control of positive Markovian jump systems with actuator saturation. e 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][15][16]. e 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. ese states can be modeled via saturation. e state saturation will not only affect the stability of the systems but also cause other fault phenomena. erefore, 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][21][22][23][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. ere 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][27][28][29]. Event-triggered strategy has many advantages such as less computation burden, less sampling time, and lower energy requirement. is strategy has been applied to multiagent systems, networked control systems, etc. [30,31]. e tool of linear matrix inequalities was employed for the event-triggered control of linear systems subject to actuator saturation [32]. In [33], an eventtriggered 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. erefore, 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 H − /H ∞ filter approach. In [39], an asynchronous l 2 -l ∞ 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 L 1 -gain asynchronous filter of positive Markovian systems was designed. ese 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. e literature [41][42][43] proposed some design approaches to the filter of positive systems. ey only considered the filter design for one class of positive systems. ese presented approaches cannot be developed for the filter issues of other hybrid systems. erefore, it is necessary to construct a unified filter design for positive systems. ese points motivate us to present the work.
is 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 eventtriggering 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. e 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. e 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 R n (R n + ) and R n×n as the sets of n-dimensional real vectors (or non-negative) and n × n-dimensional real matrices, respectively. e symbols N and N + represent the sets of nonnegative and positive integers, respectively. For a vector x ∈ R n , its 1-norm is defined by ‖x‖ 1 � n i�1 |x i |. e L 1 and ℓ 1 norms of a vector x are defined as ‖x‖ L 1 � ∞ 0 ‖x(t)‖ 1 dt and ‖x‖ ℓ 1 � ∞ t�0 ‖x(t)‖ 1 , respectively.
e L 1 and ℓ 1 spaces are denoted as ω(t): . A matrix is Metzler if all its off-diagonal elements are nonnegative. e symbol co · { } stands for the convex hull.

Problem Formulation
Consider a class of switched systems: where x(t) ∈ R n , y(t) ∈ R m , ω(t) ∈ R r + , and z(t) ∈ R s are the state, output, disturbance input belonging to an ℓ 1 space, and output to be estimated, respectively. e system matrices have appropriate dimensions. e symbol δ denotes the derivative operator in the continuous-time context (δx(t) � (d/dt)x(t), t ≥ 0) and the shift forward operator in the discrete-time context (δx(t) � x(t + 1), t ∈ N). e function σ(t) is the switching signal taking values in a finite set S � 1, 2, . . . , J { } and a switching sequence is given as in the discrete-time case. Assume that the timederivative of disturbance exists and is bounded. In order to estimate the output z(t), an asynchronous filter is designed as follows: is the switching signal of the filter, and Δ σ(t) represents the asynchronous time. e matrices A fσ′(t) , B fσ′(t) , E fσ′(t) , and F fσ′(t) 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 _ x(t) � Ax(t) + Bω(t), y(t) � Cx(t) + Dω(t), and z(t) � Ex(t) + Fω(t) is positive if and only if A is a Metzler matrix, and B≽0, C≽0, D≽0, E≽0, and F≽0. A discrete-time system is positive if and only if A≽0, B≽0, C≽0, D≽0, E≽0, and F≽0. Lemma 2 (see [1,2]). For a positive continuous-time system _ x(t) � Ax(t), the following conditions are equivalent: (i) e system matrix A is Hurwitz (ii) ere exists a vector v≻0 such that A ⊤ v≺0 Lemma 3 (see [1,2]). For a positive discrete-time system x(t + 1) � Ax(t), the following conditions are equivalent: Lemma 4 (see [1,2]). Given vectors ℘ ∈ R n and I ∈ R n , if ‖I‖ ∞ ≤ 1, then where D l is an n × n diagonal matrix with elements either 1 or 0 and in the discrete-time state space, respectively. Let the matrix H with H≽0 be a cone attract domain matrix. A symmetric polyhedron is defined as L(H) � x(t) ∈ R n : |H p x(t)| ≤ 1 in the continuous-time state space and L(H) � x(k) ∈ R n : |H p x(k)| ≤ 1 in the discrete-time state space, respectively, where H p is the p th row of the matrix H and p ∈ 1, 2, . . . , n.
Next, we introduce the event-triggering mechanism. Define the event-triggered error function: where y(t) is the output value of the event generator, y(t) � y(t l ′ ), l ∈ N, and t l ′ is the event-triggering time instance. An event-triggering condition is established based on 1-norm: where 0 < β < 1 is called event-triggering constant. Under the event-triggering condition, the asynchronous filter can be written as e filter equation (6) is constructed based on the eventtriggered mechanism, while the filter equation (2) is a timetriggered one. Replacing the output y(t) in the filter equation (2) by the output value of event generator y(t), the filter Mathematical Problems in Engineering equation (6) is obtained. e objective of this paper is to design the filter equation (6). Denote , and e(t) � z f (t) − z(t). en, we have Assume that x(t) ∈ L(H σ (t)). By Lemma 4, it derives that Definition 3 (see [11]). e system equation (8) is said to be L 1 (or ℓ 1 ) gain stable if the following two conditions hold: (i) e system equation (8) with ω(t) � 0 is asymptotically stable (ii) Under the zero initial condition, the relation holds for ω(t) ≠ 0, where c > 0 is the L 1 /ℓ 1 gain value, ϖ > 0, and ρ > 0.

Main Results
In this section, we first consider the filter design of the continuous-time system with ω(t) ≡ 0. en, the filter of the discrete-time system is proposed.

Continuous-Time Case.
Given a time interval [t l , t l+1 ), where the asynchronous time interval is [t l , t l + Δ l ) and the synchronous time interval is [t l + Δ l , t l+1 ). e q th original subsystem is active in t ∈ [t l , t l+1 ). e p th filter is active in t ∈ [t l , t l + Δ l ), and then the q th filter is active in t ∈ [t l + Δ l , t l+1 ), where t l (l � 0, 1, · · ·) is the switching time instants and Δ l is the time lag between the subsystem and the filter and Δ l < t l+1 − t l . When t ∈ [t l , t l + Δ l ), the error system can be written as where where 4

Mathematical Problems in Engineering
hold for l � 1, 2, . . . , 2 n , then under the event-triggered asynchronous filter equation (6) with and the switching law satisfies The filter error system equation (8) is positive and asymptotically stable, where Φ � I − β1 m×m , Ψ � I + β1 m×m , T − (t 0 , t) denotes the total time length of synchronous, and T + (t 0 , t) denotes the total time length of asynchronous of the switched systems. Moreover, all states starting from Proof. First, consider the positivity of the error system equation (8). For x(t 0 )≽0 and y(t 0 )≽0, it gives ‖e y (t 0 )‖ 1 ≤ β‖y(t 0 )‖ 1 by equation (5). en, we deduce that − β1 m×m y(t 0 ) ≺ e y (t 0 ) ≺ β1 m×m y(t 0 ). By equations (11) and (13), we have

Mathematical Problems in Engineering
For By equations (15a) and (15e), we have Using equation (16) gives It is easy to know that M q , A fp , and A fq are Metzler matrices. Due to E fp ≽0 and E fq ≽0, _ x(t 0 )≽0 by Lemma 1. Using recursive derivation gives _ x(t)≽0, that is to say, the error system equation (8) is positive. erefore, it holds that Next, consider the stability of the system equation (8). Choose piecewise multiple Lyapunov functions: en, Mathematical Problems in Engineering Using equations (15i) and (15j) and λ > 1, it yields By equations (15c), (15d), (15g), and (15h), it holds that us, Together with equations (15j) and (15k), it yields where ℵ � N σ (T, t 0 ). en, we have From equation (17a), we can get en, where ρ 1 and ρ 2 are the minimal and maximal elements of v (i) ∈ v (p,q) , v (q) , p ∈ S, q ∈ S . erefore, 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 Input/actuator saturation is frequently investigated since limited implementation ability of elements will lead to the saturation phenomenon. In [20][21][22][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 Mathematical Problems in Engineering 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. ese refer to state saturation, which is a new kind of saturation. eorem 1 proposes a filter design for positive switched systems in terms of linear programming. e presented design in equation (16) is different from the design approaches in [20][21][22][23].
Remark 2. Positive systems have distinct research approaches from general systems. Existing filter design approaches [34][35][36][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 eorem 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][42][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 eorem 1, the filter gains are designed as in equation (16) by following the matrix decompositionbased 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. e successful application in eorem 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, eorem 1 can be applied for the design in [41][42][43].
It is clear that condition equation (15a)-(15l) cannot be directly solved in terms of linear programming when the parameters λ, μ 1 , and μ 2 are unknown. How to choose these parameters such that equations (15a)-(15l) is feasible is key to the validity of eorem 1. Considering this point, we give Algorithm 1 to transform equations (15a)-(15l) into a linear programming form. In this algorithm, the range of μ 1 and μ 2 should be selected as large as possible to guarantee the feasibility of equations (15a)-(15l) and the parameter λ should be chosen close to 1 to obtain a lower ADT.
Next, we consider the filter design for the case ω(t) ≠ 0. e disturbance satisfies where χ is a given positive constant.

Mathematical Problems in Engineering
hold for given μ 1 > 0, μ 2 > 0, λ > 1, and any l � 1, 2, . . . , 2 n , then the filter error system equation (8) is positive and L 1 gain is stable under the event-triggered asynchronous filter equation (13) with equation (16), where Φ � I − β1 m×m , Ψ � I + β1 m×m , and the switching law satisfying where T − (t 0 , t) denotes the total time length of synchronous and T + (t 0 , t) denotes the total time length of asynchronous of the switched systems. Moreover, the system states starting from Proof. Using a similar method to eorem 1, we have By equations (40a), (40b), (40h), and (40i), it follows that

Mathematical Problems in Engineering
By equation (16), it is easy to get and F fq ΦD q − B q ≽0. us, M q , A fp , and A fq are Metzler matrices. Together with B q , E fp , E fq ≽0, and x(t 0 )≽0, it derives that the system equation (8) with ω(t) ≠ 0 is positive.

Multiplying both sides of the inequality with
It follows from Definition 2 and equation (41b) that Furthermore, Next, we present the invariance of the state. By equation . Using method of the eorem 1, we have that the states will be kept in the set ( ∪ p Λ(v (p) , 1 + cχ)) ⋃( ∪ p,q Λ(v (p,q) , 1 + cχ)) for any initial conditions starting from ∪ i (v (i) , 1 + cχ). In the following, the Zeno behavior is analyzed to avoid the continuous sampling and updating of the state under the Mathematical Problems in Engineering event-triggering mechanism. Under the event-triggering mechanism equation (5) . Without generality, assume that _ w(t) exists and is bounded. en, there exists a positive constant [ such that ‖ _ e y (t) ′ . is means that t > t l ′ . erefore, the Zeno behavior can be avoided.

Remark 4.
e condition equation (39) is introduced to ensure the bounded property of the state under the saturation restriction. From equation (39), one can find that the L 1 norm of the external disturbance w(t) is bounded. Assume that the L 1 norm is unbounded. From the last part of the proof in eorem 2, it is not hard to find that the bounded property of the state is invalid. In such a case, the saturation phenomenon cannot be handled. In future work, it is interesting to investigate the filter issue of positive systems with unbounded disturbance.
Remark 5. In [16], a filter was designed for continuous-time linear systems with sensor saturation. Four aspects should be pointed out: (i) an event-triggering mechanism is employed in this paper to design the filter of positive switched systems while the time-triggering mechanism was used in [16], (ii) the state saturation is handled in this paper, and the sensor saturation control approach in [16] cannot be developed for dealing with the state saturation problem, (iii) the filter gain design in this paper is different from the one in [16], and (iv) positive systems have distinct research approaches from general systems in [16]. Indeed, it is also clear that linear copositive Lyapunov functions and linear programming are employed in this paper, while Lacerda [16] chose a quadratic Lyapunov function and other computation method.

Remark 6.
e time-triggered filter strategy was proposed for positive systems in [41][42][43]. In most cases, it is hard to reach the continuous sample owing to limited ability of elements and high sample cost. e event-triggered filter is more practicable with respect to the time-triggered filter.
us, eorems 1 and 2 design a class of event-triggered filter for positive switched systems. Owing to the positivity requirement, the event-triggered filter of positive systems is more complex than general systems. For general systems, the error term can be easily transformed into the state term. However, such a strategy fails for positive systems. us, an interval estimation approach is introduced to transform the original system into an interval system. Finally, the positivity of the original system can be achieved by guaranteeing the positivity of the lower bound of the interval system.
Next, we present the invariance of the state. By equation (98), we have V(∞) ≤ 1 + c M k 0 ‖ω(s)‖ 1 ≤ 1 + cχ. For ω(k) � 0, Λ(v (i) , 1)⊆L(H p ). Using a similar method to eorem 3, we have that the states will be kept in the set ( ∪ p Λ(v (p) , 1 + cχ)) ⋃( ∪ p,q Λ(v (p,q) , 1 + cχ)) for any initial conditions starting from ∪ is paper proposes a unified event-triggered asynchronous filter framework for positive switched with state saturation. It should be pointed out that there are still many open issues to positive systems. On one hand, it is significant to present a low computation burden approach for dealing with the state saturation of hybrid positive systems. is paper employs the convex polytopic approach for dealing with the state saturation term. us, the saturation term is transformed into polytopic, which is dependent on 2 n vertex matrices. When the dimension of the systems is high, the corresponding computation is complex. How to present a simpler approach for reducing the computation burden is interesting. On the other hand, some novel triggering conditions can be introduced to improve the triggered mechanism in this paper. Up to now, the static event-triggering mechanism is usually used for positive systems. However, it has been verified that a dynamic event-triggered mechanism is superior to the static one. It is interesting to investigate the      dynamic event-triggered strategy of positive systems in the future work.
Remark 8. In [25], the event-triggered filter of positive systems with state saturation was investigated. is paper considers an asynchronous even-triggered filter of positive switched systems. Multiple copositive Lyapunov functions are constructed in this paper, and the ℓ 1 -gain stability for positive switched systems is achieved. Compared with positive systems, the ℓ 1 -gain stability of positive switched systems is more complex. en, an asynchronous filter is established in terms of linear programming, which is more practical than the synchronous filter in [25]. In brief, the filter in [25] is a special case of the filter designed in this paper. Additionally, a continuous-time asynchronous filter framework is also established in this paper.

Illustrative Examples
In [45,46], a state space-based water system was constructed. It should be pointed out that the water flow and the capacity of pumps is non-negative. erefore, it is more suitable to model water systems via positive systems theory. Moreover, the volume of water in the water system is limited and subject to a certain value. is is a typical saturation phenomenon. Figure 1 shows the diagram of a water supply network. e pumps are used to describe the switched rule. e water system in Figure 1 is modeled by the system equation (1), where x(t) (or, x(k)) denotes the water volume in the tank at time t (or, the k th sample instant), ω(t) (or, ω(k)) represents the wasted water from the pumps and valves, y(t) (or, y(k)) is the measured value by the sensor, and z(t) (or, z(k)) is the output to be estimated. Figure 2 shows the states of the plant and the filter, where V, V k , q in , and q out represent maximum volume, the volume at the k th sample instant, inflow quantities, and outflow quantities in the tank, respectively. Two examples are given to illustrate the effectiveness of the proposed design.

Example 2. Consider system equation (8) with
Given constants μ 1 � 0.7, μ 2 � 1.9, λ � 1.1, and c � 0.2, the event-triggering threshold is β � 0.1, and the external disturbance is x 2 (k) x 3 (k) x 1 (k) e asynchronous switch signal and the error signal e(k) of z(k) and z f (k) are given in Figures 8 and 9, respectively. e external disturbance input signal is given in Figure 10. Figure 11 shows the event-triggering release interval.

Conclusions
is paper has proposed an event-triggered asynchronous filter for positive switched systems with state saturation. A polytopic approach is introduced to deal with the state saturation term. An interval estimation approach is presented to transform the original system into an interval system. By using a matrix decomposition technique, the filter gain matrices are designed in terms of linear programming. It is shown that copositive Lyapunov functions and linear programming are more effective for solving the corresponding issues of positive systems. e obtained design is also developed for the discrete-time case.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.