H ∞ Filtering for Networked Systems with Bounded Measurement Missing

This paper is concerned with the H ∞ filtering problem for networked systems with bounded measurement missing. A switched linear system model is proposed to describe the considered filtering error system. A sufficient condition is derived for the filtering error system to be exponentially stable and achieve a prescribedH ∞ filtering performance level.The obtained condition establishes quantitative relations among theH ∞ performance level and two parameters characterizing the measurement missing, namely, the measurement missing rate bound and the maximal number of consecutive measurement missing. A convex optimization problem is presented to design the linearH ∞ filters. Finally, an illustrative example is given to show the effectiveness of the proposed results.


Introduction
A sensor network consists of spatially distributed autonomous sensors to cooperatively monitor physical or environmental conditions.The purpose of a sensor network is to provide users with the information of interest from data gathered by spatially distributed sensors, and it has found applications in a variety of areas, such as moving target localization and tracking, environment monitoring, intelligent transportation systems, and sensor-actuator networkbased control (see [1,2] and the literature therein).Thus, it is not surprising that signal estimation has been one of the most fundamental collaborative information processing problems in sensor networks [3][4][5][6][7][8][9][10]. In the environment of sensor networks, the measurements may be unavailable to the estimators intermittently.The corresponding estimation problem is usually termed estimation with missing, incomplete, or intermittent measurements and has received increasing research attention; see, for example, [11][12][13][14][15][16][17] and the references therein.
Measurement missing degrades the performance of the filtering system, and one may be concerned with the problem about how much the filtering performance declines for a certain amount of measurement missing, what is the maximal number of consecutive measurement missing that the filtering system can tolerant to guarantee the desired filtering performance, and so forth.This gives rise to the idea of revealing the relations, especially the quantitative relations, among the filtering performance level and some parameters characterizing the measurement missing, such as the missing rate/probability bound and the maximal number of consecutive measurement missing.Such relations may provide useful guidelines for designing the filtering systems with measurement missing.For example, in the sensornetwork-based filtering system, one may purposely suspend some sensor nodes intermittently to save node power without destroying the stability and desired performance level of the filtering systems; in some network-based fast sampling filtering systems, one may consider dropping certain amount of data packets to ensure that the set of the network-based filtering systems is schedulable and meanwhile to guarantee that the overall network-based filtering systems are stable and achieve prescribed filtering performance level and so forth.In these applications, establishing the relations among the filtering performance level and the parameters characterizing the measurement missing is of prior importance.Existing results on this topic can be generally classified into two frameworks, namely, the stochastic framework [18][19][20][21][22] and the deterministic framework [23].
A general structure of the filtering system with measurement missing is shown in Figure 1, where  is the measurement and ŷ is the filter input.In the stochastic framework, the measurement missing is usually described by a Bernoulli sequence or a Markov chain.For those results using the Bernoulli sequence, such as [14,15], the filter input is usually described as ŷ() = ()() + (1 − ()) ŷ( − 1) or as ŷ() = ()() by introducing the binary random variable (), where Prob{() = 1} = , and  is usually called the measurement arrival probability while 1 −  is called the measurement missing probability.For those results using Markov chain technique, such as [12,21], the measurement missing and the measurement arrival are defined as two modes of the Markov chain, and the mode transition probabilities are assumed to be known.In these ways, the stochastic parameters (either the Bernoulli sequence or the Markov chain) are incorporated into the filtering system models, and the relation between the filtering performance level and the measurement missing probability is implicitly established.Note that the maximal number of consecutive measurement missing is an important parameter that characterizes the measurement missing, and it also plays an important role in affecting the filtering performance.However, the relation between the filtering performance level and the maximal number of consecutive measurement missing is not established in all the aforementioned results using stochastic framework.Being different from the stochastic framework, the effects of the measurement missing on the filtering system are treated as time delays in the filter input in the deterministic framework, and the filtering error system with bounded measurement missing is usually described as a deterministic system with bounded timevarying delays.In this way, the relation between the filtering performance level and the maximal number of consecutive measurement missing is established.Some results on this topic can be found, for example, in [7,23], where the  ∞ filtering was investigated for linear continuous-time systems with bounded measurement missing and delays.However, the relation between the filtering performance level and the measurement missing rate/probability is not established in the existing results using deterministic framework.
By a closer inspection, it is found that the relations among the filtering performance level and the parameters characterizing the measurement missing revealed in the existing results using either the stochastic framework or the deterministic framework are quite implicit.Actually, besides of determining the maximal allowable measurement missing rate/probability bound, one may be interested more in the quantitative results about how much the filtering performance declines for a certain amount of measurement missing and for a certain extent of increase on the maximal number of consecutive measurement missing, or in other words, the quantitative relations among the performance level and the parameters characterizing the measurement missing.Specifically, this gives rise to the problem of how to express the filtering performance level as a function of the parameters characterizing the measurement missing.To the best of the authors' knowledge, such quantitative relations for the filtering systems with bounded measurement missing have not yet been established in the existing results, which motivates the presented research.
In this paper, the  ∞ filtering problem is investigated for networked systems with bounded measurement missing.The filtering error system is described as a switched linear system by using the augmentation technique and by including the numbers of consecutive measurement missing as switching parameters.Based on the obtained switched system model, a sufficient condition is derived for the filtering system to be exponentially stable and achieve a prescribed  ∞ performance level.A convex optimization problem is also presented to design the linear  ∞ filters which guarantee that the considered filtering system achieves a suboptimal  ∞ performance level.The main contributions of the paper are summarized as follows.(1) A switched system model with finite number of subsystems is established to describe the networked filtering system with bounded missing measurements.(2) A quantitative relation is established between the filtering performance and parameters characterizing the measurement missing.Specifically, the exponential decay rate of the filtering error system is given as a monotonic increasing function of the measurement missing rate bound, while the  ∞ performance level is given as a monotonic increasing function of both the measurement missing rate bound and the maximal number of consecutive measurement missing.
(3) The established relation gives quantitative results about how much the  ∞ filtering performance level declines for a certain amount of measurement missing and for a certain extent of increase on the maximal number of consecutive measurement missing.They theoretically reveal that measurement missing degrades the  ∞ filtering performance.

Modelling of the Filtering System
The considered filtering problem is shown in Figure 1, where the physical plant is described by where () ∈ R  is the system state, () ∈ R  is the measured output, () ∈ R  is the signal to be estimated, and () ∈ R  is the noise signal which belongs to  2 [0, ∞).  ,   ,   ,   , and   are constant matrices with appropriate dimensions, and   is assumed to be stable.The full-order linear filter used to estimate the signal () is given by where   () ∈ R  is the filter state, ŷ() ∈ R  is the filter input, and   () ∈ R  is the estimated signal.  ,   , and   are filter matrices to be determined.
Since the measurements may be missed during the transmission from the sensor to the filter, the filter input ŷ may not be equal to the measured output .Suppose that the number of consecutive measurement missing  is upper bounded by  ≤  and the filter input holds at its last available value if a measurement is missed, where  is a known constant.Then, we have if there are  consecutive measurement losses before the time step  + 1, where  ∈  0 = {0, 1, . . ., }.It can be seen from ( 3) that the system model of the filter varies over different sampling intervals since the number of consecutive measurement missing is time varying over different sampling periods.By defining the filtering error signal as () = () −   () and denoting we obtain the following filtering error system model: where () ∈  0 , and By applying the augmentation technique, the filtering error system is finally described as a nondelayed switched system by including the numbers of consecutive measurement missing as switching parameters.() serves as the switching signal of system (5) which contains  + 1 subsystems.The switching of the subsystems is determined by the measurement missing status.Specifically, for  = 0, 1, . . ., , we have () =  and system (5) resides in the subsystem   during the interval [,  + 1) if there is  consecutive measurement missing before time step  + 1.This is illustrated in Figure 2, where • represents the measurement that is successfully received by the filter, while ∘ stands for the one that is missed.At time step  − 1, the measurement ( − 1) is successfully received by the filter.Then, ŷ( − 1) = ( − 1), and  () resides in  0 .At time step , the measurement () is missed and the filter input holds at its previous value; that is, ŷ() takes ( − 1), and  () resides in  1 .Then, at time step  + 1, the measurement ( + 1) is missed again, and the filter input remains at (−1); that is, ŷ(+1) = (−1), and  () resides in  2 .The rest can be deduced by analogy.It is seen from the above analysis that the filtering error system  () takes on the following characteristics.
(b) Switching between the subsystems of  () occurs only when the subsystems   ( = 1, . . ., ) appear, and the number of switches of  () is determined by the total activation times of   .Moreover, at most two switches may be involved when   appears one time.
For  = 0, 1, . . .,  and 0 ≤  1 ≤  ≤  2 , let   [ 1 ,  2 ] denote the number of switches of () and   [ 1 ,  2 ] the times that the subsystem   is activated on the interval [ 1 ,  2 ].Then, it follows from the characteristics of  () that the following relation holds: Definition 1.For any switching signal () and 0 ≤  the interval The subsystem   ,  ∈ {1, . . ., } is activated when a measurement missing occurs, so the measurement missing rate on any interval [ 1 ,  2 ] can be defined as . Then, we have by relation (7) and Definitions 1 and 2 that Moreover, we have by the characteristics of  () that there exists a constant  < 1 such that the following relations are true for any  ≥ 0 and  ≥ 0: Remark 3. The considered filtering system with bounded measurement missing is finally described as a switched linear system  () with  + 1 subsystems, where  0 describes the filtering system without measurement missing, while   ( = 1, . . ., ) describes those with measurement missing.It is reasonable to see that the filtering system will achieve a better  ∞ performance level if  0 appears with higher frequency (or, in other words, the measurement missing rate is lower).This is the motivation of the proposed switched system model of the filtering error system, which may enable us to establish the quantitative relation between the  ∞ filtering performance and the measurement missing rate bound.Note that the augmentation technique is used to obtain the switched system model, which increases computational cost.
From the energy-efficient perspective, the proposed method is applicable to the case where the number of consecutive packet losses is bounded and not too large.
The following definition and assumptions are needed in the derivation of the main results.Definition 4. System ( 5) is said to be exponentially stable with decay rate  < 1 if, for any finite initial state (0) ∈ R (+2) , there exist a constant  > 0 such that ‖()‖ ≤   ‖(0)‖ holds.
The  ∞ filtering problem to be addressed in this paper is expressed as follows.

𝐻 ∞ Filtering Analysis
An  ∞ performance condition for the filtering error system (5) is presented in the following theorem.
Remark 8. Theorem 7 establishes the quantitative relations among the  ∞ filtering performance level, the measurement missing rate bound, and the maximal number of consecutive measurement missing.Specifically, the  ∞ filtering performance level (, ) is given as a monotonic increasing function of both the measurement missing rate bound  and the maximal number of consecutive measurement missing , which implies that some smaller measurement missing rate and maximal number of consecutive measurement missing lead to a better  ∞ filtering performance.
Remark 9.It is obtained from (12) that It follows from ( 22) that   () <   (0) for () = 0, which implies that the subsystem   ( ∈  0 ) is exponentially stable with decay rate .On the other hand, we have by (22) that under zero initial condition.Summing both sides of ( 23) from  = 1 to  = +∞, we obtain (1 −  2 ) −1 ∑ +∞ =0 () < 0, which yields ‖‖ 2 <  0 ‖‖ 2 .It implies that the subsystem   achieves the  ∞ performance level  0 .In summary, if (10) is true for 0 <  < 1, then the subsystem  0 is exponentially stable with decay rate  and achieves the  ∞ performance level  0 .Note that if there is no measurement missing, then  () resides in  0 , and we have  = 0,  = 0 and  = 1.In this case, () and (, ) in Theorem 7 are, respectively, reduced to  and  0 , which are just the decay rate and the  ∞ performance level of  0 , respectively.So, Theorem 7 theoretically reveals that measurement missing degrades the  ∞ filtering performance and presents the quantitative result about the effect of the measurement missing on the  ∞ filtering performance.
The condition in Theorem 10 is convex in  2 0 .So, the convex optimization problem min  s.t.
(24) and (25) with  =  2 0 (30) can be formulated to design the  ∞ filters.If  * is the solution of the optimization problem (30), then the designed filters guarantee that system (5) achieves the following  ∞ performance level: Remark 11.Note that  * (, ) is monotonic increasing on  and .Therefore,  and  should be chosen as small as possible to obtain a better  ∞ filtering performance and meanwhile to make the condition  < 1 easier to be satisfied.Nevertheless, it can be seen from ( 24) and (25) that some smaller  and  will result in a larger value of  0 and ultimately yield a larger value of  * (, ).The following algorithm provides a method on determining the parameters  and  that will result in a suboptimal  ∞ performance level  * (, ).

Illustrative Examples
Example 1.Consider a mechanical system with two masses and two springs as that studied in [7].Its state-space model is given by where  1 and  2 are the positions of masses  1 and  2 , respectively.Choose  1 = 1,  2 = 0.
Suppose that  1 is measured by a device with noise () and  2 is to be estimated by using the  ∞ filters; then we have   = [1 0 0 0] and   = [0 1 0 0].Furthermore, we choose   = 0.1.
Choose  = 1.01 and  = 0.96.Suppose that  = 2 and  = 50%.Then, by solving the optimization problem (23), In what follows, we will show the relations among the  ∞ performance level, the measurement missing rate bound, and the maximal number of consecutive measurement missing.3 is adopted, and the measurement missing is generated randomly under the constraint of the corresponding missing rate bound.It can be seen from Table 1 that the larger the missing rate bound, the worse the  ∞ filtering performance.Similarly, Table 2 shows that the larger the maximal number of consecutive measurement missing, the worse the  ∞ filtering performance.These verify the statements in Theorem 7.Moreover,   is always smaller than  * for different values of  in Table 1, and it is always smaller than  * for different values of  in Table 2, showing the effectiveness of the  ∞ filter design.On the other hand, it can also be seen from Tables 1 and 2 that  * is much larger than   for different values of  and , which indicates that there exists certain extent of conservatism in the  ∞ filter design and the estimation of the  ∞ performance level  * .

Conclusions
In this paper, the  ∞ filtering problem was investigated for networked systems with bounded measurement missing.
A switched system model was proposed to describe the considered system, which helped establish the quantitative relation between the  ∞ performance level and two parameters, namely, the measurement missing rate bound and the maximal number of consecutive measurement missing.It has been shown by the example that there exists certain extent of conservatism in the proposed  ∞ filter design, and the conservatism is partly introduced by the bounding on the switching frequency of the switched filtering error system given in (8).A tighter bounding on the switching frequency may help reduce the conservatism, which requires more detailed online information about the measurement missing status.

2 MathematicalFilterFigure 1 :
Figure 1: Diagram of the filtering problem with measurement missing.

Figure 2 :
Figure 2: An example of measurement missing.

Table 1 :
The relation between the  ∞ performance level and the measurement missing rate bound for  = 2.

Table 2 :
The relation between the  ∞ performance level and the maximal number of consecutive measurement missing for  = 50%.For  = 1.01,  = 0.96, and  = 2, the relation between the  ∞ performance level and the measurement missing rate bound is given in Table1by solving the optimization problem (30).For  = 1.01,  = 0.96, and  = 50%, the relation between the  ∞ performance level and the maximal number of consecutive measurement missing is presented in Table2by solving the optimization problem (30).In calculating the actual noise attenuation level   , the noise signal () in Figure