PaperlayerView exact revision on arXiv

Source-linked AI summary

Multiuser MISO UAV Communications in Uncertain Environments with No-fly Zones: Robust Trajectory and Resource Allocation Design

Dongfang Xu, Yan Sun, Derrick Wing Kwan Ng, Robert Schober

arXiv:1905.10731v2eess.SPcs.IT

TL;DR

Limited positioning accuracy and idealistic assumptions prevent existing resource allocation schemes from reliably providing high-data-rate services. This paper jointly optimizes UAV trajectory and beamforming under uncertainty, combining an optimal monotonic-optimization/SDP approach with a lower-complexity SCA scheme; the proposed methods save power, remain robust to key uncertainties, and support safe NFZ-avoiding trajectories.

  • Problem

    Limited positioning-module accuracy and idealistic design assumptions prevent existing resource allocation schemes from providing reliable high-data-rate communication services.

  • Method

    The paper jointly optimizes the UAV's 2-D trajectory and downlink beamformer using monotonic optimization with SDP relaxation, plus a lower-complexity iterative SCA scheme.

  • Results

    The proposed optimal and suboptimal schemes provide significant power savings versus baseline and non-robust schemes and remain robust to UAV jittering and user location uncertainty.

  • Takeaways & Limitations

    Robust design is necessary under wind uncertainty to minimize total UAV power while ensuring a secure trajectory that does not trespass no-fly zones.

Abstract

from arXiv · show

In this paper, we investigate robust resource allocation algorithm design for multiuser downlink multiple-input single-output (MISO) unmanned aerial vehicle (UAV) communication systems, where we account for the various uncertainties that are unavoidable in such systems and, if left unattended, may severely degrade system performance. We jointly optimize the two-dimensional (2-D) trajectory and the transmit beamforming vector of the UAV for minimization of the total power consumption. The algorithm design is formulated as a non-convex optimization problem taking into account the imperfect knowledge of the angle of departure (AoD) caused by UAV jittering, user location uncertainty, wind speed uncertainty, and polygonal no-fly zones (NFZs). Despite the non-convexity of the optimization problem, we solve it optimally by employing monotonic optimization theory and semidefinite programming relaxation which yields the optimal 2-D trajectory and beamforming policy. Since the developed optimal resource allocation algorithm entails a high computational complexity, we also propose a suboptimal iterative low-complexity scheme based on successive convex approximation to strike a balance between optimality and computational complexity. Our simulation results reveal not only the significant power savings enabled by the proposed algorithms compared to two baseline schemes, but also confirm their robustness with respect to UAV jittering, wind speed uncertainty, and user location uncertainty. Moreover, our results unveil that the joint presence of wind speed uncertainty and NFZs has a considerable impact on the UAV trajectory. Nevertheless, by counteracting the wind speed uncertainty with the proposed robust design, we can simultaneously minimize the total UAV power consumption and ensure a secure trajectory that does not trespass any NFZ.

I. INTRODUCTION

UAV communication systems offer flexible connectivity, but practical deployment must address jittering, uncertain user locations, wind, and no-fly zones. This paper develops robust joint trajectory and resource-allocation designs to overcome these challenges.

  • Practical challenges: Existing UAV resource-allocation schemes often assume perfectly stable flight and perfectly known user locations, limiting their practical reliability.UAV body jittering and positioning inaccuracies make these assumptions unrealistic.
  • Practical challenges: UAV jittering can create AoD estimation errors that degrade beamforming and prevent full exploitation of multiple antennas.Jittering angles may reach 10 degrees, while inaccurate user locations can add path loss to communication links.
  • Practical challenges: Wind changes UAV ground speed and planned trajectories, creating communication and flight-safety concerns that robust design must address.Wind can cause speeding or crashing if its effects are not incorporated into trajectory planning.
  • No-fly zones: NFZ constraints make trajectory design more challenging, while realistic polygonal NFZs produce disjunctive optimization problems.Earlier approaches using cylindrical NFZs cannot ensure accurate trajectories for general polygonal zones.
  • Proposed designs: The paper addresses the previously uninvestigated joint problem under AoD errors, user-location uncertainty, wind uncertainty, and polygonal NFZs.It formulates non-convex total-power minimization, solves it optimally using monotonic optimization and SDP relaxation, and proposes an SCA-based low-complexity alternative.
  • Reported findings: The proposed algorithms provide significant power savings relative to baseline schemes and robustness to UAV jittering, wind uncertainty, and user-location uncertainty.The robust design also mitigates the power impact of wind uncertainty and NFZs.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models a multiuser UAV downlink with a fixed-altitude planar trajectory, multiantenna beamforming, LoS channels, and per-slot transmissions to multiple users.

  • A. Multiuser UAV Communication System: The considered system comprises one UAV-mounted transmitter and K single-antenna ground users served simultaneously.
  • A. Multiuser UAV Communication System: The UAV uses an Mx × My uniform planar array with M = MxMy antenna elements and transmits user-specific beamformed signals.
  • A. Multiuser UAV Communication System: The UAV maintains a constant altitude above the tallest obstacles and its horizontal trajectory is represented by discrete waypoints over equal-duration time slots.
  • A. Multiuser UAV Communication System: In each time slot, the UAV simultaneously transmits K independent signals, with each signal formed by an information symbol and its corresponding beamforming vector.
  • A. Multiuser UAV Communication System: Air-to-ground links are modeled as line-of-sight channels whose antenna array response depends on the UAV-user geometry and vertical and horizontal AoDs.
  • A. Multiuser UAV Communication System: The channel model uses the UAV and user three-dimensional coordinates, with users indexed by k and the UAV’s horizontal coordinates varying by time slot.
  • A. Multiuser UAV Communication System: The received-signal model includes additive complex white Gaussian noise, and the resulting user performance is characterized by the received SINR.

B. UAV Jittering Model

The model captures UAV jittering, wind-induced trajectory deviations, and uncertain user locations through bounded deterministic uncertainty sets and a locally linearized array response.

  • B. UAV Jittering Model: Wind gusts cause UAV body jittering, which produces AoD estimation errors and imperfect AoD knowledge at the UAV.
  • B. UAV Jittering Model: The actual AoDs are represented by estimated AoDs plus unknown vertical and horizontal uncertainties contained in a bounded set with maximum variation α.
  • B. UAV Jittering Model: Because the antenna array response is nonlinear in the AoD uncertainties, the model applies a first-order Taylor expansion for tractable robust design.
  • B. UAV Jittering Model: The linearized array response is used for algorithm design, while simulations evaluate the proposed algorithm with the nonlinear array-response model.
  • B. UAV Jittering Model: Horizontal wind changes UAV ground speed through vector addition with the UAV’s horizontal velocity, while instantaneous wind speed is uncertain.
  • B. UAV Jittering Model: Wind speed uncertainty is modeled deterministically through a bounded uncertainty set around the estimated horizontal wind speed.
  • D. User Location Model: User positions are uncertain because of practical positioning limitations, so stationary ground users are represented by estimated coordinates plus horizontal location errors within bounded regions.
  • D. User Location Model: The UAV is assumed to know its own location, whereas each user’s uncertainty region has bounded radius Dk.

E. No-Fly Zone Model

The formulation combines polygonal NFZ avoidance, aerodynamic flight power, communication power, and UAV motion constraints in a robust per-slot trajectory and beamforming problem.

  • E. No-Fly Zone Model: The service area contains J polygonal NFZs, with the j-th zone having Sj sides and known geometry determined by regulation.
  • E. No-Fly Zone Model: Each polygonal NFZ is represented as the intersection of finitely many half-spaces defined by affine inequalities.
  • E. No-Fly Zone Model: Indicator constraints encode whether the UAV lies outside every NFZ by requiring at least one separating inequality for each polygon.
  • F. Aerodynamic Power Consumption: Aerodynamic power is modeled as the sum of induced, profile, and parasite power components that depend on horizontal velocity.
  • F. Aerodynamic Power Consumption: The maximum endurance speed minimizes total aerodynamic power, whereas hovering is generally not the most power-efficient operating condition.
  • G. Optimization Problem Formulation: The objective minimizes transmit, aerodynamic, and circuit power while enforcing per-antenna limits, user SINR requirements, motion constraints, and NFZ avoidance.
  • G. Optimization Problem Formulation: The constraints limit acceleration, displacement under wind uncertainty, horizontal and safety speeds, and passage through NFZs.
  • G. Optimization Problem Formulation: The resulting problem is generally intractable because it combines nonconvexity, semi-infinite constraints, and disjunctive NFZ constraints.

III. OPTIMAL SOLUTION OF THE OPTIMIZATION PROBLEM

The section converts the semi-infinite constraints into LMIs and addresses the remaining non-convex structure using slack variables, the S-procedure, and semidefinite constraints.

  • Beamforming constraints: Positive semidefinite and rank-one constraints on Wk are imposed to ensure valid beamforming representations.The beamforming matrix is constrained by Wk ⪰ 0 and Rank(Wk) ≤ 1.
  • Semi-infinite constraint transformation: Constraints C2, C4, and C6 are intractable semi-infinite constraints because variables vary continuously over uncertainty sets.
  • Semi-infinite constraint transformation: The semi-infinite constraints are transformed into linear matrix inequalities using slack variables and the S-procedure.The S-procedure converts implications between quadratic inequalities into equivalent matrix constraints under a strict-feasibility condition.
  • Residual non-convexity: The resulting formulation retains non-convexity because constraints C6 and related quadratic terms remain non-convex.

B. Transformation of the Disjunctive Constraint

The disjunctive no-fly-zone constraint is reformulated with binary auxiliary variables and then incorporated into a monotonic optimization formulation.

  • Disjunctive constraint reformulation: Disjunctive programming in constraint C7 is identified as an obstacle to solving the optimization problem.
  • Disjunctive constraint reformulation: Binary auxiliary variables lij ∈ {0, 1} transform the disjunctive constraint into mixed integer linear constraints.Theorem 1 states that the reformulation is equivalent when the binary variables satisfy the specified inequalities.
  • Disjunctive constraint reformulation: The reformulated constraint is further analyzed using slack variables, monotonicity, and convexity properties of its component constraints.Constraint C7f is monotonically increasing in t, while C7g is convex.
  • Monotonic optimization formulation: The complete problem is recast in canonical monotonic-optimization form over the intersection of a normal set G and a conormal set H.The objective is represented through auxiliary variables and the feasible set F = G ∩ H.

D. Optimal Algorithm Design

The optimal algorithm uses polyblock outer approximation and bisection-based projection to solve the monotonic formulation and obtain the globally optimal trajectory and beamforming policy.

  • Projection search: The projection of each vertex onto the feasible-set boundary is computed by bisection over a projection parameter λ.Feasibility is checked at each midpoint until λmax − λmin falls below the specified tolerance.
  • SDP relaxation: Removing the rank-one constraint yields a convex projection problem that can be solved efficiently by standard convex optimization solvers.The SDP relaxation is tight because a rank-one beamforming matrix can always be obtained when Γreqk > 0.
  • Optimality and complexity: Algorithm 1 obtains the globally optimal UAV trajectory and beamforming policy.The algorithm also provides a benchmark for evaluating suboptimal designs.
  • Optimality and complexity: The algorithm’s computational complexity increases exponentially with the number of users, making real-time operation prohibitive.This complexity motivates the subsequent low-complexity suboptimal scheme.

IV. SUBOPTIMAL SOLUTION OF THE OPTIMIZATION PROBLEM

The suboptimal design applies successive convex approximation to obtain a locally optimal solution with lower computational complexity than the optimal formulation.

  • Suboptimal SCA design: The proposed suboptimal algorithm is based on successive convex approximation to balance computational complexity and optimality.
  • Non-convexity handling: The reformulated problem remains non-convex because of the objective function and constraints C10, C12, and C14.
  • Non-convexity handling: Global underestimators replace difference-of-convex terms, producing a convex optimization problem at each SCA iteration.The resulting convex problem provides an upper bound for the reformulated problem.
  • SDP relaxation: The remaining rank-one constraint is handled through SDP relaxation, allowing each convex subproblem to be solved efficiently.Standard convex programming solvers such as CVX are used after removing constraint C10.
  • Convergence: The iterative suboptimal algorithm converges to a locally optimal solution of problem (50) in polynomial time.

SYSTEM PARAMETERS

The simulations evaluate robust UAV resource-allocation schemes under uncertain wind, user locations, AoD estimates, and polygonal NFZs. They compare optimized trajectory-and-beamforming designs with stationary or non-robust baselines using total UAV power consumption and trajectory safety as key outcomes.

  • Evaluation setup: The simulations consider users uniformly and randomly distributed within a single cell, with each user located within D_k = 20 meters of its estimated location.The cell radius is 600 meters.
  • Uncertainty settings: Unless otherwise specified, the normalized AoD estimation and wind-speed uncertainties are set to ρ_k = 0.1 and ρ_w = 0.2.The wind-speed magnitude is also varied in dedicated experiments.
  • Scenario design: The experiments compare scenarios without wind or NFZs against scenarios with wind and three randomly distributed polygonal NFZs.The wind-and-NFZ case uses T = 10 minutes.
  • Evaluation setup: The evaluation uses total UAV power consumption as its performance metric and includes RF-chain circuit power.The metric is calculated for resource-allocation schemes across the considered scenarios.
  • Trajectory behavior: Without wind and NFZs, the proposed schemes move toward the user centroid and then circle around it to reduce aerodynamic power consumption.The circling behavior exploits the lower power consumption of cruising relative to hovering for rotary-wing UAVs.
  • Trajectory behavior: With wind and NFZs, the proposed schemes detour around forbidden regions, maintain a safe boundary distance, and adapt their trajectories near the users.The trajectory balances power-efficient proximity to users with NFZ safety requirements.
  • Robustness comparison: The non-robust scheme develops a spiral trajectory under wind and crosses a polygonal NFZ, violating the safety requirement.Wind changes the ground speed in a way the non-robust design cannot fully control.

B. UAV Velocity

The velocity experiments show that the proposed schemes exploit a power-efficient cruising speed when unconstrained, while wind and NFZs require speed adaptation and periodic motion. Power consumption varies with wind, antenna count, and user count because trajectory and beamforming degrees of freedom affect aerodynamic and transmit power.

  • Velocity versus scenario: 8 m/s is maintained throughout the wind-free, NFZ-free horizon by both proposed schemes because it minimizes aerodynamic power consumption.The baseline schemes hover instead of cruising.
  • Velocity versus scenario: With wind and NFZs, the proposed UAV starts at 8 m/s, then changes speed to compensate for wind while following the safe trajectory.The trajectory circles around a rectangular NFZ, producing periodically varying ground speed.
  • Velocity versus scenario: The baseline schemes operate around 3 m/s to counteract wind and remain stationary at their desired positions.Their velocity is used to compensate for wind rather than to translate along a trajectory.
  • Power versus wind: For wind speeds above 8 m/s, the proposed schemes must use less favorable speeds to compensate for wind, increasing aerodynamic power consumption.At lower wind speeds, 8 m/s can compensate for the wind while remaining favorable aerodynamically.
  • Power versus wind: For wind-speed estimates below 6 m/s, the proposed optimal and suboptimal schemes achieve substantial power savings over both baseline schemes.Trajectory design supplies additional degrees of freedom compared with stationary baselines.
  • Power versus antennas: Increasing the number of antennas reduces power for the proposed schemes and baseline scheme 1, with diminishing gains at larger antenna counts.Additional antennas improve beamforming and multiuser-interference mitigation, while circuit power and channel hardening limit the benefit.
  • Power versus users: Power increases with the number of users because more beamforming degrees of freedom are dedicated to multiuser-interference suppression.Baseline scheme 2 benefits only slightly from additional antennas because fixed MRT cannot fully exploit them.

E. Average Total UAV Power Consumption versus Maximum Normalized AoD Estimation Error

The power study examines sensitivity to AoD error, user-location uncertainty, SINR requirements, NFZs, and robust versus non-robust design. Joint trajectory and beamforming optimization yields power savings, while uncertainty and stricter requirements increase the required total power.

  • AoD and location uncertainty: Total UAV power consumption increases monotonically with the maximum normalized AoD estimation error ρ_k.Greater AoD uncertainty makes accurate beamforming more difficult and requires higher transmit power to meet QoS requirements.
  • AoD and location uncertainty: Total UAV power consumption also increases with the user location uncertainty radius D_k.Larger uncertainty regions require less focused beamformers that cover the full possible user area.
  • Joint optimization: Joint 2-D trajectory and beamforming optimization produces considerable power savings over the baseline schemes.Trajectory selection places the UAV favorably for beamforming, while beamforming permits power-efficient flight.
  • Robust versus non-robust design: The proposed robust scheme consumes less total power than non-robust scheme 2 across the considered minimum-SINR range.Using estimated AoDs and locations as if they were exact can point a focused beamformer in the wrong direction under uncertainty.
  • SINR and NFZ effects: Average total power is monotonically nondecreasing with the minimum required SINR, and NFZs cause additional power consumption.NFZs force proposed schemes to circle around forbidden regions and force baseline scheme 1 to hover at a suboptimal position.
  • Design scope: The paper jointly optimizes 2-D trajectory and downlink beamforming while modeling AoD errors, user-location uncertainty, wind uncertainty, and polygonal NFZs.The AoD and trajectory coupling makes multi-slot joint design intractable, so the policy is optimized on a time-slot-by-time-slot basis.
  • Algorithms: Monotonic optimization with semidefinite programming relaxation solves the non-convex problem optimally, while successive convex approximation provides a lower-complexity iterative alternative.The optimal algorithm has high computational complexity and mainly serves as a benchmark for lower-complexity schemes.
  • Conclusions: The proposed schemes achieve power savings and robustness to UAV jittering and user-location uncertainty, while robust design is necessary for safe operation under wind uncertainty and NFZs.When average wind is below the UAV’s maximum endurance speed, the UAV can follow the desired trajectory with minimum aerodynamic power.

APPENDIX

The appendix proves properties of the reformulated optimization problems using binary NFZ constraints, Lagrangian duality, and monotonicity. It establishes zero duality gap and the resulting theorem conditions through separate cases.

  • NFZ reformulation: The NFZ constraint is rewritten so that at least one inequality associated with each polygonal region is satisfied.Binary variables encode which inequality is active for each NFZ.
  • Duality proof: Theorem 3 is analyzed through the primal problem and its Lagrangian dual, using weak duality and the monotonicity of the dual objective function.The dual objective is bounded above by the primal optimum and increases with the dual variable.
  • Duality proof: When the relevant dual constraint is zero, the dual solution is feasible for the primal problem and the duality gap is zero.The resulting equality yields the theorem’s optimality condition.
  • Duality proof: When the relevant dual quantity is positive, monotonic growth would make the dual objective unbounded, contradicting the finite primal objective.Therefore, the positive case cannot hold at the optimum.
Loading 1905.10731v2…