# Unveiling Disparities in Web Task Handling Between Human and Web Agent

Kihoon Son<EMAIL_ADDRESS>School of Computing, KAISTDaejeonRepublic
of Korea ,  Jinhyeon Kwon<EMAIL_ADDRESS>School of Computing,
KAISTDaejeonRepublic of Korea ,  DaEun Choi<EMAIL_ADDRESS>School of
Computing, KAISTDaejeonRepublic of Korea ,  Tae Soo Kim
<EMAIL_ADDRESS>School of Computing, KAISTDaejeonRepublic of Korea ,
Young-Ho Kim<EMAIL_ADDRESS>NAVER AI LabSeongnamRepublic of Korea ,
Sangdoo Yun<EMAIL_ADDRESS>NAVER AI LabSeongnamRepublic of Korea
and  Juho Kim<EMAIL_ADDRESS>School of Computing, KAISTDaejeonRepublic
of Korea

(2024)

###### Abstract.

With the advancement of Large-Language Models (LLMs) and Large Vision-Language
Models (LVMs), agents have shown significant capabilities in various tasks,
such as data analysis, gaming, or code generation. Recently, there has been a
surge in research on web agents, capable of performing tasks within the web
environment. However, the web poses unforeseeable scenarios, challenging the
generalizability of these agents. This study investigates the disparities
between human and web agents’ performance in web tasks (e.g., information
search) by concentrating on planning, action, and reflection aspects during
task execution. We conducted a web task study with a think-aloud protocol,
revealing distinct cognitive actions and operations on websites employed by
humans. Comparative examination of existing agent structures and human
behavior with thought processes highlighted differences in knowledge updating
and ambiguity handling when performing the task. Humans demonstrated a
propensity for exploring and modifying plans based on additional information
and investigating reasons for failure. These findings offer insights into
designing planning, reflection, and information discovery modules for web
agents and designing the capturing method for implicit human knowledge in a
web task.

Web Agent; Human Behavior; Qualitative Analysis; Knowledge Capturing; Large-
Language Model; Large Vision-Language Model

††journalyear: 2024††copyright: rightsretained††conference: CHI Conference on
Human Factors in Computing Systems; May 12, 2024; Honolulu, HI, USA††ccs:
Human-centered computing Interactive systems and tools

##  1\. Introduction

As Large-Language Models (e.g., GPT-4 (OpenAI, 2023)) and Large Vision-
Language Models (LVM; GPT-4V (202, 2023)) continue to advance, research on
agents has shown that agents are capable of performing diverse tasks to
support humans as well as in diverse formats (e.g., single or multi agents
framework). For example, LLM-based agents perform tasks such as data analysis
(Hu et al., 2024), gaming (Zhu et al., 2023), and code generation (Zhu et al.,
2023). Also, recent agent work leverages a multi-agent framework with diverse
personas to support human decision-making processes (Wu et al., 2023; Park et
al., 2023). These LLM-based agents typically incorporate four fundamental
components: Profiling, Memory, Planning, and Action (Wang et al., 2023).
Agents with this architecture operate by designing plans for tasks akin to
humans, retrieving necessary information from knowledge memory, and
subsequently executing the most appropriate actions based on the current
context (Wu et al., 2024).

Recently, with the capability of understanding graphic user interface (GUI),
web agents based on LLM and LVM have shown to be capable of performing tasks
within the web environment (Koh et al., 2024), such as shopping or finding
specific information in a social community. To successfully accomplish tasks
on the web, these agents require not only a basic understanding of graphic
user interface (GUI) but also appropriate planning on how to perform given
tasks and accurately execute an action (Cheng et al., 2024). Researchers have
shown high interest in how to provide the visual grounding (Yang et al., 2023;
Zheng et al., 2024) to the agent and generate a task plan with sub-plans (Niu
et al., 2024) and how to design an agent structure (Niu et al., 2024).

However, the web environment presents many unforeseeable challenges (e.g.,
encountering irrelevant search results or unconventional UI and interaction
design), leading to a decrease in the generalizability of agents in web tasks
(Niu et al., 2024). When faced with unexpected scenarios, humans contemplate
reasons and seek alternative solutions while planning for the next action.
However, current web agents lack modules capable of performing such functions.
Moreover, while humans can perform given tasks on unfamiliar websites through
basic UI understanding and trial and error, web agents still lack this
ability. To address this limitation, ScreenAgent (Niu et al., 2024) has
introduced a module capable of reflecting on its own actions, and FRIDAY (Wu
et al., 2024) has proposed module designs considering the biological nature of
the human brain to make an agent behave like a human in performing the task.
However, the low performance of these agents might imply the existence of
implicit knowledge or behavior patterns unique to humans on the web task to be
unpacked.

Then, what are the differences between humans and web agents in performing web
tasks? What distinguishes agents from humans in terms of planning, action, and
reflection when executing web tasks? Furthermore, what implications can be
derived from these differences for agent design? To answer these questions, we
conducted a think-aloud study where four participants conducted two different
web tasks typically executed by current web agents, such as shopping and
finding information about a post. Through a qualitative coding analysis on the
think-aloud transcripts, we identified the human’s cognitive actions (Table 1)
and operations on the web (Table 2). The tables illustrate the types of
planning humans make, the decisions they make, and how they perform rollback
in plans when something goes wrong in web tasks.

Furthermore, by comparing participants’ task execution processes with
ScreenAgent (Niu et al., 2024)’s prompting structure and its modules, we
identified that humans update their knowledge space (task- and task website-
related knowledge) by exploring and discovering new information. Our results
also reveal that humans exhibit a pattern of identifying and clarifying
ambiguous aspects of tasks through an additional information search beyond
simply performing tasks according to constructed plans. Unlike agents, humans
investigate reasons for failure or discrepancies between their knowledge and
the information they are viewing, and based on this exploration, they modify
their plans. Based on these findings, we discuss design implications for
additional modules needed in agents and how to capture implicit knowledge and
behavior patterns unique to humans during task execution processes. Our
discussion also includes the methods for capturing humans’ tacit knowledge
within the web environment to transfer this knowledge to the agent.

##  2\. Main Study

To investigate how humans execute web tasks in depth, we conducted a think-
aloud study with four participants on the two different web tasks.

###  2.1. Study Design

####  2.1.1. Task design

The tasks of study are a shopping task on the Amazon website
111https://www.amazon.com/ and an information search task on Reddit
222https://www.reddit.com/. We selected the tasks based on the VisualWebArena
paper (Koh et al., 2024) since these are the tasks commonly used in web agent
research. The two tasks are:

  * •

Find the cheapest Lavazza Gran Aroma coffee capsule on Amazon.

  * •

Find the title of the_Champion’s Hot post at r/midjourney ten months ago.

####  2.1.2. Participant

We recruited a total of four participants. Two people are familiar with the
task and the given websites, and the other two people aren’t. This recruitment
would allow us to observe how people unfamiliar with the website perceive,
understand, and respond to the given task.

####  2.1.3. Procedure

The study started with a brief introduction by the authors, followed by the
execution of the two tasks by the participants. We instructed them to think
aloud in terms of their planning, action, and reflection. Specifically, we
asked them to explain the rationale or intent behind their task planning and
action. Then, the participants conducted the two tasks in random order until
they completed or quit the given task. After conducting the two tasks, the
study finished.

###  2.2. Analysis

To analyze the think-aloud transcript data, we conducted an open coding
(Corbin and Strauss, 1990; Khandkar, 2009) on the think-aloud transcript data.

First, one author organized the transcripts while monitoring recorded study
sessions. In cases where verbal explanations lacked context (such as instances
involving pronouns like ”this” and ”that” in the think-aloud process), the
author checked the specific situation from the recorded video. Through this
process, we obtained 156 raw data from transcripts in all study sessions.

Second, the two authors jointly conducted all stages of qualitative coding.
Without further interpretation, the authors initially classified the raw data
into similar codes first and defined names for the codes. Next, the authors
reviewed the defined codes together, merging similar codes and splitting codes
if different types of raw data existed within the same code. The entire
process continued until a consensus was reached between the two authors.

Through this process, the 156 transcript raw data were classified into 39
codes. The two authors then defined 14 themes by grouping similar codes again.
Similarly, consensus was sought between the two authors in the process of
grouping and separating themes. Finally, the grouped themes were divided into
cognitive aspects of action (Table 1) and operational (Table 2) aspects
performed on the actual website when humans perform web tasks.

Finally, by assigning the themes determined through qualitative coding back to
the original raw data, we created task action sequences of cognitive actions
and operations performed by participants during web tasks. In these sequences,
each participant’s think-aloud data was assigned the themes in Table 1 and 2.
From a total of 8 sequences (4 participants and 2 tasks), we further analyzed
the most prevalent patterns by calculating sequence pairs and triples (Figure
1).

##  3\. Results

This section illustrates the results of the analysis of the think-aloud study.
Table 1 and Table 2 show the human’s cognitive actions and operation on the
web observed in the study. Based on these results, we investigated the core
difference between the recent agents.

Table 1. Qualitative coding results regarding human cognitive actions in
performing web tasks.

| Theme | Code | Count | Ratio
---|---|---|---|---
Cognitive action | Task planning | \- Establishing a non-specific initial task plan | 24 | 15.38%
\- Establishing an additional auxiliary plan to know unknown information
\- Establishing a verification plan for discerning the truth of the found
information or answer
\- Establishing an alternative plan
Task condition checking | \- Recalling the task conditions | 6 | 3.58%
\- Identifying ambiguous parts in the task conditions
Information processing | \- Collecting information helpful for performing the task | 3 | 1.92%
\- Comparing derived answers and newly discovered information
\- Comparing prior knowledge and currently observed information
Discovering | \- Discovering task-related knowledge from currently observed information | 41 | 26.28%
\- Discovering task website-related knowledge (e.g., function) through UI
understanding, intuition, or trial.
\- Discovering that the current plan is flawed based on the currently observed
information
Decision | \- Deciding on the information to search | 14 | 8.97%
\- Deciding on answer candidates
\- Deciding on whether to maintain the current plan
\- Deciding on defining the ambiguous parts in task conditions
Complete task | \- Completing the task through comparison of currently observed information with the derived answers | 7 | 4.49%
\- Completing the task by confirming satisfaction of task conditions
\- Completing the task based on prior knowledge of the task website
\- Completing the task by clarifying the vague condition
Reflection | \- Contemplating the reasons for the current plan’s failure | 6 | 3.85%
\- Contemplating the differences between prior knowledge and newly found
information
\- Contemplating whether there has been an information gain from the
information currently being viewed

Table 2. Qualitative coding results regarding human operations on a website in
performing web tasks.

|  Theme | Code | Count | Ratio
---|---|---|---|---
Operation |  Information search | \- Searching the given conditions | 12 | 7.69%
\- Search for specific information directly related to answers
\- Search for clarification of unknown part in Google
Information exploration | \- Exploring specific information by considering the task conditions or unclear parts | 16 | 10.26%
\- Exploring information to verify its irrelevance, despite initial judgment
of it being incorrect
\- Exploring the website functionalities at the beginning of the task
\- Exploring previously opened tabs that might contain useful information
Roll-back in the plan | \- Roll-backing from auxiliary plan to auxiliary plan | 9 | 5.77%
\- Roll-backing from auxiliary plan to main plan
\- Roll-backing to start point
\- Roll-backing from main plan to main plan
Start task | \- Starting task with prior knowledge like prefix of website address | 7 | 4.49%
\- Starting the task by navigating to the task website through a Google search
Trial and error | \- Trying despite uncertainty about the success of the action | 2 | 1.28%
General function | \- Click on the next step, filtering, or sorting | 7 | 4.49%
Task specific function | \- Specific task website functions like Add to cart | 2 | 1.28%

###  3.1. Cognitive Actions and Website Operations

Qualitative coding result (Table 1) reveals that in performing web tasks,
humans engage in various cognitive actions beyond task planning and
reflection. Particularly, during the task process, humans frequently discover
task-related knowledge from the information they are currently observing or
uncover functionalities of the website previously unknown to them through UI
understanding (e.g., catching a website’s functionality through UI design) and
intuition. Task-related knowledge refers to information unknown within task
conditions (e.g., the fact that Lavazza capsules come in various types), while
task website functionalities denote commands or features used within the
website (e.g., how to find a specific username on Reddit using u/).
Participants familiar with the task utilized such information as prior
knowledge, whereas those unfamiliar with the task progressed through
exploration to clarify these unclear aspects while performing the task.

From the operational aspect of tasks performed on websites (Table 2), it is
evident that humans engage in search and exploration operations more
frequently than others when encountering unknown information. Despite
participants’ high familiarity with website functions, operations such as
search and exploration were commonly employed to clarify new information
within the given conditions (e.g., types of Lavazza coffee capsules). For
instance, in a task requiring participants to find coffee capsules, even those
familiar with the website resorted to searching for related information to
understand the difference between Lavazza’s pot and coffee capsules, thus
enabling them to make a clear selection based on Amazon search results.
Additionally, participants tended to establish additional auxiliary plans for
exploring the unknown aspects rather than entirely redesigning their task
plans. They also demonstrated a propensity to formulate plans for verifying
newly acquired information. Moreover, during task execution, when the
direction of exploration went away or when it was unclear which operation to
perform, individuals exhibited a roll-back operation within the established
plan, allowing them to revert to a specific point.

###  3.2. Comparing Web Agent and Human

Figure 1. Analysis results of participant sequence data pairs and triples
based on the themes of the qualitative coding.

####  3.2.1. Human’s Discovering Action and Knowledge Update Process in Web
Task

Our study results show that the participants acquired knowledge relevant to
tasks (e.g., the difference between the types of coffee capsules) and task
websites (e.g., website function or search results pattern) through their own
UI understanding, trial and error, or intuition while performing tasks.
Furthermore, they frequently validate whether their current plan is correct or
not through these discoveries. Through this process of discovery, participants
perform rollback within their plan or additional information exploration
operations. On the cognitive action aspect, participants made decisions
related to whether to perform reflection based on the information discovered,
conduct information searches or exploration, determine answers, or maintain
the current plan.

To update the discovered knowledge during the task, the current structure of
web agents requires a framework that can distinguish and update knowledge
about the given task itself and knowledge about the task website based on the
information discovered. In the process of discovering information, web agents
might need a structure that can capture and reflect the above information in
the next decision based on task context or general website knowledge (Website
structure or UI) gained from past experiences.

####  3.2.2. Information Exploration with Trial and Error

Current web agents mainly execute actions that correspond to sub-plans once a
plan is formulated (Niu et al., 2024). From observing participants unfamiliar
with the task, we noted that they sought to clarify vague parts of the task
through exploration. An interesting point is that participants not only
explore potentially useful information but also the functionality of the task
website and sometimes inappropriate information. Ultimately, after the
exploration, they make decisions about which information is useful and which
is not. This allows the participants to expand their knowledge space related
to the task- or task website-related knowledge.

####  3.2.3. Differences of Reflection on What

ScreenAgent (Niu et al., 2024) executes a specific action for a given sub-plan
and performs reflection by observing changes on the screen before and after
execution. Through this process, it either 1) repeats the action, 2) revises
the plan, or 3) proceeds to the next step.

In contrast to this structure, our study results indicate that humans engage
in a more complex reflection process. Humans reflect not only on why a plan
failed but also on how newly acquired information differs from their existing
knowledge and whether the information gained adds new value to their knowledge
space beyond simply understanding why a plan with an action failed.

Based on this reflection, humans revise plans, perform rollbacks within plans,
or determine the direction of information exploration within the knowledge
space. Unlike current web agents, humans extend their reflection beyond
evaluating whether actions performed for sub-plans were executed correctly.
They delve into reasons for failure and possibilities for remediation and
identify additional information that may require exploration.

##  4\. Discussion

Based on the study’s results, this section explains how web agents would be
designed and discusses the limitations of think-aloud studies in capturing
implicit human behavior and knowledge.

###  4.1. Design Implications for Web Agent

Web agents must possess not only knowledge about the given task but also the
necessary knowledge within the task website. In the study, participants
demonstrated updating both forms of knowledge, unlike web agents. Beyond
simply providing task instructions as prompts, there is a need for modules
capable of understanding knowledge about the tasks themselves, comparing them
with new information discovered during tasks, and updating accordingly.
Additionally, to respond to various forms and designs of website environments
appropriately, web agents would require a combination of general web knowledge
applicable across websites and specialized web knowledge tailored to specific
websites or tasks.

In addition to incorporating modules related to knowledge, it may be possible
to design modules within the agent for human processes such as reflection,
planning, roll-back, and decision-making. The relationships between these
processes and how they interact during task execution may require further
exploration, depending on the complexity of the tasks. However, initially, it
would be essential to delineate the processes present in humans and design
them as modules within the agent. Research into how these newly designed
modules could enhance the current performance of the agent would be necessary.

###  4.2. Capturing Human’s Tacit Knowledge

In our study, we observed that people vary in the depth of their explanations
level, and it may not comprehensively capture all processes with the think-
aloud protocol (Krosnick et al., 2021). Although the study tasks were simpler
than open-ended tasks without predefined answers, such as a literature survey
or design ideation, we observed that participants missed certain aspects when
explaining their thoughts and that the level of details of explanation varied
between individuals. Particularly, it was challenging for participants to
provide clear explanations for how their current actions or explanations
related to their previous explanations, behaviors, or their prior knowledge.
From the behaviors and thought processes unique to humans, such as the process
of acquiring new information or performing tasks based on it, there is a need
to improve the think-aloud protocol to clearly capture aspects applicable to
agents.

Therefore, in this background, it is important to capture the human’s implicit
behavior pattern or tacit knowledge within a certain task process to reuse the
task-related knowledge (Son et al., 2024). This might also reveal a more
detailed implicit process, like how humans conduct the trial-and-error
processes or revise what parts of their plan are in a certain situation.
Furthermore, if we can create datasets containing even implicit patterns or
tacit knowledge and feed them to train and fine-tune agents, they would be
able to perform tasks more like humans and offer a wider range of support to
people. Through the dataset, to provide general or personalized support within
the task, we can also define the ground rules of the task knowledge that
various people utilized during the task, as well as the personalized rules or
knowledge.

##  5\. Conclusion

In conclusion, this study sheds light on the disparities observed between
human and web agents’ performance in web tasks. Through a think-aloud study of
web tasks, distinct cognitive actions and operations on websites utilized by
humans were revealed. Comparative analysis with existing agent structures
underscored differences in knowledge updating and ambiguity handling during
task execution. Humans demonstrated a propensity for adaptive planning based
on additional information and thorough investigation into reasons for failure.
These findings provide design insights into web agent architectures and the
development of methods for capturing implicit human knowledge within web
tasks. By capturing and transforming implicit behavior patterns with tacit
knowledge utilized by individuals during task execution into forms applicable
to web agents, we envision a future where web agents can provide more diverse
support for humans.

###### Acknowledgements.

This work was supported by NAVER-KAIST Hypercreative AI Center. We thank all
of our participants for engaging positively in our studies. We also thank all
of the members of KIXLAB for their helpful discussions and constructive
feedback.

## References

  * (1)
  * 202 (2023) 2023\.  GPT-4V(ision) System Card.  https://api.semanticscholar.org/CorpusID:263218031
  * Cheng et al. (2024) Kanzhi Cheng, Qiushi Sun, Yougang Chu, Fangzhi Xu, Yantao Li, Jianbing Zhang, and Zhiyong Wu. 2024.  SeeClick: Harnessing GUI Grounding for Advanced Visual GUI Agents.  _ArXiv_ abs/2401.10935 (2024).  https://api.semanticscholar.org/CorpusID:267069082
  * Corbin and Strauss (1990) Juliet Corbin and Anselm Strauss. 1990.  Grounded Theory Research: Procedures, Canons and Evaluative Criteria.  _Zeitschrift für Soziologie_ 19, 6 (Dec. 1990), 418–427.  https://doi.org/10.1515/zfsoz-1990-0602
  * Hu et al. (2024) Xueyu Hu, Ziyu Zhao, Shuang Wei, Ziwei Chai, Guoyin Wang, Xuwu Wang, Jing Su, Jingjing Xu, Ming Zhu, Yao Cheng, Jianbo Yuan, Kun Kuang, Yang Yang, Hongxia Yang, and Fei Wu. 2024.  InfiAgent-DABench: Evaluating Agents on Data Analysis Tasks.  _ArXiv_ abs/2401.05507 (2024).  https://api.semanticscholar.org/CorpusID:266933185
  * Khandkar (2009) Shahedul Huq Khandkar. 2009.  Open coding.  _University of Calgary_ 23 (2009), 2009.  http://pages.cpsc.ucalgary.ca/~saul/wiki/uploads/CPSC681/opencoding.pdf
  * Koh et al. (2024) Jing Yu Koh, Robert Lo, Lawrence Jang, Vikram Duvvur, Ming Chong Lim, Po-Yu Huang, Graham Neubig, Shuyan Zhou, Ruslan Salakhutdinov, and Daniel Fried. 2024.  VisualWebArena: Evaluating Multimodal Agents on Realistic Visual Web Tasks.  _ArXiv_ abs/2401.13649 (2024).  https://api.semanticscholar.org/CorpusID:267199749
  * Krosnick et al. (2021) Rebecca Krosnick, Fraser Anderson, Justin Matejka, Steve Oney, Walter S. Lasecki, Tovi Grossman, and George Fitzmaurice. 2021.  Think-Aloud Computing: Supporting Rich and Low-Effort Knowledge Capture. In _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_ (¡conf-loc¿, ¡city¿Yokohama¡/city¿, ¡country¿Japan¡/country¿, ¡/conf-loc¿) _(CHI ’21)_. Association for Computing Machinery, New York, NY, USA, Article 199, 13 pages.  https://doi.org/10.1145/3411764.3445066
  * Niu et al. (2024) Runliang Niu, Jindong Li, Shiqi Wang, Yali Fu, Xiyu Hu, Xueyuan Leng, He Kong, Yi Chang, and Qi Wang. 2024.  ScreenAgent: A Vision Language Model-driven Computer Control Agent.  _ArXiv_ abs/2402.07945 (2024).  https://api.semanticscholar.org/CorpusID:267636776
  * OpenAI (2023) OpenAI. 2023.  GPT-4 Technical Report.  arXiv:2303.08774 [cs.CL]
  * Park et al. (2023) Jeongeon Park, Bryan Min, Xiaojuan Ma, and Juho Kim. 2023.  ChoiceMates: Supporting Unfamiliar Online Decision-Making with Multi-Agent Conversational Interactions.  _ArXiv_ abs/2310.01331 (2023).  https://api.semanticscholar.org/CorpusID:263605868
  * Son et al. (2024) Kihoon Son, DaEun Choi, Tae Soo Kim, and Juho Kim. 2024.  Demystifying Tacit Knowledge in Graphic Design: Characteristics, Instances, Approaches, and Guidelines.  arXiv:2403.06252 [cs.HC]
  * Wang et al. (2023) Lei Wang, Chengbang Ma, Xueyang Feng, Zeyu Zhang, Hao ran Yang, Jingsen Zhang, Zhi-Yang Chen, Jiakai Tang, Xu Chen, Yankai Lin, Wayne Xin Zhao, Zhewei Wei, and Ji rong Wen. 2023.  A Survey on Large Language Model based Autonomous Agents.  _ArXiv_ abs/2308.11432 (2023).  https://api.semanticscholar.org/CorpusID:261064713
  * Wu et al. (2023) Qingyun Wu, Gagan Bansal, Jieyu Zhang, Yiran Wu, Shaokun Zhang, Erkang Zhu, Beibin Li, Li Jiang, Xiaoyun Zhang, and Chi Wang. 2023.  AutoGen: Enabling Next-Gen LLM Applications via Multi-Agent Conversation Framework.  _ArXiv_ abs/2308.08155 (2023).  https://api.semanticscholar.org/CorpusID:260925901
  * Wu et al. (2024) Zhiyong Wu, Chengcheng Han, Zichen Ding, Zhenmin Weng, Zhoumianze Liu, Shunyu Yao, Tao Yu, and Lingpeng Kong. 2024.  OS-Copilot: Towards Generalist Computer Agents with Self-Improvement.  _ArXiv_ abs/2402.07456 (2024).  https://api.semanticscholar.org/CorpusID:267626905
  * Yang et al. (2023) Jianwei Yang, Hao Zhang, Feng Li, Xueyan Zou, Chun yue Li, and Jianfeng Gao. 2023.  Set-of-Mark Prompting Unleashes Extraordinary Visual Grounding in GPT-4V.  https://api.semanticscholar.org/CorpusID:264172201
  * Zheng et al. (2024) Boyuan Zheng, Boyu Gou, Jihyung Kil, Huan Sun, and Yu Su. 2024.  GPT-4V(ision) is a Generalist Web Agent, if Grounded.  _ArXiv_ abs/2401.01614 (2024).  https://api.semanticscholar.org/CorpusID:266741821
  * Zhu et al. (2023) Andrew Zhu, Lara J. Martin, Andrew Head, and Chris Callison-Burch. 2023.  CALYPSO: LLMs as Dungeon Masters’ Assistants.  _ArXiv_ abs/2308.07540 (2023).  https://api.semanticscholar.org/CorpusID:260900406


# Special Stable Matrices and Their Non-square Counterpart

Steven W. Su 1 ∗ ∗ The co-responding author.1 College of Artificial
Intelligence and Big data for Medical Sciences, Shandong First Medical
University Shandong Academy of Medical Sciences, P. R. China.2 University of
Technology, Sydney, Australia.

###### Abstract

In this note, we discuss the extension of several important stable square
matrices, e.g., D-stable matrices, diagonal dominance matrices, Volterra-
Lyapunov stable matrices, to their corresponding non-square matrices. The
extension is motivated by some distributed control-related problems, such as
decentralized unconditional stability and decentralized integral
controllability for non-square processes. We will provide the connections of
conditions between these special square matrices and their associated non-
square counterparts. Some conjectures for these special matrices are proposed
for future research.

##  1 INTRODUCTION

In various applications, the stability of a system under different structural
variations and constraints is a major concern in relevant research areas. For
example, in the linearization of diffusion models in biological systems at
constant equilibrium, strong stable matrices are proposed [1, 11]. The field
of system control and its applications, including decentralized stability [19,
20, 11, 13, 9, 12, 6], has motivated research into the stability of various
matrices, such as P-matrices, D-stable matrices, and Volterra-Lyapunov stable
matrices, which are also significant in other areas like economics.

In the process control industry, decentralized or distributed control is often
preferred over centralized control due to its simplicity and fault tolerance.
As a result, the configuration of decentralized control has been extensively
investigated. Topics such as Decentralized Unconditional Stability and
Decentralized Controllability have been well explored [5, 4, 15, 10, 17].

In some cases, particularly in applications involving non-square systems, the
stability of some special real matrices associated with these systems must be
addressed [21, 2, 22, 18, 16]. This need has led to the extension of stability
concepts to non-square matrices. This note focuses exclusively on real
matrices, with the potential for extension to complex matrices via frequency-
based techniques.

This study investigates the connections between the stability of square
matrices and their non-square counterparts and presents conjectures for future
research.

##  2 DECENTRALIZED STABILIZATION FOR NON-SQUARE SYSTEMS AND MATRICES

The motivation for the extension of the decentralized stability results from
square matrices to non-square matrices is due to the exploration of the
decentralized unconditional stability (DUS)[15] [10] [17] of non-square
processes. A sufficient DUS condition is the D-stability alike requirement for
the system steady state gain matrix, which is a non-square real matrix for the
investigation of non-square processes. We present the D-stability-like
sufficient condition of non-square matrices here for the decentralized
unconditional stability. As we mainly concern with the investigation of real
non-square matrices in this note, we will not provide the details about how to
use this condition to ensure DUS of non-square processes. We mainly focus on
the connection of DUS conditions of the steady state transfer functions
between non-square and its related square systems. Suppose the readers are
interested in the proof of this sufficient DUS condition. In that case, we
recommend reading the existing relevant references [15] [10] [17] about the
proof of the sufficient DUS condition for square systems via singular
perturbation analysis [14].

We present the sufficient DUS condition as follows.

###### Theorem 1

For a proper and asymptotically stable non-square system
$H(s)\in\mathbb{C}^{m\times n}$, we define its steady state transfer matrix as
$A=H(0)\in\mathbb{R}^{n\times m}$. If there exists a block diagonal non-square
matrix $K\in\mathbb{C}^{n\times m}$ such that for all non-negative diagonal
matrices $E\in\mathbb{R}^{n\times n}$, and $j\in\mathcal{M}$ [7],

$Re\\{\sigma_{i}([AEK]_{j})\\}>0,$ (1)

where $\mathcal{M}$ is the index set consisting of $k$ tuples of integers in
the range $1,\cdots,m$, and $\text{Re}\\{\sigma_{i}(M)\\}$ represents the real
part of the $i$-th eigenvalue of matrix $M$,

then the nonsquare system can be controlled by a distributed integral
controller and achieve Decentralized Unconditional Stability.

Condition (17) in Theorem 1 describes the characteristic of a non-square
D-stable alike matrix. Using Theorem 1 as the starting point, we investigate
the D-stable conditions between non-square and square matrices now. To
formalize our discussion, we define the structure of the $K$ matrix. According
to this structure, we will define the squared matrices for the non-square
matrices $A$ and $E$.

First, we assume the structure of the block diagonal $K$ matrix as follows:

$K=\begin{bmatrix}k_{1,1}&\cdots&k_{1,p_{1}}&0&\cdots&0&\cdots&0&\cdots&0\\\
0&\cdots&0&k_{2,1}&\cdots&k_{2,p_{2}}&\cdots&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&0&0&\dots&0&\cdots&k_{m,1}&\cdots&k_{m,p_{m}}\end{bmatrix}^{T}$ (2)

where $\sum_{i=1}^{m}p_{i}=n$, and all the elements $k_{i,j}$ are non-
negative.

Considering the non-negative diagonal matrix
$E=diag\\{[\varepsilon_{1,1},\cdots,\varepsilon_{m,p_{m}}]\\}\,\,\,\,\,$ with
$\varepsilon_{i}\geq 0$, we have

$\displaystyle\bar{K}$ $\displaystyle=EK=$ (3)
$\displaystyle\begin{bmatrix}k_{1,1}\varepsilon_{1,1}&\cdots&k_{1,p_{1}}\varepsilon_{1,p_{1}}&0&\cdots&0&\cdots&0&\cdots&0\\\
0&\cdots&0&k_{2,1}\varepsilon_{2,1}&\cdots&k_{2,p_{2}}\varepsilon_{2,p_{2}}&\cdots&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&\cdots&0&0&\dots&0&\cdots&k_{m,1}\varepsilon_{m,1}&\cdots&k_{m,p_{m}}\varepsilon_{m,p_{m}}\end{bmatrix}^{T}$

According to the structure of $K$, we redefine the sub-indices of the $n$
columns of $A$ as follows:

$A=[\bm{a}_{1,1},\cdots,\bm{a}_{1,p_{1}},\bm{a}_{2,1},\cdots,\bm{a}_{2,p_{2}},\cdots,\cdots,\bm{a}_{m,1}\cdots,\bm{a}_{m,p_{m}}],$

then $AEK$ as shown in (17) can be expressed as

$AEK=[\sum_{i=1}^{p_{1}}\varepsilon_{1,i}k_{1,i}\bm{a}_{1,i},\sum_{i=1}^{p_{2}}\varepsilon_{2,i}k_{2,i}\bm{a}_{2,i},\cdots,\sum_{i=1}^{p_{m}}\varepsilon_{m,i}k_{m,i}\bm{a}_{m,i}].$
(4)

Now, based on the structure (2), we define the Squared Matrices for the
matrices $A$ and $K$ as follows.

###### Definition 1

(Squared Matrices) Consider a non-square matrix $A\in\mathbb{R}^{m\times n}$
and its associated block diagonal matrix $K\in\mathbb{R}^{m\times n}$, as
defined in Equation (2). We describe a procedure to construct squared matrices
from $A$ and $K$ regarding the values of $\varepsilon_{i,j}$.

First, for the case where the dimension of the squared matrix is $m$, i.e.,
$\forall i\in\\{1,2,\cdots,m\\}$, we select only one column from each
$\bm{a}_{i,j}$, where $j\in\\{1,2,\cdots,p_{i}\\}$. This corresponds to the
scenario where, for a given $i$, exactly one of the corresponding
$\varepsilon_{i,j}\neq 0$, $j\in\\{1,2,\cdots,p_{i}\\}$. We can construct
$N=\prod_{i=1}^{m}p_{i}$ such squared matrices for both $A$ and $K$, denoted
as $[A]^{m}_{s_{i}}$ and $[K]^{m}_{s_{i}}$, respectively, for
$i\in\\{1,2,\cdots,N\\}$.

Second, when the dimension of the squared matrices is less than $m$,
corresponding to the scenario where, for one or more $i\in\\{1,2,\cdots,m\\}$,
$\varepsilon_{i,j}=0$ for all $j\in\\{1,2,\cdots,p_{i}\\}$, we can construct
squared matrices of dimension $k<m$. These are denoted as $[A]^{k}_{s_{i}}$
and $[K]^{k}_{s_{i}}$.

These squared matrices are referred to as the square counterparts of the non-
square matrices $A$ and $K$.

In Definition 1, regarding the original decentralized stabilization problem,
the reduction in the dimension of matrices occurs in cases where all the
control inputs for certain columns have zero gain. In other words, for some
index $i$, if $\varepsilon_{i,j}=0$ for all $j\in{1,2,\cdots,p_{i}}$, it
corresponds to the removal of a control input from service, as discussed in
[7].

Based on the definition of the squared matrices (Definition 1), we present the
connection between square and non-square matrices.

First, we introduce two well-known stable square matrices.

###### Definition 2

[8] The matrix $A=(a_{ij})\in\mathbb{R}^{n\times n}$ is said to be

(1) D-stable if $DA$ is stable for all diagonal matrix $D>0$;

(2) Volterra-Lyapunov stable if there exists a diagonal matrix $D>0$ for which
$AD+DA^{T}>0$.

###### Definition 3

[3] A square matrix $A$ is said to be strictly column diagonally dominant if
for each row $i$, the absolute value of the diagonal element is greater than
the sum of the absolute values of the other elements in the same column:

$|a_{ii}|\geq\sum_{j>i}|a_{ij}|\quad\text{for all }i.$

###### Definition 4

A matrix $A$ is called strictly column diagonally balanced dominant if there
exists a diagonal matrix $D$ such that $|AD|+|DA^{T}|$ is strictly column
diagonally dominant. Furthermore, if all the diagonal elements of $AD+DA^{T}$
are positive then the matrix $A$ is called positive diagonally balanced
dominant.

###### Lemma 1

([8]) If a matrix $A=(a_{ij})\in\mathbb{R}^{n\times n}$ is Volterra-Lyapunov
stable, then, for any diagonal positive matrix $D>0$ and $i\in\mathcal{M}$,
where $\mathcal{M}$ is the index set consisting of $k$ tuples of integers in
the range $1,\cdots,m$,

$Re\\{\sigma[AD]_{i}\\}>0,$

which implies that $[A]_{i}$, for $i\in\mathcal{M}$, are all D-stable
matrices.

###### Definition 5

For a real nonsquare matrix $A\in\mathbb{R}^{m\times n}$, if one can find an
individual diagonal positive matrix $D_{i}>0$, so that all its $m$-order
squared matrices (see Definition 1) $[A]^{m}_{s_{i}}\in\mathbb{R}^{m\times m}$
satisfy the following LMIs,

$[A]^{m}_{s_{i}}D_{i}+D_{i}([A]^{m}_{s_{i}})^{T}>0,$

then, we call the non-square matrix $A$ an individual Volterra-Lyapunov stable
matrix.

###### Definition 6

For a real nonsquare matrix $A\in\mathbb{R}^{m\times n}$, if one can find a
diagonal positive matrix $D>0$, so that all its $m$-order squared matrices
(see Definition 1) $[A]^{m}_{s_{i}}\in\mathbb{R}^{m\times m}$ satisfy the
following LMIs,

$[A]^{m}_{s_{i}}D+D([A]^{m}_{s_{i}})^{T}>0,$

then, we call the non-square matrix $A$ a simultaneous Volterra-Lyapunov
stable matrix.

###### Definition 7

For a real nonsquare matrix $A\in\mathbb{R}^{m\times n}$, if one can find a
diagonal positive matrix $D>0$, so that

$[A]^{m}_{s_{i}}D+D([A]^{m}_{s_{i}})^{T}$

are all strictly column diagonally dominant and all their diagonal elements
are positive ( where $[A]^{m}_{s_{i}}$ is the $m$-order squared matrices (see
Definition 1) of $A$), then, we call the non-square matrix $A$ simultaneous
positive diagonally balanced dominant.

###### Note 1

It is easy to see that a non-square matrix $A$ is simultaneous Volterra-
Lyapunov stable if it is simultaneous positive diagonally balanced dominant.

###### Lemma 2

Consider $m$ groups of cards, where each group contains $p_{i}$ distinct
cards. A player can place a bet on each combination of $m$ cards, one card
from each group. The payoff for a specific combination is denoted as
$\gamma(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})>0$. We record the payoff for
each card in a constructed combination as
$\frac{\gamma(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})}{m}$. Suppose the player
desires a specific ratio $\kappa_{kj}>0$ between the $j_{th}$ card of group
$k$ and the first card of that group.

If the set of payoffs $\gamma(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})$ satisfies
the following conditions, then the player can implement their desired
strategy:

$\begin{cases}\begin{aligned} &\gamma_{(1,1,\cdots,1)}=1\\\
&\gamma_{(\zeta_{1},\zeta_{2},\cdots,\zeta_{(i-1)},j,\zeta_{(i+1)},\cdots,\zeta_{m})}=\gamma_{(\zeta_{1},\zeta_{2},\cdots,\zeta_{(i-1)},1,\zeta_{(i+1)},\cdots,\zeta_{m})}\kappa_{ij},\end{aligned}\end{cases}$
(5)

###### Proof 1

We will prove the desired strategy can be achieved by constructing a set of
payoffs that satisfies the conditions in equation (5).

1\. Normalization Condition:
First, according to Condition (5), we easily see that all
$\gamma(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})$ can be set as a positive real
value. The condition $\gamma_{(1,1,\cdots,1)}=1$ ensures that the payoff for
choosing one card from each group, where all cards are the first cards of
their respective groups, is normalized to 1. This establishes a baseline.

2\. Ratio Preservation Condition:
According to Condition (5), we easily see that all
$\gamma(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})$ can be set as a positive real
value. Then, the payoff of the first card in group $k$ is

$\frac{1}{m}(\sum_{\zeta_{1}=1}^{p_{1}}\cdots\sum_{\zeta_{k-1}}^{p_{k-1}}\sum_{\zeta_{k+1}}^{p_{k+1}}\cdots\cdots\sum_{\zeta_{m}}^{p_{m}}\gamma_{(\zeta_{1},\cdots,\zeta_{k-1},1,\zeta_{k+1},\cdots,\zeta_{m})}).$

The payoff of the $j_{th}$ card in group $k$ is

$\displaystyle\frac{1}{m}(\sum_{\zeta_{1}=1}^{p_{1}}\cdots\sum_{\zeta_{k-1}}^{p_{k-1}}\sum_{\zeta_{k+1}}^{p_{k+1}}\cdots\cdots\sum_{\zeta_{m}}^{p_{m}}\gamma_{(\zeta_{1},\cdots,\zeta_{k-1},j,\zeta_{k+1},\cdots,\zeta_{m})})$
(6) $\displaystyle=$
$\displaystyle\frac{1}{m}(\sum_{\zeta_{1}=1}^{p_{1}}\cdots\sum_{\zeta_{k-1}}^{p_{k-1}}\sum_{\zeta_{k+1}}^{p_{k+1}}\cdots\cdots\sum_{\zeta_{m}}^{p_{m}}\kappa_{kj}\gamma_{(\zeta_{1},\cdots,\zeta_{k-1},1,\zeta_{k+1},\cdots,\zeta_{m})}).$

Thus, the second equation of Condition (6) ensures that the payoff for
choosing the $j_{th}$ card of group $k$ in combination with other cards is
directly proportional to the payoff for choosing the first card of that group,
with the ratio being $\kappa_{kj}$.

###### Lemma 3

If a real nonsquare matrix $A\in\mathbb{R}^{m\times n}$ is a simultaneous
Volterra-Lyapunov stable matrix (see Definition 6), then, there exists a block
diagonal non-square matrix $K\in\mathbb{C}^{n\times m}$ such that for all non-
negative diagonal matrices $E\in\mathbb{R}^{n\times n}$, and
$j\in\mathcal{M}$, where $\mathcal{M}$ is the index set consisting of $k$
tuples of integers in the range $1,\cdots,m$,

$Re\\{\sigma_{i}([AEK]_{j})\\}>0,$ (7)

where $\text{Re}\\{\sigma_{i}(M)\\}$ represents the real part of the $i$-th
eigenvalue of matrix $M$.

###### Proof 2

Considering that $A$ is a simultaneous Volterra-Lyapunov stable matrix, the
following inequality should be true:

$[A]^{m}_{s_{i}}D+D([A]^{m}_{s_{i}})^{T}>0.$ (8)

Assume the total number of the $m_{th}$-order squared matrices
$[A]^{m}_{s_{i}}$ is $N$, i.e., $i\in\\{1,2,\cdots,N\\}$. Based on inequality
(8), we can conclude, for any $\gamma_{i}>0$, $i\in\\{1,\cdots,N\\}$, the
following inequality holds

$\displaystyle(\gamma_{1}[A]^{m}_{s_{1}}+\gamma_{2}[A]^{m}_{s_{2}}\cdots+\gamma_{N}[A]^{m}_{s_{N}})D$
$\displaystyle+$ (9) $\displaystyle
D(\gamma_{1}[A]^{m}_{s_{1}}+\gamma_{2}[A]^{m}_{s_{2}}\cdots+\gamma_{N}[A]^{m}_{s_{N}})^{T}$
$\displaystyle>0$

Based on Lemma 1, for any diagonal positive matrix $D>0$ and
$j\in\mathcal{M}$, we have

$Re\\{\sigma[(\sum_{i=1}^{N}\gamma_{i}[A]^{m}_{s_{i}})D]_{j}\\}>0.$ (10)

Without loss of generality, we only consider the proof for the $m_{th}$-order
matrices. So, we omit the subscript $j$ in the following discussions.

According to structure of $K$, as before, we redefine the sub-indices of the
$n$ columns of $A$ as follows:

$A=[\bm{a}_{1,1},\cdots,\bm{a}_{1,p_{1}},\bm{a}_{2,1},\cdots,\bm{a}_{2,p_{2}},\cdots,\cdots,\bm{a}_{m,1}\cdots,\bm{a}_{m,p_{m}}].$

Correspondingly, we redefine the sub-indices of $\gamma_{i}>0$,
$i\in\\{1,\cdots,N\\}$ in inequality (9) as follows;

$\gamma_{(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})},$ (11)

where $\zeta_{i}\in\\{1,2,\cdots,p_{i}\\}$. The coefficient
$\gamma_{(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})}$ means from each of the
following group of vectors to select one vector to correspond to a term
$\gamma_{(\zeta_{1},\zeta_{2},\cdots,\zeta_{m})}[A]^{m}_{s_{i}}$ in inequality
(10):

$\displaystyle Group\,1:$
$\displaystyle\,\,\bm{a}_{1,1},\cdots,\bm{a}_{1,p_{1}},$ (12) $\displaystyle
Group\,2:$ $\displaystyle\,\,\bm{a}_{2,1},\cdots,\bm{a}_{2,p_{2}},$
$\displaystyle\cdots,\cdots,$ $\displaystyle Group\,m:$
$\displaystyle\,\,\bm{a}_{m,1}\cdots,\bm{a}_{m,p_{m}}.$

Further assume $D=diag\\{[d_{1},d_{2},\cdots,d_{m}]\\}$. Then the core part of
the equation (10) can be expressed as follows:

$\displaystyle(\sum_{i=1}^{N}\gamma_{i}[A]^{m}_{s_{i}})D=[$ $\displaystyle
d_{1}(\sum_{j=1}^{p_{1}}(\sum_{\zeta_{2}=1}^{p_{2}}\sum_{\zeta_{3}=1}^{p_{3}}\cdots\sum_{\zeta_{m}}^{p_{m}}\gamma_{(j,\zeta_{2},\cdots,\zeta_{m})})\bm{a}_{1,j}),$
(13) $\displaystyle
d_{2}(\sum_{j=1}^{p_{2}}(\sum_{\zeta_{1}=1}^{p_{1}}\sum_{\zeta_{3}=1}^{p_{3}}\cdots\sum_{\zeta_{m}}^{p_{m}}\gamma_{(\zeta_{1},j,\zeta_{3},\cdots,\zeta_{m})})\bm{a}_{2,j}),$
$\displaystyle\cdots\cdots\cdots$ $\displaystyle
d_{m}(\sum_{j=1}^{p_{m}}(\sum_{\zeta_{1}=1}^{p_{1}}\sum_{\zeta_{2}=1}^{p_{2}}\cdots\sum_{\zeta_{(m-1)}}^{p_{(m-1)}}\gamma_{(\zeta_{1},\zeta_{2},\cdots,\zeta_{(m-1)},j)})\bm{a}_{m,j})].$

On the other hand, the matrix $AEK$ can be expressed as

$\displaystyle AEK$
$\displaystyle=[\sum_{j=1}^{p_{1}}\varepsilon_{1j}k_{1j}\bm{a}_{1,j},\sum_{j=1}^{p_{2}}\varepsilon_{2j}k_{2j}\bm{a}_{2,j},\cdots,\sum_{j=1}^{p_{m}}\varepsilon_{mj}k_{mj}\bm{a}_{m,j}]$
(14)
$\displaystyle=[\sum_{j=1}^{p_{1}}\tilde{k}_{1j}\bm{a}_{1,j},\sum_{j=1}^{p_{2}}\tilde{k}_{2j}\bm{a}_{2,j},\cdots,\sum_{j=1}^{p_{m}}\tilde{k}_{mj}\bm{a}_{m,j}]$
$\displaystyle=[\tilde{k}_{11}\sum_{j=1}^{p_{1}}\frac{\tilde{k}_{1j}}{\tilde{k}_{11}}\bm{a}_{1,j},\tilde{k}_{21}\sum_{j=1}^{p_{2}}\frac{\tilde{k}_{2j}}{\tilde{k}_{21}}\bm{a}_{2,j},\cdots,\tilde{k}_{m1}\sum_{j=1}^{p_{m}}\frac{\tilde{k}_{mj}}{\tilde{k}_{m1}}\bm{a}_{m,j}]$
$\displaystyle=[\tilde{k}_{11}\sum_{j=1}^{p_{1}}\bar{k}_{1j}\bm{a}_{1,j},\tilde{k}_{21}\sum_{j=1}^{p_{2}}\bar{k}_{2j}\bm{a}_{2,j},\cdots,\tilde{k}_{m1}\sum_{j=1}^{p_{m}}\bar{k}_{mj}\bm{a}_{m,j}].$

In the above equation, $\tilde{k}_{ij}=\varepsilon_{i,j}k_{i,j}\geq 0$, and
$\bar{k}_{ij}=\frac{\tilde{k}_{i,j}}{\tilde{k}_{i,1}}$ (considering the case
$\tilde{k}_{i,1}\neq 0$), which implies $\bar{k}_{i1}=1$.

Now, according to Lemma 2, in Equation (13), if we let $d_{i}=\tilde{k}_{i1}$
($i\in\\{1,2,\cdots,m\\}$), and

$\begin{cases}\begin{aligned} &\gamma_{(1,1,\cdots,1)}=1\\\
&\gamma_{(\zeta_{1},\zeta_{2},\cdots,\zeta_{(i-1)},j,\zeta_{(i+1)},\cdots,\zeta_{m})}=\gamma_{(\zeta_{1},\zeta_{2},\cdots,\zeta_{(i-1)},1,\zeta_{(i+1)},\cdots,\zeta_{m})}\bar{k}_{ij},\end{aligned}\end{cases}$
(15)

the left-hand side of Equation (13) equals that of Equation (14). Thus, we can
conclude, that for any non-negative diagonal matrices $E\in\mathbb{R}^{n\times
n}$, the inequality (7) is always satisfied.

###### Note 2

Lemma 3 confirmed that a non-square simultaneously Volterra-Lyapunov stable
matrix is also a non-square D-stable-like matrix. Simultaneously Volterra-
Lyapunov stable matrix implies that there exists a common diagonal Lyapunov
function for all its $m$-order squared matrices. This condition is relatively
strong, which can even guarantee the instantaneous switching stability among
all these squared matrices.

###### Note 3

The simultaneous Volterra-Lyapunov stability condition in Lemma 3 is a strong
condition especially for the case when the total number of non-squared
matrices, $N$, is big. The following theorem relaxed the simultaneous
Volterra-Lyapunov stability to individual Volterra-Lyapunov stability.

###### Lemma 4

[8] For normal matrices and in the class $\mathbb{F}=\\{A:a_{ij}\geq
0\,\,for\,\,all\,\,i\neq j\\}$, D-stability and Volterra-Lyapunoc stability
are all equivalent to stability.

###### Lemma 5

For a set of normal matrices which are also in the class
$\mathbb{F}=\\{A:a_{ij}\geq 0\,\,for\,\,all\,\,i\neq j\\}$, D-stability,
individual Volterra-Lyapunoc stability, and simultaneous Volterra-Lyapunoc
stability are all equivalent to stability.

###### Proof 3

Based on Lemma 4, this lemma can be easily proved.

###### Theorem 2

If any of the $m_{th}$-order squared matrices of $A\in\mathbb{R}^{m\times n}$
is a normal matrix and in the class $\mathbb{F}$, and also D-stable matrix,
then there exists a block diagonal non-square matrix $K\in\mathbb{C}^{n\times
m}$ such that for all non-negative diagonal matrices $E\in\mathbb{R}^{n\times
n}$, and $j\in\mathcal{M}$, where $\mathcal{M}$ is the index set consisting of
$k$ tuples of integers in the range $1,\cdots,m$,

$Re\\{\sigma_{i}([AEK]_{j})\\}>0,$ (16)

where $\text{Re}\\{\sigma_{i}(M)\\}$ represents the real part of the $i$-th
eigenvalue of matrix $M$.

###### Proof 4

Based on Lemma 5, both individual and simultaneous Volterra-Lyapunoc stability
of the matrix $A$ can be ensured. Then, Lemma 3 can directly prove this
theorem.

###### Conjecture 1

If a real nonsquare matrix $A\in\mathbb{R}^{m\times n}$ is an individual
Volterra-Lyapunov stable matrix (see Definition 5), then, there exists a block
diagonal non-square matrix $K\in\mathbb{C}^{n\times m}$ such that for all non-
negative diagonal matrices $E\in\mathbb{R}^{n\times n}$, and
$j\in\mathcal{M}$, where $\mathcal{M}$ is the index set consisting of $k$
tuples of integers in the range $1,\cdots,m$,

$Re\\{\sigma_{i}([AEK]_{j})\\}>0,$ (17)

where $\text{Re}\\{\sigma_{i}(M)\\}$ represents the real part of the $i$-th
eigenvalue of matrix $M$.

###### Note 4

If add extra conditions, the above conjecture can be proved. For example, if
the limited number of redundant inputs is only in one channel. Then, the proof
becomes straightforward. However, generally, the proof of this theorem is
complex.

##  3 CONCLUSION

In summary, we extended the concepts of key stable square matrices, such as
D-stable and Volterra-Lyapunov stable matrices, to non-square cases. This was
motivated by distributed control problems like decentralized stability and
controllability in non-square processes. We established links between these
square and non-square matrices and proposed conjectures for further research.

## References

  * CRO [1978] Three types of matrix stability.  _Linear Algebra and its Applications_ , 20(3):253–263, 1978.
  * WHE [2017] A new iterative method for solving non-square systems of linear equations.  _Journal of Computational and Applied Mathematics_ , 322:1–6, 2017.
  * KUS [2021] Generalization of the concept of diagonal dominance with applications to matrix d-stability.  _Linear Algebra and its Applications_ , 630:204–224, 2021.
  * Anderson and Clements [1981a] B. Anderson and D. Clements.  Algebraic characterization of fixed modes in decentralized control.  _Automatica_ , 17:703–712, 1981a.
  * Anderson and Clements [1981b] B. D. Anderson and D. J. Clements.  Algebraic characterization of fixed modes in decentralized control.  _Automatica_ , 17(5):703–712, 1981b.
  * Barker et al. [1978] G. P. Barker, A. Berman, and R. J. Plemmons.  Positive diagonal solutions to the lyapunov equations.  _Linear and Multilinear Algebra_ , 5(4):249–256, 1978.
  * Campo and Morari [1994] P. J. Campo and M. Morari.  Achievable closed-loop properties of systems under decentralized control: Conditions involving the steady-state gain.  _IEEE Transactions on Automatic Control_ , 39(5):932–943, 1994.
  * Cross [1978] G. Cross.  Three types of matrix stability.  _Lin. Alg. and its App._ , 20:253–263, 1978.
  * Goh [1977] B. S. Goh.  Global stability in many-species systems.  _The American Naturalist_ , 111(977):135–143, 1977.
  * Grosdidier and Morari [1986] P. Grosdidier and M. Morari.  Interaction measures for systems under decentralized control.  _Automatica_ , 22(3):309–319, 1986.
  * Hadeler [2006] K. Hadeler.  Nonlinear diffusion equations in biology.  In _Ordinary and Partial Differential Equations: Proceedings of the Fourth Conference Held at Dundee, Scotland March 30–April 2, 1976_ , pages 163–206. Springer, 2006.
  * Iggidr and Souza [2023] A. Iggidr and M. O. Souza.  On the limits of the volterra function in the lyapunov method: the anderson-may-gupta model as a cautionary example.  _Journal of Mathematical Analysis and Applications_ , 517(1):126465, 2023.
  * Johnson [1974] C. R. Johnson.  Sufficient conditions for d-stability.  _Journal of Economic Theory_ , 9(1):53–62, 1974.
  * Kokotovic et al. [1980] P. Kokotovic, J. Allemong, J. Winkelman, and J. Chow.  Singular perturbations and iterative separation of time scales.  _Automatica_ , 16:23–33, 1980.
  * Skogestad and Morari [1988] S. Skogestad and M. Morari.  Variable selection for decentralized control.  _AIChE Annual Meeting, Washington, DC_ , page Paper 128c, 1988.
  * Steentjes [2022] T. R. V. Steentjes.  Data-driven methods for distributed control of interconnected linear systems.  2022\.
  * Su et al. [2004] S. Su, J. Bao, and P. Lee.  Analysis of decentralized integral controllability for nonlinear systems.  _Comp. and Chem. Eng._ , 28:1781–1787, 2004.
  * Sujatha and Panda [2022] V. Sujatha and R. C. Panda.  A control philosophy for complex non-square chemical process.  _Journal of Loss Prevention in the Process Industries_ , 80:104900, 2022.
  * Sun [2023] Z. Sun.  A gallery of diagonal stability conditions with structured matrices (and review papers).  _arXiv preprint arXiv:2301.01272_ , 2023.
  * Wang and Davison [1973] S. Wang and E. Davison.  On the stabilization of decentralized control systems.  _IEEE Trans. Auto. Con._ , 18:473–478, 1973.
  * Zhang et al. [2017] Y. Zhang, S. Su, A. Savkin, B. Celler, and H. Nguyen.  Multiloop integral controllability analysis for nonlinear multiple-input single-output processes.  _Industrial & Engineering Chemistry Research_, 56(28):8054–8065, 2017.
  * Zhiteckii and Solovchuk [2022] L. Zhiteckii and K. Solovchuk.  Robust adaptive controls for a class of nonsquare memoryless systems.  _Advanced Control Systems: Theory and Applications_ , page 203, 2022.