1 00:00:00,550 --> 00:00:01,330 ‫Welcome back. 2 00:00:01,750 --> 00:00:10,660 ‫So I hope that you have put this differential equation in your state's best form and that's how you 3 00:00:10,660 --> 00:00:11,520 ‫should approach it. 4 00:00:12,130 --> 00:00:23,920 ‫So you take it like this, X dot equals X dot, and then you will have X double dot equals, and then 5 00:00:23,950 --> 00:00:28,630 ‫everything else will go on to the other side of the equation sine. 6 00:00:28,990 --> 00:00:38,200 ‫So you're going to have minus two times X minus three times X dot. 7 00:00:39,160 --> 00:00:51,040 ‫And then plus two times T. So your state time derivative vector will be like this X dot here and then 8 00:00:51,430 --> 00:00:58,030 ‫X double dot here equals then you will have your A matrix. 9 00:00:59,110 --> 00:01:08,950 ‫And then this will be your state vector here, you will have X here and X dot here plus, and then you 10 00:01:08,950 --> 00:01:12,730 ‫will have the B Matrix here and then. 11 00:01:13,610 --> 00:01:21,500 ‫I can put tea here, and so since you have two tea here, then that means that the B Matrix would be 12 00:01:21,500 --> 00:01:29,660 ‫like this zero here and two here, and then you're a matrix will be like this again, zero here and 13 00:01:29,660 --> 00:01:38,780 ‫one here, because your X dot equals X dot and then minus two will be here and then minus three will 14 00:01:38,780 --> 00:01:39,350 ‫be here. 15 00:01:40,170 --> 00:01:44,380 ‫So you see you still have an LTI form. 16 00:01:44,900 --> 00:01:52,430 ‫However, now you have this big matrix and that's because your differential equation is not a homogeneous 17 00:01:52,430 --> 00:01:53,360 ‫one anymore. 18 00:01:54,230 --> 00:01:57,700 ‫Now your general form is like this. 19 00:01:58,550 --> 00:02:02,420 ‫That's the form that you are familiar with. 20 00:02:03,290 --> 00:02:11,400 ‫But OK, let's go back to our error LTI differential equation, which is here and now. 21 00:02:11,420 --> 00:02:18,650 ‫This one is homogeneous and therefore you only have this a matrix here from linear algebra. 22 00:02:19,190 --> 00:02:23,300 ‫You can learn that when you have a matrix, a. 23 00:02:24,230 --> 00:02:34,730 ‫Like this, then this Matrix A can be multiplied by a special vector V, it's a special vector because 24 00:02:34,730 --> 00:02:42,830 ‫it gives you a relationship like this, a matrix times this special vector V equals. 25 00:02:43,280 --> 00:02:46,570 ‫And then I'm going to put here lambda till then. 26 00:02:47,210 --> 00:02:55,630 ‫And then an identity matrix that has the same dimension like the A matrix and then this lambda till 27 00:02:55,640 --> 00:03:03,040 ‫is just a constant and then I'm going to multiply that with this special vector again. 28 00:03:03,830 --> 00:03:06,420 ‫So in our case it will look like this. 29 00:03:07,370 --> 00:03:19,880 ‫This is your a matrix times the vector equals and now this thing here will look like this lambda tilde 30 00:03:19,880 --> 00:03:22,070 ‫here and then lambda tilde here. 31 00:03:22,070 --> 00:03:24,710 ‫You will have zero here and zero here. 32 00:03:25,460 --> 00:03:33,650 ‫And that's because you had this lambda tilde that you multiplied by an identity matrix that looked like 33 00:03:33,650 --> 00:03:34,000 ‫this. 34 00:03:34,730 --> 00:03:43,250 ‫And then you multiply all that by the special vector V when you have this kind of relationship then 35 00:03:43,760 --> 00:03:56,360 ‫the V vector is called an eigenvector and then lambda there is called and eigenvalue. 36 00:03:56,960 --> 00:04:00,290 ‫So you have eigenvectors and eigenvalues. 37 00:04:01,070 --> 00:04:04,280 ‫And this relationship here is a unique relationship. 38 00:04:04,700 --> 00:04:12,620 ‫It means that a square matrix has unique eigenvalues and eigenvectors associated with it. 39 00:04:13,550 --> 00:04:22,520 ‫The procedure to find the eigenvalues and eigenvectors for our A Matrix is to take advantage of this 40 00:04:22,520 --> 00:04:23,780 ‫unique relationship. 41 00:04:24,680 --> 00:04:29,450 ‫If this is your relationship here, then you can rewrite it like this. 42 00:04:30,470 --> 00:04:34,040 ‫The Matrix times the eigenvector minus. 43 00:04:35,000 --> 00:04:42,800 ‫Then this matrix containing the eigenvalues times the eigenvector equals zero. 44 00:04:43,880 --> 00:04:50,780 ‫Then I can factor out my eigenvectors and so I factor them out like this. 45 00:04:51,620 --> 00:04:58,250 ‫And so if I subtract this matrix from this matrix, then I will have this kind of matrix here. 46 00:04:59,290 --> 00:05:10,120 ‫Like this, and so if I take this matrix now, this difference of these two matrices and they multiplied 47 00:05:10,120 --> 00:05:13,960 ‫by the eigenvector, then it has to equal zero. 48 00:05:15,130 --> 00:05:24,250 ‫Well, more precisely, it should be a zero vector because this is a two by two matrix and this eigenvector 49 00:05:24,670 --> 00:05:26,320 ‫is two by one. 50 00:05:27,160 --> 00:05:29,610 ‫So two by two matrix. 51 00:05:30,130 --> 00:05:38,620 ‫And then if you multiply it by two by one vector, then the inner dimensions cancel out and then you 52 00:05:38,620 --> 00:05:46,140 ‫will have a vector, a zero vector, which has two rows and then one column like this. 53 00:05:47,260 --> 00:05:55,320 ‫So we have this matrix here that we multiply by the eigenvector and then we have this zero vector here. 54 00:05:55,960 --> 00:06:04,450 ‫And now if we assume that we know what K one and K two are, if we assume that we know the values for 55 00:06:04,450 --> 00:06:09,860 ‫these constants, then we would like to be able to calculate our lambda tilde. 56 00:06:10,540 --> 00:06:15,820 ‫However, the problem is that we don't know our eigenvector either. 57 00:06:16,150 --> 00:06:19,590 ‫Our eigenvector can be written like this. 58 00:06:19,780 --> 00:06:23,430 ‫It has X and Y, so they are unknown. 59 00:06:23,470 --> 00:06:26,600 ‫It has two elements, but we don't know what they are. 60 00:06:27,400 --> 00:06:32,130 ‫So it would be nice if we could get rid of them for a moment. 61 00:06:32,860 --> 00:06:35,250 ‫And so therefore, I want to show you something. 62 00:06:35,650 --> 00:06:39,010 ‫Let's take a random two by two matrix. 63 00:06:39,580 --> 00:06:46,240 ‫It has an element B, element C and D element just like this one. 64 00:06:46,570 --> 00:06:50,110 ‫But now you have random general variables here. 65 00:06:50,830 --> 00:07:01,680 ‫And then we multiply this matrix by our eigenvector X and Y and all that equals zero and zero. 66 00:07:02,440 --> 00:07:07,290 ‫And now we're going to multiply this matrix by this eigenvector. 67 00:07:07,900 --> 00:07:16,510 ‫So it's going to be eight times X plus B times Y equals zero. 68 00:07:17,020 --> 00:07:26,290 ‫And then you will have C times X plus D times Y equals zero. 69 00:07:27,310 --> 00:07:31,510 ‫Then I'm going to take this first row and I'm going to write it down like this. 70 00:07:32,170 --> 00:07:48,820 ‫Eight times X equals minus B times Y or Y equals minus A times X divided by B. 71 00:07:49,540 --> 00:07:58,390 ‫And now I'm going to take this expression here and I'm going to put it here into this equation, into 72 00:07:58,390 --> 00:08:01,540 ‫this Y variable in the second equation. 73 00:08:02,350 --> 00:08:16,810 ‫So I'm going to have three times X plus D times minus A times X and divided by B and all that equals 74 00:08:16,840 --> 00:08:17,420 ‫zero. 75 00:08:18,310 --> 00:08:36,610 ‫I can now factor out the X's and so I will have C plus D times minus A divided by B and then all that 76 00:08:37,000 --> 00:08:42,090 ‫I multiply by X and all that equals zero. 77 00:08:43,000 --> 00:08:50,650 ‫And then I can divide this side by X and I can divide this side by X and then this one will be one and 78 00:08:50,650 --> 00:08:52,520 ‫then this one will be zero. 79 00:08:53,200 --> 00:09:03,340 ‫So in total I will have C minus eight times D divided by B equals zero. 80 00:09:04,090 --> 00:09:08,260 ‫I can also rewrite this expression like this. 81 00:09:08,950 --> 00:09:19,780 ‫C equals eight times D divided by B and then I can also rewrite this expression like this. 82 00:09:20,770 --> 00:09:33,580 ‫B times C equals eight times D, and then I can rewrite this expression like this eight times D minus 83 00:09:33,970 --> 00:09:37,530 ‫B times C equals zero. 84 00:09:38,440 --> 00:09:40,180 ‫And now what is our AI? 85 00:09:40,780 --> 00:09:43,490 ‫Our AI here is this one. 86 00:09:43,780 --> 00:09:55,900 ‫So this one would be our minus lambda tilde then our D will be this one here. 87 00:09:56,910 --> 00:10:08,850 ‫So I will have K minus Lamda till the here and then we will have minus and then what is R, B will our 88 00:10:08,850 --> 00:10:10,900 ‫B is this one here. 89 00:10:11,460 --> 00:10:12,500 ‫So one. 90 00:10:13,440 --> 00:10:14,860 ‫And what is our C. 91 00:10:15,420 --> 00:10:17,880 ‫Well our C is this one here. 92 00:10:17,880 --> 00:10:22,420 ‫K one times K one equals zero. 93 00:10:23,310 --> 00:10:31,200 ‫So if we assume values for K one and K two then we can easily compute our lambda tilde. 94 00:10:32,190 --> 00:10:40,290 ‫We can easily compute our eigenvalues because essentially you will have a quadratic function here, 95 00:10:40,290 --> 00:10:50,010 ‫so Lunda tilde squared is because minus lambda till the times minus lambda till the here, then minus 96 00:10:50,010 --> 00:10:57,220 ‫K, two times lambda till there and then minus K one. 97 00:10:58,080 --> 00:11:06,150 ‫So you will have a quadratic equation and then you have formulas for computing your eigenvalues when 98 00:11:06,150 --> 00:11:08,480 ‫they are part of this quadratic equation. 99 00:11:09,300 --> 00:11:15,510 ‫But before we do that, I want to ask you something for all of you who have studied linear algebra. 100 00:11:15,660 --> 00:11:16,580 ‫Tell me something. 101 00:11:17,160 --> 00:11:20,340 ‫Does this expression seem familiar to you? 102 00:11:21,280 --> 00:11:28,990 ‫Perhaps you remember about computing the determinant of a square matrix, so if you have a square matrix, 103 00:11:28,990 --> 00:11:31,270 ‫you can compute its determinant. 104 00:11:32,140 --> 00:11:34,750 ‫The notation of the determinant is like this. 105 00:11:35,200 --> 00:11:45,550 ‫You have two bars and then inside those bars you will have A, B, C and D, and you can compute the 106 00:11:45,550 --> 00:11:46,930 ‫determinant like this. 107 00:11:47,440 --> 00:11:52,930 ‫Eight times D minus B, times C. 108 00:11:53,840 --> 00:12:03,560 ‫You see this expression, that's your determinant, and that's why the general formula for computing 109 00:12:03,560 --> 00:12:11,970 ‫the eigenvalues is taking the determinant of this matrix here and equating it to zero. 110 00:12:12,740 --> 00:12:19,280 ‫So you take the determinant of this matrix and you equate it to zero. 111 00:12:19,430 --> 00:12:25,430 ‫And then you followed this procedure and, well, then you equate it to zero. 112 00:12:26,120 --> 00:12:29,780 ‫And then through that, you will find your eigenvalues. 113 00:12:30,990 --> 00:12:41,790 ‫And so now, in order to find the eigenvalues, you have to take the determinant of this matrix and 114 00:12:41,790 --> 00:12:43,350 ‫you have to equate it to zero. 115 00:12:44,220 --> 00:12:45,720 ‫So I'm going to write it here. 116 00:12:46,920 --> 00:12:50,070 ‫So you see, that's the determinate of this matrix. 117 00:12:50,070 --> 00:12:51,680 ‫And then we equate it to zero. 118 00:12:51,960 --> 00:13:01,950 ‫And now we take the determinant of this matrix that looks like this minus lambda till there times K 119 00:13:01,980 --> 00:13:05,220 ‫two minus lambda till there. 120 00:13:05,820 --> 00:13:18,570 ‫So this element times, this element minus and now this element times this element. 121 00:13:19,380 --> 00:13:28,990 ‫So minus one times K one or you can just put K one here and that equals to zero. 122 00:13:29,730 --> 00:13:41,220 ‫And so we can open up the parentheses minus lambda till the times K two and then minus minus you will 123 00:13:41,220 --> 00:13:50,050 ‫have plus lambda till the squared and minus K one equals zero. 124 00:13:50,880 --> 00:14:01,680 ‫And so if I rearrange the terms then it will look like this lambda till the squared minus K to lambda 125 00:14:01,680 --> 00:14:07,870 ‫till they're minus K one equals zero. 126 00:14:08,760 --> 00:14:18,600 ‫And now I'm going to find out what the Lambda Tilda's are using the quadratic equations route finding 127 00:14:18,600 --> 00:14:19,340 ‫formula. 128 00:14:20,160 --> 00:14:29,280 ‫So Lambda still there and there are two of them because it's a quadratic equation, it's k two over 129 00:14:29,280 --> 00:14:46,380 ‫two and then plus minus square root K two over two and then you have squared here and then plus K one. 130 00:14:47,310 --> 00:14:57,060 ‫And look does it seem familiar to you, you had the exact same thing for your lambdas or polls when 131 00:14:57,060 --> 00:15:02,540 ‫you were computing the solution for your differential equation in a traditional way. 132 00:15:03,480 --> 00:15:12,240 ‫So this means that if you take a differential equation that is linear time invariant, and no matter 133 00:15:12,240 --> 00:15:24,030 ‫whether it's homogeneous or not, and if you put that differential equation in this form X, that vector 134 00:15:24,030 --> 00:15:33,540 ‫equals a matrix times X vector plus B matrix times the input vector. 135 00:15:34,470 --> 00:15:44,520 ‫And then if you take that A matrix here and you calculate it's eigenvalues, then those eigenvalues 136 00:15:44,850 --> 00:15:54,360 ‫which are lambda till they're they're the same good old poles of the system, the same good old power 137 00:15:54,360 --> 00:15:56,490 ‫constants lambdas. 138 00:15:57,300 --> 00:16:00,830 ‫So lambda actually equals lambda tilde. 139 00:16:01,620 --> 00:16:05,310 ‫So you see you have several names for the same thing. 140 00:16:06,440 --> 00:16:14,990 ‫If this is your general solution here, then this Lunda one and Lunda to. 141 00:16:16,170 --> 00:16:23,370 ‫They are your exponent constants here, in other words, they are your polls, in other words, they 142 00:16:23,370 --> 00:16:25,750 ‫are your eigenvalues. 143 00:16:26,730 --> 00:16:35,550 ‫So saying that you are choosing your polls or placing your polls in the lab plus domain that looked 144 00:16:35,550 --> 00:16:43,630 ‫like this, you have your real number line here and then the imaginary number line here. 145 00:16:44,490 --> 00:16:52,770 ‫So saying that you are choosing your polls or placing them here is the same thing, like saying that 146 00:16:52,770 --> 00:16:57,870 ‫you're choosing your eigenvalues and placing them here. 147 00:16:58,590 --> 00:17:08,940 ‫And after you have chosen your eigenvalues, you compute your K one and K two constants with the formulas 148 00:17:08,940 --> 00:17:10,980 ‫that we had previously derived. 149 00:17:11,850 --> 00:17:14,080 ‫So that's what I wanted to point out. 150 00:17:15,210 --> 00:17:24,180 ‫So now you know that in the control engineering context, the eigenvalues of your A matrix are your 151 00:17:24,180 --> 00:17:27,120 ‫polls, your polls of the system.