From AEOnweller@aol.com Thu Sep 5 13:20:46 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA07086; Thu, 5 Sep 96 13:20:45 EDT Received: from emout15.mail.aol.com (emout15.mx.aol.com [198.81.11.41]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id NAA23837 for ; Thu, 5 Sep 1996 13:14:35 -0400 From: AEOnweller@aol.com Received: by emout15.mail.aol.com (8.6.12/8.6.12) id NAA10198 for crose@ece.rutgers.edu; Thu, 5 Sep 1996 13:15:01 -0400 Date: Thu, 5 Sep 1996 13:15:01 -0400 Message-Id: <960905131500_516458319@emout15.mail.aol.com> To: crose@ece.rutgers.edu Subject: ghostscript for Macintosh Status: RO Hello Dr. Rose, I found a web page containing info and links to ghost-script for the mac: http://www.glyphic.com/glyphic/projects/macgs.html the actual program and assosicated file may be downloaded via ftp at: ftp://ftp.cs.wisc.edu/pub/ghost/aladdin/mac/ The above ftp site contains a readme file that tells what files are needed for different macs. Regards, Allen Onweller From crose Thu Sep 5 13:24:42 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA07101; Thu, 5 Sep 96 13:24:41 EDT Date: Thu, 5 Sep 96 13:24:41 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609051724.AA07101@MOGLI.rutgers.edu> To: AEOnweller@aol.com Subject: Re: ghostscript for Macintosh Cc: crose Status: RO THANKS MUCHO! When I get the mailing list together for 330:501, I'll post the info. Any idea about windows PCs.... I'll do the search mysefl using archie since you've now shown me that an archive is likely to exist. thanks again, Chris Rose From cripop@eden.rutgers.edu Fri Sep 6 15:07:00 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA08593; Fri, 6 Sep 96 15:07:00 EDT Received: from er5.rutgers.edu (er5.rutgers.edu [165.230.180.133]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id PAA03949 for ; Fri, 6 Sep 1996 15:00:38 -0400 Received: (from cripop@localhost) by er5.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id PAA29320 for crose@ece; Fri, 6 Sep 1996 15:01:33 -0400 From: "Dimitrie Popescu" Message-Id: <9609061501.ZM29318@er5.rutgers.edu> Date: Fri, 6 Sep 1996 15:01:32 -0400 X-Mailer: Z-Mail Lite (3.2.0 5jul94) To: crose@ece.rutgers.edu Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: RO Dear dr. Rose, I have read some problems from the first list that you gave us, and I think that problem 2 (chapter 1, page 41) has a small error in its text. It says to prove that conditions (ii)+(iii) of the definition 1 + the assumption that ro(x,x)>=0 for all x in M, should imply that ro(x,y)>=0 FOR ALL X,Y>=0!!! Maybe I am wrong, but I think that it should be "for all x,y in M", shouldn't it? I mean, there is no information that on M is defined an order relation such as "less than" or "greater than" (as for example the complex numbers). Besides, we also don't know what does 0 mean in M. Thanks in advance. Dimitrie Cristian Popescu From crose@mogli.rutgers.edu Fri Sep 6 16:11:58 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA08698; Fri, 6 Sep 96 16:11:58 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA01354; Fri, 6 Sep 96 15:58:45 EDT Date: Fri, 6 Sep 96 15:58:45 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9609061958.AA01354@mogli> To: sandhya_seshadri@MENTORG.COM Subject: Re: Refresher material Cc: crose@MOGLI.rutgers.edu Status: RO Hi, Stochastic Processes: Try the new book Roy Yates is using in 330:541 Linear Algebra: Strang is the classic... VERY understandable Probability: Try David Goodman's and Roy Yate's text Coordinate Geometry... not sure, sorry. For the Probability stuff, Roy Yates' email address is ryates@ece.rutgers.edu Cheers, Chris Rose From crose@mogli.rutgers.edu Fri Sep 6 16:12:00 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA08704; Fri, 6 Sep 96 16:11:59 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA01363; Fri, 6 Sep 96 16:02:43 EDT Date: Fri, 6 Sep 96 16:02:43 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9609062002.AA01363@mogli> To: cripop@eden.rutgers.edu Subject: problem Cc: crose@MOGLI.rutgers.edu Status: RO Hi Dimitrie, I'll take a peek at it. In the mean time, please exercise the TA on this one (if you've not already). He needs to get up to speed. If he can't answer your question (let me know soon) I'll give you the solution. But I'm going to purposely let you struggle with it for the moment. You could be right, you could be wrong! :) That's the essence of graduate study and the researcher that you'll develop into. Problems are EASY when you know they have a solution. It gets much harder when you're not even sure that the problem is stated correctly (as when you start framing research problems). Hope I didn't scare you! Don't worry, most problems I'll give WILL have solutions. CHeers, Chris Rose From crose@mogli.rutgers.edu Fri Sep 6 16:12:02 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA08708; Fri, 6 Sep 96 16:12:01 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA01401; Fri, 6 Sep 96 16:14:48 EDT Date: Fri, 6 Sep 96 16:14:48 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9609062014.AA01401@mogli> To: cripop@eden.rutgers.edu Subject: mercy Cc: crose@MOGLI.rutgers.edu Status: RO Sort of mercy anyway... Is the space M defined in problem 2 to be something in particular. Specifically, what does it mean (if anything) for x element of M to be "greater than zero" Think about that and send me a response (please exercise the TA anyway though! :) Cheers, Chris Rose From crose@mogli.rutgers.edu Mon Sep 9 15:42:08 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA11820; Mon, 9 Sep 96 15:42:08 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA03161; Mon, 9 Sep 96 15:27:37 EDT Date: Mon, 9 Sep 96 15:27:37 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9609091927.AA03161@mogli> To: ece501 Subject: update2 Cc: crose@MOGLI.rutgers.edu Status: RO Hi Folks, If you've received this message then you are on the ece501@mogli.rutgers.edu mailing list. The student count stands at 17 right now. This update was composed at about 3pm on Monday 9/9/96. The next update will be before 3pm on wednesday 9/11/96. And another on friday at about the same time. I'd like to see all approximately 40 students on the list. Otherwise, important announcements could be missed by some students (with possibly perilous results). Cheers, Chris Rose From ralf@bugeyes.rutgers.edu Tue Sep 10 15:52:54 1996 Return-Path: Received: from bugeyes.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA12984; Tue, 10 Sep 96 15:49:55 EDT Received: by bugeyes.rutgers.edu (940816.SGI.8.6.9/930416.SGI) for ece501@mogli id MAA03543; Tue, 10 Sep 1996 12:43:49 -0700 From: "Ricardo Losada" Message-Id: <9609101243.ZM3541@bugeyes.rutgers.edu> Date: Tue, 10 Sep 1996 12:43:49 -0700 X-Mailer: Z-Mail (3.2.0 26oct94 MediaMail) To: ece501@mogli.rutgers.edu Subject: TA office hours Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: RO Hey people, the following are the TA office hours: Tu: 7:30 - 8:30 pm. Th: by appointment (send me email) My info once again: Ricardo Losada ralf@ece office EE 224 Ph: 5-4996 From xxxxx@eden.rutgers.edu Tue Sep 10 15:23:15 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA12851; Tue, 10 Sep 96 15:23:14 EDT Received: from er5.rutgers.edu (er5.rutgers.edu [165.230.180.133]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id PAA02848 for ; Tue, 10 Sep 1996 15:16:10 -0400 Received: (from xxxxx@localhost) by er5.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id PAA12293 for crose@ece; Tue, 10 Sep 1996 15:17:04 -0400 Date: Tue, 10 Sep 96 15:16:59 EDT From: Mr./Ms. X To: crose@ece.rutgers.edu Subject: question Message-Id: Status: RO Prof. Rose, I am having difficulty following the test, and at times your classroom notes, since I do not understand teh meaning of many of the symbols used to denote the relationship of one set of umbers to another and so forth. From crose Tue Sep 10 15:58:53 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA13063; Tue, 10 Sep 96 15:58:51 EDT Date: Tue, 10 Sep 96 15:58:51 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609101958.AA13063@MOGLI.rutgers.edu> To: xxxxx@eden.rutgers.edu Subject: Re: question Cc: crose@MOGLI.rutgers.edu Status: RO Hi, Not sure what you mean about following the "test". Do you mean the tests of the email group for the course? As for the text (did you mean te"x"t above... then it makes sense) and notes, well, that's what office hours are for! Please come by and either the TA or I will be happy to discuss. You should try the TA first (tonite would be good) and then me. I suggest this method since I've seen this stuff quite a bit before and the further you are away from learning it, the harder it is sometimes to see points of student difficulty. If Ricardo can't answer your questions sufficiently well, then come see me and we can thrash it out between us. Looking forward to hearing from you. Cheers, Chris Rose From ralf@ece.rutgers.edu Tue Sep 10 19:57:48 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA14052; Tue, 10 Sep 96 19:54:48 EDT Received: (from ralf@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id TAA06377 for ece501@mogli; Tue, 10 Sep 1996 19:47:42 -0400 Date: Tue, 10 Sep 1996 19:47:42 -0400 From: Ricardo Losada Message-Id: <199609102347.TAA06377@ece.rutgers.edu> To: ece501@mogli.rutgers.edu Subject: TA hours Content-Length: 210 Status: RO Just to make it super-clear (since there was an email that didn't quite go through). (Or something like that). The final office hours are: Tu: 7:30 - 8:30 pm. Thu: by appointment (via email) thanks, Ricardo. From crose Thu Sep 12 20:21:33 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA17481; Thu, 12 Sep 96 20:18:29 EDT Date: Thu, 12 Sep 96 20:18:29 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609130018.AA17481@MOGLI.rutgers.edu> To: ece501 Subject: update and further explanation Status: RO Hi Folks, This is the final update on the mailing list and my first officical posting to the group as well. Tonight (Thur 9/12/96) we proved that a contraction mapping has the property that a limit point exists, is unique and is obtained by iterating the mapping from ANY starting point. I'd like to clarify part c) of the proof. In particular, in part a) we showed that all the iterative sequences starting from any point are cauchy and convergent (part b sealed up the convergent part). In part c) we'd like to know that a limit point exists. That is we need to show that there is a point x* such that A(x*) = x*. (NOT ASSUME THAT SUCH A POINT EXISTS AS I CONFUSEDLY/FATIQUEDLY STATED AFTER CLASS) So, the first step is to remember that the iterative sequence is Cauchy and therefore x_n -> x* for some x* (i.e., we showed that the sequence was convergent in part a). NOW since the mapping is BOUNDED (a contraction mapping is a subset of bounded mappings) we must have if x_n->x* then A(x_n) -> A(x*) (because a bounded mapping is continuous). The final step is to write down the iterative map x_(n+1) = A(x_n) and take the limit of each side using the above result to obtain x*=A(x*). So, if the mapping is a CONTRACTION mapping, there exists at least one point x* such that A(x*) = x*. IN ADDITION, this point x* is UNIQUE. (the last part of the proof, part d in the book). That was relatively clear, so we'll forego it here. Cheers All, Chris Rose PS: GO RUTGERS! This is first time I've been able to have school spirit for sports. The teams at my undergrad/grad school were not that good alas. Though the tiddlywinks squad was nationally ranked (I kid you not!) From crose Thu Sep 12 20:39:08 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA17549; Thu, 12 Sep 96 20:35:32 EDT Date: Thu, 12 Sep 96 20:35:32 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609130035.AA17549@MOGLI.rutgers.edu> To: ece501 Subject: homework Status: RO You should do problem set 1. If you'd like to have your work checked, you must hand it in by 1 week from today (thursday 9/19). Don't cheat yourself of true understanding by peeking at the solutions. Papers should be given to Ricardo some time before class on 9/19. Cheers, Chris Rose From ykogan@eden.rutgers.edu Sat Sep 14 13:12:33 1996 Return-Path: Received: from er5.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA19228; Sat, 14 Sep 96 13:10:35 EDT Received: (from ykogan@localhost) by er5.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id NAA09149 for ece501@MOGLI; Sat, 14 Sep 1996 13:03:49 -0400 From: "Yelenq Kogan" Message-Id: <9609141303.ZM9147@er5.rutgers.edu> Date: Sat, 14 Sep 1996 13:03:48 -0400 X-Mailer: Z-Mail Lite (3.2.0 5jul94) To: ece501@MOGLI.rutgers.edu Subject: another postcript URL @ latex2html URL Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: RO Dear Dr.Rose, The following URL address has the newest version of ghostscript program for PC: http://www.cs.wisc.edu/~ghost/index.html i guess this is the same place which Allen Onweller recommended for a Mac. There is also a site with latex-to-html program: http://cbl.leads.ac.uk/nikos/tex2html/doc/latex2html/latex2html.html it has a documentation in postscript there. Regards, Lena Kogan From crose@mogli.rutgers.edu Sat Sep 14 16:11:53 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA19323; Sat, 14 Sep 96 16:11:53 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA08551; Sat, 14 Sep 96 14:29:27 EDT Date: Sat, 14 Sep 96 14:29:27 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9609141829.AA08551@mogli> To: ykogan@eden.rutgers.edu Subject: Re: another postcript URL @ latex2html URL Cc: crose@MOGLI.rutgers.edu Status: RO LATEX TO HTML!!!!!!!?????? Thanks! I'll take a peek! If it works you will have made course material lot easier to read over the net. I don't really like the PC version of ghostview since in order to get decent resolution you seem to need the picture way to big to fit on the screen. However, it's the only way I know of that PC users can read the problem sets solutions and such. Cheers, Chris Rose From tcarvell@avionics.itt.com Mon Sep 16 13:51:33 1996 Return-Path: Received: from ittingw.itt.com by MOGLI.rutgers.edu (4.1/25-eef) id AA28231; Mon, 16 Sep 96 13:48:00 EDT Received: from ccmail.avionics.itt.com by nfsrv.avionics.itt.com; (5.65v3.2/1.1.8.2/05Sep95-0325PM) id AA04524; Mon, 16 Sep 1996 13:40:48 -0400 Received: from ccMail by CCMAIL.AVIONICS.ITT.COM (SMTPLINK V2.11 PreRelease 4) id AA842906435; Mon, 16 Sep 96 13:37:13 EST Date: Mon, 16 Sep 96 13:37:13 EST From: "Tom Carvelli" Message-Id: <9608168429.AA842906435@CCMAIL.AVIONICS.ITT.COM> To: ece501@mogli.rutgers.edu Return-Receipt-To: tcarvell@avionics.itt.com Subject: Last Thursday's Lecture Topics Status: RO What sections of chapter one did Prof Rose cover in last Thursday's class? From cripop@ece.rutgers.edu Tue Sep 17 15:35:07 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA02879; Tue, 17 Sep 96 15:32:19 EDT Received: (from cripop@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id PAA10072 for ece501@mogli.rutgers.edu; Tue, 17 Sep 1996 15:23:59 -0400 Date: Tue, 17 Sep 1996 15:23:59 -0400 From: Dimitrie Popescu Message-Id: <199609171923.PAA10072@ece.rutgers.edu> To: ece501@mogli.rutgers.edu Subject: Where is the error? Content-Length: 633 Status: RO Let us consider the mapping A(x)=x-[x] (where [x] denotes the whole part of the real no. x). The domain of A is R, and the range is (obviously) [0,1). If we consider the usual distance over R, i.e. the absolute value, we have that for all x,y elements of R |A(x)-A(y)|<=|x-y| (it is simple to see that since both A(x) and A(y) are in [0,1) the distance between them will always be <1, which is always <= |x-y|. That implies that A(x) is a bounded mapping. But, as one can easily observe from the graphical representation of A(x), this is not continuos. Then how about with boundedness => continuity? Or, where am I wrong? Dimitrie From crose Tue Sep 17 17:19:06 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA02976; Tue, 17 Sep 96 17:19:05 EDT Date: Tue, 17 Sep 96 17:19:05 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609172119.AA02976@MOGLI.rutgers.edu> To: ece501 Subject: Dimitrie's problem Cc: crose@MOGLI.rutgers.edu Status: RO Hi Folks, Dimitrie Popescu posed a problem to the group. Please think about it and post some answers. Good for the soul. It was a good question. The mapping is NOT continuous (sketch it out). So maybe you need to examine boundedness more closely, eh? Great question by the way, Dimitrie. Thanks! Cheers From crose@MOGLI.rutgers.edu Tue Sep 17 22:20:23 1996 Return-Path: Received: from qbeast.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA04848; Tue, 17 Sep 96 22:16:43 EDT Full-Name: Christopher Rose Received: by qbeast.rutgers.edu (4.1/SMI-4.1) id AA01320; Tue, 17 Sep 96 22:08:24 EDT Date: Tue, 17 Sep 96 22:08:24 EDT From: crose@MOGLI.rutgers.edu (Christopher Rose) Message-Id: <9609180208.AA01320@qbeast.rutgers.edu> To: ece501@mogli.rutgers.edu Subject: GIF! Status: RO Hi Folks, IMPORTANT NOTE If you wish to have PS1 looked over by Ricardo, you'll have to submit it in class on thursday. This procedure (submission one week after I say "go do the problem set") is in place to avoid exam time crunches which Ricardo cannot service your request in a timely fashion owing to volume and the fact that we share him with the UG linear systems course. ENHANCEMENT Thanks to a suggestion by Allen Onweller we now have problem sets and solutions in GIF format. This should be directly readable by your browser! So, if you could not bear to load a ghostscript viewer onto you PC, now you don't have to! I ran out of steam tonite trying to get the stuff working so the exams and solutions are not yet on there. The interface is also a little awkward since I simply paste gif files one after the other and don't link the image files. Cheers All, Chris Rose From cripop@ece.rutgers.edu Thu Sep 19 20:31:17 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA07784; Thu, 19 Sep 96 20:24:54 EDT Received: (from cripop@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id UAA23723 for ece501@mogli.rutgers.edu; Thu, 19 Sep 1996 20:16:12 -0400 Date: Thu, 19 Sep 1996 20:16:12 -0400 From: Dimitrie Popescu Message-Id: <199609200016.UAA23723@ece.rutgers.edu> To: ece501@mogli.rutgers.edu Subject: the error Content-Length: 464 Status: RO With the help of prof. Rose I discovered the error I was making. The mapping that I considered is not bounded. Let's take for example x=k, an integer, and y=k-eps, where eps>0 is very small. While the distance |x-y|=eps and is very small, and goes to 0 as eps goes to 0, the distance |A(x)-A(y)|>|x-y| and goes to 1 as eps goes to 0. So, the mapping is not bounded. The things I've just written are obviously if you look at the plot of this mapping Dimitrie From crose Thu Sep 19 21:00:10 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA07798; Thu, 19 Sep 96 21:00:09 EDT Date: Thu, 19 Sep 96 21:00:09 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609200100.AA07798@MOGLI.rutgers.edu> To: ece501 Subject: where are we going Cc: crose Status: RO Hi Folks, I'd like to open up a discussion of how to run lectures. My preferences is to engage you guys in the proof solving, not assume you've read the book in detail (even if you have) and seek out or at least mention the possible ways proofs can go awry. This makes for critical thinking: doing is learning as opposed to listening/watching is learning. Unfortunately, if many folks are struggling to follow what's going on (I'd say from my vantage point that at least 1/2 the class is pretty uncomfortable) it may be hard to engage in the explorative process. So, here's a question. What would you prefer for this initial math portion of the course? 1) I can stand up and give canned lectures if you need it, or 2) we can try harder to go back and forth with you folks often GUIDING what's going on rather than me simply leading. That did not work tonite since too many folks were completely lost. On a related note, it was really funny when Dirk read from the book directly, (though Dirk, I'm not sure you realized at the time why it was funny). Certainly the (usually correct) proof is in the book (and in my notes). But my simply writing down the proof and commenting as I go does not accomplish the end I have in mind. If everyone were VERY comfortable we could derive as we went and I'd not use the notes at all! So, let me know what you think. I'm a little worried looking into your eyes that if we tested tomorrow we'd be in dire straights. I'd rather not wait until the midterm to find out that we have major problems. Cheers as always, Chris Rose From crose Thu Sep 19 21:06:06 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA07805; Thu, 19 Sep 96 21:06:03 EDT Date: Thu, 19 Sep 96 21:06:03 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609200106.AA07805@MOGLI.rutgers.edu> To: cripop@ece.rutgers.edu, ece501@mogli.rutgers.edu Subject: Re: the error Cc: crose Status: RO Thanks Dimitrie, It's even easier to see (I realized later) if you use x=k and y=k-1+e where e is very close to (but less than) 1. You end up with the distance between x and y being |1-e| but the distance between A(x) and A(y) being e. Therefore, there's no value of constant Q which will have |A(x)-A(y)| = e <= Q|1-e| because |1-e| gets arbitrariliy close to zero wherease e stays near 1. Tricky question, eh? Cheers, and thanks for the formal solution Dimitrie. Chris Rose From crose@mogli.rutgers.edu Fri Sep 20 18:22:55 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA08757; Fri, 20 Sep 96 18:11:55 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA12748; Fri, 20 Sep 96 16:41:29 EDT Date: Fri, 20 Sep 96 16:41:29 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9609202041.AA12748@mogli> To: ece501@MOGLI.rutgers.edu Status: RO I GOT SOME MAIL!!!! I'm reposting the comments anonymously even though both students wanted me to include their names. I just feel it'll be easier for some folks to write if they know I'll keep their identities out of it. STUDENT (1) WRITES: Prof, I'm glad to hear that I am not the only one having problems in your course. Your suggestion about involing the class is a definite plus. One problem I have is that of relating this material to "real" things. I thought you were about to address this problem when you put on the board a circuit diagram and you were about to use what we learned to analize it. I was disappointed when you didn't because you did not have the time to go into it. Now I have an important question. Whats more important, following a schedule and doing everything on it (maybe sacrificing what one understands), or going off on a tangent as you were about to and having the class actually understand something? I don't mean to be a pain in the ass, but I really enjoy a professor going off on tangents to explain real life applications of what we learn. Isn't that the ultimate scope of learning something? Please comment on this matter. STUDENT (2) WRITES: Hi, In response to your email on how to run lectures . . . I really admired the fact that you allowed us to think about solving the proof. Often professors don't have time to allow students to go though the problem solving so they give canned lecture. I think it's more interesting when I'm given the opportunity to figure things out before simply given the answer. Although some learn more by seeing or hearing, I learn more from actually practicing something. I don't quite possess the level of analytical thinking required to solve proofs very well yet. I don't think I can learn that way of thinking unless I practiced it. The most effective professors I've had have not only taught me the subject matter, but have taught me how to think about the subject matter so that I can solve new kinds of problems. I really enjoyed this class so far and I hope the format will not change. Although things may not be obvious to me right away, I am learning to think in a new way. THE PROF IN QUESTION RESPONDS! HI (1) Thanks for your response. I DO like going off on tangents. The problem right now is that we need to get this mechanical stuff out of the way. However, I think I will go off on a few more tangents. The reason I did not, is that my quick scan of the class (excluding your face, though) was to move on and not solve a circuit (though I wanted to). In any case, as we move out of this math and more into systems stuff, there'll be more opportunities for diversion. So, unfortunately for right now we've got to trudge. Maybe I'll just survey the results and move along that way (rather than proving EVERYTHING). HI (2) I'd like to continue challenging you to do the proofs as we go, or at least think about where they come from. I also agree that the doing is important. Homework plays some role there but in-class examples also play an important role. I'll do my best to do more of these as the term wears on. As I said above for the math stuff though, we're almost done with it so I'm going to finish plunging through it so we can get to the more familiar and interesting (from the systems standpoint) stuff. ANOTHER NOTE: I'd particularly urge you (and everyone) to do the last problem on the first problem set carefully. The reason is that the mapping considered there was one of the first seen to be chaotic. The researcher was looking for a contraction and found that sometimes it was and sometimes it was not. And often when it was not, it displayed really strange behavior in that changing the starting point even the tiniest amount (and by tiny I mean ANY nonzero amount) drastically changed the system trajectory. This simply does not happen with most systems. We often rely on the seeming fact that small enough perturbations produce small changes. The analogy is the butterflyl flapping its wings in Peking which eventually causes the birth of a hurricane off the coast of Africa (REALLY SMALL CHANGES IN INITIAL CONDITIONS!). These sorts of problems where you want to know how the system behaves over time are where all the contraction/measure math come in. Well, I've blathered enough for now. Cheers to you both and thanks for the comments! Chris Rose From crose Sun Sep 22 18:28:41 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA10519; Sun, 22 Sep 96 18:26:06 EDT Date: Sun, 22 Sep 96 18:26:06 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609222226.AA10519@MOGLI.rutgers.edu> To: ece501 Subject: another comment Status: RO Hi FOlks, Yet another student responsds! ************************* Prof. Rose, Hello! As some of us have already learned (ece452), you're lectures are paced at a pretty tremendous rate. (content and activity !) The glazed eyes you might see in lecture (at least my own) are a combination of a few things. These include the effects of being exposed to some high-level abstractness. I don't think you should change the format of your lectures much. While I can often follow the material that is presented, it's often difficult to formulate any opinions/questions until I've really delved into it. Reading ahead helps, but that isn't always what happens. For me (the patterns suggest) lectures are an introduction into some of the concepts of a course. I try to follow the direction that the instructor is pointing, and pick up hints on problem solving...changing my way of thinking about a given topic. Not until I've sat down and beat on the concepts (I literally have 3 complete set of notes going into an exam, class notes, text summaries including definitions/theorems/proofs/examples written down (!) and a synopsis of the two previous...{in this case my cheat sheet} ) and completed problems can I feel _somewhat_ comfortable. I takes years to assimilate these ideas. (at least for me) So actually, I'm not a proponent of going off on a lot of tangents. There's alot of interesting topics I need introducing to and a tight schedule. (I've been hearing about "wavelet transforms" for a few years now and would enjoy a 1/2 lecture or so...time permitting of course). Personally, I'll try to read up more. Confusion and fear are great for motivation. Thanks, From MRX@caip.rutgers.edu Wed Sep 25 16:22:05 1996 Return-Path: Received: from caipfs.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA14307; Wed, 25 Sep 96 16:22:04 EDT Received: from telestar.rutgers.edu (telestar.rutgers.edu [128.6.43.4]) by caipfs.rutgers.edu (8.7.6/8.7.3) with ESMTP id QAA11385 for ; Wed, 25 Sep 1996 16:13:23 -0400 (EDT) From: Mr X X Received: (MRX@localhost) by telestar.rutgers.edu (8.6.8.1+bestmx+oldruq+newsunq/8.6.12) id QAA10900 for crose@mogli.rutgers.edu; Wed, 25 Sep 1996 16:13:22 -0400 Date: Wed, 25 Sep 1996 16:13:22 -0400 Message-Id: <199609252013.QAA10900@telestar.rutgers.edu> To: crose@mogli.rutgers.edu X-Sun-Charset: US-ASCII Status: RO In the last lecture we heared a lot about evectors. You can take them, arange as column in a matrix, check if you can find a different eigenvector for each evalue (in the generalization of the diagonalization thm) and so on. But which evector, since for each distinct evalue you have an infinity of evectors associated? In fact all this evectors differ only by a constant, so all the properties could work for no matter evector you pick. They are linearly independent no matter which evect you pick from a class associated with a distinct evalue. Usually you take the evect with norm 1 from each class (so that you can construct an othonormal basis). I make this comment because I found in my notes: "a unique evect for each eval" (generalization of the diagonalization thm), which I think should be " a different evector for each eval". (I am not sure that this is not my error, but the comment is valid anyway). Cheers, George *************prof's note***************** I agree. If I said unique e-vect for each e-val then I mis-spoke ***************************************** From crose Thu Sep 26 21:07:55 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA16331; Thu, 26 Sep 96 20:58:33 EDT Date: Thu, 26 Sep 96 20:58:33 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9609270058.AA16331@MOGLI.rutgers.edu> To: ece501 Subject: extra problem set, due tuesday Status: RO Hi Folks, These problems are worth points if you're on a grade border at the end of it all. Please submit them to me ELECTRONICALLY via email by Saturday 9/28/96 at 11:59PM. 1) You are given a tri-diagonal matrix. Its main diagonal elements are a, its upper diagonal elements are b and the lower diagonal elements are c. b,c = 0.2 and a=0.1 for example: a b 0 0 0 0 0 0 c c a b 0 0 0 0 0 0 0 c a b 0 0 0 0 0 0 0 c a b 0 0 0 0 0 0 0 c a b 0 0 0 = A 0 0 0 0 c a b 0 0 0 0 0 0 0 c a b 0 0 0 0 0 0 0 c a b b 0 0 0 0 0 0 c a You are given the iterative mapping x(n+1) = Ax(n). Is the mapping a contraction? 2) a) A demon(ess) is put in a room with N chairs and seated initially in a single chair. Being demonic it can't just sit in one chair at a time. So with the first tick of the celestial clock, it splits itself into three equal weight demons. One remains where the original demon was seated and the other two move to chairs to the left and right of the first chair. If there's no chair to the left, the left-going demon stays where it was. Likewise for the right-going demon. But demons, even smaller ones, are NEVER happy or satisfied so each smaller demon does the same splitting routine with the next tick of the celestial clock. This process is followed by each "sub-demon" for each tick of the celestial clock for all eternity. If the original demon has weight one unit. Please derive a matrix difference equation for the total weight of demons in each chair. That is, write your equation in terms of an N-vector x(n) where x_i(n) is the weight of demons sitting in chair i at tick n and x(n+1). Identify (if any) the fixed points of this mapping. You can assume x is any real N-vector for this question. Is this mapping a contraction? b) Now the chairs are arranged in a circle and the demon does not split itself but always moves one chair to the right. Write down a difference equation in matrix form for this system. What is the domain space of this mapping? What is the range space of the mapping given the domain space? Identify (if any) the fixed points of this mapping. Is this mapping a contraction? 3) What is A^N for the matrix of problem 2b? 4) What is e^(At) for the matrix of problem 2b with N=2? Happy solving! From ralf@ece.rutgers.edu Mon Oct 7 16:15:11 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11214; Mon, 7 Oct 96 16:00:41 EDT Received: (from ralf@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id PAA22712 for ece501@mogli; Mon, 7 Oct 1996 15:48:47 -0400 Date: Mon, 7 Oct 1996 15:48:47 -0400 From: Ricardo Losada Message-Id: <199610071948.PAA22712@ece.rutgers.edu> To: ece501@mogli.rutgers.edu Content-Length: 185 Status: RO Hey, this is the TA, for Jordan form you may also want to check out the appendix of the book Modern Signals and Systems by Kwakernaak and Sivan. It's the simplest I've seen. Ricardo. From ANALUCI@aol.com Tue Oct 8 11:57:07 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA12121; Tue, 8 Oct 96 11:57:06 EDT Received: from emout16.mail.aol.com (emout16.mx.aol.com [198.81.11.42]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id LAA04750 for ; Tue, 8 Oct 1996 11:45:02 -0400 From: ANALUCI@aol.com Received: by emout16.mail.aol.com (8.6.12/8.6.12) id LAA22959 for crose@ece.rutgers.edu; Tue, 8 Oct 1996 11:45:42 -0400 Date: Tue, 8 Oct 1996 11:45:42 -0400 Message-Id: <961008114542_205042671@emout16.mail.aol.com> To: crose@ece.rutgers.edu Subject: Solution Status: RO Prof. Rose, In PS2 Solution, Problem 3, we have: For A => D = [-2,4] For A' => D = [-3,5] So, if we want to combine the two we obtain lambda in [-2,4] and NOT [-3,4] (as it is in the solution). Am I right? Ana Lucia. From crose Tue Oct 8 14:13:30 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA12352; Tue, 8 Oct 96 14:06:11 EDT Date: Tue, 8 Oct 96 14:06:11 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610081806.AA12352@MOGLI.rutgers.edu> To: ANALUCI@aol.com Subject: Re: Solution Cc: ece501 Status: RO Yup, You're correct. It should be the intersection of the two regions. Will post the correction to the web.. Cheers, Chris Rose From crose Thu Oct 10 20:41:46 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA15633; Thu, 10 Oct 96 20:36:44 EDT Date: Thu, 10 Oct 96 20:36:44 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610110036.AA15633@MOGLI.rutgers.edu> To: ece501 Subject: extra points (for real!) Status: RO Hi Folks, Here's a problem. The first 10 solutions I receive get 4 extra points on their midterm exam. The next 10 get 3, the next 10 get 2 and the remaining get 1. If you don't hand it in you get ZERO! :) Also, no partial credit. You either get it all correct or all wrong! :) As for class... Unless I get strong opposition, I think I'm going to do a little more lecturing and less deriving with you. Otherwise, we'll get seriously behind... so much to do and so little time... PROBLEM PLEASE PROVIDE A PARTIAL FRACTIONS EXPANSION (IF POSSIBLE) FOR THE FOLLOWING LAPLACE TRANSFORMS AND THE ASSOCIATED TIME DOMAIN SIGNAL (assumed causal). If you can't find a partial fraction expansion, you must state why. Regardless, you must still provide a time signal 1) (s^2 + 3s - 1)/((s+2)(s+1)(s+3)(s+4)) 2) 1/(s^2+2s+1) 3) s^2/(s^2+3s+2) Please post your answers TO ME (not to the group). I will award credit on the spot! (ALL OR NONE THOUGH!) Cheers, Chris Rose From crose Thu Oct 10 20:37:53 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA15642; Thu, 10 Oct 96 20:37:53 EDT Date: Thu, 10 Oct 96 20:37:53 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610110037.AA15642@MOGLI.rutgers.edu> To: ece501 Subject: also Cc: crose@MOGLI.rutgers.edu Status: RO Cutoff date is Tuesday 10/15 (for submittnig the problems). You can also submit via hardcopy/fax if you wish (908) 445-2820 Cheers, CHris Rose From crose Thu Oct 10 21:10:36 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA15743; Thu, 10 Oct 96 21:05:13 EDT Date: Thu, 10 Oct 96 21:05:13 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610110105.AA15743@MOGLI.rutgers.edu> To: ece501 Subject: java wanted Cc: winstaff@winlab Status: RO WANTED: someone who knows how to write JAVA scripts. I want to animate a couple of lecture points with dancing equations but don't know how. Cheers, Chris Rose From crose@MOGLI Thu Oct 10 21:13:32 1996 Return-Path: Received: from liman (liman.rutgers.edu) by MOGLI.rutgers.edu (4.1/25-eef) id AA15780; Thu, 10 Oct 96 21:13:31 EDT Full-Name: Christopher Rose Received: from MOGLI.rutgers.edu by liman (SMI-8.6/SMI-SVR4) id UAA00768; Thu, 10 Oct 1996 20:53:55 -0400 Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA15743; Thu, 10 Oct 96 21:05:13 EDT Date: Thu, 10 Oct 96 21:05:13 EDT From: Christopher Rose Message-Id: <9610110105.AA15743@MOGLI.rutgers.edu> To: ece501@MOGLI Subject: java wanted Cc: winstaff@winlab.rutgers.edu Status: RO WANTED: someone who knows how to write JAVA scripts. I want to animate a couple of lecture points with dancing equations but don't know how. Cheers, Chris Rose From crose@mogli.rutgers.edu Wed Oct 16 13:42:40 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA22347; Wed, 16 Oct 96 13:42:39 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA05454; Wed, 16 Oct 96 13:28:32 EDT Date: Wed, 16 Oct 96 13:28:32 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610161728.AA05454@mogli> To: ece501@MOGLI.rutgers.edu Subject: question Cc: crose@MOGLI.rutgers.edu Status: RO Message 12: >From ANALUCI@aol.com Wed Oct 16 11:09:38 1996 From: ANALUCI@aol.com Date: Wed, 16 Oct 1996 10:56:51 -0400 To: crose@ece.rutgers.edu Subject: Question. Hello Prof Rose. In PS3, Problem 2, we have to find the fundamental matrix for a differential equation. An initial condition x(1) = ( 1,1 ) is given. What I understand is that the fundamental matrix would be the same, no matter which is the initial condition.This matrix is constructed with initial conditions equal to the basis vectors (0,...0,1,0,...,0). AM I RIGHT ? Thank you. Ana. To: ANALUCI@aol.com Subject: Re: Question. It's a problem of nomenclature. The book uses the term "fundamental matrix" for what is usually only called the transition matrix. Basically, ignore and translate fundamental to transition wherever you see it in the book. HOWEVER, in the literature, the two terms mean different things. Cheers, Chris Rose PS: I'm not really answering your question about what a fundamental matrix actually is because we're not going to use it in this course Of course, you might check other linear systems books for a definition (Kailath for example, or Chen or just about any other that treats statespace should have it). From alap@liman.Rutgers.EDU Wed Oct 16 13:57:38 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA22380; Wed, 16 Oct 96 13:57:38 EDT Received: from liman (liman.rutgers.edu [128.6.110.6]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id NAA25865 for ; Wed, 16 Oct 1996 13:44:02 -0400 Received: from random.rutgers.edu by liman (SMI-8.6/SMI-SVR4) id NAA18740; Wed, 16 Oct 1996 13:43:20 -0400 Received: by random.rutgers.edu (SMI-8.6/SMI-SVR4) id NAA23267; Wed, 16 Oct 1996 13:29:07 -0400 Date: Wed, 16 Oct 1996 13:29:07 -0400 From: alap@liman.Rutgers.EDU (Ana Lucia Pinheiro) Message-Id: <199610161729.NAA23267@random.rutgers.edu> Subject: again To: crose@ece.rutgers.edu Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Content-Md5: ILaKStgF3c7I2EJp0fN2ZA== Status: RO Prof Rose, I guess you did not understand my question. The question is not "what is" a fundamental matrix or a transition matrix, but "how" to calculate it. What I mean is that it doesn't matter the initial condition that is given in this problem (in this case x(1) = (1,1) ). If the initial condition was diferent, the transition (or fundamental) matrix would still be the same.(?) The transition matrix does not depend on the initial condition of a problem.(?) Am I right ?? Thanks Ana. ------------- Begin Forwarded Message ------------- >From crose@mogli Wed Oct 16 13:29:30 1996 Date: Wed, 16 Oct 96 13:28:32 EDT From: crose@mogli (Christopher Rose) To: ece501@MOGLI Subject: question Cc: crose@MOGLI Message 12: >From ANALUCI@aol.com Wed Oct 16 11:09:38 1996 From: ANALUCI@aol.com Date: Wed, 16 Oct 1996 10:56:51 -0400 To: crose@ece.rutgers.edu Subject: Question. Hello Prof Rose. In PS3, Problem 2, we have to find the fundamental matrix for a differential equation. An initial condition x(1) = ( 1,1 ) is given. What I understand is that the fundamental matrix would be the same, no matter which is the initial condition.This matrix is constructed with initial conditions equal to the basis vectors (0,...0,1,0,...,0). AM I RIGHT ? Thank you. Ana. To: ANALUCI@aol.com Subject: Re: Question. It's a problem of nomenclature. The book uses the term "fundamental matrix" for what is usually only called the transition matrix. Basically, ignore and translate fundamental to transition wherever you see it in the book. HOWEVER, in the literature, the two terms mean different things. Cheers, Chris Rose PS: I'm not really answering your question about what a fundamental matrix actually is because we're not going to use it in this course Of course, you might check other linear systems books for a definition (Kailath for example, or Chen or just about any other that treats statespace should have it). ------------- End Forwarded Message ------------- From crose Wed Oct 16 15:04:16 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA22454; Wed, 16 Oct 96 15:04:13 EDT Date: Wed, 16 Oct 96 15:04:13 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610161904.AA22454@MOGLI.rutgers.edu> To: alap@liman.Rutgers.EDU Subject: Re: again Cc: crose@MOGLI.rutgers.edu Status: RO Hi Ana, The transition matrix does depend on the initial time but not on the initial state. The issue in that problem was that time starts from t=1. For example, does the differential equation in that problem have a unique solution everywhere? Cheers, Chris Rose From crose Sat Oct 19 12:13:56 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA01741; Sat, 19 Oct 96 12:07:23 EDT Date: Sat, 19 Oct 96 12:07:23 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610191607.AA01741@MOGLI.rutgers.edu> To: ece501 Status: RO Hi Folks, Here are a pair of colloquies which I think illustrate my philosophy about the course and answers a couple of questions (which might be of interest before and exam :) about norms and fixed points versus limit points. QUESTION 1: On Fri, 11 Oct 1996, Christopher Rose wrote: > > Hi Folks, > > Your solutions are identical, down to the typesetting. > > Care to explain? > > For future reference, I'd prefer you not collaborate on problems I > post, unless specifically asked to do so. That way, nobody can kid > themselves about what they know. > > A group is always smarter than an individual and working in groups is > great. But unfortunately, when the exam comes, you'll have to stand > on your own. > > For this exercise, however, since it looks from my vantage point that > you innocently collaborated, please let me know who did what so I can > assign point fractions appropriately (assuming the solutions are completely > correct). > > > Cheers, > > Chris Rose > Prefessor Rose, We three guys did your problems seperately. You can trust me. I think these problems are very simple. But we actually verified our results together to make it perfect. I am unaware of its improperty. Just have a feeling of being underestimated. You can make your decision whether assign any point to me. But I should say I made them out. I am confused why I haven't received any answer to my questions? Just because I posted them to all of the students last time for your extra problems? Maybe you didn't see it or consider them stupid? RESPONSE 1: Dear Student, I'm not exactly sure about the tone of your letter. Are you upset for some reason? If so, I'm not sure why. In any case, the receipt of identical solutions, including the way they were typed, from three separate people seems to indicate that they were prepared together. Since I would have preferred for students to prepare them separately, including checking the answers, I voiced my concern to you. This is not a major issue. I'm not accusing you of cheating or any other impropriety. However, if you did collaborate I can't simply assign full credit, even if the answers are completely correct. That would be unfair to those who did not collaborate. There is no "estimation" of your ability involved. As for your not receiving comments to questions, I'm not sure to which ones you refer. The TA has just completed grading the previous extra problem submission and will be returning grades/comments shortly. For this most recent extra set dealing with Laplace transforms, grades will be returned probably by Thursday next since I have to allow everyone time to submit their solutions (tuesday is the deadline). And finally, even if your posted solutions were completely incorrect, only a poor instructor would consider them "stupid". My goal is to make sure you understand the material to the best of your ability. Supposedly "stupid" solutions offer a rare window on sources of confusion for not just the student who prepared them but for others as well. In short, there are no "stupid" solutions or questions. Cheers, Chris Rose ********************************************* QUESTION 2: Dear Prof., You said in last class that the fixed pt. is different from the limit pt. I'm confusing about it because, according to the definition, fixed pt. is a state which will not change in the future time. May I think like this"when time goes infinite, the fixed point is just the limit point". Could you give me an example in which a fixed point is not a limit point? I think the fixed point is the same as the limit point. Your answers to the problem set #1( problem 7 and 8)made these two concepts same too. In problem 1 of last midterm exam, why you chose //A//2 as the reference to decide if the mapping is contraction? Can I choose //A//1 or //A//infinit? In two dimension matrix, is //A//2 the minimum norm of all? Please give me prompt answer! RESPONSE 2: Dear Student, Please consider a pendulum using a rigid rod. Identify the fixed pointS and identify the limit point. That will answer your question. As for the choice of norm. You choose the one which gives you the answer and if that happens to be the L2 or Linf or L1 then you use whichever. The problem to which you refer (actually problem 3 I believe) is rather subtle and involved as well. The gist is that there are a number of values of the matrix components (alpha and beta) for which the mapping is a contraction (by considering the various norms). Because of its subtlety, it probably should not have been on the midterm. Please drop by on Monday to discuss if you like. I will be in from 12pm to 2pm. Cheers, Chris Rose PS: You might want take care with English idioms when sending email. "Please give me a prompt answer!" is a little rude since it could be interpreted as a command as opposed to the plea for help which it was. Since this is the second letter you've sent which skirted on the edge of rudeness, I've decided to alert you so that future misunderstandings can be avoided either with me or with others who might take umbrage more easily. From crose Sat Oct 19 12:13:57 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA01751; Sat, 19 Oct 96 12:10:37 EDT Date: Sat, 19 Oct 96 12:10:37 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610191610.AA01751@MOGLI.rutgers.edu> To: ece501 Subject: office hours!!!!! Status: RO I'll be in on monday from 12pm to 2pm. Come visit me! Cheers, Chris Rose From crose Sat Oct 19 12:32:32 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA01777; Sat, 19 Oct 96 12:32:32 EDT Date: Sat, 19 Oct 96 12:32:32 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610191632.AA01777@MOGLI.rutgers.edu> To: ece501 Subject: problem 1.17 Cc: crose Status: RO A student writes: Hi Prof. Rose, I am still having trouble getting the answer to problem 17 from S&B. I know that I must be missing something obvious because it looks like a straight forward plug and chug problem. (This is he Jordan form problem) For this problem I show lambda=2. >From example 1.19 the solution should be: f(2) (1/1!)f'(2) 0 f(2) where f(2) = e^(2t). Feel free to repost this to the group, but I have a feeling that its proably obvious to everyone else. Regards, Allen ********************* Hi Allen, The f() in question is the original f(At) = e^(At). Thus, with f(x), df/dx = e^x. In fact, all the derivatives are e^x. So actually, you have f(2t) and f'(2t) = f(2t) because you're differentiating with repect to the WHOLE ARGUMENT as opposed to the t inside. Thanks for the question. It is a SURPRISINGLY subtle point! And I'm sure you've saved hours of frightened speculation by other students by having the courage to ask about it. Cheers, Chris Rose PS: EVERYBODY KEEP THE EMAIL COMING. I'll BE CHECKING IT EVERY FEW HOURS FROM HOME. From mae@ece.rutgers.edu Sat Oct 19 13:30:26 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA01865; Sat, 19 Oct 96 13:30:25 EDT Received: (from mae@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id NAA24066 for crose@ece; Sat, 19 Oct 1996 13:15:35 -0400 Date: Sat, 19 Oct 1996 13:15:35 -0400 From: Hideki Mae Message-Id: <199610191715.NAA24066@ece.rutgers.edu> To: crose@ece.rutgers.edu Subject: closed ball Content-Length: 494 Status: RO Professor I'm not sure if I understood the soln to ch1 prob 4 correctly. By setting the Metric space M as B2 the only parts that are of any significance is a circle with center (0,0) and radius 0.5? The B1 has a bigger radius and contains the smaller circle B2. Since, M was already set to equal the B2, the only part that we are concerned with is just the region encompassed by B2 ( or M in this case). Is my reasoning correct? Is this why even though B1 is bigger, B1 is a subset of B2? From crose Sat Oct 19 15:06:39 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA01943; Sat, 19 Oct 96 15:01:14 EDT Date: Sat, 19 Oct 96 15:01:14 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610191901.AA01943@MOGLI.rutgers.edu> To: ece501 Subject: fixed points and limit points Status: RO A student writes: Professor I have a question about fixed point and limit point. For prob8 of ch.1 since A(x) = x , we can start with any number (i.e. 1, 0.5, etc.) and the iteration will return the same values. Since you can put any numbers than there is an infinite number of fixed points. The fixed points will produce iteration results that will always be the same number and hence have an infinite number of limits as well. Is this reasoning correct? ALso, a pendulum has limit point at the bottom and a fixed pt at where? IS there an example of a function that has a limit pt that is different from a fixed pt? ******************** Excellent question! Think about that pendulum. If you balance it EXACTLY at the top, there're no forces acting on the mass which means that it will go NOWHERE! However, that's not a limit point since any slight perturbation from that point will never result in the mass resting there. However, with the mass at the bottom, there's also no forces on the mass. The difference is that the mass will eventually slow down (I've been assuming friction forces) and stop at the bottom. The bottom point IS A LIMIT POINT AND A FIXED POINT whereas the top point is only a FIXED POINT of the mapping associated with the motion of the mass. Isn't that NEATO!!!!!???????? :) Now as for the identity mapping x = A(x) for all x in the domain of the mapping, it depends upon how you define limit point. We never formally defined a limit point. We only said vaguely, "it's where the system ends up after a long time... i.e., where the system settles" Well, here's a more formal definition. A limit point is a point with which is associated a nonzero measure "neighborhood". Starting the system at any point in the neighborhood will result in the system "settling" to the limit point after some sufficiently long (possibly infinite) period of time. So, the mapping x=A(x) has an infinity of FIXED POINTS but not a single limit point since if you change your starting point by even a little bit, you dont' approach the previous fixed point. That is there is no nonzero neighborhood of "attraction" about any of the fixed points. CHeers, Chris Rose From crose Sat Oct 19 15:06:42 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA01954; Sat, 19 Oct 96 15:06:41 EDT Date: Sat, 19 Oct 96 15:06:41 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610191906.AA01954@MOGLI.rutgers.edu> To: ece501 Subject: xdot = A(t)x Cc: crose Status: RO I seem to be having a problem in constructing the transition/fundamental matrix given a problem in the form dx/dt = A(t)x The problem arises when A(t) is not a constant and hence the transition matrix is not of the form e^At or A^t (for difference equations). I have looked at the solutions provided by you to problems 2.2 (Chapter 2, Question 2) and 3.1 (Chapter 3 , Question 1). The transition matrix makes sense when one sees the answer but I dont know how to go about constructing them. Also the problem 3.1 does not state the initial conditions. ******************* It'll help a lot if you take a peek at the solutions to a similar problem on last year's midterm (problem 2 I think). However, in general you're not always able to find the transition matrix. For the problems in the book, the basic idea is to just write out the differential equations implied by the matrix notation and solve them when possible. The example in the book where xdot = A(t)x (example 2.5 on page 55) just try writing out the equations for x1dot and x2dot. I think you'll see how to approach the problem once you do that. Cheers, Chris Rose From crose Sat Oct 19 15:19:01 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA01964; Sat, 19 Oct 96 15:15:43 EDT Date: Sat, 19 Oct 96 15:15:43 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610191915.AA01964@MOGLI.rutgers.edu> To: mae@ece.rutgers.edu Subject: Re: closed ball Cc: ece501 Status: RO Hi again! Yup, you got it! The purpose of that question was to illustrate that you can't just consider the metric (distance measure). You've got to look at the space as well. So to reiterate, consider a unit square with lower left corner at (0,0). This square is our metric space. Now consider a closed ball centered at (0,0) of radius 1. The ball is not a "round ball" since only the points which are IN the square are contained in B1. This does not cover the whole square. How consider a closed ball with center at (0.5,0.5) and radius sqrt(2)/2. This closed ball has radius <1 but includes the whole metric space and therefore B1 as well. So, some of our intuitive notions of inclusion and distance can be misleading when talking about non-infinite spaces! NASTY ISN'T IT!?!?!?! Once again, a distance measure does not a metric space make. YOu always have to consider the SPACE as well. Cheers, Chris Rose From ANALUCI@aol.com Sat Oct 19 16:03:51 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA02084; Sat, 19 Oct 96 16:03:50 EDT Received: from emout08.mail.aol.com (emout08.mx.aol.com [198.81.11.23]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id PAA26026 for ; Sat, 19 Oct 1996 15:48:47 -0400 From: ANALUCI@aol.com Received: by emout08.mail.aol.com (8.6.12/8.6.12) id PAA00914 for crose@ece.rutgers.edu; Sat, 19 Oct 1996 15:49:25 -0400 Date: Sat, 19 Oct 1996 15:49:25 -0400 Message-Id: <961019154924_1112510662@emout08.mail.aol.com> To: crose@ece.rutgers.edu Subject: Problem 2.18 Status: RO Hello Prof Rose, I am still thinking about problem 18.. We have xdot = A(t)x + f(t). x (to) = Xo We generate a soluiton for all "t" using equation (2.23) then we have x(t) as a function of t and to. x(t) = fctn (t,to) (*) That's OK !!! ------------------ Now, we want to find the initial condition Xo, at t=to so that the trajectory passes thru X(1) = (1,1) at t=1. So you just considered (to = 1) and (t = 0) in (*) x(t) = fcnt (t,to) and calculated x(0). You went "backwards".... As I understood we can do that because we are talking about continuos time and the fundamental matrix is always invertible. Am I right ? What I want to know is: if you start in Xo => you will get to X(1) if you start in X(1) you will get to Xo, or better, you had been in Xo before. (???) I am really confused about that (??) !!!!! Please, give me a help!!!! Thanks. Ana. From crose Sat Oct 19 16:30:54 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA02130; Sat, 19 Oct 96 16:30:53 EDT Date: Sat, 19 Oct 96 16:30:53 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610192030.AA02130@MOGLI.rutgers.edu> To: ece501 Subject: question Cc: crose@MOGLI.rutgers.edu Status: RO A student writes: *********** Hello Prof Rose, I am still thinking about problem 18.. We have xdot = A(t)x + f(t). x (to) = Xo We generate a soluiton for all "t" using equation (2.23) then we have x(t) as a function of t and to. x(t) = fctn (t,to) (*) That's OK !!! ------------------ Now, we want to find the initial condition Xo, at t=to so that the trajectory passes thru X(1) = (1,1) at t=1. So you just considered (to = 1) and (t = 0) in (*) x(t) = fcnt (t,to) and calculated x(0). You went "backwards".... As I understood we can do that because we are talking about continuos time and the fundamental matrix is always invertible. Am I right ? What I want to know is: if you start in Xo => you will get to X(1) if you start in X(1) you will get to Xo, or better, you had been in Xo before. (???) I am really confused about that (??) !!!!! Please, give me a help!!!! Thanks. Ana. ************** Yup, that's the ticket. However, you always have to be careful to make sure that the solution exists for the time at which you're trying to find it. For example, the differential equation xdot = 1/t x does not have a solution for t=0. So being given the value at t=1 and being told to find x(t=0) is nonsense. So the succinct answer is. The transition matrix for continuous time is ALWAYS INVERTIBLE BY DEFINITION. So anywhere the transition matrix exists (for start and stop times t and tau, phi(t,tau)) you can reverse the state. FOR DISCRETE TIME, however, the transition matrix IS NOT ALWAYS INVERTIBLE!!!!. Cheers and happy studying to all! Chris Rose From cripop@ece.rutgers.edu Sat Oct 19 18:17:49 1996 Return-Path: Received: from zen.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA02260; Sat, 19 Oct 96 18:12:03 EDT Received: from ece.rutgers.edu (ee.rutgers.edu [128.6.46.13]) by zen.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with SMTP id RAA21796 for ; Sat, 19 Oct 1996 17:50:59 -0400 From: Dimitrie Popescu Message-Id: <199610192150.RAA21796@zen.rutgers.edu> Subject: correction To: ece501@mogli.rutgers.edu Date: Sat, 19 Oct 1996 17:58:25 -0400 (EDT) Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-Length: 1587 Status: RO I would like to make a little correction to the formula of the fundamental matrix for solving difference equation systems. In the lecture from October 15, dr. Rose obtained: x(t_0+tau)= Phi (t_0+tau,t_0)x_0 + + sum from {k=t_0} to {t+tau-1} Phi(t_0+tau-1,k+1)f(k) which is not the same as in the book (rel 2.44/pg. 72; w/ t_0=0) By redoing the calculations we obtain: x(t_0+1) = A(t_0)x_0 + f(t_0) x(t_0+2) = A(t_0+1)A(t_0)x_0 + A(t_0+1)f(t_0) x(t_0+3) = A(t_0+2)A(t_0+1)A(t_0)x_0 + A(t_0+2)A(t_0+1)f(t_0) + + A(t_0+2)f(t_0+1) + f(t_0+2) If we denote: Phi(t_0+tau,t_0) = prod from {k=0} to {t_0+tau-1} A(t_0+k) the transition matrix from x(t_0) to x(t_0+tau), then we have, for what we wrote above (tau=3) A(t_0+2)A(t_0+1)A(t_0) = Phi(t_0+3,t_0) A(t_0+2)A(t_0+1) = Phi(t_0+3,t_0+1) A(t_0+2) = Phi(t_0+3,t_0+2) so, the general formula can be written as: x(t_0+tau)= Phi (t_0+tau,t_0)x_0 + + sum from {k=0} to {tau-1} Phi(t_0+tau,t_0+k+1)f(t_0+k) One can easily verify that for t_0=0 the formula is the same as 2.44 in the book (pag.72). Also, to have a better resemblance with the continuous-time case, we can make the substitution t_0+tau=t (the actual time instant for which we are calculating the state; since we are in the discrete case, t can only take natural values), and t_0+k=theta and we can write the formula as: x(t)= Phi (t,t_0)x_0 + sum from {theta=t_0} to {t-t_0-1} Phi(t,theta+1)f(theta) w/ Phi(t,t_0) = prod from {theta=t_0} to {t-1} A(theta) Note: theta is a dummy variable for summation, so now it can be exchanged with anything else. From cripop@ece.rutgers.edu Sat Oct 19 19:08:04 1996 Return-Path: Received: from zen.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA02301; Sat, 19 Oct 96 19:02:39 EDT Received: from ece.rutgers.edu (ee.rutgers.edu [128.6.46.13]) by zen.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with SMTP id SAA21820 for ; Sat, 19 Oct 1996 18:41:35 -0400 From: Dimitrie Popescu Message-Id: <199610192241.SAA21820@zen.rutgers.edu> To: ece501@mogli.rutgers.edu Date: Sat, 19 Oct 1996 18:49:01 -0400 (EDT) Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-Length: 2803 Status: RO In the last formula of my message, there is a small typing error, it should be sum from {theta=t_0} to {t-1} not {t-t_0-1}. So, the correct form is: x(t)= Phi (t,t_0)x_0 + sum from {theta=t_0} to {t-1} Phi(t,theta+1)f(theta) My apologizes. ===============================================================================: >From <@MOGLI.rutgers.edu:cripop@ece.rutgers.edu> Sat Oct 19 17:58:46 1996 Received: from MOGLI.rutgers.edu (mogli.rutgers.edu [128.6.46.55]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with SMTP id RAA27714; Sat, 19 Oct 1996 17:58:14 -0400 Received: from zen.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA02260; Sat, 19 Oct 96 18:12:03 EDT Received: from ece.rutgers.edu (ee.rutgers.edu [128.6.46.13]) by zen.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with SMTP id RAA21796 for ; Sat, 19 Oct 1996 17:50:59 -0400 From: Dimitrie Popescu Message-Id: <199610192150.RAA21796@zen.rutgers.edu> Subject: correction To: ece501@MOGLI.rutgers.edu Date: Sat, 19 Oct 1996 17:58:25 -0400 (EDT) Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-Length: 1587 Status: RO I would like to make a little correction to the formula of the fundamental matrix for solving difference equation systems. In the lecture from October 15, dr. Rose obtained: x(t_0+tau)= Phi (t_0+tau,t_0)x_0 + + sum from {k=t_0} to {t+tau-1} Phi(t_0+tau-1,k+1)f(k) which is not the same as in the book (rel 2.44/pg. 72; w/ t_0=0) By redoing the calculations we obtain: x(t_0+1) = A(t_0)x_0 + f(t_0) x(t_0+2) = A(t_0+1)A(t_0)x_0 + A(t_0+1)f(t_0) x(t_0+3) = A(t_0+2)A(t_0+1)A(t_0)x_0 + A(t_0+2)A(t_0+1)f(t_0) + + A(t_0+2)f(t_0+1) + f(t_0+2) If we denote: Phi(t_0+tau,t_0) = prod from {k=0} to {t_0+tau-1} A(t_0+k) the transition matrix from x(t_0) to x(t_0+tau), then we have, for what we wrote above (tau=3) A(t_0+2)A(t_0+1)A(t_0) = Phi(t_0+3,t_0) A(t_0+2)A(t_0+1) = Phi(t_0+3,t_0+1) A(t_0+2) = Phi(t_0+3,t_0+2) so, the general formula can be written as: x(t_0+tau)= Phi (t_0+tau,t_0)x_0 + + sum from {k=0} to {tau-1} Phi(t_0+tau,t_0+k+1)f(t_0+k) One can easily verify that for t_0=0 the formula is the same as 2.44 in the book (pag.72). Also, to have a better resemblance with the continuous-time case, we can make the substitution t_0+tau=t (the actual time instant for which we are calculating the state; since we are in the discrete case, t can only take natural values), and t_0+k=theta and we can write the formula as: x(t)= Phi (t,t_0)x_0 + sum from {theta=t_0} to {t-t_0-1} Phi(t,theta+1)f(theta) w/ Phi(t,t_0) = prod from {theta=t_0} to {t-1} A(theta) Note: theta is a dummy variable for summation, so now it can be exchanged with anything else. From crose@mogli.rutgers.edu Sat Oct 19 20:11:56 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA02340; Sat, 19 Oct 96 20:11:55 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA08220; Sat, 19 Oct 96 19:07:06 EDT Date: Sat, 19 Oct 96 19:07:06 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610192307.AA08220@mogli> To: mae@ece.rutgers.edu Subject: Re: p.34 Cc: crose@MOGLI.rutgers.edu, ece501@MOGLI.rutgers.edu Status: RO > I have a question about the expansion of f(A). Is equation 1.39 a form that > can be applied to any matrix A? Is it some kind of Taylor polynomial > approximantion? IS there anything special about this way of representing > f(A)? Hi, Yup, there's something special about equation 1.39. The form only works for the decomposition A = lambda I + N introduced at the top of the page (page 34). The decomposition allows you to see that exp(J) where J is a jordan block, takes a particular form (the one with f, f' f'' etc shown almost at the bottom o the page. So, yes, equation 1.39 is derived from the taylor expansion for f() BUT the specific form it reduces to is for A a jordan form block. Hope that helps... Cheers, Chris Rose From crose@mogli.rutgers.edu Sat Oct 19 20:12:34 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA02346; Sat, 19 Oct 96 20:11:57 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA08239; Sat, 19 Oct 96 19:27:07 EDT Date: Sat, 19 Oct 96 19:27:07 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610192327.AA08239@mogli> To: cripop@ece.rutgers.edu Subject: Re: correction Cc: crose@MOGLI.rutgers.edu, ece501@MOGLI.rutgers.edu Status: RO HI Dimitrie, Yup, that -1 in the first argument of the transition matrix inside the sum should not be there. Not sure how that crept in (whether it was a logical or a typographical error). In any case, an easy way to see that the -1 should not be there is to note that when k = t_0 + tau -1 we should have an identity matrix multiplying f(t_0+tau-1). That means that the transition matrix should have the same value in both arguments. IT DOES NOT if the -1 is in there. That's a dead giveaway. This basic technique (check extremal cases) is a good thing to use in your work (or on tests) as a sanity check. One further note: I'm assuming the t in the upper limit in your transcription of my board notes was a typo too. You have t + tau -1 and it should be t_0+tau-1. Board typo or your typo? Thanks for noting the difference between the board notes and the book and offering a correction, Dimitrie! Cheers, Chris Rose From mae@ece.rutgers.edu Sun Oct 20 11:20:16 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA02943; Sun, 20 Oct 96 11:20:15 EDT Received: (from mae@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id LAA03244 for crose@ece; Sun, 20 Oct 1996 11:05:15 -0400 Date: Sun, 20 Oct 1996 11:05:15 -0400 From: Hideki Mae Message-Id: <199610201505.LAA03244@ece.rutgers.edu> To: crose@ece.rutgers.edu Subject: p.55 Content-Length: 262 Status: R We were trying to figure out ex.2.5 on p.55. HOw did you get the transition mat rix? We tried it couple ways but get answers that are different from the etext book. We tried by solving the differential eqn. HBut, that doesn't seem to give the same matrix. From crose@mogli.rutgers.edu Sun Oct 20 14:11:59 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA03075; Sun, 20 Oct 96 14:11:59 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA08822; Sun, 20 Oct 96 13:04:33 EDT Date: Sun, 20 Oct 96 13:04:33 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610201704.AA08822@mogli> To: mae@ece.rutgers.edu Subject: Re: p.55 Cc: crose@MOGLI.rutgers.edu Status: R HI, Write out the differential equations in long form (i.e., expand the matrix form into a system of two differential equations in x1 and x2. Solve the one in x2 and then the solution for the one in x1 will easy. Then you use the transition matrix construction where the kth column is the solution to the initial condition e_k. Try sending me your calculations if you get stuck. Cheers, Chris Rose From mae@ece.rutgers.edu Sun Oct 20 14:20:57 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA03090; Sun, 20 Oct 96 14:20:57 EDT Received: (from mae@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id OAA04231 for crose@ece; Sun, 20 Oct 1996 14:05:56 -0400 Date: Sun, 20 Oct 1996 14:05:56 -0400 From: Hideki Mae Message-Id: <199610201805.OAA04231@ece.rutgers.edu> To: crose@ece.rutgers.edu Subject: inverse transform Content-Length: 520 Status: R I was trying to do the inverse matrix for p.80. What was the exact formula to use to find inverse matrix. is it: [a b c d] , inverse = 1/(ad-cb)[d c; b a] ? I realize you went over this in class once, but I can't find it in my notes. For p.80 it shows 1/(ad - cb)[d -c; -b a] which is different from what is on p.68 where inverse matrix is of the form : a/(ad -cb)[d c; b a] I remember that I'm suppose to multiply by -1 for certain coord. I can'r what the general formula was. From mae@ece.rutgers.edu Sun Oct 20 14:25:12 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA03096; Sun, 20 Oct 96 14:25:11 EDT Received: (from mae@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id OAA04242 for crose@mogli.rutgers.edu; Sun, 20 Oct 1996 14:10:16 -0400 Date: Sun, 20 Oct 1996 14:10:16 -0400 From: Hideki Mae Message-Id: <199610201810.OAA04242@ece.rutgers.edu> To: crose@mogli.rutgers.edu Subject: Re: p.55 Content-Length: 217 Status: R We got x1dot = tx1 + tx2 x2dot = 2tx2. We got x2 = exp(t^2). However once we plug it back into x1dot = tx1 + t*exp(t^2) we get stuck We tried differnet ways to get the soln but we can't get beyond this point. From crose@mogli.rutgers.edu Sun Oct 20 18:12:43 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA03241; Sun, 20 Oct 96 18:12:43 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA08944; Sun, 20 Oct 96 17:12:44 EDT Date: Sun, 20 Oct 96 17:12:44 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610202112.AA08944@mogli> To: ece501@MOGLI.rutgers.edu Subject: folks Cc: crose@MOGLI.rutgers.edu Status: R Hi Folks, Make sure your ssn (last four digits) appears in the list of grades, even it you did not submit a problem set. Cheers, Chris Rose From crose@mogli.rutgers.edu Sun Oct 20 18:12:47 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA03254; Sun, 20 Oct 96 18:12:47 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA08960; Sun, 20 Oct 96 17:19:31 EDT Date: Sun, 20 Oct 96 17:19:31 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610202119.AA08960@mogli> To: mae@ece.rutgers.edu Subject: Re: inverse transform Cc: crose@MOGLI.rutgers.edu Status: R Hi Hideki, yeah, it's a little painful to do the inverse, especially for larger matrices. For a 2x2 you have |a b| inverse is |c d| | d -b| |-c a| divided by (ad-bc) which is what you had. The basic idea is spelled out in Strang (a linear algebra text). I would not worry too much about this however. Cheers, Chris Rose From crose@mogli.rutgers.edu Sun Oct 20 18:12:49 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA03259; Sun, 20 Oct 96 18:12:48 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA08965; Sun, 20 Oct 96 17:22:18 EDT Date: Sun, 20 Oct 96 17:22:18 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610202122.AA08965@mogli> To: mae@ece.rutgers.edu Subject: Re: p.55 Cc: crose@MOGLI.rutgers.edu Status: R Hi again, First solve the homogeneous form. Then get the particular solution by assuming that x1 is of the form A e^(t^2). That should do the trick. In general, always first try the particular solution in the form of the drive term (in this case, something with a t^2 in the exponential). Sound like magic? Well, it is kind of. A lot of diffeq solving is simply inspired guesswork. Cheers and congratulations for keeping at it! Chris Rose From crose@mogli.rutgers.edu Sun Oct 20 18:16:24 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA03275; Sun, 20 Oct 96 18:12:53 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA08989; Sun, 20 Oct 96 17:32:42 EDT Date: Sun, 20 Oct 96 17:32:42 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610202132.AA08989@mogli> To: ece501@MOGLI.rutgers.edu Subject: new grades Status: R Hi Folks, A social security number error and a missed correct solution are included here. One of you (2674) will be disappointed to know that the missed correct solution bounced you to a 3 from a perfect 4. AAAAARRRRRRRGGGGGHHHHH!!!!! you may yell throatily! :) In any case, here's the corrected grades. Please note any ssn errors (like either your ssn last for digits do not appear at all, or suspiciously similar digits, but not quite the right ones, are on the list) ***************** SSN E1 E2 8311 0 3 1826 2 0 1525 1 2 0754 0 0 4552 0 3 2751 0 0 3022 0 4 2549 0 0 3431 1 0 9441 0 4 6372 1 4 3106 0 0 8346 0 0 4046 0 0 7111 0 0 1154 1 0 6122 3 0 4670 0 3 7058 1 3 4177 1 0 1859 0 0 2036 0 3 5645 1 0 7793 2 0 8674 0 3 5716 2 3 1899 0 0 2674 3 3 1590 1 4 0737 0 3 9700 1 4 1233 0 0 1538 0 2 1455 2 2 9928 0 4 9196 0 0 1420 0 3 4059 0 0 8656 0 0 3003 0 4 2017 0 0 From Carkas@aol.com Sun Oct 20 19:39:36 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA03383; Sun, 20 Oct 96 19:39:36 EDT Received: from emout15.mail.aol.com (emout15.mx.aol.com [198.81.11.41]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id TAA06705 for ; Sun, 20 Oct 1996 19:24:20 -0400 From: Carkas@aol.com Received: by emout15.mail.aol.com (8.6.12/8.6.12) id TAA05347 for crose@ece.rutgers.edu; Sun, 20 Oct 1996 19:25:00 -0400 Date: Sun, 20 Oct 1996 19:25:00 -0400 Message-Id: <961020192459_1313873633@emout15.mail.aol.com> To: crose@ece.rutgers.edu Subject: Questions??? Status: R Dr. Rose and/or fellow classmates: A couple questions with regard to the upcoming exam: 1) In PS3 (Chapter 2, probelm 2) we found the transition matrix of the homogeneous equation to be [ t 0] However, when we then attempt to find the transition matrix [0 t] for the same equation with a drive term added PS4 (Chapter 3, Problem 1), the "t" in the matrix mentioned above becomes "t/tau". I do not see where the scale factor tau comes from. As an aside, when you are attempting to find the solution to an equation that does contain a drive term (constant coefficient in this case), is it valid to simply ignore the drive term initially to find the transition matrix (for the homogeneous case). Then perform the necessary summation or integration including the effect of the drive term. In particular, will the transition matrix be the same in both the homogeneous and non-homogeneous case??? 2) Also, could you please explain how you derived the "A" matrix associated with the second problem of extra problem set 1 (the demoness problem). I have attempted unsuccessfully to go from the word problem given to the differential equation and the associated A matrix. Please explain the thought process here. Thank You for your help. From mae@ece.rutgers.edu Sun Oct 20 21:18:33 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA03424; Sun, 20 Oct 96 21:18:33 EDT Received: (from mae@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id VAA08364 for crose@ece; Sun, 20 Oct 1996 21:03:28 -0400 Date: Sun, 20 Oct 1996 21:03:28 -0400 From: Hideki Mae Message-Id: <199610210103.VAA08364@ece.rutgers.edu> To: crose@ece.rutgers.edu Subject: prob set 4 Content-Length: 118 Status: R For prob 1 of Set 4 is the transition matrix [t/tau 0; 0 t/tau] or is it [t 0; 0 t] which is the soln to ch2 prob.2. From crose@mogli.rutgers.edu Sun Oct 20 21:42:23 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA03476; Sun, 20 Oct 96 21:42:23 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA09138; Sun, 20 Oct 96 21:06:38 EDT Date: Sun, 20 Oct 96 21:06:38 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610210106.AA09138@mogli> To: ece501@MOGLI.rutgers.edu Subject: problem 2.6 Cc: crose@MOGLI.rutgers.edu Status: R Hi FOlks, The solution to problem 2.6 has a sign error. The terms in 0^t should have the opposite sign of what they have. By the time you read this however, the problem will almost certainly have been corrected on the web page. Mark your printed copies appropriately. Thanks is due to Ana for raising the question. Cheers, Chris Rose From crose@mogli.rutgers.edu Sun Oct 20 22:12:02 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA03510; Sun, 20 Oct 96 22:11:59 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA09191; Sun, 20 Oct 96 22:22:42 EDT Date: Sun, 20 Oct 96 22:22:42 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610210222.AA09191@mogli> To: Carkas@aol.com Subject: Re: Questions??? Cc: crose@MOGLI.rutgers.edu, ece501@MOGLI.rutgers.edu Status: R Hi Chris, For 1), you have to construct the transition matrix so that phi(t,t) = I. The t/tau comes out of that. For 2) the basic idea is that whatever the demon "mass" in a given seat, 1/3 of it goes left, 1/3 goes right and 1/3 stays put. This is independent of the number of demons residing on a given seat. For the end seats, 2/3 stays put and 1/3 goes in the direct of the next available seat. The trick, if you can call it that, is to see that the splitting of demons does not depend on the number of demons in a given seat, but only their combined weight. The other part is to see that a demon sitting at time t=0 in a particular seat is simply an initial condition on the mass distribution over the seats. In particular, if the first demon starts out at seat k, then the initial condition is e_k (as defined in the text). Also, some folks did not realize that when a demon splits, 1/3 of its WEIGHT goes in each of the directions (left, stay, right). So we don't get multiplication of demon weight (as would be the case if each demon split inot three demons of the same weight as the original). In short, we use conservation of mass (something even a demon(ess) can't avoid :) Hope that helped... Cheers, Chris Rose From MAILER-DAEMON Sun Oct 20 23:17:11 1996 Return-Path: Received: from mailbox.adm.rl.af.mil by MOGLI.rutgers.edu (4.1/25-eef) id AA03546; Sun, 20 Oct 96 23:17:10 EDT Received: from localhost (localhost) by mailbox.adm.rl.af.mil (8.7.6/8.7.6) with internal id XAA19440; Sun, 20 Oct 1996 23:05:21 -0400 (EDT) Date: Sun, 20 Oct 1996 23:05:21 -0400 (EDT) From: Mail Delivery Subsystem Subject: Warning: could not send message for past 1 hour Message-Id: <199610210305.XAA19440@mailbox.adm.rl.af.mil> To: Auto-Submitted: auto-generated (warning-timeout) Status: R ********************************************** ** THIS IS A WARNING MESSAGE ONLY ** ** YOU DO NOT NEED TO RESEND YOUR MESSAGE ** ********************************************** The original message was received at Sun, 20 Oct 1996 22:01:58 -0400 (EDT) from mogli.rutgers.edu [128.6.46.55] ----- The following addresses have delivery notifications ----- grieco@eeyore.ira.rl.af.mil. (transient failure) (expanded from: ) ----- Transcript of session follows ----- grieco@eeyore.ira.rl.af.mil.... Deferred: Connection timed out during initial connection with eeyore.ira.rl.af.mil. Warning: message still undelivered after 1 hour Will keep trying until message is 4 days old ----- Original message follows ----- Return-Path: crose@mogli.rutgers.edu Received: from MOGLI.rutgers.edu (mogli.rutgers.edu [128.6.46.55]) by mailbox.adm.rl.af.mil (8.7.6/8.7.6) with SMTP id WAA19122 for ; Sun, 20 Oct 1996 22:01:58 -0400 (EDT) Received-Date: Sun, 20 Oct 1996 22:01:58 -0400 (EDT) Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA03510; Sun, 20 Oct 96 22:11:59 EDT Received: by mogli (4.1/SMI-4.1) id AA09191; Sun, 20 Oct 96 22:22:42 EDT Date: Sun, 20 Oct 96 22:22:42 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610210222.AA09191@mogli> To: Carkas@aol.com Subject: Re: Questions??? Cc: crose@mogli.rutgers.edu, ece501@mogli.rutgers.edu Hi Chris, For 1), you have to construct the transition matrix so that phi(t,t) = I. The t/tau comes out of that. For 2) the basic idea is that whatever the demon "mass" in a given seat, 1/3 of it goes left, 1/3 goes right and 1/3 stays put. This is independent of the number of demons residing on a given seat. For the end seats, 2/3 stays put and 1/3 goes in the direct of the next available seat. The trick, if you can call it that, is to see that the splitting of demons does not depend on the number of demons in a given seat, but only their combined weight. The other part is to see that a demon sitting at time t=0 in a particular seat is simply an initial condition on the mass distribution over the seats. In particular, if the first demon starts out at seat k, then the initial condition is e_k (as defined in the text). Also, some folks did not realize that when a demon splits, 1/3 of its WEIGHT goes in each of the directions (left, stay, right). So we don't get multiplication of demon weight (as would be the case if each demon split inot three demons of the same weight as the original). In short, we use conservation of mass (something even a demon(ess) can't avoid :) Hope that helped... Cheers, Chris Rose From crose@mogli.rutgers.edu Mon Oct 21 10:11:55 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA04056; Mon, 21 Oct 96 10:11:54 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA09673; Mon, 21 Oct 96 10:05:10 EDT Date: Mon, 21 Oct 96 10:05:10 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610211405.AA09673@mogli> To: ece501@MOGLI.rutgers.edu Subject: grade corrections Cc: crose@MOGLI.rutgers.edu Status: R Hi Guys, Here's a corrected grade list. One person got credit when they shouldn't have on the second extra problem set. (they missed the delta function in the last problem). Please make sure to find your name and notify me if the last 4 digits in your ssn are either wrong or absent. Cheers, Chris Rose ************* SSN E1 E2 8311 0 3 1826 2 0 1525 1 2 0754 0 0 4552 0 3 2751 0 0 3022 0 4 2549 0 0 3431 1 0 9441 0 4 6372 1 4 3106 0 0 8346 0 0 4046 0 0 7111 0 0 1154 1 0 6122 3 0 4670 0 3 7058 1 3 4177 1 0 1859 0 0 2036 0 3 5645 1 0 7793 2 0 8674 0 3 5716 2 0 1899 0 0 2674 3 3 1590 1 4 0737 0 3 9700 1 4 1233 0 0 1538 0 2 1455 2 2 9928 0 4 9196 0 0 1420 0 3 4059 0 0 8656 0 0 3003 0 4 2017 0 0 From zhuowen@eden.rutgers.edu Mon Oct 21 13:16:09 1996 Return-Path: Received: from er7.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA04299; Mon, 21 Oct 96 13:16:09 EDT Received: (from zhuowen@localhost) by er7.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id NAA03028 for crose@mogli.rutgers.edu (Christopher Rose); Mon, 21 Oct 1996 13:02:59 -0400 Date: Mon, 21 Oct 96 13:02:56 EDT From: Wen Zhuo To: crose@mogli.rutgers.edu (Christopher Rose) Subject: Re: grade corrections In-Reply-To: Your message of Mon, 21 Oct 96 10:05:10 EDT Message-Id: Status: RO Hi, Dr. Rose What's the defination of "Convex set" and "Rank of a matrix". I cannot find it. Thank you. From grieco@eeyore Mon Oct 21 13:38:45 1996 Return-Path: Received: from uu.psi.com by MOGLI.rutgers.edu (4.1/25-eef) id AA04342; Mon, 21 Oct 96 13:38:44 EDT Received: from EEYORE.IRA.RL.AF.MIL by uu.psi.com (5.65b/4.0.061193-PSI/PSINet) via SMTP; id AA01825 for crose@mogli.rutgers.edu; Mon, 21 Oct 96 13:25:35 -0400 Received: by eeyore.ira.rl.af.mil (5.x/SMI-SVR4) id AA03597; Mon, 21 Oct 1996 13:25:33 -0400 Date: Mon, 21 Oct 1996 13:25:33 -0400 From: grieco@eeyore.ira.rl.af.mil (John J Grieco) Message-Id: <9610211725.AA03597@eeyore.ira.rl.af.mil> To: crose@mogli.rutgers.edu Subject: PS 4 #5 (ch3 prob 14) X-Sun-Charset: US-ASCII Status: RO Hi Prof. Rose, I have worked this problem and I don't get the solution as it is posted. Posted solution has: dot(z)1=0 dot(z)2=z1-z2-1 I get : dot(z)1=0 dot(z)2=z1-z2 (but no -1) Is this an error in the posted solution or am I missing something? Thanks, John From yongbai@bugeyes.rutgers.edu Mon Oct 21 16:29:17 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA04560; Mon, 21 Oct 96 16:29:16 EDT Received: from bugeyes.rutgers.edu (bugeyes.rutgers.edu [128.6.46.117]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id QAA20143 for ; Mon, 21 Oct 1996 16:14:44 -0400 Received: by bugeyes.rutgers.edu (940816.SGI.8.6.9/930416.SGI) for crose@ece.rutgers.edu id NAA14937; Mon, 21 Oct 1996 13:16:06 -0700 From: "Yong Bai" Message-Id: <9610211316.ZM14935@bugeyes.rutgers.edu> Date: Mon, 21 Oct 1996 13:16:05 -0700 X-Mailer: Z-Mail (3.2.0 26oct94 MediaMail) To: crose@ece.rutgers.edu Subject: Questions about satellite prolem Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: R Dear Prof. Rose: I read the textbook about the satellite problem, and there are two questions that I can not figure out. Would you help me to sovle them before Tuesday afternoon? 1. On Page 86 top line, why the left side of the equation is ru2 instead of u2? 2. On Page 89, the example 3.5 tells me that there is no equilibrium point for the satellite problem. But Example 3.6 says there IS equilibrium point and linearize the matrix around the equilibrium point. So how do I understand these two examples? Furthermore what is the equilibrium point for this problem? Best regards, Yong Bai -- Yong Bai From crose Mon Oct 21 16:46:09 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA04637; Mon, 21 Oct 96 16:46:08 EDT Date: Mon, 21 Oct 96 16:46:08 EDT From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9610212046.AA04637@MOGLI.rutgers.edu> To: yongbai@bugeyes.rutgers.edu Subject: Re: Questions about satellite prolem Cc: crose, ece501 Status: R HI Yong, This letter has to do with examples 3.5 and 3.6 in the text). Yong wants to know why in 3.5 they say there is not equilibrium point and then in 3.6 they go on and linearize about some funny equilibrium. 1) structural: It's r u_2 as the drive term because it's a torque about the center point. I seem to remember missing this point on the board in class. Please correct it in your notes if I did indeed neglect to put it in. 2) As for examples 3.5 and 3.6, the difference is that in 3.5 no inputs are applied (inputs are zero) and the equations are used in raw form. IN 3.6 new state variables are defined by subtracting functions from the originals. These functions are obtained by solving the differential equations for the zero input case. Thus, we're linearizing about the zero input (equilibrium) TRAJECTORY as opposed to a given state point (equilibrium). Thanks for bringing this up. It's a bit subtle. Cheers, Chris Rose From crose@mogli.rutgers.edu Mon Oct 21 16:53:28 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA04711; Mon, 21 Oct 96 16:53:27 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA09816; Mon, 21 Oct 96 16:30:31 EDT Date: Mon, 21 Oct 96 16:30:31 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610212030.AA09816@mogli> To: ece501@MOGLI.rutgers.edu Subject: problem 3.14 Cc: crose@MOGLI.rutgers.edu Status: R Linearization. The published solution should not have the -1 sitting in the equation for x2dot (problem 3.14). My error both in office hours today and last year on the solutions. The quick way to identify the error is to look at the approximation for z2dot and note that z2dot is not equal to zero at the expansion point (0,0). Since we were expanding about an equilibrium point, that's a no no. Dimitrie, mea culpa. You thought it was wrong but I confused the issue for both of us. MORAL: Stick to your guns until you're satisfied! :) Cheers, Chris Rose From crose@mogli.rutgers.edu Mon Oct 21 16:53:30 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA04717; Mon, 21 Oct 96 16:53:29 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA09829; Mon, 21 Oct 96 16:34:26 EDT Date: Mon, 21 Oct 96 16:34:26 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610212034.AA09829@mogli> To: zhuowen@eden.rutgers.edu Subject: Re: grade corrections Cc: crose@MOGLI.rutgers.edu Status: R Rank of a matrix is the number of independent rows (or columns). A convex set was defined in class (since it wasn't defined in the book) and is defined as: A set is convex if for all elements in the set x and y, z = lambda x + (1-lambda)y is also in the set for 0 <= lambda <= 1. Thus, a simple interval is convex (in R1 for example). Two disjoint intervals making up a single set is not convex. Cheers, Chris Rose From hatangad@ece.rutgers.edu Mon Oct 21 17:54:08 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA04760; Mon, 21 Oct 96 17:54:08 EDT Received: (from hatangad@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id RAA21217 for crose@MOGLI.rutgers.edu; Mon, 21 Oct 1996 17:39:35 -0400 Date: Mon, 21 Oct 1996 17:39:35 -0400 From: Sonya Hatangadi Message-Id: <199610212139.RAA21217@ece.rutgers.edu> To: crose@MOGLI.rutgers.edu Subject: Questions Content-Length: 288 Status: R In the last years midterm solution , Answer 2c, there is a note which says that the transition matrix = exp^(integral A(tau)dtau as long as A(t) and integral A(tau)dtau commute. What does it mean when we say they commute? Thanks! Sonya. From crose@mogli.rutgers.edu Mon Oct 21 20:12:04 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA04881; Mon, 21 Oct 96 20:12:04 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10024; Mon, 21 Oct 96 19:16:12 EDT Date: Mon, 21 Oct 96 19:16:12 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610212316.AA10024@mogli> To: hatangad@ece.rutgers.edu Subject: Re: Questions Cc: crose@MOGLI.rutgers.edu Status: R A and B commute if AB = BA (which is not in general the case). Cheers, Chris Rose From crose@mogli.rutgers.edu Mon Oct 21 20:12:06 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA04887; Mon, 21 Oct 96 20:12:05 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10029; Mon, 21 Oct 96 19:16:41 EDT Date: Mon, 21 Oct 96 19:16:41 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610212316.AA10029@mogli> To: ece501@MOGLI.rutgers.edu Cc: crose@MOGLI.rutgers.edu Status: R question In the last years midterm solution , Answer 2c, there is a note which says that the transition matrix = exp^(integral A(tau)dtau as long as A(t) and integral A(tau)dtau commute. What does it mean when we say they commute? Thanks! *********** A and B commute if AB = BA (which is not in general the case). Cheers, Chris Rose From crose@mogli.rutgers.edu Mon Oct 21 20:12:08 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA04892; Mon, 21 Oct 96 20:12:07 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10034; Mon, 21 Oct 96 19:17:19 EDT Date: Mon, 21 Oct 96 19:17:19 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610212317.AA10034@mogli> To: ece501@MOGLI.rutgers.edu Subject: fixed points again Cc: crose@MOGLI.rutgers.edu Status: R ...... As for fixed points, equilibrium points etc. A fixed point is a point for which and iterative mapping stays put. tThat is, if A(x) is a mapping then A(x*) = x* has x* as a fixed point. This is also called an equilibrium point. A fixed point is not necessarily a limit point since you can have "stable fixed points" which are approached from various starting points and unstable fixed points which are approached from no trajectories. The pendulum on a rigid rod is an example of this. Balanced up top: fixed but not stable (not a limit point). At rest on the bottom: fixed point and a limit point (assuming friction slows the pendulum down). An equilibrium point, fixed point, etc for a continuous time system occurs when xdot = 0. That is, you start the system off in a state from which it never leaves. Hope that helped a little, Cheers, Chris Rose From crose@mogli.rutgers.edu Mon Oct 21 22:11:56 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA04971; Mon, 21 Oct 96 22:11:55 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10144; Mon, 21 Oct 96 22:11:58 EDT Date: Mon, 21 Oct 96 22:11:58 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610220211.AA10144@mogli> To: ece501@MOGLI.rutgers.edu Subject: off the air Cc: crose@MOGLI.rutgers.edu Status: R Hi Folks, Though it may not seem like a service, I'm going off the air at 2am this evening (tonight). What I've found is that questions asked the day of the exam usually only serve to confuse issues and generate even more anxiety. So ask until you drop, but not after 2am tonight. I'll answer the last round at 9am tomorrow (approximate time), but then no more questions will be answered afterward. HOWEVER, there is absolutely NO reason you can't still post to the group and await an answer from a fellow student. In fact, in general I'd love to just preside over the "501 newsgroup" and act as a semi-moderator. I've been extremely pleased with the number of email requests for info I've been getting in the last week or so.. .. keep it up!!!! Cheers, Chris Rose From mae@eden.rutgers.edu Mon Oct 21 22:27:47 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA04984; Mon, 21 Oct 96 22:27:46 EDT Received: from er7.rutgers.edu (er7.rutgers.edu [165.230.180.135]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id WAA23441 for ; Mon, 21 Oct 1996 22:13:12 -0400 Received: (from mae@localhost) by er7.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id WAA27785 for crose@ece; Mon, 21 Oct 1996 22:14:28 -0400 Date: Mon, 21 Oct 1996 22:14:28 -0400 From: Hideki Mae Message-Id: <199610220214.WAA27785@er7.rutgers.edu> To: crose@ece.rutgers.edu Subject: midterm part 3 b. Status: R Why can't eigval not equal to 1? As long as I - A is nonsingular the resultingmatrix has form {1 - aii } in the diagonal and arbitrary elements on non-diag. parts. if you get the chara. poly. it has form (lambda - 1 + aii)*(something)... How do you get an exact value like 1 from this generalized eqn? From crose@mogli.rutgers.edu Tue Oct 22 00:11:56 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA05126; Tue, 22 Oct 96 00:11:56 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10297; Tue, 22 Oct 96 00:12:58 EDT Date: Tue, 22 Oct 96 00:12:58 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610220412.AA10297@mogli> To: mae@eden.rutgers.edu Subject: Re: midterm part 3 b. Cc: crose@MOGLI.rutgers.edu Status: R If (I-A) is singular, then there will be more than one value for of v for which (I-A)v = b for assuming arbitrary b. This means that the fixed point is not unique which is a violation of the assumption. So A can't have an eigenvalue of 1. Cheers, Chris Rose From crose@mogli.rutgers.edu Tue Oct 22 00:11:58 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA05132; Tue, 22 Oct 96 00:11:57 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10302; Tue, 22 Oct 96 00:16:22 EDT Date: Tue, 22 Oct 96 00:16:22 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610220416.AA10302@mogli> To: qhzheng@ece.rutgers.edu Subject: Re: off the air Cc: crose@MOGLI.rutgers.edu Status: R HI Quihao (KweeHow, right?) Try writing out the differential equations in x_1 and x_2 for the homogenous case (u(t) = 0). Then follow the construction for phi(t,t_0) on the top of page 53 of the text. I think that will do it for you. Good luck! :) Cheers, Chris Rose From ykogan@eden.rutgers.edu Tue Oct 22 01:52:07 1996 Return-Path: Received: from er5.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA05181; Tue, 22 Oct 96 01:52:06 EDT Received: (from ykogan@localhost) by er5.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id BAA14555 for crose@mogli.rutgers.edu; Tue, 22 Oct 1996 01:38:54 -0400 From: "Yelenq Kogan" Message-Id: <9610220138.ZM14553@er5.rutgers.edu> Date: Tue, 22 Oct 1996 01:38:53 -0400 In-Reply-To: crose@mogli.rutgers.edu (Christopher Rose) "off the air" (Oct 21, 10:11pm) References: <9610220211.AA10144@mogli> X-Mailer: Z-Mail Lite (3.2.0 5jul94) To: crose@mogli.rutgers.edu (Christopher Rose) Subject: Re: off the air Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: R Dr. Rose, I think that in problem #1 ch.3, the solution for y(t) should be: y(t) = (t/t_0 t/t_0)x_0 + integral_{t_0}^t 2t/tau u(tau) dtau. Is this correct? Thank you, Lena From crose@mogli.rutgers.edu Tue Oct 22 11:43:07 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA05490; Tue, 22 Oct 96 11:43:07 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10571; Tue, 22 Oct 96 10:57:44 EDT Date: Tue, 22 Oct 96 10:57:44 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610221457.AA10571@mogli> To: ykogan@eden.rutgers.edu Subject: Re: off the air Cc: crose@MOGLI.rutgers.edu Status: R Yup, the first term is missing 1/t_0 and you can simplify the drive term as you did. Cheers, Chris Rose *********************** Dr. Rose, I think that in problem #1 ch.3, the solution for y(t) should be: y(t) = (t/t_0 t/t_0)x_0 + integral_{t_0}^t 2t/tau u(tau) dtau. Is this correct? From crose@mogli.rutgers.edu Tue Oct 22 11:49:34 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA05495; Tue, 22 Oct 96 11:43:08 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10590; Tue, 22 Oct 96 10:58:39 EDT Date: Tue, 22 Oct 96 10:58:39 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610221458.AA10590@mogli> To: ece501@MOGLI.rutgers.edu Subject: problem 3.1 Status: RO Dr. Rose, I think that in problem #1 ch.3, the solution for y(t) should be: y(t) = (t/t_0 t/t_0)x_0 + integral_{t_0}^t 2t/tau u(tau) dtau. Is this correct? **************** Yup, the first term is missing 1/t_0 and you can simplify the drive term as you did. Cheers, Chris Rose From vpopescu@caip.rutgers.edu Tue Oct 22 20:54:59 1996 Return-Path: Received: from caipfs.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA06416; Tue, 22 Oct 96 20:54:58 EDT Received: from telestar.rutgers.edu (telestar.rutgers.edu [128.6.43.4]) by caipfs.rutgers.edu (8.7.6/8.7.3) with ESMTP id UAA06643; Tue, 22 Oct 1996 20:41:36 -0400 (EDT) From: Viorel George Popescu Received: (vpopescu@localhost) by telestar.rutgers.edu (8.6.8.1+bestmx+oldruq+newsunq/8.6.12) id UAA22043; Tue, 22 Oct 1996 20:41:27 -0400 Date: Tue, 22 Oct 1996 20:41:27 -0400 Message-Id: <199610230041.UAA22043@telestar.rutgers.edu> To: crose@mogli.rutgers.edu Subject: Re: problem 3.1 Cc: vpopescu@caip.rutgers.edu X-Sun-Charset: US-ASCII Status: R This is a quick solution to problem 1.3 The picture below show that the distance function rho(x,y)=(x-y)^2 does not satisfy the point (iii) in definition 1.1: --------------- | | | | |--------- | | | | | |-----| | | | | | | --------------- x y z (Area of the big square is bigger than the sum of the areas of smaller squares). Cheers, George From crose@mogli.rutgers.edu Tue Oct 22 23:42:04 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA06641; Tue, 22 Oct 96 23:42:03 EDT Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10947; Tue, 22 Oct 96 22:25:21 EDT Date: Tue, 22 Oct 96 22:25:21 EDT From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9610230225.AA10947@mogli> To: ece501@MOGLI.rutgers.edu Subject: problem 1.3 Cc: crose@MOGLI.rutgers.edu Status: R This is an extremely elegant solution to problem 1.3 in the text: ******** This is a quick solution to problem 1.3 The picture below show that the distance function rho(x,y)=(x-y)^2 does not satisfy the point (iii) in definition 1.1: --------------- | | | | |--------- | | | | | |-----| | | | | | | --------------- x y z (Area of the big square is bigger than the sum of the areas of smaller squares). Cheers, George ************* From crose Tue Nov 5 20:13:08 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA23595; Tue, 5 Nov 96 20:10:08 EST Date: Tue, 5 Nov 96 20:10:08 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9611060110.AA23595@MOGLI.rutgers.edu> To: ece501 Status: R SSN E2 MID 8311 3 68 1826 3 110 1525 2 101 0754 0 66 4552 3 57 2751 0 68 3022 4 108 2549 0 58 3431 0 107 9441 4 77 6372 4 98 3106 0 98 8346 0 102 4046 0 85 7111 0 85 1154 0 133 6122 0 65 4670 3 92 7058 3 102 4177 0 142 1859 0 105 2036 3 111 5645 0 133 7793 0 100 8674 3 85 5716 0 95 1899 0 102 2674 3 147 1590 4 130 0737 3 96 9700 4 95 1233 0 99 1538 2 140 1455 2 122 9928 4 100 9196 0 147 1420 3 96 4059 0 78 8656 0 78 3003 4 128 From crose Tue Nov 5 20:22:05 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA23643; Tue, 5 Nov 96 20:18:35 EST Date: Tue, 5 Nov 96 20:18:35 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9611060118.AA23643@MOGLI.rutgers.edu> To: ece501 Subject: grades Status: R I'll look over the exams and hand them back on thursday. I'm not sure how I feel yet about the letter grade (that's why I'm going to look at the exams). I'll post my feelings (or just tell you in class). By the way... AVERAGE: 100 Standard Dev. 23 High 147 Low 58 The histogram is available on the web page... Cheers, Chris Rose From crose@mogli.rutgers.edu Sun Nov 10 14:12:04 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA29948; Sun, 10 Nov 96 14:12:03 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA04346; Sun, 10 Nov 96 13:18:55 EST Date: Sun, 10 Nov 96 13:18:55 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9611101818.AA04346@mogli> To: ece501@MOGLI.rutgers.edu Subject: erratum in book Cc: crose@MOGLI.rutgers.edu Status: R >From Allen Onweller: ******************* I think there is an error on page 162 of S&B. The transfer function (eq. P21.9) looks incorrect. I think that it should be: H(s) = [K_m.K_a]/[s((sL_f + R_f)(sJ + f) + K_m.K_a.K_t)] rather than H(s) = [K_m.K_a]/[s(sL_f + R_f)(sJ + f) + K_m.K_a.K_t] The problem set solution that you provided has the correct transfer function. ***************** I agree (Chris Rose) From crose@mogli.rutgers.edu Thu Nov 14 13:42:42 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA04578; Thu, 14 Nov 96 13:42:42 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA07432; Thu, 14 Nov 96 13:55:13 EST Date: Thu, 14 Nov 96 13:55:13 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9611141855.AA07432@mogli> To: ece501@MOGLI.rutgers.edu Subject: The Nasty Book Cc: crose@MOGLI.rutgers.edu Status: R Hi Folks, I've checked and rechecked the solutions for the Wily Rotating Book. We correctly derived the energy in an inertial frame E = 0.5m(vx^2 + vy^2 + vz^2) summed over the eight bodies which make up the block. And we correctly derived the corresponding differential equations, fixed points and linearization. If you remember, the problem was that the book should have been unstable any time d>h, but we know from experiment that w comes into play as well. I've checked the equations numerically too, just in case there was some funny second order effect that squeaks in (like you need to be REALLY CLOS to the stationary point and with experiment we might have been way off). NO GO! The equations are stable for dFrom stability we go into controllability and observability which will take us into the final exam. Cheers, Chris Rose PS: Of course, I'll be banging my head against this one for some time to come until I figure out what's wrong. Yes, I could just consult a text somewhere, or yes I could ask a mechanical engineer. But I think it's better to macho (or machette if you're a woman :) out these sorts of things. For those unfamiliar with the "macho" idiom, it means to do the stereotypical male thing when confronted with an adversary: bash repeatedly until you or it drops dead. As has oft been said, "If it don't kill you, it makes you stronger!" :) :) :) From crose Tue Nov 26 19:48:39 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA18718; Tue, 26 Nov 96 19:45:13 EST Date: Tue, 26 Nov 96 19:45:13 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9611270045.AA18718@MOGLI.rutgers.edu> To: ece501 Subject: mea culpa Status: R Hi Folks, I defined v incorrectly this past class (tuesday before turkeyday). I'm not sure whether I miswrote in the previous class as well. Just in case, however, here's the correct form.... FIRST x(t1) = phi(t1,t0)x0 + regular old convolution integral with a leading phi(t1,tau) as expected. NOW HERE'S THE TWIST! We define v as, v = phi(t0,t1) [x(t1) - phi(t1,t0)x0 ] = phi(t0,t1)x(t1) - x0 = phi(t0,t1) (regular old convolution integral with leading phi(t1,tau) ) = (convolution integral with leading phi(t0,tau) because phi(t0,t1)phi(t1,tau) = phi(t0,tau) ) This gets you the distinctive controllability gramian form. Please check thursday's (11/21) notes for this error and correct if necessary. We would have had time on tuesday to catch and correct the error, but since we've got to rush a little to finish what we need to know by the end of the term we couldn't. Sorry for the confusion. Thank goodness for email! Cheers, and again, happy holidays! Chris Rose From crose Tue Nov 26 23:25:56 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA18951; Tue, 26 Nov 96 23:21:43 EST Date: Tue, 26 Nov 96 23:21:43 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9611270421.AA18951@MOGLI.rutgers.edu> To: ece501 Subject: final exam Status: R Hi Folks, Here's the 330:501 final examination information: DATE: Tuesday, December 17, 1996 TIME: 8pm-11pm PLACE: SEC-205 and SEC-208 (yes, two rooms) We'll have a proctor so there should be no problem with the splitting up. I requested two rooms because I felt it would be cramped with just one. We'll see how it goes. Maybe the rooms are bigger than I think. Cheers, Chris Rose From crose Tue Nov 26 23:25:56 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA18951; Tue, 26 Nov 96 23:21:43 EST Date: Tue, 26 Nov 96 23:21:43 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9611270421.AA18951@MOGLI.rutgers.edu> To: ece501 Subject: final exam Status: RO Hi Folks, Here's the 330:501 final examination information: DATE: Tuesday, December 17, 1996 TIME: 8pm-11pm PLACE: SEC-205 and SEC-208 (yes, two rooms) We'll have a proctor so there should be no problem with the splitting up. I requested two rooms because I felt it would be cramped with just one. We'll see how it goes. Maybe the rooms are bigger than I think. Cheers, Chris Rose From crose Fri Dec 6 12:48:10 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA00525; Fri, 6 Dec 96 12:45:29 EST Date: Fri, 6 Dec 96 12:45:29 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612061745.AA00525@MOGLI.rutgers.edu> To: ece501 Subject: update Status: R A student has already gotten $2.. that's enough for lunch! All future error postings must be submitted to the course mailing list to receive credit. That way EVERYONE gains from your observation and not just the cash recipient... Cheers From crose Fri Dec 6 17:46:16 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA00966; Fri, 6 Dec 96 17:43:14 EST Date: Fri, 6 Dec 96 17:43:14 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612062243.AA00966@MOGLI.rutgers.edu> To: ece501 Subject: problems Status: RO Hi Folks, If you've posted solution corrections to me either via email or verbally and I've aggreed with your suggestion, could you please post the correction to the mailing group... well just post the problem which needed to have a corrected solution. That'll help me a lot (and everyone else as well). Cheers All! From crose@mogli.rutgers.edu Sun Dec 8 12:15:47 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA02552; Sun, 8 Dec 96 12:12:16 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA05501; Sun, 8 Dec 96 12:04:24 EST Date: Sun, 8 Dec 96 12:04:24 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612081704.AA05501@mogli> To: ece501@MOGLI.rutgers.edu Subject: Throwing the Book Status: R Hi Folks, Here's a question and answer on the infamous rotating book problem. SYNOPSIS: we ignored an axis of rotation and that made the modeling wrong. QUESTION I don't know whether the book problem was clarified, or not, so that's why I'm writing directly to you. If you agree with my opinion, feel free to post my message on the list. So, the final conclusion that I have in my notes, is that: for h < d => alpha > 0 => unstable trajectory for h > d => alpha < 0 => stable trajectory (We have liniarized about a fixed trajetory phi = Omega t, which means constant rotation about x axis). If you take a look at the solution, with the book in your hand, and the figure in front of you, you'll see that everything is OK: h < d means the book in the position below: / ---------------------------/ / / | / / / / / / /----------------------------/ / 2d |2h / | / |----------/----------------|/ / 2w / /x-axis Rotation about x-axis won't be stable, and this is the real case, isn't it? h > d means the book in the position below: / -------------------------- / /----------------------------/ | | | | | | | | | | |2h / | | | / | | | / | | | / | / |------/--------------------|/2d / 2w / / x-axis In this case, rotation about x-axis will be stable, exactly like in the real situation. So, there was no problem with the results, only that we didn't look correctly at our pictures. P.S. If I am wrong, please let me know where am I wrong. MY ANSWER Hi What you've written (laboriously with ascii... what dedication!) is completely correct. The problem comes in when you make w < d in your first picture. The rotation is physically stable, but our equations say that it's not. The error was an oversimplification. We ignored the third axis of rotation. TRY THIS: get a pencil and place it between the book pages so that it's sticking out of the right end (in your first picture). You'll need rubber bands to hold it in place as usual. / ---------------------------/ / / | / / / / / ******************** Pencil /----------------------------/ / 2d |2h / | / |----------/----------------|/ / 2w / /x-axis Then toss the book (place the eraser side out by the way so you don't skewer yourself) and rotate it about the unstable consifiguration (again your first picture). From our modeling, the pencil should never stray ourside a vertical disk. When you do the experiment, you'll notice that the pencil wags side to side -----> which implies rotation about an axis we did not model... oh well. I've yet to find a simple way to model this problem which takes into account the effect of w. That is, it's even more complicated than what we did in class. Maybe a direct linearization would work. Cheers, Chris Rose From cripop@ece.rutgers.edu Tue Dec 10 12:10:07 1996 Return-Path: Received: from zen.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA04973; Tue, 10 Dec 96 12:07:33 EST Received: from ece.rutgers.edu (ee.rutgers.edu [128.6.46.13]) by zen.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with SMTP id MAA25823 for ; Tue, 10 Dec 1996 12:02:06 -0500 From: Dimitrie Popescu Message-Id: <199612101702.MAA25823@zen.rutgers.edu> Subject: stability of time varying systems To: ece501@mogli.rutgers.edu Date: Tue, 10 Dec 1996 12:01:56 -0500 (EST) Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-Length: 574 Status: RO We haven't discussed to much about stability of time-varying systems in the lecture, and this subject is also briefly discussed in the textbook. What I have noticed from the problem sets, and can also be obtained from the zero input response of such a system, is that it is stable if its transition matrix Phi(t,t0) is a contraction. In this case, if we start "near" an equilibrium point |x0 - x bar| < eps => |x(t) - x bar| = |Phi(t,t0)x0 - x bar| < eps1, with eps1 < eps since Phi(t,t0) is a contraction. Dr. Rose, please comment what I've stated above. Dimitrie From crose Tue Dec 10 14:06:20 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA05095; Tue, 10 Dec 96 14:01:44 EST Date: Tue, 10 Dec 96 14:01:44 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612101901.AA05095@MOGLI.rutgers.edu> To: cripop@ece.rutgers.edu, ece501@mogli.rutgers.edu Subject: Re: stability of time varying systems Status: RO Exactly correct. If Phi is a contraction, then with zero input you have no choice but for global assymptotic stability. A way to see it from your questions is Phi(t,t0) xbar = xbar (because that's the fixed point) Thus x(t) - x bar = Phi(t,t0)x0 - xbar = Phi(t,t0)(x0-xbar). Call y(t) = x(t) - xbar and you have y(t) = Phi(t,t0)y0 Phi a contraction means ||y(t)|| <= ||y0|| which implies the assumptotic condition. Cheers, Chris Rose From crose Tue Dec 10 17:25:39 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA05494; Tue, 10 Dec 96 17:22:13 EST Date: Tue, 10 Dec 96 17:22:13 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612102222.AA05494@MOGLI.rutgers.edu> To: ece501 Subject: URGENT!!!!! Status: R The time has been changed from that published for the final examination. Same rooms (sec 205 & 208) but the exam starts at 7PM (NOT 8pm)!!!!!!!!!! Spread the word! From crose Tue Dec 10 22:08:47 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA05691; Tue, 10 Dec 96 22:06:46 EST Date: Tue, 10 Dec 96 22:06:46 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612110306.AA05691@MOGLI.rutgers.edu> To: grieco@EEYORE.ira.rl.af.mil Subject: Re: Reminder- 330:501 FInal Cc: ece501 Status: R 7pm is the time. Not sure about why you got the other..... I'll check ONce again 7pm is the time on Tuesday dec 17 Room SEC 205 and 208!!! IGNORE ANYTHING ELSE!!!!! From busha@eden.rutgers.edu Wed Dec 11 14:10:34 1996 Return-Path: Received: from er5.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA06722; Wed, 11 Dec 96 14:10:33 EST Received: (from busha@localhost) by er5.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id OAA20583 for Christopher Rose ; Wed, 11 Dec 1996 14:05:21 -0500 Date: Wed, 11 Dec 96 14:05:21 EST From: Brett Bu Sha To: Christopher Rose Subject: Re: today? In-Reply-To: Your message of Tue, 12 Nov 96 12:28:39 EST Message-Id: Status: R Prof, Help me please, I've forgotten the formal definition of a singular or non-singular matrix, and there are no linear alg. books here. Brett PS If you so desire, I'ld like to be enlightened into the positive-definate, semi-definate, negative definate thing. Thanks again From busha@eden.rutgers.edu Wed Dec 11 14:10:34 1996 Return-Path: Received: from er5.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA06722; Wed, 11 Dec 96 14:10:33 EST Received: (from busha@localhost) by er5.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id OAA20583 for Christopher Rose ; Wed, 11 Dec 1996 14:05:21 -0500 Date: Wed, 11 Dec 96 14:05:21 EST From: Brett Bu Sha To: Christopher Rose Subject: Re: today? In-Reply-To: Your message of Tue, 12 Nov 96 12:28:39 EST Message-Id: Status: R Prof, Help me please, I've forgotten the formal definition of a singular or non-singular matrix, and there are no linear alg. books here. Brett PS If you so desire, I'ld like to be enlightened into the positive-definate, semi-definate, negative definate thing. Thanks again From crose Wed Dec 11 19:06:34 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA06945; Wed, 11 Dec 96 19:04:25 EST Date: Wed, 11 Dec 96 19:04:25 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612120004.AA06945@MOGLI.rutgers.edu> To: busha@eden.rutgers.edu Subject: Re: today? Cc: ece501@MOGLI.rutgers.edu Status: R Prof, Help me please, I've forgotten the formal definition of a singular or non-singular matrix, and there are no linear alg. books here. Brett PS If you so desire, I'ld like to be enlightened into the positive-definate, semi-definate, negative definate thing. Thanks again *********************** Singular matrix has at least one zero eigenvalue and is therefore uninvertible. Zero eigenvalue also implies that the rows (or columns) are not linearly independent. Nonsingular matrix has all rows (columns) linearly indep and therefore no zero eigenvalues Positive definite applies to only symmetric matrices all of whose eigenvalues are positive. For a positivie definite matrix Q we have x^T Q x > 0 for all possible x except x=0. Semidefinite implies that there might be a nonzero x such that x^TQx = 0 (which in turn implies that one of the eigenvalues must be zero). Cheers From crose@mogli.rutgers.edu Thu Dec 12 01:45:21 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA07359; Thu, 12 Dec 96 01:42:04 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA09891; Thu, 12 Dec 96 00:09:41 EST Date: Thu, 12 Dec 96 00:09:41 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612120509.AA09891@mogli> To: ece501@MOGLI.rutgers.edu Subject: hi Status: R Hi Folks, I can be in the office (and available) for questions on Thursday from about 5pm to 6pm or so. However, I must get at least one solid email request to be there beforehand. I'd prefer using the mailing list and email in general since it provides a record of our conclusions for class consumption. But I realize that face-2-face is sometimes the only way. However, I will FROWN!!!! upon questions asked during office hours which could have easily been asked via email !!!! :) Study hard! Cheers PS: Ricardo has been mumbling about a review session. Pester him into giving one! :) From ahmeda@MAILNET.ho.ATT.com Thu Dec 12 12:39:35 1996 Return-Path: Received: from cagw1.att.com by MOGLI.rutgers.edu (4.1/25-eef) id AA07741; Thu, 12 Dec 96 12:35:57 EST From: ahmeda@MAILNET.ho.att.com Received: from hoccson.ho.att.com by caig1.att.att.com (SMI-8.6/EMS-1.2 sol2) id MAA10225; Thu, 12 Dec 1996 12:26:02 -0500 Original-From: ahmeda@MAILNET.ho.ATT.com Received: by hoccson.ho.att.com (4.1/EMS-1.1.1 SunOS) id AA18878; Thu, 12 Dec 96 12:30:23 EST Received: from nj-mailnet.ho.att.com (mailnet.ho.att.com) by hoccson.ho.att.com (4.1/EMS-1.1.1 SunOS) id AA18874; Thu, 12 Dec 96 12:30:23 EST Received: by nj-mailnet.ho.att.com with SMTP (Microsoft Exchange Server Internet Mail Connector Version 4.0.993.5) id <01BBE828.43ADF370@nj-mailnet.ho.att.com>; Thu, 12 Dec 1996 12:30:31 -0500 Message-Id: Original-From: "Abdelgany, Ahmed" To: "'ece501@mogli.rutgers.edu'" Subject: RE: Symm Matrix Date: Thu, 12 Dec 1996 12:30:49 -0500 Return-Receipt-To: X-Mailer: Microsoft Exchange Server Internet Mail Connector Version 4.0.993.5 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Status: R Regarding your question about a symmetric matrics, Let me give it a shot A symmetric matrix is equal to its transpose, e.g., |a b| |b c| this is a symmetric matrix. The matrix has to be square, and has the property that its entries are mirror images of each other, i.e., a(ij) = a(ji). The inverse of the matrix (if it exists) is also symmterical now, since the matrix is equal to its transpose, the product of the two is the square of the matrix, as can be seen in the equation below. Ahmed A. >---------- >From: bsolanki@apollo.rutgers.edu[SMTP:bsolanki@apollo.rutgers.edu] >Sent: Thursday, December 12, 1996 11:59 AM >To: ece501@mogli.rutgers.edu > > > > a toss up question for anyone.... > > what constitutes a symmetric matrix? and what are its properties?? >and why are we able to say something like > >vTphi(to,tau)B(tau)BT(tau)PhiT(to,tau)v = ||BT(tau)PhiT(to,tau)v||2^2 >(pg.208) > > > thanks > > Bindu > From ralf@ece.rutgers.edu Thu Dec 12 17:43:12 1996 Return-Path: Received: from zen.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA08026; Thu, 12 Dec 96 17:39:37 EST Received: from chess.rutgers.edu (chess.rutgers.edu [128.6.46.113]) by zen.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with SMTP id RAA02207 for ; Thu, 12 Dec 1996 17:33:44 -0500 Date: Thu, 12 Dec 1996 17:33:44 -0500 From: Ricardo Losada Message-Id: <199612122233.RAA02207@zen.rutgers.edu> To: ece501@mogli.rutgers.edu Subject: review Status: R What's up people, Dr. Rose has asked me to give a review session for the course, the review will be on controllabillity, observabillity and stabillity. I plan to bring in my own worked out examples, and try to help anyone who has questions, the session would have to be tomorrow (friday 13), at a time and placed to be determined democratically. So please send your suggestions, etc. so we can set somenthing up, also at least one person must show interest in this so please at least one respond this email. I'll check later and throughout the day tomorrow for emails on this. Best luck, Ricardo. From crose Thu Dec 12 20:06:31 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA08306; Thu, 12 Dec 96 20:03:29 EST Date: Thu, 12 Dec 96 20:03:29 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612130103.AA08306@MOGLI.rutgers.edu> To: ece501 Subject: question Status: R This from a student... Could you give us some more problems, that look like those that we will receive in the exam? Almost all the problems from the textbook are of the type "plug and chug", unlike the exam problems. You've threatened us with optimal control, but the only problem that we have is the "turkey problem" in PS8. (By the way, I've tried to find a more rigurous solution to it, using partial derivatives of w(10) -the final weight of the turkey- with respect to u(t), t = 0,1,...,9 and the initial constraint for the input, that is SUM from t=0 to 9 u(t) = 10. It involves a lot of work, and actually the hypothesis that u(t) is also an nonnegative integer simplifies a lot.) ***************** I am sorry to say that I don't have time to make up new problems with solutions (other than for the final). What I'd suggest is to look hard at this year's midterm and last year's final to get a feel for the style of question I like to ask. I'll be concentrating on the latter partof the course (stability controllability observability) but anything is fair game... know something about metric spaces. I might also hit you with some Fourier transform stuff since it's very much like the laplace. In short, I'l be asking you to extend yourselves somewhat and apply what you know (on some of the problems... not all). Once again, sorry I can't be more accommodating, but time constraints have me tied to my desk doing tasks that are at least 1 month old (late). Cheers, Chris Rose From crose@qbeast.rutgers.edu Thu Dec 12 21:11:20 1996 Return-Path: Received: from qbeast.rutgers.edu by boom.rutgers.edu (4.1/SMI-4.1) id AA17189; Thu, 12 Dec 96 21:11:06 EST Received: by qbeast.rutgers.edu (4.1/SMI-4.1) id AA02815; Thu, 12 Dec 96 21:09:05 EST Date: Thu, 12 Dec 96 21:09:05 EST From: crose@qbeast.rutgers.edu (Christopher Rose) Message-Id: <9612130209.AA02815@qbeast.rutgers.edu> To: ece501@boom Subject: question Cc: `@qbeast.rutgers.edu Status: R Prof Rose, I have a very basic question. Why when we talk about Linear Time Invariant systems we relate primal with dual, in opposite to adjoint (in the controllability/observability relation) ? ************* We could in principle use either for the LTI, but the dual is easier to remember (no pesky - sign). In general however, we need the adjoint... From crose@mogli.rutgers.edu Fri Dec 13 05:11:57 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA08726; Fri, 13 Dec 96 05:11:56 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA10818; Thu, 12 Dec 96 23:56:24 EST Date: Thu, 12 Dec 96 23:56:24 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612130456.AA10818@mogli> To: ece501@MOGLI.rutgers.edu Subject: example 2.5 Cc: crose@MOGLI.rutgers.edu Status: R Try writing out the differential equations in x_1 and x_2. Solve the equation in x_2 first. Then you can solve teh equation in x_1. Then use your knowledge of transition matrix properties, in particular the construction of section 2.1.2, to get the transition matrix shown. Cheers From crose@mogli.rutgers.edu Fri Dec 13 11:44:35 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA08968; Fri, 13 Dec 96 11:42:26 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA11052; Fri, 13 Dec 96 11:14:32 EST Date: Fri, 13 Dec 96 11:14:32 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612131614.AA11052@mogli> To: busha@eden.rutgers.edu Subject: Re: today? Cc: ece501@MOGLI.rutgers.edu Status: R Jordan form: Used when eigenvalues are repeated AND there are not enough corresponding eigenvectors. Consider the matrix 1 1 0 1 Find the evals and then try to find the associated eigenvectors... you'll only be able to find 1. This particular matrix is already in jordan form. I don't like your use of "transforms" to a non-singular matrix. The Jordan form can still be singlular, in fact WILL be singular if the matrix from which it's derived is singular (has zero eigenvalues). The Jordan form has to do with missing eigenvectors, not specific eigenvalues. Stability: 0 is the stationary point. You can always translate the state variable to be centered on an equilibrium point by finding the equilibrium point xbar and then forming the new state variable x-xbar. (this comment refers to the figure on page 168 of the text). Cheers Chris Rose From crose Fri Dec 13 13:30:13 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA09074; Fri, 13 Dec 96 13:30:12 EST Date: Fri, 13 Dec 96 13:30:12 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612131830.AA09074@MOGLI.rutgers.edu> To: cripop@ece.rutgers.edu Subject: Re: problem 14 / pag. 264 Cc: crose Status: R Basically ok, The way I look at it is that if you know u(t), you can reform y(t) to get ytilde(t) = y(t) - Du. Now everything is as it was before. From crose@mogli.rutgers.edu Sat Dec 14 12:12:07 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA10208; Sat, 14 Dec 96 12:12:06 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA11780; Sat, 14 Dec 96 11:50:23 EST Date: Sat, 14 Dec 96 11:50:23 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612141650.AA11780@mogli> To: mae@ece.rutgers.edu Subject: Re: today? Cc: crose@MOGLI.rutgers.edu Status: R Semi def implies that at least one eigenvalue is zero. If any eigenvalue is zero, the matrix is singular. Cheers From crose@mogli.rutgers.edu Sat Dec 14 12:12:08 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA10214; Sat, 14 Dec 96 12:12:08 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA11785; Sat, 14 Dec 96 11:54:50 EST Date: Sat, 14 Dec 96 11:54:50 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612141654.AA11785@mogli> To: mae@ece.rutgers.edu Subject: Re: ps 6 p.1,2 Cc: crose@MOGLI.rutgers.edu Status: R The drive input is not necessarily unique. And no, uniqueness of the input has nothing to do with general observability. For observability, you observe the output and the input. So you'd know exactly what the input was. You might want to study the concepts as laid out in the book a little more closely. GOOD LUCK! As for the solutions to two problems... I'm not sure which two problems you mean so I'm at a loss. Cheers From crose Sat Dec 14 18:08:24 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA10442; Sat, 14 Dec 96 18:08:23 EST Date: Sat, 14 Dec 96 18:08:23 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612142308.AA10442@MOGLI.rutgers.edu> To: yufeng@caip.rutgers.edu Subject: The stuff Cc: crose@MOGLI.rutgers.edu Status: R The stuff we did not cover (section 4.2.2 only) will not be on the exam. What else did you think we did not cover? Cheers From mae@ece.rutgers.edu Sat Dec 14 19:30:43 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA10511; Sat, 14 Dec 96 19:30:42 EST Received: (from mae@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id TAA06441 for crose@mogli.rutgers.edu; Sat, 14 Dec 1996 19:24:22 -0500 Date: Sat, 14 Dec 1996 19:24:22 -0500 From: Hideki Mae Message-Id: <199612150024.TAA06441@ece.rutgers.edu> To: crose@mogli.rutgers.edu Subject: Re: ps 6 p.1,2 Content-Length: 506 Status: R On Ch.4 progb.2 you found the transition matrix to see if it was stable or not. THe same conclusion can be reached by finding the eig.values. Since the eig.vals were positive Re(lambda) > 0, the system is unstable. Is there any reason why you found the transition matrix rather than finding the eig.values? I guess a better way to phrase this question is: is there any restriction on when to use thm 4.5? FOr example on Ch.4 prob.1 the eig. val. depend on varibale t, can I use thm 4.5 on this prob? From crose Sat Dec 14 20:38:49 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA10541; Sat, 14 Dec 96 20:36:42 EST Date: Sat, 14 Dec 96 20:36:42 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612150136.AA10541@MOGLI.rutgers.edu> To: ece501 Subject: problem 2 chapter 4 Status: R QUESTION: On Ch.4 progb.2 you found the transition matrix to see if it was stable or not. THe same conclusion can be reached by finding the eig.values. Since the eig.vals were positive Re(lambda) > 0, the system is unstable. Is there any reason why you found the transition matrix rather than finding the eig.values? I guess a better way to phrase this question is: is there any restriction on when to use thm 4.5? FOr example on Ch.4 prob.1 the eig. val. depend on varibale t, can I use thm 4.5 on this prob? ANSWER: The eigenvalue method could have been used as well since the system is LTI. I believe the reason the transition matrix was used was because it was derived in a previous problem. BUT, you have to be careful with theorem 4.5 since not all systems which DON'T satisfy the theorem are unstable. That is, there are systems with repeated eigenvalues with real part 0 (or magnitude 1 for discrete time) that are stable (but don't satisfy theorem 4.5 because they've got repeated eigenvalues). Cheers, PS TO EVERYONE: Please post all questions to the group no matter how silly you think they are. Also if you think you have an answer to a question, please post. If there's a problem with your answer I'll let the group know. From cripop@ece.rutgers.edu Sun Dec 15 10:07:54 1996 Return-Path: Received: from zen.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11193; Sun, 15 Dec 96 10:05:49 EST Received: from ece.rutgers.edu (ee.rutgers.edu [128.6.46.13]) by zen.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with SMTP id JAA07470 for ; Sun, 15 Dec 1996 09:59:26 -0500 From: Dimitrie Popescu Message-Id: <199612151459.JAA07470@zen.rutgers.edu> To: ece501@mogli.rutgers.edu Date: Sun, 15 Dec 1996 09:59:19 -0500 (EST) Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Content-Length: 910 Status: R Dear dr. Rose, In your answer regarding problem 2 / chapter 4 you have stated the following: ------------------------------------------------------------------------------ BUT, you have to be careful with theorem 4.5 since not all systems which DON'T satisfy the theorem are unstable. That is, there are systems with repeated eigenvalues with real part 0 (or magnitude 1 for discrete time) that are stable (but don't satisfy theorem 4.5 because they've got repeated eigenvalues). _____________________________________________________________________________ Could you please give us an example of a system that has repetead eigenvalues with real part = 0 (continuous time) or magnitude = 1 (discrete time) and which is still stable? (I mean some other system than the trivial ones with A=0 (all entries of matrix are zero) in the continuous time, or A=I identity matrix in the discrete time.) Dimitrie From hatangad@ece.rutgers.edu Sun Dec 15 11:41:53 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11252; Sun, 15 Dec 96 11:39:40 EST Received: (from hatangad@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id LAA02335; Sun, 15 Dec 1996 11:33:14 -0500 Date: Sun, 15 Dec 1996 11:33:14 -0500 From: Sonya Hatangadi Message-Id: <199612151633.LAA02335@ece.rutgers.edu> To: crose@mogli.rutgers.edu Cc: ece501@mogli.rutgers.edu Subject: Transfer functions Content-Length: 272 Status: R Prof. Rose, During the lecture you had said that in a series combination of systems , the transfer function was H2(s)H1(s) and the order mattered. Since H(s) is a polynomial , H2(s)H1(s) should be the same as H1(s)H2(s). Sonya. From fam@ece.rutgers.edu Sun Dec 15 13:42:03 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11324; Sun, 15 Dec 96 13:40:12 EST Received: (from fam@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id NAA04089 for ece501@mogli; Sun, 15 Dec 1996 13:33:44 -0500 Date: Sun, 15 Dec 96 13:33:44 EST From: David Famolari To: ece501@mogli.rutgers.edu Subject: example of nontrivial repeated 0 eignevalues Message-Id: Status: R Here is what I think is a non trivial example of a system matrix with a 0 repeated eigenvalue that is still stable. 0 1 5 A= 0 0 1 0 0 5 It has an eigenvalue of 0 with multiplicity of 2. But it's jordan form will only contain jordan blocks of maximum size 1. Because the dimension of the Nullspace of A-0I = dimension of the Nullspace of A = 1. So the maximum size of a simple Jordan block for the eigenvalue = 0 is 1 and thus the system is stable. Please correct me if I'm wrong. thanks dave From fam@ece.rutgers.edu Sun Dec 15 13:50:01 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11340; Sun, 15 Dec 96 13:47:01 EST Received: (from fam@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id NAA04159 for ece501@mogli; Sun, 15 Dec 1996 13:40:33 -0500 Date: Sun, 15 Dec 1996 13:40:33 -0500 From: David Famolari Message-Id: <199612151840.NAA04159@ece.rutgers.edu> To: ece501@mogli.rutgers.edu Subject: Eigenvectors for Jordan form? Content-Length: 142 Status: R Prof. Rose, Would expect us to be able to find the generalized eigenvectors for the Jordan form? Or will finding J be enough? thanks dave From crose@mogli.rutgers.edu Sun Dec 15 15:42:19 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA11409; Sun, 15 Dec 96 15:42:18 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA12749; Sun, 15 Dec 96 14:22:50 EST Date: Sun, 15 Dec 96 14:22:50 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612151922.AA12749@mogli> To: cripop@ece.rutgers.edu Subject: others Cc: crose@MOGLI.rutgers.edu Status: R Any similarity transform of a diagonal matrix with repeated eigenvalues on the diagonal (with Re=0 or mag 1). From crose@mogli.rutgers.edu Sun Dec 15 15:42:21 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA11413; Sun, 15 Dec 96 15:42:20 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA12754; Sun, 15 Dec 96 14:23:42 EST Date: Sun, 15 Dec 96 14:23:42 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612151923.AA12754@mogli> To: hatangad@ece.rutgers.edu Subject: Re: Transfer functions Cc: crose@MOGLI.rutgers.edu Status: R Nope, H(s) is a MATRIX in general so order does matter. From mae@ece.rutgers.edu Sun Dec 15 15:46:47 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11419; Sun, 15 Dec 96 15:44:52 EST Received: (from mae@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id PAA05554 for ece501@MOGLI.rutgers.edu; Sun, 15 Dec 1996 15:38:24 -0500 Date: Sun, 15 Dec 1996 15:38:24 -0500 From: Hideki Mae Message-Id: <199612152038.PAA05554@ece.rutgers.edu> To: ece501@MOGLI.rutgers.edu Content-Length: 132 Status: R WHy is the transition mtrix for ps3 ch.2 prob.2 different from ps4 ch.3 prob.1? By solving the diff.eq. the ps3 soln seem accurate. From fam@ece.rutgers.edu Sun Dec 15 16:30:59 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11455; Sun, 15 Dec 96 16:27:22 EST Received: (from fam@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id QAA06100; Sun, 15 Dec 1996 16:20:53 -0500 Date: Sun, 15 Dec 1996 16:20:53 -0500 From: David Famolari Message-Id: <199612152120.QAA06100@ece.rutgers.edu> To: ece501@MOGLI.rutgers.edu, mae@ece.rutgers.edu Content-Length: 194 Status: R I think the transition matrices for chap.2 prob 2 and chap.3 prob.1 are the same, just that in chap2 prob2 the intial condition is explicity to=1 and in chap3 prob1 no such condition is given. From hatangad@ece.rutgers.edu Sun Dec 15 16:50:56 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11471; Sun, 15 Dec 96 16:50:54 EST Received: (from hatangad@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id QAA06416; Sun, 15 Dec 1996 16:44:25 -0500 Date: Sun, 15 Dec 1996 16:44:25 -0500 From: Sonya Hatangadi Message-Id: <199612152144.QAA06416@ece.rutgers.edu> To: crose@MOGLI.rutgers.edu Cc: ece510@MOGLI.rutgers.edu Subject: Controllability Content-Length: 545 Status: R Prof. Rose, Theorem 5.1 is a little misleading when it talks about an arbitrary state x1. I think it should say arbitrary state x1 in the range space of matrix W(t0, t1). For example, in a system where the state variables are dependent on each other, it is not possible to push the system to a state where the state variables are independent of each other. So complete controllability is when we can find an input to transfer the system between any two pints in the permissible state space. Is this correct? Sonya. From Mailer-Daemon@MOGLI.rutgers.edu Sun Dec 15 21:34:42 1996 Return-Path: Received: from boom.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11598; Sun, 15 Dec 96 21:34:41 EST Received: by boom.rutgers.edu (4.1/SMI-4.1) id AB19174; Sun, 15 Dec 96 21:29:37 EST Date: Sun, 15 Dec 96 21:29:37 EST From: Mailer-Daemon@MOGLI.rutgers.edu (Mail Delivery Subsystem) Subject: Returned mail: User unknown Message-Id: <9612160229.AB19174@boom.rutgers.edu> To: Status: R ----- Transcript of session follows ----- 421 eden: Connection refused by eden 550 Postmaster... User unknown ----- Unsent message follows ----- Return-Path: Received: from qbeast.rutgers.edu by boom.rutgers.edu (4.1/SMI-4.1) id AA17189; Thu, 12 Dec 96 21:11:06 EST Received: by qbeast.rutgers.edu (4.1/SMI-4.1) id AA02815; Thu, 12 Dec 96 21:09:05 EST Date: Thu, 12 Dec 96 21:09:05 EST From: crose@qbeast.rutgers.edu (Christopher Rose) Message-Id: <9612130209.AA02815@qbeast.rutgers.edu> To: ece501@boom Subject: question Cc: ` Prof Rose, I have a very basic question. Why when we talk about Linear Time Invariant systems we relate primal with dual, in opposite to adjoint (in the controllability/observability relation) ? ************* We could in principle use either for the LTI, but the dual is easier to remember (no pesky - sign). In general however, we need the adjoint... From Mailer-Daemon Sun Dec 15 22:29:38 1996 Return-Path: Received: by boom.rutgers.edu (4.1/SMI-4.1) id AB19179; Sun, 15 Dec 96 22:29:36 EST Date: Sun, 15 Dec 96 22:29:36 EST From: Mailer-Daemon (Mail Delivery Subsystem) Subject: Returned mail: User unknown Message-Id: <9612160329.AB19179@boom.rutgers.edu> To: crose Status: R ----- Transcript of session follows ----- 421 eden: Connection refused by eden 550 Postmaster... User unknown ----- Unsent message follows ----- Return-Path: Received: by boom.rutgers.edu (4.1/SMI-4.1) id AA17245; Thu, 12 Dec 96 21:44:44 EST Date: Thu, 12 Dec 96 21:44:44 EST From: crose (Christopher Rose) Message-Id: <9612130244.AA17245@boom.rutgers.edu> To: ece501 Subject: question from student about example QUestion about example 2.5. Anybody care to take a whack at this? ************* Hi, Dr. Rose: Could you explain how they got the transition matrix of Example 2.5 on the book? For me, it looks so 'non-unique'. From chanac@eden.rutgers.edu Sun Dec 15 23:11:47 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA11649; Sun, 15 Dec 96 23:11:46 EST Received: from er5.rutgers.edu (er5.rutgers.edu [165.230.180.133]) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with ESMTP id XAA11555 for ; Sun, 15 Dec 1996 23:05:14 -0500 Received: (from chanac@localhost) by er5.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) id XAA02728; Sun, 15 Dec 1996 23:05:45 -0500 From: "Angela Chan" Message-Id: <9612152305.ZM2726@er5.rutgers.edu> Date: Sun, 15 Dec 1996 23:05:44 -0500 X-Mailer: Z-Mail Lite (3.2.0 5jul94) To: crose@ece.rutgers.edu Subject: Questions Cc: Angela.chan@mci.com Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: R Dr Rose, I have several questions when I review the HW. May you answer my questions? - about the BIBO stability Chapter 4 prb 18 & 19, in soln, it said the eigenvalues of Matrixes are in LHP then we have BIBO. What's LHP, and also is this the other method to check the BIBO beside to use integral from t0 to tau of |tij(t,tau) dtau ? - pos definite What is pos definite? - chapter 4 prob 17, How they find the Routh Array? I didn't see any Routh Array in your note and textbook? - In pop quiz prob 4(b) They given y=[1,0,1]x , and ask u(t) and y(t) on [t0,t1] can x(t0) be uniquely determined? and in you answer said because observability matrix is full rank so, x(t0) can be determined. So is that mean when the system observable , x(t0) can be uniquely determined? - In Final exam Prob1. you ask the transition matrix is valid or not, is it use the properties of transition matrix to test them? if all 5 properties are satisfy, then transition matrix valid, if not than its not valid? - Do you have solution of the Final exam of 95? so I can check the answer Thanks Angela Chan From crose@mogli.rutgers.edu Mon Dec 16 01:42:58 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA11813; Mon, 16 Dec 96 01:42:58 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA13021; Mon, 16 Dec 96 01:13:26 EST Date: Mon, 16 Dec 96 01:13:26 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612160613.AA13021@mogli> To: ece501@mogli.rutgers.edu, mae@ece.rutgers.edu Subject: 4.1 and 2.2 (problems) Cc: crose@MOGLI.rutgers.edu Status: R  One solution is general the other is for a specific t_0 (problem 2.2 has t_0 = 1 whereas problem 3.1 specifies no particular t_0. From crose@mogli.rutgers.edu Mon Dec 16 01:43:08 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA11847; Mon, 16 Dec 96 01:43:08 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA13049; Mon, 16 Dec 96 01:25:49 EST Date: Mon, 16 Dec 96 01:25:49 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612160625.AA13049@mogli> To: ece501@mogli.rutgers.edu, fam@ece.rutgers.edu Subject: Re: example of nontrivial repeated 0 eignevalues Cc: crose@MOGLI.rutgers.edu Status: R Get rid of the 1 in the first row and I believe you. As your example stands, it's missing an eigenvector for the eigenvalue 0 (can only find one regular eigenvector for eigenvalue 0). Without the 1 in the first row there are two independent eigenvectors for eigenvalue zero. Cheers ***********************ORIGINAL QUESTION*********************** Here is what I think is a non trivial example of a system matrix with a 0 repeated eigenvalue that is still stable. 0 1 5 A= 0 0 1 0 0 5 It has an eigenvalue of 0 with multiplicity of 2. But it's jordan form will only contain jordan blocks of maximum size 1. Because the dimension of the Nullspace of A-0I = dimension of the Nullspace of A = 1. So the maximum size of a simple Jordan block for the eigenvalue = 0 is 1 and thus the system is stable. Please correct me if I'm wrong. thanks dave From crose@mogli.rutgers.edu Mon Dec 16 01:46:17 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA11852; Mon, 16 Dec 96 01:43:09 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA13083; Mon, 16 Dec 96 01:38:34 EST Date: Mon, 16 Dec 96 01:38:34 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612160638.AA13083@mogli> To: chanac@eden.rutgers.edu Subject: Re: Questions Cc: ece501@MOGLI.rutgers.edu Status: R QUESTION: Dr Rose, I have several questions when I review the HW. May you answer my questions? - about the BIBO stability Chapter 4 prb 18 & 19, in soln, it said the eigenvalues of Matrixes are in LHP then we have BIBO. What's LHP, and also is this the other method to check the BIBO beside to use integral from t0 to tau of |tij(t,tau) dtau ? - pos definite What is pos definite? - chapter 4 prob 17, How they find the Routh Array? I didn't see any Routh Array in your note and textbook? - In pop quiz prob 4(b) They given y=[1,0,1]x , and ask u(t) and y(t) on [t0,t1] can x(t0) be uniquely determined? and in you answer said because observability matrix is full rank so, x(t0) can be determined. So is that mean when the system observable , x(t0) can be uniquely determined? - In Final exam Prob1. you ask the transition matrix is valid or not, is it use the properties of transition matrix to test them? if all 5 properties are satisfy, then transition matrix valid, if not than its not valid? - Do you have solution of the Final exam of 95? so I can check the answer Thanks Angela Chan ANSWERS: 1) there's a theorem which says that if all the eigenvalues have real parts <0 then the LTI system is BIBO stable (theorem 4.13... asked you guys to read this one since we did not have time to prove it in class). Basic proof is that the transition matrix must have terms of the form At^k e^(lambda t) and if all the lambdas have real part <0 then Theorem 4.11 can be easily applied (same goes for discrete with appropriate modifications). 2) The left half plane (LHP) is the left side of the eigenvalue complex plane. The poles of a system are its eigenvalues (roots of the characteristic polynomial) 3) A positive definite matrix is a symmetric real matrix all of whose eigenvalues are positive. This also means that x^T Qx > 0 for a pos def matrix Q for any vector x that is not identically zero. 4) We did not cover the Routh array, though it's referred to. Use the Hurwitz determinants. 5) I think you need to reread the observability chapter and look over the notes if I understand your question correctly. Observability means that given the input and the output over some interval, the state at the begining of the interval can be uniquely determined. 6) If I remember taht problem, it was pretty easy to check if the transition matrix is valid (for a continous system). It must always be invertible and must reduce to the identity matrix at t=t_0. 7) No solutions to the final. Ricardo MIGHT provide some, but frankly I'd rather you simply worry about the solutions as opposed to having them in front of you. I think there's enough problems with solutions between this and last years' tests and problem sets. Cheers From crose@mogli.rutgers.edu Mon Dec 16 01:46:20 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA11830; Mon, 16 Dec 96 01:43:02 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA13031; Mon, 16 Dec 96 01:15:59 EST Date: Mon, 16 Dec 96 01:15:59 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612160615.AA13031@mogli> To: ece501@MOGLI.rutgers.edu Subject: mailing to ece501@ece Status: R Hi Folks, Please remember NOT to mail anything to ece501@ECE. That is last year's mailing list and your message won't get to me or anyone else in this semester's class. You must mail stuff to ece501@MOGLI.rutgers.edu or ece501@BOOM.rutgers.edu Cheers From crose Mon Dec 16 01:51:46 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA11877; Mon, 16 Dec 96 01:48:42 EST Date: Mon, 16 Dec 96 01:48:42 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612160648.AA11877@MOGLI.rutgers.edu> To: ece501 Subject: jordan form Status: R You will never be asked (by me) to find the Jordan form of a matrix. It will suffice to know what the Jordan form is, when it's used and how to calculate J^t or exp(Jt). From crose Mon Dec 16 01:52:09 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA11880; Mon, 16 Dec 96 01:49:15 EST Date: Mon, 16 Dec 96 01:49:15 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612160649.AA11880@MOGLI.rutgers.edu> To: ece501 Subject: 4.1 and 2.2 (problems) Status: R One solution is general the other is for a specific t_0 (problem 2.2 has t_0 = 1 whereas problem 3.1 specifies no particular t_0. From crose Mon Dec 16 01:53:22 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA11883; Mon, 16 Dec 96 01:50:24 EST Date: Mon, 16 Dec 96 01:50:24 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612160650.AA11883@MOGLI.rutgers.edu> To: ece501 Subject: example of nontrivial repeated 0 eigenvalues Status: R Get rid of the 1 in the first row and I believe you. As your example stands, it's missing an eigenvector for the eigenvalue 0 (can only find one regular eigenvector for eigenvalue 0). Without the 1 in the first row there are two independent eigenvectors for eigenvalue zero. Cheers ***********************ORIGINAL QUESTION*********************** Here is what I think is a non trivial example of a system matrix with a 0 repeated eigenvalue that is still stable. 0 1 5 A= 0 0 1 0 0 5 It has an eigenvalue of 0 with multiplicity of 2. But it's jordan form will only contain jordan blocks of maximum size 1. Because the dimension of the Nullspace of A-0I = dimension of the Nullspace of A = 1. So the maximum size of a simple Jordan block for the eigenvalue = 0 is 1 and thus the system is stable. Please correct me if I'm wrong. thanks dave From crose Mon Dec 16 01:55:18 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA11887; Mon, 16 Dec 96 01:51:07 EST Date: Mon, 16 Dec 96 01:51:07 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612160651.AA11887@MOGLI.rutgers.edu> To: ece501 Subject: controllability Status: R Nope, I don't think your argument is correct. The point of theorem 5.1 is that iff W is nonsingular then you can get from any x0 to any x1. If the system has states such that they are dependent (so you can never drive the system to some states) then W must be singular. Also, your "permissible statespace" is a funny sort of definition. The statespace is R^n for a linear system (all of it). That is, unless there are singularities in the A(t), B(t) or C(t) matrices, the solution to the diffeq is defined over R^n. Once you say "permissible" where the "permissible statespace is smaller than R^n, you're admitting uncontrollability. Cheers ***********original question*************** Prof. Rose, Theorem 5.1 is a little misleading when it talks about an arbitrary state x1. I think it should say arbitrary state x1 in the range space of matrix W(t0, t1). For example, in a system where the state variables are dependent on each other, it is not possible to push the system to a state where the state variables are independent of each other. So complete controllability is when we can find an input to transfer the system between any two pints in the permissible state space. Is this correct? From crose Mon Dec 16 01:56:28 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA11894; Mon, 16 Dec 96 01:52:14 EST Date: Mon, 16 Dec 96 01:52:14 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612160652.AA11894@MOGLI.rutgers.edu> To: ece501 Subject: notes Status: R Hi Folks, You'll be allowed 2 sheets of notes for the examination (handwritten, both sides). Cheers From crose Mon Dec 16 01:57:15 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA11903; Mon, 16 Dec 96 01:53:03 EST Date: Mon, 16 Dec 96 01:53:03 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612160653.AA11903@MOGLI.rutgers.edu> To: ece501 Status: R >From crose@mogli.rutgers.edu Mon Dec 16 01:46:17 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA11852; Mon, 16 Dec 96 01:43:09 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA13083; Mon, 16 Dec 96 01:38:34 EST Date: Mon, 16 Dec 96 01:38:34 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612160638.AA13083@mogli> To: chanac@eden.rutgers.edu Subject: Re: Questions Cc: ece501@MOGLI.rutgers.edu Status: RO QUESTION: Dr Rose, I have several questions when I review the HW. May you answer my questions? - about the BIBO stability Chapter 4 prb 18 & 19, in soln, it said the eigenvalues of Matrixes are in LHP then we have BIBO. What's LHP, and also is this the other method to check the BIBO beside to use integral from t0 to tau of |tij(t,tau) dtau ? - pos definite What is pos definite? - chapter 4 prob 17, How they find the Routh Array? I didn't see any Routh Array in your note and textbook? - In pop quiz prob 4(b) They given y=[1,0,1]x , and ask u(t) and y(t) on [t0,t1] can x(t0) be uniquely determined? and in you answer said because observability matrix is full rank so, x(t0) can be determined. So is that mean when the system observable , x(t0) can be uniquely determined? - In Final exam Prob1. you ask the transition matrix is valid or not, is it use the properties of transition matrix to test them? if all 5 properties are satisfy, then transition matrix valid, if not than its not valid? - Do you have solution of the Final exam of 95? so I can check the answer Thanks Angela Chan ANSWERS: 1) there's a theorem which says that if all the eigenvalues have real parts <0 then the LTI system is BIBO stable (theorem 4.13... asked you guys to read this one since we did not have time to prove it in class). Basic proof is that the transition matrix must have terms of the form At^k e^(lambda t) and if all the lambdas have real part <0 then Theorem 4.11 can be easily applied (same goes for discrete with appropriate modifications). 2) The left half plane (LHP) is the left side of the eigenvalue complex plane. The poles of a system are its eigenvalues (roots of the characteristic polynomial) 3) A positive definite matrix is a symmetric real matrix all of whose eigenvalues are positive. This also means that x^T Qx > 0 for a pos def matrix Q for any vector x that is not identically zero. 4) We did not cover the Routh array, though it's referred to. Use the Hurwitz determinants. 5) I think you need to reread the observability chapter and look over the notes if I understand your question correctly. Observability means that given the input and the output over some interval, the state at the begining of the interval can be uniquely determined. 6) If I remember taht problem, it was pretty easy to check if the transition matrix is valid (for a continous system). It must always be invertible and must reduce to the identity matrix at t=t_0. 7) No solutions to the final. Ricardo MIGHT provide some, but frankly I'd rather you simply worry about the solutions as opposed to having them in front of you. I think there's enough problems with solutions between this and last years' tests and problem sets. Cheers From whyuan@biomed.rutgers.edu Mon Dec 16 11:18:16 1996 Return-Path: Received: from biomed.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA12193; Mon, 16 Dec 96 11:18:15 EST Received: (from whyuan@localhost) by biomed.rutgers.edu (8.6.12+bestmx+oldruq+newsunq+grosshack/8.6.12) id KAA07653 for crose@mogli.rutgers.edu; Mon, 16 Dec 1996 10:59:31 -0500 Date: Mon, 16 Dec 1996 10:59:31 -0500 From: Weihong Yuan Message-Id: <199612161559.KAA07653@biomed.rutgers.edu> To: crose@mogli.rutgers.edu Subject: Re: mailing to ece501@ece Status: R Dr. Rose: Could you give me some suggestion on the last year's final, problem 5 a, b, c. I know that if I want to check a Lyapunov Function, I should check it's continuity, whether it is minimum at xbar and its derevitive with respect to t. But in these three paticular problems. what is x1,x2? Are they state variables? or are they differernt group of solutions? Another problem , normally how to look for a Lyapunov function? thanks weihong From bsolanki@ece.rutgers.edu Mon Dec 16 11:44:07 1996 Return-Path: Received: from zen.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA12242; Mon, 16 Dec 96 11:40:33 EST Received: from ece.rutgers.edu (ee.rutgers.edu [128.6.46.13]) by zen.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) with SMTP id LAA09230 for ; Mon, 16 Dec 1996 11:33:59 -0500 Date: Mon, 16 Dec 96 11:33:53 EST From: Bindu Solanki To: ece501@mogli.rutgers.edu Subject: final exam #4 Message-Id: Status: R how would i go about getting started to solve a problem like #4 on last years final??? i guess i find it a bit difficult because there are no numbers and just variables when we are considering the "B" matrix.... thanks.. Bindu From crose@mogli.rutgers.edu Mon Dec 16 13:46:00 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA12356; Mon, 16 Dec 96 13:45:59 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA13385; Mon, 16 Dec 96 13:39:21 EST Date: Mon, 16 Dec 96 13:39:21 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612161839.AA13385@mogli> To: whyuan@biomed.rutgers.edu Subject: Re: mailing to ece501@ece Cc: crose@MOGLI.rutgers.edu Status: RO The x_1 and x_2 are the states of the given system. You'll need to look at the properties of a Lyapunov function and figure out whether they are or are not L-functions for the given system. As for finding L-functions in general... that's hard... more or an art form, so dont' worry too much about it. CHeers, From weicong@er7.rutgers.edu Mon Dec 16 13:35:03 1996 Return-Path: Received: from eden-backend.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA12344; Mon, 16 Dec 96 13:31:47 EST Received: from er7.rutgers.edu (er7.rutgers.edu [165.230.180.135]) by eden-backend.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.12) with SMTP id NAA23323; Mon, 16 Dec 1996 13:25:42 -0500 Date: Mon, 16 Dec 1996 13:25:35 -0500 (EST) From: Wang Weicong To: Bindu Solanki Cc: ece501@MOGLI.rutgers.edu Subject: Re: final exam #4 In-Reply-To: Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO I think the first question is much difficult. The other only use A=T*^*invert T To the rank(B) problem, if rank = n, then try the Laplace transform, you get the B invert. If B is n*m, the decompose B=B1*B2, B1 has n independent rows while B2 has m independent m cols. Wang Weicong Rutgers, the State Univ. of New Jersey L.P.O. 16293 P.O.Box 5064 New Brunswick N.J. 08903 Tel:908-878-2734 On Mon, 16 Dec 1996, Bindu Solanki wrote: > > how would i go about getting started to solve a problem like #4 on > last years final??? i guess i find it a bit difficult because there are no > numbers and just variables when we are considering the "B" matrix.... > > thanks.. > > Bindu > From crose@mogli.rutgers.edu Mon Dec 16 13:50:07 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA12362; Mon, 16 Dec 96 13:46:01 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA13390; Mon, 16 Dec 96 13:39:47 EST Date: Mon, 16 Dec 96 13:39:47 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612161839.AA13390@mogli> To: ece501@MOGLI.rutgers.edu Subject: question Status: RO QUESTION Dr. Rose: Could you give me some suggestion on the last year's final, problem 5 a, b, c. I know that if I want to check a Lyapunov Function, I should check it's continuity, whether it is minimum at xbar and its derevitive with respect to t. But in these three paticular problems. what is x1,x2? Are they state variables? or are they differernt group of solutions? Another problem , normally how to look for a Lyapunov function? thanks ANSWER: The x_1 and x_2 are the states of the given system. You'll need to look at the properties of a Lyapunov function and figure out whether they are or are not L-functions for the given system. As for finding L-functions in general... that's hard... more or an art form, so dont' worry too much about it. CHeers, From crose@mogli.rutgers.edu Mon Dec 16 14:44:21 1996 Return-Path: Received: from mogli.rutgers.edu by MOGLI.rutgers.edu with UUCP (4.1/25-eef) id AA12515; Mon, 16 Dec 96 14:40:45 EST Full-Name: Christopher Rose Received: by mogli (4.1/SMI-4.1) id AA13419; Mon, 16 Dec 96 13:45:39 EST Date: Mon, 16 Dec 96 13:45:39 EST From: crose@mogli.rutgers.edu (Christopher Rose) Message-Id: <9612161845.AA13419@mogli> To: bsolanki@ee.rutgers.edu Subject: Re: final exam #4 Cc: ece501@MOGLI.rutgers.edu Status: R You have to prove that any desired output form can be obtained. The simplest way to go is through Laplace transforms. Numbers aren't necessary. Cheers From yufeng@caip.rutgers.edu Tue Dec 17 01:44:42 1996 Return-Path: Received: from caipfs.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA13042; Tue, 17 Dec 96 01:44:41 EST Received: from caip.rutgers.edu (caip.rutgers.edu [128.6.91.2]) by caipfs.rutgers.edu (8.7.6/8.7.3) with ESMTP id BAA19607 for ; Tue, 17 Dec 1996 01:38:34 -0500 (EST) From: Yufeng Liang Received: (yufeng@localhost) by caip.rutgers.edu (8.7.6/8.6.9) id BAA15146 for crose@MOGLI; Tue, 17 Dec 1996 01:38:33 -0500 (EST) Date: Tue, 17 Dec 1996 01:38:33 -0500 (EST) Message-Id: <199612170638.BAA15146@caip.rutgers.edu> To: crose@MOGLI.rutgers.edu Status: R Sorry for bugging you again. I found some other things that I don't agree with on the "answer". Problem5: a) V(x1) = 0 doesn't mean that V(x) can't be a Lyapunov function. Actually I think it is a Lyapunov function. It's continous, it has the min value x1=0, x2 = 0, from the transmittion function of the system we can know that dV/dt = (c1 * x1 - c2 * x2)^2 - c3* x2 ^ 2, where ci are positive real numbers. Then we have dV/dt <= 0. b) Not a Lyapunov function b/c min value points are non-unique. Yufeng From hatangad@ece.rutgers.edu Tue Dec 17 09:49:26 1996 Return-Path: Received: from ece.rutgers.edu by MOGLI.rutgers.edu (4.1/25-eef) id AA13266; Tue, 17 Dec 96 09:46:58 EST Received: (from hatangad@localhost) by ece.rutgers.edu (8.6.12+bestmx+oldruq+newsunq/8.6.6) id JAA01726; Tue, 17 Dec 1996 09:40:08 -0500 Date: Tue, 17 Dec 1996 09:40:08 -0500 From: Sonya Hatangadi Message-Id: <199612171440.JAA01726@ece.rutgers.edu> To: crose@mogli.rutgers.edu Cc: ece501@mogli.rutgers.edu Subject: COntrollability Content-Length: 422 Status: R In the electrical circuit example that we worked on in class (12/5/96), we said that the circuit was controllable as the controllability matrix had full rank. However, we concluded that since the state variables V1 and V2 were related to each other, ie. V1 = RC(dV2/dt) + V2 we could not take the system to a state where this equation did not hold. Is'nt this a contradiction? Sonya. From crose Tue Dec 17 10:24:58 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA13300; Tue, 17 Dec 96 10:22:17 EST Date: Tue, 17 Dec 96 10:22:17 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612171522.AA13300@MOGLI.rutgers.edu> To: hatangad@ece.rutgers.edu Subject: Re: COntrollability Cc: ece501 Status: R THe equation always holds. What controllability says is that the state (not the trajectory) can be taken to wherever you'd like. So there's not contradiction. CHeers, From crose Tue Dec 17 14:07:27 1996 Return-Path: Received: by MOGLI.rutgers.edu (4.1/25-eef) id AA13729; Tue, 17 Dec 96 14:06:36 EST Date: Tue, 17 Dec 96 14:06:36 EST From: Christopher Rose Full-Name: Christopher Rose Message-Id: <9612171906.AA13729@MOGLI.rutgers.edu> To: ANALUCI@aol.com Subject: Re: Lyapunov Cc: crose@MOGLI.rutgers.edu Status: R Your first two contentions are ok. For 3) you only need to show dV/dt < 0