 So how do we relate passivity to Lyapunov stability when we want to study the stability in this sense? So meaning that we have now a model, I have been telling you that passivity is all about inputs and outputs, but now let's say that we also consider the state of the system, okay? And we want to relate passivity with Lyapunov stability. Well, we have that if the system is passive with a certain, with a storage function or energy function V that is positive definite, then the origin for this system without inputs, because now we are talking about Lyapunov stability, so we disregard the input, then we have stability, okay? So think of again the pendulum without friction, it will just be oscillating, that's a stable behavior, it just keeps oscillating, right? If there is friction, then we will have output-strict passivity. I didn't speak about L2 and all the outputs, so forget about this. If I have output-strict passivity, that means I have friction in my system, and then the system will be asymptotically stable if in addition to being output-strictly passive, it is also serious state observable, okay? So serious state observability, I don't know if you have seen this, but it means, so serious state observability means that y equal to zero implies that x is equal to zero. So if you have worked with Lyapunov, essentially what we have in this situation of output-strict passivity is that if V is my energy equation, we will have V dot because it is output-strictly passive, we will have minus y square minus plus y or u or actually, usually it's written the other way around, plus uy, and then we forget for Lyapunov, for the purpose of Lyapunov stability, we just want to zero the input, so forget about this guy, so you have only this. Let's say that, yeah, due to this property, y is going to zero, then you just set it equal to zero and check if the state is equal to zero. We'll do that for some examples, so in case you, this doesn't ring a bell. But all this to say that, yeah, output-strictly passive systems, output-strict passivity essentially leads to asymptotic stability. Yeah, if state is strictly passive, so we have this term here, it's appearing with the whole state, then what? Then we also have asymptotic stability right away because once again, you could use this as a Lyapunov function, forget about this guy, and you will have a Lyapunov function that is negative definite. So the moral of all this is that you, if you think in terms of passivity, you have to figure out the energy function of your system and use that as a Lyapunov function essentially and then try to come up with some quadratic negative terms in your derivative, but the good thing is now you have an interpretation for this term. Now internally, Lyapunov stability, well, we saw that the interconnection of passive systems is passive, well, here because if these systems are passive, when we interconnect them, this will be passive, and due to what I just said over here, how passivity relates to Lyapunov stability, we will be able to also say things about this passive interconnection. This theorem here is taken by, it's taken from, well, it's a known resolver, it's taken from a book by Van der Schaft. It's Springer lecture notes. I can send you the complete reference. So it just essentially says that if you have these two systems that are passive, the closed loop system will passive, will be passive so you will have stability. Now it is left, of course, to determine stability of what, of which equilibrium. Well, normally when you are trying to control a system to a set point, for instance, you would like that this equilibrium be a set point that you chose, right? So the only thing that is left to do is to choose, to manipulate the energy of these systems in a way that the energy functions have minima at the equilibrium that you want to stabilize, right? So you want the energy of this guy to have a minimum but takes one and the energy of the second guy, you want it to have a minimum at another equilibrium point, right? If in addition you have output of strict passivity and the equilibrium is unique, then you will have asymptotic stability, okay? If provided that you have this property of detectability, yeah, so detectability is the following. Detectability means that y equal to zero implies that x goes to zero. So this is a weaker property than observability. We just want to be sure that basically if you know how to bring the output to zero, you want to be sure that bringing the output to zero also implies that the whole state will go to zero. If zero is the equilibrium you want to stabilize, otherwise just replace the equilibrium to the origin, okay? So yeah, I'm almost done with this set of slides. Essentially now we are, well, I presented you this a little bit fast but I will show you how this works in the other slides. But what we want to do now is passivity-based control. So in passivity-based control essentially what you have is a plant that let's suppose that it is passive, okay? Or maybe it isn't, but yeah, let's say it is passive, it has some passivity property. So you want to design a controller that you will put here in the feedback loop. And as I said, when you interconnect systems that are passive in feedback, the resulting system will be passive. If the plant is not passive initially, well, you apply a controller pro if possible to render it passive, right? So you will want this. At the end of the day anyway, what you want is to enforce the passivity or to create passivity in your system. So passivity-based control is about designing a controller in a way that the closed loop system be passive, okay? That's what you want to do. And if you want in addition really asymptotic stability of your closed loop system, you will want to be this passivity to be strict. So yeah, if you have seen these inequalities for non-linear systems, what we are going to try to do is to design you so that in closed loop you still have a passive system. So to continue speaking about passivity, what we want is to have you, you should have one component that makes the system passive. And another component that we were going to call you new just to, you know, like I did for the circuit, you just need a new input from which to you define your passive map, right? Because otherwise if you don't have this, you close the loop with some function of the state, there is no new input then you cannot speak about passivity if there is no new input, right? So you want to design you so that the closed loop system with this part here, so that will be this, should be less or equal than the new input that you're injecting, multiplied by the output, okay? In that case, you will have a passive map from the new input, okay? If you tilde in addition injects dumping through this output, then you will have this inequality here that will give you a strict passivity, okay? This on the other side is the, what is that, the energy, well, not the energy, the power in the system. If you integrate all that, you will get the energy balance equation, right? The supplied energy, the dissipated energy and the difference between the initial and final and not final, but energy at any time, too. And yeah, let me finish this set of slides with this very important theorem that says the following. This is taken from a paper by Beren Cicidori Williams, three very big names in Cicidori. And it says the following. If you have this system and it is passive and you have some energy function with respect to which it is passive and also it is zero state detectable. So I already said zero state detectability means that when y is equal to zero, that implies that x goes to zero. If you have these conditions, all you have to do is to re-inject the output into it to add the dumping and then your system will be asymptotically stable in those. So the control should look just by this, like this simple expression, just a function that is strictly contained in the first and third quadrant, right? So there. So of course, just a constant times the output will do, but you can also use a saturation. You can use many non-linear functions, right? The saturation will look like this. Bound it under and so on. So you can use many things. So let's see how passivity-based control works now. So now that we know the definitions of passivity and we know the Kalman-Yakovich Popov statement for linear systems and its equivalent version for non-linear systems, again, taken from Halil. Now we'll be working with these notations. So Lfv is the directional derivative of v in the direction of f, right? So that's partial of v with respect to x times f. That's this. The output is defined like this. So that's reminiscent of that for linear systems. This one is reminiscent of this one here. So here we have q that is positive definite and here we have a class k function of the norm of alpha. So we are asking a strict positive realness in the linear systems. It comes to asking this negative definiteness of v dot in the direction of f, okay? So basically we are asking the system to be state is strictly passive. You see, we have v dot less or equal than minus alpha plus uy. So the energy balance inequality gives me this. Supplied energy is larger than the energy dissipation through the whole state plus a constant which is actually negative minus v zero. So that's the energy at the initial time. And now, so that's what we want to work with for to use passivity to design to design controllers. This is just a reminder of passivity for basically what we saw in the previous slides with different notation to confuse you a little bit. Well, I already mentioned that this integral is known as inner product which can also be written in compact form like this. And we speak of passivity if this integral is bounded from below by a constant, a real constant. So that's basically this number here, okay? Output strict passivity, we have this guy and the state strict passivity, we have this guy. So let's see passivity based control in its general form. So this method goes back to the work of Peter Kokotovich and Hector Sussman in 89. And I don't know the story very well, but Ortega, so Kokotovich was kind of the big boss in Illinois in 89. And Ortega and Spong, Spong I think had just been recruited assistant professor and Romeo Ortega was a positive there. So yeah, Kokotovich came up with this method of passivity based control and Romeo and Spong more or less in the same time. They wrote a paper that has been cited a lot on adaptive control of robot manipulators and they coined the term passivity based control. In this paper, it's called feedback passivation or something like that. Anyway, the method is here just for the little story. And essentially it consists in the following. So they were studying systems that have this form. So these systems basically come from, you have a non-linear system, we are in the 80s. In the 80s, the people were mostly working with feedback linearization. I guess you have studied that feedback linearization. So you want to find an output and the feedback such that a coordinate transformation that brings your system into a linear form. So there are some structural conditions to be able to do that. And sometimes for a certain output, you cannot feedback linearize the system completely. So write it in linear form, but only partially. And you end up with a system that has a linear part and a non-linear part. So these guys came to this system from working with feedback linearization. So here what we have is after applying the preliminary feedback and transformation and so on, let's suppose you end up with this system. So you already applied a preliminary feedback, but you say, okay, I'm going to add another feedback to complete the stabilization task because I didn't manage to linearize this part. This is what people know in feedback linearization as zero dynamics with respect to the output Xi. So meaning that if you partially feedback linearize your system and it became like this, it is fairly easy just to design use so that if A and B is stabilizable, you just bring Xi to zero and that's great. But once you bring Xi to zero, what will happen with the zero dynamics that is the dynamics when Xi here is equal to zero. So maybe they are unstable and there is nothing you can do about it through the controller priori because well, it's just sitting there and there is no control input there. So what can you do? So these guys came up with this nice method that applies of course to a certain class of systems, not just any, a certain class of systems for which F satisfies certain properties. So we need to lay out some assumptions and definitions and stuff from this paper to understand what they did. First, they say, well, our result will hold for systems for which when we take Xi equal to zero, we know that the origin for the resulting system is gas and it's a non-minimal phase and the nonlinearity can be written in this form. Sorry, I have a doubt about this. So anyway, the nonlinearity has to be written in this form. So basically you want to, from this nonlinearity you want to extract one part that they call F zero plus the sum. So this is a bit particular of a bunch of nonlinearity that are multiplying certain outputs and they call an output feedback positive real if it has a property that there exists a feedback like this. So minus k Xi to place the poles of this guy here and an extra input such that when you apply this feedback to the linear system here, you get a strictly positive real system, meaning that the matrix A minus Bk will be Horowitz and the closed loop system with this feedback will be strictly positive real. Okay, so it will satisfy the Kalman-Jakovic problema with these for the closed loop system, right? So with the feedback matrix here. So you take A minus Bk, that should be Horowitz and plus the system should be of relative degree, relative degree one. Now, what does it mean a feedback positive real output? Well, there are a lot of definitions and it's a bit hard to read the paper, but it's a very important paper. So they call this feedback positive real output, essentially if it's an output such that the non-linearity, well, such that you can write this non-linearity like that. So they say in general terms, if f belongs to the span of y. So basically y is a vector of outputs, right? There is yi here, so there is y. So it's a vector of output. So you can consider that as if you are divide, you are splitting the non-linearity into a linear combination of a bunch of other non-linearity in which the coefficients that are multiplying your other functions are also functions of the state. So it's essentially this guy here. So again, you want the output to be composed of y1 up to ym and each of these is multiplying a non-linearity. So you have f0 plus f1, y1 plus f2, y2 plus f3, y3, et cetera. So the non-linearity has to have this special structure. And then they say, well, I'm going to call f, a stable FPR stories span the composition, essentially if you have that, the first non-linearity in f is globally a asymptotically state. So yeah, the origin for this system, okay? So x, your system is x dot equals f0 of x and psi plus y1, f1 plus up to ym, fm. Now, essentially what they are doing is assuming that this guy is already gas, so there is not, they don't have to worry about that. So basically we want to worry about the effect of these non-linearity here. And what the passivity-based control created by Kokotović is doing is defining U in a way that it takes care of the effect of the non-linearity. So the non-linearity here, defining U in a way that it takes care of the effect of these non-linearity in the system. Okay, so that's what we are aiming at. But for that they need all these assumptions. So as I was saying, this has to be gas, and not only gas, but also globally asymptotically stable uniform link psi, what does that mean? It means that we should have a Lyapunov function that is positive definite and radially bounded It's a Lyapunov function that depends also on psi, but the bounds do not depend on psi. And V should be negative definite along F0, also independently of psi. So it's a very strong condition, but yeah, you can have it for many systems. And then what? They say if we have all these conditions, essentially if F I can decompose it like this, then the system is globally smoothly stabilizable. So smoothly emphasized because again in the 80s, there were already some results using non-smooth controls and constructing Lyapunov functions that are not smooth, et cetera. So the big deal was to do all this smooth. So yeah, if your system, if your F can decompose as I said, and you have these properties, these guys say the following. Well, all you have to do is go and take that V that you already know that works for the first piece of the system that is F0. Take that F, take its derivative. The F0 term will give you something negative. And then you will have a bunch of added things that you don't know what, a priori you don't know what to do with them. Because F is F0 plus all that. So you will have this. Well, that's where this equation comes into play. The next thing you want to do is design an additional input in that system that will take care of those non-linearities. And all you have to do is to add to your Lyapunov function that you designed that you have initially for F0, right? Sorry, I wrote it as a function of X, but it's also a function of psi. So you use that function, right? No, sorry. Yeah, I think, yeah, yeah, yeah. Yeah, but the psi appears in the non-linearity. I think it should be there in V. Anyway, it's a function V that you use for this part of the system, okay? Now, you take the same Lyapunov function, you take the derivative, you have a bunch of non-linearities that appear. What you want to do next is to add a quadratically Lyapunov function in function of the state of psi, which is the system this below. And when you take the derivative of that, what will you obtain? Because you designed K for this to be Horowitz. Well, since this is Horowitz, it generates a function P, right? Metrics P that you put here, and you take the derivative of this new function. So the function you initially had plus this new quadratic function here, okay? You will obtain that the derivative is, well, is less or equal than this guy that comes from V. Now, this one, you know that it is bounded from above by a class K function. Then you have all these guys that you don't know a priori what to do with them, and then you have, of course, this guy that comes from Horowitz of these metrics. But then, because you added a new input that you are still going to design now, you will get this term here. Xy transpose times P times B times the new input, okay? And what will happen next is that you see PB by construction because your system here is strictly positive real. So it has this structure property that PB is equal to C. C being exactly the matrix that defines this output here, what you will have is that actually this term here is nothing else than the sum of all the outputs, Yi times the inputs Vi, okay? So it's essentially why transpose the new input. So you can see easily that all what you have left to do is say I'm going to take V to be equal to that with a minus sign, so because V appears there and V and Y appears here, Y appears here. So I just have to match this in this way and all these guys will go away. And you will be left only with this. As I said, this by assumption generates an alpha term and this one comes already from the orbits of that. So yeah, and all the rest you just eliminated it. And you end up with this very nice inequality. So yeah, in case you don't see the passivity here what you have is before eliminating all the terms you have this guy here, so essentially you have W dot less or equal to minus alpha 3 X, right? Then this guy minus one half psi transpose Q psi, then plus this guy plus Y transpose V, sorry it doesn't fit, plus all the bunch of things that you want to eliminate. So if you integrate this and just for the sake of argument, forget about this guy, you see that you have this passivity property between V and Y, right? So if this thing that is bothering you was not there you have a passive system. Now essentially you have a passive system with something else that is coming into your system but it's coming into your system through the right channel because it's multiplying exactly the passive output. So the perturbation that is coming into your system and so multiplying this output that is from which you have passivity so with the extra input you can actually go and kill the terms that are the perturbation that is coming into your system.