外文翻譯畢業(yè)論文

1、畢業(yè)論文（設(shè)計(jì)）外文翻譯題 目： 電氣工程畢業(yè)生使用MATLAB最優(yōu)控制的過程 系部名稱： 專業(yè)班級(jí)： 學(xué)生姓名： 學(xué) 號(hào)： 指導(dǎo)教師： 教師職稱： 講師 20 年月日電氣工程畢業(yè)生使用MATLAB最優(yōu)控制的過程 Asad Azemi 賓州法尼亞州大學(xué)大峽谷分校 電氣工程系 莫爾文 PA 193551443 Edwin Engin Yaz 美國阿肯色州大學(xué)費(fèi)耶特維爾校區(qū) 電氣工程系 AR 72701摘要控制程序設(shè)計(jì)包，比如矩陣實(shí)驗(yàn)室MATLAB、矩陣模型、C語言控制系統(tǒng)、數(shù)學(xué)非線性現(xiàn)象的語言程序系統(tǒng)等等，已經(jīng)變成了大學(xué)生和研究生系統(tǒng)課程和控制領(lǐng)域的重要材料。這些設(shè)計(jì)包記述了我們的經(jīng)驗(yàn)，賓夕法尼

2、亞大學(xué)大峽谷分校和美國阿肯色州大學(xué)的最優(yōu)控制課程就是利用這些設(shè)計(jì)包，也就是MATLAB，和它的控制系統(tǒng)，魯棒控制和線性矩陣不等式工具箱。這篇論文我們描述適用于最優(yōu)控制課程的MATLAB和它的控制系統(tǒng)工具箱的基本作用，然后我們將集中于MATLAB的特點(diǎn)，產(chǎn)生交互式仿真和用戶界面控制。在交互式仿真指導(dǎo)書的幫助下能有效的闡明因?yàn)閰?shù)的變更系統(tǒng)響應(yīng)的變化，也包括用一些例子闡明這些特點(diǎn)。討論MATLAB怎樣幫助減少計(jì)算作業(yè)分配的時(shí)間，接著是介紹如何使用圖形用戶界面，最后記錄學(xué)生的積極反應(yīng)。 簡介 賓夕法尼亞大學(xué)的電氣工程部門和阿肯色大學(xué)是以計(jì)算機(jī)輔助教育，所以計(jì)算機(jī)輔助設(shè)計(jì)包進(jìn)入了他們的課程。這些課程增

3、加這些設(shè)計(jì)包的目的是通過實(shí)習(xí)和設(shè)計(jì)的經(jīng)驗(yàn)提高學(xué)生對(duì)理論知識(shí)的理解能力。計(jì)算機(jī)輔助課程和計(jì)算機(jī)輔助設(shè)計(jì)包之前已經(jīng)被它的制作者和他的同事們?cè)囉?。這些軟件不僅在大學(xué)里使用同時(shí)也被確認(rèn)為新的教科書，而且以這些軟件為基礎(chǔ)修正了以前教科書上的習(xí)題和問題。合并這些程序包的優(yōu)點(diǎn)和缺點(diǎn)以及它的概要也進(jìn)入我們的課程，將在我們下面的文章里講述。在每個(gè)院校使用的設(shè)計(jì)包里每個(gè)節(jié)段的提綱后面是概要，下面的課程是我們總結(jié)的它的優(yōu)點(diǎn)和缺點(diǎn)。 基本優(yōu)點(diǎn)使用這些工具的主要優(yōu)點(diǎn)是：通過增加圖形和交互仿真加強(qiáng)學(xué)生對(duì)理論知識(shí)的理解，可以用筆和紙分析更復(fù)雜的系統(tǒng)，如果沒有類似的軟件的幫助老師想能夠指導(dǎo)相當(dāng)復(fù)雜的問題也是不現(xiàn)實(shí)的。學(xué)生反映

4、關(guān)于使用這些設(shè)計(jì)包基本是有利的，許多計(jì)算機(jī)輔助設(shè)計(jì)包也是值得一提的，比如MATLAB，不再局限于某一特定的歸檔，早點(diǎn)使用這些設(shè)計(jì)包對(duì)學(xué)生有好處，為了更詳細(xì)的了解它的優(yōu)點(diǎn)讀者可以參閱我們以前的文章。 基本缺點(diǎn) 使用這些程序設(shè)計(jì)包的三個(gè)缺點(diǎn)是：使用這些程序設(shè)計(jì)包計(jì)算機(jī)容易受影響，而且還要求學(xué)生或者老師學(xué)會(huì)如何使用這些計(jì)算機(jī)輔助程序設(shè)計(jì)包，并保證這些包括基準(zhǔn)線課程的設(shè)計(jì)包作為必修課程的一部分，更多詳細(xì)關(guān)于它的缺點(diǎn)的討論可以參閱我們前部分章節(jié)的內(nèi)容。 賓州法尼亞大學(xué)大峽谷分校的最有控制系統(tǒng)賓州法尼亞大學(xué)大峽谷校園是賓州法尼亞大學(xué)十八個(gè)校園的其中之一，是在費(fèi)城區(qū)域的一個(gè)研究生中心，工程師們?cè)谀抢锕ぷ鱽頋M

5、足教育的需求，利用各種各樣的仿真設(shè)計(jì)包幾乎所有的學(xué)生都變成了工程師。這些課程的目的是讓學(xué)生的數(shù)學(xué)工具參數(shù)化，動(dòng)態(tài)最優(yōu)化，并且用于設(shè)計(jì)表現(xiàn)最好的動(dòng)力系統(tǒng)。非線性系統(tǒng)課程包含下面的課題：1、 靜態(tài)最佳化2、 離散系統(tǒng)的時(shí)間最優(yōu)控制 2.1 線性二次調(diào)節(jié)器 2.2 次優(yōu)反饋的穩(wěn)態(tài)閉環(huán)控制系統(tǒng) 2.3 跟蹤問題 2.4最終狀態(tài)固定的調(diào)節(jié)閥與功能3、 最優(yōu)控制的連續(xù)系統(tǒng) 3.1 線性二次調(diào)節(jié)器 3.2 次優(yōu)反饋的穩(wěn)態(tài)閉環(huán)控制系統(tǒng) 3.3 跟蹤問題 3.4 最終狀態(tài)固定的調(diào)節(jié)閥與功能 3.5 最終時(shí)間釋放問題 3.6 約束輸入問題4、動(dòng)態(tài)規(guī)劃 4.1 離散時(shí)間 學(xué)生每周的作業(yè)包括計(jì)算機(jī)的仿真和使用，考試部

6、分也包括用計(jì)算機(jī)仿真的作業(yè)，學(xué)生可以使用學(xué)生版本的MATLAB。 阿肯色州大學(xué)最優(yōu)控制系統(tǒng) 本課程提供給本科生和研究所的學(xué)生，讓他們?yōu)檠芯拷?jīng)典的狀態(tài)空間控制技術(shù)做好充分準(zhǔn)備。這些課程的目的是讓學(xué)生的數(shù)學(xué)工具參數(shù)化，動(dòng)態(tài)最優(yōu)化，而且用于設(shè)計(jì)最好的動(dòng)力系統(tǒng)。為實(shí)現(xiàn)這一目標(biāo)，作者（E.Yaz）從1986年開始更新課程，從而下面的提綱代表性的總結(jié)了一學(xué)期的課程：1、 無約束和約束優(yōu)化問題。（一課時(shí)）2、 成本函數(shù)和參數(shù)優(yōu)化。（二課時(shí)）3、 最優(yōu)充分必要條件。（三課時(shí)）4、 連續(xù)和離散時(shí)間的動(dòng)態(tài)規(guī)劃。（四課時(shí)）5、 最小值原理。（一課時(shí)）6、 最小值原理的應(yīng)用。（三課時(shí)）7、 數(shù)值技術(shù)。（二課時(shí)）8、

7、連續(xù)和離散時(shí)間二次調(diào)節(jié)閥和它的變體。（六課時(shí)）9、 線性二次型跟蹤和抑制擾動(dòng)。（二課時(shí)）10、 線性二次調(diào)節(jié)閥的穩(wěn)健型。（二課時(shí)）11、 介紹保成本，H方程和線性矩陣不等式的方法。（三課時(shí)）12、 暴露分析模擬系統(tǒng)。（二課時(shí)） 我們一課時(shí)通常要講80分鐘，給出兩個(gè)典型的分類，其余分級(jí)根據(jù)每周布置的作業(yè)及每班學(xué)生作業(yè)的改正意見，每學(xué)期收集兩次，第一次在每學(xué)期中，第二次在每學(xué)期末，來衡量學(xué)生綜合的理解能力。家庭作業(yè)的解決方法主要利用MATLAB軟件及其它的工具箱，它多年來被推薦為指定教材及參考書。學(xué)生根據(jù)普渡大學(xué)自助餐廳學(xué)生制定的評(píng)價(jià)標(biāo)準(zhǔn)和由E.Yaz提出的非正式書面反饋得到的結(jié)論。 利用MATL

8、AB實(shí)現(xiàn)系統(tǒng)最優(yōu)控制的過程 基于上述課程綱要，MATLAB是我們發(fā)現(xiàn)的在課堂上最有用的工具，這部分我們將介紹它的特點(diǎn)。MATLAB和它的控制系統(tǒng)以及強(qiáng)健控制工具箱是理想的求解線性二次方程的調(diào)節(jié)器（H2控制）和H方程的控制問題?！發(fā)qr”指令在程序中用來解決線性二次方程問題，它的語法指令是： k,p,ev=lqr(A,B,Q,R,N) (1)K是最佳增益，p是方程式的結(jié)果，ev是最佳閉環(huán)特征值。A、B、Q、R、N分別是系統(tǒng)矩陣的輸入矩陣、描述權(quán)重矩陣、控制權(quán)重矩陣、描述交叉的權(quán)重矩陣和二次方程線性指標(biāo)的控制矩陣。一般形式的代數(shù)Katie方程可以用這個(gè)語句命令解決： X=are(A,B,C) (2

9、)當(dāng)和，而且，常量增益反饋的應(yīng)用程序，通過波特圖可以看出導(dǎo)出的系統(tǒng)函數(shù)為遞減函數(shù)。另外，奈奎斯特圖(“nyquist (A, B, C, D)” )被濾波器的單輸入系統(tǒng)導(dǎo)出。為了選擇特殊的權(quán)重矩陣 和, (3)引出了哈密頓矩陣 （4）閉環(huán)系統(tǒng)特征值就是H的穩(wěn)定特征值。自從H的特征多項(xiàng)式可以放入?yún)⒘康母壽E里，通過根軌跡圖形(“rlocus”)可以控制閉環(huán)極點(diǎn)位置。由命令“are”的卡蒂方程有助于解決在最壞的或者“極大極小”的情況下標(biāo)準(zhǔn)狀態(tài)反饋H的控制問題。在其他穩(wěn)健設(shè)計(jì)最優(yōu)控制系統(tǒng)里 （5）當(dāng) ，通常無窮大的二次方程的成本標(biāo)準(zhǔn)的最小值導(dǎo)出相同類型的H型代數(shù)方程式。最近，使用MATLAB的線性矩

10、陣不等式工具箱可以求出一些不可能追溯性分析的問題的近似解。多狀態(tài)植物能量的最小的一個(gè)上限就是一個(gè)隨機(jī)的例子 （6）當(dāng)i 的取值范圍為1至N。在運(yùn)算時(shí)用連續(xù)的線性化控制非線性系統(tǒng)會(huì)導(dǎo)致這樣的問題，盡管沒有分析解決的方案，線性矩陣不等式允許解決凸優(yōu)化的線性問題。 用MATLAB編寫交互式模擬仿真MATLAB也能制造互動(dòng)式模擬仿真，用幾何圖形仿真可以加強(qiáng)課堂講解的品質(zhì)。在互動(dòng)模擬課程的幫助下能有效的說明因?yàn)閰?shù)變化而引起的系統(tǒng)響應(yīng)的變化，這可以幫助學(xué)生更好的理解自己的課程，而且由于不需要學(xué)生自己編程序，也是的學(xué)生不受限制，即使不知道MATLAB的程序設(shè)計(jì)語言也能通過較少的時(shí)間了解它的特點(diǎn)。這個(gè)特點(diǎn)對(duì)

11、交互式數(shù)學(xué)軟件的發(fā)展起重要作用。在圖形用戶界面功能的幫助下產(chǎn)生交互式模擬仿真，圖形用戶界面由圖形對(duì)象組成，比如菜單、按鈕、列表和區(qū)域，這些對(duì)象都是有意義的。當(dāng)一個(gè)用戶選擇一個(gè)對(duì)象，他就期望有特定的行為發(fā)生。在MATLAB的環(huán)境下，使用用戶界面控制圖形用戶界面。圖一展示了最終例行程序狀態(tài)下的用戶界面最低管制能源標(biāo)量系統(tǒng)的應(yīng)用。系統(tǒng)模型和性能指標(biāo)被下式給出 （7）當(dāng)和分別表示狀態(tài)和控制輸入，r是正的權(quán)重標(biāo)量因子。這個(gè)方程式將產(chǎn)生狀態(tài)和最優(yōu)相對(duì)于時(shí)間的圖形，滑動(dòng)器允許用戶改變系統(tǒng)參數(shù)來觀察結(jié)果的變化。圖二說明在一個(gè)離散的固定在最終狀態(tài)的線性二次方程的調(diào)節(jié)器下使用用戶界面控制。系統(tǒng)模型和性能指標(biāo)被下式

12、給出 （8） 下面是最優(yōu)控制方程 被給出 被給出系統(tǒng)參數(shù)在連續(xù)的時(shí)間里被規(guī)定而且因?yàn)樵诓蓸又芷诒环蛛x，程序能產(chǎn)生輸出后的圖形控制最佳輸入和次優(yōu)調(diào)節(jié)器。T的數(shù)值和最終時(shí)間可以通過滑動(dòng)條改變。按鈕A、B、和C允許用戶改變系統(tǒng)參數(shù)，系統(tǒng)按鈕在連續(xù)和離散時(shí)間里采樣周期，復(fù)制系統(tǒng)的極點(diǎn)和最終時(shí)間。重啟按鈕根據(jù)參數(shù)的變化更新極點(diǎn)。盡管圖形用戶界面對(duì)觀察因?yàn)閰?shù)變化引起系統(tǒng)相應(yīng)的變化很有用，但是這個(gè)階段很耗費(fèi)時(shí)間而且需要掌握很深的關(guān)于MATLAB的知識(shí)。 結(jié)論在這篇論文里我們已經(jīng)提出了在賓夕法尼亞大學(xué)和阿肯色州大學(xué)的畢業(yè)生課程里使用了MATLAB和它的附屬工具箱。計(jì)算機(jī)仿真包提供了好多的好處，比如MATLA

13、B，它能增強(qiáng)我們對(duì)理論原理的理解，完成更多的更復(fù)雜的設(shè)計(jì)，增加學(xué)生的注意力，提高專業(yè)的發(fā)展。使用計(jì)算機(jī)仿真包的主要缺點(diǎn)是學(xué)生和老師需要額外的學(xué)習(xí)在容易受影響的計(jì)算機(jī)上維護(hù)和操作這些程序設(shè)計(jì)包并確保這些設(shè)計(jì)包進(jìn)入我們的基礎(chǔ)課程作為必修課程資料的一部分。在賓夕法尼亞大學(xué)里由于學(xué)生的背景各不相同，一些學(xué)生剛開始不會(huì)使用MATLAB，這就使得他們無法完成他們?nèi)康淖鳂I(yè)，用戶界面控制按鈕會(huì)減少這樣的問題，但同時(shí)也會(huì)增加老師的負(fù)擔(dān)。大部分的學(xué)生反映使用MATLAB給他們起到了很積極的作用。 參考文獻(xiàn) 1馬丁,T.W.,Azemi,王勇智教授,Hewett大學(xué)電氣工程專業(yè)施耐德,“PSpice電氣工程實(shí)驗(yàn)室

14、ASEE周年研討會(huì),1992年,高等教育出版社,第1307-1308頁。2安德魯斯博士,Azmi,A.,查爾頓,S.,Yaz E.,“電氣工程專業(yè)在仿真技術(shù)教育”方面的ASEE Gulf-Southwest節(jié)研討會(huì)會(huì)議,1994年,第82頁。3Azemi, A., Yaz, E.,“本科生非線性系統(tǒng)分析課程利用SIMULINK仿真和MATLAB軟件”第26屆教育會(huì)議前沿,第一卷,1996年,高等教育出版社，595-599頁，美國猶他州鹽湖城。4Azemi, A., Stook, C.,“本科用MATLAB計(jì)算電路課程,”第26屆教育會(huì)議前沿,第一卷，1996年,第599-603頁,美國猶他州鹽

15、湖城。5Yaz, E., Azemi, A.,“利用MATLAB在電力工程兩門研究生課程”第25次邊界教育會(huì)議，第2c6.1-2c6.4頁,1995年。6Dorf, R., Bishop, R.H. 現(xiàn)代控制系統(tǒng),第七版譯,聯(lián)經(jīng)出版事業(yè)出版社,1995年。7Saadat, H.使用MATLAB控制系統(tǒng)計(jì)算。麥格勞希爾集團(tuán)，1993年。8Strum, R., Kirk, D.現(xiàn)代線性系統(tǒng)使用MATLAB。PWS出版公司。1994。9Franklin, G., Powell, J., Workman, M.數(shù)字控制動(dòng)態(tài)系統(tǒng)。Addison-Wesley出版社,1990年。10Biship, R.H

16、.現(xiàn)代控制系統(tǒng)使用MATLAB和SIMULINK仿真分析和設(shè)計(jì)。清華大學(xué)出版社,199711Hanselman, D.C., Kuo, B.C.MATLAB工具在控制系統(tǒng)的分析和設(shè)計(jì)。Prentice Hall出版社，(第二版),1995年。12 Owens, D.H.多變量和最優(yōu)系統(tǒng)。學(xué)術(shù)出版社:倫敦,1981年。13Boyd, S., et. al.線性矩陣不等式的體制和控制原理。暹羅:費(fèi)城,賓夕法尼亞州,1994年。本文摘譯自This paper appears in: Frontiers in Education Conference, 1997. 27th Annual Confer

17、ence. 'Teaching and Learning in an Era of Change'. Proceedings.Issue Date:5-8 Nov 1997On page(s):13 - 17 vol.1Meeting Date:05 十一月 1997 - 08 十一月 1997Location: Pittsburgh, PA , USAPrint ISBN: 0-7803-4086-8References Cited:24INSPEC Accession Number:5781557Digital Object Identifier: 10.1109/FIE.

18、1997.644801 Date of Current Version: 06 八月 2002Using MATLAB in a Graduate Electrical Engineering Optimal Control CourseAsad Azemi Edwin Engin YazDepartment of Electrical Engineering Department of Electrical EngineeringPenn State University University of ArkansasGreat Valley Campus Fayetteville, AR 7

19、2701Malvern, PA 19355-1443Abstract - Control system design packages like MATLAB,MATRIXX, Control C, SIMNON, etc. have become essential ingredients of both undergraduate and graduate courses in the systems and controls area. This work describes our experience, at the Great Valley Campus of the Pennsy

20、lvania State University and the University of Arkansas, with the use of one of these packages, namely MATLAB with its Control Systems, Robust Control and LMI toolboxes in an optimal control course. In this paper we will describe those standard functions of MATLAB and Control Systems Toolbox that are

21、 most appropriate for use in an optimal control course. Next, we will focus on some special features of MATLAB that can be used to produce interactive simulations, using user interface controls. With the help of interactive simulations instructors caneffectively illustrate the change in system respo

22、nse due to parameter variations. Examples illustrating these features are included. A discussion of how MATLAB helps in reducing the amount of time spent in performing computational homework assignments and the advantages of using graphics user interface control will follow. Finally, the general pos

23、itive student reaction will be reported. IntroductionThe Electrical and Engineering Departments at Penn State University and the University of Arkansas are incorporating computer aided engineering (CAE) and computer aided design (CAD) packages into their curricula. The intent of augmenting the curri

24、culum with these packages is to enhance the students theoretical understanding of thematerial with hands on analysis and design experience. The benefits of CAE and CAD packages in the classroom have been realized by the authors and their co-workers before 1-6. The benefits of using these packages in

25、 a university setting is also confirmed by the number of new textbooks, and revisions of previously printed textbooks incorporating new exercises and problems based on these packages, such as 7-15. A summary of the advantages and disadvantages of incorporating these packages into our graduate curric

26、ula are presented below. The summary is followed by sections outlining the use of each package in each institution. A summary of the advantages and disadvantages of incorporating these packages into our curriculum are presented below. General AdvantagesThe main advantages of using these tools are: t

27、he reinforcement of student understanding of theoretical principles by means of enhanced graphical aids and interactive simulations, analysis of more complex systems that can be treated by pencil and paper, and the instructors ability to assign fairly complex design problems that otherwise would hav

28、e be unrealistic without the help of such software. Student response concerning the use of these packages is generally favorable. It is also worth mentioning that the use of many CAE packages, such as MATLAB16, are no longer limited to a a specific filed. Early exposure to these packages will benefi

29、t the students. For a more detailed discussion of this topic readers can refer to our previous works 2-4. General DisadvantagesThree of the disadvantages of using these packages are the maintenance and operation of these packages on an accessible computer system, the extra work required by students

30、(and instructors) to learn how to use CAE packages, and assuring that the packages are included in the baseline curriculum as part of the required course material. A more detailed discussion of this topic can be found in our previous works 2-4. Optimal Control Systems at Penn State Great ValleyPenn

31、State Great Valley Campus, one of the eighteen campuses of the Penn State University, is a graduate center designed to address the educational need of the working engineers in Philadelphia area. Almost all of our students are working engineers, with a wide variety of backgrounds using simulation pac

32、kages. The goals of this course are to expose the students to the mathematical tools of parametric and dynamic optimization and their uses in designing optimally behaving dynamic systems. The nonlinear systems course covers the following topics:1. Static Optimization2. Optimal Control of Discrete-Ti

33、me Systems2.1 Linear Quadratic Regulator2.2 Steady-State Closed-loop Control of Sub-Optimal Feedback2.3 The Tracking Problem2.4 Regulator with Function of Final State Fixed3. Optimal Control of Continuous-Time Systems3.1 Linear Quadratic Regulator3.2 Steady-State Closed-loop Control of Sub-Optimal F

34、eedback3.3 The Tracking Problem3.4 Regulator with Function of Final State Fixed3.5 Final-Time-Free Problem3.6 Constrained Input Problem4. Dynamic Programming4.1 Discrete-TimeSystems 14 and 15 are used in teaching the course. Students are given weekly assignments that also include computer simulation

35、/usage. Exams also include a take home part that have computer simulations. Student have access to student versions of MATLAB. Optimal Control Systems at the University of ArkansasThis course is offered to advanced undergraduate and graduate students with adequate preparation in classical and state

36、space control techniques. The goals of this course are to expose the students to the mathematical tools of parametric and dynamic optimization and their uses in designing optimally behaving dynamic systems. To accomplish these goals, the author (E. Yaz) has been updating the course materials since 1

37、986 so that the following topics are typically covered in a semester:1. Unconstrained and constrained optimization problems (1 class)2. Cost function and parametric optimization (2 classes)3. Necessary and sufficient conditions for optimality (3 classes)4. Dynamic programming in continuous- and disc

38、rete-time (3 classes)5. The minimum principle (1 class)6. Applications of the minimum principle (3 classes)7. Numerical techniques (2 classes)8. The continuous- and discrete-time quadratic regulator (LQR) and its variants (6 classes)9. Linear quadratic tracking and disturbance rejection (2 classes)1

39、0. Robustness properties of LQR (2 classes)11. Introduction to guaranteed cost, H , and linear matrix . inequality methods (3 classes)12. Exams (2 clauses) Our class is normally 80 minutes of lecture. Typically two in-class are given and the rest of the grading is based on weekly homework assignment

40、s and the student portfolio which is composed of the class notes that the student takes and the homework assignment corrections. The portfolios are collected twice, once in time to give mid-semester feedback and also, at the end of the semester, to measure the students general understanding. The maj

41、or use of the MATLAB software and its toolboxes is in homework solutions. The textbooks 14,15, 17, and 18 have been used individually over the years as the recommended textbook for the course together with reference books 19-22. The student responses that are mentioned in the conclusion are based on

42、 the standard student evaluations using the Purdue Cafeteria form and also an informal written feedback solicited by E. Yaz. Use of MATLAB in Optimal Control Systems CourseIn this section we will present those features of the MATLAB that we have found most useful in our classes, based on the aforeme

43、ntioned course outlines. The MATLAB and its Control Systems and Robust Control toolboxes are ideal for solving linear quadratic regulator (H2 Control) and H control problems. .The command “l(fā)qr” (“dqlr” in the discrete-time case) used in homework assignments to solve the linear quadratic regulator pr

44、oblems. The syntax for this command isK, p, ev = lqr (A, B, Q, R, N) (1)where K is the optimal gain, P is the solution of the algebraic Riccati equation, and “ev” are the optimal closed loop eigenvalues. The quantities A, B, Q, R, N are respectively the system matrix input matrix, state weighing mat

45、rix, control weighting matrix and the weighting in the cross product of the state and control in the quadratic performance index. A general form of the algebraic Riccati equation can be solved by commandX = are (A, B, C) (2)Where and . Moreover, since the application of the optimal constant gain fee

46、dback u = -Kx results in a linear time-invariant system description, stability margins can be found by Bode plots (“bode”). Also, the Nyquist plot (“nyquist (A, B, C, D)” )derived byKalman for single input systems. For the special choice of weighting matrices和, (3)the Hamiltonian matrix (4)is formed

47、. The closed loop system eigenvalues are just the stable eigenvalues of H . Since the characteristic polynomial for H can be put in the root locus form for the parameter “ ” by using root-locus plots (“rlocus”), one can control the closed loop pole locations. In the worst case or “minimax” the solut

48、ion of the standard state feedback H. control problem is given by a special algebraic Riccati equation that can be solved by the command “are.” In other optima robust design of control system (5)where minimization of the usual infinite horizon quadratic cost criterion leads to the same type of “H ”

49、type algebraic Riccati equation. .More recently, the MATLABs LMI toolbox 23became available, that makes the numerical solution of some problems that are not analytically traceable possible.One just example is minimizing an upper bound on the“stochastic” energy of the state of a multi-mode plant (6)

50、where i =1-N .The control of nonlinear systems with successive linearizations during the operation will lead to this kind of problems. Although, there is no known analysis solution, the LMI formulation allows a numerical solution to this convex optimization problem 24. Interactive Simulations with M

51、ATLABMATLAB is also capable of producing interactive simulations. This can enhance the quality of presentation with graphical simulations. With the help of interactive simulations instructors can effectively illustrate the change in system response due to parameter variations. This helps students ga

52、in a better understanding of the subject. Moreover, since there is no need for students to do any programming, this will allow students with limited or no knowledge of MATLAB programming to access features of MATLAB with little investments of time. This feature is essential in an interactive coursew

53、are development. Interactive simulations are produced with the help of graphical user interface (GUI) functions. The GUI is made up of graphical objects, such as menus, buttons, lists, and fields. These objects have meanings; when a user "chooses" an object there is an expectation that a c

54、ertain kind of action will take place. In MATLAB the GUI is implemented using user interface (UI) controls.Figure 1 shows the application of user interface inminimum control energy for a scalar system with fixed final state. The system model and performance index are given by (7)where and denote the

55、 state and control input,respectively. r is a positive scalar weighting factor. This program will produce plots of the state and the optimal input vs. time. The sliders will allow the user to change the system parameters and observe the resulting changes. Figure 2 illustrates the use of user interface control in a discrete-time fixed final state linear quadratic regulator. The system model and performance index are given by (8)and the optimal control law equa