# Location-aware Beamforming for MIMO-enabled UAV Communications: An Unknown
Input Observer Approach

Alireza Mohammadi1, Mehdi Rahmati2, and Hafiz Malik1
1University of Michigan–Dearborn  2Cleveland State University

###### Abstract

Numerous communications and networking challenges prevent deploying unmanned
aerial vehicles (UAVs) in extreme environments where the existing wireless
technologies are mainly ground-focused; and, as a consequence, the air-to-air
channel for UAVs is not fully covered. In this paper, a novel spatial
estimation for beamforming is proposed to address UAV-based joint sensing and
communications (JSC). The proposed spatial estimation algorithm relies on
using a delay-tolerant observer-based predictor, which can accurately predict
the positions of the target UAVs in the presence of uncertainties due to
factors such as wind gust. The solution, which uses discrete-time unknown
input observers (UIOs), reduces the joint target detection and communication
complication notably by operating on the same device and performs reliably in
the presence of channel blockage and interference. The effectiveness of the
proposed approach is demonstrated using simulation results.

###### Index Terms:

Delay-tolerant observer, joint sensing and communications, mmWave
communications, UAV network.

##  I Introduction

Thanks to the technological advancements in unmanned aerial vehicles (UAVs), a
vast number of applications ranging from mapping and surveying to search and
rescue in emergency temporary communications [1] have been enabled. Utilizing
communication-equipped agile UAVs will benefit applications such as aerial
base stations for expanding the coverage of ground cellular systems in 5G and
beyond [2]. In many of these applications such as multi-UAV cinematography [3,
4], the fast moving UAVs should maintain a reliable inter-UAV connectivity [5,
6, 7, 8]. Meeting this connectivity requirement on one hand while conducting
target detection and tracking on the other relies on coexistence, cooperation,
and joint design of sensing and communications (JSC).

The core challenges in JSC include, but are not limited to, implementing radar
and communication coexistence (RCC) system within the same band and hardware
platform, designing dual-functional radar-communication (DFRC) systems, and
combating the communication channel blockage due to geographical factors and
extreme environments [9, 10]. Despite these complications, many existing
solutions assume a fully-accessible channel that is not feasible in practical
scenarios. Accordingly, there is a need for high-quality joint design for UAV
networks in high mobility scenarios under practical constraints such as
unavailable channel information. JSC can utilize mmWave bands and multiple-
input multiple-output (MIMO) schemes for communications in order to
accommodate high-throughput applications and to provide higher degrees of
freedom [11].

In contrast to conventional precoder designs for beamformers, high-mobility
UAVs encounter numerous complications for handling air-to-air channels due to
the reduced channel coherence time and the resulting faster channel variation.
Moreover, the channel variations lead to changes in the channel path
characteristics such as path number and path direction. These channel
variations require efficient training and tracking methods [12]. If the link
remains stable, full-duplex relays can be employed on UAVs to boost the
capacity of mmWave networks [13]; however, this method requires a full
knowledge of the location of UAVs. Accordingly, measuring/predicting the
location of UAVs is crucial to beamforming whether it be digital or analog,
where the beams must be steered to reduce large-scale propagation attenuation.
Furthermore, despite the fact that beamformer design has been thoroughly
researched in the literature [14, 15], the design for JSC capable of meeting
both the sensing and communication requirements is still an active research
area.

Along the advances in JSC and mmWave UAV communications, the field of control
systems is witnessing a recent advent of algorithms for large-scale
collaborative sensor and actuator networks. Indeed, there is a re-emergence of
interest in designing state observers/estimators with the ultimate objective
of achieving state-omniscience [16, 17] at a central node and/or a collection
of distributed nodes across these sensor/actuator networks, where the goal of
such algorithms is to enable a designated network of nodes to reconstruct the
full dynamical state of the network. Therefore, the issue of information delay
and estimation of delayed data, whether the delay be network-induced or long
distance-induced, is of crucial importance in the modern communications
networks. Discrete-time unknown input observers (UIOs) are among the prominent
tools for estimating the states and unknown inputs of a given dynamical system
in the presence of delays [18, 19, 20, 21].

In this article, we will employ the UIO architecture in [20] for predicting
the location of a collection of UAVs from the previous location of them and
estimating the unknown inputs acting on the UAVs in the presence of time-
delays. The unknown input observer in [20] can provide a guaranteed level of
attenuation for bounded unknown inputs with any specified maximum estimation
delay. From this perspective, the delayed UIO in [20] has desirable properties
for our beamforming application. Indeed, the proposed UIO essentially plays
the role of a _virtual radar_ for an efficient and low-cost implementation of
JSC providing an accurate prediction of the UAV positions for the beamformer
within the communication waveform. Despite the existence of promising mmWave
FMCW radar technologies for localizing UAVs (see, e.g., [22, 23]), our
proposed UIO-based solution for localizing the UAVs becomes more desirable in
applications where there exist strict power consumption constraints such as
those on the central UAV. Furthermore, when the channel is not fully
available, our proposed method provides a robust solution for beamforming in
the presence of steering angle errors as well as combining sensing and
communications in JSC systems.

To the best of our knowledge, the only lines of work that provide a solution
for JSC using UAV location prediction is due to Liu _et al._ [24] and Peng _et
al._ [25]. In the former, a deep learning-based approach is proposed in order
to feed a beamformer while in the latter a Kalman filter is used for UAV
location prediction. These approaches differ from ours in several aspects.
First, the works in [24, 25] assume known statistical distributions for the
UAV velocity and the external unknown inputs causing perturbations in the
nominal velocity profiles of the UAVs. In our work, we do not make any
assumptions on these statistics, thanks to our UIO-based approach. Second, the
LRNet architecture in [24] is used for predicting the location of a single UAV
and the scalability of this approach to a network of UAVs is yet to be tested.
In contrast, our approach can be applied to any number of UAVs.

The rest of the paper is organized as follows. In Sect. II, we introduce our
system model and discuss our solution in details. In Sect. III, we propose our
mathematical framework and analysis for the central observer design for UAV
networks. In Sect. IV we evaluate the performance of our solution using
simulations and show the amount of gain we obtain compared to other existing
techniques. Finally, in Sect. V, we conclude the paper with final remarks and
future research directions.

Notation. Given a matrix $\mathbf{M}$, we denote its transpose by
$\mathbf{M}^{\top}$, its Hermitian by $\mathbf{M}^{H}$, its inverse by
$\mathbf{M}^{-1}$, and its conjugate by $\mathbf{M}^{*}$, respectively. We let
$\text{diag}(a_{1},\cdots,a_{N})$ denote the $N\times N$ diagonal matrix with
$a_{1},\cdots,a_{N}$ on its diagonal. For any vector
$\mathbf{a}\in\mathbb{R}^{N}$, we denote
$\|\mathbf{a}\|=\sqrt{\mathbf{a}^{\top}\mathbf{a}}$. For any matrix
$\mathbf{A}$, we let $\|\mathbf{A}\|$ denote the maximum singular value of
$\mathbf{A}$. By $\mathbf{A}\succeq\mathbf{B}$ for two symmetric matrices
$\mathbf{A}$ and $\mathbf{B}$, we mean $\mathbf{A}-\mathbf{B}$ is a positive
semi-definite matrix. For a sequence of vectors
$\\{\mathbf{a}_{k}\\}_{k=0}^{\infty}$, we denote
$\|\mathbf{a}\|_{\infty}:=\sup_{k\geq 0}\|\mathbf{a}_{k}\|$. We say a sequence
$\\{\mathbf{a}_{k}\\}\in\ell_{\infty}$ if $\|\mathbf{a}\|_{\infty}$ is
bounded. For $\\{\mathbf{a}_{k}\\}_{k=0}^{\infty}$, we let
$\limsup_{k\to\infty}\|\mathbf{a}_{k}\|$ denote the limit superior of the
sequence.

Figure 1: The schematics of the overall UIO-based location-aware beamforming
algorithm.

##  II System Model and the Proposed Solution

Communication Model. We consider a network of UAVs (see Fig.1), which consists
of a central UAV with $M_{\text{CE}}$ antennas in a massive MIMO structure
communicating with N moving UAVs each with $N_{\text{U}}$ Uniform Linear Array
(ULA) antennas on the other side of the links in which $N_{\text{U}}\leq
M_{\text{CE}}$ and the antenna elements are equally spaced at half wavelength
(see Figure 1). In practical scenarios, when the communication channel is
available, the line of sight (LoS) between the transmitter and receiver is the
dominant propagation component; otherwise, we assume that the blockage in the
channel occurs while affecting the beamformer. We assume the transmitter can
send at least one data stream per UAV, i.e., $N$ streams, simultaneously. Let
the vector $\mathbf{s}_{i,k}\in\mathbb{C}^{M_{CE}}$ be the transmitted data
from the central UAV to the $i$th UAV, $1\leq i\leq N$, at time instant $k$,
$k\geq 1$. As demonstrated in Section III, we only assume that the
kinematics/geometric information of the targets are received by the central
UAV in a cooperative manner while no reliance on the radar-based echoed
positions/velocities is assumed. This is in contrast to the conventional
methods where the known signals are sent; and, the reflected echo waves are
observed subsequently.

We assume Saleh-Valenzuela channel model [26] for mmWave communications and
the received signal model at each UAV, denoted by vector $\mathbf{r}_{i,k}$,
$1\leq i\leq N$, at time instant $k$ where
$\mathbf{r}_{i,k}\in\mathbb{C}^{N_{U}}$ and is given by

$\mathbf{r}_{i,k}=\frac{1}{\sqrt{M_{CE}N_{U}}}h_{i,k}\mathbf{b}(\theta_{i,k})\mathbf{a}^{H}(\theta_{i,k})\mathbf{s}_{i,k}+\mathbf{\nu}_{i,k}$
(1)

where $\mathbf{\nu}_{i,k}\in\mathbb{C}^{N_{U}}$ denotes the Gaussian noise
vector with zero mean and variance $\sigma^{2}$ for each sample and $h_{i,k}$
stands for the channel coefficient of the $i^{\text{th}}$ UAV at time instant
$k$. Here, for a ULA, $\mathbf{a}(\theta_{i,k})$ and
$\mathbf{b}(\theta_{i,k})$ stand for the normalized steering vectors for Angle
of Departure (AoD) at the central UAV and Angle of Arrival (AoA) at the
receiving UAV, respectively as shown below.

Steering vector on central UAV at time instant $k$: (2)
$\displaystyle\mathbf{a}(\theta_{i,k})=[1,e^{j{\frac{2\pi}{\lambda}d}\sin(\theta_{i,k})},...,e^{j{\frac{2\pi}{\lambda}d}(M_{\text{CE}}-1)\sin(\theta_{i,k})}]^{\top}\in\mathbb{C}^{M_{\text{CE}}}$
Steering vector on moving UAVs at time instant $k$:
$\displaystyle\mathbf{b}(\theta_{i,k})=[1,e^{j{\frac{2\pi}{\lambda}d}\sin(\theta_{i,k})},...,e^{j{\frac{2\pi}{\lambda}d}(N_{\text{U}}-1)\sin(\theta_{i,k})}]^{\top}\in\mathbb{C}^{N_{\text{U}}}$

where $d$ and $\lambda$ are the antenna spacing and wavelength, respectively.
The angles $\theta_{i,k}$, at each time instant $k$ are determined by the
location of the UAVs, whose dynamics are governed by a collection of kinematic
equations of motion (see “Equations of Motion of the UAV Network”).

######  Remark 1.

It should be noted that with traditional radar systems, undesirable scatters
would interfere with the received signal. We do not have such an interference
component in the received signal in the proposed technique in this research
since the sensing is performed in a novel fashion, as described in Section
III.

Equations of Motion of the UAV Network. In this article, we use the dynamical
model adopted by the UAV-based communications literature (see, e.g., [25]).
Similar kinematic models have also been utilized in UAV flocking and MPC-based
control for autonomous UAV landing (see, e.g., [27, 28]). We denote the
position of the $i$th moving UAV at time instant $k$, by
$\mathbf{u}_{i,k}:=[x_{i,k},\,y_{i,k}]^{\top}$, where $1\leq i\leq N$, and
$k\geq 0$. Then, the equations of motion read as

$\mathbf{u}_{i,k+1}=\mathbf{u}_{i,k}+\Delta
T_{i}\mathbf{v}_{i,k}+\mathbf{d}_{i,k},$ (3)

where $\mathbf{v}_{i,k}=[v_{i,k}^{x},\,v_{i,k}^{y}]^{\top}$ and
$\mathbf{d}_{i,k}=[d_{i,k}^{x},\,d_{i,k}^{y}]^{\top}$ denote the UAV average
nominal velocity and average velocity perturbation due to unknown inputs such
as wind gust at time instant $k$, respectively. Each time duration $\Delta
T_{i}$, $1\leq i\leq N$, represents the length of the time interval for
updating the position measurements associated with the $i$th UAV. In this
article, we allow these update time intervals to be different.
We concatenate the UAV positions, velocities, and unknown inputs at each time
instant $k$ into the single vectors
$\mathbf{X}_{k}:=[\mathbf{u}_{1,k}^{\top},\cdots,\mathbf{u}_{N,k}^{\top}]^{\top}\in\mathbb{R}^{2N}$,
$\mathbf{V}_{k}:=[\mathbf{v}_{1,k}^{\top},\cdots,\mathbf{v}_{N,k}^{\top}]^{\top}\in\mathbb{R}^{2N}$,
$\mathbf{\Lambda}_{k}:=[\mathbf{d}_{1,k}^{\top},\cdots,\mathbf{d}_{N,k}^{\top}]^{\top}\in\mathbb{R}^{2N}$,
respectively. Then, the aggregate equation of motion of the UAV network is

$\mathbf{X}_{k+1}=\mathbf{I}_{2N}\mathbf{X}_{k}+\mathbf{B}_{T}\mathbf{V}_{k}+\boldsymbol{\Lambda}_{k},$
(4)

where $\mathbf{B}_{T}=\text{diag}\big{(}\Delta T_{1},\Delta
T_{1},\cdots,\Delta T_{N},\Delta T_{N}\big{)}\in\mathbb{R}^{2N\times 2N}$ is
the diagonal matrix of sampling times. In this state space equation, the
vector $\mathbf{X}_{k}\in\mathbb{R}^{2N}$ denotes the state vector of the UAV
network; namely, the position of the UAVs.

We define the vector of unknown inputs as

$\mathbf{W}_{k}:=\mathbf{V}_{k}+\mathbf{B}_{T}^{-1}\boldsymbol{\Lambda}_{k}.$
(5)

This vector, which lumps the effect of the nominal velocities of the UAVs
along with perturbations due to uncertainties such as wind gust, drives the
dynamics of the UAVs. In the absence of uncertainties, we have
$\mathbf{W}_{k}=\mathbf{V}_{k}$. Hence, the discrete-time dynamics in (3) in
the aggregate are governed by

$\displaystyle\mathbf{X}_{k+1}=\mathbf{I}_{2N}\mathbf{X}_{k}+\mathbf{B}_{T}\mathbf{W}_{k},$
(6)
$\displaystyle\mathbf{Y}_{k}=\mathbf{I}_{2N}\mathbf{X}_{k}+\mathbf{D}\mathbf{W}_{k}.$

In the state space model given by (6), $\mathbf{Y}_{k}$ represents the vector
of UAV position measurements that are affected by uncertainties. If the
uncertainties do not affect the UAV vector measurements, then the matrix
$\mathbf{D}\in\mathbb{R}^{N\times 2N}$ can be set equal to $\mathbf{0}$.
According to (6), the UAV position signals that are received by the UIO do not
need to be accurate and can be affected by the vector of uncertainties
$\mathbf{W}_{k}$.

######  Remark 2.

We are assuming that either the smaller UAVs are transmitting their positions
to the central UAV via an automatic dependent surveillance-broadcast (ADS-B)
system [29, 30] or their positions can be inferred using a vision system on-
board the central UAV (see, e.g., the method in [31]).

######  Remark 3.

Unlike [24, 25], we make no assumptions on the statistics of
$\boldsymbol{\Lambda}_{k}$ and $\mathbf{V}_{k}$. In Section III, we present an
unknown input observer (UIO) algorithm for predicting the future location of
the UAVs by only using the approximate UAV position measurements given by
$\mathbf{Y}_{k}$ without relying on the knowledge of the vector of unknown
inputs $\mathbf{W}_{k}$.

Beamformer Design. Given an estimation of the angular position vector of the
UAVs in the network
$\hat{\boldsymbol{\theta}}_{k}=[\hat{\theta}_{1,k},\cdots,\hat{\theta}_{N,k}]^{\top}$
at time instant $k$ (see Section III for this estimated value), the baseband
transmit beamformer $\mathbf{F}$ is given by

$\mathbf{F}=\mathbf{A}^{*}(\hat{\boldsymbol{\theta}}_{k})\Big{(}\mathbf{A}^{\top}(\hat{\boldsymbol{\theta}}_{k})\mathbf{A}^{*}(\hat{\boldsymbol{\theta}}_{k})\Big{)}^{-1}\in\mathbb{C}^{M_{\text{CE}}\times
N},$ (7)

where
$\mathbf{A}(\hat{\boldsymbol{\theta}}_{k})=[\mathbf{a}(\hat{\theta}_{1,k}),\mathbf{a}(\hat{\theta}_{2,k}),...,\mathbf{a}(\hat{\theta}_{N,k})]\in\mathbb{R}^{M_{\text{CE}}\times
N}$ is the steering matrix at the central UAV for communicating with $N$
moving UAVs (see Figure 1). In this case, $\hat{\mathbf{s}}\in\mathbb{C}^{N}$
represents the vector of transmitting data to UAVs, which is beamformed by
${\mathbf{s}}=\mathbf{F}\hat{\mathbf{s}}$ and the result is used in (1). In
Section III, we present an observer-based algorithm that can estimate the
angular positions in (7).

######  Remark 4.

In this paper, we are assuming that the difference in the AoA of $N_{U}$
antennas in each UAV is negligible due to the small size of the UAVs.
Furthermore, because of the dominant LOS component, there are very few
reflected paths resulting in a sparse channel. Similar assumptions have been
made in [32] for mmWave positioning of vehicular networks.

##  III The Centralized UIO for UAV Networks

We adopt the predictor architecture due to Chakrabarty _et al._ [20], where
the predictor can estimate the unknown inputs acting on a given discrete time
dynamics and simultaneously predict the next location of the UAVs. In the
context of our work, the unknown inputs are the average velocities of the UAVs
and the vector of uncertainties. The dynamics of the observer/predictor in
[20] when employed for the UAV network dynamics in (6) are governed by

$\text{Prediction
step:\,}\hat{\mathbf{X}}_{k+1}=\mathbf{Q}\hat{\mathbf{X}}_{k}+\mathbf{L}\mathbf{Y}_{k},$
(8)

where $\mathbf{Q},\,\mathbf{L}\in\mathbb{R}^{2N\times 2N}$ are called the
observer state and observer gain matrices, respectively. The observer state
and gain matrices are designed by solving proper linear matrix inequalities
(LMIs). Furthermore, we define the following performance output vector

$\hat{\mathbf{Z}}_{k}=\mathbf{H}\mathbf{E}_{k},$ (9)

where $\mathbf{E}_{k}:=\hat{\mathbf{X}}_{k}-\mathbf{X}_{k}$ is the state
tracking error, and the design parameter $\mathbf{H}$ is the performance
output matrix to select subset/linear combinations of error states that are
crucial to the specific application. If all position estimation errors are of
the same importance, then one can choose $\mathbf{H}=\mathbf{I}_{2N}$.

######  Remark 5.

In this paper, we design UIOs for predicting the trajectories of UAVs in two-
dimensional settings. The developed UIO-based prediction framework can be
easily extended to three-dimensional scenarios.

According to [20], to find the state matrix $\mathbf{Q}$ and the gain matrix
$\mathbf{L}$, the following LMI system needs to be solved

$\displaystyle\begin{bmatrix}(\alpha-1)\mathbf{P}&\star&\star\\\
\mathbf{0}&-\alpha\mathbf{I}_{2N}&\star\\\
\mathbf{P}-\mathbf{Z}&\mathbf{Z}\mathbf{D}-\mathbf{P}\mathbf{B}_{T}&\mathbf{P}\end{bmatrix}\preceq\mathbf{0},\,\begin{bmatrix}\mathbf{P}&\mathbf{H}^{\top}\\\
\mathbf{H}&\mu\mathbf{I}_{2N}\end{bmatrix}\succeq\mathbf{0},$ (10)

where $\alpha\in(0,\,1)$ is fixed, and the LMI is solved for
$\mathbf{P}=\mathbf{P}^{\top}\succ\mathbf{0}$, $\mathbf{Z}$, and some desired
performance level $\gamma=\sqrt{\mu}>0$ (to be defined in the next
proposition). The $\star$ in (10) is used to avoid rewriting the symmetric
terms. Having found the matrices $\mathbf{P}$ and $\mathbf{Z}$ from the LMIs
in (10), we use

$\mathbf{L}=\mathbf{P}^{-1}\mathbf{Z},\;\mathbf{Q}=\mathbf{I}_{2N}-\mathbf{L},$
(11)

to obtain the observer state and gain matrices, respectively.

######  Remark 6.

The size of the LMI in (10) is proportional to the number of the UAVs in the
network. An efficient numerical algorithm for for finding solutions to the LMI
in (10) is due to Gahinet and Nemirovski [33]. The algorithm in [33], which
also lies at the heart of MATLAB LMI solvers, enjoys a polynomial-time
complexity.

According to the observer architecture due to Chakrabarty _et al._ [20], it is
possible to utilize the output of the predictor in (8) and the measurements
given by $\mathbf{Y}_{k}$ in (6), to find an estimate of the unknown input
vector $\hat{\mathbf{W}}_{k}$ according to

$\text{\begin{minipage}{65.04034pt} Unknown input \\\ estimation step:
\end{minipage}}\hat{\mathbf{W}}_{k}=\mathbf{G}^{\dagger}\begin{bmatrix}\hat{\mathbf{X}}_{k+1}-\hat{\mathbf{X}}_{k}\\\
\mathbf{Y}_{k}-\hat{\mathbf{X}}_{k}\end{bmatrix}$ (12)

where
$\mathbf{G}^{\dagger}=(\mathbf{G}^{\top}\mathbf{G})^{-1}\mathbf{G}^{\top}$ is
the the Moore–Penrose pseudo-inverse of the matrix
$\mathbf{G}:=[\mathbf{B}_{T}^{\top},\,\mathbf{0}_{2N}^{\top}]^{\top}$. Since
$\mathbf{G}$ is of full column rank, i.e., $\text{rank}(\mathbf{G})=2N$, the
Moore-Penrose pseudo-inverse of $\mathbf{G}$ is well-defined.

We have the following propositions regarding the state tracking error and
unknown input reconstruction performance of the observer architecture in (8)
and (12).

######  Proposition 1.

Consider the observer/predictor architecture in (8). Given the performance
matrix $\mathbf{H}$ and a fixed constant $\alpha\in(0,1)$, there exists a
bounded performance level $\gamma=\sqrt{\mu}>0$ proportional to the norm
$\|\mathbf{H}\|$, such that the system of LMIs in (10) has a solution for the
unknown matrices $\mathbf{P}=\mathbf{P}^{\top}\succ\mathbf{0}$, $\mathbf{Z}$.
Furthermore, the observer in (8) with state and gain matrices given by (11) is
$\ell_{\infty}$-stable with performance level $\gamma$ with respect to the
unknown input $\mathbf{W}_{k}$. Equivalently, the following properties are
satisfied.

  * i

If $\mathbf{W}_{k}=\mathbf{0}$, then the origin of the state tracking error
dynamics are globally uniformly exponentially stable;

  * ii

For zero initial error $\mathbf{E}_{0}$ and every bounded unknown input
sequence $\\{\mathbf{W}_{k}\\}$, we have
$\|\mathbf{Z}_{k}\|\leq\gamma\|\mathbf{W}\|_{\infty}$, where
$\|\mathbf{W}\|_{\infty}:=\sup_{k\geq 0}\|\mathbf{W}_{k}\|$;

  * iii

For every initial error $\mathbf{E}_{0}$ and every bounded unknown input
$\\{\mathbf{W}_{k}\\}$, we have
$\limsup_{k\to\infty}\|\mathbf{Z}_{k}\|\leq\gamma\gamma_{{}_{W}}$, where
$\gamma_{{}_{W}}:=\|\mathbf{W}\|_{\infty}$ is the tightest upper bound on the
sequence of unknown inputs.

###### Proof.

The pair of system matrices $(\mathbf{I}_{2N},\,\mathbf{I}_{2N})$, resulting
from the dynamics of the UAV network in (6), is detectable. Therefore, from
the proof of Theorem 3 in [20], it immediately follows that there exists a
bounded performance level $\gamma=\sqrt{\mu}>0$ proportional to the norm
$\|\mathbf{H}\|$, such that the system of LMIs in (10) has a solution. Since
the solution to the LMIs in (10) exists, it follows from Theorem 2 in [20]
that the observer in (8) with state/gain matrices given by (11) satisfies the
$\ell_{\infty}$-stability property claimed by the proposition. ∎

The following proposition provides an upper bound on unknown input estimation
error.

######  Proposition 2.

Suppose the assumptions in Proposition 1 hold and $\mathbf{H}={I}_{2N}$. Then,
the unknown input reconstruction error is bounded by
$\limsup_{k\to\infty}\|\hat{\mathbf{W}}_{k}-\mathbf{W}_{k}\|\leq
3\gamma\gamma_{{}_{W}}$, where $\gamma_{{}_{W}}:=\|\mathbf{W}\|_{\infty}$ is
the tightest upper bound on the sequence of unknown inputs.

###### Proof.

The proof follows from Theorem 5 in [20]. ∎

To compute the steering vectors that are fed to the beamformer in (7), we
utilize the position estimates provided by the UIO in (8). In particular, the
angular position vector $\theta_{i,k}$ of the $i$th UAV at the $k$th time-
instant can be easily computed from
$\hat{\mathbf{X}}_{k}=[\hat{\mathbf{u}}_{1,k}^{\top},\cdots,\hat{\mathbf{u}}_{N,k}^{\top}]^{\top}$
and the equation

$\hat{\theta}_{i,k}=\text{arccos}\frac{\hat{x}_{i,k}-x_{p}}{\|\hat{\mathbf{u}}_{i,k}-\mathbf{u}_{p}\|},$
(13)

where $\mathbf{u}_{p}$ is the position of the central UAV. The angles in (13)
can be used for constructing the steering matrix in (7).

Summary of the Proposed Location-Aware Beamformer. The proposed location-aware
beamformer can be summarized as follows (also, see Figure 1). The UIO dynamics
in (8) and (12), where the UIO observer and state gain matrices are found by
solving (10), can be used for estimating the correct position of the UAVs in
the network along with their unknown velocity profiles. The estimated
positions can then be used in (13) to compute the angular positions of the
UAVs. The estimated angular positions of the beamformers can in turn be
utilized in (7) and as a result, the model in (1) will be updated with the
beamformed data towards the desired UAV.

##  IV Simulation Studies

Figure 2: Location prediction simulation results with design parameter
$\alpha=0.5$ in (10): (top left) UAV network actual/predicted paths; (top
right) time profile of the coordinates of UAV#1. The simulation curve colors
are associated with the performance levels $\gamma_{1}=0.21$ (blue),
$\gamma_{2}=0.47$ (red), and $\gamma_{3}=0.96$ (green), respectively. The
black curves in the top left and the top right plots belong to the actual path
of the UAVs and time profile of the position coordinates of UAV#1.

In this section we present the simulation results to verify the effectiveness
of our proposed methodology. In the simulations, we assume that base drone is
hovering at the origin $\mathbf{x}_{p}=[0,\,0]^{\top}$. We consider $N\text{=
}4$ UAVs that are moving on closed curves centered at the base UAV (see Fig.
2). The duration for time instants associated with each of the four UAVs are
considered to be $\Delta T\text{= }0.15^{\text{sec}}$.

It is assumed that the UAVs, which are flying around the base UAV, traverse
perturbed circular paths that are affected by sinusoidal velocity
perturbations. It is remarked that sinusoidal motion perturbations have also
been utilized in other UAV simulation studies as well (see, e.g., [34, 35]).
In (3), the nominal velocity of the $i$th UAV is given by

$\mathbf{v}_{i,k}=R_{i}\omega\begin{bmatrix}\cos(\omega\Delta Tk+\phi_{i})\\\
-\sin(\omega\Delta Tk+\phi_{i})\end{bmatrix},$ (14)

and the velocity perturbation of the $i$th UAV is given by

$\mathbf{d}_{i,k}=\frac{R_{i}\omega}{5}\begin{bmatrix}\cos(10\omega\Delta
Tk+\phi_{i})\\\ -\sin(10\omega\Delta Tk+\phi_{i})\end{bmatrix},$ (15)

where $R_{i}$ is the radius of the nominal circle that the $i$th UAV is flying
on and $\omega=0.5\text{ rad/sec}$.

The UIO neither knows the nominal velocity of the UAVs nor does it know the
sinusoidal velocity perturbations exerted on the UAVs. As discussed in Section
II, we are assuming that either the smaller UAVs are transmitting their
positions to the central UAV via an ADS-B system or their positions can be
inferred using a vision system on-board the central UAV (see Remark 2).
However, the received position signals are not accurate and are perturbed
through the unknown input $\mathbf{W}_{k}$. In other words, the only
information available to the UIO is through the perturbed position signals,
namely, through
$\mathbf{Y}_{k}=\mathbf{I}_{8}\mathbf{X}_{k}+\mathbf{D}\mathbf{W}_{k}$. In
these simulations, $\mathbf{D}=0.5\mathbf{I}_{8}$.

Figure 3: Unknown input tracking error results with design parameter
$\alpha=0.5$ in (10): (left) estimated/actual unknown inputs acting on UAV#1;
and (right) unknown input tracking errors for UAV#1. The curve colors are
associated with the performance levels $\gamma_{1}=0.21$ (blue),
$\gamma_{2}=0.47$ (red), and $\gamma_{3}=0.96$ (green), respectively. The
black curves in the left plot belong to the actual unknown input acting on
UAV#1.

We consider a $64$-antenna MIMO on the central UAV and a $4$-antenna ULA on
the other UAVs, respectively, unless otherwise specified. The maximum range is
assumed to be less than $500~{}\rm{m}$ for an efficient mmWave communication
such that the UAVs are separated from the centeral UAV by $100~{}\rm{m}$,
$150~{}\rm{m}$, $200~{}\rm{m}$, and $250~{}\rm{m}$, respectively. The
specified angles are extracted from the UIO presented in Section III. A 16-QAM
baseband modulation within Orthogonal Frequency Division Multiplexing (OFDM)
data symbols with $234$ subcarriers and cyclic prefix length of $64$ is
considered to generate the waveform which operates at the central frequency of
$30{\text{GHz}}$ with the baseband bandwidth of $50{\text{MHz}}$. A $1/3$
Convolutional code is utilized to protect the stream against channel errors.

Figure 4: Unknown input tracking error results with different values of the
design parameter $\alpha$ and the desired performance level satisfying
$\mu=\sqrt{\gamma}\leq 0.25$. The curve colors are associated with the design
parameters $\alpha_{1}=0.5$ (blue), $\alpha_{2}=0.1$ (red), and
$\alpha_{3}=0.01$ (green), respectively. The black curves in the right plot
belong to the actual unknown input acting on UAV#1.

The proposed UIO with its prediction step in (8) and its unknown input
estimation step in (12) provides estimates of both the next position of the
UAVs and the estimated unknown velocity input vector $\mathbf{W}_{k}$ that is
driving the dynamics of the UAV network. In order to find the UIO state matrix
$\mathbf{Q}$ and the UIO gain matrix $\mathbf{L}$, we need to solve the LMIs
given by (10).

Figure 5: Spectral efficiency of the proposed method for each UAV at each time instant.  Figure 6: Spectral efficiency of each UAV plotted against different Azimuth angles.  |
---|---
(a) | (b) (c)
Figure 7: (a) The 3D response pattern of the beamformer, (b) the normalized power of the beamformer for Azimuth and Elevation cuts, and (c) the polar pattern demonstrating the directionality of the beamformer when there are $M_{CE}=64$ transmit antennas on the central UAV and $N_{U}=4$ antennas on each one of the other UAVs.  |
---|---
(a) | (b) (c)
Figure 8: As the UAVs fly around the central UAV, the beamformer dynamically alters the beam towards the desired targets according to the angular position estimates provided by the UIO, as illustrated in (a) and (b) for two different angular locations of the UAVs. The impact of increasing the number of antennas is shown in (c) on the same beamformer as Figure 7, but with $M_{CE}=128$ transmit antennas on the central UAV.  |
---|---
(a) | (b) (c)
Figure 9: The normalized power of the beamformers corresponding to the results
in Figure 8.  Figure 10: Spectral efficiency of the proposed method as
compared with the conventional scenario with no location prediction in which
the probing signal is utilized to estimate the echo for sensing.

We solved the LMIs in our simulation studies using the MATLAB CVX package
[36]. According to Proposition 1, the specific LMIs that arise in the context
of our UAV location prediction problem will necessarily admit a solution
provided that the upper bounds on the performance level $\gamma$ are not too
tight with respect to the given measurement time intervals $\Delta T_{i}$. To
solve the LMIs given by (10), we have chosen a value of $0.5$ for the constant
$\alpha$ in the LMIs. Moreover, we have chosen three different upper bounds on
the performance levels for the design parameter $\gamma$. In particular, we
solved the LMIs in (10) under the three performance level constraints of
$\mu_{1}\leq 0.05$, $\mu_{2}\leq 0.25$, and $\mu_{3}\leq 1$, where
$\gamma_{i}=\sqrt{\mu_{i}}$, $1\leq i\leq 3$. The obtained UIO state and gain
matrices in these three different scenarios are
$\mathbf{L}_{1}=0.39\mathbf{I}_{8}$, $\mathbf{Q}_{1}=0.61\mathbf{I}_{8}$,
$\mathbf{L}_{2}=0.60\mathbf{I}_{8}$, $\mathbf{Q}_{2}=0.40\mathbf{I}_{8}$,
$\mathbf{L}_{3}=0.76\mathbf{I}_{8}$, and $\mathbf{Q}_{3}=0.24\mathbf{I}_{8}$,
respectively. Furthermore, the resulting performance levels associated with
the found solutions are equal to $\gamma_{1}=0.21$, $\gamma_{2}=0.47$, and
$\gamma_{3}=0.96$, respectively (also, see “The effect of increasing
measurement time intervals”). In another simulation scenario, we solved the
LMIs in (10) under the the same performance level constraint of $\mu\leq 0.25$
and three different design parameters $\alpha$ in (10). It can be seen that by
decreasing the parameter $\alpha$ the tracking errors become smaller.

Figure 2 depicts the actual and predicted location of the UAVs in the network.
Figure 3 depicts the position and unknown input tracking error profiles
resulting from our proposed UIO-based framework. By Proposition 1, the steady
state position tracking errors are guaranteed to be bounded by
$\gamma\|\mathbf{W}\|_{\infty}$, where $\gamma$ is the performance level of
the UIO and $\|\mathbf{W}\|_{\infty}$ is the maximum perturbation amplitude.
Similarly, by Proposition 2, the steady state unknown input tracking errors
are guaranteed to be bounded by $3\gamma\|\mathbf{W}\|_{\infty}$. Figure 4
depicts unknown input tracking error results with the desired performance
level satisfying $\mu=\sqrt{\gamma}\leq 0.25$ and different values of the
design parameter given by $\alpha_{1}=0.5$, $\alpha_{2}=0.1$, and
$\alpha_{3}=0.01$, respectively.

Figure 5 displays the time profile of spectral efficiency of each UAV when the
beamformer in (7) is fed with the angular position estimates that are
generated by the UIO according to (13). Thanks to the reliable sensing
information provided by the proposed observer, the spectral efficiency is
rather resilient as shown in Figure 5. The spectral efficiency of the UAVs in
space when they are at different angular positions with respect to the central
UAV are depicted in Figure 6.

For $M_{CE}=64$ transmit antennas on the central UAV and $N_{U}=4$ antennas on
the other $4$ UAVs, the 3D response pattern of the beamformer, the normalized
power of the beamformer for Azimuth and Elevation cuts, and the polar pattern
for displaying the directionality of the beamformer are presented in Figure 7.
Since the UAVs fly around the central UAV, their angular positions vary with
time. Accordingly, the beamformer dynamically modifies the beam towards the
intended targets. Figures 8(a)-(b) depict the resulting beams in two different
time instants. Figure 8(c) depicts the effect of the number of antennas on the
same beamformer as previously plotted in Figure 7, but with $M_{CE}=128$
transmit antennas on the central UAV instead of $64$ as assumed in Figure 7.
Figure 9 shows the normalized power of the beamformers in azimuth and
elevation planes corresponding to the angles in Figure 8. Figure 10 shows the
average spectral efficiency of the proposed solution in comparison with a
conventional scenario in which a probing signal is sent by the central UAV.
The echo wave reflected by the target UAVs is received by the central UAV if
the blockage does not occur. In the case of blockage, as simulated at some
specific time instants, the average spectral efficiency in the case of
conventional echo-based scenario drops significantly, whereas the proposed
UIO-based technique exhibits high resistance to this phenomena of spectral
efficiency drop.

The effect of increasing measurement time intervals. It is expected that with
increasing the amount of interval delays $\Delta T_{i}$ in (3) for updating
the position of the UAVs, the prediction of their positions becomes more
challenging. Indeed, this has been the case in our simulations. In particular,
in another round of simulations, we started increasing the time interval
delays for the three different performance level constraints $\mu_{1}\leq
0.05$, $\mu_{2}\leq 0.25$, and $\mu_{3}\leq 1$ up to the point where the LMI
system in (10) failed to have a solution. For these three different
performance level constraints, the LMI system in (10) ceased to admit a
solution at $\Delta T=0.89$ sec, $\Delta T=1.01$ sec, and $\Delta T=1.3$ sec,
respectively.

##  V Conclusion

In this paper, we proposed a novel spatial estimation algorithm for
beamforming in UAV-based JSC. The proposed algorithm, which does not make any
_a priori_ assumptions on the statistics of the UAV velocities and unknown
inputs, relies on using UIOs that are designed by solving LMIs. Our observer-
based location prediction algorithm reduces the joint target detection and
communication complication notably by operating on the same device and
performs reliably in the presence of channel blockage and interference.

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