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Sensors 2008, 8(2), 897-909;
Abstract
: This paper is concerned with the networked estimation problem in which sensor data are transmitted over the network. In the event-driven sampling scheme known as level-crossing or send-on-delta, sensor data are transmitted to the estimator node if the difference between the current sensor value and the last transmitted one is greater than a given threshold. The event-driven sampling generally requires less transmission than the time-driven one. However, the transmission rate of the send-on-delta method becomes large when the sensor noise is large since sensor data variation becomes large due to the sensor noise. Motivated by this issue, we propose another event-driven sampling method called area-triggered in which sensor data are sent only when the integral of differences between the current sensor value and the last transmitted one is greater than a given threshold. Through theoretical analysis and simulation results, we show that in the certain cases the proposed method not only reduces data transmission rate but also improves estimation performance in comparison with the conventional event-driven method.1. Introduction
The traditional way to transmit sensor data in networked control systems is to sample sensor signals equidistant in time. This approach is widely used since analysis and design are simple and inherited from well-established control system theories. If the network speed is high and the traffic is sparse, the periodic sampling approach has many merits because network-induced delay is small and can be ignored. But when the network bandwidth is limited due to executing tasks of several nodes, time delay becomes large and randomly varying. To avoid these problems, the sensor data transmission rate should be reduced.
Recent works have discussed event-driven alternatives to traditional time-triggered sampling scheme. It has been shown to be more efficient than time-triggered one in some situations, especially in network bandwidth improvement. For instance in [1], the authors provided a comparison of time-triggered impulse control and level-triggered one where the level-triggered scheme gave lower average error for the same average rate of impulse. In [2, 3], an adjustable deadband was defined on each node to reduce network traffic. The node does not broadcast a new message if its signal is within the deadband. A level-crossing sampling scheme in association with an 1-bit coding and decoding strategy to reduce data size was introduced in [4]. This coding sampling scheme becomes highly efficient under data-rate constraints since the nodes only transmit one bit, 0 or 1, per sample. In [5], an optimal level-triggered sampling design problem was proposed in order to minimize the distortion of a filter over a finite horizon. A similar sampling scheme named send-on-delta (SOD) transmission was explored in [6-8]. By adjusting the threshold value at each sensor node, data transmission rate is reduced in order for the network bandwidth to increase and can be used for other traffic. Although the event-driven schemes mentioned above have several names, the basic principle of sampling is the same. That is, in framework of networked control system, a sensor node transmits data only if the difference Δ between the current sensor value and the last transmitted one is greater than a given threshold. Hereafter, for ease in representation we refer to them as SOD sampling.
One major issue of SOD sampling method as pointed out in [9] is that it does not detect signal oscillations or steady-state error if the difference Δ remains within the threshold range during a long time. Furthermore, in highly noisy systems, the sensor value could exceed the threshold not because the true value exceeds the threshold but because sensor noise exceeds the threshold. In that case, there could be many unnecessary sensor data transmissions.
Motivated by the perspective results of the event-based integral sampling in [9] and the networked estimation problem using SOD transmission method in [7], we propose a novel sampling scheme called area-triggered or send-on-area (SOA) in which sensor data are sent only when the integral of Δ is greater than a given threshold. The proposed SOA sampling scheme is slightly different from [9] to improve bandwidth network in case of large noise. Then, a networked estimator based on Kalman filter is formulated to estimate states of the system periodically even when the sensor nodes do not transmit data. Through theoretical analysis and simulation results, we show that in the certain cases the proposed method gives better estimation performance than the SOD method in [7].
2. Area-triggered sampling scheme
Consider a networked control system described by the linear continuous-time model:
The area-triggered sampling scheme illustrated in Fig.1b is stated as follows:
Let y_{last},_{i} (1 ≤ i ≤ p) be the last transmitted value of the i-th sensor output at instant t_{last},_{i}. The area formed by the current output y_{i}(t) and y_{last},_{i} along time:
$${\mathrm{\Pi}}_{i}(t,{t}_{\mathit{\text{last}},i})\triangleq \underset{{t}_{\mathit{\text{last}},i}}{\overset{t}{\int}}({y}_{i}(t)-{y}_{\mathit{\text{last}},i})dt.$$A new sensor value will be sent to the estimator node if the following condition is satisfied:
$$\left|{\mathrm{\Pi}}_{i}(t,{t}_{\mathit{\text{last}},i})\phantom{\rule{0.2em}{0ex}}\right|>{\alpha}_{i}$$
Note that in the SOD sampling scheme [7] shown in Fig.1a, a new sensor value will be sent if |Δ_{i}(t, t_{last},_{i})| > δ_{i}, where δ_{i} is the given threshold value.
2.1. Effect of noise on sensor data transmission rate
In this section, we assume u(t) = 0 to investigate effect of noise on the system. The difference between the current sensor value and the last transmitted one is defined by:
Inserting (4) into (5), we obtain (6):
Assuming x(t_{last},_{i}) = 0, we can derive the variance of Π_{i}(t, t_{last},_{i}) as follows
Note that the variance of Δ_{i}(t, t_{last},_{i}) in the SOD sampling scheme [7] is given by
From (7) and (8) we see that both E{Π_{i}Π_{i}′} and E{Δ_{i}Δ_{i}′} consists of two terms. The first term depends on the process noise w(t) and the second one depends on the measurement noise v(t). Therefore, we believe E{Π_{i}Π_{i}′} and (in the case of SOA) plays the same role as E{Δ_{i}Δ_{i}′} (in the case of SOD) in investigation of data transmission rate. This assumption will be verified in the next section 2.3. In the analysis in [7], we know that data transmission rate is proportional to E{Δ_{i}Δ_{i}′}. In the highly noisy systems, large R value makes E{Δ_{i}Δ_{i}′} in (8) large. It leads to increase data transmission rate when applying the SOD sampling scheme. But with the SOA sampling scheme, we can constrain the effect of R on E{Π_{i}Π_{i}′} in (7) by lowering the factor (t − t_{last},_{i}). This situation can be achieved by choosing the α_{i} threshold value such that (t − t_{last},_{i}) ≪ 1s. In other words, if the time interval of two consecutive sensor data packets is less than one second, then the SOA method can be applied. This is not a so strict condition in networked control systems.
2.2. Π_{i} computation and SOA sampling in discrete time at the sensor nodes
Let T be sampling time of the signals y_{i}(t), y_{k},_{i}(k = 1,2,3,…) be the k-th sampled value y_{i}(kT) from time the i-th sensor node transmits y_{last},_{i}. If the sensor output has no noise, y_{k},_{i} lies on the smooth curve C_{i}x(t). But under the effect of measurement noise, y_{k},_{i} will be in the vicinity of C_{i}x(t) as illustrated in Fig.2. Supposing that condition (3) is satisfied at t = t_{3},_{i}, Π_{i}(t, t_{last},_{i}) shown approximately by the slashed area is calculated:
The principle of computing the slashed area in (9) is to divide it into several small parts. Each part is a trapezoid with the height T and two sides (y_{k},_{i} − y_{last},_{i}), (y_{k}_{+1},_{i} − y_{last},_{i}) except for the first part which is a triangle. However, using (9) is inconvenient if condition (3) is not satisfied for a long time interval because the sensor node has to spend much memory on storing y_{k},_{i}. An algorithm to calculate Π_{i} and transmit sensor data in which we only use 3 memory units to store (Π_{i}, y_{last,i}, y_{0},_{i}) at every period T is proposed as follows:
2.3. Effect of noise on signal distortion
We define a performance index called squared error distortion D_{i} for each sensor node:
It is very difficult to derive an explicit expression of signal distortion in two methods SOD and SOA by theoretical analysis because it depends not only on the given threshold but also the system model and noise. Thus, in this section we evaluate the effect of measurement noise on distortion by considering an example instead.
Consider two sensor outputs y(t) = 0.1t + v(t) and y(t) = 5(1 − e^{−0.1}^{t}) + v(t), where R is varying. We will see how data transmission rate and distortion vary as R varies in both methods SOD and SOA. The evaluation is implemented with T = 0.01s in 50 seconds. The threshold values δ and α are given in 3 cases:
Case 1. δ = 0.005: small threshold
Case 2. δ = 0.01: medium threshold
Case 3. δ = 0.05: large threshold
In the above example, α is chosen according to δ such that number of data transmissions in two methods is identical as v(t) = 0 (without noise). From the results in the Fig. 3 and Fig. 4, the effect of R on data transmission rate and distortion is summarized as follows:
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When the threshold value is small as in case 1, distortion of SOA is equivalent to that of SOD but data transmission rate of SOA is smaller than that of SOD. In this case, using SOA will reduce data transmission rate.
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When the threshold value increases as in case 2, distortion of SOA is slightly smaller than that of SOD and data transmission rate of SOA is much smaller than that of SOD. In other words, we get benefit of data transmission rate reduction and a little distortion reduction from SOA method.
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When the threshold value is large as in case 3, SOA reduces not only data transmission rate but also distortion. The larger R value is, the more reduction is. This result is remarkable! It means that in the SOA method, signal distortion is not degraded along with the reduction of data transmission rate even when system noise is large. Therefore, we hope that estimation performance will be significantly improved when applying a filter.
3. State estimation with SOA Transmission method
The networked estimation problem applying SOA transmission method is depicted as follows:
Measurement outputs y_{i} (1 ≤ i ≤ p) are sampled at the period T but their data are only transmitted to the estimator node when (3) is satisfied.
For simplicity in the problem formulation, transmission delay from the sensor nodes to the estimator node is ignored.
The estimator node estimates states of the plant regularly at the period T regardless of whether or not sensor data arrive. If there is no i-th sensor data received for t > t_{last},_{i}, the estimator node considers that the measurement value of the i-th sensor output y_{i}(t) is still equal to y_{last,i} but the measurement noise increases from v_{i}(t) to v_{n,i}(t) = v_{i}(t) + Δ_{i}(t, t_{last,i}).
To formulate a state estimation problem, the bound of Δ_{i}(t, t_{last},_{i}) needs to be determined. In the next section, we will compute the bound of Δ_{i}(t, t_{last},_{i}) and then a modified Kalman filter is applied for state estimation.
3.1. Bound of Δ_{i} (t, t_{last,i})
At time t = t_{1},_{i}, if the estimator node does not receive i-th sensor data as illustrated in Fig.2, from (3) and (9) we know that:
At time t = t_{2},_{i}, if the estimator node does not receive i-th sensor data yet then:
Inserting (13) into (15) we obtain:
Computing similarly for instants t = t_{k,i}(k = 3,4,…), we always have the following inequality:
In other words, the bound of Δ_{i}(t, t_{last},_{i}) is given by
Note that Δ_{i}(t, t_{last},_{i}) = 0 at time the estimator node receives i-th sensor data. Then, the process is repeated as (12)-(17) as long as the estimator node does not receive data from the i-th sensor node yet.
3.2. State estimation
Assuming that Δ_{i}(t, t_{last},_{i}) has a uniform distribution with (18), variance of Δ_{i}(t, t_{last},_{i}) will be (6α_{i}/T)^{2} /3. Thus, if there is no i-th sensor data received for t > t_{last},_{i}, variance of measurement noise is increased from R(i, i) to R(i, i) + (6α_{i}/T)^{2} /3.
A modified Kalman filter for state estimation x̂_{k} at step k, where there is a change in the measurement update part of the discrete Kalman filter algorithm [10], is given as in the Fig.5. We use the discretized system model sampled at period T :
Q_{d} is the process noise covariance of the discretized system:
In the modified Kalman filter in Fig.5, the states of the plant are estimated regularly at every period T regardless of whether or not sensor data arrive. If i-th sensor data arrive, Δ_{i}(t,t_{last},_{i}) = 0, the modified Kalman filter acts like the conventional Kalman filter. Otherwise, it uses y_{last},_{i} as the measurement value and R̄(i,i) = R(i,i) + (6α_{i}/ T)^{2}/ 3 as measurement noise covariance for state estimation.
4. Simulation
To verify the proposed filter, we consider an example of the second-order system with step input where the output is sampled by the SOD and SOA methods:
The system parameters are given in the following two cases for performance evaluation:
Case 1. (underdampled system) M = 30, a = 5, b = 1.
Case 2. (undamped system) M = 30, a = 5, b = 0.
The simulation process is implemented for 50 seconds. In each case, we use two methods (SOD and SOA) for performance comparison. Estimation performance is evaluated by the average distortion:
The simulation results with different threshold values for the two cases are shown in Table 1 and Table 2, where α value is chosen according to δ value such that n (number of sensor data transmissions) is identical in the two methods. In both cases, we see that when δ is small (i.e. δ = 0.1, 0.3), estimation performance of SOD and SOA is almost the same. Rigorously speaking, SOD is slightly better than SOA. However, when δ is increasing, SOA method shows to be outperform significantly. For example as δ = 0.9, SOA outperforms SOD by the D̄_{1} reduction of 5 times (i.e. 0.0116 vs. 0.0536 in case 1, and 0.0017 vs. 0.0081 in case 2). As illustrated in Fig. 6 and Fig. 7, the estimation error of SOA is much smaller than that of SOD.
Through simulation results, we see that the SOA method provides better estimation performance than the SOD method. The key reason is that in the SOD method, noise-containing sensor data not only make transmission rate increase but also degrade estimation performance. Whereas, thanks to the integral block which acts as a noise filter in the SOA sampling scheme, the sensor node does not send highly noisy data, but it transmits data only when the output indeed changes value. For that reason, the transmitted sensor data in the SOA method is more reliable. This helps estimation performance better.
5. Conclusion
In this paper, the state estimation problem with SOA transmission method over network has been considered. We have shown that when increasing the threshold value to improve bandwidth network, the SOA method gives much better estimation performance than the SOD method. This is very useful in the realistic applications where sensor data transmission rate needs to be lowered due to joining of many sensor nodes or for power saving in wireless networks. The SOA method has been also proven to be more efficient than the SOD method in the highly noisy systems.
This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R01-2006-000-11334-0). The authors also would like to thank Ministry of Commerce, Industry and Energy and Ulsan Metropolitan City which partly supported this research through the Network-based Automation Research Center (NARC) at University of Ulsan.
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Table 1. Estimation performance of 2 methods with different threshold values in case 1 |
δ | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 |
α | 0.0006 | 0.0081 | 0.0302 | 0.0534 | 0.0881 |
n | 2465 | 490 | 240 | 175 | 140 |
D̄_{1} by SOD | 6.08e-4 | 0.0026 | 0.0101 | 0.0268 | 0.0536 |
D̄_{1} by SOA | 6.40e-4 | 0.0030 | 0.0061 | 0.0103 | 0.0116 |
D̄_{2} by SOD | 0.0080 | 0.0106 | 0.0176 | 0.0205 | 0.0313 |
D̄_{2} by SOA | 0.0082 | 0.0110 | 0.0139 | 0.0144 | 0.0148 |
Table 2. Estimation performance of 2 methods with different threshold values in case 2 |
δ | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 |
α | 0.0005 | 0.0038 | 0.0094 | 0.0181 | 0.0303 |
n | 3347 | 1601 | 1094 | 822 | 658 |
D̄_{1} by SOD | 3.49e-4 | 8.94e-4 | 0.0022 | 0.0043 | 0.0081 |
D̄_{1} by SOA | 3.65e-4 | 8.32e-4 | 0.0015 | 0.0016 | 0.0017 |
D̄_{2} by SOD | 0.0052 | 0.0067 | 0.0087 | 0.0109 | 0.0131 |
D̄_{2} by SOA | 0.0053 | 0.0070 | 0.0079 | 0.0080 | 0.0080 |
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