Advanced Signal Processing on Temperature Sensor Arrays for Fire Location Estimation

Transcription

1 Martin Berentsen 1, Thomas Kaiser 1, Shu Wang 2 1: Fachgebiet Nachrichtentechnische Systeme, Institut für Nachrichten- u. Kommunikationstechnik, Fakultät für Ingenieurwissenschaften, Universität Duisburg-Essen, Bismarckstrasse 81, Duisburg, berentsen@sent5.uni-duisburg.de, thomas.kaiser@uni-duisburg.de 2: Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan , P.R. China, shuwang@mail.hust.edu.cn Advanced Signal Processing on Temperature Sensor Arrays for Fire Location Estimation ABSTRACT This paper first reconsiders briefly the basics of fire source location estimation in a closed room. In this approach, temperature sensors arrays are used to extract relevant parameters out of the propagation of hot gases, caused by a fire, in order to perform location estimation. In addition to earlier publications [7],[8],[9] two ideas to increase robustness will be presented, and a faster calculation method for the required time delay estimation will be proposed. The algorithms satisfying the real-time requirements for prototype implementation. 1. Introduction To give an answer to the question: Where is a fire located?, this paper is structured into three parts. The first will give an overview about the location principle with temperature sensor arrays. These algorithms are verified by numerous fire experiments. A prototype system was also presented [9]. The second part discusses an optimization method related to the robustness of the implementation. The third part presents an idea for a numerically efficient time fast delay estimation method. This estimation is required for the location estimation of the fire source, which seems to be useful for the proposed location method. When we are talking about fire location estimation, we have to take a look at a fire in an

2 early stage. The hot gases are rising up from the fire place near the floor to the ceiling. Then the hot gases will propagate along the ceiling to the walls. Such a behavior is shown in figure 1. Due to more or less strong turbulences the shape of the temperature wavefronts ceiling wall sensor array fire floor Figure 1: fire in a closed room shown with shematic flow directions will not be perfectly circular. However, if we average in time, an almost circular shape can be expected. This observation is fundamental for the so called far field algorithm, which will be explained in the following. However, this circular behavior seems to be of limited duration. When a fire grows up, the hot gases become more turbulent, and the propagation becomes more and more of non-circular shape. Some boundary conditions must be fulfilled for the following considerations: The ceiling should be flat with a low heat conductivity. The flow current velocity v of the hot gases should be nearly constant under the ceiling in the early state of fire. The fire is near the floor level. The walls are at the same temperature. The fire is not located directly next to a wall. Other air currents, e.g. caused by a heating system, should be neglected.

3 2. Far Field Algorithm In the so-called far field scenario the propagation of the hot gases under the ceiling can be modeled as a stochastic temperature T (x,y,t) wave front of circular shape. Suppose a flat ceiling, a radius r can be introduced as r = (x x 0 ) 2 + (y y 0 ) 2, where (x 0,y 0 ) represents the coordinates of the projection from the fire to the ceiling. With the given assumptions the waveform is of circular shape, its expectation E{X} becomes only a function of time on a fixed radius E{T (x,y,t)} (x x0 ) 2 + (y y 0 ) 2 = r = f (t), r R +. Note that T (x,y,t) = T (r,t) is only a function of radius and time. Hence, we use the notation T (r,t) instead of T (x,y,t) in the following. Note also that a fixed point (x n,y n ), e.g. the location of the n-th sensor, the temperature function T (t) rn is only a function of time. Since the meaningful frequencies in the case of fire are limited up to approx. 10Hz [1], a sampling frequency of f A 100Hz seems to be more than sufficient. Assuming a rather simple model, the temperature samples from N sensors at the location (x n,y n ) can be modeled as T n (k) = S n (k) + N n (k), n = 1(1)N, (1) where S n (k) might be interpreted as a deterministic signal caused by the fire, and the noise N n (k) represents thermal noise and potential turbulences. If the range r between the fire place projection under the ceiling (x 0,y 0 ) and the location of the sensor array (x n,y n )is quite large compared to the array dimensions (d d r), the temperature wave under the ceiling can be seen as a quasi planar wavefront. In figure 2 such an array of sensors S 1,S 2,S 3,S 4, is shown in the distance r from the fire place projection under the ceiling. The wavefront is shown as quasi planar with velocity v. With knowledge of the geometric order of the sensors inside the array and the assumption o quasi planar wavefront with the velocity v, this illustrated wave reaches sensor S1

4 quasi planar wavefront y r x fire(x 0,y 0 ) v S1(x 1,y 1 ) D 12 d α d D 13 S2 S3 S4 Figure 2: geometrical arrangement of the sensor array in the far field behaviour first, then S3, then S2 and finally S4. In the following the time delay between S n and S m is denoted as D nm. Based on this signal model the location estimation problem can be reduced to a time delay estimation problem T 1 (k) = S(k) + N 1 (k) T n (k) = α n S(k k 1n ) + N n (k), n 1 with unknown delays k 1n and scaling factor α n. Observe that the deterministic signal S(k) def = S 1 (k) arrives as a delayed replica S(k k 1n ) at sensor S n. Several signal processing algorithms for time delay estimation have been investigated [2],[6]. In order to understand the principles, we will focus on the so-called SCCalgorithm (Simple Cross Correlation) only. The cross-correlation between the first and the n-th sensor output is defined as Assuming uncorrelated noise N n (k) yields R 1n (κ) = E{T 1 (k)t n (k + κ)}, n = 2(1)4. R 1n (κ) = E{S(k)S(k k 1n + κ)}, n = 2(1)4. Obviously, the maximum of R 1n (κ) occurs for κ = k 1n, since the argument becomes for arbitrary S(k) always positive. Hence, in order to estimate k 1n, only maximum of the cross

5 correlation has to be searched. Usually R 1n (κ) is estimated by averaging the temperature sample vectors from T 1 (k) and T n (k). However, because of the non stationarity o fire, a moving average ˆR 1n (κ,k) = 1 L L+m a T 1 (l)tn(l + κ), n = 2(1)4, k = m l=m M, m = 0(M)K L, κ = κ max (1)κ max. is more meaningful. Such a time-variant correlation can either be exploited to track propagation of fires, or, if the fire is becoming more and more turbulent, to stop the location estimation in case of highly dynamical and therefore unreliable results. Note that the hat indicates an estimation. The non-stationarity of the signals is taken into account by the time dependence k = m M of ˆR 1n (κ,k). The measured signals are composed of nonoverlapping blocks with length L. In order to save processing power, the estimation is only calculated each M-th time instant, where M > 1 and integer. κ max should be chosen small in order to limit the required processing power. After estimating the time delays, the mathematical relations between these delays and the parameters r, α and v are necessary. For r d, i.e. quasi planar wavefronts, it follows So the parameters α and v can be written as v k 12 f A = dcosα, v k 13 f A = dsinα. α = arctan( k 13 k 12 ), v = f A dcosα k 12, where d and f A are known a-priori. Actually, the fourth sensor S 4 seems to be redundant, but it can be exploited to further improve the estimation by averaging. ( ) ˆα = 1 3 v ˆ = f A d median ( sin ˆα ˆk 13 ; arctan( ˆk 13 ˆk 12 ) + arctan( ˆk 23 ˆk 14 ) + arctan( ˆk 24 ˆk 34 ) + π 4 cos ˆα sin ˆα cos ˆα 2sin( ˆα π ; ; ; 4 ) ; ˆk 12 ˆk 24 ˆk 34 ˆk 23, (2) ) 2cos( ˆα π 4 ). (3) Note that the median operation increases robustness against outliers. Beside ˆα a second angle is required for triangulation the location. In our case we place another sensor array in the same observation room, so that two angles ˆα 1 and ˆα 2 can be determined to ˆk 14

6 perform location estimation. In order to evaluate the quality of the estimation an error measurement is required r(k) = (x 0 ˆx 0 (k)) 2 + (y 0 ŷ 0 (k)) 2. (4) The error distance r(k) allows to investigate the proposed algorithm by real experiments. 3. Near Field Algorithm The far field algorithm fails, if the temperature wavefront is not quasi planar as assumed. Exploiting the spatial signature as an additional information is the aim of the following approach. First, we take a closer look into the the array environment, illustrated in figure 3. Observe that three triangles can be defined between the single sensors S2, S3 and S4, the sensor S1 and the fire source coordinates (x 0,y 0 ). Writing down the relations for these (x 0,y 0 ) v r α near θ S1(x S2 1,y 1 ) d d S4 S3 Figure 3: Sensor array geometric inside the circular wavefront behaviour triangles, we obtain ( v k 12 + r) 2 = r 2 + d 2 2rd cos(θ) (5) ( v k 13 + r) 2 = r 2 + d 2 2rd cos( 3π 2 θ) = r2 + d 2 + 2rd sin(θ) (6) ( v k 14 + r) 2 = r 2 + 2d 2 2 2rd cos( π 4 + θ) = 2d2 + r 2 2rd cos(θ) + 2rd sin(θ). (7)

7 Rewriting these three equations leads to ( v k 12 ) 2 d 2 = r( 2( v k 12 ) 2d cos(θ)) (8) ( v k 13 ) 2 d 2 = r( 2( v k 13 ) 2d sin(θ)) (9) ( v k 14 ) 2 2d 2 = r(2d sin(θ) 2( v k 14 ) 2d cos(θ)). (10) Equation (8), (9) and (10) yield immediately ( v k 14 ) 2 ( v k 12 ) 2 ( v k 13 ) 2 f r = a f ( a 2 ( v k 12 ) 2 + ( v k 13 ) 2 ( v k ). (11) 14 ) For the angle α near it is possible to calculate three delays; for example α near = π θ, tanθ = ( v k 13 ) 2 + 2r( v k 12 ) d 2 d 2 ( v k 12 ) 2 2r( v k 12 ). (12) However, the velocity v of the hot gases inside the array are yet unknown. For reasons of simplifying and implementability v ˆ is taken from equation (3), meaning some mixture between near- and far field approach. We further define an angle δ, which represents the maximum difference between the single estimations inside equation (2). With the definition o threshold, for example δ 10, we have found a criteria for the algorithm choice. The estimator output value ˆα will be chosen over this criteria between equation (2) and (12). With this method we do not only detect bad values from the far algorithm. We also try to increase the accuracy of the complete estimator bit Time Delay Estimation Although complexity has been reduced, the proposed time delay estimation with correlation methods still requires significant processing power. In order to further reduce complexity, a so called 1-bit algorithm is applied. It depends on to the signal shape, the array geometry and size, and the velocity v inside the array. The basic principle is illustrated in the following four figures for signals S 1 (k) and S 2 (k) taken from two sensors. The low complexity algorithm works as follows.

8 1. Step: The two signals S 1 (k) and S 2 (k) for time delay estimation has to be offset free, this can be easily achieved by high pass filtering. A/D value S1(k) S2(k) A/D value Q1(k) Q2(k) t/s t/s Figure 4: a) S 1 (k) and S 2 (k) as output of step 1 b) Q 1 (k) and Q 2 (k) as output of step 2 2. Step: The signals are 1-bit quantizied. 1 : if S n (k) 0 (positive) Q n (k) = 1 : if S n (k) < 0 (negative) After this step the sign of the delay time can be extracted by memorizing the signal changing steps. 3. Step: Now these rectangular signals Q 1 (k) and Q 2 (k) are multiplied sample by sample by each other P 12 (k) = Q 1 (k) Q 2 (k). The output function P 12 (k) includes the delay times. They have to be counted based on this function and be averaged for a fixed time window. 4. Step: Now the discrete time steps between the signal steps from +1 to -1 and -1 to +1 has to be counted. The results are shown in figure 5b. Note that this approach only requires 1-bit processing leading to efficient implementations with low power requirements.

9 1 P12(k) A/D value t/s 4 Figure 5: a) P 12 (k) out of step 3 25 k12(k) delay20 steps t/s b) k 12 (k) as delay step value 5. Conclusion With respect to a practical implementation, a low calculation power method for optimization seems to be found. The idea of using the worse estimation values from the far algorithm and recalculate them with the near algorithm seems to be practicable. Also the 1-bit time delay estimation algorithm seems to be feasible enough for an implementation. So, first these algorithms have to be verified by practical experiments. Then, for a suceeding evaluation of the prototype system these two optimizations will be taken into account. References [1] G. Cox, R. Chitty, Some Stochastic Properties of fire Plumes, Fire and Materials, vol. 6, no. 3+4, 1982 [2] G. C. Carter, Coherence and Time delay Estimation, Proceedings of the IEEE, vol. 75, no. 2, pp , February 1987 [3] L. Eikermann, Ortsbestimmung von Bränden unter der Verwendung eines Temperatursensorfeldes, Studienarbeit,Gerhard-Mercator-Universität Duisburg, 1997 [4] L. Eikermann, Weitere Untersuchungen zur Ortsbestimmung von Bränden, Diplomarbeit, Gerhard-Mercator-Universität Duisburg, 1997

10 [5] A. Telljohann Vorarbeiten zum Aufbau eines Prototypen zur Ortsbestimmung von Bränden, Studienarbeit, Gerhard-Mercator-Universität Duisburg, 1999 [6] R. Sprenger Ein erster Ansatz zur signalprozessorbasierten Brandortbestimmung, Diplomarbeit, Gerhard-Mercator-Universität Duisburg, 1999 [7] T. Kaiser, Ortsbestimmung von Bränden mit Temperatursensorgruppen, AUBE 99 proceedings, pp.52-66, 1999 [8] T. Kaiser Fire Detection with Temperature Sensor arrays, IEEE Carnahan Conference proceedings, 2000 [9] M.Berentsen, T.Kaiser: Fire location estimation using temperature sensor arrays, AUBE 01 proceedings, pp , 2001