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Robust Resource Allocation for MIMO Wireless Powered Communication Networks Based on a Non-linear EH Model

Elena Boshkovska, Derrick Wing Kwan Ng, Nikola Zlatanov, Alexander Koelpin, Robert Schober

arXiv:1609.03836v2cs.IT

TL;DR

MIMO-WPCNs need resource allocation that reflects practical non-linear energy harvesting and uncertainty in channel knowledge. The paper jointly optimizes time and power for sum-throughput and max-min objectives using transformations and one-dimensional search, achieving higher throughput than linear-EH baselines and robustness to imperfect CSI.

  • Problem

    Linear EH modeling can mismatch practical circuits and degrade resource allocation, while robust non-linear-EH allocation for MIMO-WPCNs had not been studied.

  • Method

    The paper formulates joint time-allocation and power-control problems for max-sum and max-min throughput under non-linear EH, imperfect CSI, and multiple antennas, solving them through transformations and one-dimensional search.

  • Results

    The proposed schemes achieve significant throughput gains over linear-EH baseline allocations and are robust against imperfect CSI.

  • Takeaways & Limitations

    Resource allocation designed with a practical non-linear EH model can outperform conventional linear-EH designs while addressing imperfect CSI in MIMO-WPCNs.

Abstract

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In this paper, we consider a multiple-input multiple-output wireless powered communication network (MIMO-WPCN), where multiple users harvest energy from a dedicated power station in order to be able to transmit their information signals to an information receiving station. Employing a practical non-linear energy harvesting (EH) model, we propose a joint time allocation and power control scheme, which takes into account the uncertainty regarding the channel state information (CSI) and provides robustness against imperfect CSI knowledge. In particular, we formulate two non-convex optimization problems for different objectives, namely system sum throughput maximization and maximization of the minimum individual throughput across all wireless powered users. To overcome the non-convexity, we apply several transformations along with a one-dimensional search to obtain an efficient resource allocation algorithm. Numerical results reveal that a significant performance gain can be achieved when the resource allocation is designed based on the adopted non-linear EH model instead of the conventional linear EH model. Besides, unlike a non-robust baseline scheme designed for perfect CSI, the proposed resource allocation schemes are shown to be robust against imperfect CSI knowledge.

I. INTRODUCTION

The paper addresses robust resource allocation in MIMO-WPCNs using a practical non-linear EH model and imperfect CSI. It develops schemes for throughput maximization and fairness, showing improved performance and robustness over conventional baselines.

  • Motivation: Practical RF EH circuits have highly non-linear input-output characteristics, making linear-model resource allocation potentially mismatched and performance-degrading.The paper motivates modeling end-to-end WET with practical non-linear EH behavior.
  • Research gap: Existing non-linear EH studies considered single-antenna receivers, leaving MIMO spatial multiplexing gains and practical WPCN allocation insufficiently addressed.The paper targets this gap by combining multiple-antenna transceivers with non-linear EH modeling.
  • Approach: The paper jointly designs time allocation and power control for max-sum throughput and max-min individual throughput under imperfect CSI and multiple-antenna transceivers.These objectives respectively emphasize total throughput and minimum-user performance.
  • Approach: Energy beamforming remains optimal for WET with imperfect CSI, while optimal WIT power allocation has a water-filling structure and time allocation admits an analytical solution.These structural results support the proposed resource allocation algorithms.
  • Results: The proposed schemes achieve significantly higher system performance than linear-EH baseline schemes and greater robustness than a perfect-CSI non-robust benchmark.The comparison applies to both considered design objectives.
  • Results: Comparing max-sum and max-min designs reveals a non-trivial trade-off between maximizing system sum throughput and guaranteeing user fairness.The fairness objective addresses unequal energy and throughput conditions among wireless powered users.

II. SYSTEM MODEL AND PRELIMINARIES

The MIMO-WPCN uses a harvest-then-transmit protocol: a multi-antenna power station wirelessly powers users, which then transmit information to a receiving station.

  • System architecture: The network contains a multi-antenna power station, K wireless powered users, and an information receiving station operating in a shared frequency band.The stations use time division multiple access, with NT, NUk, and NR antennas at the power station, users, and receiving station, respectively.
  • Transmission protocol: The harvest-then-transmit protocol divides each slot into downlink wireless energy transfer and uplink wireless information transfer periods.The downlink duration and each user's uplink transmission time are optimization variables.
  • Energy and information transfer: During WET, the power station sends energy signals; during WIT, users consume the harvested energy to transmit independent information signals.Each user stores the harvested energy in a sufficiently large rechargeable battery before uplink transmission.
  • Channel and receiver assumptions: The model assumes frequency-flat, slowly time-varying fading channels and perfect CSIR for coherent information decoding.The WET and WIT stations are distinct, and one multiple-antenna power station can represent connected power stations sharing resources.
  • Signal model: The downlink energy signal has covariance matrix V, while user k's uplink information signal uses covariance or precoding matrix Qk.The downlink channel is Gk and the uplink channel is Hk; received signals include additive white Gaussian noise.

B. Energy Harvesting Model

The paper replaces the conventional linear EH assumption with a practical non-linear RF-to-DC model and combines it with a deterministic imperfect-CSI model for robust design.

  • Motivation: Practical RF energy-harvesting circuits have non-linear input-output characteristics, unlike the conventional model's constant conversion efficiency.The linear model assumes harvested energy is proportional to received RF power, whereas practical conversion efficiency changes with input power.
  • Motivation: At high received powers, harvesting efficiency exhibits diminishing returns and saturation, so linear-model resource allocation may be suboptimal.The non-linear model is intended to avoid resource-allocation mismatches caused by the traditional linear model.
  • Non-linear EH model: The paper adopts a practical non-linear EH model to characterize RF-to-DC power transfer at wireless powered users during WET.The model uses a logistic function of received RF power and parameters that capture hardware non-linearities.
  • Non-linear EH model: The parameter Mk denotes the maximum harvestable power, while ak and bk capture hardware effects including sensitivity limitations and leakage currents.The EH circuit saturates when received RF power is sufficiently large.
  • Model validation: Figure 3 reports that the non-linear model closely matches measurements from a practical EH circuit and exposes limitations of the linear model.The cited discussion specifically concerns accurately modeling non-linear EH circuits.
  • Imperfect CSI model: The deterministic CSI model represents each channel as an estimate plus an uncertainty matrix bounded by a continuous uncertainty set.The bounds depend on channel coherence time, scheduling-slot duration, and channel-estimation schemes.
  • Imperfect CSI model: The uncertainty matrices capture channel-estimation errors and time variation, while their norm bounds define the maximum CSI error magnitude.The model supports robust resource allocation while isolating the algorithm from a specific channel-estimation implementation.

III. RESOURCE ALLOCATION PROBLEM FORMULATION

The resource-allocation formulation jointly optimizes WET/WIT timing and transmit covariances for robust sum-throughput maximization under non-linear harvesting and CSI uncertainty.

  • III. RESOURCE ALLOCATION PROBLEM FORMULATION: The paper formulates max-sum and max-min resource-allocation problems for the MIMO-WPCN.The max-sum objective maximizes total system throughput, while the max-min objective targets the minimum individual throughput.
  • A. Max-sum Problem Formulation: The max-sum policy jointly optimizes the time vector τ, downlink energy covariance V, and user covariance matrices Qk.τ contains the WET time τ0 and each user's uplink WIT duration τk.
  • A. Max-sum Problem Formulation: The robust constraint guarantees feasible performance for every CSI estimation error in the prescribed uncertainty set.The formulation also accounts for maximum power, maximum slot duration, circuit consumption, and power-amplifier inefficiency.
  • A. Max-sum Problem Formulation: The formulation limits each user's uplink energy by harvested energy during WET under the practical non-linear EH model.The energy constraint minimizes over all downlink CSI errors in the uncertainty set, producing a worst-case robust constraint.
  • A. Max-sum Problem Formulation: The variables τk are nonnegative, while V and Qk are constrained to be positive semi-definite Hermitian matrices.These constraints complete the physical feasibility conditions for the max-sum formulation.

B. Max-min Problem Formulation

The paper adds a fairness-oriented max-min formulation and develops transformations that make both robust non-linear-EH problems tractable through convex optimization and one-dimensional search.

  • B. Max-min Problem Formulation: Sum-throughput maximization can favor users with good channels, consuming resources that may starve users with poor channels.This motivates a separate fairness objective.
  • B. Max-min Problem Formulation: The max-min formulation maximizes the minimum throughput across wireless powered users while accounting for imperfect CSI.Its constraint set is identical to that of the sum-throughput problem.
  • IV. SOLUTION OF THE OPTIMIZATION PROBLEMS: Both optimization problems are non-convex because the energy constraint couples τk and Qk and contains a quasi-concave non-linear-EH term.The uncertainty sets additionally create infinitely many constraints and objective-function possibilities.
  • IV. SOLUTION OF THE OPTIMIZATION PROBLEMS: Several transformations are introduced to obtain tractable formulations solvable with efficient convex optimization tools.The solution procedure first develops the max-sum case and then extends the approach to the max-min problem.
  • A. Transformation of Constraint C3: A one-dimensional search over τ0 finds the optimal value and its corresponding resource-allocation policy.The fixed-τ0 convex problem is solved before selecting the best WET duration.
  • A. Transformation of Constraint C3: The continuous CSI uncertainty is handled by introducing auxiliary variables and converting the robust constraint into linear matrix inequalities.The S-Procedure supplies the implication-based transformation under its stated strict-feasibility condition.
  • A. Transformation of Constraint C3: For a fixed τ0, the reformulated objective is jointly concave in eQk and τk, and the relevant constraint set becomes convex.The auxiliary variable eQk = Qkτk decouples variables in the energy constraint.

B. Transformation of the Objective Function

The paper transforms the CSI-uncertain objective into tractable equivalent forms by exploiting invariant covariance constraints and worst-channel singular-value structure.

  • The original objective is difficult to solve efficiently because of CSI uncertainty and the Frobenius-norm formulation.
  • The objective is transformed to address the intractable CSI uncertainty embedded in its current form.
  • Under unitarily invariant covariance constraints, the robust objective becomes an equivalent optimization problem involving channel singular values.
  • The optimal transmit covariance diagonalizes the estimated channel through its singular-value decomposition.
  • The worst-channel singular values are max{bγi,k − ρk, 0} for each user and eigenmode.

C. Dual Problem Formulation and Solution

For a fixed WET duration, the transformed problem is convex and admits a structured dual solution: rank-one energy beamforming, water-filling information power, and analytical time allocation.

  • For a given τ0, the transformed optimization problem is convex, enabling dual and KKT-based solution methods.
  • The optimal energy matrix V is rank one when the problem is feasible and Pmax > 0.
  • The beamforming direction aligns with the maximum eigenmode of Γ, which depends on the estimated downlink channel.
  • Energy beamforming remains optimal despite CSI uncertainty and the non-linear EH model.
  • The optimal eigenmode power allocation has a water-filling structure.
  • Increasing CSI uncertainty requires a longer WET period and decreases the WIT period.

D. Solution of the Max-min Optimization Problem

The max-min formulation introduces a minimum-throughput variable and preserves the core solution structure while modifying power allocation to enforce fairness among users.

  • The max-min problem is transformed into an equivalent convex optimization problem using the same transformations as the sum-throughput formulation.
  • The auxiliary variable ν denotes the minimum throughput achieved by each wireless powered user.
  • The max-min power allocation uses a Lagrange multiplier associated with each user’s individual-throughput constraint.
  • Unlike the sum-throughput allocation, the max-min allocation ensures fairness among different wireless powered users.
  • The optimal time allocation retains the sum-throughput structure, while the optimal energy matrix remains rank one.

E. Overall Resource Allocation Algorithm

The proposed algorithm solves convex subproblems for a fixed WET duration, applies successive convex approximation, and searches over that duration to obtain resource allocations efficiently.

  • For a given τ0, numerical solvers such as CVX solve the resource-allocation subproblems, followed by a one-dimensional search.
  • Common convex solvers cannot directly handle the nonlinear EH constraint, motivating the successive convex approximation treatment.
  • Successive convex approximation replaces the nonlinear constraint with a local inequality based on a feasible iterative point.
  • The iterative algorithm tightens the resulting upper bound and repeats until convergence or the maximum iteration count.
  • The successive convex approximation algorithm converges to the optimum of the original formulation with polynomial-time computational complexity.
  • Less than 5 iterations were required for convergence for each channel realization in the authors’ simulations.

V. NUMERICAL RESULTS

Numerical evaluations show that the proposed resource allocation schemes improve throughput under nonlinear EH modeling and imperfect CSI, while exposing a trade-off between sum throughput and fairness.

  • Sum-throughput performance: Sum throughput increases monotonically with the power station’s maximum transmit power because users can harvest more energy for WIT.This trend holds for all considered schemes.
  • Sum-throughput performance: The proposed max-sum scheme achieves the highest sum throughput by favoring users with good channel conditions and exploiting multiuser diversity.This resource allocation can be unfair to users with poor channels.
  • EH-model comparison: Baseline schemes perform worse because linear-EH-based allocation can saturate some receivers, underutilize others, and mismatch practical nonlinear EH behavior.The resulting mismatch reduces harvested energy and uplink contributions.
  • Antenna effects: Additional antennas improve resource utilization, energy collection, spatial multiplexing, and the minimum-throughput performance.The added degrees of freedom benefit both wireless energy transfer and information transmission.
  • User scaling and fairness: The max-min scheme increases minimum individual throughput with transmit power but sacrifices sum-throughput performance to equalize users’ throughputs.Its performance is constrained by the user with the worst channel condition.
  • User scaling and fairness: Increasing the number of users raises sum throughput through multiuser diversity but decreases average minimum individual throughput because fairness constraints become more stringent.The max-min scheme is limited by users with poor channel conditions, whereas the max-sum scheme prioritizes favorable channels.
  • Robustness to imperfect CSI: As CSI quality decreases, both proposed schemes’ average sum throughput degrades; increasing channel-estimation error makes their performance approach the baseline.The non-robust scheme can allocate insufficient energy resources under imperfect CSI.
  • Time allocation: Robust and baseline schemes require different WET durations: max-min uses longer WET periods than max-sum, while robust durations increase with CSI uncertainty and eventually saturate.Saturation occurs because further WET time cannot yield high throughput for users with poor effective channels.

VI. CONCLUSIONS

The paper develops robust resource allocation for MIMO-WPCNs using a practical non-linear EH model, jointly optimizing time allocation and power control for throughput and fairness objectives. The resulting schemes are efficiently computed and outperform linear-EH baselines while remaining robust to imperfect CSI.

  • The study designs robust MIMO-WPCN resource allocation schemes based on a practical non-linear EH model.The schemes jointly account for time allocation, power control, and imperfect CSI knowledge.
  • Two objectives are optimized: system sum throughput and the minimum individual throughput among wireless powered users.
  • The resulting formulations are non-convex optimization problems solved efficiently through a one-dimensional search with a convex optimization problem at each iteration.
  • The proposed schemes achieve significant throughput gains over resource allocation schemes optimized for the traditional linear EH model.
  • The results reveal a trade-off between maximizing system throughput and ensuring fairness among wireless powered users.
  • The developed resource allocation schemes are robust against imperfect CSI knowledge.

APPENDIX

The appendix proves the structure of the optimal energy matrix using the KKT conditions and rank/null-space properties. It concludes that the optimal energy allocation is characterized by the maximum eigenvalue and associated eigenvector of matrix Γ, with full power used.

  • The proof analyzes the KKT conditions of problem (28) to derive the structure of the optimal beamforming matrix V∗.
  • The columns of V∗ lie in the null space of M∗C5, whose rank and null-space properties determine the beamforming structure.
  • The dual variable µ∗ equals the largest eigenvalue λmax of matrix Γ at the optimum.
  • The null space of M∗C5 is spanned by the unit-norm eigenvector uΓ,max associated with Γ's relevant eigenvalue.
  • The optimal solution uses the full available power, satisfying δ = Pmax and Tr(V∗) = Pmax.
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