第 55 回自動制御連合講演会 2012 年 11 月 17 日,18 日京都大学 1K403 ( ) Interpolation for the Gas Source Detection using the Parameter Estimation in a Sensor Network S. T

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第 55 回自動制御連合講演会 212 年 11 月日, 日京都大学 1K43 () Interpolation for the Gas Source Detection using the Parameter Estimation in a Sensor Network S. Tokumoto, T. Namerikawa (Keio Univ.

1 第 55 回自動制御連合講演会 212 年 11 月日, 日京都大学 1K43 () Interpolation for the Gas Source Detection using the Parameter Estimation in a Sensor Network S. Tokumoto, T. Namerikawa (Keio Univ. ) Abstract The purpose of this paper is to detect gas source which exists in the place where sensor is not arranged in the sensor network. For that purpose, interpolation of gas concentration of a strange domain is required. Therefore, we propose a novel algorithm of parameter estimation required for interpolation of gas concentration in a sensor network. Finally the proposed interpolation algorithm for gas concentration is verified via some numerical simulation results. Key Words: Sensor Network, Parameter Estimation, Interpolation 1, (WSN) 1,2).,, WSN, 3,4),,,,,., 5).,,,,, CO2. 6,7),, 6), CO2.,, CO2,, 2 Fig. 1 CO2,, CO2, Fig. 1: Purpose of Research (1) 8)., Fig. 2 c(x, t) t c(x, t) c(x, t) u(x, t) α(x, t) x x 2 s(x, t) Fig. 2: Advection-Diffusion (1) c(x, t) x, t CO2, s(x, t)/ x, t CO2, u(x, t), α(x, t) u, α 755

,.. (2)(4) c(x, t) t c(x, t ) c(x, t) (2) 1.

[ ] W k V (1) h k H E{w k x T }, E{v k x T }, E{h k x T }

c(x,t) x (u ) x (u < ) (3) c(x, t) 1 x 2 {c(x x, t) x2

2 c(x, t) { } ( u u) α 2 x x 2 c(x x, t) { } ( u u) α 2 x

c(x x, t) Z t c(x x, t) s(x, t) (6) (6), x k1 A k x k s k

k),..., c(n, k)] T R N, y k [y 1 (k), y 2 (k),.

(9) s k [s(1, k), s(2, k),..., s(n, k)] T R N.

3: Example, 2 S, (7),(8) (12), (13) c(1, k 1) c(2, k 1)

(k) y 5 (k) X k Y k Z k X k Y k Z k X k Y k Z k X k Y k Z

2 (k) v 3 (k) v 4 (k) v 5 (k) c(1, k) c(2, k) c(3, k) c(4,

2 ,.. (2)(4) c(x, t) t c(x, t ) c(x, t) (2) 1. w k, v k, h k (1), (11) {[ ] wk [ E v k w T k vk T h T ] } [ ] W k V (1) h k H E{w k x T }, E{v k x T }, E{h k x T } (11), x { c(x,t) x c(x,t) x c(x,t) c(x x,t) c(x x,t) c(x,t) x (u ) x (u < ) (3) c(x, t) 1 x 2 {c(x x, t) x2 2c(x, t) c(x x, t)} (4) (2)(4) (1) ( c(x, t ) 1 u ) 2α x x 2 c(x, t) { } ( u u) α 2 x x 2 c(x x, t) { } ( u u) α 2 x x 2 c(x x, t) s(x, t) (5), (5) c(x, t ) X t c(x, t) Y t c(x x, t) Z t c(x x, t) s(x, t) (6) (6), x k1 A k x k s k w k (7), (6) t, (7) k y k x k v k (8), x k [c(1, k), c(2, k),..., c(n, k)] T R N, y k [y 1 (k), y 2 (k),... y N (k)] T R N, y i (k) k i A k R N N, N s k1 s k h k (9) s k [s(1, k), s(2, k),..., s(n, k)] T R N. w k R N, v k R n, h k R N Fig. 3: Example, 2 S, (7),(8) (12), (13) c(1, k 1) c(2, k 1) c(3, k 1) c(4, k 1) c(5, k 1) y 1 (k) y 2 (k) y 3 (k) y 4 (k) y 5 (k) X k Y k Z k X k Y k Z k X k Y k Z k X k Y k Z k X k c(1, k) c(2, k) c(3, k) c(4, k) c(5, k) S v 1 (k) v 2 (k) v 3 (k) v 4 (k) v 5 (k) c(1, k) c(2, k) c(3, k) c(4, k) c(5, k) w 1 (k) w 2 (k) w 3 (k) w 4 (k) w 5 (k) (12) (12), c(2, k), (13) c(2, k 1) Zc(1, k) Xc(2, k) Y c(3, k) S w 2 (k) (14),, 1, 756

3 3 3.1, X k, Y k, Z k X k, Y k, Z k , (8) y k x k., (7) y k Q k p k s k w k (15) p k [X k Y k Z k ] T R 3, p k1 p k g k (16) g k, G R 3 3 Q k R N 3 y 1 (k 1) y 2 (k 1) Q k y 2 (k 1). y 1 (k 1) (). y N (k 1). y N (k 1) y N 1 (k 1) (9), (15), (16) [ ] [ ] [ ] [ ] pk1 pk I gk () s k1 s k I h k y k [ Q k I ] [ ] p k w s k (19) k (), (19) Q k [Q k I],, (2)(24) 9). [ ] [ ] ˆpk1 k ˆpk k ŝ k1 k ŝ (2) k k [ ] [ ] ˆpk k ˆpk k 1 ŝ k k ŝ k k 1 [ ]) K k (y k Q ˆpk k 1 k ŝ (21) k k 1 [ ] G P k1 k P k k H (22) P k k P k k 1 K k Q kp k k 1 (23) [ ] K k P k k 1 Q k Q T k P k k 1 Q T k W (24), ˆp k1 k p k, ˆp k k ŝ k1 k, ŝ k k (2)(24) (), (19), p k, s k Fig. 4: Problem Formulation Fig. 4 28, x 2[s] Table 1: Simulation Parameters Parameter Symbol Value Number of Sensor N 28 Dispersion Coefficients α.1[m 2 /s] Sensor Interval x 1[m] Sampling Time 1[s] System Noise w k N (,1e-3*I 28 ) Observation Noise v k N (,1e-3*I 28 ) Parameter Noise g k N (,5e-4*I 3 ) Emergence Noise h k N (,1e-3*I 28 ) Amount of Emergence s(, k) 5, I m m m Wind Speed [m/s] Fig. 5: True Value(Wind Speed),, Time[s] Fig. 6: Gas Concentration 757

X k, Y k, Z k s k ŝ k.8.8 6 6 Value of X.78.76.

Emergence (ppm) 5 4 3 2 1 Amount of Emergence

7 5 1 15 2 Fig. 7: True Value(X k ).2..16.14.

7 5 1 15 2 Fig. 8: Estimation(X k ).2..16.14.

(26), u u x (Z Y ) (27) u û Wind Speed [m/s].6.4.

13: Estimation(Wind Speed) 1 5 1 15 2 Fig.

15: Estimation(s k ) Fig. 7Fig. 12, Fig. 5 Fig.

Xc(j, t) Y c(j 1, t) Zc(j 1, t) (28) c(j 1, t

4 X k, Y k, Z k s k ŝ k Value of X Value of X Amount of Emergence (ppm) Amount of Emergence (ppm) Value of Y Value of Z Fig. 7: True Value(X k ) Fig. 9: True Value(Y k ) Fig. 11: True Value(Z k ) Value of Y Value of Z Fig. 8: Estimation(X k ) Fig. 1: Estimation(Y k ) Fig. 12: Estimation(Z k ),,. Y 2 x Z ( u u) α x 2 (25) ( u u) α 2 x x 2 (26), u u x (Z Y ) (27) u û Wind Speed [m/s] Fig. 13: Estimation(Wind Speed) Fig. 14: True Value(s k ) Fig. 15: Estimation(s k ) Fig. 7Fig. 12, Fig. 5 Fig. 13 s k Fig. 14 Fig ,, 3. 3,,, Fig. 16: Problem Formulation Fig. 16 j 2, j 1, CO2 j, j 1 (6), s(x, t), c(j, t ) Xc(j, t) Y c(j 1, t) Zc(j 1, t) (28) c(j 1, t ) Xc(j 1, t) Y c(j, t) Zc(j 2, t) (29) CO2 c(x, t ) c(x, t). c(x, t) c(x), c(j) Xc(j) Y c(j 1) Zc(j 1) (3) c(j 1) Xc(j 1) Y c(j) Zc(j 2) (31) 758

: Problem Formulation, 19 Table 2: Simulation Parameters Parameter Symbol Value Number of Sensor N 1 Number of State n 28 Dispersion Coefficient α.1[m 2 /s] Wind Speed u.

5 c(j), (j 1), (1 X)c(j) Zc(j 1) Y c(j 1) (32) Y c(j) (1 X)c(j 1) Zc(j 2) (33) X Y Z 1, c(j), (j 1) c(j) c(j 1) Y 2 Y Z Y 2 c(j 1) Y Z Z2 Z 2 Y 2 c(j 2) (34) Y Z Z2 Y 2 Y 2 c(j 1) Y Z Z2 Y Z Z 2 Y 2 c(j 2) (35) Y Z Z2. (CO2 ) c(j), c(j 1) I(j), I(j 1) 4.2. Fig. : Problem Formulation, 19 Table 2: Simulation Parameters Parameter Symbol Value Number of Sensor N 1 Number of State n 28 Dispersion Coefficient α.1[m 2 /s] Wind Speed u.5[m/s] Sensor Interval 3 x 3[m] Sampling Time 1[s] System Noise w k N (,1*I 28 ) Observation Noise v k N (,.1*I 1 ) Emergence Noise h k N (,1*I 28 ) 19, 5 19 Fig. Fig. 21 (a) Fig. : True Value (b) Fig. 2: True Value Fig. 19: Interpolation Fig. 21: Interpolation 19 Fig. Fig. 19,,, 4.3,,, Step 1. I(j), I(j 2) Xc(j 1) Y I(j 2) ZI(j) c (j 1) c (j 1), c(j 1) c (j 1) ŝ(j 1), c(j 1) j 1, ŝ(j 1). ŝ(j 1), j 1 s min ŝ(j 1) s max (36) s min, s max ŝ(j 1), j 1, 759

6 4.4 Table 2 s min, s max s min 45, s max 55. Fig. 22: Interpolation Algorithm(Step 1) Step 2. j c(j 1) Xc(j 1) Y c(j 2) Zc(j), c(j) I (j), I (j) (1 X)c(j 1) Y I(j 2) Z (37) ŝ(j) (34) I (j), Y 2 Y Z Y 2 c(j 1) (38) Y Z Z2 Z 2 Y 2 c(j 2) Y Z Z2 Y Z Y 2 ŝ(j) (39) Y Z Z2 Y 2 Y Z Y 2 Y Z Z 2 c(j 1) Z 2 Y 2 c(j 2) Y Z Z2 I(j) (4), I (j) I(j) Y Z Y 2 ŝ(j) (41) Y Z Z2. ŝ(j) s min, s max, j. 3. Fig. 23: Interpolation Algorithm(Step 2) Step 3. 2 j 2,, j 3, j Fig. 24: True Value Fig. 25: Interpolation, 5, CO2, 6),,,,, 1),,, :, (27) 2),, :,, 47-8, 649/656 (28) 3) K. Kosugi, S. Tokumoto and T. Namerikawa,Faulttolerant Sensor Network based on Fault Evaluation Matrix and Conpensation for Intermittent Observation., Proc. of 51st IEEE Conf. on Decision and Control, (212) (to be published) 4), :,, 47-8, 329/336 (211) 5),,, 28-2, 134/137, (21). 6) J. E. Weimer, B. Sinopoli, B. H. Krogh, Multiple source detection and localization in advectiondiffusion processes using wireless sensor networks., 3th, IEEE Real-Time Systems Symposium, 333/342, (29) 7) J. E. Weimer, B. Sinopoli, B. H. Krogh, A relaxation approach to dynamic sensor selection in largescale wireless sensor networks. Proceeding of IEEE International Conference on Distributed Computing Systems, 51/56, (28) 8), CFD --, (1996) 9),,, (2) 76

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