 So, we were looking at these models, I mean there are a couple more models that we will sort of you know look at. This is the anti-lock braking system model, again a very mechanical system model, yeah. So, here you look at the car as a quarter car what is called a quarter car model, yeah these are all very standard terminology in automobile system. So, you use the quarter car model to you know do a lot of design suspension braking right. So, you can do a lot of design with just the quarter car models and you expect them to work well in the real system also. So, the quarter car model looks something like this, basically it is you have again the state variables are this omega and V, which is the angular's vehicle speed and the longitudinal vehicle speed. Basically, since it is a quarter car model it is just one V. So, you just have the angular vehicle speed and the longitudinal vehicle speed and then you have the factors that are affecting these are your braking torque, yeah. This is sort of your I would say your control, yeah. And then you have this fx which is the longitudinal tire road contact force, ok. So, this is something that you that actually depends on your tire and so many other factors weight and so on and so forth. So, this is something you have to sort of arrive at for different systems differently, yeah. Then you of course, have j, m and r which are basically your inertia v of the wheel, m is the mass, r is basically the wheel radius, ok. And this fx itself as you know, the lateral force always depends on the normal force, right. So, it is basically the normal force multiplied by some coefficient, ok. So, you can see that there is actually no nonlinearity here, right. It does not look like there is any nonlinearity here. Nonlinearity actually enters here, yeah. This depends on, so basically you have the fz and then you have the mu which depends on these lambda, beta, tau and theta r, where lambda is a longitudinal slip, yeah, which is you start seeing some nonlinearity here and then you have the wheel slide slip angle and you have some road dependent parameters, alright. So, so basically this is sort of the model, so no nonlinearity here, but you start to see nonlinearity because of this expression and this expression, ok. So, again even for this quarter car model, you can see that it is a relatively complicated model, yeah, ok. So, you might need further simplifications if you are actually trying to do control design, yeah. But it is not impossible to start from here itself, ok. Because this is sort of, in this particular scenario it looks like you can sort of cancel the nonlinearity also, yeah. So, there is a possibility that you can work with even this model and do something interesting, ok, alright. So, those were all the examples that we wanted to talk about, ok. Now, one of the key things, this is where the regularity aspect of F comes in, is existence and uniqueness of solutions, ok. So, this is something that I always talk about in my nonlinear control course. This is actually material from an ODE course, yeah, but since we are talking of a nonlinear system, we sort of at least touch upon it. I am not going to prove anything here, yeah, but I am just going to give you a feel of how things can go wrong if your function is not appropriate, yeah. So, which is why more often than not we assume things like smoothness, yeah, when we give our results. Because if it is not the case, then you have to very carefully evaluate your system dynamics to see if you can even, you know, use the analysis methods that we are talking about, yeah. Designing control is a, you know, much, much further aspect. The question is whether you can even talk about, you know, analysis in the classical sense that we talk about in this course, yeah. You might need more advanced notions, alright. So, the question is when do solutions exist and are unique, okay. So, both of these are important for us, yeah. You can see this small little, you know, graphic that I sort of made here. And on the x-axis here is basically time. And on the y-axis are the system trajectories, just the state, yeah. But we don't say state because once we solve it, we say trajectories, yeah. So, this is the solution for this system given some initial condition. So, you can see that I have started with, you know, sort of two different initial conditions here, yeah. You can see that it's here and here, yeah. But somehow it seems like once I evolve these from these initial conditions, there is a meeting point here, yeah. This is an example of non-uniqueness in solutions, okay. And this is not that uncommon, by the way. So, this is an example of non-uniqueness in solutions. Because at this point, so let's forget everything that happened before, yeah. But if I initialize this system at this point, yes, please. Aren't they the same states at the same state? Absolutely. So, whenever, so your question is are they the same states at different initial conditions, yes. There is no other state. See, look at this. There is only x as the state, okay. All I'm saying is, state is just an abstraction, right. It's just a representation. I'm saying, you know, current is the state, angle of velocity, angle of position is the state. And this is an abstraction of these words, right. But when I want to solve a differential equation, I need initial conditions, yeah. So the question is what initial conditions? And depending on my system, at that particular instant, my initial conditions may be very different, yeah. Initial condition may be different. Initial time may be different. So you need both, yeah. So this is in fact, for the same state, I give two different initial conditions, yeah. And this is the sort of evolution I get. I mean, this is the solution I get, starting at this initial time t0, okay. Now if I look at this as my initial time, say, yeah. If I look at t1 as my initial time, yeah. What happens? The system can go in either side, in either direction, yeah. So there is no real clarity on where the system is going, yeah. So it's not even clear which way the system is evolving for me to be able to talk about equilibrium and stability and things like that, yeah. So this is one of the big concerns if you are going to talk about notions like stability in a sort of classical sense that we do, okay. Let's look at examples, yeah. I like these examples because they are very illustrative, yeah. So this is an example of a system where of non-existence of solutions, okay. So it's a very simple system, right. It's just x dot is equal to x cubed and I'm giving my initial time at 0 and initial condition is just x0, okay. It's not very difficult. All of you can integrate it, yeah. This is what the solution comes up to me, yeah, okay. What goes wrong? Yeah. If I have non-zero initial condition, say x0 is non-zero, then it's very evident that the solution will exist only until this is positive, right. As soon as this quantity here becomes negative, right, it's imaginary, right. There's no solution to this system, okay. So beyond a certain time and you can see this is dictated by how big time is, right. Because x0 is fixed, everything is fixed here except for time, right. So as time becomes large, in fact goes beyond this, yeah. Things get messed up. In fact, very odd thing happens at exactly at this time, the solution is actually infinity, right. Exactly at this time, solution is infinity, right. Because this is exactly going to be 0. So you know, I have an sort of an infinite jump. So you can imagine that as t gets closer to this value, you know, your solutions are going up, up, up and becoming, you know, very large numbers, right. But beyond this also, funny things happen, right. It's not like, it's not a one point escape time. It's not like only at t equal to 1 over 2x0 square, I have a problem. No, beyond 1 over 2x0 square, this is negative. So it's imaginary, so there is no solution, okay. So this is, you know, an example of non-existence of solution. So these kind of systems, why are they a problem? Why do you think they're a problem? Why? Why don't I like them as controlled guys? Why do we have issues with such systems? I mean, the non-existence, the cases of non-uniqueness can still be dealt with in some contexts, by the way. But non-existence of solutions is, you know, a very difficult, absolute no go almost. See, the problem is, you are talking about non-linear systems, okay. And at least, and in this particular course, we are still talking about asymptotic results, right. That is controls which sort of give you convergence after a large time. You may be happy with whatever happens in 10 seconds. But the theoretical result tells you that at infinity, you converse to 0, okay. Now, modern day control theory, of course, has finite time control and this and that, okay. But again, there is fixed time control. But then a lot of this such controls are non-smooth controllers, yeah. And sometimes also discontinuous controls and so on. Not discontinuous in the first derivative, but discontinuous in second-order derivative and things like that, yeah. And these such controllers typically stress your actuator a lot, yeah. If you're talking mechanical actuators, then these are very, very high-frequency controllers, yeah. So, but if you talk smooth controllers like we do here in this course, then you're talking about infinite time that you need. But here I have very, very limited time to do anything, yeah. There's no guarantee that I would have achieved anything in this time, okay. So these are sort of issues. Again, this is not a control system. Let me be clear. This is not a control system. But I'm saying even if you pose a control problem, I add a control here, yeah. It's still a problem because when I apply zero control, it escapes, yeah. It escapes, then there's no solution, yeah. So I mean, yeah, I mean it's very difficult to work with such systems. So we typically do not like systems which have non-existence beyond a certain time. These are definitely tough, tough, tough systems to handle, okay. All right, homework one, this is how the homeworks look, okay. So this is the homework one, I mean, yeah. So I want you to give one more example, okay, things like that. All right, example of non-uniqueness, yeah, I just flipped it. So it was x, x cubed, I made it x1, okay. So this is an example of a non-unique solution. And I give you an initial condition at zero time as x0 equal to zero. I start with zero initial time. So one thing should be obvious to you that this is a solution, yeah. X remaining at zero for all time is a solution, yes. But then I can also integrate this, right. I can actually integrate this is also not difficult to integrate. And you will find that the integral is this, yeah. I mean, it's not difficult to verify because I can take x dot and this comes out to be this, yeah. Now this is also a solution, right. Because this is zero at zero, notice. How do you say anything is a solution of a differential equation? It has to match at initial condition and it has to satisfy the differential equation. These are the only two requirements. Match at initial condition. So this matches at initial condition. This satisfies the differential equation. Why if I take derivative of this guy, still zero, right? And the right hand side is also zero. So this is a solution, yes. Similarly, this guy, if I take a derivative, satisfies this guy, yeah. Because I obtained this by integrating this, obviously satisfied. And at initial condition at zero time, this is zero, right? Satisfies initial condition, satisfies differential equation, yeah. But see, these are different, aren't they? If I plot it, what will happen? If I make it like this with time here, I don't know. I really don't know how to look. I know that the zero solution will look like this, yeah. And this guy will look something like this, I don't know. I really don't know. I think it will look like this. That's my guess. I don't know, yeah? Better if you plot it, yeah? Whatever. May not look, may be the other way around. But just, you can just imagine that this guy looks like this. The point is, away from zero, this is non-zero. That's the key thing to remember, right? Away from zero, this is non-zero, but this is not, yeah? So at this point, I have an issue, yeah? And notice, this point is not any point. This is the equilibrium of this system. I hope all of you know what is an equilibrium, right? Wherever the right-hand side is zero. Anyway, we'll formally talk about it. But wherever the right-hand side is zero, is the equilibrium of the system, yeah? So zero is an equilibrium, okay? So this is sort of an issue, right? You don't know, especially at equilibrium, you see that the solutions are going at two different directions. Again, an issue, right? But these are sort of not impossible to work with, especially in the modern theory of sliding mode control and finite time, fixed time control. These sort of non-unique systems with non-unique solutions are dealt with. How you deal with them is you talk about stability of all possible solutions, okay? Instead of talking about, you know, in our kind of theory, in the classical theory that we talk about, we talk about stability of an equilibrium. Here, in this more modern theory, I would say, we talk about stability of all equations, all solutions. Okay, so it's a slightly different notion, yeah? Of how you work with things. But interestingly, things don't change too much, yeah? So because typically sliding mode control, like I said, is continuous or non-smooth, results in non-uniqueness. And there you have to have notions of what is the kind of stability we are talking about, okay? All right, good. Any questions? All right, so again, homework two. Give one more example. So, this is the sort of result that we assume more or less for this course. For this course, we assume that our function f, which governs the state space model without the control that has an existence and uniqueness property, yeah? What is it? If you have the system, x dot is f t x, x is in Rn, x t 0, x 0, we assume that f is piecewise continuous in time and satisfies a global Lipschitz condition in x in the states. Looks like this. This is what is the global Lipschitz condition. This is actually a regularity condition, sort of a smoothness condition, yeah? You can see that if you have this kind of a property, it's guaranteeing differentiability, okay? So this is something more than differentiability, yeah? So this, so this is f t x minus f t y in the norm, has to be less than equal to L norm x minus y. Why is it global? Because this holds for all x y in Rn, okay? I'm not saying x y in some ball or x y in some small region around the origin or anything like that. I'm assuming global Lipschitz, which means that this holds for all x y in Rn, okay? And some constant L positive, okay? Then for any initial condition x 0, the above ODE has a unique solution for all time, okay? So this is the standard result for existence and uniqueness. You can find it in Khalil, in the Khalil's book, yeah? Maybe in Vidya Sagar's book, I don't know, but Khalil's book for sure, yeah? So very standard result. This is the sort of result we assume, okay? For this course, yeah? Because we don't like, we don't want trajectories to intersect and all that, yeah? No intersection of trajectories, okay? We don't want trajectories to intersect. We don't want trajectories to vanish and not exist. Neither is cool with us. So we assume this very, very, if you may harsh condition, yeah? Okay? So again, lot of systems may not satisfy this. In those, you will have to deal with them as very special cases, yeah? Impulse systems, I mean, if I'm thinking of dropping a ball, every time it hits the ground, you lost, yeah? Lost the smoothness, yeah? So lot of simple systems that you can think about may not satisfy this, yeah? In those cases, you just have to, why this dropping ball thing is also an important example is because all this biped, quadruped, locomotion is all based on this, yeah? There is an impulse, okay? Then how do you deal with it? So you sort of do some special case sort of a situation there, yeah? Anyway, the lot of time the control is also based on some approximation there, yeah? So in those cases, you don't think about too carefully about existence and things like that because physically somehow you know that solutions exist, yeah? So you're not too worried that you applying some control will make the solution not exist. So things like that, so those are again special cases, yeah? So whatever we do here, we assume this condition is satisfied, all right? Any questions? No? All right, I think we'll stop here.