 Hello everyone, welcome to yet another week of our NPTEL on non-linear and adaptive control. I am Srikanth Sukumar from Systems and Control IIT Bombay. So we are just entering our fifth week of this course and just to commemorate the finishing of four weeks or one third of what this course is supposed to be. I have changed our background to a very nice SpaceX satellite which is orbiting the earth. Until now we had this nice background of a rover on Mars and we said that you know you use these autonomous algorithms to you know determine where the rover is at its position and where it's supposed to go and and there are algorithms that drive the rover in a particular path all right because it cannot necessarily be remote controlled in real time from the earth. Similar situation is that for spacecraft also yeah a lot of what we do in adaptive control has to some extent been implemented in by the spacecraft community by the satellite community and so in order to maintain both the position of the spacecraft in orbit and also its orientation in orbit right which may be rather mission critical component we at several occasions required the use of adaptive control because it's not very easy to figure out several space parameters like the inertia mass etc of the spacecraft which are also of course losing fuel at a pretty good rate as they are flying yeah so anyway so this is another motivating application for us in fact a lot of my own work in my PhD in aerospace engineering was in fact using adaptive and non-linear control to spacecraft problems all right so anyway so hopefully again we will continue to see this background for a while and hopefully we can motivate ourselves to you know design develop algorithms that can drive systems such as this spacecraft all right excellent so what are we doing until now so we were actually somewhere here right we had not finished the lecture not from previous week in fact yeah and this is why I had mentioned that we should not worry too much if we are sort of moving ahead in the lecture notes because I knew that as we move forward the material will get more involved and we will start to get a little bit slower okay right right so but by now I think most of you understand what is the notion what are the notions of stability you know what is the pitfalls and analysis uh you know how to sort of we had a grip of how to use the bubble that's lemma we also saw are now looking at the Lyapunov stability theorems all right so we've already looked at the stability theorems until global uniform asymptotic stability so now we are left with only the exponential stability theorems so as you all might recall exponential stability essentially is a more advanced version of asymptotic stability I would say and why is it more advanced it is so because it actually gives you a particular rate of convergence to the equilibrium which is assumed to be the origin in almost all our problems okay so that's the idea of exponential stability so the question is what do we require yeah what do we require for exponential stability so we already have a usual assumptions right that we need to have a candidate Lyapunov function which means that it is a c1 function which is positive definite okay so this is the least that we do require all right beyond that we require v to be decrescent right because these are the properties that you already have already started to see appearing so we require v to be decrescent because exponential stability is by definition also uniform right so v is required to be decrescent further we require three class k functions all of the same order of magnitude we will talk about what this means soon enough same order of magnitude right we will talk about this soon enough let's not worry too much about what this means but we require three class k functions such that v is bounded on both ends in this way in fact although I have mentioned v is decrescent but if you look at this particular if you look at in fact let me be careful here this is not really required there is no time argument here okay there's no time argument here the third one is fine but here there is no time argument okay all right so so what we if we look at this left hand side this what does this mean this means that v is positive definite this is requiring v is to be positive definite if you look at the right side inequality this requires this implies that v is decrescent yeah so this is not actually required so if I if I I can sort of say this this is implied yeah and this guy this implies in fact I should be careful I should say it's a one way string of what this implies that this is positive definite and further the third requirement is something like this that v dot is less than equal to some negative of class k function right now this again okay so here also of course we don't need this time argument right let's be careful time argument is not needed v dot is what we have defined already yeah it is the lead derivative or the directional derivative of v right and this particular piece yeah this means what this already means implies not means but this implies negative definite plus of v of v of v dot sorry yeah so if you look at it it it seems like everything that we require here is already been mentioned here well I'm sorry everything that's required here we can look at the local version is already been mentioned here that v being positive definite is part of the candidate Lyapunov function definition right v being decrescent is mentioned here and v dot being negative definite is mentioned here so it looks like the conditions are exactly the same all right but there's a very you know nice and subtle little difference here and that is these words same order of magnitude okay this is the different same order of magnitude so here we did require class k functions to claim all of these but we didn't say anything about them being the same order but for local exponential stability we do require them to be of the same order of magnitude all right so this is important let's try to remember this okay now the global version of course you know of course if it moves from v being positive definite to v being readily unbounded and of course you will also require that v be valid over the entire domain aren't yeah unlike what's specified up top right so we require this also and we require this also that v be valid or v be defined in the entire domain and of course we need now class k r functions three class k r functions all of the same order of magnitude such that you have the same type of inequality yeah I mean I will again remove these when you're doing this comparison there is no time yeah because we are not when we do these comparisons we are not looking at x as a solution of a dynamical system but x is just a variable all right so it's not appropriate to put a time argument okay and so if this happens so if you have radial unboundedness for v yeah and you have three class k r functions then you of course you get something like global exponential stability all right okay so this is the notions of local and global exponential stability all right so let's sort of try to see what means for functions to be same order of magnitude okay so we say that two functions in reals so here we are talking about f y and g y mapping to the real numbers they're said to be the same order of magnitude if they are comparable using scalar quantities that is if there exist some scalars gamma one gamma two positive such that g is bounded on both sides by f scaled by these gamma ones and gamma two and vice versa okay and vice versa okay because if you see if this happens then the flipped version can also right because from here I can also say that so this this also implies if you notice that I can write f y bounded on both sides one over gamma one g y and something like one over gamma two g y okay so this is equivalent right these are equivalent okay so if basically one function can be bounded on both sides by the other function then the converse is also true all right should make sense yeah so two functions are said to be of the same order of magnitude if they can be compared with each other by just scalar quantities scalar constants yeah scalar constants yeah not functions of state not functions of time okay scalar constants okay so for all the above theorems so this is what it means for the same order of magnitude so for all the above theorems we say that if we satisfy any of these theorems it is said to be a Lyapunov function then that particular v is said to be a Lyapunov function if it satisfies any one of these yeah so any of one of these bullet points many bullet points then it is said to be a Lyapunov function not a candidate Lyapunov function like here but a Lyapunov function so although we don't talk about it but they do exist converse Lyapunov functions yeah and the statements always go something like this if x equal to 0 is stable for some system star then there exists v tx positive definite such that yeah positive definite and c1 such that something happens okay so this will this is what the converse statements look like yeah the converse statement essentially say that if your system is stable or asymptotically stable or exponentially stable then you're guaranteed to have v with these kind of properties with the you know with the properties that we are talking about yeah that it is it satisfies one of these bullet points okay so they do exist converse theorem yeah they do exist converse theorem but of course it's not uh it is not a constructive evidence right so uh just the existence of a converse theorem does not mean you can find a Lyapunov function yeah it's not always easy to find such a Lyapunov function and just using the fact that there exists a converse theorem yeah so this is the unfortunate truth so that's why we don't we're not really stressing too much on it or actually stating these theorems okay so there's an example that we want to do now but before we move on to the example I want us to look back at the previous examples and see if there was any system which was already exponentially stable okay so we did a few examples right if you look at uh let's see did we do a right so here no this was a harmonic oscillator which was just stable then we did okay I mean we could not talk about stability it was in fact unstable then we found a you non-uniform stable example but then we did this example right this is the this is the damped harmonic oscillator here this is a damped harmonic oscillator all right and what did we actually do we we took a Lyapunov function which is something like this right which is k x1 plus x2 squared by 2 and x1 squared by 2 alpha and we could show that v dot is also the same in some sense okay v dot also has k x1 plus x2 squared and x1 squared okay it has the same terms as this k k x1 plus x2 squared and x1 squared all right and from here we claimed global uniform asymptotic stability but the fact is that this is this also globally exponentially stable yeah why why so the first thing is that v is positive definite it satisfies all the properties right so in fact v is radially unbounded not positive definite v is radially unbounded okay and and so v is lower bounded by a class k r function similarly v dot is also radially unbounded right because it has the same terms as this if this is radially unbounded v is radially unbounded and v dot is also radially unbounded right therefore v dot also is upper bounded by a class k function right because it is a negative sign right so the upper bounded by a class k function and of course decrescence is free because there is no time argument right so therefore it is also upper bound v is also upper bounded by a class k function so v is both upper and lower bounded by a class k r function and v dot is upper bounded by the negative of a class k r function okay so this is one of the things the next thing is same order of magnitude of these functions right same order of magnitude so what are these class k r functions in fact right so if I look at what these class k r functions will be in fact I will take exactly this function itself as my class k r function okay so if you notice I had this phi 1, phi 2, phi 3 in class k r so here I will take my if I may let me make it bigger I will take my let me write in some other color actually I will take my phi 1 as half k x1 plus x2 square plus 1 over 2 alpha x2 square right which is exactly v okay sorry this is x1 so this is also the same as phi 2 and this is the same as v because I can use the same on both sides and phi 3 I will use as this k minus 1 minus k k x1 plus x2 whole square no no minus actually not in the class k function plus k over alpha x1 square okay this is what I take as my phi 3 and if you look at phi 1, phi 2, phi 3 first of all these are radially unbounded functions right this is not difficult to verify they're positive definite right how are they positive definite we've already verified this is positive definite earlier so I don't really need to do anything right because this was already done I don't want to repeat it and you can see from here now radial unboundedness is obvious because if x1, x2 go to infinity in any direction okay the only problematic direction would have been just a second only problematic direction would have been when k x1 plus x2 is zero but in that case also x1 is going to infinity still because you know x1 and x2 both have to go to infinity in some direction right so the only problematic direction would have been where this is zero but then x1 is still going to infinity so this whole thing is going to infinity so these are in fact class k r functions no questions asked all right now the fact that they're same order of magnitude is also very straightforward to verify you can see that there is related by some constants just different constants these constants are the only things that are different okay so it's very easy for me to relate phi 1, phi 2 and phi 3 by some in fact phi 1 is equal to phi 2 so the constants are one for to relate phi 1, phi 2 and phi 3 I just need to find another constant right which is very easy because they're just multiplied by some constants here okay so that's it so this is in fact turns out to be globally exponentially stable by this argument okay so we've already seen an example of global exponential stability now if you look at this guy on the other hand right this pendulum or a variation of the pendulum example in fact right if you look at this example it is uniform asymptotically so but it is not even globally uniformly asymptotically stable because this is not a radially unbounded v okay so so the question is is it locally exponentially stable at least is it at least locally exponentially stable would be the question right this is not very easy to verify this is not very easy to verify because what we have to do is now compare this guy and this guy compare this and this yeah now comparing this and this is not easy because if you look at actually we are going to compare the particular class k or k functions that bound these but even if I try to compare these two quantities see x2 is the same right it's the same function just with different gains so this is not a problem but this one okay could be a problem this one could be a problem because at x1 equal to I mean right so say at x1 equal to pi what happens let's see what happened at x1 equal to pi what happens actually we are not including pi so we might still have some chance yeah but this is not very obvious a case of same order class k functions yeah in fact there is a trigonometric formula for 1 minus cos x1 which is I think sine square x1 by 2 I believe yeah I think it's sine square x1 by 2 I will have to verify this though but but I believe it is sine square x1 by 2 but here I have sine squared x1 yeah so we still have to see if these two actually have some kind of an exponential connection with each other so I have a same order or a scalar connection with each other okay and this is not super easy to verify I can tell you that okay so it's not very clear if this is in fact locally exponentially stable or not all right for almost certainly it is not yeah all right but I will leave all of you to think about this and argue this okay it's a very interesting question is this so the the question we are asking is is this locally exponentially stable all right great now that we have seen this we want to look at this you know very nice tiny little example right so this is the system which is you know highly nonlinear right it is it actually looks like an oscillator harmonic oscillator until here but then you have some nonlinear terms multiplied by a constant c yeah some constants yeah now we take a very very simple candidate Lyapunov functions x1 squared plus x2 squared obviously this is positive definite radially unbounded and all the good things yeah and then we take the derivative as always okay we are trying to do a stability analysis so we'll take the derivative okay what happens x1 x1 dot I get I get you know x1 x1 dot and x2 x2 dot and then I simply plug in from here okay it's pretty straightforward this this basically it's a harmonic oscillator so these two terms obviously cancel out they do in all of our examples if you notice and then these two start to look the same cx1 square x1 square plus x2 square cx2 square x1 square plus x2 square so actually if you combine them you get something like this c times x1 squared plus x2 squared whole square now if you want to conclude negative definiteness of v it is the minimum thing we would want to do we want to conclude negative definiteness of v yeah first of all then we well okay let me continue then we would require c to be negative at least if c is positive then there's no scope of v dot being negative definite it should be obvious to right because this quantity is positive in like this quantity is also radially unbounded okay this quantity is also radially unbounded how you can simply see this by looking at this it's greater than equal to c x1 4 plus x2 4 but it's actually equal to this plus 2 x1 square x2 square but this is already greater than equal to 0 so I can say this is greater than equal to c x1 4 plus x2 4 which is of course radially unbounded like or the negative of it is radially unbounded right so depending on the sign of c right if c is positive well wait a second I think yeah if c is negative in fact this is not correct they should be c is negative if c is negative then v dot is actually negative definite in fact radially unbounded negative definite and therefore the equilibrium point that is the origin is globally uniformly asymptotically stable okay but if c is positive then zero is unstable we will need something called Lyapunov instability theorems okay we can actually show that the origin is unstable if c is positive yeah but we will need Lyapunov instability theorem which we did not state but that's part of your exercise okay that's part of your exercise you have to find the Lyapunov instability and instability theorem and use it to show that this system is in fact unstable at the origin if c is positive all right right now c is negative one might even ask why is it not exponentially stable if you look at this function here this is radially unbounded this is also radially unbounded if c is negative so in the negative direction so v dot minus v dot is radially unbounded right but if you look here which one what are the class kr functions so again phi 1 equal to phi 2 will still be half x1 squared plus x2 squared since there's no function of time I can directly use this this is a class kr function right and phi 3 will unfortunately be something like minus modulus of c because c is negative x14 plus x2 to the power 4 now notice that this are different powers they're similar looking polynomial terms but they have different powers so you can never compare them by a constant right because what will happen as x1 becomes larger and x2 becomes larger this will become significantly larger than these terms yeah and you cannot compare them by the same constant anymore notice I only get to choose a single constant to compare these class kr functions with yeah that is I can only oh I'm sorry I think it's right here right I only get to choose this gamma 1 and gamma 2 for all one gamma 1 and gamma 2 for all y it is not a function of y yeah but I cannot do that in this case yeah because x14 will start to dominate x1 square significantly and similarly x24 right and therefore there is no way that they are uh so they are not they are not same order yeah therefore there is no way I can get exponential stability all right but I do like global uniform asymptotic stability all right great so what have we looked at today we sort of wanted to complete our discussion on Lyapunov stability and we have done that so we were left with exponential stability theorems so we looked at local exponential stability and global exponential stability both of which augment asymptotic stability in the sense that they give you a rate of convergence and this requires us to know about similar same magnitude class k and class k are functions so we did define that yeah we also looked back at our examples to see which ones of these which ones of the examples we had already seen where in fact exponentially stable and we found that the dam harmonic oscillator was in fact exponentially stable and we also looked at a new example and you have an exercise of course and we saw that this new example of course is not exponentially stable but can be globally uniformly asymptotically stable if you have an appropriate constancy all right excellent so we will continue we will look at a slightly more involved topic here on and it basically sort of you know takes the knowledge of persistence of excitation to talk about new exponential stability theorems for linear systems and that is something that's rather useful in adaptive control and parameter convergence analysis all right all right great that's all for today thank you