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HSMAI Region Europe: Stockholm 10 November
Posted on 28. april 2016 11. november 2016 by Redaksjonen
10 November: HSMAI Region Europe invites you to Distribution & Marketing Strategy Conference
HSMAI Region Europe is holding its first European conference on Distribution and Marketing Strategy at the Operakällaren in Stockholm. The day will consist of presentations from many of the Hospitality Industry's top leaders and other inspirational thinkers (see below list) followed by interactive discussions and mingling. We will also have a 1.5 hour interactive Unconference – an interactive participant-driven session with no previously set agenda. We had a similar session at our by-invitation-only Leadership Day in France end of September and the feedback was great as everyone could share their thoughts and ideas. This is why by the request of the attendees this will be part of many of our conferences moving forward. Please note the programme can be subject to changes which will be updated on this event page.
Where: Operakällaren, Karl XII:s torg, Stockholm
When: Thursday 10 November 2016
Registration from 8 am to 9 am. The programme starts at 9 am and lasts until 5 pm, followed by wine, snacks and networking.
We will use many of the facilities of the fantastic Operakällaren venue. The conference day will start with the general session at the beautiful Operaterrassen, with lunch at the fabulous Operakällaren restaurant. The day will finish with drinks at the Brass Bar in the iconic Cafe Opera.
The charismatic and energetic Martin Creydt, Resigning COO, Scandic Hotels, is our moderator for the day. With his extensive leadership background, he is among those who knows the industry best in the Nordics. See his LinkedIn profile here!
TIME: SESSION: SPEAKER:
08:00 – 09:00 Registration and coffee/networking
09:00 – 09:15 Welcome & Update from HSMAI Region Europe
HSMAI Region Europe has grown rapidly in the past few years across Europe. There are several big projects in the making about which Ingunn will share an update.
HSMAI Europe
09:15 – 10:00 Current status of Online Distribution and thoughts on its future in the Hospitality Industry.
A great mix of academic and industry expertise, the always provocative and bright Peter will share his thoughts on what's up in the world of online distribution and its effects on you and your business. Peter O'Connor
Dean of Academic Programs
ESSEC Business School
Amongst the top experts in Hospitality distribution, digital marketing and e-commerce, Peter is a well-known speaker and author with vast expertise in the Hospitality Industry. Passionate and provocative, Peter's presentation is sure to engage the audience. Peter has been working closely with HSMAI for around 20 years. Peter was also chosen as one of the top 20 Extraordinary Minds in Sales, Marketing and Technology at the HSMAI European Awards 2015 due to his remarkable achievements in Hospitality Industry.
Peter will talk about the current status of Online Distribution and his thoughts on its future in the Hospitality Industry.
10:00 – 10:45 Guerilla Tactics: Winning in a World of Distribution Superpowers.
In order to secure the position in a hotel and travel industry that is steadily becoming more web and technology heavy, Nordic Choice Hotels have established a private company with full focus on digital and technological innovation.
Nils will discuss Nordic Choice Hotels approach on the rapidly evolving Distribution landscape. Nils Korsvoll
Vice President Distribution
Nordic Choice HotelsNils has been with Nordic Choice Hotels since 2008, holding such diverse positions as Revenue Manager, Business Analyst, Revenue Analysis Manager, Business Development Manager and Head of Future Business, all attesting to expert insight in distribution matters.
10:45 – 11:15 Coffee and networking break
11:15 – 11:45 What's going on, how are we doing? Status and future outlook of the Nordic Hospitality Industry.
New hotels opening, big acquisitions and mergers in the works … Things are moving fast in the world of Hospitality. But how are the Nordics doing? Johan will take us through the current status of Nordic Hospitality Industry and present relevant performance data as well as share his outlook on the future of the industry in the Nordics. Johan Johander
Founder and Owner
Benchmarking Alliance
Benchmarking Alliance is the leading provider of KPI benchmarking for the Nordic Hospitality Industry. Johan and Benchmarking Alliance have been members and supporters of HSMAI for many years and continue to be our local advisors in the Nordic market.
Johan will present relevant performance data and talk about the status and future outlook of the Nordic Hospitality Industry.
11:45 – 12:15 Travel & Hotel Distribution and Digital Strategies.
OTAs – can't live with them but can't live without them either?
How to build a successful distribution strategy? With an extensive background from both hotels as well as OTAs and now running a company of her own, Åsa will share her thoughts and tips on distribution and digital strategies in Hospitality. Åsa Murphy
Owner & CEO
BizStrat AB
With a background from Radisson, Expedia and most recently Nordic Choice Hotels, Åsa is a member of HSMAI Region Europe's Revenue Management Advisory Board as well as one of HSMAI Region Europe's close advisors in Sweden.
Åsa will share her insights and talk about Travel & Hotel Distribution and Digital Strategies.
12:15 – 13:15 Lunch
13:15 – 14:45 Distribution and Marketing Unconference.
A session without previously set agenda. Each attendee can raise the issues and topics they wish to discuss in smaller groups. The agenda will be formed together on the day of the event.
14:45 – 15:45 Panel discussion:
Evolving distribution landscape and its effect on the requirements for the Sales Manager of the future.
Will cold calling and fam trips still exist in 2020? What is the role of the Sales Manager moving forward? Michael Simon
General Manager Distribution EMEA and APAC
Looking after Distribution for Marriott in EMEA and ASPAC, Michael is one of the most competent Hospitality professionals on distribution and sales. Michael held various pan-European leadership positions within Sales and Marketing at Marriott. He joined HRS in 2009 before rejoining Marriott in 2014 as General Manager – Distribution for EMEA and ASPAC. Michael is a member of the HSMAI Region Europe Sales Executive Advisory Board, and was chosen as one of the top 20 Extraordinary Minds in Sales, Marketing and Technology at the HSMAI European Awards 2015 for his great achievements in the industry.
Michael will discuss the evolving distribution landscape and its affect on the requirements for the Sales Manager of the future.
Nina Nieminen
Director of Revenue Management and Distribution
SOK Travel and Hospitality Industry Chain Management (Sokos Hotels)
One of the brightest minds in all things Revenue and Distribution in Finland, Nina has an impressive background in the Hospitality Industry and is currently the Director of Revenue Management and Distribution at SOK Travel and Hospitality Industry Chain Management (Sokos Hotels). Nina is also a valued member of the HSMAI Region Europe Distribution Advisory Board that started its operation this year.
Nat Estis Green
Senior Global Solutions Engineer
Nat spearheads Global Solution Engineering for Duetto making high-level contributions across several key departments, including sales, marketing and product development. He joined Duetto in 2012, following hospitality- and technology-related roles with InterContinental Hotels Group, Kalibri Labs, Micros eCommerce and El Cid Resorts. He is additionally an advisor to the robotics company Maidbot.
Nat graduated from Cornell University and holds a Bachelors of Science from The School of Hotel Administration. He has also provided analytics support for AH&LA and HSMAI's «Distribution Channel Analysis: a Guide for Hotels» study and is CRME (Certified Revenue Management Executive) certified.
15:45 – 16:00 Wrap-up
16:00 – 19:00 Networking with drinks and snacks
Please note: HSMAI Region Europe are sponsoring students and offer them a very special rate of EUR 75.
CANCELLATION POLICY: Cancellations received three weeks prior to the conference will receive a full refund less a 75 Euro administrative fee. Cancellations received between three and one week out are entitled to a 50% refund. Cancellations received within one week will not be refunded. All cancellations must be received in writing.
"HSMAI has been very active in some parts of the Nordics for many years, and we are excited to expand on this by having our first Distribution & Marketing Strategy Conference in Stockholm this November. We are looking forward to an even stronger presence in Sweden and are thrilled to bring together our European members and the local industry in Sweden," says Ingunn Hofseth, President & CEO of HSMAI Region Europe.
"I am happy that HSMAI Region Europe has chosen Stockholm and the very nice Operakällaren venue for the first Distribution Conference. It is time to show Stockholm to the rest of Europe," says Per-Anders Pettersson, Owner of Benchmarking Alliance and a member of HSMAI Region Europe Revenue Management Advisory Board.
"With Distribution being such an important matter for the industry today, I am thrilled that HSMAI is organizing a Distribution & Marketing Strategy Conference, and I am so pleased that Stockholm will be the first city out. HSMAI is a well established organization with a lot of traction in Europe, and therefore I am also very happy to see that major chains in the Nordics now sign up for the CRME certification too," says Åsa Murphy, Owner & CEO of BizStrat AB and a member of HSMAI Region Europe Revenue Management Advisory Board.
A special thank you to our Gold Sponsor of this event:
HSMAI Region Europe Annual Gold Partners:
Photo: The Operakällaren restaurant in the Stockholm opera
Posted in HSMAI, Markedsføring, Nyheter	Tagged distribusjon, HSMAI Europe, markedsføring, Stockholm
Comfort Hotel og foodora tilbyr hotellsengleveranse av mat fra Oslo beste restauranter
Ukens navn: Kerstin Gibert


TIRR Memorial Hermann is one of the top five rehabilitation facilities in the country. TIRR, which is ranked among the top five institutions in the nation for rehabilitation, focuses on providing care to individuals who have experienced catastrophic injuries or illnesses.
TIRR is a major training site for the Baylor College of Medicine Physical Medicine and Rehabilitation residency training program. Approximately 50 percent of TIRR attending physicians are Baylor faculty. The professionals at TIRR take a comprehensive approach to improving the lives of people with disabilities by delivering outstanding clinical care, training the next generation of practitioners, and supporting cutting edge research programs.
Through Baylor College of Medicine's extensive network of affiliates and collaborators, students and trainees also have access to an exceptional depth and breadth of research training opportunities. View information regarding College of Medicine's education affiliations.


What is a P-20 STEM (Science Technology Engineering Mathematics) Council?
A community forum that provides P-20 leadership, resource development, and issues management in a systematic and strategic fashion. It ensures that all STEM programs and initiatives meet the highest and most rigorous standards and help provide a seamless educational experience for students. The Council does not develop new programs but rather enhances and links existing successful practices.
Why do we need one now?
Although the number of STEM positions available in the work world is growing at five times the rate of other occupations, student achievement in math and science is declining. We are also experiencing a very critical shortage of qualified math and science teachers. The Council tackles achievement gaps in student learning and access to advanced STEM experiences; for instance, the rate of college entry among Santa Barbara County's Latino students, including to two-year community colleges, is just 23%.


Traveling is a wonderful and sometimes odd experience. It is sometimes out of necessity, sometimes for recreation, and sometimes a little of both. Whether you travel for business or pleasure, though, you can likely benefit from some wise advice on the subject. So get ready to read some great ways to make all of your travel more pleasurable.
TIP! Do not display your personal information on a public computer. There could be malware that is tracking your personal info.
Air travel requires a bit of planning. Remember that it's easy to hit traffic on the way to the airport, especially if it's in a big city. Always do your packing the night before you travel. Get yourself completely ready to go the night before your trip. Nothing is worse than missing your flight.
Choosing an aisle seat allows you to have many more options. An aisle seat allows you to move about a bit easier, such as going to the restroom or getting items out of the overhead baggage compartment.
When you are planning for a trip, consider the nature of your trip before choosing a digital camera. If you're going backpacking, a camera with a rechargeable battery might not be the best idea. Usually, you will need a digital camera that's easy to turn on and focuses quickly in order to avoid missing once-in-a-lifetime shots.
TIP! Before you travel, make a list of what you will pack. Start as soon as you can before your trip, that way you can add your must-have items to the list as you think of them.
When you are traveling by car or plane with toddlers, you should always have interesting things available to occupy their time. Bring a lot of toys to avoid stress and hassle. You may want to consider buying a new toy or two as well, so that they have something exciting to occupy their time.
If you are getting a hotel room, inquire as to whether there are any upcoming renovations or not. You do not want to wake up to the sound of construction when vacationing. Try to avoid any renovation dates.
TIP! When traveling by car to a port prior to cruising, search for lodgings where you can park at no cost and make reservations for the night before departure. Ask the staff at the hotel where you stay about parking deals they may offer.
If you plan to travel abroad with kids, try packing a couple cookies sheets. These cookie sheets can be a flat surface to put cards or coloring books on. If you are travelling with little kids, think about bringing magnetic toys like educational numbers and letters.
TIP! Try to work in some physical activity before your flight. Being on a long flight can be incredibly boring.
When traveling overseas, it is wise to prepare for emergencies beforehand, including the loss of your U.S. passport. You can visit different websites to safeguard against this. You should visit the website of the Department of State, which will show you the locations of US Embassies in foreign countries and provide a number for contact. Print out the information and pack it along with your other items. Generally, a replacement passport can be prepared in a matter of days.
Always check expiration dates on of all of your passports. Every country has their own set of laws regarding passports. Certain countries will deny entry if the expiration date on your passport falls within a certain range. Time frames can run anywhere from three months to a year.
TIP! If you want to get the best prices available for a flight you're taking, go to the official web page for each individual airline that flies to the place where you're going. You could find a very good deal this way, and it may even beat that of sites like Travelocity.
If you plan to use traveler's checks, make sure you cash them before you go shopping or out to eat. Traveler's checks can be a pain to work with. Be safe and make sure you get local currency before buying so that you don't get shorted.
At some point, everyone takes a vacation. When you learn about travel ahead of time, the travel experience will be less stressful, more economical and more enjoyable as well. The tips you read here will help you gain the benefits of traveling and learn what it takes to be travel savvy.


If you enjoy a little mystery in your life, then our Calf Sleeve Grab Bag is the deal you've been waiting for!
Each prepackaged Grab Bag contains three pairs of top quality discontinued Calf Sleeves in randomly selected colors (13 possible colors) all in the same size, of course. And while the colors will be a surprise, the price won't. This is a one-time sale and when they're gone...they're gone!
The beauty of our compression calf sleeves is three-fold. First, they offer many of the same feel-good features as our over-the-calf socks, like true graduated compression, moisture control, and muscle support. Second, you get to mix-and-match them with different socks for all your different activities. And third, we think you'll agree, they just look great.
As with all our Grab Bag deals, we guarantee three randomly selected colors in the same size, however, bags are NOT sorted or sold by color, gender or any other criteria other than size. If you are selective about colors, please consider not ordering this Grab Bag.
All sales are final. NO special requests, returns, refunds or exchanges will be accepted. Calf Sleeves are discontinued styles and not from our current line.


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redpajama_set_name: RedPajamaGithub


B&E Boys, based in the Rossendale valley in Lancashire, required an access platform to erect a sign, do some pointing, and inspect guttering at a former mill. The task required 14 metres of outreach at 15m height to reach a chimney – and in order to reach the job site, the machine had to climb two ramps and pass through an opening just 3m high. The access platform also had to be dual fuel, so that it could work in and around the building.
Garry Buck, owner and founder of Viking Access, said, "The customer couldn't believe we got the machine in with the minimum of fuss, and that it had plenty of reach to enable the gutter repairs and other activities.


Edward Sharpe and the Magnetic Zeros played to a sold-out Cabooze Outdoor Plaza in Minneapolis last night, May 25th.
Opening up was He's My Brother She's My Sister, a real-life brother (Robert Kolar) and sister (Rachel Kolar) band. They were quite good and included a tap-dancer on the bass drum (?). They are coming back to Minneapolis to play at the 400 Bar on July 5th, which we'll probably cover because we like them.
They had a short set, starting with "Let's Go". Lots of dancing and they had some Magnetic Zeros on as guest players (drummer came on for a few songs, including "Coattails", and the piano player came on for "Can't See The Stars").
As for Edward Sharpe & the Magnetic Zeros, they came on stage with "40 Day Dream". I thought I counted 12 band members, which is a pretty large group. They did all the hits you would expect them to do, from the new one "Man On Fire" (this one got a great response from the crowd) to an extended version of "Home" (which they ended their set with).
I thought their new song, "If You Wanna" got the crowd moving, even though most of the audience have never heard it. Singer Alex Ebert explained that this is one of many songs they have written but have never recorded.
Highlight for me was whenever Ebert would interact with the audience. He knows how to work a crowd, he's always trying to invade the audience and get a chance to dance with his fans… and the audience loves it.
Edward Sharpe and the Magnetic Zeros are wrapping up their US tour in a few days. They are touring in support of their new album, Here, which is coming out May 29, 2012, via Vagrant Records and Rough Trade Records.


What's your favorite crafting activity? There's no shortage of arts and crafts hobbies to choose from. Scrapbooking, wood crafting, beading, knitting, crocheting, and painting are just a few of the ways you can get your creative juices flowing. The following stores let you do just that. These local arts and crafts stores have the best mix of supplies, tools, and workshops for your next project. Here are the best craft stores in Arizona.
You can unleash your inner artist with the help of Arizona Art Supply. This craft store has been serving Arizona artists since 1951. It's grown to encompass five locations across the state with stores in Phoenix, Tucson, Tempe, Scottsdale and Sun City. Their flagship Phoenix location holds the proud honor of being the largest art supplier in the Southwest. You'll find everything you need for crafting, drawing, painting, sculpting and more at their artist driven shops. They also have over 30,000 items for sale on their online shop on their website.
Are you looking for some Pinspiration? Make a trip to She's Crafty Boutique in Peoria. This Pinterest-inspired craft store has all the tools you need to make your favorite pins. You can choose from their collection of crafting materials to make a do-it-yourself masterpiece. She's Crafty holds open studio hours throughout the week. Their staff is on hand to assist you every step of the way as you make your favorite Pinterest creation at this fun boutique.
The art community knows all about Blick Art Materials. The chain has been arming artists with arts and craft materials since 1911. Blick may have locations nationwide, but local artists can gear up with materials at their Tempe location. As Arizona's only Blick location, the Tempe store is chock full of factory-direct art materials. Blick carries thousands of materials for virtually every kind of crafting project.
Are you ready to get crafty? You don't have to work on your latest crafting project from the confines of your living room. You can book a crafting party at the Crafting Room in Scottsdale instead. This fun event space has everything you need for your next do-it-yourself project. The Crafting Rom is a one-stop shop for everything from wood crafting to painting or scrapbooking. You can even bring snacks and alcoholic beverages for your fun-filled evening of crafting.
You can learn to spin and weave like grandma at Grandma's Spinning Wheel. This locally-owned fiber arts supply shop is filled with all sorts of yarns, needles and crochet tools for your next project. You'll find a huge selection of supplies for knitting, felting and crocheting at this Tucson shop. Grandma's spinning wheel also hosts a variety of events and workshops catered around spinning and weaving.
Looking to put the finishing touches on your latest project? That's where the Ribbon Store comes in. The Ribbon store is a Gilbert-based arts and crafts store dedicated to all things ribbons. They carry over 10,000 different ribbons and even host ribbon related workshops and classes. You can learn all about ribbon tassels, wreaths, hair bows, and necklaces at their shop. The Ribbon Store also helps customers with custom shirt orders for any style.
No arts and crafts project is complete without glitter. You can make your newest project sparkle with a trip to Art Glitter. This locally-owned glitter store has been serving the community since 1981. Today they have over 1200 colors of glitter available in their Cottonwood and Scottsdale retail locations. You can also attend their weekly Glitter 101 course for a hands-on learning experience about all things glitter.
It's not hard to find artwork stamped with a "Made in Sedona" label. Sedona is filled with stores showcasing local artwork and trinkets. You can create your own piece of Sedona art at the Bead Garden. This craft store has a huge selection of tools and supplies for bead projects. They have a large selection of classes where you can learn all about beading. The Bead Garden also has a free weekly beading table where you can use their wide-selection of beading tools for free.
University of Arizona students know all about Posner's Art Store. This locally owned and operated store is located in the heart of Wildcat country on University Boulevard. Posner's has helped the local community with arts and crafts since 1913. Today, they are committed to providing custom framing, mounting, and matting services to the Tucson community. They also carry a selection of screen-printing, paints, drawing supplies, and more.


text:
\section{Introduction}
Floating-point simulation procedures for fractional-order systems are investigated in many current research projects. Such techniques involve --- depending on the system structure --- the numerically efficient and accurate evaluation of functions of the Mittag-Leffler type (basically, if linear models are of interest), the implementation of numerically efficient and robust simulation routines based on the Gr\"unwald-Letnikov differentiation operator (for both linear and nonlinear applications) as well as Laplace domain representations. The latter approach has led to the development of frequency domain-based procedures for linear dynamic systems that are especially applicable to system identification, path planning, and feedback control synthesis for fractional-order system models. A comprehensive software equally applicable for these tasks is \textsc{crone toolbox}, covering time-domain as well as frequency-domain techniques to solve the aforementioned tasks~\cite{MALTI2015769,Podlubny,Oustaloup,Gorenflo:2014,Garrappa_2015,Gorenflo_2002}.

Besides the control-oriented \textsc{crone toolbox}, various numerical schemes for the approximation of fractional-order derivatives and a corresponding numerical integration of systems of fractional-order differential equations have become useful for a variety of engineering applications. Here, especially the {Gr\"unwald-Letnikov operator} that is based on a temporal series expansion of the fractional derivative operator is well-known and serves as the basis for studying the dynamics of engineering applications modeled by fractional-order state equations. Such applications can be found, for example, in simulation, control design, its validation and verification, the design of state-of-charge estimators and model-based aging detection for battery systems. The solution of these tasks relies on
\begin{enumerate}
	\item deriving state equations for an electric equivalent circuit model in which mostly capacitive elements are generalized by corresponding fractional-order constant phase elements,
	\item identifying suitable system parameters by means of impedance spectroscopy, and finally
	\item evaluating the resulting system models numerically for a certain finite time horizon~\cite{ZOU2018286,wang,Rauh_IFAC_2020}.
\end{enumerate}

Here, impedance spectroscopy techniques (which are based on exciting the real-life system by harmonic input signals) help to reveal fractional-order behavior which is characterized by long-term memory effects as well as by frequency domain characteristics with amplitude responses that deviate from the classical integer multiples of $\pm20\,\mathrm{dB}$ per frequency decade and, respectively, deviate from integer multiples of $\pm\frac{\pi}{2}$ in the limit values of their corresponding phase response for large angular frequencies.

The long-term memory effect of fractional-order system models also leads to a drawback concerning numerical simulations based on temporal series expansions such as the Gr\"unwald-Letnikov operator. In such cases, it is necessary to account for a large number of summands in the series expansion to capture the long-term memory effects with sufficient accuracy. Moreover, restarting a time-domain evaluation of fractional-order differential equations purely on the basis of current state information is prone to significant truncation errors. Hence, a significant amount of work has also been invested in the derivation of suitable initialization procedures as well as in the quantification of the respective truncation errors.

In many engineering applications, the number of summands that are employed during a series-based simulation is, however, often restricted to a quite small value. This inherently leads to temporal truncation errors which are, unfortunately, in many cases not quantified in a rigorous manner. This fact becomes even more challenging when system models with uncertain parameters are accounted for. Assuming that for some application at hand the influence of truncation errors can be neglected, a naive generalization of the Gr\"unwald-Letnikov operator-based series expansions to systems with interval parameters would still inevitably lead to unreasonably wide interval enclosures. This is mostly caused by the mutual couplings between various state variables which --- if characterized by finitely large interval boxes in a multi-dimensional state space --- would have to be propagated through each of the summands in the series expansion and therefore provoke the wrapping effect that is also well-known from solving initial value problems for sets of classical ordinary differential equations~\cite{Ned:2006}.

However, if it can be proven beforehand that the state equations under investigation possess the property of asymptotic stability, quasi-analytic representations of the enclosure of the systems' time responses can be defined by using Mittag-Leffler functions~\cite{Rauh_mmar_2019,Rauh_ECC_2020}. Those functions represent a generalization of classical exponential-type solutions which were originally developed for integer-order differential equations with the goal to suppress the wrapping effect in a computationally inexpensive manner for system models with a-priori proven stability properties.

For further general discussions about modeling dynamic systems by using fractional-order representations, their identification, as well as their potential use in control engineering tasks, the reader is referred to~\cite{Oustaloup}. Moreover, the paper~\cite{Rauh_Acta_2020} provides an overview of the state-of-the-art concerning already existing approaches for a verified simulation of fractional-order models. These include the definition of differential inclusions, the exploitation of cooperativity (i.e., monotonicity of solutions with respect to initial conditions) and positivity of state variables in simulation as well as state estimation, nonlinear time transformations to cast fractional-order models into equivalent integer-order representations, and, finally, also the extension of Picard iteration procedures~\cite{Lyons} to non-integer-order models.

Following the summary of required preliminaries and the problem formulation in Sec.~\ref{sec:prelim} as well as an exemplary comparison between low-dimensional fractional-order system models and significantly higher-dimensional approximations in terms of integer-order formulations in Sec.~\ref{sec:lin_frac_comparison}, a Mittag-Leffler type state enclosure technique derived in~\cite{Rauh_Acta_2020,Rauh_ECC_2020} for fractional-order models is reviewed in Sec.~\ref{sec:exponential} together with specific computational aspects for an interval-based evaluation of Mittag-Leffler type functions in Sec.~\ref{sec:interval_MLF}. In contrast to previous work of the authors, where scalar differential equations were in the focus of mostly theoretical applications, Sec.~\ref{sec:example} aims at applying the suggested iteration procedure to a non-scalar, practically motivated simplified dynamic battery model with uncertain parameters. Finally, conclusions and an outlook on future work are given in Sec.~\ref{sec:concl}.

\section{Preliminaries}\label{sec:prelim}
In this paper, a verified simulation routine for fractional-order system models that is based on a differential formulation of the Picard iteration is employed. It exploits Mittag-Leffler functions to represent guaranteed state enclosures for which the free parameters (describing a kind of decay rate of the initial conditions toward the equilibrium) are computed in an iterative manner.

Throughout this paper, Mittag-Leffler functions~\cite{Haubold,Gorenflo:2014,Garrappa_2015} are given by their two-parameter representation
\begin{equation}\label{eq:Mittag_Leffler}
E_{\nu,\beta}(\zeta) = \sum\limits_{i=0}^\infty
\frac{\zeta^i}{\Gamma\rb{\nu i + \beta}}
\end{equation}
with the general argument $\zeta \in \mathbb{C}$ as well as the parameters $\nu \in \mathbb{R}_+$ and $\beta \in \mathbb{R}$. Note, although this function is defined for each point $\zeta$ in the complex plane, we will restrict the discussion to the case of real-valued arguments and outputs of the Mittag-Leffler function.
In addition, assume that all dynamic systems in this paper are given in terms of explicit, autonomous, time-invariant\footnote{Note, the restriction to autonomous, time-invariant systems can be removed by the introduction of auxiliary state variables for the time argument as well as for time- and state-dependent expressions included in control inputs. Corresponding procedures, leading to an increase of the system dimension, were discussed, for example, in~\cite{Rauh_SCAN14} for the integer-order case.} state equations which are re-written according to Def.~\ref{def:quasi_lin} into a quasi-linear form. This reformulation enhances the efficiency of the numerical evaluation of state enclosures, especially with respect to suppressing the wrapping effect.

\begin{definition}[\label{def:quasi_lin}Quasi-linear fractional-order system model of Caputo type]
	A commensurate-order quasi-linear set of fractional-order differential equations of Caputo type~\cite{Oustaloup,Podlubny} is defined by
	\begin{equation}\label{eq:state_space_frac}
	\begin{split}
	{}_{t_0=0}\mathcal{D}_t^\nu\mathbf{x}(t) =
	{\mathbf{x}}^{\rb{\nu}}(t) & =
	{\mathbcal{f}}\rb{{\mathbf{x}}(t)} =:
	{\mathbcal{A}}\rb{{\mathbf{x}}(t)} \cdot {\mathbf{x}}(t)~,~~
	{\mathbf{x}}(t) \in \mathbb{R}^n~,~~
	{\mathbcal{A}}\rb{{\mathbf{x}}(t)} \in \mathbb{R}^{n\times n}

	~,~~
	0 < \nu <1
	\enspace ,
	\end{split}
	\end{equation}
	where $\nu$ is the non-integer derivative order and ${\mathbcal{f}}\rb{{\mathbf{x}}(t)}$ the nonlinear right-hand side of the system model re-written into a quasi-linear form; initial conditions  $\mathbf{x}(t_0 = 0)$ are specified in terms of the interval bounds
	\begin{equation}\label{eq:init_x}
	\mathbf{x}(0) \in \sqb{\mathbf{x}}(0) = \intv{\ul{\mathbf{x}}(0)}{\ol{\mathbf{x}}(0)}
	\enspace .
	\end{equation}
\end{definition}

In~(\ref{eq:init_x}), the initial state vector ${\mathbf{x}}(0)$ is described by the interval representation $\sqb{\mathbf{x}}(0)$, where the inequalities ${\ul{{x}}_i(0)}\le{\ol{{x}}_i(0)}$ hold element-wise for each vector component $i \in \{1,\ldots,n\}$.

\begin{definition}[\label{def:diag_dominant}Diagonally dominant model] Diagonally dominant quasi-linear system models are given by the state-space representation
	\begin{equation}\label{eq:diag_dom}
	\begin{split}
	{}_{0}\mathcal{D}_t^\nu\mathbf{z}(t) =
	{\mathbf{z}}^{\rb{\nu}}(t)
	& = \mathbf{f}\rb{{\mathbf{z}}(t)} =
	\rb{{\mathbf{T}}^{-1} \cdot
		{\mathbcal{A}}\rb{\mathbf{T} \cdot {\mathbf{z}}(t)} \cdot \mathbf{T}} \cdot {\mathbf{z}}(t)  =:
	{\mathbf{A}}\rb{{\mathbf{z}}(t)} \cdot {\mathbf{z}}(t)
	\end{split}
	\end{equation}
	after a suitable similarity transformation
	\begin{equation}\label{eq:trafo}
	{\mathbf{x}}(t) =
	{\mathbf{T}} \cdot {\mathbf{z}}(t)~,~~
	{\mathbf{T}} \in \mathbb{R}^{n \times n}~,~~
	{\mathbf{z}}(t) \in \mathbb{R}^{n}
	\end{equation}
	of the system model in Def.~\ref{def:quasi_lin}.
\end{definition}

\begin{remark}
	In this paper, we restrict ourselves to the case of real-valued similarity transformations in~(\ref{eq:trafo}). These transformations lead to the real-valued initial state enclosures
	\begin{equation}\label{eq:init_cond_trafo}
	\mathbf{z}(0) \in \sqb{\mathbf{z}}(0) =
	\mathbf{T}^{-1} \cdot \sqb{\mathbf{x}}(0)
	\enspace .
	\end{equation}
	According to~\cite{Rauh_SCAN14,Rauh_mmar_2019}, also complex-valued similarity transformations are possible which are advantageous for the case of systems with conjugate-complex eigenvalues and, hence, oscillatory dynamics. For both the real- and complex-valued case with system models having an eigenvalue multiplicity of one, the transformation matrix ${\mathbf{T}}$ is composed of the eigenvectors of ${\mathbcal{A}}\rb{\mathbf{x}_\mathrm{m}}$, computed at the interval midpoint $\mathbf{x}_\mathrm{m} = \frac{1}{2}\cdot \rb{{\ul{\mathbf{x}}(0)}+{\ol{\mathbf{x}}(0)}}$. Transformations applicable to systems with higher eigenvalue multiplicities were derived in~\cite{Rauh_SCAN14} for dynamic models with integer-order derivatives.
\end{remark}

\begin{remark}
	Where it is necessary for a compact notation of the iteration procedure given in the following section, it is further assumed that a translation of the state vector has been performed prior to solving the considered simulation task (in accordance with Defs.~\ref{def:quasi_lin} and~\ref{def:diag_dominant}) so that the state trajectories converge to the origin of the state space if the dynamics are asymptotically stable.
\end{remark}

\section{Comparison Between Linear Fractional-Order Models and Classical Integer-Order Approximations}\label{sec:lin_frac_comparison}

To illustrate fundamental differences between linear fractional-order differential equations and integer-order system representations, consider the dynamic system model
\begin{equation}
y^{(0.5)} (t) = -y(t) + u(t)
\end{equation}
with the input signal $u(t)$ and the system output $y(t)$. As an example, $u(t)$ and $y(t)$ may represent the total current and the voltage drop, respectively, over a parallel connection of an Ohmic resistor and a constant phase element with a maximum phase shift of $-\frac{\pi}{4}$. Such elements appear typically in the domain of electrochemical impedance spectroscopy~\cite{ZOU2018286,wang}. In the frequency domain, these so-called Warburg elements are expressed by
\begin{equation}\label{eq:ex_frac1}
F(\jmath\omega) = \frac{Y(\jmath\omega)}{U(\jmath\omega)}
=
\frac{1}{1+(\jmath \omega)^{0.5}}
\enspace.
\end{equation}

As mentioned in the introduction, parameter identification and numerical simulation of such elements are either possible by using specialized routines for fractional-order system models (as it is considered also in this paper) or by approximating them by finite-dimensional sets of ordinary differential equations. A possibility for doing so is based on firstly approximating the fractional differentiation operator by a suitable transfer function and subsequently transforming this system model back into the time domain, leading to the approximating state-space model
\begin{equation}\label{eq.approx}
\dot{\tilde{\mathbf{z}}}(t) = \mathbf{A} {\tilde{\mathbf{z}}}(t) + \mathbf{b} u(t)
\quad \text{with}
\quad
y(t) \approx \mathbf{c}^T {\tilde{\mathbf{z}}}(t) + d u(t)
\enspace .
\end{equation}
If the recursive approximation routines developed by A.~Oustaloup, cf.~\cite{Khazali,Oustaloup,817385} as well as the freely available \textsc{Matlab} implementation~\cite{Chen}, are used for that purpose, the approximations in Fig.~\ref{fig:ex_nyqist} are obtained, where the frequency range $\omega \in \intv{0.01}{100}\,\mathrm{\frac{rad}{s}}$ over which the approximation is performed, is highlighted by the vertical dotted lines in Figs.~\ref{fig_ampl} and~\ref{fig_phase}. Here, the black dashed curves denote the exact fractional-order behavior, while the red and blue models correspond to approximations of~(\ref{eq:ex_frac1}) in terms of rational transfer functions of the orders $5$ (red) and $11$ (blue), respectively.

Exemplarily, the numerical values for the approximating system matrices in~(\ref{eq.approx}) are given by
\begin{equation}
\begin{split}
\setcounter{MaxMatrixCols}{11}
\mathbf{A} &=
\scalebox{0.7}{$
\begin{bmatrix} -559.71 & -277.1 & -76.197 & -24.835 & -9.9316 & -2.4829 & -0.77609 & -0.29764 & -0.067652 & -0.017081 & -0.0039062\\ 256.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 128.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 32.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 8.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 4.0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1.0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0.25 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.125 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.03125 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.0078125 & 0 \end{bmatrix}
$},
\\
\mathbf{b}& =
\begin{bmatrix} 8.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}^T,\\
\mathbf{c}^T &=
\scalebox{0.8}{$
\begin{bmatrix} 1.7692 & 2.3998 & 1.3484 & 0.77563 & 0.48601 & 0.16983 & 0.066826 & 0.029657 & 0.0073522 & 0.0019502 & 0.00045835 \end{bmatrix}$},\\
d &= 0.030653
\enspace.
\end{split}
\raisetag{5mm}
\end{equation}
By a comparison of the frequency responses and Nyquist plots for both chosen approximation orders in Fig.~\ref{fig:ex_nyqist}, it can be concluded that all state variables of the integer-order approximation given above have a significant contribution to the approximation of the fractional-order behavior. If lower approximation orders were used, the quality of approximation deteriorates which can be seen by the oscillating nature of the red curves in Fig.~\ref{fig:ex_nyqist}. Obviously, the advantage of working directly with the fractional-order model is the significantly smaller number of required parameters and state variables.

\begin{figure}[htp]
	\centering
	\subfloat[{\label{fig_ampl}}Magnitude response.]{\resizebox{0.32 \linewidth}{!}{{\input{ex_nyquist/fig_ex_SNR20_ampl.pstex_t}}}}
	\hfill
	\subfloat[{\label{fig_phase}}Phase response.]{\resizebox{0.32 \linewidth}{!}{{\input{ex_nyquist/fig_ex_SNR20_phase.pstex_t}}}}
	\hfill
	\subfloat[{\label{fig_nyquist}}Nyquist plot with selected frequencies  $\ul{\omega}=0.01\mathrm{\frac{rad}{s}}$ ($\circ$) and $\ol{\omega}=100\mathrm{\frac{rad}{s}}$ ($\diamond$).]{\resizebox{0.32 \linewidth}{!}{{\input{ex_nyquist/fig_ex_SNR20_nyquist.pstex_t}}}}
	\caption{{\label{fig:ex_nyqist}}Frequency domain analysis of the fractional-order model~(\ref{eq:state_space_frac}) in comparison with different integer-order approximations (black dashed: fractional-order model; {\color{red}{red solid}}: 5th order approximation; {\color{blue}{blue solid}}: 11th order approximation).}
\end{figure}

\section{Mittag-Leffler Type State Enclosures for Fractional-Order Differential Equations}\label{sec:exponential}
The focus of this section is the summary of a Mittag-Leffler type state enclosure technique for sets of commensurate fractional-order models. The fundamental iteration summarized in the following Theorem~\ref{thm:MLF_encl} was first published by the authors in~\cite{Rauh_mmar_2019} and~\cite{Rauh_ECC_2020} with a further in-depth discussion of the infinite-horizon memory property that becomes crucial as soon as the integration time horizon is divided into several, finitely long temporal subslices in~\cite{Rauh_Acta_2020}.

\begin{definition}[\label{def:MLF_encl}Mittag-Leffler type state enclosure]
	The time-dependent expression
	\begin{equation}\label{eq:sol_MLF}
	{\mathbf{z}}^*(t) \in
	\mathbf{E}_{\nu,1}{\left(\left[\boldsymbol\Lambda\right] \cdot t^\nu \right)} \cdot
	\left[{\mathbf{z}}_{e}\right]\left(0\right)
	~,~~
	\mathbf{E}_{\nu,1}{\left(\left[\boldsymbol\Lambda\right] \cdot t^\nu \right)}
	:= \diag{{E}_{\nu,1}{\left(\left[\lambda_i\right] \cdot t^\nu \right)}}
	~,~~
	\sqb{{\mathbf{z}}_{e}}\left(0\right) = \sqb{{{\mathbf{z}}}_0}
	\end{equation}with the diagonal parameter matrix
	%
	$
	\left[\boldsymbol\Lambda\right] := \diag{\left[\lambda_i\right]}$, $i\in \{1,\ldots,n\}$,
	is denoted as a \emph{verified Mittag-Leffler type state enclosure} for the system model~(\ref{eq:diag_dom}) with~(\ref{eq:init_cond_trafo}) if it is determined according to Theorem~\ref{thm:MLF_encl}.
\end{definition}

\begin{theorem}[\cite{Rauh_ECC_2020,Rauh_Acta_2020,Rauh_mmar_2019}~\label{thm:MLF_encl}Iteration for Mittag-Leffler type enclosures]
	The Mittag-Leffler type state enclosure~(\ref{eq:sol_MLF}) is guaranteed to contain the set of all reachable states ${\mathbf{z}}^*(T)$ at the point of time $t=T>0$ according to
	\begin{equation}
	{\mathbf{z}}^*(T) \in
	\mathbf{E}_{\nu,1}{\left(\left[\boldsymbol\Lambda\right] \cdot T^\nu \right)} \cdot
	\left[{\mathbf{z}}_{e}\right]\left(0\right)\, ,~
	{\mathbf{z}}^*(t) \in \mathbb{R}^{n}
	\enspace ,
	\end{equation}
	if $\left[\boldsymbol\Lambda\right]$ is set to the outcome of the converging iteration
	\begin{equation}\label{eq:iteration_MLF}
	\left[{\lambda}_i\right]^{\dbr{\kappa +1}}
	:=
	\frac{
		{{f}_i}\left(
		\mathbf{E}_{\nu,1}{\left(\left[\boldsymbol\Lambda\right]^{\dbr{\kappa}} \cdot \sqb{t}^\nu\right)} \cdot
		\left[{\mathbf{z}}_{e}\right]\left(0\right)
		\right)}
	{
		{E}_{\nu,1}{\left(\left[\lambda_i\right]^{\dbr{\kappa}} \cdot \sqb{t}^\nu\right)} \cdot
		\left[{{z}}_{e,i}\right]\left(0\right)
	}
	\enspace ,
	\end{equation}
	$i\in \{1,\ldots,n\}$, with the prediction horizon $\sqb{t} = \intv{0}{T}$.
\end{theorem}

\begin{proof}
	According to~\cite{Dorjgotov,Ghosh}, the exact solution of a linear fractional-order differential equation
	\begin{equation}\label{eq:sol_frac1}
	z^{(\nu)}(t) = \lambda \cdot z(t)
	\end{equation}
	of Caputo type, for which solely initial state information at the point $t=0$ is available, is given by the analytic expression
	\begin{equation}\label{eq:sol_frac2}
	z(t) = E_{\nu,1}\rb{\lambda t^\nu} \cdot z(0)
	\enspace .
	\end{equation}
	The relation~(\ref{eq:sol_frac2}) serves as an ansatz for describing the verified state enclosures according to Def.~\ref{def:MLF_encl} by substituting it according to~\cite{Rauh_Acta_2020,Rauh_ECC_2020} into
	the differential formulation of the Picard iteration
	%
	\begin{equation}\label{eq:exp1_fde}
	\begin{split}
	{\mathbf{z}}^{\rb{\nu}}(t) & \in
	\rb{\left[\boldsymbol\Lambda\right]^{\dbr{\kappa +1}}} \cdot
	\mathbf{E}_{\nu,1}{\left(\left[\boldsymbol\Lambda\right]^{\dbr{\kappa +1}} \cdot t^\nu \right)}
	\cdot
	\left[{\mathbf{z}}_{e}\right]\left(0\right)\\
	& =
	{\mathbf{f}}\left(
	\mathbf{E}_{\nu,1}{\left(\left[\boldsymbol\Lambda\right]^{\dbr{\kappa}} \cdot t^\nu \right)} \cdot
	\left[{\mathbf{z}}_{e}\right]\left(0\right)
	\right) =:
	{\mathbf{f}}\left(\left[{\mathbf{z}}_{e}
	\right]^{\dbr{\kappa}}(\sqb{t})
	\right)
	\enspace  .
	\end{split}
	\raisetag{13mm}
	\end{equation}

\noindent Overapproximating the Mittag-Leffler type state enclosure
	$\mathbf{E}_{\nu,1}{\left(\left[\boldsymbol\Lambda\right]^{\dbr{\kappa+1}} \cdot t^\nu \right)}$ in the iteration step \mbox{$\kappa+1$} by the enclosure ${\left[{\mathbf{z}}_{e}
		\right]^{\dbr{\kappa}}(\sqb{t})}$ obtained in the previous iteration step~ $\kappa$ --- under the assumption
	that \linebreak \mbox{${\left[{\mathbf{z}}_{e}
		\right]^{\dbr{\kappa+1}}(\sqb{t})} \subseteq {\left[{\mathbf{z}}_{e}
		\right]^{\dbr{\kappa}}(\sqb{t})}$} holds due to convergence of the iteration formula ---
	 on the first line of~(\ref{eq:exp1_fde})
	leads to
	%
	\begin{equation}
	{\diag{\sqbs{\tilde{\lambda}_i}^{\dbr{\kappa +1}}}}  \cdot
	{\left[{\mathbf{z}}_{e}
		\right]^{\dbr{\kappa}}(\sqb{t})} =
	{\mathbf{f}}\left(\left[{\mathbf{z}}_{e}
	\right]^{\dbr{\kappa}}(\sqb{t})
	\right)
	\enspace .
	\end{equation}
	Solving this expression for $\sqbs{\tilde{\lambda}_i}^{\dbr{\kappa +1}}$ with subsequently renaming this parameter into $\left[{\lambda}_i\right]^{\dbr{\kappa +1}}$ completes the proof of Theorem~\ref{thm:MLF_encl}.
\end{proof}

\begin{remark}
	Quasi-linear state-space representations of fractional-order differential equations in diagonally dominant form according to Def.~\ref{def:diag_dominant} can be simulated efficiently by the symbolically simplified iteration scheme
	\begin{equation}\label{eq:MLF_mod}
	\left[{\lambda}_i\right]^{\dbr{\kappa +1}}
	:= a_{ii}\rb{\sqb{{\mathbf{z}}_{e}}^{\dbr{\kappa}}\left(\sqb{t}\right)}
	+ \sum\limits_{
		{j=1}\atop{j \neq i}
	}^n\left\{
	a_{ij}\rb{\sqb{{\mathbf{z}}_{e}}^{\dbr{\kappa}}\left(\sqb{t}\right)} \cdot
	\frac{E_{\nu,1}\rb{\left[\lambda_j\right]^{\dbr{\kappa}}\cdot \sqb{t}^\nu}}
	{E_{\nu,1}\rb{\left[\lambda_i\right]^{\dbr{\kappa}}\cdot \sqb{t}^\nu}}
	\cdot
	\frac{
		\sqb{{{z}}_{e,j}}\left(0\right)
	}
	{
		\sqb{{{z}}_{e,i}}\left(0\right)
	}
	\right\} \enspace.
	\end{equation}
	In~(\ref{eq:MLF_mod}), the quotient of two Mittag-Leffler functions cannot be simplified further in the general case. This imposes further restrictions on the numerical evaluation of this iteration scheme by means of interval techniques as shown in the following section.
\end{remark}

\begin{remark}
	For the order $\nu=1$, the iteration formula in Theorem~\ref{thm:MLF_encl} becomes identical to the solution published, for example, in~\cite{Rauh_SCAN14} due to the identity $E_{1,1}(z) \equiv e^z$.
\end{remark}

\section{Evaluating Mittag-Leffler Functions for Interval Arguments}\label{sec:interval_MLF}
As described in~\cite{Haubold}, it is possible to determine rough outer bounds for the range of a Mittag-Leffler function that is evaluated over a finitely large real-valued interval argument. These bounds result from the fact that the Mittag-Leffler function can be interpreted as a continuous interpolation between Gaussian (exponential) and Lorentzian (rational) functions. Due to this property, the bounds
\begin{equation}\label{eq:bound_ml}
\exp\rb{-\zeta} < E_{\nu,1}\rb{-\zeta} \le \frac{1}{1+\zeta}
\qquad \text{and} \qquad
\exp\rb{-\zeta^2} < E_{\nu,1}\rb{-\zeta^2} \le \frac{1}{1+\zeta^2}
\end{equation}
can be determined for $\zeta \ge 0$. A visualization of these bounds in terms of the dependency on the parameter range $\nu \in \intv{0}{1}$ for the fractional differentiation order can be found in~\cite{Rauh_Acta_2020}.

Because these bounds are usually too conservative for the interval-based evaluation of the iteration procedure presented in the previous section, floating point evaluations of the Mittag-Leffler function using the \textsc{Matlab} implementation of R.~Garrappa~\cite{Garrappa_2015} are extended in the following to obtain tighter guaranteed interval bounds for the case of real-valued arguments to which this paper is restricted.

\subsection{Interval Evaluation of the Mittag-Leffler Function with Real Arguments}
\begin{theorem}[\cite{Rauh_ECC_2020,Rauh_Acta_2020}~\label{thm:MLF_int}Interval bounds for the Mittag-Leffler  function with real arguments]\mbox{$~~$}\newline
	Interval evaluations of the two-parameter Mittag-Leffler function with real-valued arguments \linebreak$z \in \sqb{z} = \intv{\ul{z}}{\ol{z}}$ are given by
	\begin{equation}
	\begin{split}
	E_{\nu,\beta}\rb{\sqb{z}} & \in \sqb{{E}^*_{\nu,\beta}\rb{\sqb{z}}} =
	\sqb{\tilde{E}_{\nu,\beta}\rb{\sqb{z}}} +
	\frac{\epsilon}{1+  \epsilon} \cdot
	\rb{1+\left|\sqb{\tilde{E}_{\nu,\beta}\rb{\sqb{z}}}\right|} \cdot
	\intv{-1}{1}
	\end{split}
	\raisetag{13mm}
	\end{equation}
	with the tolerance value $\epsilon>0$ of a floating point evaluation of~(\ref{eq:Mittag_Leffler}) and the interval definition
	\begin{equation}\label{eq:MLF_int}
	\sqb{\tilde{E}_{\nu,\beta}\rb{\sqb{z}}} = \intv{\bigtriangledown \tilde{E}_{\nu,\beta}\rb{\ul{z}}}{\bigtriangleup \tilde{E}_{\nu,\beta}\rb{\ol{z}}}
	\end{equation}
	in which $\bigtriangledown$ and $\bigtriangleup$ denote switchings of the rounding mode toward minus and plus infinity, respectively, in a (CPU-based) floating point evaluation.
\end{theorem}

\subsection{Exploitation of Monotonicity in Interval Evaluations of the Mittag-Leffler Function}
To reduce overestimation in the interval evaluation of the iteration procedure according to Theorem~\ref{thm:MLF_encl}, monotonicity properties of the Mittag-Leffer function need to be accounted for with respect to the time $t$, the solution parameters $\lambda_i$, and the derivative order $\nu$. For a description of all corresponding monotonicity properties, the reader is referred to~\cite{Rauh_Acta_2020} and~\cite{Rauh_ECC_2020}.

\subsection{Bounding the Temporal Truncation Error due to the Infinite Memory Property of Fractional-Order Systems}\label{sec:discr_errors}
Fractional-order systems are characterized by an infinite memory of previous states~\cite{Podlubny,Oustaloup}. Therefore, the restart of the simulation of a dynamic system according to Def.~\ref{def:diag_dominant} at some point of time $t=t_{k+1}$ does not only have to account for state information $\mathbf{z}(t_{k+1})$ computed by a simulation originally initiated at some previous point $t_k < t_{k+1}$, but also the effect of a temporal truncation error that can be expressed by component-wise bounds originating from a replacement of the fractional-order derivative of order $\nu$ with a memory start at $t=t_k$ by a new starting point $t=t_{k+1}$. The corresponding derivative operators are subsequently denoted by ${}_{t_k}\mathcal{D}_t^\nu\mathbf{z}(t)$ and ${}_{t_{k+1}}\mathcal{D}_t^\nu\mathbf{z}(t)$, respectively. Note, the following Theorems~\ref{thm:mu} and~\ref{thm:contractor} are generalizations of those given in~\cite{Rauh_ECC_2020,Rauh_Acta_2020}, focusing especially on non-scalar applications.

\begin{theorem}[\label{thm:mu}Bounds for temporal truncation errors]
	Resetting the initial point of time of the integration of fractional-order models defined in Def.~\ref{def:diag_dominant} based on Theorem~\ref{thm:MLF_encl} after completion of a time interval of length $T$ requires the inflation of the right-hand side of the state equations by the symmetric interval $\intv{-\boldsymbol{\mu}}{\boldsymbol{\mu}}$ at the point $t_k+T$ with
	\begin{equation}
	\boldsymbol{\mu} :=
	\frac{{\boldsymbol{\mathbcal{Z}}} \cdot  \rb{t_k+T}^{-\nu}}{\left|\Gamma(1-\nu)\right|}
	\end{equation}
	and the component-wise defined supremum of the set of reachable states
	\begin{equation}\label{eq:sup_States}
	\mathcal{Z}_i = \sup\limits_{t \in \intv{t_0}{t_{k+1}}} \left|z_i(t)\right|
	\enspace .
	\end{equation}
	The re-initialized initial value problem is then given by
	\begin{equation}\label{eq:new_ivp}
	{}_{t_k+T}\mathcal{D}_t^\nu\mathbf{z}(t) =
	\mathbf{z}^{\rb{\nu}}(t) =
	\mathbf{f}\rb{\mathbf{z}(t)} + \intv{-\boldsymbol{\mu}}{\boldsymbol{\mu}}
	=:\tilde{\mathbf{f}}\rb{\mathbf{z}(t)}
	\end{equation}
	with the initial state enclosure ${\mathbf{z}}(t_{k}+T) \in \sqb{\mathbf{z}}(t_{k}+T)$ resulting form the solution of
	\begin{equation}
	{}_{t_0}\mathcal{D}_t^\nu\mathbf{z}(t) =
	\mathbf{z}^{\rb{\nu}}(t) =
	\mathbf{f}\rb{\mathbf{z}(t)}
	\quad
	\text{with}
	\quad
	{\mathbf{z}}(t_{0}) \in \sqb{\mathbf{z}}(t_{0})~,~~t_0=0
	\enspace .
	\end{equation}
\end{theorem}

\begin{proof}
	Theorem~\ref{thm:mu} is a consequence of the
	component-wise defined error bounds for a general fractional derivative operator of a commensurate system model on the time interval $t_k+T \le t \le t_{k+1}$ that can be computed according to
	\begin{equation}\label{eq:bounds_truncation}
	\left|{}_{t_k}\mathcal{D}_t^\nu\mathbf{z}(t) - {}_{t_k+T}\mathcal{D}_t^\nu\mathbf{z}(t)\right|
	\le
	\frac{{\mathbcal{Z}} T^{-\nu}}{\left|\Gamma(1-\nu)\right|}
	=: \boldsymbol{\mu}
	\end{equation}
	In correspondence with~\cite{Podlubny}, Eq.~(\ref{eq:bounds_truncation}) relies on the component-wise defined supremum~(\ref{eq:sup_States}) of the reachable states denoted by the vector ${\mathbcal{Z}}$.
\end{proof}

\begin{theorem}[\label{thm:contractor}Contractor for the state enclosure of fractional-order systems]
	Assume that a reference solution $\mathbf{z}(t) \in \sqb{\mathbf{z}_{\mathrm{ref}}}(t)$ has already been computed for the initial value problem with the initial point of time $t=t_k$ that is valid up to the point $t = t^* > t_k+T$
	and that the application of Theorem~\ref{thm:MLF_encl} to the re-initialized initial value problem~(\ref{eq:new_ivp}) in Theorem~\ref{thm:mu}
	with the initial point of time $t=t_k+T$ has provided the interval bounds
	$\sqb{\mathbf{z}_e}(t) =
	\exp\left(\left[\boldsymbol\Lambda\right]\cdot \rb{t-(t_k+T)}\right) \cdot
	\sqb{{\mathbf{z}}_{e}}\left(t_k+T\right)$ that are also valid up to $t = t^*$
	with the associated solution parameters $\sqb{\lambda_i}$, $i \in \{1,\ldots,n\}$. A contractor is then given by
	\begin{equation}
	\sqb{\lambda_i} := \sqb{\lambda_i} \cap \sqbs{\tilde{\lambda}_i}
	\end{equation}
	with
	\begin{equation}\label{eq:contr_lambda}
	\sqbs{\tilde{\lambda}_i} = \frac{\tilde{f}_i\rb{\sqb{\mathbf{z}_e}(\intv{t_k+T}{t^*})}
		\cap \tilde{f}_i\rb{\sqb{\mathbf{z}_{\mathrm{ref}}}(\intv{t_k+T}{t^*})}
	}{
		{\sqb{z_{e,i}}(\intv{t_k+T}{t^*})} \cap {\sqb{z_{i,\mathrm{ref}}}(\intv{t_k+T}{t^*})}
	}
	\enspace .
	\end{equation}
\end{theorem}

\begin{proof}
	The validity of Theorem~\ref{thm:contractor} is a direct consequence of the fact that both $\sqb{\mathbf{z}_{\mathrm{ref}}}(t)$ and $\sqb{\mathbf{z}_e}(t)$ are verified state enclosures according to
	$\mathbf{z}(t) \in \sqb{\mathbf{z}_{\mathrm{ref}}}(t)$ and $\mathbf{z}(t) \in \sqb{\mathbf{z}_e}(t)$ and thus have to satisfy the fractional-order differential equation in the componentwise notation~(\ref{eq:exp1_fde}). Intersecting the evaluation of~(\ref{eq:exp1_fde}) for both state enclosures and solving the result for the interval of the solution parameter $\sqbs{\tilde{\lambda}_i}$ yields the relation~(\ref{eq:contr_lambda}).
\end{proof}

\section{Numerical Examples}\label{sec:example}

\subsection{Interval Bounds for Different Integration Horizons for a Nonlinear Example}\label{sec:ex_nonlin}
Consider the nonlinear example
\begin{equation}\label{eq:example}
z^{(\nu)}(t) = p\cdot z^3(t) = p \cdot a(z(t)) \cdot z(t)
\end{equation}
with an uncertain initial state $z(0) \in \sqb{z}(0)$, the interval parameter $p \in \sqb{p}$, and the uncertain differentiation order $\nu \in \sqb{\nu}$
for the application of Theorem~\ref{thm:MLF_encl}. This example was already studied in~\cite{Rauh_ECC_2020} without a subdivision of the integration time horizon as well as without a restart of the integration according to Sec.~\ref{sec:discr_errors}. Due to the quasi-linear formulation of Eq.~(\ref{eq:example}), the modified iteration formula given in~(\ref{eq:MLF_mod}) becomes directly applicable.\\

\noindent
In the following, two cases with different amount of uncertainty are distinguished:
\begin{description}
	\item[Case~a:] $\sqb{z}(0) = \intv{0.99}{1.0}$,
	$\sqb{p}  = \intv{-2}{-1.99}$,
	$\sqb{\nu} = \intv{0.8}{0.81}$ and
	\item[Case~b:] $\sqb{z}(0) = \intv{0.5}{1.0}$,
	$\sqb{p}  = \intv{-2}{-1}$,
	$\sqb{\nu} = \intv{0.8}{0.9}$.
\end{description}

Fig.~\ref{fig:example} provides a comparison of the effect of reducing the integration step size $T$ according to the Theorems~\ref{thm:mu} and~\ref{thm:contractor}. In all presented results, an equidistant discretization step size was applied. Note, optimizing either an equidistant or adaptive integration step size will be one of the subjects of future work.
It can be seen that --- especially for \textbf{Case~a} --- the reduction of the integration step size leads to a significant reduction of overestimation in the computed state enclosures. This effect is not that much visible for \textbf{Case~b}, where step sizes smaller than $T = 2^{-2}$ lead to almost identical results, except for the very beginning of the integration time horizon. In both cases, it can be observed that the lower interval bounds of the state variable $z(t)$ are seemingly not influenced by overestimation. This can be concluded from the fact that a reduction of overestimation only occurs at the computed supremum when reducing the step size $T$.

\begin{figure}[htp]
	\centering
	\subfloat[{\label{fig_small1}}State enclosures for \textbf{Case~a}; integration step sizes \mbox{$T=1$} (blue),  $T=2^{-1}$ (red), $T=2^{-2}$ (yellow).]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_SNR_klein_1.pstex_t}}}}
	\hfill
	\subfloat[{\label{fig_large1}}State enclosures for \textbf{Case~b}; integration step sizes \mbox{$T=1$} (blue),  $T=2^{-1}$ (red), $T=2^{-2}$ (yellow).]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_SNR_gross_1.pstex_t}}}}
	\\
	\subfloat[{\label{fig_small2}}State enclosures for \textbf{Case~a}; integration step sizes \mbox{$T=2^{-2}$} (blue),  $T=2^{-3}$ (red), $T=2^{-4}$ (yellow).]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_SNR_klein_2.pstex_t}}}}
	\hfill
	\subfloat[{\label{fig_large2}}State enclosures for \textbf{Case~b}; integration step sizes \mbox{$T=2^{-2}$} (blue),  $T=2^{-3}$ (red), $T=2^{-4}$ (yellow).]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_SNR_gross_2.pstex_t}}}}
	\\
	\subfloat[{\label{fig_small3}}State enclosures for \textbf{Case~a}; integration step sizes \mbox{$T=2^{-2}$} (blue),  $T=2^{-4}$ (red), $T=2^{-5}$ (yellow).]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_SNR_klein_3.pstex_t}}}}
	\hfill
	\subfloat[{\label{fig_large3}}State enclosures for \textbf{Case~b}; integration step sizes \mbox{$T=2^{-2}$} (blue),  $T=2^{-4}$ (red), $T=2^{-5}$ (yellow).]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_SNR_gross_3.pstex_t}}}}
	\\
	\caption{{\label{fig:example}}Guaranteed state enclosures for different discretization step sizes and levels of uncertainty. }
\end{figure}

In addition, it should be pointed out that all presented state enclosures were determined under the assumption that the parameters $\sqb{p}$ and $\sqb{\nu}$ can vary arbitrarily within their respective interval bounds over the complete integration time horizon $t \in \intv{0}{1}$. As soon as more precise knowledge about the worst-case parameter variation rates becomes available, for example, if physical reasoning justifies that these values can be set to unknown but constant quantities, the state enclosures can also be tightened in \textbf{Case~b}. Then, it becomes possible to subdivide the ranges of these parameter values and to perform individual simulations for respective subdomains, where the sets of reachable states are then characterized by the convex outer hull over all individual solution tubes.

\subsection{Simplified Fractional-Order Battery Model}
As a second application scenario, consider the fractional-order battery model depicted in Fig.~\ref{fig:battery}. It describes the dynamics of the charging and discharging behavior of a Lithium-Ion battery with the help of the state of charge $\sigma(t)$ as well as with the dynamics of the exchange of charge carriers in the interior of the battery cell which is related to electrochemical double layer effects. The latter is represented with the help of the voltage drop $v_{1}(t)$ across the fractional-order constant phase element $Q$ which serves as a generalization of an ideal capacitor according to~\cite{Rauh_IFAC_2020,ANDRE20115334,ZOU2018286}. This kind of constant phase element was already motivated by the example in Sec.~\ref{sec:lin_frac_comparison}.

\begin{figure}[htp]
	\centering
	\begin{tikzpicture}[font=\scriptsize,scale=1]

	\draw[semithick,white] (0,0) to (8.5cm,0);

	\tikzmath{\rx=+0.70cm; \rh=+0.30cm;};

	\tikzmath{\qx=+0.17cm; \qy=+0.34cm; \qz=+0.17cm;};

	\tikzmath{\linx=+0.50cm; \liny=+0.50cm;};

	\tikzmath{\tex=0.20cm;};
	\tikzmath{\arrl=+0.07cm;};

	\node (vOC1) at (+0.60cm,+0.00cm) {$v_\mathrm{OC}(\sigma(t))$~~~~~~};
	\ExtractCoordinate{vOC1} 	\draw[semithick,->] (\xCoord+0.5cm,+0.35cm) to (\xCoord+0.5cm,-0.35cm) coordinate (vOC2);
	\ExtractCoordinate{vOC2} 	\draw[semithick](\xCoord+0.2cm, -0.05cm) to (\xCoord+0.6cm, -0.05cm) coordinate (vOC31);
	\ExtractCoordinate{vOC2} 	\draw[semithick](\xCoord+0.1cm,+0.05cm) to (\xCoord+0.7cm,+0.05cm) coordinate (vOC32);
	\ExtractCoordinate{vOC31}	\draw[semithick](\xCoord -0.2cm,\yCoord) to (\xCoord-0.2cm,\yCoord -0.75cm) coordinate(vOC4);
	\ExtractCoordinate{vOC32}	\draw[semithick](\xCoord -0.3cm,\yCoord) to (\xCoord-0.3cm,\yCoord+0.75cm) to (\xCoord+0.4cm,\yCoord+0.75cm) coordinate (R01);
	\node at (\xCoord+0.1cm,+0.15cm) {$+$};
	\node at (\xCoord+0.1cm, -0.15cm) {$-$};

	\ExtractCoordinate{R01}	\draw[semithick,->] 	(\xCoord-0.7cm,\yCoord) -- (\xCoord-0.1cm,\yCoord);
	\node at (\xCoord -0.4cm,\yCoord+0.25cm) {$i(t)$};
	\ExtractCoordinate{R01} 	\draw[semithick]	( \xCoord					, \yCoord				) -- ( \xCoord				, \yCoord+\rh/2	) --
	( \xCoord+\rx			, \yCoord+\rh/2	) -- ( \xCoord+\rx		, \yCoord				) coordinate (R02) --
	( \xCoord+\rx			, \yCoord -\rh/2	) -- ( \xCoord				, \yCoord -\rh/2	) --
	( \xCoord					, \yCoord				) -- cycle;
	\node at (\xCoord+\rx/2, \yCoord+\rh/2+\tex) {$R_{0}$};
	\ExtractCoordinate{R02} 	\draw[semithick]	( \xCoord					, \yCoord				) -- (\xCoord+\linx		, \yCoord				) --
	( \xCoord+\linx		, \yCoord -\liny	) -- (\xCoord+2*\linx	, \yCoord -\liny	) coordinate (R11);

	\ExtractCoordinate{R11} 	\draw[semithick]	( \xCoord					, \yCoord				) -- ( \xCoord				, \yCoord+\rh/2	) --
	( \xCoord+\rx			, \yCoord+\rh/2	) -- ( \xCoord+\rx		, \yCoord				) coordinate (R12) --
	( \xCoord+\rx			, \yCoord -\rh/2	) -- ( \xCoord				, \yCoord -\rh/2	) --
	( \xCoord					, \yCoord				) -- cycle;
	\node at (\xCoord+\rx/2, \yCoord-\rh/2-\tex) {$R$};
	\ExtractCoordinate{R12} 	\draw[semithick]	( \xCoord					, \yCoord				) -- ( \xCoord+\linx		, \yCoord				) --
	( \xCoord+\linx		, \yCoord+\liny	) coordinate(R13) -- ( \xCoord+\linx+0.7cm	, \yCoord+\liny	) --
	( \xCoord+\linx+0.7cm	, \yCoord+\liny-0.30cm	) coordinate (R21);

	\ExtractCoordinate{R21} 	\draw[semithick,<-] (\xCoord+0.00cm, -0.35cm) -- (\xCoord+0.00cm,+0.35cm); \node at (\xCoord+0.4cm,+0.00cm) {$v(t)$};
	\draw[semithick,fill=white] (\xCoord,\yCoord) circle (0.05cm) coordinate(C1);
	\draw[semithick] ( \xCoord, \yCoord -1.00cm) -- ( \xCoord, \yCoord -1.30cm) -- (vOC4);
	\draw[semithick,fill=white] (\xCoord,\yCoord -1.00cm) circle (0.05cm) coordinate(C2);
	\ExtractCoordinate{C1} 		\node at (\xCoord-0.20cm,\yCoord+0.00cm) {$+$};
	\ExtractCoordinate{C2} 		\node at (\xCoord-0.20cm,\yCoord -0.00cm) {$-$};

	\ExtractCoordinate{R02} 	\draw[semithick]	( \xCoord+\linx		, \yCoord				) -- ( \xCoord+\linx					, \yCoord+\liny	) --
	( \xCoord+\linx		, \yCoord+\liny	) -- ( \xCoord+2*\linx+\rx/2	, \yCoord+\liny	) coordinate (Q11);
	\ExtractCoordinate{Q11} 	\draw[semithick]	( \xCoord-\qz			, \yCoord-\qy/2	) -- ( \xCoord		,\yCoord) -- ( \xCoord-\qz, \yCoord+\qy/2);
	\draw[semithick]	( \xCoord-\qz+\qx	, \yCoord-\qy/2	) -- ( \xCoord+\qx,\yCoord) coordinate(Q12) -- ( \xCoord-\qz+\qx, \yCoord+\qy/2);
	\node at (\xCoord, \yCoord+\qy/2+\tex) {$Q$};
	\draw[semithick,->] ( \xCoord-0.35cm, \yCoord-\qy/2-0.10cm) -- ( \xCoord+0.35cm, \yCoord-\qy/2-0.10cm);
	\node at (\xCoord, \yCoord-\qy/2-0.3cm) {$v_{1}(t)$};
	\ExtractCoordinate{Q12} 	\draw[semithick]	( \xCoord					, \yCoord				) -- ( \xCoord+\linx+\rx/2-\qx	, \yCoord	) --
	( \xCoord+\linx+\rx/2-\qx, \yCoord-\liny ) ;
	\end{tikzpicture}\vspace{-1mm}
	\caption{\label{fig:battery}Basic fractional-order equivalent circuit model of batteries.}
\end{figure}

For the following investigation of the proposed interval-based simulation approach for fractional-order system models, assume that the state vector
\begin{equation}
\mathbf{x}(t) = \begin{bmatrix} \sigma(t) & {}_0\mathcal{D}_{t}^{0.5}\sigma(t) & v_{1}(t) \end{bmatrix}^{T}
\in \mathbb{R}^3
\end{equation}
comprises in addition to the voltage $v_1$ across the constant phase element in Fig.~\ref{fig:battery} and the state of charge also its fractional derivative of order $\nu= 0.5$. Then, following the modeling steps described in~\cite{Rauh_IFAC_2020}, where this dynamic system representation was employed for the derivation of a cooperativity-enforcing interval observer design, and generalizing the charging/discharging dynamics to
\begin{equation}
{}_0\mathcal{D}_{t}^{1}\sigma(t) = -\frac{\eta_0 \cdot i(t) + \eta_1 \cdot \sigma(t) \cdot \mathrm{sign}\rb{i(t)}}{3600 C_\mathrm{N}}
\end{equation}
 with the terminal current $i(t)$, the commensurate-order quasi-linear state equations
\begin{equation}\label{eq:state_battery}
{}_0\mathcal{D}_{t}^{0.5} \mathbf{x}(t) = \mathbcal{A}\cdot \mathbf{x}(t) + \mathbcal{b}\cdot i(t)
\end{equation}
with the system and input matrices
\begin{equation}
\mathbcal{A} = \begin{bmatrix} 0 	& \phantom{-}1 & \phantom{-} 0 \\
\frac{\eta_1 \cdot \mathrm{sign}\rb{i(t)}}{3600 C_\mathrm{N}}  & \phantom{-}0 & \phantom{-} 0 \\
0  & \phantom{-}0 & -\frac{1}{RQ} \\ \end{bmatrix}
\quad \text{and} \quad
\mathbcal{b} = \begin{bmatrix} \phantom{-}0 \\ -\frac{\eta_0}{3600 C_\mathrm{N}} \\ \phantom{-}\frac{1}{Q} \end{bmatrix}
\end{equation}
and the terminal voltage
			\begin{equation}
v(t) = \begin{bmatrix} \sum\limits_{k=0}^{4} c_{k}\sigma^{k-1}(t) && 0 && -1 \end{bmatrix}\cdot \mathbf{x}(t) + \rb{-R_{0} +d_{0}e^{d_{1}\sigma(t)}} \cdot i(t)
\end{equation}
as the system output are obtained.
The following numerical simulations are based on the system parameters summarized in Tab.~\ref{tab:parameter} which were --- except for $\eta_1$ --- identified according to the experimental data referenced in~\cite{reuter2016,Rauh_IFAC_2020} during multiple charging and discharging cycles before the occurrence of aging. Now, a linear state feedback controller with the structure presented in~\cite{Kersten2019646} for the discharging phase ($i(t)>0$) is parameterized by assigning the asymptotically stable eigenvalues $\lambda \in \{-0.0001; -0.0002; -0.4832\}$ to the closed-loop dynamics, where the last value corresponds to the fastest asymptotically stable dynamics of the open-loop system.

\newcolumntype{Y}{>{\centering\arraybackslash}X}
\begin{table}[hb] \captionsetup{width=.5\textwidth} \caption{Parameters of the Lithium-Ion battery model.} \label{tab:parameter} \centering
	\begin{tabularx}{0.85\textwidth}{Y | Y | Y | Y | Y | Y | Y}
		\hline
		$R_{0}\,[\mathrm{\Omega}]$	& $Q\,[\mathrm{F/s^{0.5}}]$ 	& $R\,[\mathrm{\Omega}]$	& $\eta_0	\,[-]$	& $\eta_1	\,[-]$	& $C_\mathrm{N}\,[\mathrm{Ah}]$		& $c_{0}\,[\mathrm{V}]$	\\
		\hline
	\mbox{$+1.7 \!\cdot\! 10^{-5}$}								& $+20.591$										& $+0.1005$									& $+1.0000$	& $+0.1000$ & $+3.1000$	 							& $+3.0607$	\\
		\hline
	\end{tabularx}

	\begin{tabularx}{0.85\textwidth}{Y | Y | Y | Y | Y | Y}
		\hline
		$c_{1}\,[\mathrm{V}]$		& $c_{2}\,[\mathrm{V}]$		& $c_{3}	\,[\mathrm{V}]$ 	& $c_{4}\,[\mathrm{V}]$		& $d_{0}\,[\mathrm{\Omega}]$ 	& $d_{1}	\,[\mathrm{-}]	$\\
		\hline
		$+3.2965$							& $-8.3942\phantom{-}$		& $+11.088$							&  $-4.8992$							& $-0.2477$									&  $-14.302$ \\
		\hline
	\end{tabularx}
\end{table}

This leads to the following closed-loop system model
\begin{equation}\label{eq:state_battery_cl}
\mathcal{D}_{t}^{0.5} \mathbf{x}(t) = \rb{\mathbcal{A}-\mathbcal{b}\mathbcal{k}^T}\cdot \mathbf{x}(t) =: \mathbcal{A}_\mathrm{C} \cdot \mathbf{x}(t) \enspace ,
\end{equation}
where the entries $\mathcal{A}_{\mathrm{C},2,1}$, $\mathcal{A}_{\mathrm{C},2,2}$, $\mathcal{A}_{\mathrm{C},2,3}$ are inflated to interval parameters by symmetric bounds of $1\%$ radius of the respective nominal quantity and $\mathcal{A}_{\mathrm{C},3,3}$ to $10\%$, respectively. The matrix of eigenvectors at the interval midpoint corresponds to the transformation matrix $\mathbf{T}$ according to Def.~\ref{def:diag_dominant}. This allows for simulating the battery model both in transformed and initial coordinates $\mathbf{z}(t)$ and $\mathbf{x}(t)$, respectively, (cf.~Defs.~\ref{def:quasi_lin} and~\ref{def:diag_dominant}) and to intersect both enclosures after transforming the model in the form~(\ref{eq:diag_dom}) back into the original state space as soon as the simulation has been completed. Note, the system matrix  $\mathbf{A} \in \sqb{\mathbf{A}}$ in the uncertain diagonally dominant model is not Metzler, so that without the additional transformation procedure presented in~\cite{Kersten2019646}, it is necessary to apply the iteration from Theorem~\ref{thm:MLF_encl}

In Fig.~\ref{fig:battery_sim}, simulations for the following two sets of initial conditions are compared. The left column of this figure corresponds to a small uncertainty level in the initial states according to
\begin{equation}
{\mathbf{x}}(0) \in
\begin{bmatrix}
\intv{\phantom{-}0.99000}{\phantom{-}1.01000} \\
\intv{-0.00101}{-0.00099} \\
\intv{\phantom{-}0.09900}{\phantom{-}0.10100}
\end{bmatrix}
\quad \text{resulting in} \quad
{\mathbf{z}}(0) \in
\begin{bmatrix}
\intv{\phantom{-}0.089139}{\phantom{-}0.091348} \\
\intv{\phantom{-}7.880509}{\phantom{-}8.120525} \\
\intv{-9.110572}{-8.890557}
\end{bmatrix}
\enspace,
\end{equation}
while the right column accounts for the larger uncertainties
\begin{equation}
{\mathbf{x}}(0) \in
\begin{bmatrix}
\intv{\phantom{-}0.9000}{\phantom{-}1.1000} \\
\intv{-0.0011}{-0.0009} \\
\intv{\phantom{-}0.0900}{\phantom{-}0.1100}
\end{bmatrix}
\quad \text{with} \quad
{\mathbf{z}}(0) \in
\begin{bmatrix}
\intv{\phantom{-}0.079204}{\phantom{-}0.101283} \\
\intv{\phantom{-}6.800442}{\phantom{-}9.200592} \\
\intv{-10.10064}{-7.900495}
\end{bmatrix}
\enspace .
\end{equation}

\begin{figure}[htp]
	\centering
	\subfloat[{\label{fig_small_SOC}}State of charge $\sigma(t)$.]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_SOC_small.pstex_t}}}}
	\hfill
	\subfloat[{\label{fig_large_SOC}}State of charge $\sigma(t)$.]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_SOC_large.pstex_t}}}}
	\\
	\subfloat[{\label{fig_small_CPE}}Voltage $v_1(t)$ across the constant phase element $Q$.]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_VCPE_small.pstex_t}}}}
	\hfill
	\subfloat[{\label{fig_large_CPE}}Voltage $v_1(t)$ across the constant phase element $Q$.]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_VCPE_large.pstex_t}}}}
	\\
	\subfloat[{\label{fig_small_out}}Battery terminal voltage $v(t)$.]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_Vout_small.pstex_t}}}}
	\hfill
	\subfloat[{\label{fig_large_out}}Battery terminal voltage $v(t)$.]{\resizebox{0.45 \linewidth}{!}{{\input{fig_SNR/fig_Vout_large.pstex_t}}}}
	\\
	\caption{{\label{fig:battery_sim}}State and output voltage enclosures for the simplified fractional-order battery model in Fig.~\ref{fig:battery}, where the enclosures for the terminal voltage are determined with the help of an interval subdivision approach to reduce overestimation caused by the dependency effect.}
\end{figure}

Here, the simulation was performed without the subdivision of the time horizon according to Theorems~\ref{thm:mu} and~\ref{thm:contractor}. Instead, time-invariant parameter enclosures $\sqb{\lambda_i}$ were determined for 10~different time horizons from the range $t \in \intv{0}{10}\,\mathrm{s}$. The endpoints of these time intervals are visible in Fig.~\ref{fig:battery_sim} as those points at which the depicted enclosures widen in a step-like manner because parameters $\sqb{\lambda_i}$ that are valid for longer time spans  need to be wider to account for cross-coupling between multiple system states.

In addition, Fig.~\ref{fig:battery_sim} visualizes that the use of tighter initial state intervals reduces the tendency of state enclosures to blow up over time if time-invariant bounds $\sqb{\lambda_i}$ are considered. Hence, future work for multi-dimensional systems which cannot be decoupled fully by the transformation according to Def.~\ref{def:diag_dominant}, needs to account for both an optimal step size control strategy with respective bounds for the temporal truncation errors and simultaneously bisectioning (or multisectioning) strategies of the initial state domains to counteract the wrapping effect especially in the initial phases of the simulation. There, stiff dynamics of the state equations may lead to a blow-up of the computed bounds if only a single transformation matrix $\mathbf{T}$ is employed for an approximate decoupling of the state equations. This, however, could be counteracted if multiple transformations are considered in parallel, where their corresponding intersection is included in the iteration procedure for determining the parameters $\sqb{\lambda_i}$ with a reduced amount of overestimation. Finally, it should be pointed out that the presented simulation results already provide an important step toward the design of interval-based state of charge estimation strategies by means of measured terminal voltages that can be intersected with the corresponding simulated bounds in Figs.~\ref{fig_small_out} and~\ref{fig_large_out} in a predictor--corrector framework. They can be incorporated in battery management systems together with parameter identification, predictive control, and interval observer synthesis.

\section{Conclusions and Outlook on Future Work}\label{sec:concl}
In this paper, extensions of an interval-based exponential enclosure technique originally developed for integer-order sets of ordinary differential equations were presented. They are based on describing verified state enclosures with the help of Mittag-Leffler functions. In such a way, a simulation-based reachability analysis, in the sense of computing guaranteed outer interval enclosures, becomes possible for explicit, continuous-time sets of fractional-order differential equations. Commonly, the subdivision of the integration time horizon into shorter time slices allows for a reduction of overestimation. To exploit this property despite the infinite horizon memory of fractional-order differential equations, it is essential to rigorously quantify temporal truncation errors.

Besides the automatic choice of optimal discretization step sizes, as a compromise between computational effort and tightness of the obtained state enclosures, future work will also address systems with oscillatory behavior, where complex-valued transformations already exist for the integer-order case~\cite{Rauh_SCAN14}.

\bibliographystyle{eptcs}

redpajama_set_name: RedPajamaArXiv


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Once you contact us with the details of your event, we work with you to select an appropriate workshop and give you a quote for the cost of instruction, materials and transportation.
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Every stitcher needs a pin cushion, or a half a dozen. Fortunately, they are easy to make and can be as simple or elaborate as the stitcher desires. Felt, embroidery floss and fiber fill are the materials used. Participants may use provided templates or design their own. A variety of stitches will be taught. In 60-90 minutes participants will have at least one pin cushion to take home. All materials provided. Recommended for ages 10-adult.
A stuffy can be anything from a basic blob to an old-fashioned doll with a wardrobe of clothing. This workshop is personalized to the participant group based on age, skill and interest. Felt, a variety of stitches, buttons and other embellishments are used to create toys, comfort objects, pillows, or whatever other use you can think of for a stuffy. In 90-120 minutes you will have a finished project. (Multiple sessions can be arranged for more elaborate projects.) All materials provided. Recommended for ages 10-adult.
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Bulky yarn and a size 11 circular needle are used to create a simple, but stylish cowl in one session. Long-tail cast on, knit and purl stitches, connecting circular work and bind off are the features taught, and the project is complete in 90-120 minutes. All materials provided. Recommended for ages 10-adult.
Bulky yarn and size 9 straight needles are used to create a pair of ribbed fingerless gloves. You learn the long-tail cast on, knit (and purl) the rectangle and sew up the seam, leaving a hole for your thumb. With focus, the pair can be completed in a two-hour workshop. All materials provided. Recommended for ages 10-adult.
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Putting a wool sweater through the washer and dryer used to be a mistake. Now it's called "fulling" and it's a way to create felt for re-cycling a sweater into other garments. In this 2-hour workshop participants will hack a felted sweater into fingerless gloves and one other item— hat, scarf, bag. Bits and pieces will be used to make flowers or other embellishments. Bring a felted sweater (one can be provided by previous arrangement). All other materials provided. Recommended for ages 10-adult.
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You have a quality credit card type wallet in which you place three different banknotes. Each is taken by out by the spectator – whilst you are not looking - and each placed in one of the spectator's pockets. You have already written down which note will be where on your prediction card which the spectator also takes from the wallet.


text:
layout: default
title: Create your own plugin for sitespeed.io
description:
keywords: plugin, sitespeed.io
nav: documentation
image: https://www.sitespeed.io/img/sitespeed-2.0-twitter.png
twitterdescription: Create your own plugin for sitespeed.io
---
[Documentation]({{site.baseurl}}/documentation/sitespeed.io/) / Plugins

# Plugins
{:.no_toc}

* Lets place the TOC here
{:toc}

Sitespeed.io uses a plugin architecture. This architecture allows you to add/remove/create your own plugins that can run additional tests on URLs or do othere things with the result, like store it to a database.

The most basic things you can do is list configured plugins (which are currently used), disable one of the default plugin, or add/enable more plugins.

## List configured plugins
You can list the plugins that will be used when you do a run:

~~~ bash
$ sitespeed.io --plugins.list https://en.wikipedia.org/wiki/Barack_Obama
~~~

And you will get a log entry that looks something like this:

~~~
...
The following plugins are enabled: assets,browsertime,coach,domains,html,screenshot
...
~~~

The default plugins lives in the [plugin folder](https://github.com/sitespeedio/sitespeed.io/tree/master/lib/plugins). This is a good starting place to look at if you wanna build your own plugin.

## Disable a plugin
You can disable default plugins if needed. For instance you may not want to output HTML and strictly send the data to Graphite.

~~~ bash
$ sitespeed.io https://www.sitespeed.io --plugins.disable html
~~~

If you want to disable multiple plugins say you don't need the html or screenshots:

~~~ bash
$ sitespeed.io https://www.sitespeed.io --plugins.disable html screenshot
~~~

At anytime if you want to verify that disabling worked, add the plugins.list to your command:

~~~ bash
$ sitespeed.io https://www.sitespeed.io --plugins.disable html screenshot --plugins.list
~~~

## Add a plugin
You can also add a plugin. This is great if you have plugins you created yourself, plugins others have created, or plugins that are not enabled by default.

There's a plugin bundled with sitespeed.io called *analysisstorer* plugin that isn't enabled by default. It stores the original JSON data from all analyzers (from Browsertime, Coach data, WebPageTest etc) to disk. You can enable this plugin like so:

~~~ bash
$ sitespeed.io https://www.sitespeed.io --plugins.load analysisstorer
~~~

If you want to run plugins that you created yourself or that are shared from others, you can either install the plugin using npm (locally) and load it by name or point out the directory where the plugin lives.

~~~ bash
$ sitespeed.io https://www.sitespeed.io --plugins.load ../my/super/plugin
~~~

## Plugins in Docker

If you run in Docker and you should. You will need to mount your plugin directory as a volume. This is the recommended best practice. Practically you should clone your repo on your server and then mount it like this.

~~~ bash
docker run -v /Users/sitespeedio/git/myplugin/:/myplugin sitespeedio/sitespeed.io -b firefox --plugins.load /myplugin -n 1 https://www.sitespeed.io/
~~~

If you want to create an image of sitespeedio with your plugins pre-baked for sharing you can also do so using the following Dockerfile.

~~~ bash
FROM sitespeedio/sitespeed.io:<insert version here>

COPY <path to your plugin> /my-custom-plugin
~~~

Then build the docker image

~~~ bash
docker build -t my-custom-sitespeedio ./plugins
~~~

Finally you can run it the same way as mentioned above without the volume mount.

~~~ bash
docker run my-custom-sitespeedio firefox --plugins.load /my-custom-plugin --my-custom-plugin.option test -n 1 https://www.sitespeed.io/
~~~

Pretty cool, huh? :-)

## How to create your own plugin
First let us know about your cool plugin! Then share it with others by publish it to npm or just use Github.

### Basic structure
Your plugin needs to follow this structure.

~~~ javascript
const path = require('path');

module.exports = {
  name() {
    // This is ... shocking news: the name of the plugin
    return path.basename(__dirname);
  },

  open(context, options) {
    // when sitespeed.io start it calls the open function once for all plugins
    // the context holds information for this specific run that
    // generated at runtime, for example you can get hold of the storageManager
    // that stores files to disk.
    // The options is the configuration supplied for the run.
  },
  processMessage(message, queue) {
    // The plugin will get all messages sent through the queue
    // and can act on specific messages by type:
    // message.type
  },
  close(options, errors) {
    // When all URLs are finished all plugins close function is called once.
    // Options are the configuration options and errors a array of errors
    // from the run.
  }
};
~~~

### name()
This is the name of your plugin. You can use the name when you want to target specific options to your plugin. If you're plugin name is *browsertime* you can make sure all your options start with that name.

### open(context, options)
The open function is called once when sitespeed.io starts, it's in this function you can initialize whatever you need within your plugin. You will get the *context* and the *options*.

The *context* holds information for this specific run that generated at runtime and looks like this:

~~~
{
  storageManager, // The storage manager is what you use to store data to disk
  dataCollection, // a shared collection of the collected data
  timestamp, // The timestamp of when you started the run
  budget, // If you run with budget, the result will be here
  log // The logger used in sitespeed.io, use it to log https://github.com/seanmonstar/intel
}
~~~

You can checkout the [StorageManager](https://github.com/sitespeedio/sitespeed.io/blob/master/lib/support/storageManager.js) and the [DataCollection](https://github.com/sitespeedio/sitespeed.io/blob/master/lib/support/dataCollection.js) to get a feel of what you can accomplish.

The *options* are the options that a user will supply in the CLI, checkout the [CLI implementation](https://github.com/sitespeedio/sitespeed.io/blob/master/lib/support/cli.js) to see all the options.

### processMessage(message, queue)
The processMessage function in your plugin is called for each and every message that is passed in the application. So what's a message you may ask? Everything is a message in sitespeed.io.:) A message contains the following information:

 * **uuid** a unique id
 * **type** the type of a message, so your plugin know if it is interested in this message type or not.
 * **timestamp** when the message was created
 * **source** who created the message
 * **data** the data that is sent in the message
 * **extras** whatever extra that you wanna include in a message

When you start the application and feed it with URLs, each URL will generate a message of type URL and will be sent to all configured plugins.

If you want to catch it, you can do something like this:

~~~
switch (message.type) {
  case 'url':
    {
      // do some analyze on the URL
    }
~~~

When you are finished analyzing the URL, your plugin can then send a message with the result, so other plugins can use it.

Here's a snippet of Browsertime sending the screenshots message (the actual screenshot is in *results.screenshots*):

~~~
const messageMaker = require('support/messageMaker');
...

queue.postMessage(make('browsertime.screenshot', results.screenshots, {
  url,
  group
}));
~~~

If you want to send messages from within your plugin, the plugin will need to depend on sitespeed and you can use require to include the message maker like this:

~~~
const messageMaker = require('sitespeed.io/lib/support/messageMaker');
~~~

### close(options, errors)
When all URLs have been analyzed, the close function is called once for each plugin. You can use this function to store the data that you collected.

## What's missing
There's no way for a plugin to tell the CLI about what type of configuration/options that are needed, but there's an [issue](https://github.com/sitespeedio/sitespeed.io/issues/1065) for that. Help us out if you have ideas!