最新雅思写作答题卡 无水印

2023年普通高等学校招生全国统一考试（新课标Ⅱ卷）英语本试卷共12页。

考试结束后，将本试卷和答题卡一并交回。

注意事项：1. 答题前，考生先将自己的姓名、准考证号码填写清楚，将条形码准确粘贴在考生信息条形码粘贴区。

2. 选择题必须使用2B铅笔填涂；非选择题必须使用0.5毫米黑色字迹的签字笔书写，字体工整、笔迹清楚。

3. 请按照题号顺序在答题卡各题目的答题区域内作答，超出答题区域书写的答案无效；在草稿纸、试卷上答题无效。

4. 作图可先使用铅笔画出，确定后必须用黑色字迹的签字笔描黑。

5.保持卡面清洁，不要折叠，不要弄破、弄皱，不准使用涂改液、修正带、刮纸刀。

第一部分听力（1—20小题）在笔试结束后进行。

第二部分阅读（共两节，满分50分）第一节（共15小题；每小题2.5分，满分37.5分）阅读下列短文，从每题所给的A、B、C、D四个选项中选出最佳选项。

AYellowstone National Park offers a variety of ranger programs throughout the park, and throughout the year. The following are descriptions of the ranger programs this summer.Experiencing Wildlife in Yellowstone (May 26 to September 2)Whether you’re hiking a backcountry trail (小径), camping, or just enjoying the park’s amazing wildlife from the road, this quick workshop is for you and your family. Learn where to look for animals and how to safely enjoy your wildlife watching experience. Meet at the Canyon Village Store.Junior Ranger Wildlife Olympics (June 5 to August 21)Kids can test their skills and compare their abilities to the animals of Yellowstone. Stay for as little or as long as your plans allow. Meet in front of the Visitor Education Center.Canyon Talks at Artist Point (June 9 to September 2)From a classic viewpoint, enjoy Lower Falls, the Yellowstone River, and the breathtaking colors of the canyon (峡谷) while learning about the area’s natural and human history. Discover why artists and photographers continue to be drawn to this special place. Meet on the lower platform at Artist Point on the South Rim Drive for this short talk.Photography Workshops (June 19 & July 10)Enhance your photography skills — join Yellowstone’s park photographer for a hands-on program to inspire new and creative ways of enjoying the beauty and wonder of Yellowstone.6/19 — Waterfalls & Wide Angles: meet at Artist Point.7/10 — Wildflowers & White Balance: meet at Washburn Trailhead in Chittenden parking area.21. Which of the four programs begins the earliest?A. Photography Workshops.B. Junior Ranger Wildlife Olympics.C. Canyon Talks at Artist Point.D. Experiencing Wildlife in Yellowstone.22. What is the short talk at Artist Point about?A. Works of famous artists.B. Protection of wild animals.C. Basic photography skills.D. History of the canyon area.23. Where will the participants meet for the July 10 photography workshop?A. Artist Point.B. Washburn Trailhead.C. Canyon Village Store.D. Visitor Education Center.BTurning soil, pulling weeds, and harvesting cabbage sound like tough work for middle and high school kids. And at first it is, says Abby Jaramillo, who with another teacher started Urban Sprouts, a school garden program at four low-income schools.The program aims to help students develop science skills, environmental awareness, and healthy lifestyles.Jaramillo’s students live in neighborhoods where fresh food and green space are not easy to find and fast food restaurants outnumber grocery stores. “The kids literally come to school with bags of snacks and large bottles of soft drinks,” she says. “They come to us thinking vegetables are awful, dirt is awful, insects are awful.” Though some are initially scared of the insects and turned off by the dirt, most are eager to try something new.Urban Sprouts’ classes, at two middle schools and two high schools, include hands-on experiments such as soil testing, flower-and-seed dissection, tastings of fresh or dried produce, and work in the garden. Several times a year, students cook the vegetables they grow, and they occasionally make salads for their entire schools.Program evaluations show that kids eat more vegetables as a result of the classes. “We have students who say they went home and talked to their parents and now they’re eating differently,” Jaramillo says.She adds that the program’s benefits go beyond nutrition. Some students get so interested in gardening that they bring home seeds to start their own vegetable gardens. Besides, working in the garden seems to have a calming effect on Jaramillo’s special education students, many of whom have emotional control issues. “They get outside,” she says, “and they feel successful.”24. What do we know about Abby Jaramillo?A. She used to be a health worker.B. She grew up in a low-income family.C. She owns a fast food restaurant.D. She is an initiator of Urban Sprouts.25. What was a problem facing Jaramillo at the start of the program?A. The kids’ parents distrusted her.B. Students had little time for her classes.C. Some kids disliked garden work.D. There was no space for schoolgardens.26. Which of the following best describes the impact of the program?A. Far-reaching.B. Predictable.C. Short-lived.D. Unidentifiable.27. What can be a suitable title for the text?A. Rescuing School GardensB. Experiencing Country LifeC. Growing Vegetable LoversD. Changing Local LandscapeCReading Art: Art for Book Lovers is a celebration of an everyday object — the book, represented here in almost three hundred artworks from museums around the world. The image of the reader appears throughout history, in art made long before books as we now know them came into being. In artists’ representations of books and reading, we see moments of shared humanity that go beyond culture and time.In this “book of books,” artworks are selected and arranged in a way that emphasizes these connections between different eras and cultures. We see scenes of children learning to read at home or at school, with the book as a focus for relations between the generations. Adults are portrayed (描绘) alone in many settings and poses — absorbed in a volume, deep in thought or lost in a moment of leisure. These scenes may have been painted hundreds of years ago, but they record moments we can all relate to.Books themselves may be used symbolically in paintings to demonstrate the intellect (才智), wealth or faith of the subject. Before the wide use of the printing press, books were treasured objects and could be works of art in their own right. More recently, as books have become inexpensive or even throwaway, artists have used them as the raw material for artworks — transforming covers, pages or even complete volumes into paintings and sculptures.Continued developments in communication technologies were once believed to make the printed page outdated. From a 21st-century point of view, the printed book is certainly ancient, but it remains as interactive as any battery-powered e-reader. Toserve its function, a book must be activated by a user: the cover opened, the pages parted, the contents reviewed, perhaps notes written down or words underlined. And in contrast to our increasingly networked lives where the information we consume is monitored and tracked, a printed book still offers the chance of a wholly private, “off-line” activity.28. Where is the text most probably taken from?A. An introduction to a book.B. An essay on the art of writing.C. A guidebook to a museum.D. A review of modern paintings.29. What are the selected artworks about?A. Wealth and intellect.B. Home and school.C. Books and reading.D. Work and leisure.30. What do the underlined words “relate to” in paragraph 2 mean?A. Understand.B. Paint.C. Seize.D. Transform.31. What does the author want to say by mentioning the e-reader?A. The printed book is not totally out of date.B. Technology has changed the way we read.C. Our lives in the 21st century are networked.D. People now rarely have the patience to read.DAs cities balloon with growth, access to nature for people living in urban areas is becoming harder to find. If you’re lucky, there might be a pocket park near where you live, but it’s unusual to find places in a city that are relatively wild.Past research has found health and wellness benefits of nature for humans, but a new study shows that wildness in urban areas is extremely important for human well-being.The research team focused on a large urban park. They surveyed several hundred park-goers, asking them to submit a written summary online of a meaningful interaction they had with nature in the park. The researchers then examined thesesubmissions, coding (编码) experiences into different categories. For example, one participant’s experience of “We sat and listened to the waves at the beach for a while” was assigned the categories “sitting at beach” and “listening to waves.”Across the 320 submissions, a pattern of categories the researchers call a “nature language” began to emerge. After the coding of all submissions, half a dozen categories were noted most often as important to visitors. These include encountering wildlife, walking along the edge of water, and following an established trail.Naming each nature experience creates a usable language, which helps people recognize and take part in the activities that are most satisfying and meaningful to them. For example, the experience of walking along the edge of water might be satisfying for a young professional on a weekend hike in the park. Back downtown during a workday, they can enjoy a more domestic form of this interaction by walking along a fountain on their lunch break.“We’re trying to generate a language that helps bring the human-nature interactions back into our daily lives. And for that to happen, we also need to protect nature so that we can interact with it,” said Peter Kahn, a senior author of the study.32. What phenomenon does the author describe at the beginning of the text?A. Pocket parks are now popular.B. Wild nature is hard to find in cities.C. Many cities are overpopulated.D. People enjoy living close to nature.33. Why did the researchers code participant submissions into categories?A. To compare different types of park-goers.B. To explain why the park attracts tourists.C. To analyze the main features of the park.D. To find patterns in the visitors’ summaries.34. What can we learn from the example given in paragraph 5?A. Walking is the best way to gain access to nature.B. Young people are too busy to interact with nature.C. The same nature experience takes different forms.D. The nature language enhances work performance.35. What should be done before we can interact with nature according to Kahn?A. Language study.B. Environmental conservation.C. Public education.D. Intercultural communication.第二节（共5小题；每小题2.5分，满分12.5分)阅读下面短文，从短文后的选项中选出可以填入空白处的最佳选项。

2023年高考英语模拟试卷注意事项：1．答卷前，考生务必将自己的姓名、准考证号、考场号和座位号填写在试题卷和答题卡上。

用2B铅笔将试卷类型（B）填涂在答题卡相应位置上。

将条形码粘贴在答题卡右上角"条形码粘贴处"。

2．作答选择题时，选出每小题答案后，用2B铅笔把答题卡上对应题目选项的答案信息点涂黑；如需改动，用橡皮擦干净后，再选涂其他答案。

答案不能答在试题卷上。

3．非选择题必须用黑色字迹的钢笔或签字笔作答，答案必须写在答题卡各题目指定区域内相应位置上；如需改动，先划掉原来的答案，然后再写上新答案；不准使用铅笔和涂改液。

不按以上要求作答无效。

4．考生必须保证答题卡的整洁。

考试结束后，请将本试卷和答题卡一并交回。

第一部分（共20小题，每小题1.5分，满分30分）1．In order to satisfy the needs of the market, the electric company has ______ its focus to tablet computers and smart mobile phones.A．switched B．shapedC．directed D．discovered2．Wild applause ______when the crowd waiting heard the announcement of the arrival of the pop star at their community.A．made out B．broke out C．let out D．worked out3．---- Which of these digital cameras do you like best?---- ______. They are both expensive and of little use.A．None B．Neither C．Nothing D．No one4．—Do you think I'm a good surfer?—Of course! I ________ you earlier．You made it look so easy，graceful even．A．am watching B．was watchingC．have watched D．had watched5．—What is your impression of your former colleague Nick?—He’s helpful, and he ________ give us a hand at work.A．must B．wouldC．may D．should6．Even though the way Lin Daiyu expresses her feelings may be ________ to a modern audience, it is rooted in her character and makes who she is.A．allergic B．foreignC．sensitive D．fundamental7．It was warm and nice that evening. I stayed awake until eleven in order to have a good look at the moon by myself.A．on average B．on purposeC．on business D．on time8．I wonder ________ the equipment will be available in ten days.A．that B．when C．whether D．where9．He had a great dearie to have a home of his own, ______ he had always lived with his grandmother.A．or B．and C．so D．for10．The problem _______he will have his college education at home or abroad remains untouched.A．how B．whether C．that D．when11．Difficulties strengthen the mind, _____ labour does the body.A．if B．asC．for D．so12．The young man has ______ experience. He probably can’t do that job.A．few B．littleC．some D．any13．---I went to see The Wandering Earth last night. It was fantastic!---You were so lucky! How I wish I _____ the ticket too.A．get B．gotC．had got D．would get14．---Are you satisfied with his school report?---Not at all. It couldn’t have be en ______.A．worse B．so badC．better D．the worst15．I wish I ________ her the news and then she wouldn’t be worried now.A．don’t tell B．didn’t tellC．haven’t told D．hadn’t told16．Many netizens are impressed with the excuse given by a teacher for quitting her job ______ she owes the world a visit.A．because B．that C．where D．why17．When the old man woke up, he found someone his TV in the living room.A．stole B．steal C．stealing D．stolen18．Mr. Smith didn’t understand _____ made his son so upset t hat evening.A．what was it B．why it was this C．how that was D．what it was that19．This was returned because the person ________ this letter was addressed had died three years ago.A．to whom B．to whichC．which D．whom20．-How can I get in touch with the travel agency, Robin?-Just surf the Internet and then call one ________ the telephone number is provided.A．with which B．in which C．of which D．by which第二部分阅读理解（满分40分）阅读下列短文，从每题所给的A、B、C、D四个选项中，选出最佳选项。

天津市滨海新区塘沽第一中学2022届高三英语毕业班复课模拟检测试题（含解析）本试卷分为第Ⅰ卷（选择题）和第Ⅱ卷（非选择题）两部分，共150分，考试用时120分钟。

第Ⅰ卷1至11页，第Ⅱ卷12至13页。

答卷前，考生务必将自己的姓名、准考号填写在答题卡上，并在规定位置粘贴考试用条形码。

答卷时，考生务必将答案涂写在答题卡上，答在试卷上的无效。

考试结束后，将本试卷和答题卡一并交回。

第Ⅰ卷注意事项：1. 每小题选出答案后，用铅笔将答题卡上对应题目的答案标号涂黑。

如需改动，用橡皮擦干净后，再选涂其他答案标号。

2. 本卷共55小题，共95分。

第一部分：英语知识运用（共两节，满分45分）第一节：单项填空（共15小题；每小题1分，满分15分）1.— Have you heard that Jay Chou is singing his latest song in the concert?—_________ Aren’t you joking with me?A. That sounds great!B. Yes I suppose it must be.C. Really?D. That’s a good idea.【答案】C【解析】【详解】考查情景交际。

句意：——你听说周杰伦在演唱会上唱他的新歌了吗？——真的吗？你不是在跟我开玩笑吧？A. That sounds great!听起来真棒！B. Yes, I suppose it must be. 是的，我想一定是这样。

C. Really?真的吗？D. That’s a good idea.好主意。

结合后文Aren’t you joking with me?可知对方不相信周杰伦在演唱会唱他的新歌这件事，因此C选项Really?“真的吗？”最符合语境。

故选C。

2.（202X·江苏）Some schools will have to make ______ in agreement with the national soccer reform.A. judgmentsB. adjustmentsC. commentsD. achievements【答案】B【解析】【详解】名词词义辨析。

2024届福建省莆田市高三下学期高考英语质量检测试题（三模）注意事项：1.答题前，考生务必将自己的姓名、考生号、考场号、座位号填写在答题卡上。

2.回答选择题时，选出每小题答案后，用铅笔把答题卡上对应题目的答案标号的黑。

如需改动，用橡皮擦干净后，再选涂其他答案标号。

回答非选择题时，将答案写在答题卡上。

写在本试卷上无效。

3.考试结束后，将本试卷和答题卡一并交回。

第一部分听力（共两节，满分30分）做题时，先将答案标在试卷上。

录音内容结束后，你将有两分钟的时间将试卷上的答案涂到答题卡上。

第一节（共5小题；每小题1.5分，满分7.5分）听下面5段对话。

每段对话后有一个小题，从题中所给的A、B、C三个选项中选出最佳选项。

听完每段对话后，你都有10秒钟的时间来回答有关小题和阅读下一小题。

每段对话仅再一遍。

例：How much is the shirt?A. £ 19.15.B. £ 9.18.C. £ 9.15.答案是C。

1. What is the man going to do?A. Hold a party.B. Go to the bookstore.C. Celebrate his birthday.2. When will the man clean up his room?A. In just half an hour.B. After seeing the movie.C. Before going to the cinema.3. How does the man usually get to his company?A. By subway.B. By car.C. By bike.4. What was the matter with Alice today?A. She did badly in the math test.B. She forgot to check her homework.C. She made many mistakes in all tests.5. How much should the woman pay in total?A. $ 30.B. $ 18.C. $ 6.第二节（共15 小题；每小题1.5分，满分22.5分）听下面5段对话或独白。

2022-2023高三上英语期末模拟试卷注意事项1．考试结束后，请将本试卷和答题卡一并交回．2．答题前，请务必将自己的姓名、准考证号用0．5毫米黑色墨水的签字笔填写在试卷及答题卡的规定位置．3．请认真核对监考员在答题卡上所粘贴的条形码上的姓名、准考证号与本人是否相符．4．作答选择题，必须用2B铅笔将答题卡上对应选项的方框涂满、涂黑；如需改动，请用橡皮擦干净后，再选涂其他答案．作答非选择题，必须用05毫米黑色墨水的签字笔在答题卡上的指定位置作答，在其他位置作答一律无效．5．如需作图，须用2B铅笔绘、写清楚，线条、符号等须加黑、加粗．第一部分（共20小题，每小题1.5分，满分30分）1．Newly-built wooden cottages line the street，___________ the old town into a dreamland.A．turned B．turningC．to turn D．having turned2．While interacting with people in India, I was ________ to a way of life completely different from my own.A．reduced B．exposed C．committed D．transferred3．People expect Shanghai Disneyland Park to offer better service than ________ of Tokyo’s.A．this B．it C．one D．that4．Regarding China-US differences on human rights issues, Hong said the two sides can enhance mutual understanding through dialogue ______ on equality and mutual respect.A．based B．to base C．basing D．base 5．I am so thrilled to have my underwater photos ______ in the National Geographic and on the cover!A．to be featured B．featured C．being featured D．to feature6．—Where on earth have they gone?—I have no idea, but I wish I .A．know B．knew C．would know D．would have known7．I’m afraid that I can’t attend Tom’s wedding party ______ next weekend.A．to be held B．being heldC．held D．is to be held8．_____ his homework in time, he had to stay up late into the night.A．Finishing B．Having finished C．To finish D．To have finished9．________ well for my job interview, I really had butterflies in my stomach on my wayto the company I had applied to.A．Having not prepared B．Not having preparedC．Not to have prepared D．To have not prepared10．As one of the most popular artists in the US, not only_______ with her hit songs, but has inspired women with her devotion to charity.A．has Beyonce influenced pop culture B．influenced pop culture has Beyonce C．Beyonce has influenced pop culture D．influenced pop culture Beyonce has 11．—My computer is out of function again.—You need to go to the customer service center and have it ________.A．to repair B．repairC．repairing D．repaired12．With the nuclear crisis worsening in Iran, the world’s attention is fixed again on________is called the Middle East.A．which B．what C．that D．it13．40 grams of meat per day is ______ people should consume in order to stay fit. A．that B．whyC．how D．what14．I guess ________ impresses me most about his painting is the colors he uses. A．who B．whichC．that D．what15．The laptops made by our company sell best, but nobody could have guessed the place in the market that they ________ 20 years ago.A．had had B．had C．were having D．were to have16．— Can you tell me something about _________ science.— OK. _ Nobel Prize in Chemistry is usually awarded to Americans.A．the; The B．/; The C．a; / D．the; /17．The doctor’s treatment has worked marvels: the patient has completely．A．repeated B．returned C．recovered D．reminded18．Could I speak to__________ is in charge of International Sales please?A．who B．whatC．whoever D．whatever19．---I saw no more than one motorcar in the shop. Will you go and buy ?---No, I’d rather find in other shops.A．it; oneB．one; itC．it; the oneD．the one; it20．on one foot with your eyes closeD．and you will soon lose your balance. A．Standing B．StoodC．To stand D．Stand第二部分阅读理解（满分40分）阅读下列短文，从每题所给的A、B、C、D四个选项中，选出最佳选项。

高中毕业班高考考前适应性模拟试卷(三)英语试题考试结束后，将本试卷和答题卡一并交回。

注意事项：1．答题前，考生先将自己的姓名、准考证号码填写清楚，将条形码准确粘贴在条形码区域内。

2．选择题必须用2B铅笔填涂；非选择题必须用0.5毫米黑色字迹的签字笔书写，字体工整、笔迹清楚。

3．请按照题号顺序在各题目的答题区域内作答，超出答题区域书写的答案无效；在草稿纸、试卷上答题无效。

4．考生必须保持答题卡的整洁，不要折叠、不要弄破、弄皱，不使用涂改液、修正带、刮纸刀。

第一部分听力（共两节，满分30分）做题时，先将答案标在试卷上。

录音内容结束后，你将有两分钟的时间将试卷上的答案转涂到答题卡上。

第一节（共5小题；每小题1.5分，满分7. 5分）听下面5段对话。

每段对话后有一个小题，从题中所给的A、B、C三个选项中选出最佳选项，并标在试卷的相应位置。

听完每段对话后，你都有10秒钟的时间来回答有关小题和阅读下一小题。

每段对话仅读一遍。

例：How much is the shirt?A.£19.15.B.£9.18.C.£9.15.答案是C。

1. What did the man forget to do?A. Pick up milk and eggs for breakfast.B. Get some noodles for dinner.C. Put the garbage downstairs.2. What are the speakers mainly talking about?A. A presentation.B. An inspiring story.C. An Austrian person.3. What will the woman probably do in ten minutes?A. Plan a party.B. Do someone a favor.C. Work on her report.4. Where does the conversation probably take place?A. In a classroom.B. In a drugstore.C. In a doctor’s office.5. Who might Cathy be?A. Bill’s friend.B. The history teacher.C. Jill’s roommate. 第二节（共15小题；每小题1.5分，满分22.5分）听下面5段对话或独白。

2025届高考10月联考英语本试卷满分150分，考试时间120分钟注意事项：1．答题前，先将自己的姓名、准考证号填写在试题卷和答题卡上，并将准考证号条形码粘贴在答题卡上的指定位置。

2．选择题的作答：每小题选出答案后，用2B铅笔把答题卡上对应题目的答案标号涂黑，写在试题卷、草稿纸和答题卡上的非答题区域均无效。

3．非选择题的作答：用签字笔直接答在答题卡上对应的答题区域内。

写在试题卷、草稿纸和答题卡上的非答题区域均无效。

4．考试结束后，请将本试题卷和答题卡一并上交。

第Ⅰ卷（选择题）第一部分听力（共两节，满分30 分）第一节（共 5 小题；每小题 1.5 分，满分7.5 分）听下面 5 段对话。

每段对话后有一个小题，从题中所给的A、B、C 三个选项中选出最佳选项。

听完每段对话后，你都有10 秒钟的时间来回答有关小题和阅读下一小题。

每段对话仅读一遍。

1. Who is on the right of the picture?A. Uncle Sam.B. Aunt Nancy.C. Cousin Chad.2. What’s the probable relationship between the speakers?A. Husband and wife.B. Friends.C. Teacher and student.3. What will the woman do first?A. Buy some paper.B. Deal with a complaint.C. Visit her son’s school.4. Where will the speakers go?A. To a cafe.B. To a bakery.C. To a bank.5. What are the speakers mainly talking about?A. A gift.B. A sports event.C. Their friend.第二节（共15 小题；每小题 1.5 分，满分22.5 分）听下面 5 段对话或独白。

雅思简介雅思（International English Language Testing System，简称IELTS，中文名为国际英语语言测试系统）是由英国文化协会（The British Council，即英国驻华大使馆/总领事馆文化教育处）、剑桥大学考试委员会（CESOL）和澳大利亚教育国际开发署（IDP Australia）共同举办的国际英语水平测试。

此项考试是为申请赴英语国家（美国、英国、澳大利亚、加拿大、新西兰等）留学、移民的非英语国家学生而设，用来评定考生运用英语的能力。

雅思考试包括四个部分，依次为听力、阅读、写作和口语，考试时间共2小时45分钟。

每一部分都独立评分，四部分得分的平均分作为考生的雅思综合得分（小数部分取舍到最近的一分或半分，即如果平均分为6.125分，雅思得分算作6分）。

成绩单上将列出考生每一部分的得分，同时给出考生的综合得分。

雅思考试满分为9分。

考试成绩在考试后十个工作日公布。

有效期为两年。

听力：首先进行的听力考试时间40分钟，题目分四个部分。

第一个部分是日常生活中会发生的对话（通常为二人），第二个部分是生活相关的独白；第三个部分是学术性的对话（通常二人以上），第四个部分为学术论文演讲，难度依次增加（亦为了再筛选精英，有时有些句子会带重口音或地道词语）。

通常前三个部分都会分成两段，分别回答不同的问题。

考生在听完每段录音后会有一小段时间复查（但因为不会重复，所以要即时写出答案）。

全部录音放完需时30分钟，剩余10分钟供考生将答案从试卷填写到答题卡上。

时间到了之后考官会把试卷收上来，并要求考生将答题卡翻过来。

阅读：时间60分钟，题目分三个部分。

每个passage大约1200-1500字，大约13-14道题，总共40道题，时间到后考官会把答题卡和试卷都收上来。

写作：阅读考试之后是写作考试，时间也是60分钟。

题目有两道，第一道是看图说明（A类）或者书信（G类），要求150字，20分钟；第二道是议论文或说明文（议论为主），要求250字，40分钟。

9Control Structures9.1IntroductionOne of the reasons for the great success of PID controllers is that they can be employed also as a basic component for more advanced control systems so that(relatively)complex control tasks can be addressed by still exploit-ing the available know-how.This chapter focuses on two control structures widely applied in industry,namely(series)cascade control and ratio control, which are still the subject of new investigations in order tofind methodologies that allow the improvement of the performance and/or to simplify the overall control system design.9.2Cascade Control9.2.1GeneralitiesIn process control applications,the rejection of load disturbances is often of main concern.In order to improve the performance for this task,the imple-mentation of a cascade control system can be considered.In a cascade control scheme the process has one input and two(or more)outputs.Indeed,in order to provide an effective disturbance rejection,an additional sensor is employed so that the fast dynamics of the process is separated as much as possible from the slow dynamics(i.e.,that with the slowest poles and the nonminimum-phase part).The typical series cascade control system is shown in Figure9.1.For the sake of simplicity,here only two nested loops are considered but the approach can be generalised to more loops.The process transfer function is denoted by P(s)=P2(s)P1(s),y1is the primary output,y2is the secondary output, C2is the secondary(or slave)controller and C1is the primary(or master) controller(it appears that the output signal of the master controller serves as the set-point for the slave controller).Analogously,the inner loop is de-2529ControlStructuresFig.9.1.Typical cascade control schemenominated as the secondary loop,while the outer loop is denominated as the primary loop.Intuitively,if P1(s)represents the slow dynamics of the process and P2(s)rep-resents the fast dynamics,the effectiveness of the cascade control system is due to the fact that disturbances affecting the(fast)secondary loop are effec-tively compensated before they affect the main process output y1.Formally, the transfer function from the load disturbance d to the process variable y1isT(s):=P1(s)P2(s)1+C2(s)P2(s)+C1(s)C2(s)P1(s)P2(s).(9.1)The characteristic equation is therefore1+C2(s)P2(s)+C1(s)C2(s)P1(s)P2(s)=0,(9.2) while,if a conventional(single-loop)feedback control is employed,the char-acteristic equation is1+C(s)P1(s)P2(s)=0,(9.3) where C(s)is the single-loop controller.When the dynamics of the secondary loop is faster than the dynamics of the primary loop,the cascade control sys-tem has improved stability characteristics and therefore a higher gain in the primary loop can be adopted.Based on this fact,it appears that the improvement in the cascade control per-formance is more significant when disturbances act in the inner loop and when the secondary sensor is placed in order to separate as far as possible the fast dynamics of the process from the slow dynamics(Krishnaswami et al.,1990). Actually,when the secondary process exhibits a significant dead time or there is an unstable(positive)zero,the use of cascade control is not useful in gen-eral(taking into account the additional cost due to the secondary sensor and to the secondary controller).As an additional advantage,the nonlinearities of the process in the inner loop are handled by that loop and therefore they are removed from the more important outer loop.In this context,the parameters of the overall control system should be selected in order to provide a tight tuning of the inner loop(with respect to the outer one).Note that the presence of an integrator in the inner loop is not strictly necessary since the null steady-state error can be assured by the outer loop. It is worth stressing that if integral action is employed both in the master and9.2Cascade Control253 in the slave controller,the integrator windup should be carefully handled.In particular,the saturation of the actuator requires that an anti-windup strat-egy(see Chapter3)is implemented for the secondary controller.However, when the secondary controller attains a limit,the primary controller acts in open-loop at the same time.A typical approach is therefore to stop the inte-gration of the primary controller when the output of the secondary controller attains its limits.This solution prevents the master controller from unneces-sarily increasing its output and therefore forcing the primary controller to be more saturated.However,a better solution is,when the output of the sec-ondary controller attains its limit,to use the secondary process output as a tracking signal for the primary controller.This solution is effective also in providing a bumpless transfer when the secondary controller switches from manual to automatic mode.The design of the overall cascade control system is usually performed byfirst tuning the secondary controller,based on the secondary process transfer func-tion(the primary loop is placed in manual mode).Then,the primary controller is tuned on the basis of the closed-loop transfer function of the secondary loop in series with the primary process transfer function(which contains the dominant dynamics,because of the tight tuning of the secondary loop).It appears that the design is performed sequentially and therefore it is more time-consuming than the design of a classical single-loop controller.There is, therefore,the need to have automatic tuning functionalities that are poten-tially able to provide a simultaneous tuning of the two controllers.9.2.2Relay Feedback Sequential Auto-tuningA technique based on the relay feedback for the automatic tuning of a cascade controller has been proposed in(Hang et al.,1994).It basically consists of applying the standard relay feedback approach(see Chapter7)first to the secondary loop(with the primary loop placed in manual mode)and then to the primary loop(with the secondary feedback controller already tuned).Actually, any tuning rule based on the ultimate gain and the ultimate frequency of the process can be applied in this context.Remarkably,the ratio of the ultimate frequencies obtained in these two steps can be adopted to assess whether a cascade controller is worth being applied (since it indicates the ratio of the speeds of the loops).Further,a refined tuning of the secondary controller can be performed(in closed-loop)by applying again the relay feedback controller to the secondary loop with the primary loop closed(i.e.,with the PID controller previously tuned that acts as a primary controller).The same refinement can be performed also on the master controller.9.2.3Relay Feedback Simultaneous Auto-tuningThe method presented in the previous section has the disadvantage that a sequential(and therefore time-consuming)tuning is actually performed.A2549ControlStructuresFig.9.2.Scheme for the on-line tuning of the cascade controller technique that allows the achievement of a simultaneous tuning of the two controllers has been proposed in(Tan et al.,2000).Therein it is assumed that the two controllers are already(roughly)tuned(mainly in order to stabilise the process).Then,the scheme shown in Figure9.2is applied in order to tune on-line the two controllers simultaneously.In particular,denote by C1,0(s) and C2,0(s)the transfer functions of the two initial controllers and byωu the ultimate frequency obtained from the experiment.A Fourier or Spectral analysis,with an appropriate weighting window,is then applied to the signalsr2and y2in order to determine T r2y2,0(jωu),where T r2y2denotes the closed-loop transfer function of the inner loop.Then,the frequency response of the secondary process atω=ωu can be derived asP2(jωu)=T r2y2,0(jωu)C2,0(jωu)(1−T r2y2,0(jωu)).(9.4)A desired frequency response¯T r2y2(jωu)has now to be specified by the user.This can be done easily by considering a prototypefirst-order-plus-dead-time (FOPDT)transfer function¯Tr2y2(s)=1T2s+1e−L2s.(9.5)The(desired)values of the dead time and of the time constant can be selected, starting from the relay feedback experiment,assuming thatT r2y2,0(jωu)=γ+jθ.(9.6) They can befixed asT2=0.5ωu1−γ2+θ2(γ2+θ2)(9.7)andL2=1ωuarccos(γ−θωu T2).(9.8)Once T r2y2(jωu)has been determined,the frequency response of the newsecondary controller atω=ωu can be calculated asC2(jωu)=¯Tr2y2(jωu)P2(jωu)1−¯T r2y2(jωu).(9.9)9.2Cascade Control255 Starting from this expression and denoting C2(jωu)asα2+jβ2,the param-eters of the PID secondary controller(in ideal form)are then determined as:K p2=α2,(9.10)T i2=−α2β2ωu,(9.11)T d2=0.25T i.(9.12) If a PI controller is employed,it can be simply set T d2=0,by keeping the same values of the proportional gain and of the integral time constant.A similar reasoning is applied in order to design the primary controller.After having applied a Fourier or Spectral analysis to the signals r1and y1sothat T r1y1,0(jωu)is derived,the frequency response of the primary process atω=ωu can be determined as:P1(jωu)=T r1y1,0(jωu)T r2y2,0(jωu)C1,0(jωu)(1−T r1y1,0(jωu)).(9.13)Then,the frequency response prototype of the primary loop atω=ωu,de-noted as¯T r1y1(jωu)is determined as for the secondary loop(see(9.6)–(9.8)).Finally,the frequency response of the new primary controller atω=ωu isC1(jωu)=¯Tr1y1(jωu)P1(jωu)¯T r2y2(jωu)1−¯T r1y1(jωu).(9.14)The PID controller parameters arefinally determined by denoting C1(jωu)as α1+jβ1and calculatingK p1=α1,(9.15)T i1=−α1β1ωu,(9.16)T d1=0.25T i.(9.17) As for the slave controller,a PI controller can be adopted simply byfixing T d1=0.In order to illustrate the methodology,the following example is provided. Consider the processesP1(s)=1(5s+1)2e−4s,(9.18)P2(s)=1s+1e−0.2s.(9.19)2569Control StructuresFig.9.3.Process variable before(dashed line)and after(solid line)the relay feed-back simultaneous auto-tuningInitially,the tuning of the two controllers isK p1=1,T i1=12,T d1=0,(9.20)andK p2=0.5,T i2=4,T d2=0.(9.21) After the application of the methodology,the parameters of the PID con-trollers are determined as(note that the derivative action of the secondary controller is not adopted):K p1=1.18,T i1=18.99,T d1=4.75,(9.22)andK p2=0.56,T i2=2.14,T d2=0.(9.23) The process variable before and after the refinement of the tuning is plotted in Figure9.3.In particular,the experiment consists of applying in both cases a unit step in the set-point signal at time t=0and a load disturbance unit step at time t=250.It appears that the performance has been improved signifi-cantly both with respect to the set-point following and to the load disturbance rejection task.However,it has to be stressed again that a(rough)tuning of the two controllers has to be performed before applying the technique.9.2Cascade Control2579.2.4Simultaneous Identification Based on Step ResponseA technique based on the step response for the simultaneous identification of the primary and of the secondary process has been presented in(Visioli and Piazzi,2006).It consists of applying a step signal to the process P(s).A FOPDT transfer function of the fast dynamics of the process can be estimated by evaluating the step response of P2(s),for example by applying the area method(see Section7.2.1).At the same time,a model for the slow dynamics of the process can be esti-mated by considering its input signal y2and its output signal y1and by apply-ing a least-squares procedure,such as the one proposed in(Sung et al.,1998) which is based on the integrated input and output signals and therefore it is inherently robust to measurement noise(see again Section7.2.1).The ob-tained(possibly high-order)model can then be reduced if a tuning rule that requires a FOPDT or a SOPDT model of the primary process is employed(see the next sections).It has to be noted that,because of the different dynamics of P2and P1,the step response of P2is indeed a sufficiently exciting signal to be adopted as an input signal for the least-squares based estimation of P1(s).9.2.5Simultaneous Tuning of the ControllersA methodology for the simultaneous tuning of the two controllers in the cascade control system has been proposed in(Lee et al.,1998a).It is based on the Internal Model Control(IMC)design methodology(Morari and Zafiriou,1989)and on the reduction of the controllers obtained by means of a Maclaurin series expansion.In particular,the secondary process transfer function can be written asP2(s)=P2m(s)P2a(s)(9.24) where P2a(s)is the all-pass portion of the transfer function containing all the nonminimmum phase dynamics(P2a(0)=1).Then,the desired inner loop transfer function is specified as¯Tr2y2(s)=P2a(s)(λ2s+1)n2(9.25)whereλ2is the user-chosen time constant of the IMCfilter and the value of n2is selected in order to make the resulting controller proper.The secondary controller can then be determined asC2(s)=P−12m(s)(λ2s+1)n2−P2a(s).(9.26)In order to approximate the controller obtained to a PID controller,the same procedure already presented in Section7.5.3can be used,i.e.,Expression (9.26)can be rewritten as2589Control StructuresC2(s)=k(s)s(9.27)and expanding C2(s)in a Maclaurin series in s:C2(s)=1sk(0)+k (0)s+k (0)2s2+···.(9.28)Expression(9.28)is indeed a PID controller in ideal form withK p2=k (0)T i2=k (0) k(0)T d2=k (0)2k (0).(9.29)An analogous procedure can be adopted for the design of the primary con-troller.By assuming that the transfer function of the inner loop is(9.25),the process model of the outer loop can be expressed asP12(s)=P1(s)P2a(s)(λ2s+1)n2(9.30)and it can be rewritten asP12(s)=P12m(s)P12a(s),(9.31) where again P12m(s)contains the invertible part of the model and P12a(s) contains the nonminimum-phase part in all-pass form.Then,the desired outer loop transfer function is specified as¯Tr1y1(s)=P12a(s)(λ1s+1)n1(9.32)and therefore the primary controller transfer function is determined asC1(s)=P−112m(s)(λ2s+1)n2P2a(s)((λ1s+1)n2−P12a(s)).(9.33)Finally,the controller transfer function can be reduced to a PID form by ap-plying again the Maclaurin series expansion.An interesting case that is worth analysing is when both processes are de-scribed by a FOPDT transfer function,namely,P1(s)=K1T1s+1e−L1s,P2(s)=K2T2s+1e−L2s.(9.34)In this case,it is9.2Cascade Control259¯Tr2y2(s)=e−L2sλ2s+1(9.35)and¯Tr1y1(s)=e−(L1+L2)sλ1s+1.(9.36)An explicit tuning rule can be therefore derived for both the primary and the secondary controller:K p2=T2+L222(λ2+L2) K2(λ2+L2)T i2=T2+L222(λ2+L2)T d2=L226(λ2+L2)⎛⎜⎜⎜⎝3−L2T2+L222(λ2+L2)⎞⎟⎟⎟⎠(9.37)andK p1=T1+λ2+(L1+L2)22(λ1+L1+L2) K1(λ1+L1+L2)T i1=T1+λ2+(L1+L2)22(λ1+L1+L2)T d1=λ2T1−(L1+L2)36(λ1+L1+L2)T1+λ2+(L1+L2)22(λ1+L1+L2)+(L1+L2)22(λ1+L1+L2).(9.38)Similarly,tuning rules can be derived also by assuming that the two processes are described by SOPDT transfer functions(Lee et al.,1998a).In any case,it can be deduced that the design phase involves the selection of the two time constantsλ1andλ2.In principle,they allow to handle the trade-offbetween aggressiveness and robustness.Actually,since the Maclaurin series expansion is eventually adopted to determine the two PID controllers, the conclusions drawn in Section7.5.5should be considered.However,based on many simulations,in(Lee et al.,1998a)the suggestion is to setλ2=0.5L2(9.39) andλ1=0.5(L1+L2).(9.40)2609Control StructuresFig.9.4.Process variable obtained with the simultaneous tuning of the controllers The following simulation result is provided in order to evaluate technique. Consider again the process whose dynamics is described by the series of the two transfer functions(9.18)–(9.19).If the method described in Section9.2.4 is applied in order to identify simultaneously the two parts of the process, two FOPDT transfer functions are derived.In particular,we obtain K2=1, T2=0.99and L2=0.21for the secondary process and K1=1,T1=7.55 and L1=6.88for the primary process.The resultingfilter time constants are therefore,according to Expressions(9.39)–(9.40),λ2=0.11andλ1=3.55. By applying the tuning rules(9.37)–(9.38),we obtain K p2=3.31,T i2=1.06, T d2=0.07,K p1=0.94,T i1=10.02,and T d1=1.89.When a unit step in the set-point signal is applied at time t=0and a load disturbance unit step is applied at time t=250,the resulting process variable is plotted in Figure 9.4.The effectiveness of the overall methodology is apparent.9.2.6Tuning of the General Cascade Control StructureThe approach presented in the previous section has been generalised in(Lee et al.,2002),where also integral and unstable processes are considered.The scheme devised exploits the presence of a set-pointfilter both in the primary and in the secondary controller(see Figure9.5).In particular,this is adopted for unstable and integrating processes and for stable processes with poles near zero,while for normal processes the same approach of the previous section has9.2Cascade Control261 to be employed(and therefore the set-pointfilters are not used).Whether a stable process is considered a normal process or a process with a pole near zero depends on the design sense.Restricting the analysis again to the case where both processes are described by FOPDT transfer functions(9.34),explicit tuning rules for the secondary and primary PID controllers result.They are(9.37)and(9.38)when normal stable processes are considered,while when stable processes with poles near zero are addressed:K p2=T2+α−λ22+L2α−12L222λ2+L2−αK2(2λ2+L2−α)T i2=T2+α−λ22+L2α−12L22 2λ2+L2−αT d2=T2α−16L32−12L22α2λ2+L2−αT2+α−λ22+L2α−12L222λ2+L2−α−λ22+L2α−12L222λ2+L2−α(9.41)andK p1=T1+λ2+β−λ21+(L1+L2)β−12(L1+L2)22λ1+L1+L2−βK1(2λ1+L1+L2−β)T i1=T1+λ2+β−λ21+(L1+L2)β−12(L1+L2)2 2λ1+L1+L2−βT d1=λ2T1+λ2β+T1β−16(L1+L2)3−12(L1+L2)2β2λ1+L1+L2−βT1+λ2+β−λ21+(L1+L2)β−12(L1+L2)22λ1+L1+L2−β−λ21+(L1+L2)β−12(L1+L2)22λ1+L1+L2−β(9.42)whereα=T2⎡⎣1−1−λ2T22e−L2T2⎤⎦(9.43)andβ=T1⎡⎣1−1−λ1T12e−L1+L2T1⎤⎦.(9.44)The set-pointfilters are selected asF2(s)=1αs+1(9.45)2629ControlStructuresFig.9.5.General cascade control structureFig.9.6.Process variable obtained with the general cascade control structureandF1(s)=1βs+1.(9.46)Similarly to the case of normal processes,the closed-loop time constantsλ1 andλ2can be chosen asλ2L2=0.5÷1(9.47)andλ1L1+L2=0.5÷1.(9.48) The same process(9.18)–(9.19)of the previous sections are used to illustrate the methodology.The technique described in Section9.2.4is applied again for the simultaneous identification of the two processes.The valuesλ2= 0.75L2=0.16andλ1=0.75(L1+L2)=5.32are selected.By applying the tuning rules(9.41)–(9.44)we obtainα=0.43,β=7.29,K p2=4.93,9.2Cascade Control263 T i2=0.51,T d2=0.07,K p1=0.93,T i1=9.75,and T d1=1.86.The process output obtained is shown in Figure9.6.It appears that a high-performance load disturbance response is achieved.With respect to the method presented in the previous section,as expected because of the presence of the set-point filters,the rise time in the set-point step response is(significantly)increased and the overshoot is reduced to zero.9.2.7Use of a Smith Predictor in the Outer LoopA scheme based on a Smith predictor has been proposed in(Kaya,2001) and it is depicted in Figure9.7.It appears that a Smith predictor scheme (Palmor,1996)is employed in order to compensate for the delay term in the primary controller.An automatic tuning procedure,based on the use of two sequential relay-feedback tests is proposed.In particular,a FOPDT transfer function isfirst estimated for the secondary process by employing, for example,an asymmetrical relay(see Section7.2.2).The parameters of a PI controller are selected consequently according to the tuning rules proposed in(Zhuang and Atherton,1993),which allows a minimisation of the ISTE integral criterion.Then,again by adopting an asymmetrical relay,the process seen by the primary controller C1is identified.In particular,a SOPDT transfer function parameters are estimated(see Section7.3.2).In this phase the Smith predictor scheme is not employed.Then,the delay free part of the model is denoted by P12and the dead time term is denoted by L.The parameters of the PID primary controller(in ideal form)arefinally selected in order to minimise again the ISTE criterion and the Smith predictor based cascade control scheme is implemented.In order to verify the effectiveness of the devised scheme,it has been tested on the same process(9.18)–(9.19)as before.The resulting PI/PID parameters are K p2=3.17,T i2=1.05,K p1=1.5,T i1=8.22,and T d1=0.91.The process output obtained is shown in Figure9.8.Obviously,the control architecture is more effective when a high normalised dead time is present in the secondary process,although it has to be stressed that mismatches in the estimation of the dead time term should be carefully handled in order to avoid a significant degradation of the overallperformance.2649Control StructuresFig.9.8.Process variable obtained with the Smith predictor based cascade control schemeFig.9.9.Two-degree-of-freedom cascade control structureThe proposed control scheme has been further developed in(Kaya et al.,2005) for stable processes and in(Kaya and Atherton,2005)for integrating and unstable processes.The tuning of the controllers is based on the so-called standard forms(Dorf and Bishop,1995),which allow the direct synthesis of controllers that minimise integral performance indexes.9.2Cascade Control265Fig.9.10.Alternative two-degree-of-freedom cascade control structure9.2.8Two Degree-of-freedom Control StructureA two-degree-of-freedom cascade control structure,aiming at decoupling the set-point tracking and the load disturbance rejection tasks has been proposed in(Liu et al.,2005).Two schemes can be implemented in this context.They are reported in Figures9.9and9.10.Note that P1m and P2m denote models of the primary and secondary processes P1and P2respectively,while P m denotes the model of the overall process P.Then F is a load disturbance estimator (indeed,it acts as a secondary controller)and C is the primary controller used for set-point tracking.From Figure9.9it can be deduced that in the nominal case,i.e.,when P1m and P2m are perfect models of P1and P2,there is an open-loop control from the set-point r to the primary output y1so that the nominal set-point response and the inner loop load disturbance response are decoupled.The scheme of Figure9.10has the advantage that an explicit model of the primary process P1is not required but,on the other side,when a load disturbance d occurs, both the load disturbance estimator F and the primary controller C concur in compensating it and therefore a performance degradation might appear. In other words,the possible performance degradation in the load disturbance response has to be accepted for a possibly easier and more effective implemen-tation of the control architecture.In any case,the design of the two controllers C and F is the same for both schemes.The design of C is based on the minimisation of the H2performance objec-tive,as in the IMC approach.The transfer functions of the two processes arerewritten asP1(s)=K1A1+(s)A1−(s)B1(s)e−L1s(9.49)andP2(s)=K2A2+(s)A2−(s)B2(s)e−L2s(9.50)2669Control Structureswhere A1+(0)=A1−(0)=B1(0)=1,A2+(0)=A2−(0)=B2(0)=1,and all zeros of A1−(s),A2−(s),B1(s)and B2(s)are located in the left half plane, while all zeros of A1+(s)and A2+(s)are located in the right half plane.The primary controller transfer function can therefore be derived asC(s)=K2B1(s)B2(s)K1K2A∗1+(s)A∗2+(s)A1−(s)A2−(s)(λc s+1)n c(9.51)where A∗1+(s)and A∗2+(s)are the complex conjugate of A1+(s)and A2+(s) respectively(i.e.,A1+(s)/A∗1+(s)and A2+(s)/A∗2+(s)are all-passfilters),λc is a tuning parameter and n c is the order of thefilter to be selected in order to make the controller transfer function proper.Indeed,λc allows the handling of the trade-offbetween aggressiveness and robustness.A good starting point is to selectλc equal to the overall time delay of the process to be controlled. The design of the load disturbance estimator F is performed by proposing the desired complementary sensitivity function of the inner loop,denoted as¯T(s). In particular,by considering again the H2optimal performance objective of the IMC theory,the expression of¯T(s)is selected as¯T(s)=1(λf s+1)n f A2+(s)A∗2+(s)e−L2s(9.52)where thefilter order n f is chosen appropriately.Thus,the expression of F(s)can be determined asF(s)=F1(s)1−F1(s)P2(s)(9.53)whereF1(s)=B2(s)K2A∗2+(s)A2−(s)(λf s+1)n f.(9.54)The design parameterλf allows again the handling of the trade-offbetween aggressiveness and robustness and it is suggested to select it,as afirst guess, equal to the secondary process estimated dead time.It is worth stressing at this point that the methodology fully exploits an ac-curate(possibly high-order)modelling of the process,but the two controllers are not of PID type in general and the overall scheme has to be implemented in a different way with respect to the standard scheme of Figure9.1.The same process(9.18)–(9.19)has been adopted to illustrate the methodol-ogy.Based again on the simultaneous identification method of Section9.2.4, the model of the primary and secondary processes are chosen respectively asP1m(s)=17.55s+1e−6.88s(9.55)andP2m(s)=10.99s+1e−0.21s.(9.56)9.3Ratio Control267Fig.9.11.Process variable obtained with the two degree-of-freedom control struc-ture of Figure9.9(solid line)and of Figure9.10(solid line)Consequently,the twofilters time constants are selected asλc=7.09and λf=0.21.It results:C(s)=(0.99s+1)(7.55s+1)(7.09s+1)2(9.57)andF1(s)=0.99s+10.21s+1.(9.58)The process variables obtained by considering the two control schemes are plotted in Figure9.11(note that a unit step load disturbance is applied again to the process at time t=250).The effectiveness of both schemes is apparent.9.3Ratio Control9.3.1GeneralitiesRatio control,which consists in keeping a constant ratio between two(or more)process variables,irrespective of possible set-point changes and load disturbances that might occur on the plant,is of concern in a variety of in-dustrial applications such as chemical dosing,water treatment,chlorination,2689Control Structuresmixing vessels and waste incinerators.For example,in combustion systems it is necessary to control accurately the air-to-fuel ratio in order to obtain a high efficiency,and in blending processes a selected ratio of differentflows has to be maintained to keep a constant product composition.In this latter case, bothflows can be controlled or,alternatively,one of them can be measured only(the so-called wildflow)and the other is regulated in order to achieve the desired ratio.Formally,denote by a the desired ratio to be kept between the values of two process variables y1and y2.For this purpose,the control scheme shown in Figure9.12(also termed series metered control)can be implemented.Each variable is controlled by two separate controllers C1and C2(typically of PI type)and the output y1of thefirst process is multiplied by a and adopted as the set-point signal of the closed-loop control system of the second process, i.e.,it is r2(t)=ay1(t).The main disadvantage of this scheme is related to its transient response to a change in the set-point r1,since the output y2is necessarily delayed with respect to y1,due to the closed-loop dynamics of the second loop.In general, the second loop is chosen as the one with the fastest dynamics.However,in order to keep the ratio close to the desired value,it might be necessary to detune thefirst loop and therefore the performance obtained in the set-point following task and in the rejection of the load disturbance d1decreases.A possible alternative scheme is the one shown in Figure9.13(termed paral-lel metered control).In this case,provided that the two closed-loop systems have the same dynamics,a high performance can be achieved in the set-point following task,but,obviously,a disturbance acting on thefirst process can cause a large error in the ratio value.For this reason,this approach has to be employed in those applications where load disturbances are unlikely to occur. Finally,it is worth remembering that in the particular case of combustion control,where a selected air-to-fuel ratio has to be maintained,it is often es-sential to prevent the occurrence of a fuel rich environment,since this might lead to a furnace explosion.In this context,the so-called cross-limiting con-trol(also known as lead-lag control)shown in Figure9.14can be adopted (Gomes,1985).The two loops are interlocked by using a low and a high se-lectors that force the fuel to follow the airflow when the set-point increases and that force the air to follow the fuel when the set-point decreases. Techniques for the improvement of these standard techniques have been re-cently proposed.They are presented in the following sections.9.3.2The Blend StationMethodologyAs already mentioned,the use of the series metered control scheme of Figure 9.12has the disadvantage that the output y2is actually delayed with respect to y1because of the closed-loop dynamics of the second loop.In order to。