 So, I wanted to talk to you about different types of models and how this the course that we are doing is relevant to models, ok. So, this is what I thought I would do today. And in the next class, I would take a specific model and then tell you how we used it in the field of computing, computer science, but I will do that in the next class. So, outline for today's class is, we are going to discuss a few examples. One is about Galileo spacecraft. How many of you have heard of this? Galileo spacecraft, can you raise your hand? Ok. So, we will discuss that. We will also discuss briefly, Jamnagar refinery. How many of you have heard of this? Ok, many of you have heard of this. How many of you are chemical engineers? And then, I would also want to talk about a model that talks about cycling from the hostel, ok. And then, hope to talk about using these models in feedback, feed forward manner. Do these terms mean anything to you? Feed forward, feedback, little bit. How many people vaguely understand this? And then, we will discuss something called time series model. I will just give a brief outline. And in the next class, I will see how to talk about it. So, outline time series feedback control, you know, most likely I will be doing this in the next class, just once again, ok. So, here is the Galileo spacecraft, ok. So, it was, it was launched by NASA to study Jupiter and its moons. It was launched on 18th October, 1989. I think all of you were born after this, right? Yes. And it arrived at Jupiter on 7th December, 1995. So, it took more than 6 years to reach Jupiter, ok. And it also functioned, ok. It was just going round and round, ok. It was to study Jupiter and its moons, ok. Let us look at the fuel that it used. By the way, it travelled 2.4 billion miles to reach the target. So, it took 6 years. It used only 67 gallons of fuel. I have taken this from NASA's website. So, 1 gallon is approximately 3.7 liters. As a result, if you just divide, you see that it has travelled 36 million miles in 1 gallon, right. So, what is that equivalent to? I am just giving an idea of, you know, this is I would say a man-made miracle. It is a miracle, ok. So, that is equivalent to 4 tablespoons of fuel to go to moon and return, ok. It is incredible. And some of the interesting facts about this rendezvous, how they calculated, judged precisely. The radio transmitter probe should probe entry, should take place in a window of 8 minute interval. How many of you are working on the satellite project here? Is there anyone who is hoping to get into the satellite project? There are some of you who are interested, right. So, for these things to happen, it should happen in a window. If you miss that, then you may have to wait for a long time. So, this, in order to create a link, proper link between transmitter and the probe and the transmitter, that entry, probe entry should take place in a 8 minute interval and it should be decided 167 days in advance. Of course, it is a 6 year travel. This fellow had already reached somewhere, but about half a year ahead of time, 6 months ahead of time, you have to decide, ok. You have to change your maneuvers and so on. So, this is 0.0033 percent error. The only thing that is allowed and it turns out that it entered 14 seconds earlier, ok. And of course, the error is very, very small, ok. So, why am I saying all this? Just see some more. So, it traveled 2.4 billion miles in 6 years and it missed the target at Jovian moon. How do you pronounce this? E E E E by only 67 miles, ok. So, in this, in this website, they are saying that it is almost like shooting an arrow from Los Angeles to New York and missing the bullseye by 4 inches. So, that is the kind of accuracy that was achieved. So, what computing resources do you think were used in this space craft? Imagine this was launched in 89, right, when the PC had just come. Now, look at this, look at its speed, ok. So, what do we, what numbers do we discuss now? How many hertz? Do we talk in mega hertz? It had princely 128 k RAM. And that too in 6 memory chips. So, I was telling you about a low cost computer that we are trying to build, right. So, we are saying that it is a low cost computer. We want to do this in about 30 dollars and it would, some of the specs are, it will have a low speed processor. Anything of the order of 500 megahertz is ok, ok. And anything of the order of half a GB is ok. Of course, it turns out that in the market you do not get anything less than 1 GB RAM nowadays, right. So, we are talking about a low cost computer that can be hopefully made in about 30 dollars, assuming that we make say 100000 pieces. Then, so this processor is on par with a 8 bit 6502 used by Apple II computers at that time. And it generated power to light and light bulbs by 70 watts. And it was, if I am not mistaken, 3 tons by this, just to give a idea of this. So, why am I talking about all this? How did they achieve all this? You know, I would say it is a miracle, really incredible, ok. Of course, now we have seen Chandrayaan, we found water and so on and so forth. How do people achieve all this? It turns out the secret of all this is mathematical model, mathematical modeling. So, of course, very simple things you would come across things like mass balance, energy balance, momentum balance, of course, control loss, predicting material properties, thermodynamics, strength of materials, so on and so forth. So, all these things are really important, ok. It turns out that many of these balance equations can be written in the form that I have in the bottom, dx by dt is f of x. So, these are all differential equation models, is that ok? Have you, I am sure you have seen these things, right, mass balance, what is the mass balance, what is the energy balance and things like that and of course, this is in the form of a transient equation or a dynamic equation. So, you have, so these things are the bedrock on which you build this entire science, ok. So, that was the Galileo spacecraft, ok. There are many, many such examples can be given. So, I thought that I would give examples from different domains, because there are people from different areas. So, I will take one more example. This is the Jamnagar refinery, many of you say that you have heard this. I have some interesting statistics, I will read them out to you. I did not, I have not put them in the part point, in this Beamer presentation. By the way, how many of you use Beamer? How many of you use Latech? Does anyone use, one person uses Latech. Do you use Beamer also for presentation? It is a great software, you may want to use that. So, Jamnagar refinery, 660,000 barrels per day, ok. And when it was established, it was the largest single stream green field refinery. So, green field means that it came from scratch, as opposed to modifying an existing one, as opposed to adding similar units of same kind together and increasing the capacity, which is something that people often do. And it was completed in a record 36 months, right, which is incredible. And this is given as the main reason for making it 40 percent cheaper than comparable units created by other people. And interestingly, no pilot plant, it is all based on simulation. Of course, in building these models, lot of experience go in, lot of experience goes in. Of course, Jamnagar refinery, the design, basic design and so on were done by a company called, does anyone know which company did this? It is a company called UOP in Chicago, no pilot plant. It is based on mathematical models, simulators, simulators for various things. And of course, mass energy balance, thermodynamics and control, ok. What I will do is, I will give some statistics about this. Unfortunately, you cannot see this. So, what I will do is, I will just read them out, because these are just statistics. So, this Jamnagar refinery is now in the process of putting up another plant right next to it. And they look at the side from space, they give this information, ok. Building the new refinery at Jamnagar involves over 200,000 engineering and supplier documents. The installation of 165,000 tons and 53 million cubic feet of concrete, 138,000 tons of structural steel equivalent to 15 Eiffel towers, ok and so on. The project site covers an area larger than London and more than 80,000 workers will be employed during the peak of construction, ok. Around the same number deployed in building the first refinery. We are talking about the second refinery of comparable size that is going to come up. What I told you till now was a previous refinery, a new refinery is going to come up, comparable size and the construction is going on right now. So, this document came in 2008 or so, ok. To overcome the shortage of skilled construction workers, reliance has set up a facility in Jamnagar to train an estimated, they are going to, they train them on site, 8,000 welders, 5,000 carpenters, 5,000 pipe fitters and a few thousand grinders and mill ride fitters. There are 80,000 workers in that site. We are talking about a really huge plant that site is larger than London, ok. Now, such a huge plant has been put up and it has been working at, I think I heard numbers like 110 percent capacity, 120 percent capacity, and so on, right. Have you heard this numbers about Jamnagar refinery? This plant is operating at 110 percent capacity, 120 percent capacity and so on. How do you explain this? Greater than the, greater than 100 percent capacity, how do you explain that? Any guess? Any guess? So, the capacities are always arrived at by assuming some safety factor, ok. So, they will say, in fact, there is something called deep bottle-necking. Essentially, what they will do is, in fact, this happens in all operational plants, whether it is power plant, whether it is refinery, oil, chemical, anything. If you take the computer, it is the same. You can think of the computer system as a plant, right. So, essentially they will look at, they will say that, ok, this plant is, it can handle, this unit can handle some 100,000, let us say, kilograms, you know, let us say per hour. Then they will say, can I increase it by 5 percent, ok. So, what will happen is, as you try to push through, you will see that something will hit the upper limit. For example, some valve may be completely open. So, they will say that, look, this pipe is blocking it, ok. Same thing will happen in the computer. You know, we are, we have so much speed, but you know, I want to increase it slightly. Can I try it? Then they will say that, maybe this particular unit, this data stream, right, is blocking it. Then they will say, ok, let me go and increase it. Go to that pipe, remove that pipe, put a new one, bigger one, ok. So, now, that will support the increased throughput, ok. Then they will say, ok, let me, can I increase it further? And then something else may give up. You may not be able to heat it. You may not be able to cool it, because you do not have enough fuel there. So, then they will say, ok, increase it by 10 percent there. So, what I am saying is, this is what is known as debottlenecking. Through this process, it is possible to increase the capacity higher than the rated capacity, ok. So, of course, people always talk, I mean, one, there is another thing about reliance. They say that, they normally build it for larger than rated capacity. So, which means from day one, it will start operating at 90 percent capacity. And then, within a month, it will go to 100 percent, then 110 percent and so on, ok. So, that is, so what is incredible is that, such a huge plant can operate, can be put up and run efficiently without building similar things based completely on mathematical modeling and nothing but these, mass balance, energy balance, whatever we do. Alright, let us take another example, something closer to what we do, ok. So, your, let us say, exam is going to start and you want to study for that till late in the night and you are in, let us say, hostel 13, ok. And you want to reach in time and it is rainy, it may rain, ok. How do I sleep? How do I study till late and get up as late as possible? May be even skip breakfast but reach the exam hall on time, ok. So, you would have, I mean, those of you would say that, I want to reach at all cost in time would say that, I need to have a good model, right. I need to have a good model. I need to, I know all the turns, I know all the slopes up and down, I know the places where it is slippery, I know all of that. I will build a mathematical model. These are all mental models, of course. Calculate and say that, I will start 7 minutes ahead of time from my hostel, from my room 10 minutes ahead of time. I will come down in 3 minutes and then 7 minutes I will be in the exam hall, ok. So, the problem with this is very difficult to predict exactly. And constructing all the things that I talked about, putting all of them is very difficult. You can use the basic models, acceleration, velocity and so on up to some extent. So, typically what we do is, we use them in a feedback form, right. That is, you may not be able to say that, you know, supposing I say that, predict exactly at what time you will reach, ok. So, then you say that, well, I guess, if there are no problems on the way, I will reach in 7 minutes at the speed, ok. Whereas, things may happen, suddenly a bus is on the way and then a car is trying to overtake and it is blocked the road, right. So, the thing is that, so we use, this is when we use the feedback form to make corrections, ok, to make corrections and then you say that, look, may be from now on I will have to travel at 60 kilometers per hour, ok. I am going to take a chance. So, you adjust your model, adjust your performance, you adjust the fuel that you use, so as to still meet the deadline, even though your model is approximate, even though your model is not predicting properly. What is important to know is that, approximate models will do in this usage, if you use it in this fashion, ok. So, typically what we do is in many of these plans, we always have some handles, some things to correct, right. So, in that sense, controls are extremely important. We talked about satellite. So, there were things to, you know, you could change the direction, you could change the fuel that is used, so on and so forth, so as to reach that window, right. Approximate models will do. This is extremely important in situations where you cannot have good models. Can you give some examples? I gave cycling example as one. Can you think of something else where you cannot have mass balance, energy balance, momentum balance and say that, I will come up with a concrete model. Any example you can think of? How does one predict rains, rainfall? We are 90 percent normal, 105 percent normal, they predict ahead of time. What model do they use? Do you have any idea? So, we use what is known as a time series model, ok. Time series models are obtained from data. How many of you have heard this time series model? I will show you what it is, what it looks like, approximately. For example, economists use them to predict the inflation, to predict monetary, to decide on monetary policies, to decide the rate of inflation, ok, to predict money supply, so on and so forth, done through time series model. Rain prediction, that uses time series model. If you want to model a computing system, you want to know how much time it will take, what is going to be the load, how many people should be allowed, how many jobs should be running at a time, ok. How many people should be allowed to access my web page, which is extremely popular, so that I maximize the throughput, but not slow it down enormously, use time series models. So, it turns out that time series model is of this form. It just says input between, so x is input, y is output, I have used n minus 1, actually I should have used a subscript here, or this n minus 1 in bracket, n minus 2 in bracket. So, these n minus 1, n minus 1, n minus 2 denote different time instance. So, I am saying that my current output, y is output, is a function of the previous input multiplied by some a 1, and one input prior to that multiplied by another unknown a 2, a 1, a 2 and so on. Now, these are the things to be obtained a 1, a 2 and so on, to be obtained from experimental data. I measure this, so now a 1, a 2 are these, ok. How does one arrive at such a model of this form that uses, that is based on the domain knowledge, ok. So, this is a time series model. By the way, I had asked a question in Moodle, I got no reply. Did anyone see it? Did you see the Moodle post? I had asked one question, how does one solve what is the meaning of equations of this form? Can you see this? How many unknowns are there? Yeah, this is the unknown, and how many equations are there? Three equations. So, what kind of mathematical model is this? What do you call it? Something in terms of some determined, so over determined model, right. There are too many equations. So, what is the meaning of this? What does it mean geometrically? Can you visualize it? Have you studied these? Yes, yes, in which course? So, suppose I have this equation as a x equals b, x equals what? How do you solve this? So, you know that method, right. You have seen this? Have you seen this least square solution? Do you know how to derive it? Did you derive this equation? Yes, no. How many people have you seen the derivation of this? Yes, raise your hand. Is it taught, but you have forgotten or it was not taught? Was it taught? It was not taught. Yes, it was taught. Derivation, you can derive it by what method? It has to be, so it can be derived by many methods. One of them is geometric method. So, I want you to visualize it, okay. Essentially what this is equivalent to is trying to fit, trying to express a vector in a 3D space by a sum of two vectors, right. Sum of two vectors means two vectors will form a plane, right. You have a plane, think of it. A plane and this is a third vector on the right hand side, okay. I will show you in the next class, okay. And then we will derive this. It is something fundamental, something fundamental to time series analysis, something fundamental to the stochastic models and so on. Whatever you do in this course has a connection, because you would try to solve equations of this form all the time, okay. So, see you on Friday.