# Event-Based State Estimation for Networked Singularly Perturbed Complex Networks

Authors: Zerong Ren
Journal: Complexity
 (2022)

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## Abstract

This paper deals with the multievent-triggering-based state estimation for a class of discrete-time networked singularly perturbed complex networks (SPCNs). A small singularly perturbed scalar is adopted to establish a discrete-time SPCNs model. To reduce the communication burdens, the data transmission between the sensor and the estimator is managed by a multievent generator function. Depending on the singularly-perturbed-based Lyapunov theory, a sufficient condition is constructed to guarantee that the estimation error is exponentially ultimately bounded in the mean square. Finally, the validity of the developed result is demonstrated by a simulation example.


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## Body

## 1. Introduction

A complex network is a set of interconnected nodes coupled by certain network topology, each node of which can be considered as a class of dynamic subsystems. Owing to its complex inherent structure, most systems in real life can be regarded simply as complex networks, including, but not limited to social networks, biological networks, power grid networks, and Internet [1–3]. Consequently, considerable research interest has been stirred over the past few decades and there has been a host of meaningful published achievements of complex networks [4–9].As far as we know, however, the complex networks with two-time scales receive little attention. However, the two-time scale case of many real-life complex networks [10–12] is continually encountered. For instance, the circuit state variables become faster than the mechanical state variables in electronic power grids, due to the difference in the time scalars on the circuit and the mechanical systems [10]. This can result in the appearance of diverse time-scale subsystems in hosts of electromechanical systems, named fast and slow dynamics. In most of the existing literature [13–15], a singularly perturbed approach is adopted to deal with the two-time scale phenomenon of these real-life systems. In other words, the fast-slow subsystem is distinguished by introducing a small singularly perturbed scalar. Hence, such complex networks can be regarded as SPCNs [16–23].What is worth mentioning is that a host of the reporting efforts [16–20] merely focuses on the synchronization phenomenon of the SPCNs. However, in some real-world scenarios, the exact state of the SPCNs on account of various factors, like the high number of nodes, disturbances in all directions, and high dimensions, is unavailable [7]. Thus, what we should pay attention to is the state estimation of the SPCNs. On the other hand, we noticed that besides [22, 23], the discrete-time SPCNs get little research concerns. The two important reasons for considering the discrete-time SPCNs are that computational simulation and network communication. Therefore, it is very necessary to investigate the state estimation of the discrete-time SPCNs.In addition, increasing attention is devoted to the event-triggered protocol (ETP), in which the current packet is released if the ETP-based triggering condition is satisfied [24–26]. Past years have witnessed an increasing interest in ETPs, including static ETPs, dynamic ETPs, and memory ETPs [27–34]. It is noted that in the above-referred ETPs, the triggering parameter is assumed to be the same for all dynamic outputs/states. Nevertheless, such an assumption is difficult to be satisfied, especially in multisensor networks, which contributes to the varying triggering parameters. In light of above-discussed phenomenon, in this work, in order to save resource consumption and solve the communication congestion, a multi-ETP is employed to deal with the large information communication among nodes of the discrete-time SPCNs, which, to some extent, promotes the current research. As such, a natural and interesting question is how to design a proper multi-ETP for discrete-time SPCNs.Based on the aforementioned observations, we try our best to develop the multi-ETP-based estimator design issue for the discrete-time SPCNs. Then, the mean square exponential bounded and state estimations are studied by using the Lyapunov function dependent on a singular perturbed parameter. In the end, a numerical example is presented to prove the effectiveness of the state estimator design method. It is worth emphasizing that even though the discrete-time SPCNs are unstable, the result of this work is still efficient. The highlights of our contributions are outlined as follows: (1) A nonlinear discrete-time SPCNs model is developed, which includes nonlinearities of complex networks and multitime scales. (2) As the study progress, the multi-ETP-based state estimation problem for the discrete-time SPCNs with nonlinearities is considered.Notation:ℕ refers to a set of all nonnegative integers. Z⊤ represents the transpose of the matrix Z. diag⋅ symbolizes the diagonal matrix. HeZ=Z⊤+Z. Im denotes the m-dimensional unit matrix. λmin⋅/λmax⋅ denotes the minimal/maximal eigenvalue. ⋅ means Euclidean vector norm. E⋅ signifies the mathematical expectation.


## 2. Problem Formulations

Consider a type of SPCNs composed ofN coupled nodes described by(1)xik+1=gϵxik+∑j=1NωijΓϵxjk+Ci,ϵνk,yik=Dixik+Eiνk,i=1,2,…,N,where xik=xis⊤kxif⊤k⊤∈ℝnx,yik=yis⊤kyif⊤k⊤∈ℝny, gϵxik=fxisk+Axifkϵhxisk+Bxifk, Γϵ=Γ11Γ12ϵΓ21ϵΓ22, Ci,ϵ=CisϵCif, Di=diagDis,Dif, Ei=EisEif, xisk∈ℝnxs, and xifk∈ℝnxfnx=nxs+nxf refer to the slow state and the fast state of node i, respectively. yisk∈ℝnys and yifk∈ℝnyfny=nys+nyf mean the measurement outputs of node i, ϵ is a singular perturbation parameter, and νk∈ℝnν signifies the bound disturbance input belonging to l20,∞, which meets νk2≤ν¯. Γϵ refers to an inner-coupling matrix with given dimensions. A, B, Ci,ϵ, Di, and Ei are known matrices with suitable dimensions.The network topologyW=N,X,W is devoted to reflect the outer coupling phenomenon of the SPCNs. N=1,2,…,N and X⊆N×N symbolize the sets of nodes and edges. For any i,j∈N, the out-coupled configuration matrix W=ωij is symmetric if ωij=ωji>0, which satisfies(2)ωii=−∑j=1,j≠iNωij,where ωij>0∀i≠j implies a connection between nodes j and i; otherwise, ωij=0.Assumption 1 (see [23]).
The nonlinear sector-valued functionsf⋅:ℝnxs⟶ℝnxs and h⋅:ℝnxs⟶ℝnxf of SPCNs (1) satisfy the following assumption:(3)fxisk−fxjsk−Λ1xisk−xjsk⊤fxisk−fxjsk−Λ2xisk−xjsk≤0,hxisk−hxjsk−ϒ1xisk−xjsk⊤hxisk−hxjsk−ϒ2xisk−xjsk≤0,where xisk, xjsk∈ℝnxs, f0=0, h0=0, Λℓ, and Υℓℓ=1,2 are constant matrices.In this paper, for the sake of saving the communication resources between the sensors and estimators, a multievent-triggered approach is presented to reduce transmission energy. The triggering instant series of nodei can be assumed as 0≤k0i<k1i<…<kti<…,i=1,2,…,N where the new transmitted instant kt+1i can be formulated as(4)kt+1i=mink∈0,ℕ|k>kti,θi−εi⊤kΦiεik<0.Withk0i=0, θi∈0,1 is the given parameter of the i-th node, Φi=diagΦis,Φif>0 is a weighting matrix of the i-th node to be determined, and εik≜yik−yikti≜εis⊤kεif⊤k⊤ with yikti referring to the latest transmitted measurement of node i. Hence, for k∈kti,kt+1i,(5)θi−εi⊤kΦiεik≥0.Remark 1.
Note that different from the existing static ETP, the multievent-triggered protocol is studied in (2). The proposed triggering protocol can be seen as a generalized framework of ETP, which cover the existing static ETP as a special case (i.e., i=1).Subsequently, based on the multievent-triggered approach, a state estimator is constructed as(6)x^ik+1=gϵx^ik+∑j=1NωijΓϵx^jk+Kiyikti−Dix^ik,i=1,2,…,N,where x^ik=x^is⊤kx^if⊤k⊤ with x^isk and x^ifk representing the state estimations of xisk and xifk, respectively. Ki=diagKis,Kif means the estimator gain of the i-th node to be judged.Let(7)e˜ik=e˜is⊤ke˜if⊤k⊤,e˜isk=xisk−x^isk,e˜ifk=xifk−x^ifk,Ak=A1⊤kA2⊤k…AN⊤k⊤e˜,ε,,B˜e˜isk=Bxisk−Bx^iskB=f,h,g˜ϵ⊤e˜isk=f˜⊤e˜iskϵh˜⊤e˜isk,g⟶ϵe˜sk=g˜ϵ⊤e˜1skg˜ϵ⊤e˜2sk…g˜ϵ⊤e˜Nsk⊤.Combining (1) and (4), the dynamics of the estimation error can be built as(8)e˜k+1=g⟶ϵe˜sk+W⊗Γϵ+IN⊗Gϵ−KDe˜k+Cϵ−KEνk+Kεk,where(9)Cϵ=C1,ϵ⊤C2,ϵ⊤…CN,ϵ⊤⊤,K=diagK1,K2,…,KNK=K,D,E=E1⊤E2⊤…EN⊤⊤,Gϵ=0A0ϵB.In the sequel, one reschedules the order of dynamic estimation errors (8). Denote T=T1×T2×⋯×TN, with Tℓ∈ℝNnx⊗Nnxℓ=1,2,…,N being row-switching elementary matrices and T being invertible. Then, one has(10)Te˜k+1=Tg⟶ϵe˜sk+TW⊗Γϵ+IN⊗Gϵ−KDT−1Te˜k+TCϵ−KEνk+TKεk,which yields(11)ek+1=Aϵek+ℐ1fesk+ϵℐ2hesk+Cϵνk+Kε¯k,where(12)ek=e˜1s⊤ke˜2s⊤k…e˜Ns⊤ke˜1f⊤ke˜2f⊤k…e˜Nf⊤k⊤,Desk=D˜⊤e˜1skD˜⊤e˜2sk…D˜⊤e˜Nsk⊤D=f,h,ε¯k=ε1s⊤kε2s⊤k…εNs⊤kε1f⊤kε2f⊤k…εNf⊤k⊤,Aϵ=−KsDs+W⊗Γ11IN⊗A+W⊗Γ12ϵW⊗Γ21ϵIN⊗B+W⊗Γ22−KfDf,ℐ1=INnxs0Nnxf×Nnxs,ℐ2=0Nnxs×NnxfINnxf,Cϵ=Cs−KsEsϵCf−KfEf,K=diagKs,Kf,Eℓ=diagE1ℓ,E2ℓ,…,ENℓE=K,D,ℓ=s,f,Fℓ=F1ℓ⊤F2ℓ⊤…FNℓ⊤⊤F=C,E,ℓ=s,f.To facilitate the derivation of the main results, the following definition and lemma are introduced.Definition 1 (see [35]).
Estimation error dynamics (7) is exponentially ultimately bounded in mean square (EUBMS), if for any solution ek with initial state e0,(13)Eek2≤αe02βk+γk,limk⟶∞γk=γ¯,holds, where α>0, β∈0,1, and γ¯>0 imply the mean square asymptotic upper bound of (11).Lemma 1 (see [23]).
Combined with Assumption1, the nonlinear functions fesk and hesk of estimation error dynamics (7) satisfy the conditions as follows:(14)e⊤kf⊤eskF1∗F2INnxsekfesk≤0,e⊤kh⊤eskH1∗H2INnxfekhesk≤0,where(15)F1=diagIN⊗Λ1⊤Λ2+Λ2⊤Λ12,0Nnxf,F2=−IN⊗Λ1+Λ220Nnxs,H1=diagIN⊗Υ1⊤Υ2+Υ2⊤Υ12,0Nnxf,H2=−IN⊗Υ1+Υ220Nnxf.Lemma 2 (see [23]).
For any matricesℋ and G and a scalar ϵ0>0, ∀ϵ∈0,ϵ0, if ℋ≤0 and ℋ+ϵ0G<0 hold, it yields ℋ+ϵ<0.


## 3. Main Results

In this section, a sufficient condition is presented to guarantee that the estimation error dynamics (7) is exponentially ultimately bounded in mean square and the desired state estimator will be designed.Theorem 1.
For givenϵ>0andc∈0,1, estimation error dynamics (7) is EUBMS, if there exist scalars λς>0ς=1,2,3 and κ>0 and matrices Pϵ>0, such that(16)−1−cPϵ−λ1F1−λ2H1∗∗∗∗−λ1F2−λ1INnxs∗∗∗−λ2H20−λ2INnxf∗∗000−λ3Φ∗1+κAϵ1+κℐ11+κϵℐ21+κK−Pϵ−1<0,where(17)Φ=diagΦ1s,Φ2s,…,ΦNs,Φ1f,Φ2f,…,ΦNf.Proof.
Firstly, construct the following Lyapunov functional candidate:(18)Vk=e⊤kPϵek.
According to (5), another equivalent form of the event-triggering condition is as follows:(19)∑i=1Nθi−ε¯⊤kΦε¯k≥0.
Along the trajectory of (11), calculating the mathematical expectation of the difference of V(k), one gains that(20)EΔVk=EVk+1−1−cVk−cVk=Ee⊤k+1Pϵek+1−1−ce⊤kPϵek−cVk≤Ee⊤k+1Pϵek+1−1−ce⊤kPϵek−cVk=EAϵek+ℐ1fesk+ϵℐ2hesk+Cϵνk+Kε¯k⊤PϵAϵek+ℐ1fesk+ϵℐ2hesk+Cϵνk+Kε¯k−1−ce⊤kPϵek−cVk≤EAϵek+ℐ1fesk+ϵℐ2hesk+Kε¯k⊤PϵAϵek+ℐ1fesk+ϵℐ2hesk+Kε¯k+ν⊤kCϵ⊤PϵCϵνk+2Aϵek+ℐ1fesk+ϵℐ2hesk+Kε¯k⊤PϵCϵνk−1−ce⊤kPϵek−cVk.
Depending on Young’s inequality, the following inequality holds:(21)2Aϵek+ℐ1fesk+ϵℐ2hesk+Kε¯k⊤PϵCϵνk≤κAϵek+ℐ1fesk+ϵℐ2hesk+Kε¯k⊤PϵAϵek+ℐ1fesk+ϵℐ2hesk+Kε¯k+1κν⊤kCϵ⊤PϵCϵνk.
Combining (19) and (21) and Lemma 1, we obtain that(22)EΔVk≤E1+κAϵek+ℐ1fesk+ϵℐ2hesk+Kε¯k⊤PϵAϵek+ℐ1fesk+ϵℐ2hesk+Kε¯k+1+1κν⊤kCϵ⊤PϵCϵνk−1−ce⊤kPϵek−cVk−λ1e⊤kf⊤eskF1∗F2INnxsekfesk−λ2e⊤kh⊤eskH1∗H2INnxfekhesk+λ3∑i=1Nθi−ε¯⊤kΦε¯k=ℵ⊤kΞϵℵk−cVk+ϱ,where(23)ℵk=e⊤kf⊤eskh⊤eskε¯⊤k⊤,ϱ=λ3∑i=1Nθi+1+1κλmaxCϵ⊤PϵCϵν¯,Ξϵ=Ξϵ11∗∗∗1+κℐ1⊤PϵAϵ−λ1F21+κℐ1⊤Pϵℐ1−λ1INnxs∗∗1+κϵℐ2⊤PϵAϵ−λ2H21+κϵℐ2⊤Pϵℐ11+κϵ2ℐ2⊤Pϵℐ2−λ2INnxf∗1+κK⊤PϵAϵ1+κK⊤Pϵℐ11+κK⊤Pϵϵℐ21+κK⊤PϵK−λ3Φ,Ξϵ11=1+κAϵ⊤PϵAϵ−1−cPϵ−λ1F1−λ2H1.
Applying the Schur complement lemma to (16), it is clear that Ξϵ<0. Consequently, one has(24)EΔVk≤−cVk+ϱ,which yields(25)EVk+1≤1−cVk+ϱ.
Then, it follows from (25) that(26)EVk≤1−ckV0+1−1−ckcϱ.
Moreover, it is easy to obtain thatVk≥λminPϵe⊤kek and V0≤λmaxPϵe⊤0e0; combining (26), it yields that(27)Eek2≤1−ckλmaxPϵλminPϵe02+1−1−ckcλminPϵϱ.
Consequently, estimation error dynamics (7) is EUBMS, andγ¯=1/cλminPϵϱ is the mean square asymptotic upper bound of (7), which completes the proof.Theorem 2.
For∀ϵ∈0,ϵ0 with the upper bound ϵ0>0 and c∈0,1, estimation error dynamics (7) is EUBMS, if there exist scalars λςς=1,2,3 and κ>0 and matrices P⌣=P⌣11∗P⌣21P⌣22,P^=P^11∗P^21P^22,Xℓ=diagXℓ1,Xℓ2,…,XℓNℓ=1,2,andK¯i℘i=1,2,…,N,℘=s,f, such that(28)P⌣≥0,P⌣+ϵ0P^>0,(29)−1−cP⌣−λ1F1−λ2H1∗∗∗∗−λ1F2−λ1INnxs∗∗∗−λ2H20−λ2INnxf∗∗000−λ3Φ∗1+κA˘1+κXℐ101+κK˜P⌣−X−X⊤≤0,(30)−1−cP⌣+ϵ0P^−λ1F1−λ2H1∗∗∗∗−λ1F2−λ1INnxs∗∗∗−λ2H20−λ2INnxf∗∗000−λ3Φ∗1+κA˘+ϵ0A^1+κXℐ11+κϵ0Xℐ21+κK˜P⌣+ϵ0P^−X−X⊤<0,where(31)X=diagX1,X2,K¯℘=diagK¯1℘,K¯2℘,…,K¯N℘,℘=s,f,A˘=−K¯sDs+X1W⊗Γ11X1IN⊗A+W⊗Γ120−K¯fDf,A^=00X2W⊗Γ21X2IN⊗B+W⊗Γ22,K˜=diagK¯s,K¯f.Moreover, the estimator gain matrices are calculated as(32)Kis=X1i−1K¯is,Kif=X2i−1K¯if,i=1,2,…,N.Proof.
From (28)–(30) and Lemma 2, it follows that for ∀ϵ∈0,ϵ0, the following conditions hold:(33)Pϵ=P⌣+ϵP^>0−1−cP⌣+ϵP^−λ1F1−λ2H1∗∗∗∗−λ1F2−λ1INnxs∗∗∗−λ2H20−λ2INnxf∗∗000−λ3Φ∗1+κA˘+ϵA^1+κXℐ11+κϵXℐ21+κK˜P⌣+ϵP^−X−X⊤<0.
Substituting (32) into (33) and noticing that −XPϵ−1X⊤≤Pϵ−HeX, one has(34)−1−cPϵ−λ1F1−λ2H1∗∗∗∗−λ1F2−λ1INnxs∗∗∗−λ2H20−λ2INnxf∗∗000−λ3Φ∗1+κXAϵ1+κXℐ11+κϵXℐ21+κXK−XPϵ−1X⊤<0.
Recalling (29), it is clear that P⌣−X−X⊤<0 with P⌣≥0. Then, X is invertible.
Premultiplying and postmultiplying (34) by diagINnx,INnxs,I2Nnxf,INny,X−1, its transposes yields (16). Therefore, inequality (16) can be guaranteed if (28)–(30) hold. This completes the proof.


## 4. Numerical Example

Similar to [23], consider SPCNs (1) with three nodes and the following parameters:(35)A=1.20.45,B=1.30.62.30.9,C1s=0.2,C2s=0.1,C3s=0.4,C1f=0.861.15,C2f=0.51.75,C3f=0.020.05,D1s=21.2,D2s=0.92.1,D3s=1.61.62,D1f=2.41,D2f=−1.40.6,D3f=2.32,E1s=0.81,E2s=0.50.1,E3s=0.4−1,E1f=0.2,E1f=0.4,E1f=0.25.The out-coupled configuration matrixW of SPCNs (1) and its inner-coupling matrix, respectively, are selected as follows:(36)Γ11=0.45,Γ12=0.20.6,Γ21=0.10.5,Γ22=0.60.20.40.25,W=−0.20.10.10.1−0.20.10.10.1−0.2.In light of Assumption1, the nonlinear vector-valued functions are chosen as(37)fxisk=0.8xisk−tanh0.4xisk,hxisk=0.3xisk−tanh0.1xisk0.4xisk−tanh0.2xisk,i=1,2,…,N,and Λ1=0.5,Λ2=0.58,ϒ1=0.20.2⊤,andϒ2=0.50.41⊤. The event-triggered thresholds are set as θ1=0.17, θ2=0.104, and θ3=0.5, and the weighting matrices are calculated as Φi=diag44.1124,44.1124,44.1124i=1,2,3. Other parameters are given as follows: ϵ0=0.0193andc=0.1.According to Theorem2, the gain matrices of state estimator (6) can be obtained as(38)Ks=0.17550.10530000000.08240.19240000000.14730.1491,Kf=0.0107000.0290000−0.017900−0.11040000.0102000.0222.The bound disturbance input is considered asνk=0.05cos2k, and the initial conditions of (1) and (6) are given as x10=0.040.010.02⊤, x20=0.030.0150.02⊤, x30=0.10.0040.004⊤, and x^10=x^20=x^30=000⊤.The simulation results are presented in Figures1–5. Figures 1–3 display the state trajectories and their estimations of three nodes, respectively. The event-based release instants and release intervals of three nodes are shown in Figure 4. Figure 5 plots the evolutions of the estimation error dynamics. It can be discovered from Figure 5 that estimation error dynamics (7) is EUBMS.Figure 1
The state trajectories and estimations of node 1.Figure 2
The state trajectories and estimations of node 2.Figure 3
The state trajectories and estimations of node 3.Figure 4
The event-based release instants and release intervals of three nodes.Figure 5
Dynamics of the estimation errorek.


## 5. Conclusions

This paper has investigated the issue of the multievent-triggered state estimation for a novel class of discrete-time nonlinear SPCNs. A discrete-time SPCN model with nonlinearities has been modeled. To alleviate energy consumption, a multievent triggered protocol is applied to regulate the communication among nodes of the SPCNs. Finally, a simulation has demonstrated the rationality, superiority, and effectiveness of the proposed method.



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