1 00:00:00,120 --> 00:00:00,570 OK. 2 00:00:01,020 --> 00:00:03,950 They have made a general introduction to the robust control. 3 00:00:04,000 --> 00:00:10,320 Let's listen now we will continue with one of the robust control methods, namely variable structure 4 00:00:10,380 --> 00:00:14,400 control or a sliding mode control 5 00:00:16,950 --> 00:00:23,640 sliding surface control, which is another name given you understand in this lesson why these names 6 00:00:23,640 --> 00:00:25,680 are given to this method. 7 00:00:26,490 --> 00:00:35,190 Indeed, there are several different robust control techniques, but we will see variable, um, variable 8 00:00:35,190 --> 00:00:41,070 surface control, variable structure control, excuse me, or the access control method. 9 00:00:42,510 --> 00:00:45,150 You can ask why this control method? 10 00:00:45,720 --> 00:00:48,630 Because firstly, it's simple to implement. 11 00:00:48,840 --> 00:00:55,530 So if you can get your goal with a simple method right to look for difficult ones, then it provides 12 00:00:55,530 --> 00:01:01,590 effective robustness against uncertainties and disturbances, which is the reason we are looking for 13 00:01:01,590 --> 00:01:02,610 a robust controller. 14 00:01:03,000 --> 00:01:09,210 Finally, as it is effective and simple to implement, it can be applied in many practical cases. 15 00:01:09,570 --> 00:01:16,920 The best way of understanding the access control method and how it works is by working or an example 16 00:01:17,370 --> 00:01:18,900 by the control algorithms. 17 00:01:18,900 --> 00:01:24,300 We try to control the dynamics of the system, namely obtain desired dynamics. 18 00:01:24,630 --> 00:01:30,720 For example, until now, by developing control algorithms, what was our purpose to make trajectory 19 00:01:30,720 --> 00:01:32,460 tracking aerodynamics? 20 00:01:32,460 --> 00:01:34,530 Asymptotically stable, wasn't it? 21 00:01:35,010 --> 00:01:41,190 So as we are working over system dynamics, let's take a dynamic system and work on it. 22 00:01:41,790 --> 00:01:46,290 Here is a second order dynamic system with positive coefficient. 23 00:01:46,680 --> 00:01:53,760 Here, I want to note that here X can be replaced with any other variable that we want to control its 24 00:01:53,760 --> 00:01:54,510 dynamics. 25 00:01:54,780 --> 00:02:00,150 For example, we can replace X with a E, namely tracking error. 26 00:02:00,480 --> 00:02:06,180 So we will get tracking error dynamics and we will try to control it as we want. 27 00:02:06,660 --> 00:02:13,020 In short, we will work on X, which is general and can be replaced with any other variable. 28 00:02:13,530 --> 00:02:20,220 OK, before developing a controller for the system, we have to know stability properties of our system, 29 00:02:20,430 --> 00:02:25,230 namely whether it is stable or unstable or asymptotically stable. 30 00:02:25,560 --> 00:02:33,510 In order to do that, let's first write system in states based form so we can easily analyze its eigenvalues 31 00:02:33,510 --> 00:02:38,730 eigenvectors and saw from these stability properties. 32 00:02:39,060 --> 00:02:43,440 We will follow standard procedure for getting system into state based form. 33 00:02:44,340 --> 00:02:49,800 So first, define X one and X two variables as this. 34 00:02:50,190 --> 00:02:53,640 These two variables are the states of our system. 35 00:02:54,030 --> 00:03:00,060 Then we can write x double dot in terms of X one and x two states. 36 00:03:00,420 --> 00:03:03,930 Finally, we can write system in matrix form as this. 37 00:03:04,260 --> 00:03:08,850 Here is the evolution equation of the states from states based formulation. 38 00:03:09,390 --> 00:03:11,920 Export equals X plus b u. 39 00:03:11,940 --> 00:03:19,680 It's very well known equation, and I think you have already know this from this equation, we can determine 40 00:03:19,680 --> 00:03:28,380 a matrix and as we don't have input you v will take B matrix as zero from this perfect. 41 00:03:28,830 --> 00:03:31,140 But we want to see the output. 42 00:03:32,160 --> 00:03:34,140 But we want to see at the output. 43 00:03:34,620 --> 00:03:35,980 But would we want to see? 44 00:03:36,000 --> 00:03:36,540 Excuse me. 45 00:03:36,690 --> 00:03:41,690 But what we want to see at the output of the system, which states do we want to analyze? 46 00:03:41,910 --> 00:03:50,100 We want to analyze both X1 and X2 or we want to observe, not analyze, but we can say, observe OK 47 00:03:50,310 --> 00:04:01,770 and put both X1 and X1, namely Mm X and X stood because we want to see evolution of the states so we 48 00:04:01,770 --> 00:04:03,960 can write output equation in this way. 49 00:04:06,360 --> 00:04:12,810 And as the states base out of the equation is like that, we can determine see matrix from here, which 50 00:04:12,810 --> 00:04:15,270 is nothing but identity matrix. 51 00:04:16,020 --> 00:04:20,640 No, we will draw vs plot of the system to analyze the evolution of the system. 52 00:04:20,640 --> 00:04:24,210 States don't let faze blow terminology intimidate you. 53 00:04:24,540 --> 00:04:30,480 It's very simple, but very, very, very useful tool to determine convergence and stability properties 54 00:04:30,480 --> 00:04:31,470 of our system. 55 00:04:31,980 --> 00:04:35,280 Let's drill and analyze phase plot of our system. 56 00:04:35,760 --> 00:04:41,940 We will do it in MATLAB, which will help us to do calculations faster and getting more initiation by 57 00:04:41,940 --> 00:04:44,880 plotting several things, as you will see in a minute. 58 00:04:45,030 --> 00:04:51,260 OK, so let's start to analyze the face of his portrait of our system. 59 00:04:52,200 --> 00:04:54,540 Um, let's do that first. 60 00:04:54,540 --> 00:04:56,970 I want to explain to you what's our. 61 00:04:58,130 --> 00:04:59,610 I want to explain. 62 00:04:59,970 --> 00:05:05,450 Wants thing to you, and this this will be the variable and you will see what's variable eight. 63 00:05:05,850 --> 00:05:14,320 And so that's why I will explain what's variable and let's see first before the anger and the phase 64 00:05:14,320 --> 00:05:15,820 plot analysis of our system. 65 00:05:16,330 --> 00:05:21,640 We know that we can know our system's stability characteristics based on its eigenvalues. 66 00:05:21,640 --> 00:05:26,650 Yes, if eigenvalues have positive real parts, what will happen? 67 00:05:26,830 --> 00:05:32,740 The system will surely diverge because our system will be explored and unstable. 68 00:05:32,980 --> 00:05:38,070 If eigenvalues have negative real parts, then our system will converge. 69 00:05:38,080 --> 00:05:39,350 It will be stable. 70 00:05:39,370 --> 00:05:40,570 It will be very beautiful. 71 00:05:40,570 --> 00:05:41,830 Everything to sing well anyway. 72 00:05:42,100 --> 00:05:50,650 And if we have a complex eigenvalues with positive result that our system trajectories will oscillate 73 00:05:50,920 --> 00:05:59,020 and explode if in opposite way, we have eigenvalues with negative at all parts but a complex them, 74 00:05:59,020 --> 00:06:03,850 they will oscillate but converge to the origin. 75 00:06:03,880 --> 00:06:06,760 OK, so let's see. 76 00:06:06,760 --> 00:06:08,110 What's our eigenvalues? 77 00:06:08,110 --> 00:06:10,500 What's the type of like in our eigenvalues? 78 00:06:10,510 --> 00:06:10,780 OK? 79 00:06:10,960 --> 00:06:14,260 What kind of trajectories our system can have? 80 00:06:14,530 --> 00:06:15,670 We want to see this. 81 00:06:16,030 --> 00:06:16,350 OK? 82 00:06:16,360 --> 00:06:21,180 In order to do this, we will analyze the overlords of our system as you can be as you. 83 00:06:21,190 --> 00:06:27,550 We have said this, our system, OK, and we want to see, but symbolically, our eigenvalues. 84 00:06:27,550 --> 00:06:31,540 OK, so let's define, see and see as symbolic variables. 85 00:06:31,540 --> 00:06:32,710 And this is our aim matrix. 86 00:06:32,710 --> 00:06:40,980 We have seen this and let's calculate the a matrix, OK before before putting a value here. 87 00:06:40,990 --> 00:06:47,820 Let's just see what's our eigenvalues so run section and let's see our The Matrix. 88 00:06:47,830 --> 00:06:50,860 OK, that's our eigenvalues, OK? 89 00:06:51,190 --> 00:06:54,310 As you can see, it is our argument is we have two eigenvalues values. 90 00:06:54,550 --> 00:06:56,530 I have written them here clearly. 91 00:06:56,530 --> 00:07:04,810 As you can see, if you see him, there is here over two, but it doesn't have here because what I have 92 00:07:04,810 --> 00:07:11,620 done, I take it and put it in inside the square, so it becomes inside the square root. 93 00:07:11,620 --> 00:07:13,660 OK, it becomes divided by four. 94 00:07:13,670 --> 00:07:17,230 So um, OK, this very simple. 95 00:07:18,010 --> 00:07:19,770 So this is our first eigenvalues. 96 00:07:19,780 --> 00:07:21,070 This is our second diagonal. 97 00:07:21,160 --> 00:07:33,490 Excuse me, let me put it here minus C and minus minus C here. 98 00:07:33,490 --> 00:07:35,140 Okay, minus perfect. 99 00:07:35,470 --> 00:07:40,040 And I have many things on my table, so that's why I read first. 100 00:07:40,450 --> 00:07:41,590 I write very slowly. 101 00:07:42,220 --> 00:07:44,850 Anyway, let's continue. 102 00:07:44,990 --> 00:07:52,000 So now what we want to see now what we want to see and we will give first. 103 00:07:52,330 --> 00:07:54,130 We will replace the C. 104 00:07:54,130 --> 00:07:58,300 OK, it's two um values of that. 105 00:07:58,300 --> 00:08:03,940 I will say, OK, we will just give first c that I built a value of eight. 106 00:08:03,940 --> 00:08:08,990 Excuse me what I say variable a not variable value of a. 107 00:08:09,310 --> 00:08:15,480 OK, you can do this saying if you can say variable because its value will change anyway, which is 108 00:08:15,490 --> 00:08:15,940 positive. 109 00:08:15,940 --> 00:08:22,360 Okay, so what will happen if let's start with months, it will be easier. 110 00:08:23,140 --> 00:08:24,730 OK, let's see. 111 00:08:24,940 --> 00:08:25,690 Is positive? 112 00:08:25,690 --> 00:08:29,860 No, what we have is put it or put it with minus eight. 113 00:08:30,010 --> 00:08:37,570 So this will happen if we put minus eight instead of C V will have plus a and plus eight. 114 00:08:37,570 --> 00:08:39,670 Yes, and a positive number. 115 00:08:39,910 --> 00:08:41,110 So what will happen here? 116 00:08:41,380 --> 00:08:43,400 C squared over four plus a. 117 00:08:43,430 --> 00:08:44,950 OK, this is positive number. 118 00:08:44,950 --> 00:08:49,360 And if we calculate the square, it will be higher than C over two. 119 00:08:49,540 --> 00:08:50,440 So what will happen? 120 00:08:50,440 --> 00:08:53,250 This eigenvalue will be less than zero. 121 00:08:53,260 --> 00:08:57,460 This is perfect because we will have stable h. 122 00:08:57,640 --> 00:09:01,270 OK, a good value like agent value. 123 00:09:02,110 --> 00:09:03,760 But what will happen in this case? 124 00:09:03,760 --> 00:09:12,190 This will be higher than zero and their unstable and unstable yes, agan value. 125 00:09:12,190 --> 00:09:17,770 Let's see whatever it is, will be higher than zero because XY squared over four plus eight will be 126 00:09:17,770 --> 00:09:23,080 higher than surely with the square root will be higher than she'll see over two. 127 00:09:23,290 --> 00:09:23,690 OK. 128 00:09:24,380 --> 00:09:27,820 Um, see our tour. 129 00:09:27,820 --> 00:09:31,090 And also we have addition here because this side is positive. 130 00:09:31,090 --> 00:09:32,230 This side is positive. 131 00:09:32,500 --> 00:09:34,870 So positive, positive will give us positive. 132 00:09:34,870 --> 00:09:40,960 So we will have positive eigenvalue, which means that our system you are like a multi unstable and 133 00:09:40,960 --> 00:09:48,550 we know that ROM on stable item value is enough to make our system unstable so our trajectories will 134 00:09:48,550 --> 00:09:49,180 explode. 135 00:09:49,210 --> 00:09:55,450 However, we know that we have one eigenvectors or one eigenvalue that's stable. 136 00:09:55,450 --> 00:09:58,260 One eigenvalue that is unstable. 137 00:09:58,270 --> 00:09:58,790 OK? 138 00:09:59,090 --> 00:09:59,860 Vs what? 139 00:09:59,970 --> 00:10:07,270 This negative, OK, when a is we when we have a minus a C is equals to minus eight. 140 00:10:07,290 --> 00:10:13,260 OK, so maybe write it in this week, then c OK, excuse me. 141 00:10:14,190 --> 00:10:17,610 C equals two equals two minus eight. 142 00:10:17,910 --> 00:10:29,030 We will have one stable who was stable, not one unstable time by road. 143 00:10:29,040 --> 00:10:30,650 I mean eigenvalues. 144 00:10:30,660 --> 00:10:37,590 OK, so overall system is unstable system OK, because we have an unstable root. 145 00:10:37,620 --> 00:10:39,980 OK, so one unstable argument. 146 00:10:39,990 --> 00:10:48,750 So what's happening if we choose a ipse as a, then this will be both of these will be minus. 147 00:10:48,990 --> 00:10:50,430 Here is the interesting thing. 148 00:10:50,820 --> 00:10:57,000 Well, if our aim is less than C squared over four, then what will happen? 149 00:10:57,240 --> 00:11:03,780 Then this square root itself will be positive and the less than what c over two. 150 00:11:03,990 --> 00:11:06,700 So this eigenvalue will be positive. 151 00:11:06,720 --> 00:11:11,220 OK, let's make it greater than zero and this will become unstable. 152 00:11:11,520 --> 00:11:17,850 So in this case, also, this will be positive because our aim is less than C squared or for this will 153 00:11:17,850 --> 00:11:18,450 be positive. 154 00:11:18,450 --> 00:11:20,730 Square root on the square root will be a positive. 155 00:11:20,880 --> 00:11:25,650 Positive plus positive will give us positive and you will have unstable eigenvalue. 156 00:11:25,920 --> 00:11:29,130 OK, so our system will be unstable. 157 00:11:29,130 --> 00:11:39,060 So when C equals to a, we will have the one variant, which is one unstable root, OK, and one unstable 158 00:11:39,390 --> 00:11:43,460 root and overall system is unstable system. 159 00:11:43,470 --> 00:11:44,340 Okay, that's perfect. 160 00:11:44,860 --> 00:11:45,510 Not perfect. 161 00:11:45,510 --> 00:11:48,630 I mean, it's a bad thing, but our system is unstable. 162 00:11:49,710 --> 00:11:53,700 So what happens when a is and OK, let me write it. 163 00:11:53,970 --> 00:12:04,230 A here is, we have said, is less than less than C a c times c OK. 164 00:12:04,710 --> 00:12:07,980 We could start with C terms. 165 00:12:08,550 --> 00:12:09,150 OK. 166 00:12:09,190 --> 00:12:12,750 E360 OK, I sorta what's happening here? 167 00:12:13,170 --> 00:12:14,460 Let me write it down here. 168 00:12:15,150 --> 00:12:16,200 OK, let's write it. 169 00:12:16,350 --> 00:12:22,260 You can understand A is less than C squared c c squared divided by four. 170 00:12:22,260 --> 00:12:22,620 OK? 171 00:12:23,860 --> 00:12:27,690 The bottom we have, we can have also c equals to. 172 00:12:27,690 --> 00:12:29,700 Um, let's see. 173 00:12:29,700 --> 00:12:30,630 Let's go to this one. 174 00:12:30,960 --> 00:12:31,770 I'm to lazy. 175 00:12:32,550 --> 00:12:38,550 C equals to a but a is less than if it is in the news. 176 00:12:38,550 --> 00:12:45,390 C What's happening here is it is less than not less than by the greater the uh, greater than the C 177 00:12:45,390 --> 00:12:46,390 squared over four. 178 00:12:46,480 --> 00:12:47,280 What will happen? 179 00:12:47,580 --> 00:12:56,010 It is greater than C squared or for them now this will be negative and the square root of negative number. 180 00:12:56,010 --> 00:12:57,430 What will what will happen? 181 00:12:57,450 --> 00:13:02,860 We will have complex number with positive real part. 182 00:13:02,880 --> 00:13:08,370 OK, so our system will be, uh, greater than zero. 183 00:13:08,370 --> 00:13:15,870 Real parts will be the eigenvalue will have real parts greater than zero, but it will be complex number. 184 00:13:15,900 --> 00:13:16,500 OK. 185 00:13:17,280 --> 00:13:25,500 And also, this road will be complex because of Y because this square is greater than axis control for 186 00:13:25,500 --> 00:13:28,680 this will be negative and the square root of negative number will be complex. 187 00:13:28,680 --> 00:13:34,190 So we will have unstable either lower in both case, but oscillator its trajectory. 188 00:13:34,200 --> 00:13:36,870 OK, oscillate 189 00:13:39,120 --> 00:13:39,810 three, possibly. 190 00:13:39,840 --> 00:13:40,110 Three. 191 00:13:40,380 --> 00:13:44,660 OK, in the same way we can write for this one, I can believe it. 192 00:13:44,670 --> 00:13:45,540 Oscillate three. 193 00:13:45,540 --> 00:13:46,260 I can rally. 194 00:13:46,260 --> 00:13:47,670 OK, oscillator trajectory. 195 00:13:47,670 --> 00:13:49,590 OK, OK. 196 00:13:49,590 --> 00:13:50,970 One unstable root. 197 00:13:51,510 --> 00:13:55,140 We have c this one key here. 198 00:13:55,680 --> 00:13:57,780 Let me write it again was unstable. 199 00:13:57,930 --> 00:13:59,910 Unstable root, unstable system. 200 00:14:00,210 --> 00:14:00,510 Eh? 201 00:14:00,510 --> 00:14:05,430 Here is oh, we write it, we write it perfect. 202 00:14:05,850 --> 00:14:07,580 But I still say is great again. 203 00:14:07,690 --> 00:14:17,690 C squared or for this will be oscillator three because of complex, um, eigenvalue OK. 204 00:14:18,000 --> 00:14:19,970 Because of complex eigenvalue. 205 00:14:19,980 --> 00:14:21,120 OK, we understood this. 206 00:14:21,360 --> 00:14:24,890 We will have two modes depending on the value of C. 207 00:14:24,900 --> 00:14:27,060 OK, we can have. 208 00:14:27,060 --> 00:14:30,810 You either have unstable or we will be unstable. 209 00:14:30,810 --> 00:14:35,010 This oscillatory behaviour or gate, depending on the value of a ibs-c. 210 00:14:35,850 --> 00:14:39,750 By changing the value of C, it will be easier or will be months. 211 00:14:39,960 --> 00:14:44,910 So let's see here what will happen in this case. 212 00:14:44,910 --> 00:14:51,180 So let's draw now phase plots of our system after we have understood there was a like and values of 213 00:14:51,180 --> 00:14:51,390 it. 214 00:14:51,690 --> 00:14:52,830 OK, let's see. 215 00:14:53,220 --> 00:14:53,790 We have. 216 00:14:54,030 --> 00:14:58,050 OK, if you will define global eight, you will see that y we define it as global. 217 00:14:58,350 --> 00:14:59,580 This is our C this. 218 00:14:59,840 --> 00:15:04,730 Were a as you can safely choose as greater than C squared over four? 219 00:15:05,110 --> 00:15:12,050 OK, we will choose it two Times Square, Dover four and IPSE as minus eight. 220 00:15:12,060 --> 00:15:15,980 So we will also thrive is a but let's choose just minus eight. 221 00:15:16,190 --> 00:15:17,830 What will happen with minus eight? 222 00:15:17,840 --> 00:15:19,220 We will have one stable route. 223 00:15:19,220 --> 00:15:20,930 We will have one unstable route. 224 00:15:20,930 --> 00:15:23,000 Our our system will be unstable. 225 00:15:23,000 --> 00:15:24,710 Overall, we will see this one. 226 00:15:24,920 --> 00:15:26,180 This is our matrix. 227 00:15:26,180 --> 00:15:28,790 OK, this is our dispense simulation time. 228 00:15:28,790 --> 00:15:37,400 OK, let's and we here make a few good Android Holden because we will do multiple plots on a good because 229 00:15:37,400 --> 00:15:39,830 we will start with different initial conditions. 230 00:15:39,830 --> 00:15:45,950 As you can see the optimize the different initial condition process, OK by using four and we will define 231 00:15:45,950 --> 00:15:47,690 our initial condition here. 232 00:15:47,930 --> 00:15:49,360 This will be our X1. 233 00:15:49,370 --> 00:15:54,440 This will be our X2 and then we will use in order to simulate our system. 234 00:15:54,440 --> 00:15:57,440 So extort equals the apex system. 235 00:15:57,440 --> 00:15:57,680 OK? 236 00:15:57,680 --> 00:16:01,100 In order to simulate this, we use or the E-40, OK. 237 00:16:01,130 --> 00:16:06,710 Here we will provide will provide function with the name of derivatives, which is defined here. 238 00:16:06,710 --> 00:16:13,100 As you can see, it takes the state as input and time, and it will give output the derivative of our 239 00:16:13,460 --> 00:16:13,820 states. 240 00:16:13,820 --> 00:16:20,840 So it will take X and we will it will call it X and give export as an output, OK? 241 00:16:21,050 --> 00:16:26,120 And now you can see why we define the as global because we will use it inside this function also. 242 00:16:26,990 --> 00:16:30,630 So we got our export, OK? 243 00:16:30,890 --> 00:16:43,100 So I mean, X1, uh, X1 and X1 and X2 saw X, X and X thought, OK, and we split them because first 244 00:16:43,100 --> 00:16:45,100 column is X. 245 00:16:45,110 --> 00:16:49,730 The second column is export and we plot our next plot. 246 00:16:49,730 --> 00:16:51,980 As we have said, what was our latest payload? 247 00:16:51,990 --> 00:16:58,210 Facebook was showing their evolving trajectory of our states. 248 00:16:58,250 --> 00:17:01,670 OK, so our state's evolution trajectory? 249 00:17:01,940 --> 00:17:06,500 OK, so that's why we plot our states with respect to each other. 250 00:17:06,500 --> 00:17:13,790 So we respect x x with respect x one with respect to X two or X with respect to extort OK with the black 251 00:17:13,790 --> 00:17:14,300 color. 252 00:17:14,480 --> 00:17:19,280 And we also plotted the initial conditions in order to see from where we have started. 253 00:17:19,550 --> 00:17:22,670 And here I am doing what it. 254 00:17:23,380 --> 00:17:29,070 This is not too much necessary, but here I just draw my eigenvectors. 255 00:17:29,090 --> 00:17:33,330 There is nothing complicated here, but you can analyze yourself also. 256 00:17:33,350 --> 00:17:39,440 This is just I get the equation of a line for eigenvectors and I just throw them. 257 00:17:39,710 --> 00:17:45,720 If you have any question about that, please let me know under comment, and I will surely answer to 258 00:17:45,720 --> 00:17:47,570 you how I did this calculation. 259 00:17:48,830 --> 00:17:55,880 But if I have a not good go to this because it will take otherwise a bit of longer time and the necessary 260 00:17:55,880 --> 00:17:56,210 time. 261 00:17:56,600 --> 00:17:56,970 OK. 262 00:17:56,990 --> 00:18:02,750 Anyway, let's simulate our system and see a phase plot of our system. 263 00:18:02,750 --> 00:18:04,040 Let's simulated. 264 00:18:05,750 --> 00:18:06,710 And that's it. 265 00:18:07,070 --> 00:18:14,720 This is our face plot of our system, and it can be seen a bit intimidating, but it is very simple. 266 00:18:14,990 --> 00:18:16,730 So this is our two eigenvectors. 267 00:18:16,730 --> 00:18:24,040 As you can see the red one and the blue one, the blue one correspond to the diverging or unstable eigenvalue. 268 00:18:24,260 --> 00:18:25,550 I don't know if it's positive. 269 00:18:26,390 --> 00:18:36,590 The Red Line belong to the stable eigenvalue, OK, namely the one which has negative real population. 270 00:18:36,860 --> 00:18:37,730 So look at that. 271 00:18:37,730 --> 00:18:40,640 Our trajectories, the trajectories, evolving trajectory. 272 00:18:40,640 --> 00:18:42,830 It's very interesting, as you can see. 273 00:18:42,830 --> 00:18:44,320 For example, let's take this one. 274 00:18:44,340 --> 00:18:45,440 OK, let's take this one. 275 00:18:45,440 --> 00:18:47,360 It will be much more easier for us. 276 00:18:47,360 --> 00:18:56,480 As you can see both of eigenvectors effects to the evolution of the system trajectories. 277 00:18:56,810 --> 00:18:58,100 So let's start from here. 278 00:18:58,520 --> 00:19:00,320 As you can see, we started from here. 279 00:19:00,320 --> 00:19:01,670 It's our initial condition. 280 00:19:01,970 --> 00:19:08,690 As you can see, we're trying to converge to the center with this stable or this stable eigenvectors. 281 00:19:08,960 --> 00:19:15,020 OK, I can make that correspond to this stable eigenvalue because as you know, if our system is stable, 282 00:19:15,020 --> 00:19:16,550 we will converge to the center. 283 00:19:16,790 --> 00:19:23,570 So if we are over there, so in the direction of red eigenvectors, we try to converge to the center. 284 00:19:23,810 --> 00:19:31,280 However, with the direction of green, excuse me, blue eigenvectors, the diversion of the Infinity-V, 285 00:19:31,730 --> 00:19:33,170 I mean, system trajectories. 286 00:19:33,170 --> 00:19:41,830 OK, so as you can see, their motion mixed motion affects today they their effects are mixed and affect 287 00:19:41,840 --> 00:19:44,030 the system trajectory. 288 00:19:44,040 --> 00:19:49,730 So our system to really try to converge in the direction of red one by the blue one. 289 00:19:49,760 --> 00:20:01,160 It has a higher magnitude, so it will make our trajectory to explode or we'll make our system unstable. 290 00:20:01,160 --> 00:20:06,500 So our tragic, tragic as it tries to convert the center, but it explodes in the direction of blue 291 00:20:06,510 --> 00:20:06,770 one. 292 00:20:06,980 --> 00:20:11,390 You can see this in other items, other system trajectories. 293 00:20:11,390 --> 00:20:17,840 Also, as you can see here, see our system tries to go in the direction of, uh, red eigenvectors 294 00:20:17,840 --> 00:20:21,110 to the center because it want to converge here there. 295 00:20:21,320 --> 00:20:28,440 Because the effect of the red eigenvectors, both blue eigenvectors will affect and make it to divert. 296 00:20:28,580 --> 00:20:28,970 OK. 297 00:20:29,180 --> 00:20:32,870 As you can see, the divergence is higher than the convergence. 298 00:20:32,870 --> 00:20:33,380 Why? 299 00:20:33,560 --> 00:20:34,460 Let's see this. 300 00:20:34,460 --> 00:20:37,220 Why this is because of the values of lambda. 301 00:20:37,460 --> 00:20:39,680 Let's see our land, the values we have. 302 00:20:40,520 --> 00:20:41,600 I mean, eigenvalues. 303 00:20:41,840 --> 00:20:45,920 As you can see, positive eigenvalue is higher than the negative I can love. 304 00:20:45,920 --> 00:20:53,150 You saw unstable root has higher magnitude than the what stable root saw. 305 00:20:53,360 --> 00:21:01,220 Our system will diverge in the higher rate than it converges to the center. 306 00:21:01,220 --> 00:21:01,850 OK? 307 00:21:02,900 --> 00:21:06,280 You can see that as you can see, the divergence rate is higher. 308 00:21:06,290 --> 00:21:07,430 It's immediate. 309 00:21:07,430 --> 00:21:15,380 They tried to diverge to the UM diverge in the direction of blue eigenvectors because the route with 310 00:21:15,380 --> 00:21:18,980 positive real part is higher than the bruit with negative real path. 311 00:21:19,020 --> 00:21:26,760 OK, so as you can see, overall, our system is diverging because as you can see, every trajectory 312 00:21:26,760 --> 00:21:32,210 of diverging to the infinity if we increase the time and make it, let me see. 313 00:21:32,210 --> 00:21:38,000 For example, two second, uh, let's see what will happen if not to second. 314 00:21:38,000 --> 00:21:43,970 But let's see one second and you will see the trajectories will continue to explode. 315 00:21:44,420 --> 00:21:47,020 Let's see, as you can see here they are explored. 316 00:21:47,030 --> 00:21:48,770 They just got infinite. 317 00:21:48,780 --> 00:21:50,540 They don't converge to there. 318 00:21:50,780 --> 00:21:56,160 As you can see, they don't converge to the origin because our system is unstable. 319 00:21:56,180 --> 00:21:56,780 Perfect. 320 00:21:57,170 --> 00:22:02,180 Let's come back to the zero point five in order to have clean fuel under. 321 00:22:03,080 --> 00:22:05,000 And OK, let me mention one thing. 322 00:22:05,000 --> 00:22:11,360 Also, as you can see, we we stated that what will happen if we start from these initial conditions? 323 00:22:11,360 --> 00:22:14,480 But I want to note three things also. 324 00:22:15,260 --> 00:22:20,450 First of all, if we start from the origin, so our initial condition in zero zero, we will stay in 325 00:22:20,450 --> 00:22:25,040 zero zero hour trajectories will not either convert, I mean, not convert. 326 00:22:25,040 --> 00:22:26,330 I mean, they will not move. 327 00:22:26,540 --> 00:22:28,070 They will not be an evolution. 328 00:22:28,250 --> 00:22:29,240 Our system will stay. 329 00:22:29,240 --> 00:22:30,560 It will stay there. 330 00:22:30,590 --> 00:22:32,510 It will not go to anywhere else. 331 00:22:32,510 --> 00:22:40,910 But what if our system is our system trajectories is on our initial condition is on blue eigenvectors. 332 00:22:41,150 --> 00:22:44,900 What we have said in blue eigenvectors system trajectories will diverge. 333 00:22:45,170 --> 00:22:54,900 So if we are on blue eigenvectors, OK, blue, I can make Terra if there is no disturbance or any estimation 334 00:22:54,900 --> 00:22:58,240 in the variable or anything else in. 335 00:22:58,250 --> 00:23:04,670 If the system is purely ideal, then our initial can. 336 00:23:05,360 --> 00:23:12,880 If we start initial tension over this blue eigenvectors, our trajectories will diverge to the infinity 337 00:23:12,900 --> 00:23:16,940 because this blue eigenvectors is unstable. 338 00:23:17,090 --> 00:23:23,120 If we start from here, it will divert this side OK, minus infinity plus infinity. 339 00:23:23,510 --> 00:23:28,460 OK, what will happen if we are over the red one? 340 00:23:29,000 --> 00:23:37,730 If we are over the red eigenvectors, then our system and trajectory will converge to their origin. 341 00:23:37,730 --> 00:23:43,280 OK, which we want because we want our system dynamics worlds to converge to the origin. 342 00:23:43,430 --> 00:23:49,170 It will slide over these eigenvectors and they converge to the Orygen, OK? 343 00:23:49,370 --> 00:23:52,520 And if we start from here, it will converge to the origin. 344 00:23:52,700 --> 00:23:57,290 So diverge in the blue eigenvectors, converge in the red. 345 00:23:57,290 --> 00:23:58,460 I can work there, OK? 346 00:23:59,330 --> 00:24:02,020 So here is the important thing. 347 00:24:02,030 --> 00:24:09,410 So we will choose as our sliding surface what's ours and what's sliding service? 348 00:24:09,450 --> 00:24:17,570 Let me briefly say that our sliding surface is the dynamics that we want to impose our system. 349 00:24:17,750 --> 00:24:23,730 The sliding surface is very important in VSS control because this is the dynamics we want to impose 350 00:24:23,730 --> 00:24:24,410 to our system. 351 00:24:24,410 --> 00:24:30,710 So we want to impose our system that try our trajectories, go over this sliding surface. 352 00:24:30,920 --> 00:24:38,990 So what we want as we want asymptotically stability, we want to choose our eigenvectors as this eigenvectors, 353 00:24:38,990 --> 00:24:40,190 a sliding surface. 354 00:24:40,430 --> 00:24:48,890 So our trajectories will come to this and we will make our trajectories to come to this sliding surface 355 00:24:48,890 --> 00:24:55,220 or this eigenvectors sliding service is not the same as eigenvectors, but in this case, we will choose 356 00:24:55,220 --> 00:25:01,910 this eigenvectors as our sliding surface because when our trajectories will reach to this eigenvectors, 357 00:25:02,120 --> 00:25:09,800 they will just slide over to the origin and they will converge, OK and converge. 358 00:25:09,830 --> 00:25:10,200 OK. 359 00:25:10,370 --> 00:25:18,200 You can even choose as a sliding surface these diverging items, but you don't want this as a sliding 360 00:25:18,200 --> 00:25:21,770 surface because you don't want all your system to explore to diverge. 361 00:25:21,950 --> 00:25:26,000 Because if your trajectories will reach this matter, they will diverge. 362 00:25:26,270 --> 00:25:34,490 But we want a smooth convergence to the origin, so we will choose our this eigenvectors azaan lot as 363 00:25:34,490 --> 00:25:37,310 a sliding surface. 364 00:25:37,490 --> 00:25:42,290 So our trajectories, system trajectories will reach there and they will just slide smoothly to the 365 00:25:42,290 --> 00:25:42,860 origin. 366 00:25:43,130 --> 00:25:51,040 And another note here is that these eigenvectors are what intrinsic to the system. 367 00:25:51,070 --> 00:25:53,540 OK, we didn't imposes them. 368 00:25:53,750 --> 00:25:58,610 We just choose this eigenvectors as our sliding surface. 369 00:25:58,880 --> 00:26:06,860 We will make our trajectories to come to this eigenvectors orbit because you can see the trajectories 370 00:26:06,860 --> 00:26:13,550 we want to diverge, but we will make them to come to this eigenvectors. 371 00:26:13,760 --> 00:26:22,340 So then there will not be any input, but they will converge themselves to the origin because the trajectories 372 00:26:22,340 --> 00:26:27,920 when trajectories is over these red eigenvectors, they will just smoothly converge at the origin so 373 00:26:27,920 --> 00:26:29,900 we will get asymptotic stability. 374 00:26:30,200 --> 00:26:34,200 OK, so that's why choosing the sliding surface correctly is very important. 375 00:26:34,220 --> 00:26:43,340 OK, now let's see one thing we have seen our characteristic, as you can see, if you say that is minus 376 00:26:43,340 --> 00:26:47,780 if you will have one stable route and an unstable route and our system will be unstable and we have 377 00:26:47,780 --> 00:26:48,260 seen that. 378 00:26:48,560 --> 00:26:53,270 So let's make it our ace plus and see what's happening. 379 00:26:53,270 --> 00:26:53,990 In this case. 380 00:26:55,730 --> 00:27:00,200 We have to have an unstable oscillator, a system. 381 00:27:00,200 --> 00:27:04,340 And as you can see, we have unstable oscillator, a system. 382 00:27:04,490 --> 00:27:11,570 As you can see, our trajectories start from there and they just spin and diverge spin and thought, 383 00:27:11,570 --> 00:27:13,010 as you can see, they are diverging. 384 00:27:13,370 --> 00:27:15,170 Whole system is diverging. 385 00:27:15,410 --> 00:27:19,880 We can make it more clear by doing increasing the time. 386 00:27:20,120 --> 00:27:24,950 Let's do it two seconds and let's see what's happening with two seconds. 387 00:27:26,040 --> 00:27:33,620 OK, as you can see, as you can see what's happening, our system now starts from here and early. 388 00:27:34,400 --> 00:27:38,270 Like in the oscillator behaviour, they are diverging to the infinity. 389 00:27:38,510 --> 00:27:42,320 They look, look, look, look, look and diverge. 390 00:27:42,320 --> 00:27:45,170 OK, as you can see, they are trying to move and diverge. 391 00:27:45,410 --> 00:27:48,860 If you increase time, let's do five seconds. 392 00:27:49,340 --> 00:27:52,280 The graph will be more complex. 393 00:27:52,280 --> 00:27:59,770 I mean, by complex, it will be, as you can see, see, let's see the you can see the spinning here, 394 00:27:59,780 --> 00:28:02,510 OK, see our trajectories spin. 395 00:28:02,780 --> 00:28:04,310 Its oscillator is spin and. 396 00:28:04,410 --> 00:28:10,880 Diverged, see, this is the phase plot, you know, it is really a very important tool, as you can 397 00:28:10,880 --> 00:28:18,680 see, by just looking this, we can see that how our systems, how our system, whether our system is 398 00:28:18,680 --> 00:28:25,130 converging or diverging and while speaking, I tried to hypnotize you, as you can see by doing this. 399 00:28:25,310 --> 00:28:25,650 Okay? 400 00:28:26,030 --> 00:28:27,990 And um. 401 00:28:28,460 --> 00:28:28,900 OK. 402 00:28:30,920 --> 00:28:34,910 I think it's enough to understand what's happening here. 403 00:28:35,180 --> 00:28:36,290 So let's continue. 404 00:28:36,710 --> 00:28:44,510 So we have seen that the dynamic system we analyze has two modes depending on the value of seek or efficient. 405 00:28:44,870 --> 00:28:53,090 Namely, if C equals to minus a, we will get this mode, which we will call Mod A, as we have seen 406 00:28:53,090 --> 00:28:53,600 before. 407 00:28:53,600 --> 00:28:57,320 In this mode, we have one stable and one unstable route. 408 00:28:57,680 --> 00:29:04,550 But one unstable route is enough to make the whole dynamic system state's evolution trajectory to explode 409 00:29:04,550 --> 00:29:06,800 or system to become unstable. 410 00:29:07,340 --> 00:29:14,330 If it was C as air, then we will get this mode, which will which will be called mode B. 411 00:29:14,720 --> 00:29:23,600 Here are two routes are not only stable, but also excuse me here two roads are not only unstable, 412 00:29:23,600 --> 00:29:25,490 but also complex. 413 00:29:26,240 --> 00:29:28,670 So we get oscillates from behavior. 414 00:29:30,380 --> 00:29:37,190 As you can see, we can obtain two different modes by changing just one parameter, which is C. 415 00:29:37,520 --> 00:29:41,480 We can change trajectory evolution at anytime we want. 416 00:29:41,480 --> 00:29:48,560 By changing the value of C as we have done the huge power, we can use it to achieve what we want, 417 00:29:48,830 --> 00:29:56,750 namely to make trajectories to go to the origin, so obtain asymptotically stable system from unstable 418 00:29:56,750 --> 00:29:57,260 system. 419 00:29:57,650 --> 00:30:05,270 So what steps we will take, we will take as control input as we can impose value on it. 420 00:30:05,840 --> 00:30:14,480 Then we will switch C from E to minus E or in the opposite way based on some control low to get asymptotic 421 00:30:14,480 --> 00:30:15,200 stability. 422 00:30:15,590 --> 00:30:16,790 What is that appropriate? 423 00:30:16,790 --> 00:30:17,600 Control low? 424 00:30:17,990 --> 00:30:25,310 As we have said before, our sliding surface choice will be this eigenvectors of the dynamic system, 425 00:30:26,120 --> 00:30:29,610 which was asymptotically stable. 426 00:30:30,410 --> 00:30:31,000 OK. 427 00:30:31,090 --> 00:30:39,980 If initial condition was, uh or on this eigenvectors, it would converge to the center, as we have 428 00:30:39,980 --> 00:30:42,860 seen during MATLAB session. 429 00:30:44,220 --> 00:30:50,670 We want to drive our trajectories or systems states to this surface, and when they will reach to this 430 00:30:50,670 --> 00:30:56,580 surface, they will just slide smoothly over this surface to the orders without any additional control 431 00:30:56,580 --> 00:30:56,970 input. 432 00:30:57,300 --> 00:31:03,990 Because as we have said before, if the states are in one of the eigenvectors, they will continue to 433 00:31:03,990 --> 00:31:11,610 evolve in this direction while either exploiting or converging depending on the eigenvectors sliding 434 00:31:11,610 --> 00:31:15,660 service will be indicated by us, which is a function of states. 435 00:31:16,080 --> 00:31:22,710 If we draw a sliding surface, the coordinate frame will be separated into positive and negative parts. 436 00:31:22,970 --> 00:31:29,310 Meanwhile, in positive parts, the valley of SW will be greater than zero and in negative parts will 437 00:31:29,310 --> 00:31:30,480 be less than zero. 438 00:31:30,780 --> 00:31:37,950 You can check the sign of SW by just picking random acts and extort values from both sides of the graph 439 00:31:38,220 --> 00:31:44,490 and plugging them into the equation of sliding surface and analyzing the sign of the output. 440 00:31:45,150 --> 00:31:54,960 OK, then we can write such control input where C is minus a ren x times X is less than zero. 441 00:31:55,290 --> 00:32:06,120 So as C is minus a m, the system dynamics will be in mode eight and if X s will be less than zero, 442 00:32:06,120 --> 00:32:09,700 then C it will be chosen as a saw. 443 00:32:09,750 --> 00:32:12,870 The system dynamics will be in mode B. 444 00:32:13,260 --> 00:32:20,760 From this picture, you can see the regions where excess is positive and where excess is negative on 445 00:32:20,760 --> 00:32:21,750 positive origins. 446 00:32:21,750 --> 00:32:29,100 Dynamics will evolve based on mode A. and on negative reviews based on mode B. 447 00:32:29,310 --> 00:32:30,690 Let's say an example. 448 00:32:33,060 --> 00:32:40,740 Let's assume that we have started from that red dot initial Poland as we are in positive origin mode. 449 00:32:40,740 --> 00:32:49,740 A meme excuse me, as we are in positive region mode e will be activated and trajectory will evolve 450 00:32:49,770 --> 00:32:52,800 based on this method, which we have seen before. 451 00:32:53,160 --> 00:32:53,790 OK. 452 00:32:53,940 --> 00:32:57,350 Trajectory evolved and restarted their negative region. 453 00:32:57,360 --> 00:32:58,050 What to do? 454 00:32:58,380 --> 00:33:01,920 Surely they will switch to the mode B in mode B. 455 00:33:01,920 --> 00:33:08,460 Trajectories will evolve in this way until they reach a sliding surface indicated by Dark Blue Line. 456 00:33:08,910 --> 00:33:16,830 And as we reach sliding surface, which we have chosen as the stable eigenvectors trajectory will evolve 457 00:33:16,830 --> 00:33:20,190 to the Orygen without any additional control input. 458 00:33:20,580 --> 00:33:26,580 This is because we do not have any disturbance and eigenvalue is intrinsic to the system. 459 00:33:27,030 --> 00:33:32,340 You can analyze this graph by yourself where we started from another region. 460 00:33:32,640 --> 00:33:33,240 Perfect. 461 00:33:33,570 --> 00:33:39,780 We get asymptotically stable system from unstable system with very simple control method. 462 00:33:40,080 --> 00:33:42,450 And this control method is robust. 463 00:33:42,630 --> 00:33:43,230 Why? 464 00:33:43,530 --> 00:33:52,920 Because when there will be uncertainties or disturbances on the system, they will try to deviate trajectory 465 00:33:52,920 --> 00:33:54,330 from sliding surface. 466 00:33:54,600 --> 00:34:02,190 So Trajectory will enter another region and control input will change based on it and will try to push 467 00:34:02,190 --> 00:34:06,210 the trajectory or system states to this sliding surface again. 468 00:34:06,540 --> 00:34:10,890 This is why another name of this control lizard is sliding more control. 469 00:34:11,160 --> 00:34:15,990 We will talk about this more detailed way in future lessons. 470 00:34:16,020 --> 00:34:23,010 OK, now let's see how we will implement our variable structure. 471 00:34:23,370 --> 00:34:32,130 Um, well, I have a structure controller um, in or slightly surface controller in MATLAB, OK, and 472 00:34:32,130 --> 00:34:36,870 see how and see how we can simulate the system, as you can see in. 473 00:34:36,930 --> 00:34:44,970 Um, while we are analysing this plot, we have used the Odyssey 45 in order to analyse our system, 474 00:34:44,970 --> 00:34:47,320 but now we will not do that. 475 00:34:47,340 --> 00:34:54,630 Instead, we will use seems similar enough in order to make our job easier. 476 00:34:54,930 --> 00:34:55,320 OK. 477 00:34:55,620 --> 00:34:59,190 So what we'll do, we will define initial condition as always. 478 00:34:59,580 --> 00:35:06,150 You know this if you will define our C as you know this, we will define our A. And we will define our 479 00:35:06,150 --> 00:35:07,350 C, OK? 480 00:35:07,590 --> 00:35:09,330 And there you know what to see. 481 00:35:09,560 --> 00:35:10,060 OK. 482 00:35:10,620 --> 00:35:16,200 Co-auteur minus C squared over four plus a square root. 483 00:35:16,230 --> 00:35:16,680 OK. 484 00:35:17,190 --> 00:35:17,660 OK. 485 00:35:17,670 --> 00:35:22,710 And this is the simulation time t equals the ten ten time units. 486 00:35:22,710 --> 00:35:25,200 We will simulate our system. 487 00:35:25,230 --> 00:35:32,670 OK, now we have every variable here, so we will simulate our system in T time units. 488 00:35:32,970 --> 00:35:35,310 And this is our simulated file. 489 00:35:36,180 --> 00:35:38,010 Let's see our simulation file. 490 00:35:38,910 --> 00:35:39,870 It's very easy. 491 00:35:39,900 --> 00:35:42,610 First, we have our state case. 492 00:35:43,870 --> 00:35:48,910 State, state, space, we hear, as you can see, we have written our aim at three zero one zero, 493 00:35:49,030 --> 00:35:57,970 see why we have done here, as you see, because as we have said, see is our input. 494 00:35:58,090 --> 00:36:03,100 OK, now we have said that see our is our input. 495 00:36:03,100 --> 00:36:12,560 So we will see, as you saw, we will take out sea valuable from a instead. 496 00:36:12,580 --> 00:36:20,530 We will multiply it with X and we will make C X as input u. 497 00:36:20,710 --> 00:36:25,090 So we will change our B instead, so we will give here one. 498 00:36:25,330 --> 00:36:33,240 So when we Colquitt X plus b, you v will get the same thing as when C was there, OK? 499 00:36:33,580 --> 00:36:38,680 If you analyze it, you will understand it's very easy. 500 00:36:39,150 --> 00:36:39,650 Okay. 501 00:36:41,110 --> 00:36:49,840 Uh, OK, because we cannot write C here, because if we let's think that we have done as before and 502 00:36:49,840 --> 00:36:55,870 write C here, what will happen in this case, we we have to change C at every time iteration, but 503 00:36:55,870 --> 00:36:59,080 in this case we will not be able to do the to do this. 504 00:36:59,320 --> 00:37:05,950 So instead, we give ipse as input here and we see X, I mean as input here. 505 00:37:06,160 --> 00:37:14,280 So we calculate X plus b u instead of just x OK by of see here anyway. 506 00:37:14,320 --> 00:37:16,030 OK, I hope that you understand it. 507 00:37:16,660 --> 00:37:24,430 So this is our system and as we have said at the output will be X and extort because we defined our 508 00:37:24,430 --> 00:37:26,990 C as identity matrix. 509 00:37:27,010 --> 00:37:33,460 OK, so we will see at the observer dotted both X and X thought here is X and years X stood with them, 510 00:37:33,520 --> 00:37:38,640 looks the multiplex where we split it. 511 00:37:38,650 --> 00:37:43,660 Our vector OK into two parts in order to have X here and X not here, OK? 512 00:37:43,900 --> 00:37:50,390 And we combine these two together in order to give as an output to our workspace. 513 00:37:50,410 --> 00:37:53,540 OK, then we will take the variables from here and analyze. 514 00:37:53,560 --> 00:37:53,920 OK. 515 00:37:54,190 --> 00:38:03,700 So as we have said, what we will calculate in our next hour, uh, how to say a sliding surface because 516 00:38:03,700 --> 00:38:08,510 based on sliding surface, we will define our input parameter C. 517 00:38:08,530 --> 00:38:10,690 OK, so let's see o. 518 00:38:11,410 --> 00:38:17,170 As we have said, our sliding surface will need X and extort. 519 00:38:17,470 --> 00:38:18,400 So we will. 520 00:38:18,640 --> 00:38:22,870 Our what was our sliding surface is minus C x plus x thought OK. 521 00:38:23,050 --> 00:38:28,750 As you can see, by doing this one, uh, we have calculated our 522 00:38:30,820 --> 00:38:38,830 um, sliding surface of our sliding surface was minus the X Plus extorts and we have done it's minus 523 00:38:38,830 --> 00:38:45,770 C x plus x thought view calculated our s and what was our input. 524 00:38:46,450 --> 00:38:55,900 As you remember, we have said that our input C will be minus E if X is Sova gets here. 525 00:38:55,900 --> 00:38:57,820 S OK, let me write it here. 526 00:38:58,120 --> 00:38:59,290 This is s what we have. 527 00:38:59,530 --> 00:39:00,850 It is our sliding surface. 528 00:39:00,880 --> 00:39:01,270 OK. 529 00:39:01,600 --> 00:39:02,680 This is our s. 530 00:39:02,890 --> 00:39:09,370 We multiply this as with X OK and the Output S6, let me write it also here. 531 00:39:09,700 --> 00:39:13,960 So s x OK or axis OK. 532 00:39:14,230 --> 00:39:18,550 This axis, based on its sign here for here we get sign. 533 00:39:18,760 --> 00:39:20,260 This will return minus one. 534 00:39:20,260 --> 00:39:25,840 If it is negative, it will return, plus one if it is positive. 535 00:39:25,840 --> 00:39:28,480 Zero otherwise, I mean, the event is zero. 536 00:39:28,810 --> 00:39:29,110 OK. 537 00:39:29,140 --> 00:39:32,080 Based on the sign of excess, we will define our h. 538 00:39:32,090 --> 00:39:34,990 So if it is negative, you will have minus eight. 539 00:39:34,990 --> 00:39:41,050 As we have said, C will be minus eight or if it is positive, ibs-c will be plus eight. 540 00:39:41,170 --> 00:39:41,860 OK, perfect. 541 00:39:41,860 --> 00:39:46,570 We get C here and there as an input. 542 00:39:46,870 --> 00:39:52,960 We VE also write its value to the input, but we will add to the workspace, but we will not analyse 543 00:39:52,960 --> 00:39:54,520 it because it's not needed. 544 00:39:54,790 --> 00:39:58,450 Even we can delete this OK, because we will not need it. 545 00:39:58,840 --> 00:39:59,320 OK. 546 00:39:59,620 --> 00:40:01,810 Here is just our C valuable. 547 00:40:02,290 --> 00:40:16,090 We multiply it with minus one because um yeah, we ve multiplied minus one because what was our state's 548 00:40:16,090 --> 00:40:17,080 space equation? 549 00:40:17,620 --> 00:40:22,340 It was in order to extort equals x plus b you. 550 00:40:22,670 --> 00:40:28,150 But our what was our a matrix? 551 00:40:28,150 --> 00:40:32,200 It was zero one minus c c OK. 552 00:40:32,380 --> 00:40:37,240 We take out minus C from a matrix. 553 00:40:37,240 --> 00:40:42,760 As we have said before, it was before their minds c because now it's input so of. 554 00:40:43,350 --> 00:40:49,290 See here we multiplied minus once obviously will have minus C v, multiply it with X. 555 00:40:49,290 --> 00:41:01,290 As you can see, we multiply it with minus X and we come we we m some it with a X, so we will have 556 00:41:01,620 --> 00:41:06,270 the same effect as a equals to zero one minus six C. 557 00:41:06,300 --> 00:41:07,950 OK, perfect. 558 00:41:09,420 --> 00:41:13,170 So this is our state space with this our system. 559 00:41:13,200 --> 00:41:13,770 OK. 560 00:41:13,800 --> 00:41:19,020 And be it robust control, sliding surface control again. 561 00:41:19,320 --> 00:41:21,420 And let's analyze the results. 562 00:41:21,420 --> 00:41:22,020 What will happen. 563 00:41:22,020 --> 00:41:25,840 So we will simulated and then we will get the outputs OK? 564 00:41:25,860 --> 00:41:29,340 As you can see, we will get first time in order to plot. 565 00:41:29,970 --> 00:41:33,680 We will get x um from the states. 566 00:41:33,690 --> 00:41:40,710 As you can see, this was our state and we get, uh, data column vector first column vector, which 567 00:41:40,710 --> 00:41:50,870 is first column, which is on the X and the second column, which is X thought, OK, then we will create 568 00:41:50,880 --> 00:42:00,060 a future and evil plot X with respect to X in order to see what's happening with our, um, what's happening 569 00:42:00,060 --> 00:42:02,580 with our stay trajectories. 570 00:42:02,580 --> 00:42:04,980 So let's do that and it's very interesting for me. 571 00:42:04,980 --> 00:42:05,400 The result? 572 00:42:05,400 --> 00:42:11,370 Oh, as you can see before, what was our system before our system was diverging? 573 00:42:11,370 --> 00:42:12,930 No, it's very interesting. 574 00:42:12,930 --> 00:42:13,980 Look, what's happening. 575 00:42:14,220 --> 00:42:15,720 This is our initial condition. 576 00:42:15,770 --> 00:42:16,050 OK? 577 00:42:16,140 --> 00:42:19,380 We are starting from minus four minus two. 578 00:42:19,650 --> 00:42:20,490 As you can see. 579 00:42:20,490 --> 00:42:26,340 First, our system is in in its approximately in region. 580 00:42:26,940 --> 00:42:28,080 It is in the region. 581 00:42:28,320 --> 00:42:38,400 I think it's in region four, OK, where the mode B is active, OK, and our system is what it is. 582 00:42:38,430 --> 00:42:44,700 Um, it tries to embody what was that must be. 583 00:42:44,970 --> 00:42:52,920 Both of our eigenvalues was negative real parts, but they were complex, so it will have oscillated 584 00:42:52,920 --> 00:42:55,290 of behaviour also unstable. 585 00:42:55,500 --> 00:43:01,140 So it will go like that our c, but they will reach to the our stable eigenvectors. 586 00:43:01,140 --> 00:43:04,320 And as you see, as you can see it, how it comes. 587 00:43:04,620 --> 00:43:07,410 I have said that it is when it reaches the island. 588 00:43:07,410 --> 00:43:11,100 This is all like, OK, first, let me see this. 589 00:43:11,130 --> 00:43:16,080 Our sliding surface, this stabilizing factor, as you can see, our trajectories reached two there 590 00:43:16,350 --> 00:43:26,730 and they just slide to the Orygen C slide to the zero y zero, as you can see from an unstable system. 591 00:43:26,730 --> 00:43:30,930 But we have to get asymptotically stable system. 592 00:43:30,940 --> 00:43:38,190 Our trajectories come to this sliding surface and they just slide to the centre without any problems 593 00:43:38,190 --> 00:43:38,790 smoothly. 594 00:43:39,040 --> 00:43:44,790 OK, let's change our initial condition and let's see what we will get. 595 00:43:46,950 --> 00:43:47,580 OK? 596 00:43:47,610 --> 00:43:49,910 As you can see, we have no different characteristic. 597 00:43:49,920 --> 00:43:50,250 OK? 598 00:43:50,550 --> 00:43:52,170 Here is our initial condition. 599 00:43:52,170 --> 00:44:00,300 It comes directly to the sliding surface and it just slide to the Odigem with the sliding surface over 600 00:44:00,300 --> 00:44:01,250 the sliding surface. 601 00:44:01,260 --> 00:44:02,190 It's very beautiful. 602 00:44:02,190 --> 00:44:02,520 Yes. 603 00:44:02,820 --> 00:44:06,030 So let's try to something else. 604 00:44:06,030 --> 00:44:09,210 I don't know whether I will be able to do that. 605 00:44:09,630 --> 00:44:14,280 Um, let's see what will happen if we choose four 606 00:44:16,590 --> 00:44:19,380 to five. 607 00:44:25,390 --> 00:44:28,330 OK, this is really interesting to see how it comes. 608 00:44:28,810 --> 00:44:30,910 It came like that, OK? 609 00:44:31,300 --> 00:44:39,490 Our system can load that and come to the eigenvectors or a sliding surface and directly slide it to 610 00:44:39,490 --> 00:44:40,750 the surface. 611 00:44:40,780 --> 00:44:41,230 OK. 612 00:44:41,920 --> 00:44:44,110 So you can change the initial condition. 613 00:44:44,120 --> 00:44:50,350 You can start from the different regions and you will see that at each and every time the system will 614 00:44:50,350 --> 00:44:52,100 be asymptotically stable. 615 00:44:52,120 --> 00:44:52,540 OK. 616 00:44:52,810 --> 00:45:00,340 And here you can see the importance of sliding surfaces because if we didn't use sliding surface correctly, 617 00:45:00,520 --> 00:45:03,790 we can by virtue OK, not converge, OK. 618 00:45:05,290 --> 00:45:10,960 We have to choose our sliding surface carefully in order to just slide over these and come to their 619 00:45:11,680 --> 00:45:12,260 origin. 620 00:45:12,280 --> 00:45:12,730 OK. 621 00:45:14,380 --> 00:45:19,810 And as you can see, this is a robust system it does not depend on from where you are starting your. 622 00:45:19,810 --> 00:45:26,360 If you think this instead of here, we would have a road. 623 00:45:27,130 --> 00:45:27,610 OK? 624 00:45:27,880 --> 00:45:30,620 And then it does doesn't depend. 625 00:45:30,640 --> 00:45:33,670 Where is our error or what's our initial error? 626 00:45:33,900 --> 00:45:41,220 OK, it will come either the state trajectories will converge to this zero and way. 627 00:45:41,470 --> 00:45:45,820 So that's why it is called robust controller and variable structure controller. 628 00:45:45,820 --> 00:45:52,210 Because as you can see, we change the our input value. 629 00:45:52,210 --> 00:45:56,590 OK, it's variable variable structure and we get different modes. 630 00:45:56,590 --> 00:46:01,150 We get different trajectories, system trajectories and we reached two there. 631 00:46:01,420 --> 00:46:04,780 And this is the origins of you get us into a stable system. 632 00:46:04,780 --> 00:46:08,260 So that's why it's called a variable structure control. 633 00:46:08,260 --> 00:46:14,890 And also, yeah, that's why it's called sliding surface control, because we are trying to make our 634 00:46:14,890 --> 00:46:21,000 system trajectories to go to the sliding surface and with the sliding surface to the origin. 635 00:46:21,880 --> 00:46:30,790 OK, if you have any problem with the code, please write me, ask me and ask me and I will try to answer 636 00:46:31,030 --> 00:46:32,500 to you, OK? 637 00:46:33,160 --> 00:46:35,290 Before concluding this list. 638 00:46:35,290 --> 00:46:37,060 And let me mention some things to you. 639 00:46:37,300 --> 00:46:44,380 Firstly, as we have said, sliding servers is nothing but the dynamics we want to impose to our system. 640 00:46:44,680 --> 00:46:48,370 So steep trajectories will evolve based on these dynamics. 641 00:46:48,640 --> 00:46:54,730 As you may notice, this makes us to choose the sliding surface very carefully to achieve our goal. 642 00:46:55,270 --> 00:47:04,060 So now let's see briefly the important steps to design these as controller first and most importantly, 643 00:47:04,060 --> 00:47:07,300 choosing appropriate sliding surface of ASX. 644 00:47:07,870 --> 00:47:13,930 Then choose appropriate control to make state trajectories evolve or a sliding surface. 645 00:47:14,290 --> 00:47:18,390 We will talk about this truly in the next few lessons here. 646 00:47:18,400 --> 00:47:21,250 I want to note one important thing. 647 00:47:21,580 --> 00:47:26,920 In previous example, eigenvectors was intrinsic for a dynamic system. 648 00:47:27,610 --> 00:47:31,810 Namely, we didn't impose it to the system as a new dynamics. 649 00:47:32,050 --> 00:47:39,730 We can understand that from this sliding surface is intrinsic to our system because in phase plots, 650 00:47:39,730 --> 00:47:46,840 when we take our initial points on this IGen vector, they have solved the slide over it. 651 00:47:47,170 --> 00:47:54,970 This is not correct when we start somewhere else or we impose some new dynamics to the system because 652 00:47:55,090 --> 00:48:01,360 state trajectories will try to leave that sliding surface because it is not intrinsic to them and they 653 00:48:01,360 --> 00:48:02,260 want to be free. 654 00:48:02,620 --> 00:48:08,650 But you have to be a dictator and force them to go over the leaving surface you want. 655 00:48:09,540 --> 00:48:12,580 Excuse me, or the sliding surface you want. 656 00:48:12,910 --> 00:48:19,390 Let me know that the basics of robust control is developed by Soviet engineers, so you can understand 657 00:48:19,390 --> 00:48:21,040 the logic behind this method. 658 00:48:21,610 --> 00:48:28,630 Additionally, if we consider disturbances and uncertainties, these will also try to divide the system 659 00:48:28,630 --> 00:48:34,720 trajectories from the sliding surface, as we have seen during introduction to the robust control. 660 00:48:35,080 --> 00:48:44,860 So you have to apply additional input m to make system trajectory stay on sliding surface, even its 661 00:48:44,860 --> 00:48:46,450 lower sliding surface.