So, in the last lecture, I had started talking about the positive real lemma. Now, I did not give the complete proof of the positive real lemma. In fact, what I had said was when you have the positive real lemma, it gives an if and only if conditions, one set of conditions is in terms of matrices from the state space representation of the system and the other set of conditions is a frequency domain condition which deals with the fact that the given transfer function is positive real and things like that. But what I showed in the last lecture was that when the set of conditions that is satisfied by the matrices, when that set of conditions are satisfied, then passivity takes place in the sense that I showed, I mean we had already discussed in the last class this thing about a system being passive is equivalent to the existence of a storage function and that the supply minus the rate of change of the storage is equal to dissipation which is a strictly positive function. And in the last lecture, I showed that when the matrix conditions are satisfied, then we do get some other matrix which is positive definite and that actually stands for the dissipation function. So, today what I will do is I will start first with positive real lemma and I will give the complete proof of the positive real lemma and then we will carry forward with the rest of the stuff. So, let me recall the statement of the positive real lemma. So, the positive real lemma, so let G s be a transfer function and let us assume that there is a minimal state realization of this transfer function which is given by x dot equal to A x plus B u, y equal to C x plus D u. Of course, because this is a minimal state representation, what that means is A B is controllable and A C is observable. Now, further we assume that this transfer function is stable, so G s is stable. Then the statement says that G s is positive real if and only if there exist three matrices. The first one P is a symmetric positive semi definite matrix. In fact, most of the times you are talking about positive matrix, a symmetric positive matrix P and two other matrices L and W such that the following three equations are satisfied and the equations are A transpose P plus P A is equal to minus L transpose L. This statement essentially says that using this positive definite matrix P and the state matrix A, you end up with this Lyapunov equation and the resulting thing is something which is negative semi definite because L transpose L is going to be positive semi definite and the minus sign will make it negative semi definite. Then P B, B being this matrix B from the state representation is equal to C transpose minus L transpose W and the last equation is W transpose W is equal to D plus D transpose. Of course, there is also this ambiguity about what we mean by positive real and what I had said in the earlier lecture is that for the time being at least we will consider a function to be positive real if Nyquist plot of that particular transfer function lies in the first and the fourth quadrant. That means the real part of the Nyquist plot is always positive and I had also said in the last lecture that of course, this definition of positive real may not necessarily be the definition that you see in all the books. Often the definition of positive real includes the fact that it is already stable and so on but what those intricacies are I will come to that a bit later and I will explain why I mean of course, there is no agreement as to what exactly the definition of positive real is but I will try and explain so that it is clear to you what are the various notions of positive real that exists and how they all related and within the epsilon neighborhood of one another. Now, in the last class, in the last lecture what I did show is the following that suppose we assume that there is a Gs and we assume this is the minimal state representation and we assume that there are matrices P and L and W such that these equations are satisfied then what it means is that the transfer function results in a system which is passive. Now, the fact that passive is equivalent to positive real is something that I have talked about but is not been completely proved. So, in some sense what I showed yesterday along with what I will show today that means these two conditions that this is equivalent to this should also prove that when you have the either these matrix conditions or the fact that something is positive real satisfied then you have a passive system. So, now let me begin the proof. So, in the beginning let me assume. So, I will prove this way that means I will assume that there exists P, L and W such that these equations are satisfied and I will show that Gs is stable and it is positive real that means the Nyquist plot is in first and the fourth quadrants. Now, if you look at this first equation this is the Lyapunov equation.  Now, it is well known that this is only possible if the matrix A is Hurwitz. So, if the matrix A is Hurwitz then the resulting transfer function is stable. So, the first equation already shows us that Gs is stable. So, all that we have to do further is show that the transfer function that you get is positive real that means it is Nyquist plot lies in the first and the fourth quadrants. So, let us start trying to do that. So, for that let me first write down what Gs is. So, Gs is really C Si minus A inverse P plus D. Let me use the symbol phi of S for Si minus A inverse. Now, what I want to show is the following that G of j omega plus G of minus j omega transpose is greater than equal to 0 for all omega. If I show this I would have shown it for the Nyquist plot but in fact what I am going to do is something more general than that what I am going to show is G of S plus G of S star transpose is greater than equal to 0. This is roughly what I would be showing. Now, how to show this? So, I will write down the expression for this and I will write down the expression for this but I will use phi of S instead of writing Si minus A. So, this expression is C phi of S plus D plus this expression will give me D transpose plus B transpose phi S star transpose phi S star transpose C transpose and now what I will do is I will make use of some equations that we already have in the positive real lemma. So, in the positive real lemma we see that D plus D transpose is W transpose W and we are assuming these things are satisfied. So, for D plus D transpose I will substitute by W transpose W. So, for this one I can write W transpose W and there are these other two terms and what I would do is I will use this particular second equation for C transpose I will substitute P B minus L transpose W. So, if I do that this particular expression becomes B transpose phi S star transpose times P B that is one expression plus I will have one more expression which will have B transpose phi S star transpose L transpose W and now similarly just like what we did for C transpose I can substitute for C it will be the transposes and so these two guys will appear as transposes here, but instead of phi S star transpose I will have phi S for those two terms. So, maybe I should just write them down. So, I will get another term B transpose P transpose phi S B and one more term which is W transpose L phi S. So, I get these five terms. Now, out of these five terms let me concentrate. So, there are these five terms that appeared let me concentrate just on this one and this one. So, let the other three terms be as they are I will just concentrate on these two.  Now, I am going to do some simplification of this. So, what I would do is this particular expression I can write it down in the following way B transpose phi S star transpose and now inside I introduce P times S i minus a and remember that we have already said that this phi of S is the inverse of S i minus a. So, these two are really inverses. So, I am effectively writing this down, but I have written the first three terms down here and then these two actually cancel followed by B and I do the same kind of thing for the other term also. So, sorry probably I should not have done it for this one, but I should have done it for this one I guess phi S. No, it does not matter. So, now for this one I can write plus plus. So, that one is the same as this one plus here I write down P transpose phi S star transpose and then I have S star i minus a transpose which is really the inverse of this. So, these two can cancel then and then I have P. So, this P transpose because P is a symmetric matrix P transpose is the same as P. So, I am just putting P here and then I have phi S B. Now, if you look at both the terms, both the terms have B transpose phi S star transpose in the left side and phi S B in the right side. So, what is inside can just be put in together and so then what you have is B transpose phi S star transpose.  And now we again go back to the positive real lemma. The first equation in the positive real lemma say that A transpose P plus P A is minus L transpose L and so we can substitute that in there. So, if you substitute that in there then this particular expression can be written as this particular term will give me S plus S star. These are just scalar. So, I can pull them out B transpose phi S star transpose P phi S B. So, I have just used up this much and then the other portion. So, P A plus A transpose P from that first equation that should be equal to minus L transpose L. So, I will substitute that minus L transpose L and therefore, I end up with B transpose phi S star transpose L transpose L phi S B. So, these two terms we picked up and we end up with these two terms. This is a term which is sort of symmetric if you like but multiplied by S plus S star and this is also something which is symmetric but what you have is that L transpose L. Now, let us go back to that previous thing where you had these five terms and we picked up these two terms and did manipulations to get what we have there. So, what I will do is into this slide I will cancel these two terms and I will add the two terms that we have got now. The two terms that we have got now are S plus S star times B transpose phi S star transpose P phi S B that is the one term and the other term is B transpose phi S star transpose L transpose L phi S B. Now, let us forget this particular, let me now not think about this one term and let us look at the other four terms and if you look at the other four terms you see this W transpose appearing in two of the terms and W appearing in two of the terms and so you can write this as a sum of squares. So, let me just write that as a sum of squares in a fresh slide. So, as a sum of squares what you get is W transpose plus B transpose phi S star transpose L transpose multiplying W plus L phi S B. So, if you multiply this out you will get W transpose W, W transpose L phi S B, B transpose phi S star L transpose W and then B transpose phi S star L transpose L phi S B. So, you would have got this, this, this and this, these four terms. Now, these four terms and then the one other thing that we have is S plus S star B transpose phi S star transpose P phi S B. So, this is the full expression that you will get and this is the expression mind you when you started out with G S plus G S star transpose this is equal to this whole expression. Now, if you look at this whole expression this is really a square. So, if this is a square this is always going to be positive and if you look here P the assumption was that P is a positive definite matrix. So, whatever is this thing this is something acting on a positive definite matrix and as a result what you have here is something positive. And if you assume that S is such that the real part of S is greater than 0 then S plus S star this is going to be positive. So, this whole thing is going to be positive. So, what we can conclude therefore is this is greater than equal to 0 and so effectively we have shown that this is greater than equal to 0. So, what we have done just now is we have started in the positive real lemma we have started with this assumption and we have gone ahead and shown that this G S is positive real. In fact, what we have shown is whatever I have been using as the definition of positive real it is something slightly more than that. So, what we have really shown is if you look at the slide it is clear what it is that we have really shown we have shown that G S plus G S star transpose is greater than equal to 0 for all S such that the real part of S is greater than 0. And in fact, this should ideally be taken as the definition of positive real. The definition of positive real in fact rather than use S greater than equal to rather than use S in the imaginary axis which means you are looking at whether the Nyquist plot is in the first or the fourth quadrant rather than use that it is this definition that means what we are really saying is that the whole of the right half plane should map under this map G S plus G S star transpose when you are taking this map from the complex plane to the complex plane whole of the right half plane should map to something which is in the first and the fourth quadrant and in that case you can call it positive real. And so if you use this definition of positive reality then in the positive real lemma in the original statement you do not have to insist that G S is stable because it has to be stable if this condition has to be satisfied. But these are intricacies and there is no general agreement about what is the exact definition. So we will leave it at that. So what we have now effectively shown is one way of this argument that means assuming that these equations are satisfied we have shown that this holds. Now we want to show the other way that means if you assume G S is stable then G S is positive real you want to show that these equations are satisfied. Now in order to do this I will have to invoke some other generic theorems that are known. One of them is the spectral factorization theorem. Now as we go along I would talk about the spectral factorization theorem. The spectral factorization theorem is a theorem which plays a central role in other fields also not just in control theory. So that is something that we require and I would also invoke a lot of things that we know about realization theory that means given a transfer function how do you realize state space representation. So let me start. So we are assuming G S is stable and G S plus G S star transpose is greater than equal to 0 for all S such that the real part of S is greater than 0. So this is our assumption. Let us look at this particular thing. Now what this means is if you specifically evaluate this particular matrix on the imaginary axis then what this tells us is G j omega plus G transpose minus j omega is greater than equal to 0 for all omega. Now what this means is that evaluated along the imaginary axis this particular quantity on the left hand side is greater than equal to 0 and of course even though I have only been talking about transfer functions these things also hold for matrices. So even though I stated this positive real lemma but before that I was only talking about transfer functions which are single input single output we can carry out I mean the positive real lemma as it stands is valid even for G S's which are not single input single output the only constraint is that this G S must be multiple input multiple output so they are square matrices the number of inputs equal to the number of outputs and G S is positive real well there is the definition of positive reality for the matrices which I have not given but I will give as soon as we finish the complete proof of positive real. So whatever I have been showing holds even for the matrix case it is not just for the Scala's case even though earlier I have only talked about the Scala situation the single input single output situation but you could also look at the multi input multi output situation and exactly the same kind of proofs go through. So the exact definition of what positive real is for multi input multi output matrices transfer functions which are matrices I have not yet given but the definition is such that it simplifies to exactly the definition that we had I mean the kind of definition that we had about the Nyquist plot that kind of definition or about the right half plane mapping to the first and the fourth quadrant well the same kind of thing does appear but that I would give just after finishing the proof of this positive real lemma. So let us finish the proof of the positive real lemma so coming back here so this is what it means and at this point I would invoke the spectral factorization theorem. So what does the spectral factorization theorem say well the spectral factorization theorem says the following that suppose you have U s which is let us assume this U s is a p cross p matrix which is positive real what we mean by a positive real matrix is something that I have not yet defined but I will define just after the complete proof of the positive real lemma but just assume I mean you could listen to this portion by just thinking of U s as a scalar single input single output case and that would be good enough.   So we are talking about spectral factorization.  So let me explain what is going on here. So we are talking about U of s which is a p cross p positive real matrix positive real well I will give you the definition shortly. But it is a positive real U so I mean instead of this p cross p one could just think of 1 cross 1 and so this is just a transfer function and it is positive real Hurwitz then there exists a r cross p well this r cross p business and the p cross p if this only appears in multi input multi output in single input single output of course this r will be p is 1 and r will also therefore be 1 1 cross 1 there exists a matrix which is Hurwitz proper rational call it V of s such that this matrix U of s plus U transpose of minus s the sum of these two matrices is equal to the product of V s with V transpose of minus s and the r comes in the matrix case essentially because when you add these two matrices they are p cross p square matrices it has a certain rank and that rank is this r and therefore this matrix V is not really a square matrix but it is so V is more like this and so V transpose is like that and so the and the product of these two matrices is a square matrix so this has p columns and r rows that is what V s is. Now if you see earlier we had said G s is stable and therefore this thing holds but you see G of j omega plus G transpose of minus j omega so if you look at the imaginary axis you are saying that in the imaginary axis it is positive definite if you think of this as matrices it is positive definite matrix. So what would happen is when you look at this sum this sum will have roots which are on the left half plane and roots which are on the right half plane and then this V transpose minus s times V s is constructed by using all in this sum using all the roots which are in the left half plane you use that to somehow construct this V s and then the V transpose minus s comes automatically because of some symmetry which exists in this sum. Now the spectral factorization is of course a very well known result and used widely for example in communication theory but we will invoke this. So because we have started out with this G s which is stable and has this property therefore now we can use this and invoke the spectral factorization theorem and by invoking the spectral factorization theorem we can say that G of s plus G transpose of minus s is equal to let me just continue calling this V V transpose minus s into V of s and now what I am going to do is I am going to start off by looking at state representations of each of these matrices. Now if you look at G of s we have already seen what the state representation of this G of s is. So it was x dot equal to A x plus B u y equal to C x plus D u. Now if this is the state representation for G of s then from here we can get the state representation of G transpose of minus s and the state representation for G transpose of minus s would be let me call the states here x 1. So x 1 dot is going to be minus A transpose x 1 plus C transpose u and y is going to be minus B transpose x 1 plus D transpose u. Now here we have G s plus G transpose minus s so it is as if these two transfer functions they are adding up which is like so they are two systems which are parallel I mean you could think of them as this is G of s this is G transpose of minus s out here you give u out here you get y. So if you now decide to look at the state representation of the full transfer function that you have on the left hand side then that is given by x x 1 dot equal to A 0 0 minus A transpose x 1 sorry x x 1 plus what you have here would be B C transpose u and y will be given by C minus B transpose x x 1 plus D plus D transpose. So what we have got here is the state representation this is the minimal state because we have said that this is the minimal state representation for G s therefore this is a minimal state representation for G transpose minus s and therefore this is a minimal state representation for what is on the left hand side. So let me just call this left hand side. So what we have is a minimal state representation for the left hand side. Now in the same way let us consider a minimal state representation for the right hand side. So, in order to construct a state representation for the right hand side, let us assume a state representation for V of s. So, let the state representation of V of s be given in the following way. So, z dot is equal to let us say f z plus g w and so w is the input for this V of s and then let me call it w 1 and then w 2 is equal to h z plus j w 1. So, if this is the state representation for V of s, then V transpose minus s has a state representation, where the matrices involved will be minus f transpose here and here you will have h transpose, here you will have minus g transpose and here you will have j transpose. But on the left hand side, we were looking at V transpose minus s times V s. So, this is like saying that w 1 is an input to V of s that way and the output of that is w 2, but this w 2 is the input to V transpose minus s and the output to that let me call it w 3. So, it is a series connection if you think about it and so I will appropriately use w 2 as the input for this for the representation of V transpose minus s. So, the input I will use as w 2, let me call the state that one I had called z, so let me call them z 1.  So, w 3 is equal to minus g transpose z 1 plus j transpose w 2. So, now what was there on the right hand side, the net thing it is a series thing. So, I will have to put both these state representations together and I will end up getting a state representation for this thing and so the state representation I will get for this is going to be z z 1 dot is equal to, so f 0 z z 1 plus g w 1 and for z 1 dot I will get minus f transpose z 1 and out here I will get a term because h transpose w 2, but w 2 is h z plus z. So, I would get here h transpose h and here I would get h transpose j and the output equation well w 3 is equal to I would have minus g transpose. So, this I have taken care of and then I have j transpose w 2, so putting that in there I will have j transpose h times z and plus j transpose j times w 1. So, this now is the minimal state representation for the right hand side. So, if you recall there was a minimal state representation for the left hand side of this equation that we obtained and we have also obtained minimal state representation of the right hand side of that equation. And so this left hand side and the right hand side ideally if they are equal that means these two minimal state representations are a similarity transform away. So, now what we would do is we would use some smart way of manipulating these matrices in such a way that we can show some relation between this and that.   So, please remember that this matrix f is a Hurwitz matrix. So, first what we are going to do is we are going to do a transformation on this second state representation that we got and the transformation that we would do on this thing is using a matrix which will help us make the system matrix here. We will convert the system matrix here into a diagonal matrix that means we will try and get rid of this h transpose h. Now, the way we get rid of this h transpose h is the following. So, we will use for the transformation a matrix T which is of the form i k 0 i and this k is not any old k, but this k is a k that would satisfy the Lyapunov equation k f plus f transpose k equal to minus h transpose h. Now, let me just revisit. This is the equation, the minimal state equation that we got for the right hand side. We want to make this matrix diagonal. In order to make this matrix diagonal also remember that this f is a Hurwitz matrix. Now, if f is a Hurwitz matrix this equation for any positive I mean f is a Hurwitz matrix if you write down this Lyapunov equation what you have on the right hand side well h transpose h this is always going to be positive semi definite. So, with a negative sign this is going to be negative semi definite and therefore, you will always get a solution k for this. So, this k is the k that you use in the similarity transformation matrix. So, if you use this k in the similarity transformation matrix. So, let us just look at what we would get we have h transpose h minus f transpose and we are going to use this. So, i 0 k i and of course, the inverse of that matrix is i minus k 0 i.  So, when you multiply this when you multiply this you get f 0 and then you have this thing multiplying this thing.   So, we have effectively managed to diagonalize that matrix by using a similarity transform t. So, now if you use this t and you diagonalize then the kind of matrices that you would get what I will do is rather than do the calculations I just write down what are the matrices that you would get. See these were the original matrices that you had and if you do this transformation using i k 0 i then the system matrices that you would get are.  So, the original matrices this is the system matrix that you had and under transformation this goes to f 0 0 minus f transpose. Then you had the input matrix as G H transpose J and under this similarity transform this will go to this will continue to be G and here you will have k G plus H transpose J then you have you have the other matrix being J transpose the C matrix or the observable matrix to be this and that will get transformed to J transpose H plus G transpose k and here you will have minus G transpose will remain as it is. And then the last one which was J transpose J the feed through matrix that will remain J transpose J. Now, this is the new set of matrices that you have and what we will now show is that you see the right hand side at these matrices and one can show that there is a similarity transform that you can do on these matrices such that you get these four matrices. So, this is what we are saying this left hand side matrix this is the right hand side matrix there is a similarity transform which will take this matrix to here this matrix to here this matrix to here and this matrix to here. I am not going to go into the details of how to construct the similarity transform, but I would just say that this similarity transform that you use you can show that it is a block diagonal matrix and so on and now it should be clear that once you do this transform whatever this gets transformed by. So, suppose you have that similarity transform acting on this to make it this then whatever this gets transformed by this will turn out to be equal to D plus D transpose. So, this J transpose J is really let me pull out the positive real lemma this last equation says D plus D transpose is equal to W transpose W this is really saying that this matrix is the same as this matrix. Now, when you do the other transforms then from equating whatever is here to whatever gets transformed here you will get this equation and the other equation about L transpose L that you will get that you will get from the similarity transform that takes a to f. So, as a result what you have shown from this construction is by this methodology and showing the similarity transform between these two you have shown that these equations are going to be satisfied and so that was the converse proof of the positive real lemma. So, it looks like I am out of time and so let me stop this lecture but what I would do in the next lecture is I would start off by giving the definition of positive real for matrices for the matrix case. We prove the positive real lemma assuming I mean initially I had only given the understanding of what positive realness is for scalar single input single output system, but of course it can be extended to matrix case. The proof of positive real lemma because the proof was dealing more with state space it really did not matter whether we were looking at the single input single output or the multi input multi output case. But what exactly the definition of positive realness is that I would talk about in the next lecture.


we're all figuring out who we are.  And that first came to study autism, because one of my colleagues mentioned the term copy and paste behavior. And I just instantly knew what that was. I had never heard that before. But but I was like, Okay, so you're telling me that other people don't constantly watch what other people are doing and like try to mirror what they're doing in the social. And then she was like, No, that's not a normal thing. So


Hello, this is Laptop Media, and today, we will show you how to open the ASUS TUF-F15 FX516 and what's inside of it. To get inside of this laptop, you have to undo 15 Philips head screws. Keep in mind that one of them is captive and will remain attached to the bottom panel, thus lifting it ever so slightly, so you can start the prying process from there. Please remember, it would be of great help to us if you just hit the like button and subscribe to our channel. That would motivate us to make even more and better videos for you. After you remove the panel, you'll see the cooling solution. It comprises 5 heat pipes, 2 common for the processor and the graphics card, and one more for each of them. Then, there is the last one, which cools the VRMs and the graphics memory. A huge improvement over last year's TUF-A15 is the battery. Our unit came equipped with a 76 Watt hours package. As you can see, there is only one Sodom slot for memory upgrades. Depending on the configuration, you get either 8 GB or 16 GB of soldered RAM, which can be expanded up to 32 GB via the aforementioned slot. Storage wise, there are 2 M.2 PCI-EX4 slots. If you'd like to see more disassembly videos of the latest notebooks we would highly appreciate if you hit the like button and subscribe to the channel.


A group under the auspices of women's situation in room Nigeria has called on Nigerians, political actors to disease from perpetrating violence against women during the electoral process. The group stated these are the one-day consultative dialogue with strategic stakeholders or the aim of raising their voices against women or violence against women, harassment and intimidation. Blast of these correspondents have been under the revised report as presented from our studios. The continuous violence against women has relegated them to the background of electoral activities. This dialogue brings key stakeholders to seek solutions to violence against women in order for their voices to be heard. The convener of women's situation in room Nigeria, Kaduna State, Kautume Abdulaziz, says 60% of women experience political and emotional violence. One of the key outcomes of this event is that women came out equivocally to talk about meeting times because sometimes some of these meetings are at night and as the Nigerian culture has it, it's very difficult for women to go out at such times. So this is one of the factors that hinders women from participating meaningfully and constructively in political competition, calling on all to support women who are capable to take on leadership roles, to support them wholeheartedly, sincerely and intentionally. The keynote speaker Bukalajaw says any form of violence is an offence as contained in the Kaduna State V.A.P. law. To create an awareness to the participants and they are really being enriched to know of all these offenses that are captured in the law. See something, say something. If you are a neighbour and your neighbour is going through violence, do not say it is not my business. It is everybody's business. They also maintain that the affirmative action on women is just on paper, stressing the need for a level playing ground for all. The inclusiveness of bringing women on board and all that but there is just something like they are saying this and another thing is happening. Even the issue of saying that they've given women free forms of 50%, I'm not down with it because where you don't contribute, you don't have a say. So it's indirectly controlling our status again in that kind of cycle. Committee of this party sometimes call for meeting 2 a.m. As a married woman, you don't expect her to be in the cycle of this man. So they should try and understand that. Let's have more women and if there is going to be meetings should be comfortable. We just want a free playing ground, not just we're not trying to say, okay, it has to be this, no. But if there should be a meeting, for goodness sake, 2 a.m., it doesn't make sense. Some of the participants commend the engagement which has broadened their knowledge. As long as women are not empowered, education first.  But because some people are not empowered, so everything will be thrown at them. Shoes that are actually stopping women from actually participating in politics and how it can be avoided and then how violence against women can actually be avoided. And for us that attend this, I will advise each and every one of us to carry this message Nigeria is the first country to run this model of a situation rule and has kept improving to ensure that all elections in the country are peaceful and women, youths play a substantive role in sustaining peace in communities, states and national levels. Updates


Alrighty, so we have another leak coming out of Six Flags Fiesta Texas now. It matches up perfectly with the teasers as well, so you have the Joker Wild Card, a giant pendulum ride coming to Six Flags Fiesta Texas. You also have them going to Great Adventure, and I'm sure we may even actually see a third one. Maybe it'll just be two, maybe it'll be three. We'll find out very shortly actually, because the time I'm recording this video I think it's like 2.30 in the morning. Well, so I don't have too much to say about this attraction, because everyone already knows about them. I think it's a great fit for Fiesta Texas. They just got single rail coaster, Raptor, and now they're getting an amazing flat ride rate after, so who could ask for more, an amazing coaster one year and then an amazing flat ride the year after. So a great fit for the park, great addition. I do have a bit of a problem with Six Flags cloning all of their attractions though, and I'll explain why in another video, but it has to do with their whole season pass program and going park to park, and I don't like when parks do that because it kind of discourages you from going park to park. So I would like to see more unique coasters coming and more unique attractions coming, and it does seem like Six Flags is starting to do that with their coasters, which is very promising, and I definitely look forward to that. And you can't really hate it on a company for cloning flat rides too much, because are there really that many choices out there for flat rides? So in the end, it's all fair game. But yeah, super exciting. And yet another leak at two in the morning between 11 p.m. and two in the morning, where you've been having leaks from Six Flags all night. Anyways, hopefully you guys enjoyed this video. Hit that like button, subscribe if you haven't, and share this video for others to see. There will be many more Six Flags reaction videos coming tomorrow and my Carolin's reaction video as well. Have a good one, guys. Bye.


Tezal Barcelona At our Tezal course in Barcelona we endeavor do everything possible to ensure that you will have an enjoyable and rewarding time in the wonderful city of Barcelona. Our Tezal course center here is home to a small and friendly team of staff which offers all course students a warm welcome and a wide variety of social and leisure activities. Our staff can arrange evening excursions to the top as bars and cafes, sailing trips and other watersports activities, skiing, in winter, wine tasting and trips to many beautiful beaches. We arrange a welcome dinner at the start of the Tezal course in Barcelona to help you get to know the other students on your course at an early stage. The Tezal course is intensive but we do everything possible to make sure that you ample free time to make the most of your time in this lovely part of the world. Barcelona is perfectly located between the warm waters of the Mediterranean Sea and the Tibor d'Abo Hills. The idyllic the lyric islands are to the east of the city, the famous resorts of the Costa Brava, Wild Coast and the towering mountains of the Pyrenees are to the north, the wine growing region lies to the west and the lovely Roman city of Tarragona and the beach town of Sitges are to the south. There is something to door somewhere to explore whichever way you turn in this charming city. Barcelona is home to the works of many of the most famous artists and architects of recent times, including the modernist spires of Gotti's Cathedral, Picasso's Cubist art and Dali surrealism. The distinctive Catalan perspective has had an impact that is felt way beyond the region. If you stroll down the most famous street in Barcelona, the Ramblas, which leads right down to the sea, you will witness grand opera houses, beautiful churches, street artists and mimes, cafes and bars galore and of course great shopping at and around the market. The centerpiece of the Gothic Quarter is its fine cathedral. The Picasso Museum, which houses the best collection of Picasso's work in Spain is nearby. It says justifiably also famous for its nightlife. It sometimes appears that every other building is a club, cafes or bar. Nightlife is Spain and in Barcelona in particular tends to start and finish late and dinner is usually served after 9pm. After dinner you can pick from a huge range of bars or cafes to continue your evenings, entertainment and those with plenty of stamina can then move on to the open air nightclubs in the Tibor Debo Hills amongst other options. With all this to offer, Barcelona is a great place to take your tesal course as well as having a great time outside of course hours. Whether you like beaches or mountains, art, or history, cafes or nightclubs, Barcelona does offer something for everyone. When you come to Orlando for your TEFL Tesal course, you will have plenty of opportunities to explore all that the city has to offer. There is a lot to do in your free time in Orlando and you will not get bored. As our center is located in the center of the city, you can easily get to anywhere in a short amount of time. You can also get advice on how to get around and what to see from our center's teacher trainers. We are working hard to provide you the best service possible and make sure that you have the best time during your stay. Orlando is especially famous for its many theme parks. Combined with the year-round excellent weather, millions of people come to Orlando to visit parks such as the Magic Kingdom, the Wizarding World of Harry Potter, Universal Studios Orlando or the Animal Kingdom. Water parks are also very popular, including Disney's Typhoon Lagoon, Wet and Wild Water Park and SeaWorld Orlando. You can use the training course in Portland as a stepping stone towards your international career and teaching English as a foreign language. The city is filled with interesting sites, including the Oregon Museum of Science and Industry, Pittock Mansion, the Pioneer Courthouse Square Oregon Zoo and Washington Park. Providence Park hosts some of the best soccer and baseball games in the region. Oregon has a wide range of natural scenery to offer. You will find splendid beaches, lush forests and tall mountains across the state. Many people visit Oregon to go surfing, whale watching, hiking and biking. Throughout the entire year, Oregon is the perfect vacation destination. Tezal Rome Rome is one of our most popular Tezal course locations. The combination of culture, history and the large number of English teaching jobs throughout Italy make it an ideal choice for a Tezal course. Our Tezal training center is in a perfect location, close to the Vatican in public transport and offers very modern training facilities and IT professional staff to ensure you have a memorable time during your course. It is easy to imagine sitting outdoors on a pleasant evening enjoying a refreshing ice cream while sitting on the world famous Spanish steps or overlooking the Colosseum and just watching as people stroll on by. This wonderful mix of the old and the new of culture and history and the unique delights that Rome offers makes it, unquestionably, one of the most beautiful, fascinating and famous cities in the world. Rome is widely known as the Eternal City and is certainly one of the most important of all European cities in terms of the sheer wealth of history, culture and religion and has been so for thousands of years. It is hard to imagine that any other city could rival the overall depth of history and beauty that can be found here with wonderful architectural structures such as the Colosseum, Treaty Fountain, Spanish Steps, the remains of the Roman Forum the imposing presence of the Vatican and the masterpiece that is Sistine Chapel and that's just beginning. However, Rome isn't only a place to experience the past. There is also a modern lively feeling to the city and it is a wonderful location in which to witness modern daily life and see how it fits uncomfortably against the backdrop of ancient culture and history. The many delightful old squares with their old fashioned cobblestone streets and courtyards are favored meeting places for both locals and tourists and are fantastic places for sitting under the warm Italian skies and people watching. Rome has a great many restaurant terraces offering authentic and sensational Italian cuisine yet at quite reasonable prices. Rome is situated along the Italian west coast and while the city is very large and spread out over an extensive area running down to the coast, the famous historical center is quite small and localized and it is certainly easy enough to visit most of the better known attractions on foot. The city, however, also has an extensive, efficient and modern public transport system for visiting further afield. Due to its close proximity to the beach, the city of Rome is ideally located for people wishing to experience a major historical city and yet at the same time have easy access to the lovely beaches of the Mediterranean. As there is a wide range of opportunities for teaching English in Italy, it is the perfect choice of location for your Tessal course. Tessal Salt Lake City As the capital and the most populous city in the state of Utah, Salt Lake City has much to offer those choosing to take their TEFL Tessal course here. The city is well known as a hub for a range of outdoor pursuits in the region and this is well demonstrated by its selection as the host of the 2002 Winter Olympics. Since the games put Salt Lake City on the world stage it has seen a dramatic increase in the number of visitors seeking to enjoy the superb winter sports facilities, not to mention the stunning natural beauty of the area. The city was first settled by Mormons in the middle of the 19th century and steadily grew with the arrival of converts to the Church of the Latter-day Saints and people looking to make their fortune prospecting for gold. However, it was the coming of the Transcontinental Railroad in 1870 that really caused the population to grow with mass migration of various groups to the city. Since its founding by the early Mormon settlers, Salt Lake City's population has gradually diversified and the city is now home to a variety of people with different religious and ethnic backgrounds, making it an ever popular city in which to live, work and study. Located in the heart of the western half of the country, Salt Lake City serves as a fantastic base from which to explore the wider region and is known as the Crossroads of the West. Surrounded by the wassach and aqua mountain ranges and the Great Salt Lake itself, the city is nestled amongst some of the most outstanding scenery in the United States. The mountains and lake provide visitors with a large number of outdoor activities to pursue throughout the year. However, skiing, hiking and rock climbing are not the only things Salt Lake City has to offer. There are also a number of other attractions including the Lagoon Amusement Park and the Clark Planetarium. Our TEFL Tezal course in Salt Lake City is the perfect choice for anyone with an interest in outdoor activities, whether it is skiing in the winter or hiking in the summer, there is plenty to do all year round. Although you do not need to be a thrill seeker to choose Salt Lake City for your courses this friendly city has something for everyone. The nation of stunning surroundings, well equipped training facilities and professional dedicated teacher trainers mean that no matter what your interests you are bound to have an entertaining and educational experience in the heart of Utah. TEZAL FUKIT TEZAL courses in FUKIT Thailand are fully accredited both internationally and also by the time Ministry of Education. We therefore offer the highest standard of TEZAL training available. The TEZAL certificate you gain after completing the course is not only valid for teaching English in Thailand but also teaching English worldwide. TEZAL, Tropical Island of FUKIT, fortunately pronounced as Pu Ket, is frequently and justifiably referred to as the Pearl of the Andaman Sea and is the largest island in Thailand. FUKIT offers something for everyone and it really is the ideal location for a TEZAL course in Thailand. FUKIT is situated in the crystal clear tropical waters of the Andaman Sea just off the southwest coast of Thailand. Although classed as an island, FUKIT is connected to the mainland by means of a bridge. FUKIT has rightly become famous all over the world as a tropical wonderland and eye water sports paradise. FUKIT offers a wide variety of terrain which includes gorgeous lengths of white sand beaches which are gently lapped by the warm and crystal clear Andaman Sea with forested cliffs and mountains for a backdrop, numerous waterfalls and a great variety of flora and fauna. However, FUKIT isn't just about the scenery and beaches as it hosts an amazing variety of water sports, attractions, nightlife and entertainment. FUKIT probably offers the widest variety of activities that can be found anywhere in Thailand. World famous diving and snorkeling facilities, sea kayaking, windsurfing, horseback riding, bungee jumping, elephant trekking and hiking through rainforests are just a few of the activities that you can take part in. The tropical island situated close to the equator the climate is tropical and as such the island offers warm hot temperatures 356 days of the year. Days are either hot and sunny or hot and rainy and while the temperature doesn't vary greatly during the year the hottest months are usually from November through March. Despite the hot climate, the sea breeze helps keep the island pleasant even during the hottest days. Thai culture is becoming increasingly known around the world and despite FUKIT's popularity as a tourist destination, you will also have the opportunity to witness genuine Thai culture and lifestyle. There are numerous attractive Buddhist temples on the island, outdoor markets can be found in many areas in and around FUKIT town. Thai people are famous for being peaceful and friendly and it is for good reason that Thailand is named the land of smiles. FUKIT is also an ideal location for visiting other provinces and regions of Thailand, as well as for traveling to interesting neighboring countries such as Vietnam, Cambodia, Laos and Thailand. FUKIT has an international airport and an extensive public transport system and so such travel is relatively quick, inexpensive and straightforward. This tropical paradise is guaranteed to please just about everyone and with the incredibly low cost of living, a tazal is a must. There is a huge variety of teaching jobs available throughout Thailand and in the whole of Asia, including some of the best-paid jobs in the EFL profession. FUKIT is one of the most popular locations to take a tazal course and you will be able to see it in Thailand. FUKIT is also available throughout Thailand and in the whole of Asia, including some of the best-paid jobs in the EFL profession. FUKIT is one of the most popular locations to take a tazal course and to find employment teaching English.


This is stock number 9719. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used SUV and truck headquarters. Remember we are the guys with the trucks on 41. In Wisconsin today we are checking out this super clean 2017 Chevy Tahoe LT. This vehicle has the 5.3 liter V8 motor. You can see it has the front sonar in the front for the forward collision warning. Has the projector lamp headlamps, the factory fog lights, and that those headlights covers are very clean. Tungsten metallic is the color. We shoot all of our videos in 1080p so if you have HD capabilities on your computer, tablet, or a smartphone device, turn them on right now cause it's like you're right here looking at the vehicle with me, factory chrome step bars. It has the 22 inch painted and polished aluminum rims. No scuffs or scrapes on these rims. Bridgestone Duler 285 45 R22 tires with right around half the tread left. Really like this Tungsten metallic color. It's a really good looking gray. Didn't see any dents or dings on that hood. We shoot all of our videos in 1080p so if you have the HD capabilities on your computer, tablet, or smartphone device, turn them on right now cause it is like you're right here looking at the vehicle with me. And definitely your best way to check out the vehicle without actually being here. Passener rim, absolutely perfect. No scuffs or scrapes on that. And as you go down this side of the vehicle you can see just how clean that body is, how reflective and mirror like that paint is. We take these HD videos so if you are far away or even if you're close by and just cannot make the trip down you can still see the vehicle, hear the vehicle, and have confidence in what you're looking at before you even get here. Has the roof rails on top. And I didn't see one single dent or ding down this side of the vehicle. Factory step bars on that side and the back tires have just as much tread as the front tires. This back rim, no scuffs or scrapes on that. Very, very nice condition. And that's just what we wanna show you is just how nice this vehicle is so that you can make the trip with the confidence that the vehicle is gonna be everything that we say it is and that there's gonna be no surprises when you get here. You can see it does have the full towing package which includes a receiver hitch and seven pin wiring. The rear gate is in excellent condition. It has a flipper glass as well as a power lift gate so you just press that button right there and that comes up for you. We do have the cover for the receiver hitch that's right here, that's in nice shape as well. You have the power fold down seats and the back seats area or back storage area I should say is in really nice condition. No rips or tears on that carpeting. And to close this door you just press that button right there and that'll come on down for you. As we go down this side of the vehicle just as clean as that passenger side no dents or dings or scuffs or scrapes down this side for full disclosure this back rim is just every bit as nice as the rest. No scuffs or scrapes on that and I didn't see any dents or dings down this side of the vehicle. You can see it has built in directional signals the blind spot monitoring. Inside you get the black leather interior with the LT package. Both of these seats are heated. No rips or tears on those seats. Power driver seat, factory floor mats wrote. This one comes with the factory brake controller power tilt and telescoping steering wheel. You have your power pedals here. Lane departure warning so it'll actually tell you if you're going outside the lines. On the road stability control backup sensors and you got your power windows power locks power mirrors power folding mirrors as well. And you have your memory driver seat down there and of course this one's going to have the Bose sound system. As we hop inside the vehicle here you can see that this one has 28,128 miles the instrument cluster is very nice and clean. Heated leather wrap steering wheel that's in excellent condition. You have your forward collision warning and cruise controls right there. Heated steering wheel button and your Bluetooth and audio controls right there. Comes with the six speed automatic transmission. This one has the Chevy MyLink touch screen radio. You can see it has a projection so you do get the Android auto and Apple CarPlay capabilities. This screen actually goes up for storage in there and there is a little USB plug in there as well if you wanted to charge your phone or something. Has your heated seats right here your dual climate control and this is also where your backup camera shows up. AM, FM, Sirius XM radio capabilities. You have your CD player right there as well. You have the on start in the rear view mirror. You have your home link buttons as well as your power sunroof buttons and you do get that power sunroof right there. Headliner is nice and clean on this vehicle all the way through. And the passenger seat is in really nice shape as well. No rips or tears on that seat. Smells very clean in this vehicle. I don't think it's ever been smoked in and this one does have remote start. We'll take a quick look at the back seats and then we will check out under the hood. This one also has the wireless cell phone charge pad right there. As we check out the back seats you can see it has the quad seating. Both of these seats are heated. You have the latch child safety system. No rips or tears on those seats. And you have your rear air controls as well. You also get a power outlet back here. I believe that's 110, 150 watt and carpeting and floor mats are very clean as well. Third row seats are in excellent condition. No rips or tears back here and the carpeting is in nice shape too. We'll take a quick look under the hood. I'd like to thank you personally for checking out the video today. Hopefully from this HD video you will have been able to tell just how clean this Tahoe is inside and out all the way around. Under the hood we have the 5.3 liter V8 motor. Engine bay is very clean, runs very smooth. This Tahoe has been fully safety and inspected by our service shop has a fresh oil and filter change. All the fluids have been checked and topped off. And this vehicle is 100% ready to go. To see more pictures of this Tahoe or one of our other 400 new and used cars, trucks, SUVs, minivans, wranglers, you name it, we gotta go to our website, www.summitauto.com. Full pictures and descriptions of every single vehicle on our lot. Videos of every single used vehicle that we have all at summitauto.com. If you wanna check out just our used inventory you can go to truckson41.com. That's truckson41.com. Takes you right to our used inventory page. And if you wanna make this super clean Tahoe years today give us a call right now,<PHONE_NUMBER>. Ask for one of our sales associates who are standing by to take your call to make this Tahoe years today. Once again that number is<PHONE_NUMBER>. And if you'd like to check out more HD videos you can go to youtube.com. Remember to like, subscribe and share on this vehicle and all the videos that you see there. In fact in a second you will see a link to subscribe to our YouTube channel on your left. A link to more used inventory videos like this one on your right. If you have not been to our website on the bottom a link to this vehicle on our website click those check us out. And we really look forward to helping with this super clean 2017 Chevy Tahoe LT. Thanks again.


Selendi News, the Senate, will debate what it describes as the invasion of the residents of the Supreme Court's judge, Justice Miry Adili, by yet to be identified security operatives when it resumes plenary on November 9. The chairman, Senate Committee on Judiciary, Senator Oqbe'emi Bamedili, announced this to journalists during an oversight visit to the Chief Justice of Nigeria on Monday in Abuja. Senate Oqbe'emi Bamedili said the Senate cannot pretend to be unaware of the embarrassing situation of the invasion of the residents of Justice Miry Adili on Friday night and it will discuss it as a matter of urgent national importance on resumption of legislative proceedings.