In this paper, the optimal energy scheduling problem is investigated in the sensor network with energy sharing and imperfect feedback information. Each sensor is equipped with an energy harvesting module and a transceiver device for the energy sharing between adjacent sensors. Sensors can transmit information and receive packet receipt acknowledgments from the remote estimator via erroneous feedback channels. The goal of this paper is to determine the energies used for data transmission and for sharing between sensors to ensure high-quality estimation performance. To address the challenges posed by imperfect feedback, a novel approach is first proposed for the synergistic estimation of the probability distribution of feedback information for each sensor. Subsequently, a joint error covariance estimation method based on the concept of information state is introduced. By this method, the impact of other sensors' transmission power on the overall transmission process is effectively accounted for. As a result, the proposed energy scheduling problem with uncertain feedback is converted into a Markov decision process (MDP) with perfect information, which becomes more amenable to analysis and solution. The Bellman dynamic programming (DP) equation is employed to obtain the optimal solution. In situations where feedback information may be missing, the relative value iteration algorithm is utilized to solve the Bellman equation, through which the optimal energy allocation strategy is derived. Ultimately, the structural results and a numerical simulation verify the performance of the proposed energy allocation policy.
Zhiping JU
,
Lijun GUO
,
Guoliang WEI
,
Jiajia LI
. ENERGY SCHEDULING IN MULTI-SENSOR ESTIMATION OVER PACKET DROPPING CHANNELS WITH IMPERFECT ACKNOWLEDGMENTS AND ENERGY CONSTRAINTS[J]. Acta mathematica scientia, Series B, 2026
, 46(2)
: 957
-970
.
DOI: 10.1007/s10473-026-0222-2
[1] Akyildiz I F, Su W, Sankarasubramaniam Y, et al. A survey on sensor networks. IEEE Commun Mag, 2002, 40(8): 102-114
[2] Duarte-Melo E J, Liu M. Data-gathering wireless sensor networks: Organization and capacity. Comput Netw, 2003, 43(4): 519-537
[3] Chong C Y, Kumar S. Sensor networks: Evolution, opportunities and challenges. Proc IEEE, 2003, 91(8): 1247-1256
[4] Benfradj A, Thaljaoui A, Moulahi T, et al. Integration of artificial intelligence (AI) with sensor networks: Trends, challenges,future directions. J King Saud Univ-Comput Inf Sci, 2024, 36(1): Art 101892
[5] Cefai E J, Coombes M, O'Boy D. Intelligent wireless sensor network sensor selection and clustering for tracking unmanned aerial vehicles. Sensors, 2025, 25(2): Art 402
[6] Priyadarshi R. Exploring machine learning solutions for overcoming challenges in IoT-based wireless sensor network routing: A comprehensive review. Wirel Netw, 2024, 30(4): 2647-2673
[7] Jones C E, Sivalingam K M, Agrawal P, et al. A survey of energy efficient network protocols for wireless networks. Wirel Netw, 2001, 7(4): 343-358
[8] Goldsmith A J, Wicker S B. Design challenges for energy-constrained ad hoc wireless networks. IEEE Wirel Commun, 2002, 9(4): 8-27
[9] Zhang L, Quevedo F D E, Lau V, et al. Power control of an energy harvesting sensor for remote state estimation. IEEE Trans Autom Control, 2017, 62(1): 277-290
[10] Yang T, Zhou S, Litak G, et al. Recent advances in correlation and integration between vibration control, energy harvesting and monitoring. Nonlinear Dyn, 2023, 111: 20525-20562
[11] Peng L H, Cao X H, Sun C Y. Optimal transmit power allocation for an energy-harvesting sensor in wireless cyber-physical systems. IEEE Trans Cybern, 2021, 51(2): 779-788
[12] Varshney L R.Transporting information and energy simultaneously//2008 IEEE International Symposium on Information Theory Proceedings. 2008: 1612-1616
[13] Zhang R, Ho C K. MIMO broadcasting for simultaneous wireless information and power transfer. IEEE Trans Wirel Commun, 2013, 12(5): 1989-2001
[14] Liu X, Goldsmith A J.Kalman filtering with partial observation losses//2004 43rd IEEE Conference on Decision and Control. 2004: 4180-4186
[15] Sharma V, Mukherji U, Joseph V, et al. Optimal energy management policies for energy harvesting sensor nodes. IEEE Trans Wirel Commun, 2010, 9(4): 1326-1336
[16] Kurs A, Karalis A, Moffatt R, et al. Wireless power transfer via strongly coupled magnetic resonances. Science, 2007, 317(5834): 83-86
[17] Qiu X, Wang X. An adaptive event-triggered secondary regulation strategy for microgrids with communication link faults. Nonlinear Dyn, 2023, 111: 18013-18026
[18] Huang K, Lau V K N. Enabling wireless power transfer in cellular networks: architecture, modeling and deployment. IEEE Trans Wirel Commun, 2014, 13(2): 902-912
[19] Fouladgar A M, Simeone O. On the transfer of information and energy in multi-user systems. IEEE Commun Lett, 2012, 16(11): 1733-1736
[20] Huang K, Larsson E. Simultaneous information and power transfer for broadband wireless systems. IEEE Trans Signal Process, 2013, 61(23): 5972-5986
[21] Quevedo D E, Ahlen A, Leong A S, et al. On Kalman filtering over fading wireless channels with controlled transmission powers. Automatica, 2012, 48(7): 1306-1316
[22] Li Y Z, Mehr A S, Chen T W. Multi-sensor transmission power control for remote estimation through a SINR-based communication channel. Automatica, 2019, 101: 78-86
[23] Leong A S, Dey S.Power allocation for error covariance minimization in Kalman filtering over packet dropping links//2012 IEEE 51st Annual Conference on Decision and Control. 2012: 3335-3340
[24] Nourian M, Leong A S, Dey S.Optimal energy allocation for Kalman filtering over packet dropping links with energy harvesting constraints//5st IFAC Workshop on Distributed Estimation and Control in Networked Systems. 2013: 261-268
[25] Nourian M, Leong A S, Dey S. Optimal energy allocation for Kalman filtering over packet dropping links with imperfect acknowledgments and energy harvesting constraints. IEEE Trans Autom Control, 2013, 59(8): 2128-2143
[26] Sinopoli B, Schenato L, Franceschetti M, et al. Kalman filtering with intermittent observations. IEEE Trans Autom Control, 2004, 49(9): 1453-1464
[27] Kumar P R, Varaiya P.Stochastic Systems: Estimation, Identification and Adaptive Control. Englewood Cliffs, NJ: Prentice-Hall, 1986
[28] Bertsekas D P.Dynamic Programming and Optimal Control. Vol 1. Belmont, MA: Athena Scientific, 1995
[29] Yu H, Bertsekas D P.Discretized approximations for POMDP with average cost//20th Conference on Uncertainty in Artifical Intelligence. 2004: 619-627
[30] Ho C K, Zhang R. Optimal energy allocation for wireless communications with energy harvesting constraints. IEEE Trans Signal Process, 2012, 60(9): 4808-4818
[31] Topkis D M.Supermodularity and Complementarity. Princeton: Princeton Univ Press, 2001
