Nezar M. Alyazidi and Magdi S. Mahmoud

Abstract—This paper addresses an infinite horizon distributed HHH2/HHH∞filtering for discrete-time systems under conditions of bounded power and white stochastic signals. The filter algorithm is designed by computing a pair of gains namely the estimator and the coupling. Herein, we implement a filter to estimate unknown parameters such that the closed-loop multi-sensor accomplishes the desired performances of the proposed HHH2 and H∞H∞H∞schemes over a finite horizon. A switched strategy is implemented to switch between the states once the operation conditions have changed due to disturbances. It is shown that the stability of the overall filtering-error system with HHH2/HHH∞performance can be established if a piecewise-quadratic Lyapunov function is properly constructed. A simulation example is given to show the effectiveness of the proposed approach.

I. INTRODUCTION

DISTRIBUTED estimation is an important problem and has received a considerable focus in academia and industrial. Distributed estimation/filtering schemes are being frequently utilized for distributed multi-sensors due to their remarkable characteristics,such as flexibility,robustness,easy maintenance and diagnosis. A sensor network architecture typically comprises of spatially distributed sensing nodes,in which each node is collaborating with neighborhoods to perform the main tasks [1]-[4]. In the multi-sensor network,several communication links are emerging to enhance the network performance employing reliable routing techniques.Each individual sensor/filter within the network can estimate the dynamic of states in terms of its observations and its neighboring nodes observations as well. In particular, the system dynamics are often subjected to any sort of uncertainties.These uncertainties may be stochastic white noise or boundedpower signals.DistributedH2filter is immensely implemented to tackle well-known stochastic white signals by minimizing the performance cost measure. However, with bounded-power signals, a distributedH∞filter/estimator can be designed to tackle such uncertainties. Also, a mixedH2/H∞controller provides a powerful tool that can sustain the robustness and optimality for the uncertain plant.

A survey on distributed Kalman estimations was reported in[5], in which various sorts of distributed Kalman filters have been clearly evaluated. The features of these Kalman filters were excellently reviewed and compared. The uncertainties in the system dynamics were considered to be stochastic processes with well-known statistical features in [5]. Centralized Kalman filtering/estimation has been utilized in distributed multi-sensor structure to handle disturbances and to estimate unknown parameters. This structure can provide a desired performance for a wide range of applications. However, once the centralized control node drops out, the entire system fails.To avoid such situation, advanced estimation structures (distributed/decentralized) are beneficially performed by employing local filters/controllers[6]-[8].The proposed structure has an effective design that comprises of distributed filter units, in which the failure of a single unit can be technically avoided by reconfiguration of the system structure, such as in [9]-[11].

In [12], the stability of the filtering/estimation problem is sustained by claiming that the communication topology was time-invariant local information structure [13]. Additionally,a distributed Kalman filtering has been immensely utilized to treat stochastic signals with the assumptions that the system parameters and their statistical natures are well-known.However,the system parameters are not exactly identified and the statistical nature may be unknown, where the modeling of identification errors has considerable impacts on the classical Kalman filtering as well as stochastic disturbance [14].

Over years, optimal control schemes suchH2controllers and robust schemes mainlyH∞controllers have been immensely introduced by several researchers [15]-[16].H2controller was smoothly utilized to tackle impulse disturbances and to ensure linear quadratic controllers performance, but it can not sustain the robustness in terms of irregular systems and parametric perturbations [17]. Typically, theH2is an optimal control methodology that has capability to minimize the objective cost of tracking error and actuation signal.MixedH2/H∞scheme has a supremacy over optimal and robust schemes in engineering practice because it can sustain the optimality and robustness against disturbances [18].H∞controller is applied to reduce the influence of disturbances.The desired of designing a mixedH2/H∞architecture is that theH2controller sustains the optimality in the occurrence of the worst-case disturbance, and its influence must be reduced to a tolerance level. The mixedH2/H∞schemes can provide performance better than theH2andH∞schemes when they are implemented alone [18]-[19]. A mixedH2/H∞controller was studied for stochastic plants in the occurrence of dependent disturbances [18]-[19]. TheH2/H∞scheme was firstly investigated in[20]as multi-objective optimization,and it has been broadly implemented in variety of applications[18], [19], [21]. Also, anH2/H∞frame was investigated for a linear perturbed model with a suboptimal controller utilizing a reduced order technique.

Authors in [22] have introduced a distributedH2/H∞filtering in continuous time domain in the presence of bounded and white disturbances. In our work, we are focusing on anH2/H∞filtering design for discrete-time switched systems in the presence of bounded and white disturbances.

In general, the operation points of power systems are changing faster due to electronic elements such as in wind turbine, or the operation points may change slowly due to different disturbance and, herein, a switching methodology can be utilized to switch between filter algorithms to estimate the dynamic of the state subject to the current operation points to accomplish the prescribed performance.In this paper,a linearized small signal wind turbine model is tested, in which the mixed filtering algorithm is implemented to estimate the unknown parameters and to attenuate the impact of the disturbances [23].

This paper builds on the foregoing literatures and extends them further to the distributedH∞filtering problem for discrete-time systems. The contribution of this paper is as follows:

1) It provides complete results of the infinite horizon distributedH2/H∞filtering design problem for discrete-time systems with bounded power stochastic signals.

2) It formulates the filtering design solution as a two-step procedure, which presents a convenient computational load.

3) A switched strategy is developed to sustain the stability of the system by switching the state according to the change in the operation conditions.

4) It demonstrates the performance evaluation of the filter under external disturbances.

The paper is arranged as follows.Section II introduces some basic definitions.Section III states the problem description and provides the mathematical foundation of the filtering problem.In Section IV, the distributed filtering problem and related issues are developed. Section V shows the practicability of the proposed filter scheme via computer simulation.

II. PRELIMINARIES

In this section, we present definitions that are needed in further development.

A. Definitions

Definition 1:Letnbe an integer anduk ∈Rqbe a discretetime real stochastic vector sequence associated with

as the auto-correlation function.

Definition 2:The Fourier transform ofRuu(n),or the power spectral density ofuk, is given by

Definition 3:The 2-norm||x||2used for vector sequencexk ∈Rnis defined as

The setL2refers to the space of mean square summable infinite vector sequences with

In the sequel, a stationary stochastic vector signal is said to have bounded power ifRuu(n)andSωexist and this imply that

III. PROBLEM STATEMENT

Consider a dynamical model of a linear plant for distributed nodes over a communication topology:

wherexk ∈Rndenotes the state,vk ∈Rm,wk ∈Rpare a bounded-power stationary signal and white noise respectively. (A,B,B1w) are constant matrices, andCi ∈Rq×n,Ei ∈Rq×m,Ewi ∈Rq×pare measurements stationary signals with bounded-power and white noise signals, respectively.is the measurement of nodei. LetZkbe the measurement vector to be estimated. Whereσikrepresents the switching signal among the states.The system(5)is observed by a sensor(node), which is one of distributedNnodes. In this structure,we assume a proposed communication framework that enables two-direction of information to follow.

In this work, estimation/filtering units are utilized to estimate observationsxkfor each sensor. For instance, sensori,may use its own observations and its neighbors observations to estimate its dynamic to make an action. We can get the estimator law for each nodei.

Remark 1:The designedH2/H∞filtering sustains the robustness under the influence of worst bounded stochastic signals by means ofH∞filtering and sustains the optimality under the influence of white stochastic signals by minimizingH2measure of the performance. In this sense, the states and the neighborhood information are not available to the estimator unit. As result, this mixed filtering scheme comprises of two phases. In the first phase, the estimator algorithm is required to estimate the unknown terms and coupled gains can be determined by solving the candidate Riccati equations. In the second stage, The coupled gains can sustain the stability of the closed loop systems.

The estimation error is defined byThen we can deduce the error dynamic as follows:

whereA11, A12,toA1nrepresent a dynamic of each sensor,B11, B12,toB1nrepresent a control input of each sensor,B1w1, B1w2,toB1wnrepresent a input of each sensor, andG11, G12,toG1nare the coupling gains. Here, we can consider the difference of the estimates among neighbors as

For the givenγ ＞0 andβ ＞0,we design the gain matricesKiandGisuch that:

To study the performance ofH∞in the presence of external disturbancewk

2) For anyif E[e(0)]=0

It can be seen that, when the external disturbancewk ≡0,system (5) is stable. This corresponds to minimizing the following norm

for the worst disturbancewk.

To tackle the aforementioned problems, we consider the following standard assumptions:

Assumption 1:All distributed sensors are connected through a network, in which communication constraints will not consider over the present work.

Assumption 2:The pair (Ci,Ai) is detectable.

Discrete-time algebraic Riccati equation (DARE) is in general utilized in optimal control and filtering issues when the model has full ranking matrices as [24].

Assumption 3:

has full row rank for allω ∈R

Assumption 4:

has full row rank for allω ∈R

andare non-singular.

IV. MAIN RESULTS

The aim of this work is to develop a distributed filtering scheme such that the closed loop system is asymptotically stable and satisfies the prescribed necessary conditions.

A. Filtering Design

In this section, we consider a two-step procedure to the filtering design process:

First, we calculate the expected of estimation error as.

Using above (12), we deduce that:

Lemma 1:Consider system (12). If E[e(0)]=0, then there exist positive-definite stabilizing solutionsandSi ≥0 to a discrete algebraic Riccati equation (DARE):

The filter gain given by:

is investigated such that the closed loop system is guaranteed asymptotically stable.

Proof:The aim of localH∞filtering on each sensor is to tackle the impact of estimation error, and both disturbances.For analyzing theH∞performance, we consider a candidate cost function:

whereγ ＞0, and the Lyapunov quadratic function is given by:

Using (12), we arrive at:

If there is no disturbance , i.e.,wik ≡0, we obtain:

which means that the estimation error converges to zero for 0≤i ≤n. Hence, the closed loop systemis asymptotically stable. In this sense, it turns out that (10) is satisfied.

whereS1kis the solution of the Lyapunov equation:

To handle the distributed filtering problem by satisfying (10),we minimize the constrained systems (13) and (21) to obtain the estimator gainKi.To avoid the coupling problem by using(21), we consider the following equation

The following result pertaining to the solution of the Lyapunov equation stands out:

Lemma 2:If S1k andare the solutions of(21)and(22),respectively then tr

To analyze the stability of the system, we examine the following state dynamics:

By selecting a candidate Lyapunov function,

It can be seen that the termsS1k+Sik ≥0,and.

The previous lemma demonstrates thatis limited byS1k.By minimizing the constrained systems(13)and(22),

Introduce the performance criterion:

By figuring out the minimization issue minKi(tr(S1k)) for the constrained systems (13) and (26), the following result is established

Proof:Doing a little algebra on (27) shows that

Based on foregoing lemmas,we can establish the following theorem:

Theorem 1:Consider the model(5)subject to the associated Assumptions 1-5. If there exist positive definite stabilizing solutionsPik,Si,to (13) and (26)-(29), respectively,then the distributed filters/estimators in (6) guarantee that the(10)is feasible.Moreover,we havewhere the estimator in (6) and gainsKi,Giare given in (14) and(30), respectively.

Proof:By employing Lemma 2, the closed loop systemis asymptotically stable and the developed estimators ensure that (10) is feasible. By Lemmas

In Theorem 1, the matricesandγsolving the coupled matrix (14)-(30) are readily obtained.

Theorem 2:Consider the given model (5) subject to associated Assumptions 1-5, IfEi= 0 and there exist positive definite stabilizing solutionsPikandSi,andPiito (13)and (26)-(29), then the distributed estimators in (6) ensure thatis asymptotically stable, (10) is feasible, andwhere the estimator gainis defined in(6)and the controller gainKiis detailed as follows:

We can observe that Theorem 2 has more relaxed conditions, and it considersandin which it simplifies the computing of the local observer and coupled gains, however, the disturbance is commonly presented in the observations.

Consider reliable communications channels over the network where the information can be exchanged regularly.Thus Assumptions 1-5 are required. In what follows, we direct attention to the situation where Assumptions 1-5 can be relaxed. For sensori, we let the vectorbe a collection of the measurementyiand its neighbors measurementyiforcan be written in the following form

Then relaxed assumptions are given as:

Assumption 6:The pairis detectable.

Assumption 7:

Assumption 8:

Assumption 9:is non-singular.

In terms of the observations(32),we haveandWe assumeandUsing aforementioned assumptions,we deduce that:

Theorem 3:Consider the model (5) subject to Assumptions 1 and 6-9. If there exist matrix solutionsPik ＞0 to (13)andSi ≥0, andto (26)-(29) respectively, then the distributed estimators in (6) guarantee that the closed-loop matrixis asymptotically stable, (10) is achieved.

Theorem 4:Consider the model given in (5) subject to Assumptions 1 and 6-9. IfEi= 0 and there exist positive solutionsPik ＞0 to (13) andSi ≥0,andto(26)-(29)respectively,then the distributed estimators in(6)sustain that the closed loop matrixis asymptotically stable, (10) is achieved.

B. Relaxed H2/H∞Filtering

In this sense, Lemma 2 can be rephrased as:

Lemma 4:For system (12), if E[e(0)]=0, then there exist positive-definite stabilizing solutionsandSi ≥0 to the DARE:

We examine the gain of the form:

The filter gain is selected so that the closed loop systemis asymptotically stable.

Proof:Let= E(eik) andVik=We obtain

Using above equations, we deduce:

Ifwe deduce:

Here, we study theH2performance (7). Since the closed loop matrixis Hurwitz, we can deduce that:

LetSidenote the solution of the subsequent Lyapunov equation

Lemma 5:IfSiis the solution of (38), thenwhereis the solution of

Since the closed-loop systemis Hurwitz,Siis positive semi-definite. Sinceis a positive definite solution of (33), we assure thatAiis generally Hurwitz.Utilizing equations (38) and (39), we deduce:

LetSirepresent the solution of the subsequent equation:

Deploying the consequence from (33) and (42), we have:

Lemma 6:Consider system(12).If E[ei(0)]=0,then there exists a positive-definite stabilizing solutionandSi ≥0 to a discrete-time Algebraic Riccati equation (DARE):

It follows from Lemma 6 that we can minimize cost function under (43). The Lyapunov function is given by:

Simple computation yields:

By minimizingtrSi, we setand deduce the gain matrix:

To fulfill the previous outcomes,we introduce the following theorem:

Theorem 5:Consider system(5).Given Assumptions(1-5),if

wherewith2i=has a positive definite stabilizing solution,the distributed filter(6)with the estimator gain designed by (47) and the coupling gain:

Then the closed-loop systemis Hurwitz, (10) is satisfied, and

Proof:The gainKiis given by (47), the equation in (48)can be rearranged as (43). Subsequently from the proof of Lemma 4, it can be clearly noted thatis Hurwitz.This means that(10)is satisfied.The(50)has been developed in Lemma 4. ■

Remark 2:From Theorem 5,it can be seen that the estimator gainKirelies on the solution of the DARE (43). Theorem 5 can provide overcomes easier than Theorem 1. Additionally,theH2performance in Theorem 5 is smaller than Theorem 1. In these sense, the former theorem can provide handy computing forH2performance and gain matrices.

If own and neighbor measurements are available for filtering/estimation unit, we deduce:

Theorem 6:For the given system (5), subject to associated Assumptions (1-9), if

V. NUMERICAL SIMULATION

In this section,we give a simulation example to illustrate the effectiveness of the developed methods. Consider the model(5) described as follows. In this example, we applied the proposed scheme for a discretized wind turbine mode. The original wind turbine model is given in [25], as:

The measurements of distributed sensors are :

And the disturbance is given as:

In this framework, each sensor (node) has access to information from different neighbors. Furthermore, it can be substantiated that Assumptions(1-5)are achieved.The developed distributedH2/H∞filters are successfully implemented to tackle the relaxedH2/H∞in the presence of external disturbances.

By selectingand, and carrying out DARE(48), we deduce

The estimator gains are given:

While the coupling gains:

Figs.1-3 show the efficiency of the proposed filtering scheme against the bounded disturbance. It is noticed from Fig.3 that the error system is asymptotically stable when,and the filtering scheme has the capability to attenuate the bounded disturbance impacts. Figs.1 and 2 represent the actual and estimated states for one sensor. It can also be confirmed that the closed-loop matrixis Hurwitz.TheH2performance is bounded by

Fig.1. Dynamic response of actual states.

VI. CONCLUSION

AnH2/H∞filtering technique is proposed to estimate states and to attenuate the influence of external disturbances of discrete-time systems. Under the condition of stochastic and bounded power signals, the filtering strategyH2/H∞is selected based on the two step computing procedure to determine the estimator and coupling gains. A switched strategy is implemented to switch the states once the operation conditions have changed. Then, the stability of the overall system in terms ofH2/H∞performance is established using candidate switched Lyapunov functions.Simulation example is given to show the effectiveness of the proposed approach. The proposed filtering structure showed robustness against the stochastic and bounded power signals.

Fig.2. Dynamic response of estimation states.

Fig.3. Dynamic response of estimation error.