Abstract
This paper is concerned with the 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 prescribed filtering performance level. The obtained condition establishes quantitative relations among the 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 linear filters. Finally, an illustrative example is given to show the effectiveness of the proposed results.
1. 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 network-based 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 [3–10]. 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, [11–17] 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 sensor-network-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 [18–22] 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 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 or as by introducing the binary random variable , where , and is usually called the measurement arrival probability while 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 time-varying 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.
2. Modelling of the Filtering System
The considered filtering problem is shown in Figure 1, where the physical plant is described by where is the system state, is the measured output, is the signal to be estimated, and is the noise signal which belongs to . , , , , 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 is the filter state, is the filter input, and 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 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 , where . 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 , 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 subsystems. The switching of the subsystems is determined by the measurement missing status. Specifically, for , we have and system (5) resides in the subsystem during the interval if there is consecutive measurement missing before time step . 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 , the measurement is successfully received by the filter. Then, , and resides in . At time step , the measurement is missed and the filter input holds at its previous value; that is, takes , and resides in . Then, at time step , the measurement is missed again, and the filter input remains at ; that is, , and resides in . The rest can be deduced by analogy. It is seen from the above analysis that the filtering error system takes on the following characteristics.(a)For , subsystem always appears after the subsystem , and subsystem always appears before or after .(b)Switching between the subsystems of occurs only when the subsystems () 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 and , let denote the number of switches of and the times that the subsystem is activated on the interval . Then, it follows from the characteristics of that the following relation holds:
Definition 1. For any switching signal and , let denote the number of switches of on the interval . Then, is called the switching frequency of on the interval .
Definition 2. For and , let . Then is called the occurrence rate of the subsystem on the interval .
The subsystem , is activated when a measurement missing occurs, so the measurement missing rate on any interval 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 such that the following relations are true for any and :
Remark 3. The considered filtering system with bounded measurement missing is finally described as a switched linear system with subsystems, where describes the filtering system without measurement missing, while () describes those with measurement missing. It is reasonable to see that the filtering system will achieve a better performance level if 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 if, for any finite initial state , there exist a constant such that holds.
Assumption 5. For any and , is upper bounded by a constant measurement missing rate bound ; that is, , where .
Assumption 6. and , for all .
The filtering problem to be addressed in this paper is expressed as follows.
Problem HFBMM ( Filtering with Bounded Measurement Missing). For the filtering problem in Figure 1 and a given system (1), determine the matrices , , and in filter (2), such that the system (5) with is exponentially stable, and the estimation error satisfies a prescribed performance level (i.e., ) under zero initial conditions (i.e., ) for all admissible bounded measurement missing.
3. Filtering Analysis
An performance condition for the filtering error system (5) is presented in the following theorem.
Theorem 7. For given scalars and satisfying , if there exist matrices , , and a scalar , such that the following inequalities hold: then system (5) is exponentially stable with decay rate and achieves a prescribed performance level , where is the measurement missing rate bound.
Proof. Choose the Lyapunov function and define . For , it follows from (5) and (10) that
where . For any given integer , let , denotes the switching instants of the switching signal on the interval . Then, we have by (11) that
It follows from (12) and (13) that
where
First, we consider the exponential stability of system (5) for . It follows from (8), (14), , and that
It is reasonable in practice that the time horizon is larger than . So, it can be further obtained from (16) by assumption 1 that
which yields , where and . and represent the maximum and the minimum eigenvalues of the matrix , respectively. On the other hand, and guarantee that . Thus, it is concluded by Definition 4 that system (5) is exponentially stable with decay rate .
Next, to prove the performance, we consider . Then, under zero initial condition, we have, by (14) and the fact that , that , and thus
It follows from Definition 1 that , which leads to
We have by (18) and (19) that
Summing both sides of (20) from to and changing the order of the summation, taking (9) into account, we obtain
It follows from (21) that , which implies that the filtering error system (5) achieves the performance level . The proof is completed.
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 for , which implies that the subsystem () 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 to , we obtain , which yields . It implies that the subsystem achieves the performance level . In summary, if (10) is true for , then the subsystem is exponentially stable with decay rate and achieves the performance level . Note that if there is no measurement missing, then resides in , and we have , and . In this case, and in Theorem 7 are, respectively, reduced to and , which are just the decay rate and the performance level of , 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.
4. Filter Design
Theorem 10. Consider the filtering problem in Figure 1. For given scalars and satisfying , if there exist matrices , , , , , , , , , and of appropriate dimensions and a scalar , such that the following linear matrix inequalities hold:then the filters of form (2) guarantee that the filtering error system (5) is exponentially stable with decay rate and achieves a prescribed performance level , and the filter matrices are given by where and are given in Theorem 7, and
Proof. By Lemma 1 in [24], (10) is true if and only if there exists a matrix such that the following inequality holds: On the other hand, (24) indicates that is nonsingular, and thus we can always find square and nonsingular matrices and satisfying . Now, introduce the variables and , and let Then, pre- and postmultiplying (28) and (11) by and , respectively, following some routine matrix manipulations, yield (24) and (25). Furthermore, denote the filter transfer function from to by . Substituting the filter matrices with (29) and taking into account, we obtain . So, the filter matrices are given by (26). Then, it follows from Theorem 7 that Theorem 10 is true. The proof is completed.
The condition in Theorem 10 is convex in . So, the convex optimization problem 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 easier to be satisfied. Nevertheless, it can be seen from (24) and (25) that some smaller and will result in a larger value of 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 .
Algorithm 12.
Step 1. Choose some sufficiently large initial and such that holds. Set and . Solve the optimization problem (30) and calculate for given and .
Step 2. Decrease by a certain step length; say, ; that is, set . Then, solve the optimization problem (30) and calculate for given and .
Step 3. If , then return to Step 2. Otherwise, exit and set .
Step 4. Decrease by a certain step length; say, ; that is, set . Then, solve the optimization problem (30) and calculate for the obtained and given and .
Step 5. If , then return to Step 4. Otherwise, exit and set . Then, and are the desired values of the parameters and , respectively, and is the desired suboptimal performance level for given and .
5. Illustrative Examples
Example 5. Consider a mechanical system with two masses and two springs as that studied in [7]. Its state-space model is given by
where and are the positions of masses and , respectively. Choose , , , and , and set ; then we obtain the discrete-time system model (1) with
Suppose that is measured by a device with noise and is to be estimated by using the filters; then we have and . Furthermore, we choose .
Choose and . Suppose that and . Then, by solving the optimization problem (23), we obtain the performance level with the following filter gain matrices:
In the simulation, the noise signal is assumed to be uniformly distributed with for the interval . The measurement missing is generated randomly and is shown in Figure 3(b), where the measurement missing rate is . The trajectories of and are depicted in Figure 3(a). By calculation, we obtain from the results in Figure 3(a) that , showing the effectiveness of the filter design. 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. For , , and , the relation between the performance level and the measurement missing rate bound is given in Table 1 by solving the optimization problem (30). For , , and , the relation between the performance level and the maximal number of consecutive measurement missing is presented in Table 2 by solving the optimization problem (30). In calculating the actual noise attenuation level , the noise signal in Figure 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 .
(a)
(b)
(c)
6. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work is partially supported by Zhejiang Provincial Natural Science Foundation under Grants nos. LQ12F03014 and Y13F010013.