Co-design of jump estimators and transmission policies for wireless multi-hop networks with fading channels

Automatica 81 (2017) 68–74

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Co-design of jump estimators and transmission policies for wireless multi-hop networks with fading channels✩ Daniel Dolz a , Daniel Quevedo b , Ignacio Peñarrocha a , Alex Leong c , Roberto Sanchis a a

Department of Industrial System Engineering and Design, Universitat Jaume I, Castellón, Spain

b

Faculty of Electrical Engineering and Information Technology (EIM-E), Paderborn University, Paderborn, Germany

c

Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Vic. 3010, Australia

article

info

Article history: Received 30 April 2015 Received in revised form 23 January 2017 Accepted 28 February 2017

Keywords: State estimation Wireless networks Jump linear systems Power control

abstract We study transmission power budget minimization of battery-powered nodes in a remote state estimation problem over multi-hop wireless networks. Communication links between nodes are subject to fading, thereby generating random dropouts. Relay nodes help to transmit measurements from distributed sensors to an estimator node. Hopping through each relay node introduces a unit delay. Motivated by the need for estimators with low computational and implementation cost, we propose a jump estimator whose modes depend on a Markovian parameter that describes measurement transmission outcomes over a finite interval. It is well known that transmission power helps to increase the reliability of measurement transmissions, at the expense of reducing the life-time of the nodes’ battery. Motivated by this, we derive a tractable iterative procedure, based on semi-definite programming, to design a finite set of filter gains, and associated power control laws to minimize the energy budget while guaranteeing an estimation performance level. This procedure allows us to tradeoff the complexity of the filter implementation with performance and energy use. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Wireless communication technologies have considerably improved in recent years in terms of reliability and transmission rates, favoring their use for control and estimation purposes (Chen, Johansson, Olariu, Paschalidis, & Stojmenovic, 2011). However, wireless links are subject to channel fading that may lead to timevarying delays and packet dropouts (Hespanha, Naghshtabrizi, & Xu, 2007) that must be taken into account when designing networked control systems. Considering remote estimation over networks, Kalman filter (KF) approaches (with time-varying gains) may yield optimal performance (for linear dynamical systems), but possibly at the expense of notable implementation and computational complexity

✩ This work has been funded by projects TEC2015-69155-R from MICINN, PI15734, E-2015-15 and P1·1B2015-42 from Universitat Jaume I. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hideaki Ishii under the direction of Editor Christos G. Cassandras. E-mail addresses: [email protected] (D. Dolz), [email protected] (D. Quevedo), [email protected] (I. Peñarrocha), [email protected] (A. Leong), [email protected] (R. Sanchis).

http://dx.doi.org/10.1016/j.automatica.2017.03.006 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

(e.g. Schenato, 2008). Motivated by alleviating the computing requirements, we extend the proposal in Smith and Seiler (2003) that used a jump linear estimator whose gains depend on the history of measurement transmission outcomes for a system with one sensor and without delays. Our approach was initially presented in Dolz, Quevedo, Peñarrocha and Sanchis (2014). Here, we extend it for its use in multisensor schemes with delays, and show the feasibility conditions and a design procedure to reduce the complexity of the filter. A higher transmission power leads to lower dropout probabilities, which improves estimation performance but shorten battery life-time, what encourages to design both the estimator and the transmission policy (Nourian, Leong, & Dey, 2014; Quevedo, Ahlén, Leong, & Dey, 2012; Shi & Xie, 2012). These works present methodologies to minimize the estimation error using a KF while limiting the energy use considering only the sensor and estimator nodes. In this work, we focus on multi-hop wireless networks where some nodes (relays) consciously help to transmit the information from the source to the final destination, and where node data broadcasts are more likely to be acquired from nearby nodes, a more general topology than the two-hop network presented in Shi, Jia, Mo, and Sinopoli (2011). In our recent articles (Leong & Quevedo, 2013; Quevedo, Østergaard, & Ahlén, 2014), we studied

D. Dolz et al. / Automatica 81 (2017) 68–74

the KF estimation and power control problem through multi-hop fading networks. In the current work, we will assume that hopping through each relay introduces an additional unit delay on the data, an effect that was neglected in the previous works. In this paper, we study the transmission power budget minimization of wireless self-powered nodes in a remote state estimation problem for multi-sensor systems over multi-hop networks. Wireless links are subject to fading leading to random dropouts; hopping through each intermediate node introduces an additional unit delay. We describe this via a finite measurement outcome parameter taken as a finite Markov chain and, based on the network average behavior, we propose a jump linear filter structure. As a difference w.r.t. Smith and Seiler (2003), we use convex optimization in the design of the filter, which allows us to include constraints to fix the number of gains of the jump filter, leading to a trade-off between filter complexity and estimation performance. We characterize this compromise and give some insights on how to reduce the filter complexity via Lagrange multipliers. We study the co-design problem of minimizing the power budget while guaranteeing a prescribed estimation performance. Since this optimization is non-linear, we derive a greedy algorithm that solves iteratively semi-definite programming problems in order to obtain the set of filter gains and the power transmission laws.

We consider a LTI discrete-time system defined by ys [k] = cs x[k] + vs [k],

a single time-stamped packet and broadcast it once (without retransmissions) at the same time. Only nodes within a lower layer will accept the transmission (i.e., from d1 -layer to d2 -layer with d1 > d2 ), establishing wireless links. The rest of the nodes in the same or higher layers ignore the reception. Thus, a node may receive multiple measurement packets from higher layer nodes and may forward this information to various lower layer nodes. We denote the entire set of wireless links as I, and a single link as (Na , Nl ) ∈ I. When the dedicated transmission time slot is over, the following set of nodes starts to transmit. After all nodes have attempted to communicate (and before the sampling period has passed), the estimator uses all the received information at instant k to run the state estimation algorithm to be presented in Section 4. While each sensor transmits the current sampled output, each relay node transmits at instant k only the acquired data at k − 1. The transmission protocol implies that communicating through each relay layer introduces an additional unit delay. Direct transmissions to the estimator node do not introduce delays. Thus, a measurement being transmitted at time k by sensor node Ns ∈ Nd+1 may arrive at the estimator node with an end-to-end delay of up to d time steps, depending on the number of intermediate layers visited. The estimator node discards measurements already received. 3. Transmission outcome model

2. Remote estimation over a multi-hop network

x[k + 1] = A x[k] + B w[k],

69

(1)

n

where x[k] ∈ R is the system state, ys [k] ∈ R is the sth measured output (s = 1, . . . , ny ), w[k] ∈ Rnw is the state disturbance assumed to be a Gaussian signal of zero mean and (known) covariance E{w[k] w[k]T } = W , and vs [k] ∈ R is the sth sensor’s measurement noise considered as an independent zero mean Gaussian signal with (known) variance E{vs [k]2 } = σs2 . For further reference, we define y[k] , [y1 [k] . . . yny [k]]T . Also, we assume the pair

(A, C ) to be detectable, where C = [c1T . . . cnTy ]T .

In this work, we study the remote estimation of the system states (1) where the received measurements at the estimator node arrive through an unreliable multi-hop wireless network with fading channels and known topology. We assume that multiple sensors sample the system outputs synchronously and send them independently through the network to a centralized estimator. We assume that nodes cannot send and receive at the same time and there is no interference between them. Moreover, we assume that nodes are time-driven and synchronized. Here, we consider multi-hop wireless networks that can be described via an acyclic directed graph. We denote the set of network nodes by N = {N1 , . . . , NM , NM +1 } with M > ny being the number of transmitter nodes. N1 to Nny are the sensor nodes, Nny +1 to NM are the relay nodes and NM +1 refers to the estimator node. While relay nodes are used to retransmit data, sensors can only send their own samples. The network topology is classified and ordered by layers depending on the maximum number of hops (longest path) for a transmission to arrive at the estimator from each node. We assume that the number of different layers is bounded by d¯ + 2 and thus, the maximum number of hops is d¯ + 1. The set of nodes in the d-layer is denoted by Nd , {Na ∈ Nd : |(Na , NM +1 )| = d} ⊂ N where |(Na , NM +1 )| stands for the maximum number of hops from Na to the estimator node. The 0-layer contains only the estimator node, the (d¯ + 1)-layer includes only sensor nodes, and all other layers may comprise either relay nodes (intermediate nodes that help to retransmit the data) or sensors. At each instant k, a set of nodes (that transmit in different frequency bands) aggregate all their available measurements in

To model the unreliable transmission through the available wireless links (Na , Nl ) ∈ I, we introduce the binary variable γa,l [k] that takes value 1 if Nl receives a packet from Na at k and 0 otherwise. Throughout the first part of this work, we assume that each γa,l [k] is an i.i.d. stochastic process. The probability of successfully acquiring a transmitted packet is given by

βa,l , Pr{γa,l [k] = 1},

a, l ∈ {1, . . . , }.

(2)

We denote by τs [k] ∈ N the delay experienced by the kth measurement from sensor s when accepted at the estimator node. Thus, τs [k] = d means that ys [k] is accepted, i.e., for the first time received by the estimator at time k + d. The instance τs [k] > d states that the measurement may still be acquired with an induced delay greater than d. Since the number of hops is bounded by ¯ i.e., τs [k] ∈ d¯ + 1, the maximum possible end-to-end delay is d, {0, 1, . . . , d¯ }. Thus, τs [k] > d¯ means that ys [k] is lost. Let us use Γsk,d to enumerate the Boolean combinations (logical ‘‘and’’ and ‘‘or’’ operations) of variables γa,l that define the possible paths a measurement from sensor Ns sent at time k may take to reach and be accepted by the estimator node with a given delay d, i.e., with τs [k] = d. The possible node-to-node transmission outcomes leading to τs [k] > d are denoted by ks,d and can be obtained by the negation of the  disjunction of the corresponding 

Γsk,d , i.e.,

k s ,d

=¬

d

δ=0

Γsk,δ .

Considering the network model described above, the available information at the estimator node at time k are the pairs ¯ where (ms,d [k], αs,d [k]) for all s = 1, . . . , ny and d = 0, . . . , d, ms,d [k] = αs,d [k] ys [k − d] and αs,d [k] is a binary variable that takes value 1 if ys [k − d] is received at time k, and 0 otherwise. When αs,d [k] = 1, the measurement sent at time k − d from sensor Ns has experienced a delay of τs [k − d] = d. If ys [k − d] has not yet arrived at time k, then ms,d [k] = 0. Since delayed copies (already received measurements with a higher delay) are discarded, αs,d [k] is equal to zero if αs,d−δ [k −δ] = 1 for some integer δ ∈ {1, . . . , d}. Let us now introduce a vector θs,d [k] which models the successful reception of ys [k − d] during the interval {k − d, k − d + 1, . . . , k}:

 θs,d [k] = αs,0 [k − d]

αs,1 [k − d + 1]

···

 αs,d [k] .

(3)

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D. Dolz et al. / Automatica 81 (2017) 68–74

4. Markovian jump filter

¯ [k] = [m ¯ 1 [k]T · · · m ¯ ny [k]T ]T , v¯ [k] = [¯v1 [k] · · · v¯ n¯ y [k]]T where m

To take into account the reception of delayed measurements up ¯ we propose to use an aggregated model to d,

with v¯ s [k] = [vs [k] · · · vs [k − d¯ ]], and c¯s = [¯cs,0 . . . c¯s,d¯ ]T with c¯s,d = [01×n·d cs 01×n·(d¯ −d) ]T (and cs as defined in (1)) are the

x¯ [k + 1] = A¯ x¯ [k] + B¯ w[k],

covariance E{¯v [k]¯v [k]T } = V =

(4)

 T T

with x¯ [k] = x[k]T · · · x[k − d¯ ] and (A¯ , B¯ ) appropriate augmented matrices. We denote by τ¯ ∈ N a parameter that allows to adjust the length to look back in time. If τ¯ = d¯ we consider all the possible delayed measurements. If τ¯ < d¯ we narrow the historical interval to just take into account acquired measurements with a ¯ while if τ¯ > d¯ we allow to look further back in delay lower than d, time, even if no measurements with a higher delay than d¯ will be received. Then, concatenating vectors θs,d [k] in (3) for d = 0, . . . , τ¯ and for all sensors, we have



 θ [k] = θ1,0 [k]

···

θ1,d¯ [k]

θ2,0 [k]

···

θny ,τ¯ [k]



(5)

(τ¯ +1)(τ¯ +2)

where θ [k] is a binary column vector, of length nθ = ny , 2 representing the full set of measurements successfully received τ¯ from k − τ¯ to k and fulfilling the conditions δ=d¯ +1 αs,δ [t −δ+ d] =

τ¯

0 and ∥θs,d [k]∥1 ≤ 1, where ∥θs,d [k]∥1 = δ=0 αs,δ [t − δ + d]. These conditions describe the fact that measurements only arrive with a delay up to d¯ and that delayed copies are discarded. The possible occurrences of θ[k] lie within a finite set, i.e., θ[   k] ∈ ny

¯ Θ = {ϑ0 , ϑ1 , . . . , ϑr } with r = (d¯ + 2)!(d¯ + 2)⌈τ¯ −d⌉ − 1 and being ⌈·⌉ the operator that rounds its argument to the nearest positive integer (including zero) towards infinity. Each ϑi (for i = 0, . . . , r) represents one of the possible combinations

of the historical measurement transmission outcomes. From the definition of θ [k], and taking into account the fact that the occurrences of αs,d [k] depend on node-to-node transmission outcomes (defined by γa,l [k], and which are assumed i.i.d., see (2)), we can conclude that θ [k] is a homogeneous Markov process. We shall assume that it is ergodic. The following result shows how to obtain the transition probabilities of θ[k] (see Dolz, Quevedo, Peñarrocha & Sanchis, 2017 for the proof). Lemma 1. The elements of the transition probability matrix Λ = [pi,j ] of θ [k] with pi,j , Pr{θ [k] = ϑj θ [k − 1] = ϑi } can be computed as follows: pi,j = Pr {ϕ(k, j) ∧ ϕ(k − 1, i)} /Pr {ϕ(k − 1, i)}

(6)

with



 

ϕ(k, j) = 

{s,d,δ}∈D1 (ϑj )

Γsk,δ−d



 



  {s,d}∈D2 (ϑj )

k−d s,d

,

(7)

d¯

αs,d [k] . The possible values of α[k] belong to a set of the form α[k] ∈ Ξ = {η0 , η1 , . . . , ηq } with q = 2n¯ y − 1 and where ηi (for i = 1, . . . , q) d=0

denotes each possible combination. ¯ s [k], we express the availability of the Finally, using vector m measurements at time k from sensor s sent from k − d¯ to k as ¯ s [k] = [ms,0 [k] . . . ms,d¯ [k]]T . Using α[k], we rewrite the received m measurement information at time k as follows:

¯ [k] = α[k] C¯ x¯ [k] + v¯ [k] m 



d=0

σs2 .

(9)

x˜ [k] = (I − L[k]α[k]C¯ ) A¯ x˜ [k − 1] + B¯ w[k − 1] − L[k]α[k]¯v [k].





One of the aims of this work is to compute gain matrices L[k] to obtain acceptable estimation performance while requiring low computing and storage capabilities. Using predefined gain filters (Dolz, Peñarrocha & Sanchis, 2016; Smith & Seiler, 2003) instead of KF (Schenato, 2008) helps to alleviate the on-line computational burden, and also allows for dealing with e.g. uncertain systems. In the current work, we extend results in Smith and Seiler (2003) to include multisensor transmission and delayed measurements and we use a convex optimization procedure to obtain the filter gains. We propose the following adaptation law: L(θ[k]) =



if ψ(θ [k]) = 0, if θ [k] = ϑi , ψ(ϑi ) ̸= 0.

0 Li

(10)

We will compute the gain matrices off-line leading to the finite set L(θ[k]) ∈ L = {L0 , . . . , Lr }. 5. Filter design To design the filter, we first note that the Markov chain {θ [k]} has a stationary distribution (due to ergodicity) that satisfies π = π Λ, where π = [π1 , . . . , πr ] and πi = Pr{θ [k] = ϑi } are the probabilities of being at a given state. Based on this, the next result characterizes the evolution of the state estimation error covariance matrix. Theorem 2 (Dolz et al., 2014). Let Pj [k] = E{˜x[k]˜x[k]T |θ [k] = ϑj } (with j = 1, . . . , r) be the modal state estimation error covariance matrix updated at time k with information θ[k] = ϑj . We then have r 

pi,j

i=0 r 

 πi  ¯ ¯ B¯ T )FjT Fj (APi [k − 1]A¯ T + BW πj

pi,j

πi Xj VXjT πj

(11)

with Fj = I − Lj ψ(ϑj )C¯ and Xj = Lj ψ(ϑj ). Furthermore, the expected value of the estimation error covariance matrix is given state r by E{˜x[k]˜x[k]T } = j=0 Pj [k]πj .

Let us now define the measurement availability matrix at   time s=1

d¯

When no measurement is available, the estimator is run in open loop. Otherwise, the state estimation is updated with the updating gain matrix L[k]. Considering (4) and (8)–(9), the dynamics of the estimation error, defined as x˜ [k] = x¯ [k] − xˆ¯ [k], is

i=0

D2 (ϑj ) , {s, d : θ[k] = ϑj , ∥θs,d [k]∥1 = 0}.

ny

s=1

¯ [k] − α[k]C¯ A¯ xˆ¯ [k − 1]). xˆ¯ [k] = A¯ xˆ¯ [k − 1] + L[k](m

+

D1 (ϑj ) , {s, d, δ : θ[k] = ϑj , δ ≤ d¯ , αs,δ [k − d + δ] = 1},

ny

We propose the state estimation algorithm

Pj [k] =

s = 1, . . . , ny , d = 0, . . . , τ¯ , δ = 0, . . . , d and

k as follows: α[k] = ψ(θ [k]) ,

rows of matrix C¯ . In (8), v¯ [k] is the measurement noise vector with  

(8)

The above theorem defines a recursion on the modal covariance matrices in (11), that we write as Pj [k] = Ej {P [k − 1]} where P [k] , (P0 [k], . . . , Pr [k]) and Ej {·} is the linear operator over all the modal covariance matrices that gives (11). Thus, we write the full recursion (for j = 0, . . . , r) as P [k] = E{P [k − 1]} where E{·} , (E0 {·}, . . . , Er {·}). To compute a steady state solution, one must address the problem of finding a set of filter gains that satisfy the Riccati equation E{P [k]} = P [k]. Han, Zhang, and Fu (2013) and Smith and Seiler (2003) show how to obtain the explicit values of the gains when there is a different gain for each state of the Markov chain. However, the methods in Han et al. (2013) and

D. Dolz et al. / Automatica 81 (2017) 68–74

Smith and Seiler (2003) cannot be applied when the filter shares the same gain for different modes of the Markov chain (i.e., Li = Lj for some i ̸= j). Thus, Han et al. (2013) and Smith and Seiler (2003) do not allow us to explore the trade-offs between storage complexity and estimation performance. To overcome this issue, we adopt the following optimization problem: minimize L,P

T {P } s.t. E{P } − P ≼ 0

(12)

In this case, we obtain

λ=

r 

r      πq Zq C¯ T Mi − Mj + K πq Yi,q − Yj,q

q =0

r

with K = Mi =



(15)

q=0 q =0

Zq [k − 1]C¯ T Mi + Mj



pq,i ψ(ϑi ) ,

 r

q =0

Yj,q + Yi,q

−1

,

¯ B¯ , Zq = A¯ P¯ q A¯ + BW

T

T

T

i∈K

with P , (P0 , . . . , Pr ), T {P } = tr j=0 Pj πj and where E{P } − P ≼ 0 denotes that Ej {P } − Pj ≼ 0 for all j = 1, . . . , r. We showed in Dolz et al. (2014) that the feasibility of problem (12) is a sufficient condition to guarantee the boundedness of E{˜x[k]˜x[k]T }. The iteration P [k] = E{P [k − 1]} converges to the unique positive semi-definite solution P for the given L, obtained both from (12). Let us now state some necessary conditions which must be satisfied in order for (12) to have a solution (see Dolz et al., 2017 for the proof).

r



Theorem 3. For problem (12) to have a solution, the transition probabilities of θ[k] (and thus the node-to-node successful transmission probabilities in (2)) must fulfill the following constraints: pjj · ρ(A¯ )2 − 1 ≤ 0, pjj · ρ(A¯ i )2 − 1 < 0,

∀j : ψ(ϑj ) = 0,

(13a)

∀j : ψ(ϑj ) = ηi , i ∈ ND,

(13b)

where pij are the probabilities defined in (6), ρ(A¯ ) denotes the spectral ¯ ND is the set of reception scenarios ηi from which radius of matrix A, (A¯ , ηi C¯ ) is non-detectable, and ρ(A¯ i ) is the spectral radius of the unobservable subspace of A¯ from the reception scenario ηi .1 We can reduce the storage complexity of the filter (at the cost of a worse performance) by reducing the measurement outcome history τ¯ taken into account or by imposing the same filter gain for different states θk . The aim is to obtain the lowest number of different filter gains that lead to a given prescribed filter performance T {P }. In order to use one unique gain for a couple of states, one must set constraints like Li − Lj = 0 for some i ̸= j over L. Thanks to the convergence property P = E{P }, we can rewrite problem (12) with the corresponding equality constraints as minimize L ,P

71

T {E{P }}

s.t. Li = Lj , i ̸= j.

(14)

One can solve the previous optimization problem for all the possible pairings and then choose the sharings that guarantee the prescribed performance with the lower number of different gains. This procedure is highly consuming and other approaches must be explored. Thus, we firstly solve the unconstrained problem and then look for pairs of modes that can share the gain without affecting too much the achieved performance. Each possible equality constraint to be added to the problem will affect the achieved performance T {P } in a different amount, so only the pairings less harmful for the achieved performance will be kept for further recursions. In order to avoid solving the full optimization problem when searching for new pairings, we measure the sensitivity of each constraint to be added to take that decision. For a fixed τ¯ , let us assume that we have fixed Li − Lj = 0, and let us quantify the effect of perturbing this equality over T {P } by means of the Lagrange multipliers λ from Λ(L, P , λ) = T {E{P }} + vec(λ)T vec(Li − Lj ). Vector λ is obtained solving the homogeneous equations resulting from the partial derivatives of Λ(L, P , λ) w.r.t (L, P , λ), where vec(·) is the vectorization operator.

1 A¯ = O T A¯ O , where O is the unobservable subspace i  T  O = ker (ηi C¯ )T . (ηi C¯ A¯ )T · · · (ηi C¯ A¯ n−1 )T

Yi,q =



pq,i ψ(ϑi )Xq ψ(ϑi )T ,

Xq = C¯ Zq C¯ T + V

i∈K

where K is a set containing the index of the states ϑi that already share Li (similarly with Lj ) and P¯ q is the modal covariance of the state ϑq obtained with the additional constraint Li = Lj . As we want to avoid the complete computation of the optimization problem before choosing a pairing, P¯ q is not available. In order to make use of this information just for pairing purposes, we can approximately quantify the Lagrange multipliers using the modal covariance P¯ q obtained in a previous step, free of this constraint. Therefore, we incorporate new constraints in each iteration by evaluating sequentially ∥λ∥2 for each new possible pair. 6. Transmission over fading channels So far, we have assumed that the node-to-node successful transmission probabilities in (2), used to compute the predefined estimator gains, were known and time-invariant. However, in wireless networks with fading channels, the probability of successfully acquiring at Nl a transmitted packet from Na depends on the fading channel gain ha,l [k] ∈ Ωa,l ⊂ R≥0 and on the transmission power ua [k] ∈ [0, u¯ ] as per (cf. (2)) Pr{γa,l [k] = 1|ha,l [k] = h, ua [k] = u} = fa,l (h u)

(16)

where the function fa,l is monotonically increasing and differentiable, and depends on the modulation scheme employed. Let us aggregate in vector Ha [k] ∈ Ωa the outgoing fading channel gains from node Na , i.e., Ha [k] = [ha,l [k] . . . ha,m [k]]T with Ωa = Ωa,l × · · · × Ωa,m where {(Na , Nl ), . . . , (Na , Nm )} ⊂ I, and assume that each node Na knows Ha [k]. We shall focus on local power control policies of the form ua [k] = κa (Ha [k]), where κa : Ωa → [0, u¯ ] is a parameterized and integrable function over Ωa . To use the model in (2), one could seek to control the power to reach the same constant successful transmission probability at each instant (which might not be achieved). Alternatively, βa,l can also be treated as the average behavior of the communication channel over an infinity-time window. For further references, we denote by Ua the set that contains the parameters of κa (·). We shall next show how to retrieve these values. The fading in channel (Na , Nl ) is a stochastic process that might be correlated with other channels (representing some spatial correlation) and that we assume to have a time-invariant distribution. With this, we denote by ga (Ha ) the joint probability density function of Ha [k] = Ha which is considered to be known. Then, we can compute the transmission probability βa,l as

βa,l =



ga (Ha ) fa,l ha,l κa (Ha ) dHa .



Ωa



(17)

7. Co-design Transmitter nodes are often self-powered, and thus, preserving battery life is an important concern. Motivated by this, we will next show how to compute off-line the parameters that define the power control policies in ua [k] = κa (Ha [k]) and a set of filter gains guaranteeing the performance of the estimator. Our aim is to guarantee a certain estimation performance γP so that the network transmission power usage is minimized. Let us characterize the

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D. Dolz et al. / Automatica 81 (2017) 68–74

Fig. 1. Multi-hop network.

power budget by J (U ) = a=1 µa E{ua } where µa ∈ R and the expected value(average) of the transmission power of each node Na is E{ua } = Ω ga (Ha )κa (Ha )dHa . Then, the synthesis problem a can be formulated as follows:

M

minimize

J (U )

subject to

T {P } ≤ γP ,

L,P ,U

(18)

8. Numerical studies We consider the unstable system

E{P } − P ≼ 0, (17), ∀a = 1 , . . . , M .

ua [k] ∈ [0, u¯ ],

can be reduced, in Step 3, we first translate the effect of the reduction of the power budget to a single node. This leads to as many power transmission policies (control parameters Uia ) as there are nodes. To obtain the new power control law characterized by Uia , we solve (19) where we only modify the transmission policy of Na such that the successful transmission rate is maximized, whilst fulfilling the new power budget. Then, with the new set of power control parameters Uia for Na and the already existing Uim , the algorithm computes the transition probabilities of the reception scenario model θ[k] and verifies the fulfillment of the filter existence necessary conditions developed in Theorem 3. If the latter holds, then we design the state estimator by solving (12) including the predefined gain equality constraints. Once this has been done for the M nodes, in Step 4 the proposed heuristic selects the solution with the lowest estimation performance index γa . The algorithm ends when all the obtained γa are higher than the prescribed upperbound γP .



This is a nonlinear optimization problem, as the average node-tonode transmission probabilities (17) depend on the power control strategies. As a fast way to obtain a possibly suboptimal solution, in this work we propose the use of a greedy algorithm that never comes back to previous incumbent solutions to change the search path, and globally optimal solutions are not guaranteed.

A=

 C =



maximize Ua

βa,l

(19)

l : (Na ,Nl )∈I M

subject to

Ji −

 m=1 m̸=a

µm E{uim } − µa E{uia } = 0,

ua [k] ∈ [0, u¯ ],

∀a = 1, . . . , M

with βa,l as defined in (17). If this problem has no solution, then set γa ← ∞. Otherwise, compute the transition probabilities given in (6) and check conditions (13). If they are not fulfilled, then set γa ← ∞. Otherwise, solve optimization problem (12) including the corresponding gain equality constraints leading to the desired |L|. If the problem is infeasible, then set γa ← ∞. Otherwise, store Pa , La and set γa ← T {Pa }. i−1 Step 4. Set a ← arg min γa . If γa ≤ γP , then set Uim ← Um a

for all m = ̸ a; store Ui = {Ui1 , . . . , UiM }, P i = Pai and i i L = La , and go to Step 2. Otherwise, exit with the best solution found in iteration i − 1. The algorithm starts in Step 1 by considering the most favorable power control policy, i.e., where the probabilities of receiving packets are highest (highest transmission powers). Then, at each iteration (Step 2, 3 and 4), it tries to reduce the power budget J (U ) while the feasibility of the problem (18) is preserved. Each time it

0.53 0.72

 −0.1 , 1.05  0.39 . 0.35

0.01 0.01

 B=

0.13 , 0.08



The state disturbance and sensor noise covariances are 0.26 −0.003

 W =

Step 1. Set i ← 0. For a given u¯ , choose M sets of power control parameters U0a , a = 1, . . . , M to maximize each power transmission E{ua }. For given constants µa , define index M 0 J0 , a=1 µa E{ua }. Choose a small positive parameter value ξ . Step 2. Set i ← i + 1 and J i ← J i−1 − ξ . For a = 1 to M repeat Step 3, then go to Step 4. Step 3. Set Uim ← Uim−1 for all m ̸= a with m = 1, . . . , M. Obtain Uia as the power control parameter set resulting from the optimization problem

1.05 0.74

 −0.003 , 0.25

 2   σ1 0.0086 = . 0.0079 σ22

Measurements are acquired through the multi-hop network in Fig. 1 that may induce up to a unit delay in the end-to-end transmission. Thus, the number of states of the Markov chain θ [k] is |Θ | = ((1 + 2)!)2 = 36. Nodes transmit using BPSK modulation (see Quevedo et al., 2012) with b = 4 bits and a transmission power bounded by u¯ = 10. We consider correlated fading channels with h1,4 [t ] = (h1,3 [t ]+ h3,4 [t ])/100 and h2,4 [t ] = (h2,3 [t ]+ h3,4 [t ])/10 where h1,3 [t ], h2,3 [t ], h3,4 [t ] follow an independent exponential distribution (Rayleigh fading) with means h¯ 1,3 = 1, h¯ 2,3 = 0.3 ¯ and  h3,4 = 0.5. We denote the estimation performance index tr

35

i=0

Cx Pi CxT πi

as γ , where Cx = [In 0n×(n·d¯ ) ] selects the co-

variance corresponding to x[k] − xˆ [k|k]. First, let us assume that each node uses a constant power u1 [k] = u2 [k] = u3 [k] = 5. Under this scenario, the presence of a relay helps to improve the performance index obtained while solving (12) (without gain equality constraints) from γ = 0.112 to γ ⋆ = 0.037, where γ ⋆ was retrieved with 33 different gains. In this case, the estimation performance index obtained with the KF is γKal = 0.034 (where γKal = tr(Cx E{˜x[k]˜x[k]T }CxT )), which is 7% lower. However, the KF needs to execute up to 976 floating-point operations at every time instant, while the off-line method only requires at most 64 (independent of the gain grouping strategy). We can further reduce the filter complexity (storage requirements) using the algorithm at the end of Section 5. Fig. 2 explores the achieved trade-offs showing that the proposed decision rule leads to the best estimation performances. Also, we notice that we can reduce the filter complexity from 33 to 14 different gains without affecting the estimation performance index. Using the greedy algorithm developed in Section 7, we now explore the design trade-offs between power budget and estimation performance when all the filter gains are different. For that purpose, we focus on constant (C1) and saturated inverted channel gain (C2) transmission power policies (see Quevedo et al., 2012). Strategy C2 is implemented using the information of

D. Dolz et al. / Automatica 81 (2017) 68–74

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Fig. 2. Achievable trade-offs. Left: Estimation performance vs. filter complexity, where γ ⋆ = 0.037. Right: Power budget vs. estimation performance for constant (C1) and saturated inverted channel gain (C2) transmission power policies, where J ◦ = 30 and γ ◦ = 0.023.

the realizations h1,3 [t ], h2,3 [t ] and h3,4 [t ] whereas C1 only uses statistical information. Fig. 2 shows that at the expense of only a 10% deterioration of the best estimation performance, we can reduce the power budget by 25%. Note that, for the given example, power control strategy C2 only reduces the power budget by up to 4% with respect to C1 at the expense of requiring the value of the channel fading gain at each instant. 9. Conclusions In this work, we developed a model for multi-hop networked estimation with fading channels. Random dropouts are generated due to fading links, while delays are introduced while hopping through relay nodes. We introduced a finite Markovian process that captures the network behavior by keeping track of a finite number of measurement transmission outcomes. Using the average network behavior, we conceived a jump filter whose complexity can be selected as a trade-off between storage requirements and estimation performance. To keep the network operation powerefficient, we designed power policies and filter gains to minimize the power budget while guaranteeing a certain estimation error performance index. The design is carried out with an iterative procedure based on semi-definite programming. Numerical results suggest that (1) intermediate relays help to reduce the power budget for prescribed estimation performance, (2) increasing estimator complexity allows in general for power savings, and (3) not too many different filter gains are needed to achieve similar estimation performance than KF approach while offering a much lower computational burden. Future research may include design of topology reconfiguration methods and distributed power control strategies. References Chen, J., Johansson, K. H., Olariu, S., Paschalidis, I. Ch., & Stojmenovic, I. (2011). Guest editorial special issue on wireless sensor and actuator networks. IEEE Transactions on Automatic Control, 56(10), 2244–2246. Dolz, D., Peñarrocha, I., & Sanchis, R. (2016). Jump state estimation with multiple sensors with packet dropping and delaying channels. International Journal of Systems Science, 47(4), 982–993. Dolz, D., Quevedo, D.E., Peñarrocha, I., & Sanchis, R. (2014). Performance vs complexity trade-offs for Markovian networked jump estimators. In 19th IFAC world congress (pp. 7412–7417). Dolz, D., Quevedo, D.E., Peñarrocha, I., & Sanchis, R. (2017). Co-design of jump estimators and transmission policies for wireless multi-hop networks with fading channels [Extended]. Available at Arxiv. Han, C., Zhang, H., & Fu, M. (2013). Optimal filtering for networked systems with Markovian communication delays. Automatica, 49(10), 3097–3104. Hespanha, J. P., Naghshtabrizi, P., & Xu, Y. (2007). A survey of recent results in networked control systems. Proceedings of the IEEE, 95(1), 138–162. Leong, A.S., & Quevedo, D.E. (2013). On the use of a relay for Kalman filtering over packet dropping links. In Proc. Amer. contr. conf., Washington, DC. Nourian, M., Leong, A. S., & Dey, S. (2014). Optimal energy allocation for Kalman filtering over packet dropping links with imperfect acknowledgments and energy harvesting constraints. IEEE Transactions on Automatic Control, 59(8), 2128–2143.

Quevedo, D. E., Ahlén, A., Leong, A. S., & Dey, S. (2012). On Kalman filtering over fading wireless channels with controlled transmission powers. Automatica, 48(7), 1306–1316. Quevedo, D. E., Østergaard, J., & Ahlén, A. (2014). Power control and coding formulation for state estimation with wireless sensors. IEEE Transactions on Control Systems Technology, 22(2), 413–427. Schenato, L. (2008). Optimal estimation in networked control systems subject to random delay and packet drop. IEEE Transactions on Automatic Control, 53(5), 1311–1317. Shi, L., Jia, Q.-S., Mo, Y., & Sinopoli, B. (2011). Sensor scheduling over a packetdelaying network. Automatica, 47(5), 1089–1092. Shi, L., & Xie, L. (2012). Optimal sensor power scheduling for state estimation of Gaussian Markov systems over a packet-dropping network. IEEE Transactions on Signal Processing, 60(5), 2701–2705. Smith, S. C., & Seiler, P. (2003). Estimation with lossy measurements: jump estimators for jump systems. IEEE Transactions on Automatic Control, 48(12), 2163–2171.

Daniel Dolz was born in Castellón, Spain in 1988. He received his M.Sc. degree in industrial engineering in 2011 and his Ph.D. in industrial technologies in 2014 from the Universitat Jaume I, Spain. He also holds from 2011 a M.Sc. degree in automatic and electronic engineering from INSA Toulouse, France. Currently, he is a system engineer in the digitization department at Procter and Gamble, Euskirchen (Germany). He was with the Department of Industrial Systems Engineering and Design at the Universitat Jaume I from 2011 to 2016. His research interests include estimation, fault diagnosis and control over networks, and wireless sensor networks. Daniel Quevedo (S’1997–M’05–SM’14) is the head of the chair of automatic control (Regelungs-und Automatisierungstechnik) at Paderborn University, Germany. He received Ingeniero Civil Electrónico and M.Sc. degrees from the Universidad Técnica Federico Santa María, Chile, in 2000. In 2005, he was awarded the Ph.D. degree from the University of Newcastle in Australia. He was supported by a full scholarship from the alumni association during his time at the Universidad Técnica Federico Santa María and received several university-wide prizes upon graduating. He received the IEEE Conference on Decision and Control Best Student Paper Award in 2003 and was also a finalist in 2002. In 2009, he was awarded a five-year Research Fellowship from the Australian Research Council. He is an associate editor of the IEEE Control Systems Magazine, an editor of the International Journal of Robust and Nonlinear Control, Steering Committee Member of the IEEE Internet of Things Initiative, and serves as the chair of the IEEE Control Systems Society Technical Committee on Networks & Communication Systems. His research interests are in control of networked systems and of power converters. Ignacio Peñarrocha was born in Castelló, Spain in 1978. He received the M.Sc. degree in industrial engineering from the Universitat Jaume I de Castelló (UJI), Spain, in 2002, and his Ph.D. in computering and control engineering from Universitat Politècnica de València (UPV), Spain, in 2006. He has been working since 2004 at the Universitat Jaume I de Castelló. His current position is as an associate professor at the Department of Industrial Systems Engineering and Design. He has participated in several local and national research projects. His research interests include identification, estimation, fault diagnosis and control over networks, and fault tolerant control of wind turbines and wind farms.

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Alex Leong (M’08) was born in Macau in 1980. He received the B.S. degree in mathematics, the B.E. degree in electrical engineering in 2003, and the Ph.D. degree in electrical engineering in 2008, all from the, University of Melbourne, Australia. He is currently a research associate at Paderborn University, Germany. He was with the Department of Electrical and Electronic Engineering, University of Melbourne, from 2008 to 2015. His research interests include networked control systems, signal processing for sensor networks, and statistical signal processing. He was the recipient of the L.R. East Medal from Engineers Australia in 2003, an Australian Postdoctoral Fellowship from the Australian Research Council in 2009, and a Discovery Early Career Researcher Award from the Australian Research Council in 2012.

Roberto Sanchis was born in Genovés, Valencia, Spain in 1968. He received the M.Sc. degree in electrical engineering in 1993 and his Ph.D. in control engineering in 1999 from the Polytechnic University of Valencia (UPV), Spain. He was awarded the first national prize for university graduation in 1993. During 1994 and 1995, he was a teaching assistant at the Systems and Control Engineering department of the UPV. He has been working since 1996 at the University Jaume I of Castellón, His current position is an associate professor at the Department of Industrial Systems Engineering and Design. His research interests include scarce-data and networked based estimation and control, PID tuning methods and event based estimation and control. He has also worked in some industrial applications, especially in the control of waste water treatment plants. He has authored a set of free tools for system identification and for PID controller tuning (https://sites.google.com/a/uji.es/freepidtools).