The families of 12 severely epileptic children have written to The Times to condemn new guidelines for medical cannabis.
In a joint letter, the parents say they face the "cruel and ridiculous" prospect of being forced to travel abroad for treatment even though the drugs are now technically legal in the UK.
The Home Office announced in October that doctors would be able to prescribe medical cannabis to patients after a review prompted by the high-profile cases of Alfie Dingley and Billy Caldwell.
The signatories of the letter, including Hannah Deacon, Alfie Dingley's mother, are concerned that advice issued by two professional bodies to clinicians regarding the new system is overly restrictive and will deter medics from recommending cannabis-derived products.


The introduction of Minimum Energy Efficiency Standards has brought EPC ratings sharply into focus as an important factor in leasing and investment decisions. However, as EPCs predominately designed to rate modern buildings, they do not always account for the traditional characteristics of heritage properties or propose the most appropriate energy efficiency recommendations.
This report has been developed in collaboration with Sturgis Carbon Profiling, to support owners of heritage properties in meeting compliance requirements of Minimum Energy Efficiency Standards, whilst simultaneously recognising the special requirements of heritage properties.


I'm looking forward to speaking to the members of the North Penn Arts Alliance on Wednesday February 6, starting at 7:00 PM. The group meets at the William Trego Arts Center, 125 N Main Street in North Wales, PA 19454. The Trego Center is in an old Church with a parking lot behind the church off Second Street.
I'm getting a slide show together of my children's book illustrations and gag cartoons, and the public is invited to this free talk. See you there!
Addendum: If you plan to come tonight to my talk, and have not been to the Trego Art Center before – it is located in a building attached to St. Luke's United Church of Christ, 125 N. Main St., North Wales. The easiest way to get into the Center is, coming down N. Main St., turn east on E. Walnut St. , go one block and turn left onto N. 2nd St. About halfway down that (1-way) street you will see the back of a large church on the left; park in that parking lot, and the modern-looking addition attached to the right side of the church is the entrance into the Trego Center. Go in the door and to the right and the room is the 2nd on the left.
This entry was posted in book illustration, business illustrations, cartoon, children's book illustration, Illustrations, magazine illustration, News, pat achilles and tagged cartooning, illustration, illustration talk, north penn arts alliance, pat achilles by patachilles. Bookmark the permalink.
Pat, will anyone be videoing this? 🙂 Wish we were closer.
You know, I thought about that today, Cathy – I don't think so, but that gives me the idea, maybe I'll do a brief video myself, it's supposed to be good to have video on your website, social media etc. Wouldn't be hard to do . . .


text:
Граф Станислав Станиславович Потоцкий (; 1787—1831) — российский военачальник эпохи наполеоновских войн, генерал-майор, генерал-адъютант, , обер-церемониймейстер.

Биография
Родился в 1787 году; представитель польского магнатского рода Потоцких, сын Станислава Щенсного Потоцкого.

В шестилетнем возрасте был записан на военную службу в Конный лейб-гвардии полк с увольнением до «указанных лет», но в 1796 году был исключен из списков полка. С 14 мая 1803 года — поручик Лейб-кирасирского Её Величества полка; в 1804 году вновь был зачислен в Конный лейб-гвардии полк.

С 1805 года находился в заграничном походе русской армии, под непосредственной командой цесаревича Константина Павловича; под Аустерлицем он получил за отличие орден Святой Анны 3-й степени. В 1807 году (29 мая) был под Гейльсбергом, исполняя различные поручения цесаревича; под Фридландом (2 июня) Потоцкий заслужил орден Святого Владимира 4-й степени и золотую шпагу с надписью «За храбрость». В том же году Потоцкий был произведён в штаб-ротмистры, а 14 января 1809 года, уже в чине ротмистра, вышел в отставку; 18 апреля пожалован камергером, но вскоре (4 августа того же года) снова был принят в военную службу подполковником по армейской кавалерии и участвовал походе в Молдавию. Назначенный, 20 декабря 1810 года, флигель-адъютантом к Его Императорскому Величеству, Потоцкий был переведен в Преображенский лейб-гвардии полк и 15 сентября 1811 года получил чин полковника.

Принимал участие в Отечественной войне 1812 года и Войне шестой коалиции, в ходе которой за Битву народов был 8 октября 1813 года награждён орденом Святого Георгия 4-го класса:

Был произведён в генерал-майоры 15 сентября 1813 года; в 1814 году, состоя непосредственно при Александре I, участвовал в сражении при Бриенне, за отличие в котором получил баварский военный орден Максимилиана Иосифа, а за сражение при Аренсе — орден Св. Анны 1-й степени. В кампанию 1815 года он состоял при баварской армии.

С 1 июля 1817 года граф Потоцкий был назначен генерал-адъютантом и оставался в этом звании до 6 сентября 1822 года, когда, по домашним обстоятельствам, вышел в отставку. В это время он числился дежурным генералом Главного Штаба войск польской армии и жил в Варшаве. Спустя четыре года (25 мая 1826 года) он был пожалован в чин тайного советника и назначен обер-церемониймейстером Высочайшего Двора; 24 мая 1829 года награждён Орденом Белого орла.

Потоцкий был очень богат: в Подольской и Киевской губерниях у него насчитывалось 8000 душ крестьян.

Станислав Станиславович Потоцкий умер в Санкт-Петербурге  года. Узнав о его смерти, император Николай писал князю Паскевичу: «Потерял я также бедного генерала Потоцкого, которого любил и уважал, как друга, но сей (умер) не от холеры, а от камня в печени».

Был похоронен на Смоленском лютеранском кладбище. В июне 1858 года прах был перенесен в фамильный склеп в крипте храма Посещения пресвятой девой Марией Елизаветы, расположенного на территории устроенного в 1856 году Выборгского римско-католического кладбища в Санкт-Петербурге.

Семья

Был женат с 1813 года на вдове княгине Екатерине Ксаверьевне Сангушко (1781—1820), урождённой графине Браницкой, сестре Е. К. Воронцовой и В. К. Браницкого. Её жизнь была короткой: заболев, скончалась в Петербурге в феврале 1820 года. К. Я. Булгаков сообщал брату:
Оставила мужу одну дочь:
 Александра (1818—1892), с 1840 года жена графа Августа Потоцкого (1806—1867).

Примечания

Литература

Станислав
Графы России
Генерал-адъютанты (Российская империя)
Генерал-майоры (Российская империя)
Российские командиры Наполеоновских и Революционных войн
Обер-церемониймейстеры (Российская империя)
Перезахороненные со Смоленского лютеранского кладбища
Похороненные в храме Посещения Девой Марией Елизаветы в Санкт-Петербурге
Участники сражения при Аустерлице
Участники сражения при Гейльсберге
Участники сражения под Фридландом
Участники сражения при Лейпциге
Участники сражения при Бриенне

redpajama_set_name: RedPajamaWikipedia


text:
Home Space Astronomy
Quasar measurements suggest the universe is expanding faster than we thought
We might need some new physics to explain how this is possible.
in Astronomy, Astrophysics, Space
Dark energy density seems to be increasing over time, and we're not really sure why.
The universe is a weird place and it just got a bit weirder. Image credits: NASA / Hubble.
Discovering that the universe is expanding was one of the biggest turning points in astronomy, and science in general. We don't know how big the universe is, we don't even know if it's infinite or not, but we're pretty certain that it's expanding. The first evidence emerged in the 1920s, when Alexander Friedmann derived a set of equations known as the Friedmann equations, showing that the universe might expand. The theory really picked up steam a few years later, when Edwin Hubble found that some galaxies appear to be moving away from us.
Hubble also found that not only is the universe is expanding but its expansion is accelerating. This seemed stunning at the time. Not only is the universe getting bigger, but it's getting bigger, faster. Hubble calculated a universal expansion rate of 500 km/s/Megaparsec, with one megaparsec being equivalent to 3.3 million light years. So for every 3.3 million light-years farther away, the matter where you are is moving away 500 km faster — every second. Subsequent measurements have greatly refined and reduced this value, but there is still some controversy and uncertainty. Most studies, however, agree that the universal expansion rate is around 70km/s/Megaparsec.
But it gets even weirder. Ironically, the universal expansion rate is also called Hubble constant — when it's anything but constant. Not only do different measurements come up with slightly different values, but when you look in different parts of the universe, you'll also find different expansion rates.
For instance, the nearby universe, as measured by telescopes like Hubble and Gaia, seems to sport a value of 73 km/s/Mpc. Meanwhile, when the Planck telescope looked towards the distant universe, it came back with a value of just under 70 km/s/Mpc. So Hubble's constant seems to vary both in time and in space — so much for being a constant.
This is where the new study comes in — but instead of clearing things up, it adds even more mystery.
The brightest of the brightest
Artistic depiction of a distant quasar. Image credits: ESO/M. Kornmesser.
Hubble's initial studies, like many subsequent measurements, were based on something called redshift. Essentially, as light travels from its original source to us, space is stretched, and the wavelength itself is stretched. This stretch shifts the wavelength towards the redder parts of the spectrum — hence the name "redshift".
Now, if you want to look at something very distant, you want a very powerful light source. In the new study, researchers focused on the brightest sources of light: quasars.
In a stellar twist of fate, the brightest sources of light go hand in hand with the darkest objects in the universe: supermassive black holes. These black holes, believed to lie at the center of all galaxies, are surrounded by a gaseous accretion disk. As gas falls toward the black hole, energy is released in the form of electromagnetic radiation with incredible power. This phenomenon is called a quasar, and some quasars are thousands of times brighter than the entire Milky Way, which is exactly what you want in this type of study. Quasars are spread across the universe, making them ideal target for multiple measurements.
Astronomers from Durham University in the UK and the Universita degli Studi di Firenze in Italy used observations from 1,600 quasars to calculate the expansion rate of the Universe up to about one billion years after its birth.
When you look at something that's one light year away, you're essentially looking into the past and seeing that object as it was one year ago. In this case, the astronomers look around 12 billion years into the past. The strange thing was that the values they found for expansion rates 12 billion years ago were similar to expansion rates reported by previous studies looking at areas from some 8 billion years ago. In other words, two different epochs had the same expansion rate, when really they shouldn't. There's nothing in our current arsenal of cosmological knowledge that could convincingly explain that.
"When we combine the quasar sample, which spans almost 12 billion years of cosmic history, with the more local sample of type-Ia supernovas, covering only the past eight billion years or so, we find similar results in the overlapping epochs," said Dr. Elisabeta Lusso of Durham University, in a statement.
"However, in the earlier phases that we can only probe with quasars, we find a discrepancy between the observed evolution of the Universe and what we would predict based on the standard cosmological model."
Of course, this is still an early study, which will be thoroughly investigated and replicated — but if it is confirmed, then astrophysicists will have a lot of digging to do to find an explanation.
However, one possible solution, still speculative at this point. could have something to do with the elusive dark energy, a theoretical form of energy postulated to act in opposition to gravity and to occupy the entire universe. Lead author Dr Guido Risaliti, of the Università degli Studi di Firenze, concludes:
"One of the possible solutions to the expansion of the early Universe would be to invoke an evolving dark energy, with a density that increases as time goes by.
The study "Cosmological constraints from the Hubble diagram of quasars at high redshifts" was published in Nature Astronomy.
Tags: Hubble expansionquasarsupernova

redpajama_set_name: RedPajamaCommonCrawl


We are 27 Peel Organizations that are partnered together to provide resources and consultation to individuals who have a developmental disability and/or dual diagnosis in times of crisis.
There are a variety of community services available to you and your family in the Peel Region that may be of support and help to prevent crisis.
Crisis is the onset of an emotional disturbance or situational distress which may be cumulative and result in the sudden breakdown of an individual's ability to cope. Such crisis may result in putting the person at risk of potential harm to themselves or others.


This soft, stretchy hood is made of lycra spandex, in a bright neon UV glow lightning print. It is made of stretchy lycra spandex, so you can put a lot of hair or dreads in the back.
This can also be made in almost any other fabric shown in our shop, if you would like a different color please contact!


It was revealed last week that Cambridge Analytica abused personal information from 50 million Facebook accounts in early 2014 to build a system to profile individual American voters for the 2016 presidential election. The goal was to then target the users with personalised political advertisements attacking Hilary Clinton and loving The Donald. It's still not clear whether this was illegal or merely repugnant.
Most people are focussing on the fact that Cambridge Analytica was headed at the time by Steve Bannon, which provides yet another malodorous link to Trump. Facebook's share price is down about 12% but so far there has been no accountability apart from the inevitable class action litigation lawyers circling. What matters the most here is that we are becoming de-sensitized to data breaches like this.
$300M of Etherium permanently lost. Hey, it's just crypto and it wasn't mine, so who cares?
Do you know anyone who lost sleep over 143 million Americans and 100,000 Canadians that were exposed by Equifax's massive data breach.
Every Yahoo account was compromised in 2013, which Yahoo did not figure out until 2017. That was 3 billion accounts. You likely had one of those accounts. Did you complain about it?
Citibank failed to protect the personal data (including birthdates and Social Security numbers) of approximately 146,000 customers who filed for bankruptcy between 2007 and 2011. That's adding insult to injury.
40 million Target customers were exposed in 2013. The remedial cost to Target, not including the class action litigation, was roughly $252M. Did you join the class to get your rightful piece of the settlement?
$81 million stolen from the Bank of Bangladesh by compromising the Swift system in 2016. This was the second time Swift was used as a medium of theft. But hey, that could never happen over here in the civilized world, right?
Look at the lists here and here and here for some of the largest data breaches of all time. How many of these do you remember, or care about?
Even worse, according to the Online Trust Alliance in its terrifying Cybersecurity and Breach Trends Report from January of this year, is that 93% of these breaches were self-inflicted and easily preventable. Apathy is our real enemy.
And next up are the assaults from Artificial Intelligence.
AI spans a broad area. A Nest WiFi-enabled thermostat can self-regulate if it feels the sun directly on it rather than air in the home environment – is that 'intelligent' or just good programming? Cruise control on your car? A video game that gets harder the further you go and that learns your favourite moves? Neural networks? Deep learning? The hated robo-advisor? Predictive weather analysis? Smart tokens in the ICO universe?
AI is just a software operating in a hardware environment, but somehow it has gained noble status. Perhaps it's the use of the word "intelligence" that lulls us into thinking that the software is actually alive.
It's not. It's just software, a compendium of zeros and ones that open and close circuits inside chips. Software is vulnerable to coding errors, intentional or negligent. It's vulnerable to breakdowns in its hardware. And it's entirely vulnerable to malicious third parties for cryptojacking.
Our courts and insurers will have to address who becomes liable when those things go wrong. The worse situation is where software causes death, like earlier this week when a self-driving car killed a woman in Tempe, Arizona. Elaine Herzberg was walking her bicycle when she was hit by a vehicle in autonomous mode going 40 km/h. It doesn't take a crystal ball to see Mr. Herzberg is the first of many such deaths.
Who will carry the financial burden of the error when smart tokens co-ordinate a contract for one billion rolls of toilet paper when the intention was for 100 rolls of paper towel? Is this contract law or negligence? Can you contract out of liability? Medical diagnostic software misses an obvious cause resulting in patient death? Who pays the repair bills when Skynet finally goes live and the Terminator kicks in your door?
Vernor Vinge's 1993 short paper The Coming Tehnological Singularity is a marvel of literature that manages to inspire and terrify at the same time. Should something we created actually develop its own intelligence, the pace at which technology would from that point develop would be inconceivable to humans. The human era would be over.
Back to the breaches, both malicious and self-inflicted. Incompetence and thievery have been with humanity for recorded history. The first trojan horse was the serpent surreptitiously attacking the Old Testament God by way of his human creations and an apple. Sadly, we do need various levels of government to help us defend ourselves. This will require some levels of regulation, even if unwanted.
The CryptoCrowd may not like it, but regulation is needed and it's coming. At least there seems to be some regulatory recognition that data is a different world requiring a different set of regulatory parameters. See for example the British Columbia Securities Commission's 2018 outreach efforts seeking innovation while maintaining confidence in the capital markets.
This apathy is a strange mindset, especially since the business world otherwise takes confidentiality seriously. We sign confidentiality agreements and NDA's. We expect our employees to leave our IP at the office. Securities laws exist to prevent insider trading and to protect the dignity of the market. Larger boards have committees specializing in privacy and data protection. There are few things more valuable to any company than the integrity of its data.
So we should be outraged by these ongoing assaults on us, our data and our companies. We should be in the streets, with torches and pitchforks, demanding that heads roll and attackers be found. Instead, we shrug and say "What can we do? I'm just one helpless person. The government will protect us." That only goes so far.
We have to use what the government gives us. CASL (Canada's AntiSpam Legislation) is a horribly mis-named piece of legislation that has teeth. It codifies an individual's right to control the inbox. It isn't about spam, it's about your digital liberty.
The GDPR is the European Union's approach, and it's a good one. A prior article explaining GDPR is here. Recent recommendations from House of Commons Standing Committee on Access to Information, Privacy and Ethics indicate that Canada will adopt an approach similar to GDPR to give you the tools to protect yourself. So use them.
Ultimately, it's up to you. Be vigilant. Protect your local network. Follow good protocols. Don't be sloppy. And be angry over every breach. Demand accountability. Next time it could be you.


The Window Covering Safety Council (WCSC) and U.S. Consumer Product Safety Commission (CPSC) are urging parents and caregivers to check all window coverings for exposed or dangling cords that could pose a strangulation hazard to infants and young children.
Both government and industry safety officials say only cordless window coverings, or those with inaccessible cords, should be used in homes with young children. To heighten public awareness of window-cord dangers, the Council and CPSC have again declared October as National Window Covering Safety Month.
According to information provided by the CPSC, since 1990 more than 200 infants and young children have died from strangling in window cords.
The Window Covering Safety Council's month long, nationwide campaign is designed to increase consumer awareness of potential window-cord hazards, as well as to urge parents and caregivers of young children to only use cordless window products in homes with young children and to replace all window coverings in the home made before 2001 with today's safer products.
"Window cord strangulations are tragic, but preventable," said CPSC Chairman Inez Tenenbaum. "In stores across the country, parents and consumers can find a variety of cordless blinds and shades, as well as window coverings with inaccessible cords. Going cordless is a smarter and safer approach in homes with young children." Older window coverings should be retrofitted or replaced with today's safer products.
Make sure tasseled pull cords are adjusted to be as short as possible.
Continuous-loop pull cords on draperies and vertical blinds should be pulled tight and anchored to the floor or wall with a tension device.
Be sure cord stops are properly installed and adjusted to limit movement on inner cords on blinds and shades.
To learn more about window-cord safety, or to order free retrofit kits for older window coverings,visit the Window Covering Safety Council's website at www.windowcoverings.org or call toll-free at 1-800-506-4636. Parents and caregivers can also learn more about window covering safety by connecting with WCSC on Facebook and Twitter.
The Window Covering Safety Council (WCSC) is a coalition of major U.S. manufacturers, importers and retailers of window coverings dedicated to educating consumers about window cords safety. The Council also assists and supports its members in the industry's ongoing efforts to encourage the use of cordless products in homes with young children, its redesign of corded products and to support the national ANSI/WCMA standard for corded window coverings.WCSC's activities in no way constitute an assumption of any legal duty owed by its members or any other entity.
Impossible not to blame myself for tragedy Tory minister refuses to back mum's call for blind ban.


Looking for Danby at Marine Park Appliances in Brooklyn, NY?
Marine Park Appliances in Brooklyn, NY is an authorized dealer of Danby Products. Danby was founded in 1947, as a family owned manufacturer of small electrical appliances. Danby opened its U.S. home office in Findlay, Ohio in 1992 as business expanded across North America. Danby's reputation as a leader in serving the appliance market has been achieved by producing innovative, quality products at competitive pricing. All of Danby's products meet and exceed all environmental safety standards and offer UL, CSA, NRTL, and ETL recognized standards approvals.
So if you are looking for Danby products in Brooklyn, New York City, Queens, Manhattan, Bronx, Elmont, Valley Stream, Long Beach, Newark, Jersey City and Hoboken, or if you have any questions about Danby products, please feel free to call us at (718) 513-1808 or simply stop by Marine Park Appliances at any time and we would be glad to help you.


These Regulations are prescribed in accordance with Paragraph 3, Article 14 of the Occupational Safety and Health Act (hereinafter referred to as the "Act").
1.Hazardous chemicals as defined in subparagraph 3, Paragraph 1, Article 29 and subparagraph 5, Paragraph 1, Article 30, as specified in Appendix 1.
III.Others designated and published by the central competent authority.
The term "handling" referred to in these Regulations means the act of manufacturing, importing, supplying of priority management chemicals, or providing such for workers to use or handle .
The term "handlers" referred to in these Regulations mean the manufacturers, importers, suppliers or employers who engage in the aforementioned acts.
8.Others designated by the central competent authority.
For priority management chemicals designated and published as controlled chemicals by the central competent authority, the handlers shall comply with the Regulations Governing Designating and Handling Permission of Controlled Chemicals.
3.Other information designated by the central competent authority.
2.If the number of workers is less than 100, the handler shall file the report within 18 months of the publication day made by the central competent authority.
The periodic updates of the first Paragraph shall be made between April and September each year following the year for which the report was filed.
A handler filing the report and making annually periodic updates of the preceding Article shall register thereof on the designated information website in accordance with the methods published by the central competent authority.
When the central competent authority deems it necessary to supplement other handling information to assess the exposure risk of the priority management chemicals, it may designate the handler, within the time limit specified, to complete the supplementary handling information form, as specified in Appendix 5, and register thereof on the designated information website.
3.Others as designated by the central competent authority.
2.Where the handler is notified to make correction within a prescribed time period for incorrect or incomplete information, but fails to do so.
3.Where the handler does not comply in accordance with Article 6 to Article 10, and fails to make correction within a prescribed time period.
The handler, who submit false or misleading information for recordation shall be punished in accordance with Article 43 of the Act.
These Regulations shall take effect from January 1, 2015.


Besides being some of the most alluring objects of beauty the world over, flowers are also one of the most popular images which are photographed. There are many ways to capture a gorgeous bouquet of flowers or single, perfect bloom. In the world of digital photography, the cost is minimal once you have a camera. Even a simple point-and-shoot model can produce excellence, as half the success can be measured in composition, flower subject and lighting.
Flowers are beautiful, but their beauty often shortchanges them in some ways. That is, most people think of flowers as things you arrange and place on tables to sit and look pretty.


It is a recurring thread in many modern pantomimes that the titular heroines, from Sleeping Beauty to Cinderella, are among the least served in terms of scripted character. That's certainly the case with Aylesbury's version of Snow White and the Seven Dwarfs, where the characters of Muddles, Dolly the Nurse and the Wicked Queen transcend all others.
That is in itself no bad thing: local radio personality Andy Collins is a veteran of Aylesbury pantomime, having played the fool rôle for several years and with great aplomb. Reuniting with him is drag artist and singer La Voix, returning after last year's triumphant performance as Widow Twankey in Aladdin to once again play the dame.
Having a dame who can not only sing, but sing well, adds a sheen that many commercial pantomimes miss out on. From her opening number (the theme from Fame, reworded to an emphatic 'Dame!'), La Voix stamps her mark on the show, and her double act with Collins is easy and assured, radiating warmth that the audience cannot fail to absorb.
As the Wicked Queen, Su Pollard throws herself into the villainous role. While Andrew Ryan's script is not the sharpest, Pollard's commitment to song and dance – especially in her transformation sequence – helps elevate the role. Even when forced into a series of outlandish costumes in her quest to woo Jon Moses' charming Prince, Pollard keeps up the character work. It is a shame, then, that her ultimate comeuppance is relatively low key, which does an injustice to character and actor alike.
Amid all these larger-than-life performances, Snow White herself becomes a bit player in her own story, but recent graduate Jenna Innes delivers the goods in her first pantomime role. Innes demonstrates a powerful voice that deserves to be heard more widely. Despite being saddled with a backstory as a scullery maid – an irrelevant plot detail that belongs more in Cinderella than Snow White, and which gets forgotten after the show's initial scenes – Innes nonetheless convinces as the princess for whom everybody risks the queen's wrath.
Despite many questions about the play's script, the key points of a traditional panto are present, from the "ghosties and ghoulies" skit to the raucous reworking of The Twelve Days of Christmas – which, after several years, has escalated into a literal battle between Collins and his audience, such that some canny members of the audience had umbrellas on standby ready for when the super-soaker water pistol fight commenced.
And that is the key to Aylesbury pantos: solidly reliable fun that entertains and amuses. Other years' efforts may have been stronger in some areas, but the commitment to the genre remains constant.


text:
\section{Introduction} \vspace{-0.075cm}
State estimation is crucial in several research fields such as fault detection and isolation \cite{combastel2008fault}, localization problem \cite{jaulin2009nonlinear} and state-feedback control \cite{dahleh19871}. In such settings, Bayesian/stochastic estimation approaches such as particle or Kalman filtering can be applied if distributions/stochastic descriptions of uncertainties are known. However, in bounded-error settings where distribution-free set-valued uncertainties are considered, guaranteed sets of true values of state trajectories which are compatible/consistent with constraints/observations are desired. Obtaining the exact characterization of such sets that contain the evolution of the system states is very complicated and mostly intractable \cite{kieffer2004guaranteed}, hence developing set-theoretic approaches to tractably derive enclosures to such sets, while trying to make the enclosing sets as tight as possible, is critical.

\emph{Literature review.} In the context of bounded-error settings, where dynamical systems are subject to distribution-free and bounded uncertainties, several seminal studies have proposed set-membership state estimation approaches for discrete-time constrained systems, to compute enclosing sets to all possible system trajectories \cite{chisci1996recursive,le2013zonotopic,rego2020guaranteed}. A well-known strategy, which is common to all these methods is finding an enclosing set to the image set of the dynamics vector field, i.e., \emph{propagation/prediction} step, as well as refining the obtained propagated set by finding an enclosure to its intersection with the set of states that are compatible/consistent with the observation/measurements, i.e., \emph{update} step.

In case of linear systems with polytopic initial set, it is theoretically shown that tight (exact) enclosures can be obtained \cite{girard2008efficient}. However, even for linear systems, the computational complexity of polytopic propagation is extensive and grows dramatically with time \cite{shamma1999set}. Hence, simpler sets such as parallelotopes \cite{vicino1996sequential,chisci1996recursive}, ellipsoids \cite{khajenejad2019simultaneous,polyak2004ellipsoidal,khajenejadasimultaneous}, intervals \cite{zheng2016design,khajenejad2020simultaneousinterval1,wang2015interval,khajenejad2020simultaneousinterval2} or zonotopes \cite{le2013zonotopic,combastel2015merging} have been used to characterize the enclosures. However, structural limitations of these sets sometimes leads to conservative enclosures. To address this, the work in \cite{scott2016constrained} introduced \emph{constrained zonotpes} to ease some of the limitations imposed by zonotopes, while \emph{zonotope bundles} were proposed in \cite{althoff2011zonotope} to describe the intersection of zonotopes without explicit computations.

Regarding nonlinear systems, obtaining efficient set-valued estimates is still very challenging, contrary to the linear case. A classical approach to tackle this problem has been to use interval arithmetic-based inclusion functions \cite{moore2009introduction} to propagate the current enclosing sets through the nonlinear dynamics and then to apply interval-based set inversion techniques (e.g., SIVIA) to find upper approximations for the set of compatibles states with the current measurements \cite{jaulin2009nonlinear,jaulin2016inner}. These approaches are computationally very efficient, but unfortunately, due to the nature of interval arithmetic, the resultant bounds are mostly conservative.

Alternatively, given linear observation functions, zonotopic propagation methods have been developed in \cite{combastel2005state,alamo2005guaranteed,alamo2008set}, based on the first order Taylor expansion, the mean value extension or DC programming. However, significant errors are caused in update step due to the symmetry of zonotopes, even for linear measurements \cite{scott2016constrained}. More recently, the interesting work in \cite{rego2020guaranteed} proposed constrained zonotopic propagation and update algorithms for discrete-time nonlinear systems with linear observation functions, based on  mean value and first order Taylor extensions.

 \emph{Contributions.} This paper  proposes novel methods for recursive state estimation (consisting of propagation and update steps) using polytopes (equivalently, constrained zonotopes or zonotope bundles) for nonlinear bounded-error discrete-time systems with nonlinear observation functions.  Leveraging remainder-form mixed-monotone decomposition functions \cite{khajenejad2021tightremainder} and following the standard propagation and update approach, this paper bridges the gap between constrained zonotope (CZ)/zonotope bundle (ZB)-based set-valued state estimation and nonlinear observation/constrained functions. In particular, for the propagation step, we transform the prior ZB/CZ's into the space of CZ/ZB generators, which are interval-valued, and further transform
 the vector field into two components, one that is proven to attain tight image sets, as well as a linear remainder function, for which a family of  remainder-form mixed-monotone decomposition functions \cite{khajenejad2021tightremainder} can be obtained. Each of the decomposition functions produce enclosures of the state trajectories and thus, we can intersect them to obtain the desired propagated ZB/CZ enclosures.

Moreover, we show that a similar idea, i.e., transformation from the state+uncertainty space to the space of generators of CZ/ZB's, can be used for the update step to find a family of enclosures to the \emph{generalized nonlinear intersection} of the propagated set with the set of states that is compatible with the observations, where the final enclosures are proven to be ZB/CZ's. Furthermore, we prove that the mean value extension approach used in \cite{rego2020guaranteed} to enclose a multiplication of an interval matrix to a constrained zonotope, {can also be} leveraged for the update step when the observation function is nonlinear. Finally, we compare our proposed approaches together and with the mean value extension-based approach in \cite{rego2020guaranteed}, {implementing it on} two examples, one with a linear and the other with a nonlinear observation function.
\section{Preliminaries}
In this section, we briefly introduce some of the main concepts that we use throughout the paper, as well as some important existing results that will be used for deriving our main results and for comparison.

{\emph{{Notation}.}} $\mathbb{N},\mathbb{N}_a, \mathbb{R}^n$ and $\mathbb{R}^{m \times n}$ denote the set of positive integers, the first $a$ positive integers, the $n$-dimensional Euclidean space and the space of $m$ by $n$ real matrices, respectively. For $\mathcal{Z},\mathcal{W} \subset \mathbb{R}^n, R \in \mathbb{R}^{m \times n}, \mathcal{Y} \subset \mathbb{R}^m$, and $\mu:\mathbb{R}^n \to \mathbb{R}^m$, $R\mathcal{Z}\triangleq \{Rz|z \in \mathcal{Z}\}, \mathcal{Z} \oplus \mathcal{W} \triangleq \{z+w | z \in \mathcal{Z},w \in \mathcal{W} \}$, $\mathcal{Z} \ominus \mathcal{W} \triangleq \{z-w | z \in \mathcal{Z},w \in \mathcal{W} \}$, $\mu(\mathcal{Z}) \triangleq \{\mu(z) | z \in \mathcal{Z}\}$ and $\mathcal{Z} \cup_{\mu} \mathcal{Y} \triangleq \{z \in \mathcal{Z} | \mu(z) \in \mathcal{Y} \}$ denote the linear mapping, Minkowski sum, set subtraction, general (nonlinear) mapping and generalized (nonlinear) intersection, respectively. Moreover, the transpose,
Moore-Penrose pseudoinverse and $(i,j)$-th element
of $R$ are given by $R^\top$,
$R^\dagger$ and $R_{ij}$,
 while
its row support is $r=\textstyle{\mathrm{rowsupp}}(R) \in \mathbb{R}^m$, where $r_i=0$ if the $i$-th row of $R$ is zero and $r_i=1$ otherwise, $\forall i \in \mathbb{N}_m$. Furthermore, $\mathbb{B}^n_{\infty} \triangleq \{z \in \mathbb{R}^n | \|z\|_{\infty} \leq 1\}$ and $\mathbf{0}_n$ denote the $\infty$-norm hyperball and the zero vector in $\mathbb{R}^n$, respectively. For $z \in \mathbb{R}^n$, $\textstyle{\mathrm{diag}}(z)$ is a diagonal matrix in $\mathbb{R}^{n \times n}$, with its diagonal elements being the corresponding elements of $z$. $\langle \cdot,\cdot\rangle$ denotes the inner product operator.
\begin{defn}[Intervals, Polytopes, Constrained Zonotopes (CZ) and Zonotope Bundles (ZB)]\label{defn:CZ_ZB}
A set $\mathcal{Z} \subset \mathbb{R}^n$ is a(n) (i) interval, (ii) polytope, (iii) constrained zonotope (CZ), (iv) zonotope bundle (ZB), if
\begin{enumerate}[(i)]
\item $\exists \underline{z},\overline{z} \in \mathbb{R}^n$ such that $\mathcal{Z}=[\underline{z},\overline{z}] \triangleq \{z \in \mathbb{R}^n | \underline{z} \leq z \leq \overline{z}\}$. An interval matrix can be defined similarly, in an element-wise manner.
\item $\exists A_p \in \mathbb{R}^{n_p \times n},b_p \in \mathbb{R}^{n_p}$ such that $\mathcal{Z}=\{A_p,b_p\}_P \triangleq \{ z \in \mathbb{R}^n |A_pz\leq b_p \}$.
\item $\exists \tilde{G} \in \mathbb{R}^{n \times n_g}, \tilde{c}\in \mathbb{R}^{n},\tilde{A} \in \mathbb{R}^{n_c \times n_g},\tilde{b} \in \mathbb{R}^{n_c}$ such that $\mathcal{Z}=\{\tilde{G},\tilde{c},\tilde{A},\tilde{b}\}_{CZ} \triangleq \{ \tilde{G}\xi+\tilde{c} |\xi \in \mathbb{B}^{n_g},\tilde{A}\xi=\tilde{b} \}$. $n_g$ and $n_c$ are called the number of CZ's generators and constraints, respectively.
\item $\mathcal{Z}$ can be represented as an intersection of $S \in \mathbb{N}$ zonotopes, i.e., $\exists \{{G}_s \in \mathbb{R}^{n \times \hat{n}_s},{c}_s \in \mathbb{R}^n\}_{s=1}^S$ such that $\mathcal{Z}=\bigcap\limits_{s=1}^S\{{G}_s,{c}_s\}_{Z} \triangleq \bigcap\limits_{s=1}^S \{{G}_s\zeta+{c}_s |\zeta \in \mathbb{B}^{\hat{n}_g}\}$, with $\hat{n}_s,s=1,\dots,S$, being called the number of generators for each zonotope.
\end{enumerate}
It is worth mentioning that using CORA2020 \cite{althoff2020cora}, if a set $\mathcal{Z}$ has each of the polytopic, CZ or ZB representations, it can be equivalently and tightly transformed into the other two representations. This is represented thorough this paper as:  $$\textstyle \mathcal{Z} =\{A_p,b_p\}_P \equiv \{\tilde{G},\tilde{c},\tilde{A},\tilde{b}\}_{CZ} \equiv \bigcap\limits_{s=1}^S\{{G}_s,{c}_s\}_{Z}.$$
\end{defn}
\begin{prop}\label{prop:interval_cen_rep}
Consider an interval vector $\mathbb{IZ} \triangleq [\underline{z},\overline{z}] \subset \mathbb{IR}^{n}$ and an interval matrix $\mathbb{J} \in \mathbb{IR}^{n \times m}$. Then,
$\mathbb{IZ}$ and $\mathbb{J}$ can be equivalently represented as
\begin{align}
\mathbb{IZ} &\triangleq [\underline{z},\overline{z}] \equiv \textstyle{\mathrm{mid}}(\mathbb{IZ})\oplus \frac{1}{2}\textstyle{\mathrm{diag}}(\textstyle{\mathrm{diam}}(\mathbb{IZ}))\mathbb{B}^n_{\infty},\label{eq:interv1} \\
\mathbb{J} &\triangleq [\underline{J},\overline{J}] \equiv \textstyle{\mathrm{mid}}(\mathbb{J})\oplus \mathbb{J}_{\Delta}\label{eq:interv2},
\end{align}
where for $q \in \{\mathbb{IZ},\mathbb{J}\}$, $\textstyle{\mathrm{mid}}(q) \triangleq \frac{1}{2}(\overline{z}+\underline{z})$, $\textstyle{\mathrm{diam}}(q) \triangleq (\overline{z}-\underline{z})$, and
$\mathbb{J}_{\Delta} \in \mathbb{IR}^{n \times m}$ is an interval matrix that is defined as $[\mathbb{J}_{\Delta}]_{ij} \triangleq \frac{1}{2}[-\textstyle{\mathrm{diam}}(\mathbb{J})_{ij} \ \textstyle{\mathrm{diam}}(\mathbb{J})_{ij}], \forall i \in \mathbb{N}_{n},\forall j \in \mathbb{N}_{m}$.
\end{prop}
\begin{proof}
To prove \eqref{eq:interv1}, consider $z \in \mathbb{IZ} \Leftrightarrow \underline{z} \leq z \leq \overline{z} \Leftrightarrow \underline{z}-\textstyle{\mathrm{mid}}(\mathbb{IZ}) \leq z-\textstyle{\mathrm{mid}}(\mathbb{IZ}) \leq \overline{z}-\textstyle{\mathrm{mid}}(\mathbb{IZ})\Leftrightarrow -\frac{1}{2}\textstyle{\mathrm{diam}}(\mathbb{IZ}) \leq z-\textstyle{\mathrm{mid}}(\mathbb{IZ}) \leq \frac{1}{2}\textstyle{\mathrm{diam}}(\mathbb{IZ})\Leftrightarrow \textstyle{\mathrm{mid}}(\mathbb{IZ})-\frac{1}{2}\textstyle{\mathrm{diam}}(\mathbb{IZ}) \leq z \leq \frac{1}{2}\textstyle{\mathrm{diam}}(\mathbb{IZ})+\textstyle{\mathrm{mid}}(\mathbb{IZ})\Leftrightarrow \exists \xi \in \mathbb{B}^n_{\infty}, s.t.\ z=\textstyle{\mathrm{mid}}(\mathbb{IZ})+\frac{1}{2}\textstyle{\mathrm{diag}}(\textstyle{\mathrm{diam}}(\mathbb{IZ}))\xi \Leftrightarrow z \in \textstyle{\mathrm{mid}}(\mathbb{IZ})\oplus \frac{1}{2}\textstyle{\mathrm{diag}}(\textstyle{\mathrm{diam}}(\mathbb{IZ}))\mathbb{B}^n_{\infty}$. The result in \eqref{eq:interv2} is a straightforward extension of \eqref{eq:interv1}.
\end{proof}
\begin{prop}\cite[Theorem 1]{rego2020guaranteed}\label{prop:Rego_CZ_bounding}
Let $\mathcal{X}=\{G,c,A,b\}_{CZ} \subset \mathbb{R}^m$ be a constrained zonotope with $n_g$ generators and $n_c$ constraints, and $\mathbb{J} \in \mathbb{IR}^{n \times m}$ be an interval matrix. Consider the set $S=\mathbb{J}\mathcal{X} \triangleq \{Jx | J \in \mathbb{J}, x \in \mathcal{X} \} \subset \mathbb{R}^n$. Let $\overline{\mathcal{X}}=\{\overline{G},\overline{c}\}_Z$ be a zonotope satisfying $\mathcal{X} \subseteq \overline{\mathcal{X}}$ and $\overline{c} \in \mathbb{R}^{\overline{n}_g}$. Let $\mathbf{m} \in \mathbb{R}^n$ be an interval vector such that $\mathbf{m} \supset (\mathbb{J}-\textstyle{\mathrm{mid}}(\mathbb{J}))\overline{c}$ and $\textstyle{\mathrm{mid}}(\mathbf{m})=\mathbf{0}_n$. Let $P \in \mathbb{R}^{n \times n}$ be a diagonal matrix defined as follows. $\forall i=1,\dots,n$:
\begin{align}\label{eq:Rego_P}
P_{ii}=\frac{1}{2}\textstyle{\mathrm{diam}}(\mathbf{m}_i)+\frac{1}{2}\sum_{j=1}^{\overline{n}_g} \sum_{k=1}^m \textstyle{\mathrm{diam}}(\mathbb{J}_{ik})|\overline{G}_{kj}|.
\end{align}
Then, $\mathcal{S} \subseteq \textstyle{\mathrm{mid}}(\mathbb{J})\mathcal{X} \oplus P\mathbb{B}^n_{\infty}$
\begin{align}\label{eq:CZ_bounding}
\quad \quad \quad =\{\begin{bmatrix}\textstyle{\mathrm{mid}}(\mathbb{J})G & P\end{bmatrix}\hspace{-.1cm}, \textstyle{\mathrm{mid}}(\mathbb{J})c, \begin{bmatrix} A & 0_{n_g \times n}\end{bmatrix}, b \}_{CZ}.
\end{align}
\end{prop}
\begin{prop}[RRSR Propagation Approach] \cite[Theorem 2]{rego2020guaranteed}\label{prop:Rego_propagation}
Let $f:\mathbb{R}^n \times \mathbb{R}^{n_w} \to \mathbb{R}^n$ be continuously differentiable and $\nabla_xf$ denote the gradient of $f$ with respect to its first argument. Let $\mathcal{X}=\{G_x,c_x,A_x,b_x\}_{CZ} \subset \mathbb{R}^n$ and $\mathcal{W} \subset \mathbb{R}^{n_w}$ be constrained zonotopes. Choose any $h \in \mathcal{X}$. If $\mathcal{Z}$ is a constrained zonotope such that $f(h,\mathcal{W}) \subseteq \mathcal{Z}$ and $\mathbb{J} \in \mathbb{IR}^{n \times n}$ is an interval matrix satisfying $\nabla^\top_x f(\mathcal{X},\mathcal{W}) \subseteq \mathbb{J}$, then
\begin{align}\label{eq:Rego_prediction}
f(\mathcal{X},\mathcal{W}) \subseteq \mathcal{Z} \oplus \textstyle{\mathrm{mid}}(\mathbb{J})(\mathcal{X}-h) \oplus \tilde{P}\mathbb{B}^n_{\infty},
\end{align}
where $\tilde{P}$ can be computed through \eqref{eq:Rego_P}, using $\mathbb{J}$ and an enclosing zonotope to $\mathcal{X}-h$.
\end{prop}
\begin{defn}[Mixed-Monotone (One-Sided) Decomposition Functions For Discrete-Time Systems]\label{defn:dec_functions}
\cite[Definitions 3,4]{khajenejad2021tightremainder}
A mapping $f_d:\mathcal{Z} \times \mathcal{Z} \subset \mathbb{R}^{2n} \to \mathbb{R}^m$ is a discrete-time mixed-monotone decomposition function with respect to $f:\mathcal{Z} \subset \mathbb{R}^n \to \mathbb{R}^m$, over the set $\mathcal{Z}$, if it satisfies the following:
$f_d(x,x)=f(x),x \geq x' \Rightarrow f_d(x,y) \geq f_d(x',y), y \geq y' \Rightarrow f_d(x,y) \leq f_d(x,y'), \forall x,y,x',y' \in \mathcal{Z}$. Further, if there exists two mixed-monotone mappings $\overline{f}_d,\underline{f}_d:\mathcal{Z} \times \mathcal{Z} \to \mathbb{R}^m$, such that for any $\underline{z},z,\overline{z} \in \mathcal{Z}$, the following holds: $\underline{z} \leq z \leq \overline{z} \Rightarrow \underline{f}_d(\underline{z},\overline{z}) \leq f(z) \leq \overline{f}_d(\overline{z},\underline{z})$, then $\overline{f}_d$ and $\underline{f}_d$ are called upper and lower decomposition functions for $f$ over $\mathcal{Z}$,  respectively.
\end{defn}
It is trivial to see that $\forall x \in [\underline{x},\overline{x}], f_d(\underline{x},\overline{x}) \leq f(x) \leq f_d(\overline{x},\underline{x})$, where $f_d$ is a decomposition function of $f$.
\begin{prop}[Tight and Tractable Remainder-Form Upper and Lower Decomposition Decomposition Functions]\cite[Theorems 1,2,3 ]{khajenejad2021tightremainder}\label{prop:Remainder_DF}
Consider a locally Lipschitz vector field ${f}_i:\mathbb{IZ} \triangleq [\underline{z},\overline{z}] \subseteq \mathbb{IR}^{n_{z}} \to \mathbb{R}$. Let
$\mathbb{N}_{n_{z}} \triangleq \{1,\dots,n_{z}\}$ and $\overline{{J}}^{\tilde{f}}_i,\underline{{J}}^{\tilde{f}}_i \in \mathbb{R}^{n_{z}}$ denote the upper and lower bounds for the Jacobian matrix (vector) of ${f}_i$ over $\mathbb{IZ}$. Suppose that Assumption \ref{assumption:mix-lip} in Section \ref{sec:formulation} holds. Then, $f_i(\cdot)$ admits a family of mixed-monotone remainder-form decomposition functions denoted as $\{{f}_{d,i}(z,\hat{z};m,h(\cdot))\}_{\mathbf{m} \in \mathbf{M}_i,h(\cdot) \in \mathcal{H}_{\mathbf{M}^c_i}}$, that is parametrized by a set of supporting vectors $\mathbf{m} \in \mathbf{M}^c_i$
\begin{align}

 \nonumber \mathbf{m} \in \mathbf{M}^c_i \triangleq \{&\mathbf{m} \in \mathbb{R}^ {n_z} | \mathbf{m}_j = \min(\underline{{J}}^f_{ij},{0}) \ \lor \\
 &\mathbf{m}_j = \max(\overline{{J}}^f_{i,j},{0})
  ,
 \forall j \in \mathbb{N}_{n_z} \}.\label{eq:Mj}

  \end{align}
and a locally Lipschitz remainder function $h(\cdot) \in \mathcal{H}_{M^c_i}$, where
{\begin{align}\label{eq:decomp1}
\hspace{-.3cm}f_{d,i}(z,\hat{z};\mathbf{m},h(\cdot))\hspace{-.1cm}=\hspace{-.1cm} h(\zeta_{\mathbf{m}}(\hat{z},{z}))\hspace{-.1cm}+\hspace{-.1cm}f_i(\zeta_{\mathbf{m}}(z,\hat{z}))\hspace{-.05cm}) \hspace{-.1cm}-\hspace{-.1cm}h_i(\zeta_{\mathbf{m}}(z,\hat{z})),
\end{align}}
$\zeta_{\mathbf{m}}(z, \hat{z})=[\zeta_{\mathbf{m},1}(z,\hat{z}),\dots,\zeta_{\mathbf{m},n_z}(z,\hat{z})]^\top$, $\forall j \in \mathbb{N}_{n_z}$:
\begin{align}\label{eq:corners}

{\zeta}_{\mathbf{m},j}(z,{\hat{z}})\hspace{-.1cm}=\hspace{-.1cm}\begin{cases} \hat{z}_j, \ \text{if} \ \mathbf{m}_j \hspace{-.1cm}=\hspace{-.1cm} \max(\overline{{J}}^f_{i,j},{0}) \\
{z}_j, \ \text{if} \ \mathbf{m}_j \hspace{-.1cm}=\hspace{-.1cm} \min(\underline{{J}}^f_{i,j},{0})
   \end{cases},
 \end{align}
 and $\mathcal{H}_{\mathbb{M}_i} \triangleq \{h:\mathbb{IZ} \to \mathbb{R} | [\underline{J}^h(z),\overline{J}^h_C(z)] \subseteq \mathbb{M}_i,\forall z \in \mathbb{IZ} \}$.

 Moreover, search for the tightest mixed-monotone upper and lower  remainder-form decomposition functions in the form of \eqref{eq:decomp1} can be equivalently restricted to the set of ``linear remainders", parametrized by $\mathbf{m} \in \mathbf{M}^c_i$, i.e.,

 linear remainders $\{{h}(\cdot)\}_{\mathbf{m} \in \mathbf{M}^c_i}=\{{\langle}\mathbf{m}^i,\cdot{\rangle}\}_{\mathbf{m} \in \mathbf{M}^c_i}$.

\end{prop}
\begin{cor}\label{cor:supproting_vectors}
Consider a locally Lipschitz mapping $\tilde{f}(\cdot):{\mathbb{I}\Xi} \triangleq [\underline{\xi},\overline{\xi}] \subseteq \mathbb{IR}^{n_{\xi}} \to \mathbb{R}^{n_x}$ that satisfies the assumptions in Proposition \ref{prop:Remainder_DF}. Let us define: $\mathbb{N}_{n_{x}} \triangleq \{1,\dots,n_{x}\}$ and
\begin{align}\label{eq:JSS_maker}
  \mathbf{H}_{\tilde{f}}  \triangleq  \{H  \in  \mathbb{R}^{ n_{x} \times n_{\xi}} | H^\top_{i,:}\in \mathbf{M}^c_i, \forall i \in \mathbb{N}_{n_{x}}  \},
\end{align}
where $\mathbf{M}^c_i$ is defined in \eqref{eq:Mj}. Then, $\forall \xi \in \mathbb{I}\Xi,\forall H \in \mathbf{H}^{\tilde{f}}, \tilde{g}^H(\xi)\triangleq \tilde{f}(\xi)-H\xi $ is proven to be a Jacobian sign-stable (JSS) function, i.e., {$\forall i \in \mathbb{N}_{n_x}, \forall j \in \mathbb{N}_{n_z}, J^{H}_{ij}(\xi)\triangleq \frac{\partial f_i}{\partial \xi_j}(\xi)\geq 0, \forall \xi \in  \mathbb{I}\Xi$ or $J^{H}_{ij}(\xi)\triangleq \frac{\partial \tilde{g}^H_i}{\partial \xi_j}(\xi)\leq 0, \forall \xi \in  \mathbb{I}{\Xi}$.}. Consequently, $\tilde{g}^H(\cdot)$ can be tightly bounded in each dimension $i \in \mathbb{N}_{n_x}$ by remainder-form decomposition functions $\tilde{g}_{d,i}(\cdot,\cdot;H^\top_{i,:},\langle H^\top_{i,:},\cdot \rangle)$, constructed using \eqref{eq:decomp1}--\eqref{eq:corners}, as follows:
\begin{align*}
 \tilde{g}_{d,i}(\underline{\xi},\overline{\xi};H^\top_{i,:},\langle H^\top_{i,:},\cdot \rangle)  \leq \tilde{g}_i(\xi) \leq  \tilde{g}_{d,i}(\overline{\xi},\underline{\xi};H^\top_{i,:},\langle H^\top_{i,:},\cdot \rangle),
\end{align*}
where, by \cite[Lemma 3]{khajenejad2021tightremainder} and defining $m \triangleq H^\top_{i,:}$, we obtain $\tilde{g}_{d,i}(\overline{\xi},\underline{\xi};{m},\langle {m},\cdot\rangle)=\tilde{f}_i(\zeta^+_m)+{m}^\top(\zeta^-_{{m}}-\zeta^+_{{m}})$, $\tilde{g}_{d,i}(\underline{\xi},\overline{\xi};{m}^\top,\langle {m},\cdot\rangle)=\tilde{f}_i(\zeta^-_{{m}})+{m}^\top(\zeta^+_{{m}}-\zeta^-_{{m}})$, $\zeta^+_m \triangleq \zeta_m({\xi},\underline{\xi})$, $\zeta^-_m \triangleq \zeta_m(\underline{\xi},\overline{\xi})$, with $\zeta_m(\cdot,\cdot)$ given in \eqref{eq:corners}.
\begin{proof}
The proof follows the lines of the proof of \cite[Lemma 1, Proposition 10 and Corollary 2]{khajenejad2021tightremainder}.
\end{proof}
\end{cor}
\section{Problem Formulation} \label{sec:Problem} \label{sec:formulation}
\vspace{-0.1cm}
\noindent\textbf{\emph{System Assumptions.}}
Consider the following bounded-error nonlinear constrained discrete-time system:
\begin{align} \label{eq:system}
\begin{array}{ll}
x_{k+1}&=\hat{f}(x_k,w_k,u_k)=f(z_k), \\
\hat{\mu}(x_k, u_k)&=\mu(x_k) \in \mathcal{Y}_k, \ x_0 \in \hat{\mathcal{X}}_0,w_k \in \mathcal{W}_k,  \end{array}
\end{align}
where $z_k \triangleq [x^\top_k   w^\top_k]^\top$, $x_k \in \mathbb{R}^{n_x}$ is the state vector, $w_k \in \mathcal{W}_k \subset \mathbb{R}^{n_w}$ is the augmentation of all the exogenous uncertain inputs, e.g., bounded process disturbance/noise and internal uncertainties such as uncertain parameters and $u_k \in \mathcal{U}_k \subseteq \mathbb{R}^{n_u}$ is the \emph{known} input signal. Furthermore,
$f :\mathbb{R}^{n_z} \to \mathbb{R}^{n_x}$ (with $n_z \triangleq n_x+n_w$) and $\mu : \mathbb{R}^{n_x}  \to  \mathbb{R}^{n_{\mu}}$
are nonlinear state vector field and observation/constraint mapping, respectively, which are well-defined, given $\hat{f}(\cdot,\cdot)$ and $\hat{\mu}(\cdot,\cdot)$, as well as the fact that $u_k$ is known. {Note that the mapping $\mu(\cdot)$ along with the set $\mathcal{Y}_k$ characterize all the existing and/or a prior known or even manufactured/redundant constraints over the states, observations and measurement noise signals or uncertain parameters at time step $k$.}.

The initial state estimate $\hat{\mathcal{X}}_0$ is assumed to be a known set a prior.
Moreover, we assume the following assumptions.
\begin{assumption}\label{ass:polytopic_uncertainty}
The sets $\hat{\mathcal{X}}_0$, as well as $\mathcal{W}_k,\mathcal{U}_k, \forall k \geq 0$ are prior known polytopes, or equivalently constrained zonotopes or zonotope bundles (cf. Definition \ref{defn:CZ_ZB}).
\end{assumption}

\begin{assumption}\label{assumption:mix-lip}
The nonlinear vector fields $f(\cdot)$ and $\mu(\cdot)$ are locally Lipschitz on their domains. Consequently, they are differentiable and have bounded Jacobian matrix elements, almost everywhere. We further assume that given any $\mathcal{Z} \subset \mathbb{R}^{n_z}$ and $\mathcal{X} \subset \mathbb{R}^{n_x}$, some upper and lower bounds for all elements of Jacobian matrices for $f(\cdot)$ and $\mu(\cdot)$ over $\mathcal{Z}$ and $\mathcal{X}$ are available or can be computed. In other words, $\exists \underline{{J}}^f,\overline{{J}}^f \in \mathbb{R}^{n_x \times n_z},\underline{{J}}^\mu,\overline{{J}}^\mu \in \mathbb{R}^{n_{\mu} \times n_x}$, such that: $\underline{{J}}^f \leq {{J}}^f(z) \leq \overline{{J}}^f,\underline{{J}}^{\mu} \leq {{J}}^{\mu}(x) \leq \overline{{J}}^{\mu},\forall z \in {\mathcal{Z}},\forall x \in \mathcal{X}$, where ${J}^f(z)$ and ${J}^{\mu}(x)$ denote the Jacobian matrices of the mappings $f(\cdot)$ and $\mu(\cdot)$ at the points $z$ and $x$, respectively.
\end{assumption}
In this paper, we aim to propose novel set-membership approaches for obtaining set-valued state estimates for bounded-error nonlinear systems in the form of \eqref{eq:system}. More formally, proceeding the well-known two-step i) Propagation (prediction), ii) update (refinement) approach and given the initial uncertainty set $\hat{\mathcal{X}}_0$, we seek to solve the following corresponding problems at each time step $k \in \mathbb{N}$ (where $\mathcal{Z}_k \triangleq \mathcal{X}^u_k \times \mathcal{U}_k, \mathcal{X}^p_0 \triangleq \hat{\mathcal{X}}_0$):
\begin{problem}[Propagation]\label{prob:propagation}
Find the ``propagated set" $\mathcal{X}^p_k$ that satisfies
\begin{align}\label{eq:propagation}
f(\mathcal{Z}_{k-1})\hspace{-.1cm}\triangleq \hspace{-.1cm}\{\hat{f}(x,w,u_{k-1})|x \hspace{-.1cm} \in \hspace{-.1cm}\mathcal{X}^u_{k-1}, w \in \mathcal{W}_k \} \subseteq \mathcal{X}^p_k.
\end{align}
\end{problem}
\begin{problem}[Update]\label{prob:update}
Find the updated set $\mathcal{X}^u_k$ that satisfies
\begin{align}\label{eq:update}
\mathcal{X}^p_k \cap_{\mu} \mathcal{Y}_k \triangleq \{x \in \mathcal{X}^p_k | \mu(x) \in \mathcal{Y}_k \}  \subseteq \mathcal{X}^u_k.
\end{align}
\end{problem}
\section{Main Results} \label{sec:main}
To address the above problems, we propose set-theoretic approaches which are based on our recently developed tight remainder-form mixed-monotone decomposition functions \cite{khajenejad2021tightremainder} (also cf. Proposition \ref{prop:Remainder_DF} and Corollary \ref{cor:supproting_vectors}), to compute supersets of i) $f(\mathcal{Z})$, and ii) $\mathcal{Z}_f \cap_{\mu} \mathcal{Z}_{\mu}$, given locally Lipschitz functions $f,\mu$ that satisfy Assumption \ref{assumption:mix-lip} and arbitrary sets $\mathcal{Z},\mathcal{Z}_f,\mathcal{Z}_{\mu}$ that satisfy Assumption \ref{ass:polytopic_uncertainty}. Then, armed with the aforementioned proposed set-membership approaches, we sequentially apply the one-step operations in \eqref{eq:propagation} and \eqref{eq:update}
at each time step $k \in \mathbb{N}$ to system \eqref{eq:system}, with the purpose of  computing i) the propagated set $\mathcal{X}^p_k$ that satisfies the inclusion in  \eqref{eq:propagation},
and ii) the updated set $\mathcal{X}^u_k$ that satisfies the inclusion in \eqref{eq:update}, respectively.
\subsection{Decomposition-Based ZB/CZ propagation} \label{sec:propagation}
In this section, we address Problem \ref{prob:propagation}, assuming that the initial set is a zonotope bundle (Lemma \ref{lem:ZB_propagation}) or a constraint zonotope (Lemma \ref{lem:CZ_propagation}). The main idea is to ``lift up" the initial ZB/CZ's from the $z$-space, i.e., the space of augmented state $x$ and process uncertainty $w$, to intervals in the $\xi$-space, i.e., the space of ZB/CZ generators. Then, based on our recent results in \cite{khajenejad2021tightremainder}, we decompose the transformed vector fields in the $\xi$-space into two components, a Jacobian sign stable (JSS) and a linear remainder mapping (cf. Corollary \ref{cor:supproting_vectors}). Finally, we apply our recently developed family of mixed-monotone remainder-form decomposition functions to find enclosures to the JSS components, with interval domains, which are proven to be tight by Corollary \ref{cor:supproting_vectors}. Using these tight obtained bounds and thanks to linearity of the remainders, we show that by augmenting and intersecting all the obtained enclosures, the resultant set is a ZB/CZ. We formally summarize our proposed Decomposition-Based ZB/CZ approach through the following Lemmas \ref{lem:ZB_propagation} and \ref{lem:CZ_propagation}.
\begin{lem}[Decomposition-Based ZB Propagation]\label{lem:ZB_propagation}
Let $f:\mathcal{Z}\subset \mathbb{R}^{n_z}  \to \mathbb{R}^{n_x}$ satisfies Assumption \ref{assumption:mix-lip}. Let $\mathcal{Z}$ be a ZB in $\mathbb{R}^{n_z}$, i.e., $\mathcal{Z}=\bigcap\limits_{s=1}^{S} \{G_{s},c_{s}\}_Z$, and $\forall s \in \mathbb{N}_{S} \triangleq \{1,\dots,S\}$, $n_s$ be the number of generators of the corresponding zonotope. Then, the following set inclusion holds:
\begin{align}
f(\mathcal{Z}) \subseteq \mathcal{ZB}_f \triangleq\bigcap\limits_{s=1}^{S} \bigcap\limits_{H_s \in \mathbf{H}_{\tilde{f}_s}}\{G^{H_s}_{s},c^{H_s}_{s}\}_Z, \label{eq:ZB_prediction}
\end{align}
where $G^{H_s}_{s} \triangleq [H_s \  \frac{1}{2}\textstyle{\mathrm{diag}}(\overline{g}^{H_s}_{s}-\underline{g}^{H_s}_{s})], \ c^{H_s}_{s} \triangleq  \frac{1}{2}(\overline{g}^{H_s}_{s}+\underline{g}^{H_s}_{s})$,
\begin{align}
& \overline{g}^{H_s}_{s,i} \triangleq \label{eq:ov_g} g^{s}_{i,d}(\mathbf{1}_{n_{s}},-\mathbf{1}_{n_{s}};{H_s^\top}_{(i,:)},\langle {H_s^\top}_{(i,:)},\cdot\rangle),\\
&\underline{g}^{H_s}_{s,i} \triangleq \label{eq:und_g} g^{s}_{i,d}(-\mathbf{1}_{n_{s}},\mathbf{1}_{n_{s}};{H_s^\top}_{(i,:)},\langle {H_s^\top}_{(i,:)},\cdot\rangle),
\end{align}
$g^{s}_{i,d}(\cdot,\cdot;{H_s^\top}_{(i,:)},\langle {H_s^\top}_{(i,:)},\cdot\rangle)$ is the tight mixed-monotone decomposition function (cf. Proposition \ref{prop:Remainder_DF}) for the JSS mapping $g^{H_s}_{s,i}(\xi) \triangleq \tilde{f}_{s,i}(\xi)-\langle {H_s^\top}_{(i,:)}, \xi \rangle : \mathbb{B}^{n_s}_{\infty} \to \mathbb{R}^{n_x}$, $\mathbf{H}_{\tilde{f}_s}$ is defined in Corollary \ref{cor:supproting_vectors} (with  the  corresponding  function  being $\tilde{f}_s$)
and $\tilde{f}_{s}(\xi) \triangleq f({c}_s+{G}_s\xi)$.
\end{lem}
\begin{proof}
To show \eqref{eq:ZB_prediction}, $\forall s \in \mathbb{N}_S$, consider the zonotope $\mathcal{Z}_s \triangleq \{G_s,c_s\}_Z \triangleq \{z=G_s\xi+c_s|\xi \in \mathbb{B}^{n_s}_{\infty}\}$ and let us define $\tilde{f}_s(\xi):\mathbb{B}^{n_s}_{\infty} \to \mathbb{R}^{n_x} \triangleq f(G_s\xi+c_s)$ that implies
\begin{align}\label{eq:set_equiv}
f(\mathcal{Z}_s) \subseteq \tilde{f}_s(\mathbb{B}^{n_s}_{\infty}), \forall s \in \mathbb{N}_S .
\end{align}
On the other hand, note that by Corollary \ref{cor:supproting_vectors}, $\forall H_s \in \mathbf{H}_{\tilde{f}_s}$, $\tilde{f}_s(\cdot)$ can be decomposed as
\begin{align}\label{eq:decomposed}
\tilde{f}_s(\xi)\hspace{-.1cm} =\hspace{-.1cm}g^{H_s}_s(\xi)\hspace{-.1cm}+\hspace{-.1cm}H_s\xi, \forall s \in \mathbb{N}_S, \forall \xi \in \mathbb{B}^{n_s}_{\infty},\forall H_s \in \mathbf{H}_{\tilde{f}_s}
\end{align}
where $g^{H_s}_s(\xi)$ is a JSS function in $\mathbb{B}^{n_s}_{\infty}$ and $\mathbf{H}_{\tilde{f}_s}$ can be computed from \eqref{eq:JSS_maker}, with the corresponding function being $\tilde{f}_s$. Now \eqref{eq:set_equiv} and \eqref{eq:decomposed} together imply:
\begin{align}\label{eq:sum_decomp}
f(\mathcal{Z}_s) \subseteq g^{H_s}_s(\mathbb{B}^{n_s}_{\infty})\oplus H_s\mathbb{B}^{n_s}_{\infty}, \forall s \in \mathbb{N}_S, \forall H_s \in \mathbf{H}_{\tilde{f}_s}.
\end{align}
Again, it follows from Corollary \ref{cor:supproting_vectors} and the fact that $g^{H_s}_s(\xi)$ is a JSS function that in each dimension $i \in \mathbb{N}_{n_x}$, $g^{H_s}_{s,i}(\xi)$ can be tightly bounded as $\underline {g}^{H_s}_{s,i}\leq g^{H_s}_{s,i}(\xi) \leq \overline {g}^{H_s}_{s,i}, \forall \xi \in \mathbb{B}^{n_s}_{\infty}, \forall H_s \in \mathbf{H}_{\tilde{f}_s}  $, with $\overline {g}^{H_s}_{s,i},\underline {g}^{H_s}_{s,i}$ given in \eqref{eq:ov_g} and \eqref{eq:und_g}, respectively. Augmenting all these $\mathbb{N}_{nx}$ one-dimensional inequalities yields the following set inclusion for all $s \in \mathbb{N}_{S}$ and all $H_s \in \mathbf{H}_{\tilde{f}_s} $:
${g}^{H_s}_{s}(\mathbb{B}^{n_s}_{\infty}) \subseteq [\underline {g}^{H_s}_{s},\overline {g}^{H_s}_{s}]=\frac{1}{2}((\underline {g}^{H_s}_{s}+\overline {g}^{H_s}_{s})\oplus \textstyle{\mathrm{diag}}(\overline {g}^{H_s}_{s}-\underline {g}^{H_s}_{s})\mathbb{B}^{n_x}_{\infty})$,
where the last equality follows from Proposition \ref{prop:interval_cen_rep}. This, \eqref{eq:sum_decomp} and the fact that the inclusion in \eqref{eq:sum_decomp} holds for all $s \in \mathbb{N}_{S}$ and all $H_s \in \mathbf{H}_{\tilde{f}_s} $ and hence for the intersection of all of them, return the result in \eqref{eq:ZB_prediction}.
\end{proof}\vspace{-.3cm}
\begin{lem}[Decomposition-Based CZ Propagation]\label{lem:CZ_propagation}
Let $f:\mathcal{Z}\subset \mathbb{R}^{n_z}  \to \mathbb{R}^{n_x}$ satisfies Assumption \ref{assumption:mix-lip}. Let $\mathcal{Z}$ be a CZ in $\mathbb{R}^{n_z}$, i.e., $\mathcal{Z}= \{\tilde{G},\tilde{c},\tilde{A},\tilde{b}\}_{CZ}$, and $n_g$ be the number of generators of $\mathcal{Z}$. Then, the following set inclusion holds:
\begin{align}
f(\mathcal{Z}) \subseteq \mathcal{CZ}_f \triangleq \bigcap\limits_{H \in \mathbf{H}_{\tilde{f}}} \{\tilde{G}^H,\tilde{c}^H,\mathbb{A},\tilde{b}\}_{CZ},\label{eq:CZ_prediction}
\end{align}
where $\tilde{G}^H \triangleq [H \ \  \frac{1}{2}\textstyle{\mathrm{diag}}(\overline{g}^H-\underline{g}^H)], \mathbb{A} \triangleq [\tilde{A} \ \ 0_{n_g \times n_x}]$,
\begin{align}
 \overline{g}^H_{i} &\triangleq \label{eq:ov_g_cz} \tilde{g}_{i,d}(\overline{\mathbf{l}}_{n_{g}},\underline{\mathbf{l}}_{n_{g}};H^\top_{i,:},\langle H^\top_{i,:},\cdot\rangle),\tilde{c}^H \hspace{-.1cm}\triangleq \hspace{-.1cm} \frac{1}{2}(\overline{g}^H+\underline{g}^H),\\
\underline{g}^H_{i} &\triangleq \label{eq:und_g_cz} \tilde{g}_{i,d}(\underline{\mathbf{l}}_{n_{g}},\overline{\mathbf{l}}_{n_{g}};H^\top_{i,:},\langle H^\top_{i,:},\cdot\rangle), \\
\nonumber \overline{\mathbf{l}}_{n_{g}} & \triangleq \min(\mathbf{1}_{n_g},\tilde{A}^\dagger\tilde{b}+\kappa \mathbf{r}_{n_g}),\underline{\mathbf{l}}_{n_{g}} \hspace{-.1cm} \triangleq \hspace{-.1cm} \max(-\mathbf{1}_{n_g},\tilde{A}^\dagger\tilde{b}\hspace{-.1cm}-\hspace{-.1cm}\kappa \mathbf{r}_{n_g})
\end{align}
$\tilde{g}_{i,d}(\cdot,\cdot;H^\top_{i,:},\langle H^\top_{i,:},\cdot\rangle)$ is the tight mixed-monotone decomposition function (cf. Proposition \ref{prop:Remainder_DF}) for the JSS mapping $\tilde{g}_{i}(\xi) \triangleq \tilde{f}_{i}(\xi)-\langle H^\top_{i,:}, \xi \rangle : \mathbb{B}^{n_g}_{\infty} \to \mathbb{R}^{n_x}$, $\mathbf{H}_{\tilde{f}}$ is defined in Corollary \ref{cor:supproting_vectors} and $\tilde{f}(\xi) \triangleq f(\tilde{c}+\tilde{G}\xi)$, $\mathbf{r}_{n_g} \triangleq \textstyle{\mathrm{rowsupp}}(I_{n_g}-\tilde{A}^\dagger \tilde{A})$ and $\kappa$ is a very large positive real number (infinity).
\end{lem}
\begin{proof}
To prove the inclusion in \eqref{eq:CZ_prediction}, consider the constrained zonotope representation of the set $\mathcal{Z}$, i.e., $\mathcal{Z} \triangleq \{\tilde{G},\tilde{c},\tilde{A},\tilde{b}\}_{CZ} \triangleq \{z=\tilde{G}\xi+c|\xi \in \mathbb{B}^{n_g}_{\infty}, \tilde{A}\xi=\tilde{b}\}$. Using similar notation as in the proof of Lemma \ref{lem:ZB_propagation}, let us define $\tilde{f}(\xi):\mathbb{B}^{n_g}_{\infty} \to \mathbb{R}^{n_x} \triangleq f(\tilde{G}\xi+\tilde{c})$ that consequently return
\begin{align}\label{eq:set_equiv_cz}
f({\mathcal{Z}}) \subseteq \{\tilde{f}(\xi)|\xi \in \mathbb{B}^{n_g}_{\infty},\tilde{A}\xi=\tilde{b} \} .
\end{align}
Note that by \cite[Theorem 2]{james1978generalised}, $\tilde{A}\xi=\tilde{b} \Rightarrow \xi \in \mathbb{I}{\Xi} \triangleq [\tilde{A}^\dagger\tilde{b}-\kappa \mathbf{r}_{n_g},\tilde{A}^\dagger\tilde{b}-\kappa \mathbf{r}_{n_g}]$, where $\mathbf{r}_{n_g} \triangleq \textstyle{\mathrm{rowsupp}}(I_{n_g}-\tilde{A}^\dagger \tilde{A})$ and $\kappa$ is a very large positive real number. This, in addition to the fact that $\xi \in \mathbb{B}^{n_g}_{\infty}$ (cf. \eqref{eq:set_equiv_cz}), imply that $\xi \in \mathbb{I}\tilde{\Xi}\triangleq \mathbb{I}{\Xi} \cap \mathbb{B}^{n_g}_{\infty}=[\underline{\mathbf{l}}_{n_g},\overline{\mathbf{l}}_{n_g}]$, where $\underline{\mathbf{l}}_{n_g},\overline{\mathbf{l}}_{n_g}$ are defined below \eqref{eq:und_g_cz}.
On the other hand, similar to the proof of Lemma \ref{lem:ZB_propagation}, using Corollary \ref{cor:supproting_vectors} we conclude that $\forall H \in \mathbf{H}_{\tilde{f}}$, $\tilde{f}(\cdot)$ can be decomposed as
\begin{align}\label{eq:decomposed_cz}
\begin{array}{rl}
\tilde{f}(\xi) &=\tilde{g}^H(\xi)+H\xi,  \ \ \ \forall H \in \mathbf{H}_{\tilde{f}},\forall \xi \in \mathbb{I}\tilde{\Xi}, \\
\Rightarrow \tilde{f}(\mathbb{I}\tilde{\Xi}) &\subseteq \tilde{g}^H(\mathbb{I}\tilde{\Xi})\oplus H\mathbb{I}\tilde{\Xi}, \forall H \in \mathbf{H}_{\tilde{f}},
\end{array}
\end{align}
where $\tilde{g}^H(\xi)$ is a JSS function in $\mathbb{I}\tilde{\Xi}$ and $\mathbf{H}_{\tilde{f}}$ is given in \eqref{eq:JSS_maker}.
Also, by Corollary \ref{cor:supproting_vectors},
in each dimension $i \in \mathbb{N}_{n_x}$, $\tilde{g}^H_{i}(\xi)$ can be tightly bounded as $\underline {{g}}^H_{i}\leq \tilde{g}^H_{i}(\xi) \leq \overline {{g}}^H_{i}, \forall \xi \in \mathbb{I}\tilde{\Xi}, \forall H \in \mathbf{H}_{\tilde{f}} $, with $\overline{{g}}^H_{i},\underline{{g}}^H_{i}$ given in \eqref{eq:ov_g_cz} and \eqref{eq:und_g_cz}, respectively. Augmenting all these $\mathbb{N}_{nx}$ one-dimensional inequalities, as well as Proposition \ref{prop:interval_cen_rep} yield the following set inclusion: $\forall H \in \mathbf{H}_{\tilde{f}} $:
\begin{align*
\tilde{g}^H(\mathbb{I}\tilde{\Xi})\hspace{-.1cm} \subseteq\hspace{-.1cm} [\underline {g}^H,\overline {g}^H]\hspace{-.1cm}=\hspace{-.1cm}\frac{1}{2}((\underline {g}^H+\overline {g}^H)\oplus \textstyle{\mathrm{diag}}(\overline {g}^H-\underline {g}^H)\mathbb{B}^{n_x}_{\infty}).
\end{align*}
This, \eqref{eq:set_equiv_cz}, \eqref{eq:decomposed_cz} and the fact that the inclusion in \eqref{eq:decomposed_cz} holds for  all $H \in \mathbf{H}_{\tilde{f}}$ and hence for the intersection of all of them, return: $ \forall H \in \mathbf{H}_{\tilde{f}}: f({\mathcal{Z}}) \subseteq \{H\xi+\textstyle{\mathrm{diag}}(\overline {g}^H-\underline {g}^H)\theta+\frac{1}{2}((\underline {g}^H+\overline {g}^H)|\xi \in \mathbb{B}^{n_g}_{\infty},\theta \in \mathbb{B}^{n_x}_{\infty}, \tilde{A}\xi=\tilde{b}\}$, where the set on the right hand side of the inclusion is equivalent to the intersection of the CZs {on} the right hand side of \eqref{eq:CZ_prediction}.
\end{proof}
Finally, for further improvement, we can take the intersection of the resultant propagated sets in Lemmas \ref{lem:ZB_propagation} and \ref{lem:CZ_propagation}.   This, is formally summarized in the following Theorem \ref{thm:ZB_CZ_propagation}.
\begin{thm}[Decomposition-Based ZB/CZ Propagation]\label{thm:ZB_CZ_propagation}
Suppose all the assumptions in Lemmas \ref{lem:ZB_propagation} and \ref{lem:CZ_propagation} hold. Then, $f(\mathcal{Z}) \subseteq \mathcal{ZB}_f \cap \mathcal{CZ}_f$, where $\mathcal{ZB}_f,\mathcal{CZ}_f$ are computed in Lemmas \ref{lem:ZB_propagation} and \ref{lem:CZ_propagation}, respectively.
\end{thm}
\begin{proof}
It follows from Lemmas \ref{lem:ZB_propagation} and \ref{lem:CZ_propagation} that $f(\mathcal{Z}) \subseteq \mathcal{ZB}_f $ and $f(\mathcal{Z}) \subseteq \mathcal{CZ}_f$, and so $f(\mathcal{Z}) \subseteq \mathcal{ZB}_f \cap \mathcal{CZ}_f$.
\end{proof}
\subsection{Decomposition-Based CZ/ZB update} \label{sec:update}
In this section, we address Problem \ref{prob:update}, given a locally Lipschitz nonlinear vector field $\mu(\cdot)$ and assuming that the initial propagated and the observation/constraint sets are zonotope bundles (Lemma \ref{lem:ZB_update}) or constraint zonotopes (Lemma \ref{lem:CZ_update}). Using similar idea as in Section \ref{sec:propagation}, i.e, lifting to the space of generators, decomposing the transformed observation function into a JSS and a linear component, applying the tight remainder-form decomposition functions \cite{khajenejad2021tightremainder} to bound the JSS component, augmenting and intersecting, as well as taking the advantage of linear remainder functions, we obtain ZB/CZ enclosures to the nonlinear generalized intersection in \eqref{eq:update}. The results of this section are summarized in Lemmas \ref{lem:ZB_update} and \ref{lem:CZ_update} and Theorem \ref{thm:ZB_CZ_update}.

\begin{lem}[Decomposition-Based ZB Update]\label{lem:ZB_update}
Let $\mu: \mathbb{R}^{n_x}  \to \mathbb{R}^{n_{\mu}}$ satisfies Assumption \ref{assumption:mix-lip}. Let $\mathcal{Z}_f \subset \mathbb{R}^{n_{x}}$ and $Z_{\mu} \subset \mathbb{R}^{n_{\mu}}$ be two ZB sets, i.e.,
$\mathcal{Z}_f=\mathcal{ZB}_f=\bigcap\limits_{r=1}^{R} \{G_f^{r},c_f^{r}\}_Z$ and $\mathcal{Z}_{\mu}=\mathcal{ZB}_{\mu}=\bigcap\limits_{t=1}^{T} \{G_{\mu}^{t},c_{\mu}^{t}\}_Z$,
and $\forall r \in \mathbb{N}_{R} \triangleq \{1,\dots,R\},\forall t \in \mathbb{N}_{T} \triangleq \{1,\dots,T\}$, $n_r,n_t$ be the number of generators of the corresponding zonotopes, respectively.
Then, the following set inclusion holds:
\begin{align} \label{eq:ZB_update}
\mathcal{ZB}_f \cap_{\mu} \mathcal{ZB}_{\mu} \hspace{-.1cm}\subseteq \hspace{-.1cm} \mathcal{ZB}_u \hspace{-.1cm} \triangleq \hspace{-.1cm}\bigcap\limits_{r=1}^R\bigcap\limits_{t=1}^T\bigcap\limits_{Q_r \in \mathbf{Q}_{\tilde{\mu}_r}} \{\hat{G}^t_{r},\hat{c}_{r},\hat{A}^{Q_r}_{r,t},\hat{b}^{Q_r}_{r,t}\}_{CZ},
\end{align}
where $ \hat{G}^t_{r} \triangleq [{G}^r_f \ \mathbf{0}^t],\hat{c}_r \triangleq c^r_f, \ \hat{b}^{Q_r}_{r,t} \triangleq c^t_{\mu}-\frac{1}{2}(\overline{p}^{Q_r}_r+\underline{p}^{Q_r}_r)$,
\begin{align}
\nonumber \hat{A}^{Q_r}_{r,t} &\triangleq \begin{bmatrix} Q_r & -G^t_{\mu} & \frac{1}{2}\textstyle{\mathrm{diag}}(\overline{p}^{Q_r}_r-\underline{p}^{Q_r}_r) \end{bmatrix}, \\
 \overline{p}^{Q_r}_{r,i} &\triangleq  p^{r}_{i,d}(\mathbf{1}_{n_{r}},-\mathbf{1}_{n_{r}};Q^\top_{r(i,:)},\langle \label{eq:over_p_zb} Q^\top_{r(i,:)},\cdot\rangle), \\
 \underline{p}^{Q_r}_{r,i} &\triangleq p^{r}_{i,d}(-\mathbf{1}_{n_{r}},\mathbf{1}_{n_{r}};Q^\top_{r(i,:)},\langle Q^\top_{r(i,:)},\cdot\rangle),\label{eq:under_p_zb}
\end{align}
 $p^{r}_{i,d}(\cdot,\cdot;Q_r,\langle Q^\top_{r(i,:)},\cdot\rangle)$ is the tight mixed-monotone decomposition function (cf. Proposition \ref{prop:Remainder_DF}) for the JSS mapping $p^{Q_r}_{r,i}(\alpha) \triangleq \tilde{\mu}_{r,i}(\alpha)-\langle Q^\top_{r(i,:)},\alpha\rangle:\mathbb{B}^{n_r}_{\infty} \to \mathbb{R}^{n_{\mu}}$, $\mathbf{Q}_{\tilde{\mu}_r}$ is defined similar to $\mathbf{H}_{f}$ in Corollary \ref{cor:supproting_vectors} (with the corresponding function being $\tilde{\mu}_r(\alpha) \triangleq \mu(c^r_{f}+G^r_{f}\alpha)$) and $\mathbf{0}^t$ is a zero matrix in $\mathbb{R}^{n_{x} \times (n_{t}+n_{\mu})}$.
\end{lem}
\begin{proof}
Suppose $z \in \mathcal{ZB}_f \cap_{\mu} \mathcal{ZB}_{\mu}$. Then by definition of the operator $\cap_{\mu}$ (cf. \eqref{eq:update}), $z \in \mathcal{ZB}_f$ and $\mu(z) \in \mathcal{ZB}_{\mu}$. The former implies that $\forall r \in \mathbb{N}_{R}, \exists \alpha \in \mathbb{B}^{n_{r}}_{\infty}$ such that $z=G^r_f\alpha+c^r_f$, while it follows from the latter that $\mu(z)=\mu(G^r_f\alpha+c^r_f) \triangleq \tilde {\mu}_r(\alpha) \in \mathcal{ZB}_{\mu} \Rightarrow \forall t \in \mathbb{N}_T, \exists \zeta \in \mathbb{B}^{n_t}_{\infty}$, such that $\tilde{\mu}_r(\alpha)=c^t_{\mu}+G^t_{\mu}\zeta$.
Putting these two results in a set representation form, we obtain:
\begin{align} \label{eq:int_cz}
z \hspace{-.1cm}\in \hspace{-.1cm} \bigcap\limits_{r=1}^R\hspace{-.1cm}\bigcap\limits_{t=1}^T\{G^r_f\alpha\hspace{-.1cm}+\hspace{-.1cm}c^r_f|\tilde{\mu}_r(\alpha)\hspace{-.1cm}=\hspace{-.1cm}c^t_{\mu}\hspace{-.1cm}+\hspace{-.1cm}G^t_{\mu}\zeta,\alpha \hspace{-.1cm}\in\hspace{-.1cm} \mathbb{B}^{n_{r}}_{\infty},\zeta \hspace{-.1cm}\in\hspace{-.1cm} \mathbb{B}^{n_t}_{\infty} \}
\end{align}
On the other hand, using Corollary \ref{cor:supproting_vectors}, $\tilde{\mu}_r(\cdot)$ can be decomposed into a JSS and a linear mapping as follows: $\forall r \in \mathbb{N}_R,\forall Q_r \in \mathbf{Q}_{\tilde{\mu}_r}, \forall \alpha \in \mathbb{B}^{n_{r}}_{\infty}$:
\begin{align}\label{eq:dec_zb_update}
\tilde{\mu}_r(\alpha)=p^{Q_r}_r(\alpha)+Q_r\alpha.
\end{align}
Moreover, by the same corollary, the JSS component $p^{Q_r}_r(\cdot)$ is tightly bounded as follows: $\forall i \in \mathbb{N}_{n_{\mu}},\forall Q_r \in \mathbf{Q}_{\tilde{\mu}_r}, \underline{p}^{Q_r}_{r,i} \leq p^{Q_r}_{r.i}(\alpha) \leq  \underline{p}^{Q_r}_{r,i}, \forall \alpha \in \mathbb{B}^{n_{r}}_{\infty}$, with $\underline{p}^{Q_r}_{r,i},\overline{p}^{Q_r}_{r,i}$ given in \eqref{eq:over_p_zb} and \eqref{eq:under_p_zb}, respectively. This, as well as \eqref{eq:dec_zb_update} and Proposition \ref{prop:interval_cen_rep} imply $\forall r \in \mathbb{N}_R,\forall Q_r \in \mathbf{Q}_{\tilde{\mu}_r}, \forall \alpha \in \mathbb{B}^{n_{r}}_{\infty}, \exists \theta \in \mathbb{B}^{n_{\mu}}_{\infty}$ such that $\tilde{\mu}_r(\alpha)=\frac{1}{2}(\underline{p}^{Q_r}_{r,i}+\overline{p}^{Q_r}_{r,i})+\frac{1}{2}\textstyle{\mathrm{diag}}(\overline{p}^{Q_r}_{r,i}-\underline{p}^{Q_r}_{r,i})\theta+Q_r\alpha$, that along with \eqref{eq:int_cz} returns $z \in  \bigcap\limits_{r=1}^R\bigcap\limits_{t=1}^T\bigcap\limits_{Q_t \in \mathbf{Q}_{\tilde{\mu}_t}} \{G^r_f\alpha+c^r_f|\frac{1}{2}(\underline{p}^{Q_r}_{r,i}+\overline{p}^{Q_r}_{r,i})+\frac{1}{2}\textstyle{\mathrm{diag}}(\overline{p}^{Q_r}_{r,i}-\underline{p}^{Q_r}_{r,i})\theta+Q_r\alpha=c^t_{\mu}+G^t_{\mu}\zeta,\alpha \in \mathbb{B}^{n_{r}}_{\infty},\zeta \in \mathbb{B}^{n_t}_{\infty}, \theta \in \mathbb{B}^{n_{\mu}}_{\infty} \}  $, where the set on the right hand side is equivalent to the one on the right hand side of \eqref{eq:ZB_update}.
\end{proof}
\begin{lem}[Decomposition-Based CZ Update]\label{lem:CZ_update}
Let $\mu: \mathbb{R}^{n_x}  \to \mathbb{R}^{n_{\mu}}$ satisfies Assumption \ref{assumption:mix-lip}. Let $\mathcal{Z}_f \subset \mathbb{R}^{n_{x}}$ and $\mathcal{Z}_{\mu} \subset \mathbb{R}^{n_{\mu}}$ be two CZ sets, i.e.,
$\mathcal{Z}_f=\mathcal{CZ}_f= \{\tilde{G}_f,\tilde{c}_f,\tilde{A}_f,\tilde{b}_f\}_{CZ}$ and $\mathcal{Z}_{\mu}=\mathcal{CZ}_{\mu}= \{\tilde{G}_{\mu},\tilde{c}_{\mu},\tilde{A}_{\mu},\tilde{b}_{\mu}\}_{CZ}$,
and $n_c,n_{\tau}$ be the number of generators of $\mathcal{Z}_f,\mathcal{Z}_{\mu}$, respectively. Then, the following set inclusion holds:
\begin{align} \label{eq:CZ_update}
\mathcal{CZ}_f \cap_{\mu} \mathcal{CZ}_{\mu} \subseteq \mathcal{CZ}_u \triangleq \bigcap\limits_{\Omega \in \mathbf{\Omega}_{{\lambda}}} \{\mathbb{G},\tilde{c}_{f},\mathbb{A}_{\Omega},\tilde{b}_{\Omega}\}_{CZ},
\end{align}
where $\mathbb{G} \triangleq [\tilde{G}_f \ {0} \ 0], \ \tilde{b}_{\Omega} \triangleq [\tilde{b}^\top_f \ \tilde{b}^\top_{\mu} \ (\tilde{c}_f-\frac{1}{2}(\overline{\nu}^{\Omega}+\underline{\nu}^{\Omega}))^\top]^\top$,
\begin{align}
\nonumber \mathbb{A}_{\Omega} &\triangleq \begin{bmatrix} \tilde{A}_f & 0 & 0 \\ 0 & \tilde{A}_{\mu} & \\ \Omega & -\tilde{G}_{\mu} & \frac{1}{2}\textstyle{\mathrm{diag}}(\overline{\nu}^{\Omega}-\underline{\nu}^{\Omega}) \end{bmatrix}, \\
 \overline{\nu}^{\Omega}_{i} &\triangleq  {\nu}_{i,d}(\overline{\mathbf{l}}_{n_{c}},\underline{\mathbf{l}}_{n_{c}};\Omega^\top_{(i,:)},\langle \label{eq:over_p_cz} \Omega^\top_{(i,:)},\cdot\rangle), \\
 \underline{\nu}^{\Omega}_{i} &\triangleq {\nu}_{i,d}(\underline{\mathbf{l}}_{n_{c}},\overline{\mathbf{l}}_{n_{c}};\Omega^\top_{(i,:)},\langle \Omega^\top_{(i,:)},\cdot\rangle),\label{eq:under_p_cz}\\
 \nonumber \overline{\mathbf{l}}_{n_{c}} & \triangleq \hspace{-.1cm} \min(\mathbf{1}_{n_c},\tilde{A}_f^\dagger\tilde{b}_f+\kappa \mathbf{r}_{n_c}),\underline{\mathbf{l}}_{n_{c}} \hspace{-.15cm} \triangleq \hspace{-.1cm} \max(-\mathbf{1}_{n_c},\tilde{A}_f^\dagger\tilde{b}_f\hspace{-.1cm}-\hspace{-.1cm}\kappa \mathbf{r}_{n_c}\hspace{-.05cm})
\end{align}
 ${\nu}_{i,d}(\cdot,\cdot;{\Omega}^\top_{(i,:)},\langle \Omega^\top_{(i,:)},\cdot\rangle)$ is the tight mixed-monotone decomposition function (cf. Proposition \ref{prop:Remainder_DF}) for the JSS mapping $\nu^{\Omega}_{i}(\beta) \triangleq {\lambda}_{i}(\beta)-\langle \Omega^\top_{(i,:)},\beta\rangle:\mathbb{B}^{n_c}_{\infty} \to \mathbb{R}^{n_{\mu}}$, $\mathbf{\Omega}_{{\lambda}}$ is defined similar to $\mathbf{H}_{f}$ in Corollary \ref{cor:supproting_vectors} (with the corresponding function being ${\lambda}(\beta) \triangleq \mu(\tilde{c}_{f}+\tilde{G}_{f}\beta)$), $\mathbf{r}_{n_c} \triangleq \textstyle{\mathrm{rowsupp}}(I_{n_c}-\tilde{A}_f^\dagger \tilde{A}_f)$ and $\kappa$ is a very large positive real number (infinity).
\end{lem}
\begin{proof}
Suppose $z \in \mathcal{CZ}_f \cap_{\mu} \mathcal{CZ}_{\mu}$. Then by definition of the operator $\cap_{\mu}$ (cf. \eqref{eq:update}), $z \in \mathcal{CZ}_f$ and $\mu(z) \in \mathcal{CZ}_{\mu}$. The former implies that $ \exists \beta \in \mathbb{B}^{n_{c}}_{\infty}$ such that $\tilde{A}_f \beta=\tilde{b}_f \land z=\tilde{G}_f\beta+\tilde{c}_f$, while it follows from the latter that $\mu(z)=\mu(\tilde{G}_f\beta+\tilde{c}_f) \triangleq  {\lambda}(\beta) \in \mathcal{CZ}_{\mu} \Rightarrow , \exists \gamma \in \mathbb{B}^{n_{\tau}}_{\infty}$, such that $\tilde{A}_{\mu}\gamma=\tilde{b}_{\mu} \land {\lambda}(\beta)=\tilde{c}_{\mu}+\tilde{G}_{\mu}\gamma$.
Putting these two results into a set representation form, we obtain:
{\small
\begin{align} \label{eq:int_cz2}
z \hspace{-.1cm}\in \hspace{-.1cm} \{\tilde{G}_f\beta\hspace{-.1cm}+\hspace{-.1cm}\tilde{c}_f|\lambda(\beta)\hspace{-.1cm}=\hspace{-.1cm}\tilde{c}_{\mu}\hspace{-.1cm}+\hspace{-.1cm}\tilde{G}_{\mu}\gamma,\hspace{-.1cm} \tilde{A}_f \beta\hspace{-.1cm}=\hspace{-.1cm}\tilde{b}_f,\tilde{A}_{\mu}\gamma\hspace{-.1cm}=\hspace{-.1cm}\tilde{b}_{\mu},
\beta \hspace{-.1cm}\in\hspace{-.1cm} \mathbb{B}^{n_{c}}_{\infty},\gamma \hspace{-.1cm}\in\hspace{-.1cm} \mathbb{B}^{n_{\tau}}_{\infty} \}
\end{align}
}
On the other hand, using Corollary \ref{cor:supproting_vectors}, $\lambda(\cdot)$ can be decomposed into a JSS and a linear mapping as follows:
\begin{align}\label{eq:dec_cz_update}
\forall \Omega \in \mathbf{\Omega}_{\lambda}, \forall \beta \in \mathbb{B}^{n_{c}}_{\infty}:\lambda(\beta)={\nu}^{\Omega}(\beta)+\Omega\beta.
\end{align}
Further, note that by \cite[Theorem 2]{james1978generalised}, $\tilde{A}_f\beta=\tilde{b}_f \Rightarrow \beta \in \mathbb{IB} \triangleq [\tilde{A}_f^\dagger\tilde{b}_f-\kappa \mathbf{r}_{n_c},\tilde{A}_f^\dagger\tilde{b}_f-\kappa \mathbf{r}_{n_c}]$, where $\mathbf{r}_{n_c} \triangleq \textstyle{\mathrm{rowsupp}}(I_{n_c}-\tilde{A}_f^\dagger \tilde{A}_f)$ and $\kappa$ is a very large positive real number, which given that $\beta \in \mathbb{B}^{n_c}_{\infty}$, results is $\beta \in \mathbb{IB} \cap \mathbb{B}^{n_c}_{\infty}=[\underline{\mathbf{l}}_{n_{c}},\overline{\mathbf{l}}_{n_{c}}]$. This and Corollary \ref{cor:supproting_vectors} imply that the JSS component ${\nu}^{\Omega}(\cdot)$ is tightly bounded as follows: $\forall i \in \mathbb{N}_{n_{\mu}},\forall \Omega \in \mathbf{\Omega}_{\lambda}, \underline{\nu}^{\Omega}_{i} \leq {\nu}^{\Omega}_{i}(\beta) \leq  \underline{\nu}^{\Omega}_{i}, \forall \beta \in \mathbb{B}^{n_{c}}_{\infty}$, with $\underline{\nu}^{\Omega}_{i},\overline{\nu}^{\Omega}_{i}$ given in \eqref{eq:over_p_cz} and \eqref{eq:under_p_cz}, respectively. This, \eqref{eq:dec_cz_update} and Proposition \ref{prop:interval_cen_rep} imply $\forall \Omega \in \mathbf{\Omega}_{\lambda}, \forall \beta \in \mathbb{B}^{n_{c}}_{\infty}, \exists \rho \in \mathbb{B}^{n_{\mu}}_{\infty}$ such that $\lambda(\beta)=\frac{1}{2}(\underline{\nu}^{\Omega}+\overline{\nu}^{\Omega})+\frac{1}{2}\textstyle{\mathrm{diag}}(\overline{\nu}^{\Omega}-\underline{\nu}^{\Omega})\rho+\Omega\alpha$, which along with \eqref{eq:int_cz2} returns $z \in  \bigcap\limits_{\Omega \in \mathbf{\Omega}_{\lambda}} \{\tilde{G}_f\beta+\tilde{c}_f|\frac{1}{2}(\underline{\nu}^{\Omega}+\overline{\nu}^{\Omega})+\frac{1}{2}\textstyle{\mathrm{diag}}(\overline{\nu}^{\Omega}-\underline{\nu}^{\Omega})\rho+\Omega\beta=\tilde{c}_{\mu}+\tilde{G}_{\mu}\gamma,\tilde{A}_f\beta=\tilde{b}_f,\tilde{A}_{\mu}\gamma=\tilde{b}_{\mu},\beta \in \mathbb{B}^{n_{c}}_{\infty},\gamma \in \mathbb{B}^{n_{\tau}}_{\infty}, \rho \in \mathbb{B}^{n_{\mu}}_{\infty} \}  $, where the set on the right hand side is equivalent to the one on the right hand side of \eqref{eq:ZB_update}.
\end{proof}\vspace{-.2cm}
We conclude this subsection by combining the results in Lemmas \ref{lem:ZB_update} and \ref{lem:CZ_update} via the following Theorem \ref{thm:ZB_CZ_update}.
\begin{thm}[Decomposition-Based ZB/CZ Update]\label{thm:ZB_CZ_update}
Suppose all the assumptions in Lemmas \ref{lem:ZB_update} and \ref{lem:CZ_update} hold. Then $$\mathcal{Z}_f \cap_{\mu} \mathcal{Z}_{\mu} \subseteq \mathcal{ZB}_u \cap \mathcal{CZ}_u,$$ where $\mathcal{ZB}_u,\mathcal{CZ}_u$ are given in Lemmas \ref{lem:ZB_update} and \ref{lem:CZ_update}, respectively.
\end{thm}
\begin{proof}
By Lemmas \ref{lem:ZB_update} and \ref{lem:CZ_update}: $\mathcal{Z}_f \cap_{\mu} \mathcal{Z}_{\mu} \subseteq \mathcal{ZB}_u $ and $\mathcal{Z}_f \cap_{\mu} \mathcal{Z}_{\mu} \subseteq \mathcal{CZ}_u$, and hence $\mathcal{Z}_f \cap_{\mu} \mathcal{Z}_{\mu} \subseteq \mathcal{ZB}_u \cap \mathcal{CZ}_u$.
\end{proof}
\subsection{Modifications to The Approach in \cite{rego2020guaranteed}}
The purpose of this subsection is twofold. i) We make a potential refinement/improvement to the propagation approach in \cite[Theorem 2]{rego2020guaranteed} (recapped in Proposition \ref{prop:Rego_propagation}) through the following Proposition \ref{prop:Rego_propagation_refinement}, by applying our previously developed remainder-form decomposition functions to compute potentially tighter enclosing intervals to Jacobian matrix of $f(\cdot)$. ii) We propose an update method via Lemma \ref{lem:Rego_update}, that is based on the ``CZ-inclusion" introduced in \cite[Theorem 1]{rego2020guaranteed} (recapped in Proposition \ref{prop:Rego_CZ_bounding}). The proposed update method is applicable to general nonlinear observation functions (similar to the proposed methods in Lemmas \ref{lem:ZB_update} and \ref{lem:CZ_update}), as opposed to the update (i.e, linear intersection) approach in \cite{rego2020guaranteed} that is only applicable when the observation function is linear.
\begin{prop}[Refinement to The Propagation Approach in \cite {rego2020guaranteed}]\label{prop:Rego_propagation_refinement}
Suppose all the assumptions in Proposition \ref{prop:Remainder_DF} (i.e, \cite[Theorem 2]{rego2020guaranteed}) hold. Then the set inclusion in \eqref{eq:Rego_prediction} holds also with replacing $\mathbb{J}$ with $\tilde{\mathbb{J}}$ (or the best (tightest) of them), where $\tilde{\mathbb{J}}$ is an enclosing interval to $g(x)\triangleq \nabla^\top_x f(X,W)$ that can be computed by applying Proposition \ref{prop:Remainder_DF} to the function $g(\cdot)$.
\end{prop}
\begin{proof}
Directly follows from Proposition \ref{prop:Remainder_DF}.
\end{proof}
\begin{lem}[Update Based on ``CZ-Inclusion" in \cite {rego2020guaranteed}]\label{lem:Rego_update}
Suppose all the assumptions in Lemma \ref{lem:CZ_update} hold. Let $x_0 \in \mathcal{CZ}_f$ and $\mathbb{J}^{\mu},\mathbb{J}^{\mu}_{\Delta} \in \mathbb{R}^{n_{\mu} \times n_x}$ be interval matrices satisfying $J^\mu(\mathcal{CZ}_f) \subseteq \mathbb{J}^{\mu}$ and $\forall i \in \mathbb{N}_{n_{\mu}},\forall j \in \mathbb{N}_{n_{x}},[\mathbb{J}^{\mu}_{\Delta}]_{ij} \triangleq \frac{1}{2}\begin{bmatrix} - \textstyle{\mathrm{diam}}(\mathbb{J}^{\mu})_{ij} & \textstyle{\mathrm{diam}}(\mathbb{J}^{\mu})_{ij} \end{bmatrix}$, where $J^{\mu}$ denotes the Jacobian of $\mu(\cdot)$. Let $\overline{\mathcal{Z}}_{f}=\{\overline{G}^{f},\overline{c}^{f}\}_Z$ be a zonotope satisfying $\mathcal{CZ}_f \ominus x_0 \subseteq \overline{\mathcal{Z}}_f$, with $\overline{c}^f \in \mathbb{R}^{\overline{n}}$. Let $\mathbf{m}^{\mu} \in \mathbb{R}^{n_{\mu}}$ be an interval vector such that $\mathbf{m}^{\mu} \supset \mathbb{J}_{\Delta}^{\mu}\overline{c}^f$ and $\textstyle{\mathrm{mid}}(\mathbf{m}^{\mu})=\mathbf{0}_{n_{\mu}}$. Let $P^{\mu} \in \mathbb{R}^{n_{\mu} \times n_{\mu}}$ be a diagonal matrix defined as follows: $\forall i=1,\dots,n_{\mu}$:
\begin{align}\label{eq:Rego_P_update}
P^{\mu}_{ii}\hspace{-.1cm}=\hspace{-.1cm}\frac{1}{2}\textstyle{\mathrm{diam}}(\mathbf{m}^{\mu})_i\hspace{-.1cm}+\hspace{-.1cm}\frac{1}{2}\sum_{j=1}^{\overline{n}} \sum_{k=1}^{n_x} \textstyle{\mathrm{diam}}(\mathbb{J}^{\mu}_{\Delta})_{ik}|\overline{G}^f_{kj}|.
\end{align}
Then, the following set inclusion holds:
\begin{align} \label{eq:Rego_update}
\mathcal{CZ}_f \cap_{\mu} \mathcal{CZ}_{\mu} \subseteq \mathcal{CZ}^R_u \triangleq \{{G}_u,{c}_u,{A}_{u},{b}_{u}\}_{CZ},
\end{align}
where $ {b}_{u}\hspace{-.1cm} \triangleq\hspace{-.1cm} [(\tilde{c}_{\mu}\hspace{-.1cm}-\mu(x_0)+\textstyle{\mathrm{mid}}(\mathbb{J}^{\mu})(x_0-\tilde{c}_f)-c_R)^\top \ \tilde{b}^\top_{f} \ \tilde{b}^\top_{\mu} \ {b}^\top_{R} ]^\top$
\begin{align}
\nonumber {A}_{u} &\triangleq \begin{bmatrix} \textstyle{\mathrm{mid}}(\mathbb{J}^{\mu})\tilde{G}_f & -\tilde{G}_{\mu} & G_R \\ \tilde{A}_f & 0 & 0 \\ 0 & \tilde{A}_{\mu} & 0 \\ 0 & 0 & A_R \end{bmatrix}, \ {G}_u \triangleq [\tilde{G}_f \ {0} \ 0], \\
  G_R &\triangleq [0 \ P^{\mu}], \ c_R \triangleq 0, \ A_R \triangleq [\tilde{A}_f \ 0], \ b_R \triangleq \tilde{b}_f.\label{eq:Rego_param}

\end{align}
\end{lem}
\begin{proof}
Suppose $z \in \mathcal{CZ}_f \cap_{\mu} \mathcal{CZ}_{\mu}$. Then by definition of the operator $\cap_{\mu}$ (cf. \eqref{eq:update}), $z \in \mathcal{CZ}_f$ and $\mu(z) \in \mathcal{CZ}_{\mu}$. Note that by Proposition \ref{prop:interval_cen_rep} and the mean value theorem, $z \in \mathcal{CZ}_f \Rightarrow \mu(z) \in \mu(\mathcal{CZ}_f) \subseteq \mu(x_0)\oplus \mathbb{J}^{\mu}(\mathcal{CZ}_f \ominus x_0)=\mu(x_0)\oplus (\textstyle{\mathrm{mid}}(\mathbb{J}^{\mu})+\mathbb{J}^{\mu}_{\Delta})(\mathcal{CZ}_f \ominus x_0)=$
\begin{align}\label{eq:MVT}
\mu(x_0)-\textstyle{\mathrm{mid}}(\mathbb{J}^{\mu})\oplus \textstyle{\mathrm{mid}}(\mathbb{J}^{\mu})\mathcal{CZ}_f\oplus \mathbb{J}^{\mu}_{\Delta}(\mathcal{CZ}_f \ominus x_0).
\end{align}
On the other hand, by Proposition \ref{prop:Rego_CZ_bounding} :
\begin{align}\label{Rego_inclusion}
\mathbb{J}^{\mu}_{\Delta}(\mathcal{CZ}_f \ominus x_0) \subseteq \mathcal{CZ}_R \triangleq \{G_R,c_R,A_R,b_R\}_{CZ},
\end{align}
with $G_R,c_R,A_R,b_R$ given in \eqref{eq:Rego_P_update} and \eqref{eq:Rego_param} (note that $\textstyle{\mathrm{mid}}(\mathbb{J}^{\mu}_{\Delta})=0$ by its definition) and where $\mathcal{CZ}_R$ has $n_R$ generators. Then, the facts that $z \in \mathcal{CZ}_f \triangleq \{\tilde{G}_f\beta+\tilde{c}_f |\tilde{A}_f \beta=\tilde{b}_f,\beta\in \mathbb{B}^{n_{c}}_{\infty}\}$, $\mu(z) \in \mathcal{CZ}_{\mu} \triangleq \{\tilde{G}_{\mu}\gamma+\tilde{c}_{\mu} |\tilde{A}_{\mu} \gamma=\tilde{b}_{\mu},\gamma \in \mathbb{B}^{n_{\tau}}_{\infty}\}$, \eqref{eq:MVT} and \eqref{Rego_inclusion} imply that
$z \in  \{\tilde{G}_f\beta+\tilde{c}_f|\tilde{c}_{\mu}+\tilde{G}_{\mu}\gamma=\mu(x_0)+\textstyle{\mathrm{mid}}(\mathbb{J}^{\mu})(\tilde{c}_f-x_0)+\textstyle{\mathrm{mid}}(\mathbb{J}^{\mu})\beta+C_R+G_R\xi_R, \tilde{A}_f \beta=\tilde{b}_f,\tilde{A}_{\mu}\gamma=\tilde{b}_{\mu},A_R \xi_R=b_R,
\beta\in \mathbb{B}^{n_{c}}_{\infty},\gamma \in \mathbb{B}^{n_{\tau}}_{\infty},\xi_R \in \mathbb{B}^{n_{R}}_{\infty}\}$, where the set on the right hand side is equivalent to the CZ on the right hand side of \eqref{eq:Rego_update}.
\end{proof}

 \section{Simulations} \label{sec:examples}\vspace{-0.05cm}
In this section we compare the performance of five approaches to guaranteed state estimation: i) {RRSR}, i.e., the mean value extension-based propagation introduced in \cite{rego2020guaranteed} (recapped in Proposition \ref{prop:Rego_propagation}) in addition to the update approach in \cite{rego2020guaranteed} for the case when the observation function is linear (e.g., Example I below) and its extension in Lemma \ref{lem:Rego_update} to nonlinear measurements (e.g., Example II below), ii) {D-RRSR}, i.e, a modification to RRSR where the bounds for Jacobian matrices are computed using the reminder-form decomposition functions (cf. Proposition \ref{prop:Rego_propagation_refinement}), iii) {D-ZB}, i.e., decomposition-based propagation and update with ZBs (cf. Lemmas \ref{lem:ZB_propagation} and \ref{lem:ZB_update}), iv) {D-CZ}, i.e., decomposition-based propagation and update with CZs (cf. Lemmas \ref{lem:CZ_propagation} and \ref{lem:CZ_update}) and v) {COMB}, i.e., a combination of i)--v) via intersection.
 \subsection{Example I}
Consider the following discrete-time system \cite[Section]{rego2020guaranteed}
\begin{align*
\begin{array}{rll}
x_{1,k}&=3x_{1,k-1}-\frac{x_{1,k-1}^2}{7}-\frac{4x_{1,k-1}x_{2,k-1}}{4+x_{1,k-1}}+w_{1,k-1},\\
x_{2,k}&=-2x_{2,k-1}+\frac{3x_{1,k-1}x_{2,k-1}}{4+x_{1,k-1}}+w_{2,k-1},\\
\begin{bmatrix}
y_{1,k}\\
y_{2,k}
\end{bmatrix}&=
\begin{bmatrix}
1 & 0\\
-1 & 1\end{bmatrix}
\begin{bmatrix}
x_{1,k}\\
x_{2,k}\end{bmatrix}+
\begin{bmatrix}
v_{1,k}\\
v_{2,k}
\end{bmatrix},
\end{array}
\end{align*}
with $\left\lVert w_k\right\rVert_\infty \leq 0.1$, $\left\lVert v_k\right\rVert_\infty \leq 0.4$,
and an initial zonotopic enclose for the initial state:
    $ \mathcal{X}_0=\{\begin{bmatrix}
     0.1& 0.2& -0.1\\
     0.1&0.1&0
     \end{bmatrix},
     \begin{bmatrix}
     0.5\\0.5
     \end{bmatrix}\}$.

\begin{figure}[h]
\includegraphics[scale=0.152,trim=27mm 0mm 5mm 5mm,clip]{FigureRegoLatest3.eps}
\caption{Results from the first five time steps of set-valued state estimation, using five different approaches, noticeable from the label on the figure. Black dots are obtained from uniform sampling of $X_0$ propagated through the vector field in Example I\label{Ex_Rego}}
\end{figure}
As can be seen from Figure \ref{Ex_Rego}, {D-ZB} provides less conservative enclosures compared to the other individual approaches, and further, the {COMB} approach results in significant improvement, taking the advantage of intersection. Moreover, a comparison between corresponding average times and the enclosure set volumes to the five approaches shown in Table \ref{Table_Rego}. As it can be observed, D-ZB gives the fastest responses, while combination of all, as expected, took the longest, while RRSR and D-RRSR took approximately the same time on average. In terms of volume, D-ZB and D-RRSR generate the least conservative enclosures compared to the other individual approaches, while a further improvement is obtained using the the intersection of all the individual approaches, i.e., COMB.
\begin{table}[H]
\captionsetup{font=small}
\caption{Average total times (second) and average total volumes ($10^{-4}$) at each time step of the estimators is calculated. Each average is taken over $50$ separate simulations.}

\begin{centering}
\tabcolsep=0.11cm
\begin{tabular}{|l|l l l l l l|}

\hline
\multicolumn{2}{|c}{Methods:\quad  \quad \quad \quad } & $k=0$ & $k=1$ & $k=2$ & $k=3$ &$k=4$\tabularnewline
\hline
\multirow{2}{*}{RRSR} & Time: & $0.087$ & $0.250$ & $0.193$ & $0.196$ & $0.204$\\
 & Vol.: & $0.5196$ & $2.3522$ & $3.6224$ & $3.0243$ & $1.8968$\tabularnewline
\hline
\multirow{2}{*}{D-RRSR} & Time: & $0.022$ & $0.225$ & $0.181$ & $0.198$ & $0.201$\tabularnewline
 & Vol.: & $0.5196$ & $2.1062$ & $3.2979$ & $1.3811$ & $0.6320$\tabularnewline
\hline
\multirow{2}{*}{D-ZB} & Time: & $0.192$ & $0.095$ & $0.091$ & $0.091$ & $0.123$\tabularnewline
 & Vol.: & $0.5196$ & $0.7974$ & $0.9918$ & $0.8316$ & $0.6170$\tabularnewline
\hline
\multirow{2}{*}{D-CZ} & Time: & $0.065$ & $2.825$ & $2.920$ & $2.118$ & $3.318$\tabularnewline
 & Vol.: & $0.5196$ & $1.8857$ & $2.1534$ & $1.7924$ & $1.3764$\tabularnewline
\hline
\multirow{2}{*}{COMB} & Time: & $0.063$ & $6.193$ & $6.882$ & $6.278$ & $6.951$\tabularnewline
 & Vol.: & $0.5196$ & $1.4226$ & $1.8399$ & $1.0038$ & $0.6387$\tabularnewline
\hline
\end{tabular}
\label{Table_Rego}
\par\end{centering}
\end{table}
  \subsection{Example II (The Unicycle System)}
 Now consider the following discretized unicycle-like mobile robot system \cite{chen2018nonlinear} in the form of \eqref{eq:system}:
  \begin{align}\label{eq:unicyclemodel}
  \begin{array}{rlll}
      s_{x,k+1}&=s_{x,k}+T_0\phi_{w}\cos(\theta_k)+w_{1,k},\\
       s_{y,k+1}&=s_{y,k}+T_0\phi_{w}\sin (\theta_{k})+w_{2,k},\\
      \theta_{k+1}&=\theta_k+T_0\phi_{\theta}+w_{3,k},\\
      y_k&=[d_{1,k} \ \phi_{1,k} \ d_{2,k} \ \phi_{2,k}]^\top +v_k,
      \end{array}
  \end{align}
where $x_k \triangleq [s_{x,k} \ s_{y,k} \ \theta_k]^\top$, $w_k=[w_{x,k} \ w_{y,k} \ w_{\theta,k}]^\top, \phi_{\omega,k}=0.3, \phi_{\theta,k}=0.15,  w_{x,k}=0.2(0.5\rho_{x_{1,k}}-0.3), w_{y,k}=0.2(0.3\rho_{x_{2,k}}-0.2)$ and $w_{\theta,k}=0.2(0.6\rho_{x_{3,k}}-0.4)$, with $\rho_{x_{l,k}} \in [0,1]$ $(l=1,2,3)$ and initial state $x_0=[0.1 \ 0.2 \ 1]^\top$. Moreover, $\forall i \in \{1,2\}$, $d_{i,k}=\sqrt{(s_{x_i}-s_{x,k})^2+(s_{y_i}-s_{y,k})^2}$ and $\phi_{i,k}=\theta_k-\arctan(\frac{s_{y_i}-s_{y,k}}{s_{x_i}-s_{x,k}})$, with $s_{x_i} , s_{y_i}$ being two known values. Furthermore, $v_{1,k}=0.02 \rho_{{y_1},k}-0.01$, $v_{2,k}=0.03 \rho_{{y_2},k}-0.01$, $v_{3,k}=0.03 \rho_{{y_3},k}-0.02$, $v_{4,k}=0.05 \rho_{{y_4},k}-0.03$ and $\rho_{{y_k},k} \in [0,1]$ $(k=1,2,3,4)$. Applying all the methods, one can observe from Figure \ref{Ex_Unicycle}
that the resultant set estimates are very comparable for all the different five approaches. In terms of computation time, Table \ref{Table_Unicycle} shows that D-CZ takes minimum computation time followed by RRSR, D-RRSR, COMB and D-ZB. In terms of set volumes, the COMB approach takes minimum volume followed by D-ZB, D-CZ, RRSR and D-RRSR. Note that the computation time for D-ZB's is exceptionally large, presumably because the conversion of a polytope to a zonotope bundle in CORA usually results in a higher number of zonotopes than the needed minimal  number of zonotopes in the zonotope bundle.
The reduction of this number of zonotopes in the bundle could be an interesting topic to explore in the future, which could significantly decrease the computation time of our D-ZB approach.
  \begin{figure}[h]

\includegraphics[scale=0.152,trim=20mm 0mm 5mm 5mm,clip]{FigureUnicycleLatest2.eps}\vspace{-.15cm}
\caption{Results from the first five time steps of set-valued state estimation, using five different approaches, noticeable from the label on the figure. Black dots are obtained from uniform sampling of $X_0$ propagated via \eqref{eq:unicyclemodel}}\label{Ex_Unicycle}
\end{figure}

\begin{table}
\captionsetup{font=small}
\caption{Average total times (second) and average total volumes ($10^{-4}$) at each time step of the estimators is calculated. Each average is taken over $20$ separate simulations.}

\centering{}%
\tabcolsep=0.11cm
\begin{tabular}{|l| l l l l l l|}
\hline
\multicolumn{2}{|c}{Methods:  \quad \quad \quad } & $k=0$ & $k=1$ &$k=2$& $k=3$ & $k=4$\tabularnewline
\hline
\multirow{2}{*}{RRSR} & Time: & $0.772$ & $4.256$ & $4.188$ & $2.995$ & $3.675$\tabularnewline
 & Vol.: & $1.7309$ & $1.0293$ & $5.528$ & $1.377$ & $1.413$\tabularnewline
\hline
\multirow{2}{*}{D-RRSR} & Time: & $1.669$ & $42.90$ & $45.57$ & $28.64$ & $50.53$\tabularnewline
 & Vol.: & $1.279$ & $9.717$ & $5.654$ & $1.978$ & $1.888$\tabularnewline
\hline
\multirow{2}{*}{D-ZB} & Time: & $1.397$ & $34.20$ & $163.08$ & $147.75$ & $131.75$\tabularnewline
 & Vol.: & $0.042$ & $0.030$ & $0.013$ & $0.043$ & $0.042$\tabularnewline
\hline
\multirow{2}{*}{D-CZ} & Time: & $0.602$ & $2.182$ & $2.016$ & $2.228$ & $2.591$\tabularnewline
 & Vol.: & $0.073$ & $0.072$ & $0.030$ & $1.261$ & $0.091$\tabularnewline
\hline
\multirow{2}{*}{COMB} & Time: & $0.236$ & $34.90$ & $65.50$ & $62.37$ & $57.73$\tabularnewline
 & Vol.: & $0.041$ & $0.028$ & $0.012$ & $0.041$ & $0.040$\tabularnewline
\hline
\end{tabular}
\label{Table_Unicycle}
\end{table}
\vspace{-.4cm}
\section{Conclusion} \label{sec:conclusion}
New set-membership methods were presented in this paper for   state estimation in   bounded-error discrete-time nonlinear systems, subject to   nonlinear observations/constraints. By transforming ZB/CZ uncertainty sets from the state space  to  the space of ZB/CZ's generators, our recently developed tight remainder-form mixed-monotone decomposition functions were applied to compute enclosures that were guaranteed to enclose the set of the state trajectories of the system. Further, remainder-form decomposition functions were leveraged to bound the nonlinear observation function to derive the updated set, i.e., to return enclosures to intersection of the propagated set and the set of states that are consistent with the measurements. Finally, the mean value extension-based approach in \cite{rego2020guaranteed} was also generalized to compute the updated set for nonlinear observations.

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redpajama_set_name: RedPajamaArXiv