1 00:00:00,500 --> 00:00:01,300 ‫Welcome back. 2 00:00:01,820 --> 00:00:10,190 ‫In the world of differential equations, there is a type of differential equations called a linear time 3 00:00:10,190 --> 00:00:20,770 ‫invariant second order differential equation, so an LTI second order differential equation. 4 00:00:21,530 --> 00:00:24,860 ‫And this is its general form. 5 00:00:25,580 --> 00:00:26,960 ‫That's how it looks like. 6 00:00:27,790 --> 00:00:32,480 ‫Now, before we go any further, let's first take a simpler example. 7 00:00:33,080 --> 00:00:43,700 ‫Let's say that we have a typical equation in which Y equals F as a function of X, and let's say that 8 00:00:43,700 --> 00:00:51,610 ‫F as a function of X equals three X squared plus two. 9 00:00:52,370 --> 00:00:59,690 ‫So that means that Y equals three X squared plus two. 10 00:01:00,420 --> 00:01:02,870 ‫It's a typical function, right. 11 00:01:03,740 --> 00:01:15,050 ‫And so your ex is an independent variable and your wife is a dependent variable, so your Y is a function 12 00:01:15,050 --> 00:01:24,740 ‫of X, your Y depends on X, you put some kind of number into the X variable and then that will give 13 00:01:24,740 --> 00:01:27,950 ‫you a value for your Y variable. 14 00:01:28,760 --> 00:01:39,140 ‫A differential equation, however, is an equation in which the dependent variable Y not only depends 15 00:01:39,140 --> 00:01:49,190 ‫on the independent variable X, but it also depends on the change of Y with respect to X. 16 00:01:50,210 --> 00:02:01,760 ‫DUI, the ex and the derivative can be first order or even a second order, or it can even be a third 17 00:02:01,760 --> 00:02:05,130 ‫order or a fourth order, et cetera. 18 00:02:05,560 --> 00:02:06,220 ‫Right. 19 00:02:07,070 --> 00:02:16,610 ‫The important thing is to have at least one term that contains the derivative of Y with respect to X, 20 00:02:17,120 --> 00:02:24,310 ‫regardless of its order, whether its first order or second or third order, it doesn't matter. 21 00:02:25,010 --> 00:02:27,460 ‫And then it's a differential equation. 22 00:02:28,160 --> 00:02:37,070 ‫You can also rewrite this general differential equation form like this queue times y, which is this 23 00:02:37,070 --> 00:02:50,570 ‫one here, equals F as a function of X minus P, D, Y, the X, which is this one that you put on the 24 00:02:50,570 --> 00:03:00,470 ‫other side of the equation sine and then minus the second derivative of Y with respect to X and by the 25 00:03:00,470 --> 00:03:07,060 ‫way, the P and Q letters, they're simply random constants. 26 00:03:07,070 --> 00:03:13,300 ‫So there's some kind of numbers, they are not functions but there some kind of constants. 27 00:03:14,120 --> 00:03:16,040 ‫And now you take this. 28 00:03:16,040 --> 00:03:29,870 ‫Q And you put it here, then also here and then also here, because now you can express y variable in 29 00:03:29,870 --> 00:03:31,310 ‫terms of everything else. 30 00:03:32,360 --> 00:03:43,790 ‫And if our F as a function of X equals three times X squared plus two, then that means that Y equals 31 00:03:44,810 --> 00:03:58,580 ‫three times X squared plus two divided by Q then minus P divided by Q times this derivative of Y with 32 00:03:58,580 --> 00:04:09,470 ‫respect to X and then minus one over Q times the second derivative of Y with respect to X.. 33 00:04:10,320 --> 00:04:23,690 ‫So you see that in this case Y is a function of X, Y, the X and the second derivative of Y with respect 34 00:04:23,690 --> 00:04:34,820 ‫to X and because of this and this, you know that it's a differential equation since in our case the 35 00:04:34,820 --> 00:04:48,230 ‫high is derivative, is the second derivative, then we call it an LTI second order differential equation. 36 00:04:49,370 --> 00:04:53,630 ‫It's second order because the highest derivative is the second derivative. 37 00:04:54,500 --> 00:05:03,050 ‫Now if F as a function of X equals a zero, not this function any more, three times X squared plus 38 00:05:03,050 --> 00:05:08,300 ‫two, but instead it equals zero, then. 39 00:05:09,300 --> 00:05:14,050 ‫Your differential equation will look like this. 40 00:05:14,220 --> 00:05:23,810 ‫This would be its general form, and in that case, why only depends on its own derivatives. 41 00:05:24,600 --> 00:05:38,190 ‫And in that case, we say that this differential equation is a homogeneous LTI second order differential 42 00:05:38,190 --> 00:05:38,850 ‫equation. 43 00:05:39,660 --> 00:05:44,190 ‫And now what does it mean when we talk about solving a differential equation? 44 00:05:44,730 --> 00:05:55,750 ‫Solving a differential equation means that you go from this form here to this form here. 45 00:05:56,010 --> 00:05:56,700 ‫All right. 46 00:05:57,570 --> 00:06:10,290 ‫So you have y as a function of X, D.U.I., D, X, and then the second derivative of Y with respect 47 00:06:10,290 --> 00:06:12,870 ‫to X, and that equals something. 48 00:06:12,870 --> 00:06:16,610 ‫You have some kind of equation just like here. 49 00:06:17,460 --> 00:06:26,310 ‫And if you solve a differential equation, then your solution will be a function and that function will 50 00:06:26,310 --> 00:06:30,180 ‫get rid of these derivative independent variables. 51 00:06:30,900 --> 00:06:39,870 ‫And then you will have y as a function of X equals something, some kind of function that will only 52 00:06:39,870 --> 00:06:43,200 ‫contain the X independent variable. 53 00:06:43,410 --> 00:06:48,500 ‫It will not contain the derivatives as independent variables anymore. 54 00:06:49,320 --> 00:06:56,100 ‫So remember, the solution for a differential equation is not just a number, it is a function. 55 00:06:57,030 --> 00:07:05,550 ‫You get rid of the derivatives and you express y only in terms of X, and that's what solving differential 56 00:07:05,550 --> 00:07:07,050 ‫equations is all about. 57 00:07:08,210 --> 00:07:17,870 ‫And now let's go back to our control inputs now that you know how a homogeneous LTI second order differential 58 00:07:17,870 --> 00:07:25,980 ‫equation looks like, which can be written like this, let's rewrite it like this. 59 00:07:26,720 --> 00:07:33,860 ‫Now we're going to keep the second derivative of Y with respect to X on one side of the equation sine 60 00:07:34,730 --> 00:07:37,760 ‫and the rest will be on the other side of the equation sine. 61 00:07:38,650 --> 00:07:48,310 ‫So I've put y here as the first term and then I've put the X here as the second term and then why is 62 00:07:48,310 --> 00:07:52,120 ‫multiplied now by negative Q and the Y? 63 00:07:52,120 --> 00:07:59,710 ‫The X is multiplied by negative P because we put them on the other side of the equation sine and now 64 00:07:59,920 --> 00:08:02,560 ‫I'm going to write our control loss. 65 00:08:03,010 --> 00:08:17,680 ‫So you X equals this equation K one times X plus K two times E dot x then this is for use of Y and this 66 00:08:17,680 --> 00:08:18,980 ‫is for use of Z. 67 00:08:19,810 --> 00:08:28,350 ‫And so now I ask you, what are you X, you Y and you Z. 68 00:08:29,140 --> 00:08:41,800 ‫If you look at this differential equation here and and these three control laws, then tell me what 69 00:08:41,800 --> 00:08:46,460 ‫are you up X use of Y and use up Z. 70 00:08:47,410 --> 00:08:49,160 ‫So let it be your exercise. 71 00:08:49,450 --> 00:08:53,480 ‫Think about it and I will see you in the next video. 72 00:08:54,160 --> 00:08:55,000 ‫Thank you very much.