 So, let's just formalize a little bit what we have seen for these elementary examples into some definitions of passivity that you find in the literature and that are useful in passivity-based control. So, if we consider a generic system, again, just a black box with inputs and outputs and we denote E as its energy, we are going to assume that it is bounded. We are going to say that the system is passive if the energy balance equation looks like this. So, available energy equals initial energy plus whatever I supplied into my system. We are going to say that the system is output is strictly passive if the energy balance equation looks like this. So, not only I have the initial energy and the supplied energy, but some energy was dissipated and just went somewhere else, I don't know where, but just into heat or whatever as I explained before. So, there is this component of dissipated energy with a minus sign. That means that I have some damping or friction if you want in my system. So, we will call it output strictly passive. So, notice that we have a coefficient here that is positive and we have the square of the output in this integral. That's what this is. And then we are going to call it input strictly passive if we have a similar balance equation, but instead of this, what we had before with the output square here, we have actually the input. Actually, this is the kind of system that will be very useful later. It's the kind of system that you want to have when you are doing some control because it means that you have dissipation in your system and you are able to stabilize it. A very important statement actually, the fundamental theorem of passivity is that when you take two systems that are passive and you interconnect them in feedback like here. So, let's say it's passive from this input to this output and this one is passive from here to here. And you interconnect them in feedback. You will again recover a system that is passive from the new input which would be the vector of these two guys to a new output that would be the vector of these two guys. And you can keep interconnecting blocks in feedback and you will conserve the passivity property. And as we see, this can be very useful to construct your controllers, always trying to maintain this passivity property or rendering the system passive if it is not in the beginning. So, if the systems Sigma 1 and Sigma 2 are passive, the interconnected system is passive meaning the map from U to Y is passive. So, we write that as using this funny arrow like this, that's in LaTeX, that's maps too. So, we are seeing the system as a map as an operator that maps once again transforms inputs into outputs. So, that's why we write it like that. So, once again the electrical circuit, let's see what happens to this circuit. We already saw that if we have a resistive element, we actually have this integral of I square R, sitting there with a minus sign. So, meaning that system satisfies this balance equation. So, my circuit, my RLC circuit is output of the circuit passive. So, what can I do with a system that is output of the strictly passive? Well, I can for instance, interconnect it with another system that is passive. In this case, if I just use another resistive element, this resistive element you can consider as input is strictly passive because it's just a constant. So, the input here, I mean the output on this side will be of course just RCI. So, you can see that this is of course input strictly passive because the input and the output are basically the same just the scale by some constant. So, it comes to connecting these over here, right? So, I'm adding this resistor into my circuit and now I will have a new input. I will be measuring the voltage from here from this terminal. So, this is what I'm doing from the block diagram perspective. I have this output, I pass it through this new resistor, the current and I inject everything back into my circuit and now I have a new and new input. So, this is how we deal with the passivity and we go constructing passivity blocks, right? So, we start with some passivity block, we make a feedback interconnection with another passivity block and we define a new passive map from a new input to probably a new output. In this case, the same output. All what we did here was adding to add a resistor. So, what will happen with the Kirchhoff's law is that there will be this new term here appearing. So, the voltage in the new resistor that I added and it will just go to the other side and we add up to RP. So, what I did with this is of course I added more dissipation into my system. But I added more with, so I used P for plant, right? There was already some dissipation there but let's suppose that this is a controlled term. So, I added, I chose RC and that's like modifying the circuit with this. So, I'm adding dissipation into my system. That's what I have done with this feedback here of RC, right? So, now my new energy balance equation will look like this. And I will have that the available energy equals the initial energy of course. So, potential energy in the capacitor plus kinetic energy in the inductor that were already in the system. And I just added a resistor to increase the dissipation in my circuit. I could probably also add a capacitor and modify the potential energy, right? I could also modify that. Yeah, yeah, yeah. So, if you connect a system that is just, you will definitely conserve the passivity property that you had initially, right? So, with this feedback, you will probably be adding more passivity or just conserving what you had. So, if you have passive and you interconnect with passive, you will have passive, right? But if you add some, so let's, okay, let's see it this way. Imagine you don't have RP. So, initially you don't have the dissipation. So, you don't have this term here. That system would only be passive, right? But then when you inject this dissipation, you are adding, you are enforcing the passivity and then you will have output-strict passivity, right? This is the output, yeah? So, this energy equation is telling you that the system after the interconnection is output-strictly passive. Yeah, so you can enforce the passivity property with the feedback. So, the energy of the circuit is the same with or without feedback. That's the energy. But by adding a purely resistive element, we just added dissipation. We added friction. Imagine if it were a pendulum, previously I showed you an example. Without friction, then you come and add friction. Now you have output-strict passivity. So, you enforce the passivity. The addition of an inductance would change the kinetic energy. You could also go and do that. And you could also add a capacitor and modify the potential energy. So, you can do a lot of things. You can modify, reshape the energy the way you want and then add dissipation the way you want. We will see how by manipulating these things, you can control the system very nicely. The resulting system is still passive because once again the fundamental theorem of passivity tells me that the interconnection of passive systems is passive. So, what happens with if I now I interconnect two systems that are different in nature, right? A mechanical and an electrical system. Well, actually the motor, this is supposed to be a motor, it's electromechanical. Well, let's just suppose it's just an electrical system. So, before we said that the input to the pendulum was torque, right? But this torque is actually coming from somewhere, probably from my hand or preferably from some actuator from a motor. So, this motor has an output which is going to be the torque that is going to be injected into the pendulum. But it obviously needs to have an input so that it transforms the energy to move the pendulum. And that input will be typically a voltage, right? You apply a voltage into the motor, you make it spin and generate a torque that moves the pendulum, right? We all know that. If we look at the equations, we will have the following. We have the pendulum equation with a torque that is being injected. And this torque is coming from the motor, right? So, the motor sees this torque. So, this term appears in both equations, here with positive sign, here with negative sign. Because in the equation of the motor, so this is the equation of the motor, this is supposed to be the inertia. And this is the resistance in the friction, sorry, in the rotor. And this is the generated electro, what do you call it, magnetic feedback torque or something. The torque that the motor is generating. And the motor sees the torque tau L as a load, right? Because for the motor, it's a load with the pendulum. It's a load. So, you have this term there and there. So, this is how these two systems get to be interconnected. And we go and look at the energy balance equations for these guys. So, basically, we integrate on both sides of both equations. We will have an energy balance equation for the motor, which will be, of course, the energy available at some moment will be equal to the available energy at the beginning. Whatever was dissipated in, because we have friction in the rotor. And then the supplied energy, so basically the energy that is entering into the motor. Well, there is, of course, what I supply into the motor to make it spin, that comes here. But there is the load. So, I inject some energy into it, but some energy goes into moving the pendulum. So, the supplied energy will look like this. And for the pendulum, the pendulum is all happy because it only sees energy coming in. So, there is supplied energy there and there is some initial energy and there is then available energy. So, basically, the available energy equals whatever is supplied through, I mean, coming from the motor plus the potential energy that was over there already. Now, the nice thing about this is that the whole thing, when I, the system interconnected, will also have some energy, right? And this energy will be basically just the sum of this plus that. So, now we have the sum of the energy in the motor plus the energy in the pendulum, that will be the energy of the whole thing. And what will happen is that if the pendulum didn't have friction, it doesn't matter because this guy will just contribute to there. And the energy balance equation for the closed loop system, for the interconnected system will look like this. Now, we have this term of energy dissipation and we have this term of supplied energy coming from all the way from the input voltage that is here and now my output is still there. Okay, so now I have an interconnected system and I have passivity, actually I will have output-strict passivity due to this from the map V to Q dot. And this is, so once again this is because the motor is output-strictly passive with input being this difference. The pendulum is passive, so I'm interconnected a passive system with an output-strictly passive system. So the resulting interconnection will be output-strictly passive due to this guy, right? So I recover this term in the total energy balance equation. So, again the feedback interconnection of two passive systems yields a passive system. A passive system with input-strict passive feedback yields an output-strictly passive feedback. Yeah, interconnected it will become output-strictly passive. That's the case that they showed you with the electrical circuit with the additional resistor, right? The output is Q dot, yeah. The output is Q dot. The output is Q dot. Yeah, there is no passivity with respect to Q. The output is Q dot. Velocity. And yeah, one thing to remember is that passivity is conserved when you interconnect the systems in feedback. Okay, where is that feedback? Yeah, feedback. If you interconnect them in cascade, so the output of one goes as input into the other and so on. But there is no feedback loop. You don't have passivity necessarily, okay, from here to whatever output you choose there. We are talking about interconnections in feedback form. I will not do input-output stability. Let me just tell you a little bit about this because I want to show you how we can use passivity for control. But for that, I first need to tell you about a couple of things in the steel of passivity theory. So in linear systems, we have this concept of positive realness, which I believe was introduced by Popov, a Romanian mathematician, who studied a lot equations, linear equations, right, and was figuring out properties of these linear equations. And one property that I think he actually came up with this concept of positive realness is the following. We say that the system, a transfer function G of S for a linear system with A, B, C and D matrices is said to be positive real if the real part of the transfer function is larger or equal than zero. The nice thing about this is that a positive real system is passive. And there is another important concept for linear systems, which is called a strict positive realness. So a strict positive realness is the property that the real part of the transfer function with S shifted a little bit, right, so epsilon is a small number but positive. That should be larger equal to zero. So basically you are asking, in this case, you are asking, in the first case, you are asking that the Nyquist plot is somewhere here, right, so in all this zone. And in the other case, it has to be like, sorry, in the other place, right, so the positive real. In the first case, you are asking that the Nyquist be here, right, and in the other case, you are asking that the Nyquist be strictly separated from the vertical axis and we call it strictly positive real. Now strict positive realness is a very strong property for a linear system to have. It basically means that the system is strictly passive in the whole state. I didn't speak about state-strict passivity, but you can just think of this as having the, in the energy balance equation, instead of having a q dot, you will have here the whole state, right, so x, okay. So both, you know, we're both input and output strictly passive. Now the nice thing about strictly positive real systems, we will see that with the up and down functions, it will probably be clearer, is that we have this very nice lemma that is known as Kalman Jacobovich Popov, especially because I think the story goes that Popov proved the one implication, Jacobovich proved both, Kalman proposed different proof. Each of them work separately on this result. I think Kalman published it first, but Jacobovich apparently had it first, but then as many good Russian mathematicians just put it in the drawer and went to solve something else. So anyway, so it is called the KYP lemma, it's a very famous lemma, and it says that a system is a strictly positive real, if and only if, so a linear system like that. If and only if, yeah, there exists a matrix P, matrices L and W such that all these equations hold, all these equalities hold. So in the case when there is no fit through, I mean, directly from the input to the output, which would be this term here, then these equations just boil down to these equations here, okay. So we have that ATP plus PA equals minus Q and this structural condition here. Now this one may be quite familiar to you because it just, you know that if A is Hurwitz for a system, you can always find for any Q positive definite, you can find P also positive definite, so such that you have this equation, it's called the Lyapunov equation, right. But here we have something else, we also have a relation between, so B is the thing that is multiplying the inputs and C is what defines the output, yeah. So this condition gives a structural property between the, about a structural property of the system. And actually this condition means that the relative degree of my system of G is one, at most it can be one. It will be zero if there is this fit through, but if we are in this case down here, then it can only be one, yeah. So this is a relative degree condition and this is basically a condition that says that A should be Hurwitz, right, so it should be stable. Yeah, so that's another thing to know, passive system necessarily is of relative degree either one or zero. It cannot, system of higher relative degree than that cannot be passive. That's me back in the day with Professor Jakubowicz, I'm very proud of this, of this visit, just after my PhD. And this is the piece of paper that he used to announce me as Professor Antonio Lurie, I had just finished my PhD. So yeah, it's sitting there, I will not do these slides, now I will do it in the other set of slides, so I will just pass that. Yeah, one nice thing about passivity is that you can also use it, as I said, you can use it for linear or for nonlinear systems, because again it's a concept of input-output systems, right. So for nonlinear systems, yeah, let me go here, this is like a nonlinear version of the KYP lemma. And basically it says, well this statement I took it from Halil, but the result is by the original papers are, if you are interested, you should look at the Hill and Moylan. I think Hill was a student of Moylan, but I'm not sure. I'm talking about a series of papers in 76 and 80 and there is probably one more intersections. So he essentially introduced passivity for nonlinear systems. And these expressions, as you can see, they are really like a nonlinear version of the previous ones. So essentially you want to have a storage function or call it an energy function such that the inner product of the partial of v times f is less or equal to zero, less or equal to zero and you have this inequality here. So basically the first one is like having ATP plus PA equal to zero or less or equal to zero. And this is, if we had an equality here, this would be the condition that we saw. I think it was, yeah, probably it was like this, right. And in this case, as you can see, we have a quadratic term here. So this is like having ATP plus PA equals minus y square. And here we have again basically the structural condition. You already saw this. So yeah, this is the nonlinear version of this thing here. So this is the inequality of what is below, right? And this formulation here is again, is from Halil, but the original result goes way back to the 70s. As we will see, we can use these to also use passivity for nonlinear systems, not just for linear systems. But it's nice because you can kind of in a way see linear systems as if they were nonlinear, as if they were linear because of these input output properties. And think of energy terms and so on.