# Optical Character Recognition of 19th Century Classical Commentaries: the
Current State of Affairs††thanks: This version of the article corresponds to
the accepted manuscript. For the final published version see
https://doi.org/10.1145/3476887.3476911.

Matteo Romanello

Université de Lausanne

<EMAIL_ADDRESS>
Sven Najem-Meyer

EPFL

<EMAIL_ADDRESS>
Bruce Robertson

Mount Allison University

<EMAIL_ADDRESS>

###### Abstract

Together with critical editions and translations, commentaries are one of the
main genres of publication in literary and textual scholarship, and have a
century-long tradition. Yet, the exploitation of thousands of digitized
historical commentaries was hitherto hindered by the poor quality of Optical
Character Recognition (OCR), especially on commentaries to Greek texts. In
this paper, we evaluate the performances of two pipelines suitable for the OCR
of historical classical commentaries. Our results show that Kraken + Ciaconna
reaches a substantially lower character error rate (CER) than Tesseract/OCR-D
on commentary sections with high density of polytonic Greek text (average CER
7% vs. 13%), while Tesseract/OCR-D is slightly more accurate than Kraken +
Ciaconna on text sections written predominantly in Latin script (average CER
8.2% vs. 8.4%). As part of this paper, we also release GT4HistComment, a small
dataset with OCR ground truth for 19th classical commentaries and Pogretra, a
large collection of training data and pre-trained models for a wide variety of
ancient Greek typefaces.

##  1 Introduction

Figure 1: Example pages from 19th century commentaries on Sophocles’ Ajax. The
commentaries are by (from left to right): Lobeck (1835), Schneidewin (1853),
Campbell (1881), Jebb (1896) and Wecklein (1894).

Recent large-scale digitization initiatives have greatly increased the access
to cultural heritage collections. For researchers, however, their benefits
remain highly dependent on the quality of their automatic transcription.
Modern and contemporary historians, for example, are now in a relatively good
position to exploit large collections of digitized newspapers, thanks to the
reasonable accuracy reached by optical character recognition (OCR) systems on
these materials (see e.g. [22]). On the contrary, classicists have long been
confronted to a paradoxical situation, where large amounts of digitized
sources were openly available online, but where the poor performances of
polytonic Greek OCR hampered their exploitation. This situation has started to
change thanks to dramatic improvements on polytonic Greek OCR over the last
decade. These advancements have enabled new ambitious projects such as the
Free First Thousand Years of Greek (FF1KG) [13], an effort allied with the
more extensive Open Greek & Latin Project111https://opengreekandlatin.org/
(OGLP), which aims to create large open collections of digital editions of
Greek and Latin primary sources. However, one type of historical document that
has remained hitherto marginal—both to these efforts and their concomitant
improvements—are classical commentaries.

Scholarly commentaries are publications dedicated to the in-depth analysis and
explanation of literary works, and are aimed at facilitating the reading and
understanding of a given text. Moreover, they constitute a notable example of
historical documents that present considerable challenges for OCR systems.
However, because of the lack of ground truth data [19], they are rarely
considered by builders of OCR systems. Commentaries on Classical Greek and
Latin literature present the typical difficulties of historical OCR; they have
complex layouts, often with multiple columns and rows of text, and
additionally contain Latin and Greek scripts mixed together within the same
page and often even within the same line of text. As a result, the OCR quality
of commentaries available in open repositories such as the Internet Archive
renders these materials inadequate, most of the times, for use in research
contexts, despite their recognized potential [9].

In this paper we investigate the current state of affairs with regard to the
OCR of historical Classical commentaries. We evaluate and compare the quality
of OCR obtained with two open source OCR pipelines—OCR-D and OGLP’s
Ciaconna—on a new, multilingual ground truth dataset of 19th century
commentaries. We begin by presenting prior works related to the challenges of
recognizing polytonic Greek and of historical OCR (Section 2). In Section 3 we
briefly present the Ajax Multi-Commentary project, which constitutes the
context for the work presented in this paper, and in Section 4 we describe the
OCR ground truth datasets that we release as part of this work. In Section 5
we present the results of our experiments, and discuss the quality of OCR we
achieved on these materials in the light of their intended future usage. We
conclude the paper with some remarks on open challenges to be addressed by
future work.

##  2 Related Work

Substantial efforts have been made to bring polytonic Greek OCR to a level of
accuracy that would make its manual correction less costly than double-entry
keying. Initial research in this area is characterized by the heavy use of OCR
post-processing techniques, and had to deal with the lack of open source
engines able to outperform ABBYY FineReader on texts with mixed Latin and
polytonic Greek scripts, such as critical editions [1]. To tackle this issue,
ground truth data was created [25] to improve the ability of the open source
engine Tesseract (version 2) to recognize polytonic Greek, and especially to
deal with Greek diacritics (i.e. accents, breathing marks and their
combinations). Another proposed solution [17] consisted in using a k-nearest
neighbors algorithm for character classification on top of the character
segmentation and recognition performed by ABBYY, with substantial improvements
in character recognition accuracy over ABBYY’s output (94.2% vs. 68.7%).

From an OCR perspective, one of the main challenges of polytonic Greek OCR is
the presence of diacritics, which results in a very large number of character
classes to recognize (over 270). Furthermore, being very small, diacritics are
often the source of errors. An approach to deal with this issue was proposed
by Gatos et al. [7] and consisted in combining multiple recognition modules,
each of them dedicated to a specific class of characters or diacritics. Their
OCR framework reached a character accuracy of 90% and a word accuracy of 63%
on Greek texts. Another open source OCR software that was adapted to work on
polytonic Greek is Gamera. A substantially extended version of Gamera became
the basis for Rigaudon [16], a processing pipeline developed for large scale
OCR that employs image pre-processing and OCR post-processing to improve the
accuracy of recognition, leading to an average character accuracy of 96%.
Finally, unlike the majority of previously discussed OCR systems, which
require character segmentation, Katsouros et al. [11] have developed a
segmentation-free method that uses Hidden Markov Models (HMMs) to OCR text
lines, reaching an average character accuracy of 92.39% on the GRPOLY-DB
dataset [8].

More recently, deep neural architectures have proved to be very effective for
OCR. Recurrent neural networks with Long Short Term Memory (LSTM), such as
implemented in OCRopus 2222https://github.com/ocropus/ocropy2 and
3333https://github.com/NVlabs/ocropus3, led to good results both on historical
documents in English and German Fraktur script [2], and on historical
polytonic Greek [18]. As shown by Robertson [15], these results can be
improved even further with post-processing. This approach was implemented in
Ciaconna444https://github.com/brobertson/ciaconna, which applies spellcheckers
and word dehyphenation to OCRopus’ output.

Moreover, hybrid neural architectures combining LSTMs with a convolutional
layer were shown to improve LSTMs’ results, for instance on early printed
books (15th-16th century). Using Calamari555https://github.com/Calamari-OCR
[27] with a hybrid architecture, Wick et al. [26] report average improvements
on CERs of 29.3%, 34.9% and 43.6% when using respectively 60, 100 and 1,000
ground truth lines for training. Another open source implementation of this
hybrid architecture is provided by Kraken [12]. On a sample of historical
documents including polytonic Greek, the framework reached mean character
accuracy rates ranging from 96.2% to 99.5% Finally, neural architectures for
Handwriting Text Recognition (HTR) demonstrated their effectiveness also when
applied to historical OCR. Ströbel et al. [22] trained the HTR+ model of the
Transkribus framework [24] and reached an average character recognition
accuracy of 99.5% on a historical German-language newspaper printed in black
letter.

In addition to the development and improvement of OCR methods, recent efforts
were devoted to the creation of platforms and frameworks aimed at facilitating
working with OCR and at bringing the benefits of the latest advancements in
OCR technologies to a larger public. Three main tools are worth mentioning in
the context of historical Classical commentaries. First,
Lace666https://github.com/brobertson/Lace2 is a GUI-based platform for the
visualization and manual correction of OCR output, designed for automatic
conversion to TEI-encoded XML. Second, OCR4all [14] is an open source OCR tool
explicitly developed for users with no prior technical background, and
especially those working on the earliest printed books. It implements an
iterative workflow that allows for rapidly training very accurate OCR models
for specific publications or publication series. Lastly, OCR-D [5, 6] is a
workflow-oriented, modular platform integrating several OCR engines into a
common architecture. Unlike OCR4all, OCR-D was developed for a technical
audience, such as staff working in the digitization units of cultural heritage
institutions. Being an integrated platform, OCR-D is particularly suited for
experimenting with various OCR tools and techniques, and exploring the effects
of pre- and post-processing steps.777For detailed information about the OCR-D
pipeline, we refer the reader to its extensive documentation available at
https://ocr-d.de/ (in English and German).

##  3 The Ajax Multi-Commentary Project

Scholarly commentaries do not only constitute reference works of absolute
importance for any in-depth study of a text. They are also a fossilized
testimony of reading and understanding a literary text as a historically-
situated intellectual process. As such, commentaries can be used to study,
from a history of science perspective, the history and evolution of this genre
and of its exegetic practices. This is precisely the goal of Ajax Multi-
Commentary project, which forms the context for the work presented in this
paper.

The project focuses as a case study on the commentaries published about
Sophocles’ Ajax, a tragedy on the suicide of the Greek hero Ajax during the
Trojan War. In order to enable an epistemological study of Ajax’s
commentaries, a digital multi-commentary is being developed in order to read,
compare and analyze the entire commentary tradition of this tragedy. Layout
detection and OCR of the digitized commentaries constitute the first necessary
steps for the creation of multi-commentary interfaces. Further processing
steps will entail the linking of the commentary section (or lemma) with the
portion of text it refers to, the alignment of commentaries with one another,
as well as the semi-automatic enrichment of the commentary sections (e.g. by
means of named entity processing and citation mining).

##  4 Ground Truth Datasets

While datasets such as GT4HistOCR [21, 20] (Latin script) and GRPOLY-DB [8]
(polytonic Greek script) do exist, there are no ground truth datasets with
mixed Latin and polytonic Greek scripts that could be used for the evaluation
of historical commentaries OCR. To start filling this gap, we release the OCR
ground truth dataset for historical commentaries used for evaluation as well
as the ground truth data for polytonic Greek that was used to train the Kraken
models presented in Section 5.2.2.

###  4.1 OCR GT for Historical Commentaries

GT4HistComment is a dataset of OCR ground truth for historical commentaries,
created from the public domain subset of Ajax’s
commentaries.888https://github.com/AjaxMultiCommentary/GT-commentaries-OCR. It
consists of five 19th century commentaries written in German, English, and
Latin. Its main goal is to enable the evaluation of the OCR quality,
especially in the light of further NLP processing that will be carried out
during the Ajax Multi-Commentary project (see Section 3).

Figure 2: Example lines from selected commentaries on Sophocles’ Ajax: a)
Lobeck, b) Schneidewin, c) Campbell, d) Jebb and e) Wecklein.

The five commentaries in this dataset represent a variety of different layouts
with varying degrees of complexity (see Fig. 1). They feature multiple
typefaces for the Latin and polytonic Greek scripts (Fig. 2). The commentaries
by Jebb (1896) and Campbell (1881) are written in English, Schneidewin (1853)
and Wecklein (1894) in German, and Lobeck (1835) in Latin. All use serif fonts
for Latin characters, with the exception of Wecklein, which is partly typeset
in black letter. For Greek characters, the English commentaries use Porson (or
Porson-like) fonts, whereas the others use a non-Teubner italicized font.

To create this dataset, the OCR output of the Kraken + Ciaconna pipeline
(Section 5.2.2) was manually corrected by a pool of annotators using the Lace
platform. For the evaluation set, we selected a sample of five pages (2081±621
words) from each commentary.999 Detailed statistics (line, word and char-
counts) can be found in the dataset’s README.md file. We deliberately decided
against random sampling to ensure that this sample is representative of the
different layout types (e.g. introduction, table of contents, commentary,
etc.) and of their respective frequency within each commentary. Additionally
to these, ten pages (2956±18 words) of ground truth were created only for
selected commentaries and used to retrain the base Kraken + Ciaconna models
(see Section 5.1).

Figure 3: The main layout elements of a scholarly commentary page.

In order to ensure the quality of the ground truth datasets, an additional
verification of all transcriptions made in Lace was carried out by an
annotator on line-by-line pairs of image and corresponding text. Moreover, we
annotated each page with OLR information; pages are divided into regions, and
each region is assigned a type. OLR annotations were initially performed
within Lace and subsequently refined with the help of the VGG Image Annotator
(VIA) [4], according to a taxonomy that we devised specifically for
commentaries. One of the usages of this fine-grained OLR annotation was to
produce a detailed breakdown of character error rates according to region
types (see Section 5.4).

While an in-depth discussion of our taxonomy goes beyond the scope of this
paper, it is worth providing a brief presentation of the coarse-grained
classes we defined (see Fig. 3). These distinguish between the original Greek
texts of the work being commented upon (Primary text), the commentary sections
(Commentary), the commentator’s translation of the commented text
(Translation), the section containing information about manuscript readings
and editorial conjectures (Critical Apparatus), the paratextual elements, such
as table of contents, appendices, indices, footnotes, introductory and
prefatory materials (Paratext) and finally page numbers, line numbers
alongside the primary text, as well as line numbers in the critical apparatus
and commentary sections (Number).

For evaluation purposes (see Table 1), certain regions were grouped as follows
: _Commentary-like Texts_ comprises commentaries and footnotes; _Low Greek
Texts_ , comprises Latin script areas such as introduction, preface and
translation; finally, Structured Texts groups together appendices,
bibliography, indices, titles and tables of contents.

###  4.2 Polytonic Greek Training Data from Historic Texts (Pogretra)

Besides GT4HistComment, we release a substantial, carefully verified portion
of the data produced by OGLP’s editing process as the Pogretra
dataset101010Pogretra is published at https://doi.org/10.5281/zenodo.4774200.,
which constitutes the initial basis of the recognition of these commentaries
in the Kraken + Ciaconna pipeline.

In addition to ground truth, this dataset includes several Kraken classifiers,
each corresponding to a broad family of ancient Greek typefaces, as
experiments to generate a Kraken classifier combining all historical typefaces
did not yield satisfactory results. Of the two classifiers used in this study,
the Porson classifier is well suited to typefaces common in the English-
speaking world from the late 19th century to present. Very different is the
highly serifed family of Greek typefaces that became conventional in German-
speaking countries of the 19th century. Given this substantial difference,
Pogretra provides a separate classifier for this typeface family, called
‘German-serifs’. Because most pages of ancient Greek also comprise important
Latin-script content (such as the critical apparatus), these classifiers
recognize Latin characters as well. The Porson Kraken classifier used in this
paper is trained on 6,607 lines of data comprising 79,823 words. The German-
serifs Kraken classifier is trained on 25,365 lines of data and 218,806 words.

##  5 Experiments

Table 1: Global and regional weight-averaged results (±std) of tested
pipelines. CER scores ($0\leq CER\leq 1$) are aggregated according to the
following region groups (as explained in Section 4.1): Greek Texts (Greek),
Commentary-like (Comm.), Low Greek Texts (Low Greek), Critical Apparatus (App.
Crit.), Structured Texts (Struct.) and Numbers.

Region | Global | Greek | Comm. | Low-Greek | App. Crit. | Struct. | Numbers
---|---|---|---|---|---|---|---
Nb. of chars (% Greek) | 51186 (29%) | 6657 (92%) | 23825 (23%) | 13322 (2%) | 2062 (43%) | 3371 (34%) | 693 (0%)
Metric | F1 | CER | WER | CER | CER | CER | CER | CER | CER
Calamari GT4Hist | .61±.05 | .30±.04 | .39±.05 | .84±.05 | .27±.04 | .05±.04 | .46±.12 | .34±.01 | .23±.26
Tesseract | .82±.04 | .07±.02 | .16±.03 | .13±.05 | .08±.02 | .01±.00 | .12±.01 | .07±.01 | .13±.13
Kraken+Ciaconna | .82±.02 | .08±.02 | .18±.02 | .07±.04 | .11±.05 | .04±.01 | .07±.00 | .07±.02 | .13±.17

Table 2: F1-Score, CER ($0\leq CER\leq 1$) and normalized Levenstein distance
(NLD) per commentary.

Commentary | Lobeck | Schneidewin | Campbell | Jebb | Wecklein
---|---|---|---|---|---
Metric | F1 | CER | NLD | F1 | CER | NLD | F1 | CER | NLD | F1 | CER | NLD | F1 | CER | NLD
Calamari GT4Hist | .52 | .37 | .63 | .61 | .28 | .72 | .67 | .27 | .73 | .63 | .31 | .69 | .59 | .32 | .68
Tesseract | .76 | .11 | .89 | .82 | .08 | .92 | .87 | .05 | .95 | .80 | .08 | .92 | .82 | .05 | .95
Kraken+Ciaconna (base) | .75 | .11 | .89 | .68 | .17 | .83 | .83 | .07 | .93 | .78 | .12 | .88 | .83 | .05 | .95
Kraken+Ciaconna (retrained) | .81 | .09 | .91 | .82 | .09 | .91 | - | - | - | .82 | .09 | .91 | - | - | -

In this section we present and discuss the evaluation results for two OCR
pipelines: Tesseract/OCR-D (LSTM), and Kraken + Ciaconna (CNN-LSTM, same as
Calamari). The Gamera software111111https://gamera.informatik.hsnr.de/, which
provides support for polytonic Greek via the GreekOCR Toolkit add-on, was not
considered as it proved less flexible than neural network-based pipelines due
to its finite state machine-based architecture. It also requires a more
laborious training process and its results are not superior.

###  5.1 Comparing base and retrained models

We first investigated the benefits of adding 10 manually corrected pages to
the baseline training material described in Section 4.2. We performed this
experiment on the three commentaries by Lobeck, Schneidewin and Jebb. The
first two were selected as the respective base Kraken models were found to
perform particularly poorly (Lace accuracy121212We used the accuracy score
provided by Lace as an initial indicator of OCR quality, which corresponds to
the ratio of document tokens that are found in the dictionaries included in
Lace. ¡ 60%). Jebb, instead, was chosen as its training data could be reused
also for Campbell’s commentary, which uses identical typefaces. The
improvement brought by retrained models varies substantially across
commentaries. In fact, the CER of the retrained vs. base models for Lobeck,
Schneidewin and Jebb decreases by 1.2%, 8% and 2.9% respectively (see Table
2).

###  5.2 Comparing OCR pipelines

####  5.2.1 Tesseract/OCR-D

Two series of experiments were conducted within the framework of OCR-D. First,
we evaluated the performances of Calamari trained on GT4HistOCR [21]. This
model, however, was not trained on polytonic Greek data. Secondly, we tried
two workflows using several default models from Tesseract in a confidence-
based voting system. For each commentary, we took the model best suited for
its language (English, Latin or German) and combined it with the default model
for polytonic Greek. For both Schneidewin and Wecklein, we added a model
trained on GT4HistOCR. Finally, for Wecklein only, we added three fraktur
models to the confidence-based voting system131313For the detailed list of
Tesseract’s models and training data, see https://tesseract-
ocr.github.io/tessdoc/#traineddata-files-1.. The first workflow used Tesseract
to segment and recognize binarized images. In the second workflow, images were
segmented and pre-processed before being passed to Tesseract. However, since
Tesseract could not be prevented from resegmenting words in its own
fashion141414See https://github.com/OCR-D/ocrd-website/issues/215., this
workflow performed poorly (results not reported).

####  5.2.2 Ciaconna

The Ciaconna OCR pipeline is based on Kraken and, occasionally, Tesseract.
Training material is provided by the constantly expanding data from OGLP. As
explained in Section 5.1, additional training data can be produced manually.
Training uses NFC unicode normalization mode, which produces slightly better
results, and data augmentation on scant data (less than 300 lines). To
generate a classifier for Fraktur typefaces, we added the GT4HistOCR ‘RIDGES’
material to Pogretra’s german-serifs training set.151515The GT4HistOCR
collection is available at https://zenodo.org/record/1344132.

In terms of OCR post-processing, Ciaconna applies dehyphenation and two kinds
of polytonic Greek spellcheck to the OCR output. Dehyphenation aims to match
every line ending in a hyphen with the first ‘word’ of the next line, even if
marginal line numbers intercede. When successful, the dehyphenated form is
stored for later use. Ciaconna’s first spellcheck uses a list of special Greek
words which admit only one configuration of diacritics. Words from the output
containing more than five character are then de-accented and matched with de-
accented word-forms from the list. Successful matches are then correctly re-
accented. The second spellcheck compares the OCR output with a list of 1.4
million unique Greek words. Ciaconna allows the replacement of up to one pair
of commonly confused characters (or composed characters).

###  5.3 Evaluation setting

Due to the layout complexity of commentaries, different OCR-systems can output
different reading orders of a same page, thus making it impractical to use a
purely text-based approach for evaluation. We therefore computed character
error rates (CER) by using coordinates-based alignment of words, which allowed
for a fast, region-based evaluation without need for carefully aligned text
and line-images.161616Combined diacritic-main forms (NFC unicode normalization
mode) were used. Our OCR evaluator is available at
https://github.com/AjaxMultiCommentary/oclr. For comparison with other
research on historical OCR and on the impact of OCR quality on downstream NLP
tasks, we report scores according to two additional metrics: PRImA TextEval’s
[3] bag-of-words F1-score and normalized Levenshtein distance between ground
truth and OCR output.

###  5.4 Results and Discussion

The goal of the evaluation is twofold. Firstly, it aims to assess the
strengths and weaknesses of each pipeline, both globally and regionally.
Secondly, it seeks to determine whether the obtained OCR quality is acceptable
for the type of analysis and further NLP steps planned in the AjMC project.

In Table 1 we report the results of the evaluation both at the global level
and for groups of layout region types, showing the effect of polytonic Greek
texts on the overall accuracy of each OCR pipeline. Unsurprisingly, an OCR
pipeline without specific training for this script, such as Calamari GT4Hist
in our experiment, has high CERs in all sections that have a considerable
proportion of Greek characters. While Tesseract and Kraken + Ciaconna perform
similarly on text consisting of numbers or structured texts (e.g. indices,
appendices, table of contents, etc.), the main differences in CERs are
observed in text regions with a substantial presence of Greek characters.
While Tesseract outperforms Kraken + Ciaconna in regions written mostly in
Latin script, the latter remains more reliable in page regions such as primary
text and critical apparatus, which are written predominantly in Greek script
and/or are characterized by the presence of words that are usually not found
in language models (e.g. people and place names, abbreviated Greek words).

Moreover, the OCR quality deserves to be briefly discussed in the light of the
upcoming NLP-steps to be performed on the OCRed texts. In fact, researchers
have recently investigated the impact of OCR quality on downstream NLP tasks.
Hill and Hengchen [10] found that texts with an F-score higher than 80% can be
reliably used for analysis such as topic modelling, collocations, vector space
analysis and authorial attribution. van Strien et al. [23] investigated the
impact of OCR noise on sentence segmentation, named entity recognition and
depedency parsing. They found that the least impact of OCR noise is observed
on documents where the Normalized Levenshtein Distance (between ground truth
and OCR output) is higher than .9. As reported in Table 2, the OCR of all
considered commentaries falls into the quality band where, according to both
studies, further processing and analyses can be carried out with reliable
results.

##  6 Conclusions and further work

In this paper we evaluated the performances of Tesseract/OCR-D and Kraken +
Ciaconna OCR pipelines on a dataset of 19th century classical commentaries. We
released also ground truth data and corresponding Kraken models produced by
the Open Greek & Latin project. It is hoped that these two datasets will spark
further research on historical documents with mixed languages, and will lead
to processing-ready versions of the thousands of commentaries available in the
public domain.

As it emerges from our evaluation results, Tesseract/OCR-D and Kraken +
Ciaconna can be seen as two rather complementary pipelines with regards to the
OCR of historical commentaries. The former performs better on texts written
mostly in Latin script, while the latter excels on text with a large
proportion of polytonic Greek, thanks also to extensive training data and the
use of OCR post-processing. Based on these findings, we plan to explore how
leveraging layout classification to perform OCR can lead to improved
performances on this type of materials. In fact, classifying layout regions
would allow for feeding different segments of the page to different OCR models
and then combining these outputs in an ensemble-like system.

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# LiQuiD-MIMO Radar: Distributed MIMO Radar with Low-Bit Quantization

Yikun Xiang, Feng Xi, Shengyao Chen Y. Xiang, F. Xi, and S. Chen are with the
Department of Electronic Engineering, Nanjing University of Science and
Technology, Nanjing 210094 China (email: xifeng@njust.edu.cn). This work was
supported in part by the Natural Science Foundation of Jiangsu Province,
China, under grant No.BK20221486.

###### Abstract

Distributed MIMO radar is known to achieve superior sensing performance by
employing widely separated antennas. However, it is challenging to implement a
low-complexity distributed MIMO radar due to the complex operations at both
the receivers and the fusion center. This work proposed a low-bit quantized
distributed MIMO (LiQuiD-MIMO) radar to significantly reduce the burden of
signal acquisition and data transmission. In the LiQuiD-MIMO radar, the
widely-separated receivers are restricted to operating with low-resolution
ADCs and deliver the low-bit quantized data to the fusion center. At the
fusion center, the induced quantization distortion is explicitly compensated
via digital processing. By exploiting the inherent structure of our problem, a
quantized version of the robust principal component analysis (RPCA) problem is
formulated to simultaneously recover the low-rank target information matrices
as well as the sparse data transmission errors. The least squares-based method
is then employed to estimate the targets’ positions and velocities from the
recovered target information matrices. Numerical experiments demonstrate that
the proposed LiQuiD-MIMO radar, configured with the developed algorithm, can
achieve accurate target parameter estimation.

###### Index Terms:

Distributed MIMO radar, low-bit quantization, low-rank matrix, robust
principal component analysis.

##  I Introduction

Over the last few years, multiple-input-multiple-output (MIMO) radar [1, 2, 3]
has attracted significant attention in civil and military applications, due to
its superior performance for target detection and localization. Depending on
the antenna configurations, two typical MIMO radar architectures are
developed, namely collocated MIMO radar [2] and distributed MIMO radar [3].
The antennas in collocated MIMO radar are closely spaced while those are
widely separated in distributed MIMO radar. In this paper, we mainly focus on
distributed MIMO radar.

In distributed MIMO radar [3], the spatial diversity is exploited to improve
the target detection performance, since each pair of transmit (Tx) and receive
(Rx) antennas observes a target from different angles. The main challenge of
distributed MIMO radar is to jointly process the signals received by those
widely separated antennas. Typically, in a distributed MIMO radar, each
receiver is equipped with limited hardware for local processing. To jointly
process all the data collected by each receiver, the results of local
processing have to be delivered to a fusion center to make the final decision.
However, with the widespread application of large-scale arrays, the complexity
of such a MIMO radar system and the corresponding data transmission and
processing becomes prohibitively high. The aim of this paper is to develop a
novel architecture to cope with the system complexity, data transmission
volume, and energy consumption of such complicated systems.

The leading approach in the literature to reduce the complexity of a MIMO
radar system is to utilize compressive sensing [4] to reduce the sampling rate
[5, 6], the number of antennas [7], and the amount of samples [8, 9]. These
works mainly focus on the sampling aspect of signal acquisition and ignore the
effect of quantization, assuming the use of high-resolution analog-to-digital
converters (ADCs). Unfortunately, the energy consumption and cost of an ADC
grow exponentially with the bit depth [10].

Recent years have witnessed a growing interest in signal acquisition operating
with low-resolution ADCs. The major issue in low-bit quantization is
developing signal processing methodologies to compensate for the distortion
induced in quantization. A variety of signal acquisition architectures as well
as signal processing algorithms built upon low-resolution ADCs have been
proposed in a multitude of different applications, including radars [11, 12,
13, 14], communications [15, 16], joint radar-communication systems [17, 18],
array signal processing [19, 20, 21], and so on. In the context of MIMO
radars, the works [22, 23, 24] proposed to use one-bit or low-bit quantization
to reduce the system complexity and modified the signal processing to account
for low-bit quantized observations. However, these works focus on collocated
MIMO radar, which are not suitable for distributed MIMO radar due to the
totally different architectures.

In this paper, we propose a new distributed MIMO radar architecture, referred
to as the Low-bit Quantized Distributed MIMO (LiQuiD-MIMO) radar, in which
each receiver quantizes the received signal via low-resolution ADCs, and then
transmits the low-bit quantized data to the fusion center for sophisticated
processing. Owing to the low-bit quantization, the cost, energy consumption,
and hardware complexity at each receiver are largely reduced, and the amount
of data transmitted from the receiver located remotely to the fusion center is
greatly decreased. By exploiting the low-rank structure of the target
information matrix (TIM) and the sparsity of the data transmission error
(DTE), we formulate the target estimation problem at the fusion center as a
quantized version of robust principal component analysis (RPCA) problem [25],
referred to as QRPCA problem. To deal with the QRPCA problem, an accelerated
proximal gradient (APG)-based algorithm is proposed to iteratively recover the
TIM and DTE matrices. Finally, the least squares (LS)-based method is applied
to combine all the information collected by each pair of Tx-Rx antennas and
output the position and velocity estimation of the targets.

The rest of the paper is organized as follows: Section II introduces the
LiQuiD-MIMO radar system. Section III formulates the QRPCA problem and
develops the APG-based algorithm to solve it. The target parameter estimation
from the solutions to the QRPCA problem is detailed in Section IV. Sections V
and VI provide numerical simulations and concluding remarks, respectively.

##  II LiQuiD-MIMO Radar Model

In this section, we first describe the signal model for our distributed MIMO
radar system and then introduce the sampling and quantization scheme with low-
resolution ADCs in the LiQuiD-MIMO radar.

###  II-A Signal Model

Consider a distributed MIMO radar system equipped with $M_{t}$ transmit
antennas and $M_{r}$ receive antennas, which are widely separated in a two-
dimensional (2-D) Cartesian coordinate system. The positions of the $m$th
transmitter antenna and the $n$th receiver antenna are denoted as
$\mathbf{p}_{t}^{(m)}=[x_{t}^{(m)},y_{t}^{(m)}]^{T}$ and
$\mathbf{p}_{r}^{(n)}=[x_{r}^{(n)},y_{r}^{(n)}]^{T}$, respectively. A set of
$M_{t}$ narrowband and orthogonal waveforms, denoted as
$s_{1}(t),\cdots,s_{M_{t}}(t)$, are transmitted in pulses, with a pulse
repetition interval (PRI) $T_{\text{PRI}}$. We assume that each coherent
processing interval (CPI) includes $Q$ pulses, i.e., the $m$th transmitted
signal during one CPI is $\sum_{q=0}^{Q-1}s_{m}(t-qT_{\text{PRI}})$.

To guarantee the orthogonality of the transmitted waveforms, we consider
orthogonality achieved using frequency division multiple access (FDMA)
signaling. In FDMA, the carrier frequency of the $m$th transmitted signal is
$f_{m}=f_{0}+(m-1)\Delta f$, $m=1,\cdots,M_{t}$, where $f_{0}$ is the
reference carrier frequency and $\Delta f$ is the frequency increment across
transmit antennas, satisfying that $\Delta f\geq B_{0}$ with $B_{0}$ being the
bandwidth of $s_{m}(t)$.

We assume that all the targets are distributed in the same 2-D plane where the
transmit and receive antennas are located. Without loss of generality, we can
extend the analysis in this paper to the three-dimensional (3-D) case. Suppose
there are a total of $K$ targets located in the radar’s unambiguous time-
frequency region. The $k$th target is located at the position
$\mathbf{p}^{(k)}=[x_{k},y_{k}]^{T}$, moving at a velocity of
$\mathbf{v}^{(k)}=[v_{x}^{(k)},v_{y}^{(k)}]^{T}$. In the distributed MIMO
radar, each of the $M_{t}$ transmitted signals are reflected by the targets
and collected by each of the $M_{r}$ receive antennas. For the $k$th target,
the radar echoes from the $m$th transmit antenna to the $n$th receive antenna
are characterized by the following parameters: its reflection coefficient
$\tilde{\beta}_{mn}^{(k)}$, the time delay $\tau_{mn}^{(k)}$, and the Doppler
frequency $f_{mn}^{(k)}$. We assume that these parameters are constant during
one CPI. Accodding to the geometric relationship, $\tau_{mn}^{(k)}$ and
$f_{mn}^{(k)}$ are, respectively, determined by the following equations:

$\tau_{mn}^{(k)}=\frac{\|\textbf{p}^{(k)}-\textbf{p}_{t}^{(k)}\|+\|\textbf{p}^{(k)}-\textbf{p}_{r}^{(k)}\|}{c},$
(1)
$f_{mn}^{(k)}=\frac{f_{m}}{c}\left(\frac{\langle\textbf{v}^{(k)},\textbf{p}^{(k)}-\textbf{p}_{t}^{(m)}\rangle}{\|\textbf{p}^{(k)}-\textbf{p}_{t}^{(m)}\|}+\frac{\langle\textbf{v}^{(k)},\textbf{p}^{(k)}-\textbf{p}_{r}^{(m)}\rangle}{\|\textbf{p}^{(k)}-\textbf{p}_{r}^{(m)}\|}\right),$
(2)

where $c$ is the speed of light.

Then the received signal at the $n$th receive antenna can be expressed as

$\begin{split}y_{n}(t)=\sum_{q=0}^{Q-1}\sum_{m=1}^{M_{t}}\sum_{k=1}^{K}\tilde{\beta}_{mn}^{(k)}s_{m}(t-\tau_{mn}^{(k)}-qT_{\rm{PRI}})\\\
\times e^{j2\pi(f_{m}+f_{mn}^{(k)})(t-\tau_{mn}^{(k)})}+w_{n}(t)\end{split}$
(3)

where $w_{n}(t)$ is the additive spatio-temporally white, zero mean Gaussian
noise with variance $\sigma_{n}^{2}$. After demodulation and channel
separation, the received signal at the $n$th receiver from the $m$th transmit
antenna is

$\begin{split}y_{mn}(t)=\sum_{q=0}^{Q-1}\sum_{k=1}^{K}\beta_{mn}^{(k)}s_{m}(t-\tau_{mn}^{(k)}-qT_{\rm{PRI}})\\\
\times e^{j2\pi f_{mn}^{(k)}qT_{\rm{PRI}}}+w_{mn}(t)\end{split}$ (4)

where
$\beta_{mn}^{(k)}=\tilde{\beta}_{mn}^{(k)}e^{-j2\pi(f_{m}+f_{mn}^{(k)})\tau_{mn}^{(k)}}$,
and $w_{mn}(t)$ denotes the noise in the corresponding transmission channel.
In the above, we use the approximation $e^{j2\pi f_{mn}^{(k)}qt}\approx
e^{j2\pi f_{mn}^{(k)}qT_{\rm{PRI}}}$ which follows the assumption that the
targets are slow moving and have low acceleration.

###  II-B Sampling and Quantization with Low-Resolution ADCs

In classic distributed MIMO radar, the received signal $y_{mn}(t)$ is sampled
at the Nyquist rate and quantized by high-resolution ADCs, such that the
quantization distortion is ignored in the subsequential signal processing. Let
$T_{s}$ be the Nyquist sampling interval. For each pulse received, we collect
$L_{mn}$ unambiguous samples, where
$L_{mn}=\lfloor\frac{T_{p}+\tau_{mn}^{\text{(max)}}}{T_{s}}\rfloor$ with
$T_{p}$ and $\tau_{mn}^{\text{(max)}}$ denoting the pulse duration and maximum
possible time-delay of targets, respectively. For simplicity, we assume
$T_{p}=NT_{s}$. Then the samples of the time delayed signal
$s_{m}(t-\tau_{mn}^{(k)})$ can be equivalently represented as
$\mathbf{C}_{L_{mn}^{(k)}}^{T}\mathbf{s}_{m}$, where
$\mathbf{C}_{L_{mn}^{(k)}}=\left[\mathbf{0}_{N\times
L_{mn}^{(k)}}\quad\mathbf{I}_{N}\quad\mathbf{0}_{N\times(L_{mn}-N-L_{mn}^{(k)})}\right]\in\mathbb{C}^{N\times
L_{mn}}$ with $L_{mn}^{(k)}=\lfloor\frac{\tau_{mn}^{(k)}}{T_{s}}\rfloor$, and
$\mathbf{s}_{m}\in\mathbb{C}^{N}$ is the sampled $m$th transmit waveform.

After the Nyquist sampling and high-resolution quantization, the data
collected during the $q$th pulse can be expressed as

$\mathbf{y}_{mn}^{(q)}=\mathbf{x}_{mn}^{(q)}+\mathbf{w}_{mn}^{(q)},\quad
q=0,...,Q-1,$ (5)

where $\mathbf{x}_{mn}^{(q)},\mathbf{w}_{mn}^{(q)}\in\mathbb{C}^{L_{mn}}$ are
the the sampled signal vector and noise vector, respectively. According to
(4), $\mathbf{x}_{mn}^{(q)}$ can be equivalently represented as

$\mathbf{x}_{mn}^{(q)}=\sum_{k=1}^{K}\beta_{mn}^{(k)}e^{j2\pi
f_{mn}^{(k)}qT_{\rm{PRI}}}\mathbf{C}_{L_{mn}^{(k)}}^{T}\mathbf{s}_{m}=\mathbf{A}_{mn}\boldsymbol{\Lambda}_{mn}\mathbf{b}_{mn}^{(q)},$
(6)

where
$\mathbf{A}_{mn}=\left[\mathbf{C}_{L_{mn}^{(1)}}^{T}\mathbf{s}_{m},...,\mathbf{C}_{L_{mn}^{(K)}}^{T}\mathbf{s}_{m}\right]\in\mathbb{C}^{L_{mn}\times
K}$, $\mathbf{b}_{mn}^{(q)}=\left[e^{j2\pi
f_{mn}^{(1)}qT_{\rm{PRI}}},\cdots,e^{j2\pi
f_{mn}^{(K)}qT_{\rm{PRI}}}\right]^{T}\in\mathbb{C}^{K}$, and
$\boldsymbol{\Lambda}_{mn}=\operatorname{diag}\\{[\beta_{mn}^{(1)},\cdots,\beta_{mn}^{(K)}]\\}$
is a $K\times K$ diagonal matrix.

Stacking the data $\\{\mathbf{y}_{mn}^{(q)}\\}_{q=0}^{Q-1}$ of all the $Q$
pulses into a $L_{mn}\times Q$ matrix, we have

$\mathbf{Y}_{mn}=\mathbf{X}_{mn}+\mathbf{W}_{mn},$ (7)

where
$\mathbf{X}_{mn}=\mathbf{A}_{mn}\boldsymbol{\Lambda}_{mn}\mathbf{B}_{mn}$ with
$\mathbf{B}_{mn}=[\mathbf{b}_{mn}^{(0)},\cdots,\mathbf{b}_{mn}^{(Q-1)}]$.
Since $\mathbf{X}_{mn}$ is determined by the unknown target parameters, we
refer to it as the target information matrix (TIM).

Finally, the $n$th receive antenna will send the data set
$\\{\mathbf{Y}_{mn}\\}_{m=1}^{M_{t}}$ to the fusion center for target
parameter estimation. Thus, there are totally $M_{t}M_{r}$ raw data matrices
being sent, i.e., the amount of data is $M_{t}M_{r}L_{mn}Q\tilde{b}$ if each
data is quantized into $\tilde{b}$ bits. In classic distributed MIMO radar, to
avoid the effect of quantization distortion, the typical value of $\tilde{b}$
is $10\sim 16$, leading to a large amount of communication traffic between the
receive antennas and the fusion center. However, in the LiQuiD-MIMO radar, we
propose to use the low-resolution ADCs rather than the high-resolution ADCs
such that the system complexity and the data volume that has to be sent to the
fusion center will be largely reduced.

Let
$\mathcal{Q}_{C}^{\gamma,b}(\cdot)=\mathcal{Q}^{\gamma,b}(\Re\\{\cdot\\})+j\mathcal{Q}^{\gamma,b}(\Im\\{\cdot\\})$
be an element-wise complex-valued quantization operator with
$\mathcal{Q}^{\gamma,b}(\cdot)$ given as:

$\mathcal{Q}^{\gamma,b}(x)=\left\\{\begin{array}[]{ll}-\gamma+\frac{2\gamma}{b}(l+\frac{1}{2}),&\begin{aligned}
&x-l\frac{2\gamma}{b}+\gamma\in[0,\frac{2\gamma}{b}],\\\
&l\in\\{0,1,\cdots,b-1\\}\end{aligned}\\\
\rm{sign}(x)(\gamma-\frac{\gamma}{b}),&\lvert
x\rvert>\gamma,\end{array}\right.$ (8)

where $\gamma$ is the dynamic range of the quantizer, $b$ is the number of
quantization levels, i.e., $\tilde{b}=\log_{2}b$. In practice, $\gamma$ is
appropriately chosen to avoid inducing additional distortion due to saturation
[26]. With low-resolution ADCs, the value $\tilde{b}$ is much smaller than
that with high-resolution ADCs. An extreme case is $\tilde{b}=1$, i.e., 1-bit
quantization.

Then, in the LiQuiD-MIMO radar, the $\tilde{b}$-bit quantized data is sent to
the fusion center. By considering the induced error during data transmission,
the received data at the fusion center can be written as

$\mathbf{Z}_{mn}=Q^{\gamma,b}_{C}(\mathbf{Y}_{mn})+\tilde{\mathbf{T}}_{mn}=Q^{\gamma,b}_{C}(\mathbf{X}_{mn}+\mathbf{W}_{mn})+\tilde{\mathbf{T}}_{mn},$
(9)

where $\tilde{\mathbf{T}}_{mn}$ models the data transmission error (DTE).

##  III Quantized Robust PCA (QRPCA) for LiQuiD-MIMO Radar

In this section, we first formulate the problem of recovering the target
information matrix (TIM) as a quantized robust PCA (QRPCA) problem, then
develop an accelerated proximal gradient (APG)-based algorithm to solve the
formulated QRPCA problem.

###  III-A QRPCA Problem Formulation

According to (7), the TIM matrix $\mathbf{X}_{mn}$ can be decomposed into
$\mathbf{X}_{mn}=\mathbf{A}_{mn}\boldsymbol{\Lambda}_{mn}\mathbf{B}_{mn}$. It
has been shown in [27] that $\textbf{X}_{mn}$ is of low rank, and its rank
depends on the number of targets with different distances or different
velocities. In addition, the DTE $\tilde{\mathbf{T}}_{mn}$ is generally sparse
since the bit error rate (BER) is generally quite low. In addition, the
received data $\mathbf{Z}_{mn}$ can be equivalently represented as

$\mathbf{Z}_{mn}=Q^{\gamma,b}_{C}(\mathbf{X}_{mn}+\mathbf{T}_{mn}+\mathbf{W}_{mn}),$
(10)

where $\mathbf{T}_{mn}$ is an equivalent sparse DTE before quantization.

For ease of notation, we will omit the subscript $mn$ in the remainder of this
section. By exploiting the classic RPCA formulation, we can simultaneously
recover the low-rank matrix $\mathbf{X}$ and the sparse matrix $\mathbf{T}$ by
solving the following quantized version of RPCA problem, referred to as the
QRPCA problem:

$\min_{\mathbf{X},\mathbf{T}}\quad\frac{1}{2}D(\mathbf{Z},\mathbf{X}+\mathbf{T})+\mu\|\mathbf{X}\|_{\ast}+\lambda\|\mathbf{T}\|_{1},$
(11)

where $D(\cdot,\cdot)$ is some similarity metric which measures the similarity
between the quantized data and the unquantized data, and $\mu$ and $\lambda$
denote the regularized parameters.

According to the definition of quantization operator in (8), it can be derived
that the quantization data $\mathbf{Z}$ determines the upper bound and the
lower bound of the unquantization data $\mathbf{Y}$, i.e.,

$\begin{split}-\frac{\Delta}{2}\leq\Re\\{\mathbf{Y}-\mathbf{Z}\\}\leq\frac{\Delta}{2}\\\
-\frac{\Delta}{2}\leq\Im\\{\mathbf{Y}-\mathbf{Z}\\}\leq\frac{\Delta}{2}\end{split}$
(12)

where $\Delta=\frac{2\gamma}{b}$ denotes the quantization interval. By
considering the unknown noise distortion on the quantized data, we define the
similarity metric function $D(\cdot,\cdot)$ in (11) as follows:

$\begin{split}&D(\mathbf{Z},\mathbf{X}+\mathbf{T})=\\\
&\left\|\rho\left(\left[\Re\\{\mathbf{X}+\mathbf{T}-\mathbf{Z}\\}+\frac{\Delta}{2};\Im\\{\mathbf{X}+\mathbf{T}-\textbf{Z}\\}+\frac{\Delta}{2}\right]\right)\right\|_{F}^{2}\\\
+&\left\|\rho\left(\left[\Re\\{\mathbf{Z}-\mathbf{X}-\mathbf{T}\\}+\frac{\Delta}{2};\Im\\{\mathbf{Z}-\mathbf{X}-\mathbf{T}\\}+\frac{\Delta}{2}\right]\right)\right\|_{F}^{2},\end{split}$
(13)

where $\rho(\cdot)$ is an element-wise function with $\rho(x)=\max\\{-x,0\\}$.

###  III-B Accelerated Proximal Gradient Algorithm for QRPCA Problem

Now we introduce an accelerated proximal gradient (APG)-based algorithm [28]
for solving the QRPCA problem (11). Since the QRPCA problem is convex, the
interior point method [29] can be used here. However, it requires high
computational complexity and cannot be applied to large-scale problems. For
the real-time implementation of target detection in radar applications, we
propose the computationally efficient APG-QRPCA algorithm.

According to the idea of APG [28], we reformulate the objective function in
(11) as

$\min_{\mathbf{X},\mathbf{T}}\quad
h(\mathbf{X},\mathbf{T})+g(\mathbf{X},\mathbf{T}),$ (14)

where
$h(\mathbf{X},\mathbf{T})\triangleq\mu\|\mathbf{X}\|_{\ast}+\lambda\|\mathbf{T}\|_{1}$
and
$g(\mathbf{X},\mathbf{T})\triangleq\frac{1}{2}D(\mathbf{Z},\mathbf{X}+\mathbf{T})$.
During each of the iterations, the algorithm first updates the matrices
$\mathbf{X}_{l}$ and $\mathbf{T}_{l}$ to $\bar{\mathbf{X}}_{l}$ and
$\bar{\mathbf{T}}_{l}$, respectively, by carrying some momentum from previous
iterations, and then performs the following computation:

$\mathbf{X}_{p}=\bar{\mathbf{X}}_{l}-\delta\nabla_{\mathbf{X}}g(\bar{\mathbf{X}}_{l},\bar{\mathbf{T}}_{l}),$
(15)
$\mathbf{T}_{p}=\bar{\mathbf{T}}_{l}-\delta\nabla_{\mathbf{T}}g(\bar{\mathbf{X}}_{l},\bar{\mathbf{T}}_{l}),$
(16)
$(\mathbf{X}_{l+1},\mathbf{T}_{l+1})=\text{prox}_{h,\delta}(\mathbf{X}_{p},\mathbf{T}_{p}),$
(17)

where $\delta$ is the step size, $\nabla_{\mathbf{X}}g(\mathbf{X},\mathbf{T})$
and $\nabla_{\mathbf{T}}g(\mathbf{X},\mathbf{T})$, respectively, denote the
gradients of $g(\mathbf{X},\mathbf{T})$ with respect to $\mathbf{X}$ and
$\mathbf{T}$, i.e.,

$\begin{split}&\nabla_{\mathbf{X}}g(\mathbf{X},\mathbf{T})=\nabla_{\mathbf{T}}g(\mathbf{X},\mathbf{T})\\\
=&\left[\rho\left(\Re(\mathbf{Z}-\mathbf{X}-\mathbf{T})+\frac{\Delta}{2}\right)-\rho\left(\Re(\mathbf{X}+\mathbf{T}-\mathbf{Z})+\frac{\Delta}{2}\right)\right]\\\
+j&\left[\rho\left(\Im(\mathbf{Z}-\mathbf{X}-\mathbf{T})+\frac{\Delta}{2}\right)-\rho\left(\Im(\mathbf{X}+\mathbf{T}-\mathbf{Z})+\frac{\Delta}{2}\right)\right].\end{split}$
(18)

The function $\text{prox}_{h,\delta}(\mathbf{X}_{p},\mathbf{T}_{p})$ in (17)
denotes the proximal mapping which solves the following problem:

$\begin{split}&\text{prox}_{h,\delta}(\mathbf{X}_{p},\mathbf{T}_{p})=\\\
&\arg\min_{\mathbf{X},\mathbf{T}}\frac{1}{2\delta}\|\mathbf{X}+\mathbf{T}-\mathbf{X}_{p}-\mathbf{T}_{p}\|_{F}^{2}+h(\mathbf{X},\mathbf{T}).\end{split}$
(19)

By solving the problem (19), the proximal mapping can be split into the
following two subproblems, which have closed-form expressions:

$\mathbf{X}_{l+1}=\arg\min_{\mathbf{X}}\left\\{\mu\|\mathbf{X}\|_{*}+\frac{1}{2\delta}\|\mathbf{X}-\mathbf{X}_{p}\|^{2}_{F}\right\\}=\mathbf{U}_{p}\mathcal{S}_{\mu\delta}(\boldsymbol{\Sigma}_{p})\mathbf{V}_{p}^{T}$
(20)
$\textbf{T}_{l+1}=\arg\min_{\mathbf{T}}\left\\{\lambda\|\mathbf{T}\|_{1}+\frac{1}{2\delta}\|\mathbf{T}-\mathbf{T}_{p}\|^{2}_{F}\right\\}=\mathcal{S}_{\lambda\delta}(\mathbf{T}_{p})$
(21)

where $\mathbf{X}_{p}=\mathbf{U}_{p}\boldsymbol{\Sigma}_{p}\mathbf{V}_{p}^{T}$
is the singular value decomposition of $\mathbf{X}_{p}$, and
$\mathcal{S}_{\varepsilon}(\cdot)$ is the element-wise soft-thresholding
operator:

$\mathcal{S}_{\varepsilon}(x)=\left\\{\begin{aligned}
x-\varepsilon,\quad&x>\varepsilon\\\ x+\varepsilon,\quad&x<-\varepsilon\\\
0,\quad&\rm others.\end{aligned}\right.$ (22)

Finally, we can obtain the estimates of the TIM matrix $\mathbf{X}$ and the
DTE matrix $\mathbf{T}$, denoted as $\hat{\mathbf{X}}$ and $\hat{\mathbf{T}}$,
when the iterations are ended.

The details of the proposed APG-QRPCA algorithm are given in Algorithm 1.

Algorithm 1 APG-QRPCA Algorithm

0: $\mathbf{Z}$, $\mu,\lambda,\delta,\text{MaxIter}$

1: Initialization:
$\mathbf{X}_{1}=\mathbf{X}_{0}=0;\mathbf{T}_{1}=\mathbf{T}_{0}=0;\zeta_{0}=1;l=1$

2: while $l\leq\text{MaxIter}$ or not convergence do

3: $\zeta_{l}=\frac{1+\sqrt{4\zeta_{l-1}^{2}+1}}{2}$;

4:
$\bar{\mathbf{X}}_{l}=\mathbf{X}_{l}+\frac{\zeta_{l-1}-1}{\zeta_{l}}(\mathbf{X}_{l}-\mathbf{X}_{l-1}),\bar{\mathbf{T}}_{l}=\mathbf{T}_{l}+\frac{\zeta_{l-1}-1}{\zeta_{l}}(\mathbf{T}_{l}-\mathbf{T}_{l-1})$;

5:
$\mathbf{X}_{p}=\bar{\mathbf{X}}_{l}-\delta\nabla_{\mathbf{X}}g(\bar{\mathbf{X}}_{l},\bar{\mathbf{T}}_{l}),\mathbf{T}_{p}=\bar{\mathbf{T}}_{l}-\delta\nabla_{\mathbf{T}}g(\bar{\mathbf{X}}_{l},\bar{\mathbf{T}}_{l})$;

6:
$(\mathbf{U}_{p},\boldsymbol{\Sigma}_{p},\mathbf{V}_{p})=\text{svd}(\mathbf{X}_{p}),\mathbf{X}_{l+1}=\mathbf{U}_{p}\mathcal{S}_{\mu\delta}(\boldsymbol{\Sigma}_{p})\mathbf{V}_{p}^{T},\mathbf{T}_{l+1}=\mathcal{S}_{\lambda\delta}(\mathbf{T}_{p})$;

7: $l=l+1$

8: end while

8: $\hat{\mathbf{X}}=\mathbf{X}_{l}$, $\hat{\mathbf{T}}=\mathbf{T}_{l}$

##  IV Least Square (LS)-based Target Parameter Estimation

By using the proposed APG-QRPCA algorithm, the $M_{t}M_{r}$ TIM matrices
$\\{\mathbf{X}_{mn},m=1,\cdots,M_{t},n=1,\cdots,M_{r}\\}$ can be recovered at
the fusion center, which can then be used to estimate the unknown target
parameters $\\{\mathbf{p}^{(k)},\mathbf{v}^{(k)}\\}_{k=1}^{K}$. There are a
variety of classical signal processing techniques can be used here, including
least squares (LS) [30], maximum likelihood (ML)[31], as well as sparse
reconstruction methods[32]. In the following, we introduce a sequential LS
method to sequentially estimate the position and velocity parameters.

Let
$\boldsymbol{\theta}=\\{\boldsymbol{\theta}_{p},\boldsymbol{\theta}_{v}\\}$ be
the unknown target parameters, where
$\boldsymbol{\theta}_{p}=\\{\mathbf{p}^{(k)}\\}_{k=1}^{K}$ and
$\boldsymbol{\theta}_{v}=\\{\mathbf{v}^{(k)}\\}_{k=1}^{K}$ denote the position
parameters and velocity parameters, respectively. According to (7),
$\boldsymbol{\theta}_{p}$ and $\boldsymbol{\theta}_{v}$ are implicitly
determined by the matrices $\mathbf{A}_{mn}$ and $\mathbf{B}_{mn}$,
respectively. For the sake of convenience, we use the notations
$\mathbf{A}_{mn}(\boldsymbol{\theta}_{p})$ and
$\mathbf{B}_{mn}(\boldsymbol{\theta}_{v})$ in the following discussion. The
target parameter estimation problem can be formulated as the following LS
problem:

$\hat{\boldsymbol{\theta}}=\arg\min_{\boldsymbol{\theta}}\sum_{m=1}^{M_{t}}\sum_{n=1}^{M_{r}}\|\hat{\mathbf{X}}_{mn}-\mathbf{A}_{mn}(\boldsymbol{\theta}_{p})\boldsymbol{\Lambda}_{mn}\mathbf{B}_{mn}(\boldsymbol{\theta}_{v})\|_{2}^{2}.$
(23)

which is a computationally demanding problem, requiring searching all possible
parameters over high-dimensional space. Therefore, we separate (23) into two
sequential subproblems.

Since the LS solution of the matrix
$\boldsymbol{\Lambda}_{mn}\mathbf{B}_{mn}(\boldsymbol{\theta}_{v})$ is
$(\mathbf{A}_{mn}^{H}(\boldsymbol{\theta}_{p})\mathbf{A}_{mn}(\boldsymbol{\theta}_{p}))^{-1}\mathbf{A}_{mn}^{H}(\boldsymbol{\theta}_{p})\hat{\mathbf{X}}_{mn}$,
we can formulate the following position estimation problem:

$\hat{\boldsymbol{\theta}}_{p}=\arg\min_{\boldsymbol{\theta}_{p}}\sum_{m=1}^{M_{t}}\sum_{n=1}^{M_{r}}\|\mathbf{P}_{p,mn}^{\perp}(\boldsymbol{\theta}_{p})\hat{\mathbf{X}}_{mn}\|_{2}^{2},$
(24)

where
$\mathbf{P}_{p,mn}^{\perp}(\boldsymbol{\theta}_{p})=\mathbf{I}-\mathbf{A}_{mn}(\boldsymbol{\theta}_{p})(\mathbf{A}_{mn}^{H}(\boldsymbol{\theta}_{p})\mathbf{A}_{mn}(\boldsymbol{\theta}_{p}))^{-1}\mathbf{A}_{mn}^{H}(\boldsymbol{\theta}_{p})$.
The above problem can be solved by searching over the two-dimensional plane
where the targets are located.

Once the position parameters are estimated, we also get an estimate of the
matrix $\boldsymbol{\Lambda}_{mn}\mathbf{B}_{mn}(\boldsymbol{\theta}_{v})$,
denoted as $\hat{\boldsymbol{\Xi}}_{mn}$. Then, by performing FFT over each
row of $\hat{\boldsymbol{\Xi}}_{mn}$, the Doppler frequency $f_{mn}^{(k)}$ of
each target corresponding to each pair of Tx-Rx antennas can be estimated,
denoted as $\hat{f}_{mn}^{(k)}$. Then the velocity parameters can then be
estimated by solving the following LS problem

$\hat{\boldsymbol{\theta}}_{v}=\arg\min_{\boldsymbol{\theta}_{v}}\sum_{m=1}^{M_{t}}\sum_{n=1}^{M_{r}}\sum_{k=1}^{K}(\hat{f}_{mn}^{(k)}-f_{mn}^{(k)}(\mathbf{v}_{k}))^{2},$
(25)

where the function $f_{mn}^{(k)}(\mathbf{v}_{k})$ is given by (2).

Figure 1: Performance of recovering the low-rank matrix $\mathbf{X}_{mn}$
versus different SNRs.

##  V Numerical Results

Now we evaluate the performance of our proposed algorithm for the LiQuiD-MIMO
radar system. Throughout the simulations, we consider a MIMO radar equipped
with $M_{t}=3$ transmit antennas and $M_{r}=10$ receive antennas uniformly
distributed on the concentric circles with radius $5$km and $3$km,
respectively. The reference carrier frequency parameters $f_{0}=5$GHz and the
frequency increment $\Delta f=50$MHz. One CPI consists of $Q=128$ pulses with
$T_{\rm{PRI}}=0.5$ms and $T_{p}=6.4\mu\rm{s}$. The Hadamard sequence is used
as the baseband waveform with $B_{0}=10$MHz. At the fusion center, $1\%$ of
the received data is assumed to be received incorrectly, leading to a sparse
DTE matrix. We consider the scenario that one target is located at
$\mathbf{p}^{(1)}=[1100,1100]^{T}$m with $\mathbf{v}^{(1)}=[10,10]^{T}$ m/s.
The performance of the classic RPCA algorithm without considering the effect
of quantization is also demonstrated as a benchmark, denoted as “No Quan.”
method.

Fig.1 and Fig.2 depict the relative errors in recovering the low-rank matrix
$\mathbf{X}_{mn}$ and the sparse matrix $\mathbf{T}_{mn}$, respectively, with
respect to different SNR values, by relying on the proposed APG-QRPCA
algorithm. It is shown that it is possible to simultaneously recover the
matrices $\mathbf{X}_{mn}$ and $\mathbf{T}_{mn}$ from the low-bit quantized
data. When the SNR is less than $20$dB, the performance of 6-bit quantization
is very close to that without quantization, proving the effectiveness of low-
bit quantization. However, there exists a performance gap between the low-bit
quantization and that without quantization when SNR is higher than $20$dB.

Figure 2: Performance of recovering the sparse matrix $\mathbf{T}_{mn}$ versus
different SNRs.

(a) Position estimation

(b) Velocity estimation.

Figure 3: The results of the target parameter estimation for a target at
$\mathbf{p}=[1100,1100]^{T}$m with velocity $\mathbf{v}=[10,10]^{T}$m/s, by
using $4$-bit quantization at $\text{SNR}=20$dB.

After recovering the set of matrices
$\\{\mathbf{X}_{mn},m=1,\cdots,M_{t},n=1,\cdots,M_{r}\\}$, we now show the
performance of target parameter estimation using the LS-based method proposed
in Section IV. Fig.3 shows the results of position estimation and velocity
estimation at $\text{SNR}=20$dB by solving the problems (24) and (25),
respectively. It can be seen that the position and velocity of the target can
be accurately estimated by using the LS-based method.

##  VI Conclusion

In this work we proposed a low-bit quantized distributed MIMO radar system,
which significantly reduces the system complexity of the widely-separated
receivers and the data transmission volume between the receivers and the
fusion center. The main challenge in such LiQuiD-MIMO radar is to deal with
the quantization distortion induced by low-resolution ADCs. By exploiting the
low-rank structure of the target information matrix and the sparsity of the
data transmission error, we formulated a quantized version of RPCA problem,
referred to as QRPCA problem, and developed an APG-based algorithm, refered to
as APG-QRPCA algorithm, to simultaneously recover the infinite-precision
target information matrix and the errors during data transmission. Simulation
results demonstrate that it is feasible to implement a distributed MIMO radar
system by employing low-resolution ADCs. The analysis of the performance bound
of the proposed LiQuiD-MIMO radar will be our future work.

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#  Letter Stabilization with Prescribed Instant
via Lyapunov Method

Jiyuan Kuang, Yabin Gao, Yizhuo Sun, Jiahui Wang, Aohua Liu, Yue Zhao,
Jianxing Liu Jiyuan Kuang, Yabin Gao, Yizhuo Sun, Aohua Liu, and Jianxing Liu
are with the department of Control Science and Engineering, Harbin Institute
of Technology, Harbin 150001, China. (Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>jx.liu@hit.edu.cn). Jiahui Wang is with the College of
Intelligent System Science and Engineering, Harbin Engineering University,
Harbin 150001, China. (Email: jiahuiwang@hebut.edu.cn). Corresponding author:
Jianxing Liu.

Dear Editor,

This letter investigates the prescribed-instant stabilization problem for
high-order integrator systems. In anothor word, the settling time under the
presented controller is independent of the initial conditions and equals the
prescribed time instant. The controller is designed with the concept of
backstepping. A strict proof based on the Lyapunov method is presented to
clamp the settling time to the prescribed time instant from both the left and
right sides. This proof serves as an example to present a general framework to
verify the designed stabilization property. It should be emphasized that the
prescribed-time stability (PSTS) [1] can only prescribe the upper bound of the
settling time and is different from this work. The detailed argumentation will
be presented after a brief review of the existing important research.

Traditional asymptotic stability ensures the system states converge to
equilibrium as time goes to infinity. Since the system states actually never
reach an equilibrium, the separation principle must be rigorously
substantiated [2]. Finite-time stability (FNTS) guarantees that states
convergence happens in a finite time but mostly depending on parameters and
initial conditions [3]. By using a finite time differentiator or observer, the
correct information can be estimated after a finite time [4]. This makes it
easier to get the closed-loop system stability. However, this finite time
increases as the initial values of system states increase, and there are no
uniform bounds. To solve this problem, one way is to estimate the settling
time in some frequently used finite-time stabilization algorithms, such as
super-twisting algorithms [5]. Another way is developing some new algorithms
that can ensure a uniform bound of the settling time. The fixed-time stability
(FXTS) guarantees the settling time to be bounded by a constant, which however
is not explicit and is determined by the controller parameters [6, 7]. So, it
is complex to calculate every parameter according to the desired bound of
settling time [8]. Moreover, the settling time in FXTS is very conservative.

The prescribed-time stability (PSTS) ensures the system states converge to
zero in a prescribed time $T_{p}$, where $T_{p}$ is an explicit parameter of
the controller [9]. Some results of PSTS even show a pre-specified settling
time (in simulations at least) [10, 11]. However, their corresponding
theoretical analysis cannot explain this fact, except for some first-order
systems. A detailed analysis can be seen in Remark 1 and 2.

Up to now, only a few works with strict proofs forced the settling time to an
arbitrarily selected time instant. For example, the work in [12] ensured this
property by using a novel sliding mode control. The corresponding proof was
demonstrated through a detailed analysis of the infinitesimal order of each
state. The work in [13] designed a controller based on the backstepping
method. Reduction to absurdity was utilized to verify the exact settling time.
However, these methods of proof are circumscribed and can not be generalized
easily.

This letter considers $n$-order integrator systems, of which the settling time
under the presented controller is exactly the prescribed time instant. A
corresponding proof based on the Lyapunov method provides a general framework
to verify the exact settling time. Moreover, this framework can also help to
decrease potential conservativeness in the settling time of traditional PSTS.

Problem Statement: Consider the following system

$\dot{x}=g(x,u),$

where $g:\mathbb{R}^{n}\times\mathbb{R}\rightarrow\mathbb{R}^{n}$,
$x\in\mathbb{R}^{n}$ denotes the states, and $u\in\mathbb{R}$ is the control
variable. Consider the control variable as $u(x,t,\eta)$, where
$\eta\in\mathbb{R}^{m}$ denotes the parameters. We can obtain the closed-loop
system in (1) with initial value $x(t_{0})=x_{0}$. The initial time $t_{0}=0$
is default in this letter.

$\dot{x}=f(x,t,\eta):=g(x,u(x,t,\eta)).$ (1)

###### Definition 1

[13] If for any physically possible positive number $T_{p}$, there exists
parameters $\eta$ such that the system settling time $T(x_{0})$ can be
prescribed as $T(x_{0})=T_{p},\forall x_{0}\in\mathbb{R}^{n}$. Then, the
origin of the system (1) is said to be prescribed-instant stable (PSIS).

Consider the following linear system:

$\left\\{\begin{array}[]{lr}\dot{x}_{i}=x_{i+1},i=1,...,n-1,&\\\
\dot{x}_{n}=u.\end{array}\right.$ (2)

Since the PSIS has already been defined and proved in [13], the main
contribution of this letter is presenting a Lyapunov method to verify the
controller can ensure the system in (2) is PSIS.

The expressions of Theorem 2 in [1] and Theorem 1 in [11] may mislead the
readers to think that the PSTS also ensures $T(x_{0})=T_{p}$. To clear the
air, it is urgent to emphasize the following fact.

###### Remark 1

The proof in [1] is one of the main thought of proof of PSTS. The key step is
to obtain the following inequality:

$\frac{\text{d}V(t)}{\text{d}t}\leq-2k\mu(t)V(t),\;k>0,\;t\in[0,T_{p}),$ (3)

where $V$ is a Lyapunov function of the controlled system and

$\mu(t)=\frac{T_{p}^{m+n}}{(T_{p}-t)^{m+n}},\;t\in[0,T_{p}).$ (4)

If the formula in (3) is equality, there is no doubt that the settling time
equals the prescribed time $T_{p}$. However, for a high-order system, it is
difficult to obtain equality of (3). As a result, $V(t)$ is reset to zero in
the prescribed time _no longer than_ $T_{p}$ irrespectively of the initial
value $V(t_{0})\in\mathbb{R}_{\geq 0}$ (Section 2 in [14]). We have
$T(x_{0})\leq T_{p}$. A similar problem also exists in [10], whose equation
(27) in Theorem 1 is also an inequality.

###### Remark 2

Another thought of proof of PSTS in some research such as [11] is based on
some time scale transformation from $t\in[0,T_{p})$ to $\tau\in[0,+\infty)$.
The most familiar transformation is

$\left\\{\begin{array}[]{lr}\tau=-\text{ln}(\frac{T_{p}-t}{T_{p}}),&\\\
t=T_{p}(1-\text{e}^{-\tau}).\end{array}\right.$ (5)

Since $\frac{\text{d}\tau}{\text{d}t}=\frac{1}{T_{p}-t}$, we have two
functions equivalent to each other:

$\begin{cases}\frac{\text{d}V(t)}{\text{d}t}=-\frac{1}{T_{p}-t}V(t),,&t\in[0,T_{p});\\\
\frac{\text{d}V(t(\tau))}{\text{d}\tau}=\frac{\text{d}V(t)}{\text{d}t}\frac{\text{d}t}{\text{d}\tau}=-V(t(\tau)),&\tau\in[0,+\infty).\end{cases}$
(6)

If one can prove the Lyapunov function $V(t(\tau))$ converges exponentially,
or $V(t(\tau))\rightarrow 0$ as $\tau\rightarrow+\infty$, the conclusion is
definitely obtained that $V(t)\rightarrow 0$ just as $t\rightarrow T_{p}$.
However, Lemma 2 in [11] only shows
$\frac{\text{d}V(t(\tau))}{\text{d}\tau}\leq-V(t(\tau))$. There stands a
chance that
$\frac{\text{d}V(t(\tau))}{\text{d}\tau}\leq-V^{1.5}(t(\tau))-V^{0.5}(t(\tau))$.
As a result, $V(t(\tau))=0,\forall\tau\geq\pi$. Since $V(t(\tau))$ converges
to zero before $\tau\rightarrow+\infty$, $V(t)$ converges to zero before
$t=T_{p}$, i.e., $T(x_{0})\leq T_{p}$.

Although some simulations of PSTS have obtained $T(x_{0})=T_{p}$, we have
clearly shown that the corresponding proofs of the PSTS are not sufficient to
obtain this result.

###### Remark 3

The work in [15] defined the properties of PSTS (PSIS) as free-will weak
(strong) arbitrary time stability, correspondingly. It recognized that any
single inequality of derivative from the Lyapunov function could not obtain
PSIS directly. However, the free-will strong arbitrary time stability, which
is consistent with the presented PSIS, can be established as long as

$\frac{\text{d}V}{\text{d}t}=-\frac{k(1-\text{e}^{-V})}{(T_{p}-t)},\;k>1,\;t\in[0,T_{p}).$
(7)

Although the proof of free-will strong arbitrary time stability for high-order
systems remains open, yet this proof can be completed once the following
inequalities are considered,

$\begin{cases}\dot{V}_{1}\leq\frac{-k_{1}(1-\text{e}^{-V_{1}})}{(T_{p}-t)},&k_{1}>1,t\in[0,T_{p});\\\
\dot{V}_{2}\geq\frac{-k_{2}(1-\text{e}^{-V_{2}})}{(T_{p}-t)},&k_{2}>k_{1},t\in[0,T_{p}).\end{cases}$

Specifically, one can obtain the PSIS by limiting the settling time from both
the left and right sides of Lyapunov function derivative formula. This is the
main thought of the PSIS presented in this letter.

To realize the above-mentioned thought, one can find a differential function
whose solution converges to zero just at the prescribed instant $T_{p}$.
Definition 5 of [13] presented a series of such functions named reference
convergence differential functions (RCDFs).

###### Example 1

Let us see some typical RCDFs ($\psi$) given in [13]:

$\begin{cases}\dot{v}_{1}=-\psi_{v_{1}}=-\frac{\eta(v_{1}^{2}+1)\text{arctan}(v_{1})}{T_{p}-t},&v_{1}(t)=\text{tan}(T_{p}-t)^{\eta};\\\
\dot{v}_{2}=-\psi_{v_{2}}=-\frac{\eta
v_{2}}{T_{p}-t},&v_{2}(t)=(T_{p}-t)^{\eta};\\\
\dot{v}_{3}=-\psi_{v_{3}}=-\frac{\eta(1-\text{e}^{-|v_{3}|})}{T_{p}-t}\text{sign}(v_{3}),&v_{3}(t)=\text{ln}(1+(T_{p}-t)^{\eta}).\end{cases}$

It is noted that $\psi(v,t,T_{p},\eta)$ can be written as
$\frac{\eta\zeta(v)}{T_{p}-t}$, and $\zeta(v)$ has the same sign as $v$.
Moreover, $\zeta(v)=O(v)$ (infinitesimal of the same order), and

$\lim\limits_{t\to T_{p}}\psi(v,t,T_{p},\eta)=\lim\limits_{t\to
T_{p}}\frac{v(t)-0}{t-T_{p}}\sim(T_{p}-t)^{\eta-1},\forall\eta>1.$

In addition, Claims and Lemmas in [13] provide other RCDFs may help to promote
the proof of PSTS in existing research to obtain PSIS.

Main Results:

Controller Design: The controller is designed with backstepping method and is
presented as a recursive form. The desired value of $x_{1}$ is $x_{1,d}=c$,
where $c$ is a constant. The recurrence relation ($i\geq 2$) is a little
different from that in [13],

$x_{i+1,d}=\dot{x}_{i,d}-z_{i-1}-\psi_{i},$ (8)

where $z_{i}=x_{i}-x_{i,d}$ and $x_{2,d}=-\psi_{1}$. It is noted that
$\psi_{i}(z_{i},t,T_{p},\eta_{i})$ belongs to the same RCDF. To prevent the
singularity problem of the control signal at $t=T_{p}$, $\eta_{i}$ in
$\psi_{i}$ is designed to satisfy $\eta_{i}>n+1-i,i=1,2,...,n$. For the system
(2),

$u=\begin{cases}x_{n+1,d},&{0\leq t<T_{p}},\\\ 0,&{T_{p}\leq t}.\end{cases}$
(9)

So, the dynamics of each state’s tracking error when $t\in[0,T_{p})$ is:

$\left\\{\begin{array}[]{lr}\dot{z}_{1}=z_{2}-\psi_{1},\\\
\dot{z}_{i}=z_{i+1}-z_{i-1}-\psi_{i},i=2,3,...,n-1,\\\
\dot{z}_{n}=-z_{n-1}-\psi_{n}.\end{array}\right.$ (10)

In the following, we will present the PSIS of the transformed system (10), and
then obtain the PSIS of the system (2).

Prescribed-instant Stability:

###### Lemma 1

Suppose the function $h(x):\mathbb{R}\rightarrow\mathbb{R}$ is concave on the
interval $I$ and $\lambda_{1},\lambda_{2},...,\lambda_{n}$ satisfy
$\sum_{i=1}^{n}\lambda_{i}=1$. Then,

$\sum_{i=1}^{n}\lambda_{i}h(x_{i})\leq
h(\sum_{i=1}^{n}\lambda_{i}x_{i}),\;\forall x_{i}\in I.$ (11)

Especially, if $\lambda_{1}=\lambda_{2}=...=\lambda_{n}=\frac{1}{n}$, we have

$\frac{\sum_{i=1}^{n}h(x_{i})}{n}\leq
h(\frac{\sum_{i=1}^{n}x_{i}}{n}),\;\forall x_{i}\in I.$ (12)

This lemma is the so-called Jenson inequality. A geometric proof is given in
the following.

_P_ roof: Suppose $x_{1}\leq x_{2}\leq...\leq x_{n}$, connecting
$(x_{1},h(x_{1}))$, $(x_{2},h(x_{2})),...,(x_{n},h(x_{n}))$ in turn can form a
convex polygon. As shown in Fig. 1, the point
$(\sum_{i=1}^{n}\lambda_{i}x_{i},\sum_{i=1}^{n}\lambda_{i}h(x_{i}))$ is a
convex combination of the convex polygon vertex. It’s vertical coordinate is
definitly smaller than $h(\sum_{i=1}^{n}\lambda_{i}x_{i})$, i.e.,
$\sum_{i=1}^{n}\lambda_{i}h(x_{i})\leq h(\sum_{i=1}^{n}\lambda_{i}x_{i})$.
This completes the proof.

Figure 1: Abridged general view of the proof for Lemma 1.

###### Theorem 1

The origin of the system (2) under the controller (9) is PSIS with prescribed
time instant $T_{p}$. The control signal $u(t)$ converges to zero at
$t=T_{p}$, and $x(t)=u(t)=0,\;\forall t\geq T_{p}$.

_P_ roof: As long as the controller is designed as the equation (9), the
dynamics of $z_{i}$ is given by the equation (10). Choosing the Lyapunov
function as $V_{n}=\sum_{i=1}^{n}z_{i}^{2}$. According to equation (10),

$\dot{V}_{n}=-\sum\nolimits_{i=1}^{n}2z_{i}\psi_{i}(z_{i},t,T_{p},\eta_{i})=-\sum\nolimits_{i=1}^{n}\frac{2\eta_{i}|z_{i}|\zeta(|z_{i}|)}{T_{p}-t}.$
(13)

Denote the vector ${\bf z}=(z_{1},z_{2},...,z_{n})$. According to Lemma 1, the
time derivative of $V_{n}$ satisfies,

$\begin{split}\dot{V}_{n}&\leq-\frac{2n\text{min}(\eta_{1},\eta_{2},...,\eta_{n})\frac{||z||_{1}}{n}\zeta(\frac{||z||_{1}}{n})}{T_{p}-t}\\\
&\leq-\frac{2n\text{min}(\eta_{1},\eta_{2},...,\eta_{n})\frac{\sqrt{V_{n}}}{n}\zeta(\frac{\sqrt{V_{n}}}{n})}{T_{p}-t}.\end{split}$
(14)

Let $a_{1}=\frac{\sqrt{V_{n}}}{n}$ of which the derivative is

$\dot{a}_{1}=-\frac{\dot{V}_{n}}{2n\sqrt{V_{n}}}\leq-\frac{\text{min}(\eta_{1},\eta_{2},...,\eta_{n})\zeta(a_{1})}{n(T_{p}-t)},\;(V_{n}\neq
0,\forall z_{i}\neq 0).$ (15)

This means $a_{1}$ converges to zero before or at $t=T_{p}$, as well as
$V_{n}$. One can obtain that $z_{i}$ converges to zero before or at $t=T_{p}$.

Another inequality of $V_{n}$ is given by

$\begin{split}\dot{V}_{n}&\geq-\frac{2(\eta_{1}+\eta_{2}+...+\eta_{n})||z||_{\infty}\zeta(||z||_{\infty})}{T_{p}-t}\\\
&\geq-\frac{2(\eta_{1}+\eta_{2}+...+\eta_{n})\sqrt{V_{n}}\zeta(\sqrt{V_{n}})}{T_{p}-t}.\end{split}$
(16)

Let $a_{2}=\sqrt{V_{n}}$ of which the derivative is

$\dot{a}_{2}=-\frac{\dot{V}_{n}}{2\sqrt{V_{n}}}\geq-\frac{(\eta_{1}+\eta_{2}+...+\eta_{n})\zeta(a_{2})}{T_{p}-t},\;(V_{n}\neq
0,\forall z_{i}\neq 0).$ (17)

This means $a_{2}$ does not converge to zero before $t=T_{p}$, as well as
$V_{n}$. Hence, $V_{n}$ converges to zero at $t=T_{p}$. Meanwhile, $z_{1}$ to
$z_{n}$ converge to zero.

Since $z_{i}\rightarrow 0$ as $t\rightarrow T_{p}$, by combining systems (2),
(8), and (9), one can obtain the following relationship as $t\rightarrow
T_{p}$

$\left\\{\begin{array}[]{lr}x_{2}=x_{2,d}=-\psi_{1}=u_{1},&\\\
x_{3}=x_{3,d}=-\dot{\psi_{1}}-z_{1}-\psi_{2}=u_{2},&\\\
x_{4}=x_{4,d}=-\ddot{\psi_{1}}-\dot{z_{1}}-\dot{\psi_{2}}-z_{2}-\psi_{3}=u_{3},&\\\
\vdots&\\\
x_{n+1}=x_{n+1,d}=\dot{x}_{n,d}-z_{n-1}-\psi_{n}=u_{n}.\end{array}\right.$
(18)

Each equation in (18) contains $\psi_{i}$, $z_{i}$, and their derivatives, and
$\psi_{1}$ is derivatives the most times. For example, $u_{n}$ contains
$\psi_{1}^{(n-1)}$ and $\psi_{i}^{(n-i)}$. As mentioned in Example 1,
$\psi\sim(T-t)^{\eta-1}$, $\psi^{(n-1)}\sim(T_{p}-t)^{\eta-n}$ as
$t\rightarrow T_{p}$. As long as the parameters are selected as
$\eta_{i}>n+1-i$, the derivative of each $\psi_{i}$ will tend to zero as
$t\rightarrow T_{p}$. Hence, the controller (9) will tends to zero as
$t\rightarrow T_{p}$, and so do the states of the system (2). Because
$u(t)=0,\;\forall t\geq T_{p}$, we have $x(t)=0,\;\forall t\geq T_{p}$.
Therefore, the system (2) is PSIS with the prescribed time instant $t=T_{p}$.
This completes the proof.

Numerical example: Consider a simple pendulum system:

$\left\\{\begin{array}[]{lr}\dot{x}_{1}=x_{2},&\\\
\dot{x}_{2}=-\frac{g}{l}\text{sin}x_{1}-\frac{k}{m}x_{2}+\frac{1}{ml^{2}}T,\end{array}\right.$

where $x_{1}$ denotes the angle, and $T$ is the torque. Moreover,
$l=0.5\;\text{m}$, $m=0.1\;\text{kg}$, $\text{g}=9.81\;\text{m/s}^{2}$, and
$k=0.01$. We set the initial values as $x_{1}(0)=0.09\,\text{rad}$ and
$x_{2}(0)=0.1\,\text{rad}/\text{s}$. The control objective is making
$x_{1}=x_{1,d}=0.15\;\text{rad}$ at $T_{p}=0.5\;\text{s}$, i.e.,
$z_{1}=x_{1}-0.15$. Let
$T=ml^{2}(\frac{g}{l}\text{sin}x_{1}+\frac{k}{m}x_{2}+u)$. Then,
$\dot{x}_{1}=x_{2},\dot{x}_{2}=u.$

Figure 2: Trajectories of (a) $x_{1}$ (angle), and $x_{2}$ (angular velocity);
(b) $u$ (equivalent control), and $T$ (torque applied to the pendulum).

Choosing the RCDFs as:

$\psi_{1}=\frac{\eta_{1}z_{1}}{T_{p}-t},\eta_{1}=3;\;\psi_{2}=\frac{\eta_{2}z_{2}}{T_{p}-t},\eta_{2}=2.$

According to the equations (8) and (9), the specific controller is:

$u=\begin{cases}-\frac{3x_{2}}{T_{p}-t}-\frac{3z_{1}}{(T_{p}-t)^{2}}-z_{1}-\frac{6z_{1}}{(T_{p}-t)^{2}}-\frac{2x_{2}}{T_{p}-t},&{0\leq
t<T_{p}};\\\ 0,&{T_{p}\leq t}.\end{cases}$

As presented in Fig. 2, each state is stabilized to the desired value at
$t=T_{p}=0.5\,\text{s}$. One characteristic of the PSIS is presented in Fig. 2
(b), i.e., $u$ strikes zero at $t=T_{p}=0.5\,\text{s}$.

Conclusion: This letter provides a proof framework based on the Lyapunov
method to ensure the real convergence time of a high-order integrator system
equals the prescribed time instant. Therefore, the settling time in this work
is irrelevant to the initial conditions and can be any physically feasible
assigned time instant. A simulation with a simple pendulum system has verified
the results of this approach. Extending the proposed method to get PSIS for
the systems with disturbance and saturation is a consequential topic in the
future.

Acknowledgments: This work was supported in part by the National Natural
Science Foundation of China under Grant 62022030, Grant 62033005, and Grant
62103118; in part by the China Postdoctoral Science Foundation under Grant
2021T140160 and Grant 2021M700037; in part by the Fundamental Research Funds
for the Central Universities under Grant HIT.OCEF. 2021005; and in part by the
Self-Planned Task of State Key Laboratory of Advanced Welding and Joining
(HIT).

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