METHOD FOR DETERMINING A GAS CONCENTRATION FROM A GROUP OF SENSORS

Abstract

A method for determining a measure of a gas concentration from a group of at least two non- dispersive infrared, NDIR, gas sensors (S1-SN) is described. The method comprises the steps of obtaining, at a processing unit (1), from each NDIR gas sensor (S1-SN) a measure of a gas concentration as a belief function Pi(x), which provides a probability as a function of the sensed light intensity at a specific wavelength, merging, in the processing unit, the belief functions Pi(x) to a merged belief function P(x). A computer program performing the method is also described.

Claims

1. A method for determining a measure of a gas concentration from a group of at least two non-dispersive infrared, NDIR, gas sensors (S1-SN), wherein the method comprises the steps of obtaining, at a processing unit, from each NDIR gas sensor (S1-SN) a measure of a gas concentration as a belief function P.sub.i(x), which is a probability as a function of the measure of the gas concentration at a specific wavelength, and merging, in the processing unit, the belief functions P.sub.i(x) to a merged belief function P(x).

2. The method according to claim 1, also comprising the step of calibrating each NDIR gas sensor using the merged belief function P(x).

3. The method according to claim 1, wherein the processing unit is a central processing unit (1) which is in communication with each one of the NDIR gas sensors (S1-SN).

4. The method according to claim 1, wherein the merging of the belief functions Pi(x) is performed using the Dempster-Shafer rules for merging.

5. The method according to claim 1, wherein the merging of the belief functions Pi(x) comprises calculation of a weighted average P(x) of the belief functions P.sub.i(x), wherein the weight of each belief function is dependent on the distance W(P.sub.i(x), P.sub.j(x)) between each belief function P.sub.i(x) and the other belief functions P.sub.j,.sub.j≠i(x) such that an increased distance W.sub.2(P.sub.i(x), P.sub.j(x)) of a belief function P.sub.i(x) from the other belief functions P.sub.j,.sub.j≠i(x) results in a decreased weight of the belief function P.sub.i(x).

6. The method according to claim 5 wherein the distance between two belief functions P.sub.i(x), P.sub.j(x) used in the weighting of the belief functions P.sub.i(x) is determined as the Wasserstein distance W.sub.p, where W.sub.p, p≥1 for two probability measures P.sub.i and P.sub.j defined on the gas concentration range is given by $W_p\left(P_i,P_j\right)={\left(\underset{\left({\widehat{P}}_i,{\widehat{P}}_j\right)\in \text{Γ}\left(P_i,P_j\right)}{\inf }\mathbb{E}\left\{d{\left({\widehat{P}}_i,{\widehat{P}}_j\right)}^p\right\}\right)}^{1/p}$ wherein Γ(Pi, Pj) denotes the set of joint probability measures P.sub.ij defined on the gas concentration range, with marginals Pi and Pj and d denotes the distance of the gas concentration values from the corresponding random variables,.

7. The method according to claim 6, wherein the support degree Supp(P.sub.i(x)) of a given belief function P.sub.i(x) is calculated as $Supp\left(P_i\left(x\right)\right)={\displaystyle \underset{j=1,j\ne i}{\overset{N}{.Math.}}S\left(P_i\left(x\right),P_j\left(x\right)\right)}$ wherein $S\left(P_i\left(x\right),P_j\left(x\right)\right)=1-{\widehat{W}}_2\left(P_i\left(x\right),P_j\left(x\right)\right)$ wherein ${\widehat{W}}_2\left(P_i\left(x\right),P_j\left(x\right)\right)=\frac{2\times W_2\left(P_i\left(x\right),P_j\left(x\right)\right)}{{\displaystyle {.Math.}_i{\displaystyle {.Math.}_j{W}_2\left(P_i\left(x\right),P_j\left(x\right)\right)}}},$ wherein the weighting factor α.sub.i is expressed as $\frac{Supp\left(P_i\left(x\right)\right)}{{\displaystyle {.Math.}_{i=1}^{N}Supp\left(P_i\left(x\right)\right)}}$ and wherein the weighting factor α.sub.i is used in the merging of the belief functions Pi(x).

8. The method according to claim 1, wherein the method also comprises the step of selecting the NDIR gas sensors (S1-SN) in the group from a plurality of NDIR gas sensors, wherein each NDIR gas sensor (S1-SN) of the plurality of NDIR gas sensors is related to a present position, and wherein the NDIR gas sensors (S1-SN) are selected based on their present position.

9. The method according to claim 8, wherein the NDIR gas sensors (S1-SN) are also related to historic positions and wherein the NDIR gas sensors (S1-SN) are also selected based on their historic positions.

10. The method according to claim 8, wherein the NDIR gas sensors are also selected based on the gas concentration measurements of the NDIR gas sensors (S1-SN).

11. The method according to 8, wherein the selection of NDIR gas sensors (S1-SN) is performed repeatedly over time.

12. A computer program for determining a gas concentration from a group of at least two non-dispersive infrared, NDIR, gas sensors (S1-SN), comprising instructions which, when executed by a processor in a processing unit causes the processing unit to control the processing unit to carry out the method according to claim 1.

13. A processing unit configured for determining a gas concentration from a group of at least two non-dispersive infrared gas, NDIR, sensors (S1-SN), wherein the processing unit is configured to obtain from each NDIR gas sensor (S1-SN) a measure of gas concentration as a belief function, which is a probability as a function of the measure of the gas concentration at a specific wavelength, and merging, in the processing unit the belief functions P.sub.i(x) to a merged belief function P(x).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0034] FIG. 1 shows system with a group of gas sensors which are in communication with a processing unit which is configured to perform the method according to the present invention.

[0035] FIG. 2 shows different belief functions provided from the gas sensors in the system of FIG. 1.

[0036] FIGS. 3a and 3b shows the merged belief function resulting from merging the belief functions in FIG. 4 using Dempster-Shafer rules for merging and weighted averages, respectively.

[0037] FIG. 4 shows another example of different belief functions provided from the gas sensors in the system of FIG. 1.

[0038] FIGS. 5a and 5b shows the merged belief function resulting from merging the belief functions in FIG. 4 using Dempster-Shafer rules for merging and weighted averages, respectively.

[0039] FIG. 6 shows schematically a gas sensor according to an embodiment of the present invention.

[0040] FIG. 7 shows a plurality of cars which each comprises a gas sensor.

DETAILED DESCRIPTION

[0041] In the following detailed description of the invention similar features in the different figures will be denoted with the same reference numeral.

[0042] FIG. 1 shows a system with a group of N gas sensors S1-SN which are in communication with a processing unit which is configured to perform the method according to the present invention. All gas sensors preferably communicate wirelessly with the processing unit. According to an alternative embodiment the processing unit 1 may be integrated in at least one of the gas sensors. According to another alternative a processing unit 1 may be integrated in all of the gas sensors S1-SN. According to this alternative embodiment all gas sensors communicate with each other and each processing unit performs the method according to an embodiment of the invention.

[0043] According to the embodiment with a separate processing unit 1, the processing unit obtains from each NDIR gas sensor a measure of a gas concentration as a belief function, which provides a probability as a function of the sensed light intensity measured at a specific wavelength, providing the belief functions from each one of the NDIR gas sensors to a processing unit, and merging, in the processing unit 1, the belief functions P.sub.i(x) to a merged belief function P(x). The merged belief function P(x) provides a more reliable measure of the gas concentration. In order to further improve future measurements of the gas concentration the processing unit 1 may provide the merged belief function to each one of the gas sensors S1-SN for calibration of the gas sensors S1-SN.

[0044] FIG. 2 shows schematically a gas sensor S1-SN according to an embodiment of the present invention. The gas sensor S1-SN comprises an NDIR sensing unit 2, a processing unit 1 and a communication interface 3. The processing unit 1 obtains a measure of the gas concentration from the NDIR sensing unit 2 and controls the sending of the measure of the gas concentration from the gas sensor S1-SN to the other gas sensors S1-SN. The gas sensor S1-SN also comprises a positioning sensor 4 such as a, e.g., GPS sensor, which enables the gas sensor S1-SN to determine its position.

[0045] FIG. 3 shows five different belief functions P.sub.1(x)-P.sub.5(x) provided from the gas sensors S1-SN in the system of FIG. 1. Each belief function P.sub.1(x)-P.sub.5(x) shows the probability as a function of CO.sub.2 concentration. As can be seen in FIG. 3 the belief functions P.sub.1(x)-P.sub.5(x) extend over a large concentration span.

[0046] FIGS. 4a and 4b shows the merged belief function resulting from merging the belief functions in FIG. 3 using Dempster-Shafer rules for merging and weighted averages, respectively. Both different merging techniques result in similar resulting merged belief function P(x). Both merging techniques result in a probability peak at 482 ppm of CO.sub.2.

[0047] FIG. 5 shows another example of different belief functions P.sub.1(x)-P.sub.5(x) provided from the gas sensors in the system of FIG. 1. Each belief function P.sub.1(x)-P.sub.5(x) shows the probability as a function of CO.sub.2 concentration. As can be seen in FIG. 3 the belief functions P.sub.1(x)-P.sub.5(x) extend over a large concentration span. As can be seen in FIG. 5 the fifth belief function P.sub.5(x) is visibly shifted to lower CO.sub.2 concentrations than the other belief functions P.sub.1(x)-P.sub.4(x).

[0048] FIGS. 6a and 6b shows the merged belief function resulting from merging the belief functions in FIG. 5 using the Dempster-Shafer rules for merging and weighted averages, respectively. As can be seen in FIG. 6 the merging of the belief functions P.sub.1(x)-P.sub.5(x) using the Dempster-Shafer rules results in a merged belief function P(x) with a clear probability peak at 479 ppm in CO.sub.2 concentration. The merging of the of the belief functions P.sub.1(x)-P.sub.5(x) using weighted averages results in a merged belief function P(x) with a clear probability peak at 482 ppm in CO.sub.2 concentration. Thus, when merging using weighted averages the effect of the fifth belief function Ps(x) is diminished. It is probable that the fifth gas sensor has drifted and provides an erroneous result. Thus, it is favourable to diminish the effect of the fifth gas sensor Sensors.

[0049] In the following it will be described in more detail how a merged belief function P(x) may be achieved and calculated.

[0050] The distance between two belief functions P.sub.i(x), P.sub.j(x) used in the weighting of the belief functions P.sub.i(x) is determined as the Wasserstein distance W.sub.p, where W.sub.p, p≥1 for two probability measures P.sub.i and P.sub.j defined on the gas concentration range is given by

[00007]$W_p\left(P_i,P_j\right)={\left(\underset{\left({\widehat{P}}_i,{\widehat{P}}_j\right)\in \text{Γ}\left(P_i,P_j\right)}{\inf }\mathbb{E}\left\{d{\left({\widehat{P}}_i,{\widehat{P}}_j\right)}^p\right\}\right)}^{1/p}$

wherein Γ(Pi, Pj) denotes the set of joint probability measures P.sub.ij defined on the gas concentration range, with marginals Pi and Pj and d denotes the distance of the gas concentration values from the corresponding random variables. It is favourable to use the Wasserstein distance in the calculations as this discriminates belief functions which differ from the majority of belief functions which measures well the similarity.

[0051] The support degree of a given belief function may be calculated as

[00008]$Supp\left(P_i\left(x\right)\right)={\displaystyle \underset{j=1,j\ne i}{\overset{N}{.Math.}}S\left(P_i\left(x\right),P_j\left(x\right)\right)}$

wherein

[00009]$S\left(P_i\left(x\right),P_j\left(x\right)\right)=1-{\widehat{W}}_2\left(P_i\left(x\right),P_j\left(x\right)\right)$

and wherein

[00010]${\widehat{W}}_2\left(P_i\left(x\right),P_j\left(x\right)\right)=\frac{2\times W_2\left(P_i\left(x\right),P_j\left(x\right)\right)}{{\displaystyle {.Math.}_i{\displaystyle {.Math.}_j{W}_2\left(P_i\left(x\right),P_j\left(x\right)\right)}}}.$

[0052] The corresponding weighting factor of belief function Pi is then obtained after normalization, where the weighting factor α.sub.i is expressed as

[00011]$\frac{Supp\left(P_i\left(x\right)\right)}{{\displaystyle {.Math.}_{i=1}^{N}Supp\left(P_i\left(x\right)\right)}}.$

[0053] The weighted average of all the N belief functions can be expressed

[00012]$\widehat{P}\left(x\right)={\displaystyle {.Math.}_{i=1}^N{\alpha }_i{P}_i\left(x\right)}$

which is used to compute the final belief function for each sensor. There are alternative ways to use the weighted average to compute the merged belief, for instance it is possible to use the Dempster-Shafer rule applied to the weighted average belief function with itself N-1 times.

[0054] FIG. 7 shows a plurality of vehicles 5, which each comprises a gas sensor S1-SN (FIG. 2), at an intersection between two streets 6, 7. In the embodiment shown in FIG. 7 a processing unit 1, comprising a processor 10, is arranged at a fixed position which may be remote to the intersection. The processing unit 1 is connected to a communication interface 3 which communicate with the communication interfaces of the gas sensors S1-SN in the vehicles 5. The processing unit 1 selects the vehicles 5 which are positioned in the middle of the intersection within the dashed rectangle 8. The positions are determined using the positioning sensor 4 in each gas sensor S1-SN. After having received the belief functions P.sub.i(x) from each gas sensor S1-SN the processing unit 1 may calculate a merged belief function P(x) as has been described above.

[0055] The above described embodiments may be amended in many ways without departing from the scope of the invention which is limited only by the appended claims.