text:
\begin{document}

\title{Performance and Design of Consensus on Matrix-Weighted and Time Scaled Graphs}

\author{Dillon R. Foight, \IEEEmembership{Student Member, IEEE,} Mathias Hudoba de Badyn, \IEEEmembership{Member, IEEE,}\\ and
        Mehran Mesbahi, \IEEEmembership{Fellow, IEEE}
\thanks{Manuscript received November 22, 2019; revised March 18, 2020 and May 12, 2020; accepted May 26, 2020. This work was supported by the National Science Foundation (grant number DGE- 1762114), the National Sciences and Engineering Research Council of Canada (funding reference number CGSD2-502554-2017), the U.S. Army Research Laboratory and the U.S. Army Research Office (contract number W911NF-13-1-0340), and the U.S. Air Force Office of Scientific Research (grant number FA9550-16-1-0022). \textit{(D. Foight and M. Hudoba de Badyn are co-first authors.) (Corresponding author: D. Foight.)} A preliminary version of this work appears in the Proceedings of the 58th IEEE Conference on Decision and Control~\cite{Foight2019a}. }
\thanks{The authors are with the William E. Boeing Department of Aeronautics and Astronautics at the University of Washington, Seattle, WA 98195, USA.
  MHdB is now with the Automatic Control Laboratory at ETH Z\"{urich}, 8092 Z\"{u}rich, Switzerland. e-mails: \texttt{\{dfoight,hudomath,mesbahi\}@uw.edu}. \textcopyright IEEE 2020}
}

\markboth{Transactions on Control of Networked Systems,~Vol.~xx, No.~x, August~20xx}
{Foight, Hudoba de Badyn, \& Mesbahi: Time Scale and Matrix Weight Design for Consensus}

\maketitle

\begin{abstract}
  In this paper, we consider the $\Htwo$-norm of networked systems with multi-time scale consensus dynamics and vector-valued agent states.
  This allows us to explore how measurement and process  noise affect consensus on matrix-weighted graphs by examining edge-state consensus.
  In particular, we highlight an interesting case where the influences of the weighting and scaling on the $\Htwo$ norm can be separated in the design problem.
  We then consider optimization algorithms for updating the time scale parameters and matrix weights in order to minimize network response to injected noise.
  Finally, we present an application to formation control for multi-vehicle systems.
\end{abstract}

\IEEEpeerreviewmaketitle

\input{./1-intro.tex}
\input{./2-math-prelim.tex}
\input{./3-graphcomp.tex}
\input{./4-performance.tex}
\input{./5-example.tex}

\section{Conclusion}
\label{sec:conclusion}

In this paper we have developed a framework for investigating noise-driven consensus on matrix-weighted and time scaled graphs, with the aim of minimizing the $\Htwo$ performance of such systems.

We identified a convenient choice of noise covariances that allows for the separation of the performance contributions from the edge weighting and time scales.
This allowed us to investigate the optimal assignment of time scale parameters and edge weights to promote network resilience.
Finally, we applied these results to a flocking example, where we observed that applying a time scale assignment and edge weight update in response to an injection of noise results in less perturbation of the agent states compared to the non-updated case, or optimizing over only the edges or only the time scales.

In real-world networked dynamical problems adversarial noise will likely not be applied to all nodes within the network as was taken to be the case in this work.
Thus, a useful extension of this work is to investigate what results can be found when the noise injection is limited to a subset of nodes, or in the matrix weighted case, a subset of substates.
Furthermore, interesting problems such as leader selection in noise driven consensus networks have previously been investigated, so these areas are also potential directions for future work by extending those related topics to the time scaled and matrix weighted case.

\section*{Acknowledgments}
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

\bibliographystyle{IEEEtran}
\bibliography{references}

\input{6-appendix.tex}

\begin{IEEEbiography}
[{\includegraphics[width=1in,height=1.25in,clip]{./figs/dillon_small.eps}}]
{Dillon Foight} received a B.S. degree in space physics (astrophysics) from Embry-Riddle Aeronautical University, Prescott, AZ, USA in 2009, and worked as a science mission planner for the Chandra Space Telescope until 2015. He is currently a graduate student in the William E. Boeing Department of Aeronautics \& Astronautics at the University of Washington, Seattle, WA, USA. As a member of the Robotics, Aerospace, and Information Networks (RAIN) Lab, he focuses on analysis, control, and influence of networks featuring multi-time scale behavior. He is the recipient of the National Science Foundation Graduate Research Fellowship (2017).
\end{IEEEbiography}
\vskip 0pt plus -1fil
\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip]{./figs/HudobaDeHeadColour.eps}}]
{Mathias Hudoba de Badyn}
is a postdoctoral scholar in the Automatic Control Laboratory at the Swiss Federal Institute of Technology in Z\"{u}rich.
He received his Ph.D.~degree in the William E. Boeing Department of Aeronautics and Astronautics, and an M.Sc.~degree in the Department of Mathematics in 2019 at the University of Washington.
In 2014, he graduated from the University of British Columbia with a BSc in Combined Honours in Physics and Mathematics.
He held an NSERC PGS-D (CGS-D offered) from 2017-2019, and a University of Washington College of Engineering Dean's Fellowship from 2014-2015.
His research interests include the analysis and control of networked dynamical systems, with applications to autonomous air and space multi-vehicle systems.
\end{IEEEbiography}
\vskip 0pt plus -1fil
\begin{IEEEbiography}
[{\includegraphics[width=1in,height=1.25in,clip]{./figs/MesbahiHead.eps}}]
{Mehran Mesbahi}
is a Professor of Aeronautics \& Astronautics and an Adjunct Professor of Mathematics and Electrical Engineering at the University of Washington.
He received his Ph.D. from USC in 1996.
He was a member of the Guidance, Navigation, and Analysis group at Jet Propulsion Laboratory from 1996-2000 and an Assistant Professor of Aerospace Engineering and Mechanics at the University of Minnesota from 2000-2002.
He is currently the Director of the Robotics, Aerospace, and Information Networks (RAIN) Laboratory and the Executive Director of the Joint Center for Aerospace Technology Innovation.
His research interests are distributed and networked systems, autonomous aerospace and robotic systems, and the intersection between data science, networks, autonomy, and control theory, with multi-disciplinary applications.
\end{IEEEbiography}

\end{document}

config: arxiv


TITLE: Finding a unique continuous $g$ on $[0,1]$ satisfying $g(x) = \frac{1}{2}g\left(\frac{x+1}{2}\right) + f(x)$ for given $f$
QUESTION [5 upvotes]: Let $f$ be a given continuous function on $[0,1]$. How do you prove that there is a unique continuous function $g$ on $[0,1]$ satisfying $$g(x) = \frac{1}{2}g\left(\frac{x+1}{2}\right) + f(x)$$ for all $x\in[0,1]$?

REPLY [1 votes]: The functions $\hat g(t):=g(1-t)$ and $\hat f(t):=f(1-t)$ together satisfy the functional equation
$$\hat g(t)=\hat f(t)+{1\over2}\>\hat g\!\left({t\over2}\right)\ .\tag{1}$$
In particular one has $\hat g(0)=2\hat f(0)$. Iterating $(1)$ we obtain the condition
$$\hat g(t)=\sum_{k=0}^n {\hat f(2^{-k}t)\over 2^k}+{1\over2^{n+1}}\hat g(2^{-(n+1)}t)\qquad (n\geq0)\ .$$
for $\hat g$.  Letting $n\to\infty$ we can conclude that necessarily
$$\hat g(t)=\sum_{k=0}^\infty {\hat f(2^{-k}t)\over 2^k}\ .$$
It is easy to check that this $\hat g$ is continuous and indeed satisfies $(1)$. It follows that the originally given functional equation has the unique solution
$$g(x)=\hat g(1-x)=\sum_{k=0}^\infty{f\bigl(1-2^{-k}(1-x)\bigr)\over 2^k}\ .$$


TITLE: Group operation is well defined
QUESTION [1 upvotes]: One of the most common manipulations performed when working with group equations is left or right multiplication, i.e. if you have a group $G$ with $a,b,c \in G$ and you have something of the form $a = b$, we can then say that $ac = bc$ and $ca = cb$. How can one prove this statement from the basic axioms of group theory? I have proved it under the assumption of left and right cancellation properties, but the only proofs I have seen of those properties relies on this fact.

REPLY [2 votes]: I'm not sure what part of other answers are not satisfying, so let me break this down to basic set theoretic concepts and see if that helps.
Formally, a group is defined to be a set $G$ together with a function $\phi : G \times G \to G$ (the group operation) satisfying some axioms (which won't be used for this problem). We usually use the shorter "binary operation notation"
$$ab = \phi(a,b), \quad a,b \in G
$$
but let me stick with the function notation for the moment.
Suppose we are given $a,b,c \in G$.
If $a=b$ then we have equality of the ordered pairs $(a,c)=(b,c) \in G \times G$ (this is one of the basic properties of ordered pairs, part of set theory).
Since $\phi$ is a function, we therefore have equality of the following values of $\phi$, namely $\phi(a,c)=\phi(b,c) \in G$ (this is one of the basic properties of a function, also part of set theory).
Returning now to binary operation notation, it follows that $ac=bc$.


TITLE: Proof checking : Show that $\bigcap\limits_{n=1}^{\infty}\left(-\frac{1}{n},\frac{1}{n}\right)= \{0\}. $
QUESTION [3 upvotes]: Can anyone check my proof please? Thank you.

Show that $$\bigcap\limits_{n=1}^{\infty}\left(-\frac{1}{n},\frac{1}{n}\right)= \{0\}.$$

Let take an arbitrary $x \in \textbf{R}$ such that $|x|\leq 1/n$  for all $n$ in $\textbf{N}$.
Then $$ -\frac{1}{n}\leq x\leq\frac{1}{n}.$$
We can now say that $$x \in \bigcap\limits_{n=1}^{\infty}\left(-\frac{1}{n},\frac{1}{n}\right) .$$
We will now show that $x=0$.
We have: $$ -\frac{1}{n}\leq x \leq\frac{1}{n}.$$
Let's now take the limit $n\to\infty$:
$$\lim_{n\to\infty}-\frac{1}{n} \leq x \leq \lim_{n\to\infty}\frac{1}{n}.$$
$$0\leq x \leq0.$$
By the BigMac theorem, we conclude that $x=0$.

REPLY [1 votes]: Let's now take the limit $n\to\infty$:$$lim_{n\to\infty}-\frac{1}{n}<x< lim_{n\to\infty}\frac{1}{n}.$$

This part is incorrect and should be
$$
\lim_{n\to\infty}-\frac{1}{n}\leq x\leq \lim_{n\to\infty}\frac{1}{n}$$
which implies that
$$
0\leq x\leq 0
$$
and thus $x=0$.

Also, note that $0<x<0$ means $0<x$ and $x<0$, which is a false statement whatever $x$ is. I have only seen "Big Mac Theorem" before as a stronger statement of the so called "Hamburger's Theorem".


TITLE: Is non-standard analysis worth learning?
QUESTION [152 upvotes]: As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides some formalism to this type of calculus.
So, do you guys think it is a subject worth learning? Is it a branch that is growing and becoming interconnected with other branches of math? Does it make calculus any easier?

REPLY [3 votes]: As a student of calculus that had the perennial trouble with epsilon-delta proofs and manipulations before trying my hand at non-standard calculus, perhaps I can provide a bit of perspective:
I still have trouble with epsilon-delta proofs of for derivative formulae. I think that arises from a lack of familiarity with the rigorous handling of intervals, neighborhoods, sequences and the like and I imagine that a more in-depth treatment of the subject would remedy this.
Infinitesimal calculus (as I have seen it so far) seems like a straightforward extension of the properties of "regular" numbers to "really small" numbers (infinitesimals) and "really big" numbers (infinites). The principles used for working with hyperreal numbers pretty much guarantee that an infinitesimal will behave in the same way you'd expect a "really small" real number to behave, while infinites behave as you'd expect "really big" real numbers to behave.
This makes the proofs of the basic differentiation formulae much easier both to understand and to read.
I also disagree with the complaint that one needs a good grasp of set theory and mathematical logic to be able to work with hyperreals. In this sense I think of hyperreals the same way I think of imaginary numbers, or sausages. While it can be illuminating to know how the sausage is made, one does not need to know that to find the sausage tasty. As a tutor of lower-level mathematics I can say that many, if not most of the students can handle imaginary numbers just fine without knowing how to construct the complex number system.
In short, I find non-standard calculus worth learning because it makes the arguments more understandable, easier to follow and more intuitive. However, if you are already comfortable with epsilon-delta calculus I don't know if you'd find it of much use.