Thank you for joining me for Psychic Medium, Tony Green. I am super excited about today's show. First, yes, I do private sessions. And if you would like a private session, please go to my website, t-o-n-i-g dot info-i-n-f-o. Second, there are imitators out there. People pretending to be me. I know some of them, ha, ha, ha. I know, seriously, there are scammers out there. So be careful, only use the link in the description on blog talk or YouTube to get to my website. And it's kind of, it's not a normal website. It's dot info, so that should be a little bit helpful. And on my website, you can see all the services I offer and as well as the books that I have authored and written and authored, published. And then what is coming up, everything I have coming up. If you are watching me on Rude Rangers TV, on any of the streaming channels on YouTube, on any of the streaming channels that I am on or on WSES. Thank you so much. I'm so grateful for you viewing me. If you're watching me on YouTube live, please say hi in the chat or the comments and let me know that you're joining us here. If you have a question, you can call in<PHONE_NUMBER>. Now I go live for sure every Monday at noon. During the week, I also do, I try to do a second live show. This week on Wednesday, I'll be doing another relationship show with a co-host. My co-host this week will be Dylan. And then next Wednesday, my co-host will be Rudy from Rude Rangers TV. Now I need to tell you, these guys are coming with some really good info for us. They're being extremely honest. So if you are in a relationship, if you're in a situation, if you do not know what's going on or is this going someplace or what can I do? This is the way my partner is behaving. What can I do? Definitely call in on Wednesday. And you can get my psychic ability answer and then Dylan or Rudy's man answer. Is that the right, even the right way for me to say that? I don't know, their man answer. And trust me, Rudy from Rude Rangers, I believe he's married with children and he's had probably more than one relationship. But you know what, these guys have friends too. They're not only pulling from their own personal experience, they watch the way other men behave and they will be honest with you. As Dylan asked me a couple of weeks ago before the first time he was on the show, he said, how honest can I be? And I said, brutally, I said, I don't want you on the show unless you're gonna be brutally honest with people with your answers because dating is like, whoo, it's like being in a war zone sometimes. I mean, the games people play and that goes male, female, that is not a gender specific thing anymore. And you know, I just wanna say this and this is an 18 over show. If you are out there on dating apps or dating or without apps, wherever you are out there doing what you're doing, I find it, and this is not a judgment, please don't come at me saying you don't understand, I do understand, I get it. I do, you know, I get it. I'm a full grown woman, okay? I get things, okay? I understand them and I'm not judging. This is more of a PSA, please cover up, protect up because I cannot tell you the number of people calling me upset because they are getting some STDs and they're the kind that stick with you for life, okay? Like these, there might be things to help you but there's nothing to make them poof, go away. Now you would think in today's day and age, people would be aware that even if you are suiting Charlie Brown up and Charlie Brown is downtown Charlie Brown, even if you are covering that baby up, excuse my covering, covering that baby up, you can still spread STDs, okay? We are full grown adults here, I know some people are not aware of this. They get shocked when I say this, have a talk with your doctor.  There are certain things that can be spread even if you are using condom. So please, for the love of all that is good in this world, protect yourself and again, not a judgment, just a statement, maybe don't be doing things with things that you don't know about, I don't know, and that's not a judgment, that's just a protect you thing, your energy because when you're intimate with a person, you're transmitting energy between the two of you and if they've been intimate with somebody right before, this is, and guys, this is how women know if you cheat. Straight up and women, men, it goes both ways, it's not only one sided. If a man is with a woman and they're in a relationship and that guy steps out and comes back to his woman, we feel that other person's energy, we know. Before you kiss us on the phone, we will feel that other person's energy, yes, we will. If we are in tune with you, we will know.  Just because we didn't talk about it or say something to you so you could lie to us and tell us that didn't happen and we're crazy, doesn't mean we didn't know, we know.  I'm just saying, I'm bringing facts today. I don't even know why I'm bringing all of these facts but I am. Okay, let's go to a happier subject because I'm sure people do not wanna want me to talk about this for the whole show. Anyway, Wednesday's show is all about this stuff. You can call in and ask questions about is this the one? Will we be together? Why is this person behaving in this way? Hey Todd, how are you? I hope you're having a good day. Thank you for joining me here. Okay, I'm gonna take my first caller. If you do wanna call in, the number is 845-277-914. I will be answering questions. If I call your area code, please, if you have me on speakerphone or earbuds, take me off, put the phone to your ear because the connection might not be clear. And remember, this is going out on TV stations and I wanna have the best audio connection possible. Also have one exact question or connection to the other side that you would like to connect with. And an exact connection to the other side is like Judy, my mom, Bill, my uncle. Not anyone on the other side that wants to talk with me. Okay, here we go, 310. You are the first person in the call log. How can I help you today? What's up? What's shaking? New York. Hey, Ben. So I have a two-part job question. The first part is last Tuesday review on Zoom. With the competitor that I'm working for right now, and I'm wondering if you see me getting that job to work. My boss called me into the office and said that I'm locking and tagging it out. And I told him that's not true. I told him what happened. I told him my site that I was using a long, room with a long handle and not once did any part of my body become in contact with any of the moving parts of the machine. So they suspended me pending investigation. They should have taken one to three weeks. So I'm wondering if you see me getting cleared. Okay, Ben. And getting my job back or... Here's what I'm gonna give you. The interview, I'm feeling like that's not as strong as it could be, but I am also getting that you need to keep looking. Definitely keep looking, keep you are gonna find something and you're gonna find something within a month. I'm going to tell you, whatever happened with that machine thing, here's the thing. Just keep looking. I feel like within a week, you're gonna get an interview for a very good company and within a month, you will, if you want, have employment someplace else. Having said that, I'm gonna do the first clearing of the show that we are all moving forward and I get a no. So we're gonna start that clearing. We are stuck, which is pretty much the same thing. I get a yes. So we're gonna clear that. And then for those of us looking for work or wanting to switch jobs that we are finding the perfect employment for us, and I'm getting a no. So we're gonna start clearing that. Who, because energetically, you are kind of the word they're using is stifled when it comes to work. And I want you to be in a place where you're open and you know there's a lot of opportunities coming in and moving forward. I do feel like you definitely need to keep looking. I feel like there is something coming in. It might not be the thing that you went for, but you definitely will find something very soon. Thank you so much for calling in. I'm gonna go directly to the next caller and the next caller is 386. Hey, 386, this is Tony. How can I help you? Call, I called Thursday like you told me to for the relationship reading and I didn't get in in time and the show ended. So I'm calling back again. So is it okay if I touch on that subject? Absolutely, bring it. How can I help you? Okay, before about this relationship and you told me some very good advice. Well, he went totally AWOL starting fights all day last Monday and I broke it off with him Monday night. And as I was exiting, he was gonna go home and tell me it's not over, we're not done fighting. And I'm like, I'm going home. I just exited and I out and no, I won't be coming back and I won't be seeing you again. But you're wonderful people and it's been a pleasure knowing you. And, you know, his mom started crying and grabbed my arm and dad started this way. He insults me, he devalues me, he criticizes me, he yells and screams at me constantly and I'm done. He came out maniac.  His parents were screaming at him to stop. They're 90 years old. And they just kept crying and screaming. Anyway, long story to go. Okay, right here, right here. I just, you know what? Here's what I'm gonna say. First and foremost, you know, there's a lot I wanna say about this and on your part, Mark's part, on everything.  You just exit. You don't, if somebody has a hair a while, no, you exit way before it gets to this point. At the first sign somebody has is having a bad day and they wanna make it your bad day go. Now, I'm not saying this is, Mark, I'm gonna tell you a few things that we have learned with time. Sometimes when someone is trying to start a fight with us, it's for one of two reasons. They did something and they're trying to take it out on us, they're trying, they feel guilty and they're trying to take it out on us, or B, they want to run and go do something and have a little break up or break time so they can run and do something and think in their little P immature brain that, oh, we were on a break, so it was okay. But if we turn around and do that to them, that's not happening. But that's actually what I wanted to talk to you about because I think, I'm wondering if that's why you did it because when he started screaming and he had there, I felt his parents when they were broke up. Wait, wait, wait, wait, wait.  I know your story is very important and you wanna tell your story, but here's what happens. We tell the story and we diffuse our emotions and the more we tell our story, the more we diffuse those emotions and the more normal that story becomes in our head. It just becomes a story instead of the trauma that this was, the trauma to his parents, the trauma that he literally put hands on you. He put hands on you. Girl, that should have been the headline. That's what I wanted to say to you. No, stop, that should, no, stop.  No, wait, I don't even, I'm gonna tell you something. I don't even need the story about what you said. He said the moment he did that, you should have been dialing 911. I don't care how long you know him. I don't care his mama, daddy were there. I don't give a happy hoo-hoo. He put hands on you. The whole rest of the story might be, okay, he's having a bad day, but this guy's been an issue for more than a minute. Okay, he's been an issue for more, way more than a minute. So you get to a point now where his words are not working. He clearly has anger issues. He clearly tries to incite drama and fighting with you. And I'm going to say this is not only your story, this is a story for a lot of people listening and it goes both ways. There are a lot of drama mamas out there as well as drama daddies, okay? This isn't like only men start drama or only women. This goes both ways.  Dude put hands on you. Just because he didn't swing on you, are you going to wait for that next? I mean, come on, that's redunculus. That's that, you would never tolerate that for somebody you love. And yet you're telling me how all day he was trying to start an argument and how his parents were so traumatized and you're not even aware of all the trauma. His parents just suffered watching that whole mess when you should have just walked out early in the morning. Now I'm not going to blame you for a man putting hands on you. I am going to say there is something with you that is blocking that out. There is something with you that puts you in a bit of denial about that because that was not your lead. That was not the headline of this. That to me is almost frightening that you couldn't see that as the headline to the question that you didn't, you didn't even get to your question yet.  And I don't even know if there's a question here because, and let me tell you sweet girl, if the question is, should I go back?  No, you shouldn't have been there in the first place. You should have been long gone, overdone with poof, magic, you are disappeared out my life, okay? There are things we can question in a relationship. There are things we can kind of say, should this, shouldn't I? This guy disappeared for a week. I'm gonna find out what that's about. Somebody put hands on you, you gotta go. You gotta go. And since the two of you don't even live together yet, and I'm not minimizing your feelings because you do not live together, okay? I am not minimizing your feelings because you do not live together. However, the fact that you do not live together makes it easier for you to hatch at this situation, just like, you know, again, poof, you gone. I did some magic and you disappeared now. I said poof, I created a little imaginary story and now we're done.  If you go through something like this with a partner, I don't care if a woman lays hands on you just because it's a woman, it doesn't make it right. If it happens once, it will happen again. Don't talk to sister, brother, mama, daddy, friend, whomever and tell the story because as you tell the story and people start to, and I've had so much of this and I'm gonna tell you right now, as you tell the story, in your mind, you're normalizing it. You're making it more normal and it's not as shocking and traumatic. Second, as you tell the story, people are gonna try like I was doing, trying to tell you how wrong this is and you're more invested in telling the story and then as soon as we say something bad about the person who's harming you, your mind and its human nature is automatically gonna pick up what's good about this person and defend them and that is human nature and it's a fact. It is a fact. You pull those good things because you're not ready for that ending. It's just a fact, it's psychological, it's a fact, okay? We've all done it, maybe not when somebody's laid hands on us but we go back to, but he's really a good person and this was just a one-time thing and I don't know how to get out. We own a business together or we live together or we got kids together. Whatever the reasoning, the justification is, if you're gonna tell this story, do it in a therapist's office, a doctor's office or a police station where action can be taken to resolve this situation. My answer to any question that you have right now and not that your story isn't important but the outcome of what you do next is so much more important than what happened and to me I'm going to tell you, I need you to write down word for word everything that happened from the first argument he tried to start to the laying hands on you and wherever that went and then you decide if you need to press charges against him for putting hands on you and I strongly recommend you do not only for you but for any other woman he might get involved with because you probably are not the first and you will not be the last and I see cap guys I date. First thing I do is I look up their criminal history and if there's anything on there that's, oh, I skip it, I'm not the girl that goes, I can fix that, he beat his last girlfriend, it was probably her fault, she was probably crazy, uh-uh, that's a him problem, that's not a you problem, that's a him problem, okay? He obviously has some serious anger issues that need to be resolved and the second thing is if you do call the police and it does go to court and it should, he might be ordered to get some anger management classes or counseling and take a good look at himself and understand that, yeah, guess what? I have some issues, I need to work on them, I do have anger. Now, I know chances are you might not do any of this, that's not my place to tell you you have to, these are options available to you.  If he has anything of yours at his house, you don't need it, buy it new. You don't need your toothbrush, your sweater, your favorite dish, go buy a new one. Don't let that be a reason to reconnect. Walk on, just walk on. If you cannot protect yourself, you cannot be in this, well, nobody should be in this relationship, okay, this dynamic. But clearly, this is a person who doesn't understand boundaries, they don't understand how to protect you, they don't understand that their need to, their anger is an issue. Now, there's so much more I could say about this and I'm not going to, I'm going to say the answer to any question that you have is walk on, leave it, let it go, don't even think and anybody that you ask about this is going to give you the exact same answer from the police to a therapist to a psychic to any friend unless they are used to being abused and think abuse is okay. Abuse is never okay. Emotional, mental, psychological, physical, financial abuse, it's never okay and it's never right. And there's never, there's never a valid excuse or reason for it.  No, I do, I do. My first relationship when I was very young, he was extremely abusive. I mean, physically abusive and it didn't start him being extremely abusive. It started very slow with the yelling, with the arguing to the point where I don't think I should say that on the air. It's an 18 over show. Nope, but I will say it is one of my near death experiences but there was a lot of abuse. And finally, I created an exit plan and got out. So I know, because I did live with this person. So I do know, I know what I'm talking about here. The first time is just that, the first time. It's not the last time. And sometimes even with therapy, it's not the last time. So sweetie, do not go back for your toothbrush. Do not go back for your sweater. Do not go back for his parents. They've been traumatized enough by this situation. Walk on, do yourself a solid, think enough about yourself to protect yourself. Even if he never laid hands on you again, the fact that he verbally abuses you all the time, walk on and let it be done. That's all you can do at this point. Okay, everybody, thank you so much. I'm sorry that that took up so much of my show but that is a very important subject and that doesn't only go for physical abuse, emotional, psychological. If you're in it, find a way to get out, create an exit strategy and walk on because your life is worth so much more than that. I love y'all so very, very much until Wednesday with the relationship show. Have an amazing week and create miracles if in your life. And I'll see you in the next video. Bye.


Hello and welcome to the liquidity marker video, in this video we'll be going over how to use the liquidity marker. To add the liquidity marked bookmap, simply go to the bookmap knowledge base, the addons tab, liquidity markers, and then just follow the written guide on how to install it. Once you have it installed, now it's time to use it. There are two modes, a max liquidity level and a liquidity changes. First, we'll go over the liquidity changes. Within the changes, you have a threshold, a max liquidity changes for more than I've selected 500. By default, this is set 20. You can also auto-calculate it. I would rather prefer manual calculation after you've spent time looking and reading the charts. As this will give you a much granular control than an auto-calculation, but each their own. What this will do is, for instance this you see a plus 531, for instance if I change this to 531 or 532, this will no longer show up as my threshold to show a marker is set to 532. Since I have it set to 500, it shows it. So any number below 500 in changing the liquidity will not show. As I have purposely said, do not display that. Next is filter markers near BBL. BBL stands for best bid and offer. All this will do is, for instance, I have a tick size of 10. That means each one represents 10 units of price. And if I go into my settings, it says 6 ticks away from the level. That means it will not account for anything from 1, 2, 3, 4, 5, 6. From here on out, it will start displaying changes in liquidity. If I were to change this, this will bring it closer or further away. Can you go on if I went up or down? Next is number of markers displayed. If you want to display less on your charts, you can do so. I chose to leave this uncheck mark as I want to see everything. Because I do find it that it's not too much, it's not too overbearing. If you start having more things in your charts, you might get overbearing and that is in up to the end user to define what is overbearing. Again, check marking this on and simply going up and below will change how much I get to see at a given point. And checking this will allow me to see everything. Lastly is showing increasing and decreasing spikes. I highly recommend having both of these on as it would make sense to show when liquidity is increasing and decreasing. Now we have that and we'll let that load in. Once that loads in, we'll go over the max liquidity level. Under max liquidity level, we can see that the liquidity marker of more than 10 for the threshold. This is just the sensitivity that we can think of as a sensitivity. If you're in a less volatile environment, having this lower will be more helpful. Let's say you're in a very volatile environment and you see rapid changes very often. Then you want to change the threshold to a higher number by default set to 20. So let's go ahead and change this back and increase it. And the last point is the number of levels to space 5. This will just say the highest liquidity on this chart is 91 here, 103 here and so on. It won't show you increases and decreases. If I find it slightly disadvantageous, then the change is in liquidity, which shows you a lot more. Again, you might find this to be too much information and find it hard to read. But all this sort of suggests is in this instance here, there was an addition of 1k bitcoin over 3 points and a minus of 3k bitcoin and those were six separate events. The right marker is events and the left marker is contracts or the amount of bitcoin or whatever you're trading. As you can see here, there was a minus, but then a lower addition. This tells you that someone wanted to remove their bids here and keep it closed here, but at a less increasing rate. As soon as we came into here, someone took their bids off as they didn't want to get failed and price did not want to check this. That covers the basics of liquidity marker. If you have any questions, leave them in the discord or the comment section and just ask anyone in the bookmap team to clarify things. Hopefully you found this informational. Other than that, have a lovely day and I will see you guys in the next video. Peace.


All right, everybody, we're back here. Dave Vellante with Rob Streche. We're in Las Vegas, and I said, John Sferrier's actually in New York City. I said he was in DC before, but anyway, he's at Mongo, got a lot going on at theCUBE.net. This week, next week, next week's data week. We got Databricks, we got Snowflake. Right now we have one of the crown jewels of HPE's portfolio. It was an acquisition company they made in 2015, Aruba. It's been a, more than a home run. It's been a double grand slam. Phil Matrimus here is the executive VP and the GM of the Aruba networking business at Hewlett Packard Enterprise. Thanks for coming to theCUBE. Make it some time. Busy week this week. Yeah, thanks for inviting me. Well, appreciate you hanging in there with us. So what's shaking this week? I mean, wow. What's happening? Yeah, I mean, as you said, we've been kind of in the news a little bit from an Aruba perspective, making inroads into the market. And obviously we're kind of famous for Wi-Fi. I think that's kind of what most people know us for. And we've been in the Wi-Fi business for over 20 years and we've shipped, I think it's now getting close to 28 million Wi-Fi access points around the world. And if you lined all of those up, by the way, you'd get, I think it's 55 billion square feet of coverage from all of those access points. So that was Wi-Fi. So that's what we're famous for. We then moved into switching. So campus, land switching. And in particular, we launched our own platform which is called the CX platform. And that's done phenomenally well. So we think we're probably going to hit about $2 billion worth of sales in October this year. And then the third area that we recently became famous for was that of SD-WAN, where we acquired a company called Silver Peak three or four years ago. And that's been a great acquisition for us. And now we've got 3,600 customers on that platform. So they're the things we're kind of famous for. And then what we've been talking about this week is our expansion into three new areas. So the first area has been data center networks. So if you take what we do in campus and land switching, it's about 80 or 90% of what you would need in order to build out a data center switch. So in the last 12 months or so, we've been heavily building extra features and functionality to give ourselves a good run into the data center market. And we've got some pretty good early successes there. And then the other two areas that we're expanding into, one is an area of security. So networks and security are coming together. You may have heard of the SASE framework. So we had two of the five elements with kind of network firewall and SD-WAN through Silver Peak. And then we bought an Israeli software company called Axis Security about two months ago. And they give us ZTNA, Swig, and Caspis. We get a full SASE offering. And then the final area of expansion is the area of Private 5G, where we bought a company called Athanet, who are based in Northern Italy and they're specialists in Private 5G. So that's what we've been talking about, those famous for those three areas and expanding in the other three. I'm smiling because we were at MWC this year on theCUBE. And we were talking and Adele just announced a partnership with Athanet. And then right after that, you guys bought the company. It's a bit awkward. Well, so we asked them, we said, okay, well, how do you feel about that? Are you still going to work with them? And they said, well, you know what? If they work with us, we'll work with them. So you'll still work with them, right? Yeah, absolutely, absolutely. It's all about the customer, right? Yeah, yeah, that's exactly right, yeah, exactly. Well, so it's been interesting to see, first of all, the success of the, I guess you call it the intelligent edge business. Yes. Aka, Aruba powers that. And then to see how the IP in Aruba has stretched across Green Lake, Antonio has taken an executive out of your organization, Tom Black, put him into storage. So to sort of drive that, I was down at April 4th, they had a big storage announcement. There was a lot of Aruba IP inside of that. So it seems like this is this main spring of innovation coming out of Aruba. Yeah, I think that's right. Yeah, I mean, I think Antonio, he was obviously the big supporter and the brains actually behind the Aruba acquisition. And what he did when he bought Aruba, I'm not sure if you know the story, but he folded HP's networking assets under Aruba. Which was that three-com acquisition? Yeah, it was a valiant try, but it really didn't move the market. Yeah, I know, exactly right. But I mean, it shows what trustee had in the management capability and management team within Aruba to hold the HP networking group in there. And I think Aruba has been a very innovative company. They developed the Aruba central platform, which has now got 2.7 million devices connected to it, 230,000 different users. And I think Antonio could recognize that that was a great platform to build on and stretch across the broader company's range of offers. And that's obviously what spawned the HP GreenLake platform, the HP GreenLake strategy. So yeah, I mean, good for Antonio for kind of spotting that innovation. So you talked about security and networking coming together. Yeah. I don't necessarily think of HPE as having a big security business that's driving a lot of revenue. Can that now change or does it just sort of embedded inside of the networking? Look, I think what we see in the market is SD-WAN decisions, so network decisions, I don't know, three years ago, I think probably 90% of the decision was made by the network department. Now it's definitely more of a joint decision between network and security. And so we're not trying to focus on the entirely broad security market because it's a big market, isn't it? But where we are focused is in secure connectivity. So really targeting those five elements of the Sassi framework. But yeah, we feel good about our opportunity in this space. And what Gartner's say, and we kind of believe them, is that right now, organizations are potentially picking five different suppliers for the five elements of the Sassi framework. And they believe that three years from now, 60% of organizations will want to buy all five elements from one player. And so we think there's a great opportunity for HPE-Ruba to be that player in the market. So that was the kind of rationale behind the acquisition. How do you square that with the obvious, presence of so many tools out there that many could be partners of yours or actually in your customer base? How do you rationalize that? Yeah, I mean, so Silver Peak in particular has done a great job of partnering with a range of different companies and lots of security companies. And that will continue. Because for some customers, they're not necessarily going to want to buy all these elements from one platform provider and they're going to want to mix and match. And where that's the case, we're fine to support that. We're all about open technology, working with a broader ecosystem. So we're comfortable with that as well. Yeah, I mean, it seems like even Tom, when he went over to the storage, is doing that with Altaira.  Exactly right. And we just make sure it's very easy. So our SD-WAN platform is pre-integrated with some of those large security vendors. But hopefully customers decide that they want to buy all elements from one platform provider, such as ourselves. And we should be in a good position if that's the case. I mean, it's too complicated right now. And if you can be a consolidator, why not? Yeah, exactly right. What about as a service in networking? It's been sort of slower to develop. Yeah, I think so. I mean, we led the market. And everyone says, oh, we led the market. But we did lead the market, I think, in this space. So we did some very large NAS, Network-as-a-Service contracts over the last couple of years, more than 30 big contracts. And those have worked well for us and for the customer as well, by the way. So we're happy, the customers happy as well. And then more recently, what we've been doing is trying to broaden our coverage of the market by coming up with more standardized NAS offers. And we've launched 10 standardized offers into the market in the last 12 months. So they're very repeatable. We know what we're delivering. The customer knows what they're getting, very replicable. And that allows us to scale out through partners and hit a broader base of customers. So we're excited about the NAS opportunity. I think I agree with you that maybe the demand is softer than we originally thought. It does appeal for customers who find it difficult to get CAPEX budgets. So they like a predictable OPEC spend. And it also works for customers who are interested in HPE Aruba managing more of their network. And actually, if you speak to most network managers and ask them what you spend your time doing, in 85% of cases, they're spending their time managing the day-to-day faults on the network. So they don't get a chance to think about the future. So anything you can do as part of a managed service to kind of free up more time, there obviously all is for that. So, you know, we do that in conjunction with NAS, but also investing in the platform, which is Aruba Central, and really adding to the AI capabilities in that platform. And it would seem that that's a, to me, having network as a service, you really do need that data center core. And because people are looking not just to do network as a service over here and over here, and then be missing on the core there. Yeah, exactly right, exactly right. And that's where we think we've got a differentiated position because now we have these broad range of offers. We talked about, you know, the Wi-Fi, the campus switching, we talked about SD-WAN, we talked about data sensors, we talked about private 5G and security, but they're all linked together under one platform, which is the Aruba Central platform. And when I talk to customers and ask them about where they're frustrated with our competitors, it's typically because some of our competitors have different products connected to different platforms. So even if they go on an end-to-end journey with one of our competitors, they still finish up with multiple platforms. Yeah, with HP, Aruba, it's all one platform. It would seem that, like also, and that the private 5G and the fact that there is overlap in the Wi-Fi space, there's the transition, it seems like a natural fit. Are you seeing that a lot of people are saying, hey, you know, the 5G works here, but I moved 13 feet to the right and I'm blocked by a cement or something like that. Is that a big piece of where Aruba Central's going to actually play? Is helping them balance that? Yeah, exactly right. So we see some customers with use cases where they believe private 5G is better than Wi-Fi. Right, and some examples would be outdoor use cases, mines, battlefields actually. So defense forces are really interested in booting up a private 5G network for the battlefield, right? And so these sorts of scenarios, private 5G works better because you get more coverage and also it supports higher speed applications as well. So we wanted to make sure that we could serve the needs of all of our customers and that was behind the Athena acquisition. And Athena's a company that was founded by people who used to work in Ericsson. You know, they've got a great team in Northern Italy. They've got real customers around the world. So it's a perfect fit for us. When you think about that unification trend, that single view, and as you fill out your portfolio, what about the networking cloud? I mean, sometimes we call it the super cloud that spans whether it's on-prem or mini-cloud, out to the edge. Is that a vision that is, first of all, is it technically feasible? And is it sort of something that you think you'll pursue? Yeah, yeah, yeah. I mean, so absolutely. And that was all behind the silver peek acquisition. You know, we and Antonio's very clear on this. You know, it's kind of like a multi-cloud world. Isn't it hybrid cloud, private cloud, AWS, GCP, all of these different clouds that exist for customers and they need an intelligent network to link all of that together. And that's exactly what the silver peek platform does every day, we're linking customers with all of these different clouds over one common network. And it's not like somebody goes and buys the super cloud, it's just it's an evolution that occurs over time. Exactly right, exactly right. And they continue to add to it. Yeah, and then so you're connecting the cloud, you're connecting employees, you're connecting suppliers, you're connecting data centers, campus offices, all coming together over one common network. How about, I got to ask you a more tactical question, backlog, right, you guys, and what I see is the problem with your backlog is you're booking so much business. You've got such great bookings and it's like, you can't catch up. I mean, it's a great problem to have, but it's a problem. Yeah, I spent it in the reverse, isn't it? Yeah, no, I mean, obviously it's testament to the fact that we're doing something right to get all of these good bookings. But yeah, you're exactly right. We did, in common with most of our competitors, by the way, we did build up a substantial backlog because of the supply chain constraints that we all had last year, but we're pleased to say that they are clearing and we took some of the steps as well, by the way, to redesign products. So where we had a product that was constrained by a particular component, we would redesign the product to cut that component out. And so that's all helping matters. And then now the supply chain's really starting to hum again, so. It reminds me, when I see tanker with, back in the day, you could tanker with cars, right? And then you'd take them apart and put them back together again, and it would start, but then you'd have these parts and you weren't sure what they did. I presume that's not what you did to your customers, but. It is kind of weird though, in one of the scenarios, there was this part that was really difficult to get a hold of, so we redesigned, and these things take time, by the way. It takes a couple of months and you've got to get re-certified and re-sized. And by the time we get there, I think this kind of takes three or four months. That part that was constrained is then freely available, as it's always the case, isn't it? But you guys have managed it brilliantly, and congratulations on the success of the business unit, and it seems like there's real tailwinds right now for you. Yeah, yeah, yeah, we grow our revenues by north of 50% last year, it's a year-on-year, which is incredible, isn't it? And yeah, so that's a testament to the hard work of the team. And your operating margins are the envy of the industry, really. And obviously, other groups, we said Tom Black, you take that Aruba Mojo and double those storage margins. Well, he's the father of a lot of this as well, by the way. I mean, obviously the CX platform is basically his baby, so every time I see Tom, I give him a pat on the back and thank him for all this hard work. I think all the benefits of all of his hard work. Tom Black did a great job building the platforms for us, so he's a great engineer, as I'm sure you know. And all the engineers love him, by the way. You mentioned his name, and they're all like, oh, he's like a hero. He can get into it, he loves to go deep, and so it's fun. He completely gets into it. Some good more stories there. Well, Phil, thanks for stopping by theCUBE. Really appreciate it. We know it's been super busy week, and thanks for making the time for us. Thanks for having me on. Ready to speak to you both. All right, keep it right there. Rob Stretchy and I will be back to wrap up. In case you missed it, stay tuned. We're going to do the three minute wrap right back on theCUBE at HPE Discover 23 from Las Vegas.


Let us take a quick review of what we have done in the previous lecture and then we will take it forward to model development which are control relevant.  Now I am using this notation here Y belong Rn which means X is a vector which is n dimensional U is a vector which is m dimensional R is a vector Y is a vector measurements are R dimension there are R measurements there are n states there are m inputs and there are D disturbances. So we are looking at a system which is multi variable in general okay then we want to have a local perturbation model this local perturbation model is developed in the neighborhood of some steady state operating point this is denoted here by X bar U bar D bar well if you notice here I am using capital letters to denote all the variables capital letters here indicate that these are absolute values okay we will move on to deviation variables I will start using small letters for deviation variables so these are absolute values and then I want to do multi variable tell the series expansion in the neighborhood of the operating point okay the first question is bar these are dependent variables independent inputs are only disturbances and manipulated variables okay if I specify levels of U bar and D bar using the non-linear differential equation I can solve for the steady state I can solve for f of X bar U bar D bar equal to 0 I do this using Newton Raphson method or Newton's method or some iterative method for solving non-linear algebraic equations steady state is nothing but when you convert this differential equation and view it at steady state you have a set of non-linear algebraic equations which have to be solved simultaneously. So that can be just done using standard techniques for non-linear equation solving and then for given D bar and U bar you can find out X bar this is using a well known method such as Newton's method having found X bar U bar D bar we want to use multi variable tell the series expansion so what is this multi variable tell a series expansion this is my multi variable tell a series expansion I am going to expand the right hand side I am going to expand the right hand side in the neighborhood of X bar U bar and D bar okay now here f is a function vector f is a vector of functions we know what it is partial derivative of with respect to X is going to be a matrix it is going to be a matrix what is the dimension of this matrix f is a function vector there are n functions n differential equations and X has a dimension n it will be n x n this will be n x n mind you when you write this equation you have to be very very careful about the order in which variables appear I cannot write X t minus X bar I cannot change this order of I cannot change order of this matrix and this vector this is a vector this is a matrix okay so matrix times vector it has to be that way you cannot even for even by mistake change the order what is the meaning of this X bar U bar D bar written here it means partial derivatives are yeah evaluated at X bar U bar D bar once you fix this X bar U bar D bar and once you evaluate this partial derivatives at a particular point this is a constant matrix okay same is the case with doh f by doh u doh f by doh u is derivative of function vector with respect to inputs this is a matrix which is n x m and you have a matrix which is n x d which is doh f by doh d so there are two there are three terms appearing here two because of inputs one because of the state itself just look at this equation it tells you that this is not a algebraic equation this is a differential equation a differential equation has a memory of past okay if you try to analyze the terms in this equation it has three terms okay on the right hand side it says that the current derivative current rate of change is function of what has happened in the past where does the past information come from X is the past information okay whatever has happened from time 0 to current time t is getting captured through xt or xt minus X bar in this case perturbation what is happening new what is the new input that is coming in that is U and t okay so it just says that rate of change of X or perturbations in or small perturbations in X is governed by this particular equation equation to I have just written the steady state equation I am going to subtract basically at steady state we know that f of X bar U bar d bar is 0 okay so actually you are subtracting 0 from both sides but I am subtracting equation one from equation two okay and I will get the perturbation model I will get a perturbation which is X minus X bar okay when I subtract this equation from this equation f will vanish from both sides okay on the left hand side I will get X minus X bar okay and right hand side will be again X minus X bar U minus U bar and D minus D U bar this term will vanish when I subtract okay in the same way I am going to linearize the map or the measurement model measurement model is Y are measured variables which are some functions of states okay I have taken the most general case where G is some non-linear function okay and I am developing a perturbation models so small yt is a perturbation from the steady state Y bar which is given by dou G by dou X dou G by dou X will be a vector which is r cross n there are n states I have just given these equations these matrices here for your convenience so these are n cross n n cross m n cross d and r cross n matrices these are evaluated at the steady state operating point so once you evaluate them at a particular operating point you have fixed matrices okay so I am going to it is inconvenient to work with those dou X dou F by dou X and all that I am going to switch to a simplified notation ABTD you open any text book and control they actually start with this model they do not tell you how you got this model they just start with this model X dot is equal to ax plus bu plus hd and so on so but actually these are perturbation variables the small perturbations in neighborhood of the operating point and the connection with the models that come from physics that you know from your engineering background is actually through Taylor series expansion linear approximation local approximation and so on okay so as far as controller design is concerned I am going to use this simplified local linear model why linear differentiations were very well understood even if they are multivariable I can solve them I can manipulate them I can play with them you can solve them analytically okay and that is going to be the strength when you develop linear control theory well what is my aim ultimately what is what is the aim of control get to the set point somebody else can tempt safety okay I will put it in little more abstract term I want to shape the dynamics okay I want to shape the dynamics the way I want sometimes I want to maintain at certain point sometimes I want to move from a point a to point b right where is the trouble here in this equation one trouble is disturbance okay one trouble is disturbance what is it that you have in your hand is you so I want to shape I want to choose you in such a way that output y has a desired transient behavior okay so I want to manipulate and if I want to manipulate I should be able to you know turn around this equation the way I want that is possible very I mean that becomes very easy when you are differential equations it is not that you cannot do it with nonlinear differential equations well for us to go there would take at least two more courses to reach the point where we start looking at nonlinear differential equation manipulations ES we cannot many times capture H so I will be actually dealing with it much more elaborately in next few lectures so right now what I am saying is that see in this particular case assumption is that you have a good model coming from physics availability in that case I can find out H matrix okay well when you do not have a model what you do will answer that question later right now we are considering a scenario where you have a good first principle model you have a good mechanistic model and this is possible for some systems see it is not that I mean some systems in say robotics okay when is the equations that come from mechanics can be written quite precisely okay so it is not that I mean this course the way we are going to go about is not related to any particular domain though it says advance process control you know it is advanced control of course so in some cases you have these models in some cases you can write those you can find out H matrix starting from mechanistic model okay well we have seen this matrix for quadruple tank setup and then actually this is the steady state operating conditions which I have put them on the corner right corner the steady state y bar and u bar is nothing but four levels and two inputs which are given there okay and if I actually linearize substitute values I get these three matrices ABC there is no H there is no disturbance consider right now so there are three matrices that we get and then we said well we can actually convert this into a transfer function matrix I just wanted to map this to a transfer function matrix the model that we have is actually as this model that we have got here is called as a state space model okay it is a linear differential equation it tells you something about what is happening inside through the states okay but all the states may not be measured in this particular case out of four levels only two levels are measured okay so we know we know here this model is more powerful than the input output model tells you something about things that are not measured now the next thing is you know you know is through Laplace transforms if I assume that initiative 0 and if I take Laplace transforms we did this last time we get this transfer function matrix y which relates to u and to d and for this particular case I calculated a transfer function matrix this turns out to be this two cross two matrix two level measured h1 h2 to level lower levels and two voltage inputs you can see the transfer function matrix tells you that level one is affected by both you know v1 and v2 level two is affected by both u1 and v2 okay matrix is full in reality when you go to a complex system you will have many measurements you will have many inputs and typically the matrix will not be you know diagonal it will be full many things affect many things it is very difficult to find a system in which only one input affects only one output very very rare okay and we started looking at this computer oriented models and we said we have to worry about two things one is measurements are sampled okay you have you know a regular interval at which the measurements are coming to a computer and I am going to denote them by yk that is my notation actually yk means measurement perturbation variable y available at time kt kt is k is the sampling instant I start from say 0 1 2 3 4 gap between two samples could be one second could be five seconds could be ten milliseconds depends upon the system which you are considering and then inputs I said are going to be piecewise constant that is because I am sending inputs my from my computer my computer can only generate sequence of numbers my computer cannot generate a continuous signal I need a device called as d2 a converter digital to unlock converter and this to unlock converter will reconstruct a continuous time signal a wall control wall or a stepper motor or whatever is your actuating element will receive a continuous signal it will not receive you know pulses or you will not receive impulses finite impulses it will receive a continuous time signal. So now I need to adjust my model to the reality that I have this differential equation and I want to develop a computer oriented model okay a computer oriented model which in which we convert the differential equation into a difference equation what kind of difference equation I want to hop in time okay I want to go from time instant 1 to 2 2 to 3 3 to 4 okay every time in my computer I am going to get one sample at you know let us say 5 seconds interval I want a model that relates what happened now to the next instant what happened in the next instant to instant after that okay so convert a differential equation into a difference equation what are the constraints or what is the at let us now let us not worry about time 0 let us look at a situation where we are standing at some sampling instant okay at current time okay and I want to I have the I know the state value at current time which is xk I am going to denote it as xk here this is a short term notation for x at time tk where tk is nothing but k times t capital T here is the sampling interval okay capital T here is sampling interval so sampling interval will depend upon how to choose the sampling interval is somewhat complex business and there is a theorem called Shannon sampling theorem which applied you can probably go back and read in the references that I have given I do not want to spend time on Shannon sampling theorem here but what is important I will tell you qualitatively you should sample fast enough so that you do not miss out the major features of your signal okay what is this you know from system to system it is different okay in a furnace which is very very slow it might be sufficient to sample every one minute you do not miss too much if you take furnace temperature measurement every one minute because furnace has a very slow dynamics okay automobile you need samples every you know 50 milliseconds or 100 milliseconds okay we were controlling a fuel cell we needed samples at 10 milliseconds so it depends upon what is it what is so this T could be you know whatever depending upon the system and as a engineer you have to take a call what should be the sampling interval okay so digital control system the signals arrive okay a measurement arrives at instant time kt and input is sent out at to the system at time kt so let us call it instant k sampling instant k so I want to convert a differential equation into difference equation I am going to use a notation here xk is nothing but the state of the system at instant k okay it is a short hand notation yk is the output measurement at instant k uk is the input sent out to the plant at instant k okay now how do I convert well I want you to remember something that you are done in your first year of engineering remember integrating factors how do you integrate this differential so you take a xt on the left hand side then you have an integrating factor e to the power minus at right then you multiply both sides by integrating factor and then you integrate both the sides I am going to do the same thing except I have to do it for a vector differential equation okay so I need something I need something that is equivalent to an integrating factor but now I do not have a scalar in this equation I have just put here a scalar equation a is a scalar b is a scalar okay this is a very simple system it is very easy to integrate this differential equation all of you know how to do it I need to define something equivalent to an integrating factor I need something equivalent to this what is nice thing about e to the power at or e to the power t if you differentiate what will you get back it is a very nice it is a very nice signal continuous time signal if you differentiate e to the power t you will get back e to the power t okay I am going to introduce what is called as max matrix exponential now by analogy you can see how it is defined see the factor here one here is replaced by I identity matrix okay then you have a times t t is a scalar of course well it is convention to write it like this probably when you do scalar and matrix multiplication you should not write scalar after the matrix you should write t a but books sometimes write interchangeably okay now look at this newly introduced matrix exponential well let me tell you something e to the power at is defined as limit of this infinite sequence on the right hand side just like e to the power you know scalar at is defined as this is definition of e to the power at now I need to find out what is the derivative of this particular particular you know matrix exponential just take derivative on the right hand side okay you see what I have done here I can take out a and derivative of e to the power at turns out to be e to the power at minus a first of all do not forget e to the power at is a matrix even though we write it as e to the power at and e is a scalar e to the power at is a matrix the definition is I plus at times a square t square so this is a matrix so when I differentiate this I get e to the power at post multiplied by a matrix you cannot write pre-multiplied it is post multiplied by how am I going to use this okay now I am going to go the same way that what we do for the scalar case I am going to multiply both the sides by the integrating factor okay is everyone with me on this I am p multiplied by in this case you can actually in this case you can in my derivation it is convenient to write a in this particular case you can write a or either way well what I want to stress is when you probably I over stressed it when you deal with these things you have to be very very careful you know order order is not to be missed okay now I am rearranging this equation I have just taken e to the power at on the left hand side okay so the second step is everyone with me on this yeah and now I am going to integrate integrate from where to where I am standing at sampling instant k I am just worried about going from k to k plus 1 okay I just want to model that relates what is happening now with the next sampling instant okay that is all I am worried about so I am going to integrate from time k t to k plus 1 t t is of course the sampling interval okay now here the right hand side integral can be simplified because what we know is within the sampling interval my inputs are constant piecewise constant okay within the sampling interval my inputs are piecewise constant so I am able to take this u k t outside and then I only have to integrate e to the power a times b dt okay this integral I have to compute okay is everyone with me on this d of look at what is here this is d of this when I integrate this I will get I will get you know e to the power at x t I applied two limits I have skipped one in between step is anyone has doubt with this I have just applied two limits I have taken the integral applied two limits and moved one part to the right side okay yeah it is a tau yeah yeah that is a typo yeah please correct it this is e to the power a tau not at in the second term it is important it is quite important what you say is quite important okay now or either a tau or I have to change to dt either of the two okay I think changing to dt will help instead of changing everything to tau okay so is this step here okay I have just integrated the left hand side applied the two limits okay and rearrange right now what I am going to do is I am going to multiply both the sides by e to the power minus a k plus 1 t okay moment I do that see what remains here here it is minus a kt this is plus a k plus 1 t so the difference will be only e to the power at that is what remains okay here my integral changes to this is it okay I am just going slow because you should you know understand the steps is this clear yeah I am going to multiply plus a both sides now a to the power k plus 1 t k plus 1 is a fixed value so I can take it inside the integral without making any it is minus oh this should be oh there is one more typo here yeah you should have yeah in this in this equation I should have e to the power mine you should have minus a t that is better is that okay now yeah okay so far we have reached this point is this now consistent yeah so I need to integrate I need to find out e to the power capital a t I need to integrate this right hand side equation well now how do you do that I am going to do some change of variable I am changing I am defining tau as k plus 1 t minus small t so d tau will become minus d t and then with a little bit of algebra you can show that the integral can be changed to 0 to t e to the power a tau b d tau okay so with this with this particular modification you can change this to okay so nice thing about nice thing about these two matrices is that here and here k has disappeared okay this is only function of b a and sampling interval okay it is only function of b a and sampling interval so now I need to worry about I am going to define two more matrices phi and gamma okay and I am going to use this phi and gamma throughout to denote this linear difference equation model this is my linear difference equation model which is computer oriented computer relevant okay digital control relevant whatever you want to call it output map does not change there is no integration output map is a measurement map is a static map y t is equal to c times x t so y k at instant k is c times x k so that does not change I do not do anything for changing the measurement model I only have to change the differential equation into difference equation okay so this phi gamma matrices are defined as exponential e to the power a t where t is the sampling interval and gamma is this complex integral that we have listed here and we will now talk about how to compute these integrals well just a reminder that piecewise constant inputs it only holds for manipulated variables the disturbances are not piecewise constant what is going out of my computer is piecewise constant okay so developing a model for disturbances which are not really piecewise constant is a tricky business when it comes to computer oriented modeling we are going to look at it this tricky business but let me just pre-empt let me just say that it is not an easy task okay how do I compute lot of algebra I assume that all of you are familiar with this grand equation you can I am going to take a special case first then I will talk about how to do it for general case through another method why I am doing it this way is because many times it is easy to get some insights let us assume that matrix A that we get after linearization of non-linear differential equation let us assume that it is eigenvalues or it is eigenvectors are such that they are linearly independent if they are linearly independent you can diagonalize matrix matrix diagonalization is a very very powerful tool it is used all over in applied mathematics in engineering what I am going to do is I am going to A as matrix psi capital lambda psi inverse capital lambda is a diagonal matrix it has all the eigenvalues appearing on the diagonal okay psi is a matrix in which you keep eigenvectors next to each other this psi is a matrix in which this eigenvector is a column vector okay v1 is the column is a eigenvector for eigenvalue lambda 1 v2 is for lambda 2 there are n eigenvectors I am just keeping them next to each other this equation talks about diagonalization of matrix A okay is this clear it is okay I am going to use this equation to compute matrix exponential it makes it very very easy just a reminder what is scalar case okay this is my e to the power at I want to compute this I am going to just rewrite this is everyone with me on this what is a square see we have this so a square is psi lambda psi inverse into psi lambda psi inverse which is psi lambda square psi inverse okay so likewise I can write a to power n as psi lambda power n psi inverse okay this is a trick which I am going to use so this is the trick which I am going to use to convert I can just pull out psi and psi inverse I get this I plus this into you know the matrix inside the brackets is very nice why it is nice because lambda is a diagonal matrix okay lambda is a diagonal matrix so what is e to the power capital lambda t e to the power capital lambda t is nothing but I plus lambda t plus now look at this matrix if you actually compute this matrix okay look at this diagonal terms diagonal terms will be this is a diagonal matrix this is a diagonal matrix this is a diagonal matrix okay I am just combining that into one big matrix this one but matrix is one plus lambda one t plus lambda one square but this is nothing but e to the power lambda one right and so on so for each one of them that is e to the power lambda t so this is very nice so which means you know computing matrix exponential is not that difficult if you know Eigen values and Eigen vectors you just compute Eigen vectors you compute e to the power lambda t you are there well computing gamma once you know how to compute this computing gamma also becomes very easy okay is everyone clear about this is this fine any difficulties no okay so computing this matrix is inside matrix here the one inside this now once you have Eigen vectors decided this size is a constant matrix then computing these integrals is not at all difficult it is very very easy you just have to integrate each term separately I am talking about an alternate method of doing the same thing how do I get this phi gamma well I have this differential equation and I am going to now go through Laplace transforms okay I show how to do it in time domain let us also look at Laplace transform if you are comfortable with that okay take Laplace transforms on both sides okay the difference is when you take Laplace transform of a differential equation you will get x not okay what is x not in this case no it is not at time 0 it is at time k I am solving differential equation from time k to time k plus 1 so here instead of x0 I am getting xk we assume that x0 or 0 but xk is not equal to 0 I cannot ignore it okay I have to take it into consideration so this equation I have this xk appearing which cannot be ignored of course this brings in information about the past into the system okay how am I going to take a Laplace transform of u which is starting at time kt and going up to time k plus 1t okay I am modelled as a pulse I modelled as a pulse so pulse of duration t is entering the system okay so it is like a forward step and then a delayed step in the negative direction so you are starting from 0 going to uk coming back to uk minus 1 so that is how it is modelled okay so to go back to time domain I should take Laplace inverse and if I take this Laplace inverse I will get ? if I go to time domain from this equation I will get ? e to the power at is nothing but Laplace inverse of si minus a inverse okay si minus a inverse yeah that is because of pulse I am looking at input as a pulse square pulse okay and my gamma would be e to the power t integral 0 to t you have two integrals coming here into picture and then combined will give you this particular integral I will just show it for one particular example yeah yeah x of t you will get x at time you will get x of t and then you find out x at time k plus 1t okay I have skipped in between two three steps because you cannot go to that detail you know you have to fill in the blanks okay you will actually when you take Laplace inverse you will get x of t then you put t equal to k plus 1t and then you will get the next equation okay is that clear yeah okay let us take a simplified model this is a simple problem from Astrou and Wittenmark one of the bench one of the classical books in the control so let us not worry about what is x and what is y this is a DC motor model you know just look at the algebra that we want to learn now okay I want to compute ? so I should compute si minus a inverse now these kind of matrices probably you have not dealt some of you at least may not have dealt with earlier I am going to do algebra on these matrices as if elements are numbers okay. So how do you find out matrix inverse you take adjoint you take you know determinant I have done the same just look here this matrix si minus a inverse okay in fact you will notice lot of similarity with eigenvalue equation and si minus a what is eigenvalue how do you find eigenvalue ? i minus a okay so just keep that in mind they are not going to be turn out that they are same things okay so if I actually do this algebra of si minus a inverse for this particular matrix I will get this matrix here on the right hand side one upon and then I will take a Laplace inverse now how do you take Laplace inverse element by element you take Laplace inverse that is very easy if I do that Laplace inverse of that I will get this equation I choose one particular sampling interval T let us say one second point what can it is a DC motor maybe .001 second then I will get these elements you know right now I am not computed for a specific T but you will get this element this is how I compute this is a how I what about ? for ? what I do is I write a general expression for ? okay e to the power a T and then I integrate for integrate I will get this right hand side here is this clear is an example here this just go over it okay this is how I can compute now when you are going to do it actually the example system use this one of these whichever you are comfortable with but actually when we the main thing in this course is going to be you know using MATLAB programs and MATLAB you do not have to do all these equations writing is a MATLAB command called C2D continuous to discrete you just give matrix ABC matrices it will convert for you and will give you ? ? C you do not have to do any of these complex computations when it comes to a real big problem so yeah not KT K is gone see first of all ? and ? are not functions of K at all they are functions of only AB and sampling interval there see you will get only the difference you will get only the difference between the two time instances you will never get K itself I mean variant systems very very simple okay well when matrix A is invertible you can do this formula for ? okay so if A has linearly independent eigen vectors does not mean it is invertible it may have a zero eigen value it may not be invertible but if it has linearly independent eigen if it has all eigen values which are non zero then you have a simplified formula for computing ? okay quadruple time setup what I did of course is asked MATLAB to do this I just gave ABC matrices and MATLAB just gives me this through C2D I do not have to do bigger and bigger matrices you cannot start writing your own programs you just use MATLAB to convert I get these numbers this is a now I am to go and introduce something which is going to help me throughout the course I have to use some kind of transforms to denote to do some of the algebra that I need to do later on and for that I am going to use something called a shift operator a time shift operator Q is going to denote time shift operator okay now what are we considering here we are considering a discrete time signal okay a discrete time signal xk a discrete time signal xk is a collection of vectors I have put it in curly brackets okay so see when you write sin t where t goes from some you know 0 to some value or 0 to infinity what does it indicate does it indicate one value of the sin it indicates the entire function going for a time goes from 0 to infinity it is the entire time function from 0 to infinity in this case I am considering discrete time signals so I have to look at sequences I have to look at sequences these are sequences of vectors okay k is the index I have sequences of vectors I am going to define a shift operator Q okay Q is a time shift operator when Q is on x I will get one time in future you see this here when Q operates on this signal xk I will get signal xk plus 1 there is one okay so you get the shifted sequence in general when I operate okay so I shifted sequence I get a shifted sequence make this correction though it appears small conceptually means a lot okay so when Q operates Q is a shift operator you know I want to look at signals which are shifted in time now I am going to develop something called as a Q transfer function okay I am going to develop a Q transfer function why I need transfer functions you know progressively will become clear to you why I need transfer functions so xk plus 1 I am going to now skip writing curly braces okay it is understood we are dealing with sequences every time I am not going to write curly braces so I am going to taking a Q transform so xk plus 1 is nothing but Q times xk is this step clear with this definition xk plus 1 is Q times xk Q transform of xk is nothing but xk it is Q to the power 0 okay you are not shifting forward you are not shifting backward is the same sequence okay so this equation this equation I can rearrange as Q i minus 5 okay times xk okay is equal to gamma times uk okay and then y is equal to y is equal to c of xk I have just eliminated xk is Q i minus 5 inverse times uk I have just substituted that here okay I got this I got this equation which is between only input and output okay y is equal to gq into uk okay is this clear any doubts here no okay so this new animal is called as pulse transfer function pulse transfer function matrix we are going to deal with two different notations one is Q transform or Q operator shift operator and Z transforms and well we have to use both the reasons are different remember here I am still in time domain my signals are in time domain okay I have not gone to frequency domain when you do Laplace transform what happens you start working in frequency domain right Laplace transform you start working in frequency domain here this Q operator is only time shift operator I am shift I am just representing into time okay suppose I do not want I do not want to work with x I do not want internal variables state variables I just want to work with input and output okay I just want to look at major levels two voltages which are going to the full tank setup I am not I am not all four levels so I have got rid of you know the states now my model is only between measured outputs and okay now what does this Q how does it help me well in one lecture I am trying to pack too many concepts but well with some practice you will get what is if I do this if I do this calculations see I already have I already have five matrix I already have gamma matrix I have calculated that for quadruple tank setup so I can just do this calculation again MATLAB will help me do this calculations but for simple system two cross two system you can do it by hand it is not so difficult okay do you want to try this for the model that we derived maybe we should do that let us go back to the DC motor model so you have this phi and gamma do you want to try converting let us let us put this as you know phi equal to you know this is alpha 1 minus alpha 0 1 this gamma matrix is of the form 1 minus alpha t minus 1 plus alpha right okay so just try to compute what will be this matrix this is q i minus phi so this will be q 0 0 q minus alpha 0 1 minus alpha 1 this just try it just do it no so this is q minus alpha 0 q minus 1 and minus 1 minus alpha and then you have to find out inverse of this okay so I want to find out q i minus phi inverse how will you find inverse of this yeah so find out cofactor find out determinant 1 upon determinant times cofactor transpose that will give you adjoint transpose or adjoint adjoint transpose adjoint transpose so that will give you the you know and then you know you can you can multiply this by multiply pre-multiply by c and by gamma okay what will transfer function matrix in this case will it be 1 cross 1 2 cross 2 what will it be in dc motor how many outputs are there in dc motor in dc motor there is only one output there is only one input you will get a scalar transfer function q transfer function okay you get a scalar q transfer function what is the relevance why am I so much so I did this calculations for finding out the q transfer function between h 1 h 2 b 1 and v 2 if I have a Laplace transfer function if I have a Laplace transfer function what does it indicate in time domain what does it indicate in time domain let us say let us say well I want to give an analogy because we learn much better much faster with analogy see if I have this Laplace transfer function say y s y us is equal to b 1 s plus b 0 a 2 s square plus a 1 s plus 1 what does this indicate what does this tell you what is the time domain equivalent I multiply both sides I will get a 2 s square plus a 1 s plus 1 into y of s is equal to b 1 s plus 1 into u of s okay now this one is equivalent to a differential equation okay so a 2 s square y will give me d 2 y y d t square so this is what was achieved when you go from time domain to Laplace domain my differential operator becomes an algebraic operator in terms of s okay in some sense well even in time domain people use two notations sometimes they use p instead of s and p is defined as d by dt okay and you can write a p transfer function p transfer function is b 1 p plus b 0 and denominator will be p square a 2 p square and so on okay when you write a p transfer function you will not write y and y s and us we will write y t and u t you are in time domain okay so the advantage that I got when I went from you know time domain to Laplace domain algebraic expression okay I could do manipulation okay in controller design and all that so the same advantage I want to gain in discrete domain that is why I have put this q operator so now this gives me a relationship between inputs and outputs and I am going to convert this analogously to time domain difference equation okay I want to convert the transfer function what happened in the Laplace transform case I had a transfer function between input and output okay I could convert the transfer function into a differential equation they were actually one and the same only they were different representations transfer function Laplace transfer function and the second order differential equation which I wrote okay let us just go back here this transfer function and this differential equation are not different okay is the same thing with two different clothes you know it is just change of drapery it is not that you are looking at two different entities so it is only the convenience of algebra that is why we went to Laplace domain so coming back here I want to be use the same advantage here I want to go back to time domain difference equation okay I have just picked up one of them right now let us assume let us go back here let us go back here h1 is affected by what does this matrix tell you perturbation in level 1 is influenced by both perturbations in voltage 1 and perturbations in voltage through how does it affect through this you know transfer function matrix q transfer function matrix let make a simplifying assumption that perturbation v1 is 0 okay let me make a simplifying assumption only v2 is changing I want to find out how h1 is affected by v2 okay so I am just looking at the sub component of this model h1 affected by v2 v1 is 0 okay now I have just divided here if you look here I converted this from equation in q to equation in q-1 okay just algebra what I do next there is a small error here okay well what is q-1 time shift by 1 what is q-2 time shift by 2 okay so I am going to convert this into a linear difference equation okay this linear differential this linear difference equation here and this q transfer function are equivalent no difference there is different representations the same advantage that I get doing algebra of Laplace transfer function I want to derive here using this q shift operator okay there is no other purpose but this gives me relationship between only the input and output okay yeah basically I want to write current see I have done it in 10 to the power-1 that is because I want to write k is the current instant okay so my convention is going to be k is current k-1 is past k-2 is 2 in past k plus 1 is future okay so when I am writing a difference equation particularly for the behavior at current time instant see I cannot well we will be talking sometime later about future predictions but right now I want to write in terms of what is current in terms of what has happened in the past that is why I have put it like this okay I could have written converted into difference as h k plus 2 and all that but k is my present k plus 1 is future k plus 2 is 2 in future k-1 is 1 in past k-2 is 2 in past and so on so I want to write a difference equation which talks about current equal to something in the past actually this difference equation very nicely tells you about the dynamical systems okay it tells you that current what is happening now okay is effect of what has happened in the past okay see level 1 at time instant k current instant is the linear combination of four terms here on a difference equation it has the memory of two levels in the past h k-1 h1 k-1 h1 k-2 weighted by those two coefficients are just weighting factors okay then it is also influenced by the new inputs what are the new inputs b2k but remember if I if my computer sends out a signal v2k at instant k it will not have an instantaneous effect on y there is always a delay of 1 if I take an action its effect will be seen one sample later okay one sample later our model was x k plus 1 is function of uk okay what you do now will have effect one sample like one sample later so all dynamical systems have memory and this is evident from this equation I get a linear difference equation model here is everyone with me on this is this clear why I went to Q transforms because I could now just talk about relationship between an output and an input okay that is that is the reason now should I do it in this class or should I not okay let us do in the next class frequency domain we are doing controller design we need Nyquist plots I do not think you fondly remember Nyquist plots and Bode plots but you have to use Nyquist plots when you do control my course is mostly going to be in time domain so you will understand a lot of things frequency domain is minimal but there is no way you can escape frequency domain you have in some cases you need frequency domain interpretations which are very nice and there is no other way of going about so we have to touch frequency domain so you have to study this Q transform is only a time domain arrangement I would say a time domain arrangement you talk about shift in signal and then you deal with this using time shift operators difference equation representation here I have to talk about a Z transform okay now Z transform is defined as a Z transform is defined for a signal starting from time 0 to time infinity okay we are dealing with signals which are time 0 to time infinity right see when I start my level control system I will start getting level measurements 1 2 3 4 5 6 and if I do not stop I am going to time infinity right virtually I am going to time infinity so I want to talk about signals which are 0 to infinity is some kind of idealization here in reality you will stop the plant after one day or few days or whatever now here Z is a complex variable I wanted to understand Z transform at a working level not from the complex algebra viewpoint or complex analysis viewpoint take this with a pinch of salt right we are not going to compute those counter integrals in this course but inverse transform once you define transform we have to define inverse transform transform is I will give you analogy for transforms you have some signal say X okay and many times you want to work with some kind of a transform signal so you multiply this by a matrix A which is invertible and you get a new vector Y okay so this is a transform I transform every X to a vector Y by pre-multiplying by invertible matrix A okay and then how do I recover X you know A inverse Y okay transforms are useful they help you you know sometimes simplify the equation okay I will give an example we are looking at this DX by DT is equal to AX we also have BU right now ignore BU just for a second okay actually we have this when we are looking at equations we have B times U let us assume for the time being this is 0 okay now my A is a full matrix okay N cross N matrix my A is N matrix let us assume that this is diagonalizable if it is diagonalizable then I can write ? ? inverse right so I am going to write this equation as DX by DT is equal to ? ? ? inverse X everyone with me on this a little bit of algebra will tell me that ? inverse DX by DT is equal to ? times ? inverse X fine everyone with me on this so this term is nothing but D ? inverse X by DT is equal to ? ? inverse X okay you might wonder why why I am writing like this I am going to define a new variable called Z so this is my Z new vector called Z okay so what is this Z Z is a vector which is obtained by ? inverse X how do you get X back ? Z fine this is my transformation this is my transformation okay now with this new definition what happens is DX by DT which is A times X okay with this Z defined as ? inverse X will give me DZ by DT well this Z here is not Z transform it just a vector which is into ? Z what is ? it is a diagonal matrix there are zeros here there are zeros here what is the advantage of looking at this differential equation through Z over through X here this is nothing but DZI by DT is equal to ? I ZI where ZI is the ith element I get n differential equations which are decoupled which are not related to each other if A is a full matrix which has you know all many non zero elements then solving the original problem is much more difficult than solving this problem and I can solve for Z and I can go back to X using inverse transform right I can go back to X through inverse transform so transforms help transforms really help in solving problems what is Laplace transform what is how it is defined if I have a signal FT integral 0 to e to the power well conceptually this is not different from you know X is a vector when operated by operator ? inverse gives me vector Z okay what is equivalent to matrix here integral it is called integral kernel this integral kernel is equivalent to this multiplying by a matrix okay why do we get integral kernel here we are dealing with functions not finite dimensional vectors we are dealing with a function FT is a function from time 0 to time infinity FT is a function so conceptually these two are similar equations they are not different they are one and the same conceptually okay so yeah no no I want you to understand the philosophy of transforming what I am saying is that why do we transform one is okay algebraically similarity transformation but both are invertible transformations yeah so just draw an analogy one is of course in finite dimensions the other one is infinite dimensions okay so just draw an analogy just to understand what is happening in transforms then see what was what was the point here the point here was when I transform it is easy to work with a transform signal okay it was easy to solve the differential equation and transform domain and then you could go back to the original domain you could do some algebra in transform domain go back to the original domain that was the idea so when I am going to define this transforms the idea is that I should be able to do some things simplified simple algebra which is in the transform domain and then move back to the original domain so this is my Z transform my Z transform is defined as you know fk where fk here is the time domain signal at time instant k and g to the power minus k z here is a complex variable it is related to Laplace variable through this equation z is equal to e to the power ts as I said I am not going to go too deep into this you can probably refer to Astroman Wittenmark or Franklin and Pavel I have given these books in my I will be giving this references as a part of my lecture notes so well just like Laplace transform has some nice properties Z transform also has some nice properties one is linearity if you add two signals and take its Z transform its equivalent to linear combination of individual Z transforms property that is going to be used very often by us is time shift okay Z transform of a time shift signal is nothing but z to the power minus you will find for practical purposes q and z are similar okay as far as to difference equation is concerned they turn out to be quite similar not identical but quite similar so you have Z transform of a signal which is shifted in the past and you have Z transform of a signal which is shifted in future okay so I have given here the expressions the derivations are not so difficult and you can find them in a standard textbook well what do you have in Laplace transform you study initial value theorem same things are here you have final value theorem and initial value theorem and so on what is going to be most important for us is this particular time shift you take Z transform of a time shifted signal you will get this expression you will get z to the power n and you will get z to the power minus n when it is shifted in the past okay just some examples of Z transforms if you take this simple signal you know a step function then I can take alpha out and then take you know summation of one that this is so this in any digital control book you will have Z transform tables and which is similar to what you have in for Laplace transforms for ramp and for sinusoid and then you can go to a book on digital control you will find listing of all these transform the way I am going to use this well I will again revisit this in my next class but I will just preempt what I want to do I want to take a Z transform of my difference equation okay and I want to get a Z transfer function matrix just like S transfer function matrix I want to get a Z transfer function matrix when you have this Q thing why do you want Z because Z is related to e to the power T S where S is complex number and then you can draw frequency response using that interpretation that is not possible with time shift in time shift you are working in time domain okay when you define these operators at some point we are going to use frequency domain interpretation and that is why we need this Z here okay so my Z transform is I am going to take a Z transform of both the sides and this equation will finally yield something which looks very very similar to the Q transform nothing different okay final expressions are going to look not different from Q transform just Q is replaced by Z philosophically they are completely different in one case you are working with time domain in other case you are working with frequency domain expressions turn out to be similar okay so I can find out pulse transfer function for my system by converting my difference equation into pulse transfer function and I will visit this again in my next class in the beginning but I get pulse transfer function all that I have done is same thing except Q is replaced by Z here okay but notice I do not have any more here you know H1 of K and H2 of K that is H1 of Z H2 of Z and V1 of Z V2 of Z and this is my Z domain pulse transfer function this has been obtained using linearization of a first principle model okay so I will stop here and then we will proceed from here in the next class.