Robust Trajectory and Transmit Power Optimization for Secure UAV-Enabled Cognitive Radio Networks

The explosive increase of wide-band service requirements and the unprecedented proliferation of mobile devices result in a severe spectrum scarcity issue. In order to alleviate the spectrum crunch and to meet the increasing demand for high data rates, cognitive radio (CR) that aims to realize spectrum sharing between a primary network and a secondary network has been proposed to improve the spectral efficiency [1], [2]. Specifically, secondary users (SUs) coexist with primary users (PUs) in the same frequency spectrum while ensuring that PUs can tolerate the interference caused by SUs. Due to the promised high-spectral efficiency brought by CR, it has been widely investigated in traditional terrestrial communication networks such as cellular networks, wireless sensor networks, and relaying networks [3], [4]. However, in CR networks, the transmit power of a secondary base station should be restricted in order to guarantee the quality of service (QoS) of the PUs. What’s worse, the harsh channel fading conditions of terrestrial communication networks further degrade the performance of SUs. Thus, it is of utmost importance to study how to improve the performance of SUs in CR networks.

Recently, unmanned aerial vehicles (UAVs) communications have been extensively investigated for wireless communications in both military and civil applications due to the advantages of UAVs in terms of highly controllable mobility and flight flexibility [5]-[7]. Compared with terrestrial CR networks, which are dominated by non-line-of-sight (NLoS) links due to multipaths and blockages, UAV-enabled CR networks can flexibly deploy UAVs in the air according to the actual environment [8]. In particular, the communication performance of ground nodes can be greatly improved since the air-to-ground channel is dominated by line-of-sight (LoS) with generally a small path loss exponent [9]. Moreover, the LoS channel facilitates the establishment of high data rate communications between UAVs and ground nodes. In practice, the throughput of UAV-enabled CR networks can be significantly improved by designing the UAV’s trajectory that exploits the extra design degrees of freedom [10], [11]. Thus, it is promising to exploit UAVs to establish CR networks that can improve the performance of SUs. However, due to the existence of strong LoS links from UAVs to PUs in UAV-enabled CR networks, the interference caused by the cognitive UAV transmitters to PUs is also increased substantially [12]. Thus, this is a major challenge for the design of the UAV’s trajectory and transmission power in UAV-enabled CR networks.

Moreover, due to the broadcast nature of wireless communications and the existence of LoS channels, potential eavesdroppers (Eves) on the ground are more prone to illegally intercept confidential information sent to legitimate recipients. It results in a significant security challenge for UAV-enabled CR networks. Besides conventional cryptography encryption methods enabling secure information transmission, physical-layer security technology has emerged as a promising alternative approach to guarantee the security of wireless communication systems by exploiting the physical layer characteristics of wireless channels [13]-[17]. In [18], the physical layer security of a UAV network was studied to jointly design the trajectory and transmission power of the UAV in order to maximize the minimum average secrecy rate of all users. As a significant metric for physical layer security, it was shown that the secrecy rate is closely related to channel conditions [19]. Thus, it is necessary to design the trajectory in UAV-enabled CR networks to improve the secure performance of SUs.

To unlock the potential of UAV-enabled communication systems, joint UAV’s trajectory and transmit power design has been studied under different scenarios [29]-[33]. However, these works assumed perfect channel estimations, which depends on the availability of accurate estimation for the locations of ground nodes. In practice, it is extremely difficult to acquire the exact location of PUs/Eves due to the existence of location estimation errors and quantization errors. Especially in UAV-enabled CR networks, it is challenging for a UAV to keep tracking the exact location of PUs/Eves via a camera or a synthetic aperture radar due to the energy limitation of the UAV [20]. As a result, the trajectory design and power allocation based on the assumption of perfect location information may lead to a large performance loss for UAV-enabled CR networks in practice. Recently, although the robust trajectory and transmit power designs have been considered in [16], [17], only the worst-case scenario has been studied. Specifically, in the worst-case scenario, the location errors are modeled by bounded sets. However, the bounded location error model usually leads to a conservative resource allocation which underestimates the actual communication performance of UAV systems. Thus, in the considered UAV-enabled CR network, we not only focus on the robust design based on the bounded location error model, but also consider the robust design based on the probabilistic location error model. For the latter model, the location errors are captured by a probability distribution [21], and outage probability constraints are proposed to replace the excessively conservative worst-case constraints adopted in the existing robust designs. Under this model, UAV-enabled CR networks can achieve better performance. In particular, the latter robust design is more suitable for scenarios with delay-sensitive communication devices [27]. Our proposed two robust UAV’s trajectory and transmit power optimization schemes in this paper are promising to improve the secure communication performance of SUs against the channel estimation error and offer a better protection to the performance of PUs. Moreover, the studied fair performance comparison between the two proposed robust designs is helpful for the performance analysis in UAV-enabled CR networks with practically imperfect channel state information (CSI).

In this paper, we focus on designing robust trajectory and transmit power for secure UAV-enabled CR networks under two practical cases that the locations of Eves and PUs cannot be accurately obtained. Two practical estimation error models are considered, namely, the bounded location error model and the probabilistic location error model. There is one SU, multiple Eves, and multiple PUs in the networks. It is assumed that the UAV knows the accurate location of the SU, but only knows the approximate regions where PUs and Eves are located. Our goal is to maximize the average secrecy rate subject to the UAV’s maximum mobility constraint, the UAV’s transmit power constraint, and the average interference power constraint. To the best of our knowledge, this is the first work that studies the robust design of secure UAV-enabled CR networks under these two location error models. Although the bounded location error model has been studied in [16], [17], and the probabilistic location error model has been studied in [21], the proposed schemes in [16], [17] are only applicable to conventional wireless terrestrial communication networks and cannot work in UAV-enabled CR networks, and the physical-layer security issue in CR networks has not been considered in [21], and thus the security of the UAV-enabled network cannot be guaranteed. Different from existing works, the main contributions of this paper are summarized as follows.

• •

  The robust trajectory and transmit power design problem is first studied in the secure UAV-enabled CR networks under the bounded location error model. The considered worst-case robust average secrecy rate maximization (WCR-ASRM) problem is intractable due to its non-convexity and the existence of semi-infinite constraints. To tackle the intractability, we propose a suboptimal algorithm to iteratively solve an approximation problem obtained by using the successive convex approximation (SCA) method and $\mathcal{S}$-Procedure.

• •

  An outage-constrained robust average secrecy rate maximization (OCR-ASRM) problem is formulated for secure UAV-enabled CR networks under the probabilistic location error model. The relationship between the two location error models is studied and highlighted. The Bernstein-type inequalities are exploited to approximate the probabilistic constraints which are difficult to handle since they have no closed-from expressions. The original problem is tackled by an iterative algorithm based on the SCA method and a suboptimal solution is obtained.

• •

  Simulation results and theoretical derivations show that the robust trajectory and transmit power design under the probabilistic location error model achieves a higher average secrecy rate and a lower algorithm complexity compared to that under the bounded location error model. Besides, our proposed robust schemes can achieve a larger average secrecy rate compared to benchmark schemes.

The rest of this paper is organized as follows. In Section II, the system model is presented. Section III presents a robust trajectory and transmit power design problem under the bounded location error model. Section IV presents a robust trajectory and transmit power design problem under the probabilistic location error model. In Section V, simulation results are presented. Finally, this paper is concluded in Section VI.

Notations: Boldface capital letters and boldface lower case letters represent matrixes and vectors, respectively. ${\mathbb{C}^{M{\times}N}}$, $\mathbb{H}^{N}$, $\mathbb{R}$, and $\mathbf{I}_{n}$ denote the set of $M$-by-$N$ complex matrixs, the set of $N$-by-$N$ Hermitian matrices, the set of all real numbers, and a $n\times n$ identity matrix, respectively. $\mathbf{x}^{T}$ represents the transpose of a vector $\mathbf{x}$. $\Re\{\mathbf{x}\}$ denotes the real part of vector $\mathbf{x}$. $\left\|\cdot\right\|$ denotes the Euclidean norm of a vector. $\mathbf{x}\sim\mathbb{N}(\bm{\mu},\bm{\Sigma})$ means that $\mathbf{x}$ is a real-valued random vector following a Gaussian distribution with mean $\bm{\mu}$ and covariance matrix $\bm{\Sigma}$. $[a]^{+}$ denotes the function $\max(a,0)$. ${{\nabla}^{2}}f\left(x,y\right)$ denotes a square matrix composed of second-order partial derivatives of the multivariate function $f\left(x,y\right)$. $\mathbf{A}{\succeq}{\mathbf{0}}$ means that $\mathbf{A}$ is a Hermitian positive semi-definite matrix. $\text{Tr}(\mathbf{A})$ and $\text{vec}(\mathbf{A})$ denote the trace operation and the vectorization, respectively.

According to the works in [16] and [17], in practice, the estimation errors for the location of PUs and Eves can be modeled as the bounded location error model. The bounded location error model for the location of the $l$th PU is given as

and the bounded location error model for the location of the $k$th Eve is given as

where ${{\mathbf{\bar{q}}}_{l}}$ and ${{\mathbf{\bar{e}}}_{k}}$ denote the estimates of the location vectors ${{\mathbf{q}}_{l}}$ and ${{\mathbf{e}}_{k}}$, respectively; $\Delta{{\mathbf{q}}_{l}}$ and $\Delta{{\mathbf{e}}_{k}}$ denote the location estimation errors of ${{\mathbf{q}}_{l}}$ and ${{\mathbf{e}}_{k}}$, respectively; ${{\Psi}_{l}}$ and ${{\Omega}_{k}}$ represent the uncertainty regions of ${{\mathbf{q}}_{l}}$ and ${{\mathbf{e}}_{k}}$, respectively; ${{\omega}_{l}}$ and ${{\xi}_{k}}$ denote the radii of the uncertainty regions ${{\Psi}_{l}}$ and ${{\Omega}_{k}}$, respectively.

Based on the bounded location error model, the WCR-ASRM problem subject to the UAV mobility constraint, the transmit power constraint, and the average interference power constraint is formulated as

where $\mathcal{P}$ and $\mathcal{Q}$ denote the sets of variables $\left\{P\left[n\right]\right\}_{n=1}^{N}$ and $\left\{\mathbf{q}\left[n\right]\right\}_{n=1}^{N}$, respectively. (10b) can guarantee that the average interference power caused by the UAV to each PU does not exceed the IT threshold for all location estimation errors satisfying (8b). Owing to the nonlinear objective function and the non-convex constraint in (10b), the original optimization problem (10) is non-convex and challenging to solve. Moreover, the infinite inequality constraint caused by the uncertain region complicates the problem. Note that the operator $[\cdot]^{+}$ in the objective function can be safely omitted without affecting the optimal value of the optimization problem. For more relevant proof details, please refer to the work in [16].

In general, the formulated problem is non-convex and intractable. As a compromise approach, we aim to design a suboptimal algorithm to achieve an efficient solution. To this end, slack variables ${{\alpha}_{n}}$, ${{\beta}_{n}}$, and ${{\gamma}_{l,n}}$, where $n\in\mathcal{N}$ and $l\in\mathcal{L}$, are introduced. Thus, (10) can be equivalently expressed as

where $\bm{\alpha}$, $\bm{\beta}$, and $\bm{\gamma}$ denote the sets of slack variables $\{{\alpha}_{n}\}_{n=1}^{N}$, $\{{\beta}_{n}\}_{n=1}^{N}$, and $\{\{{\gamma}_{l,n}\}_{l=1}^{L}\}_{n=1}^{N}$, respectively. Note that (11d) and (11e) are infinite inequality constraints due to the uncertainty of the location information. In order to tackle them, the $\mathcal{S}$-Procedure is introduced as follows.

Lemma 1 ($\mathcal{S}$-Procedure) [37]: Let ${{f}_{j}}\left(\mathbf{x}\right)={{\mathbf{x}}^{T}}{{\mathbf{A}}_{j}}\mathbf{x}+2\Re\left\{\mathbf{b}_{j}^{T}\mathbf{x}\right\}+{{c}_{j}}$, $j\in\left\{1,2\right\}$, where $\mathbf{x}\in{\mathbb{C}^{N\times 1}}$, ${{\mathbf{A}}_{j}}\in{{\mathbb{H}}^{N}}$, ${{\mathbf{b}}_{j}}\in{\mathbb{C}^{N\times 1}}$, and ${{c}_{j}}\in\mathbb{R}$. Then, the expression ${{f}_{1}}\left(\mathbf{x}\right)\leq 0\Rightarrow{{f}_{2}}\left(\mathbf{x}\right)\leq 0$ holds if and only if there exists a $\mu\geq\text{0}$ such that

provided that there exists a vector ${\mathbf{\hat{x}}}$ such that ${{f}_{\text{1}}}\left({\mathbf{\hat{x}}}\right)<0$.

By applying the $\mathcal{S}$-Procedure and introducing slack variables $\theta_{k,n}$, $\chi_{l,n}$, $\lambda_{k,n}$, and $\mu_{l,n}$, where $n\in\mathcal{N}$, $l\in\mathcal{L}$, and $k\in\mathcal{K}$, (11) can be expressed equivalently as

where ${{c}_{k,n}}$ and ${{c}_{l,n}}$ are given as, respectively,

where $\bm{\theta}$, $\bm{\chi}$, $\bm{\lambda}$, and $\bm{\mu}$ denote the sets of slack variables $\{\{\theta_{k,n}\}_{k=1}^{K}\}_{n=1}^{N}$, $\{\{\chi_{l,n}\}_{l=1}^{L}\}_{n=1}^{N}$, $\{\{\lambda_{k,n}\}_{k=1}^{K}\}_{n=1}^{N}$, and $\{\{{\mu}_{l,n}\}_{l=1}^{L}\}_{n=1}^{N}$, respectively. Thus, there are finite numbers of inequality constraints in problem (13). However, (13) is still non-convex due to the existence of variables coupling in (11c), (13b), (13c), (13d), and (13e). To solve this difficulty, we introduce the slack variables, $\tau\left[n\right]={{\left(P\left[n\right]\right)}^{-1}}$ and ${{\varphi}_{n}}={{2}^{{{\beta}_{n}}}}-1$, where $n\in\mathcal{N}$. Then, the problem in (13) can be rewritten as

where $\bm{\tau}$ and $\bm{\varphi}$ denote the sets of the slack variables $\{\tau_{n}\}_{n=1}^{N}$ and $\{\varphi_{n}\}_{n=1}^{N}$, respectively. Note that (15b) and (15c) are convex constraints. However, (15d) is a non-convex constraint which is verified in the following proposition.

Proposition 1 : Let $f\left(x,y\right)\triangleq\frac{1}{xy}$, where $x>0$ and $y>0$. Then, $f\left(x,y\right)$ is jointly convex with respect to $x$ and $y$.

Meanwhile, the objective function in (15a) is non-concave, since $-{{\log}_{2}}\left({{\varphi}_{n}}+1\right)$ is convex. Moreover, constraints (13b) and (13d) are non-convex due to the existence of the second-order variables $x^{2}[n]$ and $y^{2}[n]$ in $c_{k,n}$ and $c_{l,n}$. Fortunately, the above functions are convex which are lower bounded by the corresponding first-order Taylor approximations. For a given set of feasible points, $\left\{\tilde{\tau}\left[n\right],\mathbf{\tilde{q}}\left[n\right],{{{\tilde{\alpha}}}_{n}},{{{\tilde{\varphi}}}_{n}}\right\}_{n=1}^{N}$, inequalities (16) shown at the bottom of this page must hold $\forall n\in\mathcal{N}$. Then, the lower bound value of (15) can be obtained by solving the following problem:

where $\tilde{c}_{k,n}$ an $\tilde{c}_{l,n}$ are respectively given as, respectively,

Note that (17) is a convex optimization problem, which can be efficiently solved by using the standard convex optimization method, e.g., the interior-point method [37]. Then, Algorithm 1 based on the SCA method is proposed to tighten the first-order Taylor approximations. In this case, the suboptimal solutions of the original optimization problem (10), $\mathcal{P}^{*}$ and $\mathcal{Q}^{*}$, can be obtained by iteratively solving (17).

Remark 1: Due to the inequalities in (16b), (16c), and (16d), the left hand side (LHS) of (15d) is a lower bound of (17b), and the LHS of (13b) and (13d) are upper bounds for (17c) and (17d), respectively. In other words, (17b), (17c), and (17d) hold implying (15d), (13b), and (13d) hold, respectively. Hence, the solution of problem (17) is a feasible suboptimal solution of problem (15).

Remark 2: The convergence of Algorithm 1 is guaranteed. Since (16a) is a lower bound of $-{{\log}_{2}}\left({{\varphi}_{n}}+1\right)$, the objective function of (17) is a lower bound of the objective function of (15). Note that the objective value of (15) is equal to that of (17) only at feasible points $\left\{\tilde{\tau}\left[n\right],\mathbf{\tilde{q}}\left[n\right],{{{\tilde{\alpha}}}_{n}},{{{\tilde{\varphi}}}_{n}}\right\}_{n=1}^{N}$, and the objective value of (15) is larger than that of (17) with the solution of (17). Thus, the objective value of (15) with the solution of (17) is no less than that with the solution $\left\{\tilde{\tau}\left[n\right],\mathbf{\tilde{q}}\left[n\right],{{{\tilde{\alpha}}}_{n}},{{{\tilde{\varphi}}}_{n}}\right\}_{n=1}^{N}$. It means that the objective value of (17) and (15) is non-descending over iteration.

Remark 3: In step 1 of Algorithm 1, the convex optimization problem in (17) can be solved by using the interior-point method (IPM) [37]. According to [38], the computational complexity of Algorithm 1 using the IPM can be divided into three parts, namely, the required numbers of iterations of the SCA method, the iteration complexity, and the per-iteration computation cost. It is assumed that the maximum number of iterations of the SCA method is $T_{1}$ and the accuracy of the iteration is $\epsilon$. Problem (17) has $(K+L)N$ linear matrix inequalities (LMIs) of size 3, $2(K+L+2)N+L+1$ LMIs of size 1. The number of decision variables $m$ is on the order of $(K+L)N$, e.g., $m=\mathcal{O}((K+L)N)$, where $\mathcal{O}(\cdot)$ is the big-O notation. Thus, the total complexity of Algorithm 1 under the bounded location error model is given by (19), as shown at the bottom of this page. Note that the proposed Algorithm 1 has a polynomial time computational complexity which is suitable for practical implementation.

According to the work in [21], it is assumed that the locations of each PU and each Eve are stochastic and follow the Gaussian distribution due to the estimation errors. Specifically, $x$ and $y$ coordinates are independent and identically distributed (i.i.d.) Gaussian random variables. The Gaussian location error models for the location of the $l$th PU and the $k$th Eve, are given as, respectively,

where ${{\mathbf{\bar{q}}}_{l}}$ and ${{\mathbf{\bar{e}}}_{k}}$ denote the estimated location; $\Delta{{\mathbf{q}}_{l}}$ and $\Delta{{\mathbf{e}}_{k}}$ denote the location estimation errors; ${{\varpi}^{2}_{l}}$ and $\varepsilon_{k}^{2}$ are variances of the corresponding location estimation errors.

Based on the probabilistic location error model, the OCR-ASRM problem is studied under the outage probabilistic constraints which is formulated as

where ${\rho}\in(0,1]$ denotes the maximum outage probability associated with the transmission rate of all Eves and ${\phi}\in(0,1]$ denotes the maximum outage probability associated with the interference power of all PUs. Since the location estimation errors of Eves are probabilistic, the problem of maximizing the average secrecy rate (7a) can be transformed into a problem that maximizes (21a) while satisfying the corresponding outage probability constraint (21b) by introducing an auxiliary variable ${\beta}_{n}$. (21c) guarantees that the outage probability of the interference power of the PUs being larger than ${\gamma}_{l,n}$ is less than ${\phi}$. Note that the probability term in (21b) can be regarded as the coupling of probability terms among Eves, which is intractable and complicated. In order to decouple the joint probabilistic constraint into tractable terms, by exploiting the independence between the locations of Eves, we have the following implication:

where $\bar{\rho}=1-{{\left(1-\rho\right)}^{{1}/{K}}}$. (22b) means that the outage probability of mutual information for each Eve being larger than ${\beta}_{n}$ should be lower than $\bar{\rho}$. Then, by introducing the slack variables $\alpha_{n}$, $n\in\mathcal{N}$, a lower bound value of problem (21) can be obtained by solving the following problem:

For a fair comparison between the performance of the WCR-ASRM design and the OCR-ASRM design, the radii of uncertainty regions, ${\Psi}_{l}$ and ${\Omega}_{k}$, are chosen as [39]

where $F_{\chi_{2}^{2}}^{-1}\left(\cdot\right)$ denotes the inverse cumulative distribution function of a Chi-square random variable with 2 degrees of freedom. Thus, the uncertainty radius ${{\omega}_{l}}$ can satisfy $\text{P}{{\text{r}}_{\left\{\Delta{{\mathbf{q}}_{l}}\right\}}}\left\{{{\left\|\Delta{{\mathbf{q}}_{l}}\right\|}^{2}}\leq\omega_{l}^{2}\right\}=1-\phi$ and the uncertainty radius ${{\xi}_{k}}$ can satisfy $\text{P}{{\text{r}}_{\left\{\Delta{{\mathbf{e}}_{k}}\right\}}}\left\{{{\left\|\Delta{{\mathbf{e}}_{k}}\right\|}^{2}}\leq\xi_{k}^{2}\right\}=1-\bar{\rho}$. By choosing ${{\omega}_{l}}$ and ${{\xi}_{k}}$, if constraints (11d) and (11e) are satisfied, constraints (23b) and (21c) must be satisfied. Thus, the WCR-ASRM problem (11) can be considered as a safe approximation to the OCR-ASRM problem (23). Note that the closed-form expressions of constraints (21c) and (23b) are difficult to derive directly due to the outage probability. To convert the probability constraints into deterministic forms, the Bernstein-type inequality is applied to provide a safe approximation of problem (23), which is summarized as follows.

Lemma 2 (The Bernstein-type Inequality) [40]: Define $\mathbf{A}\in{{\mathbb{H}}^{N}}$, $\mathbf{x}\sim\mathbb{N}\left(\mathbf{0},\mathbf{I}\right)$, ${{\mathbf{b}}}\in{\mathbb{C}^{N\times 1}}$, ${{c}}\in\mathbb{R}$, and ${\rho}\in(0,1]$, the following implication holds

where ${{\upsilon}_{1}}$ and ${{\upsilon}_{2}}$ are slack variables.

By applying Lemma 2, an approximate expression of the outage interference power constraint (21c) can be given as

where ${{\eta}_{l,n}}$ and ${{\zeta}_{l,n}}$ are slack variables; ${{\mathbf{A}}_{l}}=\varpi_{l}^{2}{{\mathbf{I}}_{2}}$, ${{\mathbf{b}}_{n}}={{\varpi}_{l}}\left({{{\mathbf{\bar{q}}}}_{l}}-\mathbf{q}\left[n\right]\right)$, and

Note that the first part of (26c) always holds since ${{\zeta}_{l,n}}+\varpi_{l}^{2}\geq 0$. Similar to the outage interference power constraint, the outage transmission rate constraints of eavesdroppers (23b) can be approximated as

where ${{\upsilon}_{k,n}}$ and ${{\varsigma}_{k,n}}$ are slack variables; ${{\mathbf{A}}_{k}}=\varepsilon_{k}^{2}{{\mathbf{I}}_{2}}$, ${{\mathbf{b}}_{k,n}}={{\varepsilon}_{k}}\left({{{\mathbf{\bar{e}}}}_{k}}-\mathbf{q}\left[n\right]\right)$, and

Note that the first part of (28c) always holds since ${{\varsigma}_{k,n}}+\varepsilon_{k}^{2}\geq 0$. Thus, these LMIs in the first part of (26c) and (28c) can be omitted. To tackle the coupling of variables in (23c), (27), and (29), we introduce slack variables $\tau[n]={{\left(P\left[n\right]\right)}^{-1}}$ and $\varphi_{n}={{2}^{{{\beta}_{n}}}}-1$. Thus, the approximation problem for (23) can be rewritten as