Recently, wireless communication employing unmanned aerial vehicles (UAVs) has received much attention as a promising approach for offering ubiquitous real-time high data-rate communication services –
. Compared to conventional cellular systems, which depend on a fixed terrestrial infrastructure, UAV-assisted communication systems can flexibly deploy UAV-mounted transceivers to a target area to provide on-demand connectivity. For instance, in the case of natural disasters and disease outbreaks, UAVs can be employed as flying base stations to offer temporary communication links in a timely manner. Moreover, benefiting from their high mobility and maneuverability, UAVs are able to adapt their positions and trajectories based on the environment and the terrain which results in extra degrees of freedom for potential system performance improvement. In , the authors proposed a new framework for joint power allocation and UAV trajectory optimization to maximize the system throughput of a mobile relaying system. The authors of  considered a multicarrier solar-powered UAV communication system and proposed the joint design of power and subcarrier allocation and three-dimensional (3-D) UAV positioning for maximization of the system sum throughput. In , the authors studied UAV trajectory design for maximization of the energy-efficiency of a UAV communication system. Besides, secure UAV communications was studied in  where the trajectory of a UAV and its transmit power were jointly optimized to maximize the system secrecy rate. However, – assumed that the channel state information (CSI) of the users was perfectly known at the UAV which may not hold in practice.
In practical UAV communication systems, UAV-mounted transceivers flying in the air may encounter strong wind gusts, which leads to random body jittering with respect to angular movements 
. The estimation accuracy of the angle of departure (AoD) between the UAV and the users is impaired by this jittering which results in non-negligible AoD estimation errors. Moreover, due to the weather conditions and electromagnetic interference, the information about the user location may be imperfect at the UAV . As a result, additional path loss resulting from user location uncertainty may impair the communication links between the UAV and users. Thus, perfect CSI knowledge of the users cannot be guarantee at the UAV, and the system performance is degraded due to the imperfectness of the CSI . On the other hand, multiple antennas performing beamforming can be employed to improve spectral efficiency in future multiuser communication systems. However, the results in – are valid for single-antenna UAVs and are not applicable to multiple-antenna UAVs, since the positioning of the UAV is coupled with the beamformer design. To the best of the authors’ knowledge, optimal resource allocation design for multiuser multiple-antenna UAV communication systems in the presence of imperfect CSI has not been investigated in the literature yet.
In this paper, we address the above issues. To this end, the resource allocation algorithm design is formulated as a non-convex optimization problem for minimization of the total transmit power of a downlink (DL) UAV communication system taking into account the quality-of-service (QoS) requirements of the users and imperfect CSI knowledge of the links between the UAV and the users. Thereby, we linearize the antenna array response (AAR) with respect to the AoD estimation errors, since these errors are expected to be small in practice. The formulated non-convex problem is solved optimally by applying transformations and semidefinite programming (SDP) relaxation.
Ii System and CSI Models
In this section, we present the system and CSI models for multiuser DL UAV communication. However, first we introduce some notation.
In this paper, matrices and vectors are denoted by boldface capital and lower case letters, respectively.and denote the sets of all real-valued and complex-valued matrices, respectively. denotes the set of all Hermitian matrices. denotes the
dimensional identity matrix.and represent the absolute value of a complex scalar and the Euclidean norm of a vector, respectively. and denote the transpose of vector and the conjugate transpose of vector , respectively. denotes a diagonal matrix whose diagonal entries are . and are the rank and the trace of matrix , respectively. means matrix is positive semidefinite. denotes the Hadamard product of two matrices and having the same dimensions. denotes statistical expectation.
indicates that random variable
is circularly symmetric complex Gaussian distributed with mean
and variance. means “defined as”. denotes the gradient vector of function , i.e., its components are the partial derivatives of . represents the -th order derivative of at .
Ii-B Multiuser UAV Communication System
The considered multiuser DL UAV communication system model consists of one rotary-wing UAV-mounted transmitter and users, cf. Figure 1. The UAV-mounted transmitter is equipped with antenna elements, and the antenna elements are equally spaced forming a uniform linear array (ULA). Moreover, the flight height of the UAV is fixed at to avoid obstacles. Besides, we assume that all users are single-antenna devices. For convenience, we define the set of all users as .
In each scheduling time slot, the UAV transmits independent signals simultaneously to the DL users. Specifically, the transmit signal vector to desired user is given by
where and represent the information symbol for user and the corresponding beamforming vector, respectively. Without loss of generality, we assume , .
In this paper, we assume that the air-to-ground links between the UAV and the users are line-of-sight (LoS) channels. In practice, since UAVs fly in the air such that scatterers are encountered with a low probability, the communication links between the UAV and the ground users are typically LoS-dominated. In particular, the channel vector between the UAV and user is modelled as 
where is a constant with being the wavelength of the center frequency of the carrier. and denote the 3-D Cartesian coordinates of the UAV and user , respectively. Moreover, is the average channel power gain between the UAV and user . Besides, represents the AAR between the UAV and user and is given by 
where is the AoD of the path between the ULA and DL user , and is the separation distance between the antennas equipped at the ULA.
Therefore, the received signal at user is given by
where captures the joint effect of the background noise and the thermal noise at the receive antenna of user . We model as additive complex Gaussian noise with variance , i.e., .
Ii-C Channel State Information Uncertainty
In practice, the stability of the UAV in the air is affected by the random nature of wind gusts . In particular, UAVs suffer from body jittering in the presence of strong wind, and the flight behaviour of the UAV changes with respect to the pitch, yaw, and roll angles , cf. Figure 1. Moreover, varying pitch angles capture the main impact of UAV jittering, since horizontal wind gusts in the lower troposphere are the dominant cause of UAV jittering . As a result, the estimation of the AoD is influenced by the varying pitch angle. In fact, due to the randomness of wind gusts, the onboard sensors of the UAV may not be able to measure the exact pitch angle. Hence, AoD estimation errors occur which leads to imperfect AoD knowledge at the UAV. To capture this effect, we adopt a deterministic model for the resulting AoD uncertainty . Specifically, the AoD between the ULA and DL user , i.e., , is modelled as
where and represent the estimated AoD between the ULA and user and the unknown AoD uncertainty, respectively, cf. Figure 2. Besides, the continuous set contains all possible AoD uncertainties with bounded maximum pitch variation 111In practice, the pitch angle varies between rad to rad .. In practice, the value of depends on the climatic conditions and the UAV model . Then, the imperfect AAR is given by
We note that is a nonlinear function with respect to , which complicates robust resource allocation algorithm design. To tackle this problem, and since the are generally small, for a given , we approximate by applying a first order Taylor series expansion:
Then, the AAR between the UAV and user is modeled as
where and are defined as
respectively. We note that and are the AAR estimate of user and the corresponding linearized AAR uncertainty, respectively.
Remark 1: We note that the linearized AAR model in (7) is employed since is small in practice and to make resource allocation design tractable. In our simulations, we adopt the nonlinear AAR model in (6) to evaluate the proposed resource allocation algorithm.
On the other hand, the user location information at the UAV, provided e.g. by GPS , may be also imperfect due to radio signal interference, satellite shadowing, and atmospheric impairments222In practice, positioning errors in forth-generation long-term evolution (4G LTE) networks are typically in the range from 10 meters to 200 meters, depending on the adopted positioning protocol .. Thus, in this paper, we also take into account the user location uncertainty for robust resource allocation algorithm design. Specifically, since we assume that all users are on the ground, their coordinates are all set to 0. Then, the coordinates of user are modelled as
respectively, where and are the user location estimates available at the UAV, and and denote the respective location uncertainties. Furthermore, we assume that the UAV knows its own location perfectly. In fact, thanks to onboard multi-sensor systems and advanced positioning strategies for UAVs, the positioning accuracy of UAVs can be improved to centimeter level . To simplify notation, we define
where vectors , , , and include the coordinates of the UAV, the actual coordinates of user , the estimated coordinates of user , and the uncertainties of user , respectively. Then, the 3-D Cartesian coordinates of the UAV and user can be expressed equivalently as
and the distance between the UAV and user can be rewritten as
Besides, we define set to collect all possible location uncertainties of user as follows
where is the radius of the circular uncertainty region, whose value depends on the positioning accuracy.
Iii Problem Formulation and Solution
In this section, we formulate the joint power and two-dimensional (2-D) positioning optimization problem for the considered UAV communication system after defining the adopted system performance metric. Then, we solve the resulting problem optimally via SDP relaxation.
Iii-a Problem Formulation
The received signal-to-interference-plus-noise ratio (SINR) of user is given by
In practice, the endurance of the UAVs is restricted by the limited onboard battery capacity . Thus, power-efficient communication for minimization of the UAV transmit power is of utmost importance for UAV-assisted communication systems. Hence, in this paper, we aim to minimize the total UAV transmit power while meeting the QoS requirements of all DL users. The optimal power allocation and 2-D positioning policy for the UAV can be obtained by solving the following optimization problem:
where denotes the -entry of a matrix. Constraint C1 constrains the transmit power of the -th antenna element of the UAV to not exceed the maximum power allowance . In practice, the transmit power of each antenna element is limited individually by the corresponding power amplifier in the analog front-end. Constraint C2 ensures that the QoS requirements of all users are satisfied, and is the minimum SINR required by user for reliable information decoding.
We note that the optimization problem in (19) is non-convex because of the non-convexity of constraint C2. General systematic methods for solving non-convex optimization problems are not known. In addition, constraint C2 involves infinitely many inequality constraints which makes robust resource allocation algorithm design intractable. However, in the next subsection, we will show that the resulting problem can be solved optimally via SDP relaxation.
Iii-B Solution of the Optimization Problem
In this subsection, the problem in (19) is reformulated into an equivalent form, and then the semi-infinite constraint C2 is transformed into linear matrix inequality (LMI) constraints. Finally, we employ SDP relaxation to recast the considered problem as a convex optimization problem, which allows us to solve it optimally in an efficient manner.
To facilitate SDP relaxation, we define , , , and rewrite the problem in (19) as
where , , and in constraints C3 and C4 are imposed to ensure that holds after optimization. Moreover, we note that constraint C2 is a semi-infinite constraint, as the coupled uncertainty variables and are continuous in the sets and , respectively. In order to transform constraint C2 into a tractable form, we first decouple the uncertainty variables by multiplying simultaneously the numerator and the denominator of the fractional term with . Then, we introduce a scalar slack variable , and rewrite constraint C2 equivalently as
Next, we introduce a lemma which can be used to transform constraints C2a and C2b into LMIs with a finite number of constraints.
Lemma 1 (S-Procedure ): Let a function , , , be defined as
where , , and . Then, the implication holds if and only if there exists a such that
provided that there exists a point such that .
Using Lemma 1, the following implication can be obtained: C2a holds if and only if there exist such that,
We apply Lemma 1 to C2b and obtain an equivalent LMI constraint:
Now, the only obstacle to solving problem (20) efficiently is the rank-one constraint C4. To handle this problem, we employ SDP relaxation by removing constraint C4, and the considered problem becomes an SDP which is given by
Theorem 1: If , an optimal rank-one beamforming matrix in (29) can always be obtained.
Proof: Please refer to the Appendix.
Theorem 1 unveils that the optimal beamforming matrix, , is rank-one, and hence, allows the extraction of the optimal beamforming vector , despite the imperfect AoD knowledge and the user location uncertainty at the UAV.
Remark 2: In this paper, to make the resource allocation design tractable, we design the beamforming vectors for the linearized AAR model in (10). However, this approximation may lead to a violation of the QoS constraint C2 for the actual nonlinear AAR model in (6). To circumvent this problem, we solve (29) for slightly higher minimum required SINRs , where is a small positive number, which is chosen sufficiently large to ensure that C2 is also met for the nonlinear AAR model.
|Carrier center frequency and bandwidth||GHz and kHz|
|ULA antenna element separation,||meter|
|UAV fixed flight altitude,||meters|
|DL user noise power,||dBm|
|UAV maximum per-antenna transmit power,||dBm|
|Minimum required SINR at user ,||dB|
Iv Simulation Results
In this section, the performance of the proposed resource allocation scheme is investigated via simulations. The simulation parameters are listed in Table I. Specifically, there are users which are uniformly and randomly distributed within a single cell of radius meters. The UAV location coordinates , the estimated user location coordinates , and the estimated AoD between the UAV and user , , are known at the UAV. The location uncertainty area of user is assumed to be a circle with a radius meters, unless specified otherwise. For ease of presentation, in the sequel, we define the maximum normalized estimation error of the AoD between the UAV and user as , where , . Moreover, we employ the nonlinear AAR model in (6) for all simulations. We choose dB for all results shown as this ensured that the desired SINR is achieved for the proposed scheme in all considered cases. Besides, the results presented in this section are obtained by averaging over 1000 channel realizations.
To illustrate the power savings achieved by the proposed scheme, we compare with two baseline schemes. For baseline scheme 1, we adopt zero-forcing beamforming (ZF-BF) at the UAV such that multiuser interference is avoided at the users. Specifically, based on the estimated AoD, the direction of beamforming vector for desired user is fixed and lays in the null space of all the other users’ channels. Then, we jointly optimize the UAV coordinates and the power allocated to under the SDP formulation subject to constraints , , and C3 as in (29). For baseline scheme 2, we employ maximum ratio transmission (MRT), i.e., we set the beamforming vector as , where and are the allocated power and the channel vector of the -th user, respectively. Then, the allocated power and the UAV 2-D positioning vector are jointly optimized for problem (29) subject to constraints , , and C3. In addition, since for most channel realizations the baseline schemes can not simultaneously fulfill the per-antenna power constraint and the QoS requirements, we omit constraint C1 for both baseline schemes to obtain feasible solutions.
In Figure 3, we study the average total transmit power versus the maximum normalized AoD estimation error, , for users, minimum required user SINRs dB, and different numbers of transmit antennas at the UAV, . As can be observed, the average total transmit powers for the proposed scheme and the baseline schemes increase monotonically with increasing . This can be explained by the fact that, as the AoD estimation error increases, it is more difficult for the UAV to perform accurate DL beamforming. Hence, the UAV has to transmit the information signal with higher power to meet the QoS requirements of the users. Moreover, a significant amount of transmit power can be saved by increasing the number of UAV antennas. This is due to the fact that the extra degrees of freedom provided by the additional antennas facilitate a more power efficient resource allocation. On the other hand, the two baseline schemes require a significantly higher total transmit power compared to the proposed scheme. In particular, for the two baseline schemes, the UAV transmitter cannot fully exploit the available degrees of freedom since the beamforming vector is partially fixed.
Figure 4 illustrates the average total transmit power versus the minimum required user SINRs, , for users and different maximum normalized channel estimation errors, . The UAV has transmit antennas. As expected, the average total transmit power of the proposed resource allocation scheme is monotonically nondecreasing with respect to the minimum SINR threshold . This is due to the fact that to meet a larger minimum required SINR in constraint , the UAV has to transmit with higher power. Moreover, it can be observed that the total transmit power for the proposed scheme increases with increasing . In fact, with increasing user location uncertainty, the UAV is forced to use a less focused beamformer to cover a larger area such that a higher transmit power is needed to satisfy the users’ QoS requirements. In Figure 4, we also show the average total transmit power of a non-robust scheme. In particular, for the non-robust scheme a similar optimization problem as in (29) is formulated but the estimated AoD and user location are treated as perfect. Then, using the actual AoDs and user locations, the transmit power allocated to the beamforming vectors is increased until the QoS requirements of the users are satisfied. Both the non-robust scheme and the two baseline schemes result in a higher total transmit power compared to the proposed robust scheme for the entire considered range of .
The robust resource allocation design for multiuser DL UAV communication systems was studied in this paper. We formulated the algorithm design as an optimization problem for minimization of the total UAV transmit power taking into account the QoS requirements of the users, the AoD imperfectness caused by UAV jittering, and a user location uncertainty. Thereby, the AAR was linearized with respect to the AoD estimation error. Due to the intractability of the resulting non-convex problem, we transformed it into an equivalent problem by replacing its semi-infinite constraints with LMI constraints. Subsequently, the reformulated problem was solved optimally by employing SDP relaxation. The approximation error introduced by the linearization of the AAR was accounted for by a small increase of the minimum required SINR. Our simulation results revealed dramatic power savings enabled by the proposed robust scheme compared to two baseline schemes. Besides, the robustness of the proposed scheme with respect to UAV jittering and user location uncertainty was confirmed.
Appendix- Proof of Theorem 1
We can verify that the relaxed problem in (29) is jointly convex with respect to the optimization variables and the Slater’s condition is satisfied. As a result, strong duality holds, and the optimal solution of the primal problem can be obtained by solving the dual problem. Thus, we first write the Lagrangian function of the problem in (29) in terms of beamforming matrix as follows:
where denotes the collection of primal and dual variables and constants that do not affect the proof. The are the Lagrange multipliers associated with constraint C1. Matrix is the Lagrange multiplier associated with constraint . Matrix is the Lagrange multiplier associated with the positive semidefinite constraint C3. Therefore, the dual problem of (29) is given by
Then, we study the structure of the optimal of dual problem (29) based on the Karush-Kuhn-Tucker (KKT) conditions. The KKT conditions for the optimal are given by
and is defined as .
Next, we reveal that is a positive semidefinite matrix by contradiction. Specifically, if is a negative definite matrix, then from (35), must be a full-rank positive definite matrix. Considering the KKT condition in (33), this implies which cannot to be the optimal solution for . Therefore, we focus on the case where is a positive semidefinite matrix in the rest of the proof. Due to the KKT condition in (32), which indicates that matrix is also positive semidefinite, we have
denotes the maximum eigenvalue of matrix. Reviewing the KKT condition in (35), for the case where , we can see that matrix turns into a positive definite matrix with full rank. Again, this leads to which contradicts the positive minimum required SINR . Thus, for the optimal solution, the maximum eigenvalue of matrix must fulfill . Since the users are randomly distributed within UAV’s service area, the case where multiple eigenvalues have the same value occurs with probability zero. Hence, we focus on the case where has a unique maximum eigenvalue which leads to . Moreover, in order to obtain a bounded optimal dual solution, we span the null space of by a vector , i.e., , where
is the unit-norm eigenvector of matrixcorresponding to the maximum eigenvalue . As a result, for , the optimal beamforming matrix satisfies and can be expressed as
where is a parameter which guarantees that the per-antenna transmit power satisfies constraint C1.
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