Due to the high flexibility and mobility of unmanned aerial vehicles (UAVs) offered to wireless communication systems, several interesting applications of UAV have been proposed , such as mobile base stations , mobile relays , and mobile data collections , etc. In practice, the total energy budget for maintaining stable flight and communication is limited by the onboard battery capacity. Hence, energy efficiency has become an important figure of merit for UAV-based communications. For example, the authors in  studied the energy efficiency maximization for wireless sensor networks via jointly optimizing the weak up schedule of sensor nodes and UAV’s trajectory. Yet, the flight power consumption of the system was not considered which contributes a significant portion of total power consumption in the systems. Besides, a UAV trajectory design was developed to optimize the system energy efficiency in . However, the investigation of variable speed as well as transmit power allocation strategy for communications were not conducted which plays an important role for the design of energy-efficient UAV systems. In , the authors compared the delivery ratio and average delay of UAV-based wireless communication systems with constant speed, variable speed, and adaptive speed of the UAV, for reducing the system energy consumption. Yet, the study was limited to the case of multiple sensors deployed in a specific environment and their results cannot be applied to the general case with different system topologies. Furthermore, although orthogonal frequency division multiple access (OFDMA) has been commonly adopted in conventional communication systems, an energy-efficient trajectory and resource allocation design enabling secure UAV-OFDMA wireless communication systems has not been reported in the literature yet.
Meanwhile, since the line-of-sight (LoS) paths dominate the air-to-ground communication channels, UAV-based communications are susceptible to potential eavesdropping. Thus, there is an emerging need for designing secure UAV-based communication. For instance, the authors in  proposed a joint power allocation and trajectory design to maximize the secrecy rate in both uplink and downlink systems. In , secure energy efficiency maximization for UAV-based relaying systems was studied. However, both works only considered the case of single-user and the proposed designs in [7, 8] are not applicable to the case of multiple users. Besides, the availability of the eavesdropper location was assumed in [7, 8], which is generally over optimistic. Although  studied the resource allocation design for secure UAV systems by taking into account the imperfect channel state information (CSI) of an eavesdropper, the energy efficiency of such systems is still an unknown.
In this paper, we tackle the aforementioned problems via optimizing the trajectory and resource allocation strategy for energy-efficient secure UAV-OFDMA systems with multiple legitimate users and the existence of a potential eavesdropper. Particularly, the malicious eavesdropper is located at an uncertain region between the UAV’s initial location and its destination. By exploiting the high flexibility of the UAV, one can either reduce its transmit power or fly away from the uncertain region centered at the eavesdropper to guarantee secure communications for legitimate users. We aim to propose a joint design of resource allocation and trajectory to maximize the system energy efficiency while considering the maximum tolerable signal leakage of the eavesdropper and the minimum individual user data rate requirement. An iterative algorithm is proposed to achieve a suboptimal solution of the design problem. Simulation results unveil that the performance of our proposed algorithm offers a considerable system energy efficiency compared to a baseline scheme adopting a straight flight trajectory and a constant speed.
Ii System Model
A UAV-based OFDMA communication system is considered which consists of a UAV serving as a transmitter, legitimate users, and a potential eavesdropper, as shown in Fig. 1. All the transmitter and receivers are single-antenna devices. We assume that the total bandwidth and the time duration of the system are divided equally into subcarriers and time slots, respectively. In this system, we assume that the UAV flies at a constant altitude and all the ground nodes remain steady for time slots. The distance between the UAV and user at time slot is given by
where represents the location of the ground user , and represents the horizontal location of the UAV at time slot . Similarly, the distance between the UAV and the potential eavesdropper at time slot can be modeled by
denotes the estimated location of the eavesdropper anddenotes the estimation error of . The estimation error satisfies
where is the radius of the uncertain circular region surrounding the estimated location of the eavesdropper.
To facilitate the design of energy-efficient resource allocation, the system power consumption is defined as follows. The flight power consumption for rotary-wing UAV at time slot with respect to (w.r.t.) velocity is given by :
where the notations and the physical meanings of the variables in (4) are summarized in Table I. The total power consumption at time slot in Joules-per-second (J/sec) includes the communication power and flight power consumptions which can be modeled as
where denote the power allocation of user on subcarrier at time slot and denotes the constant circuit power consumption. Variable represents that subcarrier is assigned to user at time slot . Otherwise, .
We assume that both channels from the UAV to users and from the UAV to the eavesdropper are dominated by LoS links. Thus, the channel power gain between the UAV and user in time slot can be characterized by the commonly adopted free-space path loss model [2, 10, 11] which is given by
where represents the channel power gain at a reference distance of 1 meter. The data rate for user on subcarrier at time slot is given by
where represents the subcarrier bandwidth and is the power spectral density of the additive white Gaussian noise (AWGN). On the other hand, the signal-to-noise ratio (SNR) leakage between the UAV and the potential eavesdropper on subcarrier for user at time slot is given by
Thus, the system energy efficiency in bits-per-Joule (bits/J) is defined as
where the user scheduling variable set as , the transmit power variable set as , the UAV’s trajectory variable set as , and the velocity variable set as .
|Blade angular velocity in radians/second|
|Rotor radius in meter|
|Air density in|
|Rotor solidity in|
|Rotor disc area in|
|Blade profile power in hovering status in W|
|Induced power in hovering status in W|
|Mean rotor induced velocity in forwarding flight in m/s|
|Fuselage drag ratio|
Iii Problem Formulation
The energy-efficient design of user scheduling, power allocation, UAV’s trajectory, and flight velocity is formulated as the following optimization problem:
Note that and are user scheduling constraints such that each subcarrier at each time slot can be assigned to at most one user111The extension to non-orthogonal multiple access [12, 13] and massive multiple-input multiple-output [14, 15] will be considered in our future work. to avoid multiple access interference. is the non-negative power constraint. in is the peak transmit power at each time slot. in is the maximum limitation for total power consumption at each time slot. in denotes the minimum required individual user data rate. in is the maximum tolerable SNR threshold for the potential eavesdropper in eavesdropping the information of user on subcarrier . Note that constraint takes into account the location uncertainty of the potential eavesdropper. and indicate the required UAV’s initial and final locations, respectively. draws the connections between the velocity of the UAV and its displacement at two consecutive time slots. is the UAV’s the maximum flight velocity constraint. in constraint is the maximum allowable acceleration in a given time slot. Note that the flight velocity of a UAV can be expressed as a function of its trajectory for a given constant time slot duration . Yet, expressing the flight power consumption as a function of trajectory would complicate the resource allocation design. Therefore, we introduce the flight velocity variable to simplify the problem formulation.
In the considered system, secure communication can be guaranteed when holds. In particular, the parameters and can be chosen by the system operator to provide flexibility in designing resource allocation algorithms for different applications requiring different levels of communication security and the adopted formulation has been widely adopted, e.g. .
Iv Problem Solution
The formulated problem in (10) is non-convex, which generally cannot be solved efficiently by conventional convex optimization methods. To facilitate a low computational complexity design of resource allocation and trajectory, we divide the problem (10) into two sub-problems and solve them iteratively to achieve a sub-optimal solution using the alternating optimization approach . In particular, sub-problem 1 aims to optimize the user scheduling and the transmit power allocation for a given feasible UAV’s trajectory and flight velocity . On the other hand, sub-problem 2 aims to optimize the UAV’s trajectory and flight velocity under a given feasible user scheduling and transmit power allocation . Now, we first study the solution of sub-problem 1.
Iv-a Sub-problem 1: Optimizing User Scheduling and Transmit Power Allocation
For a given UAV’s trajectory and flight velocity , we can express sub-problem 1 as the following optimization problem:
In order to solve sub-problem 1 in (11), we introduce an auxiliary variable , and the transformed problem is given by
Note that since the trajectory of the UAV is given for sub-problem 1, the minimum distance between the UAV and the potential eavesdropper is known. In other words, is a constant for a given uncertain area of the eavesdropper. The main obstacle in solving (12) arises from the binary user scheduling constraint and the objective function in fractional form. First, we handle the binary constraint. In particular, we follow the approach as in , and relax the subcarrier variable such that it is a real value between and , i.e.,
Meanwhile, the relaxed version of serves as a time-sharing factor for user in utilizing subcarrier at time slot . Note that the relaxation is asymptotically tight even if the number of subcarriers is small, e.g. subcarriers .
Then, we tackle the fractional-form objective function. Let be the maximum system energy efficiency of sub-problem 1 which is given by
where and are the sets of the optimal user scheduling and power allocation, respectively. is the feasible solution set spanned by constraints –. Now, by applying the fractional programming theory , the objective function of (12) from a fractional form can be equivalent transformed into a subtractive form. More importantly, the optimal value of can be achieved if and only if
for and .
Therefore, we can apply the iterative Dinkelbach method  to solve (12). In particular, for the -th iteration for sub-problem 1 and a given intermediate value , we need to solve a convex optimization as follows:
where is the optimal solution of (18) for a given . Then, the intermediate energy efficiency value should be updated as for each iteration of the Dinkelbach method until convergence222Note that the convergence of the Dinkelbach method is guaranteed if the problem in (18) can be solved optimally ..
In the following, we discuss the solution development for solving (18). Since problem (18) is jointly convex w.r.t. user scheduling as well as transmit power allocation . Also, it satisfies the Slater’s constraint qualification. Therefore, the strong duality holds and the duality gap is zero. Hence, solving the dual problem is equivalent to solving the primal problem of sub-problem 1 in (18). Although we can directly solve (18) via numerical convex program solvers, e.g. CVX, it does not shed light on important system design insights such as the impact of optimization variables on the system performance. To this end, we focus on the resource allocation design for solving the dual problem. Now, we first derive the Lagrangian function of (18):
where , , , , and denote the Lagrange multipliers for constraints , , , , and , respectively. Constraints and will be considered in the Karush-Kuhn-Tucker (KKT) conditions when deriving the optimal solution in the following. Then, the dual problem of (18) is given by
Subsequently, the dual problem is solved iteratively via dual decomposition. In particular, the dual problem is decomposed into two nested layers: Layer 1, maximizing the Lagrangian over user scheduling and power allocation in (20), given the Lagrange multipliers , , , , and ; Layer 2, minimizing the Lagrangian function over , , , , and in (20), for a given user scheduling and power allocation .
Solution of Layer 1 (Power Allocation and User Scheduling): We assume that and denote the optimal solutions of sub-problem 1. Then, the optimal power allocation for user on subcarrier at time slot is given by
where . The optimal power allocation in (21) is the classic multiuser water-filling solution. The water-levels for different users, i.e., , are generally different on different subcarrier and time slot . In particular, on one hand, the Lagrange multiplier forces the UAV to increase the transmit power to satisfy the minimum required individual user data rate of the system. On the other hand, the Lagrange multiplier adjusts the water-level such that the maximum SNR leakage constraint in can be satisfied. Besides, to find the optimal subcarrier allocation, we take the derivative of the Lagrangian function w.r.t. which yields
In fact, denotes the marginal benefit of the system performance improvement when subcarrier is allocated to user at time slot . As (22) is independent of , due to constraint , the optimal user scheduling for each subcarrier and time slot is given by
Solution of Layer 2 (Master Problem): To solve Layer 2 master minimization problem in (20), the gradient method is adopted and the Lagrange multipliers can be updated by
where is the iteration index for sub-problem 1 and , are step sizes satisfying the infinite travel condition . Then, the updated Lagrangian multipliers in (24)–(27) are used for solving the Layer 1 sub-problem in (20) via updating the resource allocation policies . Since the user scheduling and power allocation variables are finite and non-decreasing over iterations for solving the problem, the convergence of the proposed algorithm to the optimal solution of sub-problem 1 is guaranteed. The proposed Algorithm for sub-problem 1 is summarized in Algorithm 1.
Iv-B Sub-problem 2: Optimizing UAV’s Trajectory and Flight Velocity
For given user scheduling and transmit power allocation , we can express sub-problem 2 as
The problem in (28) is non-convex and non-convexity arises from the objective function and constraint . To facilitate the derivation of solution, we introduce a slack variables to transform the problem into the following equivalent form:
It can be proved that (28) and (29) are equivalent as the inequality constraint is active at optimal solution of (29). Then, we handle the location uncertainty of the eavesdropper by rewriting constraint as:
Note that the location uncertainty introduces an infinite number of constraints in . To circumvent the difficulty, we apply the -Procedure  and transform into a finite number of linear matrix inequalities (LMIs) constraints. In particular, if there exists a variable such that
Note that in constraint (33) is non-convex. To design a tractable resource allocation, the successive convex approximation (SCA) is applied [5, 7, 21]. In particular, for a given feasible solution in the -th main loop iteration for sub-problem 2, since , we obtain a lower bound of equation (33), which is given by