 So, why do we do nonlinear dynamical systems and control is because very very important most systems that we see around us are nonlinear and if you are doing if you want to study stability, if you want to understand that behavior is evolution asymptotically or the transient behavior you need to understand how to analyze these systems. So, that is sort of the first relatively big half yeah we learn techniques so everything we do in this course is deterministic we are not doing anything stochastic here. So, it is a deterministic course and we of course want to introduce a little bit of design. So, we especially in the second half of the semester you will see more and more of doing design methods of doing design and then examples of the design yeah. So, that by the end of this course if you have you know some example system yeah for example, if you have a robotic system if you have an electrical system anything yeah and you have a fairly you know good fidelity model for this then you can actually use some of the methods that you have learned here in order to do a control design alright. Mostly of course this is a very fundamental course for you know control theory practitioners a control theorist and control theory practitioners industry folks. So, of course a lot of SISCON students show up here because it is a core course yeah but it is also of a lot of interest for anybody who is working in the general area of dynamical system. So, I get a lot of students from aerospace mechanical I myself from an aerospace background so a lot of what I do is you know applying these theories to aerospace systems you know things like satellites attitude control you know spacecraft on orbits I mean my own background is that I did my aerospace engineering PhD from the University of Texas at Austin. So, it is one of the nicer programs in aerospace in the US I would say and Texas was also a very good place to be and we had a lot of exposure to not just doing a lot of theory but we also got to interact with you know folks from NASA air force research labs who were sort of coming in with their own projects you know and their own research which was sort of very real problems yeah and you know so it would typically work like it works here the faculty has a project and you typically get put on the project as a PhD student and then you contribute bits and pieces to it yeah of course you are not just running the whole thing but we did a little bit of satellite control design for a few of these agencies okay. So, lot of lot of applications in recent times of course I have seen applications in I have seen people work on applications in a smart grid control yeah I mean in Germany Siemens actively sponsors these projects with the smart grid control because in Europe a lot of this I believe this you know energy generation locally is very very highly encouraged so a lot of you know local folks do energy generation and then they supply to the grid but then the problem with these is that everybody is generating at their own pace at their own frequency so the parameters and everything is off right so you can just join things to a grid and expect a uniform frequency. So here you have in applications for non-linear systems in terms of synchronization so it is like multi system multi agent cooperative system sort of a problem where you are you know you are talking about how to synchronize the frequencies you know how to synchronize some of the power outputs yeah so that when you join one thing into the grid you know the grid does not start to fail so lot of applications now of course there are folks who are doing biological systems biological kind of application reaction rate during this covid pandemic there were a lot of models again I am not sure how successful those were but there are applications now more modern applications to you know reaction rate models which is sort of the infectious disease spreading models then there is a please come forward then there is lot of applications to you know more recently social systems yeah for example if you are talking about advertising and marketing alright there is a lot of applications there where a particular marketer wants to know which agents to influence for example there may be I mean a lot of you are on social media platforms right like Instagram so you see that there are a few people who have you know millions of followers so these folks are influences so if they say that you know by this toothpaste your teeth will be perfect forever then people believe them so it's like so typically these advertising agencies and these marketing managers they now want model based decisions right on who to target which particular you know which particular influencer to target or which particular you know social agent to target so that their campaign you know goes forward much more efficiently yeah so here also there is in applications of nonlinear dynamical systems yeah nonlinear dynamical systems to look at the evolution right once I introduce an opinion in one corner of the graph of a very very huge network of social graph there is you know what's the outcome and asymptotically after a long time you want to see how many new people got influenced to buy this toothpaste and the other end of a spectrum is how to influence where to introduce the new yeah so so things like these alright so let's see the syllabus is from as you can see 2021 I will fix it and then upload it to the moodle platform don't worry about it so the structure is pretty standard what we have here it's just lecture and credit we don't have tutorial and practice included here anyway we may have extra sessions you know with the ta is depending on how you know we think we are progressing if you want to learn more if you want to sort of you know you have trouble understanding some particular bit of material then we have some sessions but they are not officially included okay this is also wrong yeah like I said 2021 I'm going to fix all this you know the sections are Wednesday Friday 11 to 12 30 right here okay so this is what is going to be office as I'll actually put a formal set of office hours yeah not by appointment so this is also something I will sort of introduce yeah there'll be a formal office hours during the week and so that if any of you have doubts you can just walk in during that time I will be available during that time so you can walk in and ask your doubts no need for an appointment yeah if you want of course separately to talk to me then of course you will alright ta is also I will make this change soon yeah we just have met Ray and Pallavi STS not so we put all right this is one of the key things I sort of expect although I know we have newly joined students who probably don't exactly have this background but we sort of stretch this okay so the general expectation is that there is a graduate level competence in ordinary differential equation so you must have seen some kind of a state space ODE course okay so that you know even the models shouldn't look like you know unfamiliar and foreign to you all right so because that's what we work with we start with models yeah okay so if you're unsure of our academic preparation please talk to me after the class yeah this is the first class so usually a lot of people show up and anyway so please talk to me after the class if you have so I expect some basic MATLAB or Python or some alternate programming experience because see I do some bit of design aspect also in the second half so there I can assure you if you unless you do some bit of you know hands-on coding and things like that at least you don't get a good feel for you know whether what you did work well or not okay so so I expect some basic MATLAB or some programming experience yeah all right again if you're unsure please talk to me and if a lot of you are not on the same page with terms of programming then of course we may have to chuck it but I prefer not yeah you will also enjoy a little bit of a programming all right you see some real systems and then working under the controllers you design yeah and then you learn also things like things that we don't talk about this like how to tune the gains and I mean it's still a little bit at home yeah yeah all right okay great the topics this is the very very broad overview of the topics okay we may choose not to do some of them depending on how much time we have yeah but this is a generally broad overview more or less you can assume that we will cover at least 90% or more of this okay so it's not going to change significantly like I said this is a very fundamental course so unfortunately I don't have a lot of freedom in you know talking about a lot of new areas okay so if we do have time of course we will but usually be a fundamental course so the methods that we talk about are by now classical okay not classical in the sense of 1800s but classical in the sense that everybody knows this in the community yeah we just learn it better here okay so first we start with some non-linear systems introduction example some preliminaries yeah which all of you will require which will sort of set up notation okay for how things will look how the mathematical notation will look all right then we will have we start immediately with the Lyapunov stability okay so this is the fundamental stability notion for non-linear systems like for linear system well again these notions are also pretty much valid for linear systems okay so there's no real difference as such the only thing is that linear systems you are used to working with input-output stability yeah so you hardly talk about there is notions of internal stability right and there you pretty much characterize it with what do you characterize it by the way how do you characterize internal stability for linear systems eigenvalues yeah just write the matrix x dot is a x and just check the eigenvalues of the system if they are on the negative left of plane whatever I mean the real parts are negative you are good okay so so this is how you evaluate but that's a evaluation method yeah that's not a definition all right it's an evaluation method and not a definition so let's be very clear definition still remain the same okay so you you see a lot more evaluation methods when you talk about linear systems you hardly look at the definition most of you when you did linear systems you would not have seen definitions you would have seen tests like you know check if the poles are on the left half plane and things like that you would not have talked about what exactly stability okay so Lyapunov stability definitions are the universal standard for what is stability okay how do we characterize what exactly is the notion that we are talking about so things immediately get mathematical here yeah so welcome to the course very quickly all right then of course we talk about Lyapunov theorems yeah these are the tests for nonlinear systems okay in nonlinear systems notions of eigenvalues and all that no questions about it because you cannot write it as a matrix a constant matrix okay and so if you even even if you have a linear time varying system I hope all of you understand that just checking the eigenvalues of a linear time varying system does not tell you anything about stability yeah even if for all time the time varying matrix that you have has negative eigenvalues negative real eigenvalues it doesn't guarantee stability okay this is a well-known fact yeah so for time varying linear systems also these eigenvalues test do not work you know the way you expect them okay so there is no question of them working for nonlinear systems yeah I mean obviously not of course there is possibility of doing linearizations and things like that we do not talk about linearizations okay we directly test stability of nonlinear systems via Lyapunov theorems okay that's the idea all right then we talk about invariance theorems which are give a little bit more flexibility in terms of sort of an extension for our Lyapunov theorems then we'll talk about input output stability this is sort of the linear system equivalent yeah also has its advantages in nonlinear systems in modern nonlinear systems there is notion of notions of input to state stability yeah so ISS results and input output stability these are important when you're talking about disturbances okay so any real system is affected by disturbances yeah whatever I do in theory you cannot expect that you know the real system is going to have a very similar sort of an outcome you know you may apply the same control yeah you design a control for a quadcopter and you put it on a quadcopter it will not behave how you think it is expected yeah these are all effects of disturbances okay of course there is no there's actuator saturation and so many other things but all of these can be clubbed as disturbances if you want okay so input output stability is an important notion it sort of connects linear system notions yeah then the design part yeah which I'm hoping a lot of you will be excited about yeah so the first is Lyapunov redesign where we use the notions of Lyapunov functions to design controllers we learn how to do that okay then we have one of the most most powerful techniques of designing Lyapunov functions so once you design a Lyapunov function of course you can use Lyapunov redesign to get a controller right so this is called back stepping design so it's like a it's a step-by-step way of coming up with a Lyapunov function so it's a rather powerful method I talk about it even in adaptive control course because back stepping is the sort of central idea for designing controllers there then we have feedback linearization which is one of the oldest ways of designing non-linear controllers and then finally we have passivity and energy shaping which I have clearly said is subject to time availability okay so again depending on how I feel I may add some more design elements yeah the the fundamental analysis elements are standard there's no change all of you have to learn this if you have to even talk about design and talk about non-linear system stability and so on yeah so this there will be hardly any change in this first section but in the design part we can choose to do a little bit more here and there depending on what our interests are and what is the time that okay so these are the references very standard okay Khalil's book by non-linear systems for the invariance design very very good book very good reference for that Vidya Sagar's book is one of the most mathematically precise non-linear systems analysis book yeah then all the geometric notions feedback linearization all of these are best explained in Alberto Isidori's book yeah very good book again and then you have this Christik the KKK book so it's the Christik Kanalakopolis and Kokotovic book for non-linear adaptive control since it's so difficult to say I just call it the KKK book so usually I will tell you which book I am somehow referring material from but of course as you will see I have I have hand written notes and things like that so so maybe in the future become printed notes but right now it's hand written notes so mostly we will have this sort of these are the sort of key references here okay very good books all of them very good books I mean over time if you want to be in this area of non-linear control I think you should own all these books yeah over time I'm not saying you have to buy it now or I'm saying these are really really good books to have as references for life yeah good