Hello everyone. Welcome to yet another session of our NPTEL on nonlinear and adaptive control. I am Srikanth Sakumar from Systems and Control IIT Bombay. So we are into the eighth week of this course on nonlinear adaptive control and we are already looking at several algorithms, we've already looked at several algorithms which will help us to control autonomously systems such as what we see in the background of a satellite orbiting the Earth. We are also now able to design algorithms against uncertainties in such systems. So what we were doing until last time is basically completing the proof for model reference adaptive control. So model reference adaptive control was a paradigm for adaptive control of linear systems which is a very, very popular and very, very well known and very well cited and even used sort of set of methods in adaptive control. And the basic difference from what we have been doing before this is that here we actually track a state that comes out of a reference model than just a reference signal. So the picture looked somewhat like this. So that is the big difference here that you have a reference model instead of just a reference state. And everything else is sort of similar. In terms of the analysis, the big difference that we saw was that we had unknowns which were actually matrices. So we started with A star and sorry, A and B being unknowns. And then of course we moved it to sort of redesigned it so that the unknowns became these K star and L star which were still matrices. And because we were dealing with matrices and matrix unknowns, we also learned a rather novel way of designing Lyapunov candidate functions for matrix states or matrix unknowns. So here the parameter error term instead of just being a quadratic or a square of the vector norm is in fact a kind of a matrix Frobenius norm, weighted matrix Frobenius norm. And it is defined in terms of the trace function. So this was the big difference. And in order to sort of complete the proof, we also needed to use some very, very interesting and useful trace properties. So this is something that I hope all of you will remember always because these are very, very useful in manipulating any kind of Lyapunov candidate in any other context. If you see trace functions, then it's very, very useful to manipulate these trace functions. And we can use these equalities in order to do that. All right. And of course, then we designed our update law. As usual, it was set a certainty equivalent adaptation. So all was as usual, other than these particular features. Okay. So now we are sort of ready to move into the next week's lectures. Again, please do not get confused. The sort of classification of the week is also in a sense to provide homework and just to organize material. As in when we do complete the material for a particular week, we will move on to what we call next week lectures. So we are in week number eight. We are already going to start looking at week nine lectures. Yeah. So I really hope that this doesn't cause any confusion. Should be pretty straightforward. This is just for us to, you know, organize the homework in a proper way. Yeah. So this week nine is titled adaptive integrator back stepping and extended matching design. So we will do a couple of things in this set of lectures. The first is we sort of generalize this adaptive integrator back stepping that we did a couple of weeks ago. So not the week number eight, but week number seven lectures, right? In week number seven lectures, we saw integrator back stepping for the first time. We also saw how to do a back stepping design for the unmatched case. So what we will do here is to generalize that idea. Okay. And then we look at a different method of doing adaptive integrator back stepping so that we don't have to resort to over parameterization. All right. So that's the agenda for week nine lecture notes. Yeah. So let's continue. Right. So let's suppose that for the system. Right. So now suppose we are, since we are generalizing, so we are essentially taking vector states and all that. Okay. So, well, I mean, to be honest, let me mark this first before we start. So this is actually lecture eight point three. All right. This is lecture eight point three. Okay. So, so we'll recall is this week seven. I mean, I'm going to go here to week seven. We had a system like this. Right. Where we learned how to do back stepping for unmatched parameters. Right. And we know how we did it. We started with the first system then assume that there is an unknown parameter. So the desired quantity was designed with a cap. Right. And then we of course had a Lyapunov kind of design. But then when we went to the second Lyapunov function and we tried to choose a control, there was a problem because the control is not implemented. But the control is not implementable due to the presence of this theta here. Right. So control contains an extra desired dot, which can, which brings up a theta, which is not implementable. So obviously, we don't want to use a theta hat again, because we already have specified an update law and we can't do it. It will create an error in the analysis. So instead of theta hat, we put in a new estimate new hat. Right. So we put in a new estimate new hat for the same quantity theta. We have two estimates, theta hat and new hat. Right. And so with that, we declare this new variable as X2 hat, X2 desired hat dot instead of X2 desired dot. And then we continue this analysis with this new mu hat. And then of course, you have a candidate Lyapunov function, which also contains a mu tilde on top of the theta tilde. Okay. So essentially, because we have two parameters, two estimates for the same parameter, we have two terms corresponding to the parameter errors. Right. So there are two different parameters. All right. So that essentially was what we were doing. I mean, we then specify mu hat dot and all that in the standard way. Yeah. If you don't remember, please go back and revise what we sort of did in this lecture. So what we want to do today, at least in the beginning is that we want to look at a generalization of this, then the states are not scalar states. Yeah, because most in most problem more often than not, your states will be vectors. Yeah. So it doesn't make any sense to just restrict ourselves to scalar states. And doing this will also give you a very fair idea that the analysis methods don't change significantly when you have a vector state instead of a scalar state. Okay. So it's not like things become too complicated. Okay. So that's rather nice. So this is another thing that I want all of you to get used to. Because suppose I think of any robotic application, say a two-armed manipulator or think a quadrotor, a quadrotor has six degrees of freedom, right? So three rotational and three translational. So corresponding to that, there are 12 states because three position states, three translational velocity states, three angular position states, three angular velocity states. So there are 12 states. So it's vectors, right? So we describe the equations in terms of vector equations, right? Similarly for a two-joint manipulator, right? For a two-joint manipulator, so you can think of the shoulder and elbow, right? If I have a two-joint manipulator, so the shoulder has two states corresponding to the shoulder angle and the shoulder angular velocity. And the elbow has two states corresponding to the elbow angle and elbow angular velocity. Right? So therefore, this is again a vector. Okay. So it doesn't make sense always to look at scalar states because we might end up having vector states. More often than not. More often than not. Yeah. So that is the purpose. Two things. One, you want to generalize two vector states, right? And of course, this particular method to vector states. And second, we also want to sort of get a feel for how to deal with vectors in Lyapunov analysis because in many cases the states are going to be vectors. Yeah. And you will see they're not significantly different. Great. Now that we have this preamble. So suppose we have this system.  Now the states are in some rn. The unknown is in some vector rp. It's a vector rp. So p-dimensional vector. State is n-dimensional. Right? And you can find the corresponding dimensions of f and f and all that. And g. Right? And the control input u is assumed to be an m-dimensional vector. Okay. So there are three different dimensions. The states are n-dimensional. Theta is p-dimensional. u is m-dimensional. Okay. And of course the f and small f and whatever and also g actually, also g. Yeah. Are assumed to be sufficiently smooth and have all the nice properties. Okay. So that, you know, we can take differentials if we want and so on and so forth. Right? So suppose for this system, there exists an adaptive controller. Okay. That is, what is an adaptive controller? There are two pieces to an adaptive controller. First is a control law u, a specification for what the actuator is to generate. And this depends on x and an estimate of the state, theta hat. And further there is an update law for theta hat. That is a theta hat dot, which again depends on the state and theta hat possibly. Right? So this is what constitutes an adaptive controller. Two pieces. Right? A control law u and a parameter update law theta hat. Okay. So suppose that there exists such an adaptive controller for this system and a smooth v function. That is a, which is essentially upon a function, smooth v, which takes the state and the parameters. It's radially unbounded in x and theta tilde. Right? Such that if I take v dot, right? So what is this? This quantity is just, this quantity is just v dot. Okay. This quantity is just v dot. Right? Just taking, because v is a function of x and theta tilde. So first I take partial with respect to x. Then I multiply it with x dot. Right? Because x dot now contains the control. This gets substituted here. So this is the closed loop system. This gets substituted here. So del v del x fx plus capital fx theta plus gx alpha. Okay? So this is del v del x x dot plus del v del theta tilde, theta tilde dot. Okay? All right. So let's see. You have to be careful here. We have to define. Let's see. Let me see if we have done this correctly. Okay. I think, I think this is, this should be, actually this should be theta tilde dot. Okay. This is not theta hat dot, but a theta tilde dot. It doesn't matter because I mean, we usually are defining theta tilde using theta hat. So theta tilde is defined as theta minus theta hat typically. So theta tilde dot is equal to minus theta hat dot. Yeah. Because theta is assumed to be a constant. Right? So basically this doesn't change anything. I mean, if you, I mean, it just changed the sign, but that's important because that's the sign used here. Right? So this is theta tilde dot, not theta hat dot. So it's a derivative of the parameter error. But remember the right side cannot depend on theta tilde because theta tilde is unknown. We keep that in mind. But then this theta tilde dot is just minus theta hat dot. So if you know theta tilde dot, you know minus theta hat dot also. Okay. Great. So, so this is just the derivative of v along the closed loop trajectories. Right? So this is a function of x and theta tilde. So partial of v with respect to x times x dot and partial of v with respect to theta tilde times theta tilde dot. Right? And the assumption is that for this, for this particular v, v dot along the system trajectories is less than equal to minus w x theta tilde, which is a negative semi definite function. So minus of w is negative semi definite. Okay. So this is the assumption. Right? So this is the assumption. Right? We start with this kind of a system. Right? And we have an adaptive controller and a corresponding v. Okay. Yeah. And a corresponding v says that v dot turns out to be less than equal to a negative definite function. Okay. So that's the idea. Then we go on to add an integrator. Before we do that, I wanted to compare this with our, sorry, our system here. If you look at this guy, the first piece. Right? Where you have this system. Just look at this. Just look at this system. And if you think of x2 as the control, just like we did in backstipping. If you think of x2 as the control. Right? And then you do have this control law, which contains the theta hat and an update law. Right? With a corresponding v says that v dot is negative semi definite. All right? So we exactly have this. Right? So what's the situation then? Our first system is of the form this in fact I have written it here. Right? Where this is actually the control. And this is the control law. Right? Which is a function of x1 and theta hat. And of course there's an update law, which is theta hat dot. There is a v function, which is this and v dot equals or less than equals minus w would also depend on theta hat. But the point is it is negative semi definite. Yeah? The key point to remember is that it's negative semi definite. So we exactly have the same setup already available for the scalar state case, scalar state scalar control case. So we are exactly generalizing that set. Okay? Exactly generalizing that particular setup. So we assume that we have now a vector system. We have an adaptive controller for this vector system such that there also exists a smooth v says that v dot is negative semi definite. Okay? So now if we add an integrator, which is essentially what's been done here. Right? If you look at this, we think of this as the control, but actually you have an additional state x2 dot is u. Essentially we have an integrator. So that's what we say. Now, if we add an integrator, so the control here, so the structure is slightly more general of course, but still we do the same thing. If we add an integrator, that is the control is replaced by this sign and then the side is u. So the control is at the next level. Okay? Then what do we do? All right? So this is where the integrator back stepping was used earlier. And we want to of course, generalize it to this vector case. Right? So one thing should be evident that this size is also an RM. It has to be the same dimension as the control. Right? Because otherwise this is not a feasible situation. Okay? So if size is a different dimension, then there can be problems. So we assume that size is the same dimension as the control. Okay? So then the claim is that we can still construct a Lyapunov function, which allows us to compute an adaptive controller, which is going to guarantee closed loop signals remain bounded and also that somehow the back stepping error and W go to 0, ST goes to 0. Because this is the best we can do anyway. But even in the scalar case, we prove that W goes to 0. What was W? In this case, W is this. So we prove X1 goes to 0. That's the best we can do. Right? And of course, we also prove that the back stepping error goes to 0. So we prove these two things. That's what we want. Okay? Even in this scalar case. So that's what we claim here also. We want to claim here at least, right? That this W goes to 0 and the back stepping error, which is psi minus the, because the psi is the state now and cannot always be made exactly equal to alpha, but in steady state it can be. So psi has to follow alpha and that is the back stepping error. And that is also going to be driven to be 0. Okay? What is this magical Lyapunov function? Nothing very magical to be honest. All right? So we call this V bar now and it has several states. It has X and psi, which are now the new states. It has the earlier parameter error theta tilde and it has the new parameter error. New parameter theta bar. Okay? Remember, there we used a mu, right? Here, of course we used a mu, right? Let's see. Yeah. Here we used a mu. Okay? The mu hat. So here we are using theta bar. Okay? Just a difference in notation. All right? So it's a function of four quantities now. X, which was always there, theta tilde, which was already there for the one state system. And now we have psi and theta bar corresponding to the integrator and the new over parameterization. Okay? How do we construct it? Take the same V, which was just a function of X and theta hat. Then we add a backstipping error term standard. Then we add a term corresponding to the new parameter error. Okay? The theta bar is a new parameter estimate. So I just add a term corresponding to that. Now, notice that because this is a vector, I have taken care that this is written as a norm. Okay? So this is written as a norm. This is no longer just square of psi minus alpha. It is, in fact, norm squared of psi minus alpha. And it is very standard to use the Euclidean norm, all the two norms. So this is denoted as Z, of course, like is mentioned here. Right? So this term is actually one of Z transpose Z. And this is the Euclidean norm squared. Yeah. And similarly, I construct a very similar looking term for the theta minus theta bar. That is a new parameter error also. Just that I add an adaptation gain. I always add an adaptation gain. This is the adaptation gain. Okay? So let me see. Unfortunately, we have used the same gamma here. Okay? This is not the same. Okay? Let's be clear on that. This is like a, let me call it an S inverse. Okay? So S is some constant positive definite matrix. So this is not the same as that gamma. Okay? That gamma was the... This gamma is actually the update law for the parameter theta hat. Okay? This is the different parameter, different matrix S.  It actually controls how fast or slow your adaptation with them. Okay? All right. Great. So instead we have this sort of a new kind of Lyapunov function for this integrator system with vector states. Right? So therefore we have norms appearing here. We have transposers appearing here. All right? Now we are going to, of course, take the derivative. Right? I mean, our claim is that this is a good Lyapunov function. Okay? This is our claim. Now, of course, we want to verify this claim. Right? So, let's see. Okay? Let's see. So the dynamics... First we want to do... The first thing we want to do is try to dynamics in the new variables, which is x and z now. Right? Because we introduced a back-stepping variable. It's not x and psi, but it's x and z. Okay? So we just do that. So first... So x dot is fx plus fx theta plus gx times a u. Right? And u can be written as... What? I'm sorry. It's not u. This is in fact... I'm sorry. This is my back. It's not u, but this is psi. Right? So this is what is the dynamics. Right? So and psi can be written as z plus alpha. That's what I do. Okay? And what is z dot? Z dot is just the derivative of this, which is psi dot, which is u. And then minus alpha dot. So then I have minus of alpha dot. So all of this is minus of alpha dot. Okay? Alpha is a function of two variables, x and theta hat. So first I take the derivative with respect to x. Okay? So del alpha del x times an x dot. And x dot is just plugged from here. Okay? So this guy is right here. Okay? And then minus del alpha del theta hat times theta hat dot. Okay? Yeah. Times theta hat dot. So what is theta hat dot? So this is where we have to be careful. Right? So this is actually... Now let me be careful. This term should be minus del alpha del theta hat theta hat dot. Remember, alpha cannot depend on theta tinder. Because it was the first stage control. It can depend only on theta hat because theta tinder is unknown. Okay? So, and remember from the previous derivation, theta hat dot is actually equal to minus theta tinder dot. So it is equal to minus gamma. Okay? Minus gamma. So I have to be careful. This is actually going to be equal to plus del alpha del theta hat gamma x theta hat. There's a plus sign and not a minus sign. So I need to be careful here. This is actually going to be a plus sign. Okay? This is actually going to be a plus sign. Okay? I hope that is clear. So what have I done? I have simply written the dynamics. It's just a lot of bookkeeping again. All right? I have simply written the dynamics in terms of the new variables x and z. This is exactly what I was doing for the scalar case also. All I'm doing here is being a little bit more careful because I have vectors that are involved. And that's it. Okay? So here I have x dot which was fx plus cap fx theta plus gx times psi. And psi is just replaced in terms of the back stepping error variable as z plus alpha. Okay? Then I have z dot which is psi dot minus alpha dot. Yeah? So when I say dot it is a derivative with respect to time. And so I take psi dot which is u and minus alpha dot. So there are two terms. Del alpha del x minus del alpha del x x dot minus del alpha del theta hat theta hat dot. So there are two terms. And then the first term is therefore del alpha del x and x dot which is substituted from here. And then I have del alpha del theta hat theta hat dot which brings in a negative gamma. So this becomes a positive. So I have fixed this sign. So when I write it in this form, again, remember that what's the dimension of z? So it's important that we remember all this. So x is in... So x dot belongs... Well, I will write it here. So z... x is in Rn, of course, I don't need to write it. But z is the same dimension as the control. So it is in Rn. Right? So this is Rn. Now, what is the dimension of this quantity? This is Rn. So if I look at thing in the bracket, this is simply x dot. So this is equal to x dot. And this, of course, belongs to Rn. I need to be careful here. Now, so what is the dimension of del alpha del x? Therefore, what has to be the dimension of del alpha del x? It has to be... It has to be Rm cross n. To be consistent with this expression, right? Because the right hand side has to be Rm. And this is in Rn. Therefore, this has to be an m cross n matrix. Now, how does this happen? Similarly, before we go on, gamma here is from theta tilde dot or theta hat dot whatever and theta was in Rp. Right? Rp. There are p unknown parameters. Therefore, this guy has to belong to Rm cross p. Okay? So how does this happen? So look at what is alpha. Alpha is a function of xn theta hat. Okay? What do I know about dimension of alpha? I know that alpha, if I may write it carefully, alpha is in fact a map from x states. So Rn and p states, which is Rp. So that is theta hat states to what? It is the same dimension as phi. So it is n, Rm. Okay? So now if I take partial of alpha with respect to only these states, it is obvious that I will get an m by n matrix. Okay? When I take partial of an m vector with respect to only n of the states, I will get an m by n matrix. Similarly, if I take the partial of an m vector with respect to p states, then I will get an m by p matrix. So this is all consistent. Okay? So just keep this in mind that all these partials are now Jacobians. What we call Jacobians. And they're all matrices. All right? Excellent. So what is it that we did today? We sort of started to generalize the adaptive integrator backstepping that we looked at in week number seven to the vector case. And we are sort of trying to understand the differences when there is the vector case or the norms appear, the transposes appear, and then there's the Jacobian and all these notions. So we have to do very careful bookkeeping. But as of now, I hope you've seen that the methods are not distinctly different just because the states become vectors or something like that. All right? So of course we'll continue working on this generalization of adaptive integrator backstepping to the vector case in the subsequent session also. All right? All right. Great. I'll see you again next time. Thanks.


I am noticing a sharp increase in both the IQ and the sanity of the remaining subscribers. Let me venture a guess, all the conspiracy theories have finally left us alone. Thank God for that, even though I am not a believer. So, today we are going to slaughter another holy cow. And the holy cow is closure. Is closure good for you? Everyone and his dog will tell you, yes, closure is great for you. You should pursue closure. You should even try to communicate with your abuser to obtain closure. Regrettably, like most information online, this is nonsense. Long debunked by psychological studies. This is the channel where you are exposed to real psychology in the making. And not to, you know, media hype or self-styled experts. So today we are going to tear apart closure, its adherence, its proponents and its promoters. Let's start by asking, where did the idea of closure come from? Where else it came from Gestalt therapy or Gestalt psychology? In Gestalt psychology, in the early writings of Gestalt theories, they suggested that the mind perceives images and processes them in a highly specific way. One of the main Gestalt principles is that the mind seeks closure. So if you see an image of a circle and the circle is incomplete, the mind will still perceive it as a circle. It will close the circle. It will generate closure. So closure in original Gestalt psychology had to do with image processing, nothing to do with relationships. And then a group of self-styled, self-proclaimed experts, self-styled, self-proclaimed gurus, borrowed this principle from Gestalt psychology and corrupted it beyond recognition. Now these people say that we process life the same way that we process images. When we have something that is not closed, for example, an unresolved trauma from the past, we need to close it. We need to resolve it in order to properly process it and to be able to move on. So inappropriately, the principle of image closure in Gestalt psychology was expanded upon and extrapolated to include trauma, relationships, and everything else under the sun frankly, political processes, you name it. And of course immediately based on that, there was a cottage industry. People started to make money exactly as happened in narcissism. When all kinds of people discovered narcissism, they declare themselves experts in order to make money off the gullible. So the same happened with closure. There were therapeutic techniques which attended to closure and ensured closure. Although no one had defined closure actually to this very day.  But everyone is talking about it with great assurance and authoritativeness. And so there's the technique of the empty chair. It's when the participants imagine the source of the frustration, the source of the unresolved trauma. For example, an abusive parent, a departed lover, an ex-spouse and so on and so forth. So you imagine this person sitting in an empty chair opposite you and you speak to the chair as though it were that missing person. So this is the empty chair technique. I have a whole video dedicated to it. Now the technique itself is very interesting and elicits healing in some ways and to a limited extent. But it does not generate closure. Anyone who tells you otherwise is a con artist. So the empty chair technique provides temporary release. We have multiple studies that show that the empty chair technique has no long-term emotional effects, no long-term emotional release and no effect on processing trauma. The pain remains and is reawakened usually after a few hours. So that's an example of a therapeutic technique borrowed from Gestalt by the way which had been abused to actually promise the gullible outcomes which can never ever materialize, which are not embedded in the technique itself. This is false advertising and closure to a large degree is hype coupled with false advertising. There's a happy ending narrative which is essentially Holywoodian. It's a Hollywood thing. It's not in psychology. It's not happy ending in psychology. There's no ending in psychology actually. We don't recognize end points in psychology. Everything is in flux. Everything is a process. Everything leads to something else and everything else leads to another thing. So there are no end points. And the methods of Gestalt are very controversial although some of them do produce results like the empty chair. They're very controversial because Gestalt does promise the equivalent of closure or the equivalent of a happy ending. So popular culture and pop psychology and self-styled artists who are basically hacks and con artists and so on jumped on the bandwagon and are promising you all this. And now in talk shows, you know, you see all kinds of lovers or spouses or parents with children and so on and so forth. They're on stage, they confront each other and then they burst out crying and disintegrate on stage and then they reunite and hug and all is well and they exit left happily ever after. It's addictive to see this. It's very appealing. We want to believe in happy endings. We want to believe that the world holds in store some promise but nothing is more pernicious than hope. I've been saying it for two years now. Hope is the worst invention ever. And so closure is closely allied with the concept of hope. What is the truth you ask? What is the truth about closure? Well, start with the psychotherapist Ashley Davis Bush. She wrote the book Hope and Healing for Transcending Loss. She argues that the pop psychology brand of closure is actually not achievable and even to some extent non-productive. She says, the thing about pain, the idea that we can completely end emotional pain is false. You can ease the pain, you can learn how to cope better, you can temporarily forget about the pain but in reality it's not that the pain ends but it changes over time. Many experts today and all psychological, all real psychologists, they would tell you that closure is a myth and not only a myth but a pernicious counterproductive myth, destructive in some cases. Now closure provides hope, the idea of closure provides hope and provides comfort.  So had you experienced a tragic loss? Had you undergone a traumatic breakup? Closure is the solution for you. That's a piece of advice that family, friends, popular media will give you, they're all movies based on closure. When we mourn, when we go to a funeral, it's a form of closure. It's a ritual of heartbreak, a ritual of pain but death is final. Death does provide an end point, not so with traumatic breakups for example because your ex is still alive and well and dating others and so on. So it's nice to believe, it's malignant optimism. It's pernicious hope to believe that there's some series of steps you can take and there are not too many and you can achieve this magical state. Your pain will vanish, bad memories will evaporate, and so on and so forth and you'll be happy ever after. And this is seriously dangerous. It leads you astray. The real focus should be on other things. When you attempt to move on, that's not easy and if you disperse the scarce energy that you do have, you're depleted after a trauma or a breakup and so on and you have very little energy and if you misuse this energy, if you allocate this energy wrongly, you're left without any energy to do the real work, the what you really must do. So the advice is get rid of things, objects, remove physical reminders that trigger you. I don't know, your lover's possessions, your ex-spouses, things, get rid of them, write a goodbye letter, scatter the ashes, find forgiveness in your heart, get answers from your abuser, et cetera, et cetera. These are all closure techniques and they are premised on the main message of Western civilization. Everything has a solution and if we only put our minds to it, everything ends happily. Well, as COVID-19 is teaching us, some things don't have a solution and some end many things, do not end happily. The life is not clean cut, there's no finality and there is no foreseeable end to pain. Life is about suffering and loss, suffering and loss of the main and probably exclusive engines of personal growth and development. We evolve through suffering, we progress through loss. And the absence of these or when we try to shunt them aside, shunt them and hide them and avoid them, we never grow, we never evolve. And that guarantees a constant victimhood stance and a pain that cannot be mitigated or ameliorated, it just transforms and shape shifts so the pain can become anxiety or the pain can become depression. We need to confront the pain, head on, accept it, embrace it, live with it. There's a belief that you have to take action, you have to initiate, you have to make sure that you obtain closure. It's all in your hands and of course this puts an enormous honest and enormous pressure on the victim, for example of abuse or the survivor of trauma. It's like, what's wrong with you? Move on, confront your abuser, do this, do that. And if you're not doing these things, something's wrong with you. But this provides an illusion of control when a situation where there's actually no control and it depletes further the exhausted victim or survivor. Mariana Boccaova had written in psychology today that closure in breakups is about finding answers. She says, through closure, we can restructure our past, present and future in a healthy way through understanding what went wrong and reconfiguring our story accordingly. I beg to differ, not one word of this sentence, of this paragraph is substantiated by research or studies and most of what she had said is starkly contradicted by everything we had learned in the last 20 years. Additionally, it's a very tall order, listen again. You can restructure your past, present and future in a healthy way. This is asking too much. This is adding to the stress and the pressure. This is not helpful at all. Getting answers has nothing to do with your pain. No matter how many answers you get, it's not gonna ameliorate your pain, mitigate it. If you find out the true reason for a breakup, this reason is not gonna lessen your pain or take care of it. On the contrary, by the way, the majority of cases is gonna increase your pain. It could be even more traumatizing, depending on what the reason is, of course. Ending pain is a myth. It's an unattainable goal. No serious psychology is trying to accomplish this. Searching for a permanent closure to emotional pain closes you off to healthier ways of processing grief and sadness, the Kubler-Ross cycle. You must not close the door on growth and positive growth.  It's an essential part of life to experience the full range of sadness, dysphoria, pain, agony, suffering. These are honest emotions. They are necessary. You can't experience joy and happiness if you have had no experience of the other negative affectivity emotions. So, when you're experiencing pain, to some extent, you need to wallow in it. You need to immerse yourself in your pain, to let go in the pain, to become friends with the pain, to make friends with the pain, to accommodate and accept the pain as an inevitable part of who you are and of your life, and to let the pain teach you how to grow out of it and into a better person. Also, pain teaches you lessons. Pain is a great educator. The idea of closure is harmful. Clinical social worker, Bob Livingstone, wrote that the pressure to achieve closure and its inevitable failure in the long run can worsen the feeling of loss. Repressing your emotions, he says, is dangerous and unsustainable. How should we approach traumatic loss? Bush says that when she sees clients that grieve over a lost one or a loved one, she doesn't offer closure. She offers healing. She offers growth. Surviving spouses, for example, they need to carry the painful memories because they're precious, but they need to learn to carry this pain in a positive way, not in a negative way, not in a growth retarding or growth stunting way, but in a growth promoting way. They need to live with a lost love. They need to embrace and accept loss. Bush encourages her clients to keep the memories, not to get rid of them. You must learn how to let the memories fortify you instead of weigh you down, she says. And so, coming back full circle to the issue of death. In Western civilization, we reject death. We try to avoid it. We're terrified by it. We hide it. We gray it out in media reports. There's no death, our world is sanitized. When we come across a virus, we deny its existence because we don't want to confront the fact that we are mortal and we are all about to be done. We're here temporarily. And so, this rejection of death has a lot to do with the concept of closure. This closure is about rejecting emotional death. Sociologist Nancy Burns argues that the popularity of the idea of closure among loss specifically is largely embedded in the commercial interests of funeral home directors and wrongful death attorneys who offer the appealing emotional state to their clients. Closure in many ways keeps you stuck in the past. If you try to obtain closure from your abuser, you are still in a relationship with your abuser. Nevermind how limited.   You actually believe that your abuser holds the key to your healing growth and development. You adopt the perception that without your abuser you're unable to move on. Everyone worries about being stuck in the past and everyone advises against being stuck in the past, but closure is the quintessential move of remaining stuck in the past. So we must advocate against closure if we really promote moving forward. Moving forward, remaining emotionally alive, experiencing all your feelings, however painful. You know, this is the way you fall in love again only when you had processed the pain of having lost a loved one. Intense grief is the way our psychology copes with loss, pain, trauma, suffering, deep feelings. Deep feelings and memories attendant upon these emotions are treasures to be cherished. You don't need to sacrifice these things. They're a gift. Some elements of closure do exist, of course. You sign divorce papers. You move out of an apartment that you had shared.  These are concrete ways which symbolize finality, but finality is not closure. It's not the same. Closure implies finality without emotions. Finality is healthy. Although not always obtainable. Closure is not. Things get complicated. Our past relationships and experiences are always with us. No matter how hard you try to get rid of these memories and pains and suffering and torment, you can't. You just can't.  You are augmenting and enhancing your anxiety and depression. And a much more productive way is to deal with this negative, this negative affectivity, bad emotions, horrible memories, to deal with them. And by dealing with them, you're honoring yourself and you're extracting wisdom from these things. Because they're supposed to teach you something. So don't think about a relationship as a hotel room and the end of the relationship as a closing door or an ending chapter or the end of a movie. Think of it as a leg in your journey, as a layover, as a part of your path, as a process. You carry baggage with you. The baggage could be unpleasant and heavy and discomforting and huge, but you have to drag this package with you. And you have to drag it because you're on a journey, you're in a trip. And the only way to lighten the load is to integrate it with yourself. When this baggage is no longer external, but part of who you are, you travel light or you travel lighter. Research had demonstrated that writing is a good idea. Journaling is a good idea. It does not lead to closure. There's no such thing as closure. But it does help you to process your emotions and to put them to good use. So journaling is a very good idea. There are quite a few studies about journaling and I just type journaling studies or journaling research and you will find numerous studies. Generally in all these studies, we discovered that after a writing exercise or a journaling exercise, there was no increase in negative emotions among people who embraced the pain, who regarded it as a positive thing. There was even a boost in positive outcomes, including rise in confidence, self-esteem, empowerment, comfort, optimism, gratitude, wisdom. So if you accept and embrace and integrate everything that had happened to you, good or bad, negative or positive, painful or not, you say, this is me, this is me. I'm gonna write about it, I'm gonna journal about it, but I'm gonna do it in a positive way. I'm going to try to remember the positive aspects, even of the pain. This pain is teaching me something or the pain is indicative that the relationship is over and I'm not gonna experience more pain, et cetera. So try to find the positive aspects. If you do journaling and writing is useful, but if you are journaling and writing just to vent or to carp or to complain or to whine, it's bad for you. It's gonna drag you down. Venting, by the way, is one of the worst possible things to do. So I have a video about this. So journaling should not be a form of written or codified venting, it should be about finding the positive. Closure is not possible, it's not desirable, it's not necessary. This is the consensus in psychology today. Anyone who tells you otherwise has no idea what they're talking about, which unfortunately makes the overwhelming vast majority of self-styled experts on this platform.


What is going on guys welcome back to the channel and I appreciate you being here today I bring you another apex legends video and today I bring you a great location to spawn in and get that easy elite loot But before we go any further guys if you do enjoy the video and would like to show you support You can by hitting that like button and if you are new around here and enjoy the content be sure to subscribe Okay So this video is for newcomers to the game and there are plenty of you out there as the game is literally Brand-new respawn drop this on a dime surprised many many people Crazy thing is it's a great great game. Yeah, many are disappointed. It is an air more like Titanfall But hey, it is what it is plus it's free to play you can't get much better than that people So if you're watching this video wondering what this game is thinking that looks good You can get it for absolutely free zero cost on PC Xbox or PlayStation Just search apex legends within them stores. It's as simple as that So if the game is entirely brand-new to you check out this video right here, which will explain many things you need to know about the game Apex legends a tactical squad-based battle royale that has galvanized to the outlands It is the ultimate test of survival you win or you die You tried you died But this spectacle is nothing without the legends From larger than life characters like the famed Gibraltar whose presence on the battlefield can always be felt To Mirage the fan favorite jokester whose command of holo technology makes him a master of deception Well, sorry, did I confuse you? There's the mysterious hunter bloodhound who quietly stalks their prey We talk of victory and slaughter And the tormented rape who has learned to trust the voices in her head a side-effect of phasing between infinite dimensions looking for me But even the most experienced legends must rely on their squad to achieve victory Who might best compliment Bangalore a weapons expert and offensive powerhouse not your day shooter Is it the combat medic lifeline whose healing abilities and emergency supplies can tip the scales in her favor? Or is she better suited with pathfinder the scouting specialist armed with a grappling hook and zip line Or should she take her chances with caustic This sociopath relishes the final moments of his victims as they fall prey to his toxic chemicals Everything dies Located deep in the outlands King's Canyon was once a rich thriving territory with natural resources The varied landscape offers a wide range of tactical challenges Legends can easily find themselves fully exposed in the long sight lines of the cascades and shattered forests While the iconic Leviathan skeleton provides an epic backdrop in skull town and Thunderdome Here I go It all starts with the drop Each squad is assigned a jump master When the jump master chooses to drop the whole squad exits the drop ship together If you're looking to gear up quickly dropping into one of the marked hot zones is your best bet to find some higher tier loot But be warned you could find yourself in a fight almost immediately The ultimate test is to secure one of the roving supply ships If you're skilled enough to land on one of these hovering giants some elite gear may be waiting for you The apex arena features a robust arsenal of weapons and gear from agile SMGs And heavy hitting shotguns To deadly light machine guns And energy weapons There's something for everyone Finding the right gear is just as important. A single pickup can be the difference between victory and death And people look out for those larger backpacks. You'd hate to run out of ammo in a fight Strategy and communication are essential to a squad success Context sensitive smart comms allow legends to quickly share information with their squad flatline here Thanks If you find yourself cornered or in an area that's already been picked clean Locate one of the jump towers Squads can zip line up and redeploy into a new zone And always be mindful of the encroaching ring. It can easily spoil your plans Or used wisely can be the key to victory Keeping your entire squad alive is crucial If a squadmate goes down move quickly to revive them and get back in the fight You'll be all right If they do expire all hope is not lost You'll have a short window to recover the fallen teammates banner card If you manage to collect it locate the nearest respawn beacon and call them back into the fight If be warned you'll be heavily exposed and your returning teammate will have no gear when they arrive So make sure you're prepared. Thank you The competition is all about survival, but the true glory comes from a legend style As you climb higher in the ranks a host of customizations and cosmetic loot will become available to you Use these to deck out your apparel and weapons And of course show the whole arena what you're about with your legend banner The competition continues to attract new aspiring legends from all corners of the frontier Who knows where the next champion will come from? And with seasonal events bringing new challenges and opportunities We've only seen the beginning of what's in store for those brave enough to step foot into the arena Shoot this A legend once said it's a simple game They kill you They're better You kill them You're better So there you got to learn much about the characters the map and so forth Now as I said in my previous video this game is smooth as heck It feels as fluid as Titanfall thought which is a great thing The ui is amazing and it's easy to navigate the way armor and weapons work as well It's also great to find in attachments Which truly affect the way a weapon feels armor upgrades. We can sponge more than normal damage Characters having specific super abilities or ultimate abilities and the fact that there's so much to own too It's a great great game. It's kind of a mix between blackout and overwatch It just plays way more fluid than blackout did but also isn't as character demanding as overwatch was Or picking the right characters for your style of play to match your team is recommended And this is why I say it kind of reminds me of overwatch, but the game is absolutely awesome Early days yet though, but it's definitely after a good start now watching that video Which is basically an all you need to know about the game You would have heard her comment on supply ships and if you land on these from spawn You have a great chance of getting that high elite gear, but hey, she makes it sound easy It isn't and rightly so Firstly you have to time your jump perfectly as the ship is always moving and secondly once you land there You have to pull up with other teams thinking the same thing The place gets absolutely hectic and it's only a small area a few times. I've landed up there It's already been occupied by people and they've already stolen the best loot So you just stand no chance But there is another spot which you can use which in my opinion You have a much better chance of getting elite and high level loot And that's landing on a marked blue zone on the map as you spawn in Also known as the hut zone the hut zone consists of a minor blue circle on your map This can be located anywhere under the flight path of the ship across the other side of the map Absolutely anywhere this hut zone in my opinion though is a much better place to go and try your luck in getting that high level loot Now just like the ship sometimes teams will land here But if you're quick enough and get there first you can bag yourself an elite weapon Now it isn't always a guarantee that the best royalty loot was born here It was just a chance, but it will tell you on your screen There is loot in this hut zone But even so you still have a better chance of getting here getting that loot and surviving Then fighting it out on a 50 foot by 50 foot ship with 10 other teams Yes, that sounds fun But odds are if you don't get there first you ain't living very long So yeah, I recommend you try to land in a hut zone and looking for that high level loot because it isn't marked on the map I mean the hut zone is marked, but the actual loot isn't marked on the map So it's in a hidden random place. So even if you land there late, there's still a chance you will find it And the first time we landed in this hut zone my teammate found an elite weapon Elite weapons can fully maxed out with attachments and they are just monstrous and you will see that on screen now So yes people If it's elite loot you want you have two options the ship or the hut zone I mean there's a rare chance they will spawn elsewhere, but it is a very rare indeed I've already made a video showcasing the best spawn spots For easy and little plenty of loot traits and that video you'll find linked within a video description If you do want to check it out I do recommend that spot though If you're a newcomer to the game and you want to get used to the way things work Armour weapons attachments and so forth and like I said that video will be found linked within a video description Basically a spawn in location when there's about 10 or 15 loot traits Which can relate with your team and get fully kitted out on gear Armour weapons and so forth, you know it goes people But yeah guys just a quick video for the newcomers to the game and how you get that elite loot If you are a newcomer to the game and this helps you out leaving a like It really does help me out if you're new around here and you want to see more apex legend videos Let me know down below in that comment section guys, but hopefully you enjoyed the video and hopefully I will see you on that next one


Good morning everyone. Welcome back to another apples and tiara's vlog. I realized that I haven't vlogged with you guys in a while so I thought my student teacher's gone today. I'll just pull up the camera. I'll talk to you guys. We'll just catch up and hang out. How about that? Today is Thursday, February 23rd. And I just got to school. As you probably will notice, Cash is not here. He's with Scott right now. And the whereabouts of both of them is kind of a secret still. So I'm not gonna share it with you yet, but I do have a pretty big announcement coming up in the next couple of videos. So today is just a regular Thursday. It's been really windy and rainy down here in the valley. And for the last few days, we've had some pretty chronic indoor recess. So I'm hoping that today, we don't have a ton of rainfall or water precipitation. Hopefully we don't have a lot of that because I know my kids are antsy. They want to get outside. And I'm a little bit antsy for them. I want them to get outside. So I'm gonna hope that all goes well in the atmosphere today for me. I got these new liquid IV packs. They're the new immunity support drink. So it's immunity support and hydration and they're good. It's the wild berry blend. If you guys didn't know, I'm actually a brand, not ambassador. Well, I guess I'm an ambassador for liquid IV. So I do get a coupon code to share with you guys. And then they just send me a pack of these every couple months. So if you're interested in saving a little bit of money on the liquid IV website, and a lot of people buy liquid IV either from Costco, Amazon, sometimes people pick it up in the small packs that like target a Walmart, but you can get like a whole bag. Let me show you. Like this big bag on the liquid IV website. And typically my coupon codes are between like 30, 25, 25 to 30% off, sometimes 20% off. But you know what? 20% off is still a good deal compared to painful price. So I will put my liquid IV information down in the description right at the top so that you guys can get a little discount on your liquid IV. Okay. So what time is it? It is 7.15. I'm gonna get everything set up in here and then I'll walk you guys through what my fourth grade science students are doing today. My sweet, sweet kiddos brought me a pack of like fidget and anxiety tools for Valentine's Day. I was like, oh honey, you know my heart. He and I both always share about how our ADHD, you know, works and how each one of us is different and how each of us have different like triggers and different things that our brains do. And because both of us have ADHD, it's really nice to be able to talk to him and share how I've managed to overcome my ADHD and how I don't let it hinder me. And so, so sweet of him.  So I've got him kind of like stacked up in the back of my room here so that if I need it or if he needs it, he can ask me and he can come back and borrow one. Okay. Another thing I wanted to share but I first wanna put you guys on my tripod. Okay. So my fourth graders are super duper obsessed with crocs lately. Like they're all wearing crocs. They love the croc charms. They're kind of obsessed. We have our own little like Friday is where crocs and socks stay. So we have been kind of riding this whole croc train for a while. So I saw on this TikTok that I follow, learning is magical or magically learning. I'll have to put the link right here. But she was showing how she bought croc charms for her treasure box. And I absolutely love the idea because my kids are obsessed with crocs. But some kids don't have them. And so she was showing that she actually purchased these like croc charm bracelets. So I actually went and spent a little bit of money to update my treasure box. Cause right now I have some meme stickers in there which I mean they're kind of cool but I also have like this huge bag of like water bottle stickers that I keep in here and the kids when they purchase prize box, they can choose a sticker. Or I have some little mini like erasers in here. Sometimes I'll get special pencils from like the school and I'll put those in there. But the latest installment of the prize box has been these croc charm bracelets and the croc charm which I bought on Amazon. I think I got like 200 or something of them for like $10. So I'm like, okay, yeah, this definitely needs to happen. So what I'm gonna do is I'm gonna empty all my new croc charms into my croc charm baggie here. And then I'm gonna stock the bracelet section because I've got a lot of kids that really want the croc charms but they don't wear crocs. So if they want to purchase treasure box it's 40 dojo points. They can spend 40 points on the bracelet and then another 40 points on the croc charm or they can do one at a time. I always tell the kids, if you have crocs, skip the bracelet but if you don't, hi honey, then get the croc charms. So anyway, that is how I've kind of boosted up my treasure box and I have had to raise the price because my kids have been getting a lot of dojo points. So I'm trying to show them what supply and demand is, what inflation is. I'm just trying to keep it real, you know what I mean? Like keep it like it's going on in the world. So anyway, I am gonna move my treasure box just over here. I'll show you guys how I kind of like rearranged my space a little bit and then that way I can kind of show you my new area. So I was kind of, I had myself parked back here for the longest time like my teacher space was back here and I just found that a lot of students have recently been trying to sit here because it's my desk and they've been kind of sitting back here and for the most part, they function well like they get their work done and stuff like that. So I actually wanted to open that back table up to be another continent. So as you guys know, I organize my table groups by continents. So I have North and South America, Europe and Africa and then I have Asia and Australia and then I have this little desk, which we call Antarctica. But if I could potentially, I'm actually thinking about switching up the layout, pushing all six of these main tables back a little bit, pulling that table up front, calling it Antarctica and it can be a space for my students that need to sit kind of outside of their groups, like maybe they need to just sit alone or for whatever reason they're mad at their teammates and they need to get up and move. I'm thinking if I can pull that to the front, then that can be a space for those students without being in the back where they're messing around. So I think I'm gonna do that this morning is just kind of push everything back. And then I'm still gonna keep all my stuff that's back there, back there, like that stuff that I don't really use all the time. Everything that I use daily, I've kind of moved up here and so far it's been fine. And I'm really not gonna change anything else in this room but I do wanna try out this whole other table sort of thing. So I think what I'm gonna do is get myself set up for the day and then before my students get here, which I have about 30 minutes, I think it'll be fine, I'm gonna just kind of rearrange things which isn't rearranging, I'm just pushing things around. So I'll show you the end result. All right, so this is what we're looking at here. We've got, we're gonna call this Antarctica and then the rest of the tables fit just fine in here. They just go all the way back here instead of that. So I have more room in the front, which is gonna be great because then, you know, I really don't use a lot of the stuff that's on here, I might move a couple things up front like I'll probably move that basket up front but like realistically, I don't really get into here a lot and then this just kind of holds my stuff. So I think I'm gonna just stick with this for a little bit, see if I like it and then this way I have more room up front too. Like I can do demonstrations up here. I can even use this table for demonstrations and then I moved my inbox up front too. It'll be up here. So I'll have to tell kids about that today. Work on today. So I have this pack that I made a while ago. Basically what it is, it is a sorting activity where the kiddos take little cards that have been cut out like this and they are properties of seismic waves, different seismic waves. So the kid's job is to organize the table based on information from this passage and what they've learned about seismic waves. So this is really just a reminder but basically they're going to fill out the table with the different properties of the seismic wave. They'll call me over to verify that they've done it right and then there is a digital sorter that they will do on their Chromebooks by themselves. So they'll get to do it once with a team and then the second time they do it, they're gonna do it completely alone. They have these notes in their notebooks. They have drawings on theirs. They drew different waves and then they actually wrote the words in here and this was all after investigations and labs and things like that. So that is what they're gonna do today and then if they finish any of it early, they will be working on finished or like unfinished work that needs to be done, redone or some sort of enrichment activity on their Chromebook.


Hello, I'm Megan Tomeo, Engineering Librarian at Ohio University. This short video will show you how to find subject specific databases such as Psychinfo, Science Citation Index, or LexisNexis. Start at the library's homepage at ohio.edu slash library. Click on the third green tab, Databases. From here there are a couple of options. If you know the title of the database that you want to search, enter it in. So let's look for Kempendex. Clicking on the title of the database will get you access to that database. And now you can enter in your topic. Starting over from the library's homepage and the green Databases tab, let's look at another option. If you are unsure of the title of a database, you can browse by subject. There are a variety of options for finding a database on this page. The first two allow you to browse or search by title, but since we don't know a title, we're going to use the third and fourth option either looking for a database by keyword or by subject. If I was looking for a statistical database, I could enter in statistics as my keyword. Note, this is not the point where you enter in your topic that you were looking for. From the list of databases that come up, once I find the database that I am looking for, Statista, in this case, I click on the title and now here is the point where I would enter in my topic. Back on the database page, I have another option. I could look at the list here underneath databases by subject and I can see that there is a category for statistical sources. Again, I'm given a list of databases and I could find Statista within this list. Again, I would be able to enter in my topic here. If you need any assistance, please do not hesitate to ask us.


A major Soviet-era dam in the Russian-controlled part of southern Ukraine was breached on June 6, unleashing floodwaters across the war zone in what both Ukraine and Russia said was an intentional attack by the others' forces. Drone footage published on Tuesday showed water surging through the remains of the dam with some damaged structures. Reuters was able to locate the drone video showing the Novokokovka Dam matching building signage, road layout and structures around the dam as seen on file and satellite images. Reuters was not able to confirm when the video was filmed. The video has also been shared by the Ukrainian President Volodymyr Zelensky. The dam, 30 meters tall and 3.2 kilometers long and which holds water equal to the Great Salt Lake in the U.S. state of Utah, was built in 1956 on the Dniepero River as part of the Kokovka Hydroelectric Power Plant.