Robust Distributed H ∞ Filtering for Nonlinear Systems with Sensor Saturations and Fractional Uncertainties with Digital Simulation

This paper is concerned with the robust distributed filtering problem for nonlinear systems subject to sensor saturations and fractional parameter uncertainties. A sufficient condition is derived for the filtering error system to reach the required performance in terms of recursive linear matrix inequality method. An iterative algorithm is then proposed to obtain the filter parameters recursively by solving the corresponding linear matrix inequality. A numerical example is presented to show the effectiveness of the proposed method.


Introduction
The sensor networks have gradually received more and more research interest in the past decades.A typical example is the power grids which have evolved over the past century from a series of small independent community based systems to large-scale and complex systems involving many kinds of intelligent electronic devices [1].As is well known, the information collected by the sensor networks need to be further processed before used for diagnosis, control, optimization, and other transactions for the smart grids, which requires much more advanced methodologies that are more sensitive, reliable, and economic than those applied in the traditional centralized electricity networks.Due to its clear engineering insights, recently, sensor networks have gained increasing research interests during the past decades in various branches of theoretical research and industrial applications.In particular, the distributed filtering over sensor networks has been ongoing research issue that attracts special attention from researchers in the area.Different from traditional filtering techniques based on single or centralized structured/located sensors, the information available for the filter design on an individual node of the sensor network is not only from its own measurement but also from its neighboring sensors' measurements according to the given topology [2].As such, the objective of filtering based on a sensor network can be achieved in a distributed yet collaborative way.Such a problem is usually referred to as the distributed filtering problem.It is noticed that one of the main challenges in designing distributed filters lies in how to cope with the complicated couplings between one sensor and its neighboring sensors by reflecting such couplings in the filter structure specification.
It is worth mentioning that in real-world applications, due to a variety of reasons, such as abrupt changes of working conditions, internal/external disturbances, erosions, equipment aging, the sensor outputs are usually corrupted by these different disadvantages.Such phenomena are always referred to as incomplete information.Up to now, filtering problems with incomplete information have been widely studied and a lot of results have been reported in the literature; see [3][4][5][6][7][8][9][10][11][12][13][14] for some latest publications.As far as the distributed filtering problems subject to incomplete information are concerned, some recent representative work can be summarized as follows.In [15], the distributed filtering problem was solved when randomly occurring saturations and successive packet dropouts appear over sensor networks.In [16], distributed  ∞ filtering problem was studied for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks.In [17], distributed filtering for a class of time-varying systems was discussed over sensor networks with quantization errors and successive packet dropouts.The distributed  ∞ state estimation with stochastic parameters and nonlinearities through sensor networks was solved in [18] over the finite-horizon.With randomly varying nonlinearities and missing measurements, [19] developed a distributed filtering method to achieve the performance requirements.In [20], distributed state estimation problem for uncertain Markov-type sensor networks with modedependent distributed delays was solved.Using linear matrix inequality approach, the distributed filters were given in [21,22], respectively, for different types of stochastic systems.However, it is worth mentioning that, up to now, the distributed filtering problem has not been studied for nonlinear time-varying systems subject to randomly occurring sensor failures.
Motivated by the above discussion, in this paper, it is the objective to design a robust filter for a class of discrete time-varying nonlinear stochastic systems subject to sensor saturations and fractional uncertainties.The contribution of this paper is twofold: (i) for the first time, the filtering problem is discussed while taking both sensor saturations and fractional uncertainties into consideration.(ii) An iterative algorithm is developed to solve the robust  ∞ filtering problem and then seek the desired filter parameters step by step.The rest of this paper is organized as follows: Section 2 formulates the distributed event-based filter design problem for nonlinear discrete time-varying stochastic system.The main results are presented in Section 3 where sufficient conditions for the existence of the desired filter are given in terms of recursive linear matrix inequalities.Section 4 gives a numerical example.Section 5 is the conclusion.

Problem of Formulation
In this paper, it is assumed that the sensor network has  sensor nodes which are distributed in the space according to a specific interconnection topology characterized by a directed graph G = (V, E, L), where V = {1, 2, . . ., } denotes the set of sensor nodes, E ⊆ V × V is the set of edges, and L = [  ] × is the nonnegative adjacency matrix associated with the edges of the graph, that is,   > 0 ⇔ edge(, ) ∈ E, which means that there is information transmission from sensor  to sensor .If (, ) ∈ E, then node  is called one of the neighbors of node .For all  ∈ V, denote N  ≜ { ∈ V | (, ) ∈ E}, which means that, in the sensor network, sensor node  can receive the information from its neighboring nodes  ∈ N  according to the given network topology.
Let us consider the discrete-time nonlinear stochastic system with  sensors defined on  ∈ [0, ]: where   ,  , ,   , and   represent the state, the measured output of the th sensor, estimated output, and disturbance belonging to  2 , respectively.  ,  , ,   ,  , , and   are known real time-varying matrices with appropriate dimensions.
Assumption 1.The nonlinear function   (  ) is assumed to obey the following constraint: where   are known positive definite matrix sequence with appropriate dimensions describing the shape of the ellipsoids with 0 being the center of the ellipsoids and  > 0 is a known positive scalar.
The nonlinear function   (  ) stands for the nonlinearity that is unknown, bounded, and deterministic but reside within an ellipsoidal set.Such a type of nonlinearity is usually occurring in practical engineering practice and is always a main origin for the degradation of the system performance.
Assumption 2 (see [23]).Δ  has the linear fractional form as follows: with    <  and Σ   Σ  < , where , , and  are known constant matrices and Σ  denotes the unknown matrix functions with Lebesgue measurable elements.
for some real matrices Θ 1 and Θ 2 , where Θ = Θ 2 − Θ 1 is a symmetric positive definite matrix.In this case, we say that As discussed in [23], there exist matrices  1 and  2 such that 0 ≤  1 <  ≤  2 ; the sensor saturation function ( ,   ) can be written as where   ( ,   ) is a nonlinear vector-valued function satisfying the sector-bounded condition with Θ 1 = 0 and Θ 2 = .
In this case,   ( ,   ) can be represented as follows: where The objective of this paper is to design a filter with the following form for system (1): where x, is the state estimate, ẑ is the output to be estimated, and  , and  , are the filter parameters to be determined.

Main Results
Before giving the main results, some useful lemmas are firstly introduced.
Theorem 7. Given a  ∞ performance  > 0 and the initial matrix Ξ > 0. Let the filter parameters G  and H  be given.If there exist three sequences of positive scalars  1 ,  2 (0 ≤  ≤ ), and a sequence of positive definite matrices   (0 ≤  ≤  + 1) satisfying   0  0  0 ≤  2   0 Ξ 0 such that the following matrix inequality where is feasible, then  ∞ performance can be satisfied. Proof.Defining then the filtering error system can be expressed by Defining where Adding the zero term to both sides of (23), one can get where Next, by summing up (26) on both sides from 0 to  with respect to , one can have which is equivalent to From ( 29), since  +1 > 0 and the initial condition where with S  = diag{  ,   , . . .,   }.
Moreover, it also can be obtained from the sensor fault constraint (6) that where By Schur Complement Equivalence Lemma, the matrix inequality (18) can be rewritten as follows: By Lemma 4,    Λ    ≤ 0 can be satisfied, which means that the required  ∞ performance can be satisfied.The proof is complete.Remark 8.By means of iterative matrix inequality method, Theorem 7 presents a sufficient condition for filtering error system to achieve the desired  ∞ specification subject to the sensor saturations.Moreover, notice that the obtained condition is expressed by certain matrix inequalities including fractional parameter uncertainties Δ  ; we cannot solve the matrix inequality directly to obtain the explicit parametric expression of the required filter parameters G and H.As such, in the following, we will provide a theorem with which the addressed problem can be solved and the filter structure can then be obtained step by step.

Filter Design
Theorem 9. Let the  ∞ performance  > 0 and the initial condition matrix   > 0 be given.If there exist a sequence of matrices G  (0 ≤  ≤ ), a sequence of matrices H  (0 ≤  ≤ ), three sequences of positive scalars  1 ,  2 , and   (0 ≤  ≤ ), a sequence of positive definite matrices   (0 ≤  ≤  + 1), and a sequence of positive definite matrices   (0 ≤  ≤  + 1) satisfying where with X  =  −1  and Z  =  −1  , and  is the solution to the following optimization problem min then the addressed  ∞ filtering problem is solvable and the desired filter parameters at each time  can be obtained by solving the corresponding matrix inequality.
Proof.The proof is quite straightforward based on Theorem 7. First, define One can obtain that the matrix inequality ( 18) is equivalent to the following inequality: In what follows, we will try to prove that inequality (38) can be satisfied if inequality (35) is fulfilled.To this end, rewrite (38) as follows: where From Lemma 6, it follows that Furthermore, one can obtain that which, by Schur Complement Lemma, is equivalent to the linear matrix inequality (35).Thus, inequality (18) is satisfied if (35) is met, and, according to Theorem 7, the required  ∞ filtering problem is solved.
Remark 10.Theorem 9 outlines the methodology with which the addressed robust  ∞ distributed filtering problem is solved for nonlinear systems subject to sensor saturations and fractional uncertainties.An iterative linear matrix inequality approach has been developed to calculate the desired filter parameters step by step.It should be pointed out that it will inevitably lead to the larger computing burdens.However, the algorithm we proposed in this paper is capable of dealing with the time-varying case, and the increase in computing burden can be largely reduced with the fast development of modern computer technology.In what follows, we will present a computing algorithm with which the numerical values of filter parameters can be obtained in each time step.
Step 1. Set the required , the initial condition Ξ, the initial  0 , and .Choose properly the values of , , and .Select the initial values for X 0 and Z 0 satisfying the initial condition constraint.
The  ∞ Filter Design Algorithm gives a recursive way to obtain the numerical values of the desired filter parameters at each time point .It should be pointed out that the existence of the filter is expressed as the feasibility of certain linear matrix inequalities that can be solved forward in time.The possible research topic in the future is to consider more performance indices and give the filtering schemes that are able to satisfy multiple requirements simultaneously.

Simulation Example
In this section, an illustrative example is presented to show the effectiveness of the proposed  ∞ Filtering Design Algorithm.
Let us consider the nonlinear system (1) with the parameters given below:

Conclusion
The problem of distributed  ∞ filtering for nonlinear stochastic systems in the presence of sensor saturations and fractional uncertainties.A sufficient condition is proposed under which the required  ∞ performance can be satisfied even if the randomly occurring sensor failures happen.A recursive algorithm is given in order to obtain the required filter parameters iteratively.Finally, an illustrative example is presented to show the effectiveness of the proposed method.Noting that the nonlinearities considered in this paper are relatively simpler, one of the possible future research directions may be the robust distributed  ∞ filtering problems with much more complex nonlinearities.

Figure 2 :
Figure 2: The state  2 and its estimate.