Network Topology Reconfiguration for State Estimation Over Sensor Networks With Correlated Packet Drops

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1 Preprints of the 9th World Congress The International Federation of Automatic Control Networ Topology Reconfiguration for State Estimation Over Sensor Networs With Correlated Pacet Drops Alex S. Leong Daniel E. Quevedo Anders Ahlén Karl H. Johansson Department of Electrical and Electronic Engineering, University of Melbourne, Australia ( School of Electrical Engineering and Computer Science, University of Newcastle, Australia ( Signals and Systems, Uppsala University, Uppsala, Sweden ( ACCESS Linnaeus Centre, School of Electrial Engineering, Royal Institute of Technology, Stocholm, Sweden ( Abstract: This paper studies the problem of optimal networ topology reconfiguration in sensor networs for state estimation. Multiple sensors mae observations of a process, which are then transmitted, possibly via intermediate sensors, to a central gateway. Transmission over each lin can experience pacet drops. The time-varying wireless networ environment is modelled by the notion of a networ state as in Quevedo et al. (03a). For each networ state, different networ configurations can be used, which govern the networ topology and routing of pacets. Switching between different configurations incurs a cost, in that unwanted lins will need to be removed before the establishment of new lins, leading to a transient time in which some lins may not be available. The problem is to determine the optimal networ configurations to use in each networ state, in order to minimize an expected error covariance measure that taes into account the cost of reconfiguration. A simpler suboptimal method which minimizes the upper bound of the expected error is also proposed, which in numerical simulations gives essentially identical results to the optimal method.. INTRODUCTION Wireless sensor networs, which consist of a number of small and inexpensive sensors connected together via wireless lins, have had many applications in recent years, for instance in environmental and industrial monitoring. The problem of estimation using wireless sensor networs has been an active research area, due to the unreliable nature of wireless lins and the associated stability and performance issues. The modelling of the wireless lin as a pacet dropping lin, in which transmission of pacets is assumed to be dropped or lost if the lin is of poor quality, is common in the control literature. Kalman filtering for a single sensor over a pacet dropping lin has been considered in e.g. Sinopoli et al. (004); Huang and Dey (007); Epstein et al. (008); Schenato (008), to name a few. Extensions to multiple sensors include Liu and Goldsmith (004); Gupta et al. (009b), and to sensor networs with various different architectures such as Gupta et al. (009a); Chiuso and Schenato (0); Quevedo et al. (0, 03b). Kalman filtering over networs with tree structures include Shi (009); Mo et al. (0), which don t consider pacet drops, and Quevedo et al. (03a) which does. The wor of Quevedo et al. (03a) introduces the notion of a networ state, which models time variations in the wireless environment, for example due to moving machines and robots in a factory. This networ state process can be either a Marov chain or a semi- Marov process, which was also considered for a single sensor case in Censi (0). In Quevedo et al. (03a) the networ topology, i.e. which sensors communicate to each other and how pacets are routed through the networ, is assumed to be fixed even over different networ states. In practice, if in a certain networ state some lins are of poor quality, e.g. a robot in a factory is blocing the line of sight between two sensors, then the sensors can possibly bypass these lins by seeing different paths through the networ, as is often done in networing by rerouting, see Bertseas and Gallager (99); Kurose and Ross (0). In this paper, we consider the problem of determining the optimal networ topology configurations to use in each networ state. The problem is complicated by the fact that networ topology reconfigurations do not occur instantly, but may incur a cost, see Basaran et al. (007); Krasteva et al. (0); Ramarishnan et al. (0) for examples of different cost functions. The cost of reconfiguration we consider in this paper is that in changing from one configuration to another, unwanted lins will need to be removed before new lins can be established (Basaran et al. (007)). This leads to a transient time where some lins may not be available, leading to poor performance. We optimize an expected error covariance measure over the possible networ configurations, taing into account this transient state when switching between different configurations. Computation of expected error covariances can be computationally demanding, so we also consider optimizing an upper bound to the expected error covariance. The paper is organized as follows. The system model is described in Section. The optimal networ reconfiguration prob- Copyright 04 IFAC 553

2 lem is stated in Section 3, with computational issues discussed in Section 3.3 and a suboptimal method given in Section 3.4. A numerical example is studied in detail in Section 4.. SYSTEM MODEL The process is a discrete time linear system of the form x( + ) = Ax() + w() where x() R n and w() N(0, Q) is i.i.d. over time. The process is observed by M sensors, with measurements y m () = C m x() + v m (), m =,..., M where y m () R lm, and v m () N(0, R m ) are i.i.d. over time. We assume that {w} and {v m }, m =,..., M are mutually independent. We mae the standard assumption that (A, C) is detectable and (A, Q / ) is stabilizable, where C col(c,..., C M ). given tree, there is a unique path from each node S m to the gateway S 0, denoted by Path(S m ). We will call the set of edges and the set of nodes along this path by Edge(Path(S m )) and Node(Path(S m )) respectively. In this paper, due to changes in the wireless environment, the networ topology formed by the set of active lins can change over time because of reconfigurations of the networ. As in Quevedo et al. (03a), changes in the environment will be modelled by the notion of a networ state process Ξ() B {,,... B }, which is time-varying. As motivation for this idea, consider Fig., which plots some fading channel measurements taen at a rolling mill at Sandvi in Sweden. Mobile machines cause infrequent but substantial variations in the expected channel gains. Received Signal Strength Indicator. Sensor networ model We consider the case where the sensors are connected together to form a sensor networ with a gateway/fusion center. The sensor networ in general is assumed to have a mesh structure. Sensor measurements are to be transmitted to the gateway which runs a Kalman filter. The paths used in transmitting from the sensors to the gateway are usually computed using routing algorithms. We assume that the lins which are utilized in the set of routes from the sensors to the gateway, which we denote as the set of active lins, has a tree structure (i.e. has no cycles). In this model, the set of active lins can be described using a directed graph with nodes/vertices {S 0, S,..., S M }, where the root node S 0 denotes the gateway, and S m, m =,..., M denote the sensors. See Fig. for an example with nine nodes (eight sensors and a gateway). Each sensor aggregates its own measurement to the received Fig.. Sensor networ with nine nodes. The set of active lins represented by arrows forms a tree, while the dotted lines represent inactive lins. pacets from incoming nodes and transmits the resulting pacet to a single destination node. We assume that transmissions can occur over a faster time scale than the process, as is typical in the industrial wireless sensor networs standard WirelessHART (HART Communication Foundation (009a)), thus delays experienced in travelling through the networ will be ignored. Following the notation of Quevedo et al. (03a), we call the node that sensor S m transmits to the parent of S m, denoted by Par(S m ). The outgoing lin/edge from each of the nodes will be denoted as E m = (S m, Par(S m )), m =,..., M. For a For instance, this will be the case when using shortest path type routing algorithms (Bertseas and Gallager (99)). Fig.. Channel measurements taen at a rolling mill We will assume in this paper that {Ξ} is a discrete-time semi- Marov process (Ross (996)), to model situations where networ state transitions do not necessarily have to occur at every time instant. The times of transitions between different networ states is denoted by K { l }, with 0 = 0, and 0 < <... all integers. The holding times (the amount of time spent in a particular state between transitions) are defined as l l+ l. We assume that the holding times have finite support, thus l max, l By the semi-marov property, we have P(Ξ( l+ ) = j, l = δ Ξ( 0 ),..., Ξ( l ), Ξ( l ) = i, 0,..., l ) = P(Ξ( l+ ) = j Ξ( l ) = i)p( l = δ Ξ( l ) = i) = q ij ψ i (δ), ( l, δ, i, j) K N B B where q ij P(Ξ( l+ ) = j Ξ( l ) = i) are the transition probabilities of the embedded Marov chain, and ψ i (δ) P( l = δ Ξ( l ) = i) () are the conditional probabilities of the holding times. The networ configuration π() at time determines which nodes each sensor will receive from and forward to, i.e. the set of routes from the sensors to the gateway. The set of all possible configurations is denoted by Π = {,,..., Π }. Depending on channel lin conditions, some configurations may not be feasible in certain networ states. We call Π j Π the set of possible configurations when the networ state is equal to j. Thus (Ξ(), π()) B Π j for Ξ() = j. We assume that 5533

3 the set of all possible configurations have been precomputed and are nown at the gateway. This could be the case in some industrial applications where the number of sensors is relatively small. For instance, WirelessHART provides redundancy by maintaining multiple sets of routes which the networ can switch to at different time instances (HART Communication Foundation (009b)). Define the random variables γ m (), m =,..., M by {, transmission via lin Em at time is γ m () = successful 0, otherwise and the corresponding lin success probabilities by φ m (j,p) P(γ m () = Ξ() = j, π() = p) We will assume in this paper that, conditioned on a networ state, {γ m } are i.i.d. Bernoulli processes, with {γ m } independent of {γ n } for m n. Note that in this model there is temporal correlation, in that the pacet reception probabilies can be different in different networ states. The special case of i.i.d. Bernoulli processes (no temporal correlation) is often considered in the literature, see e.g. Sinopoli et al. (004); Liu and Goldsmith (004); Epstein et al. (008); Schenato (008). Marovian pacet losses as studied in e.g. Huang and Dey (007); You et al. (0) can also be regarded as a special case of this model, see Quevedo et al. (03a) for details.. Kalman filter at Gateway Define the random variables θ m (), m =,..., M by {, transmission via Path(Sm ) at time is θ m () = successful 0, otherwise Due to the fact that the set of active lins forms a tree, we have θ m () = γ i () E i Edge(Path(S m)) and P(θ m () = Ξ() = j, π() = p) = E i Edge(Path(S m)) φ i (j,p). Let θ() [ θ ()... θ M () ] T, θ ()y () θ ()C y()., C(). θ M ()y M () θ M ()C M The Kalman filtering equations can then be written as: ˆx( + ) = Aˆx( ) + K()(y() C()ˆx( )) P ( + ) = AP ( )A T +Q K()C()P ( )A T where R diag(r,..., R M ) and K() AP ( )C() T( C()P ( )C() T + R ). In the sequel, we will use the shorthand P () P ( ). 3. NETWORK RECONFIGURATION As stated in the previous section, the networ states model changes in the wireless environment, with changes in networ Another possible form of redundancy is by transmitting the same information along multiple paths. However, due to large overheads this is usually not implemented in practice. () states occurring at the random transition times 0,,,.... Due to changes in the wireless environment, the pacet reception probabilies of existing lins can change, and there could even be a complete loss of connectivity in some lins. The purpose of the present wor is to illustrate how to compensate for changes in the wireless environment through networ reconfiguration. At each new transition time instant l when the networ state changes, we wish to choose a new configuration π( l ) in order to minimize an expected error covariance performance measure. However, networ configuration changes are not immediate and there is often a cost involved in reconfiguring the networ, as explained below. 3. Reconfiguration Issues In what follows, we will use a similar cost of reconfiguration as in Basaran et al. (007), where in changing from one configuration to another, unwanted lins will need to be removed before the establishment of new lins. We will refer to this as a transient state. Thus there is a transient time or reconfiguration time T 0 where some lins will not be available, resulting in poor transitory performance of the Kalman filter. Example. Consider for example the networ configuations shown in Fig. 3 (see Section 4 for the full set of networ configurations). In reconfiguring from networ configuration to networ configuration 3, the lins from sensor 3 to sensor, and from sensor 4 to sensor, will first need to be removed, leading to the transient state shown in Fig. 3 where sensors 3 and 4 do not have connectivity to the rest of the networ for some time T. Similarly, reconfiguring from networ configuration 3 to configuration will also lead to the same transient state. Fig. 3. Transient state when reconfiguring between two networ configurations, see Example. The value of T is dependent on the underlying communication technology. For instance, in IEEE 80. the time needed to reroute a wireless networ could be on the order of seconds (Pham et al. (007)). On the order hand, in WirelessHART which actively maintains multiple routes that can be switched at different time instances (HART Communication Foundation (009b)), it might be more appropriate to tae T = 0. See also Basaran et al. (007) and the references therein for the reconfiguration time of optical networs. Therefore, in choosing which new configuration to use, there is potentially a tradeoff between a configuration that gives good performance (after it is fully reconfigured) but requires many lin changes, versus a configuration that has fewer lin changes but slightly 5534

4 poorer performance. We will state the problem formally in the next subsection. 3. Optimal networ reconfiguration At each transition instant l K, we see to find a networ configuration π( l ) which is to be held until the next transition instant l+ K, and which minimizes the expected state estimation covariance over this holding period. Here we will assume that the gateway has nowledge of the current error covariance P ( l ), the old networ configuration π( l ), and the current networ state Ξ( l ). For ease of exposition, we introduce the aggregated process U( l ) ( P ( l ), Ξ( l ), π( l ) ), l K. (3) In terms of U( l ), the new configuration π( l ) Π j when Ξ( l ) = j is found via the following optimization: π( l ) = arg min V(U( l ), π), where π Π j { l } (4) V(U( l ), π) = E tr P ( l + i) U( l ). i= The quantity V(U( l ), π) amounts to the expected trace of the error covariance over the holding time l, when the configuration π is used. In (4), the expectation is taen over both the pacet loss processes (which affect the Kalman filter recursions ()) and the random holding times (using ()). Following the model in Section., the networ state Ξ( l ) determines the distribution of the holding time and thereby the upper limit of the sum in (4), see (); differences between the decision variable π and the previous configuration π( l ) determine which lins would be moved to a transient state. In particular, the terms E {P ( l + i) U( l )} (5) are computed based on whether the networ is still in the transient mode (if i T ) or has been fully reconfigured (if i > T ), with the expectation over the discrete variables {θ( l ),..., θ( l + i )}. Remar. Given the semi-marov networ model adopted, the reconfiguration strategy (4) yields that the process U( l ) at the transition instants l K is Marovian. This opens the possibility of analyzing estimator stability (see Sinopoli et al. (004)) by adapting the methods developed in Quevedo et al. (03a). These results will be detailed in future wor. 3.3 Computational Aspects In principle, minimization of (4) can be carried out by checing the values of V(U( l ), π) for each of the different configurations π Π j. However, computation of the expectations (5) involves considering the values of P ( l + i) for all possible combinations of {θ( l ),... θ( l + i )}, with the number of possibilities being O( Mi ) in general. In particular, computing E {P ( l + max ) U( l )} will have a complexity of O( M max ). Thus, for large holding times, minimization of (4) is computationally intensive. 3.4 Suboptimal networ reconfiguration To address the computational issues outlined above, we propose to adopt a suboptimal approach wherein, using U( l ) defined as in (3), the new configuration π( l ) Π j is obtained via π( l ) = arg min W(U( l ), π), where π Π j W(U( l ), π) = max δ= i= δ tr Y ( l + i)p{ l = δ Ξ( l ) = j}, (6) Here, the sequence {Y ( l + ), Y ( l + ),... } is given by: Y ( + ) = AY ()A T + Q E[AY ()C() T (C()Y ()C() T +R) C()Y ()A T ] (7) for l, with initial condition Y ( l ) = P ( l ), and where the expectation is with respect to the random matrix C(). Again, (7) is computed taing into account whether the networ is still in the transient mode or has been fully reconfigured. The following result, lining the optimal and the sub-optimal approaches is easy to show: Lemma. The sequence Y () is an upper bound to E{P () U( l )} for l. Proof Define g (X) = AXA T + Q E[AX C T ( C X C T + R) C XA T ] where C is a random matrix having the same distribution as C(). Lemma is proved by using the fact that g (.) is concave in X, and induction. The concavity of g (.) is shown by using similar techniques as in Sinopoli et al. (004); Dey et al. (009). The details are omitted for brevity. Upper bounding sequences of the form (7) are much easier to compute than the expected error covariance when the holding times are large. Furthermore, the bounds generally seem to be quite tight, see, e.g., Leong and Quevedo (03). In the next section we will loo at a numerical example, where we will see that the new configurations obtained using the suboptimal method are essentially identical to the configurations obtained using the optimal scheme. 4. SIMULATION STUDY We consider a simple example with four sensor nodes. The set of all networ configurations are shown in Fig. 4. There are two networ states, with networ configurations and possible in networ state (so that Π = {, }), and networ configurations and 3 possible when in networ state (so that Π = {, 3}). The reconfiguration time is taen to be fixed at T =. The pacet reception probabilities for the lins in each of the networ configurations are: φ (,) = 0.5, φ (,) = 0.5, φ 3 (,) = 0., φ 4 (,) = 0.5 φ (,) = 0.5, φ (,) = 0.5, φ 3 (,) = 0.8, φ 4 (,) = 0.5 φ (,) = 0.5, φ (,) = 0.5, φ 3 (,) = 0.5, φ 4 (,) = 0. φ (,3) = 0.5, φ (,3) = 0.5, φ 3 (,3) = 0.5, φ 4 (,3) = 0.8 (8) Networ state could correspond to the case where there is a robot between sensor nodes and 3, giving a low probability of pacet reception of 0. for the direct lin (from sensor 3 to sensor ) in networ configuration, while in networ configuration sensor 3 will instead transmit to sensor with a higher pacet reception probability of 0.8. Similarly networ state will correspond to the case where the robot is now situated between sensors and

5 0 with reconfiguration no reconfiguration 8 Tr(P(+)) Fig. 4. Networ configurations for example of Section 4 The holding times have the following distribution: P( l = Ξ( l ) = ) = P( l = Ξ( l ) = ) = 0. P( l = Ξ( l ) = ) = P( l = Ξ( l ) = ) = 0. P( l = 3 Ξ( l ) = ) = P( l = 3 Ξ( l ) = ) = 0. P( l = 4 Ξ( l ) = ) = P( l = 4 Ξ( l ) = ) = 0.7 The transition probabilities for the embedded Marov chain {Ξ( l )}, l K are P(Ξ( l+ ) = Ξ( l ) = ) = q = 0.5 P(Ξ( l+ ) = Ξ( l ) = ) = q = 0.5 P(Ξ( l+ ) = Ξ( l ) = ) = q = 0.5 P(Ξ( l+ ) = Ξ( l ) = ) = q = 0.5 We consider a system with parameters [ ]. 0. A =, Q = [ C = C = C 3 = C 4 = [ ], R = R = 0, R 3 = R 4 = 0.. The differences in the sensor measurement noise covariances correspond to the situation where the process to be observed is located much closer to sensors 3 and 4, than to sensors and. In Fig. 5 we plot the simulated values of tr P ( + ) obtained by solving the networ reconfiguration problem (4). We also include the case where only networ configuration is used when in both networ states and, which can be regarded as the case of no reconfiguration. We see that there are times where the optimal reconfiguration has error covariance that is either larger or smaller than the case of no reconfiguration, illustrating the tradeoff mentioned at the end of Section 3.. From Monte Carlo simulations, the trace of the average error covariance is around.56, whereas the case of no reconfiguration is around.87, which amounts to a performance gain of about 0%. Fig. 6 illustrates the corresponding networ states Ξ() and Fig. 7 the corresponding networ configurations π() used at each time instance. For this example, the networ configurations obtained using the suboptimal method of Section 3.4 by solving problem (6) is identical to Fig. 7. It seems that for the pacet reception probabilities given in (8), networ configuration will not be chosen. However, different behaviour can be observed by modifying these probabilites. In Fig. 8 we plot the simulation run of the networ configurations used at each time instance, with the same networ states as in Fig. 6, but now with φ 3 (,) = 0.45 and φ 4 (,) = 0.45, so that the probability of pacet reception in these two lins (for these networ states) is ], Fig. 5. Error covariances at different time instances Networ state Fig. 6. Networ states at different time instances Networ configuration Fig. 7. Networ configurations at different time instances using both proposed methods, given success probabilities as in (8). increased. Simlarly, in Fig. 9 we plot the networ configurations used when φ 3 (,) = 0.6 and φ 4 (,) = 0.6. Again, the networ configurations obtained using the suboptimal method of Section 3.4 are identical to Fig. 8 and Fig. 9. We see that as φ 3 (,) and φ 4 (,) are increased, the networ is less liely to reconfigure. The cost of reconfiguration which causes lins to be lost in the transient state leads to the networ not changing its topology. 5. CONCLUSION We have presented a networ topology reconfiguration method for state estimation in sensor networs. Networ reconfigura- 5536

6 Networ configuration Fig. 8. Networ configurations at different time instances: φ 3 (,) = 0.45 and φ 4 (,) = 0.45 Networ configuration Fig. 9. Networ configurations at different time instances: φ 3 (,) = 0.6 and φ 4 (,) = 0.6 tions are triggered when the wireless environment, modelled by the notion of a networ state, changes. The optimization of an expected error performance measure which taes into account the cost of reconfiguration has been studied, and a less computationally intensive suboptimal method proposed. Future wor will include the derivation of stability conditions for the estimation scheme with reconfigurations, and consideration of more general fading channel distributions. REFERENCES Basaran, E., Llorca, J., Milner, S.D., and Davis, C.C. (007). Topology reconfiguration with successive approximations. In Proc. MILCOM. Orlando, FL. Bertseas, D. and Gallager, R. (99). Data Networs. Prentice-Hall, New Jersey, nd edition. Censi, A. (0). Kalman filtering with intermittent observations: Convergence for semi-marov chains and an intrinsic performance measure. IEEE Trans. Autom. Control, 56(), Chiuso, A. and Schenato, L. (0). Information fusion strategies and performance bounds in pacet-drop networs. Automatica, 47, Dey, S., Leong, A.S., and Evans, J.S. (009). Kalman filtering with faded measurements. Automatica, 45(0), Epstein, M., Shi, L., Tiwari, A., and Murray, R.M. (008). Probabilistic performance of state estimation across a lossy networ. Automatica, 44, Gupta, V., Dana, A.F., Hespanha, J.P., Murray, R.M., and Hassibi, B. (009a). Data transmission over networs for estimation and control. IEEE Trans. Autom. Control, 54(8), Gupta, V., Martins, N.C., and Baras, J.S. (009b). Optimal output feedbac control using two remote sensors over erasure channels. IEEE Trans. Autom. Control, 54(7), HART Communication Foundation (009a). Control with WirelessHART. HART Communication Foundation (009b). System redundancy with WirelessHART. Huang, M. and Dey, S. (007). Stability of Kalman filtering with Marovian pacet losses. Automatica, 43, Krasteva, Y.E., Portilla, J., de la Torre, E., and Riesgo, T. (0). Embedded runtime reconfigurable nodes for wireless sensor networs applications. IEEE Sensors Journal, (9), Kurose, J.F. and Ross, K. (0). Computer Networing: A Top-Down Aproach. Pearson, 6th edition. Leong, A.S. and Quevedo, D.E. (03). On the use of a relay for Kalman filtering over pacet dropping lins. In Proc. ACC. Washington, DC. Liu, X. and Goldsmith, A.J. (004). Kalman filtering with partial observation losses. In Proc. IEEE Conf. Decision and Control, Bahamas. Mo, Y., Garone, E., Casavola, A., and Sinopoli, B. (0). Stochastic sensor scheduling for energy constrained estimation in multi-hop wireless sensor networs. IEEE Trans. Autom. Control, 56(0), Pham, V., Larsen, E., Øvsthus, K., Engelstad, P., and Kure, Ø. (007). Rerouting time and queueing in proactive ad hoc networs. In Proc. IPCCC, New Orleans, LA. Quevedo, D.E., Ahlén, A., and Johansson, K.H. (03a). State estimation over sensor networs with correlated wireless fading channels. IEEE Trans. Autom. Control, 58(3), Quevedo, D.E., Ahlén, A., Leong, A.S., and Dey, S. (0). On Kalman filtering over fading wireless channels with controlled transmission powers. Automatica, 48(7), Quevedo, D.E., Østergaard, J., and Ahlén, A. (03b). Power control and coding formulation for state estimation with wireless sensors. IEEE Trans. Control Syst. Technol. To be published. Available at Ramarishnan, R., Ram, N.S., and Alheyasat, O.A. (0). A cost aware reconfiguration technique for recovery in wireless mesh networs. In Proc. ICRTIT, Chennai, India. Ross, S.M. (996). Stochastic Processes. John Wiley & Sons, New Yor, nd edition. Schenato, L. (008). Optimal estimation in networed control systems subject to random delay and pacet drop. IEEE Trans. Autom. Control, 53(5), Shi, L. (009). Kalman filtering over graphs: Theory and applications. IEEE Trans. Autom. Control, 54(9), Sinopoli, B., Schenato, L., Franceschetti, M., Poolla, K., Jordan, M.I., and Sastry, S.S. (004). Kalman filtering with intermittent observations. IEEE Trans. Autom. Control, 49(9), You, K., Fu, M., and Xie, L. (0). Mean square stability for Kalman filtering with Marovian pacet losses. Automatica, 47(),