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Heparin Advisor - Vanderbilt University Medical Center PPT
Presentation Summary : Nurse Managed Heparin Protocol View History Feature Reviewed by LM 4/15/14 Nurse Managed Heparin Protocol The Heparin Advisor is a decision support tool that allows ...
Source : http://www.mc.vanderbilt.edu/documents/sss2/files/HeparinAdvisor7_09.ppt
Presentation Summary : HEPARIN INDUCED THROMBOCYTOPENIA (HIT) Roy R. Danks, DO, FACOS ARGATROBAN IN HIT ACCP RECOMMENDATIONS Bolus: None Continuous infusion: Normal organ function: 2 mcg/kg ...
Source : http://www.facos.org/imis15/Public/Files/Meetings/IDR/2014/Danks%203.ppt
Dalteparin (Fragmin®) - University of Pennsylvania Health System PPT
Presentation Summary : Dalteparin (Fragmin®) Anticoagulation Inpatient Management (AIM) Team 2009 Objectives Formulary update Understand dalteparin Mechanism of action Indications ...
Source : http://www.uphs.upenn.edu/ppmc_emergency/PPMC%20Bookmarks/Dalteparin%20(Fragmin)/DalteparinFinal111909.ppt
IV Dosage Calculations Tutorial.pptx - Faculty Site Listing PPT
Presentation Summary : mL) = Flow Rate. Total Infusion Time (in hours) (in . mL /h) ... mL. How many units of heparin is the patient receiving each hour? *This is a reverse calculation, ...
Source : http://faculty.mccneb.edu/awwinter/IV%20Dosage%20Calculations%20Tutorial.pptx
Heparin-Induced Thrombocytopenia - Centers for Disease ... PPT
Presentation Summary : Heparin-Induced Thrombocytopenia Lawrence Rice, MD Chief, Division of Hematology Clinical Chief, Hem/Onc Service The Methodist Hospital Weill Cornell Medical College
Source : http://www.cdc.gov/nchs/ppt/icd9/att4_Rice_Sep07.ppt
Presentation Summary : Changes to Heparin Infusion Documentation: Name Change to "Heparin (IV Med Drip)" In addition to Units/kg/hr and Unit/hr. options, drop down is now a multi-select ...
Source : http://www.mc.vanderbilt.edu/documents/HED/files/Hep_Drip_changes_10_16_12.pptx
Understanding Heparin-Induced Thrombocytopenia (HIT ... PPT
Presentation Summary : Historical and Clinical Perspectives Heparin-Induced Thrombocytopenia Heparin is a widely used anticoagulant drug for the treatment and prevention of thromboembolic ...
Source : http://www.coaginfo.com/files/ss5/understanding_Heparin_induced_thrombocytopenia.ppt
Acute Coronary Syndromes - Tulane University PPT
Presentation Summary : Definitions Acute coronary syndrome is defined as myocardial ... may be superior to continuous infusion of Heparin LMWH Trials FRISC TIMI IIB ESSENCE INTERACT EVET ...
Source : http://tulane.edu/som/tuhvi/upload/studentpresentation08-04-03.ppt
UNFRACTIONATED HEPARIN - Yola PPT
Presentation Summary : Title: UNFRACTIONATED HEPARIN Author: kiko ontalan Last modified by * Created Date: 10/11/2006 12:23:56 AM Document presentation format: On-screen Show
Source : http://im-mmc.synthasite.com/resources/UNFRACTIONATED%20HEPARIN%20in%20Hemodialysis%20-%20Ramon%20C%20Mora,MD.ppt
Presentation Summary : Maintain patency of CRRT circuit. Minimize patient complications of anticoagulation therapies. ... Titrate heparin infusion to maintain ACT. range of 180-220 seconds.
Source : http://www.pcrrt.com/talks/talks2008/Eding%20Nursing%20Workshop.ppt
Presentation Summary : TPA infusion started at 1mg/hr. Heparin infusion through the side arm of the vascular sheath at 500 units/hr. Patient transferred to ICU for ... (May-Thurner ...
Source : http://www.freewebs.com/livoniaradiology/May%20thurner.ppt
Anticoagulation Bridging Decision Support (Ver 3) ... PPT
Presentation Summary : If patient is undergoing a surgery or procedure with low risk for bleeding, consider bridging with heparin infusion, therapeutic enoxaparin, ...
Source : http://excellence.acforum.org/sites/default/files/Anticoagulation%20Bridging%20Decision%20Support%20%28Ver%20%203%29%202012_1.ppt
C-06_Dailey.ppt - Advocate Health Care PPT
Presentation Summary : Continuous heparin infusions require monitoring of aPTT values and appropriate adjustments in the infusion rate. If the heparin is infusing through the central ...
Source : http://www.advocatehealth.com/documents/clinicalresearch/poster2011/C-06_Dailey.ppt
Misinterpreted Physician's Prescriptions PPT
Presentation Summary : ... affix an auxiliary label to the bottle and segregate the concentrated solution from the other strengths ... to heparin infusion bags Use the patient's ...
Source : http://www.pharmacist.com/sites/default/files/files/Lecture9HighAlertDrugs2.ppt
Common Drug Dosing and Monitoring Problems PPT
Presentation Summary : Common Drugs - Dosing and Monitoring PHENYTOIN Toxic? Me? With My Therapeutic Window? Introduction This module is designed to help with your learning around some ...
Source : http://www.smd.qmul.ac.uk/prescribeskills/CommonDrugDosing_MonitoringProblems.ppt
ICU Cardiac Rehab: Heading 'em Up and Moving 'em Out PPT
Presentation Summary : Do NOT discontinue or decrease the infusion ... (vasopressors and ... Bolus Dose (units) aPTT (seconds) *** Rate change is based on Heparin infusion of ...
Source : http://vmw-disccnet.dheweb.duke.edu/heart_research/InternationNurses/Teaching,%20Technology,%20Teamwork4_26_07.ppt
Understanding Heparin-Induced Thrombocytopenia (HIT ... PPT
Presentation Summary : Diovision of Cardiology Beth Israel Deaconess Medical Center Boston Heparin-Induced Thrombocytopenia "Heparin as a Cause of Thrombosis" and "The HIT Syndromes"
Source : http://clintrialresults.org/Slides/HIT.THEPVD.ORG.ppt
Intelligent Use of Anticoagulants - University of South Florida PPT
Presentation Summary : ... angiography Treatment Rigorous resuscitation IV heparin anticoagulation ... Study Design Enoxaparin sodium 1mg/kg q12h SC Adjusted-dose heparin infusion ...
Source : http://www.usfvascularsurgery.com/images/uploaded/univsofl/Intelligent%20Use%20of%20Anticoagulants.ppt
Argatroban as an Alternative to Heparin for Vascular and ... PPT
Presentation Summary : Argatroban as an Alternative to Heparin for Vascular and Cardiovascular Surgery in Patients with a ... The infusion was stopped just prior to removal of the cross ...
Source : http://aprilschmidtsportfolio.weebly.com/uploads/1/1/8/4/11847764/argatroban_as_an_alternative_to_heparin_for_vascular.pptx


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David Lerner Basic Cap Sleeve Dress
Curved seams enhance the body-con look of a little black dress from David Lerner. Hidden back zip and cap sleeves. Unlined.
- Fabric: Jersey.
- 86% nylon/14% spandex.
- Wash cold.
- Made in the USA.
- Length: 30in / 76cm, from shoulder
Sold Out
Up to 25% Off: GOBIG14
Mixed lace and diagonal trim bring a bold look to this form-fitting David Lerner dress. Soft V neckline. Sheer long sleeves. Neutral mesh lining. Fabric: Lace. Shell: 85% nylon/15% elastane. Trim: 80% rayon/20% polyester. Lining: 100% polyester. Wash cold. Made in the USA. Measurements Length: 33.5in / 85cm, from shoulder Measurements from size S
David Lerner Pieced Lace Dress $285 $142.50
Curved seams bring a sexy fit to this David Lerner dress. Cutouts detail the raglan seams, and soft leather panels trim the long sleeves. Exposed back zip. Unlined. Fabric: Stretch jersey. Shell: 88% nylon/12% spandex. Trim: 100% leather. Dry clean. Made in the USA. Measurements Length: 33in / 84cm, from shoulder Measurements from size S
Leather trims add downtown edge to this body-conforming David Lerner dress. Piping trims the curved seams. Exposed back zip. Mesh lining. Fabric: Mid-weight jersey. Shell: 92% nylon/8% spandex. Trim: 100% lambskin. Dry clean. Made in the USA. Measurements Length: 39in / 99cm, from shoulder Measurements from size S
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From £ 9.90
Turn your road trip into a culinary experience. Quite simply, this book is destined to reach camper van bible status. Ideal for families, singlies, foodies, festival-goers and outdoor lovers.
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Status: New - SAY GOODBYE TO ROUGHING IT. AND HELLO TO A NEW KIND OF FREEDOM. WITH OVER 80 FABULOUSLY TASTY RECIPES YOU CAN COOK ON JUST TWO RINGS AND A FEW ...
Postage & Packaging: £ 4.36 | Availability: Usually dispatched within 1-2 business days...


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Have you ever wanted to be on a TV Game Show? Which one would you want to be on?
TV Game Show 101
Winning on a TV Game Show
Now Playing: Come To The River
Artist: Rhett Walker Band
On-Air: Scott & Kelli
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text: STEP UP ON SECOND JOB DESCRIPTION SENIOR PROJECT MANAGER SUMMARY: Step Up on Second is dedicated to long-term support of people in recovery and their families, offering quality housing, educational, social and work experience. The Senior Project Manager is responsible for real estate activities related to the acquisition and development of affordable and market rate housing. Our ideal candidate is a bright, self starter who is charmingly persuasive, communicating with integrity and enthusiasm. The Senior Project Manager will be responsible for all activities related to the development of 2-4 affordable multifamily housing developments with the support of an additional project manager. DUTIES: Primary duties include, but are not limited to, the following: Prepare financial pro forma analyses for affordable housing developments. Obtains project financing, including preparing financing applications, creating deal structures, coordinating negotiation of legal agreements for financing and equity syndication, and maintaining knowledge of current financing programs. Prepare loan, grant and Low Income Housing Tax Credit applications. Attend job meetings, prepare construction draws and monitor construction progress. Taking charge of new development, from site acquisition through entitlement approval and financing commitments, to construction completion and successful lease up. Preparing regular project budgets, cash flow projections, and project schedules. Leading loan closings, including working with attorneys, lenders, investors, and escrow and title companies in the coordination and delivery of due diligence, legal and financial documents. Successfully completing applications in response to Requests for Proposals (RFPs). Provide leadership to project development team, including: architect, construction manager, contractor, attorney, environmental and other consultants. Assists Accounting and Support with project transition by preparing documentation with essential project information for ongoing operation. Other related duties as required. REPORTS TO: Housing Director HOURS: 40 hours per week CLASSIFICATION: Exempt SALARY: $75- 82K QUALIFICATIONS: Applicants should have at least three years experience in the development and/or financing of Affordable Housing. The ideal candidate will have managed at least two housing development from site acquisition through permanent loan closing. Applicant should have experience and/or demonstrated abilities in the following areas: Masters preferred, but not required. Excellent written and verbal communication skills. Self-motivated and ability to work under pressure in an entrepreneurial environment. STEP UP IS AN EQUAL OPPORTUNITY EMPLOYER STEP UP ON SECOND JOB DESCRIPTION Preparation of loan, grant and Low Income Housing Tax Credit applications. Ability to work effectively with peers and public and private lenders. Preparation and analysis of pro forma financials for low income multifamily housing developments. Proven real estate finance and structuring expertise. Able to create project pro formas and articulate the case for a specific project. Should have both an intuitive and a detailed sense of the dynamics, risks, and structured financings of each transaction. This position is characterized by a high degree of initiative, responsibility, accountability and ability to work congenially with a wide variety of individuals, community based organizations and governmental entities. Ability to operate a motor vehicle. The ability to work collaboratively within a small team environment to maintain harmony in an established team. This person must be able to handle the entire development process with minimal day to day supervision, although he/she will communicate quickly, openly, and fully on matters needing attention from the Director and others in the company. Must possess the confidence to hold his/her own internally. Equally important, has the confidence to ask questions and get direction when required. Possesses excellent people skills; has the ability to be productive and likable, and knows how to create win-win relationships with colleagues and external parties. Strong overall interpersonal skills. Team player with "can do" attitude and a containable ego. Able to work independently while also contributing in a team environment. Ability to embrace and adhere to the organization's mission, policies and procedures and operating standards. Temperament and cultural sensitivity required to work effectively in Santa Monica. A sense of humor. The position will be filled as soon as possible. Please e-mail resume to firstname.lastname@example.com or fax to (424) 214-5060. Unfortunately, we are unable to personally speak with everyone who contacts us. Should there be a potential match to our current needs, you will be contacted for an interview. STEP UP IS AN EQUAL OPPORTUNITY EMPLOYER


text:
Extended Stay America - Dallas - Lewisville
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Story Marketing Goods and services
... you should be able to get higher conversion into the insuarnce services that you're promoting as long as you write ceratin articles in a wide fashion putting more emphasis on the page content a simple optimization. The distribution of articles literally lets you successful sooner in comparison to the manual methods of article marketing. This should inspire you to do more jotting down work and help you adapt to the different guidelines of every ...
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... are simply not that many around. If you don't live close enough to a field to hold your party there, the next best option is to hold your own, however this will require an expansive enough land space to play. Fortunately, any forests or tremendous open up area shall do which are fairly simple quite enough to locate. When website hosting your paintball business, the cheapest way is to 'BYOPBG' (Take Your Paintball Firearm). Delivering just about every single dose of ...
Don't fly American Air
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FBI releases phone recordings in Sheehan case
Federal prosecutors are presenting phone call recordings as evidence in the case against Daniel Sheehan who has been charged with extortion and using a destructive device in the commission of extortion. Sheehan allegedly threatened to blow up three Home Depots on Long Island in 2012 unless he was paid money. Audio has been edited for profanity. Handout: FBI (June 12, 2013)
Algieri's rise from Paramount to Pacquiao Smithtown town resident criticizes town animal shelter 2 dead after wrong-way crash Car crashes through front door of house Algieri: 'Manny didn't give me a lot of holes' Driver seriously injured in Lindenhurst accident Chaminade 48, St. John the Baptist 29 Bellone: Suffolk County to help snow-battered Buffalo 3 arrested in Mastic standoff Man sentenced in fatal beating of 4-year-old Cops: Pedestrian hit, killed by truck on LIE Eastbound LIE reopened in Melville after traffic accident


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Mountain West goes to division setup, title game
With 12 football schools next season after Boise State and San Diego State backed out of commitments to join the Big East, the Mountain West will have two six-team divisions, the Mountain and the West. The winners of the divisions will play in a conference title game on Dec. 7, 2013. The team with the highest BCS ranking will be the host.
The Mountain will include Air Force, Boise State, Colorado State, New Mexico, Utah State and Wyoming. The West will consist of Fresno State, Hawaii, Nevada, San Diego State, San Jose State and UNLV. The divisions were determined according to geography and rivalries.
Each team will play five divisional games and three opponents from the other division.
The conference also announced that there will not be a division setup for basketball.
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Drivers discover dead body along highway in Oklahoma City
Authorities make arrests after setting up DUI checkpoints across metro
Late night earthquakes rattle Oklahoma
National Geographic calls Oklahoma City a must-see place to visit in 2015
Teen son cited in La. crash that killed 5 family members traveling to Disney World


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October 12, 2012 The Charlotte Bobcats take on the New Orleans Hornets in an NBA exhibition game.
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Raymond Sessions runs the offense for the Charlotte Bobcats against the New Orleans Hornets at an NBA exhibition game at the North Charleston Coliseum on Thursday October 11, 2012. (Wade Spees/postandcourier.com)
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October 1, 2008
| Article
We know that selling retail can increase our bottom line, but for so many of us, selling retail — particularly non-nail items — is a foreboding hurdle. The question to ask is not should I sell retail, but how do I sell retail.
September 1, 2008
| Product
New Metallic Trapezoid Bags from Action Bag grab attention with their luxurious sheen and unique shape.
July 1, 2008
| Article
Salon consultant Bryan Durocher of Durocher Enterprises offers a way to make opening a dialog with clients just a little bit easier.
June 1, 2008
| Article
In this occasional series, we follow along as Adrienne, a mother of two who just moved with her husband to North Carolina, works to rebuild her nail business from scratch. She's working with Heather, a certified salon success coach, to help her succeed. If you missed the first installment, see NAILS February issue or visit The Coaching Chronicles blog (updated twice weekly) at http://blogs.nailsmag.com/coach.
March 1, 2008
| Article
We asked NAILS readers what new retail products they've recently introduced. Far and away the most popular move was to add a new polish line from the top professional brands. Cuticle care products and lotions were also popular choices. Foot care follows in fourth place, then skin care, with Entity's Nanovive Skin Revival System a particular favorite.
February 1, 2008
| Article
Five marketing tips to push your salon to achieve extraordinary results.
November 1, 2007
| Article
Says our employee representative: Meeting more of a client's needs through retail will make her more likely to stay loyal and to send referrals.
Everyone talks about the great promise of retailing, but to hear about the challenges from someone in the trenches is a reality check.
September 1, 2007
| Article
What if, instead of calling things like cuticle oils and nail strengtheners "retail products" and suggesting that you "retail them," we call them "take-home products" that you "recommend"?
November 1, 2006
| Article
A career in the nail industry has its fair share of rewards, but with it can come issues that challenge even the most die-hard professional. From finding it difficult to retail products to being allergic to chemicals in the salon, it seems almost every nail tech has some concerns regarding her occupation.
July 1, 2006
| Article
As a nail tech, you know a lot your clients don't know — from trends in the industry to what products will help with skin and nail problems. But don't keep it to yourself! By using your knowledge to educate clients, you could significantly impact your bottom line. Learn how to use your power of persuasion to meet clients' needs — and to boost your retail sales.
October 1, 2005
| Article
Need a good reason to get more business education? Salon owner Millie Haynam highlights the importance of not just relying on your technical prowess. From marketing and money management to people skills and PR, business skills are vital to making your salon a success.
June 1, 2005
| Article
Coach advises nail tech that to serve her primary customers, she'll have to change her hours to accommodate them.
If nail techs won't retail nail products, manufacturers will have to sell them in alternative outlets.
January 1, 2004
| Article
In a commission-based salon or spa, technicians sometimes assume the owner is banking the other half of their commission. Opening the salon's books will open employees' eyes to the vital need to retail for the salon's survival.
24 Holiday Retail Items24 photos
Show [Us] Your...Creative Space8 photos
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text:
Find an Internship or Volunteer in Bellevue, Washington ( WA )
If you're looking for internships or volunteer work in Bellevue, Washington or even to get a summer job, you've come to the right place! With over 30,000 around the county, chances are that GrooveJob.com has the right job for you and wants to be your only source for finding internships and volunteer positions in Bellevue, Washington.
We've got plenty of intern and volunteer work in Bellevue, Washington. Check out the list of employers below who have positions in Bellevue. Remember to check back with Groovejob.com on a regular basis. We add new internships and volunteer jobs listings to our database every week, so you'll find new opportunities every time you check back with us.
|Browse by Employer (Click On Logos)
|Click for Intern or Volunteer at SEAWORLD ADVENTURE PARK
|Click for Intern or Volunteer at INROADS
Jobs by Title
Sales and Management Intern - Fashion Industry Opportunity
…often said that experience is life's best teacher, and as a Buckle Intern , you'll get exactly that-a challenging, hands-on retail...
Bellevue, WA
Fall PR Intern - Microsoft Account
…6, and run through Friday, December 19. WITH A WAGGENER EDSTROM COMMUNICATIONS INTERNSHIP , YOU WILL HAVE THE OPPORTUNITY TO: *...
Bellevue, WA
Branch Intern (Finance and Economics)
Branch Intern (Finance and Economics) - Bellevue, WA Job ID: 20142407-9403 Description: Scottrade is seeking an energetic, career-minded...
Bellevue, WA
HR Intern
…wide variety of people HRnovations is an equal opportunity employer. Keyword: intern , internship , human resources, Bellevue,...
Bellevue, WA
Panda Express - Restaurant Manager Internship - NorthWest North
…leaders that have the interest and ability to grow with us. The Panda Internship Program for Restaurant Operations is designed to give...
Issaquah, WA
(Contract) Tax Preparer/ Intern
…below. As a Tax Quality Analyst you will be responsible for applying knowledge of tax law and tax filing processes to review and test...
Renton, WA
Summer Intern
…classes. Completes a project assigned through management at the beginning of the internship . With the guidance of an assigned mentor,...
Issaquah, WA
College Sports - Baseball Internship
…Athletics is currently accepting resumes for 2015. Students interested in an internship should email their resume to General Manager...
Seattle, WA
Sports Media Company - Intern - Winter
…college or university. Able to provide a letter from that college or university stating that the student will be granted credit for the ...
Seattle, WA
Intern - TV Production Winter 2015 - Entertainment - TV
…We no longer have openings for fall 2014 Production interns.) This is a paid intern position, 15 hours per week, January through...
Seattle, WA
Intern - Winter 2015 - Entertainment - Radio
Job Description Internship Description: The iHeartMedia Internship Program has been designed to provide qualified college and...
Seattle, WA
Corporate Technology Internship 2015 - Engineering - Fashion Industry Opportunity
…new experiences that delight both our online and in-store customers. As aNordstrom creative Intern , you'll play a dual role -...
Seattle, WA
Software Development (CMS) Internship , Corporate - Spring 2015 - Entertainment and Media Industry Opportunity
…Disney's DTSS Engineering Services Group, in Seattle, develops applications and services for all domestic and international business...
Seattle, WA
Software Engineering Internship , Corporate - Spring 2015 - Entertainment and Media Industry Opportunity
…and solutions to meet the tactical needs of our customer's business. This is an intern position for a future software engineer who...
Seattle, WA
Food Scientist - Summer Intern
JOB DESCRIPTION JOB TITLE: Food Scientist – Summer Intern STATUS: Hourly, Non - Exempt LOCATION: Seattle, WA area (Tukwila) SCHEDULE:...
Seattle, WA
Intern - Web Production Winter 2015 - Entertainment - TV
…https://home.eease.adp.com/recruit/?id=11341171 Application deadline is December 31, 2014. This is a paid intern position, 10-15 hours...
Seattle, WA
Spring 2015 Intern /Co-op, Project Manager
Position Type: Intern FLSA Status: Non-Exempt Position Summary: Complete a three, six, nine or twelve month session between or in...
Seattle, WA
Intern , Index Reconstitution - Finance and Investment Industry Opportunity
…manner as well as organizing and maintaining electronic and hard copy files. This internship will begin in February and end on May 31,...
Seattle, WA
Weather Intern - Entertainment - Digital - TV
…* Create weather graphics * Write forecast for website * Other duties may be assigned. Sinclair Broadcast Group, Inc. is proud to be an...
Seattle, WA
Campaign Operations Internship - Spring 2015 - Entertainment - Digital
…* Manages pre and post-sale client related tasks * Creates reports and analyzes advertising campaign performance data (from DoubleClick...
Seattle, WA
|<< First < Previous 1 2 Next > Last >>


text:
News in Brief: A Washington Roundup
NAEP Test to Stress Content of Science
The science portion of the National Assessment of Educational Progress will place a greater emphasis on scientific content and facts, as opposed to the conducting of investigations, as the result of a new blueprint for the exam approved this month.
The revamped framework, or outline for science content on NAEP, will increase the proportion of questions focused on factual scientific knowledge and application to 60 percent, up from 45 percent now. The National Assessment Governing Board, the federal panel that sets NAEP policy, approved the framework Nov. 18.
Once a new science test is designed to match the framework, students in the 4th, 8th, and 12th grades will begin taking it in 2009. The current framework has been in place since 1996. For the first time, the new science test also will assess students on technological design, generally defined as the connections between science and technology, engineering, and real-world applications. Some advocates had opposed including that concept on the exam, fearing it would leave less room for the testing of core scientific principles.
The new test also covers the topic of evolution, but makes no mention of supposed alternatives to it.
Vol. 25, Issue 13, Page 28
Sponsored Whitepapers
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Nov. 27 (Bloomberg) -- On today's "Chart Attack," Tkng Capital Partners CIO Neil Grossman and Bloomberg's Adam Johnson look at U.S. debt. They speak on Bloomberg Television's "Street Smart." (Source: Bloomberg)
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October 23, 2013 – The Houston Business Journal announced on Wednesday, October 16th that Revention, Inc. has been chosen for the publication's annual Best Places to Work award, a highly sought-after accolade within Houston's business community.
2013 proved to be HBJ's most competitive year ever for Best Places to Work. Companies were selected not only on the basis of being top performers in their industries, but were also evaluated for the amount of value they place on their talented employees. To obtain this information, employees of the nominated companies were surveyed by a third party firm, Quantum Workplace, which issued each company a score based on employees' anonymous answers. All employees of the nominated companies were given the opportunity to take the confidential survey online.
Employees evaluated their company in areas such as goals and leadership, communication and support, advancement opportunity and skill development, recognition and compensation, tolerance and flexibility and embracing innovation and ideas. Being a Best Place to Work also involves giving back to the community, so this characteristic was assessed, along with the importance placed on fun and a deep sense of teamwork in the workplace.
Each firm was required to have scored at least an 85 on its employee survey to be named a finalist, and even then HBJ still had more companies on their list than ever before. The list of finalists was then narrowed down to 100 by HBJ based on industry performance, as well emphasis on team work and philanthropy.
Revention equates company success with employee satisfaction and is proud to have been named a Best Place to Work by HBJ. Revention's company culture promotes fun, laughter, and sharing. The result is a cohesive team unit that continuously produces exceptional work. Office perks include an arcade room, on-site professional massage therapist, great benefits, travel opportunities, frequent company gatherings, weekly catered events and a lively work atmosphere.
Companies: Revention
Papa John's new Red Kettle Cookie raises funds for Salvation Army campaign
Let's Pizza vending machine ready for U.S. debut
Canadian 'Express Entry' program will remove caps on cooks, managers
How to get millennials to stay committed to your brand
Pizza Hut franchisee: High confidence in brand relaunch despite minimal testing
Pizzeria Planning: Designing and Maintaining an Efficient Pizza Kitchen
Experimenting with Pizza Flavors
Case Study: Dairy Queen® Franchisee
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The Wrecking Yard Salutes Dimebag Darrell on His Birthday! [VIDEO]
Today marks "Dimebag" Darrell Abbott's 45th birthday.
Today marks "Dimebag" Darrell Abbott's 45th birthday.
I can't wait for the return of Suicide Silence when they hit the Hub on Aug. 18th.
When people ask me who I think to the greatest guitar players are of all time, Dimebag Darrell is the first that comes to my mind. I could go on and on about Dime's style of play. So it's nice when a legitimate guitar player takes time to show some Dime secrets
Check out this song from Sean Yseult of White Zombie. This was recorded on tour in 1992. And the most important part is that it isn't a White Zombie track. It was recorded by Sean, Phil Anselmo and Dimebag Darrell!
As always The Cockfight tugs at your metal heart strings and we throw 2 bands in the drag pit and let you vote! This week it was the battle of The Metal Ballad's Pantera's "Suicide Note pt. 1" VS Down's "Stone The Crow."
Have you ever been to a strip bar and seen a girl on that brass pole gyrating in a way that appeared she was having a seizure? That's because she didn't pick the right song to swing those hips to.
Whether you need the best stripper songs for your occupation or just plain listening pleasure, I hope you have fun with today's list
So, Gibson recently conducted a poll of the Top 50 Metal Songs of All Time. We saw huge flaws in the rankings in the poll and revised the top 15. You tell us who's list is better after the break.
Down will be hitting the road, unfortunately only one Texas date has been scheduled, March 6th in Houston. The band features ex-Pantera members Phil Anselmo and Rex Brown. Learn a little metal history and check out the swampy greatness of "Stone The Crow".


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Teresita Fernández Named 2013 Aspen Award for Art Recipient
The Aspen Art Museum has announced the selection of renowned contemporary artist Teresita Fernández as the recipient of the museum's 2013 Aspen Award for Art. The award will be presented on Friday, August 2, 2013, during the museum's 9th annual ArtCrush summer benefit gala. The Aspen Award for Art is given each year to an artist who has made a significant contribution to the field of contemporary art. Past artist-honorees are: Tom Sachs (2012), Roni Horn (2011), Marilyn Minter (2010), Fred Tomaselli (2009), Ed Ruscha (2008), Jim Hodges (2007), Tony Feher (2006), and Richard Tuttle (2005).
On Friday, August 2, 2013, the Aspen Art Museum will host its annual summer benefit gala, with over 350 of the world's top art collectors, wine connoisseurs, artists, gallery owners, curators, celebrities, and global business leaders, who will come together in Aspen for three days of events to celebrate the Aspen Art Museum. The hugely successful 2012 AAM ArtCrush summer benefit raised $1.8 million to fund the Museum's educational programming and exhibitions. The ten-lot 2012 ArtCrush Live Auction of contemporary art raised a total of $770,000 and the seventy-lot Silent Auction art auction garnered an additional $774,000. ArtCrush is chaired by veteran Chair and AAM National Council member Amy Phelan.
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In a control system for controlling a plant system or process (22), a model based problem is formulated and an explicit control solution is constructed off-line such that a parametric profile (24) techniques (26). As a result the system can operate in real-time with significantly reduced computational burden.
1. A method of deriving an explicit control solution for operation of a system, the system having a control variable and a parameter that can be measured or estimated, comprising the steps of constructing, off-line, a system operation model based on said control variable and measurable parameter, and constructing, off-line, an optimised explicit control solution for said control variable and parameter using parametric optimisation.
2. A method as claimed in
3. A method as claimed in
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5. A method as claimed in claims 3 in which the MPC is formulated as a Quadratic Program.
6. A method as claimed in
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10. A method as claimed in
11. A system as claimed in
12. A method as claimed in
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15. A method as claimed in
16. A method of controlling operation of a system comprising the steps of measuring or estimating, on-line, at a first time interval a parameter of the system, obtaining a control variable based on the parameter from a pre-defined parametric optimised explicit control solution and inputting the control variable to the system.
17. A method as claimed in claims 16 in which the steps of measuring or estimating the on-line parameter and deriving the control variable are repeated at subsequent time intervals.
18. A method as claimed in
19. A method as claimed in
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21. A method as claimed in
22. A method as claimed in
23. A method for controlling operation of a system according to an explicit control solution having a disturbance estimate comprising the steps of adjusting the control solution in on-line control operation to correct the disturbance estimate.
24. A control system for controlling operation of a system having a control variable and a state that can be measured or estimated, the control system including a system state measurement or estimation element, a data storage element storing a predetermined parametric optimised explicit control solution, a processor arranged to obtain a control variable from said data storage element control solution based on a system state value and a controller arranged to input the obtained control variable to the system.
25. A computer program for implementing a method as claimed in
26. A computer readable medium storing a computer program as claimed in claims 25.
27. A data storage device storing a control solution derived by a method as claimed in
28. A method or system or medium or device substantially as herein described and as illustrated with reference to the figures.
29. A method as claimed in
30. A method as claimed in
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The invention relates to improved process control, in particular improved model based process control.
Modern automated processes such as manufacturing processes are implemented under strict and generally highly complex control regimes both to maximise efficiency and avoid error or failure. In their simplest form these regimes are typically based on a feedback system whereby a parameter of the process is monitored and fed to a controller as a control parameter for comparison with a target value, the state of the process being altered as a function of the control parameter to track the target value.
One known and widely adopted control regime is model predictive control (MPC) which is described in �Model Predictive Control; past, present and future.� Morari, M. and J. H. Lee (1999) Comput. Chem. Engng. 23, 667-682 which is incorporated herein by reference. The basic principle is illustrated in
Referring to
This operation is carried out on-line and in real time, and is illustrated in
Mathematically, the process or plant model can be represented, in state-space, by the equations:
The equations represent mathematically how at a time t in a process the future state of the process at a discrete, subsequent time interval t+1 is predicted as a function of the measured state x at time t and the control variables applied at time t�this calculation is then iterated forward, still at time t, for the predicted state at time t+2 based on the calculation for time t+1 and so forth. At time t, in addition, the process output is a direct function of the instantaneous process state. The process output is calculated at equation (2) to ensure it does not exceed the ouput constraints.
It has been shown that the iteration up to a time t+Ny can be posed in the form of the following problem to be optimised (for example, see Morari and Lee paper identified above):
Equation (3) is the iterated expansion of equation (1). The first term on the right hand comprises a quadratic function of the predicted state variables at time t+Ny, using the vector transpose x′ and constant matrix P. The second term is a summation of all the intermediate predicted state variables from time t to time t+Ny−1 using the vector transpose x′ and constant matrix Q and, over the same range, of corresponding control variables u again in quadratic form with constant matrix R. In both cases each successive state value xt+k+1 is derived from the previous state value x1+k using equation 1. Each value of ut+k is derived as a function of the predicted state value for that time and gain matrix K. The constraints for y and u are controlled up to a time Nc. The function J is thus constructed.
The left hand side of equation (3) requires that an optimal solution is selected from the solution space by minimising the function J of the state x(t) and the vector U of control/input variables for all time intervals. The function J is minimised in control variable space, that is, the optimal value of U is found satisfying equation (3) including the various conditions in the �subject to� field such that J is minimum. The resulting values of u(t) and x(t) are then representable as shown in
A significant problem arises in practical implementation of MPC as the processing power required for real time calculation is extremely high. Certainly if a large number of variables are required, complex control functions needed for complex processes or extremely short time intervals are desired, the burden can become cost, energy or even space inefficient. In addition, as the solution is created on-line and tailored only to instantaneous conditions, it provides no quantitative data on whether the model itself is appropriate.
Various attempts have been made to improve MPC implementation. For example in U.S. Pat. No. 6,056,781 to Wassick et al, in addition to a plant or process model, a disturbance model represents deviations from the plant model. In this version, if anything, the real time processing burden is enhanced. U.S. Pat. No. 5,347,446 to Iino et al describes a system taking into account constraints on the input or manipulated variables as well as constraints on controlled variables and change rates of the controlled variables. Again, the processing burden remains a significant concern.
U.S. Pat. No. 6,064,809 to Braatz et. al. addresses the problem of processing burden by approximating off-line the constraint set, that is; the limits or constraints on the input or manipulated variables as a mathematical construct off-line, in the form of an iteratively derived ellipsoid, effectively providing a simplified replacement for equation 3 above. However this simplified construct is then no more than an approximation which is scaled and then solved on-line in conventional MPC fashion. Accordingly Braatz et al. approached the problem by improving or simplifying the modeling process, but still requires significant on-line computational power.
According to the invention there is provided a method of deriving an explicit control solution for operation of a system, the system having a control variable and a measurable parameter, comprising the steps of constructing, off-line, a system operation model based on said control variable and measurable parameter, and constructing, off-line, an optimised explicit control solution for said control variable and measurable parameter using parametric optimisation. Because the solution is explicit and constructed off-line, on-line control of the system can be speeded up significantly using the solution. In addition the explicit solution can be reviewed off-line to provide insights into the control system/control model itself.
The explicit control solution is preferably stored on a data storage device such as a chip and the system operation model is preferably a model predictive control (MPC) model. The explicit control solution is preferably constructed in the continuous time domain allowing more accurate control. The MPC is preferably formulated as a Quadratic Program.
The control solution preferably comprises a co-ordinate space of at least one control variable value region comprising a control variable substitution value or a control variable value as a function of a measurable parameter value, the coordinate being the measurable parameter value. As a result a simple and manipulable parametric profile is provided. The measurable parameter may comprise a measurable state or a system disturbance. The method may further comprise the step of analysing the control solution and constructing a revised system model based on said analysis.
According to the invention there is further provided a method of controlling operation of a system comprising the steps of measuring, on-line, at a first time interval a parameter of the system, obtaining a control variable based on the measured parameter from a pre-defined parametric optimised explicit control solution and inputting the control variable to the system.
The steps of measuring the on-line state and deriving the control variable may be repeated at subsequent time intervals. The control solution preferably comprises a co-ordinate space of at least one control variable value region comprising a control variable substitution value or a control variable as a function of a measured state value, the co-ordinate being the measured state value.
According to the invention there is further provided a control system for controlling operation of a system having a control variable and a measurable state, the control system including a system state measurement element, a data storage element storing a predetermined parametric optimised explicit control solution profile, a processor arranged to obtain a control variable from said data storage element control solution based on a measured system state value and a controller arranged to input the obtained control variable to the system.
The method is preferably implemented in a computer program which can be stored in any appropriate computer readable medium. A data storage device is provided according to the invention storing a control solution derived by a method as herein described.
Embodiments of the invention will now be described by way of example with reference to the drawings, of which:
The present invention provides an alternative, improved real time solution to the problems of model-based control for example using model predictive control. In particular, by applying the technique generally known as �parametric optimisation� to the process model the invention allows control variables to be derived off-line for all operational parameters that will be encountered. The explicit analytical control solution thus constructed is embedded in a chip or other data store as a function of the operational parameters such that the process model does not have to be optimised at each time interval, hence reducing the processing burden significantly. Instead the control variables are immediately available as a function of the operational parameters and/or in look-up table form. Yet further the solution is viewable off-line and offers information on the problem to be solved.
Parametric Optimisation
The techniques of parametric or optimisation or parametric programming will be well known to the skilled person and the general principles are set out in Gal, T. (1995) �Postoptimal Analyses, Parametric Programming and Related Topics�, de Gruyter, New York and Dua, V. and E. N. Pistikopoulos (1999) �Algorithms for the Solution of Multiparametric Mixed-Integer Nonlinear Optimization Problems�, Ind. Eng. Chem. Res. 38, 3976-3987 which are incorporated herein by reference. In essence parametric optimisation is a mathematical technique by which a problem is first formulated as an optimisation problem, an objective function and optimisation variables are obtained as a function of parameters in the optimisation problem, and the region in the parameter space where the functions are valid are also obtained. In the most general form the parametric optimisation problem can be formulated as:
where y is vector of 0-1 binary variables, x is a vector of continuous variables, f is a scalar, continuously differentiable function of x, g is a vector of continuously differentiable functions of x, b and d are constant vectors, E and F are constant matrices, θ is a vector of parameters, θmin and θmax are the vectors of lower and upper bounds on θ, and X and Θ are compact and convex polyhedral sets of dimensions n and s respectively.
This most general formulation requires minimisation of z, a function of optimization or control variables, x and y, to obtain them as a function of parameters which can be for example measurable states or system disturbances represented by vector θ. The generalised formulation covers cases where the control variables are continuous value vectors x, or binary value vectors y. The parameter values θ are bounded by minimum and maximum constraints. In the example discussed below the parameter values are measurable states.
Solutions are known for various special cases. For example where there is no control variable y (ie no binary variables), f(x) is quadratic and convex in x, where f(x) is defined to be convex if for any x1,x2ε X and 0≦α≦1, f[(1−α)x1+αx2]≦(1−α)f(x1)+α(x2), and g(x) is linear in x, the problem is reduced to one of a multiparametric Quadratic Program (mp-QP). The solution to this case is discussed in Dua et al. �An algorithm for the solution of multi-parametric quadratic programs� (1999) tech. Rep. D00.7, Centre for Process Systems engineering, Imperial College London, which is incorporated herein by reference. An example of the form of the solution to an mp-QP problem set out in Table 1 is now discussed.
As can be seen the function f(x) is a quadratic function of variables x1, x2. Each of x1 and x2 are further constrained dependent on state parameters θ1, θ2, which in turn are bounded. The function is minimised in control variable x1, x2 space, ie the values of x1, x2 are found for which f(x) is optimised.
The solution generated according to the Dua et. al. reference above takes the form of the plot in a coordinate space θ shown in
Feasible region 1
Feasible region 2
Feasible region 3
Feasible region 4
It can be seen that for regions 1, x1 and x2 have fixed values, whereas for regions 2 to 4 x1 and x2 are expressed as functions of the coordinates (θ1, θ2)�it will be noted that x1 and x2 will thus vary within a given region dependent on their position in that region.
Another special case is where f(x) and g(x) are linear in x. The problem is then presented as a multiparametric Mixed Integer Linear Program (mp-MILP) for which algorithms have been developed as discussed in Acevedo J. and Pistikopoulos E. N. �A Multiparametric Programming Approach for Linear Process Engineering Problems under Uncertainty� (1997) Ind. Eng. Chem. Res. 36, 717-728 which is incorporated herein by reference.
An mp-MILP problem is set out in Table 2
Feasible region 1
Feasible region 2
Feasible region 3
Binary vector y=[0 1]
Feasible region 4
Binary vector y=[1 1]
Feasible region 5
It will further be recognised that the presentation of the solution in this simple, explicit form offers insights into the control process as a whole. Most significantly, it indicates the extent to which the state measurements are required for process control. For example in more complex processes having a large number of measured states it is entirely possible that the explicit solution will turn out to be independent of one or more of the measured states as a result of which the process control hardware can be designed or modified to eliminate the corresponding state measurement device and the software to eliminate the processing of related measured state data so rendering the process control scheme for efficient and economic operation at all levels.
The skilled person will be aware of other special cases for which the parametric optimization solution is known and further discussion of those is thus not required here.
Application of Parametric Optimisation Techniques
Using these techniques a model-based problem can be optimised according to the present invention using parametric optimisation providing control actions as a function of measurements from a plant to be controlled, by defining the control variables as the optimisation variables and the state of the plant as parameters. Then, during real time operation, as soon as the measurements from the plant are obtained, this function is evaluated to obtain the control actions using the on-line optimisation routine designated generally 20 in
According to this approach a model-based problem to be optimised is formulated for the plant/system/process 22 in a known manner. An explicit control solution, parametric profile 24, is constructed off-line using parametric optimisation techniques 26. In the simplified scheme shown, the profile is represented as a two-dimensional plot of state variables as input and control variables as output, but it will be appreciated that the profile can be multi-dimensional and include function evaluation dependent on the state variable defined region as discussed above with reference to
Returning to the MPC problem of equation (3); the function J(U, x(t)) can be formulated as a Quadratic Programming (QP) problem by expanding the formula set out in equation (3) for xt+k+1 at time t as the summation:
This term is substituted into equation (3) to yield by simple algebraic manipulation the QP Problem:
The QP problem represented by equation (6) can be formulated as the mp-QP again using simple algebraic manipulation:
It will be appreciated that the above derivation is performed based on predictions for discrete future intervals t, t+1 . . . t+k i.e. in the discrete time domain, but similar techniques can be applied to obtain the mp-QP problem in the continuous time domain, i.e. continuing to take measurements at discrete intervals t, but predicting from those measurements future continuous profiles of the state variable at a time t+k where the intermediate intervals t+1 . . . t+k−1 are infinitesimal.
In this case equation (6) is rewritten as:
initial condition (I.C.): x(to)=xo(θ,v)
represent the 2-norm of the state and control variables from the origin, whereas the terms ∥P2x(tf)∥ and
represent the 1- or ∞-norms of the state and control deviations from the set point. The time horizon tf of this problem is considered finite. The point constraints are imposed at Nf discrete instants within the horizon. The optimisation variables, v, are called �control variables�. The initial conditions set out in the �subject to� field comprise the control problem to be solved.
Note that in (8), we have integrals and differential equations in the continuous model, whereas in (3) we had summation in the objective function and discretized model. Problem (8) is described in (i) Anderson, B. D. and J. B. Moore (1989) �Optimal Control: Linear Quadratic Methods�, Prentice Hall, Englewood Cliffs, N.J., (ii) Bryson, A. E. and Y. Ho (1975) �Applied Optimal Control�, Taylor and Francis, New York, and (iii) Kwakernak, H. and R. Sivan (1972) �Linear Optimal Control Systems�, Wiley Interscience, New York all of which are incorporated herein by reference.
The result of a parametric analysis on problem (8) is to express the optimal objective (φ) and the optimisation variables (v) as a function of the parameters (θ) for the complete space of the parameters variations. This can be achieved by transforming problem (8) to a form that can be solved with the current techniques of steady state parametric programming. Such a form in this case is the mp-QP formulation.
By adapting control vector parameterisation for solving the optimal control problem (e.g. �Off-line Computation of Optimum Controls for Plate Distillation Column� Pollard and Sargent, 1970 Automatica 6, 59-76, incorporated herein by reference), the optimisation problem can be considered only in the reduced space of v variables. This of course requires the explicit mapping of the x variables that appear in the constraints and the objective as a function of v,θ. This expression can be obtained by solving the linear ODE system in equation (8). The solution of this system is exact and in the case of time-invariant parameters and optimisation variables, is given by the analytical expression:
eAt is the exponent of a matrix that is defined through eigenvalue or other decomposition analysis methods (�Matrix Computation�, Second Edition Colub and Van Loan 1990, The John Hopkins University Press).
The integral evaluation in equation (9) requires that the form of the profile of the uncertainty and control variables over time be specified. Control variables are usually represented as piecewise polynomial functions over time where the values of the coefficients in the polynomial are determined during the optimization (Vassiliadis, V. S., R. W. H. Sargent and C. C. Pantelides (1994) �Solution of a Class of Multistage Dynamic Optimization Problems. 1. Problems without Path Constraints.� Ind. Eng. Chem. Res. 33, 2111-2122 incorporated herein by reference). Note that if the order of the polynomial is assumed to be zero the control profiles correspond to a piecewise constant discretization. For the case of uncertainty, θ, it is described by using periodic harmonic oscillations, i.e. sinusoidal functions over time, of uncertain amplitude or mean value (Kookos, I. K and J. D. Perkins (2000) �An Algorithm for Simultaneous Process Design and Control�, Tech. Rep. A019, Centre for Process Systems Eng., Imperial College, London incorporated herein by reference). Note that more complex, forms of periodic functions can be reduced to a linear combination of sinusoidal functions by using Fourier analysis. By substituting (9) into (8) and hence eliminating the system of differential equations the problem is transformed as follows:
This methodology transforms the path constraints that have to be satisfied throughout the complete time-horizon to an equivalent series of point constraints that hold only at particular time points and therefore, are inherently finite.
Hence the time in the path constraints of (10) is substituted by an appropriate set of fixed time instants tk, k=1 . . . K. Consequently the constraints become a linear function of the control variables and parameters only. Thus the equivalent mathematical description of problem (10) is the following:
Problem (11) is now a multiparametric quadratic program (mp-QP) and its solution provides the control variables as a function of the state variables and uncertain parameters, similarly as in the discrete time case.
As a result, known parametric optimisation techniques for mp-QP problems can be applied to MPC problems in the discrete or continuous time domain to obtain the full solution set for z and hence u as an affine function of x of the form:
A limitation of the system described above is that it does not take into account model inaccuracies and constant disturbances except to the extent that the disturbances can be modeled as additional parameters. Accordingly inaccurate forecasting of the process behavior can lead to infeasabilities such as off-specification production, excessive waste disposal or even hazardous plant operation. Below there are discussed two general methodologies for compensating for disturbances.
In real processes there are usually disturbances due to sensor malfunctioning, degrading performance of the process or system due to catalyst fouling, noise, human error or mistake, cost or model inaccuracies, amongst other possible factors. These disturbances cannot always be measured and accordingly it is desirable to take such disturbances into account in deriving the optimal control law.
Rewriting equation (3) above, an appropriate formulation for deriving an explicit model-based control law for a process system is as follows:
By considering the current states x, as parameters, problem (13) is recast as a multiparametric quadratic program (mp-QP) of the form given in (7). The solution of that problem corresponds to an explicit control law for the system that expresses the optimum control action in terms of the current states. The mathematical form of this parametric controller, similar to that given in (12), is as follows:
The model-based parametric controller described in the previous paragraph does not account for the impact of persistent slowly varying unmeasured disturbances and uncertainties of the type discussed above that either enter the system as inputs or derive from modelling errors and time varying process characteristics. These disturbances result in an offset, i.e. a permanent deviation, of the output variables from their target values and additionally may cause constraint violations. As discussed below, according to the invention two possible solution techniques are introduced: Proportional Integral (PI) control, and origin re-adjustment. The suitable model readjustment according to disturbance estimates, enables the recursive on-line adaptation of the model based controller to the current conditions.
The reason for the off-set is that the model has not predicted the presence of the disturbance.
Following the fist approach according to the invention, to eliminate the off-set an integral state is incorporated in the model-based optimisation framework. This provides off-line derived parametric proportional integral controller (�Parametric PI�). As discussed in more detail below, in particular an integral state is added into the dynamic system, and is also added in the objective function with a particular waiting penalty. The updated problem is solved again to obtain the model-based parametric control law discussed above.
In conventional feedback control schemes, Seborg et al., (1989) �Process Dynamics & Control�, Wiley & Sons, integral action is incorporated for attenuating any permanent deviation of the output variables from their set-points due to the presence of disturbances. This gives rise to the so-called proportional integral controller (PI-controller). Here, the incorporation of the integral action for the same purpose is achieved by introducing an integral state that is equal to the anticipated deviations of the output from its reference point, usually the origin. This state is augmented as an additional quadratic penalty in the objective function (Kwakernaak and Sivan, 1972) �Linear Optimal Control Systems�, John Wiley & Sons Ltd. This technique has also been adopted by the Generic Model Control framework (Lee and Sullivan, 1988 �Generic Model Control (GMC)� Comput. Chem. Eng. 12(6), 573-580), but in that control scheme no state constraints or other model characteristics are considered and on-line optimization is required for deriving the control actions. Equation (13) after the incorporation of the integral state is modified as follows:
Equation (16) and the associated conditions effectively introduce the additional state xq as an additional term in the objective function and as an additional condition dependent on the preceding integral state value and the respective output.
By eliminating the equalities associated with the dynamic system and considering the current pure and integral states as parameters a multiparametric quadratic program similar to equation (7) is formulated. The solution of this mp-QP corresponds to the following piecewise affine control law:
The piecewise affine multivariable controller represented by equation (4) contains a proportional part acxt(t*) with respect to the states, an integral part dcxqt in terms of the outputs and a bias bc, hence it is a parametric proportional integral controller (Parametric PI). This control law is asymptotically stable, thus it guarantees no steady state offset from the target point on condition that (i) the dimension of the controls is larger or equal to the output dimension q≧m, which is the prevailing case in control problems (ii) the open-loop transfer matrix defined from the equation: H(s)=B1)(sI−A1)−IA2+B2 posses no zeros at the origin and (iii) the quadratic cost matrix Q1 that penalizes the integral error is positive-definite.
Similarly, derivative action can be accommodated in the parametric control law. An algebraic variable is introduced in the formulation that is equal to the difference between the current and the preceding output variable. This variable is penalized in the objective function as follows:
The solution of this problem leads to a parametric controller featuring explicit derivative action, i.e. a parametric proportional integral derivative (parametric PID) controller results. Unlike the integral action incorporation, here, the states of the system remain unaltered. Since the derivative function penalizes the rate of change of the output trajectories, it has the advantage of achieving faster settling times and hence improved disturbance rejection (Seborg, et al., 1989 ibid).
This system does not necessitate the existence of a disturbance or uncertainty estimator since this is taken into consideration implicitly via integral action. The method does not modify the prediction model according to the current disturbance realization. In order to prevent the resultant model based controller failing to locate accurately and promptly the constraint bounds leading to possible constraint violations when operating dynamically near the feasible region boundaries, the known technique of back-off is applied, where the output set-point is moved away from the constraints (e.g. Narraway and Perkins, 1993). Integral and derivative action can be accounted for separately giving rise to parametric PI and parametric PD controllers respectively, or they can be used simultaneously resulting in a parametric PID controller. The parametric PID controller is not obtained by purely including extra terms in pre-derived control expressions, but is based on systematically modifying the original model-based optimal control formulation and solving the resulting parametric optimization problem with implicit additional terms.
Accordingly, the solution space comprises one or more regions each with an explicit control law but taking into account disturbances by optimising for all PI/PID actions and obtaining the control action solution explicitly as a function of the states, the integral xq and the derivatives yd of the outputs. This allows systematic enhancement of the control law as opposed to existing solutions which are based on ad-hoc integral/derivative control tuning.
The second approach is a model readjustment approach that enables the recursive on-line adaptation of the model based controller to current conditions. According to this system, as described in more detail below, depending on an estimate of the disturbance a new optimum steady state point can be identified in terms of the states and controls. The explicit control law derived off-line by parametric optimisation as described in equations (3) and (4) is modified to incorporate the presence of the disturbance in an on-line operation, in effect shifting the solution space.
In known systems, the incorporation of a disturbance regulator in an MPC controller is performed by estimating the disturbance values and updating the model accordingly to ensure nonzero output target tracking (Li et al.,1989 �A state space formulation for model predictive control� AIChE J. 35(2), 241-248; Muske and Rawlings, 1993 �Model Predictive Control with Linear Models� AIChE J. 3a(2), 262-287; Loeblein and Perkins, 1999 �Structural design for on-line process optimisation �I. Dynamic Economics of MPC� AIChE J. 45(5) 1018-1029). A problem with these known solutions is that, in effect, control law is obtained as a result of which the solution can be predicted only by repetitive solution on-line of optimization problems. As a result useful information concerning the model and process is lost.
According to the invention, a mechanism is generated for updating the model on-line and modifying accordingly the derived control law. A distinction is made between two systems (i) one being the real process plant and (ii) the other comprising a model that represents an estimate of the process behavior, as shown in Table 3:
Once a disturbance enters into the system, at time k, there is an anticipated discrepancy z between the measured output ŷ and the predicted output y. This is computed one interval after the disturbance has propagated i.e. at k+1:
Based on the assumption that the disturbance that generated the discrepancy retains the same value in the following interval its estimation is evaluated via rearranging to provide:
Based on the disturbance value a new steady state point [xs vs] is computed. If the dimension of output variables y is equal to the dimension of input controls v and no input and state constraints are violated in the new steady state, then this is done as follows:
After the computation of the new appropriate steady state, the control law is shifted as follows:
The derivation of the modified control law (24) is based on the assumption that (i) no pure state or mixed input, output, state constraints are violated during its implementation and (ii) that no input bounds are exceeded, for example, (vt|0vmax) after shifting on-line the steady state. If either of these conditions occur, the optimum control policy is determined by incorporating the disturbances wt explicitly in the parametric quadratic optimization problem (13) giving rise to the formulation:
In problem (25) both the current states xt and the disturbances wt are treated as parameters that remain constant throughout the receding horizon. The control law that is obtained by solving this problem, has a piecewise affine form that includes a feedforward element dc w t:
The disturbances values wt are updated according to their estimation (equation 20) every time a new output measurement is available. Next, they are provided regularly along with the current state values to the explicit control law described in (26) and the new control sequence is computed. As a result there is no need for information about the disturbance to be obtained via measurements, allowing a more realistic disturbance estimator to be introduced.
As a result the control law is first obtained and an origin readjustment is introduced onto it in the form of a direct disturbance estimate as a result of which the disturbance does riot always need to be included as a parameter. As discussed above, the disturbance is obtained on-line and the control law updated by specifying the disturbance term more and more accurately as additional information is obtained concerning the disturbance, using algebraic manipulation as set out in equations (19) to (26). The resulting solution is more computationally attractive for off-line processing as a specific disturbance parameter does not need to be incorporated into the parametric programming. The on-line processing is, in practice, extremely straightforward and quick.
Referring to
Set point tracking is treated in a similar fashion by including a variational term yref the right band side of equation (21), i.e.: replace
It will be appreciated that the invention can be put into effect in any appropriate way. For example the algorithms involved may be implemented in any appropriate software, for example using the MATLAB software package sold by The Math Works Inc. A specific example of the techniques used in a real control problem is set out below.
An example of parametric optimisation as applied to MPC in relation to a conventional process is now described with reference to
The products are collected at the top of the reactor separator 44 from outlet line 50 and the spent catalyst is recycled back to the regenerator 42 via return line 52. At the bottom of the regenerator 42 air is fed at rate Fa controlled by a valve 53 via a line 54 to burn the coke and regenerate the spent catalyst. Exhaust gas 60 escapes the regenerator 42 to a flue.
For this problem the control variables are Fs and Fa and the state variables are Crc, the mass fraction of coke in the regenerator 42 and Trg, the temperature in the catalyst bed in the regenerator. The target values for the state variables are:
The optimisation problem therefore is to manipulate the control variables to attain these state values using the continuous time model described above for an mp-QP using equation (8) rewritten as:
Equation (27) hence represents a function to be minimised formulated in a manner that will be well known to the skilled person and based, here, on the work in the Hovd & Skogestad reference identified above.
Matrices A,B,D,H are derived from the model process and are given in the Hovd and Skogestad reference (see Annex A to the present description). The matrices in the cost function Q, R are considered as (Pantas, A. V. (1999) Robustness and Economic Considerations in the Design of Multivariable Optimal Controllers, PhD Thesis, Imperial College, University of London):
where to=0 min, t1=1.5 min, t3 =4.5 min and t 4=tf=6 min. For this control description the analytical solution of the dynamic system (27) provides the following state time profiles:
Once these state functions (29) are substituted in the objective and the constraints of problem (27) the only free optimisation variables become the control elements: u1,u2,u3,u4. The mathematical form of the resulting problem is:
The time varying terms S2, S3, S4, Sf, W2, W3, W4, Wf, are a obtained from the problem matrices, A, B, Q, R, P via matrix multiplications. By evaluating the matrix time integrals in equation (30) the objective is expressed explicitly as a quadratic time-invariant function of the current state x, and the control elements ui only. The same does not apply however, to the constraints that are an explicit function of the control elements, the current state and the time. The time dependence of the constraints renders problem (30) infinite dimensional. This difficulty is surpassed by applying the methodology for converting the infinite number of path time-varying constraints to an equivalent finite set of point constraints described in the text earlier. Hence, time t in the exponentials of equation (30) is substituted by an appropriate set of time fixed instants tk, k=1, . . . K. Consequently, the constraints become a linear function of the control elements and the current states only. Thus, the equivalent mathematical description of problem after appropriate algebraic manipulations (30) is the following:
Problem (31) is a multiparametric quadratic program that is solved using the well-established mp-QP algorithm (set out in the Dua et al. reference identified above) and the matrix and other values for the solution to the specific example are provided in the annex. The solution of this problem provides an explicit mapping of the control action as a function of the current state. This mapping represents the optimal control law for the process system. The control law is partitioned to 39 regions, as shown in
The parametric solution shown in
Initial State
In the first case the state parameters progress from a starting point 70 for (Crc,Trg) at (4.166�103,930.8) to the target point 72 at (5.207�10−3,965.4). It will be seen that the starting point 70 falls in region 15 which is described by the following constraints:
Reverting to
To demonstrate the disturbance rejection features of the present invention we consider an example concerned with determining the control policy for an evaporator process shown in
Next, the problem is rearranged to include integral action in the optimum control scheme. The problem is formulated as in equation (16) and hence, a parametric proportional integral controller is derived. The integral weight in the objective function is:
For testing the capabilities of this controller in presence of disturbances, a sequence of step disturbances is considered in F1 and C1 as shown in
As an example of the PI Parametric Controller, Control Law for region CR20 is given by
Control Law:
Region Boundaries:
These last equations are disclosed in more detail in Chen, H. and F. Allgower (1998). A Quasi-Infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability. Automatica 34(10), 1205-1217 and Scokaert, P. O. M. and J. B. Rawlings (1998). Constrained Linear Quadratic Regulation. IEEE Trans. Aut. Cont. 43(8) 1163-1169.
Applying parametric optimisation to the above problem reduces the on-line computational burden as discussed in detail above.
Then, the controls are parameterised using a set of time-variant optimisation variables. The most common form of control parameterisation is to assume a piecewise polynomial function for each control variable over a finite number of time elements. For convenience low order Lagrange polynomials are employed. Thus, over the element i in time the jth control variable is written as:
Equation (7a) holds for t ε└tt−1,t1┘; M is the order of the polynomial approximations and index i=1, . . . Nq runs over all the control elements; u1ji are the time invariant variables to be determined by the solution of the optimisation problem. Expression (7a) is substituted in (6a) yielding the linear dependence of the states on the set of time invariant control elements. However, the constraints of the optimisation, that in general hold over the complete time horizon, remain infinite dimensional, resulting in general in a semi-infinite dimensional parametric optimisation problem (Hettich and Kortanek, 1993).
I Initialise a finite number of interior point constraints at times t*ε Tk, where k denotes the number of iterations. The constraints are imposed only on these points. For this fixed set the parametric optimisation problem, (1a)-(5a) is transformed to a finite dimensional multiparametric quadratic program (mp-QP):
II Solve the parametric optimisation problem (mp-QP) for this particular set of interior constraints. The parametric solution provides a set of piecewise affine expressions for the manipulated inputs in terms of the states that hold over a set of critical state-space regions.
III Now, based on the parametric solution, test whether the same constraints are satisfied ∀t+ε[α,β] and ∀xε X.
IV If some of the constraints are violated at particular points tk* εTk* then augment these points into Step I, i.e. T1={Tk∪Tk*}, then set k=k+1 and continue until the constraints are satisfied ∀tε[α,β].
In Annex B it is proved that the decomposition method (I-IV) ensures convergence and optimality. The explicit control law for the system (1a)-(5a) is derived from the solution of the last mp-QP in Step II. The mathematical description of this control law is:
The solution of the multiparametric dynamic optimisation (mp-DO) problems (1a)-(5a) in the presence of only state constraints (C2=null matrix) can be approached alternatively, by applying sensitivity analysis (Fiacco and Ishizuka, 1990) to the stationary conditions of the embedded dynamic optimisation problem. The Karish-Kuhn-Tucker optimality conditions for the optimal control problem (1)-(5) for fixed parameters x* and a fixed set of interior point constraints C1�x(tk)≦0 are (Malanowski and Maurer, 2001):
The solution of the multiple-point boundary value problem (10a)-(13a) is uniquely determined from the evaluation of the set of (i) the adjoint variables at the inception point λ(t*)=λo, (ii) the lagrange multipliers vk and (iii) the state variables at the final point x(tf)=xf. Thus, the derivation of expressions for the optimal finite set ξ={λovk,xf} as a function of the state variables leads explicitly to the optimal control profile in terms of the state variables v(x*,t). For this purpose the transition matrix technique is employed here (Bryson and Ho, 1975), where first a joint ordinary differential system is constructed containing both the adjoint and the state equations:
where matrix Matk11. ij consists of the (1:n) x (1:n) elements of matrix Matkij. Matrices Matk12 ij, Matk21 ij, Matk22 ij are defined similarly. Following that notation, the sufficient optimality conditions with respect to the set ξ={λo,vkxf} are recast as shown in Table 1a. Matrix I in Table la denotes the identity matrix of dimensions n x n. Next, Theorem 3.1 enables the derivation of the control functions v(t,x*) in terms of the current state conditions x*.
The affine expressions for ξ(x*) hold over a particular critical region CR in the state-space where the set of active constraints remains unaltered. Hence, it can be proved via the solution of the dynamic system (15a), the jump conditions (12a) (Bryson and Ho, 1975) and equation (11a) that (i) the control law v(x*,t) is a continuous piecewise linear function in terms of x* and (ii) {circumflex over (Φ)}(x*) is a continuous, convex and piecewise quadratic function.
The identification of the critical regions (CR) in the state space where the affine control expressions hold is performed via the feasibility and optimality conditions. Feasibility is ensured by substituting ξ(x*) into the inactive inequality set encapsulated in (19a) and forcing it to be less than zero, whereas optimality is represented by {tilde over (v)}(x*)≧0. This set of state inequalities define the state boundaries of each one of the critical regions:
The proposed solution procedure does not require any appriori discretisation of the controls, therefore, the length of the measurement sampling interval is independent of the optimal control expressions.
A simple numerical example is presented to demonstrate the features of the newly developed parametric controller for continuous systems. Here, only the method based on calculus of variations will be demonstrated because of space limitation reasons. Consider a SISO plant control example corresponding to the following numerical values of the problem matrices (1a)-(5a):
The Fiacco and Ishizuka reference is Fiacco, A. V. and Y. Ishizuka (1990). Sensitivity and Stability analysis for nonlinear programming. Annals of Oper. Res. 27, 215-237 and the Bryson and Ho reference is Bryson, A E. and Y. Ho (1975). Appl. Opt. Cont. The Hettich & Kortanek reference is (1993) Semi-infinite Programming: Theory, Methods & Applications, Siam Review 35(3), 380-429. The Malanowski Maurer reference is (2001) Sensitivity Analysis for Optimal Control Problems subject to Higher Order State Constraints Ann. Op. Res. 101, 43-73.
As a result an exact solution of problem is obtained since time is not discretised and no hypothesis is made about the polynomial nature of the control variables. However it should be born in mind in the particular methodology applied, constraints on control variables are not considered as it is assumed that matrix C2 in equation (5) contains only zero elements and that constraints on state variables exist at certain discrete times, not continuously for complete time intervals. It will be appreciated, however, that using appropriate mathematical techniques these limitations can be avoided. An alternative way of compensating for disturbance is now discussed. As evident from the above treatment, if one assumes that there are no disturbances in the model and derives the explicit controller, and then tries to implement it on a real system which includes disturbances then there may be constraint violations and hence infeasible operation. In overview, the approach set out in the following discussion solves this problem by finding the value of the disturbance that could result in the worst-case violation of the constraints and then deriving the explicit controller for a problem including this disturbance. It can be shown that if the system is feasible for the worst-case then it will be feasible for any value of the disturbance. An advantage of this approach is that on-line adjustment of the control solution to correct the disturbance estimate is not required. As a result a global if somewhat less optimised control solution is obtained, but this is mitigated by a reduction in the on-line computational burden, bearing in mind that the disturbance is no longer a co-ordinate of the solution space. In the following discussion disturbance attenuation is achieved for the case both of slowly varying disturbances and for bounded uncertainties by including in the parametric controller design, an explicit feasibility constraint that ensures feasible operation for all possible uncertain parameter realisations. It will be appreciated that the approach can be extended to other types of disturbances and problem-types using appropriate mathematical methodology.
Reverting to equation (17) the controller is asymptotically stable, thus it guarantees no steady state offset from the target point in the presence of constant disturbances θ ε Θ ⊂
Due to space limitations the theorem is proved here only for the case where the output trajectory yt|k lies in a critical region where none of the constraints g is active {circumflex over (x)} ε {tilde over (C)}R. Consider for simplicity B2=0, the proof is valid for B2≠0 with minor modifications. Since the target point lies in the unconstrained region, according to condition (iv) the control brings the plant into an invariant set within this region. The control law for the unconstraint case can be proved to be of the form v=−F1�x−F2.xq. If we substitute this law into the plant {yk=B1xk+B2vk, xk+1=A1xk+A2vk+Wθk} and take the z− transform of the dynamic system we will eventually obtain:
The main advantage of the parametric PI controller (17) is that it does not necessitate the existence of a disturbance or uncertainty estimator since this is taken into consideration implicitly via integral action. If the target point does not belong to the feasible region ŷ≠0 ∈ Y then the equilibrium point ŷ≠0 in terms of the control driven outputs will lie on the boundaries of the feasible region. Then Theorem 2.1 still holds provided that for the evaluation of the integral states, the error of the outputs is shifted according to the modified equilibrium point, i.e.
In this work, feasibility analysis theory (Grossmann et al., 1983) is utilised for deriving the robust parametric controller in a similar fashion to the multiperiod approach Kakalis et al., (2001); However, here, a more compact optimisation problem is formulated that surpasses in terms of constraint satisfaction. The system is considered to be perturbed by a vector of unknown input affine time-varying uncertainties θ that are assumed to belong to a compact polyhedral set θεΘ⊂
[g j(x t|k ,v t|k)≦0|x t|k+1 =x t|k A 1 x t|k +A 2 v t|k +Wθ t|k ,x t|0 x*]
Optimisation over the single nominal uncertainty scenario results the following mathematical formulation:
s.t. ∀θεΘ(∀jεJ|g j(x*,v t|k,θt|k)≦0;j=1, . . . J;k=0,1,2, . . . N;v t|k εV;θεΘ]) (6b)
The constraint in (6) is denoted as the feasibility constraint since it ensures that the single control law derived from this problem satisfies all the constraints for all the possible bounded uncertainty scenarios. The feasibility constraint represents an infinite set of constraints since the inequalities are defined for every possible value of θεΘ. Hence, problem (6) is in general a semi-infinite program that is equivalent to the following bilevel program:
Problem (7b), (8b) is a bilevel program that has as constraints an embedded max max optimisation problem. The next steps will enable us to solve parametrically the embedded optimisation (8b) in terms of vt|k, x* and replace it in (7b)-(8b) as a set of linear equations. This is done as follows:
The importance of expressing the underlying optimisation problem (8b) as a set of inequalities is that we can now write the overall problem (7b)-(8b) as a single-level program:
Problem (11b) is loaded with only {circumflex over (N)}reg≦J feasibility constraints in addition to the constraints in (13). This is in addition to the constraints in (13). This is in contrast to the multiperiod approach where J constraints are included for each period. Note also that the maximum number of additional constraints is independent of the number of bounded uncertainties: w. Whereas in the approach of Campo and Morari (1987), 22w scenarios are examined giving rise to 22w J constraints that have to be incorporated.
An example is presented to demonstrate the features of this newly developed control design methodology. The problem is concerned with deriving the explicit robust control law for an evaporator process studied in Kookos and Perkins, 2001. The controlled outputs are the pressure P2 and the concentration C2 of the product stream that are also the states of the system. Their nominal reference values are: P2=50.57 KPa; Ĉ2=25%. The manipulating inputs are the pressure of the utility stream P100 and the cooling water flow F200. The active output constraint when operating around the nominal point is: C2≦25% whereas additional constraints are also imposed on the inputs. The feed flowrate Fj and composition Cj are considered as a disturbances.
For deriving the robust control law, first the feasibility problem (8b) is considered with a horizon N=3, considering Fi and Ci as deterministic bounded uncertainties varying within 0.1 kg/min and 0.1% respectively. 12 feasibility functions ψ(vt|k,X*) are derived. A model-based receding horizon optimal control problem of the form of (11b) is then formulated. By treating the current states as parameters the optimal control problem is recast as an mp-QP and its solution features a robust parametric controller (robust parco). The state space partition of this controller is shown in
The references set out in the above passage are as follows:
It will be seen that, by applying parametric optimisation the controller need only carry out simple substitution or function evaluation at each stage rather than complex optimisation calculations, the off-line optimisation stage allowing construction of a simple plot against state variables of a plurality of regions each representing a simple solution.
The parametric profile for a given plant, system or process as derived off-line from a model based control strategy such as MPC can then be embedded in software or hardware in an appropriate data store such as a chip, disk, a memory device or so forth. The solution can be embedded in the form of a simple look-up table against state variable values in relation to substition-type solutions (ie control variables constant in a given solution reason) and simple function-based algorithms in function evaluation regions or, evidently, as a combination of the two.
It will be appreciated that any form of process, plant or system capable of being modeled can have a control strategy constructed off-line, within the capabilities of parametric optimisation techniques currently available. However it will be further appreciated that as additional techniques are developed including the solution of the generalised problem at equation (4) above, the skilled person can apply these techniques equally based on the teachings of the present specification.
The techniques according to the invention can be applied to complex systems including any number of state and control variables, and the solution space being dimensioned accordingly. The measurement intervals and prediction horizon (ie the time period over which the state variable predictions are formulated) can be of any appropriate duration dependent only on the off-line processing capability available.
Although the specific description concerns the techniques of the present invention as applied to model predictive control, they can be applied equally to other control schemes including hybrid control in which MPC is extended to include mixed integer variables (i.e. both continuous variables and (0, 1) binary variables) planning under certainty, where performance criteria and an optimal process plan is obtained as a function of uncertain parameters and robustness analysis in model based control applications where robustness as a measure of the ability to absorb uncertainty or variations can be obtained as a function of the uncertainty. Other model based control applications can be optimised using the present invention including more conventional feedback based systems such as proportional integral (PI) systems and so forth.
The form of a general parametric semi-infinite program (Hettich and Kortanek, 1983) is:
Assume that in program, (P) (i) f and h are convex with respect to z and (ii) f, h are finite over
for every possible realisation of θεΘ.
Note that problem (1a)-(7a) falls into the general category of (P). In our case f is a convex quadratic function and h is a linear function with respect of the optimisation variables. Hence, according to (5.1) ∃ a finite set of discrete time points such that by imposing the constraints (5a) on those points the solution of the original parametric dynamic optimisation problem is equivalent to the solution of the resulting multiparametric quadratic problem. This practically means that if the feasibility test in Step III has always a solution, the decomposition method (Steps I-IV) converges, guaranteeing optimality of problem (1a)-(5a).
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