Finite-horizon H ∞ state estimation for time-varying complex networks based on the outputs of partial nodes

In this paper, the partial-nodes-based resilient filtering problem for a class of discrete time-varying complex networks is investigated. In order to reduce the effect of imprecision of filter parameters on estimation performance, a set of resilient filters is proposed. The measurement output from all network nodes may not be available in the actual system, but only from a fraction of nodes. The state estimators are designed for the time-varying complex network based on partial nodes to make the estimation error achieve the performance constraint over a finite horizon. By employing the completing-the-square technique and the backward recursive Riccati difference equations, the sufficient conditions for the existence of the estimator are derived. Then the gain of the estimator is calculated. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.


Introduction
A complex network is composed of a series of nodes with certain dynamic performance, which are connected with each other through a network topology. There are many examples of complex networks in various natural and man-made systems, such as the World Wide Web, genetic networks, power grids and social networks (Albert & Barabási, 2002;Boccaletti et al., 2006;Costa et al., 2011Costa et al., , 2007Khafaf & Jalili, 2019;Pagani & Aiello, 2013). In the past several decades, the dynamic analysis of complex networks has received extensive attention, such as state estimation (Ding et al., 2012;H. Li, 2013;Sheng et al., 2017;Zou et al., 2017), synchronization (Adu-Gyamfi et al., 2018;Tang et al., 2014;X. Wang et al., 2020) and stability (C. Zhang & Han, 2019). However, owing to the enormous number of nodes and complicated topology of complex networks, it is generally impossible to obtain the information of all nodes directly. Hence, in order to analyse the trajectory of node state change, it is very valuable to use the available measurement output for state estimation.
For decades, filters (Ding et al., 2015;X. M. Li et al., 2020;Liang et al., 2014;Sheng, Niu, Zou, et al., 2018;Z. Wang et al., 2013;Zhao et al., 2018;Zou et al., 2019aZou et al., , 2019bZou, Wang, Hu, et al., 2020; have been widely applied in control, signal processing, target tracking and other engineering fields because they CONTACT Li Sheng shengli@upc.edu.cn can utilize the available measurement output signals for state estimation. Among the existing filters, H ∞ filter can provide a bound for the worst situation estimation error without statistical information of noise, so it has been a focal point in dynamic analysis. For time-varying systems, compared with moving horizon (Zou et al., 2019a;Zou, Wang, Hu, et al., 2020;, it is more meaningful for on-line implementation to study state estimation in finite horizon. In recent years, there have been mainly two methods for finite-horizon filtering, including linear matrix inequality method (X. M. Li et al., 2020;Liang et al., 2014;Zhao et al., 2018) and Riccati difference equations (RDEs) method (Ding et al., 2015;Sheng, Niu, Zou, et al., 2018;Z. Wang et al., 2013). For instance, a periodic neural network over multiple fading channels was considered in X. M. Li et al. (2020) and the parameters of the estimator were calculated by solving the recursive linear matrix inequality (RLMI). In Sheng, Niu, Zou, et al. (2018), a new coupled recursive Riccati difference equations method was proposed to solve the filtering problem of complex networks with multiplicative noise and random coupling strength.
In reality, the filter cannot achieve the effect accurately in some cases. Due to some physical reasons such as digital to analog conversion, rounding error, limited precision or internal noise (D. Zhang et al., 2014), the implementation of digital state estimator will have errors. That is, its actual gain may fluctuate or vary compared with the expected gain (Yu et al., 2016). Based on this fact, desired filter is insensitive to some gain errors, in other words, the designed filter is resilient (or non fragile). In the past few years, resilient filters have been widely studied in sensor networks (Sheng, Niu, & Gao, 2018;L. Zhang et al., 2017), complex networks (W. Li et al., 2018;Yang et al., 2019) and neural networks (Hou et al., 2016;Sakthivel et al., 2015). For example, in the literature (L. Zhang et al., 2017), the problem of non-fragile H ∞ filtering for large-scale power systems based on sensor networks is studied. In Yang et al. (2019), the resilient state estimation problem for a class of time-delay complex networks with stochastic communication protocol (SCP) is discussed.
Most of the state estimation problems for complex networks mentioned above are based on the assumption that the measurement output signals from all the network nodes are available. However, in reality, because of the harsh physical environment or limited communication resources, not all the measurement outputs of the nodes can be accessed directly. Moreover, due to the large scale of nodes in practical complex networks, some nodes may have similar functions, and the information of such nodes may be redundant for state estimation. Therefore, it is of application significance to estimate the states by measuring outputs from certain network nodes. This is also known as the partial-nodes-based (PNB) estimation problem. Recently, Liu has done pioneering work on the state estimation of delayed complex networks . Han extended it to sensor networks and studied the H ∞ consensus filtering algorithm based on partial nodes for time-varying sensor networks (Han et al., 2017). Nevertheless, finite-horizon H ∞ state estimation of complex networks based on partial nodes has not been fully investigated yet, which is the motivation of this paper.
Based on the above discussion, the objective of this paper is to investigate the resilient filtering problem of time-varying complex networks based on the measurement output of partial nodes. The main contributions of this paper are as follows.
(1) Considering that the measurement output of complex networks may not be available for all nodes, but come from partial nodes, this can reflect the reality more closely. (2) In view of the harmful influence caused by inaccurate filter parameters, a set of resilient filters is designed to eliminate it. (3) A new parameter variable is designed to facilitate subsequent calculation, which includes disturbance, filter parameter uncertainty and nonlinear uncertainty. By means of the recursive RDEs approach, the estimation error can satisfy finite-horizon H ∞ performance constraint.
R is the space of all real numbers. R n and R m×n represent the set of all n-dimensional vectors and the set of all m × n real matrices, respectively. A > 0 (A ≥ 0) means that A is a real symmetric positive definite (positive semidefinite) matrix. A T stands for the transpose of a matrix A, and A † ∈ R m×n respects the Moore-Penrose pseudo inverse of A ∈ R n×m . I denotes the unit matrix with appropriate dimension and diag{· · · } represents a diagonal matrix. ⊗ is the operation of Kronecker product. x describes the Euclidean norm of a vector x. E{x} respects the mathematical expectation of the stochastic variable x. For simplicity, sym{A} stands for A + A T . A F denotes the Frobenius norm of a matrix, i.e. A F = (trace(A T A)) 1 2 .

Problem formulation and preliminaries
Consider the following time-varying complex network consisting of M coupling nodes: where x i (k) ∈ R n , z i (k) ∈ R n z denote the state vector, regulated output of the ith node, respectively. v i (k) ∈ R n v is the disturbance input which belongs to l 2 ([0, N], ∈ R n v ).
= diag{λ 1 , λ 2 , . . . , λ n } ≥ 0 is an inner-coupling matrix linking the jth state variable if λ j = 0. W = [w ij ] M×M is the outer coupled matrix satisfying w ij ≥ 0(i = j) and , and E i (k) are known real matrices with appropriate dimensions. f (·) : [0, N] × R n → R n is a nonlinear vectorvalued function. Suppose that the nonlinear function f (·) satisfies the sector-bounded condition: where U 1 (k) and U 2 (k) are real matrices with appropriate dimensions and satisfy U 1 (k) ≤ U 2 (k) for all k.
As mentioned in the introduction, one of the purposes of this paper is to investigate the PNB state estimation of the complex network (1), which the measurement output of the network only comes from a fraction of its nodes. On this basis, we reorder the nodes of system (1). Considering the universality, it is assumed that we can access the measurement output of the first l 0 nodes. The known measurement output expressions of these nodes are as follows: where y i (k) ∈ R n y (1 ≤ n y ≤ n) is the measurement output of the ith node. C i (k) and H i (k) are known real matrices with appropriate dimensions. In order to estimate the state of complex network (1) and consider the imprecision of filter parameters, resilient filters of the ith node of (1) are constructed as follows: wherex i (k) ∈ R n ,ẑ i (k) ∈ R n z are the estimation of state and regulated output on ith node, respectively. K i (k) ∈ R n×n y is the filter parameter to be designed. K i (k) ∈ R n×n y represents the uncertainties of filter parameter concerning K i (k), and it is assumed that it can be described as where S i (k), T i (k) are known real-valued matrices with proper dimensions, and , the dynamic estimation error can be derived from (1) and (4) as follows: From (2), we have, which can be rewritten as It can be inferred that there exists at least a function . Therefore, the sector-bound condition can be converted to the sectorbound uncertainties expressed as (11) and (12), the combination of (1) and (7) generates the following augmented system: wherẽ -(14) can be written as following forms: wherē Our objective of this paper is to find the sequence of filter parameter matrices K i (k), such that the estimation error outputz(k) satisfies the following H ∞ performance requirement. The main objective is to obtain the filter parameter matrixK(k) so that output estimation error z(k) can satisfy the H ∞ performance constraint over finite horizon. Given the disturbance attenuation level γ > 0 and positive definite matrix > 0. If the following inequality holds, then the system satisfies the H ∞ performance requirement, Redefining H ∞ performance requirement for system (15), whereT Lemma 2.1: Considering the performance requirements of (16) and (17), it can be inferred that J ≤J.

Based onL T (k)L(k)) ≤ I,¯ T (k)¯ (k)
≤ I, we have J −J ≤ 0, which means that J ≤J. In this sense, performance index (17) can be used instead of (16) to find the estimator gain.

Main results
In this section, by utilizing the stochastic analysis method and the complete square approach, the sufficient conditions for system (15) to satisfy the H ∞ performance constraint are obtained, and the estimator gain K i (k) is solved.
Next, the parameter K i (k) is calculated in the worst situation. Suppose the worst situation is At this point, the augmented system (15) can be rewritten as Moreover, a cost functional is constructed as follows for the sake of the estimator gain: Theorem 3.2: For the complex network (1) with the estimator (4), given a disturbance attenuation level γ > 0, two positive scalars ε 1 (k) > 0, and ε 2 (k) > 0, and a positive definite matrix > 0. The augmented system (15) satisfies the H ∞ performance constraint defined in (17) over a finitehorizon [0, N], if there exists a solution (P(k), R(k),K(k)) to the following coupled backward RDEs: subject to Proof: On the basis of Theorem 3.1, define Adding the zero term to the right side of (41), one has Applying the completing-the-square technique, we have where ρ * (k) = −1 (k + 1)Ī T ( (k) + I) T R(k + 1)Ā(k)η(k).
Furthermore, it can be deduced that In the light of condition (36), one obtains If the matrixK(k) satisfies Equation (40), then the cost of H(K(k), ω * (k)) can be minimized, which completes the proof.
Finally, according to (40), we will determine the estimator parameter K i (k), i = 1, 2, . . . , l 0 . where It can be inferred from the above formula that As a result, K i (k) is the block matrix from row n + 1 to 2n in the block diagonal matrixK i (k), On the basis of Theorems 3.1 and 3.2, the method of solving state estimator gain is summarized as Algorithm 1.
In this way, online state estimation can be realized by software.

Algorithm 1
The estimator design algorithm.

Numerical example
In this section, a numerical example is presented to illustrate the effectiveness of the proposed method. Consider the complex network (1) with 4 nodes, and the parameters are given as follows: The nonlinear function is selected as which implies that Thus, it can be concluded that The disturbance input v i (k) obeys uniform distribution over [−0.2 0.2]. It is assumed that the measured output of two nodes is known, i.e. l 0 = 2. Other parameters are shown below (Table 1) In this example, set theH ∞ performance level γ = 1.3, parameter ε 1 = 0.4, ε 2 = 0.6. The initial value of the state  The simulation results are shown in Figures 1-5. Figures 1-4 depict the curves of two states of node 1 and node 2 and their estimations. Figure 5 shows the variation curves of estimation errors, which indicate that the state estimator designed by this algorithm is effective. From (16), the H ∞ performance constraint can be calculated from (E{ N k=0 z(k) 2 }) 1 2 /(E{ N k=0 v(k) 2 + η T (0) η(0)}) 1 2 = 0.1801 < γ = 1.3.

Conclusion
The problem of H ∞ state estimation for time-varying complex networks over finite horizon has been investigated. Because of the possible physical environment, the measurement output may not be measured from all nodes. Thus the measurement outputs for state tracking in this paper come from partial nodes. Moreover, because of the inaccuracy of the parameters in practical application, the estimator parameters may deviate from the expected results, so resilient filters have been designed to eliminate its adverse effects. By using the stochastic analysis method and the complete square method, the sufficient conditions for the estimation error to satisfy the H ∞ performance constraint have been obtained. Then, the filter gain has been calculated by solving backward RDEs. Finally, a numerical simulation example has been given to demonstrate the effectiveness of the proposed state estimation method.

Disclosure statement
No potential conflict of interest was reported by the author(s).