Design of Event-Triggered Asynchronous H∞ Filter for Switched Systems Using the Sampled-Data Approach

The design of networked switched systems with event-based communication is attractive due to its potential to save bandwidth and energy. However, ensuring the stability and performance of networked systems with event-triggered communication and asynchronous switching is challenging due to their time-varying nature. This paper presents a novel sampled-data approach to design event-triggered asynchronous <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}_{\infty} $ </tex-math></inline-formula> filters for networked switched systems. Unlike most existing event-based filtering results, which either design the event-triggering scheme only or co-design the event-triggering condition and the filter, we consider that the event-triggering policy is predefined and synthesize the filter. We model the estimation error system as an event-triggered switched system with time delay and non-uniform sampling. By implementing a delay-dependent multiple Lyapunov method, we derive sufficient conditions to ensure the global asymptotic stability of the filtering error system and an <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}_{\infty} $ </tex-math></inline-formula> performance level. The efficacy of the proposed design technique and the superiority of the filter performance is illustrated by numerical examples and by comparing the performance with a recent result.

Networked control is a control scheme where measurement feedback and control actions route through a communication network. Such a scheme offers many advantages, such as low installation cost, reduced maintenance cost, and flexible system structure [9]. Networked control of switched dynamical The associate editor coordinating the review of this manuscript and approving it for publication was Engang Tian . systems is an active research area, with many researchers addressing problems in this area, see for example [10] and [11]. Existing results on networked control of switched systems either consider problems in the continuous time or use a periodic sampling scheme [12]. However, periodic sampling often results in the wastage of network resources. An alternative to periodic sampling is event-based sampling, which saves energy and communication resources [13]. Recent approaches to event-triggered control use adaptive event-triggering schemes that offer a further reduction in the usage of network resources [14], [15], [16].
Filtering or state estimation problem is an important one in control systems. A reason for that is many advance control algorithms are based on state feedback. However, not all state variables can be measured. A filter or a state estimator then provides an estimate of the state variables for feedback.
Monitoring systems also use an estimate of the state variables. Two popular algorithms for filtering or state estimation in control systems are Kalman filtering and H ∞ filtering [17]. Conventional implementation of these algorithms use periodic sampling. This work is focused on event-based H ∞ filtering or state estimation.
Event-based filtering for networked switched systems has been addressed in [6], [18], [19], [20], and [21]. In [6], an asynchronous finite-time stable filter is devised for continuous-time switched systems. In [18], an event-based H ∞ filtering problem is addressed for networked switching systems in continuous time domain. Finite-time stability and H ∞ performance of the filtering error system are proved by using a delay-dependent Lyapunov functional. In [19], an event-triggered H ∞ filter is designed to ensure stability in the presence of packet disorders and maintain an H ∞ performance level. In [20], an event-triggered fault detection H ∞ filter is designed. A discrete-time event-triggered H ∞ filter is designed in [21]. By using multiple Lyapunov function approach, sufficient conditions are derived for exponential stability and H ∞ performance of the error system. It is pertinent to note that all these works follow the co-design framework where the event-triggering conditions and filter parameters are designed simultaneously [22]. However, there are situations where an event-generator is predefined, such as hardware-based event triggers, and it is not possible to redesign them. In such situations, the goal is to design the filter parameters such that the filtering error system remains stable and satisfies the given performance criterion. To the author's best knowledge, this problem has not been addressed in the literature.
To fill this research gap, we propose a technique for the event-based H ∞ filter design where an event-triggering policy is pre-defined. This type of filter design problem can be addressed using the sampled-data systems framework [23]. In particular, we view the event-based filter as a sampled-data system with non-uniformly sampled measurements [24]. The filtering error system can then be modeled as a switched dynamical systems with time delay to cater for the non-uniform sampling of measurement. The stability and performance of the error system can be analysed using the tools developed for switched time-delay systems.
The main contributions of this paper are as follows.
1) A novel technique to design an event-triggered H ∞ filter is proposed for networked switched systems when an event-triggering policy is predefined. We show that the estimation error system can be modelled as switched system with time delay and non-unform sampling. 2) We show utilization of a delay-dependent multiple Lyapunov functional to develop sufficient exponential stability criterion of the error system and the H ∞ performance.
The rest of this paper is organized as follows. In the next section, the problem is formulated. In the following section, the key results are presented to analyse filter performance and design it. Then, two simulation examples are given to show the efficacy of the presented approach. Finally, the paper is concluded in the last section.
Notation: In this paper, R denotes the set of real numbers and Z denotes the set of integers. R n is the n-dimensional Euclidean space. For a matrix P ∈ R n×n , P > 0 means P is positive definite, P T denotes the transpose of matrix P, and l 2 [0, ∞) represents sequences with finite 2-norm ∥.∥ 2 . For x ∈ R n and S ∈ R n×n , ∥x∥ 2 S = x T Sx is the weighted norm of x. The acronym LTI stands for linear, time-invariant.

II. PROBLEM FORMULATION
Consider the filtering scenario as shown in Figure 1. The components of the filtering system are described as follows.

A. SYSTEM
The system is described by a discrete LTI model This model is obtained by sampling the original continuoustime state-space model of the switched system at sampling instants t k , where t k = kh, k ∈ Z, x k = x(t k ) ∈ R n is the state of the system, y k = y(t k ) ∈ R q is the output of the system, and w k = w(t k ) ∈ R p is the disturbance signal. We assume that w k ∈ l 2 [0, ∞), where l 2 is the space of finite-energy sequences. A sequence {f k } k≥0 belongs to l 2 if . . , N } represents the switching rule and N ≥ 1 represents the number of subsystems. At any instant t k , θ k depends on k or x k or some other switching rule. For all θ k ∈ I N , we have the following matrices A θ k , B θ k , C θ k and D θ k with compatible dimensions. Furthermore, the matrices corresponding to θ k = i ∈ I N are denoted by In this paper, we study switched systems of the form in (1) with Average Dwell Time (ADT) switching. The system can switch from one mode to another arbitrarily, but the switching is slow. The dwell time between mode switches is no less than the ADT. Definition 1: For k > k 0 , let N θ (k, k 0 ) denotes the number of switchings of θ k over the interval [k 0 , k). If there exist VOLUME 11, 2023 N 0 > 0 and τ a > 0 such that then N 0 and τ a are called the chatter bound and average dwell time of the switching signal θ k .

B. EVENT-TRIGGERED SAMPLING
The measurement y k is sampled at sampling instants t k and is fed to the event-generator. The event-generator uses a predefined criterion to determine whether to communicate the measurement or not. Let {τ k } k≥0 , k ∈ Z be the instants when the measurement is transmitted. The time instants τ k are called event triggering instants. The event triggering instants are decided according to a given event triggering policy L(x k , y k , k).
with τ 0 = 0 and M ∈ Z. The constant M is chosen in such a way that M ≤ τ a , i.e., there is at most one switching in event-triggering instant. Notice that τ k forms an increasing sequence, that is τ 0 < τ 1 < τ 2 < . . . < τ k , therefore, the Zeno phenomenon does not occur. We assume that Assumption 1: τ k = mt k where m, k ≥ 0 and m, k ∈ Z.
Assumption 1 implies that the event-triggering instants commensurate with the measurement sampling instants. This is a realistic assumption because the event-generator activates at each measurement sampling instants to decide whether to discard or transmit the measurement. Assumption 2 implies that the measurement from the process will be transmitted at least after M periods. Event-triggered sampling with an upper bound on the sampling interval has already been used in the literature, see for example [21]. This sampling technique is a hybridization of the two sampling techniques; the event triggered-sampling and self-triggered sampling [25]. It provides a reasonable compromise between the utilization of network bandwidth and system performance by ensuring that fresh data is communicated to the filter after M sampling intervals.
The event-triggering policy L(x k , y k , k) could be a function of state, output, or time. Various event-triggered sampling policies have been proposed in the literature [22]. Some examples are: where 0 < σ < 1 is a scalar, e k = x τ k − x k ( or y τ k − y k ) is the error since last transmission, α, and γ are suitable functions.

C. FILTER
We take the filter structure to bê wherex k ∈ R n is the state of the filter,ȳ k ∈ R p is the input to the filter andŷ k ∈ R q is the filter output. The matrices , and D f θ ′ k are the filter design parameters and θ ′ k is the switching signal of the filter. Again, for the sake of brevity, we denote the filter parameters The switching signals θ k and θ ′ k are in general not synchronized. There is generally some lag between the switching of the system mode and the corresponding filter. Let T max be the maximum lag between θ k and θ k ′ , that is, A larger value of T max means that the activated system and filter modes are not synchronised for longer time. Also, due the event-triggered sampling, the input to the filter is intermittent and non-uniform. That is A longer interval may cause the estimation error to diverge due to the presence of disturbance and asynchronous switching. This inherently presents a compromise between the duration of the event-triggering interval and disturbance attenuation to maintain the state estimation quality at a desired level. From the sampled-data control theory point of view, the event-triggered filtering problem with a specified event-triggering policy can be viewed as a filtering problem with non-uniform sampling [24].

D. ERROR SYSTEM
Letz = y k -ŷ k denotes the estimation error. From assumption 2, the input to the filter during each transmission interval will beȳ The transmission interval can be partitioned as . Different system and filter modes are activated when k ∈ [τ k , τ k + T max ). Let θ k = i, θ ′ k = j, and φ k = 0, the dynamics of the filtering error system will be x k+1 When k ∈ [τ k + T max , τ k+1 ), the system and filter modes will be synchronized. Let θ k = θ ′ k = i, and φ k = 0, the dynamics of the filtering error system will be When φ k = 1, θ k = i, and θ ′ k = j, the dynamics of the error system will be Similarly, when φ k = 1, θ k = θ ′ k = i, the dynamics of the error system will be In general, the dynamics of the error system can be written as where Notice that the error system is a switched time-delay system where the time-dependent delay φ k takes values in the interval I φ .

E. PROBLEM STATEMENT
In this paper, we consider the following problem: Given the system in (1) and event-triggered sampling in (3), satisfying assumptions 1 and 2, design the filter parameters in (4), such that the error system in (7) is • globally uniformly asymptotically stable withw k = 0, • when initially relaxed, the H ∞ -norm of the error system is less than γ , where γ > 0 is a positive scalar. Definition 2 [26]: Whenw k = 0, the system in (7) is said to be globally uniformly asymptotically stable if, for all switching signals θ k and θ ′ k , its solutions satisfy . . , k 0 and ∥ψ l ∥ = sup k 0 −M ≤l≤k 0 ∥ψ(l)∥. VOLUME 11, 2023 Definition 3: The error system in (7) is said to have H ∞ performance level γ > 0, if it is globally asymptotically stable and under zero initial conditions, the l 2 gain satisfies When there is no event-triggered sampling, we can write the system in (7) as The stability and H ∞ performance of this system can be analyzed by using the following results from [26] and [5]. Lemma 1: Consider the error system in (8) with w k = 0 and given 0 < α < 1, µ ≥ 1, and β ≥ 0. Let there be C 1 functions V θ k : R n → R, θ k ∈ I N , and κ 1 and κ 2 belonging then the estimation error system in (8) is globally uniformly asymptotically stable for any θ k with ADT τ a given as Lemma 2: Consider the error system in (8) and let 0 < α < 1, µ ≥ 1, and β ≥ 0 be given constants. Suppose there exist positive definite functions where (z k , w k ) ≡z T kz k −γ 2 i w T k w k , then the switched system is globally asymptotically stable for any θ k satisfying (2) and has an l 2 -gain no greater that γ * = max{ . Note that the Lyapunov function can increase with a bounded rate during the asynchronous periods. This allows to capture the impact of asynchronous switching between the system and filter modes on the error system stability and H ∞ performance. Lemma 1 and Lemma 2 consider the impact of asynchronous switching, but do not consider the impact of event-triggered sampling. Our goal in this manuscript is to present improved filter design criteria that incorporate both the aforementioned phenomena.

III. MAIN RESULTS
In this section, we present the main results. First, we develop the conditions under which the filtering error system will be globally asymptotically stable and will have the desired H ∞ performance level. This is accomplished by using a delay-dependent multiple Lyapunov functional. Next, we convert these analysis conditions to synthesis conditions by appropriately partitioning the Lyapunov matrices.
A. H ∞ FILTER PERFORMANCE ANALYSIS Theorem 1: Consider 0 < α < 1, µ ≥ 1, and β ≥ 0 are specified, the error dynamics in (7) are globally uniformly asymptotically stable and has H ∞ performance γ * , where γ * = max{ if one can find matrices P i > 0 and Q r,i > 0 for all i, j ∈ I N , i ̸ = j, and r ∈ I φ such that P i ≤ µP j , Q r,i ≤ µQ r,j and the conditions in (9) and (10) hold wherẽ VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Proof: First of all the asymptotic stability of (7) is proved by considering the following switched Lyapunov function Withw k = 0, the system in (7) becomes For θ k = i and θ k+1 = j, taking the increment of the Lyapunov function V i (η k ) = V i (η k+1 ) − V i (η k ) along the state path of the estimation error system, we obtain With further manipulation, we can write where T . If the inequalities in (9) and (10) hold, thenÃ T ijr P jÃijr + ir ≤ 0 andÃ T ijr P jÃijr + ir ≤ 0 for i, j ∈ I N and r ∈ I φ . Therefore, the error system will be asymptotically stable. Now for k ∈ [τ k , τ k + T max ) Using (12)- (14), we can write where (9) and (10) hold, then from (15) This completes the proof.
The inequalities in (9) and (10) involves the matrix product of different system modes. It becomes hard to convert them into design conditions. This difficulty can be overcome by using the approach given in [27].
Lemma 3: The filtering error dynamics in (7) are asymptotically stable and have ∥z k ∥ 2 ≤ γ * ∥w k ∥ 2 where γ * = max{ √ T max −1 γ i }, if one can find P i > 0 and Q r,i > 0, R i for all i, j ∈ I N , i ̸ = j and r ∈ I φ such that P i ≤ µP j , Q r,i ≤ µQ r,j and the conditions in (16) and (17) hold: Proof: We prove (16) only. The proof of (17) is similar. Note that VOLUME 11, 2023 If the inequality in (17) holds, then Pre-and post-multiplying the above equation with diag{R −1 i , I , I , I } and diag{R −T i , I , I , I } and then pre-and post-multiplying with diag{P i , I , I , I } and diag{P i , I , I , I }, yields (9), thus the proof is concluded.

B. H ∞ FILTER DESIGN
In this section, we provide a theorem that establishes conditions such that a solution to the H ∞ filtering problem exists, and a filter can be designed.
Theorem 2: Consider 0 < α < 1, µ ≥ 1, and β ≥ 0 for the system in (1) be given, if one can matrices P 1i > 0, P 3i > 0, Q 1r,i > 0, Q 3r,i > 0, and P 2i , ϒ ijr and ϒ ir are partitions of ijr , ir , and ϵ r , defined in (9). Then, the error system in (7) will be asymptotically stable with H ∞ performance level γ * = max{ The corresponding filter parameters are given by Proof: Take matrices P i , R i and Q r,i in (16) as Inserting them in (16), we get Inserting these expressions, we get (18). The inequality in (19) can be derived similarly. This completes the proof.

IV. SIMULATION RESULTS
In this section, we give two examples to show the utility of the proposed technique. Example 1 demonstrate the effectiveness of the technique in reducing the utilization of network bandwidth while Example 2 compares the performance of the proposed technique with an existing result.

A. EXAMPLE 1: EVENT-TRIGGERED FILTER DESIGN FOR PWM-DRIVEN DC-DC BOOST CONVERTER
Consider the Pulse Width Modulation (PWM)-driven DC-DC boost converter studied in [4], [28], and [21]. A schematic of the converter circuit is shown in Figure 2. A PWM signal drives the switch ξ (t) with a period of T seconds. The circuit has a source voltage e(t), resistance R, capacitance C, and inductance L. Also, i L (t) is the current through the inductor, e C (t) is the voltage across the capacitor, i o (t) is the current through the resistor, and v o (t) is the voltage across the resistor. The dynamics of this circuit are governed by the following differential equations.
where s = t.T , C 1 = CT −1 , and L 1 = LT −1 . The dynamics in (20) can be written in switched form aṡ The switching signal is defined as Using similar approach as in [28], we write matrices A c 1 and A c 2 in the normalized form. The parameter α controls the rate of decrease of the Lyapunov during the matched (synchronous) period and β controls the rate of increase during the mismatch (asynchronous) period. For slow switching, α and β have small values. The parameter µ controls the increase in Lyapunov function at switching instants. We choose µ to ensure that the value of multiple Lyapunov functional forms a decreasing sequence. The attained value of H ∞ performance level is γ * = 2.4689. The event-triggered sampling policy is taken as where e k = y(k) − y(τ k ) and η = 0.2. Let the disturbance input w k be w k = 0.1 exp(−0.04k) sin(0.1πk). Figure 3 shows event-triggered sampling of the system measurement. The corresponding event-triggering instants are shown in Figure 4. As seen in Figure 4, the measurement may be transmitted after one sampling period or two sampling periods, if the event-triggering condition is satisfied, However, it is certainly transmitted after three sampling periods. VOLUME 11, 2023    Also, as expected, the number of measurement transmissions is reduced as the filtering error converges to zero. There are 73 data transmissions during 100 sampling intervals. That means that the data transmission is reduced by 27%. The estimation error response is shown in Figure 5. Clearly, the error converges to zero. The corresponding system and filter modes are shown in Figure 6. There is a maximum lag of two sampling intervals between the system and filter modes. The switching signal has an average dwell time of 2. The convergence of the error under average dwell time switching and the significant reduction in data transmission rate demonstrate that the approach is effective. This system was considered in [21] where the authors designed a switched H ∞ filter for this system with M = 3 (cf. τ d = 3 in [21]). Their attainable weighted H ∞ performance level was γ = 6.8761. For the same system, using Theorem 2 with α = 0.02, β = 0.25, µ = 1.02, and T max = 0, we design the filter parameters and the same event-triggered sampling policy as in [21].
L(x k , y k , k) = e T k i e k − y T k i y k with 1 = 1.4690, 2 = 1.4989 1 = 0.5096, 2 = 0.5001 Fig. 7 shows the actual system output and outputs estimated by the proposed filter and the filter in [21]. It can be seen that the output estimated by the proposed filter is closer to the actual system output.   The inputs to the filters is shown in Figure 8. The duration of inter-event data transmission intervals are shown in Fig. 9.  The duration varies from 0 to 2 indicating that the measurement may be transmitted after one, two, or three sampling periods, if the event-triggering condition is satisfied or if the system is self-triggered. Also, as expected, the number of measurement is reduced as the actual and filtered outputs converge. The corresponding system modes are shown in Figure 10.
To quantify the H ∞ performance of both filters, we compute the performance index as Figure 11 shows a plot of the H ∞ performance index γ for both filters. The performance index of the proposed filter is about half of the performance index of the filter in [21]. The computed performance indices of both filter in 60 samples simulation are shown in Table 1. Both filters have performance indices smaller than their computed worst-case bounds; however, the computed bound for the proposed filter is much small than the filter in [21]. These results clearly demonstrate that the filter designed using the proposed tech- nique exhibits improved performance as compared to the existing results.

V. CONCLUSION
We have presented a technique for event-triggered H ∞ filtering when an event-triggering policy already exists or is prespecified. We have shown that this problem can be viewed as a state estimation problem with non-uniform sampling. Thus, the filtering error system is modelled as a delay-dependent switched system with non-uniformly sampled measurements. By using the multiple Lyapunov method, the filter design conditions are given as linear matrix inequalities that can be easily solved using modern solvers. In the end, two numerical examples are given to show the efficacy of the proposed approach and to compare results with the existing ones in the literature. The approach is applicable to event-triggered systems where an event-generator is predefined, such as systems with hardware-based event-generators. A limitation of the method is that the parameter M is chosen manually to adapt to the behavior of the event-generator. In the future, we plan to extend this work to systems with modeling uncertainties and automatic selection of the parameter M .