The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain Observations
This paper considers the problems of the posterior Cramér–Rao bound and sensor selection for multi-sensor nonlinear systems with uncertain observations. In order to effectively overcome the difficulties caused by uncertainty, we investigate two methods to derive the posterior Cramér–Rao bound. The f...
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MDPI AG
2018-04-01
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| Series: | Sensors |
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| Online Access: | http://www.mdpi.com/1424-8220/18/4/1103 |
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| author | Zhiguo Wang Xiaojing Shen Ping Wang Yunmin Zhu |
| author_facet | Zhiguo Wang Xiaojing Shen Ping Wang Yunmin Zhu |
| author_sort | Zhiguo Wang |
| collection | DOAJ |
| description | This paper considers the problems of the posterior Cramér–Rao bound and sensor selection for multi-sensor nonlinear systems with uncertain observations. In order to effectively overcome the difficulties caused by uncertainty, we investigate two methods to derive the posterior Cramér–Rao bound. The first method is based on the recursive formula of the Cramér–Rao bound and the Gaussian mixture model. Nevertheless, it needs to compute a complex integral based on the joint probability density function of the sensor measurements and the target state. The computation burden of this method is relatively high, especially in large sensor networks. Inspired by the idea of the expectation maximization algorithm, the second method is to introduce some 0–1 latent variables to deal with the Gaussian mixture model. Since the regular condition of the posterior Cramér–Rao bound is unsatisfied for the discrete uncertain system, we use some continuous variables to approximate the discrete latent variables. Then, a new Cramér–Rao bound can be achieved by a limiting process of the Cramér–Rao bound of the continuous system. It avoids the complex integral, which can reduce the computation burden. Based on the new posterior Cramér–Rao bound, the optimal solution of the sensor selection problem can be derived analytically. Thus, it can be used to deal with the sensor selection of a large-scale sensor networks. Two typical numerical examples verify the effectiveness of the proposed methods. |
| first_indexed | 2024-04-14T02:21:47Z |
| format | Article |
| id | doaj.art-37860e44aed1405cbf4e7a2fac11be6a |
| institution | Directory Open Access Journal |
| issn | 1424-8220 |
| language | English |
| last_indexed | 2024-04-14T02:21:47Z |
| publishDate | 2018-04-01 |
| publisher | MDPI AG |
| record_format | Article |
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| spelling | doaj.art-37860e44aed1405cbf4e7a2fac11be6a2022-12-22T02:18:01ZengMDPI AGSensors1424-82202018-04-01184110310.3390/s18041103s18041103The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain ObservationsZhiguo Wang0Xiaojing Shen1Ping Wang2Yunmin Zhu3School of Mathematics, Sichuan University, Chengdu 610064, ChinaSchool of Mathematics, Sichuan University, Chengdu 610064, ChinaSchool of Mathematics, Sichuan University, Chengdu 610064, ChinaSchool of Mathematics, Sichuan University, Chengdu 610064, ChinaThis paper considers the problems of the posterior Cramér–Rao bound and sensor selection for multi-sensor nonlinear systems with uncertain observations. In order to effectively overcome the difficulties caused by uncertainty, we investigate two methods to derive the posterior Cramér–Rao bound. The first method is based on the recursive formula of the Cramér–Rao bound and the Gaussian mixture model. Nevertheless, it needs to compute a complex integral based on the joint probability density function of the sensor measurements and the target state. The computation burden of this method is relatively high, especially in large sensor networks. Inspired by the idea of the expectation maximization algorithm, the second method is to introduce some 0–1 latent variables to deal with the Gaussian mixture model. Since the regular condition of the posterior Cramér–Rao bound is unsatisfied for the discrete uncertain system, we use some continuous variables to approximate the discrete latent variables. Then, a new Cramér–Rao bound can be achieved by a limiting process of the Cramér–Rao bound of the continuous system. It avoids the complex integral, which can reduce the computation burden. Based on the new posterior Cramér–Rao bound, the optimal solution of the sensor selection problem can be derived analytically. Thus, it can be used to deal with the sensor selection of a large-scale sensor networks. Two typical numerical examples verify the effectiveness of the proposed methods.http://www.mdpi.com/1424-8220/18/4/1103Cramér–Rao boundsensor selectionuncertain measurementtarget tracking |
| spellingShingle | Zhiguo Wang Xiaojing Shen Ping Wang Yunmin Zhu The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain Observations Sensors Cramér–Rao bound sensor selection uncertain measurement target tracking |
| title | The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain Observations |
| title_full | The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain Observations |
| title_fullStr | The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain Observations |
| title_full_unstemmed | The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain Observations |
| title_short | The Cramér–Rao Bounds and Sensor Selection for Nonlinear Systems with Uncertain Observations |
| title_sort | cramer rao bounds and sensor selection for nonlinear systems with uncertain observations |
| topic | Cramér–Rao bound sensor selection uncertain measurement target tracking |
| url | http://www.mdpi.com/1424-8220/18/4/1103 |
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