Data-Driven Power Control for State Estimation: A Bayesian Inference Approach
Junfeng Wu, Yuzhe Li, Daniel E. Quevedo, Vincent Lau, Ling Shi
DData-Driven Power Control for State Estimation: A BayesianInference Approach (cid:63)
Junfeng Wu a , Yuzhe Li a , Daniel E. Quevedo b , Vincent Lau a , Ling Shi a a Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong. b School of Electronic Engineering and Computer Science, University of Newcastle, Australia.
Abstract
We consider sensor transmission power control for state estimation, using a Bayesian inference approach. A sensor node sendsits local state estimate to a remote estimator over an unreliable wireless communication channel with random data packetdrops. As related to packet dropout rate, transmission power is chosen by the sensor based on the relative importance ofthe local state estimate. The proposed power controller is proved to preserve Gaussianity of local estimate innovation, whichenables us to obtain a closed-form solution of the expected state estimation error covariance. Comparisons with alternativenon-data-driven controllers demonstrate performance improvement using our approach.
Key words:
Kalman filtering, Transmission power control, State estimation, Packet losses, Bayesian inference
Wireless networked systems have a wide spectrum of ap-plications in smart grid, environment monitoring, intel-ligent transportation, etc. State estimation is a key en-abling technology where the sensor(s) and the estimatorcommunicate over a wireless network. Energy conserva-tion is a crucial issue as most wireless sensors use on-board batteries which are difficult to replace and typ-ically are expected to work for years without replace-ment. Thus power control becomes crucial. In this work,we consider sensor transmission power control for re-mote state estimation over a packet-dropping network.Transmission power control in state estimation scenariohas been considered from different perspectives. Someworks took transmission costs as constant. Shi et al. [1]assumed sensors to have two energy modes, allowing itto send data to a remote estimator over an unreliablechannel either using a high or low transmission powerlevel. The optimal power controller is to minimize the (cid:63)
The work of J. Wu, Y. Li and L. Shi is supported by a HKRGC GRF grant 618612.
Email addresses: [email protected] (Junfeng Wu), [email protected] (Yuzhe Li), [email protected] (Daniel E.Quevedo), [email protected] (Vincent Lau), [email protected] (Ling Shi). expected terminal estimation error at the remote esti-mator subject to an energy constraint. Similar works canalso be found in [2, 3]. Meanwhile, some literature hastaken channel conditions into account. Quevedo et al. [4]studied state estimation over fading channels. They pro-posed a predictive control algorithm, where power andcookbooks are determined in an online fashion basedon the undergoing estimation error covariance and thechannel gain predictions. More related works can beenseen in [5–7].An important issue which has not been taken seriously inmost works is that the transmission power assignment,as a tool to control the accessibility of information to thereceiver, should be determined not only by the underly-ing channel condition and the desired estimation perfor-mance, but also by the transmitted information itself.In [4] and [5], the authors failed to associate transmis-sion power with data to be sent. The plant states areused to determine the transmission power in [8]. In thiscase, lost packets signal the receiver of the state infor-mation. To avoid computation difficulty, the signalinginformation is discarded.In this paper, we focus on how to adapt the transmis-sion power to the measurements of plant state and howto exploit information contained in the lost packets. Wepropose a data-driven power controller, which utilizes
Preprint submitted to Automatica 4 October 2018 a r X i v : . [ c s . S Y ] M a r ifferent transmission power levels to send the local esti-mate according to a quadratic function of a key param-eter called “incremental innovation” which is evaluatedby the sensor at each time slot. By doing this, even whendata dropouts occur, the remote estimator can utilizethe additional signaling information to refine the pos-terior probability density of the estimation error by aBayesian inference technique (see [9]), therefore derivingthe MMSE estimate. It compensates the deteriorated es-timation performance caused by packet losses. To facili-tate analysis, we assume that a baseline power controllerhas already been established based on different factorswith regard to different settings, such as the requirementof estimation performance as in [1] or the channel condi-tions as in [4,5,7]. We are devoted to developing a powercontroller that embellishes this baseline controller byadapting the transmission power to the measurementssuch that the averaged power with respect to all possiblevalues taken by the measurements does not exceed thatof the baseline power controller. The proposed powercontroller, driven by online measurements, can run ontop of non-data-driven power controllers, which resultsin hierarchical power control mechanisms. Then exten-sion to a time-varying power baseline is established inSection 4.4. Note that a related controller was first pro-posed in [10], but as a special case of the controller inthis work. The main contributions of the present workare summarized as follows.(1) We propose a data-driven power control strat-egy for state estimation with packet losses, whichadapts the transmission power to the measuredplant states.(2) We prove that the proposed power controller pre-serves Gaussianity of the local innovation. It sim-plifies derivation of the MMSE estimate and leadsto a closed-form expression of the expected stateestimation error covariance.(3) We present a tuning method for parameter design.Despite of its sub-optimality, the controller is shownto perform not worse than an alternative non-data-driven one.The remainder of this paper is organized as follows.In Sections 2 and 3, we give mathematical models ofthe considered system and introduce the data-driventransmission power controller. In Section 4, we presentthe MMSE estimate at the remote estimator and asub-optimal power controller that minimizes an upperbound of the remote estimation error. In Section 5, com-parisons with alternative non-data-driven controllersdemonstrate performance improvement using our ap-proach. Section 6 presents concluding remarks. Notation : N (and N + ) is the set of nonnegative (andpositive) integers. S n + is the cone of n by n positivesemi-definite matrices. For a matrix X , λ i ( X ) is the i thsmallest nonzero eigenvalue. We abuse notations det( X )and X − , which are used, in case of a singular matrix X , to denote the pseudo-determinant and the Moore- Penrose pseudoinverse. δ ij is the Dirac delta function,i.e., δ ij equals to 1 when i = j and 0 otherwise. The no-tation pdf( x , x ) represents the probability density func-tion (pdf) of a random variable x taking value at x . Consider a linear time-invariant (LTI) system: x k +1 = Ax k + w k , (1) y k = Cx k + v k , (2)where k ∈ N , x k ∈ R n is the system state vector at time k , y k ∈ R m is the measurement obtained by the sensor,the state noise w k ∈ R n and observation noise v k ∈ R m are zero-mean i.i.d. Gaussian noises with E [ w k w (cid:48) j ] = δ kj Q ( Q (cid:23) E [ v k ( v j ) (cid:48) ] = δ kj R ( R (cid:31) E [ w k ( v j ) (cid:48) ] =0 ∀ j, k ∈ N . The initial state x is a zero-mean Gaus-sian random vector with covariance Π (cid:23) w k and v k . ( A, C ) is assumed to be de-tectable and (
A, Q / ) is assumed to be stabilizable. Fur-thermore, we assume A is Hurtwitz. Sensing Unit Preprocessor Remote Estimator Dynamic Tx Power Controller Process Wireless Channel RF Unit Sensor
Fig. 1. The system architecture.
Hovareshti et al. [11] illustrated that utilization of thecomputation capabilities of wireless sensors may im-prove the system performance significantly. Equippedwith such “smart sensors”, the sensor locally runs aKalman filter to produce the MMSE estimate ˆ x sk of thestate x k based on all the measurements collected upto time k , i.e., y k (cid:44) { y , ..., y k } , and then transmitsits local estimate to the remote estimator. Denote thesensor’s local MMSE state estimate, the corresponding Since we focus on remote state estimation in this paper,for any practically working systems (to be monitored alone), A has to be Hurwitz. Otherwise, the system state will gounbounded and there is no real sensing device which cantrack an unbounded state trajectory. Adding a control inputto regulate the system state for an unstable A and studyingits associated stability issue will be beyond the scope of thispaper and will be left as our future work. x sk , e sk and P sk ,respectively, i.e., ˆ x sk (cid:44) E [ x k | y k ], e sk (cid:44) x k − ˆ x sk and P sk (cid:44) E [( x k − ˆ x sk )( x k − ˆ x sk ) (cid:48) | y k ]. Standard Kalmanfiltering analysis suggests that these quantities can becalculated recursively (cf., [12]), where the recursionstarts from ˆ x s = 0 and P s = Π (cid:23)
0. Since P sk convergesto a steady-state value exponentially fast (cf., [12]), weassume that the sensor’s local Kalman filter has enteredthe steady state, that is, P sk = P (cid:23) ∀ k ∈ N , This as-sumption simplifies our subsequent analysis and results,such as Theorem 4.8 and Proposition 4.17. The data are sent to the remote estimator over an Addi-tive White Gaussian Noise (AWGN) channel using theQuadrature Amplitude Modulation (QAM) whereby ˆ x sk is quantized into K bits and mapped to one of 2 K avail-able QAM symbols. For simplicity, the following as-sumptions are made:A.1: The channel noise is independent of w k and v k .A.2: K is large enough so that quantization effect is neg-ligible when analyzing the performance of the re-mote estimator.A.3: The remote estimator can detect symbol errors .Only the data arriving error-free are regarded asbeing successfully received; otherwise they are re-garded as dropout.These assumptions are commonly used in communica-tion and control theories (cf., [4, 5, 8, 13, 14]). For ex-ample, Fu and Souza [14] demonstrated that the esti-mation quality improvement (in terms of reduction ofthe remote estimation error) achieved by increasing thenumber K of the quantization bits is marginal when K is sufficiently large (in their example K only needs tobe greater or equal to 4. Based on A.3, the communica-tion channel can be characterized by a random process { γ k } k ∈ N + , where γ k = (cid:26) , if ˆ x sk arrives error-free at time k ,0 , otherwise,initialized with γ = 1. Denote γ k (cid:44) { γ , . . . , γ k } . Let ω k ∈ [0 , + ∞ ) be the transmission power for the QAMsymbol at time k . We adopt the wireless communicationchannel model used in [10], and have Pr ( γ k = 0 | ω k ) = q ω k , where q is given by q (cid:44) exp( − α/ ( N W )) ∈ (0 , ,N is the AWGN noise power spectral density, W is thechannel bandwidth, and α ∈ (0 ,
1] is a constant that de-pends on the specific modulation being used. To send lo-cal estimates to the remote estimator, the sensor chooses QAM is a common modulation scheme widely used inIEEE 802.11g/n as well as 3G and LTE systems, due to itshigh bandwidth efficiency. In practice, symbol errors can be detected via a cyclicredundancy check (CRC) code. from a continuum of available power levels ω k (cid:62)
0, seeFig. 1. Note that different power levels lead to differentdropout rates, thereby affecting estimation performance.
Define I k as the information available to the remote es-timator up to time k , i.e.,I k = { γ ˆ x s , γ ˆ x s , ..., γ k ˆ x sk } ∪ { γ k } . (3)Denote ˆ x k and P k as the remote estimator’s own MMSEstate estimate and the corresponding estimation errorcovariance, i.e., ˆ x k (cid:44) E [ x k | I k ] and P k (cid:44) E [( x k − ˆ x k )( x k − ˆ x k ) (cid:48) | I k ], where expectations are taken with respect to afixed power controller. We assume that the remote esti-mator feedbacks acknowledgements γ k before time k +1.Such setups are common especially when the remote es-timator (gateway) is an energy-abundant device. Thisenergy asymmetry allows the estimator to trade energycost for estimation accuracy. Our strategy uses the measurements to assign transmis-sion power level efficiently. As focusing on how the powercontroller utilize the sensor’s real-time data, to simplifydiscussion, we assume a constant power baseline ¯ ω inthis section. We define θ (cid:44) { θ k } k ∈ N + as a transmissionpower controller over the entire time horizon, where θ k is a mapping from y k and γ k to ω k . Before proceedingto study θ , let us first briefly explain the idea of data-driven power control mechanism. Define τ ( k ) ∈ N + asthe holding time since the most recent time when theremote estimator received the data from the sensor, i.e., τ ( k ) (cid:44) k − max (cid:54) t (cid:54) k − { t : γ t = 1 } . (4)We interchange τ ( k ) with τ when the underlying timeindex is clear from the context. Define ε k as the incre-mental innovation in the sensor local state estimate com-pared to time k − τ , the previous reception instant, i.e., ε k = ˆ x sk − A τ ˆ x sk − τ . (5) Lemma 3.1 E [ e sk ε (cid:48) k | I k − , γ k = 0] = 0 ∀ k ∈ N + . Proof:
The result follows from noting that E [ e sk ε (cid:48) k | I k − , γ k = 0] = E [ E [ e sk ε (cid:48) k | y k , γ k ] | I k − , γ k = 0]= E [ E [ e sk | y k ] ε (cid:48) k | I k − , γ k = 0] = 0 , where the second equality holds because e sk is independentof γ k , and the last equality holds since E [ e sk | y k ] = 0 . ε k = 0, then the sensor generates a lo-cal estimate, ˆ x sk identical to the prediction A τ ˆ x sk − τ . Wewould say that, for the remote estimator, the “value”of information contained in ˆ x sk is null. As ε k becomeslarger, ˆ x sk has an increasing drift from the prediction A τ ˆ x sk − τ and the importance of the sensor sending ˆ x sk thereby raises. Motivated by these observations, we de-fine a stationary power controller, θ ef : ε k → ω k , asan increasing function of ε k . To fit the above observa-tions, we introduce a quadratic function of ε k given by C ( ε k , Q ) (cid:44) ε k (cid:48) Q ε k , where Q ∈ S n + is a weight matrix.According to Lemma 3.1, the covariance of ε k is a func-tion of τ ( k ). Therefore we specify τ ( k ) for the index of Q and construct the following controller: θ ef : { ω k = N W α C ( ε k , Q τ ) + ω } . (6)In contrast to (6), most non-data-driven transmissionpower controllers (i.e., [4,5]) use a given power ¯ ω regard-less of what value ε k takes. Note that in (6) a constantterm ω is added after C ( ε k , Q τ ). If one sets Q τ = 0,then the transmission with the baseline power controller ω = ¯ ω is a special case of the proposed transmissionpower controller. As for Q τ (cid:54) = 0, the transmission poweris a constant ω if C ( ε k Q τ ) = 0; otherwise it is adaptedaccording to C ( ε k , Q τ ). Compared with a related con-troller proposed earlier in [10], θ ef in (6) is more gen-eral at least from two aspects: 1) we introduce a weightmatrix Q τ to highlight the roles of different entries of ε k ; 2) it allows the sensor to transmit using a stan-dard power ω even if C ( ε k , Q τ ) = 0, which includes anon-data-driven power transmission as a special case.As shown later in Lemma 4.4, given I k − , ε k is zero-mean Gaussian with a covariance Σ τ depending on τ ( k ),i.e., ( ε k | I k − ) ∼ N (0 , Σ τ ) . For convenience of our subse-quent analysis, we define a new parameter Ψ τ satisfyingΨ τ (cid:44) (cid:0) Q k + Σ − τ (cid:1) − , where Σ τ (cid:23) Ψ τ (cid:23)
0. We now listthe main problems considered in the remainder of thiswork,(1) Under θ ef defined in (6), what is the MMSE es-timate and its associated estimation error covari-ance?(2) What value should Q τ (or Ψ τ ) take in order tominimize, E [ P k ], the expected estimation error atthe remote estimator?The solution to the first problem is presented in Sec-tion 4.2. A sub-optimal solution to the second one isgiven in Section 4.3 in view of the difficulty of the opti-mization problem.Before proceeding, we note that in previous works suchas [8] the difficulty of using the information containedin lost packets, i.e., γ k = 0, when computing the MMSEestimate of the plant state has been acknowledged. Onetypically discards such information as was done in [8] or resorts to approximations, e.g., treating a truncatedGaussian distribution as a Gaussian distribution as wasdone in [15]. These approaches either lead to conserva-tive results (due to the unutilized information) or inac-curate results (due to approximations). Our method, onother hand, makes use of the information contained inthe event γ k = 0 to improve the estimation performance.The associated MMSE estimate, relying on no approxi-mation techniques, is derived in a closed-form. For any Σ (cid:23)
U, D ∈ R n × n such that Σ = U DU (cid:48) , where U is uni-tary, whose columns are right eigenvectors of Σ, and D (cid:44) (cid:34) ∆ 00 0 (cid:35) , where ∆ is a diagonal matrix generated bythe corresponding nonzero eigenvalues of Σ. Let Σ / (cid:44) U √ D . Then Σ = Σ / (cid:0) Σ / (cid:1) (cid:48) . Generally speaking, an n -dimensioned random vector x ∼ N ( µ, Σ), does not have a pdf with respect to theLebesgue measure on R n if some entries in x degener-ate to almost surely constant random variables. To workwith such vectors, one can instead consider Lebesguemeasure in the rank(Σ)-dimension affine subspace: Ω (cid:44) { µ + Σ / z : z ∈ R n } , with respect to which x has apdf pdf( x , x ) = √ σ exp (cid:0) − ( x − µ ) (cid:48) Σ − ( x − µ ) (cid:1) , where σ = (2 π ) rank(Σ) det(Σ). Without loss of generality, inthe remainder of this paper, for a random variable x ∼N (0 , Σ) with a singular Σ, the pdf of x means the prob-ability density on Ω. Note that the Moore-Penrose pseu-doinverse of Σ is unique and given byΣ − = U (cid:34) ∆ −
00 0 (cid:35) U (cid:48) , (7)and that the pseudo-determinant of Σ equals to the prod-uct of all nonzero eigenvalues of Σ.Consider the power control law θ ef defined in (6). In or-der to guarantee that ω k is always nonnegative for anyvalue ε k , the difference of Ψ − τ and Σ − τ needs to beat least positive semi-definite, i.e., two conditions mustbe simultaneously satisfied, which are Σ τ (cid:23) Ψ τ andΨ − τ (cid:23) Σ − τ . The following lemma provides a necessarycondition that Ψ τ needs to satisfy. Lemma 4.1
Suppose Σ and Ψ satisfy Σ (cid:23) Ψ and Ψ − (cid:23) Σ − . Then rank(Ψ) = rank(Σ) (8) and Im(Σ / ) = Im(Ψ / ) , (9)4 here Im( X ) is the image of X . Proof:
Since Σ (cid:23) Ψ , it is true that rank(Σ) ≥ rank(Ψ) .To verify (8) , suppose that rank(Σ) > rank(Ψ) . Thenfrom (7), rank(Σ − ) > rank(Ψ − ) , which contradictswith Ψ − (cid:23) Σ − . To prove (9) , let us denote rank(Ψ) (cid:44) r and assume there is a set of vectors W (cid:44) { w , . . . , w r } such that Im(Ψ / ) = span ( { w , . . . , w r } ) . Suppose
Im(Σ / ) (cid:54) = Im(Ψ / ) . Then there exists a vector in W (without loss of generality, let it be w ), and a vector w ∈ Ker (cid:0) (Σ / ) (cid:48) (cid:1) where the operator Ker( X ) is thekernel of a matrix X , such that w (cid:48) w (cid:54) = 0 . It leads tothe fact that w (cid:54)∈ Ker (cid:0) (Ψ / ) (cid:48) (cid:1) . We in turn have w (cid:48) Σ / (cid:16) Σ / (cid:17) (cid:48) w = 0 while w (cid:48) Ψ / (cid:16) Ψ / (cid:17) (cid:48) w > , which contradicts with Σ (cid:23) Ψ . For convenience, denote n τ (cid:44) rank(Σ τ ) = rank(Ψ τ ),Ω τ (cid:44) Im(Σ / τ ) = Im(Ψ / τ ) and Φ τ (cid:44) (cid:16) Σ / τ (cid:17) (cid:48) Ψ − τ Σ / τ . One has next lemma, the proof provided in the Ap-pendix.
Lemma 4.2
The rank of Φ τ equals that of Σ τ (or Ψ τ ),i.e., rank(Φ τ ) = n τ . Example 4.3
Two matrices are provided below as asimple example for n = 3 , Σ τ = , and Ψ τ = − − . We can verify that n τ = 2 , Σ τ (cid:23) Ψ τ , Ψ − τ (cid:23) Σ − τ , (8),and Lemma 4.1 holds.4.2 MMSE State Estimate In general, the posterior distribution of ε k fails tomaintain Gaussianity without analog-amplitude obser-vations. The defect is especially common for quantizedKalman filtering and Gaussian filters, where it is tack-led by Gaussian approximation [12, 16, 17]. By contrast,the following lemma shows that, using θ ef in (6), thedistribution of ε k conditioned on I k − , γ k = 0 is Gaus-sian. The proof, similar to that of Lemma 3.5 in [10], isomitted. Lemma 4.4
Under θ ef defined in (6) , given I k − , ε k follows a Gaussian distribution: ( ε k | I k − ) ∼ N (0 , Σ τ ) , where Σ τ is given by the following recursion: Σ τ = A Ψ τ − A (cid:48) + (cid:0) h ( P ) − P (cid:1) , (10) with Ψ = 0 . It is also true that, given γ k = 0 and I k − , ( ε k | I k − , γ k = 0) ∼ N (0 , Ψ τ ) . Proposition 4.5
Under θ ef defined in (6) , given I k − , the packet drop rate at time k is given by Pr( γ k = 0 | I k − ) = √ det(Σ τ )det(Ψ − τ ) exp (cid:16) − αN W ω (cid:17) . We denote the packet arrival rate as p τ (cid:44) − Pr( γ k =0 | I k − ) , where the subscript τ is to emphasize that itdepends on Σ τ and Ψ τ . To ensure that the averagedtransmission power with respect to different valuestaken by the measurement in θ does not exceed ¯ ω , i.e., E [ ω k | I k − ] ≤ ¯ ω , we require the following result. Lemma 4.6
Under θ ef (6) , given I k − , the relation be-tween E [ ω k | I k − ] and Ψ τ , and ω is given by E [ ω k | I k − ] = N W α (cid:0) Tr(Σ τ Ψ − τ ) − n τ (cid:1) + ω. (11) Proof:
From Lemma 4.4, we know that ( ε k | I k − ) ∼N (0 , Σ τ ) . Under θ ef , we have: E [ ω k | I k − ] = E [ E [ ω k | ε k ] | I k − ]= N W α E (cid:2) ε (cid:48) k (cid:0) Ψ − τ − Σ − τ (cid:1) ε k (cid:12)(cid:12) I k − ] + ω = N W α Tr (cid:0) E [ ε k ε (cid:48) k | I k − ] (Ψ − τ − Σ − τ ) (cid:1) + ω = N W α (cid:0) Tr(Σ τ Ψ − τ ) − n τ (cid:1) + ω. With θ ef defined in (6), the remote estimator computes x k and P k according to the following two theorems. Theorem 4.7
Under θ ef (6) , the remote estimator com-putes ˆ x k as ˆ x k = (cid:40) ˆ x sk , if γ k = 1 ,A τ ˆ x sk − τ , if γ k = 0 , (12) where ˆ x sk is updated as ˆ x sk = A τ ˆ x sk − τ + ε k when γ k = 1 . Proof:
When γ k = 1 , the result is straightforward since ˆ x sk is the MMSE estimate of x k given y k . Now consider γ k = 0 . The tower rule gives E [ x k | I k − , γ k = 0] = E [ E [ x k | y k , γ k ] | I k − , γ k = 0]= E (cid:2) A τ ˆ x sk − τ + ε k | I k − , γ k = 0 (cid:3) = A τ ˆ x sk − τ + E [ ε k | I k − , γ k = 0] . Lemma 4.4 leads to E [ ε k | I k − , γ k = 0] = 0 . heorem 4.8 Under θ ef (6) , P k at the remote estimatoris updated as P k = (cid:40) P , if γ k = 1 ,P + Ψ τ , if γ k = 0 . (13) Proof:
When γ k = 1 the result is straightforward. Weonly prove the case when γ k = 0 . E [( x k − ˆ x k )( x k − ˆ x k ) (cid:48) | I k − , γ k = 0]= E (cid:2) ( x k − A τ ˆ x sk − τ )( x k − A τ ˆ x sk − τ ) (cid:48) | I k − , γ k = 0 (cid:3) = E [ E [( e sk + ε k )( · ) (cid:48) | y k , γ k ] | I k − , γ k = 0]= E [( e sk )( e sk ) (cid:48) | y k ] + E [( ε k )( ε k ) (cid:48) | I k − , γ k = 0]= P + Ψ τ , where the third equality is due to Lemma 3.1 and the lastone is from Lemma 4.4. Remark 4.9
Under a baseline power controller with aconstant power control ¯ ω , the remote estimator’s esti-mate still obeys the recursion (12) ; however, the estima-tion error covariance is updated differently: P k = P when γ k = 1 , and P k = h ( P k − ) = Σ τ when γ k = 0 ). Notethat although the obtained estimates under the two powercontrollers are the same, their different estimation er-ror covariance matrices suggest different confident levelswith which the remote estimator trusts the obtained es-timate: with the data-driven power controller, it is moreconvinced that the obtained estimate is close to the realstate while less convinced with a non-data-driven powercontroller.4.3 Selection of Design Parameters The performances of θ ef for different Ψ τ ’s are difficult tocompare in general. However, for Σ τ and Ψ τ , there mustexist a real number (cid:15) τ ∈ (0 ,
1] such that Ψ τ (cid:22) (cid:15) τ Σ τ andΨ τ (cid:54)(cid:22) (cid:15) Σ τ , ∀ (cid:15) < (cid:15) τ . Observe thatΦ τ = (cid:16) Σ / τ (cid:17) (cid:48) Ψ − τ Σ / τ (cid:23) (cid:15) τ (cid:34) I n τ
00 0 (cid:35) , which yields (cid:15) τ = λ (Φ τ ) . In light of (10), we furtherhave Ψ τ (cid:22) (cid:15) τ Σ τ = (cid:15) τ ( A Ψ τ − A (cid:48) + Σ ) . According toProposition 4.5, it can be seen given τ ( k ) = τ that E [ P k | τ ( k ) = τ ] has an upper bound: E [ P k | τ ( k ) = τ ] (cid:22) P + (1 − p τ ) (cid:15) τ ( A Ψ τ − A (cid:48) + Σ ) . Instead of minimizing E [ P k ], we minimize its upper bound which is equivalentto minimize (1 − p τ ) (cid:15) τ . Iterating over time, one eventu-ally needs to minimize (1 − p τ ) (cid:15) τ for any τ ( k ) ∈ N + atany k ∈ N + . To this end, we propose to assign param-eters of θ ef in (6) as the solution to the following opti-mization problem: Problem 4.10 min Ψ τ , Σ τ ,ω (cid:0) det(Σ τ )det(Ψ − τ ) (cid:1) / λ (Φ τ ) exp (cid:20) − αN W ω (cid:21) , s . t . N W α (cid:0) Tr(Σ τ Ψ − τ ) − n τ (cid:1) + ω ≤ ¯ ω. The constraint is imposed by (11). To solve Prob-lem 4.10, we first note that Tr(Σ τ Ψ − τ ) = Tr(Φ τ ) . However, for any matrix
X, Y ∈ R n × n , det( XY ) =det( X )det( Y ) does not hold in general since det( X )means X ’s pseudo-determinant (in case X is singular).Fortunately, this property still holds for Σ τ and Ψ − τ .The proof is given in the Appendix. Lemma 4.11
Suppose Σ τ and Ψ τ satisfy Σ τ (cid:23) Ψ τ (cid:23) and Ψ − τ (cid:23) Σ − τ . Then det(Σ τ )det(Ψ − τ ) = det(Φ τ ) . From linear algebra, det(Φ τ ) = (cid:81) n τ i =1 λ i (Φ τ ), andTr(Φ τ ) = (cid:80) n τ i =1 λ i (Φ τ ) . We simply write λ i (Φ τ ) as λ i ( τ ), and denote the nonzero eigenvalues of Φ τ byΛ τ (cid:44) [ λ ( τ ) , . . . , λ n τ ( τ ) ]. Then Problem 4.10 can berecast as Problem 4.12 min Λ τ ,ω λ ( τ ) (cid:81) n τ i =1 λ i ( τ ) / exp (cid:20) − αN W ω (cid:21) , (14)s . t . N W α (cid:34) n τ (cid:88) i =1 λ i ( τ ) − n τ (cid:35) + ω = ¯ ω, ω ≥ ≤ λ ( τ ) ≤ λ j ( τ ) , ∀ j = 2 , . . . , n τ . Lemma 4.13
Let Λ ∗ τ be the optimal solution to Prob-lem 4.12. Then Λ ∗ τ satisfies λ ( τ ) ∗ = λ ( τ ) ∗ = · · · = λ n τ ( τ ) ∗ . (15) Proof:
Suppose that Λ is the optimal solution to Prob-lem 4.12 but does not satisfy (15) . We will show thatthere must exist another vector, which is different from Λ and has a smaller cost function (14) . Let (cid:80) n τ i =1 λ i = c where c is a positive constant. Due to the fact that λ in Λ is the minimum eigenvalue of Φ τ and the inequal-ity of arithmetic and geometric means, we have λ ≤ cn τ and (cid:81) n τ i =1 λ i ≤ (cid:16) cn τ (cid:17) n τ , the equalities simultaneouslysatisfied when λ i = cn τ , ∀ i = 1 , . . . , n τ . Thus, Λ =[ cn τ , . . . , cn τ ] results in a smaller value of (14) , which con-tradicts with the assumption and completes the proof. The following lemma is a result of Lemma 4.13. Its proofis presented in the Appendix.6 emma 4.14 If ¯ ω > N Wα , then the optimal solution toProblem 4.12 is ω = ¯ ω − N Wα and Λ ∗ τ = [1 + 2 n τ , . . . , n τ ] . (16) Otherwise, if ¯ ω ≤ N Wα , the optimizer is ω = 0 and Λ ∗ τ = [1 + 2 α ¯ ωn τ N W , . . . , α ¯ ωn τ N W ] . (17)Denote by θ ∗ ef the transmission power associated withthe solution to Problem 4.12. Then we have the followingtheorem. It can be readily verified from Lemma 4.14. Theorem 4.15 If ¯ ω > N Wα , then θ ∗ ef is given by θ ∗ ef : { ω k = N Wαn τ ε (cid:48) k Σ − τ ε k + ¯ ω − N Wα } , where Σ τ +1 = n τ n τ +2 A Σ τ A (cid:48) + h ( P ) − P with Σ = 0 .Otherwise, if ¯ ω ≤ N Wα , θ ∗ ef is given by θ ∗ ef : { ω k = ¯ ωn τ ε (cid:48) k Σ − τ ε k } , where Σ τ +1 = n τ N Wn τ N W +2 α ¯ ω A Σ τ A (cid:48) + h ( P ) − P . Remark 4.16
A non-data-driven baseline power con-troller with a constant power level ¯ ω is feasible to Prob-lem 4.10. Since θ ∗ ef is the optimal solution, it has notworse state estimation performance compared with thealternative non-data-driven power controller. Numeri-cal examples in Section 5 demonstrate performance im-provements using θ ∗ ef compared with the non-data-drivenpower controller. The following proposition shows that the rank of Σ τ canbe calculated offline. The proof is given in the Appendix. Proposition 4.17
Consider the θ ∗ ef given in Theo-rem 4.15, for any τ ∈ N + , n τ can be calculated as: n τ = rank( h τ ( P ) − P ) . In particular, when τ ≥ n , thedimension of x , n τ becomes a constant which is givenby: n τ = rank( h n ( P ) − P ) , ∀ τ ≥ n. In many cases, the base-line power controller changesover time with respect to different settings. For example,in [4], block fading channels were taken into account. Todeal with a time-varying channel power gain h k , a pre-dictive power control algorithm was established, which The term “channel power gain” means the square of themagnitude of the complex channel. determines the transmission power level, bit rates andcodebooks used by the sensors. The algorithm in [4] re-quires that the receiver (i.e, the remote estimator) runsa channel gain predictor, see e.g., [18]. A key observa-tion is that the data-driven controller proposed in thepresent work can be readily adapted to situations wherethe baseline controller provides time-varying power lev-els ¯ w k . In fact, by solving Problem 4.12 for a time-varying power baseline ¯ ω k , we obtain the optimal solu-tion θ ∗ ef as follows: If ¯ ω k > N Wα , then θ ∗ ef is given by θ ∗ ef : { ω k = N Wαn τ ε (cid:48) k Σ − k ε k + ¯ ω k − N Wα } and Ψ k = n τ n τ +2 Σ k − . Otherwise, if ¯ ω k ≤ N Wα , θ ∗ ef isgiven by θ ∗ ef : { ω k = ¯ ωn τ ε (cid:48) k Σ − k ε k } and Ψ k = n τ N Wn τ N W +2 α ¯ ω Σ k . In both cases, Σ k = (1 − γ k − ) A Ψ k − A (cid:48) + h ( P ) − P . Note that Σ k , Ψ k and Φ k are calculated similar to Σ τ , Ψ τ and Φ τ given in The-orem 4.15. To reduce the sensor’s computational load,the sensor only needs to calculate the quadratic form ε (cid:48) k Σ − τ ε k , while the rest of the paraments are updatedand then sent to the sensor by the estimator. Note thatcalculating ε (cid:48) k Σ − τ ε k has a complexity of O ( n ). Consider a system with parameters as follows: A = (cid:34) .
99 0 . . . (cid:35) , C = (cid:34) . . (cid:35) , R = Q = I × . We firstassume that θ has a constant power baseline ¯ ω = 5 and N Wα = 3 < ¯ ω. In Section 5.2, a time-varying powerbaseline is considered. ¯ ω J ( θ ) θ θ Fig. 2. Empirical estimation covariance provided by con-trollers θ ∗ ef ( θ ) and θ as a function of energy constraint ¯ ω . We compare our proposed schedule θ ∗ ef (denoted as θ )with a constant baseline power controller within the en- Following assumptions commonly made in the literature,see, e.g., [4, 7], in the sequel we shall assume that the chan-nel gain h k is available via the one-step ahead channel gainpredictor. θ : { ω k = ¯ ω } ). Define J k ( θ ) = k (cid:80) ki =1 Tr ( E [ P i ]) as the empirical approxima-tion (via 100000 Monte Carlo simulations) of the aver-age expected state error covariance (denoted as J ( θ )).We choose J ( θ ) as an approximation of J ( θ ).Fig. 2 shows that θ ∗ ef leads to a better system perfor-mance when compared to θ under the same energy con-straint. In practice, wireless communication channels typicallycomprise fading often assumed to be Rayleigh [19], i.e.,the channel power gain h k is exponentially distributedwith pdf( h k ) = h exp ( − h k h ) , where h k (cid:62) h isthe mean of h k . Truncated channel inversion transmitpower controllers have been studied in several works [5,7, 20], where the transmission power is the inversion of h k , with a truncated boundary. In this subsection, weuse the baseline power determined by truncated channelgain inversion Denote the truncated channel inversiontransmission power controller as θ : ω k = (cid:40) vh k , h k > h (cid:63) , vh (cid:63) , otherwise . (18)where v and h (cid:63) are design parameters. Consider the caseof h = 1 and set h (cid:63) = 5. Based on the results in [5], wecan choose v to meet the energy constraint. Fig. 3 sug-gests that θ ∗ ef leads to better system performance whencompared with θ . Fig. 4 shows the comparison given aspecific realization of channel power gains. k J k ( θ ) θ θ Fig. 3. Comparison of θ ∗ ef ( θ ) and θ under Rayleigh fading. We proposed a data-driven transmission power con-troller for remote state estimation, which adjusts thesensor’s transmission power according to its real-timemeasurements. Then we proved that the proposed powercontroller preserves Gaussianity of the incremental in-novation and provided a closed-form expression of theexpected state estimation error covariance. a tuning
Fig. 4. Comparison of θ ∗ ef ( θ ) and θ given a specific realiza-tion of channel power gains. method for parameter design was presented to guar-antee that the data-driven power controller not worseperformance than the alternative non-data-driven ones.Comparisons were conducted to illustrate estimationperformance improvement. Appendix
Proof of Lemma 4.2:
To verify the clain, it suffices toshow that rank(Φ τ ) ≥ n τ . Suppose that rank(Φ τ ) = r < n τ . Since Φ τ (cid:23)
0, there must exist exactly n − r mutually orthogonal vectors e , . . . , e n − r such that e i (cid:48) Φ τ e i = 0 , for i = 1 , . . . , n − r. Denote the unit vec-tor with only the ( n τ + j )th entry being 1 by i j , thatis, i j = [ 0 , . . . , , (cid:124) (cid:123)(cid:122) (cid:125) n τ + j , , . . . , (cid:48) . Since i j (cid:48) Φ τ i j = 0 , j =1 , . . . , n − n τ , without loss of generality, let e j = i j . As weassume that e n − r is orthogonal to e j , j = 1 , . . . , n − n τ , it is true that D / τ e n − r (cid:54) = 0. Since U τ is nonsingu-lar and Ker( U τ ) = { } , we have e (cid:44) Σ / τ e n − r (cid:54) = 0.We then observe that e (cid:48) Ψ − τ e = e n − r (cid:48) Φ τ e n − r = 0 , and e (cid:48) Σ − τ e = e n − r (cid:48) (cid:34) I n τ
00 0 (cid:35) e n − r > , which contradictswith Ψ − τ (cid:23) Σ − τ . (cid:4) Proof of Lemma 4.11:
By definition, it is easy to see thatdet(Σ τ )det(Ψ − τ ) = (cid:81) n τ i =1 λ i (Σ τ ) λ i (Ψ τ ) − . Thereforewe only need to prove det(Φ τ ) = (cid:81) n τ i =1 λ i (Σ τ ) λ i (Ψ τ ) − . Observe that Σ τ and Ψ τ can be factorized as Σ τ = U τ (cid:34) ∆ τ
00 0 (cid:35) U τ (cid:48) and Ψ τ = V τ (cid:34) Θ τ
00 0 (cid:35) V τ (cid:48) , where∆ τ and Θ τ are diagonal matrices generated respec-tively by the nonzero eigenvalues of Σ τ and Ψ τ . For i = 1 , . . . , n τ , u i and v i are the eigenvectors asso-ciated with λ i (Σ τ ) and λ i (Ψ τ ). In addition, U τ =[ u , . . . , u n τ , , . . . , V τ = [ v , . . . , v n τ , , . . . , τ can be written as Φ τ = (cid:34) M τ
00 0 (cid:35) , where M τ =8 τ / ˜ U (cid:48) τ ˜ V τ Θ τ − ˜ V (cid:48) τ ˜ U τ ∆ τ / ∈ S n τ + , ˜ U τ = [ u , . . . , u n τ ]and ˜ V τ = [ v , . . . , v n τ ]. According to Lemma 4.3, M τ is nonsingular, so det(Φ τ ) = det( M τ ). SinceIm(Σ τ ) = Im(Ψ τ ) from (9), there exists a unitarymatrix V such that ˜ V τ = ˜ U τ V . Thus, det( M τ ) =det (cid:16) ∆ τ / V Θ τ − V (cid:48) ∆ τ / (cid:17) = det (cid:0) ∆ τ Θ τ − (cid:1) , whichcompletes the proof. (cid:4) Proof of Lemma 4.14:
According to Lemma 4.13, we set λ ( τ ) = · · · = λ n τ ( τ ) = λ τ . Logarithm does not changethe monotonicity of (14). Problem 4.12 is consequentlytransformed tomin λ τ ,ω − αN W ω − ( n τ λ τ , (19)s.t. n τ N W α ( λ τ −
1) + ω = ¯ ω, ω ≥ . Substituting ω = − αN W ¯ ω + n τ ( λ τ − − ( n τ + 1) ln λ τ into (19) and taking derivative, it yields that the min-imum of (19) is attained at λ τ = 1 + n τ . Meanwhile ω needs to be nonnegative, so the optimal solution toProblem 4.10 is (16) if ¯ ω > N Wα or (17) otherwise. (cid:4) Proof of Proposition 4.17:
Consider a matrix Σ = (cid:80) τi =1 ρ i (cid:0) h i ( P ) − h i − ( P ) (cid:1) with ρ i ∈ (0 , ρ Σ / ρ A Σ / · · · ρ τ A τ − Σ / ][ · ] (cid:48) )= Im([ ρ Σ / ρ A Σ / · · · ρ τ A τ − Σ / ])= Im([ Σ / A Σ / · · · A τ − Σ / ])= Im([ Σ / A Σ / · · · A τ − Σ / ][ · ] (cid:48) )= Im( h τ ( P ) − P ) , (20)which leads to the first assertion. By the Cayley-Hamilton theorem, we have A k = − a ( k ) A n − − a ( k ) A n − −· · ·− a n ( k ) I, ∀ k ≥ n, where a ( k ) , . . . , a n ( k )are coefficients of the characteristic polynomial of A .When τ ≥ n + 1, we haveIm([ Σ / A Σ / · · · A τ − Σ / ][ · ] (cid:48) )= Im([ Σ / A Σ / · · · − a ( τ − A n − Σ / − a ( τ − A n − Σ / − · · · − a n ( τ − / ]) , The last assertion follows from the reasoning usedin (20). (cid:4)
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