1 00:00:00,120 --> 00:00:00,480 OK. 2 00:00:01,230 --> 00:00:05,100 I hope that you have already have a clear idea about robust control. 3 00:00:05,490 --> 00:00:13,680 We have seen how simple and effective was its technique to make systems stable and robust asymptotically 4 00:00:13,680 --> 00:00:14,190 stable. 5 00:00:14,650 --> 00:00:20,730 However, until now, we did not consider estimations in the parameters or external disturbances, which 6 00:00:20,730 --> 00:00:24,300 are the key problems that we want to solve with the robust control. 7 00:00:24,930 --> 00:00:31,140 Let's now try to include these effects also and see what kind of problems we have to handle. 8 00:00:32,460 --> 00:00:38,790 Firstly, as I have said already, we have to consider some practical cases because, as you know, 9 00:00:38,790 --> 00:00:42,420 practical cases are always different from ideal conditions. 10 00:00:42,820 --> 00:00:50,550 We have to consider Consi the uncertainties in the system, then we have to consider external disturbances 11 00:00:50,550 --> 00:00:53,370 that can exist and effect the system. 12 00:00:53,850 --> 00:01:01,140 And finally, but important when considering the keys, when the sliding surface we have chosen is not 13 00:01:01,140 --> 00:01:05,250 intrinsic to the system, but it is impulse to the system by us. 14 00:01:05,640 --> 00:01:11,820 Robust control technique has been developed to solve these issues, and it can handle all of the above 15 00:01:11,820 --> 00:01:12,930 problems easily. 16 00:01:13,350 --> 00:01:19,980 But we have some constraints that we have to obey in order to get effective robustness against disturbances 17 00:01:20,220 --> 00:01:29,340 and also uncertainty in parameters and also the problems when we will undergo during imposing sliding 18 00:01:29,340 --> 00:01:29,940 surface. 19 00:01:30,180 --> 00:01:33,150 First, the value of the disturbance has to be bounded. 20 00:01:33,600 --> 00:01:40,080 Yes, we may not know exact valley of the disturbance, but we have to know the upper bound for it because 21 00:01:40,080 --> 00:01:46,050 robust control reject disturbances by applying much higher input than the value of disturbance. 22 00:01:46,560 --> 00:01:52,470 Otherwise, the symbols would be greater than the input and input control will not be effective against 23 00:01:52,470 --> 00:01:53,070 disturbance. 24 00:01:53,310 --> 00:01:59,220 So we have to know the upper bound of the disturbance value to design input control. 25 00:01:59,610 --> 00:02:03,260 Additionally, the rate of change of the disturbance has to be bounded. 26 00:02:03,270 --> 00:02:07,260 Also, we will see why this is important in the next lesson. 27 00:02:07,590 --> 00:02:09,960 But keep this condition also in mind. 28 00:02:10,320 --> 00:02:15,870 So we have said we have to consider some practical cases that we have mentioned above. 29 00:02:16,350 --> 00:02:22,230 These considerations will require a robust controller to apply additional input control even when the 30 00:02:22,230 --> 00:02:27,180 trajectories of system states are over this sliding surface. 31 00:02:27,600 --> 00:02:35,640 Because disturbances and inherent sliding surface will cause the trajectories to deviate from sliding 32 00:02:35,640 --> 00:02:42,240 surface and additional control input has to be applied to keep to keep them over sliding surface. 33 00:02:43,650 --> 00:02:48,030 Then these considerations will also cause a significant problem. 34 00:02:48,210 --> 00:02:54,280 That is one of the biggest, biggest drawbacks of the robust control, and this problem is chartering. 35 00:02:54,630 --> 00:03:00,510 We will see this problem and how to decrease its effect, as we have done in previous lesson. 36 00:03:00,870 --> 00:03:04,560 The best way to understand these concepts is to take an example. 37 00:03:04,920 --> 00:03:09,000 Here was the example dynamic system that we will analyze here. 38 00:03:09,090 --> 00:03:13,590 Safety is nothing but external disturbance and duty is nothing. 39 00:03:13,590 --> 00:03:20,970 But the control input system is second order dynamic system and its asymptotically stable as long as 40 00:03:20,970 --> 00:03:24,090 a and B coefficients are greater than zero. 41 00:03:24,540 --> 00:03:27,950 So you can ask this question here, OK? 42 00:03:28,140 --> 00:03:30,750 But our system is older than some thought goes stable. 43 00:03:31,110 --> 00:03:32,310 What do you want from it? 44 00:03:32,910 --> 00:03:37,230 Yes, it's asymptotically stable, but we have to concede there are two cases here. 45 00:03:37,440 --> 00:03:39,960 Firstly, it cannot be stable. 46 00:03:40,470 --> 00:03:42,760 It mine might not be OK. 47 00:03:42,760 --> 00:03:49,800 It may not be stable when there is disturbances and uncertainties in the parameters, as we have seen 48 00:03:49,800 --> 00:03:52,680 during the introduction to the robust controller. 49 00:03:53,520 --> 00:03:57,660 Secondly, we may want different kind of convergence dynamics. 50 00:03:57,840 --> 00:03:59,790 Yes, we may want it. 51 00:04:00,360 --> 00:04:05,010 What if we want faster convergence to the origin or slower convergence? 52 00:04:05,250 --> 00:04:12,570 We can make it possible by imposing new sliding surface that represent desired dynamics, and we will 53 00:04:12,570 --> 00:04:16,950 derive start to drive state trajectories to this surface. 54 00:04:17,190 --> 00:04:24,840 OK, let's convert the system into state space form so we can draw its face, blood and on the low stability 55 00:04:24,840 --> 00:04:25,560 properties. 56 00:04:25,950 --> 00:04:32,280 Before doing that, I want to note one thing here is the system diagram with input you and disturbance 57 00:04:32,340 --> 00:04:35,640 see and plant of GSI, which is our system. 58 00:04:36,360 --> 00:04:42,450 Here we can see some disturbance and input term you to get new input term of you, dash. 59 00:04:42,990 --> 00:04:46,440 We will use this as an input to our system. 60 00:04:46,710 --> 00:04:49,680 So let's get the system in state based form. 61 00:04:50,130 --> 00:04:59,400 First, we will define X one and X two as y and derivative of Y, then the right y double dot into y 62 00:04:59,400 --> 00:04:59,760 double. 63 00:05:00,030 --> 00:05:07,230 In terms of X one and X two states after that, we can write equation in states based form where we 64 00:05:07,230 --> 00:05:09,600 can obtain A and B mattresses. 65 00:05:10,050 --> 00:05:14,430 And here is the output equation from which we can determine see matrix. 66 00:05:14,700 --> 00:05:15,300 OK. 67 00:05:15,540 --> 00:05:21,210 We converted the system equation into stable, excuse me, into state space form. 68 00:05:21,690 --> 00:05:25,410 Now we can drill phase of the system and analyze it. 69 00:05:25,830 --> 00:05:33,270 Let's switch to MATLAB in order to realize that, OK, after we have got a matrix, let's say we converted 70 00:05:33,270 --> 00:05:35,310 our system to state space form. 71 00:05:35,340 --> 00:05:38,280 Let's analyze the first portrait of our system. 72 00:05:38,670 --> 00:05:42,890 We will take our input you and also disturbance as zero. 73 00:05:42,930 --> 00:05:47,400 So that's why View X not equals the X plus b you. 74 00:05:47,410 --> 00:05:55,350 But as we don't have you or our you, that equals to zero because we say that in this case, we will 75 00:05:55,350 --> 00:05:58,680 choose you as zero and disturbance as zero. 76 00:05:59,100 --> 00:06:02,700 So this will have on the X slot equals a X. 77 00:06:03,210 --> 00:06:06,120 So let's see what's happening here. 78 00:06:06,540 --> 00:06:14,250 As we have said, we will choose our A, B or F issuance as a positive definite and OK. 79 00:06:14,970 --> 00:06:19,410 I mean it indefinite in order our system to be asymptotically stable. 80 00:06:19,710 --> 00:06:20,190 OK. 81 00:06:21,030 --> 00:06:25,440 So you will you can ask, what's what about this C? 82 00:06:25,680 --> 00:06:26,130 C? 83 00:06:26,790 --> 00:06:27,840 Don't worry about that. 84 00:06:28,110 --> 00:06:30,060 We will see in a minute. 85 00:06:30,660 --> 00:06:32,200 This is for sliding surface. 86 00:06:32,220 --> 00:06:32,530 OK. 87 00:06:32,550 --> 00:06:34,050 The conversion for sliding surface. 88 00:06:34,230 --> 00:06:41,100 So I will also draw here this sliding surface that we will choose in the next minutes. 89 00:06:41,140 --> 00:06:41,570 OK. 90 00:06:42,370 --> 00:06:45,300 So this is for sliding surface. 91 00:06:45,330 --> 00:06:45,810 OK. 92 00:06:46,710 --> 00:06:54,180 Let me call that, uh, core efficient of, uh, sliding surface. 93 00:06:54,210 --> 00:06:54,570 OK. 94 00:06:55,530 --> 00:06:57,530 So this is our a matrix. 95 00:06:57,540 --> 00:06:57,830 OK? 96 00:06:57,840 --> 00:07:03,450 I cannot to explain this too much because in previous listen, we have seen about the phase plots. 97 00:07:03,690 --> 00:07:04,980 So our a matrix. 98 00:07:05,010 --> 00:07:05,520 OK. 99 00:07:05,820 --> 00:07:08,670 This is the time span and we will simulate our system. 100 00:07:08,940 --> 00:07:11,130 And you can ask all of these angles. 101 00:07:12,120 --> 00:07:18,270 I will do initial conditions as a let me show you in this, as you can see the initial conditions as 102 00:07:18,630 --> 00:07:20,430 in round the weight of the inner circle. 103 00:07:20,610 --> 00:07:25,830 So that's why I need angles in order to draw the initial points. 104 00:07:25,830 --> 00:07:32,820 So choose the initial points as a circle, OK, with radius of two from the center, I mean, zero zero 105 00:07:32,820 --> 00:07:33,660 point OK. 106 00:07:33,660 --> 00:07:42,230 And here, as you can see, if we get our initial conditions by there, this is nothing important here. 107 00:07:42,240 --> 00:07:44,640 We just x by coordinates for the initial conditions. 108 00:07:44,940 --> 00:07:46,080 OK, in a circle. 109 00:07:46,800 --> 00:07:51,360 CMOs t to and our customers T to in order to get X and Y angles. 110 00:07:51,750 --> 00:07:52,200 Mm hmm. 111 00:07:52,410 --> 00:07:52,710 OK. 112 00:07:52,740 --> 00:07:59,520 After we have got our initial conditions, OK, this is our initial conditions that we weren't OK with 113 00:07:59,520 --> 00:08:00,600 x y coordinates. 114 00:08:00,840 --> 00:08:02,830 If you want, we can see it in this way. 115 00:08:02,850 --> 00:08:06,030 Let's run it and here we will stop. 116 00:08:06,030 --> 00:08:08,470 And let's see what's our initial conditions. 117 00:08:08,490 --> 00:08:08,910 OK? 118 00:08:09,630 --> 00:08:16,160 As you can see, this is x y x y x y for each of the initial conditions. 119 00:08:16,170 --> 00:08:22,510 OK, so let's quit it, and let's continue. 120 00:08:22,530 --> 00:08:29,730 So here we are creating new figure and we will, as in the previous case, we vote for each case, for 121 00:08:29,730 --> 00:08:33,420 each initial condition, the system trajectory, OK, stay trajectory. 122 00:08:33,630 --> 00:08:40,770 So we start from one and until two, the four looping over each of the initial conditions. 123 00:08:41,100 --> 00:08:46,620 We again use all the F45 to simulate our system dynamic system. 124 00:08:46,620 --> 00:08:54,990 We give it the time span, initial conditions, OK, and then we then extract, extolled and extend 125 00:08:54,990 --> 00:08:55,380 out. 126 00:08:55,440 --> 00:08:59,430 In this case, it will be shortly y o y de out, OK. 127 00:08:59,760 --> 00:09:04,440 Um, and so X1 and X2 states OK. 128 00:09:04,680 --> 00:09:10,320 And we will plot them with black color, and then we will put those initial conditions OK. 129 00:09:10,590 --> 00:09:12,060 And here what I am doing. 130 00:09:12,060 --> 00:09:14,010 I'm calculating the eigenvalues. 131 00:09:14,340 --> 00:09:25,380 And after that, um, what I am doing is just to throw this eigenvalues here, as I have said previously, 132 00:09:25,380 --> 00:09:30,600 and also the imposed the sliding surface. 133 00:09:30,630 --> 00:09:30,960 OK. 134 00:09:31,170 --> 00:09:37,920 And here are the derivative function to calculate the UM in order to calculate which will be which will 135 00:09:37,920 --> 00:09:41,220 be used inside or the effort, you know, with our system. 136 00:09:41,400 --> 00:09:44,670 So our extrude equals the eight times x. 137 00:09:44,680 --> 00:09:45,510 Okay, perfect. 138 00:09:45,720 --> 00:09:47,490 Not before doing anything else. 139 00:09:47,490 --> 00:09:48,930 Let's analyze our system. 140 00:09:48,940 --> 00:09:56,010 We have said that if we choose a and B positive definite OK, we will have us in particular stable system. 141 00:09:56,010 --> 00:09:57,240 Let's see if we have. 142 00:09:57,570 --> 00:09:58,050 OK. 143 00:09:58,110 --> 00:09:59,880 This is our aim ethics and let. 144 00:09:59,940 --> 00:10:01,560 See our eigenvalues. 145 00:10:01,590 --> 00:10:09,690 OK, let me just do this is like a this eigenvalues equals to egg is OK. 146 00:10:09,880 --> 00:10:10,080 Eight. 147 00:10:10,740 --> 00:10:11,290 OK. 148 00:10:11,310 --> 00:10:13,680 As you can see, this is our eigenvalues. 149 00:10:13,920 --> 00:10:18,330 As you can see, we have two eigenvalues with negative real part. 150 00:10:18,570 --> 00:10:24,270 So our system should be asymptotically stable one that there will not be any oscillations. 151 00:10:24,510 --> 00:10:28,230 I'm OK because there will not be any. 152 00:10:30,030 --> 00:10:33,180 They don't because I mean, these are not complex. 153 00:10:33,210 --> 00:10:34,980 OK, let's see. 154 00:10:35,250 --> 00:10:44,500 Let's on the West, I can, as you can see, if one eigenvalues absolute value is higher than another 155 00:10:44,500 --> 00:10:49,380 about another eigenvalues, absolute volume. 156 00:10:49,620 --> 00:10:57,450 As you can see in the dynamic equations solution, these eigenvalues will be over the exponential OK. 157 00:10:57,630 --> 00:11:05,370 So these eigenvalues absolute value determines how fast the system will converge. 158 00:11:05,370 --> 00:11:13,200 As you can see, we have to eigenvalue, so we will have to eigenvectors OK, corresponding to this 159 00:11:13,200 --> 00:11:13,890 eigenvectors. 160 00:11:13,890 --> 00:11:19,860 OK, this one and this one, I mean, this is the coordinates for like a vector or this like vector 161 00:11:19,860 --> 00:11:28,350 belongs to the first eigenvalue and this eigenvectors belongs to the second eigen value added over this 162 00:11:28,350 --> 00:11:31,650 eigenvalue, the convergence will be faster. 163 00:11:31,680 --> 00:11:32,250 Why? 164 00:11:32,520 --> 00:11:38,150 Because I mean, over this eigenvectors, the converges in the direction of this ikem vector will become 165 00:11:38,160 --> 00:11:42,120 a Commodus will be high at Y because of the M. 166 00:11:42,360 --> 00:11:45,410 Absolute value of the eigenvalue in this case is higher. 167 00:11:45,450 --> 00:11:54,090 Corresponding ogunmola is higher, so exponential or minus five will cause minus five times t ok. 168 00:11:54,360 --> 00:12:03,450 E over minus five t is the convergence will be faster than E or minus zero point three T. 169 00:12:03,510 --> 00:12:03,920 OK. 170 00:12:04,170 --> 00:12:09,360 So over the first eigenvectors, this convergence will be slow. 171 00:12:09,360 --> 00:12:12,660 Over the second eigenvectors, the convergence will be higher. 172 00:12:12,870 --> 00:12:16,190 OK, let's now see what will happen. 173 00:12:16,220 --> 00:12:17,730 What's our diagram? 174 00:12:17,730 --> 00:12:20,160 As you can see, this is our phase plot. 175 00:12:20,490 --> 00:12:25,430 This red line is imposed sliding surface. 176 00:12:25,440 --> 00:12:27,300 OK, as you can see, this is not 177 00:12:30,240 --> 00:12:37,590 intrinsic to the system because as you can see, it is this sliding service is different from the eigenvectors. 178 00:12:37,590 --> 00:12:38,300 You can see it. 179 00:12:38,310 --> 00:12:39,960 Red eigenvectors, OK? 180 00:12:40,200 --> 00:12:41,790 And the interesting sic. 181 00:12:42,210 --> 00:12:44,320 Let's look at this, OK? 182 00:12:44,340 --> 00:12:53,100 The interesting thing is here, as you can see, the convergence is very high in this eigenvectors because 183 00:12:53,100 --> 00:12:58,680 this Feigen vector, OK, as you can see, this eigenvectors corresponds to the eigenvalue with higher 184 00:12:58,680 --> 00:12:59,880 absolute value. 185 00:13:00,120 --> 00:13:00,750 OK. 186 00:13:01,020 --> 00:13:07,680 So that's why, as you can see, the convergence or it is very hard boiled, the second one is lower, 187 00:13:07,680 --> 00:13:12,390 OK, it comes the convergence or the second one or this one is slower. 188 00:13:12,600 --> 00:13:16,200 And finally, over this eigenvectors, it converges to the zero. 189 00:13:16,560 --> 00:13:19,170 Well, OK, you can see the same in this way. 190 00:13:19,860 --> 00:13:25,680 The direction over this eigenvectors, which has higher absolute valley, which corresponds to the higher 191 00:13:25,680 --> 00:13:25,860 ups. 192 00:13:26,040 --> 00:13:31,200 Well, the eigenvalue and the convergence rate is higher or higher in this India. 193 00:13:31,200 --> 00:13:36,960 For this eigenvectors, which corresponds to the lower absolute valley eigenvalue, the convergence 194 00:13:36,960 --> 00:13:38,250 rate is slower. 195 00:13:38,250 --> 00:13:45,750 So as you can see, our system anyway is asymptotically stable because it converges to the zero. 196 00:13:45,980 --> 00:13:46,340 OK. 197 00:13:46,710 --> 00:13:53,580 So let's now try to change the values and let's see something different. 198 00:13:54,540 --> 00:14:03,320 OK, let's now give eight and give them to the excuse me, fifteen, OK. 199 00:14:03,330 --> 00:14:06,060 Fifteen to the be core efficient. 200 00:14:06,060 --> 00:14:08,310 And now let's first. 201 00:14:08,910 --> 00:14:14,730 Unless as we as this positive definite our system A and B questions are positive. 202 00:14:14,730 --> 00:14:17,310 Definite, I mean, they are higher than zero. 203 00:14:17,620 --> 00:14:22,140 Um, the system should be asymptotically stable. 204 00:14:22,140 --> 00:14:23,270 And let's check that. 205 00:14:23,280 --> 00:14:32,550 And as you can see, if we do this and let's check again, like a menu of a as you can see, we have 206 00:14:32,550 --> 00:14:38,610 eigenvalue of minus three and minus five with negative real parts, so our system will be asymptotically 207 00:14:38,880 --> 00:14:39,490 stable. 208 00:14:39,510 --> 00:14:39,990 OK. 209 00:14:40,740 --> 00:14:44,060 And let's check what will be the face of our system. 210 00:14:44,070 --> 00:14:44,500 OK? 211 00:14:44,520 --> 00:14:47,660 Here is an interesting phase plot we have. 212 00:14:47,670 --> 00:14:54,190 Okay, let's make them bit like, you know my -- because I'm there, but maybe we don't need it's 213 00:14:54,210 --> 00:14:54,930 enough today. 214 00:14:55,080 --> 00:14:55,380 OK? 215 00:14:55,590 --> 00:14:59,760 As you can see, this is our import sliding service and this. 216 00:14:59,900 --> 00:15:02,650 Are a different eigen vectors, OK? 217 00:15:02,930 --> 00:15:05,910 As you can see again over. 218 00:15:06,580 --> 00:15:13,850 That's OK, we have cleaned it, but as our one of our eigenvalues is higher than the another in terms 219 00:15:13,850 --> 00:15:16,720 of absolute value, the convergence reigned over. 220 00:15:16,730 --> 00:15:22,190 Erm, the one which has higher absolute value will be higher than the other one. 221 00:15:22,290 --> 00:15:22,730 OK. 222 00:15:24,560 --> 00:15:32,980 But anyway, our system is asymptotically stable and it converges to the centre. 223 00:15:33,230 --> 00:15:41,500 Let's check what will happen if we give negative values so our system has to be unstable. 224 00:15:41,660 --> 00:15:44,180 OK, let's check first eigenvalues. 225 00:15:44,180 --> 00:15:48,470 OK, let's check first eigenvalues and let's see what we will get. 226 00:15:48,950 --> 00:15:49,530 OK? 227 00:15:49,550 --> 00:15:53,300 As you can see, our eigenvalues has positive neutral parts. 228 00:15:53,330 --> 00:15:56,690 OK, so well, what does it mean? 229 00:15:57,680 --> 00:16:00,660 It means that our system will be asymptotic. 230 00:16:00,920 --> 00:16:02,240 Excuse me, unstable. 231 00:16:02,420 --> 00:16:04,290 Let's look at the first plot. 232 00:16:04,310 --> 00:16:05,500 It's interesting for me. 233 00:16:05,510 --> 00:16:11,240 OK, now, as you can see, our trajectories explode to the infinity. 234 00:16:11,240 --> 00:16:15,500 As you can see, they are starting from this initial conditions and all of them are exploding. 235 00:16:15,770 --> 00:16:16,540 Can you see it? 236 00:16:16,550 --> 00:16:18,830 They are exploding to the infinity. 237 00:16:19,130 --> 00:16:27,050 They are not converging to the zero because our system is not asymptotically stable but unstable. 238 00:16:27,080 --> 00:16:27,590 OK. 239 00:16:28,670 --> 00:16:36,110 And you will see that even when there when we apply the robust control or even our system is unstable, 240 00:16:36,930 --> 00:16:44,090 a robust controller will help will make the system asymptotically stable, as we have seen previously. 241 00:16:44,780 --> 00:16:45,350 OK. 242 00:16:45,620 --> 00:16:46,010 Hmm. 243 00:16:48,020 --> 00:16:48,470 OK. 244 00:16:49,040 --> 00:16:51,110 Let's continue with our presentation. 245 00:16:51,170 --> 00:16:57,440 As we have said, in order to implement robust control of successfully to conditions of A and B have 246 00:16:57,440 --> 00:16:58,070 to be met. 247 00:16:58,730 --> 00:17:05,390 We can formally these two conditions like that we are dealt a zero and delta are the bones up her bones. 248 00:17:06,860 --> 00:17:11,810 As always, our goal is to make the system dynamics to be asymptotically stable. 249 00:17:12,140 --> 00:17:15,260 Let's choose the sliding surface, which is given like that. 250 00:17:15,980 --> 00:17:22,250 C has to be greater than zero because otherwise sliding surface will be unstable dynamics. 251 00:17:22,580 --> 00:17:26,570 Let's see how we can write sliding surface equation in this way. 252 00:17:27,170 --> 00:17:34,640 If we solve this simple or the E or differential equation, we will get y as this. 253 00:17:35,090 --> 00:17:40,220 From this formula, you can clearly see why converges only when C is positive. 254 00:17:40,550 --> 00:17:47,060 So a sliding surface B have imposed provide us asymptotic stability and form the formula. 255 00:17:47,060 --> 00:17:53,390 You can see that we can control convergence rate by controlling the value of C parameter. 256 00:17:54,080 --> 00:18:00,200 Here are important things to note about sliding surface we have chosen from the Formula one point zero. 257 00:18:00,230 --> 00:18:07,550 You can see that sliding surface doesn't include anything related to the system parameters that can 258 00:18:07,550 --> 00:18:08,720 be uncertain. 259 00:18:09,200 --> 00:18:16,640 It only includes Parameter C, which is controlled by us so we can control desired dynamics by changing 260 00:18:16,640 --> 00:18:21,560 parameter C and get faster or slower convergence rates. 261 00:18:22,010 --> 00:18:27,830 And the uncertainty will not affect the sliding surface dynamics as it doesn't include. 262 00:18:27,830 --> 00:18:35,830 These parameters and disturbance will not affect if we have obeyed conditions A and B, so we achieved 263 00:18:35,850 --> 00:18:37,460 robustness in control. 264 00:18:38,030 --> 00:18:45,590 However, sliding surface here is not intrinsic to the system dynamics because we have imposed its source. 265 00:18:45,590 --> 00:18:52,430 The trajectories will try to leave sliding surface and we have to apply continuously control input in 266 00:18:52,430 --> 00:18:55,160 order to keep them in the sliding surface. 267 00:18:55,190 --> 00:19:01,430 OK, now the important question after choosing sliding surface is how to design an input control that 268 00:19:01,430 --> 00:19:08,150 will drive state trajectories to the sliding surface and keep them over a sliding surface. 269 00:19:08,720 --> 00:19:12,620 Let's plot aerodynamics here in the plot. 270 00:19:12,620 --> 00:19:19,460 It is written Accent X thought, but please replace them with E and E Dot, where E is nothing but the 271 00:19:19,460 --> 00:19:23,600 difference between Y desired OK, which we will see. 272 00:19:23,600 --> 00:19:30,200 For example, if y desire to be something, for example, to read, you will see in my simulation much 273 00:19:30,200 --> 00:19:30,770 more clearly. 274 00:19:31,280 --> 00:19:37,690 Y desired and y and e dot is difference between y of desired and wider. 275 00:19:38,030 --> 00:19:41,600 So we want aerodynamics to convert zero. 276 00:19:41,930 --> 00:19:43,160 Not why? 277 00:19:43,700 --> 00:19:53,780 OK, we want E to converge zero, not y to converge zero because we want y to converge to y desired 278 00:19:54,080 --> 00:19:59,090 and in order to y to converge to y desired e should converge to. 279 00:19:59,170 --> 00:20:03,680 Zero, because he is nothing but why me, why decide minus why? 280 00:20:05,050 --> 00:20:12,160 OK, let's assume that trajectory is in positive, but so we have to apply input to get it back to the 281 00:20:12,160 --> 00:20:13,240 sliding surface. 282 00:20:13,510 --> 00:20:19,690 The scene is when the trajectories in negative part of the negative region and the input to drive it 283 00:20:19,690 --> 00:20:21,400 to the sliding surface again. 284 00:20:21,970 --> 00:20:28,380 The control input of you is given in this Formula one point one. 285 00:20:28,570 --> 00:20:33,850 Will this input wheel drives the trajectory into the sliding surface successfully? 286 00:20:34,240 --> 00:20:41,740 Here, ASG end function is signaling function, which returns one if it is positive, minus one if it's 287 00:20:41,740 --> 00:20:43,750 negative and zero if it is zero. 288 00:20:44,230 --> 00:20:51,610 You can see why this input is effective by just finding solution to our example dynamic system, which 289 00:20:51,610 --> 00:20:52,360 we have seen. 290 00:20:53,880 --> 00:20:56,260 OK, how to find this solution? 291 00:20:56,260 --> 00:20:57,310 I'm sorry you can do it. 292 00:20:58,750 --> 00:21:00,460 Even with MATLAB, it's very easy. 293 00:21:00,520 --> 00:21:05,920 If you cannot, please notify me and please write me and I will write it. 294 00:21:08,840 --> 00:21:10,080 For you, OK? 295 00:21:10,130 --> 00:21:16,260 Even the best ways I will attach the solution for it into the paper. 296 00:21:16,740 --> 00:21:18,740 And I will attach to this video. 297 00:21:19,040 --> 00:21:26,270 So you will you will see that, OK, you will see that if input is positive, it will increase whi, 298 00:21:26,330 --> 00:21:28,400 which will make it to reach wide. 299 00:21:28,850 --> 00:21:34,310 Yes, because firstly, we will start with Y, which is less than zero. 300 00:21:34,550 --> 00:21:35,090 Oh, OK. 301 00:21:36,410 --> 00:21:41,750 And so the aerodynamic or error trajectory will be in the positive side. 302 00:21:42,050 --> 00:21:46,550 So we need to increase VI, so we will choose you as positive. 303 00:21:46,550 --> 00:21:55,910 So it will increase y so y will reach to the Y d, so it will also error trajectory will decrease to 304 00:21:56,570 --> 00:21:57,020 zero. 305 00:21:57,560 --> 00:22:04,250 And when you is negative, it will decrease y, which will make it again closer to divide when it is 306 00:22:04,250 --> 00:22:08,930 higher than Y, then, as you can see from Equation 1.1. 307 00:22:08,960 --> 00:22:15,500 Control input is like switching action, namely it jumps from K to minus K and opposite, depending 308 00:22:15,500 --> 00:22:23,090 on which region is the trajectory and based on condition A. We will choose the value of K, which has 309 00:22:23,090 --> 00:22:27,200 to be higher than bill to zero new upper bound of disturbance. 310 00:22:27,470 --> 00:22:28,640 Be careful about that. 311 00:22:28,970 --> 00:22:33,050 We will see what will happen if we not obey this condition. 312 00:22:33,650 --> 00:22:36,690 Let's jump to the MATLAB and analyze the controller. 313 00:22:36,710 --> 00:22:37,130 OK. 314 00:22:38,120 --> 00:22:45,860 Let's continue with the MATLAB and try to see how we will simulate our controller robust controller 315 00:22:45,860 --> 00:22:47,090 that we have developed. 316 00:22:47,120 --> 00:22:47,570 OK. 317 00:22:47,960 --> 00:22:51,710 So here I have written what I will do. 318 00:22:51,710 --> 00:22:53,030 I have to show to you. 319 00:22:53,030 --> 00:22:57,560 So it's not related to this is our code. 320 00:22:59,060 --> 00:23:03,320 We have done in the previous lesson how to do the controller. 321 00:23:03,320 --> 00:23:12,920 And as you know, we have done the general controller part in the Symfonisk and we have used this month 322 00:23:12,920 --> 00:23:19,820 of script and MATLAB script for the plotting, the output potatoes and we will do the same here. 323 00:23:20,060 --> 00:23:23,320 So let's see how we developed the controller. 324 00:23:23,330 --> 00:23:25,750 As you can see, this is our state space. 325 00:23:25,850 --> 00:23:35,000 This power plant dynamics system, which was described by example both plus a extort plus uh yes. 326 00:23:35,240 --> 00:23:42,650 Extort, excuse me, my double dose plus avoiders plus b y equals the you dash, which was the sum of 327 00:23:43,250 --> 00:23:43,940 society. 328 00:23:43,970 --> 00:23:48,950 I mean, it was beside the disturbance, plus you, our input controller. 329 00:23:48,950 --> 00:23:52,250 So let's write each of these. 330 00:23:52,580 --> 00:23:53,900 We have seen how we will. 331 00:23:54,510 --> 00:23:55,190 What was our. 332 00:23:55,920 --> 00:23:58,660 That is our so repeat this source. 333 00:23:58,670 --> 00:24:06,320 See, yeah, we want to see both X1 and X2 states of the output and D will be nothing but zeros and 334 00:24:06,320 --> 00:24:08,990 initial condition as an initial condition. 335 00:24:09,290 --> 00:24:13,040 Um, we have put, uh, zero zero. 336 00:24:13,290 --> 00:24:13,710 OK. 337 00:24:13,880 --> 00:24:18,620 So we will always start with y zero and y equals the zero. 338 00:24:18,770 --> 00:24:19,190 OK. 339 00:24:19,580 --> 00:24:21,230 Because our initial condition zero zero. 340 00:24:21,560 --> 00:24:23,510 So then we get the outputs. 341 00:24:23,510 --> 00:24:27,320 I mean, X1 and X2 here, the output states, OK. 342 00:24:28,040 --> 00:24:35,510 And what we will try to find here is we will try to control, okay, this is the interesting part. 343 00:24:35,720 --> 00:24:49,070 We will know, eg try to control the error dynamics again, not directly the system dynamics itself. 344 00:24:49,070 --> 00:24:58,310 I mean, y double dose plus, uh, a white dot plus um b b y x. 345 00:24:58,850 --> 00:24:59,720 Yeah, be y. 346 00:25:00,560 --> 00:25:09,110 But we will get here the error of the system and we will use this error in order to calculate our, 347 00:25:10,960 --> 00:25:20,270 uh, to calculate our um sliding surface OC in and calculation of our sliding surface. 348 00:25:20,570 --> 00:25:29,060 OK, and after that, if you will give this as an input to our system, OK, as we have done in the 349 00:25:29,510 --> 00:25:30,470 previous case. 350 00:25:30,680 --> 00:25:36,330 So a sliding surface we VE imposed. 351 00:25:36,530 --> 00:25:43,970 So what we have done, we have defined our sliding surface as what sliding surface us vi plus c y equals 352 00:25:43,970 --> 00:25:44,450 the zero. 353 00:25:44,480 --> 00:25:44,720 Yes. 354 00:25:44,720 --> 00:25:51,110 And we have said that if C is positive definite, then our system will be our sliding surface will be 355 00:25:51,110 --> 00:25:52,360 asymptotically stable. 356 00:25:52,370 --> 00:25:53,120 Okay, perfect. 357 00:25:53,360 --> 00:25:56,330 But in this case, what we want to make the 358 00:25:59,150 --> 00:26:06,170 asymptotically stable our error because, for example, we will give that OK, I want our y desire to 359 00:26:06,170 --> 00:26:07,730 be, for example, to. 360 00:26:07,990 --> 00:26:18,820 So now hope we will know that we have reached why desired by doing by by dropping this plot of not why, 361 00:26:18,820 --> 00:26:21,180 but our error, OK? 362 00:26:21,570 --> 00:26:25,510 And yet we will become and plot our way also. 363 00:26:26,410 --> 00:26:34,480 Yes, but we will generally throw our error phase plots or we will see that what happens with our y 364 00:26:34,480 --> 00:26:40,340 minus y desired OK and with y not minus y desired. 365 00:26:40,370 --> 00:26:40,750 OK. 366 00:26:41,320 --> 00:26:44,350 So if we reach our desired point or not? 367 00:26:44,390 --> 00:26:46,930 OK, if error reaches zero zero. 368 00:26:47,090 --> 00:26:47,620 OK. 369 00:26:47,860 --> 00:26:50,230 If error reaches zero zero, what does it mean? 370 00:26:50,380 --> 00:26:56,770 Because we want error dynamics to be asymptotically stable, OK, because we don't want y to be asymptotically 371 00:26:56,770 --> 00:26:57,100 stable. 372 00:26:57,100 --> 00:27:04,360 Because if because y desired, for example, will be two and not equal to zero, then at the same trajectory, 373 00:27:04,450 --> 00:27:08,450 uh, for example, y has to converge to two and not zero. 374 00:27:08,480 --> 00:27:08,830 OK. 375 00:27:08,980 --> 00:27:13,090 But would we want to be zero E's aerodynamics? 376 00:27:13,090 --> 00:27:23,020 Because if E and it becomes zero, then our Y will reach a wide and Vidot will reach y desired thought 377 00:27:23,020 --> 00:27:23,830 OK design. 378 00:27:23,830 --> 00:27:29,230 We will get desired position and desired velocity or something else where what y means it depends on 379 00:27:29,230 --> 00:27:29,920 the situation. 380 00:27:30,190 --> 00:27:30,550 OK. 381 00:27:31,810 --> 00:27:33,370 I hope that I may. 382 00:27:33,760 --> 00:27:35,290 I could make it clear for you. 383 00:27:35,530 --> 00:27:39,280 So here we will conclude our error how people calculate our error. 384 00:27:39,860 --> 00:27:46,450 We have avoided at the output of our system, OK, because x one was y and X two was y dot OK. 385 00:27:46,690 --> 00:27:58,360 After that, we are just subtracting y the subsets abstracting away from Y and avoid the I mean, this 386 00:27:58,360 --> 00:28:05,400 is y desired dot of y dot desired minus y d on in order to calculate E and e dot again. 387 00:28:05,650 --> 00:28:08,470 And we will miss now. 388 00:28:08,590 --> 00:28:15,550 We will combine them so by a vector come Canongate under the will give us an output, our error output, 389 00:28:15,550 --> 00:28:17,360 which we will throw in the phase plot. 390 00:28:17,380 --> 00:28:17,680 OK. 391 00:28:18,800 --> 00:28:19,180 OK. 392 00:28:20,370 --> 00:28:27,480 So and then what we will do is we will give this one in order to calculate our sliding surface. 393 00:28:27,490 --> 00:28:28,900 What was our sliding surface? 394 00:28:29,080 --> 00:28:32,950 Our sliding surface was y dot plus my equals to zero bust. 395 00:28:33,610 --> 00:28:39,720 But in this case, instead of why we will have error because we want aerodynamics to go to zero. 396 00:28:39,730 --> 00:28:42,930 Yes, because we want e note plus c e equals the zero. 397 00:28:42,940 --> 00:28:44,260 OK, be careful about that. 398 00:28:44,500 --> 00:28:48,690 We want aerodynamics to go zero b we don't want y to go zero. 399 00:28:48,700 --> 00:28:51,610 B we want y to go to y desired. 400 00:28:52,360 --> 00:28:58,120 I'm sorry that repeating this, but I want you to be careful and I want you to understand it. 401 00:28:58,750 --> 00:28:59,200 OK. 402 00:29:00,670 --> 00:29:05,480 I hope I can explain to you it clearly. 403 00:29:05,550 --> 00:29:09,730 OK, then what we are doing instead of why we have E here. 404 00:29:09,730 --> 00:29:17,230 So why c will be e times C and we will summit instead of why don't we will have e dot Aviva's sum here 405 00:29:17,230 --> 00:29:23,140 together and we will get here in e dot plus c e equals to zero. 406 00:29:23,170 --> 00:29:23,620 OK. 407 00:29:23,920 --> 00:29:31,480 And this will be and what we have said about our control algorithm vs say that our controller control 408 00:29:31,480 --> 00:29:37,450 algorithm will be you will be input will be equals minus key times. 409 00:29:37,450 --> 00:29:39,350 Signal function, yeah. 410 00:29:39,430 --> 00:29:45,380 And assign function of X as a sliding surface opposite to the sliding surface direction. 411 00:29:45,400 --> 00:29:51,790 OK, so we get here, as you can see sine function, OK, and we multiply by K. 412 00:29:51,790 --> 00:29:59,610 As we have said previously, we multiply by and this is our input and this is our sliding surface here, 413 00:29:59,630 --> 00:30:00,130 OK? 414 00:30:00,760 --> 00:30:05,620 We draw our sliding surface here and we draw our input also here. 415 00:30:05,650 --> 00:30:06,190 OK. 416 00:30:06,250 --> 00:30:09,070 You just give it as an input to our system. 417 00:30:10,360 --> 00:30:13,420 Here we also have disturbance term. 418 00:30:13,450 --> 00:30:13,760 OK. 419 00:30:13,810 --> 00:30:15,950 This is a step disturbance. 420 00:30:15,970 --> 00:30:16,300 OK. 421 00:30:16,480 --> 00:30:21,490 So for example, we will define here at which time, for example, if our simulation is two seconds, 422 00:30:21,700 --> 00:30:22,890 we will say that OK. 423 00:30:22,930 --> 00:30:24,670 At one second, OK? 424 00:30:24,910 --> 00:30:29,890 So a zero at one second, please apply with this amplitude. 425 00:30:29,890 --> 00:30:37,270 I'm sorry with this amplitude step function with sampling time of SW is that this is the sampling time 426 00:30:37,270 --> 00:30:38,970 of our disturbance. 427 00:30:38,980 --> 00:30:39,370 OK? 428 00:30:39,750 --> 00:30:41,810 Uh, don't worry about this one. 429 00:30:41,830 --> 00:30:42,160 OK. 430 00:30:42,520 --> 00:30:47,800 And this is just in order to get, um, beautiful. 431 00:30:48,610 --> 00:30:51,570 How can I say beautiful step function? 432 00:30:51,580 --> 00:30:56,200 OK with our how higher a lower sample time, OK? 433 00:30:56,710 --> 00:31:02,110 Our sampling frequency will be higher, and so we will sample our step function in a better way. 434 00:31:02,140 --> 00:31:02,440 OK. 435 00:31:02,470 --> 00:31:04,330 And this is no advantage. 436 00:31:04,600 --> 00:31:06,790 Let's then just go to that. 437 00:31:08,580 --> 00:31:13,170 Go to our simulation plotting here. 438 00:31:13,440 --> 00:31:16,920 And let's start by making this one. 439 00:31:17,190 --> 00:31:21,090 First of all, we will define our eight as eight as 15. 440 00:31:21,120 --> 00:31:23,860 OK, so these are positive, definite. 441 00:31:23,880 --> 00:31:31,170 So our function, I mean, our core efficiencies are positive. 442 00:31:31,380 --> 00:31:34,890 And so our system will be asymptotically stable, OK? 443 00:31:35,910 --> 00:31:37,560 And C equals the 10. 444 00:31:37,620 --> 00:31:42,300 We don't we do it in our sliding surface to be asymptotically stable. 445 00:31:42,300 --> 00:31:44,610 So our error will converge. 446 00:31:44,610 --> 00:31:46,470 Aerodynamics will be asymptotically stable. 447 00:31:46,470 --> 00:31:52,320 Our key is one step and look at here we have white desire equals the two. 448 00:31:52,330 --> 00:31:58,230 Again, we want our why variable to go to why desired of two. 449 00:31:58,450 --> 00:31:58,760 OK. 450 00:31:58,770 --> 00:32:01,440 And we want our why don't desire to go? 451 00:32:01,740 --> 00:32:02,270 Excuse me? 452 00:32:02,280 --> 00:32:05,260 Why not to go to y to decide which is zero? 453 00:32:05,490 --> 00:32:09,060 So this is no problem because our initial conditions were zero zero. 454 00:32:09,060 --> 00:32:16,050 So my desire that will not change and the vivo added no disturbance. 455 00:32:16,050 --> 00:32:17,550 So let's make it zero. 456 00:32:17,610 --> 00:32:19,800 We don't want any disturbance, OK? 457 00:32:20,280 --> 00:32:24,990 And this is our some control of sampling time. 458 00:32:24,990 --> 00:32:26,040 We will talk about it. 459 00:32:26,580 --> 00:32:28,350 And let's simulate our function. 460 00:32:28,350 --> 00:32:35,970 But before simulating, let's well, let me talk about one thing here, and we put our desire to avoid 461 00:32:36,130 --> 00:32:39,180 S2, so we have to 462 00:32:41,880 --> 00:32:43,670 give some additional game. 463 00:32:43,680 --> 00:32:52,870 OK, so our one part of our key will be to use the in order to achieve this y desired. 464 00:32:52,890 --> 00:32:53,370 OK. 465 00:32:54,180 --> 00:32:54,630 Why? 466 00:32:55,170 --> 00:32:58,800 Because of this one, let's say this is our dynamic system. 467 00:32:58,800 --> 00:32:59,940 OK, not. 468 00:32:59,940 --> 00:33:00,620 Let's say this. 469 00:33:00,630 --> 00:33:02,100 This was our dynamic system. 470 00:33:02,340 --> 00:33:05,400 Let's get its transfer function, how we can do that. 471 00:33:05,400 --> 00:33:07,620 We will first dollar plus transform. 472 00:33:07,860 --> 00:33:15,810 Then our transfer function was G equals the y-yes over us, where y is our output, you is our input 473 00:33:15,810 --> 00:33:20,210 and we can get our GS equals two from this equation. 474 00:33:20,220 --> 00:33:27,180 By manipulating, we will get these equals the one or s squared plus s plus b and the hope we can find 475 00:33:27,180 --> 00:33:32,550 our the C game of our system by its transfer function. 476 00:33:32,550 --> 00:33:38,640 B by just putting instead of s, we put this zero and we get what's our um? 477 00:33:40,390 --> 00:33:48,960 And it transfer function again is one or b again so we can write what we can write our output, what 478 00:33:48,960 --> 00:33:53,220 we want our output to be or we want our output to be two. 479 00:33:53,250 --> 00:34:01,200 Yes, divided by let me just what's our gain was run over B, which we can write one over our gain. 480 00:34:01,350 --> 00:34:03,860 In this case, our B is 15. 481 00:34:03,870 --> 00:34:04,260 Yeah. 482 00:34:05,460 --> 00:34:10,260 OK, let's just plug these numbers like that one. 483 00:34:10,260 --> 00:34:14,930 Our B is, let's put it in a symbolic way and we will change. 484 00:34:14,940 --> 00:34:21,270 So one of the B, which our game equals two to not two. 485 00:34:21,610 --> 00:34:23,220 This will be why desire? 486 00:34:23,280 --> 00:34:29,120 OK, which one we want to give y desires over our input? 487 00:34:29,130 --> 00:34:29,590 OK. 488 00:34:30,120 --> 00:34:30,690 Like this? 489 00:34:30,930 --> 00:34:40,110 So from here, you can see that in order to achieve what we want to achieve is instead we want to achieve 490 00:34:40,110 --> 00:34:40,440 here. 491 00:34:40,650 --> 00:34:40,980 Two. 492 00:34:41,010 --> 00:34:41,400 Yes. 493 00:34:41,610 --> 00:34:43,670 Why we want our Y desires to be two. 494 00:34:43,680 --> 00:34:45,540 So what we will the right one overall. 495 00:34:45,810 --> 00:34:47,110 What's our B? 496 00:34:47,130 --> 00:34:52,500 It's 15 equals the why this was our why design is to divide it by u. 497 00:34:52,740 --> 00:34:57,130 So from here, we will see that our u equals the surface. 498 00:34:57,130 --> 00:35:00,260 So just serve two of our inputs. 499 00:35:00,270 --> 00:35:04,410 OK, which is our input in this case is 150. 500 00:35:04,530 --> 00:35:05,550 OK, 150. 501 00:35:06,210 --> 00:35:17,190 The certainty of this 150 will be used in order to get up in order to get the video of two. 502 00:35:17,220 --> 00:35:17,700 OK. 503 00:35:17,940 --> 00:35:24,480 So in order to make the error to convert to zero, if our input would be less than this, for example, 504 00:35:24,480 --> 00:35:32,940 twenty five, then we would not have enough input in order to make y desired in order to achieve our 505 00:35:32,940 --> 00:35:33,390 desired. 506 00:35:33,420 --> 00:35:34,650 OK, be careful about that. 507 00:35:34,650 --> 00:35:41,400 But as we have 150 and as we have no disturbance, OK, because disturbance also can. 508 00:35:41,790 --> 00:35:44,880 Also, we'll take something about, uh, from the K, our input. 509 00:35:44,880 --> 00:35:47,790 OK, as we are, disturbance is zero. 510 00:35:47,790 --> 00:35:53,040 We will just use our input to achieve desired vida. 511 00:35:53,220 --> 00:35:58,520 OK, so let's simulate our system and let's see the drawings, OK? 512 00:35:58,830 --> 00:36:06,370 You know, as you can see, our system is this is error time and this is e dot and this is E! 513 00:36:06,420 --> 00:36:07,230 OK, so. 514 00:36:07,780 --> 00:36:12,940 Is here today, as you can see, both of our error and error thoughts. 515 00:36:13,060 --> 00:36:18,430 So velocita error and I mean my velocity error, I mean, why not? 516 00:36:18,790 --> 00:36:19,210 OK. 517 00:36:19,540 --> 00:36:23,680 And there why both reaches to the desired value? 518 00:36:23,890 --> 00:36:28,420 So because our E is zero, this means that we have achieved aerodynamic. 519 00:36:29,170 --> 00:36:32,750 Excuse me, e d of E of zero. 520 00:36:32,780 --> 00:36:33,190 OK. 521 00:36:33,220 --> 00:36:35,380 And we have achieved e d of zero. 522 00:36:35,590 --> 00:36:39,340 As you can see, we are starting from here as you can see our error. 523 00:36:40,090 --> 00:36:41,290 And what was our Y design? 524 00:36:41,470 --> 00:36:45,030 Our y desired was to OK, what was our initial point? 525 00:36:45,030 --> 00:36:47,860 That was why it was zero? 526 00:36:48,070 --> 00:36:49,600 Well, what's our initial error? 527 00:36:49,600 --> 00:36:50,170 It is two. 528 00:36:50,170 --> 00:36:50,920 And let's see. 529 00:36:50,920 --> 00:36:53,260 Yes, it's to our E is two. 530 00:36:53,470 --> 00:37:02,500 And as our initial velocity was zero and the Y desired dot is zero zero one zero view, this will give 531 00:37:02,950 --> 00:37:04,540 us e dot of zero. 532 00:37:04,750 --> 00:37:14,230 So we are starting from zero two or two zero and we are converging and they're converging to this zero 533 00:37:14,230 --> 00:37:14,560 zero. 534 00:37:14,590 --> 00:37:16,840 I mean, asymptotically stable. 535 00:37:17,290 --> 00:37:19,990 Our aerodynamics is asymptotically stable to the origin. 536 00:37:20,620 --> 00:37:22,330 This is our sliding surface. 537 00:37:22,570 --> 00:37:26,080 As you can see, it starts from, uh, twenty. 538 00:37:26,370 --> 00:37:28,450 OK, why start from today? 539 00:37:28,450 --> 00:37:33,520 Because what was our um, what was our sliding surface equation? 540 00:37:33,520 --> 00:37:35,890 It was white dot plus c y OK. 541 00:37:36,130 --> 00:37:38,230 Initially white, it was zero over. 542 00:37:38,230 --> 00:37:39,370 C is 10. 543 00:37:39,370 --> 00:37:41,020 And what was our initial y point? 544 00:37:41,020 --> 00:37:41,950 It was two. 545 00:37:41,950 --> 00:37:43,630 Yes, two times 10. 546 00:37:43,630 --> 00:37:47,320 It's 20 and then it reaches to the zero hour sliding surface. 547 00:37:47,320 --> 00:37:47,770 Why? 548 00:37:47,980 --> 00:37:51,260 Because we want our sliding surface to have zero value. 549 00:37:51,280 --> 00:37:54,310 OK, so that's why we have reached to the zero. 550 00:37:54,580 --> 00:37:55,090 That's perfect. 551 00:37:55,210 --> 00:37:55,720 Oh, OK. 552 00:37:55,930 --> 00:37:58,210 Excuse me, I have closed. 553 00:37:58,330 --> 00:37:59,620 Oh my gosh, not this one. 554 00:37:59,890 --> 00:38:01,810 I have closed the one important figure. 555 00:38:02,080 --> 00:38:02,590 I don't. 556 00:38:02,590 --> 00:38:05,150 When I do the OK, I don't want it to be closed. 557 00:38:05,190 --> 00:38:06,640 Yeah, we can close this one. 558 00:38:06,650 --> 00:38:09,120 OK, the important thing is our input. 559 00:38:09,130 --> 00:38:11,020 Let me take a make a bit bigger. 560 00:38:11,260 --> 00:38:11,590 OK. 561 00:38:11,680 --> 00:38:17,470 You can see here, and let's put them here, and let's compare what's happening here. 562 00:38:17,710 --> 00:38:27,400 As you can see, we are first applying 150 constant input here and the 150 because our trajectories 563 00:38:27,400 --> 00:38:30,730 is on there, as you can see positive side of the sliding. 564 00:38:30,730 --> 00:38:33,730 So these are sliding sims on our trajectories one 50. 565 00:38:33,910 --> 00:38:36,160 So our input doesn't need to change. 566 00:38:36,160 --> 00:38:37,270 OK, it comes columns. 567 00:38:37,270 --> 00:38:39,100 Collins comes close to this sliding surface. 568 00:38:39,130 --> 00:38:41,820 OK, so that's why it's constant. 569 00:38:41,830 --> 00:38:46,300 OK, 150 is our input OK? 570 00:38:46,480 --> 00:38:47,170 What happens? 571 00:38:47,170 --> 00:38:48,430 Then what happens? 572 00:38:48,430 --> 00:38:51,700 Then it switches to that. 573 00:38:53,320 --> 00:38:54,160 You know what happens? 574 00:38:54,160 --> 00:38:56,770 It directly switches to that. 575 00:38:58,780 --> 00:39:05,080 Where it switches, OK, then OK, it's in here is sampling. 576 00:39:05,080 --> 00:39:07,090 Time is high, so it's not. 577 00:39:07,210 --> 00:39:10,150 Excuse me, it's something flick was this high something time is low. 578 00:39:10,390 --> 00:39:15,520 So as you can see, then maybe I can make it clear for you. 579 00:39:15,620 --> 00:39:22,570 I me this one, as we have said before, as there is something frequency is finite, that there will 580 00:39:22,810 --> 00:39:30,220 be chattering filament, OK, and so input will make the as we have reached a sliding surface, the 581 00:39:30,220 --> 00:39:33,640 input will make the trajectory to leave a bit a little bit. 582 00:39:33,640 --> 00:39:37,300 We cannot see it a little bit the sliding surface. 583 00:39:37,300 --> 00:39:39,640 So that's what input will directly change. 584 00:39:39,640 --> 00:39:47,650 Its input will change its switch to the negative 150, OK, and it will start to oscillate as we have 585 00:39:47,650 --> 00:39:49,780 seen input will input. 586 00:39:49,990 --> 00:39:54,280 Not only input will make switching OK act. 587 00:39:54,580 --> 00:39:59,470 It is less accessible as you can see it switching action. 588 00:40:00,160 --> 00:40:05,170 Yes, because this is yeah, as the frequency sampling frequency is very high. 589 00:40:05,170 --> 00:40:08,590 So you see it in a solid way, but it is not solid. 590 00:40:08,590 --> 00:40:18,100 It's very little little little squares, OK, like square squares, signals, OK, because it switches 591 00:40:18,100 --> 00:40:21,460 from 150 to minus 150, 160 to minus 150. 592 00:40:21,730 --> 00:40:29,560 There is chattering cinnamon because our inputs excuse me, our sampling time sampling frequency is 593 00:40:29,560 --> 00:40:31,810 not infinite but finite. 594 00:40:32,080 --> 00:40:39,730 OK, you can see the chattering here more clearly, and we will see the show this um, in a minute much 595 00:40:39,730 --> 00:40:41,950 more clearly, we will talk about it anyway. 596 00:40:42,310 --> 00:40:44,950 We have seen this one also. 597 00:40:45,340 --> 00:40:52,660 Oh, OK, we I think we have talked about this and there, as you can see, we are going to the region 598 00:40:52,840 --> 00:41:00,760 where the front OK, here, if you reach the sliding surface at this second, OK, and then we are starting 599 00:41:00,760 --> 00:41:02,140 to just switch. 600 00:41:03,280 --> 00:41:05,680 So let's continue. 601 00:41:05,710 --> 00:41:06,790 OK, what is that? 602 00:41:06,790 --> 00:41:07,030 Oh. 603 00:41:07,540 --> 00:41:14,370 This is what we value, as you can see, our way values starts from zero and it reaches where we desired 604 00:41:14,380 --> 00:41:15,070 off to. 605 00:41:15,110 --> 00:41:21,760 If you want wide desired wealth to OK, this is our error error converges to zero. 606 00:41:22,120 --> 00:41:24,400 Uh, okay. 607 00:41:24,670 --> 00:41:29,560 So now let's let's take this example. 608 00:41:29,770 --> 00:41:38,010 What will happen if we put a disturbance of minus 40, for example, we put a disturbance of minus 40? 609 00:41:38,210 --> 00:41:38,530 Yeah. 610 00:41:41,110 --> 00:41:44,350 What will happen if you put a disturbance of minus 40? 611 00:41:44,360 --> 00:41:47,710 Let's just run this and it will. 612 00:41:48,700 --> 00:41:52,510 The disturbance will be applied one second at one second, OK. 613 00:41:52,690 --> 00:41:55,870 So overall simulation is to second and at one second. 614 00:41:56,350 --> 00:42:02,410 At first, second, the disturbance will be applied and our gain is still 150mm. 615 00:42:02,650 --> 00:42:03,940 Let's see what will happen. 616 00:42:04,550 --> 00:42:07,580 Um, as you can see, we have achieved perfect result. 617 00:42:07,600 --> 00:42:08,000 OK. 618 00:42:08,020 --> 00:42:08,500 Why? 619 00:42:08,770 --> 00:42:12,520 Because the robust control of it is enough, OK? 620 00:42:12,790 --> 00:42:21,720 We have forty four disturbance and we use also 30 of the input for the for the what for. 621 00:42:22,180 --> 00:42:24,090 In order to achieve the desired of two. 622 00:42:24,310 --> 00:42:31,090 So just 70, our input has to be so our key has to be greater than 70. 623 00:42:31,270 --> 00:42:38,020 In order to achieve what we want and as it is greater than 70, we can achieve what we want. 624 00:42:38,020 --> 00:42:44,590 So as you can see, disturbance doesn't affect, didn't affect the stability of our system and our error 625 00:42:44,590 --> 00:42:48,760 converges to zero without any problem. 626 00:42:48,760 --> 00:42:51,610 And let, OK, let's continue. 627 00:42:51,610 --> 00:43:01,720 Then let me just show the this is the way it reaches to, as we have said, no problem with this one. 628 00:43:01,820 --> 00:43:02,270 OK. 629 00:43:02,530 --> 00:43:07,840 This is our as you can see our error of the excuse me, our disturbance. 630 00:43:07,840 --> 00:43:18,470 At one second, we apply the minus 40 of disturbance, OK, and this is error which reaches to zero. 631 00:43:18,820 --> 00:43:19,240 OK. 632 00:43:19,530 --> 00:43:20,200 To zero. 633 00:43:20,200 --> 00:43:24,170 And um, this is just our sliding service. 634 00:43:24,220 --> 00:43:26,470 As you can see, we are reaching here. 635 00:43:27,640 --> 00:43:32,140 The sliding surface of zero is not a problem. 636 00:43:32,440 --> 00:43:32,890 OK. 637 00:43:33,220 --> 00:43:33,700 OK. 638 00:43:33,790 --> 00:43:42,160 So, um, let me show you let me just make it in 40 and let's see what will happen. 639 00:43:43,210 --> 00:43:45,290 And this should not be anything changing. 640 00:43:45,310 --> 00:43:45,590 OK? 641 00:43:45,610 --> 00:43:47,380 As you can see, there is nothing. 642 00:43:48,010 --> 00:43:48,760 Nothing changed. 643 00:43:49,150 --> 00:43:56,290 Again, um, everything works as expected, and this is our again input. 644 00:43:56,860 --> 00:43:59,800 Until here we are trying to reach the sliding surface. 645 00:43:59,800 --> 00:44:01,450 And then you switch switches happen. 646 00:44:01,450 --> 00:44:08,190 And as you can see, the switching action increases when disturbance is applied. 647 00:44:08,200 --> 00:44:16,960 OK, and we need more switching in order in order to get rid of the disturbance, OK? 648 00:44:17,560 --> 00:44:22,450 The switching happens more OK because it deviates the trajectory more so. 649 00:44:22,450 --> 00:44:26,500 That's why we have this action lit under. 650 00:44:26,920 --> 00:44:34,090 Let's see what will happen if we make our game less than 70, for example. 651 00:44:34,090 --> 00:44:35,290 Let's make it fifty. 652 00:44:35,790 --> 00:44:38,830 OK, let's check it now. 653 00:44:38,900 --> 00:44:39,640 What will happen? 654 00:44:39,640 --> 00:44:41,440 So our input is not enough. 655 00:44:41,770 --> 00:44:43,950 It's less than 70. 656 00:44:43,960 --> 00:44:47,260 OK, so let's see what's happening. 657 00:44:47,570 --> 00:44:53,560 And as you can see, our trajectory not reaches their origin, but it deviates from it. 658 00:44:53,590 --> 00:44:53,980 OK? 659 00:44:54,220 --> 00:44:56,860 Our aerodynamics deviates from that. 660 00:44:57,310 --> 00:44:59,820 It's not asymptotically stable. 661 00:44:59,830 --> 00:45:00,280 OK? 662 00:45:01,660 --> 00:45:06,640 It hurts and becomes unstable because error. 663 00:45:06,880 --> 00:45:07,420 Excuse me. 664 00:45:07,420 --> 00:45:13,540 Because again, key can is not enough to get rid of the disturbance. 665 00:45:13,840 --> 00:45:14,260 OK. 666 00:45:14,350 --> 00:45:15,470 This is our why. 667 00:45:15,490 --> 00:45:23,560 As you can see, it reaches two, but again, it deviates from it because um, and the gain is not enough 668 00:45:23,560 --> 00:45:26,130 to get rid of the disturbance. 669 00:45:26,140 --> 00:45:26,590 OK? 670 00:45:26,940 --> 00:45:27,370 OK. 671 00:45:27,430 --> 00:45:33,190 As you can see here, why is reaching to at almost one second? 672 00:45:33,460 --> 00:45:38,500 And the applied disturbance at one second, and as you can see it, deviates from it. 673 00:45:38,510 --> 00:45:41,560 So that's why the input is like that, OK? 674 00:45:41,800 --> 00:45:45,700 It because it deviates to the as I have closed it here. 675 00:45:46,120 --> 00:45:48,850 Oh my gosh, I'm stupid the way I did closed. 676 00:45:49,090 --> 00:45:56,140 But anyway, you have seen that the trajectory was deviating toward where to the positive side. 677 00:45:56,140 --> 00:46:06,220 So it applies that game in order to get rid of it, in order to make it back to go back to where to 678 00:46:06,250 --> 00:46:07,450 the sliding surface of. 679 00:46:07,950 --> 00:46:14,850 But it's as you can see, the trajectory continues to deviate because the input is not enough. 680 00:46:15,690 --> 00:46:16,100 Okay. 681 00:46:18,120 --> 00:46:24,780 So always remember our gains should be enough, so it shouldn't be higher than the absolute value of 682 00:46:24,780 --> 00:46:26,160 our total disturbances. 683 00:46:26,190 --> 00:46:29,310 OK, let's do it now. 684 00:46:29,730 --> 00:46:36,130 Our system unstable again by putting minus eight here and making it again one shift. 685 00:46:36,450 --> 00:46:37,070 OK? 686 00:46:37,140 --> 00:46:39,930 And let's make the disturbance a zero. 687 00:46:40,720 --> 00:46:42,690 OK, we have band disturbance zero. 688 00:46:43,020 --> 00:46:44,460 Let's check what will happen. 689 00:46:44,910 --> 00:46:53,250 Our system, as we have seen this previously in previous video, we have seen that this will be as a 690 00:46:53,310 --> 00:46:53,790 equals. 691 00:46:53,790 --> 00:46:56,670 The minus eight equals the minus eight. 692 00:46:56,850 --> 00:46:58,410 Our system will be unstable. 693 00:46:58,410 --> 00:47:02,610 And let's see if robust control is enough to make our system stable. 694 00:47:02,880 --> 00:47:07,140 Yes, as you can see, our system became stable, OK? 695 00:47:07,500 --> 00:47:09,060 As you can see in the. 696 00:47:10,790 --> 00:47:18,140 Input make it a game converts to this zero zero, so our aerodynamics is asymptotically stable, OK, 697 00:47:18,170 --> 00:47:25,040 and this is our sliding surface and this is our input, as you can see this our input again, it switches 698 00:47:25,310 --> 00:47:27,350 very fast in order to make the 699 00:47:30,470 --> 00:47:35,510 drive, the trajectory or to the sliding surface, OK? 700 00:47:36,740 --> 00:47:42,410 As you can see, our error converges to its thrust from took because our way deep first these two and 701 00:47:42,410 --> 00:47:45,080 it converges to zero as time passes. 702 00:47:45,440 --> 00:47:45,710 OK. 703 00:47:47,240 --> 00:47:53,570 So as you can see and hear, even in the system is unstable. 704 00:47:53,570 --> 00:47:57,470 Our aerodynamics became stable. 705 00:47:57,470 --> 00:48:05,180 We make our aerodynamics asymptotically stable using using robust control techniques. 706 00:48:05,360 --> 00:48:09,590 Perfect virtual sliding surface and designed control input. 707 00:48:09,770 --> 00:48:12,950 But we have to analyse this controller in two cases. 708 00:48:13,310 --> 00:48:13,780 Why? 709 00:48:14,210 --> 00:48:17,120 Because our control input is like switching action. 710 00:48:17,510 --> 00:48:20,300 Let's go deeper and you will understand what I mean. 711 00:48:20,900 --> 00:48:26,720 First, we will analyse system in continuous the mean which is ideal, namely control action. 712 00:48:26,720 --> 00:48:32,270 You, in other words, switching action computes in an inside night frequency. 713 00:48:32,690 --> 00:48:40,250 In other words, as the system and the system is continuous, sampling frequency is infinite and we 714 00:48:40,250 --> 00:48:47,960 can detect the deviation from sliding surface instantly and input will change based on that instantly. 715 00:48:48,230 --> 00:48:52,550 And so deviation will be corrected instantly. 716 00:48:54,140 --> 00:49:00,320 So we will have zero oscillation at output variable y because it's always over a sliding surface. 717 00:49:01,220 --> 00:49:04,310 But we control the robot with controllers, aren't we? 718 00:49:04,940 --> 00:49:09,560 And they have finite something frequency as they are discrete time systems. 719 00:49:10,020 --> 00:49:11,380 Now we have a problem. 720 00:49:11,390 --> 00:49:17,930 France, a sampling frequency is finite, switching action commutes in a finite frequency, so trajectory 721 00:49:17,930 --> 00:49:23,780 deviation from sliding surface is detected late and the corresponding action is taken late. 722 00:49:24,020 --> 00:49:27,890 And this latency depends on the sampling time of the controller. 723 00:49:28,670 --> 00:49:33,460 So we will have non-zero amplitude oscillations at the art variable. 724 00:49:33,470 --> 00:49:38,390 Why this is called chattering phenomenon, and we will analyze it more deeply in a minute. 725 00:49:39,080 --> 00:49:45,440 The amplitude of these oscillations is proportional to the gain K in the control, input and sampling 726 00:49:45,440 --> 00:49:49,310 period T, as depicted in Eq. one point two. 727 00:49:49,820 --> 00:49:56,900 This is because as gain K is, a higher trajectory will be directed to the sliding surface faster with 728 00:49:56,900 --> 00:50:01,340 higher energy and it will reach pass sliding surface more. 729 00:50:01,730 --> 00:50:07,880 If you add to it also slow sampling time, then the situation will be worse because the deviation from 730 00:50:07,880 --> 00:50:13,130 sliding surface will be detected late, so oscillation amplitudes will be higher. 731 00:50:13,490 --> 00:50:17,750 Think it in this way instead of trajectory, you put yourself in the plot. 732 00:50:17,990 --> 00:50:24,620 If you are in positive region, your friend will push you towards the negative region and your friend 733 00:50:24,650 --> 00:50:27,710 has something to add in each sample time. 734 00:50:27,920 --> 00:50:32,840 It will check in which region you are and will take appropriate action. 735 00:50:33,350 --> 00:50:38,020 If he pushes you with high power, then you will reach faster. 736 00:50:38,060 --> 00:50:44,450 The sliding surface But as sampling time of your friend is finite, he will not be aware that you have 737 00:50:44,450 --> 00:50:48,530 reached and passed the sliding surface in the next sampling. 738 00:50:48,710 --> 00:50:55,070 He will check and see, Oh, you will have reached and lift the sliding surface and he will switch and 739 00:50:55,070 --> 00:50:56,810 push you toward the opposite side. 740 00:50:57,410 --> 00:51:03,470 If you gently push you instead of high power, then you will leave less the sliding surface during the 741 00:51:03,470 --> 00:51:04,790 given sampling time. 742 00:51:05,240 --> 00:51:11,180 So less sampling time and lower m k are better. 743 00:51:12,110 --> 00:51:15,510 Excuse me, by less sampling time? 744 00:51:15,800 --> 00:51:19,940 Yeah, I mean, unless something triggers a high frequency. 745 00:51:20,270 --> 00:51:23,120 So, uh, the latency is low. 746 00:51:23,510 --> 00:51:26,450 But why we don't always choose lower. 747 00:51:26,990 --> 00:51:29,090 This is because of condition. 748 00:51:29,090 --> 00:51:33,020 A K has to be higher than the disturbance upper bound. 749 00:51:33,260 --> 00:51:37,370 If disturbance is high, then the K will be high also. 750 00:51:37,820 --> 00:51:45,650 And in order to have less sampling time, you have to invest more money and get more powerful controller, 751 00:51:46,040 --> 00:51:48,710 which will also surely consume more power. 752 00:51:49,190 --> 00:51:53,180 Let's try to understand charging better by visualization. 753 00:51:53,330 --> 00:51:55,490 Here is the plot and initial condition. 754 00:51:55,880 --> 00:52:00,800 We start to drive state trajectories toward sliding surface until they reach it. 755 00:52:01,310 --> 00:52:07,700 However, because of finite sampling time again, by finite, I mean greater than zero. 756 00:52:08,240 --> 00:52:09,750 OK, sampling time control. 757 00:52:09,880 --> 00:52:16,420 Will still apply, control, input and so tragic that trajectory will leave the sliding surface sampling 758 00:52:16,420 --> 00:52:24,400 time reaches and controller checks in this region is a trajectory and sees that it is in negative region, 759 00:52:24,670 --> 00:52:29,950 so it switches and pushes the trajectory to the other side. 760 00:52:31,600 --> 00:52:37,090 But again, due to the finite sampling time, it continues to apply input even after the trajectory 761 00:52:37,090 --> 00:52:46,000 reaches the sliding surface, then the same thing happened as before, and it continues like that until 762 00:52:46,000 --> 00:52:47,980 it reaches to the origin. 763 00:52:48,340 --> 00:52:54,370 You can see the chattering in the plot instead of smoothly going to the origin. 764 00:52:55,330 --> 00:53:01,320 Now, let's see what happens if we increase sampling time again. 765 00:53:01,330 --> 00:53:09,820 Input drive trajectory again, input drives trajectory to the sliding surface, but because of finite 766 00:53:09,820 --> 00:53:12,640 sampling time, controller continues to apply. 767 00:53:12,670 --> 00:53:18,670 Input and trajectory leaves sliding surface as in this case, sampling time is higher. 768 00:53:18,670 --> 00:53:20,530 Deviation also becomes more. 769 00:53:20,950 --> 00:53:27,610 After sampling, time reaches control of chips and sees the trajectory is in negative region, so applies 770 00:53:27,610 --> 00:53:32,050 opposite input and again, because of finite sampling time. 771 00:53:32,500 --> 00:53:39,220 Trajectory reaches the sliding surface and leaves it and the process continues like that. 772 00:53:39,580 --> 00:53:42,670 I didn't throw fully, but you understand what will happen. 773 00:53:43,420 --> 00:53:47,460 It will converge to this until Odigem. 774 00:53:47,650 --> 00:53:55,570 If we summarize as key becomes larger, oscillation amplitude at the output also increases and sampling 775 00:53:55,570 --> 00:53:57,010 time T increases. 776 00:53:57,010 --> 00:54:03,970 Oscillation amplitude also increases at the output, so we want controller with high sampling frequency 777 00:54:05,200 --> 00:54:11,650 chattering, indeed not avoidable problem and causes some problems like stream on the structure of the 778 00:54:11,650 --> 00:54:18,070 robot because switching action could change in the direction of control inputs, but structure has to 779 00:54:18,070 --> 00:54:20,470 be able to cope with that instant changes. 780 00:54:21,220 --> 00:54:28,120 Additionally, this fast switching action will create noise because of changing actuators, so we have 781 00:54:28,120 --> 00:54:31,230 to find some methods to handle this problem. 782 00:54:31,240 --> 00:54:35,440 I mean, charging problem and we will see them on the next lesson. 783 00:54:35,860 --> 00:54:40,390 Now let's jump into the MATLAB and observe chattering in practice. 784 00:54:40,450 --> 00:54:49,410 OK, now let's continue with our chattering Fenimore and Feinerman and see what's happening if we can 785 00:54:49,420 --> 00:54:55,830 say, yeah, what will happen if we change the our systems? 786 00:54:56,380 --> 00:54:57,820 Something time or time? 787 00:54:58,060 --> 00:55:04,630 How we can change in most of the sampling time, let's we will come here, we will go to model settings 788 00:55:04,630 --> 00:55:14,200 and here you can make it the configuration solver and solver selection a fixed step. 789 00:55:14,200 --> 00:55:21,820 You will make it not, uh, variables that are fixed up and you will hear will put your sampling time 790 00:55:21,820 --> 00:55:28,570 OK, and it will define your sampling frequency because sampling frequency is nothing but one or sampling 791 00:55:28,570 --> 00:55:28,980 period. 792 00:55:28,990 --> 00:55:31,910 OK, and here is our controller sampling time. 793 00:55:31,930 --> 00:55:32,380 OK. 794 00:55:33,370 --> 00:55:33,760 OK. 795 00:55:33,820 --> 00:55:42,280 Why I did this sampling time for input block because it has to be always less than the sampling time 796 00:55:42,280 --> 00:55:45,040 of the FSD. 797 00:55:45,050 --> 00:55:46,960 OK, so sampling time of the controller. 798 00:55:46,990 --> 00:55:52,360 So let's see if it's at this sampling under. 799 00:55:52,390 --> 00:55:55,240 Let's see what will happen with our chattering. 800 00:55:55,240 --> 00:56:01,120 We have um, we don't have disturbance now is zero disturbance and our system is stable. 801 00:56:01,120 --> 00:56:07,900 As you can see, this is, um, these are not interesting and this is our input as you can see how it 802 00:56:09,970 --> 00:56:11,740 switches very, very fast. 803 00:56:11,740 --> 00:56:19,870 As it seems, input seems to us as solid block, OK, because the sampling is very a sampling frequency 804 00:56:19,900 --> 00:56:27,280 is very high, so our input commutes very high and you can see our output. 805 00:56:27,290 --> 00:56:30,610 Why there is no chattering here. 806 00:56:30,920 --> 00:56:34,660 Surely there is chattering, but it seems very smooth to us, OK? 807 00:56:34,840 --> 00:56:36,730 And also, you can see it from here. 808 00:56:36,730 --> 00:56:38,300 As you can see, it is very smooth. 809 00:56:38,320 --> 00:56:41,470 There is like no chattering, but indeed there is chattering. 810 00:56:41,470 --> 00:56:46,690 If we indeed let me again make it, maybe when I zoom, we can see the chattering. 811 00:56:46,960 --> 00:56:47,920 But let me try. 812 00:56:48,310 --> 00:56:49,800 Maybe we can see. 813 00:56:49,810 --> 00:56:53,600 Yes, as you can see, there is chatting, but it's very, very small. 814 00:56:53,620 --> 00:56:54,130 OK. 815 00:56:54,250 --> 00:57:00,940 Let's now check what will happen if we decrease our sampling time and let's decrease. 816 00:57:00,940 --> 00:57:04,630 Let's not stick with this one and make it run. 817 00:57:05,350 --> 00:57:06,320 And let's see. 818 00:57:06,340 --> 00:57:06,890 OK? 819 00:57:06,910 --> 00:57:09,710 As you can see, it clearly can see. 820 00:57:10,410 --> 00:57:21,660 Chattering, OK, as you can see, as our sampling time increases, OK, we increase before I say, 821 00:57:22,050 --> 00:57:25,830 let's decrease, but I'm sorry this was a false word. 822 00:57:26,200 --> 00:57:26,700 OK. 823 00:57:27,840 --> 00:57:35,540 We are increasing the sampling time, OK, we increase the sampling time and as you can see, if we 824 00:57:35,550 --> 00:57:37,660 can see clearly, the chattering Fentiman. 825 00:57:38,090 --> 00:57:43,020 OK, here is that with OK, we even don't need to zoom. 826 00:57:43,020 --> 00:57:47,070 We can see the chattering and also we can see the charging and output. 827 00:57:47,070 --> 00:57:49,260 Let's see the output. 828 00:57:49,260 --> 00:57:55,520 OK, OK, we can see the uh, oscillations at the output, OK? 829 00:57:55,730 --> 00:57:58,230 And I want to show you also the input. 830 00:57:58,230 --> 00:58:01,350 I forgot it, but let me show you the input. 831 00:58:01,350 --> 00:58:09,300 As you can see here, the input is not as solid as before, OK, because it commutes now with less frequency. 832 00:58:10,980 --> 00:58:16,290 OK, let's not check what will happen if we make it, uh, zero point zero one. 833 00:58:16,650 --> 00:58:18,960 Let me increase the sampling time. 834 00:58:22,200 --> 00:58:22,860 OK. 835 00:58:23,290 --> 00:58:25,050 And let's check what will happen. 836 00:58:26,700 --> 00:58:27,220 OK. 837 00:58:27,270 --> 00:58:30,710 And we can clearly see here the chattering. 838 00:58:30,720 --> 00:58:31,140 OK? 839 00:58:31,170 --> 00:58:33,420 We can clearly see that chattering. 840 00:58:33,540 --> 00:58:34,050 OK. 841 00:58:35,220 --> 00:58:41,490 And you can see the input with your bare eyes. 842 00:58:41,550 --> 00:58:42,060 OK. 843 00:58:42,590 --> 00:58:46,320 And there you can see the oscillations at the output, OK? 844 00:58:46,350 --> 00:58:50,760 As you can see, these are very high oscillations in the output. 845 00:58:50,760 --> 00:58:51,050 OK? 846 00:58:51,170 --> 00:58:53,190 You can see that oscillations at the output. 847 00:58:54,300 --> 00:58:55,000 OK. 848 00:58:56,070 --> 00:59:02,750 So as you can see how chattering affects to the UM, to our problem. 849 00:59:02,760 --> 00:59:03,640 OK, so Sam. 850 00:59:04,080 --> 00:59:09,880 So as a whole, sampling time affects our um chattering from. 851 00:59:10,500 --> 00:59:12,300 I don't know if I am. 852 00:59:12,960 --> 00:59:14,250 Let me just I didn't 853 00:59:17,070 --> 00:59:21,970 try it, but let's check if we increase going to high. 854 00:59:21,990 --> 00:59:25,710 What will happen if we get chattering phenomenon there, Sean? 855 00:59:27,120 --> 00:59:28,320 Let's check. 856 00:59:28,440 --> 00:59:29,550 OK. 857 00:59:29,910 --> 00:59:33,900 And as you can see as you go, let's let's try it. 858 00:59:34,350 --> 00:59:35,610 Let's try in this way. 859 00:59:35,950 --> 00:59:36,540 OK. 860 00:59:37,260 --> 00:59:38,970 This is in this way, OK? 861 00:59:39,000 --> 00:59:42,330 Let's see what will happen if we decrease will be. 862 00:59:43,590 --> 00:59:45,130 Will it be the same? 863 00:59:46,080 --> 00:59:50,910 But you know, we will typically make with 150, and let's check. 864 59:52.870 --> 1:00:00.460 OK, surely, surely you can see the difference, OK, in the previous case, as you can see the chattering 865 1:00:00.460 --> 1:00:02.830 phone numbers much clearer. 866 1:00:02.960 --> 1:00:08.830 OK, we could see with our bare eyes the chattering. 867 1:00:08.830 --> 1:00:15.790 So as you can see the increase, the even, let's increase like that and see and we can see the OK as 868 1:00:15.790 --> 1:00:19.960 you can see the chattering clearly, as you can see, it affects gain. 869 1:00:19.960 --> 1:00:27.970 And also the sampling time affects to our to the chattering and it creates oscillations at the output.