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This paper is concerned with a networked estimation problem in which sensor data are transmitted over the network. In the event-based sampling scheme known as level-crossing or send-on-delta (SOD), 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. Event-based sampling has been shown to be more efficient than the time-triggered one in some situations, especially in network bandwidth improvement. However, it cannot detect packet dropout situations because data transmission and reception do not use a periodical time-stamp mechanism as found in time-triggered sampling systems. Motivated by this issue, we propose a modified event-based sampling scheme called modified SOD in which sensor data are sent when either the change of sensor output exceeds a given threshold or the time elapses more than a given interval. Through simulation results, we show that the proposed modified SOD sampling significantly improves estimation performance when packet dropouts happen.

Recent works have discussed event-driven alternatives to traditional time-triggered sampling schemes. It has been shown to be more efficient than time-triggered one in some situations, especially in network bandwidth improvement. In [

However, analysis and simulation in the the works on event-driven sampling scheme were performed under ideal communication network conditions: no delays or packet dropouts are assumed, but in realistic applications, network induced delays and packet losses do happen.

The issues of network delays and packet dropouts in time-triggered systems have been addressed and solved by researchers in [_{2} filtering problems associated respectively with possible delay of one sampling period, uncertain observations and multiple packet dropouts are studied under a unified framework. The H_{2}-norm of systems with stochastic parameters is defined and computed via a Lyapunov equation and a steady-state filter is designed via an LMI approach. In [

In conventional event-based sampling systems, also called send-on-delta (SOD) sampling [

Consider a networked control system described by the linear continuous-time model:
^{n}^{p}

The modified SOD sampling scheme illustrated in

Let _{last,i}_{last,i}_{y,i}_{t,i}

Using the modified SOD sampling scheme above we will obtain some benefits. Firstly, the estimator can detect signal oscillations or steady-state error if the difference of output value remains within the threshold range during a long time. Secondly, the estimator can detect multiple packet dropouts if it does not receive sensor data within the interval (0, _{t,i}

However, this scheme has one disadvantage that sensor data transmission rate will be increased due to condition (2b). If _{t,i}_{t,i}_{t,i}

The estimator node detects packet dropouts of _{last,i}_{t,i}_{last,i}_{t,i}

_{last,i}_{i}δ_{t,i}_{i}_{last,i}

Note that the estimator just detects “at least” _{i}_{i}_{last,i}_{last,i}_{t,i}_{last,i}_{t,i}_{last,i}_{last,i}_{t,i}_{last,i}_{t,i}_{t,i}_{t,i}

The networked estimation problem applying modified SOD transmission method can be described as follows:

Measurement output _{i}

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 _{last,i}_{i}δ_{t,i}_{i}_{last,i}_{i}_{n,i}_{i}_{i}_{last,i}

Note that if _{i}_{i}_{last,i}_{i}_{n,i}_{i}

We know from _{i}_{last,i}_{y,i}_{y,i}

For general cases, as shown in _{i}

Note that (3) is also applied to the case of no packet dropout [_{i}_{i}_{last,i}_{i}_{last,i}

Therefore, if there is no _{last,i}_{i}_{y,i}^{2}/3.

A modified Kalman filter for state estimation _{k}_{d}_{last}

In the modified Kalman filter in _{i}_{last,i}_{last,i}_{i}_{y,i}^{2}/3 as measurement noise covariance for state estimation.

As stated in [_{i}_{i}

As mentioned in Section 3, _{t,i}_{t,i}_{t,i}_{t,i}_{t,i}

The total sensor data transmission rate caused by condition (2b) in a time unit:

Let _{i}_{i}_{i}_{i}_{i}_{y,i}_{t,i}_{i}_{t,i}

The average number of packet dropouts in the conventional SOD sampling per a time unit:

In the proposed SOD sampling, the average number of packet dropouts within the time interval _{t,i}

We know from Section 4.1 that the larger number of consecutive packet dropouts is, the larger measurement noise covariance is. Measurement noise covariance is largest if _{i}_{i}

In this section, _{t,i}

The estimation performance in this case can be computed from the following discrete algebraic Riccati equation:

Note that (10) does not provide the actual estimation error covariance of the filter. The main purpose of (10) is to evaluate how _{t,i}_{t,i}

The solution of (10) is denoted by _{t,i}_{i}

_{t}_{0} is the upper bound error covariance with given value _{y,i}_{0} is also the estimation performance of the conventional SOD. _{t,i}

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 modified SOD sampling:

Choose _{t,1}, _{t,2} of (11) along with _{y,i}_{i}_{t,i}_{y,i}_{i}_{y,i}_{t,i}_{i}_{t,i}

_{y,1} = _{y,2} = 0.5, _{1}, _{2} are varying 5%, 10%, 15%, 20%. Estimation error is evaluated by:
_{i}_{i}

In _{1} = _{2} = 0.05, the total number of sensor data transmissions in the modified SOD (# 137) is just slightly greater than that in conventional SOD (# 126) but the estimation error is reduced so much ((e_{1} = 0.0075, e_{2} = 0.0096) compared to (e_{1} = 0.0383, e_{2} = 0.0167)).

_{1} = _{2} = 0.05, _{y,1} = _{y,2} = 0.5, _{t,1} = 4.12, _{t,2} = 4.69. The boundry of _{1} in the modified SOD filter (SODa) is much smaller than that in the conventional SOD filter. _{1} = 7, n_{2} = 7) compared to (n_{1} = 101, n_{2} = 36)]. When the modified SOD sampling is applied, the total number of sensor data transmissions is slightly increased, but the estimation error is significantly reduced. Therefore, the modified SOD sampling significantly improves estimation performance with only a little increase in the data transmission rate.

Notice that if we just consider the transmission condition (

As illustrated in

In case the output changes fast, it is obvious that ignoring packet dropout will introduce extremely incorrect result because we still use the wrong old measurement noise value even when we do not know how much the output value changes.

In this paper, the state estimation problem with modified SOD transmission method over networks, in which an event-based sampling is combined with a time-triggered sampling to detect packet loss situations, has been considered. We have shown that when using the proposed modified SOD filter, estimation performance is significantly improved with a small increase in sensor data transmission. If multiple packet dropouts happen, the estimator node will detect and compensate for them with an amount of additive measurement noise to improve estimation performance. This method is very useful for networks where data transmission is unreliable due to noise.

This work was supported by the Korea Research Foundation Grant D00059 (I00048). The second author would like to thank Ministry of Knowledge Economy and Ulsan Metropolitan City which supported this research through the Network-based Automation Research Center (NARC) at the University of Ulsan

Principle of SOD and modified SOD sampling schemes.

Multiple packet dropout detection.

Measurement noise increased due to multiple packet dropouts.

Structure of the modified Kalman filter.

_{t,1} of (11) along with _{y,1} and _{1}.

_{t,2} of (11) along with _{y,2} and _{2}.

Estimation error in two filters as _{1} = _{2} = 0.05.

Instants the sensor node transmits data due to condition (2b).

Estimation error along with packet loss rate in two filters.

_{1} = _{2} |
||||
---|---|---|---|---|

n (SOD) | n_{1} = 95 | |||

n_{2} = 31 | ||||

_{t,i} |
_{t,1} = 4.12 |
_{t,1} = 2.08 |
_{t,1} = 1.73 |
_{t,1} = 1.52 |

_{t,2} = 4.69 |
_{t,2} = 2.31 |
_{t,2} = 1.91 |
_{t,2} = 1.66 | |

n (modified SOD) | n_{1} = 101 |
n_{1} = 109 |
n_{1} = 112 |
n_{1} = 115 |

n_{2} = 36 |
n_{2} = 44 |
n_{2} = 47 |
n_{2} = 50 | |

e (SOD) | e_{1} = 0.0383 |
e_{1} = 0.0384 |
e_{1} = 0.0386 |
e_{1} = 0.0391 |

e_{2} = 0.0167 |
e_{2} = 0.0168 |
e_{2} = 0.0169 |
e_{2} = 0.0172 | |

e (modified SOD) | e_{1} = 0.0075 |
e_{1} = 0.0064 |
e_{1} = 0.0039 |
e_{1} = 0.0020 |

e_{2} = 0.0096 |
e_{2} = 0.0089 |
e_{2} = 0.0082 |
e_{2} = 0.0069 |