 Welcome to another session of non-linear control SC602. So, we had started off with defining passivity as basically this guy right a system is said to be passive if there is a storage function you know such that the derivative of the storage function has this less than equal to u transpose y thing yeah. So, so this was what we had instead. So, this is the only thing is that this is sort of a little bit later in what he covered it came up a little bit later in what he covered that is all that is really the only thing yeah. So, so anyway so let me sort of try to go back to a wow this is a let us see if I can go to 1. So, if you see if you remember we he started off talking about all the energy balance which is actually the real motivation for passivity. Basically if you looked at how we discussed passivity you would understand that almost every system that you can think of is can be thought of as a passive system if you if you do not you know sort of you know keep injecting too much energy into the system yourself yeah and if you are very careful about choosing the inputs and outputs that is the only real you know trick here input is anyway typically fixed you cannot really decide what is your input of the system. But you can always try to determine an output with respect to which you will have passivity. So, the basic equation was just this that the energy available is the initial energy and then there is some dissipativity and supplied energy. So, this is essentially what this is and if you look at this sort of expression the way he talked about it yeah is that the energy difference if you take this to the left hand side the energy difference is less than equal to some inner product of input and output right essentially that is what it is energy difference if I take this to the left hand side is less than equal to the inner product of input and output. This could be any input output right I mean or it could be any inner product also in this case it turned out to be the integral functions and so on. So, this is an inner product in the function space yeah, but it can be any inner product we have also looked at. So, this is what I also mentioned actually it is not different yeah v dot v dot is what it is just the difference right yeah if you look at think of v dot what is this v t minus v 0 divided by delta t right if you think of it like that. So, this is just the difference and it is just the u transpose y or if you think of it differently if I integrate both sides then you have a inner product on the right hand side typical function inner product and on the left hand side you have v t minus v 0 right. So, it is almost looks like the energy loss right is essentially upper bounded by the inner product of input and output ok. So, the definitions that we talked about are not different yeah it is just that he came from a different motivation and then we looked at several examples and probably you got confused I have no idea where you got confused, but that is essentially what it is right it is simply just the motivation is pretty much simple. Then of course, he talked about these are very useful concepts that with this is why you can keep extending this notion of passivity to larger and larger systems ok you can add interconnections. So, essentially what he said is that if you interconnect two passive systems then you retain the passivity property ok and this is a pretty cool thing yeah because you know we also we also discussed if you remember that if you have passivity right and the 0 state observability which you already defined right then you can actually design a controller very easily which is a function of just the output the output with respect to your which you have passivity ok. So, we discussed this right we this is basically the system is passive with some storage function and it is 0 state observable then you are then any feedback of this form with these properties globally asymptotically stabilizes the system the origin ok. So, it is a very cool thing right. So, for passive systems I have something nice ok. So, in this Khalil's text or whatever I am covering I am always talking about passivity. So, if you notice Antonio also spoke about strict passivity by the way. So, where you have actual dissipation happening if you have actually a dissipation happening then it is strict passivity otherwise it is just passivity which is I mean either of them is good enough why is either of them good enough because I can always use the feedback term to introduce this strict passivity ok. So, that is a pretty the transition from passivity to strict passivity only requires me to introduce some it is as good as introducing a negative definite term with this feedback see because if you see it is u transpose y it is less than equal to u transpose y if I make u yeah as some negative function of y right you can already see that it gives me a negative term here right ok. So, I have some dissipation right and therefore, I will get to the strict passivity version yeah I can always say that my control is some new control minus a function of y yeah which will give me a negative definite term here like a y transpose y if you may yeah and another control v transpose y with respect to a new control v yeah. So, I will have again with respect to the control v and y strict passivity ok. So, going from passivity to strict passivity very easy yeah not difficult at all. So, whenever I do a feedback interconnection of this passive systems I retain the passivity property and that is a very powerful sort of result to have yeah, but of course, he mentioned that if you are in cascade it is not necessarily passive and all that, but let us not worry about that I mean I will let the TAs get into that stuff right which is yeah I mean this material was a little bit more involved because of how it is stated not actually not very complicated it is just more involved in the in the in the sense that how it is stated ok. So, this is again the same idea you have these feedback interconnection of passive systems you retain passivity ok which means inherently that I can globally asymptotically stabilize some system equilibrium the only question now is what is that equilibrium right because I connected two different systems right. So, one system could have one equilibrium other one could have other equilibrium right and in energy terms this is the same as. So, this is very well known by the way that equilibrium when you are looking at system which you know sort of conserve energy and or Lagrangian Hamiltonian system equilibrium is just the minima of potential energy ok this is standard remember this even if you do not understand this it is almost like saying that I have my potential energy like this and this is where my equilibrium is yeah this is very standard ok is you may not have seen this, but for mechanical systems folks this is the standard idea that equilibrium is actually a minima of the potential energy of the system yeah you can think of it I mean if I drop a ball what is the equilibrium it is the ground where the potential energy is 0 systems want to minimize their potential energy ok that is why I mean yeah turbine whatever everything works because of that right you drop water it goes down yeah basically all systems work like this yeah this is nature nature's equilibrium yeah this is nature's equilibrium. So, you see this is also turns out to be the equilibrium in the sense of stability for stability also this turns out to be the equilibrium of the system ok. So, this is very standard for mechanical systems that equilibrium turns out to be a potential energy minima that is why he keeps talking about energy terms ok. So, you have you have energy functions corresponding to these systems which you can think of as Lyapunov functions maybe if you yeah and you look at the minima's of these there are two different minima's ok. Then if both of them have unique minima at 0's right. So, basically what he is saying is that if you have output strict passivity then and 0 state detectability we already know what these properties are we said 0 state observability he is weakened it to 0 state detectability let us not worry about that yeah he actually stated it much more simply than I did I sort of gave you a more complicated version of what does 0 state observability mean he stated it very simply saying that if y equal to 0 x equal to 0 yeah this is what is 0 state observability yeah I also stated the same slightly in a slightly more complicated way that h equal to 0 set can contain only the 0 trajectory yeah then the only then the only trajectory in this set is the trivial trajectory yeah I stated more like the Lassal invariance that you are used to because all of this passivity based results require use of application of the Lassal invariant that is why I wrote it in this form otherwise what he said is good enough if you remember we also connected this with the linear system observability ok very cleanly connected with linear system yeah please yes this one it would be strict less than that is it huh ok it is strict less than and then there is multiple versions of it if you remember I will go to this other lecture he had the first slide or second slide at this slide yeah the beta was actually the energy available so you can think of this as V dot you can think of this as V dot passivity is just less than equal to right the strict passivity is or output strict passivity let me be more formal output strict passivity is this available energy or this V dot is less than equal to this minus this guy so there is a dissipation what is this term is basically dissipation term you just saw that example right you saw electrical systems and so on here when output strict passivity the dissipation is a function of purely the output input strict passivity function of purely the input the dissipation is purely a function of the input this is an unusual case I have no idea I never know how such a dissipation would happen this has to be some kind of dissipation in your actuator itself so I don't know how else I would get this kind of a dissipation and finally you have state strict passivity where this dissipation is now a function of all states this is obviously the strongest sort of result ok and this beta thing is whatever he used this notation beta but actually it is the energy available like e t minus e 0 so it is or v of t minus v 0 if you want to think in term of Lyapunov function it is v of t minus v 0 yeah we we have used the Lyapunov notation because we are used to it yeah that is about it ok so lot of there is pretty clean connections here yeah so don't worry about I mean don't think that he you know suddenly went into something too crazy or new he did not ok the other thing is and we sort of go forward here we saw multiple notions of getting to passivity right of course you may not start be able to start with a passive output you may not have one or you may not know one ok though the first is if you do have a passive output we already proved that you can get this nice feedback and zero state observability will ensure that you have global SM dot X stability great we proved this result ok so the first obvious thing was choosing a care choosing carefully an output so that you have passivity right so that's what this is right in fact he also has this if you notice see what if you remember he talked about the KYP for the nonlinear systems that's exactly this output is chosen as partial of v with respect to X yeah and then there is a transpose like LGV is what LGV is nothing but so if I take the transpose to the other side ok what did we do just look at this exactly the same thing he said G is what multiplies the control this is how you choose the output that will give you passivity that's exactly what he said yeah here ok exactly what you did yeah because as soon as you do this you will get V dot as minus alpha X plus U transpose Y yeah this is actually state strict passivity right because this alpha is now you know whatever class K function so it is negative in the state itself not just the output and the input or input it is actually negative definite in the state yeah because it's a class K function and then you have U transpose Y which is what you wanted right because you chose why is this because you chose why is this yeah you chose an output it may not be a real measurement or anything like that we discuss this it may be some fake unreal quantity but the purpose here is to just design a control now yeah it's not actually to focus on what is this why and so if you don't try to think of questions like you know why this this this is not a real measurement how even meant to measure something funny like this you're probably not yeah this is just a way of designing a control we figured out a output why so that you get passivity so this was the first method of inducing passivity remember by output selection that's what we covered okay and that's what he also covered when he spoke about the KYP lemma for linear systems and corresponding KYP lemma for the nonlinear systems no difference all right great then we went to the important concept of feedback passivation this was a case where you may not have passivity in the original system but you construct a feedback so that your system is passive in the new input and some output okay so in such a case you're already given an input and output and you are claiming that if you construct there exists such a feedback such that if I plug in this feedback and notice V is the new control if I plug in this feedback here I get this system right and the claim and the assumption is that with this V and with this why the system is passive this is the assumption okay then you know that then it then you say that the system is feedback passive and as long as soon as you get passivity with this new control V and Y you're done right you can again apply the same theorem that you had that the control V can now be some function of Y and you will get global asymptotic stability under some simple conditions okay so so this is what was feedback passivity where did he cover feedback passivity that is where you guys got super lost this one this is where he covered feedback passivity he considered a special case okay not he considered but that's where Kokotovich and Sussmann started he considered a simpler case if you see here there is no output yet correct there was no output at all whereas we we already said there is an output and then we construct a feedback so that you make the input output system feedback pass or passive right there is no output here at all how did he go about it he first said that let me say that I can decompose this guy in this form okay that is it is somehow linear in some output of this system okay for this he introduced some definitions and whatever I'm going to completely ignore the definition because they really scare you and confuse you yeah all this FPR and this and that the basic idea is that he assumed that the nonlinear system can be decomposed as a linear sum of the outputs of the linear system okay pretty serious assumption by the way yeah on the structure of the system okay you are saying that notice why doesn't even appear here okay but why is some output of the linear system some C matrix okay what am I saying I am saying that using this why I can decompose this f into this where there is some drift type term and then there is some you know you can think of it as control type term or an output type term I would say yeah where this y is basically this and output of this guy okay and there are some really nice assumptions on this output already what are the nice assumptions that this output is making this system making the linear system this guy with this is passive this is an assumption okay passivity of this linear system that is with this psi and with this y is passive okay so to get this passivity you invoke what is called the Kalman Yakovov which pop off the KYP lemma okay basically it's just a condition on passivity so these are the conditions yeah because C is what governs the output you decide okay and K is the feedback okay so basically if you have these two conditions it guarantees that this system is passive with this output okay and not just that this output allows you to linearly specify or linearly parameterize the drift or the or the dynamics of the X system okay these are all assumptions all assumptions okay so that's why I kept asking him oh god these are very scary assumptions I mean where do you even get them satisfied yeah but that's why he actually showed an example of a robot system where you can actually have these assumptions yeah and these these definitions FPR output FPR spanned all this is just to say what I just said nothing these are nothing special okay so and then FPR stable decomposition basically all that is saying is that you you are able to decompose it you have the linear system with this Y to be already passive on top of it you are saying that if psi is 0 then this system F0 X0 yeah X dot is F0 X0 is actually globally asymptotically stable already okay so several several assumptions that are there yeah so so this system with psi equal to 0 is already asymptotically stable okay so under these assumptions you are designing a control you to stabilize this whole system okay alright so what have we assumed that the linear system with some output Y is globally asymptotically stable actually it's this is what I think there was an error they should be F0 this should not be F yeah the linear system with the output Y is passive okay the nonlinear system can be decomposed in terms of the outputs of the linear system okay and the drift of the nonlinear system is globally asymptotically stabilizing if the linear system states are 0 okay under these assumptions you can design a you know strict passive control and so on and so forth I mean so that's I am not going into the details of this it's complicated it's confusing not complicated but it's essentially doing what I'm doing I'm also doing saying the same thing in a in a more general setting actually okay in a slightly more general setting I'm you're doing the same thing what they are doing is also feedback passivation yeah the only difference is they also have see why they say it's a cascade is because you see there is a the passivity is only assumed on the linear system yeah and then there is a nonlinear system which has no control in it right the nonlinear system has no control variable at all yeah where does the control for the nonlinear system come in it comes in through this guy you can sort of think of it as coming through this guy yeah but then you're also assuming nice things on the nonlinear system already that this is somehow stabilizing yeah and then you have this so this is why it's called a cascade because you're sort of connecting a nonlinear system a linear system in cascade to a nonlinear system okay this is not a feedback interconnection this is actually a cascade connection yeah yeah I think he used the term cascade somewhere I don't know cascade this is cascade why because output of this system feeds into this system because this f x i is basically summation of sorry f 0 x i plus summation of y i f i yeah so this is actually in cascade yeah cascade means output of one goes into the input of other feedback is the circuit circuitous route right output doesn't go directly to the yeah so there is a cycle there so here there is no cycle yeah if you notice the the linear system is not getting any feedback from the nonlinear system it is free of the nonlinear system right linear system is free of the nonlinear system so what does what does he do he says that if you have the you have all the nice assumption basic conjecture is that if you can make psi go to 0 fast enough then you can stabilize okay so so be I mean not this one but but actually this is a counter example that's a counter example I apologize so basically what if you notice what he will do is in this sort of an example all he does is he let's see he has a stabilizing term let me try to point it out I know this is again getting confusing I'm sure yeah this is easier yeah he makes it into a sort of feedback interconnection just by the way he designs the controller if you see for the linear system he constructed a new feedback there was original feedback was what you correct so he said you will be minus bk plus v this is like the feedback passivation type thing that he's doing yeah and that my minus bk went in here and then the v remained here all right and then what is he going to do he's going to set the v as this guy what is this term do you understand what this term is why do I set v as this what is this we just saw this somewhere in output passivation it is just you see if you think of y as the control it is just lgv right if you think of y as the control this is exactly ngv so he is essentially setting this to that okay that's it once you have passivity you are just choosing the correct output right and he chose this as his output in some sense he's just setting this v as you know lgv type of a term okay here it shows up in the feedback instead of showing up in the output yeah because you are doing this feedback passivation type thing that's it but if you see the structure of these things are exactly what you are doing yeah this is exactly lgv if you assume y as the control okay for this system if y is the control then this is exactly the output that you would choose to make the system passive and in order to ensure that this is the output yeah you choose the v appropriately that's it it is just a smart trick so that you get this sort of an output using this sort of a feedback structure now okay and the math of course is you can make it go through very comfortably it's not a problem yeah so this is how the general passivity based control looks like yeah where you have a system with which you have a linear system in a nonlinear system with which you have a linear system in cascade yeah and under some whatever under several assumptions you sort of get this passivation yeah here we said we had a slightly simpler setting yeah let me see we also get to the next step by the way I think we also covered this robot example right a little bit and I think we were somewhere here right we were somewhere here and then there was this exercise right yeah anyway the exercise will be set today and you will know when numerical exercises in the sense you will have to do some simulations also now okay because you have a robot they have to make the robot move all right now if you yeah not experiment just simulations yeah if you notice the next step was exactly this cascade connection with passive system this is what we are going to look at now in our context which he presented sort of in the last lecture so we will sort of look at it again all right