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This paper is concerned with improving performance of a state estimation problem over a network in which a send-on-delta (SOD) transmission method is used. The SOD method requires that a sensor node transmit data to the estimator node only if its measurement value changes more than a given specified _{i}_{i}_{i}_{i}

Recently, interest in the study of networked control system (NCS) and sensor networks has increased widely due to its low cost, high flexibility, simple installation and maintenance [

One of the most interesting problems is how to reduce network bandwidth when there are many nodes on the network. This can be achieved by reducing either data packet size or data packet transmission rate. In [

The purpose of this paper is to extend our work on the modified Kalman filter employing a SOD transmission method [

Consider a networked control system illustrated as in ^{n}^{p}

The following assumptions are made on the data transmission over network:

Measurement outputs are s _{i}(1 ≤ _{i}

For simplicity in problem formulation, any transmission delay from the sensor nodes to the estimator node is ignored.

The estimator node estimates the states of the plant regularly at period _{i}_{i}

Let the last received value of _{last,i}_{last,i}_{last,i}_{i}_{last,i}_{i}_{n,i}_{i}_{i}_{last,i}_{i}_{last,i}

In [_{i}_{last,i}_{(}_{i,i}_{)} is smaller than
_{(}_{i,i}_{)} is the (_{i}_{last,i}_{i}_{i}_{i}_{last,i}_{i}_{last,i}_{i}^{2}. When the measurement covariance is large, then Δ_{i}_{last,i}

To demonstrate this argument, consider an example of the output response of an 2^{nd} order system with step input shown in _{n}^{2} and ^{2}, respectively. Obviously, the waveform of _{n}_{n}

Therefore, the assumption in [^{2}. Furthermore, in realistic systems, measurement noise is normally small, while

Now we compute the mean value and variance of _{n,i}_{n,i}_{i}

We see that variance of _{n,i}

In this section, we propose a method to adaptively use _{n,i}

Firstly, we compute the mean value of _{n,i}_{last}_{−1}_{,i}_{last,i}_{last}_{−1}_{,i}_{last,i}_{i}_{last,i}_{last,i}_{i}_{last,i}_{i}_{last,i}_{last,i}_{last,i}_{i}

Therefore, mean value of the new measurement noise _{n,i}

Once _{n,i}_{last,i}

Now we investigate the system response to determine when _{i}_{i}_{i}_{i}_{i}_{i}_{i}^{2} and ^{2}.

A modified Kalman filter for state estimation at step ^{2}. Basic principle of Kalman filters could be found in [

_{d}

_{last}

and

In the modified filter above, vector _{n}_{last}_{last}

The _{i}_{i}_{i}

It is difficult to derive an explicit expression of performance improvement. However, we could say that the error covariance _{k}_{k}

To verify the proposed filter, we consider an example of the step response of a second-order system where the output is sampled by the SOD method:

The system parameters are given in 3 following cases for performance evaluation:

M = 30, a =10, b = 10 : overdamped system

M = 30, a =5, b = 1 : underdamped system

M = 30, a =5, b = 0 : undamped system

The simulation process is implemented for 50 seconds, and _{1}_{=5000}.

Estimation errors for different ^{2} is satisfied, are given in

The results also show that the filter using _{last}_{last}_{last}

The proposed filter overcomes this situation by applying the algorithm

In this paper, the state estimation problem with SOD transmission method over the network has been considered. The main objective of this paper is how to reduce the estimation performance degradation when the SOD transmission method is used. With given

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 financial support from post-BK(Brain Korea) 21 program. This work was supported by grant No. R01-2006-000-11334-0 from the Basic Research Program of the Korea Science & Engineering Foundation.

Configuration of a networked control system

Effect of _{n}

Structure of the modified Kalman filter loop

Estimation error of 3 filters (

Estimation error of 3 filters (

Estimation error of 3 filters (

Estimation error with different

454 | 268 | 185 | 141 | |

Filter using (3) | 2.5643e-004 | 3.2061e-004 | 4.6965e-004 | 5.6528e-004 |

Filter using (4) | 1.9581e-004 | 2.0124e-004 | 3.0587e-004 | 3.5756e-004 |

Proposed filter | 1.9581e-004 | 2.0123e-004 | 3.0588e-004 | 3.5755e-004 |

Estimation error with different

1168 | 657 | 484 | 370 | |

Filter using (3) | 0.9644e-004 | 2.7051e-004 | 4.1805e-004 | 5.5656e-004 |

Filter using (4) | 1.7065e-004 | 2.9365e-004 | 4.6972e-004 | 5.5783e-004 |

Proposed filter | 0.9468e-004 | 2.6787e-004 | 4.1551e-004 | 5.5166e-004 |

Estimation error with different

4055 | 3091 | 1939 | 1700 | |

Filter using (3) | 4.1432e-005 | 5.1109e-005 | 4.1805e-005 | 6.5436e-005 |

Filter using (4) | 4.4961e-005 | 6.6618e-005 | 4.6972e-005 | 8.2118e-005 |

Proposed filter | 3.9019e-005 | 4.6125e-005 | 4.1751e-005 | 6.0537e-005 |