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

Authors: Linmin Li; Peng Lin; Junfeng Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## 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-triggeredl1-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.


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## Body

## 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 H∞ 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 H−/H∞ filter approach. In [39], an asynchronous l2-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 L1-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: Section2 gives the problem formulation and preliminaries, Section 3 addresses main results, Section 4 provides two illustrative examples, and Section 5 concludes the paper.Notations: denoteℜnℜ+n and ℜn×n as the sets of n-dimensional real vectors (or non-negative) and n×n-dimensional real matrices, respectively. The symbols ℕ and ℕ+ represent the sets of nonnegative and positive integers, respectively. For a vector x∈ℜn, its 1-norm is defined by x1=∑i=1nxi. The L1 and ℓ1 norms of a vector x are defined as xL1=∫0∞xt1dt and xℓ1=∑t=0∞xt1, respectively. The L1 and ℓ1 spaces are denoted as ωt:ωtL1<∞ and ωk:ωkℓ1<∞. Let 1n=1,…,1︸n⊤ and 1ni=0,…,0︸i−1,1,0,…,0︸n−i⊤, and 1n×n be an n×n matrix whose elements are all one. Matrix I is the identity matrix with appropriate dimension. For a vector v∈ℜn, vi is the i th component of v and v≻0±0 means that vi>0vi≥0. Given a matrix A=aij∈ℜn×m, aij is the i th row j th column element of A. A≻0≻0 means that aij>0aij≥0. Similarly, A≽BA≺¯B means that aij≥bijaij≤bij. A matrix is Metzler if all its off-diagonal elements are nonnegative. The symbol co⋅ stands for the convex hull.


## 2. Problem Formulation

Consider a class of switched systems:(1)δxt=satAσtxt+Bσtωt,yt=Cσtxt+Dσtωt,zt=Eσtxt+Fσtωt,where xt∈ℜn, yt∈ℜm, ωt∈ℜ+r, and zt∈ℜs are the state, output, disturbance input belonging to an ℓ1 space, and output to be estimated, respectively. The system matrices have appropriate dimensions. The symbol δ denotes the derivative operator in the continuous-time context δxt=d/dtxt,t≥0 and the shift forward operator in the discrete-time context δxt=xt+1,t∈ℕ. The function σt is the switching signal taking values in a finite set S=1,2,…,J and a switching sequence is given as 0≤t0≤t1≤⋯. The function sat⋅:ℜm⟶ℜm is the saturation function with satu=satu1,satu2,…,satum⊤, where satuϱ=signuϱminuϱ,1,ϱ=1,2,…,m. Assume that Aσt is Metzler matrix and Bσt≥0, Cσt≽0, Dσt≽0, Eσt≽0, Fσt≽0 in the continuous-time case and that Aσt≽0,Bσt≽0,Cσt≽0,Dσt≽0,Eσt≽0,Fσt≽0 in the discrete-time case. Assume that the time-derivative of disturbance exists and is bounded. In order to estimate the output zt, an asynchronous filter is designed as follows:(2)δxft=Afσ′txft+Bfσ′tyt,zft=Efσ′txft+Ffσ′tyt,where xft is the filter state, zft is the estimate output, σ′t=σt−Δσt is the switching signal of the filter, and Δσt represents the asynchronous time. The matrices Afσ′t, Bfσ′t, Efσ′t, and Ffσ′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 systemx˙t=Axt+Bωt,yt=Cxt+Dωt,andzt=Ext+Fωt 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 xt+1=Axt+Bωt,yt=Cxt+Dωt,zt=Ext+Fωt 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 systemx˙t=Axt, the following conditions are equivalent:(i)
The system matrixA is Hurwitz(ii)
There exists a vectorv≻0 such that A⊤v≺0Lemma 3 (see [1, 2]).
For a positive discrete-time systemxt+1=Axt, the following conditions are equivalent:(i)
The system matrixA is Schur(ii)
There exists a vectorv≻0 such that A−I⊤v≺0Lemma 4 (see [1, 2]).
Given vectors℘∈ℜn and ℑ∈ℜn, if ℑ∞≤1, then(3)sat℘=coD1℘+D1−ℑ,D2℘+D2−ℑ,…,D2n℘+D2n−ℑ,where Dl is an n×n diagonal matrix with elements either 1 or 0 and Dl=I−Dl−, l=1,2,3,…,2n. Then, sat℘=∑l=12nℏlDl℘+Dl−ℑ.
A cone domainΩv,1 is defined as Ωv,1=xt∈ℜ+n:x⊤tv≤1 in the continuous-time state space and Ωv,1=xk∈ℜ+n:x⊤kv≤1 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 LH=xt∈Rn:Hpxt≤1 in the continuous-time state space and LH=xk∈Rn:Hpxk≤1 in the discrete-time state space, respectively, where Hp is the pth row of the matrix H and p∈1,2,…,n.Definition 2.
(see [44]). For a switching signal σt and 0≤t1≤t2, let Nσt,t0 denote the switching number of σt. If the condition Nσt1,t2≤N0+t2−t1/τa holds, then τa is called average dwell time of σt and N0≥0 is the chatter bound.
Next, we introduce the event-triggering mechanism. Define the event-triggered error function:(4)eyt=y^t−yt,t∈tl′,tl+1′,where y^t is the output value of the event generator, y^t=ytl′,l∈ℕ, and tl′ is the event-triggering time instance. An event-triggering condition is established based on 1-norm:(5)eyt1>βyt1,where 0<β<1 is called event-triggering constant. Under the event-triggering condition, the asynchronous filter can be written as(6)δxft=Afσ′txft+Bfσ′ty^t,zft=Efσ′txft+Ffσ′ty^t.
The filter equation (6) is constructed based on the event-triggered mechanism, while the filter equation (2) is a time-triggered one. Replacing the output yt in the filter equation (2) by the output value of event generator y^t, the filter equation (6) is obtained. The objective of this paper is to design the filter equation (6).
Denotex˜t=x⊤txe⊤t⊤, xet=xft−xt, and et=zft−zt. Then, we have(7)δx˜t=satAσtxt+BσtωtAfσ′txft+Bfσ′ty^t−satAσtxt−Bσtωt,et=Efσ′txft+Ffσ′ty^t−Eσtxt−Fσtωt.
Assume thatxt∈LHσt. By Lemma 4, it derives that(8)δx˜t=A˜σ′tσtx˜t+B˜σ′tσtωt+C˜σ′teyt,et=E˜σ′tσtx˜t+F˜σ′tσtωt+Ffσ′teyt,where(9)A˜σ′tσt=Mσt0Afσ′t+Bfσ′tCσt−MσtAfσ′t,B˜σ′tσt=BσtBfσ′tDσt−Bσt,C˜σ′t=0Bfσ′t,E˜σ′tσt=Efσ′t+Ffσ′tCσt−EσtEfσ′t,F˜σ′tσt=Ffσ′tDσt−Fσt,where Mσt=∑l=12nℏlDlAσt+Dl−Hσt with ∑l=12nℏl=1 and 0≤ℏl≤1.Definition 3.
(see [11]). The system equation (8) is said to be L1 (or ℓ1) gain stable if the following two conditions hold:(i)
The system equation (8) with ωt=0 is asymptotically stable(ii)
Under the zero initial condition, the relation(10a)ϖ∫0∞e−ρtet1dt≤γ∫0∞ωt1dtin the continuous−time case,(10b)ϖ∑0∞e−ρkek1≤γ∑0∞ωk1in the discrete−time case,holds forωt≠0, where γ>0 is the L1/ℓ1 gain value, ϖ>0, and ρ>0.


## 3. Main Results

In this section, we first consider the filter design of the continuous-time system withωt≡0. Then, the filter of the discrete-time system is proposed.


### 3.1. Continuous-Time Case

Given a time intervaltl,tl+1, where the asynchronous time interval is tl,tl+Δl and the synchronous time interval is tl+Δl,tl+1. The q th original subsystem is active in t∈tl,tl+1. The p th filter is active in t∈tl,tl+Δl, and then the q th filter is active in t∈tl+Δl,tl+1, where tll=0,1,⋯ is the switching time instants and Δl is the time lag between the subsystem and the filter and Δl<tl+1−tl. When t∈tl,tl+Δl, the error system can be written as(11)x˜˙t=A˜pqx˜t+B˜pqωt+C˜peyt,et=E˜pqx˜t+F˜pqωt+Ffpeyt,where(12)A˜pq=Mq0Afp+BfpCq−MqAfp,B˜pq=BqBfpDq−Bq,C˜p=0Bfp,E˜pq=Efp+FfpCq−EqEfp,F˜pq=FfpDq−Fq.Whent∈tl+Δl,tl+1, it follows that(13)x˜˙t=A˜qx˜t+B˜qωt+C˜qeyt,et=E˜qx˜t+F˜qωt+Ffqeyt,where(14)A˜q=Mq0Afq+BfqCq−MqAfq,B˜q=BqBfqDq−Bq,C˜q=0Bfq,E˜q=Efq+FfqCq−EqEfq,F˜q=FfqDq−Fq.Theorem 1.
If there exist constantsμ1>0;μ2>0;α>0;ε>0;λ>1; ℜn vectors gp,q≻0, hp,q≻0, gq≻0, hq≻0, θpı≻0, θp≻0, ξpı≻0, and ξp≻0; and ℜm vectors ηpı≻0, ηp≻0, ςpı≻0, and ςp≻0 such that(15a)∑ı=1n1nıθpı⊤1n⊤hp−αI+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≽0,(15b)∑j=1s1sjξpjT+∑j=1s1pjjςpjTΦCq−Eq≽0,(15c)Mq⊤gp,q+λθp−λε1n+λCq⊤Ψ⊤ηp−λMq⊤hp−μ2gp,q≺0,(15d)λθp−λε1n−μ2hp,q≺0,(15e)∑ı=1n1nıθqı⊤1n⊤hq−αI1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≽0,(15f)∑j=1s1sjξpjT+∑j=1s1pjjςpjTΦCq−Eq≽0,(15g)Mq⊤gq+θq−ε1n+Cq⊤Ψ⊤ηq−Mq⊤hq+μ1gq≺0,(15h)θq−ε1n+μ1hq≺0,(15i)θpı≺θp,ηpı≺ηp,α≥nε,(15j)hp≺¯λhp,q,hp≺¯λhq,p,hp,q≺¯λhp,hq,p≺¯λhp,(15k)gp≺¯λgp,q,gp≺¯λgq,p,gp,q≺¯λgp,gq,p≺¯λgp,(15l)−gq≺¯Hp⊤≺¯gq,−hq≺¯Hp⊤≺¯hq,−gp,q≺¯Hp⊤≺¯gp,q,−hp,q≺¯Hp⊤≺¯hp,q.hold for l=1,2,…,2n, then under the event-triggered asynchronous filter equation (6) with(16)Afp=∑ı=1n1nıθpı⊤−αI1n⊤hp,Bfp=∑ı=1n1nıηpı⊤1n⊤hp,Efp=∑j=1s1sjξijT,Ffp=∑j=1s1sjςpjT,and the switching law satisfies(17a)T−t0,tT+t0,t≥μ2+μ1μ1−μ1∗,μ1∗∈0,μ1,(17b)τa≥τa∗=2lnλμ1∗.
The filter error system equation (8) is positive and asymptotically stable, where Φ=I−β1m×m, Ψ=I+β1m×m, T−t0,t denotes the total time length of synchronous, and T+t0,t denotes the total time length of asynchronous of the switched systems. Moreover, all states starting from xt0∈Λvi,1 will remain inside the cone set ∪iΛvi,1, where vi∈vp,q,vq for p,q∈S×S.Proof.
First, consider the positivity of the error system equation (8). For xt0≽0 and yt0≽0, it gives eyt01≤βyt01 by equation (5). Then, we deduce that −β1m×myt0≺¯eyt0≺¯β1m×myt0. By equations (11) and (13), we have(18)x˜˙t0≽A˜1pqx˜t0,et0≽E˜1pqx˜t0,for t∈tl,tl+Δl, where(19)A˜1pq=Mq0Afp+BfpΦCq−MqAfp,E˜1pq=Efp+FfpΦCq−EqEfp,(20)x˜˙t0≽A˜1qx˜t0,et0≽E˜1qx˜t0.
Fort∈tl+Δl,tl+1, where(21)A˜1q=Mq0Afq+BfqΦCq−MqAfq,E˜1q=Efq+FfqΦCq−EqEfq.
By equations (15a) and (15e), we have(22)∑ı=1n1nıθpı⊤−αI1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≽0,∑ı=1n1nıθqı⊤−αI1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≽0.
Using equation (16) gives Afp+BfpΦCq−Mql≽0. Thus, Afp+BfpΦCq−∑l=12nℏlMql≽0 and ∑l=12nℏlAfp+BfpΦCq−Mql≽0, which implies that Afp+BfpΦCq−Mq≽0. Similarly, we can also obtain Afq+BfqΦCq−Mq≽0. By equations (15b) and (15f), we get that Efp+FfpΦCq−Eq≽0 and Efq+FfqΦCq−Eq≽0. It is easy to know that Mq, Afp, and Afq are Metzler matrices. Due to Efp≽0 and Efq≽0, x˜˙t0≽0 by Lemma 1. Using recursive derivation gives x˜˙t≽0, that is to say, the error system equation (8) is positive. Therefore, it holds that(23)−β1m×myt≺¯eyt≺¯β1m×myt.
Next, consider the stability of the system equation (8). Choose piecewise multiple Lyapunov functions:(24)Vt=x˜⊤tvp,q,t∈tl,tl+Δl,x˜⊤tvq,t∈tl+Δl,tl+1,where vp,q=gp,qhp,q and vq=gqhq. By equation (23), we have(25)x˜˙t≺¯A˜2pqx˜t,et≺¯E˜2pqx˜t,for t∈tl,tl+Δl, where(26)A˜2pq=Mq0Afp+BfpΨCq−MqAfp,E˜2pq=Efp+FfpΨCq−EqEfp,(27)x˜˙t≺¯A˜2qx˜t,et≺¯E˜2qx˜t,for t∈tl+Δl,tl+1, where(28)A˜2q=Mq0Afq+BfqΨCq−MqAfq,E˜2q=Efq+FfqΨCq−EqEfq.
Then,(29)V˙t≤x˜t⊤A˜2pq⊤vp,q,t∈tl,tl+Δl,x˜t⊤A˜2q⊤vq,t∈tl+Δl,tl+1.
Moreover, we have(30)V˙t≤x˜⊤tMq⊤gp,q+Afp⊤+Cq⊤Ψ⊤Bfp⊤−Mq⊤hp,qAfp⊤hp,q,t∈tl,tl+Δl,x˜⊤tMq⊤gq+Afq⊤+Cq⊤Ψ⊤Bfq⊤−Mq⊤hqAfq⊤hq,t∈tl+Δl,tl+1.
Using equations (15i) and (15j) and λ>1, it yields(31)V˙t≤x˜⊤tMq⊤gp,q+λθp−λε1n+λCq⊤Ψ⊤ηp−λMq⊤hpλθp−ε1n,t∈tl,tl+Δl,x˜⊤tMq⊤gq+θq−ε1n+Cq⊤Ψ⊤ηq−Mq⊤hqθq−ε1n,t∈tl+Δl,tl+1.
By equations (15c), (15d), (15g), and (15h), it holds that V˙t≤μ2Vt,t∈tl,tl+Δl,−μ1Vt,t∈tl+Δl,tl+1.
Thus,(32)Vt≤eμ2t−tlVtl,t∈tl,tl+Δl,e−μ1t−tl−ΔlVtl+Δl,t∈tl+Δl,tl+1.
Together with equations (15j) and (15k), it yields(33)Vt≤λeμ2t−tlVtl−,t∈tl,tl+Δl,λe−μ1t−tl−ΔlVtl+Δl−,t∈tl+Δl,tl+1.
ForT∈tNσT,t0+ΔNσT,t0,tNσT,t0+1, we get(34)Vσtℵ+1T≤λe−μ1T−tℵ−ΔℵVσtℵ+Δℵtℵ+Δℵ≤λ2e−μ1T−tℵ−Δℵeμ2ΔℵVσtℵtℵ≤λ3e−μ1T−tℵ−1−Δℵ−Δℵ−1eμ2ΔℵVσtℵ−1+Δℵ−1tℵ−1+Δℵ−1≤⋯≤λ2ℵe−μ1T−t0−Δℵ−Δℵ−1−⋯−Δ1eμ2Δℵ+Δℵ−1+⋯+Δ1Vσt0t0,where ℵ=NσT,t0. Then, we have(35)Vσtℵ+1T≤λ2ℵe−μ1T−t0,Teμ2T+t0,TVσt0t0≤λ2ℵe−μ1T−t0eμ1+μ2T+t0,TVσt0t0≤e2N0lnλe2lnλ/τa−μ1T−t0eμ1+μ2T+t0,TVσt0t0.
From equation (17a), we can get(36)μ1+μ2T+t0,T≤μ1−μ1∗T−t0.
Furthermore, it follows thatVσtℵ+1T≤e2N0lnλe2lnλ/τa−μ∗T−t0Vσt0t0. By Definition 2 and equation (17b), it gives(37)VT≤e2N0lnλe−μ1∗−2lnλ/τaT−t0Vt0.
Then,(38)x˜T1≤ρ22N0lnλρ1e−μ1∗−2lnλ/τaT−t0x˜t01,where ρ1 and ρ2 are the minimal and maximal elements of vi∈vp,q,vq,p∈S,q∈S. 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 satisfyingx˜t0∈Λvi,1, that is, x˜⊤t0vi≤1. By equation (38), x˜⊤tvi≤x˜⊤t0vi≤1. So, x˜t∈Λvi,1. Given H˜p=HpHp, we have H˜plx˜t=x⊤tHpl⊤+xe⊤tHpl⊤≤x˜⊤tvi≤1, where H˜pl is the l th row of H˜p. By xt0≽0, xet0≽0, and equation (15l), it follows that Hplxt≤H˜plx˜t≤1, which leads to x˜t∈∪pΛvp,1∪∪p,qΛvp,q,1. Then, xt∈LHp.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 λ,μ1, and μ2 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 μ1 and μ2 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.
Algorithm 1: Searching parameters for Theorem 1.
Input: Ap,Bp, Cp,Dp, Ep, Fp, p∈S, λ1>1, ℏ>0, ν>0, μ¯>μ¯>0;Output: λ,μ1,μ2;DefineΘ=∅;Main:λ=λ1,μ2=μ¯;whileλ≤λ1+ν doifμ2≤μ¯thenμ1=μ¯;repeatif equations (15a)–(15l) is feasible thenSaveλ,μ1,μ2toΘ;endμ1←μ1+ℏuntilμ1>μ¯;μ2←μ2+ℏendλ←λ+ℏ;end
Next, we consider the filter design for the caseωt≠0. The disturbance satisfies(39)∫t0∞ω⊤sds≤χ,where χ is a given positive constant.Theorem 2.
If there exist constantsμ1>0;μ2>0;α>0;ε>0;λ>1; ℜn vectors gp,q≻0, hp,q≻0, gq≻0, hq≻0, θpı≻0, θp≻0, ξpı≻0, and ξp≻0; and ℜm vectors ηpı≻0, ηp≻0, ςpı≻0, and ςp≻0 such that(40a)∑ı=1n1nıθpı⊤−αI+∑ı=1n1nıηpı⊤ΦCq−1n⊤hpMql≽0,(40b)∑i=1n1niηΦDq−1nThpBq≽0,(40c)∑j=1s1sjξpiT+∑j=1s1sjςpjTΦCq−Eq≽0,(40d)∑j=1s1sjςpj⊤ΦDq−Bq≽0,(40e)Mq⊤gp,q+λθp−λε1n+λCq⊤Ψηp−λMq⊤hp+Efp⊤1s+Cq⊤ΨFfp⊤1s−Eq⊤1s−μ2gp,q≺¯0,(40f)λθp−λε1n+Efp⊤1s−μ2hp,q≺¯0,(40g)Bq⊤gp,q+λDq⊤Ψηp−λBq⊤hp+Dq⊤ΨFfp⊤1s−Fq⊤1s−γ1m≺¯0,(40h)∑ı=1n1nıθqı⊤−αI+∑ı=1n1nıηqı⊤ΦCq−1n⊤hqMql±0,(40i)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40j)λθp−λε1n+Efp⊤1s−μ2hp,q≺¯0,(40k)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40l)Mq⊤gp,q+λθp−λε1n+λCq⊤Ψηp−λMq⊤hp+Efp⊤1s+Cq⊤ΨFfp⊤1s−Eq⊤1s−μ2gp,q≺¯0,(40m)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40n)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40o)θpı≺θp,ηpı≺ηp,α≥nε,(40p)hp≺¯λhp,q,hp≺¯λhq,p,hp,q≺¯λhp,hq,p≺¯λhp,(40q)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40r)Mq⊤gp,q+λθp−λε1n+λCq⊤Ψηp−λMq⊤hp+Efp⊤1s+Cq⊤ΨFfp⊤1s−Eq⊤1s−μ2gp,q≺¯0,hold for given μ1>0,μ2>0,λ>1, and any l=1,2,…,2n, then the filter error system equation (8) is positive and L1 gain is stable under the event-triggered asynchronous filter equation (13) with equation (16), where Φ=I−β1m×m, Ψ=I+β1m×m, and the switching law satisfying(41a)T−t0,tT+t0,t≥μ2+μ1μ1−μ1∗,μ1∗∈0,μ1,(41b)τa≥τa∗=2lnλ+μ1+μ2Δmμ1∗,where T−t0,t denotes the total time length of synchronous and T+t0,t denotes the total time length of asynchronous of the switched systems. Moreover, the system states starting from ∪iΛvi,1 will remain inside the set ∪iΛvi,1+γχ.Proof.
Using a similar method to Theorem1, we have(42)x˜˙t0≽A˜1pqx˜t0+B˜1pqωt0,et0≽E˜1pqx˜t0+F˜1pqωt0.
Fort∈tl,tl+Δl, where(43)A˜1pq=Mq0Afp+BfpΦCq−MqAfp,B˜1pq=BqBfpΦDq−Bq,E˜1pq=Efp+FfpΦCq−EqEfp,F˜1pq=FfpΦDq−Bq,(44)x˜˙t0≥A˜1qx˜t0+B˜1qωt0,et0≥E˜1qx˜t0+F˜1qωt0.
Fort∈tl+Δl,tl+1, where(45)A˜1q=Mq0Afq+BfqΦCq−MqAfq,B˜1q=BqBfqΦDq−Bq,(46)E˜1q=Efq+FfqΦCq−EqEfq,F˜1q=FfqΦDq−Bq.
By equations (40a), (40b), (40h), and (40i), it follows that(47)∑ı=1n1nıθpı⊤1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≥0,∑ı=1n1nıηpı⊤1n⊤hpΦDq−Fq≥0,∑ı=1n1nıθqı⊤1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≥0,∑ı=1n1nıηqı⊤1n⊤hqΦDq−Fq≥0.
By equation (16), it is easy to get Afp+BfpΦCq−Mql≽0. Furthermore, Afp+BfpΦCq−∑l=12nℏlMql±0 and ∑l=12nℏlAfp+BfpΦCq−Mql≽0, which implies that Afp+BfpΦCq−Mq≽0. Then, BfpΦDq−Bq≽0,Efp+FfpΦCq−Eq≽0 and FfpΦDq−Bq≽0. Similarly, we can obtain Afq+BfqΦCq−Mq≽0,BfqΦDq−Bq≽0,Efq+FfqΦCq−Eq≽0 and FfqΦDq−Bq≽0. Thus, Mq, Afp, and Afq are Metzler matrices. Together with Bq,Efp,Efq≽0, and x˜t0≽0, it derives that the system equation (8) with ωt≠0 is positive.
Next, consider the stability of the system equation (8). From equation (23), we have(48)x˜˙t≺¯A˜2pqx˜t+B˜2pqωt,et≺¯E˜2pqx˜t+F˜2pqωt.
Fort∈tl,tl+Δl, where(49)A˜2pq=Mq0Afp+BfpΨCq−MqAfp,B˜2pq=BqBfpΨDq−Bq,E˜2pq=Efp+FfpΨCq−EqEfp,F˜2pq=FfpΨDq−Bq.
Whent∈tl+Δ1,tl+1, we have(50)x˜˙t≺¯A˜2qx˜t+B˜2qωt,et≺¯E˜2qx˜t+F˜2qωt,where(51)A˜2q=Mq0Afq+BfqΨCq−MqAfq,B˜2q=BqBfqΨDq−Bq,E˜2q=Efq+FfqΨCq−EqEfq,F˜2q=FfqΨDq−Fq.
Choose the same Lyapunov function as Theorem1. From equations (48), (50), and (23), we get(52)V˙t≤x˜⊤tMq⊤gp,q+Afp⊤+Cq⊤Ψ⊤Bfp⊤−Mq⊤hp,qAfp⊤hp,q+ω⊤tBq⊤gp,qDq⊤ΨBfp⊤−Bq⊤hp,q,t∈tl,tl+Δl,x˜⊤tMq⊤gq+Afq⊤+Cq⊤Ψ⊤Bfq⊤−Mq⊤hqAfq⊤hq+ω⊤tBq⊤gqDq⊤ΨBfq⊤−Bq⊤hq,t∈tl+Δl,tl+1.
DefineΓt=et1−γωt1. By equations (40e)–(40g) and 40l–(40n), we have(53)V˙t≤μ2Vt−Γt,t∈tl,tl+Δl,−μ1Vt−Γt,t∈tl+Δl,tl+1.
Integrating both sides of equation (53) gives(54)Vt≤eμ2t−tlVtl−∫tlteμ2t−sΓsds,t∈tl,tl+Δl,e−μ1t−tl−ΔlVtl+Δl−∫tl+Δlte−μ1t−sΓsds,t∈tl+Δl,tl+1,where Γs=es1−γωs1. For T∈tNσT,t0+ΔNσT,t0,tNσT,t0+1, using equations (40p) and (40q) and (54) follows that(55)Vσtℵ+1T≤λe−μ1T−tℵ−ΔℵVσtℵ+Δℵtℵ+Δℵ−∫tℵ+ΔℵTe−μ1T−sΓsds≤λ3e−μ1T−tℵ−1−Δℵ−Δℵ−1eμ2ΔℵVσtℵ−1+Δℵ−1tℵ−1+Δℵ−1−λ2eμ2Δℵ∫tℵ−1+Δℵ−1tℵe−μ1T−Δℵ−sΓsds−λe−μ1T−tℵ−Δℵ∫tℵtℵ+Δℵeμ2tℵ+Δℵ−sΓsds−∫tℵ+ΔℵTe−μ1T−sΓsds≤⋯≤e2ℵlnλe−μ1T−t0+μ1+μ2ℵΔmVσt0t0−∫t0Teμ2Nσs,TΔme−μ1T−t0−Nσs,TΔme2Nσs,TlnλΓsds,where ℵ=Nσt0,T and Δm is the maximum of time lag. Under zero initial conditions, we have(56)0≤−∫t0Teμ2Nσs,TΔme−μ1T−t0−Nσs,TΔme2Nσs,TlnλΓsds.
FromΓs=es1−γωs1, we have(57)∫t0Teμ2Nσs,TΔme−μ1T−t0−Nσs,TΔme2Nσs,Tlnλes1ds≤γ∫t0Teμ2Nσs,TΔme−μ1T−t0−Nσs,TΔme2Nσs,Tlnλws1ds.
Multiplying both sides of the inequality withe−μ1+μ2Nσt0,TΔme−2Nσt0,Tlnλ, we get(58)∫t0Te−μ1T−t0−μ1+μ2Nσt0,sΔm−2Nσt0,slnλes1ds≤γ∫t0Te−μ1T−t0−μ1+μ2Nσt0,sΔmωs1ds.
It follows from Definition2 and equation (41b) that(59)Nσt0,s≤N0+μ1∗s−t02lnλ+μ1+μ2Δm.
Then,(60)∫t0Te−μ1T−t0−μ1+μ2Δm+2lnλN0+μ1∗s−t02lnλ+μ1+μ2Δmes1ds≤γ∫t0Te−μ1T−toωs1ds.
Furthermore,(61)e−μ1+μ2Δm+2lnλN0∫t0Te−μ1∗s−t0es1ds≤γ∫t0Tωs1ds.
Integrating the above inequality at0,∞, it gives ϖ∫0∞e−ρtet1dt≤γ∫0∞ωt1dt, where ρ=μ1∗ and ϖ=e−μ1+μ2Δm+2lnλN0. By Definition 3, the system equation (8) achieves a prescribed L1-gain performance level γ.
Next, we present the invariance of the state. By equation (55), letting T⟶∞, we have V∞≤1+∫t0Tω⊤sds≤1+γχ. For ωt=0, Λvi,1⊆LHp. Using method of the Theorem 1, we have that the states will be kept in the set ∪pΛvp,1+γχ∪∪p,qΛvp,q,1+γχ for any initial conditions starting from ∪ivi,1+γχ.
In the following, the Zeno behavior is analyzed to avoid the continuous sampling and updating of the state under the event-triggering mechanism. Under the event-triggering mechanism equation (5), it gives e˙yt=−y˙t=−Cσtx˙t−Dσtw˙t,t∈tl′,tl+1′. Without generality, assume that w˙t exists and is bounded. Then, there exists a positive constant ℧ such that e˙yt1<℧ holds for t∈tl′,tl+1′. Then, eyt1<℧t−tl′ with eytl′1=0 and t∈tl′,tl+1′. This implies that t−tl′>eyt1/℧. Owing to xt≠0 and wt≠0, it is not hard to obtain eyt1>0 for t⟶tl+1′. This means that t>tl′. Therefore, the Zeno behavior can be avoided.Remark 4.
The 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 L1 norm of the external disturbance wt is bounded. Assume that the L1 norm is unbounded. From the last part of the proof in Theorem 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.
The time-triggered filter strategy was proposed for positive systems in [41–43]. In most cases, it is hard to reach the continuous sample owing to limited ability of elements and high sample cost. The event-triggered filter is more practicable with respect to the time-triggered filter. Thus, Theorems 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. Thus, 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.


### 3.2. Discrete-Time Case

For the discrete-time case, define the asynchronous time intervalkl,kl+Δl and the synchronous time interval kl+Δl,kl+1. In the interval kl,kl+1, the q th subsystems is active. In the interval kl,kl+Δl, the p th filter is active. In the interval kl+Δl,kl+1, the q th filter is active, where kl,l=0,1,⋯ is the switching time instant and Δl is the time lag between the two subsystems and Δl<kl+1−kl. When k∈kl,kl+Δl, the error system can be written as(62)x˜k+1=A˜pqx˜k+B˜pqωk+C˜peyk,ek=E˜pqx˜k+F˜pqωk+Ffpeyk,where(63)A˜pq=Mq0Afp+BfpCq−MqAfp,B˜pq=BqBfpDq−Bq,C˜p=0Bfp,E˜pq=Efp+FfpCq−EqEfp,F˜pq=FfpDq−Fq.Whenk∈kl+Δl,kl+1,(64)x˜k+1=A˜qx˜k+B˜qωk+C˜qeyk,ek=E˜qx˜k+F˜qωk+Ffqeyk,where(65)A˜q=Mq0Afq+BfqCq−MqAfq,B˜q=BqBfqDq−Bq,C˜q=0Bfq,E˜q=Efq+FfqCq−EqEfq,F˜q=FfqDq−Fq.Theorem 3.
If there exist constants0<μ1<1;μ2>1;λ>1; ℜn vectors gp,q≻0, hp,q≻0, gq≻0, hq≻0, θpı≻0, θp≻0, ξpı≻0, and ξp≻0; and ℜm vectors ηpı≻0, ηp≻0, ςpı≻0, and ςp≻0 such that(66a)∑ı=1n1nıθpı⊤1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≽0,(66b)∑j=1s1sjξpjT+∑j=1s1pjjςpjTΦCq−Eq≽0,(66c)Mq⊤gp,q+λθp+λCq⊤Ψ⊤ηp−λMq⊤hp−μ2gp,q≺0,(66d)λθp−μ2hp,q≺0,(66e)∑ı=1n1nıθqı⊤1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≽0,(66f)∑j=1s1sjξqiT+∑j=1s1sjςqjTΦCq−Eq≽0,(66g)Mq⊤gq+θq+Cq⊤Ψ⊤ηq−Mq⊤hq−μ1gq≺0,(66h)θq−μ1hq≺0,(66i)θpı≺θp,ηpı≺ηp,(66j)hp≺¯λhp,q,hp≺¯λhq,p,hp,q≺¯λhp,hq,p≺¯λhp,(66k)gp≺¯λgp,q,gp≺¯λgq,p,gp,q≺¯λgp,gq,p≺¯λgp,(66l)−gq≺¯Hp⊤≺¯gq,−hq≺¯Hp⊤≺¯hq,−gp,q≺¯Hp⊤≺¯gp,q,−hp,q≺¯Hp⊤≺¯hp,q,hold for l=1,2,…,2n, then the filter error system equation (8) with the event-triggered asynchronous filter equation (6) satisfying(67)Afp=∑ı=1n1nıθpı⊤1n⊤hp,Bfp=∑ı=1n1nıηpı⊤1n⊤hp,Efp=∑j=1s1sjξpjT,Ffp=∑j=1s1sjςpjT,is positive and stable, where Φ=I−β1m×m, Ψ=I+β1m×m, and the switching law satisfies(68a)T−k0,kT+k0,k≥lnμ2−lnμ1lnμ1∗−lnμ1,μ1∗∈μ1,1,(68b)τa≥τa∗=2lnλ−lnμ1∗,where T−k0,k denotes the total time length in the synchronous case and T+k0,k denotes the total time length in the asynchronous case. Moreover, all states will remain inside the cone set ∪pΛvp,1∪∪p,qΛvp,q,1 for xk0∈Λvi,1.Proof.
Forxk0±0 and yk0±0, it follows from equation (5) that −β1m×myk0≺¯eyk0≺¯β1m×myk0. By equations (62) and (64), we have(69)x˜k0+1≥A˜1pqx˜k0,ek0≥E˜1pqx˜k0,for k∈kl,kl+Δl, where(70)A˜1pq=Mq0Afp+BfpΦCq−MqAfp,E˜1pq=Efp+FfpΦCq−EqEfp.
Fork∈kl+Δl,kl+1,(71)x˜k0+1≽A˜1qx˜k0,ek0≽E˜1qx˜k0,where(72)A˜1q=Mq0Afq+BfqΦCq−MqAfq,E˜1q=Efq+FfqΦCq−EqEfq.
From equations (67a) and (67b), we have(73)∑ı=1n1nıθpı⊤1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≽0,∑ı=1n1nıθqı⊤1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≽0.
Then, we obtainAfp+BfpΦCq−Mql≽0. From ∑l=12nℏlAfp+BfpΦCq−Mql≽0 and Afp+BfpΦCq−Mq≽0. Similarly, Afq+BfqΦCq−Mq≽0. Together with equations (66b) and (66f), it follows that Efp+FfpΦCq−Eq≽0 and Efq+FfqΦCq−Eq≽0. It is easy to obtain Mq≽0,Afp≽0,Afq≽0,Efp≽0,andEfq≽0. By Theorem 1, the positivity of the system equation (8) is achieved. Thus,(74)−β1m×myk≺¯eyk≺¯β1m×myk.
Choose a multiple Lyapunov function candidate:(75)Vk=x˜k⊤vp,q,k∈kl,kl+Δl,x˜k⊤vq,k∈kl+Δl,kl+1,where vp,q=gp,qhp,q and vq=gqhq. From equation (74), we have(76)x˜k≺¯A˜2pqx˜k−1,ek≺¯E˜2pqx˜k,for k∈kl,kl+Δl, where A˜2pq=Mq0Afp+BfpΨCq−MqAfp,E˜2pq=Efp+FfpΨCq−EqEfp, and(77)x˜k≺¯A˜2qx˜k−1,ek≺¯E˜2qx˜k,for k∈kl+Δl,kl+1, where A˜2q=Mq0Afq+BfqΨCq−MqAfqandE˜2q=Efq+FfqΨCq−EqEfq. Then,(78)Vk≤x˜⊤k−1A˜2pq⊤vp,q,k∈kl,kl+Δl,x˜⊤k−1A˜2q⊤vq,k∈kl+Δl,kl+1.
Then,(79)Vk≤x˜⊤k−1Mq⊤gp,q+Afp⊤+Cq⊤Ψ⊤Bfp⊤−Mq⊤hp,qAfp⊤hp,q,k∈kl,kl+Δl,x˜⊤k−1Mq⊤gq+Afq⊤+Cq⊤Ψ⊤Bfq⊤−Mq⊤hqAfq⊤hq,k∈kl+Δl,kl+1.
Using equations (66i)–(66k) and λ>1, it can be derived that(80)Vk≤x˜⊤k−1Mq⊤gp,q+λθp−λhp+λCq⊤Ψ⊤ηp−λMq⊤hpλθp−λhp,k∈kl,kl+Δl,x˜⊤k−1Mq⊤gq+θq−hq+Cq⊤Ψ⊤ηq−Mq⊤hqθq−hq,k∈kl+Δl,kl+1.
By equations (66c), (66d), (66g), and (66h), we have(81)Vk≤μ2k−klVkl,k∈kl,kl+Δl,μ1k−kl−ΔlVkl+Δl,k∈kl+Δl,kl+1.
Together with equations (66j) and (66k) it yields that(82)Vk≤λμ2k−klVkl,k∈kl−1+Δl−1,klλμ1k−kl−ΔlVkl+Δl,k∈kl,kl+Δl.
ForT∈kNσT,k0+ΔNσT,k0,kNσT,k0+1, we can obtain(83)Vσkℵ+1T≤λμ1T−kℵ−ΔℵVσkℵ+Δℵkℵ+Δℵ≤λ2μ1T−kℵ−Δℵμ2ΔℵVσkℵkℵ≤⋯≤λ2ℵμ1T−k0−Δℵ−Δℵ−1−⋯−Δ1μ2Δℵ+Δℵ−1+⋯+Δ1Vσk0k0,where ℵ=NσT,k0. It can follow from equation (68a) that lnμ2−lnμ1T+k0,T≤lnμ1∗−lnμ1T−k0. By Definition 2 and equations (66a) and (83), it gives VT≤e2N0lnλelnμ1∗+2lnλ/τaT−k0Vk0. Then,(84)x˜T1≤ρ2e2N0lnλρ1elnμ1∗+2lnλ/τaT−k0x˜k01,where ρ1 and ρ2 are the minimal and maximal elements of vi∈vp,q,vq, respectively.
Finally, we provide the invariance of the state. Give any initial conditions satisfyingx˜⊤k0∈Λvi,1, that is, x˜⊤k0vi≤1. By equation (84), x˜⊤kvi≤x˜⊤k0vi≤1. So, x˜k∈Λvi,1. Given H˜p=HpHp, we have H˜plx˜k=x⊤kHpl⊤+xe⊤kHpl⊤≤x˜⊤tvi≤1, where H˜pl is the l th row of H˜p. By xk0≽0, xek0≽0, and equation (66l), it gives Hplxk≤H˜plx˜k≤1, which leads to x˜k∈∪pΛvp,1∪∪p,qΛvp,q,1. Then, xk∈LHp.
Next, we consider the filter design for the caseωk≠0. The disturbance is assumed to be located in the set:(85)W=ωk≽0:ωℓ1<∞.Theorem 4.
If there exist constants0<μ1<1;μ2>1;λ>1; ℜn vectors gp,q≻0, hp,q≻0, gq≻0, hq≻0, θpı≻0, θp≻0, ξpı≻0, and ξp≻0, and ℜm vectors ηpı≻0, ηp≻0, ςpı≻0,andςp≻0 such that(86a)∑ı=1n1nıθpı⊤+∑ı=1n1nıηpı⊤ΦCq−1n⊤hpMql≽0,(86b)∑i=1n1nıηpı⊤ΦDq−1n⊤hpBq≽0,(86c)∑j=1s1sjξpjT+∑j=1s1sjςpjTΦCq−Eq≽0,(86d)∑j=1s1sjςpjTΦDq−Fq≽0,(86e)Mq⊤gp,q+λθp+λCq⊤Ψηp−λMq⊤hp+Efp⊤1s+Cq⊤ΨFfp⊤1s−Eq⊤1s−μ2gp,q≺¯0,(86f)λθp+Efp⊤1s−μ2hp,q≺¯0,(86g)Bq⊤gp,q+λDq⊤Ψηp−λBq⊤hp+Dq⊤ΨFfp⊤1s−Fq⊤1s−γ1m≺¯0,(86h)∑ı=1n1nıθqı⊤+∑ı=1n1nıηqı⊤ΦCq−1n⊤hqMql≽0,(86i)∑ı=1n1nıηqı⊤ΦDq1n⊤hq−Bq≽0,(86j)∑j=1s1sjξqiT+∑j=1s1sjςqjTΦCq−Eq≽0,(86k)∑j=1s1sjςqjTΦDq−Fq≽0,(86l)Mq⊤gq+θq+Cq⊤Ψηq−Mq⊤hq+Efq⊤1s+Cq⊤ΨFfq⊤1s−Eq⊤1s−μ1gq≺¯0,(86m)θq+Efq⊤1s−μ1hq≺¯0,(86n)Bq⊤gq+Dq⊤Ψηq−Bq⊤hq+Dq⊤ΨFfq⊤1s−Fq⊤1s−γ1m≺¯0,(86o)θpı≺θp,ηpı≺ηp,(86p)hp≺¯λhp,q,hp≺¯λhq,p,hp,q≺¯λhp,hq,p≺¯λhp,(86q)gp≺¯λgp,q,gp≺¯λgq,p,gp,q≺¯λgp,gq,p≺¯λgp,(86r)−gq≺¯Hp⊤≺¯gq,−hq≺¯Hp⊤≺¯hq,−gp,q≺¯Hp⊤≺¯gp,q,−hp,q≺¯Hp⊤≺¯hp,q,hold for l=1,2,…,2n, then the filter error system equation (8) with equations (68a) and (68b) is positive and ℓ1 gain stable, where Φ=I−β1m×m and Ψ=I+β1m×m, and the switching law satisfies(87a)T−k0,kT+k0,k≥lnμ2−lnμ1lnμ1∗−lnμ1,μ1∗∈μ1,1,(87b)τa≥τa∗=2lnλ+lnμ2−lnμ1Δm−lnμ1∗,where T−k0,k denotes the total time length in the synchronous case and T+k0,k denotes the total time length in the asynchronous case. Moreover, the system states starting from ∪iΛvi,1+γχ will remain inside the set ∪pΛvp,1+γχ∪∪p,qΛvp,q,1+γχ with N0=0.Proof.
By Theorem3, it holds that(88)x˜k0+1≽A˜1pqx˜k0+B˜1pqωk0,ek0≽E˜1pqx˜k0+F˜1pqωk0,for k∈kl,kl+Δl, where(89)A˜1pq=Mq0Afp+BfpΦCq−MqAfp,B˜1pq=BqBfpΦDq−Bq,E˜1pq=Efp+FfpΦCq−EqEfp,F˜1pq=FfpΦDq−Fq.
Fork∈kl+Δl,kl+1,(90)x˜k0+1≽A˜1qx˜k0+B˜1qωk0,ek0≽E˜1qx˜k0+F˜1qωk0,where(91)A˜1q=Mq0Afq+BfqΦCq−MqAfq,B˜1q=BqBfqΦDq−Bq,E˜1q=Efq+FfqΦCq−EqEfq,F˜1q=FfqΦDq−Fq.
By equations (86a), (86b), (86h), and (86i), we have(92)∑ı=1n1nıθpı⊤1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≥0,∑ı=1n1nıηpı⊤1n⊤hpΦDq−Fq≥0,∑ı=1n1nıθqı⊤1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≥0,∑ı=1n1nıηqı⊤1n⊤hqΦDq−Fq≥0.
By equations (68a) and (68b), it follows that Afp+BfpΦCq−Mql≽0. From ∑l=12nℏlAfp+BfpΦCq−Mql≽0, we can obtain Afp+BfpΦCq−Mq≽0. Similarly, Afq+BfqΦCq−Mq≽0. In addition, it is clear that BfpΦDq−Bq≽0, Efp+FfpΦCq−Eq≽0, FfpΦDq−Fq≽0, BfqΦDq−Bq≽0, Efq+FfqΦCq−Eq≽0, and FfqΦDq−Fq≽0. It is easy to get that Mq≻0,Afp≻0,Afq≻0,Bq≻0,Efp≻0,andEfq≻0. Then, the system equation (8) with ωt≠0 is positive.
From equation (74), we have(93)x˜k+1≺¯A˜2pqx˜k+B˜2pqωk,ek≺¯E˜2pqx˜k+F˜2pqωk,for k∈kl,kl+Δl. When k∈kl+Δ1,kl+1, we have(94)x˜k+1≺¯A˜2qx˜k+B˜2qωk,ek≺¯E˜2qx˜k+F˜2qωk,where E˜2pq=Efp+FfpΨCq−EqEfp,F2pq=FfpΨDq−Fq,E˜2q=Efq+FfqΨCq−EqEfq,F2q=FfqΨDq−Fq.
Choose the same Lyapunov function as Theorem3. From equations (93), (94), and (74), we get(95)Vk+1≤x˜⊤kMq⊤gp,q+Afp⊤+Cq⊤Ψ⊤Bfp⊤−Mq⊤hp,qAfp⊤hp,q+ω⊤kBq⊤gp,qDq⊤ΨBfp⊤−Bq⊤hp,q,k∈kl,kl+Δl,x˜⊤kMq⊤gq+Afq⊤+Cq⊤Ψ⊤Bfq⊤−Mq⊤hqAfq⊤hq+ω⊤kBq⊤gqDq⊤ΨBfq⊤−Bq⊤hq,k∈kl+Δl,kl+1.
LetΓk=ek1−γωk1. From equations (86e)–(86g) and (86l)–(86n), it follows that(96)Vk≤μ2Vk−1−Γk−1,k∈kl,kl+Δl,μ1Vk−1−Γk−1,k∈kl+Δl,kl+1.
Thus,(97)Vk≤μ2k−klVkl−∑s=klk−1μ2k−s−1Γs,k∈kl,kl+Δl,μ1k−kl−ΔlVkl+Δl−∑s=kl+Δlk−1μ1k−s−1Γs,k∈kl+Δl,kl+1),where Γs=e˜s1−γωs1. For T∈kNσk0,T+ΔNσk0,T,kNσk0,T+1,(98)VT≤μ1T−kNσk0,T−ΔNσk0,TVkNσk0,T+ΔNσk0,T−∑s=kNσk0,T+ΔNσk0,TT−1μ1T−s−1Γs≤λeT−kNσk0,T−ΔNσk0,Tlnμ1eΔNσk0,Tlnμ2VkNσk0,T−λeT−kNσk0,T−ΔNσk0,Tlnμ1×∑s=kNσk0,TkNσk0,T+ΔNσk0,T−1ekNσk0,T+ΔNσk0,T−s−1lnμ2Γs−∑s=kNσk0,T+ΔNσk0,TT−1eT−s−1lnμ1≤⋯≤e2Nσk0,TlnλeT−k0−Nσk0,TΔmlnμ1eNσk0,TΔmlnμ2Vk0−∑k0T−1e2Nσ\s,T−1lnλeNσs,T−1Δmlnμ2eT−1−k0−Nσs,T−1Δmlnμ1Γs,where Δm denotes the maximum of time lag. Under zero initial condition, we have(99)0≤−∑k0=0Me2Nσs,MlnλeNσs,MΔmlnμ2eM−k0−Nσs,MΔmlnμ1Γs,where M=T−1. By Γs=es1−γωs1, it holds that(100)∑k0Me2Nσs,MlnλeNσs,MΔmlnμ2eM−k0−Nσs,MΔmlnμ1es1≤γ∑k0Me2Nσs,MlnλeNσs,MΔmlnμ2eM−k0−Nσs,MΔmlnμ1ωs1.
Multiplying both sides of the inequality with(101)∑k0Melnμ1M−k0+lnμ1−lnμ2Nσk0,sΔm−2Nσk0,slnλes1≤γ∑k0Melnμ1M−k0+lnμ1−lnμ2Nσk0,sΔmωs1.
It follows from Definition2 and equation (87b) that Nσk0,s≤N0−lnμ1∗s−k0/2lnλ+lnμ2−lnμ1Δm. Then,(102)∑k0Melnμ1M−k0−2lnλ+lnμ2−lnμ1ΔmN0−lnμ1∗s−k02lnλ+lnμ2−lnμ1Δmes1≤γ∑k0Melnμ1M−k0ωs1.
Furthermore,(103)e−2lnλ+lnμ2−lnμ1ΔmN0∑k0Melnμ1∗s−k0‖es‖1≤γ∑k0M‖ωs‖1.
Summing the above inequality from 0 to∞, it follows that ϖ∑0∞eρkek1≤γ∑0∞ωk1, where ρ=lnμ1∗ and ϖ=e−2lnλ+lnμ2−lnμ1ΔmN0. By Definition 3, the system equation (8) achieves a prescribed ℓ1-gain performance level γ.
Next, we present the invariance of the state. By equation (98), we have V∞≤1+γ∑k0Mωs1≤1+γχ. For ωk=0, Λvi,1⊆LHp. Using a similar method to Theorem 3, we have that the states will be kept in the set ∪pΛvp,1+γχ∪∪p,qΛvp,q,1+γχ for any initial conditions starting from ∪ivi,1+γχ with ωk≠0.Remark 7.
This 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. This paper employs the convex polytopic approach for dealing with the state saturation term. Thus, the saturation term is transformed into polytopic, which is dependent on2n 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. This paper considers an asynchronous even-triggered filter of positive switched systems. Multiple copositive Lyapunov functions are constructed in this paper, and the ℓ1-gain stability for positive switched systems is achieved. Compared with positive systems, the ℓ1-gain stability of positive switched systems is more complex. Then, 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.


## 3.1. Continuous-Time Case

Given a time intervaltl,tl+1, where the asynchronous time interval is tl,tl+Δl and the synchronous time interval is tl+Δl,tl+1. The q th original subsystem is active in t∈tl,tl+1. The p th filter is active in t∈tl,tl+Δl, and then the q th filter is active in t∈tl+Δl,tl+1, where tll=0,1,⋯ is the switching time instants and Δl is the time lag between the subsystem and the filter and Δl<tl+1−tl. When t∈tl,tl+Δl, the error system can be written as(11)x˜˙t=A˜pqx˜t+B˜pqωt+C˜peyt,et=E˜pqx˜t+F˜pqωt+Ffpeyt,where(12)A˜pq=Mq0Afp+BfpCq−MqAfp,B˜pq=BqBfpDq−Bq,C˜p=0Bfp,E˜pq=Efp+FfpCq−EqEfp,F˜pq=FfpDq−Fq.Whent∈tl+Δl,tl+1, it follows that(13)x˜˙t=A˜qx˜t+B˜qωt+C˜qeyt,et=E˜qx˜t+F˜qωt+Ffqeyt,where(14)A˜q=Mq0Afq+BfqCq−MqAfq,B˜q=BqBfqDq−Bq,C˜q=0Bfq,E˜q=Efq+FfqCq−EqEfq,F˜q=FfqDq−Fq.Theorem 1.
If there exist constantsμ1>0;μ2>0;α>0;ε>0;λ>1; ℜn vectors gp,q≻0, hp,q≻0, gq≻0, hq≻0, θpı≻0, θp≻0, ξpı≻0, and ξp≻0; and ℜm vectors ηpı≻0, ηp≻0, ςpı≻0, and ςp≻0 such that(15a)∑ı=1n1nıθpı⊤1n⊤hp−αI+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≽0,(15b)∑j=1s1sjξpjT+∑j=1s1pjjςpjTΦCq−Eq≽0,(15c)Mq⊤gp,q+λθp−λε1n+λCq⊤Ψ⊤ηp−λMq⊤hp−μ2gp,q≺0,(15d)λθp−λε1n−μ2hp,q≺0,(15e)∑ı=1n1nıθqı⊤1n⊤hq−αI1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≽0,(15f)∑j=1s1sjξpjT+∑j=1s1pjjςpjTΦCq−Eq≽0,(15g)Mq⊤gq+θq−ε1n+Cq⊤Ψ⊤ηq−Mq⊤hq+μ1gq≺0,(15h)θq−ε1n+μ1hq≺0,(15i)θpı≺θp,ηpı≺ηp,α≥nε,(15j)hp≺¯λhp,q,hp≺¯λhq,p,hp,q≺¯λhp,hq,p≺¯λhp,(15k)gp≺¯λgp,q,gp≺¯λgq,p,gp,q≺¯λgp,gq,p≺¯λgp,(15l)−gq≺¯Hp⊤≺¯gq,−hq≺¯Hp⊤≺¯hq,−gp,q≺¯Hp⊤≺¯gp,q,−hp,q≺¯Hp⊤≺¯hp,q.hold for l=1,2,…,2n, then under the event-triggered asynchronous filter equation (6) with(16)Afp=∑ı=1n1nıθpı⊤−αI1n⊤hp,Bfp=∑ı=1n1nıηpı⊤1n⊤hp,Efp=∑j=1s1sjξijT,Ffp=∑j=1s1sjςpjT,and the switching law satisfies(17a)T−t0,tT+t0,t≥μ2+μ1μ1−μ1∗,μ1∗∈0,μ1,(17b)τa≥τa∗=2lnλμ1∗.
The filter error system equation (8) is positive and asymptotically stable, where Φ=I−β1m×m, Ψ=I+β1m×m, T−t0,t denotes the total time length of synchronous, and T+t0,t denotes the total time length of asynchronous of the switched systems. Moreover, all states starting from xt0∈Λvi,1 will remain inside the cone set ∪iΛvi,1, where vi∈vp,q,vq for p,q∈S×S.Proof.
First, consider the positivity of the error system equation (8). For xt0≽0 and yt0≽0, it gives eyt01≤βyt01 by equation (5). Then, we deduce that −β1m×myt0≺¯eyt0≺¯β1m×myt0. By equations (11) and (13), we have(18)x˜˙t0≽A˜1pqx˜t0,et0≽E˜1pqx˜t0,for t∈tl,tl+Δl, where(19)A˜1pq=Mq0Afp+BfpΦCq−MqAfp,E˜1pq=Efp+FfpΦCq−EqEfp,(20)x˜˙t0≽A˜1qx˜t0,et0≽E˜1qx˜t0.
Fort∈tl+Δl,tl+1, where(21)A˜1q=Mq0Afq+BfqΦCq−MqAfq,E˜1q=Efq+FfqΦCq−EqEfq.
By equations (15a) and (15e), we have(22)∑ı=1n1nıθpı⊤−αI1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≽0,∑ı=1n1nıθqı⊤−αI1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≽0.
Using equation (16) gives Afp+BfpΦCq−Mql≽0. Thus, Afp+BfpΦCq−∑l=12nℏlMql≽0 and ∑l=12nℏlAfp+BfpΦCq−Mql≽0, which implies that Afp+BfpΦCq−Mq≽0. Similarly, we can also obtain Afq+BfqΦCq−Mq≽0. By equations (15b) and (15f), we get that Efp+FfpΦCq−Eq≽0 and Efq+FfqΦCq−Eq≽0. It is easy to know that Mq, Afp, and Afq are Metzler matrices. Due to Efp≽0 and Efq≽0, x˜˙t0≽0 by Lemma 1. Using recursive derivation gives x˜˙t≽0, that is to say, the error system equation (8) is positive. Therefore, it holds that(23)−β1m×myt≺¯eyt≺¯β1m×myt.
Next, consider the stability of the system equation (8). Choose piecewise multiple Lyapunov functions:(24)Vt=x˜⊤tvp,q,t∈tl,tl+Δl,x˜⊤tvq,t∈tl+Δl,tl+1,where vp,q=gp,qhp,q and vq=gqhq. By equation (23), we have(25)x˜˙t≺¯A˜2pqx˜t,et≺¯E˜2pqx˜t,for t∈tl,tl+Δl, where(26)A˜2pq=Mq0Afp+BfpΨCq−MqAfp,E˜2pq=Efp+FfpΨCq−EqEfp,(27)x˜˙t≺¯A˜2qx˜t,et≺¯E˜2qx˜t,for t∈tl+Δl,tl+1, where(28)A˜2q=Mq0Afq+BfqΨCq−MqAfq,E˜2q=Efq+FfqΨCq−EqEfq.
Then,(29)V˙t≤x˜t⊤A˜2pq⊤vp,q,t∈tl,tl+Δl,x˜t⊤A˜2q⊤vq,t∈tl+Δl,tl+1.
Moreover, we have(30)V˙t≤x˜⊤tMq⊤gp,q+Afp⊤+Cq⊤Ψ⊤Bfp⊤−Mq⊤hp,qAfp⊤hp,q,t∈tl,tl+Δl,x˜⊤tMq⊤gq+Afq⊤+Cq⊤Ψ⊤Bfq⊤−Mq⊤hqAfq⊤hq,t∈tl+Δl,tl+1.
Using equations (15i) and (15j) and λ>1, it yields(31)V˙t≤x˜⊤tMq⊤gp,q+λθp−λε1n+λCq⊤Ψ⊤ηp−λMq⊤hpλθp−ε1n,t∈tl,tl+Δl,x˜⊤tMq⊤gq+θq−ε1n+Cq⊤Ψ⊤ηq−Mq⊤hqθq−ε1n,t∈tl+Δl,tl+1.
By equations (15c), (15d), (15g), and (15h), it holds that V˙t≤μ2Vt,t∈tl,tl+Δl,−μ1Vt,t∈tl+Δl,tl+1.
Thus,(32)Vt≤eμ2t−tlVtl,t∈tl,tl+Δl,e−μ1t−tl−ΔlVtl+Δl,t∈tl+Δl,tl+1.
Together with equations (15j) and (15k), it yields(33)Vt≤λeμ2t−tlVtl−,t∈tl,tl+Δl,λe−μ1t−tl−ΔlVtl+Δl−,t∈tl+Δl,tl+1.
ForT∈tNσT,t0+ΔNσT,t0,tNσT,t0+1, we get(34)Vσtℵ+1T≤λe−μ1T−tℵ−ΔℵVσtℵ+Δℵtℵ+Δℵ≤λ2e−μ1T−tℵ−Δℵeμ2ΔℵVσtℵtℵ≤λ3e−μ1T−tℵ−1−Δℵ−Δℵ−1eμ2ΔℵVσtℵ−1+Δℵ−1tℵ−1+Δℵ−1≤⋯≤λ2ℵe−μ1T−t0−Δℵ−Δℵ−1−⋯−Δ1eμ2Δℵ+Δℵ−1+⋯+Δ1Vσt0t0,where ℵ=NσT,t0. Then, we have(35)Vσtℵ+1T≤λ2ℵe−μ1T−t0,Teμ2T+t0,TVσt0t0≤λ2ℵe−μ1T−t0eμ1+μ2T+t0,TVσt0t0≤e2N0lnλe2lnλ/τa−μ1T−t0eμ1+μ2T+t0,TVσt0t0.
From equation (17a), we can get(36)μ1+μ2T+t0,T≤μ1−μ1∗T−t0.
Furthermore, it follows thatVσtℵ+1T≤e2N0lnλe2lnλ/τa−μ∗T−t0Vσt0t0. By Definition 2 and equation (17b), it gives(37)VT≤e2N0lnλe−μ1∗−2lnλ/τaT−t0Vt0.
Then,(38)x˜T1≤ρ22N0lnλρ1e−μ1∗−2lnλ/τaT−t0x˜t01,where ρ1 and ρ2 are the minimal and maximal elements of vi∈vp,q,vq,p∈S,q∈S. 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 satisfyingx˜t0∈Λvi,1, that is, x˜⊤t0vi≤1. By equation (38), x˜⊤tvi≤x˜⊤t0vi≤1. So, x˜t∈Λvi,1. Given H˜p=HpHp, we have H˜plx˜t=x⊤tHpl⊤+xe⊤tHpl⊤≤x˜⊤tvi≤1, where H˜pl is the l th row of H˜p. By xt0≽0, xet0≽0, and equation (15l), it follows that Hplxt≤H˜plx˜t≤1, which leads to x˜t∈∪pΛvp,1∪∪p,qΛvp,q,1. Then, xt∈LHp.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 λ,μ1, and μ2 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 μ1 and μ2 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.
Algorithm 1: Searching parameters for Theorem 1.
Input: Ap,Bp, Cp,Dp, Ep, Fp, p∈S, λ1>1, ℏ>0, ν>0, μ¯>μ¯>0;Output: λ,μ1,μ2;DefineΘ=∅;Main:λ=λ1,μ2=μ¯;whileλ≤λ1+ν doifμ2≤μ¯thenμ1=μ¯;repeatif equations (15a)–(15l) is feasible thenSaveλ,μ1,μ2toΘ;endμ1←μ1+ℏuntilμ1>μ¯;μ2←μ2+ℏendλ←λ+ℏ;end
Next, we consider the filter design for the caseωt≠0. The disturbance satisfies(39)∫t0∞ω⊤sds≤χ,where χ is a given positive constant.Theorem 2.
If there exist constantsμ1>0;μ2>0;α>0;ε>0;λ>1; ℜn vectors gp,q≻0, hp,q≻0, gq≻0, hq≻0, θpı≻0, θp≻0, ξpı≻0, and ξp≻0; and ℜm vectors ηpı≻0, ηp≻0, ςpı≻0, and ςp≻0 such that(40a)∑ı=1n1nıθpı⊤−αI+∑ı=1n1nıηpı⊤ΦCq−1n⊤hpMql≽0,(40b)∑i=1n1niηΦDq−1nThpBq≽0,(40c)∑j=1s1sjξpiT+∑j=1s1sjςpjTΦCq−Eq≽0,(40d)∑j=1s1sjςpj⊤ΦDq−Bq≽0,(40e)Mq⊤gp,q+λθp−λε1n+λCq⊤Ψηp−λMq⊤hp+Efp⊤1s+Cq⊤ΨFfp⊤1s−Eq⊤1s−μ2gp,q≺¯0,(40f)λθp−λε1n+Efp⊤1s−μ2hp,q≺¯0,(40g)Bq⊤gp,q+λDq⊤Ψηp−λBq⊤hp+Dq⊤ΨFfp⊤1s−Fq⊤1s−γ1m≺¯0,(40h)∑ı=1n1nıθqı⊤−αI+∑ı=1n1nıηqı⊤ΦCq−1n⊤hqMql±0,(40i)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40j)λθp−λε1n+Efp⊤1s−μ2hp,q≺¯0,(40k)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40l)Mq⊤gp,q+λθp−λε1n+λCq⊤Ψηp−λMq⊤hp+Efp⊤1s+Cq⊤ΨFfp⊤1s−Eq⊤1s−μ2gp,q≺¯0,(40m)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40n)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40o)θpı≺θp,ηpı≺ηp,α≥nε,(40p)hp≺¯λhp,q,hp≺¯λhq,p,hp,q≺¯λhp,hq,p≺¯λhp,(40q)∑ı=1n1nıηqı⊤ΦDq−1n⊤hqBq≽0,(40r)Mq⊤gp,q+λθp−λε1n+λCq⊤Ψηp−λMq⊤hp+Efp⊤1s+Cq⊤ΨFfp⊤1s−Eq⊤1s−μ2gp,q≺¯0,hold for given μ1>0,μ2>0,λ>1, and any l=1,2,…,2n, then the filter error system equation (8) is positive and L1 gain is stable under the event-triggered asynchronous filter equation (13) with equation (16), where Φ=I−β1m×m, Ψ=I+β1m×m, and the switching law satisfying(41a)T−t0,tT+t0,t≥μ2+μ1μ1−μ1∗,μ1∗∈0,μ1,(41b)τa≥τa∗=2lnλ+μ1+μ2Δmμ1∗,where T−t0,t denotes the total time length of synchronous and T+t0,t denotes the total time length of asynchronous of the switched systems. Moreover, the system states starting from ∪iΛvi,1 will remain inside the set ∪iΛvi,1+γχ.Proof.
Using a similar method to Theorem1, we have(42)x˜˙t0≽A˜1pqx˜t0+B˜1pqωt0,et0≽E˜1pqx˜t0+F˜1pqωt0.
Fort∈tl,tl+Δl, where(43)A˜1pq=Mq0Afp+BfpΦCq−MqAfp,B˜1pq=BqBfpΦDq−Bq,E˜1pq=Efp+FfpΦCq−EqEfp,F˜1pq=FfpΦDq−Bq,(44)x˜˙t0≥A˜1qx˜t0+B˜1qωt0,et0≥E˜1qx˜t0+F˜1qωt0.
Fort∈tl+Δl,tl+1, where(45)A˜1q=Mq0Afq+BfqΦCq−MqAfq,B˜1q=BqBfqΦDq−Bq,(46)E˜1q=Efq+FfqΦCq−EqEfq,F˜1q=FfqΦDq−Bq.
By equations (40a), (40b), (40h), and (40i), it follows that(47)∑ı=1n1nıθpı⊤1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≥0,∑ı=1n1nıηpı⊤1n⊤hpΦDq−Fq≥0,∑ı=1n1nıθqı⊤1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≥0,∑ı=1n1nıηqı⊤1n⊤hqΦDq−Fq≥0.
By equation (16), it is easy to get Afp+BfpΦCq−Mql≽0. Furthermore, Afp+BfpΦCq−∑l=12nℏlMql±0 and ∑l=12nℏlAfp+BfpΦCq−Mql≽0, which implies that Afp+BfpΦCq−Mq≽0. Then, BfpΦDq−Bq≽0,Efp+FfpΦCq−Eq≽0 and FfpΦDq−Bq≽0. Similarly, we can obtain Afq+BfqΦCq−Mq≽0,BfqΦDq−Bq≽0,Efq+FfqΦCq−Eq≽0 and FfqΦDq−Bq≽0. Thus, Mq, Afp, and Afq are Metzler matrices. Together with Bq,Efp,Efq≽0, and x˜t0≽0, it derives that the system equation (8) with ωt≠0 is positive.
Next, consider the stability of the system equation (8). From equation (23), we have(48)x˜˙t≺¯A˜2pqx˜t+B˜2pqωt,et≺¯E˜2pqx˜t+F˜2pqωt.
Fort∈tl,tl+Δl, where(49)A˜2pq=Mq0Afp+BfpΨCq−MqAfp,B˜2pq=BqBfpΨDq−Bq,E˜2pq=Efp+FfpΨCq−EqEfp,F˜2pq=FfpΨDq−Bq.
Whent∈tl+Δ1,tl+1, we have(50)x˜˙t≺¯A˜2qx˜t+B˜2qωt,et≺¯E˜2qx˜t+F˜2qωt,where(51)A˜2q=Mq0Afq+BfqΨCq−MqAfq,B˜2q=BqBfqΨDq−Bq,E˜2q=Efq+FfqΨCq−EqEfq,F˜2q=FfqΨDq−Fq.
Choose the same Lyapunov function as Theorem1. From equations (48), (50), and (23), we get(52)V˙t≤x˜⊤tMq⊤gp,q+Afp⊤+Cq⊤Ψ⊤Bfp⊤−Mq⊤hp,qAfp⊤hp,q+ω⊤tBq⊤gp,qDq⊤ΨBfp⊤−Bq⊤hp,q,t∈tl,tl+Δl,x˜⊤tMq⊤gq+Afq⊤+Cq⊤Ψ⊤Bfq⊤−Mq⊤hqAfq⊤hq+ω⊤tBq⊤gqDq⊤ΨBfq⊤−Bq⊤hq,t∈tl+Δl,tl+1.
DefineΓt=et1−γωt1. By equations (40e)–(40g) and 40l–(40n), we have(53)V˙t≤μ2Vt−Γt,t∈tl,tl+Δl,−μ1Vt−Γt,t∈tl+Δl,tl+1.
Integrating both sides of equation (53) gives(54)Vt≤eμ2t−tlVtl−∫tlteμ2t−sΓsds,t∈tl,tl+Δl,e−μ1t−tl−ΔlVtl+Δl−∫tl+Δlte−μ1t−sΓsds,t∈tl+Δl,tl+1,where Γs=es1−γωs1. For T∈tNσT,t0+ΔNσT,t0,tNσT,t0+1, using equations (40p) and (40q) and (54) follows that(55)Vσtℵ+1T≤λe−μ1T−tℵ−ΔℵVσtℵ+Δℵtℵ+Δℵ−∫tℵ+ΔℵTe−μ1T−sΓsds≤λ3e−μ1T−tℵ−1−Δℵ−Δℵ−1eμ2ΔℵVσtℵ−1+Δℵ−1tℵ−1+Δℵ−1−λ2eμ2Δℵ∫tℵ−1+Δℵ−1tℵe−μ1T−Δℵ−sΓsds−λe−μ1T−tℵ−Δℵ∫tℵtℵ+Δℵeμ2tℵ+Δℵ−sΓsds−∫tℵ+ΔℵTe−μ1T−sΓsds≤⋯≤e2ℵlnλe−μ1T−t0+μ1+μ2ℵΔmVσt0t0−∫t0Teμ2Nσs,TΔme−μ1T−t0−Nσs,TΔme2Nσs,TlnλΓsds,where ℵ=Nσt0,T and Δm is the maximum of time lag. Under zero initial conditions, we have(56)0≤−∫t0Teμ2Nσs,TΔme−μ1T−t0−Nσs,TΔme2Nσs,TlnλΓsds.
FromΓs=es1−γωs1, we have(57)∫t0Teμ2Nσs,TΔme−μ1T−t0−Nσs,TΔme2Nσs,Tlnλes1ds≤γ∫t0Teμ2Nσs,TΔme−μ1T−t0−Nσs,TΔme2Nσs,Tlnλws1ds.
Multiplying both sides of the inequality withe−μ1+μ2Nσt0,TΔme−2Nσt0,Tlnλ, we get(58)∫t0Te−μ1T−t0−μ1+μ2Nσt0,sΔm−2Nσt0,slnλes1ds≤γ∫t0Te−μ1T−t0−μ1+μ2Nσt0,sΔmωs1ds.
It follows from Definition2 and equation (41b) that(59)Nσt0,s≤N0+μ1∗s−t02lnλ+μ1+μ2Δm.
Then,(60)∫t0Te−μ1T−t0−μ1+μ2Δm+2lnλN0+μ1∗s−t02lnλ+μ1+μ2Δmes1ds≤γ∫t0Te−μ1T−toωs1ds.
Furthermore,(61)e−μ1+μ2Δm+2lnλN0∫t0Te−μ1∗s−t0es1ds≤γ∫t0Tωs1ds.
Integrating the above inequality at0,∞, it gives ϖ∫0∞e−ρtet1dt≤γ∫0∞ωt1dt, where ρ=μ1∗ and ϖ=e−μ1+μ2Δm+2lnλN0. By Definition 3, the system equation (8) achieves a prescribed L1-gain performance level γ.
Next, we present the invariance of the state. By equation (55), letting T⟶∞, we have V∞≤1+∫t0Tω⊤sds≤1+γχ. For ωt=0, Λvi,1⊆LHp. Using method of the Theorem 1, we have that the states will be kept in the set ∪pΛvp,1+γχ∪∪p,qΛvp,q,1+γχ for any initial conditions starting from ∪ivi,1+γχ.
In the following, the Zeno behavior is analyzed to avoid the continuous sampling and updating of the state under the event-triggering mechanism. Under the event-triggering mechanism equation (5), it gives e˙yt=−y˙t=−Cσtx˙t−Dσtw˙t,t∈tl′,tl+1′. Without generality, assume that w˙t exists and is bounded. Then, there exists a positive constant ℧ such that e˙yt1<℧ holds for t∈tl′,tl+1′. Then, eyt1<℧t−tl′ with eytl′1=0 and t∈tl′,tl+1′. This implies that t−tl′>eyt1/℧. Owing to xt≠0 and wt≠0, it is not hard to obtain eyt1>0 for t⟶tl+1′. This means that t>tl′. Therefore, the Zeno behavior can be avoided.Remark 4.
The 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 L1 norm of the external disturbance wt is bounded. Assume that the L1 norm is unbounded. From the last part of the proof in Theorem 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.
The time-triggered filter strategy was proposed for positive systems in [41–43]. In most cases, it is hard to reach the continuous sample owing to limited ability of elements and high sample cost. The event-triggered filter is more practicable with respect to the time-triggered filter. Thus, Theorems 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. Thus, 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.


## 3.2. Discrete-Time Case

For the discrete-time case, define the asynchronous time intervalkl,kl+Δl and the synchronous time interval kl+Δl,kl+1. In the interval kl,kl+1, the q th subsystems is active. In the interval kl,kl+Δl, the p th filter is active. In the interval kl+Δl,kl+1, the q th filter is active, where kl,l=0,1,⋯ is the switching time instant and Δl is the time lag between the two subsystems and Δl<kl+1−kl. When k∈kl,kl+Δl, the error system can be written as(62)x˜k+1=A˜pqx˜k+B˜pqωk+C˜peyk,ek=E˜pqx˜k+F˜pqωk+Ffpeyk,where(63)A˜pq=Mq0Afp+BfpCq−MqAfp,B˜pq=BqBfpDq−Bq,C˜p=0Bfp,E˜pq=Efp+FfpCq−EqEfp,F˜pq=FfpDq−Fq.Whenk∈kl+Δl,kl+1,(64)x˜k+1=A˜qx˜k+B˜qωk+C˜qeyk,ek=E˜qx˜k+F˜qωk+Ffqeyk,where(65)A˜q=Mq0Afq+BfqCq−MqAfq,B˜q=BqBfqDq−Bq,C˜q=0Bfq,E˜q=Efq+FfqCq−EqEfq,F˜q=FfqDq−Fq.Theorem 3.
If there exist constants0<μ1<1;μ2>1;λ>1; ℜn vectors gp,q≻0, hp,q≻0, gq≻0, hq≻0, θpı≻0, θp≻0, ξpı≻0, and ξp≻0; and ℜm vectors ηpı≻0, ηp≻0, ςpı≻0, and ςp≻0 such that(66a)∑ı=1n1nıθpı⊤1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≽0,(66b)∑j=1s1sjξpjT+∑j=1s1pjjςpjTΦCq−Eq≽0,(66c)Mq⊤gp,q+λθp+λCq⊤Ψ⊤ηp−λMq⊤hp−μ2gp,q≺0,(66d)λθp−μ2hp,q≺0,(66e)∑ı=1n1nıθqı⊤1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≽0,(66f)∑j=1s1sjξqiT+∑j=1s1sjςqjTΦCq−Eq≽0,(66g)Mq⊤gq+θq+Cq⊤Ψ⊤ηq−Mq⊤hq−μ1gq≺0,(66h)θq−μ1hq≺0,(66i)θpı≺θp,ηpı≺ηp,(66j)hp≺¯λhp,q,hp≺¯λhq,p,hp,q≺¯λhp,hq,p≺¯λhp,(66k)gp≺¯λgp,q,gp≺¯λgq,p,gp,q≺¯λgp,gq,p≺¯λgp,(66l)−gq≺¯Hp⊤≺¯gq,−hq≺¯Hp⊤≺¯hq,−gp,q≺¯Hp⊤≺¯gp,q,−hp,q≺¯Hp⊤≺¯hp,q,hold for l=1,2,…,2n, then the filter error system equation (8) with the event-triggered asynchronous filter equation (6) satisfying(67)Afp=∑ı=1n1nıθpı⊤1n⊤hp,Bfp=∑ı=1n1nıηpı⊤1n⊤hp,Efp=∑j=1s1sjξpjT,Ffp=∑j=1s1sjςpjT,is positive and stable, where Φ=I−β1m×m, Ψ=I+β1m×m, and the switching law satisfies(68a)T−k0,kT+k0,k≥lnμ2−lnμ1lnμ1∗−lnμ1,μ1∗∈μ1,1,(68b)τa≥τa∗=2lnλ−lnμ1∗,where T−k0,k denotes the total time length in the synchronous case and T+k0,k denotes the total time length in the asynchronous case. Moreover, all states will remain inside the cone set ∪pΛvp,1∪∪p,qΛvp,q,1 for xk0∈Λvi,1.Proof.
Forxk0±0 and yk0±0, it follows from equation (5) that −β1m×myk0≺¯eyk0≺¯β1m×myk0. By equations (62) and (64), we have(69)x˜k0+1≥A˜1pqx˜k0,ek0≥E˜1pqx˜k0,for k∈kl,kl+Δl, where(70)A˜1pq=Mq0Afp+BfpΦCq−MqAfp,E˜1pq=Efp+FfpΦCq−EqEfp.
Fork∈kl+Δl,kl+1,(71)x˜k0+1≽A˜1qx˜k0,ek0≽E˜1qx˜k0,where(72)A˜1q=Mq0Afq+BfqΦCq−MqAfq,E˜1q=Efq+FfqΦCq−EqEfq.
From equations (67a) and (67b), we have(73)∑ı=1n1nıθpı⊤1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≽0,∑ı=1n1nıθqı⊤1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≽0.
Then, we obtainAfp+BfpΦCq−Mql≽0. From ∑l=12nℏlAfp+BfpΦCq−Mql≽0 and Afp+BfpΦCq−Mq≽0. Similarly, Afq+BfqΦCq−Mq≽0. Together with equations (66b) and (66f), it follows that Efp+FfpΦCq−Eq≽0 and Efq+FfqΦCq−Eq≽0. It is easy to obtain Mq≽0,Afp≽0,Afq≽0,Efp≽0,andEfq≽0. By Theorem 1, the positivity of the system equation (8) is achieved. Thus,(74)−β1m×myk≺¯eyk≺¯β1m×myk.
Choose a multiple Lyapunov function candidate:(75)Vk=x˜k⊤vp,q,k∈kl,kl+Δl,x˜k⊤vq,k∈kl+Δl,kl+1,where vp,q=gp,qhp,q and vq=gqhq. From equation (74), we have(76)x˜k≺¯A˜2pqx˜k−1,ek≺¯E˜2pqx˜k,for k∈kl,kl+Δl, where A˜2pq=Mq0Afp+BfpΨCq−MqAfp,E˜2pq=Efp+FfpΨCq−EqEfp, and(77)x˜k≺¯A˜2qx˜k−1,ek≺¯E˜2qx˜k,for k∈kl+Δl,kl+1, where A˜2q=Mq0Afq+BfqΨCq−MqAfqandE˜2q=Efq+FfqΨCq−EqEfq. Then,(78)Vk≤x˜⊤k−1A˜2pq⊤vp,q,k∈kl,kl+Δl,x˜⊤k−1A˜2q⊤vq,k∈kl+Δl,kl+1.
Then,(79)Vk≤x˜⊤k−1Mq⊤gp,q+Afp⊤+Cq⊤Ψ⊤Bfp⊤−Mq⊤hp,qAfp⊤hp,q,k∈kl,kl+Δl,x˜⊤k−1Mq⊤gq+Afq⊤+Cq⊤Ψ⊤Bfq⊤−Mq⊤hqAfq⊤hq,k∈kl+Δl,kl+1.
Using equations (66i)–(66k) and λ>1, it can be derived that(80)Vk≤x˜⊤k−1Mq⊤gp,q+λθp−λhp+λCq⊤Ψ⊤ηp−λMq⊤hpλθp−λhp,k∈kl,kl+Δl,x˜⊤k−1Mq⊤gq+θq−hq+Cq⊤Ψ⊤ηq−Mq⊤hqθq−hq,k∈kl+Δl,kl+1.
By equations (66c), (66d), (66g), and (66h), we have(81)Vk≤μ2k−klVkl,k∈kl,kl+Δl,μ1k−kl−ΔlVkl+Δl,k∈kl+Δl,kl+1.
Together with equations (66j) and (66k) it yields that(82)Vk≤λμ2k−klVkl,k∈kl−1+Δl−1,klλμ1k−kl−ΔlVkl+Δl,k∈kl,kl+Δl.
ForT∈kNσT,k0+ΔNσT,k0,kNσT,k0+1, we can obtain(83)Vσkℵ+1T≤λμ1T−kℵ−ΔℵVσkℵ+Δℵkℵ+Δℵ≤λ2μ1T−kℵ−Δℵμ2ΔℵVσkℵkℵ≤⋯≤λ2ℵμ1T−k0−Δℵ−Δℵ−1−⋯−Δ1μ2Δℵ+Δℵ−1+⋯+Δ1Vσk0k0,where ℵ=NσT,k0. It can follow from equation (68a) that lnμ2−lnμ1T+k0,T≤lnμ1∗−lnμ1T−k0. By Definition 2 and equations (66a) and (83), it gives VT≤e2N0lnλelnμ1∗+2lnλ/τaT−k0Vk0. Then,(84)x˜T1≤ρ2e2N0lnλρ1elnμ1∗+2lnλ/τaT−k0x˜k01,where ρ1 and ρ2 are the minimal and maximal elements of vi∈vp,q,vq, respectively.
Finally, we provide the invariance of the state. Give any initial conditions satisfyingx˜⊤k0∈Λvi,1, that is, x˜⊤k0vi≤1. By equation (84), x˜⊤kvi≤x˜⊤k0vi≤1. So, x˜k∈Λvi,1. Given H˜p=HpHp, we have H˜plx˜k=x⊤kHpl⊤+xe⊤kHpl⊤≤x˜⊤tvi≤1, where H˜pl is the l th row of H˜p. By xk0≽0, xek0≽0, and equation (66l), it gives Hplxk≤H˜plx˜k≤1, which leads to x˜k∈∪pΛvp,1∪∪p,qΛvp,q,1. Then, xk∈LHp.
Next, we consider the filter design for the caseωk≠0. The disturbance is assumed to be located in the set:(85)W=ωk≽0:ωℓ1<∞.Theorem 4.
If there exist constants0<μ1<1;μ2>1;λ>1; ℜn vectors gp,q≻0, hp,q≻0, gq≻0, hq≻0, θpı≻0, θp≻0, ξpı≻0, and ξp≻0, and ℜm vectors ηpı≻0, ηp≻0, ςpı≻0,andςp≻0 such that(86a)∑ı=1n1nıθpı⊤+∑ı=1n1nıηpı⊤ΦCq−1n⊤hpMql≽0,(86b)∑i=1n1nıηpı⊤ΦDq−1n⊤hpBq≽0,(86c)∑j=1s1sjξpjT+∑j=1s1sjςpjTΦCq−Eq≽0,(86d)∑j=1s1sjςpjTΦDq−Fq≽0,(86e)Mq⊤gp,q+λθp+λCq⊤Ψηp−λMq⊤hp+Efp⊤1s+Cq⊤ΨFfp⊤1s−Eq⊤1s−μ2gp,q≺¯0,(86f)λθp+Efp⊤1s−μ2hp,q≺¯0,(86g)Bq⊤gp,q+λDq⊤Ψηp−λBq⊤hp+Dq⊤ΨFfp⊤1s−Fq⊤1s−γ1m≺¯0,(86h)∑ı=1n1nıθqı⊤+∑ı=1n1nıηqı⊤ΦCq−1n⊤hqMql≽0,(86i)∑ı=1n1nıηqı⊤ΦDq1n⊤hq−Bq≽0,(86j)∑j=1s1sjξqiT+∑j=1s1sjςqjTΦCq−Eq≽0,(86k)∑j=1s1sjςqjTΦDq−Fq≽0,(86l)Mq⊤gq+θq+Cq⊤Ψηq−Mq⊤hq+Efq⊤1s+Cq⊤ΨFfq⊤1s−Eq⊤1s−μ1gq≺¯0,(86m)θq+Efq⊤1s−μ1hq≺¯0,(86n)Bq⊤gq+Dq⊤Ψηq−Bq⊤hq+Dq⊤ΨFfq⊤1s−Fq⊤1s−γ1m≺¯0,(86o)θpı≺θp,ηpı≺ηp,(86p)hp≺¯λhp,q,hp≺¯λhq,p,hp,q≺¯λhp,hq,p≺¯λhp,(86q)gp≺¯λgp,q,gp≺¯λgq,p,gp,q≺¯λgp,gq,p≺¯λgp,(86r)−gq≺¯Hp⊤≺¯gq,−hq≺¯Hp⊤≺¯hq,−gp,q≺¯Hp⊤≺¯gp,q,−hp,q≺¯Hp⊤≺¯hp,q,hold for l=1,2,…,2n, then the filter error system equation (8) with equations (68a) and (68b) is positive and ℓ1 gain stable, where Φ=I−β1m×m and Ψ=I+β1m×m, and the switching law satisfies(87a)T−k0,kT+k0,k≥lnμ2−lnμ1lnμ1∗−lnμ1,μ1∗∈μ1,1,(87b)τa≥τa∗=2lnλ+lnμ2−lnμ1Δm−lnμ1∗,where T−k0,k denotes the total time length in the synchronous case and T+k0,k denotes the total time length in the asynchronous case. Moreover, the system states starting from ∪iΛvi,1+γχ will remain inside the set ∪pΛvp,1+γχ∪∪p,qΛvp,q,1+γχ with N0=0.Proof.
By Theorem3, it holds that(88)x˜k0+1≽A˜1pqx˜k0+B˜1pqωk0,ek0≽E˜1pqx˜k0+F˜1pqωk0,for k∈kl,kl+Δl, where(89)A˜1pq=Mq0Afp+BfpΦCq−MqAfp,B˜1pq=BqBfpΦDq−Bq,E˜1pq=Efp+FfpΦCq−EqEfp,F˜1pq=FfpΦDq−Fq.
Fork∈kl+Δl,kl+1,(90)x˜k0+1≽A˜1qx˜k0+B˜1qωk0,ek0≽E˜1qx˜k0+F˜1qωk0,where(91)A˜1q=Mq0Afq+BfqΦCq−MqAfq,B˜1q=BqBfqΦDq−Bq,E˜1q=Efq+FfqΦCq−EqEfq,F˜1q=FfqΦDq−Fq.
By equations (86a), (86b), (86h), and (86i), we have(92)∑ı=1n1nıθpı⊤1n⊤hp+∑ı=1n1nıηpı⊤ΦCq1n⊤hp−Mql≥0,∑ı=1n1nıηpı⊤1n⊤hpΦDq−Fq≥0,∑ı=1n1nıθqı⊤1n⊤hq+∑ı=1n1nıηqı⊤ΦCq1n⊤hq−Mql≥0,∑ı=1n1nıηqı⊤1n⊤hqΦDq−Fq≥0.
By equations (68a) and (68b), it follows that Afp+BfpΦCq−Mql≽0. From ∑l=12nℏlAfp+BfpΦCq−Mql≽0, we can obtain Afp+BfpΦCq−Mq≽0. Similarly, Afq+BfqΦCq−Mq≽0. In addition, it is clear that BfpΦDq−Bq≽0, Efp+FfpΦCq−Eq≽0, FfpΦDq−Fq≽0, BfqΦDq−Bq≽0, Efq+FfqΦCq−Eq≽0, and FfqΦDq−Fq≽0. It is easy to get that Mq≻0,Afp≻0,Afq≻0,Bq≻0,Efp≻0,andEfq≻0. Then, the system equation (8) with ωt≠0 is positive.
From equation (74), we have(93)x˜k+1≺¯A˜2pqx˜k+B˜2pqωk,ek≺¯E˜2pqx˜k+F˜2pqωk,for k∈kl,kl+Δl. When k∈kl+Δ1,kl+1, we have(94)x˜k+1≺¯A˜2qx˜k+B˜2qωk,ek≺¯E˜2qx˜k+F˜2qωk,where E˜2pq=Efp+FfpΨCq−EqEfp,F2pq=FfpΨDq−Fq,E˜2q=Efq+FfqΨCq−EqEfq,F2q=FfqΨDq−Fq.
Choose the same Lyapunov function as Theorem3. From equations (93), (94), and (74), we get(95)Vk+1≤x˜⊤kMq⊤gp,q+Afp⊤+Cq⊤Ψ⊤Bfp⊤−Mq⊤hp,qAfp⊤hp,q+ω⊤kBq⊤gp,qDq⊤ΨBfp⊤−Bq⊤hp,q,k∈kl,kl+Δl,x˜⊤kMq⊤gq+Afq⊤+Cq⊤Ψ⊤Bfq⊤−Mq⊤hqAfq⊤hq+ω⊤kBq⊤gqDq⊤ΨBfq⊤−Bq⊤hq,k∈kl+Δl,kl+1.
LetΓk=ek1−γωk1. From equations (86e)–(86g) and (86l)–(86n), it follows that(96)Vk≤μ2Vk−1−Γk−1,k∈kl,kl+Δl,μ1Vk−1−Γk−1,k∈kl+Δl,kl+1.
Thus,(97)Vk≤μ2k−klVkl−∑s=klk−1μ2k−s−1Γs,k∈kl,kl+Δl,μ1k−kl−ΔlVkl+Δl−∑s=kl+Δlk−1μ1k−s−1Γs,k∈kl+Δl,kl+1),where Γs=e˜s1−γωs1. For T∈kNσk0,T+ΔNσk0,T,kNσk0,T+1,(98)VT≤μ1T−kNσk0,T−ΔNσk0,TVkNσk0,T+ΔNσk0,T−∑s=kNσk0,T+ΔNσk0,TT−1μ1T−s−1Γs≤λeT−kNσk0,T−ΔNσk0,Tlnμ1eΔNσk0,Tlnμ2VkNσk0,T−λeT−kNσk0,T−ΔNσk0,Tlnμ1×∑s=kNσk0,TkNσk0,T+ΔNσk0,T−1ekNσk0,T+ΔNσk0,T−s−1lnμ2Γs−∑s=kNσk0,T+ΔNσk0,TT−1eT−s−1lnμ1≤⋯≤e2Nσk0,TlnλeT−k0−Nσk0,TΔmlnμ1eNσk0,TΔmlnμ2Vk0−∑k0T−1e2Nσ\s,T−1lnλeNσs,T−1Δmlnμ2eT−1−k0−Nσs,T−1Δmlnμ1Γs,where Δm denotes the maximum of time lag. Under zero initial condition, we have(99)0≤−∑k0=0Me2Nσs,MlnλeNσs,MΔmlnμ2eM−k0−Nσs,MΔmlnμ1Γs,where M=T−1. By Γs=es1−γωs1, it holds that(100)∑k0Me2Nσs,MlnλeNσs,MΔmlnμ2eM−k0−Nσs,MΔmlnμ1es1≤γ∑k0Me2Nσs,MlnλeNσs,MΔmlnμ2eM−k0−Nσs,MΔmlnμ1ωs1.
Multiplying both sides of the inequality with(101)∑k0Melnμ1M−k0+lnμ1−lnμ2Nσk0,sΔm−2Nσk0,slnλes1≤γ∑k0Melnμ1M−k0+lnμ1−lnμ2Nσk0,sΔmωs1.
It follows from Definition2 and equation (87b) that Nσk0,s≤N0−lnμ1∗s−k0/2lnλ+lnμ2−lnμ1Δm. Then,(102)∑k0Melnμ1M−k0−2lnλ+lnμ2−lnμ1ΔmN0−lnμ1∗s−k02lnλ+lnμ2−lnμ1Δmes1≤γ∑k0Melnμ1M−k0ωs1.
Furthermore,(103)e−2lnλ+lnμ2−lnμ1ΔmN0∑k0Melnμ1∗s−k0‖es‖1≤γ∑k0M‖ωs‖1.
Summing the above inequality from 0 to∞, it follows that ϖ∑0∞eρkek1≤γ∑0∞ωk1, where ρ=lnμ1∗ and ϖ=e−2lnλ+lnμ2−lnμ1ΔmN0. By Definition 3, the system equation (8) achieves a prescribed ℓ1-gain performance level γ.
Next, we present the invariance of the state. By equation (98), we have V∞≤1+γ∑k0Mωs1≤1+γχ. For ωk=0, Λvi,1⊆LHp. Using a similar method to Theorem 3, we have that the states will be kept in the set ∪pΛvp,1+γχ∪∪p,qΛvp,q,1+γχ for any initial conditions starting from ∪ivi,1+γχ with ωk≠0.Remark 7.
This 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. This paper employs the convex polytopic approach for dealing with the state saturation term. Thus, the saturation term is transformed into polytopic, which is dependent on2n 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. This paper considers an asynchronous even-triggered filter of positive switched systems. Multiple copositive Lyapunov functions are constructed in this paper, and the ℓ1-gain stability for positive switched systems is achieved. Compared with positive systems, the ℓ1-gain stability of positive switched systems is more complex. Then, 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.


## 4. 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. Therefore, 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. This is a typical saturation phenomenon. Figure 1 shows the diagram of a water supply network. The pumps are used to describe the switched rule. The water system in Figure 1 is modeled by the system equation (1), where xt (or, xk) denotes the water volume in the tank at time t (or, the k th sample instant), ωt (or, ωk) represents the wasted water from the pumps and valves, yt (or, yk) is the measured value by the sensor, and zt (or, zk) is the output to be estimated. Figure 2 shows the states of the plant and the filter, where V, Vk, qin, and qout 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.Figure 1
A mode of water system.Figure 2
The states of the plant and the filter.Example 1.
Consider system equation (8) with(104)A1=−1.70.10.30−1.50.70.50−1.6,B1=0.020.010.030.010.010.03,C1=0.010.010.010.010.010.01,D1=0.010.010.010.01,E1=0.010.010.01,F1=0.010.01,A2=−1.60.70.30−1.80.700.1−1.7,B2=0.020.010.020.020.030.01,C2=0.020.030.02000.02,D2=0.020.010.010.01,E2=0.010.010.01,F2=0.010.01.
Letα=1.8, μ1=0.1, μ2=0.3, λ=1.1, and γ=0.2. The event-triggered parameter is β=0.1, and the external disturbance input is wt=0.012e−0.02tsin0.4t+1+10.012e−0.03tcos0.3t+1+1⊤. By Theorem 2, the filter gain matrices are(105)Af1=−1.70790.09210.09210.0921−1.70790.09210.09210.0921−1.6691,Bf1=56.991251.491251.491251.491239.649251.4912,Ef1=0.00700.00700.0070,Ff1=0.77500.4750,Af2=−1.66910.08690.08690.0869−1.71310.08690.08690.0869−1.7131,Bf2=54.571349.071354.571349.071349.071349.0713,Ef2=0.00700.00700.0070,Ff2=0.78750.5875.
Figures3 and 4 are the asynchronous switch signal and the error of zt and zft, respectively. The external disturbance input signal is given in Figure 5. Figure 6 is the event-triggering release interval. Figure 7 shows the simulation that includes the cone domain and the state trajectories.Figure 3
The switching signal withADT=2 and Δm=0.2.Figure 4
The error signalet.Figure 5
The external disturbance input signal in Theorem2.Figure 6
Release instants and release intervals.Figure 7
The cone domain and the state trajectories.Example 2.
Consider system equation (8) with(106)A1=0.30.3000.60.1000.4,B1=0.020.010.030.010.010.03,C1=0.010.010.010.010.010.01,D1=0.010.010.010.01,E1=0.020.010.01,F1=0.010.02,A2=0.50.7000.40000.4,B2=0.020.010.020.020.030.01,C2=0.020.030.02000.02,D2=0.020.010.010.01,E2=0.010.020.01,F2=0.020.01.
Given constantsμ1=0.7,μ2=1.9,λ=1.1,andγ=0.2, the event-triggering threshold is β=0.1, and the external disturbance is wk=0.012e−0.02ksin0.4k+1+10.012e−0.03kcos0.3k+1+1⊤. By Theorem 1, the filter gain matrices are(107)Af1=0.33330.35120.33330.33330.35600.33330.33330.33330.3333,Bf1=0.91670.33330.33332.16671.62750.3333,Ef1=0.01000.01000.0100,Ff1=0.62921.9958,Af2=0.31150.33180.31150.31150.35600.31150.31150.31150.3115,Bf2=1.63640.31150.31152.18851.63640.3115,Ef2=0.01000.01000.0100,Ff2=0.63921.9958.
The asynchronous switch signal and the error signalek of zk and zfk are given in Figures 8 and 9, respectively. The external disturbance input signal is given in Figure 10. Figure 11 shows the event-triggering release interval. Figure 12 shows the simulations of the cone domain and the state trajectories.Figure 8
The witching signal withADT=2 and Δm=0.2.Figure 9
The error signalek.Figure 10
The external disturbance input signal.Figure 11
Release instants and release intervals.Figure 12
The cone domain and the state trajectories.


## 5. Conclusions

This 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. The obtained design is also developed for the discrete-time case.



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