On Kalman filtering over fading wireless channels with controlled transmission powers

Automatica 48 (2012) 1306–1316

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Brief paper

On Kalman filtering over fading wireless channels with controlled transmission powers✩ Daniel E. Quevedo a,1 , Anders Ahlén b , Alex S. Leong c , Subhrakanti Dey c a

School of Electrical Engineering & Computer Science, The University of Newcastle, NSW 2308, Australia

b

Department of Engineering Sciences, Signals and Systems, Uppsala University, P. O. Box 534, SE-751 21 Uppsala, Sweden

c

Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, VIC 3052, Australia

article

info

Article history: Received 27 March 2011 Received in revised form 9 September 2011 Accepted 13 November 2011 Available online 30 April 2012 Keywords: State estimation Fading channels Packet dropouts Stochastic stability Power control

abstract We study stochastic stability of centralized Kalman filtering for linear time-varying systems equipped with wireless sensors. Transmission is over fading channels where variable channel gains are counteracted by power control to alleviate the effects of packet drops. We establish sufficient conditions for the expected value of the Kalman filter covariance matrix to be exponentially bounded in norm. The conditions obtained are then used to formulate stabilizing power control policies which minimize the total sensor power budget. In deriving the optimal power control laws, both statistical channel information and full channel information are considered. The effect of system instability on the power budget is also investigated for both these cases. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction The interest in estimation and control over wireless communication channels has increased tremendously. The rapid evolution of wireless sensor networks, see e.g., Gharavi and Kumar (2003), Havinga, Hou, and Zhao (2004), Ilyas, Mahgoub, and Kelly (2004), and Shen, Zhang, and Caiming Qiu (2007), has made wireless sensors (WSs) cheap and reliable enough to be brought into commercial use. WSs offer several advantages for industrial control systems, such as, flexibility, low cost, and fast deployment. Furthermore, with WSs electrical contact problems, often regarded as a nuisance in wired systems, are no longer an issue. Furthermore, WSs and actuators can be placed where wires cannot go, or where power sockets are unavailable. The use of wireless communications for state estimation and closed loop control, however, also

✩ This research was supported under Australian Research Council’s Discovery Projects funding scheme (project number DP0988601). A preliminary version of parts of this work appears in Quevedo, Ahlén, Leong, and Dey (2011). The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (D.E. Quevedo), [email protected] (A. Ahlén), [email protected] (A.S. Leong), [email protected] (S. Dey). 1 Tel.: +61 2 49217072; fax: +61 2 49601712.

0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.03.025

brings new challenges in system design. In fact, wireless communication channels are frequently subject to time-varying fading and interference, which may lead to packet errors; see, e.g., Hespanha, Naghshtabrizi, and Xu (2007), Proakis (1995), and Schenato, Sinopoli, Franceschetti, Poolla, and Sastry (2007). A key aspect is that fading channels can partially be compensated for by adjusting the transmission power levels; higher transmission power giving less packet errors. Power control has been shown to be a key enabling technology whenever wireless links are used, constituting a versatile means to guarantee quality of service to individual users (Buche & Kushner, 2002; Charalambous, Djouadi, & Denic, 2005; Gunnarsson & Gustafsson, 2003; Hanly & Tse, 1999; Huang, Caines, & Malhamé, 2004). Since in most WS applications, energy is severely limited, power control design involves trading energy consumption for accuracy (Caire, Taricco, & Biglieri, 1999; Pantazis & Vergados, 2007; Park, Fischione, & Johansson, 2008; Xiao, Cui, Luo, & Goldsmith, 2006; Zhang, Poor, & Chiang, 2008). Kalman filters are widespread in estimation and control applications and the effect of random measurement losses on filter stability has received considerable attention in the recent literature. In particular, Sinopoli et al. (2004) studied a singlelink estimation setup for LTI plants where dropout processes are independent and identically distributed (i.i.d.). Thus, in this modeling paradigm it is assumed that the probability of drops is the same from one transmission to the next. By using a fixed-point argument, Sinopoli et al. (2004) established that there exists a

D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

critical dropout probability value which separates situations where the expected value of the covariance matrix remains bounded from instances where it diverges, see also Kluge, Reif, and Brokate (2010), Liu and Goldsmith (2004) and Plarre and Bullo (2009).2 The latter work examines a state estimation architecture with two channels affected by i.i.d. dropouts. The case of Kalman filtering with a single link and where the dropouts are described by a timehomogeneous two-state Markov chain was investigated in Huang and Dey (2007), Jin, Gupta, and Murray (2006), Shi et al. (2010) and Xie and Xie (2008), whereas in the recent work (Censi, 2011) also semi-Markov chains are considered. Given the fundamental importance of power control in wireless communications, it is surprising that the role of power control on stability of Kalman filters with random packet dropouts has not yet been investigated. In fact, to the best of the authors’ knowledge, the only works which have examined power control design for Kalman filtering with packet dropouts are our own; see, e.g., Østergaard, Quevedo, and Ahlén (2010a) and Quevedo, Ahlén, and Østergaard (2010b). These works examine the trade-off between energy use and estimation accuracy, but do not investigate stability issues. In the present work we study state estimation for linear timevarying systems via a Kalman filter performing optimal centralized state estimation at a single gateway. To keep WSs simple, only raw measurements are transmitted to the gateway; c.f., Gupta, Martins, and Baras (2009), Hespanha et al. (2007), Schenato (2008) and Shi et al. (2010). The wireless transmission channels are block-fading and generate random packet loss. The packet loss probabilities depend, in a nonlinear fashion, upon the timevarying channel gains and power levels used by the sensors. The gateway has knowledge on whether received data is error-free or not. Our main purpose is to investigate the role of power control of the transmission power levels to be used by each WS. We establish sufficient conditions which ensure that the Kalman filter covariance matrix is exponentially bounded in norm. The conditions obtained are then used to formulate stabilizing optimal power allocation laws which minimize the total power used by the sensors. The remainder of this paper is organized as follows. Section 2 introduces the system setup and revises basic aspects on state estimation with packet drops and power control. A sufficient condition for stochastic stability of the estimator is then obtained in Section 3. Section 4 studies decentralized power control laws. Robustness issues are investigated in Section 5. The stability results are used to perform optimum power allocation in Section 6. Numerical studies are included in Section 7. Section 8 draws conclusions. All proofs of technical results are included in appendices. Notation. We write R≥0 for the nonnegative real numbers and N0 for {0, 1, 2, . . .}. For sequences, we use {ν}N0 for {ν(0), ν(1), . . .}, and {ν}k0 for {ν(0), . . . , ν(k)}. Given a vector v , diag(v) is the

√

corresponding diagonal matrix; its Euclidean norm is |v| = v T v , where the superscript T refers to transposition. The eigenvalues of a square matrix M are denoted via eigs(M ), its trace via trM, and its  spectral norm by ∥M ∥ , max eigs(M T M ). If M is positive definite (semidefinite), then we write M ≻ 0 (M ≽ 0). The L × L identity matrix is denoted via IL ; its ℓ-th row is denoted via δLℓ . Given two

Fig. 1. State estimation with transmission power control of 3 WSs.

sets A and B, A \ B , {a ∈ A: a ̸∈ B}. The indicator function of a set A is denoted via 1A , i.e., 1A (a) =



1 0

if a ∈ A, if a ̸∈ A.

(1)

A function θ : R≥0 → R≥0 is of class-K , if it is continuous, zero at zero and strictly increasing. We write Pr{Ω | ∆} for the conditional probability of Ω given ∆ and Pr{Ω } for the unconditional probability. The cumulative distribution function (cdf) of a random variable ν is Fν (v) , Pr{ν ≤ v}. The expected value  of ν is, thus, given by the Lebesgue–Stieltjes integral E{ν} = Λ zdFν (z ), where Λ is the support of ν ; E{ν | ∆} refers to conditional expectation. A random variable ν , which is Gaussian with mean value m and covariance matrix Γ ≻ 0 is denoted ν ∼ G(m, Γ ). At times, we use the same notation for random variables and their realizations. What is meant will depend on the context. 2. The system setup Consider a linear time-varying n-dimensional plant model x(k + 1) = A(k)x(k) + w(k) ∈ Rn ,

x(0) ∼ G(x0 , P0 ).

(2)

The driving noise process {w}N0 is white, where each w(k) ∼

G(0, Q (k)).

To remotely estimate the state sequence {x}N0 , L ≥ n wireless sensors are used. Each sensor ℓ ∈ {1, 2, . . . , L} provides a noisy measurement sequence, say {yℓ }N0 , where yℓ (k) = Cℓ x(k) + vℓ (k) ∈ Rmℓ ,

mℓ ∈ N

(3)

and {vℓ }N0 are white with vℓ (k) ∼ G(0, Rℓ (k)). Throughout this work, we assume that {A}N0 , {Q }N0 and {Rℓ (k): k ∈ N0 , ℓ ∈ {1, 2, . . . , L}} are deterministic and bounded sequences, which are known at the gateway. The L measurements in (3) are to be transmitted to a single gateway via wireless links. Upon receipt, the signals are used to remotely estimate the state of the system (2); see Fig. 1. 3

2.1. Effects of channel fading In the present work, we will adopt a block-fading channel gain model, which is a common information theoretic model for fading wireless channels where the channel power gains remain invariant over a block (equivalent to the coherence time of the channel) and may change from block to block (Caire et al., 1999). We furthermore assume single measurements yℓ (k) to be a packet, thus, the block length equals the packet transmission time. The associated channel gain processes are denoted via hℓ (k) ∈ Ωℓ ⊆ R≥0 ,

2 If rather than the expected value of the covariance matrix, other metrics are used, then different critical dropout probabilities will be obtained; see also Censi (2011), Huang and Dey (2007) and Rohr, Marelli, and Fu (2010). Alternatively, the works (Censi, 2011; Epstein, Shi, Tiwari, & Murray, 2008; Kar, Sinopoli, & Moura, 2012; Shi, Epstein, & Murray, 2010) directly examine the distribution of the covariance matrix.

1307

ℓ ∈ {1, 2, . . . , L}.

(4)

3 In addition to measurement noise, v (k) may also describe quantization ℓ effects, here modeled as Gaussian and introducing possibly time-varying distortion (Msechu, Roumeliotis, Ribeiro, & Giannakis, 2008; Quevedo et al., 2010b).

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D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

Since the L links are wireless, transmission errors are likely. Faulty packets will be discarded when estimating {x}N0 ; c.f. Mostofi and Murray (2009). To model transmission effects we introduce the L binary stochastic arrival processes {γℓ }N0 , where:

γℓ (k) =



if yℓ (k) arrives error-free at time k, if yℓ (k) does not arrive error-free at time k.

1 0

max

where uℓ

(5)

(6)

where the functions fℓ (·): R≥0 → [0, 1] are monotonically increasing and differentiable, and depend upon the modulation scheme employed; see, e.g., Proakis (1995) and Quevedo et al. (2010b). We introduce {γ }N0 , {h}N0 and {u}N0 , where, ∀k ∈ N0 :

  γ1 (k) γ2 (k)  γ (k) ,   ..  , .

h1 (k) h2 (k)

 h(k) ,  

γL (k)  u1 (k) u2 (k)  u(k) ,   ..  . . uL (k)



..  , . hL (k)



(7)

fℓ (hu) =

√ hu

1

√ 0

2π

b −t 2 /2

e

dt

Fhℓ (k) (h) =

T

hˆ L (k)

···

(9)

4

are channel gain estimates. These satisfy

ℓ ∈ {1, . . . , L},

(10)

where ϵℓ (k) are the relative estimation errors, with support Λℓ ⊂ R. If we define

 ϵ(k) , ϵ1 (k)

ϵ2 (k)

···

ϵL (k)

T

∈Λ

(11)

then (9) can be written in compact form as hˆ (k) = (IL + diag (ϵ(k))) h(k).

(12)

2.3. Kalman filtering with packet drop-outs Throughout this work, we will assume that the packets transmitted from the sensors to the gateway incorporate error detection coding (Proakis, 1995). Hence, the gateway knows whether received packets are correct or not. Thus, at any time k, past and present realizations of the process {γ }N0 are available at the gateway. For state estimation purposes, the system amounts to sampling (2), (3) with stochastic output matrix

1 − e−λℓ h , 0,

y(k) , C (k)x(k) + v1 (k)T



v2 (k)T

vL (k)T

···

T

.

The conditional probability distribution of x(k), given {γ }k0 and correctly received packets with sensor measurements up to time k, i.e., {y}k0 , is Gaussian. Consequently,



xˇ (k) , E x(k) | {y, γ }k0−1



P (k) , E

where λℓ > 0. For other transmission schemes, see Dey, Leong, and Evans (2009). 



x(k) − xˇ (k)

 T

x(k) − xˇ (k)



 | {y, γ }k0−1 ,

satisfy the Kalman filter recursions (see, e.g., Kumar and Varaiya (1986)): xˆ (k) = xˇ (k) + K (k) y(k) − C (k)ˇx(k)





xˇ (k) = A(k − 1)ˆx(k − 1)

2.2. Control of transmission powers The main purpose of this work is to study the role of power control for state estimation over wireless channels. It follows from (6) that one can improve transmission reliability and, thus, state estimation accuracy for a given wireless propagation environment, by increasing the transmission powers. However, with WSs, it is of fundamental importance to save energy. Thus, power control design involves a trade-off between transmission error probabilities (and, thus, state estimation accuracy) and energy consumption; see also Quevedo et al. (2010b). We focus on nonlinear feedback policies which are implemented at the gateway, see Fig. 1, and are of the form

κℓ : Ω1 × · · · × ΩL → [0, umax ℓ ] hˆ (k) → uℓ (k) = κℓ (hˆ (k)),

and measurements



if h ∈ R≥0 , if h < 0,

(13)

γL (k)CL

xˆ (k) , E x(k) | {y, γ }k0

and the gains, hℓ (k), are exponentially distributed with cdf



hˆ 2 (k)

  γ1 (k)C1 γ2 (k)C2   C (k) = C (γ (k)) ,   ..  , .

Example 1 (BPSK with Rayleigh Fading). Suppose that binary phase shift keying transmission over i.i.d. block fading additive white Gaussian noise channels is used. Each sensor measurement consists of a packet containing b bits. Since the bit errors are independent within a packet, the success probabilities can be written as, see Proakis (1995),





Λ , Λ1 × · · · ΛL

is the peak power level. More precisely, we have

Pr {γℓ (k) = 1 | hℓ (k) = h, uℓ (k) = u} = fℓ (hu),

hˆ (k) , hˆ 1 (k)

hˆ ℓ (k) = (1 + ϵℓ (k))hℓ (k),

The success probabilities of {γℓ }N0 are determined by the propagation environment and transmission power uℓ (k) ∈ [0, umax ℓ ],

where

(8)

(14)

P (k + 1) = A(k) (In − K (k)C (k)) P (k)A(k)T + Q (k) with initial values P (0) = P0 and xˆ (−1) = x0 , and where

 −1

K (k) , P (k)C (k)T C (k)P (k)C (k)T + R(k)



R(k) , diag (R1 (k), R2 (k), . . . , RL (k)) ≻ 0. Note that C (k) takes one of 2L possible values. The distribution of C (k) depends upon the current channel gains hℓ (k) and the power levels used; see (6) and (13). Thus, {C }N0 is a random process, the recursion (14) is stochastic and the error covariance {P }N0 becomes a random process.

4 For example, one could use hˆ (k) = E{h(k)}; see also, Ekman (2002).

D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

1309

3. Stability with power controlled radio-links

using (1), and where

Due to random packet dropouts, {P }N0 in (14) will in general not converge to a unique value as k → ∞. Depending upon properties of (2) and {γ }N0 , E{∥P (k)∥} may remain bounded or diverge. This issue of stability of the Kalman filter is of fundamental importance to characterize estimation quality. The special case of Kalman filtering for LTI systems with one or two links and where arrival processes are i.i.d. Bernoulli with fixed success probabilities has been extensively studied; see, e.g., Liu and Goldsmith (2004) and Schenato et al. (2007). A data-fusion strategy with multiple sensors is studied in Xia, Shang, Chen, and Liu (2009). We next examine stability of the estimator for more complex cases where the packet dropout processes are time-varying and governed by power controlled fading channels. For that purpose, we make the following assumption.

J , Γ ∈ {0, 1}L ⊂ RL : C (Γ ) has column-rank n ,

Assumption 2. The joint process {ξ }N0 , where

ξ (k) =

h(k) , ϵ(k)





k ∈ N0

(15)

is white, i.e., the random variables {ξ (k): k ∈ N0 } are independent (but not necessarily identically distributed).  Channel gains are temporally uncorrelated for fading channels for which coherence times (times for which the channel remains invariant) are shorter than the packet transmission times. This assumption will hold in many practical applications, and was considered, e.g., also in Mostofi and Murray (2009). If Assumption 2 holds, then {γℓ }N0 are white. In addition, it follows from (6) and (8) that Pr{γℓ (k) = 1} =

 Ω ×Λ

ϕℓ (ξ )dFξ (k) (ξ ),

(16)

where Ω , Ω1 × · · · × ΩL and

  ϕℓ (ξ ) = fℓ δLℓ hκℓ ((IL + diag(ϵ))h)   ξ T = hT ϵ T , h ∈ Ω , ϵ ∈ Λ.

(17)

Theorem 3, stated below, establishes sufficient conditions for convergence of E{∥P (k)∥}, k ∈ N0 . To state our result, we introduce the random process {s}N0 , where s(k) =



1, 0,

if C (k) has full column-rank, otherwise.

(18)

Theorem 3. Consider state estimation with power controlled radio links as described in Section 2 and suppose that Assumption 2 holds. If there exists ρ ∈ [0, 1), such that: Pr{s(k) = 1} ≥ 1 −

ρ , ∥A(k)∥2

for some α, β ∈ R≥0 .

∀k ∈ N 0 ,

(19)

∀k ∈ N 0

(20)



(21)

(22)

see (13). Alas, Pr{s(k) = 1} may be hard to quantify since, due to power control, the individual transmission outcomes γℓ (k), ℓ ∈ {1, 2, . . . , L} may be correlated. However, as stated in Lemma 5 below, if the transmission outcomes satisfy the following assumption, then Pr{s(k) = 1} can be stated in terms of the conditional success probabilities ϕℓ . Assumption 4. The transmission outcomes γℓ (k) are conditionally independent given (h(k), u(k)), i.e., Pr γℓ (k) = i, γp (k) = j | h(k) = h, u(k) = u





= Pr {γℓ (k) = i | h(k) = h, u(k) = u}   × Pr γp (k) = j | h(k) = h, u(k) = u ,

(23)

for all k ∈ N0 , ∀i, j ∈ {0, 1} and ∀ℓ, p in {1, 2, . . . , L}.



Lemma 5. Suppose that Assumptions 2 and 4 hold, then Pr{s(k) = 1} =



L  

1 − δLℓ Γ (1 − ϕℓ (ξ ))

Ω ×Λ Γ ∈J ℓ=1



 + δLℓ Γ ϕℓ (ξ ) dFξ (k) (ξ ),

(24)

where ϕℓ (ξ ) are defined in (17) and J, in (22). Remark 6. As already discussed in Section 1, situations where success probabilities are the same from one instant to the next have been extensively studied in the literature. In particular, (Kar et al., 2012; Kluge et al., 2010; Shi et al., 2010) have mainly focused on the case where Cℓ has full rank. Our result in Theorem 3 complements existing stability results, in that it considers transmission over multiple power controlled wireless channels with time-varying (but white) success probabilities. Naturally, since Theorem 3 is applicable for time-varying systems, it requires strong conditions, namely (19) needs to hold at all instances k ∈ N0 .  Remark 7. It follows from (18) that Theorem 3 disregards the role of measurement updates where C (k) is not full-rank toward reducing the estimation error covariance. Overcoming this limitation remains a topic for future research.  Example 8. Consider a plant model (2) with state dimension n = 2 and L = 2 sensors, with m1 = 1, m2 = 2, and where C1 = 1

0 ,





1 C2 = 1



0 H⇒ C (k) = 1



γ1 (k) γ2 (k) γ2 (k)

0 0 . γ2 (k)



Thus, J = [0 1]T , [1 1]T and s(k) = γ2 (k). If we suppose that the controller has access to perfect channel estimates  and that Assumption 2 holds, then Pr{s(k) = 1} = Ω ×{0}



Under the hypotheses of the above theorem, (a specific form of) stochastic stability of Kalman filtering when used for state estimation over wireless fading channels with power control of the form (8) will hold. The process s(k) is a logical function of γ (k). In fact, we have s(k) = 1J (γ (k)),





then, {P }N0 is exponentially bounded in norm, i.e., E {∥P (k)∥} ≤ αρ k + β,





ϕ2 (ξ )dFξ (k) (ξ ), where ξ = [h1 h2 ϵ1 ϵ2 ]T , and ξ (k) = [h1 (k) h2 (k) ϵ1 (k) ϵ2 (k)]T . Thus, the sufficient   condition for stability established in Theorem 3 becomes: Ω f2 [0 1]hκ2 (h) dFh(k) (h) ≥ 1 − ρ/∥A(k)∥2 , ∀k ∈ N0 , where h = [h1 h2 ]T and h(k) = [h1 (k) h2 (k)]T . 

In the sequel, we will take a closer look at how Theorem 3 can be used in various power control architectures.

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D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

4. Decentralized control policies A special instance of the class of power control policies considered in (8) is that of decentralized control. Here, the power levels of each of the L transmitters are calculated at the gateway, but depend only upon the channel gain estimate of their links to the gateway: uℓ (k) = κℓ (hˆ ℓ (k)),

∀k ∈ N0 , ℓ ∈ {1, 2, . . . , L};

(25)

see (10). Such control laws are, in general, easier to implement than centralized ones of the form (8), especially when the number of transmitters is large. In the decentralized case (25), ϕℓ (ξ ) in (17) becomes

  ϕℓ (ξ ) = fℓ δLℓ hκℓ ((1 + δLℓ ϵ)δLℓ h)

Example 10. Consider the configuration in Fig. 1 with L = 3 scalar sensors and state dimension n = 2. Suppose that the matrix C (k) ∈ R3×2 in (13) has full column-rank if and only if at least two measurements are received. Then, see (21),



J = [0

1]T , [1

1

Pr{s(k) = 1} =

 Ω ×Λ

Ωℓ ×Λℓ

+ Ω ×Λ

 + Ω ×Λ



h (k) ξℓ (k) = ℓ , ϵℓ (k)



  hℓ

ϵℓ

,

(28)

Lemma 9. Suppose that Assumptions 2 and 4 hold, that the control laws are decentralized, see (25), and define

ψ(Φ (k)) ,

ΦL (k)

···



(29)

1 − δLℓ Γ (1 − Φℓ (k)) + δLℓ Γ Φℓ (k) ,





Γ ∈J ℓ=1

where J is as in (22). Then Pr{s(k) = 1} = ψ(Φ (k)).



A particularly simple decentralized control policy consists in using saturated constant gain power controllers for each channel. Here, the aim is to provide a constant success probability by inverting the channel gain, provided sufficient power is available. Such a control policy is described via:

 Kℓ /hˆ ℓ (k) κℓ (hˆ ℓ (k)) , max uℓ

if hˆ ℓ (k) ≥ Kℓ /umax ℓ , if hˆ ℓ (k) < Kℓ /umax ℓ ,

(30)

where K1 , K2 , . . . , KL ∈ R≥0 . Consequently, (27) gives that if Assumption 2 holds, then Pr{γℓ (k) = 1} =

 Σℓ

fℓ (Kℓ /(1 + ϵℓ )) dFξℓ (k) (ξℓ )

 + (Ωℓ ×Λℓ )\Σℓ

fℓ (hℓ umax ℓ )dFξℓ (k) (ξℓ ),

(31)

ϵℓ ]T ∈ (Ωℓ × Λℓ ): (1 + ϵℓ )hℓ ≥ Kℓ /umax where Σℓ . ℓ The above expressions can be used for robust control design, i.e., control design which uses knowledge of the cdf Fϵℓ (k) .  = [ hℓ

1]T



ϕ1 (ξ )(1 − ϕ2 (ξ ))ϕ3 (ξ )dFξ (k) (ξ ) ϕ1 (ξ )ϕ2 (ξ )(1 − ϕ3 (ξ ))dFξ (k) (ξ ) ϕ1 (ξ )ϕ2 (ξ )ϕ3 (ξ )dFξ (k) (ξ ).

(32)

Pr{s(k) = 1}

ℓ ∈ {1, . . . , L}.

As shown in Lemma 9 given below, in the decentralized case, Pr{s(k) = 1} becomes a function of the unconditional link probabilities Pr{γℓ (k) = 1} and the result in Lemma 5 can be further simplified. Note that, as in the centralized case (8), with decentralized control, the resulting stability condition (19) involves, in general, more than one channel.

L  

1

If the control laws and channel estimators are decentralized, then ϕℓ (ξ ) takes the form (26) and (32) simplifies to

ξℓ =

Φℓ (k) , Pr{γℓ (k) = 1}  Φ (k) , Φ1 (k) Φ1 (k)

Ω ×Λ

(27)

where



0]T , [1

(1 − ϕ1 (ξ ))ϕ2 (ξ )ϕ3 (ξ )dFξ (k) (ξ )



(26)

fℓ (hℓ κℓ ((1 + ϵℓ )hℓ )) dFξℓ (k) (ξℓ ),

1

Lemma 5 establishes that, if Assumptions 2 and 4 hold,

+



1]T , [1

s(k) = (1 − γ1 (k))γ2 (k)γ3 (k) + γ1 (k)(1 − γ2 (k))γ3 (k) + γ1 (k)γ2 (k)(1 − γ3 (k)) + γ1 (k)γ2 (k)γ3 (k).

so that, in contrast to (16) we have Pr{γℓ (k) = 1} =

0



 = Ω1 ×Λ1

(1 − ϕ1 (ξ1 ))dFξ1 (k) (ξ1 )

 × Ω3 ×Λ3

 Ω2 ×Λ2

ϕ2 (ξ2 )dFξ2 (k) (ξ2 )

ϕ3 (ξ3 )dFξ3 (k) (ξ3 ) + · · · + · · · + · · ·

= (1 − Φ1 (k))Φ2 (k)Φ3 (k) + Φ1 (k)(1 − Φ2 (k))Φ3 (k) + Φ1 (k)Φ2 (k)(1 − Φ3 (k)) + Φ1 (k)Φ2 (k)Φ3 (k), in agreement with Lemma 9.



5. Robustness to uncertainty in channel gains We will next turn our attention to nominal designs, i.e., where the control policy is chosen such that stability is guaranteed under the assumption that the channel estimates are error-free. Thus, the control laws κℓ are chosen by only taking into account the nominal quantities:

  ϕˆ ℓ (h) , fℓ δLℓ hκℓ (h)  ˆ ℓ (k) , Φ ϕˆ ℓ (h)dFh(k) (h) Ω  ˆ (k) , Φ ˆ 1 (k) Φ ˆ 2 (k) · · · Φ

(33)

 ˆ L (k) . Φ

Whilst assuming that channel gains are exactly known is fairly common in the wireless communication community, see also Dey et al. (2009) and Mostofi and Murray (2009), it is certainly of interest to investigate robustness of nominal designs to channel ˆ (k) is estimation errors. In particular, with estimation errors, Φ only an approximation of the actual success probabilities Φ (k) as defined in (16) and (29). Robust stability, as per Theorem 3, in the presence of channel estimation errors depends upon the success probabilities of the individual channels. A simple bound is established in Theorem 12 given below, which applies to cases where the functions fℓ and κℓ are continuous. Assumption 11. There exist class-K functions θℓ and σℓ , ℓ ∈ {1, 2, . . . , L} such that the following are satisfied:

|fℓ (ν1 ) − fℓ (ν2 )| ≤ θℓ (|ν1 − ν2 |) |κℓ (υ1 ) − κℓ (υ2 )| ≤ σℓ (|υ1 − υ2 |) , for all ν1 , ν2 ∈ R≥0 and for all υ1 , υ2 ∈ Ωℓ .

(34)

D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

The result in Theorem 3 motivates one to minimize the nominal average sum power

Theorem 12. Suppose that Assumption 11 holds, then

ˆ ℓ (k) − ∆ℓ (k), ∀ℓ ∈ {1, 2, . . . , L}, Φℓ (k) ≥ Φ (35)    ℓ where ∆ℓ (k) , Ω ×Λ θℓ δL hσℓ (|diag(ϵ)h|) dFξ (k) (ξ ). If, in addition, κℓ is decentralized (and in (34), one takes υ1 , υ2 ∈ Ωℓ ), then, see (29),   ˆ (k) − ∆(k) , Pr{s(k) = 1} ≥ ψ Φ (36)   where ∆(k) , ∆1 (k) · · · ∆L (k) .  The bound established in Theorem 12 can be combined with Theorem 3 to analyze robust stability of nominal control laws. As shown below, if κℓ are decentralized and non-increasing, then smaller values for ∆ℓ (k), thus giving tighter bounds in (35) and (36), can be found. Lemma 13. If Assumption 11 holds and κℓ are decentralized and non-increasing, then (35) and (36) hold with

∆ℓ (k) =

∆′ℓ (k) ,

 Ωℓ ×Λ+ ℓ

where Λ+ ℓ , Λℓ ∩ R≥0 .



The following corollary to Theorem 12 can be used for robust design, i.e., control design which uses Fξ (k) to obtain stabilizing control in the presence of channel estimation errors. It uses the following definition:





c ∈[0,1]

U¯ (k) ,

L  

Ω

ℓ=1

κℓ (h)dFh(k) (h),

(37)

6.1. Constant power levels The simplest decentralized power control method consists in using constant power levels {u1 , . . . , uL }. By Lemma 9, the optimization problem for the nominal case becomes L 

min

(u1 ,...,uL )

uℓ ,

ˆ (k)) ≥ 1 − s.t. ψ(Φ

ℓ=1



f (h u )dFhℓ (k) (hℓ ); Ωℓ ℓ ℓ ℓ

  ˆ (k) ≥ 1 − ψ Φ

ρ˜ , ∥A(k)∥2

∀k ∈ N0 ,

(38)

(41)

see (33).

ˆ (k)) = ˆ gives that ψ(Φ ℓ=1 Φℓ (k). By taking logarithm of the constraint in (41), we obtain the equivalent problem L

min

L 

s.t.

L 

uℓ

ℓ=1

   ˆ ℓ (k) ≥ log 1 − log Φ

ℓ=1

Corollary 14. Suppose that the hypotheses of Theorems 3 and 12 are satisfied, and that L decentralized control policies have been designed such that, for some ρ˜ ∈ [0, 1),

ρ , ∥A(k)∥2

Example 16. Consider a simple architecture with scalar sensors, where L = n, and where the vectors {C1 , . . . , CL } are linearly independent. Thus, C (k) has full rank, if and only if γℓ(k) = 1, ∀ℓ ∈ {1, 2, . . . , L}. In this case J = [1 1 · · · ]T and (29)

{u1 ,...,uL }

where ∇ψ(χ ) is the gradient of ψ at χ , ∆(k) is as in Theorem 12, ˆ (k) is defined in (33). and Φ

(40)

subject to (19). In what follows we investigate three nominally stabilizing power control laws which minimize (40) and are of increasing complexity.

ˆ ℓ (k) = where Φ

θℓ (hℓ σℓ (ϵℓ hℓ )) dFξℓ (k) (ξℓ ),

ˆ (k) − c ∆(k) ∆(k)T , D(k) , max ∇ψ Φ

1311

ρ ∥A(k)∥2

(42)



.

The following result shows that under BPSK transmission and Rayleigh fading, in this simple case, the optimal power levels can be found with standard optimization software. Lemma 17. Consider the communication setup in Example 1. Then (42) is a convex optimization problem. 

i.e., the nominal system is stable; see Lemma 9 and Theorem 3. If

ρ˜ < 1 − ∥A(k)∥2 D(k),

∀k ∈ N 0 ,

then (20) holds for some ρ ∈ [0, 1).

(39) 

ˆ (k) giving ρ˜ < 1 in (38) have inherent Thus, designs which use Φ robustness with respect to small estimation errors. Example 15. Suppose that L = 2 and that Pr{s(k) = 1} = Φ1 (k)Φ2 (k), both packets need to be successfully received for C (k) ˆ to  have full-rank.  Direct  calculations provide ∇ψ(Φ (k)− c ∆(k)) = ˆ 2 (k)∆1 (k) + ˆ ˆ ∆ ( k ) ∆ ( k ) . Thus, D(k) = Φ Φ2 (k) Φ1 (k) − c 2 1

ˆ 1 (k)∆2 (k) can be used in (39) as a sufficient condition for robust Φ stochastic stability of the Kalman filter with decentralized power control.  6. Optimal power control strategies In general, state estimation accuracy can be improved by increasing the power on the channels being in a fade. However, such a simple strategy may often waste energy. A better idea is to allocate only as much power to the sensors as is necessary to attain desired performance. A lower bound would then be the allocation of power to the sensors that is sufficient to maintain stability.

6.2. Saturated constant gain power control We will next investigate the saturated power control law of (30), Here, we set Ωℓ = R≥0 for all ℓ ∈ {1, . . . , L}. In this case it follows directly from (29) and (31) that

ˆ ℓ (k) = Φ

Kℓ /umax ℓ

 0

fℓ (hℓ umax ℓ )dFhℓ (k) (hℓ )

∞

 +

Kℓ /umax ℓ

fℓ (Kℓ )dFhℓ (k) (hℓ ),

(43)

whereas the average sum power becomes U¯ (k) =

L 



Kℓ /umax ℓ



 0

ℓ=1



∞

+ Kℓ /umax ℓ

Kℓ hℓ

umax ℓ dFhℓ (k) (hℓ )

 dFhℓ (k) (hℓ ) .

(44)

It is easy to see that, since fℓ are monotonically increasing, for fixed ˆ ℓ (k) and ψ(Φ ˆ (k)) maximum peak power levels {umax , . . . , umax }, Φ 1 L are increasing functions of the gains {K1 , . . . , KL }. Thus, a necessary

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D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

condition on the maximum peak power levels so that (19) can be satisfied is that

ψ(Υ (k)) ≥ 1 − ρ/∥A(k)∥2 ,

for some ρ ∈ [0, 1),

Υℓ (k) ,

1−

∞

 0

fℓ (hℓ umax ℓ )dFhℓ (k) (hℓ ).

On the other hand, the average sum power U¯ (k) in (44) is also an increasing function of the gains and maximum peak power levels. Consequently, when designing constant gain saturated power control laws, there exists a trade-off between energy use and satisfaction of the sufficient condition for stability established in Theorem 3. If {umax , umax , . . . , umax } are given, then the design problem 1 L 2 consists in choosing the control gains {K1 , . . . , KL }. The optimization problem in (40) leads to: min U¯ (k),

{K1 ,...,KL }

ˆ (k)) ≥ 1 − ρ/∥A(k)∥2 , s.t. ψ(Φ

(46)

ˆ ℓ (k) as in (43). This problem is, in general, non-convex, with Φ since the objective function U¯ (k) is actually a concave function of { K1 , . . . , KL } . As an alternative to (46), one may also seek to optimize both {K1 , . . . , KL } and {umax , . . . , umax }, which is, again a non-convex L 1 optimization problem. In our numerical results documented in Section 7, these two optimization problems are addressed by using the NMinimize routine in Mathematica to find locally optimal solutions. 6.3. Centralized control With centralized control as in (8), the power levels uℓ (k) depend on the instantaneous channel power gains h(k). The optimization problem in (40) can then be stated as, see (24), min

  L

{κ1 ,...,κL } Ω ℓ=1

s.t.

κℓ (h)dFh(k) (h) (47)

ρ Ξℓ (h, Γ )dFh(k) (h) ≥ 1 − ∥ A ( k)∥2 Γ ∈J ℓ=1

  L Ω

where Ξℓ (h, Γ ) , (1 − δLℓ Γ )(1 − ϕˆ ℓ (h)) + δLℓ Γ ϕˆ ℓ (h) and ϕˆ ℓ (h) is as in (33). The optimization problem (47) is not convex. To compute (sub-optimal) solutions, we use the generalized Karush–Kuhn–Tucker (KKT) conditions (Luenberger, 1969), which are necessary conditions for optimality. The Lagrangian is

L(κ1 , κ2 , . . . , κL , ν)



ρ = κℓ (h) − ν Ξℓ (h, Γ ) − 1 − ∥ A ( k)∥2 Γ ∈J ℓ=1 ℓ=1 L 

L 





.

Thus, the generalized KKT conditions become5

 1 − ν fℓ′ (hℓ κℓ (h))hℓ

  L Ω Γ ∈J ℓ=1

Ξj (h, Γ )dFh(k) (h) = 1 −

ν ≥ 0,

5 We write h for δ L h. ℓ ℓ

j̸=ℓ

ρ , ∥A(k)∥2

ν fℓ′ (hℓ κℓ (ν, h))hℓ

   ℓ Ξj (h, Γ ) = 0 (2δL Γ − 1) Γ ∈J

j̸=ℓ

numerically for {κ1 (ν, h), . . . , κL (ν, h)}. Amongst all possible values of ν , we then find a value ν ∗ that satisfies the constraint with equality. A sub-optimal solution to the centralized control design problem (47) is then given by {κ1 (ν ∗ , h), . . . , κL (ν ∗ , h)}. Remark 18. As the optimization problem is not convex, the generalized KKT conditions (48) are necessary but not sufficient conditions for optimality. As an example consider the case of L = 1 channel, J = {1}, and BPSK transmission with b = 16 bits per packet; see Example 1. The generalized KKT conditions give that κ1 (ν, h1 ) should satisfy 1 − ν f1′ (h1 κ1 (ν, h1 ))h1 = 0.

(49)

Now suppose we have ν = 10, and h1 = 1. Then we find that κ1 (ν, h1 ) = 9.713 × 10−7 , κ1 (ν, h1 ) = 0.7438 and κ1 (ν, h1 ) = 4.9324 all satisfy (49). Thus, (49) cannot be solved uniquely.  7. Numerical results To illustrate properties of the power allocation strategies discussed above, we next present some numerical results. 7.1. The two-channel topology Consider the estimation architecture described in Example 16, with L = n = 2 Rayleigh fading channels with λ1 = 1 and λ2 = 2, and BPSK transmission with b = 4 bits; see Example 1. Fig. 2 depicts the average sum power required as a function of ∥A(k)∥2 /ρ for the following control strategies: optimized constant power levels (see Section 6.1), saturated constant gain power control for given saturation values (see Section 6.2), and (sub)optimal centralized control (see Section 6.3). Clearly, centralized power control outperforms the simpler decentralized schemes. Interestingly, for small values of ∥A(k)∥2 /ρ , the saturated gain power controllers (with umax = umax = umax ) perform worse 1 2 than using constant power levels, designed as per (42); for larger values of ∥A(k)∥2 /ρ , the saturated gain controllers perform better. One possible interpretation of this observation could be that the stability condition is easier to be met when ∥A(k)∥2 /ρ is small, whereas doing channel inversion in the saturated gain controller will result in increased power usage. However, as ∥A(k)∥2 /ρ approaches the maximum value where the stability condition can still be satisfied, see (45), the saturated gain power controllers again perform worse, since as Kℓ → ∞, the policy (30) approaches a policy with constant powers {umax , . . . , umax }, but which are not L 1 necessarily the optimal constant power levels. 7.2. Estimation with three sensors

   ℓ (2δL Γ − 1) Ξj (h, Γ ) = 0, Γ ∈J



(45)

where

  Υ1 (k)   Υ (k) =  ...  , ΥL (k)

where ℓ ∈ {1, . . . , L}. For given ν and h, we can solve the system of L equations

(48)

We next consider the topology in Fig. 1 with BPSK transmission and Rayleigh fading channels with λ1 = 1, λ2 = 2 and λ3 = 1.5; see Example 10. Fig. 3 compares the average sum power required under the three power control schemes examined in Section 6. It is easy to see that, if the power levels are allowed to depend upon the actual channel gains, then less sum power is needed to preserve the stability condition given in Theorem 3. The performance of the saturated gain power controller lies between the case with static power levels and the case with channel gain dependent power levels. Note that static power levels can be regarded as a

D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

1313

can significantly outperform the allocation of static powers which depend upon the channel statistics only. Future work could include further investigating optimal power control policies given different information available at the gateway. Also of interest would be the incorporation of source coding effects and allowing for correlated wireless channel models. Preliminary results on the latter have been documented in Quevedo, Ahlén, and Johansson (2011). It is also of interest to consider more general wireless sensor network architectures, including those where the sensors have local processing capability as studied, e.g., in Gupta et al. (2009), Hespanha et al. (2007), Msechu et al. (2008), Østergaard et al. (2010a), Schenato (2008), Shi et al. (2010) and Speranzon, Fischione, Johansson, and Sangiovanni-Vincentelli (2008). Appendix A. Proof of Theorem 3

Fig. 2. Average sum power required to meet the stability condition (19) for the two-channel topology.

By Assumption 2, {ξ }N0 defined in (15) is white. Thus, it follows from (6), (8) and (13) that {C }N0 are independent, so that {P }N0 in (14) is Markovian; see, e.g., Kumar and Varaiya (1986, chapter 2). We next adopt a stochastic Lyapunov function approach with candidate function Vk , trP (k) ≥ 0, ∀k ∈ N0 , to show that there exist β ′ ∈ (0, ∞) and ρ ∈ [0, 1) such that E{Vk+1 |P (k)} ≤ ρ Vk + β ′ ,

∀k ∈ N 0 .

(A.1)

For that purpose, we condition on s(k) and write: E{Vk+1 |P (k)} = E {E{Vk+1 |P (k), s(k)}}

= E{Vk+1 |P (k), s(k) = 1}Pr{s(k) = 1} + E{Vk+1 |P (k), s(k) = 0}Pr{s(k) = 0}.

(A.2)

For s(k) = 1, C (k) has full column-rank and a simple state prediction is x¯ (k + 1) = A(k)(C (k)T C (k))−1 C (k)T y(k). Since Q (k), R(k) and A(k) are assumed bounded, the covariance of x(k + 1)−¯x(k + 1) can be uniformly bounded by WIn , for some W ∈ (0, ∞). Since the Kalman filter gives the minimum conditional estimation error covariance and the expectation and the trace operators commute, it holds that E{Vk+1 |P (k), s(k) = 1} ≤ nW . Thus, E{Vk+1 |P (k), s(k) = 1}Pr{s(k) = 1} ≤ nW . Fig. 3. Average sum power required to meet the stability condition (19) for the topology in Fig. 1.

special case of the saturated gain controller by setting Kℓ = ∞, which explains why the jointly optimized saturated gain power controller will perform better than the case with static power levels. Note that, as the number of bits per packet, b, and ∥A(k)∥2 /ρ increase, the performance gains of using time-varying power levels become more pronounced. For example, in the situation examined, if ∥A(k)∥2 /ρ = 4 and b = 8 bits, then using optimized constant power levels requires more than 50% additional average sum power than using centralized (sub)-optimal power control. 8. Conclusions We have studied stability of a Kalman filter receiving sensor data from a linear time-varying system over block fading wireless channels governed by power control. By the use of stochastic stability methods, we have established conditions on system parameters which ensure that the Kalman filter covariance matrix is exponentially bounded in norm. The stability condition obtained was then used as a constraint to obtain the minimum averaged sum power for different types of power control policies. Numerical studies indicate that channel gain dependent power allocation

(A.3)

If s(k) = 0, then P (k + 1) can be upper bounded by the covariance matrix of the estimation error resulting from the worst case, i.e., where γℓ (k) = 0, for all ℓ ∈ {1, 2, . . . , L}: E{Vk+1 |P (k), s(k) = 0} ≤ E{Vk+1 |P (k), C (k) = 0}

  = tr A(k)P (k)A(k)T + Q (k) ≤ ∥A(k)∥2 Vk + q,

(A.4)

where we have used (14) and Bernstein (2009, Fact 8.12.29) and where q , maxk∈N0 trQ (k). If we now substitute (A.3) and (A.4) into (A.2) and set β ′ = Wn + q, then we obtain E{Vk+1 |P (k)} ≤ Pr{s(k) = 0}∥A(k)∥2 Vk + β ′ and (19) gives (A.1). It now follows from Meyn (1989, Proposition 3.2), that E {Vk |P (0)} ≤ ρ k V0 + β ′ /(1 − ρ). Since P (k) ≽ 0, we have Vk = trP (k) ≥ ∥P (k)∥ for all k ∈ N0 .  Appendix B. Proof of Lemma 5 By (21), the event {s(k) = 1} can be written as the sum of mutually exclusive events:

{s(k) = 1} =

 Γ ∈J

{γ (k) = Γ } .

(B.1)

1314

D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

Now, Assumption 4 and Eq. (6) give that, for all Γ ∈ J,

Φℓ (k) ≥

 Ωℓ ×(Λℓ \Λ+ ℓ)

Pr{γ (k) = Γ | h(k) = h, u(k) = u}

=

L  

 +

1 − δLℓ Γ (1 − fℓ (hℓ uℓ )) + δLℓ Γ fℓ (hℓ uℓ ) ,





Ωℓ ×Λ+ ℓ

ℓ=1

ˆ ℓ (k) =Φ

where hℓ = δLℓ h and uℓ = δLℓ u. Expression (B.1) thus gives: Pr{s(k) = 1 | h(k) = h, u(k) = u}

=

L  

Ωℓ ×Λ+ ℓ

The result follows from substitution of the control policy (8)–(11), Assumption 2 and the law of total probability. 

 Ωℓ ×Λ+ ℓ

≥

Appendix C. Proof of Lemma 9

Ωℓ ×Λ+ ℓ

Ω ×Λ Γ ∈J ℓ=1

− Ωℓ ×Λ+ ℓ

ˆ ℓ (k) =Φ

1 − δLℓ Γ (1 − ϕℓ (ξℓ ))



Λ+ ℓ

−

 + δL Γ ϕℓ (ξℓ ) dFξℓ (k) (ξℓ ) L     = 1 − δLℓ Γ (1 − ϕℓ (ξℓ ))

θℓ (hℓ σℓ (|ϵℓ hℓ |)) dFξℓ (k) (ξℓ )





ℓ

Γ ∈J ℓ=1

(E.1)

fℓ (hℓ κℓ (hℓ )) dFξℓ (k) (ξℓ )

 Since the control laws and channel estimates are decentralized, Eq. (24) of Lemma 5 allows us to write: Pr{s(k) = 1} =

fℓ (hℓ κℓ ((1 + ϵℓ )hℓ )) dFξℓ (k) (ξℓ ).

fℓ (hℓ κℓ ((1 + ϵℓ )hℓ )) dFξℓ (k) (ξ )



L  

dFϵℓ (k)

On the other hand, (D.1) provides that

Γ ∈J ℓ=1



Λℓ \Λ+ ℓ



 1 − δLℓ Γ (1 − fℓ (hℓ uℓ )) + δLℓ Γ fℓ (hℓ uℓ ) .

fℓ (hℓ κℓ ((1 + ϵℓ )hℓ )) dFξℓ (k) (ξℓ )



+



fℓ (hℓ κℓ (hℓ )) dFξℓ (k) (ξℓ )

Ωℓ ×Λ+ ℓ

dFϵℓ (k)

θℓ (hℓ σℓ (ϵℓ hℓ )) dFξℓ (k) (ξℓ ).

(E.2)

Substitution of (E.2) into (E.1) shows that (35) holds with ∆ℓ (k) = ∆′ℓ (k). This proves the result. 

Ωℓ ×Λℓ

 + δLℓ Γ ϕℓ (ξℓ ) dFξℓ (k) (ξℓ ).

Appendix F. Proof of Corollary 14

Use of (27) and (29), thus gives Pr{s(k) = 1} =

L   1 − Φℓ (k), Φℓ (k), Γ ∈J ℓ=1

This proves the result.

Expressions (36) and (38) give that

if δLℓ Γ = 0 if δLℓ Γ = 1.









ˆ (k) − ∆(k) − ψ(Φ ˆ (k)). ˆ (k) ≥ ψ Φ Pr{s(k) = 1} − ψ Φ Now, ψ in (29) is differentiable. Thus, we can use the mean-value theorem to obtain:





Appendix D. Proof of Theorem 12



ˆ (k) ≥ −∇ψ(Φ ˜ (k))∆(k)T , Pr{s(k) = 1} − ψ Φ ˜ (k) = Φ ˆ (k) − c ∆(k), for some c ∈ [0, 1]. Use of (37) and where Φ (38), therefore provides

By Assumption 11, one can bound

  ℓ    fℓ δ hκℓ (h) − fℓ δ ℓ hκℓ ((IL + diag(ϵ))h)  L L   ≤ θℓ δLℓ h |κℓ (h) − κℓ ((IL + diag(ϵ))h)|   ≤ θℓ δLℓ hσℓ (|h − (IL + diag(ϵ))h|)   = θℓ δLℓ hσℓ (|diag(ϵ)h|) ,

Thus, the sufficient condition for exponential boundedness established in Theorem 3 is satisfied if ∃ρ ∈ [0, 1) such that

which directly gives

1 − ρ/∥ ˜ A(k)∥2 − D(k) ≥ 1 − ρ/∥A(k)∥2 ,

  fℓ δLℓ hκℓ ((IL + diag(ϵ))h)     ≥ fℓ δLℓ hκℓ (h) − θℓ δLℓ hσℓ (|diag(ϵ)h|) .

Pr{s(k) = 1} ≥ 1 −

ρ˜ − D(k). ∥A(k)∥2

which is equivalent to (39). (D.1)

The bound (35) now follows from integrating (D.1) with respect to  dFξ (k) and noting that Λ dFϵ(k) = 1. For decentralized control policies, (36) follows directly from (35) upon noting that Pr{s(k) = 1} is minimized when Pr{γℓ (k) = 1} are at their lowest values.  Appendix E. Proof of Lemma 13 For non-increasing decentralized control policies κℓ and since fℓ are non-decreasing, fℓ (hℓ κℓ ((1 + ϵℓ )hℓ )) ≥ fℓ (hℓ κℓ (hℓ )), ∀ϵℓ ≤ 0, i.e., underestimating the channel gains does not reduce the success probabilities. Thus,



Appendix G. Proof of Lemma 17 Since the objective function is linear, (42) is convex if each of ˆ ℓ (k) are log-concave functions of uℓ . By properties the functions Φ of log-concave functions (see, e.g., Boyd and Vandenberghe (2004, p. 106)), a sufficient condition for this is that both the functions f˜ℓ (h, u) , fℓ (hu) and the derivatives of Fhℓ (k) (h) are log-concave functions in (h, u). For BPSK transmission with Rayleigh fading as presented in Example 1, the log-concavity of the derivative of Fhℓ (k) (h), namely λℓ e−λℓ h is obvious. Since log-concave functions are closed under multiplication, it suffices to show that g (h, u) , √

√  (1/ 2π ) 0

hu

2 e−t /2 dt is log-concave in (h, u) for h, u ∈ R≥0 . Now

D.E. Quevedo et al. / Automatica 48 (2012) 1306–1316

the second order condition for log-concavity says that a function is log-concave if and only if

∇ g ∇ g T − g ∇ 2g   2 ∂ 2g ∂g  ∂ h − g ∂ h2  =  ∂g ∂g ∂ 2g −g ∂u ∂h ∂ u∂ h

 ∂g ∂g ∂ 2g −g ∂h ∂u ∂ h∂ u    ≥ 0.  2 2  ∂g ∂ g −g 2 ∂u ∂u

After some computations, we can find that



∂g ∂h

2 −g

∂ 2g e−hu u2 = ∂ h2 16π (hu)3/2  √ √ × 2 hu + ehu/2 2π (1 + hu)     × 1 + erf hu/2 ≥ 0, √

where erf(x) , (2/ π )

x 0

2

e−t dt and

     2 ∂g 2 ∂ 2g ∂ 2g ∂g ∂g ∂ 2g −g 2 −g 2 − −g ∂h ∂u ∂u ∂h ∂u ∂ h∂ u    √ e−3hu/2 1 + erf hu/2 = 2 32π hu    √ hu/2 × e π hu + 2π hu + ehu/2 π hu erf hu/2 ≥ 0.



∂g ∂h

2

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Daniel E. Quevedo received Ingeniero Civil Electrónico and Magister en Ingeniería Electrónica degrees from the Universidad Técnica Federico Santa María, Valparaíso, Chile in 2000. In 2005, he received the Ph.D. degree from The University of Newcastle, Australia, where he is currently a research academic. He has been a visiting researcher at various institutions, including Uppsala University, Sweden, KTH Stockholm, Sweden, Aalborg University, Denmark, Kyoto University, Japan, and INRIA Grenoble, France. Dr. Quevedo 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 an Australian Research Fellowship. His research interests cover several areas of automatic control, signal processing, communications, and power electronics.

Anders Ahlén is a full Professor and holds the chair in Signal Processing at Uppsala University where he is also the head of the Signals and Systems Division of The Department of Engineering Sciences. He was born in Kalmar, Sweden, and received the Ph.D. degree in Automatic Control from Uppsala University. He was with the Systems and Control Group, Uppsala University from 1984 to 1992 as an Assistant and Associate Professor in Automatic Control. During 1991 he was a visiting researcher at the Department of Electrical and Computer Engineering, The University of Newcastle, Australia. He was a visiting professor at the same university in 2008. In 1992 he was appointed Associate Professor of Signal Processing at Uppsala University. During 2001–2004 he was the CEO of Dirac Research AB, a company offering state-of-the-art audio signal processing solutions. He is currently the chairman of the board of the same company. Since 2007 he has been a member of the Uppsala VINN Excellence Center for Wireless Sensor Networks, WISENET. His research interests, which include Signal Processing, Communications and Control, are currently focused on Signal Processing for Wireless Communications, Wireless Systems Beyond 3G, Wireless Sensor Networks, Wireless Control, and Audio Signal Processing.

From 1998 to 2004 he was the Editor of Signal and Modulation Design for the IEEE Transactions on Communications.

Alex S. Leong was born in Macau in 1980. He received the B.S. degree in Mathematics and B.E. degree in Electrical Engineering in 2003, and the Ph.D. degree in Electrical Engineering in 2008, all from the University of Melbourne, Parkville, Australia. He is currently a research fellow in the Department of Electrical and Electronic Engineering, University of Melbourne. His research interests include statistical signal processing, signal processing for sensor networks, and networked control systems. Dr. Leong was the recipient of the L. R. East Medal from Engineers Australia in 2003, and an Australian Postdoctoral Fellowship from the Australian Research Council in 2009.

Subhrakanti Dey was born in Calcutta, India, in 1968. He received the B.Tech. and M.Tech. degrees from the Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur, India, in 1991 and 1993, respectively, and the Ph.D. degree from the Department of Systems Engineering, Research School of Information Sciences and Engineering, Australian National University, Canberra, Australia, in 1996. He has been with the Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Australia, since February 2000, where he is currently a full Professor. From September 1995 to September 1997 and September 1998 to February 2000, he was a postdoctoral Research Fellow with the Department of Systems Engineering, Australian National University. From September 1997 to September 1998, he was a postdoctoral Research Associate with the Institute for Systems Research, University of Maryland, College Park. His current research interests include networked control systems, wireless communications and networks, signal processing for sensor networks, and stochastic and adaptive estimation and control. Prof. Dey currently serves on the Editorial Board of Elsevier Systems and Control Letters. He was also an Associate Editor for the IEEE Transactions on Signal Processing and the IEEE Transactions on Automatic Control. He is a Senior Member of IEEE.