# Fairness Enhancement of UAV Systems with Hybrid Active-Passive RIS

Nhan Thanh Nguyen, , Van-Dinh Nguyen, , Hieu Van Nguyen, , Qingqing Wu, ,
Antti Tölli, ,
Symeon Chatzinotas, , and Markku Juntti This research was supported by Academy
of Finland under 6G Flagship (grant 346208), EERA Project (grant 332362),
Infotech Oulu under EEWA Project, and Research Council of Finland Fellowship
(grant 354901). The work of V.-D. Nguyen was supported in part by the
VinUniversity Seed Grant Program. The work of Q. Wu was supported by NSFC
62371289, NSFC 62331022 Guangdong science and technology program under grant
2022A0505050011 and FDCT under Grant 0119/2020/A3. The work of S. Chatzinotas
was supported by the Luxembourg National Research Fund via project 5G-SKY,
ref. FNR/C19/IS/13713801/5G-Sky and project RISOTTI, ref.
FNR/C20/IS/14773976/RISOTTI. A short version of this paper was presented in
the IEEE International Conference on Communications (ICC), 2022, Seoul, South
Korea. Corresponding author: Nhan Thanh Nguyen.N. T. Nguyen, A. Tölli, and M.
Juntti are with Centre for Wireless Communications, University of Oulu,
P.O.Box 4500, FI-90014, Finland (e-mail:{nhan.nguyen, antti.tolli,
markku.juntti}@oulu.fi).V.-D. Nguyen is with the College of Engineering and
Computer Science and also with the Center for Environmental Intelligence,
VinUniversity, Vinhomes Ocean Park, Hanoi 100000, Vietnam (e-mail:
dinh.nv2@vinuni.edu.vn).H. V. Nguyen is with the Faculty of Electronic and
Telecommunication Engineering, The University of Danang, University of Science
and Technology, Da Nang 50000, Vietnam (email: nvhieu@dut.udn.vn).Q. Wu is
with the Department of Electronic Engineering, Shanghai Jiao Tong University,
200240, China (e-mail: qingqingwu@sjtu.edu.cn).S. Chatzinotas is with the
Interdisciplinary Centre for Security, Reliability and Trust (SnT), University
of Luxembourg, L-1855 Luxembourg, (email: symeon.chatzinotas@uni.lu).

###### Abstract

We consider unmanned aerial vehicle (UAV)-enabled wireless systems where
downlink communications between a multi-antenna UAV and multiple users are
assisted by a hybrid active-passive reconfigurable intelligent surface (RIS).
We aim at a fairness design of two typical UAV-enabled networks, namely the
static-UAV network where the UAV is deployed at a fixed location to serve all
users at the same time, and the mobile-UAV network which employs the time
division multiple access protocol. In both networks, our goal is to maximize
the minimum rate among users through jointly optimizing the UAV’s
location/trajectory, transmit beamformer, and RIS coefficients. The resulting
problems are highly nonconvex due to a strong coupling between the involved
variables. We develop efficient algorithms based on block coordinate ascend
and successive convex approximation to effectively solve these problems in an
iterative manner. In particular, in the optimization of the mobile-UAV
network, closed-form solutions to the transmit beamformer and RIS passive
coefficients are derived. Numerical results show that a hybrid RIS equipped
with only $4$ active elements and a power budget of $0$ dBm offers an
improvement of $38\%-63\%$ in minimum rate, while that achieved by a passive
RIS is only about $15\%$, with the same total number of elements.

###### Index Terms:

Beamforming, hybrid active-passive RIS, UAV-enabled communications, trajectory
design, successive convex approximation

##  I Introduction

Unmanned aerial vehicle (UAV)-enabled communications systems have attracted
growing interest in both the industry and research community thanks to their
controllable mobility and cost-effectiveness [1]. However, the blockage of
line-of-sight (LoS) links and severe path loss caused by long-distance
transmissions and obstacles, especially in complex urban environments, may
cause significant system performance degradation [2]. A recently emerging
technology called reconfigurable intelligent surfaces (RISs) can be deployed
to reflect signals to favorable directions and offer passive beamforming gains
to greatly improve system performance [3, 4, 5, 6, 7, 8, 9]. However, the main
barrier to deploying conventional RISs is that its pure passive reflecting
gain may be insufficient to overcome the double path loss in reflecting
channels [10]. This can happen in harsh environments such as when the UAV
flies at a high altitude, or when multiple users are distributed at the edge
of the network. A recently introduced variant of the passive RIS called the
hybrid active-passive RIS, where a few elements are activated to amplify
incident signals [11], can be a promising solution to overcome the
aforementioned challenges.

###  I-A Related Works

RIS-aided UAV communications have been widely studied recently [2, 12, 13, 14,
15, 16, 17, 18, 19, 20]. Specifically, Cao et al. [2] designed an adaptive
RIS-aided transmission protocol to maximize the system throughput via both
numerical method and machine learning. In [12] and [14], a joint design of the
UAV’s trajectory and RIS passive beamforming was proposed to maximize the
system throughput. The works in [21, 13, 16] focused on designing the secure
communication protocol for UAV systems with the presence of eavesdroppers.
More specifically, Li et al. [21] developed an efficient algorithm based on
alternating optimization (AO) to design a robust solution for the UAV’s
trajectory, RIS’s passive beamforming, and transmit power of the legitimate
transmitters under imperfect channel state information (CSI). On the other
hand, Guo et al. [16] proposed effective solutions to the UAV’s trajectory,
UAV’s beamformer, and RIS coefficients based on deep reinforcement learning. A
similar design was considered in [13] for a complex urban environment where
ideal LoS links are unavailable in the air-ground channel. Moreover, the rate
performance of non-orthogonal multiple access [22, 23], ultra-reliable and
low-latency [24], millimeter-wave [15], and THz [18] RIS-aided UAV systems
were also investigated.

The deployment of low-power consumption RISs also brings benefits in terms of
power consumption and energy efficiency to the system [22, 19, 17, 25, 26]. In
particular, Jeon et al. [25] considered installing the RIS on an aerial
platform to exploit rich LoS communications for an energy-efficient aerial
backhaul system. Similarly, Diamanti et al. [26] considered the RIS-aided UAV-
enabled integrated access and backhaul network targeting at energy-efficient
operations, whereas the work [27] focused on minimizing the age-of-information
to maintain the freshness of reflected information. In the aforementioned
studies, the common task is to jointly design the UAV’s trajectory/location
and transmit beamforming/power allocation as well as the RIS reflecting
coefficients. Due to highly coupled design variables and unit modules of RIS
coefficients, the formulated problems are often solved via the block
coordinate descent/ascent (BCD/BCA) and successive approximation (SCA) [14,
21, 23, 24, 13]. Furthermore, leveraging deep neural networks [2, 19, 16, 22,
28] and deep reinforcement learning [19, 29, 30] can offer significant
performance improvement and complexity reduction.

It is noted that all the above works considered passive RISs, which only
provide passive reflecting gains. Recently, the hybrid active-passive RIS
architecture has been introduced [31, 32, 33] to overcome the inherent
limitations of passive RISs, especially in harsh transmission scenarios such
as in low signal-to-noise ratio (SNR) regime and/or the severe path loss in
reflecting channels. The key idea of the hybrid RIS is to employ a few active
elements to enable both reflecting and amplifying gains simultaneously at the
RIS. As a result, the hybrid RIS is capable of mitigating the effects of the
double path loss and significantly improving the system performance in terms
of spectral efficiency (SE) [33, 34, 35, 36, 37, 38, 39], secrecy rate [40,
41], harvested energy [42], and reliability [43]. Moreover, active elements
with RF chains in [31, 32, 44] can be leveraged for processing of incident
signals and channel estimation at RISs.

The above discussed advantages of hybrid RISs can also be reaped by fully
active RISs [45, 46, 9], which, however, require higher power consumption,
hardware, and computational cost. Furthermore, based on the recent studies on
the performance gains of the hybrid and active RISs [11, 33, 36, 45], a small
number of active elements are sufficient to ensure satisfactory improvement in
the SE performance, and excessive use of active elements can cause a
significant loss in the SE and EE when the active power budget is limited.
Therefore, hybrid RISs can offer a good SE–EE tradeoff while maintaining
reasonable hardware costs. It is noted that the hybrid RIS can serve as the
conventional passive or active RISs when its number of active elements is set
to zero or the number of available RIS elements, respectively.

###  I-B Contributions

In most of the previous works, RIS is often deployed on the building facade,
which is in the vicinity of either BS or user equipments (UEs) to exploit LoS
communications [47, 10, 48]. However, in multi-user air-ground communications
systems, this favorable deployment becomes challenging and impractical. On one
hand, it is difficult to guarantee the short distance between the RIS and the
UAV due to the UAV’s high mobility and altitude. On the other hand, when
multiple UEs are distributed in a large area, the LoS channels from the RIS to
all UEs cannot be guaranteed, especially in complex urban areas [13].
Therefore, the performance gain offered by the passive RIS may be severely
limited in UAV-enabled networks. To overcome the aforementioned challenges and
motivated by the potential performance improvement of the hybrid active-
passive RIS, we consider in this work hybrid RIS-aided UAV communications
systems. The hybrid RIS is practically realized by a recently introduced
technology called reflection amplifier [49]. The deployment of the hybrid RIS
benefits the UAV systems in the following aspects. First, without requiring a
large number of elements, a single hybrid RIS with amplifying gains can
efficiently compensate for the severe path loss on the reflecting channels,
resulting in a significant throughout improvement. Secondly, the hybrid RIS
offers a great performance gain with respect to conventional systems (without
RIS or with passive RIS) in low and moderate power regimes [11]. This
guarantees improved system sum throughput and max-min fairness, especially for
the UAV with limited battery capacity.

To reap the above-mentioned benefits of the hybrid RIS in UAV air-ground
downlink communications, this work considers two typical UAV-enabled systems.
In the first one, a tethered UAV is placed at a fixed location and serves all
UEs at the same time. In the second scenario, the UAV employs the time
division multiple access (TDMA) protocol to serve each UE at a time slot of a
certain time period. Although the use of TDMA may cause additional latency, it
allows to fully exploit the high and controllable mobility of the UAV.
Specifically, by optimizing the trajectory and user scheduling, the
communication distance between the UAV and ground users can be signiﬁcantly
shortened, enhancing the air-ground communications performance [1, 50]. In
both scenarios, we focus on the fairness design to maximize the minimum rate
among all UEs. Our specific contributions are summarized as follows:

  * •

We propose the deployments of the hybrid active-passive RIS to assist the air-
ground communications in the two typical UAV-enabled networks mentioned above.
These have the potential in mitigating the limitations of the conventional
systems, i.e., the blockage of LoS links in UAV systems without RIS and the
double path loss in reflecting channels with the passive RIS. The resultant
signal models are significantly different from that of the conventional
passive RIS. Specifically, the signals received at UEs in hybrid RIS-aided
systems include not only the signal transmitted from the UAV, but also the
noise and self-interference (SI) amplified from the RIS’s active elements.
Furthermore, we consider the case that the UAV is equipped with multiple
antennas to further enhance the system performance.

  * •

We formulate two max-min rate problems in two considered networks, where the
UAV’s location or trajectory, transmit beamformers, and RIS
amplifying/reflecting coefficients are jointly optimized to guarantee the
system fairness. Although the formulated problems have the same fairness
objective, their solutions are significantly different due to the different
system models. Both problems are practical and appealing but have not been
considered in the literature, according to the authors’ best knowledge. In
particular, they are more challenging compared to those with conventional
passive RISs due to additional constraints on the power consumption of active
elements and the introduction of the amplified noise/SI at these elements.
Combining tools from BCA and SCA methods, we first decompose the original
problems into subproblems and derive efficient approximation functions to
tackle the nonconvex constraints. Two efficient iterative algorithms are then
developed to obtain the locally optimal solutions to the formulated problems.

  * •

Finally, we provide extensive simulation results to validate the effectiveness
of the proposed algorithms as well as the performance of the hybrid RIS. They
reveal that the hybrid RIS offers remarkable performance improvement compared
to conventional systems without RIS and with passive RIS, especially in
extreme conditions such as low SNR regime, severe path loss in reflecting
channels, and limited power budget at the UAV. The results also show that the
full potential of the hybrid RIS can be achieved by employing the TDMA
protocol.

###  I-C Organization and Notations

The rest of the paper is organized as follows. In Section II, we present the
system model and formulate the max-min rate problems. The solutions of these
problems are given in Sections III and IV. Numerical results are provided in
Section V. Finally, Section VI concludes the paper.

Notations: Throughout this paper, numbers, vectors, and matrices are denoted
by lower-case, bold-face lower-case, and bold-face upper-case letters,
respectively. Furthermore, $(\cdot)^{*}$,
$(\cdot)^{\scriptscriptstyle\mathsf{T}}$, and
$(\cdot)^{\scriptscriptstyle\mathsf{H}}$ denote the conjugate of a complex
number, the transpose and the conjugate transpose of a matrix or vector,
respectively. Matrix ${\mathbf{I}}_{N}$ is the identity matrix of size
$N\times N$, $\mathrm{diag}\\{a_{1},\ldots,a_{N}\\}$ represents a diagonal
matrix with diagonal entries $a_{1},\ldots,a_{N}$, whereas
$\mathrm{diag}\left\\{{\mathbf{A}}\right\\}$ returns the diagonal elements of
matrix ${\mathbf{A}}$. In addition, $\left|\cdot\right|$ denotes the absolute
value of a scalar, $\left\lVert\cdot\right\rVert_{\mathcal{F}}$ denotes the
Frobenius norm of a matrix, $\angle x$ represents the phase of complex number
$x$, and $\mathrm{sgn}(x)$ returns the sign of real number $x$. Finally,
$\circ$ represents a Hadamard product.

##  II System Model and Problem Formulations

We consider a UAV-enabled wireless network where a UAV serves $K$ single-
antenna UEs, and the air-ground communication is assisted by a hybrid active-
passive RIS installed on the building facade at a certain altitude. The UAV is
equipped with $N_{t}$ antennas and is assumed to fly at a fixed altitude of
$z_{0}$ m. Let $\mathcal{K}\triangleq\\{1,2,\ldots,K\\}$ denote the set of
UEs.

###  II-A Hybrid RIS and Channel Model

####  II-A1 Hybrid RIS

The RIS is equipped with $N$ elements, among which $N_{\mathrm{a}}$ elements
are active. To guarantee a minimal increase in power consumption and hardware
cost of the hybrid RIS, a few active elements are employed. Let
$\mathcal{A}\subset\\{1,2,\ldots,N_{\mathrm{a}}\\}$ be the set of the RIS
active elements with $\left|\mathcal{A}\right|=N_{\mathrm{a}}$. The RIS active
elements can potentially be realized by low-power reflection amplifiers [49].
We refer readers to [45, 11, 49] for more details on the reflection amplifier-
based active RIS. It is seen that the fully passive RIS (i.e. with
$N_{\mathrm{a}}=0$) is just a special case of the hybrid RIS. Therefore, we
will use the general term “RIS” for the discussion in the system model, while
specific terms “passive RIS” or “hybrid RIS” are used for comparisons.

Let $\alpha_{n}$ denote the coefficient associated with the $n$th element of
the RIS. It can be expressed as
$\alpha_{n}=\left|\alpha_{n}\right|e^{j\theta_{n}}$, where $\theta_{n}$ and
$\left|\alpha_{n}\right|$ represent the phase shift and amplitude of the
$n$-th coefficient of the RIS, respectively, with $\theta_{n}\in[0,2\pi)$
[47], $\left|\alpha_{n}\right|\leq 1,n\notin\mathcal{A}$, and
$\left|\alpha_{n}\right|\leq a_{\mathrm{max}}$ for $n\in\mathcal{A}$. Here,
$a_{\mathrm{max}}$ is the maximum gain that an active load can provide, which
is up to $40$ dB if active elements are realized by reflection amplifiers [45,
49]. We note here that to achieve interference cancelation in multi-user
systems, the reflection amplitude of the RIS passive elements may not
necessarily be unity [47]. Let
$\boldsymbol{\Upsilon}=\mathrm{diag}\left\\{\alpha_{1},\ldots,\alpha_{N}\right\\}\in\mathbb{C}^{N\times
N}$ be the diagonal matrix of the RIS coefficients. We define an additive
decomposition for it as
$\boldsymbol{\Upsilon}=\boldsymbol{\Phi}+\boldsymbol{\Psi}$, where
$\boldsymbol{\Psi}=\mathbbm{1}^{\mathcal{A}}_{N}\circ\boldsymbol{\Upsilon}$
and
$\boldsymbol{\Phi}=\left({\mathbf{I}}_{N}-\mathbbm{1}^{\mathcal{A}}_{N}\right)\circ\boldsymbol{\Upsilon}$
contain the active and passive coefficients, respectively. Here,
$\mathbbm{1}^{\mathcal{A}}_{N}$ is an $N\times N$ diagonal matrix whose non-
zero elements are all unity and have positions determined by $\mathcal{A}$.

####  II-A2 Channel Model

Let ${\mathbf{v}}$, ${\mathbf{r}}$, and ${\mathbf{u}}_{k}$ denote the
locations of UAV, RIS, and UE $k$, respectively. We denote by
${\mathbf{h}}_{0,k}\in\mathbb{C}^{N_{t}\times 1}$,
${\mathbf{H}}_{1}\in\mathbb{C}^{N\times N_{t}}$ and
${\mathbf{h}}_{2,k}\in\mathbb{C}^{N\times 1}$ the channels between UAV and UE
$k$, between UAV and RIS, and between RIS and UE $k$, respectively.
Furthermore, we denote by
$\\{{\mathbf{g}}_{0,k}^{\scriptscriptstyle\mathsf{H}},{\mathbf{G}}_{1},{\mathbf{g}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\\}$
the corresponding small-scale fading channels of
$\\{{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}},{\mathbf{H}}_{1},{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\\}$.
As a result, we can write [21, 2]

$\displaystyle{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}$
$\displaystyle=\zeta_{0}^{\frac{1}{2}}\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\frac{\epsilon_{0}}{2}}{\mathbf{g}}_{0,k}^{\scriptscriptstyle\mathsf{H}},$
(1a) $\displaystyle{\mathbf{H}}_{1}$
$\displaystyle=\zeta_{0}^{\frac{1}{2}}\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\frac{\epsilon_{1}}{2}}{\mathbf{G}}_{1},$
(1b) $\displaystyle{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}$
$\displaystyle=\zeta_{0}^{\frac{1}{2}}\left\lVert{\mathbf{r}}-{\mathbf{u}}_{k}\right\rVert^{-\frac{\epsilon_{2}}{2}}{\mathbf{g}}_{2,k}^{\scriptscriptstyle\mathsf{H}},$
(1c)

where $\zeta_{0}$ is the path loss at the reference distance of $1$ m, and
$\\{\epsilon_{0},\epsilon_{1},\epsilon_{2}\\}$ are the corresponding path loss
exponents. The effective channel between the UAV and UE $k$ can be written as

$\displaystyle{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}={\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}{\mathbf{H}}_{1}.$
(2)

###  II-B Signal Models

(a) Static UAV

(b) Mobile UAV

Fig. 1: (a) The UAV is deployed at a fixed location and serves all UEs at the
same time. (b) The UAV flies along a trajectory and employs TDMA to serve each
UE at a time slot.

In this work, we consider two typical scenarios of UAV-enabled communications
systems, as illustrated in Fig. 1. In Fig. 1(a), the UAV is assumed to be
deployed at a fixed position and serves all UEs at the same time [23]. In
contrast, in Fig. 1(b), the UAV employs the TDMA transmission protocol and
flies over consecutive time slots to communicate with at most one user at each
time slot [1]. The signal models of these considered systems are explained in
detail in the following.

####  II-B1 Static UAV

Denote by $s_{k}$ and ${\mathbf{w}}_{k}\in\mathbb{C}^{N_{t}\times 1}$ the
transmitted symbol and the beamforming vector intended for UE $k$, with
$\mathbb{E}\\{\left|s_{k}\right|^{2}\\}=1$. The transmitted signal from the
UAV can be expressed as
${\mathbf{x}}=\sum_{k=1}^{K}{\mathbf{w}}_{k}s_{k}\in\mathbb{C}^{N_{t}\times
1}$. The UAV’s transmit power constraint as
$p^{\mathrm{uav}}=\sum_{k=1}^{K}\left\lVert{\mathbf{w}}_{k}\right\rVert^{2}\leq
p_{\mathrm{max}}^{\mathrm{uav}}$, where $p_{\mathrm{max}}^{\mathrm{uav}}$ is
the UAV’s power budget. The received signal at UE $k$ can be expressed as

$\displaystyle
y_{k}=\underbrace{{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{k}s_{k}}_{\text{desired
signal}}+\underbrace{\sum_{j\neq
k}^{K}{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j}s_{j}}_{\text{interference}}+\underbrace{{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Psi}{\mathbf{n}}_{\mathrm{r}}}_{\text{RIS
effective noise}}+\underbrace{n_{\mathrm{u}}}_{\text{noise at
UE}\leavevmode\nobreak\ k},$ (3)

where $n_{\mathrm{u}}\sim\mathcal{CN}(0,\sigma_{\mathrm{u}}^{2})$ and
${\mathbf{n}}_{\mathrm{r}}\sim\mathcal{CN}(\boldsymbol{0},\mathbbm{1}^{\mathcal{A}}_{N}\circ\sigma_{\mathrm{r}}^{2}{\mathbf{I}}_{N})$
are the additive white Gaussian noise (AWGN) at UE $k$ and the total effective
noise including the AWGN noise and residual SI caused by the RIS active
elements, respectively. This SI is due to the dull-duplex operation of the
RIS’s active elements. Thus, the term
${\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Psi}{\mathbf{n}}_{\mathrm{r}}+n_{\mathrm{u}}$
in (3) is the aggregated noise at UE $k$. Note that there are numerous sources
of imperfections in active elements, and it has been shown in [51] that the
residual SI can be eliminated to be as low as 1 dB over the noise floor
independent of the transmit power and the number of array elements. Therefore,
we adopt the AWGN to model the RIS residual SI which is proportional to the
noise power, as in [52, 11].

####  II-B2 Mobile UAV

In this scenario, the UAV flies within a given period $\Delta t$, which is
divided into $T$ consecutive equal time slots $\\{1,\ldots,T\\}$, each is of
length $\delta_{t}=\frac{\Delta t}{T}$. Here, $\delta_{t}$ is assumed to be
sufficiently small for which a UAV’s location approximately remain unchanged
even when it flies at a maximum speed $v_{\mathrm{max}}$ [1]. The trajectory
of the UAV can be denoted as a sequence $\\{{\mathbf{v}}[t]\\}_{t=1}^{T}$,
where
${\mathbf{v}}[t]=[x_{\mathrm{v}}[t],y_{\mathrm{v}}[t],z_{0}],t=1,\ldots,T,$
represents the location of the UAV at time slot $t$. Similarly, time index
$[t]$ will be added to the system model to appropriately reflect the TDMA
transmission. Specifically, in this scenario, $s_{k}[t]$ represents the
transmit symbol intended for UE $k$;
$\\{{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}[t],{\mathbf{H}}_{1}[t],{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\\}$
represent the UAV-UE $k$, UAV-RIS, and RIS-UE $k$ channels, associated with
small-scale fading channels
$\\{{\mathbf{g}}_{0,k}^{\scriptscriptstyle\mathsf{H}}[t],{\mathbf{G}}_{1}[t],{\mathbf{g}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\\}$,
respectively; and
$\\{\boldsymbol{\Upsilon}[t],\boldsymbol{\Phi}[t],\boldsymbol{\Psi}[t]\\}$
represent the matrices of reflecting/amplifying coefficients
$\alpha_{n}[t]=\left|\alpha_{n}[t]\right|e^{j\theta_{n}[t]}$ of the RIS at
time slot $t$. We note that the channels between the RIS and UE $k$, i.e.,
${\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}$ and
${\mathbf{g}}_{2,k}^{\scriptscriptstyle\mathsf{H}}$, remains unchanged over
time slots because the RIS is deployed at a fixed location, while the UEs are
assumed to have slow mobility. As a result, the effective channel at time slot
$t$ is expressed as

$\displaystyle{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}[t]={\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}[t]+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}[t]{\mathbf{H}}_{1}[t].$
(4)

Let $b_{k}[t]\in\\{0,1\\}$ indicate whether or not UE $k$ is scheduled for
transmission at time slot $t$, i.e. $b_{k}[t]=1$ implies that UE $k$ is served
at time slot $t$; otherwise, $b_{k}[t]=0$. We assume that at each time slot,
the UAV serves at most one UE; thus, $\sum\nolimits_{k=1}^{K}b_{k}[t]\leq
1,\forall t$. If UE $k$ is scheduled for transmission at time slot $t$, its
received signal can be modeled as

$\displaystyle y_{k}[t]$
$\displaystyle={\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{w}}[t]s_{k}[t]+\left({\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}[t]\boldsymbol{\Psi}[t]{\mathbf{n}}_{\mathrm{r}}+n_{\mathrm{u}}\right),$
(5)

where ${\mathbf{w}}[t]$ is the beamforming vector at the UAV subject to
$\left\lVert{\mathbf{w}}[t]\right\rVert^{2}\leq
p_{\mathrm{max}}^{\mathrm{uav}},\forall t$, and
${\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}[t]\boldsymbol{\Psi}[t]{\mathbf{n}}_{\mathrm{r}}+n_{\mathrm{u}}$
is the aggregated noise at UE $k$th at time slot $t$, with
${\mathbf{n}}_{\mathrm{r}}$ and $n_{\mathrm{u}}$ having the same distributions
as those in (3). Here, without loss of generality, we assume the same AWGN
noise ${\mathbf{n}}_{\mathrm{r}}$ and $n_{\mathrm{u}}$ for all time slots
because their distributions are unchanged. This assumption is for ease of
exposition and obviously does not affect the solutions to the problems of
interest. Furthermore, we assume the availability of perfect CSI for the
design and optimization of both static and mobile UAV networks [22, 12, 13].
We recognize that this assumption limits the practicality of the design.
However, in this work, we mainly focus on demonstrating the fundamental
performance benefits of the hybrid RIS architecture in assisting UAV air-
ground communications.

###  II-C Problem Formulation

Our goal is to maximize the minimum rate of UEs in both scenarios discussed
previously by jointly optimizing the location/trajectory and transmit
beamforming of the UAV as well as the RIS beamforming coefficients. The
different signal models in the two scenarios (i.e. (3) and (5)) lead to
different optimization problems which will be elaborated next.

####  II-C1 Joint UAV’s Placement and Beamforming Optimization Problem

Based on (3), the achievable rate of UE $k$ (in nats/s/Hz) can be expressed as

$\displaystyle\mathcal{R}_{k}^{\mathrm{loc}}=\log\left(1+\frac{\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{k}\rvert^{2}}{\sum\nolimits_{j\neq
k}\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j}\rvert^{2}+\sigma_{\mathrm{r}}^{2}\lVert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Psi}\rVert^{2}+\sigma_{\mathrm{u}}^{2}}\right).$
(6)

Furthermore, note that the RIS amplifies both incident signal
${\mathbf{H}}_{1}{\mathbf{x}}$ and total noise and residual SI
${\mathbf{n}}_{\mathrm{r}}$. Thus, the transmit power of active elements of
the RIS is computed as

$\displaystyle p^{\mathrm{ris}}$
$\displaystyle=\mathrm{trace}\left(\boldsymbol{\Psi}\mathbb{E}\left\\{{\mathbf{n}}_{\mathrm{r}}{\mathbf{n}}_{\mathrm{r}}^{\scriptscriptstyle\mathsf{H}}+{\mathbf{H}}_{1}{\mathbf{x}}{\mathbf{x}}^{\scriptscriptstyle\mathsf{H}}{\mathbf{H}}_{1}^{\scriptscriptstyle\mathsf{H}}\right\\}\boldsymbol{\Psi}^{\scriptscriptstyle\mathsf{H}}\right)$
$\displaystyle\stackrel{{\scriptstyle\text{(a)}}}{{=}}\sigma_{\mathrm{r}}^{2}\left\lVert\boldsymbol{\Psi}\right\rVert_{\mathcal{F}}^{2}+\left\lVert\boldsymbol{\Psi}{\mathbf{H}}_{1}\right\rVert_{\mathcal{F}}^{2}\sum_{k=1}^{K}\left\lVert{\mathbf{w}}_{k}\right\rVert^{2}\stackrel{{\scriptstyle\text{(b)}}}{{=}}\sum_{n\in\mathcal{A}}\left|\alpha_{n}\right|^{2}\xi_{n},$
(7)

where equality (a) follows the fact that ${\mathbf{n}}_{\mathrm{r}}$ has zero
mean and is independent of ${\mathbf{H}}_{1}{\mathbf{x}}$; in (7),
$\xi_{n}\triangleq\sigma_{\mathrm{r}}^{2}+\left\lVert{\mathbf{h}}_{1,n}\right\rVert^{2}\sum_{k=1}^{K}\left\lVert{\mathbf{w}}_{k}\right\rVert^{2}$,
with ${\mathbf{h}}_{1,n}$ being the $n$th row of ${\mathbf{H}}_{1}$, and
equality (b) follows the diagonal structure of $\boldsymbol{\Psi}$ whose non-
zero elements are in $\mathcal{A}$ only. The total transmit power at the RIS
is constrained as $p^{\mathrm{ris}}\leq p_{\mathrm{max}}^{\mathrm{ris}}$,
where $p_{\mathrm{max}}^{\mathrm{ris}}$ is the power budget of the RIS. As a
result, the max-min rate problem can be formulated as:

$\displaystyle(\mathcal{P}_{\mathrm{loc}}):\
\underset{\begin{subarray}{c}{\mathbf{v}},\\{{\mathbf{w}}_{k}\\},\\{\alpha_{n}\\}\end{subarray}}{\textrm{maximize}}\quad$
$\displaystyle\underset{k\in\mathcal{K}}{\textrm{min}}\\{\mathcal{R}_{k}^{\mathrm{loc}}\\}$
(8a) subject to
$\displaystyle\sum\nolimits_{k=1}^{K}\left\lVert{\mathbf{w}}_{k}\right\rVert^{2}\leq
p_{\mathrm{max}}^{\mathrm{uav}},$ (8b) $\displaystyle 0\leq\theta_{n}\leq
2\pi,\ \forall n,$ (8c) $\displaystyle\left|\alpha_{n}\right|\leq 1,\ \forall
n\notin\mathcal{A},$ (8d) $\displaystyle\left|\alpha_{n}\right|\leq
a_{\mathrm{max}},\ \forall n\in\mathcal{A},$ (8e)
$\displaystyle\sum\nolimits_{n\in\mathcal{A}}\left|\alpha_{n}\right|^{2}\xi_{n}\leq
p_{\mathrm{max}}^{\mathrm{ris}},$ (8f)

where constraints (8c)–(8f) are design constrains of the hybrid RIS. It is
noted that in (8e), only active elements ($n\in\mathcal{A}$) can amplify the
signals with amplification gains restricted by $a_{\mathrm{max}}$ [45].

####  II-C2 Joint UAV’s Trajectory Planning and Beamforming Optimization

From the signal model in (5), the average rate (in nats/s/Hz) of UE $k$ over
all $T$ time slots is given as

$\displaystyle\mathcal{R}_{k}^{\mathrm{tra}}=\frac{1}{T}\sum_{t=1}^{T}b_{k}[t]\log\left(1+\frac{\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{w}}[t]\rvert^{2}}{\sigma_{\mathrm{r}}^{2}\lVert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}[t]\boldsymbol{\Psi}[t]\rVert^{2}+\sigma_{\mathrm{u}}^{2}}\right).$
(9)

Following a similar derivation as in (7), the transmit power of the RIS at
time slot $t$ can be given as
$p^{\mathrm{ris}}[t]=\sum_{n\in\mathcal{A}}\left|\alpha_{n}[t]\right|^{2}\xi_{n}[t],$
where
$\xi_{n}[t]\triangleq\sigma_{\mathrm{r}}^{2}+\left\lVert{\mathbf{h}}_{1,n}[t]\right\rVert^{2}\left\lVert{\mathbf{w}}[t]\right\rVert^{2}$.
Thus, the max-min rate problem in this scenario can be formulated as

$\displaystyle(\mathcal{P}_{\mathrm{tra}}):\underset{\begin{subarray}{c}\\{b_{k}[t]\\},\\{{\mathbf{v}}[t]\\},\\\
\\{{\mathbf{w}}[t]\\},\\{\alpha_{n}[t]\\}\end{subarray}}{\textrm{maximize}}\quad$
$\displaystyle\underset{k\in\mathcal{K}}{\textrm{min}}\\{\mathcal{R}_{k}^{\mathrm{tra}}\\}$
(10a) subject to $\displaystyle{\mathbf{v}}[1]={\mathbf{v}}[T],\ $ (10b)
$\displaystyle\left\lVert{\mathbf{v}}[t+1]-{\mathbf{v}}[t]\right\rVert^{2}\leq
d_{\mathrm{max}}^{2},$ $\displaystyle\hskip 42.67912ptt=1,\ldots,T-1,$ (10c)
$\displaystyle b_{k}[t]\in\\{0,1\\},\ \forall k,t,$ (10d)
$\displaystyle\sum\nolimits_{k=1}^{K}b_{k}[t]\leq 1,\ \forall t,$ (10e)
$\displaystyle\left\lVert{\mathbf{w}}[t]\right\rVert^{2}\leq
p_{\mathrm{max}}^{\mathrm{uav}},\ \forall t,$ (10f) $\displaystyle
0\leq\theta_{n}[t]\leq 2\pi,\ \forall n,t,$ (10g)
$\displaystyle\left|\alpha_{n}[t]\right|\leq 1,\ n\notin\mathcal{A},\forall
t,$ (10h) $\displaystyle\left|\alpha_{n}[t]\right|\leq a_{\mathrm{max}},\
n\in\mathcal{A},\forall t,$ (10i)
$\displaystyle\sum_{n\in\mathcal{A}}\left|\alpha_{n}[t]\right|^{2}\xi_{n}[t]\leq
p_{\mathrm{max}}^{\mathrm{ris}},\ \forall t.$ (10j)

In constraints (10b) and (10c), it is required that the UAV returns to its
initial location at the end of each period $\Delta t$, and the maximum
horizontal distance it can travel in each time slot is
$d_{\mathrm{max}}=\delta_{t}v_{\mathrm{max}}$. Similar to (8c)–(8f),
constraints (10g)–(10j) are constrains associated with the hybrid RIS in all
the time slots.

Both problems $(\mathcal{P}_{\mathrm{loc}})$ in (8) and
$(\mathcal{P}_{\mathrm{tra}})$ in (10) are nonconvex, which are difficult to
solve optimally. In fact, even finding a feasible point to these problems is
rather challenging due to a strong coupling between the optimization variables
in the objective function as well as constraint (8f) and (10j). Furthermore,
problem $(\mathcal{P}_{\mathrm{tra}})$ is a mixed-integer nonconvex
programming due to integer constraints (10e), (10d). To tackle the above
challenges, we first provide the following lemma with some useful
approximations. Then, efficient solutions to problems
$(\mathcal{P}_{\mathrm{loc}})$ and $(\mathcal{P}_{\mathrm{tra}})$ will be
presented in Sections III and IV, respectively.

###### Lemma 1

Consider the following concave power, quadratic, bilinear, and quadratic-over-
linear functions:

$\displaystyle f_{\mathtt{pow}}(x;c)\triangleq-x^{c},\
x\in\mathbb{R}_{++},c>1\text{\leavevmode\nobreak\ or\leavevmode\nobreak\
}c<0,$ $\displaystyle
f_{\mathtt{qua}}({\mathbf{x}};{\mathbf{c}})\triangleq-\left\lVert{\mathbf{x}}-{\mathbf{c}}\right\rVert^{2},\
{\mathbf{x}},{\mathbf{c}}\in\mathbb{C}^{n},$ $\displaystyle
f_{\mathtt{bil}}(x,y,\delta_{s})\triangleq\delta_{s}xy,\
(x,y)\in\mathbb{R}_{++}^{2},\delta_{s}=\pm 1,$ $\displaystyle
f_{\mathtt{qol}}({\mathbf{x}},y;{\mathbf{C}})\triangleq-\frac{{\mathbf{x}}^{\scriptscriptstyle\mathsf{H}}{\mathbf{C}}{\mathbf{x}}}{y},\
{\mathbf{x}}\in\mathbb{C}^{n},y\in\mathbb{R}_{++},{\mathbf{C}}\in\mathbb{C}^{n\times
n},$

respectively. Their convex upper bounds can be found as [53, 1]:

$\displaystyle f_{\mathtt{pow}}(x;c)\leq
F_{\mathtt{pow}}(x;c,x_{0})\triangleq\left(c-1\right)x_{0}^{c}-cx_{0}^{c-1}x,$
(11) $\displaystyle f_{\mathtt{qua}}({\mathbf{x}};{\mathbf{c}})\leq
F_{\mathtt{qua}}({\mathbf{x}};{\mathbf{c}},{\mathbf{x}}_{0})\triangleq
2({\mathbf{c}}-{\mathbf{x}}_{0})^{\scriptscriptstyle\mathsf{T}}({\mathbf{x}}-{\mathbf{x}}_{0})$
$\displaystyle\hskip
156.49014pt-\lVert{\mathbf{x}}_{0}-{\mathbf{c}}\rVert^{2},$ (12)
$\displaystyle f_{\mathtt{bil}}(x,y;1)\leq
F_{\mathtt{bil}}(x,y;1,x_{0},y_{0})\triangleq\frac{1}{2}\left(\frac{y_{0}}{x_{0}}x^{2}+\frac{x_{0}}{y_{0}}y^{2}\right),$
(13) $\displaystyle f_{\mathtt{bil}}(x,y;-1)\leq
F_{\mathtt{bil}}(x,y;-1,x_{0},y_{0})\triangleq\frac{1}{4}(x-y)^{2}$
$\displaystyle\hskip
64.01869pt+\frac{1}{4}(x_{0}+y_{0})^{2}-\frac{1}{2}(x_{0}+y_{0})(x+y),$ (14)
$\displaystyle f_{\mathtt{qol}}({\mathbf{x}},y)\leq
F_{\mathtt{qol}}({\mathbf{x}},y;{\mathbf{x}}_{0},y_{0})\triangleq\frac{{\mathbf{x}}_{0}^{\scriptscriptstyle\mathsf{H}}{\mathbf{C}}{\mathbf{x}}_{0}}{y_{0}^{2}}y-\frac{2\mathfrak{R}{\left({\mathbf{x}}_{0}^{\scriptscriptstyle\mathsf{H}}{\mathbf{C}}{\mathbf{x}}\right)}}{y_{0}},$
(15)

respectively, based on the first-order Taylor approximations around feasible
points $x_{0}$ for (11), ${\mathbf{x}}_{0}$ for (12), $(x_{0},y_{0})$ for (13)
and (14), and $({\mathbf{x}}_{0},y_{0})$ for (15), respectively. The proof is
conceptually simple but requires some algebra, which is omitted in this work.

##  III Joint UAV’s Placement and Beamforming Optimization in
$(\mathcal{P}_{\mathrm{loc}})$

To solve problem $(\mathcal{P}_{\mathrm{loc}})$ in (8), we first introduce
variable $\tau$ to bypass the non-smooth of (8a), which leads to the following
epigraph form:

$\displaystyle\underset{\begin{subarray}{c}\tau,{\mathbf{v}},\\{{\mathbf{w}}_{k}\\},\\{\alpha_{n}\\}\end{subarray}}{\textrm{maximize}}\quad$
$\displaystyle\tau$ (16a) subject to
$\displaystyle\mathcal{R}_{k}^{\mathrm{loc}}\geq\tau,\ \forall k,$ (16b)
$\displaystyle\eqref{cons_puav_loc}-\eqref{cons_pris_loc}.$ (16c)

Next, we utilize the SCA framework, wherein each iteration utilizes the BCA
approach to decouple (16) into three distinct subproblems, namely the UAV’s
location optimization, and UAV and RIS beamforming designs. The detailed
solutions are elaborated in the following subsections.

###  III-A UAV Transmit Beamforming Design

With given $({\mathbf{v}},\\{\alpha_{n}\\})$, the beamformers
$\\{{\mathbf{w}}_{k}\\}$ at the UAV can be obtained by solving the following
problem:

$\displaystyle\underset{\tau,\\{{\mathbf{w}}_{k}\\}}{\textrm{maximize}}\
\tau,\ \textrm{subject to}\
\eqref{cons_puav_loc},\eqref{cons_pris_loc},\eqref{cons_min_rate_loc},$ (17)

where constraint (16b) is nonconvex. To overcome this, we introduce slack
variables $\\{\gamma_{k}\\}$ as the lower bound of the signal-to-interference-
plus-noise ratio (SINR) term in (6) and transform (16b) to the following set
of constraints:

$\displaystyle\log(1+\gamma_{k})$ $\displaystyle\geq\tau,\ \forall k,$ (18a)
$\displaystyle\frac{\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{k}\rvert^{2}}{\sum\nolimits_{j\neq
k}\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j}\rvert^{2}+\sigma_{k}^{2}}$
$\displaystyle\geq\gamma_{k},\ \forall k,$ (18b)

where
$\sigma_{k}^{2}\triangleq\sigma_{\mathrm{r}}^{2}\lVert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Psi}\rVert^{2}+\sigma_{\mathrm{u}}^{2}$
is independent of $\\{{\mathbf{w}}_{k}\\}$. Let

$\displaystyle{\mathbf{w}}$
$\displaystyle\triangleq[{\mathbf{w}}_{1}^{\scriptscriptstyle\mathsf{T}},\ldots,{\mathbf{w}}_{K}^{\scriptscriptstyle\mathsf{T}}]^{\scriptscriptstyle\mathsf{T}},$
$\displaystyle\hat{{\mathbf{H}}}_{k}$
$\displaystyle\triangleq\mathrm{blkdiag}\left\\{\boldsymbol{0},\ldots,\boldsymbol{0},{\mathbf{h}}_{k}{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}},\boldsymbol{0},\ldots,\boldsymbol{0}\right\\},$
$\displaystyle\bar{{\mathbf{H}}}_{k}$
$\displaystyle\triangleq\mathrm{blkdiag}\left\\{{\mathbf{h}}_{k}{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}},\ldots,{\mathbf{h}}_{k}{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}},\boldsymbol{0},{\mathbf{h}}_{k}{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}},\ldots,{\mathbf{h}}_{k}{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}\right\\},$

with ${\mathbf{w}}\in\mathbb{C}^{KN_{t}\times 1}$ and
$\hat{{\mathbf{H}}}_{k},\bar{{\mathbf{H}}}_{k}\in\mathbb{C}^{KN_{t}\times
KN_{t}}$. Then, we can rewrite constraint (18b) as
$\frac{{\mathbf{w}}^{\scriptscriptstyle\mathsf{H}}\hat{{\mathbf{H}}}_{k}{\mathbf{w}}}{{\mathbf{w}}^{\scriptscriptstyle\mathsf{H}}\bar{{\mathbf{H}}}_{k}{\mathbf{w}}+\sigma_{k}^{2}}\geq\gamma_{k},\forall
k$, which is equivalent to
${\mathbf{w}}^{\scriptscriptstyle\mathsf{H}}\bar{{\mathbf{H}}}_{k}{\mathbf{w}}+\sigma_{k}^{2}-f_{\mathtt{qol}}({\mathbf{w}},\gamma_{k};\hat{{\mathbf{H}}}_{k})\leq
0,\forall k,$ where $f_{\mathtt{qol}}(\cdot;\cdot)$ is defined in Lemma 1. By
using approximation (15) around the feasible point
$({\mathbf{w}}^{(i)},\gamma_{k}^{(i)})$ found at iteration $i$ of the proposed
iterative algorithm presented shortly, this constraint is approximated as

$\displaystyle{\mathbf{w}}^{\scriptscriptstyle\mathsf{H}}\bar{{\mathbf{H}}}_{k}{\mathbf{w}}+\sigma_{k}^{2}+F_{\mathtt{qol}}({\mathbf{w}},\gamma_{k};{\mathbf{w}}^{(i)},\gamma_{k}^{(i)})\leq
0,\ \forall k.$ (19)

Given the introduction of ${\mathbf{w}}$, we have
$\sum_{k=1}^{K}\left\lVert{\mathbf{w}}_{k}\right\rVert^{2}=\left\lVert{\mathbf{w}}\right\rVert^{2}$,
and constraints (8b) and (8f) become

$\displaystyle\left\lVert{\mathbf{w}}\right\rVert^{2}\leq
p_{\mathrm{max}}^{\mathrm{uav}},\ \text{and}\
\sum_{n\in\mathcal{A}}\left|\alpha_{n}\right|^{2}\left(\sigma_{\mathrm{r}}^{2}+\left\lVert{\mathbf{h}}_{1,n}\right\rVert^{2}\left\lVert{\mathbf{w}}\right\rVert^{2}\right)\leq
p_{\mathrm{max}}^{\mathrm{ris}},$ (20)

respectively. Finally, we can approximate problem (17) by the following convex
program at iteration $i$:

$\displaystyle\underset{\tau,{\mathbf{w}},\\{\gamma_{k}\\}}{\textrm{maximize}}\quad\tau,\
\textrm{subject to}\
\eqref{cons_min_rate_BF1},\eqref{cons_min_rate_BF2_2},\eqref{cons_ris_power_1}.$
(21)

###  III-B Optimization of the UAV’s Location

For given $(\\{{\mathbf{w}}_{k}\\},\\{\alpha_{n}\\})$, the UAV’s location can
be optimized by solving the following problem with respect to
$\\{\tau,{\mathbf{v}}\\}$:

$\displaystyle\underset{\tau,{\mathbf{v}}}{\textrm{maximize}}\ \tau,\
\textrm{subject to}\
\eqref{cons_pris_loc},\eqref{cons_traj_1p},\eqref{cons_min_rate_loc}.$ (22)

The challenge in solving (22) is that variable ${\mathbf{v}}$ is currently
hidden in all the constraints. To overcome this, we first expand the
expression of $\mathcal{R}_{k}^{\mathrm{loc}}$ in (6) to show the role of
${\mathbf{v}}$. Specifically, from (1) and (2), we can rewrite

$\displaystyle\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j}\rvert^{2}=\lvert{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j}+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}{\mathbf{H}}_{1}{\mathbf{w}}_{j}\rvert^{2}$
$\displaystyle=\lvert{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j}\rvert^{2}+\lvert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}{\mathbf{H}}_{1}{\mathbf{w}}_{j}\rvert^{2}+2\mathfrak{R}{({\mathbf{w}}_{j}^{\scriptscriptstyle\mathsf{H}}{\mathbf{h}}_{0,k}{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}{\mathbf{H}}_{1}{\mathbf{w}}_{j})}$
$\displaystyle=c_{0,kj}\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}}+c_{1,kj}\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}}$
$\displaystyle\qquad+c_{2,kj}\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}/2}\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2},\forall
k,j,$ (23)

where
$c_{0,kj}\triangleq\zeta_{0}\lvert{\mathbf{g}}_{0,k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j}\rvert^{2}$,
$c_{1,kj}\triangleq\zeta_{0}\lvert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}{\mathbf{G}}_{1}{\mathbf{w}}_{j}\rvert^{2}$,
and $c_{2,kj}\triangleq
2\zeta_{0}\mathfrak{R}{({\mathbf{w}}_{j}^{\scriptscriptstyle\mathsf{H}}{\mathbf{g}}_{0,k}{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}{\mathbf{G}}_{1}{\mathbf{w}}_{j})}$
are all constants with respect to ${\mathbf{v}}$. Thus, we can rewrite
$\mathcal{R}_{k}^{\mathrm{loc}}$ as (24) (at the top of the next page).

$\displaystyle\mathcal{R}_{k}^{\mathrm{loc}}=\log\left(1+\frac{c_{0,kk}\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}}+c_{1,kk}\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}}+c_{2,kk}\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}/2}\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2}}{\sigma_{k}^{2}+\underset{{j\neq
k}}{\sum}c_{0,kj}\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}}+c_{1,kj}\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}}+c_{2,kj}\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}/2}\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2}}\right).$
(24)

This complicated form of $\mathcal{R}_{k}^{\mathrm{loc}}$ clearly exposes the
nonconvexity as well as the challenges in addressing constraint (16b). In
particular, we note in (24) that while $c_{0,kj},c_{1,kj}>0,\forall k,j$,
$c_{2,kj}$ can be either positive or negative, making it very difficult to
address the nonconvexity of (16b). Regarding this, let us introduce set of
slack variables
$\boldsymbol{\mathcal{V}}\triangleq\\{\\{\hat{v}_{0,k}\\},\\{\bar{v}_{0,k}\\},\hat{v}_{1},\bar{v}_{1}\\}$
satisfying

$\displaystyle\hat{v}_{0,k}$
$\displaystyle\leq\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}/2},$
(25a) $\displaystyle\bar{v}_{0,k}$
$\displaystyle\geq\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}/2},\forall
k,$ (25b) $\displaystyle\hat{v}_{1}$
$\displaystyle\leq\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2},$
(25c) $\displaystyle\bar{v}_{1}$
$\displaystyle\geq\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2}.$
(25d)

Furthermore, we introduce slack variables $\\{\hat{a}_{k},\bar{a}_{kj}\\}$
such that

$\displaystyle\hat{a}_{k}$
$\displaystyle\leq\begin{cases*}c_{0,kk}\hat{v}_{0,k}^{2}+c_{1,kk}\hat{v}_{1}^{2}+\left|c_{2,kk}\right|\hat{v}_{0,k}\hat{v}_{1},\
\text{if\leavevmode\nobreak\ }c_{2,kk}>0\\\
c_{0,kk}\hat{v}_{0,k}^{2}+c_{1,kk}\hat{v}_{1}^{2}-\left|c_{2,kk}\right|\bar{v}_{0,k}\bar{v}_{1},\
\text{otherwise}\end{cases*},$ $\displaystyle\hskip 184.9429pt\forall k,$
(26a) $\displaystyle\bar{a}_{kj}$
$\displaystyle\geq\begin{cases*}c_{0,kj}\bar{v}_{0,k}^{2}+c_{1,kj}\bar{v}_{1}^{2}+\left|c_{2,kj}\right|\bar{v}_{0,k}\bar{v}_{1},\
\text{if\leavevmode\nobreak\ }c_{2,kj}>0\\\
c_{0,kj}\bar{v}_{0,k}^{2}+c_{1,kj}\bar{v}_{1}^{2}-\left|c_{2,kj}\right|\hat{v}_{0,k}\hat{v}_{1},\
\text{otherwise}\end{cases*},$ $\displaystyle\hskip 177.82971pt\forall k,j.$
(26b)

Thus, (16b) is equivalent to set of constraints (25)–(26b) and

$\displaystyle\log\Bigl{(}1+\frac{\hat{a}_{k}}{\sigma_{k}^{2}+\sum_{j\neq
k}\bar{a}_{kj}}\Bigr{)}\geq\tau,\forall k,$ (27)

which are still nonconvex, but more tractable. To address (27), we rewrite it
as $\log(\sigma_{k}^{2}+\hat{a}_{k}+\sum\nolimits_{j\neq
k}\bar{a}_{kj})\geq\tau+\tilde{r}_{k},\forall k$, where
$\tilde{r}_{k}=\log(\sigma_{k}^{2}+\sum\nolimits_{j\neq k}\bar{a}_{kj})$. By
applying the first-order Taylor approximation around $\bar{a}_{kj}^{(i)}$ of
the concave function $\tilde{r}_{k}$, its convex upper bound can be found as

$\displaystyle\tilde{r}_{k}\leq\tilde{r}_{\mathrm{ub},k}^{(i)}\triangleq\log\Big{(}\sigma_{k}^{2}+\sum\nolimits_{j\neq
k}\bar{a}_{kj}^{(i)}\Big{)}+\frac{\underset{j\neq
k}{\sum}\bar{a}_{kj}-\bar{a}_{kj}^{(i)}}{\sigma_{k}^{2}+\underset{j\neq
k}{\sum}\bar{a}_{kj}^{(i)}}.$

As a result, (27) is transformed to the following convex constraint:

$\displaystyle\log\Big{(}\sigma_{k}^{2}+\hat{a}_{k}+\sum\nolimits_{j\neq
k}\bar{a}_{kj}\Big{)}\geq\tau+\tilde{r}_{\mathrm{ub},k}^{(i)},\forall k.$ (28)

Next, we apply inequalities (11)–(13) around points
$\\{\hat{v}_{0,k}^{(i)},\bar{v}_{0,k}^{(i)},\hat{v}_{1}^{(i)},\bar{v}_{1}^{(i)}\\}$
to approximate nonconvex constraints (25)–(26b) as

$\displaystyle\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert+F_{\mathtt{pow}}(\hat{v}_{0,k};-2/\epsilon_{0},\hat{v}_{0,k}^{(i)})\leq
0,\ \forall k,$ (29a) $\displaystyle
F_{\mathtt{qua}}({\mathbf{v}};{\mathbf{u}}_{k},{\mathbf{v}}^{(i)})\leq
1/F_{\mathtt{pow}}(\bar{v}_{0,k};4/\epsilon_{0},\bar{v}_{0,k}^{(i)}),\ \forall
k,$ (29b)
$\displaystyle\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert+F_{\mathtt{pow}}(\hat{v}_{1};-2/\epsilon_{1},\hat{v}_{1}^{(i)})\leq
0,$ (29c) $\displaystyle
F_{\mathtt{qua}}({\mathbf{v}};{\mathbf{r}},{\mathbf{v}}^{(i)})\leq
1/F_{\mathtt{pow}}(\bar{v}_{1};4/\epsilon_{1},\bar{v}_{1}^{(i)}),$ (29d)
$\displaystyle\begin{cases*}\hat{a}_{k}+c_{0,kk}F_{\mathtt{qua}}(\hat{v}_{0,k};0,\hat{v}_{0,k}^{(i)})+c_{1,kk}F_{\mathtt{qua}}(\hat{v}_{1};0,\hat{v}_{1}^{(i)})\\\
\hskip 85.35826pt+\left|c_{2,kk}\right|\hat{F}_{\mathtt{bil},kk}^{(i)}\leq
0,\text{if\leavevmode\nobreak\ }c_{2,kk}>0\\\
\hat{a}_{k}+c_{0,kk}F_{\mathtt{qua}}(\hat{v}_{0,k};0,\hat{v}_{0,k}^{(i)})+c_{1,kk}F_{\mathtt{qua}}(\hat{v}_{1};0,\hat{v}_{1}^{(i)})\\\
\hskip 99.58464pt-\left|c_{2,kk}\right|\bar{F}_{\mathtt{bil},kk}^{(i)}\leq
0,\text{otherwise}\end{cases*},$ $\displaystyle\hskip 199.16928pt\forall k,$
(29e) $\displaystyle\begin{cases*}\bar{a}_{kj}\geq
c_{0,kj}\bar{v}_{0,k}^{2}+c_{1,kj}\bar{v}_{1}^{2}+\left|c_{2,kj}\right|\bar{F}_{\mathtt{bil},kj}^{(i)},\
\text{if\leavevmode\nobreak\ }c_{2,kj}>0\\\ \bar{a}_{kj}\geq
c_{0,kj}\bar{v}_{0,k}^{2}+c_{1,kj}\bar{v}_{1}^{2}-\left|c_{2,kj}\right|\hat{F}_{\mathtt{bil},kj}^{(i)},\
\text{otherwise}\end{cases*},$ $\displaystyle\hskip 170.71652pt\forall k,j\neq
k,$ (29f)

where $\hat{F}_{\mathtt{bil},kj}^{(i)}\triangleq
F_{\mathtt{bil}}(\hat{v}_{0,k},\hat{v}_{1};\mathrm{sgn}(c_{2,kj}),\hat{v}_{0,k}^{(i)},\hat{v}_{1}^{(i)})$
and $\bar{F}_{\mathtt{bil},kj}^{(i)}\triangleq
F_{\mathtt{bil}}(\bar{v}_{0,k},\bar{v}_{1};\mathrm{sgn}(c_{2,kj}),\bar{v}_{0,k}^{(i)},\bar{v}_{1}^{(i)})$,
$\forall k,j$. Constraints (29a)–(29f) are convex assuming that
$\epsilon_{0},\epsilon_{1}<4$. Similarly, by writing
$\xi_{n}=\sigma_{\mathrm{r}}^{2}+c_{3,n}(\left\lVert{\mathbf{v}}-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2})^{2}$,
where
$c_{3,n}\triangleq\zeta_{0}\left\lVert{\mathbf{g}}_{1,n}\right\rVert^{2}\sum_{k=1}^{K}\left\lVert{\mathbf{w}}_{k}\right\rVert^{2}$
is independent of ${\mathbf{v}}$, we can rewrite (8f) equivalently as set
constraints (29d) and

$\displaystyle\sum\nolimits_{n\in\mathcal{A}}\left|\alpha_{n}\right|^{2}\Big{(}\sigma_{\mathrm{r}}^{2}+\sum\nolimits_{k=1}^{K}c_{3,n}\bar{v}_{1}^{2}\Big{)}\leq
p_{\mathrm{max}}^{\mathrm{ris}}.$ (30)

In summary, we can approximate problem (22) at iteration $i$ by the following
convex program:

$\displaystyle\underset{\tau,{\mathbf{v}},\boldsymbol{\mathcal{V}}}{\textrm{maximize}}\
\tau,\ \textrm{subject to}\
\eqref{cons_traj_1p},\eqref{cons_min_ratep_approx},\eqref{cons_v0hat_loc_approx}-\eqref{cons_pris_loc_approx}.$
(31)

###  III-C Optimization of RIS Beamforming Coefficients

Given $({\mathbf{v}},\\{{\mathbf{w}}_{k}\\})$, $\\{\alpha_{n}\\}$ can be
optimized by solving the following problem:

$\displaystyle\underset{\begin{subarray}{c}\tau,\\{\alpha_{n}\\}\end{subarray}}{\textrm{maximize}}\
\tau,\ \textrm{subject to}\
\eqref{cons_min_rate_loc},\eqref{cons_ris_phase_loc}-\eqref{cons_pris_loc}.$
(32)

It is easy to see that constraints (8c)-(8f) are convex with respect to
$\\{\alpha_{n}\\}$, except (16b). Furthermore, the optimization variable
$\\{\alpha_{n}\\}$ has not been exposed in the current form of (16b). As the
first step to solve (32), we express $\mathcal{R}_{k}^{\mathrm{loc}}$ as a
function of $\\{\alpha_{n}\\}$ in the following derivations. For ease of
exposition, we denote
$\bar{h}_{0,kj}\triangleq{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j},\forall
k,j$, and
$\bar{{\mathbf{h}}}_{1,j}\triangleq{\mathbf{H}}_{1}{\mathbf{w}}_{j},\forall
j$, which yields
${\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{k}=\bar{h}_{0,kk}+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}\bar{{\mathbf{h}}}_{1,k},\
{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}{\mathbf{w}}_{j}=\bar{h}_{0,kj}+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}\bar{{\mathbf{h}}}_{1,j}$.
Then, we can write the SINR term in (6) as

$\displaystyle\frac{\lvert\bar{h}_{0,kk}+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}\bar{{\mathbf{h}}}_{1,k}\rvert^{2}}{\underset{{j\neq
k}}{\sum}\lvert\bar{h}_{0,kj}+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}\bar{{\mathbf{h}}}_{1,j}\rvert^{2}+\sigma_{\mathrm{r}}^{2}\lVert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Psi}\rVert^{2}+\sigma_{\mathrm{u}}^{2}}\triangleq\frac{N_{k}}{D_{k}}.$

Let
$\boldsymbol{\alpha}\triangleq[\alpha_{1},\ldots,\alpha_{N}]^{\scriptscriptstyle\mathsf{T}}\in\mathbb{C}^{N\times
1}$,
$\tilde{{\mathbf{H}}}_{2,k}\triangleq\mathrm{diag}\\{{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\\}\in\mathbb{C}^{N\times
N}$, and
$\tilde{{\mathbf{h}}}_{12,kj}\triangleq\tilde{{\mathbf{H}}}_{2,k}\bar{{\mathbf{h}}}_{1,j}\in\mathbb{C}^{N\times
1}$. Then, we have

$\displaystyle\bar{h}_{0,kk}+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}\bar{{\mathbf{h}}}_{1,k}$
$\displaystyle=\bar{h}_{0,kk}+\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{h}}}_{12,kk}$
$\displaystyle\bar{h}_{0,kj}+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}\bar{{\mathbf{h}}}_{1,j}$
$\displaystyle=\bar{h}_{0,kj}+\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{h}}}_{12,kj},$
$\displaystyle{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Psi}$
$\displaystyle=\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{T}}\mathbbm{1}^{\mathcal{A}}_{N}\tilde{{\mathbf{H}}}_{2,k}.$

With several algebraic manipulations, we obtain

$\displaystyle N_{k}$
$\displaystyle=\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}{\mathbf{Q}}_{k}\boldsymbol{\alpha}+2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}{\mathbf{q}}_{k}\right)}+e_{k},$
$\displaystyle D_{k}$
$\displaystyle=\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{Q}}}_{k}\boldsymbol{\alpha}+2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{q}}}_{k}\right)}+\tilde{e}_{k},$

where
${\mathbf{Q}}_{k}=\tilde{{\mathbf{h}}}_{12,kk}^{*}\tilde{{\mathbf{h}}}_{12,kk}^{\scriptscriptstyle\mathsf{T}}$,
${\mathbf{q}}_{k}=\tilde{{\mathbf{h}}}_{12,kk}^{*}\bar{h}_{0,kk}$,
$e_{k}=\left|\bar{h}_{0,kk}\right|^{2}$,
$\tilde{{\mathbf{Q}}}_{k}=\sigma_{\mathrm{r}}^{2}\mathbbm{1}^{\mathcal{A}}_{N}\tilde{{\mathbf{H}}}_{2,k}^{*}\tilde{{\mathbf{H}}}_{2,k}^{\scriptscriptstyle\mathsf{T}}\mathbbm{1}^{\mathcal{A}}_{N}+\sum_{j\neq
k}\tilde{{\mathbf{h}}}_{12,kj}^{*}\tilde{{\mathbf{h}}}_{12,kj}^{\scriptscriptstyle\mathsf{T}}$,
$\tilde{{\mathbf{q}}}_{k}=\sum_{j\neq
k}\tilde{{\mathbf{h}}}_{12,kj}^{*}\bar{h}_{0,kj}$, and
$\tilde{e}_{k}=\sigma_{\mathrm{u}}^{2}+\sum_{j\neq
k}\left|\bar{h}_{0,kj}\right|^{2}$. As a result, $\boldsymbol{\alpha}$ is
exposed in the following form of $\mathcal{R}_{k}^{\mathrm{loc}}$:

$\displaystyle\mathcal{R}_{k}^{\mathrm{loc}}=\log\left(1+\frac{\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}{\mathbf{Q}}_{k}\boldsymbol{\alpha}+2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}{\mathbf{q}}_{k}\right)}+e_{k}}{\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{Q}}}_{k}\boldsymbol{\alpha}+2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{q}}}_{k}\right)}+\tilde{e}_{k}}\right).$

With this form, we can rewrite (16b) as

$\displaystyle\bar{r}_{k}\geq\tau+\tilde{r}_{k},$ (33)

where
$\bar{r}_{k}\triangleq\log\Big{(}\lVert\bar{{\mathbf{Q}}}_{k}^{\frac{1}{2}}\boldsymbol{\alpha}\rVert^{2}+2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\bar{{\mathbf{q}}}_{k}\right)}+\bar{e}_{k}\Big{)}$,
$\tilde{r}_{k}\triangleq\log(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{Q}}}_{k}\boldsymbol{\alpha}+2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{q}}}_{k}\right)}+\tilde{e}_{k})$,
with $\bar{{\mathbf{Q}}}_{k}={\mathbf{Q}}_{k}+\tilde{{\mathbf{Q}}}_{k}$,
$\bar{{\mathbf{q}}}_{k}={\mathbf{q}}_{k}+\tilde{{\mathbf{q}}}_{k}$,
$\bar{e}_{k}=e_{k}+\tilde{e}_{k}$, and note that $\bar{{\mathbf{Q}}}_{k}$ is
positive semidefinite. These two are nonconvex with respect to
$\boldsymbol{\alpha}$. However, by noting from approximation (12) that
$\lVert\bar{{\mathbf{Q}}}_{k}^{\frac{1}{2}}\boldsymbol{\alpha}\rVert^{2}=-f_{\mathtt{qua}}(\bar{{\mathbf{Q}}}_{k}^{\frac{1}{2}}\boldsymbol{\alpha};\boldsymbol{0})\geq-
F_{\mathtt{qua}}(\bar{{\mathbf{Q}}}_{k}^{\frac{1}{2}}\boldsymbol{\alpha};\boldsymbol{0},\bar{{\mathbf{Q}}}_{k}^{\frac{1}{2}}\boldsymbol{\alpha}^{(i)})$,
a concave lower bound of $\bar{r}_{k}$ can be found as

$\displaystyle\bar{r}_{k}\geq\bar{r}_{\mathrm{lb},k}^{(i)}\triangleq\log\Big{(}2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\bar{{\mathbf{q}}}_{k}\right)}-F_{\mathtt{qua}}(\bar{{\mathbf{Q}}}_{k}^{\frac{1}{2}}\boldsymbol{\alpha};\boldsymbol{0},\bar{{\mathbf{Q}}}_{k}^{\frac{1}{2}}\boldsymbol{\alpha}^{(i)})+\bar{e}_{k}\Big{)}.$

To tackle the nonconvexity of $\tilde{r}_{k}$, we introduce slack variables
$\vartheta_{k}$ such that
$\vartheta_{k}\geq\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{Q}}}_{k}\boldsymbol{\alpha}+2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{q}}}_{k}\right)}$.
With this introduction of $\vartheta_{k}$, we can write
$\tilde{r}_{k}=\log(\vartheta_{k}+\tilde{e}_{k})$. Thus, an upper bound of
$\tilde{r}_{k}$ can be found as
$\tilde{r}_{k}\leq\tilde{r}_{\mathrm{ub},k}^{(i)}\triangleq\log\Big{(}\vartheta_{k}^{(i)}+\tilde{e}_{k}\Big{)}+\frac{\vartheta_{k}-\vartheta_{k}^{(i)}}{\vartheta_{k}^{(i)}+\tilde{e}_{k}}$,
based on the first-order Taylor approximation around $\vartheta_{k}^{(i)}$. As
a result, constraint (33) can be approximated by the following set of convex
constraints:

$\displaystyle\bar{r}_{\mathrm{lb},k}^{(i)}$
$\displaystyle\geq\tau+\tilde{r}_{\mathrm{ub},k}^{(i)},$ (34)
$\displaystyle\forall k,\ \vartheta_{k}$
$\displaystyle\geq\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{Q}}}_{k}\boldsymbol{\alpha}+2\mathfrak{R}{\left(\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{q}}}_{k}\right)},\
\forall k.$ (35)

Furthermore, we can rewrite (8f) as

$\displaystyle\boldsymbol{\alpha}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Xi}\boldsymbol{\alpha}\leq
p_{\mathrm{max}}^{\mathrm{ris}},$ (36)

where
$\boldsymbol{\Xi}=\mathrm{diag}\\{\tilde{\xi}_{1},\ldots,\tilde{\xi}_{N}\\}$
with $\tilde{\xi}_{n}=\xi_{n}$ for $n\in\mathcal{A}$, and $\tilde{\xi}_{n}=0$,
otherwise. Finally, problem (32) can be approximated by the following convex
program at iteration $i$:

$\displaystyle\underset{\begin{subarray}{c}\tau,\boldsymbol{\alpha},\\{\vartheta_{k}\\}\end{subarray}}{\textrm{maximize}}\
\tau,\ \textrm{subject to}\
\eqref{cons_ris_phase_loc}-\eqref{cons_active_modul_loc},\eqref{cons_min_rate_loc_2b},\eqref{cons_pris_loc2}.$
(37)

###  III-D Overall Algorithm

We summarize the BCA-SCA-based iterative algorithm to jointly optimize the
UAV’s transmit beamforming and locations as well as the RIS beamforming in
Algorithm 1. In Step 1, the initial solutions to ${\mathbf{v}}^{(0)}$,
$\\{{\mathbf{w}}_{k}^{(0)}\\}$, $\\{\alpha_{n}^{(0)}\\}$, and
$\boldsymbol{\mathcal{V}}^{(0)}$ are generated. In Steps 2–7, subproblems
(21), (31), and (37) are alternatively solved and ${\mathbf{v}}^{(i)}$,
$\\{{\mathbf{w}}_{k}^{(i)}\\}$, $\\{\alpha_{n}^{(i)}\\}$, and
$\boldsymbol{\mathcal{V}}^{(i)}$ are updated after each iteration until the
objective value converges. The complexities required to solve subproblems
(21), (31) and (37) are $\mathcal{O}(\sqrt{2}K^{3.5}N_{t}^{3})$,
$\mathcal{O}(8K^{4})$, and $\mathcal{O}(\sqrt{2N+2K}(N+K)^{3})$, respectively.
Therefore, the total complexity of Algorithm 1 is

$\displaystyle\mathcal{C}_{\mathrm{loc}}=\mathcal{O}\left(\mathcal{I}_{\mathrm{loc}}\left(\sqrt{2}K^{3.5}N_{t}^{3}+8K^{4}+\sqrt{2}(N+K)^{3.5}\right)\right),$

where $\mathcal{I}_{\mathrm{loc}}$ is the number of iterations in Algorithm 1
until convergence. Considering that $N\gg K,N_{t}$, the complexity of this
algorithm can be approximated as $\mathcal{O}(N^{3.5})$, which is required for
the optimization of RIS coefficients.

Algorithm 1 Proposed Iterative Algorithm to Solve Problem
$(\mathcal{P}_{\mathrm{loc}})$ in (8)

1: Initialization: Set $i=0$. Generate initial values ${\mathbf{v}}^{(0)}$,
$\\{{\mathbf{w}}_{k}^{(0)}\\}$ and $\\{\alpha_{n}^{(0)}\\}$. Initialize
$\\{\gamma_{k}^{(0)}\\}$, $\\{\vartheta_{k}^{(0)}\\}$, and
$\boldsymbol{\mathcal{V}}^{(0)}$ by considering equalities of (18b), (35), and
(25), respectively.

2: repeat

3: Solve problem (21) with given $\\{{\mathbf{v}}^{(i)}\\}$,
$\alpha_{n}^{(i)}$, and $\\{\gamma_{k}^{(i)}\\}$ to obtain solutions
$\\{{\mathbf{w}}_{k}^{\star}\\}$ and $\\{\gamma_{k}^{\star}\\}$. Set
${\mathbf{w}}_{k}^{(i+1)}={\mathbf{w}}_{k}^{\star}$ and
$\gamma_{k}^{(i+1)}=\gamma_{k}^{\star},\forall k$.

4: Solve problem (31) with given $\\{{\mathbf{w}}_{k}^{(i+1)}\\}$,
$\\{\alpha_{n}^{(i)}\\}$, and $\boldsymbol{\mathcal{V}}^{(i)}$ to obtain
solutions ${\mathbf{v}}^{\star}$ and $\boldsymbol{\mathcal{V}}^{\star}$. Set
${\mathbf{v}}^{(i+1)}={\mathbf{v}}^{\star}$ and
$\boldsymbol{\mathcal{V}}^{(i+1)}=\boldsymbol{\mathcal{V}}^{\star}$.

5: Solve problem (37) with given $\\{{\mathbf{w}}_{k}^{(i+1)}\\}$,
${\mathbf{v}}^{(i+1)}$, and $\\{\vartheta_{k}^{(i)}\\}$ to obtain solutions
$\boldsymbol{\alpha}^{\star}$ and $\\{\vartheta_{k}^{\star}\\}$. Set
$\boldsymbol{\alpha}^{(i+1)}=\boldsymbol{\alpha}^{\star}$ and
$\gamma_{k}^{(i+1)}=\gamma_{k}^{\star},\forall k$.

6: Update $i=i+1$.

7: until convergence.

Algorithm 1 solve original problem (8) via convex sub-problems (21), (31), and
(37), combining that the early update of the partial set of variables in each
iteration guarantees feasible points and non-decreasing objective values.
Owing to the concreteness and convexity of the feasible sets in the sub-
problems, the proposed algorithm remains the convergence of the inner
approximation method. Let us denote by
$\tau_{{\mathbf{w}}}({\mathbf{w}}^{(i)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})$
and
$\tau_{{\mathbf{w}}}^{(i)}({\mathbf{w}}^{(i)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})$
the objective values of the original subproblem (17) and the approximate
convex subproblem (21) obtained at iteration $i$ with respect to
${\mathbf{w}}$, respectively. Following the SCA principles [54], it is true
that

$\displaystyle\tau_{{\mathbf{w}}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})\geq\tau_{{\mathbf{w}}}^{(i)}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})$
$\displaystyle\qquad\geq\tau_{{\mathbf{w}}}^{(i)}({\mathbf{w}}^{(i)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})=\tau_{{\mathbf{w}}}({\mathbf{w}}^{(i)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)}).$

We note that
$\tau_{{\mathbf{w}}}^{(i)}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})$
$>\tau_{{\mathbf{w}}}^{(i)}({\mathbf{w}}^{(i)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})$
and
$\tau_{{\mathbf{w}}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})>\tau_{{\mathbf{w}}}({\mathbf{w}}^{(i)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})$
whenever ${\mathbf{w}}^{(i)}\neq{\mathbf{w}}^{(i+1)}$. Similarly for problems
(22) and (32), we can prove that

$\displaystyle\tau_{{\mathbf{v}}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})=\tau^{(i)}_{{\mathbf{v}}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})$
$\displaystyle\tau_{{\mathbf{v}}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i+1)},\boldsymbol{\alpha}^{(i)})>\tau_{{\mathbf{v}}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)}),$

as far as ${\mathbf{v}}^{(i)}\neq{\mathbf{v}}^{(i+1)}$, and
$\tau_{\boldsymbol{\alpha}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i+1)},\boldsymbol{\alpha}^{(i)})=\tau^{(i)}_{\boldsymbol{\alpha}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i+1)},\boldsymbol{\alpha}^{(i)})$
and
$\tau_{\boldsymbol{\alpha}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i+1)},\boldsymbol{\alpha}^{(i+1)})>\tau_{\boldsymbol{\alpha}}({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i+1)},\boldsymbol{\alpha}^{(i)})$
as far as $\boldsymbol{\alpha}^{(i)}\neq\boldsymbol{\alpha}^{(i+1)}$. Thus, we
can readily show that
$\tau({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i+1)},\boldsymbol{\alpha}^{(i+1)})$
$>\tau({\mathbf{w}}^{(i)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})$,
which indicates that the objective value of problem (8) is non-decreasing
after each iteration. Because it is upper bounded by a finite value, Algorithm
1 is guaranteed to converge. Utilizing the non-decreasing property of the
objective values over iterations, the convergence is determined based on the
difference between the objective values in successive iterations. More
specifically, the algorithm converges if
$\tau({\mathbf{w}}^{(i+1)},{\mathbf{v}}^{(i+1)},\boldsymbol{\alpha}^{(i+1)})-\tau({\mathbf{w}}^{(i)},{\mathbf{v}}^{(i)},\boldsymbol{\alpha}^{(i)})<\varepsilon$,
where $\varepsilon$ is the convergence stopping criteria.

##  IV Joint UAV’s Trajectory Planning and Beamforming Optimization in
$(\mathcal{P}_{\mathrm{tra}})$

Similar to (16), $(\mathcal{P}_{\mathrm{tra}})$ in (10) has the following
epigraph form:

$\displaystyle\underset{\begin{subarray}{c}\tau,\\{b_{k}[t]\\},\\{{\mathbf{v}}[t]\\},\\\
\\{{\mathbf{w}}[t]\\},\\{\alpha_{n}[t]\\}\end{subarray}}{\textrm{maximize}}\quad$
$\displaystyle\tau$ (38a) subject to
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}b_{k}[t]\mathcal{R}_{k}^{\mathrm{tra}}[t]\geq\tau,\forall
k,$ (38b) $\displaystyle\eqref{cons_traj_1p}-\eqref{cons_pris_tra},$ (38c)

where

$\displaystyle\mathcal{R}_{k}^{\mathrm{tra}}[t]=\log\left(1+\frac{\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{w}}[t]\rvert^{2}}{\sigma_{\mathrm{r}}^{2}\lVert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}[t]\boldsymbol{\Psi}[t]\rVert^{2}+\sigma_{\mathrm{u}}^{2}}\right).$
(39)

To solve (38), we leverage the BCA framework and consider four sub-problems,
including the UE scheduling, UAV’s trajectory, beamforming optimization, and
hybrid RIS beamforming design. However, compared to problem (16), the numbers
of variables in each subproblems of (38) significantly increase, which are
linear with $T$. Therefore, a direct application of SCA would result in an
extremely high computational complexity. To overcome this challenge as well as
the mathematical difficulties of (38), we provide an efficient solution with
low complexity in the following subsections.

###  IV-A UE Scheduling Design

We first aim at solving $\\{b_{k}[t]\\}$ while fixing
$(\\{{\mathbf{v}}[t]\\},\\{{\mathbf{w}}[t]\\},\\{\alpha_{n}[t]\\})$. To bypass
the difficulty of binary constraints (10d), we relax binary variables
$\\{b_{k}[t]\\}$ into continuous one, i.e. $b_{k}[t]\in[0,1],\forall k,t$ [1].
Thus, the UE scheduling can be optimized by solving the following convex
linear program:

$\displaystyle\underset{\tau,\\{b_{k}[t]\\}}{\textrm{maximize}}\ \tau,\
\textrm{subject to}\
\eqref{cons_scheduling_3p},\eqref{cons_scheduling_1p},\eqref{cons_min_rate},0\leq
b_{k}[t]\leq 1,\ \forall k,t.$ (40)

###  IV-B Optimization of UAV Beamforming

Given $(\\{b_{k}[t]\\},\\{{\mathbf{v}}[t]\\},\\{\alpha_{n}[t]\\})$, the
optimal UAV beamformers $\\{{\mathbf{w}}[t]\\}$ are found by solving

$\displaystyle\underset{\tau,\\{{\mathbf{w}}[t]\\}}{\textrm{maximize}}\ \tau,\
\textrm{subject to}\
\eqref{cons_puav_tra},\eqref{cons_pris_tra},\eqref{cons_min_rate}.$ (41)

Towards an appealing application, we will find a closed-form solution to this
problem that provides more insights into the system design and requires much
lower complexity. To this end, we rewrite constraint (38b) as
$\frac{1}{T}\sum_{t=1}^{T}b_{k}[t]\log\left(1+\frac{\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{w}}[t]\rvert^{2}}{\sigma_{k}^{2}[t]}\right)\geq\tau,\forall
k,$ where
$\sigma_{k}^{2}[t]\triangleq\sigma_{\mathrm{r}}^{2}\lVert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}[t]\boldsymbol{\Psi}[t]\rVert^{2}+\sigma_{\mathrm{u}}^{2}$
is constant with respect to $\\{{\mathbf{w}}[t]\\}$. The left hand side (LHS)
of this constraint monotonically increases with
$\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{w}}[t]\rvert^{2}$.
Thus, the optimal beamforming vector admits the following form

$\displaystyle{\mathbf{w}}[t]^{\star}=\sqrt{p[t]^{\star}}\frac{{\mathbf{h}}_{k}[t]}{\left\lVert{\mathbf{h}}_{k}[t]\right\rVert},\
\forall t,$ (42)

where $p[t]^{\star}$ is the optimal transmit power of the UAV at time slot $t$
and $p[t]^{\star}\leq p_{\mathrm{max}}^{\mathrm{uav}}$. We note the fact that
$\left\lVert{\mathbf{w}}[t]^{\star}\right\rVert^{2}=p[t]^{\star}$ and rewrite
constraints (38b), (10f), and (10j) equivalently as

$\displaystyle\frac{1}{T}\sum_{t=1}^{T}b_{k}[t]\log\left(1+\frac{p[t]^{\star}\left\lVert{\mathbf{h}}_{k}\right\rVert^{2}}{\sigma_{k}^{2}[t]}\right)\geq\tau,\forall
k,$ $\displaystyle 0\leq p[t]^{\star}\leq
p_{\mathrm{max}}^{\mathrm{uav}},\forall t,$
$\displaystyle\sum_{n\in\mathcal{A}}\left|\alpha_{n}[t]\right|^{2}\left(\sigma_{\mathrm{r}}^{2}+\left\lVert{\mathbf{h}}_{1,n}[t]\right\rVert^{2}p[t]^{\star}\right)\leq
p_{\mathrm{max}}^{\mathrm{ris}},\forall t,$

respectively. These reveal the optimal power allocation

$\displaystyle
p^{\star}[t]=\min\left\\{p_{\mathrm{max}}^{\mathrm{uav}},\frac{p_{\mathrm{max}}^{\mathrm{ris}}-\sum_{n\in\mathcal{A}}\left|\alpha_{n}[t]\right|^{2}\sigma_{\mathrm{r}}^{2}}{\sum_{n\in\mathcal{A}}\left\lVert{\mathbf{h}}_{1,n}[t]\right\rVert^{2}}\right\\},\
\forall t.$ (43)

Finally, $\\{{\mathbf{w}}[t]^{\star}\\}$ is reconstructed by (42).

###  IV-C Optimization of the UAVs’ Trajectory

With given $(\\{b_{k}[t]\\},\\{{\mathbf{w}}[t]\\},\\{\alpha_{n}[t]\\})$, the
UAV’s trajectory can be optimized by solving the following problem with
respect to $\\{\tau,\\{{\mathbf{v}}[t]\\}\\}$:

$\displaystyle\underset{\tau,\\{{\mathbf{v}}[t]\\}}{\textrm{maximize}}\ \tau,\
\textrm{subject to}\
\eqref{cons_traj_1p},\eqref{cons_traj_2p},\eqref{cons_pris_tra},\eqref{cons_min_rate},$
(44)

where constraints (10c), (10j) and (38b) are nonconvex. Furthermore,
$\\{{\mathbf{v}}[t]\\}$ is hidden in all these constraints. To tackle these
challenges, we transform constraints (10j) and (38b) into equivalent
constraints but more tractable forms.

First, we expand the expression of $\mathcal{R}_{k}^{\mathrm{tra}}[t]$ in (6)
to show its dependence on ${\mathbf{v}}$. By a similar derivation as in (23),
we obtain

$\displaystyle\lvert{\mathbf{h}}_{k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{w}}[t]\rvert^{2}$
$\displaystyle=\bar{c}_{0,k}[t]\left\lVert{\mathbf{v}}[t]-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}}+\bar{c}_{1,k}[t]\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert^{-\epsilon_{1}}$
$\displaystyle\qquad+\bar{c}_{2,k}[t]\left\lVert{\mathbf{v}}[t]-{\mathbf{u}}_{k}\right\rVert^{\frac{-\epsilon_{0}}{2}}\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert^{\frac{-\epsilon_{1}}{2}},$

where

$\displaystyle\bar{c}_{0,k}[t]$
$\displaystyle\triangleq\zeta_{0}\lvert{\mathbf{g}}_{0,k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{w}}[t]\rvert^{2},$
$\displaystyle\bar{c}_{1,k}[t]$
$\displaystyle\triangleq\zeta_{0}\lvert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}[t]{\mathbf{G}}_{1}[t]{\mathbf{w}}[t]\rvert^{2},$
$\displaystyle\bar{c}_{2,k}[t]$ $\displaystyle\triangleq
2\zeta_{0}\mathfrak{R}{\left({\mathbf{w}}_{k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{g}}_{0,k}[t]{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Upsilon}[t]{\mathbf{G}}_{1}[t]{\mathbf{w}}[t]\right)},$

are constants with respect to variable ${\mathbf{v}}[t]$. Thus,
$\mathcal{R}_{k}^{\mathrm{tra}}[t]$ in (9) can be rewritten as

$\displaystyle\mathcal{R}_{k}^{\mathrm{tra}}[t]=$
$\displaystyle\log\Big{(}1+\rho_{k}[t]\Big{(}\bar{c}_{0,k}[t]\left\lVert{\mathbf{v}}[t]-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}}$
$\displaystyle\qquad+\bar{c}_{1,k}[t]\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert^{-\epsilon_{1}}$
$\displaystyle\qquad+\bar{c}_{2,k}[t]\left\lVert{\mathbf{v}}[t]-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}/2}\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2}\Big{)}\Big{)},$

where
$\rho_{k}[t]\triangleq\left(\sigma_{\mathrm{r}}^{2}\lVert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Psi}[t]\rVert^{2}+\sigma_{\mathrm{u}}^{2}\right)^{-1}$
is constant with $\\{{\mathbf{v}}[t]\\}$. It is observed that although
$\mathcal{R}_{k}^{\mathrm{tra}}[t]$ has a simpler form than
$\mathcal{R}_{k}^{\mathrm{loc}}$ in (24), it inherits the mathematically
intractable challenges of $\mathcal{R}_{k}^{\mathrm{loc}}$. Specifically, it
is still a very complicated function of $\\{{\mathbf{v}}[t]\\}$ with the note
that while $\bar{c}_{0,k}[t]>0$, $\bar{c}_{1,k}[t]>0,\forall k,t$, the sign of
$\bar{c}_{2,k}[t]$ varies depending on $k$ and $t$. We apply a similar
approach presented in problem (22) to tackle constraint (38b). Specifically,
let us introduce slack variables $\\{\hat{v}_{0,k}[t]\\}$ and
$\\{\hat{v}_{1}[t]\\}$, satisfying

$\displaystyle\hat{v}_{0,k}[t]$
$\displaystyle\leq\left\lVert{\mathbf{v}}[t]-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}/2},\
\forall k,t,$ $\displaystyle\hat{v}_{1}[t]$
$\displaystyle\leq\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2},\
\forall t,$ (45)

and $\\{\bar{v}_{0,k}[t]\\}$, $\\{\bar{v}_{1}[t]\\}$ satisfying

$\displaystyle\begin{cases*}\bar{v}_{0,k}[t]=\hat{v}_{0,k}[t],\text{if\leavevmode\nobreak\
}\bar{c}_{2,k}[t]\geq 0\\\
\bar{v}_{0,k}[t]\geq\left\lVert{\mathbf{v}}[t]-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}/2},\text{otherwise}\end{cases*},\forall
k,t,$
$\displaystyle\begin{cases*}\bar{v}_{1}[t]=\hat{v}_{1}[t],\text{if\leavevmode\nobreak\
}\bar{c}_{2,k}[t]\geq 0\\\
\bar{v}_{1}[t]\geq\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2},\text{otherwise}\end{cases*},\forall
t.$ (46)

$\mathcal{R}_{k}^{\mathrm{tra}}[t]$ is then lower bounded as

$\displaystyle\mathcal{R}_{k}^{\mathrm{tra}}[t]$
$\displaystyle\geq\log\Big{(}1+\rho_{k}[t]\Big{(}\bar{c}_{0,k}[t](\hat{v}_{0,k}[t])^{2}+\bar{c}_{1,k}[t](\hat{v}_{1}[t])^{2}$
$\displaystyle\hskip
113.81102pt+\bar{c}_{2,k}[t]\bar{v}_{0,k}[t]\bar{v}_{1}[t]\Big{)}\Big{)}.$

To further simplify it, we introduce slack variables $\\{\hat{a}_{k}[t]\\}$
satisfying

$\displaystyle\hat{a}_{k}[t]$
$\displaystyle\leq\begin{cases*}\bar{c}_{0,k}[t](\hat{v}_{0,k}[t])^{2}+\bar{c}_{1,k}[t](\hat{v}_{1}[t])^{2}\\\
\hskip
28.45274pt+\left|\bar{c}_{2,k}[t]\right|\bar{v}_{0,k}[t]\bar{v}_{1}[t],\
\text{if\leavevmode\nobreak\ }\bar{c}_{2,k}[t]\geq 0\\\
\bar{c}_{0,k}[t](\hat{v}_{0,k}[t])^{2}+\bar{c}_{1,k}[t](\hat{v}_{1}[t])^{2}\\\
\hskip
28.45274pt-\left|\bar{c}_{2,k}[t]\right|\bar{v}_{0,k}[t]\bar{v}_{1}[t],\
\text{otherwise}\end{cases*},\ \forall k,t.$ (47)

Thus, (38b) is transformed to the set of constraints (45)-(47) and

$\displaystyle\frac{1}{T}\sum_{t=1}^{T}b_{k}[t]\log\Bigl{(}1+\rho_{k}[t]\hat{a}_{k}[t]\Bigr{)}\geq\tau,\forall
k,$ (48)

which is convex with respect to $\\{\hat{a}_{k}[t]\\}$. To handle the
nonconvex constraints (45)-(47), we apply the inequalities (11)–(14) to
convexify them as

$\displaystyle\left\lVert{\mathbf{v}}[t]-{\mathbf{u}}_{k}\right\rVert+F_{\mathtt{pow}}(\hat{v}_{0,k}[t];-2/\epsilon_{0},\hat{v}_{0,k}[t]^{(i)})\leq
0,\forall k,t,$ (49a)
$\displaystyle\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert+F_{\mathtt{pow}}(\hat{v}_{1}[t];-2/\epsilon_{1},\hat{v}_{1}[t]^{(i)})\leq
0,\ \forall t,$ (49b)
$\displaystyle\begin{cases*}\bar{v}_{0,k}[t]=\hat{v}_{0,k}[t],\
\text{if\leavevmode\nobreak\ }\bar{c}_{2,k}[t]\geq 0\\\
F_{\mathtt{qua}}({\mathbf{v}}[t];{\mathbf{u}}_{k},{\mathbf{v}}[t]^{(i)})\leq
1/F_{\mathtt{pow}}(\bar{v}_{0,k}[t];4/\epsilon_{0},\bar{v}_{0,k}[t]^{(i)}),\end{cases*}$
$\displaystyle\hskip 149.37697pt\text{otherwise},\forall k,t,$ (49c)
$\displaystyle\begin{cases*}\bar{v}_{1}[t]=\hat{v}_{1}[t],\
\text{if\leavevmode\nobreak\ }\bar{c}_{2,k}[t]\geq 0\\\
F_{\mathtt{qua}}({\mathbf{v}}[t];{\mathbf{r}},{\mathbf{v}}[t]^{(i)})\leq
1/F_{\mathtt{pow}}(\bar{v}_{1}[t];4/\epsilon_{1},\bar{v}_{1}[t]^{(i)}),\end{cases*},$
$\displaystyle\hskip 156.49014pt\text{otherwise},\forall t,$ (49d)
$\displaystyle\hat{a}_{k}[t]\leq\bar{c}_{0,k}[t]F_{\mathtt{qua}}(\hat{v}_{0,k}[t];0,\hat{v}_{0,k}[t]^{(i)})$
$\displaystyle\hskip
42.67912pt+\bar{c}_{1,k}[t]F_{\mathtt{qua}}(\hat{v}_{1}[t];0,\hat{v}_{1}[t]^{(i)})$
$\displaystyle\hskip
42.67912pt+\left|\bar{c}_{2,k}[t]\right|F_{\mathtt{bil}}(\bar{v}_{0,k}[t],\bar{v}_{1}[t];$
$\displaystyle\hskip
56.9055pt\mathrm{sgn}(\bar{c}_{2,k}[t]),\bar{v}_{0,k}[t]^{(i)},\bar{v}_{1}[t]^{(i)}),\
\forall k,t,$ (49e)

around
$\\{\hat{v}_{0,k}[t]^{(i)},\hat{v}_{1}[t]^{(i)},\bar{v}_{0,k}[t]^{(i)},\bar{v}_{1}[t]^{(i)},\hat{a}_{k}[t]^{(i)},\tilde{v}_{1}[t]^{(i)}\\}$.

To address constraint (10j), we first rewrite $\xi_{n}[t]$ as
$\xi_{n}[t]=\sigma_{\mathrm{r}}^{2}+c_{3,n}[t]\left(\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert^{-\frac{\epsilon_{1}}{2}}\right)^{2},$
where
$c_{3,n}[t]\triangleq\zeta_{0}\left\lVert{\mathbf{g}}_{1,n}[t]\right\rVert^{2}\sum_{k=1}^{K}\left\lVert{\mathbf{w}}[t]\right\rVert^{2}$
is independent of $\\{{\mathbf{v}}[t]\\}$. Thus, slack variables
$\\{\tilde{v}_{1}[t]\\}$ satisfying
$\tilde{v}_{1}[t]\geq\left\lVert{\mathbf{v}}[t]-{\mathbf{r}}\right\rVert^{-\epsilon_{1}/2}$
are introduced to transform (8f) to the set of following convex constraints:

$\displaystyle
F_{\mathtt{qua}}({\mathbf{v}}[t];{\mathbf{r}},{\mathbf{v}}[t]^{(i)})\leq\frac{1}{F_{\mathtt{pow}}(\tilde{v}_{1}[t];4/\epsilon_{1},\tilde{v}_{1}[t]^{(i)})},$
$\displaystyle\sum_{n\in\mathcal{A}}\left|\alpha_{n}[t]\right|^{2}\left(\sigma_{\mathrm{r}}^{2}+c_{3,n}[t](\tilde{v}_{1}[t])^{2}\right)\leq
p_{\mathrm{max}}^{\mathrm{ris}},\forall t.$ (50)

In summary, we can approximate problem (10) at iteration $i$ by the following
convex program:

$\displaystyle\underset{\tau,\\{{\mathbf{v}}[t]\\},\boldsymbol{\mathcal{V}}}{\textrm{maximize}}\
\tau,\ \textrm{subject to}\
\eqref{cons_traj_1p},\eqref{cons_traj_2p},\eqref{cons_min_ratep_tra},\eqref{cons_v0hat_tra_2}-\eqref{def_f_hat_2},\eqref{eq_pris_approx}$
(51)

where
$\boldsymbol{\mathcal{V}}[t]\triangleq\\{\hat{v}_{0,k}[t],\hat{v}_{1}[t],\bar{v}_{0,k}[t],\bar{v}_{1}[t],\hat{a}_{k}[t],\tilde{v}_{1}[t]\\}$.

###  IV-D Optimization of Hybrid RIS Coefficients

Finally, we complete the solution to problem (38) by solving
$\\{\alpha_{n}[t]\\}$ in the following problem:

$\displaystyle\underset{\begin{subarray}{c}\tau,\\{\alpha_{n}[t]\\}\end{subarray}}{\textrm{maximize}}\
\tau,\textrm{subject to}\
\eqref{cons_ris_phase_tra}-\eqref{cons_pris_tra},\eqref{cons_min_rate},$ (52)

with given $(\\{b_{k}[t]\\},\\{{\mathbf{v}}[t]\\},\\{{\mathbf{w}}[t]\\})$.
Note that constraint (38b) is nonconvex. To examine the role of
$\\{\alpha_{n}\\}$ and to make (38b) more tractable, we substitute (2) into
(6). Furthermore, by recalling that
$\boldsymbol{\Upsilon}[t]=\boldsymbol{\Phi}[t]+\boldsymbol{\Psi}[t]$, we can
rewrite (39) as
$\mathcal{R}_{k}^{\mathrm{tra}}[t]=\log\left(1+\frac{\left|\bar{h}_{0,k}[t]+{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}(\boldsymbol{\Psi}[t]+\boldsymbol{\Phi}[t])\bar{{\mathbf{h}}}_{1,k}[t]\right|^{2}}{\sigma_{\mathrm{r}}^{2}\lVert{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Psi}[t]\rVert^{2}+\sigma_{\mathrm{u}}^{2}}\right),$
where
$\bar{h}_{0,k}[t]\triangleq{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}[t]{\mathbf{w}}[t]$,
$\bar{{\mathbf{h}}}_{1,k}[t]\triangleq{\mathbf{H}}_{1}[t]{\mathbf{w}}[t]$;
furthermore, we note that
$\tilde{{\mathbf{H}}}_{2,k}=\mathrm{diag}\\{{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\\}$,
and $\boldsymbol{\Phi}[t]$ and $\boldsymbol{\Psi}[t]$ are diagonal matrices
containing only the passive and active coefficients, respectively. Let
$\boldsymbol{\psi}[t]\triangleq\mathrm{diag}\left\\{\boldsymbol{\Psi}[t]\right\\}\in\mathbb{C}^{N\times
1}$,
$\boldsymbol{\phi}[t]\triangleq\mathrm{diag}\left\\{\boldsymbol{\Phi}[t]\right\\}\in\mathbb{C}^{N\times
1}$ and
$\tilde{{\mathbf{h}}}_{12,k}[t]\triangleq\tilde{{\mathbf{H}}}_{2,k}\bar{{\mathbf{h}}}_{1,k}[t]$.
Following the development in Section III-C, we can rewrite
$\mathcal{R}_{k}^{\mathrm{tra}}[t]$ as

$\displaystyle\mathcal{R}_{k}^{\mathrm{tra}}[t]=\log\left(1+\frac{\left|\bar{h}_{0,k}[t]+\left(\boldsymbol{\psi}[t]+\boldsymbol{\phi}[t]\right)^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{h}}}_{12,k}[t]\right|^{2}}{\sigma_{\mathrm{r}}^{2}\lVert\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{H}}}_{2,k}\rVert^{2}+\sigma_{\mathrm{u}}^{2}}\right).$
(53)

###### Remark 1

For the conventional passive RIS, i.e., when $\mathcal{A}=\emptyset$ and
$\boldsymbol{\psi}[t]=\boldsymbol{0}$, the denominator of the SINR term in
(53) is $\sigma_{u}^{2}$. Therefore, the optimal solutions to
$\\{\alpha_{n}[t]\\}$ are those maximizing the received signal power at the UE
scheduled at time slot $t$, given as
$\alpha_{n}[t]^{\star}=e^{j\theta_{n}[t]^{\star}},\
\text{where\leavevmode\nobreak\
}\theta_{n}[t]^{\star}=\angle\bar{h}_{0,k}[t]-\angle\tilde{h}_{12,kn}[t],\forall
n,t,$ with $\tilde{h}_{12,kn}$ being the $n$th element of
$\tilde{{\mathbf{h}}}_{12,k}$. However, for the hybrid RIS, this solution is
not valid anymore due to the additional noise
$\sigma_{\mathrm{r}}^{2}\lVert\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{H}}}_{2,k}\rVert^{2}$.

The solution to problem (52) can be found by jointly optimizing RIS passive
and active coefficients, as done for (32). However, the resultant solution
requires a complexity of at least $\mathcal{O}(N^{3.5})$, which is high due to
a large value of $N$. To overcome this, based on Remark 1, we decouple the
optimization of active and passive elements of the RIS. More specifically,
with given $\boldsymbol{\psi}$, the solution to $\boldsymbol{\phi}$ is
obtained by solving the following problem

$\displaystyle\underset{\begin{subarray}{c}\tau,\boldsymbol{\phi}\end{subarray}}{\textrm{maximize}}\
\tau,\textrm{subject to}\
\eqref{cons_ris_phase_tra},\eqref{cons_passive_modul_tra},\eqref{cons_min_rate},$
(54)

and then, the solution to $\psi$ is obtained by solving

$\displaystyle\underset{\begin{subarray}{c}\tau,\boldsymbol{\psi}\end{subarray}}{\textrm{maximize}}\
\tau,\textrm{subject to}\
\eqref{cons_ris_phase_tra},\eqref{cons_active_modul_tra},\eqref{cons_pris_tra},\eqref{cons_min_rate}.$
(55)

This approach allows solving (52) with much lower complexity as (54) admits a
closed-form solution, while the number of variables in (55), i.e., the number
of active coefficients to optimize, is just $N_{\mathrm{a}}\ll N$. We
elaborate these solutions in the following.

####  IV-D1 Solution to Problem (54)

It is clear from (53) that with given $\\{\boldsymbol{\psi}[t]\\}$, the
optimal solutions to the passive coefficients are

$\displaystyle\phi_{n}[t]^{\star}=e^{j\theta_{n}[t]^{\star}},\ \forall
n\notin\mathcal{A},\forall t.$ (56)

where
$\theta_{n}[t]^{\star}=\angle\left(\bar{h}_{0,k}[t]+\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{h}}}_{12,k}[t]\right)-\angle\tilde{h}_{12,kn}[t]$.

####  IV-D2 Solution to Problem (55)

We first aim at tackling the nonconvexity of constraint (38b). Let us denote
$\bar{\bar{h}}_{0,k}[t]\triangleq\bar{h}_{0,k}[t]+\boldsymbol{\phi}[t]^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{h}}}_{12,k}[t]$.
Then, by several algebraic operations, we can express the numerator and
denominator of the SINR term in (53) as
$\lvert\bar{h}_{0,k}[t]+\left(\boldsymbol{\psi}[t]+\boldsymbol{\phi}[t]\right)^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{h}}}_{12,k}[t]\rvert^{2}=\lvert\bar{\bar{h}}_{0,k}[t]+\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{h}}}_{12,k}[t]\rvert^{2}=\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}{\mathbf{D}}_{k}[t]\boldsymbol{\psi}[t]+2\mathfrak{R}{\left(\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}{\mathbf{d}}_{k}[t]\right)}+z_{k}[t]$
where
${\mathbf{D}}_{k}[t]=\tilde{{\mathbf{h}}}_{12,k}^{*}[t]\tilde{{\mathbf{h}}}_{12,k}^{\scriptscriptstyle\mathsf{T}}[t]$,
${\mathbf{d}}_{k}[t]=\tilde{{\mathbf{h}}}_{12,k}^{*}[t]\bar{\bar{h}}_{0,k}[t]$,
and $z_{k}=\lvert\bar{\bar{h}}_{0,kk}\rvert^{2}$, and
$\sigma_{\mathrm{r}}^{2}\lVert\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{T}}\tilde{{\mathbf{H}}}_{2,k}\rVert^{2}+\sigma_{\mathrm{u}}^{2}=\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{D}}}_{k}\boldsymbol{\psi}[t]+\sigma_{\mathrm{u}}^{2}$,
where
$\tilde{{\mathbf{D}}}_{k}\triangleq\tilde{{\mathbf{H}}}_{2,k}\tilde{{\mathbf{H}}}_{2,k}^{\scriptscriptstyle\mathsf{H}}$.
Note that ${\mathbf{D}}_{k}[t],{\mathbf{q}}_{k}[t],\tilde{{\mathbf{D}}}_{k}$,
and $z_{k}[t]$ are all constant with respect to $\boldsymbol{\psi}[t]$. Now,
$\mathcal{R}_{k}^{\mathrm{tra}}[t]$ can be represented as

$\displaystyle\mathcal{R}_{k}^{\mathrm{tra}}[t]\\!$
$\displaystyle=\\!\log\left(\\!1+\\!\frac{\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}{\mathbf{D}}_{k}[t]\boldsymbol{\psi}[t]+2\mathfrak{R}{\left(\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}{\mathbf{d}}_{k}[t]\right)}+z_{k}[t]}{\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{D}}}_{k}\boldsymbol{\psi}[t]+\sigma_{\mathrm{u}}^{2}}\right)$
$\displaystyle=\bar{r}_{k}[t]-\tilde{r}_{k}[t],$

where
$\bar{r}_{k}[t]\triangleq\log\left(\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}\bar{{\mathbf{D}}}_{k}[t]\boldsymbol{\psi}[t]+2\mathfrak{R}{\left(\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}{\mathbf{d}}_{k}[t]\right)}+\bar{z}_{k}[t]\right)$,
and
$\tilde{r}_{k}[t]\triangleq\frac{1}{T}\sum_{t=1}^{T}b_{k}[t]\log\Big{(}\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}$
$\tilde{{\mathbf{D}}}_{k}\boldsymbol{\psi}[t]+\sigma_{\mathrm{u}}^{2}\Big{)}$,
with $\bar{{\mathbf{D}}}_{k}[t]={\mathbf{D}}_{k}[t]+\tilde{{\mathbf{D}}}_{k}$,
and $\bar{z}_{k}[t]=z_{k}[t]+\sigma_{\mathrm{u}}^{2}$. Thus, constraint (38b)
can be rewritten as $\bar{r}_{k}[t]\geq\tau+\tilde{r}_{k}[t],\forall k,t,$
which has the same form as (33). Therefore, by applying similar
transformations as in (33)–(35), we can transform it to the set of convex
constraints:

$\displaystyle\bar{r}_{\mathrm{lb},k}^{(i)}[t]\geq\tau+\tilde{r}_{\mathrm{ub},k}^{(i)}[t],\
\vartheta_{k}[t]\geq\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}\tilde{{\mathbf{D}}}_{k}\boldsymbol{\psi}[t],\
\forall k,t,$ (57)

where
$\bar{r}_{\mathrm{lb},k}^{(i)}[t]\triangleq\log\Big{(}2\mathfrak{R}{\left(\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}{\mathbf{d}}_{k}[t]\right)}+\bar{z}_{k}[t]-F_{\mathtt{qua}}(\bar{{\mathbf{D}}}_{k}^{\frac{1}{2}}[t]\boldsymbol{\psi}[t];\boldsymbol{0},\bar{{\mathbf{D}}}_{k}^{\frac{1}{2}}[t]\boldsymbol{\psi}[t]^{(i)})\Big{)}$
and
$\tilde{r}_{\mathrm{ub},k}^{(i)}[t]\triangleq\log\Big{(}\vartheta_{k}[t]^{(i)}+\sigma_{\mathrm{u}}^{2}\Big{)}+\frac{\vartheta_{k}[t]-\vartheta_{k}[t]^{(i)}}{\vartheta_{k}[t]^{(i)}+\sigma_{\mathrm{u}}^{2}}.$
Furthermore, similar to (36), (10j) is equivalent to the quadratic convex
constraint

$\displaystyle\boldsymbol{\psi}[t]^{\scriptscriptstyle\mathsf{H}}\boldsymbol{\Xi}[t]\boldsymbol{\psi}[t]\leq
p_{\mathrm{max}}^{\mathrm{ris}},\forall t,$ (58)

where
$\boldsymbol{\Xi}[t]=\mathrm{diag}\left\\{\tilde{\xi}_{1}[t],\ldots,\tilde{\xi}_{N}[t]\right\\}$,
with $\tilde{\xi}_{n}[t]=\xi_{n}[t],n\in\mathcal{A}$ and
$\tilde{\xi}_{n}[t]=0,n\notin\mathcal{A}$. Finally, problem (55) at iteration
$i$ can be approximated by the following convex program:

$\displaystyle\underset{\begin{subarray}{c}\tau,\boldsymbol{\psi}[t],\\{\vartheta_{k}[t]\\}\end{subarray}}{\textrm{maximize}}\
\tau,\ \textrm{subject to}\
\eqref{cons_ris_phase_tra},\eqref{cons_active_modul_tra},\eqref{cons_min_rate_2b}-\eqref{cons_pris_tra_1}.$
(59)

Once solutions $\\{\phi_{n}[t]^{\star}\\}$ and $\\{\psi_{n}[t]^{\star}\\}$ are
found, $\\{\alpha_{n}[t]^{\star}\\}$ is readily obtained by

$\displaystyle\alpha_{n}[t]^{\star}=\phi_{n}[t]^{\star}+\psi_{n}[t]^{\star}.$
(60)

###  IV-E Overall Algorithm

The joint optimization of the UAV’s transmit beamforming, trajectory planning
and the RIS beamforming design is outlined in Algorithm 2. Specifically, the
initial solutions to $\\{b_{k}[t]^{(0)}\\}$, $\\{{\mathbf{w}}[t]^{(0)}\\}$,
$\\{{\mathbf{v}}[t]^{(0)}\\}$, $\\{\alpha_{n}[t]^{(0)}\\}$, and
$\\{\boldsymbol{\mathcal{V}}[t]^{(0)}\\}$ are first generated in Step 1. Then,
in Steps 2–10, subproblems (40), (41), (51), (54) and (59) are alternatively
solved. In each iteration, $\\{b_{k}[t]^{(i)}\\}$,
$\\{{\mathbf{w}}[t]^{(i)}\\}$, $\\{{\mathbf{v}}[t]^{(i)}\\}$,
$\\{\alpha_{n}[t]^{(i)}\\}$ and $\\{\boldsymbol{\mathcal{V}}[t]^{(i)}\\}$ are
updated until convergence. We note that the solutions to (41) and (54), i.e.,
$\\{{\mathbf{w}}[t]^{(i)}\\}$ and $\\{\alpha_{n}[t]^{(i)}\\},\forall
n\notin\mathcal{A}$, are obtained based on their closed-forms with simple
mathematical operations (scalar multiplications and dot products). Therefore,
most of the complexity of Algorithm 2 comes from solving subproblems (40),
(51) and (59). These require complexities of
$\mathcal{O}(\sqrt{2KT+K+T}K^{3}T^{3})$,
$\mathcal{O}(\sqrt{3KT+K+5T}(3KT+6T)^{3})$, and
$\mathcal{O}(\sqrt{N(N_{\mathrm{a}}+2K+T)}N_{\mathrm{a}}^{3}T^{3})$,
respectively. Therefore, the total complexity of Algorithm 2 is

$\displaystyle\mathcal{C}_{\mathrm{tra}}$
$\displaystyle=\mathcal{O}\big{(}\mathcal{I}_{\mathrm{tra}}\Big{(}\sqrt{2KT+K+T}K^{3}T^{3}+\sqrt{3KT+K+5T}$
$\displaystyle\qquad\times(3KT+6T)^{3}+\sqrt{N(N_{\mathrm{a}}+2K+T)}N_{\mathrm{a}}^{3}T^{3}\big{)}\Big{)},$

where $\mathcal{I}_{\mathrm{tra}}$ is the number of iterations until
convergence. It is observed that the major complexity of Algorithm 2 comes
from the optimization over $T$ time slots. The monotonic convergence the
objective value in Algorithm 2 can be shown similarly as in Algorithm 1.

Algorithm 2 Proposed Iterative Algorithm to Solve Problem
$(\mathcal{P}_{\mathrm{tra}})$ in (10)

1: Initialization: Set $i=0$. Generate initial values $\\{b_{k}[t]^{(0)}\\}$,
$\\{{\mathbf{w}}[t]^{(0)}\\}$, $\\{{\mathbf{v}}[t]^{(0)}\\}$, and
$\\{\psi_{n}[t]^{(0)}\\}$. Obtain $\\{\phi_{n}[t]^{(0)}\\}$ based on (56) and
obtain $\\{\alpha_{n}[t]^{(0)}\\}$ based on (60). Initialize
$\\{\boldsymbol{\mathcal{V}}[t]^{(0)}\\}$ and $\\{\vartheta_{k}[t]^{(0)}\\}$
by setting (45)–(46) and (57) to be equalities.

2: repeat

3: Solve problem (40) for given $\\{{\mathbf{w}}[t]^{(i)}\\}$,
$\\{{\mathbf{v}}[t]^{(i)}\\}$, $\\{\alpha_{n}[t]^{(i)}\\}$ to obtain solution
$\\{b_{k}[t]^{\star}\\}$. Set $\\{b_{k}[t]^{(i+1)}\\}=\\{b_{k}[t]^{\star}\\}$.

4: Obtain $\\{{\mathbf{w}}[t]^{\star}\\}$ based on (42) and (43). Set
$\\{{\mathbf{w}}[t]^{(i+1)}\\}=\\{{\mathbf{w}}[t]^{\star}\\}$.

5: Solve problem (51) with given $\\{b_{k}[t]^{(i+1)}\\}$,
$\\{{\mathbf{w}}[t]^{(i+1)}\\}$, $\\{\alpha_{n}[t]^{(i)}\\}$,
$\\{\boldsymbol{\mathcal{V}}[t]^{(i)}\\}$ to obtain solution
$\\{{\mathbf{v}}[t]^{\star}\\}$ and
$\\{\boldsymbol{\mathcal{V}}[t]^{\star}\\}$. Set
$\\{{\mathbf{v}}[t]^{(i+1)}\\}=\\{{\mathbf{v}}[t]^{\star}\\}$ and
$\\{\boldsymbol{\mathcal{V}}[t]^{(i+1)}\\}=\\{\boldsymbol{\mathcal{V}}[t]^{\star}\\}$.

6: Obtain $\\{\phi_{n}[t]^{\star}\\}$ based on (56). Set
$\\{\phi_{n}[t]^{(i+1)}\\}=\\{\phi_{n}[t]^{\star}\\}$.

7: Solve problem (59) with given $\\{b_{k}[t]^{(i+1)}\\}$,
$\\{{\mathbf{w}}[t]^{(i+1)}\\}$, $\\{{\mathbf{v}}[t]^{(i+1)}\\}$,
$\\{\phi_{n}[t]^{(i+1)}\\}$, $\\{\vartheta_{k}[t]^{(i)}\\}$, and
$\\{\psi_{n}[t]^{(i)}\\}$ to obtain solutions
$\\{\boldsymbol{\psi}[t]^{\star}\\}$ and $\\{\vartheta_{k}[t]^{\star}\\}$. Set
$\\{\boldsymbol{\psi}[t]^{(i+1)}\\}=\\{\boldsymbol{\psi}[t]^{\star}\\}$ and
$\\{\vartheta_{k}[t]^{(i+1)}\\}=\\{\vartheta_{k}[t]^{\star}\\}$.

8: Obtain $\\{\alpha_{n}[t]^{\star}\\}$ based on (60). Set
$\\{\alpha_{n}[t]^{(i+1)}\\}=\\{\alpha_{n}[t]^{\star}\\}$.

9: Update $i=i+1$.

10: until Convergence.

##  V Simulation Results

In this section, numerical results are provided to validate the Algorithms 1
and 2 as well as the performance of the hybrid RIS. We assume that the UAV
flies at a fixed altitude of $z_{0}=100$ m, the UEs are randomly and uniformly
distributed in an area of $D\times D$ $\text{m}^{2}$, and the RIS is deployed
at $(D/2,D,50)$ in a three-dimensional (3D) coordinate system, as will be
illustrated latter in Fig. 3. We set $\delta_{t}=0.1$ s and
$v_{\mathrm{max}}=50$ m/s.

The small-scale fading channels
$\\{{\mathbf{g}}_{0,k},{\mathbf{G}}_{1},{\mathbf{g}}_{2,k}\\},\forall k$ are
modeled as follows. First, we assume that ${\mathbf{g}}_{0,k}$ follows the
Rayleigh fading model, i.e., $g_{0,kt}\sim\mathcal{CN}(0,1),t=1,\ldots,N_{t}$,
$\forall k$. This is because the LoS links between UAV and ground UEs are
easily blocked by high buildings, while the scattering components are often
extensive, especially in the low-to-medium frequency band and complex urban
environments [21, 2, 12, 55]. In contrast, the channels between RIS and UEs,
i.e., ${\mathbf{g}}_{2,k},\forall k$, are assumed to follow the Rician fading
model [21, 2] with Rician factor $\kappa$, i.e.,
${\mathbf{g}}_{2,k}=\sqrt{\frac{\kappa}{\kappa+1}}{\mathbf{g}}_{2,k}^{\mathrm{LoS}}+\sqrt{\frac{1}{\kappa+1}}{\mathbf{g}}_{2,k}^{\mathrm{NLoS}},\forall
k$, where
${\mathbf{g}}_{2,k}^{\mathrm{NLoS}}\sim\mathcal{CN}(\boldsymbol{0},{\mathbf{I}}_{N})$
represents the non-LoS (NLoS) channels, and
${\mathbf{g}}_{2,k}^{\mathrm{LoS}}$ represents the deterministic LoS component
of ${\mathbf{g}}_{2,k}$. The channel between UAV and RIS, ${\mathbf{G}}_{1}$,
is modeled as LoS deterministic channel because both UAV and RIS are located
at certain heights so that the channel between them is barely blocked by any
obstacles [21, 2]. Furthermore, we assume a uniform planar array (UPA) of
$N=N_{x}\times N_{y}$ elements for the RIS, with $\\{N_{x},N_{y}\\}$ being the
numbers of elements on each row and column, respectively. Thus, the $n$th
elements of ${\mathbf{G}}_{1}$ and ${\mathbf{g}}_{2,k}^{\mathrm{LoS}}$ can be
given as $g_{1,n}=\exp\left(j\frac{2\pi
d_{0}}{\lambda}f\left(n,\phi_{1,}^{\mathrm{azi}},\phi_{1,}^{\mathrm{ele}}\right)\right)$
and $g^{\mathrm{LoS}}_{2,kn}=\exp\left(j\frac{2\pi
d_{0}}{\lambda}f\left(n,\phi_{2,k}^{\mathrm{azi}},\phi_{2,k}^{\mathrm{ele}})\right)\right)$,
respectively [48], where $d_{0}$ and $\lambda$ are the antenna separation and
carrier wavelength, respectively;
$\\{\phi_{1,}^{\mathrm{azi}},\phi_{1,}^{\mathrm{ele}}\\}$ and
$\\{\phi_{2,k}^{\mathrm{azi}},\phi_{2,k}^{\mathrm{ele}}\\}$ denote the azimuth
and elevation angle of-arrival/departure (AoA/AoD) of the RIS associated with
the UAV-RIS and RIS-UE channels, respectively; and,
$f(n,\phi,\phi^{\prime})\triangleq\lfloor\frac{n}{N_{x}}\rfloor\sin\phi\sin\phi^{\prime}+\left(n-\lfloor\frac{n}{N_{x}}\rfloor
N_{x}\right)\sin\phi\cos\phi^{\prime}$ [48]. Then,
$\\{{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}},{\mathbf{H}}_{1},{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\\}$
are obtained based on (1).

In the following simulations, we set $\zeta_{0}=-30$ dB,
$\\{\epsilon_{0},\epsilon_{1},\epsilon_{2}\\}=\\{3.2,2.0,2.2\\}$ and
$\kappa=10$ [23]. The system bandwidth is set to $20$ MHz, corresponding to
$\sigma^{2}_{\mathrm{u}}=-80$ dBm. The total power of noise and residual SI of
the RIS is computed as
$\sigma^{2}_{\mathrm{r}}=(\eta+1)\sigma^{2}_{\mathrm{u}}$, with $\eta=1$ dB
reflecting the possible residual SI caused by active elements operating in the
full-duplex mode [52, 11]. The positions of RIS active elements are fixed to
$\mathcal{A}=\\{1,\ldots,N_{\mathrm{a}}\\}$. To solve the convex subproblems,
we use the modeling toolbox YALMIP with solver MOSEK. The convergence stopping
criteria is set to $\varepsilon=10^{-4}$. For initialization of Algorithm 1,
we set ${\mathbf{v}}^{(0)}=[D/2,D/2,z_{0}]$, i.e., above the center of the
area,
${\mathbf{w}}_{k}^{(0)}=\sqrt{\frac{p_{\mathrm{max}}^{\mathrm{uav}}}{K}}\frac{{\mathbf{g}}_{0,k}}{\lVert{\mathbf{g}}_{0,k}\rVert},\forall
k$, and $\alpha_{n}^{(0)}=re^{j\theta_{n}^{(0)}}$, with $\theta_{n}^{(0)}$
being randomly generated, $\forall n$, and $r$ satisfying (8f). For Algorithm
2, $\\{b_{k}[t]\\}$ are simply initialized as $b_{1}[t]^{(0)}=1,\forall t$ and
$b_{j}[t]^{(0)}=0,\forall t,j>1$, i.e., UE $1$ is scheduled for
initialization, and thus, ${\mathbf{w}}[t]^{(0)}$ is set to
$\sqrt{p_{\mathrm{max}}^{\mathrm{uav}}}\frac{{\mathbf{g}}_{0,1}[t]}{\lVert{\mathbf{g}}_{0,1}[t]\rVert}$.
Similar to $\alpha_{n}^{(0)}$ in Algorithm 1, we initialize
$\psi_{n}[t]^{(0)}=re^{j\theta_{n}[t]^{(0)}}$ with random phase shifts
$\\{\theta_{n}[t]^{(0)}\\}$ and $r$ satisfying (10j). Furthermore,
${\mathbf{v}}[t]^{(0)}$ is initialized based on the low-complexity circular
trajectory scheme with details being presented in [1].

Table I: Simulation parameters for the static- and mobile-UAV systems. Parameters | Values | Parameters | Values
---|---|---|---
Deployment area ($D\times D$) |  $D=\\{50,200\\}$ m |  Path loss at $1$ m |  $\zeta_{0}=-30$ dB
RIS’s location | $(D/2,D,50)$ | Path loss exponents | $\\{\epsilon_{0},\epsilon_{1},\epsilon_{2}\\}=\\{3.2,2.0,2.2\\}$
UAV’s altitude |  $z_{0}=100$ m | Rician factor | $\kappa=10$
Time slot’s length |  $\delta_{t}=0.1$ s | System bandwidth |  $20$ MHz
UAV’s maximum speed |  $v_{\max}=50$ m/s | Noise power |  $\sigma_{\mathrm{u}}^{2}=-80$ dBm

###  V-A Convergence of Algorithms 1 and 2

(a) Algorithm 1

(b) Algorithm 2, $T=50$

Fig. 2: Convergence of Algorithms 1 and 2 with $K=4$, $N_{t}=2$,
$N_{\mathrm{a}}=2$, $N=32$, $D=200$ m,
$p_{\mathrm{max}}^{\mathrm{uav}}=\\{20,30\\}$ dBm and
$p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm.

We first show in Fig. 2 the convergence of Algorithms 1 and 2 with $K=4$,
$N_{t}=2$, $N_{\mathrm{a}}=2$, $N=32$, $D=200$ m,
$p_{\mathrm{max}}^{\mathrm{uav}}=\\{20,30\\}$ dBm,
$p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm and $T=50$. It is observed in Fig.
2(a) that Algorithm 1 converges after about eight to ten iterations. Whereas
Algorithm 2 requires about five to eight iterations to converge, which is
slightly faster than Algorithm 1. The convergences are similar for both cases
$p_{\mathrm{max}}^{\mathrm{uav}}=\\{20,30\\}$ dBm. Furthermore, it is clear
that Algorithm 2 achieves a larger minimum rate (0.7 nats/s/Hz with
$p_{\mathrm{max}}^{\mathrm{uav}}=20$ dBm) compared to those obtained in
Algorithm 1 (0.3 nats/s/Hz with $p_{\mathrm{max}}^{\mathrm{uav}}=20$ dBm).
This is attributed to the fact that by employing the TDMA protocol, the high
mobility of the UAV and the high active amplifying gain of the hybrid RIS can
be fully exploited. Furthermore, it is seen that Algorithms 1 and 2 can
achieve relatively good performance at the first iteration, which is due to
the efficient initialization of the UAV’s position and trajectory. Indeed,
initially deploying the UAV right above the center of the area in the static-
UAV network and employing the initial circular trajectory in the mobile-UAV
network [1] enables good fairness performance among the UEs. In particular,
such initialization is also close to the optimized UAV locations/trajectory,
as will be further seen in the next figure.

###  V-B Location Deployment and Trajectory of the UAV

In Fig. 3, we show the optimized locations and trajectories of the UAV in the
horizontal view of three systems, namely, without RIS, with the passive RIS,
and with the hybrid RIS, for $D=200$, $N_{\mathrm{a}}=\\{2,4,8\\}$, and
$p_{\mathrm{max}}^{\mathrm{uav}}=20$ dBm; the other parameters are set the
same as those in Fig. 2. The locations of the UAV are shown for $20$ channels
in Fig. 3(a). It is observed that without the RIS and with the passive RIS,
the UAV (black circles and blue triangles) generally deploys in between the
UEs, and in some cases, right above one of the UEs. However, when being aided
by the hybrid RIS, the UAV (red squares, diamonds, and hexagrams) also deploys
near the RIS. There are some cases where the UAV flies right above the hybrid
RISs with $N_{\mathrm{a}}=\\{4,8\\}$. This verifies that the hybrid RIS
significantly improves the reflecting channels, so that the UAV does not
necessarily always deploy in between or close to the UEs to reduce the path
loss on the direct channels, but it can also deploy near the RIS to enjoy the
enhanced communications. In Fig. 3(b), we show the trajectories of the UAV
employing the TDMA protocol with $T=\\{50,100\\}$ for five channels. A common
observation on the trajectories of all the compared schemes is that as $T$
increases, the UAV exploits its mobility to enlarge and adjust its trajectory
to move closer to all the UEs. As $T$ becomes sufficiently large, e.g.,
$T=100$, the UAV visit very close to all the UEs. In particular, with the
presence of the passive or hybrid RISs, the UAV flies closer to the RIS
compared to the case without RIS, similar to the location deployment in Fig.
3(a).

(a) UAV’s location in 20 channels

(b) UAV’s trajectory in $T=50$ time slots

Fig. 3: UAV’s locations and trajectory with $K=4$, $N_{t}=2$,
$N_{\mathrm{a}}=\\{2,4,8\\}$, $N=32$, $D=200$ m,
$p_{\mathrm{max}}^{\mathrm{uav}}=20$ dBm, and
$p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm.

###  V-C Performance Improvement of the Hybrid RIS

In this section, we show simulation results on the minimum rate versus various
parameters. The positions of the UEs and the RIS are the same as those in Fig.
3.

####  V-C1 Static-UAV network without TDMA

We first show the minimum rate for $p_{\mathrm{max}}^{\mathrm{uav}}=[0,30]$
dBm and $N_{\mathrm{a}}=\\{2,4,8\\}$, and $D=\\{200,50\\}$ in Figs. 4(a) and
4(b), respectively. We note that in the hybrid RIS-aided UAV system, the
hybrid RIS requires an additional power budget of
$p_{\mathrm{max}}^{\mathrm{ris}}$. Therefore, we also consider the case that
the power budget at the UAV in this system is reduced by
$p_{\mathrm{max}}^{\mathrm{ris}}$ to be
$p_{\mathrm{max}}^{\mathrm{uav}}-p_{\mathrm{max}}^{\mathrm{ris}}$ in Fig.
4(a). We note that this is fair in the sense that all the compared schemes
have the same total power budget, but it does not necessarily reflect the
practical deployment. From Fig. 4, the following observations are made:

  * •

The conventional passive RIS deployed in the $200\times
200\leavevmode\nobreak\ \text{m}^{2}$ (Fig. 4(a)) area provides only marginal
performance improvement due to the severe double path loss on the reflecting
channels. However, when the area is shrunken to $50\times
50\leavevmode\nobreak\ \text{m}^{2}$ (but the UEs and RIS distribution are
unchanged), its gain becomes more significant, as seen in Fig. 4(b).

  * •

With $N_{\mathrm{a}}=2$, the hybrid RIS performs only slightly better than the
passive RIS, but with $N_{\mathrm{a}}=\\{4,8\\}$, it attains significant
performance gains. For example, at $p_{\mathrm{max}}^{\mathrm{uav}}=20$ dBm,
the hybrid RISs with $N_{\mathrm{a}}=4$ achieve $\\{33.33\%,38.33\%\\}$
improvement in Figs. 4(a) and 4(b), respectively, while those of the passive
RIS are only $\\{3.70\%,13.80\%\\}$. In particular, the gain is more
significant at low $p_{\mathrm{max}}^{\mathrm{uav}}$ as power constraint (8f)
shows that a smaller $p_{\mathrm{max}}^{\mathrm{uav}}$ leads to larger
$\left|\alpha_{n}\right|,\forall n\in\mathcal{A}$. This makes the hybrid RIS
important in UAV systems since the capacity of the UAV’s battery is generally
limited.

  * •

The performance gains of the hybrid RIS with $N_{\mathrm{a}}=\\{4,8\\}$ are
still significant even when the UAV has a reduced power budget of
$p_{\mathrm{max}}^{\mathrm{uav}}-p_{\mathrm{max}}^{\mathrm{ris}}$.
Specifically, it is observed in Fig. 4(a) that reducing the UAV’s power budget
only cause a slight performance degradation at low
$p_{\mathrm{max}}^{\mathrm{uav}}$ (we note here that because we assume
$p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm, the dotted curves must start from a
point $p_{\mathrm{max}}^{\mathrm{uav}}>0$ dBm). This is reasonable because at
moderate and high $p_{\mathrm{max}}^{\mathrm{uav}}$ ($\geq 5$ dBm), reducing
an amount of $p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm almost does not cause any
significant impact on the system performance.

(a) $D=200$ m

(b) $D=50$ m

Fig. 4: Minimum rate of the UEs versus $p_{\mathrm{max}}^{\mathrm{uav}}$ with
$K=4$, $N_{t}=2$, $N_{\mathrm{a}}=\\{2,4,8\\}$, $N=32$, $D=\\{50,200\\}$ m,
$p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm, and
$p_{\mathrm{max}}^{\mathrm{uav}}=[0,30]$ dBm.

In Figs. 5(a) and 5(b), we show the minimum rate of the hybrid RIS for
different values of $p_{\mathrm{max}}^{\mathrm{ris}}$ and $N$, respectively,
both with $p_{\mathrm{max}}^{\mathrm{uav}}=20$ dBm. It is obvious from Fig.
5(a) that the performance of the system without RIS and with passive RIS is
unchanged with $p_{\mathrm{max}}^{\mathrm{ris}}$. The performance of hybrid
RISs with $N_{\mathrm{a}}=\\{4,8\\}$ first increases, peaks at
$p_{\mathrm{max}}^{\mathrm{ris}}=5$ dBm and then is likely to saturate at high
$p_{\mathrm{max}}^{\mathrm{ris}}$. This is because the amplitudes of all the
active elements are restricted by constraint (8e). Furthermore, it should be
noted that in this system, an active element with a larger amplitude does not
always result in a performance improvement because it also amplifies inter-
symbol interference. Regarding the performance versus $N$, it is observed in
Fig. 5(b) that deploying more elements in the passive RIS results in improved
performance, which is more significant than doing the same for the hybrid RIS.
Specifically, as $N$ increases, the performance of the passive RIS increases
faster than that of the hybrid RIS. This shows that active elements play a
more important role in the RISs of smaller sizes.

(a) Minimum rate vs. $p_{\mathrm{max}}^{\mathrm{ris}}$

(b) Minimum rate vs. $N$

Fig. 5: Minimum rate of the UEs versus $p_{\mathrm{max}}^{\mathrm{ris}}$ (Fig.
(a)) and versus $N$ (Fig. (b)) with $K=4$, $N_{t}=2$,
$N_{\mathrm{a}}=\\{2,4,8\\}$, $D=200$ m, $p_{\mathrm{max}}^{\mathrm{uav}}=20$
dBm, (a) $N=32$, $p_{\mathrm{max}}^{\mathrm{ris}}\in[-10,20]$ dBm, (b)
$N\in[8,128]$, $p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm.

####  V-C2 UAV network with TDMA

(a) $D=200$ m, $p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm

(b) $D=50$ m, $p_{\mathrm{max}}^{\mathrm{ris}}=-5$ dBm

Fig. 6: Minimum rate of the UEs versus $p_{\mathrm{max}}^{\mathrm{uav}}$ with
$K=4$, $N_{t}=2$, $N_{\mathrm{a}}=\\{2,4,8\\}$, $N=32$, $D=\\{50,200\\}$ m,
$p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm,
$p_{\mathrm{max}}^{\mathrm{uav}}=[0,30]$ dBm.

In this section, we investigate the performance of the network in which the
UAV employs the TDMA protocol. In Fig. 6, the minimum rate is shown for
$T=50$, and the other parameters are the same as those in Fig. 4. Similar to
Fig. 4, it is clear that the performance gain of the passive RIS is improved
when it is deployed closer to the UEs (in Fig. 6(b)), and the hybrid RIS with
$N_{\mathrm{a}}=\\{4,8\\}$ provides significant performance improvement,
especially at low $p_{\mathrm{max}}^{\mathrm{uav}}$. However in this scenario,
i.e., with TDMA, the hybrid RIS with only $N_{\mathrm{a}}=2$ active elements
also performs very well to achieve up to $\\{55.31\%,62.90\%\\}$ improvement
at $p_{\mathrm{max}}^{\mathrm{uav}}=20$ dBm for $D=\\{200,50\\}$,
respectively, compared to only $\\{8.51\%,14.52\%\\}$ attained by the passive
RIS. We note that this remarkable improvement was not seen for
$N_{\mathrm{a}}=2$ in Fig. 4. This is because when there is no inter-symbol
interference, active elements can serve with their maximum amplifying gain to
enhance the system performance. More specifically, our numerical results show
that for the same power budgets, $\left|\alpha_{n}\right|,n\in\mathcal{A}$ in
this scenario is much larger than those without TDMA. In particular, in the
area $50\times 50\leavevmode\nobreak\ \text{m}^{2}$ (Fig. 6(b)), the hybrid
RIS offers remarkable performance improvement with a power budget of only $-5$
dBm. It is also seen from Fig. 6(a) that when the UAV transmit with a reduced
power budget, i.e.,
$p_{\mathrm{max}}^{\mathrm{uav}}-p_{\mathrm{max}}^{\mathrm{ris}}$, the
performance loss at low $p_{\mathrm{max}}^{\mathrm{uav}}$ is more significant
than those seen in Fig. 4. This is because when TDMA is applied, the
performance is mostly decided by the received signal power. However, the
performance loss is negligible for $p_{\mathrm{max}}^{\mathrm{uav}}\geq 5$
dBm.

###  V-D Performance of the Hybrid RIS under Imperfect CSI

In this section, we examine the performance of UAV systems in the presence of
CSI error. We denote by
$\\{\hat{{\mathbf{h}}}_{0,k}^{\scriptscriptstyle\mathsf{H}},\hat{{\mathbf{H}}}_{1},\hat{{\mathbf{h}}}_{2,k}^{H}\\}$
the imperfect channel estimates of
$\\{{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}},{\mathbf{H}}_{1},{\mathbf{h}}_{2,k}^{\scriptscriptstyle\mathsf{H}}\\}$,
respectively. Here, $\hat{{\mathbf{h}}}_{0,k}^{\scriptscriptstyle\mathsf{H}}$
can be modeled as
$\hat{{\mathbf{h}}}_{0,k}^{\scriptscriptstyle\mathsf{H}}={\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}-\Delta{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}$,
where $\Delta{\mathbf{h}}_{0,k}^{\scriptscriptstyle\mathsf{H}}$ represents the
CSI error whose entries have distributions
$\mathcal{CN}(0,\hat{\zeta}_{0,k}\epsilon_{0,k}^{2})$ with
$\hat{\zeta}_{0,k}=\zeta_{0}\left\lVert{\mathbf{v}}-{\mathbf{u}}_{k}\right\rVert^{-\epsilon_{0}}$
being the large-scale coefficient and $\epsilon_{0,k}$ being the CSI
uncertainty level of
$\hat{{\mathbf{h}}}_{0,k}^{\scriptscriptstyle\mathsf{H}}$. Here, we assume
that the CSI error only occurs in small-scale fading channels because the
large-scale coefficients can be easily estimated with high accuracy [11]. The
channels $\hat{{\mathbf{H}}}_{1}$ and $\hat{{\mathbf{h}}}_{2,k}^{H}$ are
modeled similarly.

(a) Static-UAV system

(b) Mobile-UAV system

Fig. 7: Minimum rate of the UEs in the hybrid RIS-aided static- and mobile-UAV
systems in the presence of imperfect CSI with $K=4$, $N_{t}=2$,
$N_{\mathrm{a}}=4$, $N=32$, $D=50$ m, $p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm,
$T=50$, and $\epsilon=\\{0,0.05,0.1,0.2,0.5\\}$.

In Fig. 7, we show the minimum rate performance of the considered UAV systems
aided by the hybrid and passive RISs under CSI errors. We set $K=4$,
$N_{t}=2$, $N_{\mathrm{a}}=4$, $N=32$, $D=50$ m,
$p_{\mathrm{max}}^{\mathrm{ris}}=0$ dBm, and $T=50$. Without loss of
generality, we assume all the imperfect channel estimates have the same
uncertainty level, which is set to $\epsilon\in\\{0,0.05,0.1,0.2,0.4\\}$. It
is observed that as $\epsilon$ increases, the UAV systems have performance
degradation, which is more significant in the static-UAV scenario (in Fig.
7(a)) than the mobile-UAV scenario (in Fig. 7(b)). This is because the former
is vulnerable to the interference in multiuser communications. In contrast,
due to the high mobility of the UAV and the TDMA protocol, the performance of
the mobile-UAV network depends more on the large-scale channels rather than
the erroneous small-scale ones. Furthermore, it is promising to observe that
despite the loss due to the CSI errors, the hybrid RIS still offers remarkable
performance improvement with respect to the passive RIS, especially at low
$p_{\mathrm{uav}}^{\mathrm{max}}$. Particularly, the mobile-UAV system aided
by the hybrid RIS under severe CSI errors (e.g., $\epsilon=\\{0.1,0.2,0.4\\}$)
still performs far better than when being aided by the passive RIS with small
CSI errors (e.g., $\epsilon=0.05$), as seen in Fig. 7(b).

In Figs. 3–7, we have numerically demonstrated the performance benefits of
deploying hybrid RISs in two typical UAV-enabled networks, i.e., the static-
and mobile-UAV networks. We remark the following important observations:

  * •

When the systems are aided by the passive RIS or no RIS, the UAV generally
flights close to the UEs. However, with the introduction of active RIS
elements (i.e., when the hybrid RIS is deployed), the UAV moves toward the RIS
to enjoy enhanced communications.

  * •

Compared to the purely passive RIS, the hybrid one offers a significant max-
min fairness enhancement, especially with a low or moderate power budget at
the UAV. Thus, hybrid RISs are beneficial for practical deployments of UAVs.

  * •

The performance gains of the hybrid RIS are more clearly seen when the number
of active elements, i.e., $N_{\mathrm{a}}$, and/or the RIS power budget, i.e.,
$p_{\mathrm{max}}^{\mathrm{ris}}$, increase. However, the gains are upper
bounded when $p_{\mathrm{max}}^{\mathrm{ris}}$ is sufficiently large. In such
scenarios, it is more beneficial to deploy the RIS with more active elements
rather than with a higher power budget.

##  VI Conclusion

In this paper, we have proposed two novel hybrid RIS-assisted UAV
communications systems. Toward a fairness design of these systems, our goal is
to maximize the minimum rate among users through jointly optimizing the
location/trajectory, and transmit beamforming of the UAV and RIS
reflecting/amplifying coefficients. The formulated problems inherit the
challenges of conventional passive RIS design due to non-convexity and
strongly coupling variables while facing additional difficulties in the design
of active coefficients under power constraints. We have developed efficient
solutions to solve the considered problem by leveraging BCA and SCA
approaches. Finally, we have provided extensive numerical results to
demonstrate the efficacy of the proposed algorithms. They have also revealed
the remarkable performance improvement of the hybrid RIS compared with
existing schemes. The improvement is particularly significant when the UAV has
a limited power budget, making the deployment of the hybrid RIS important
because the UAV’s battery capacity is generally limited. For future studies,
leveraging learning capabilities of deep neural networks and deep
reinforcement learning for RIS-assisted UAV system designs is a potential
extension of this work.

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