Tuesday at Y2 for Fuchannel is where you can find us across all those social media handles at Michelle Lashira is where you can find me across all my social. In this particular section we dive into another interview that looks at innovation in digital business in studio. I'm joined with Fabrice Mugabe, who is the CEO of Mugra EA Limited. Thank you very much for creating time to be with us Fabrice. Thank you for having me. You look dashing by the way. Thank you. Happy Tuesday. Thank you. So I introduce yourself. Who is Fabrice? What do you do? Yeah, my name is Fabrice Mugabe. As you mentioned, I'm a founder of Mugra EA Limited. where b it's a software company that we deal with software development and digital content creations. All right, fantastic. So what is your educational background looking like? Bel länger, yo kwa p doubleshacked jera, такая ni na ubali, ba mteguli yvie wanakinu. Resi wai qiwa즈 n Cafe situ percent hiru para ubuhu h견 propagandu. Kapa pense o GSA? Salpye hamba na ubazeike 없이 high ufandu ni makataba ideas ya kanatwa na zondo. When you say market ideas, it's an idea that anyone could invest in. You may be having an idea, but as long as it doesn't fit in market, then it won't be an innovation. Right, and how important is it for business owners to digitize and have space in the Internet space, that is? Yeah, it is important because we are moving digital. Now we are in 21st century, so you see, everything is moving to digital. I remember I didn't used to have phones. Now we are having smartphones, so that's the first advantage of moving. We will continue to digital platforms. And for any persone who are watching this conversation and they admire and desire to get into business, they are always looking for that idea that will stand out that it will be illaib pointe o ambao. That is just to be known and make in the business world. What is the process looking like when it comes to the formulation of an idea? There are a lot but I would mention 2. The one that is common is looking ahead. By looking ahead we mean look for a working idea. The working idea means the idea that we suit the market. Like there are a lot of people here for example around this area. Now what are we solving to that people? That's one. Two, we can talk of embrace the failure. By embracing I mean if a failure would have been a person or to give him a hug. Why? Because we learn from our mistakes. If you cannot make a mistake technically you cannot run. So when you are moving from analogue to digital and when you want to digitalize your market definitely you have to face those challenges. And how do I identify a gap in the market? By talking to people that's one way because those people that we talk to they already are there in market. Now you are bringing them what you are solving to them. So when you want to catch or to feel that gap definitely you need to talk to them and then you cut across your ego. Because there is a way that you might think I'm thinking this way and other person is thinking that way. The moment you have your thought in mind and that person has another thought in mind you come and then when you talk to them you feel like you can cover this gap. There's nothing that is needed to be done. Now that's how you can embrace that technology. Now you say that if this person is lacking ABC now how can I use technology to help that person? That gap.  It takes nothing except your time of course, patience. Again it needs some later skills. Like technological skills yes when you are innovating on technology you have a business skills yes you have to have a later skills on business marketing and you also need to have marketing skills. Because you want to innovate on something that will help people note you as a person. That's how you should think as an innovator. The solution what are you bringing to the table is to help this amount of people. Alright and how can we use the digital space to just increase revenues when it comes to our businesses? Yeah make sure you are always updated or say this like you update your software you have to be always up to date because today in technology nothing lasts over three months. Is this tomorrow there is that? Our new trends coming up every single day. Every single day that's right. And how can we when you mention something important which is the fact when it comes to this businesses and just having a space in the internet world the digital space. So how can we increase our visibility in this particular space and take us to this particular option for to increase our visibility in the business? I can't say that you can avoid the hackers but there is a way you can avoid for you being hacked. That's one way because if I come to your digital platform and find it like the reviews are not good why would I come to you again? So the first thing you need to do you need to secure your platforms. The moment you have secured your platform of course you have also secured your people because in this era what matters to you a lot is what are we bringing to the people. So I'd say first secure your area. By securing area I mean secure all your platforms or anything that you are using to reach the people so that the reviews will be good. If the reviews are bad and comments definitely you will lose the market. So take us what is such engine optimization and its role when it comes to just putting my business out there promoting what I do. Well, such engine optimization most of the people they know it as SEO but what it normally means is the moment you are looking for something it pops up fast. Sometimes we use it as those ads that pops up. So for example I am looking for a phone. If I am searching a phone I get an iPhone. That name that came first on top of the technical search engine optimization on any such engine that we are having by now we only know Google as a search engine but there are other search engines that maybe let me not mention all of them to provide a few times.  So now what determines such engines rankings? Come again? What determines such engines rankings? Reviews. Because if you type in a particular name or even a particular brand they come up really quick compared to others. So what determines that, the ranks? Yes, the ranks will depend on how many people are visiting your website and how many people are visiting your products and how often do they buy if you are selling something online. For example, if I am selling this glass how often do I sell, how people do talk to me how people react to me. Interact to whatever you are doing. So make sure your reach is high that's what we call reach and hits on the website. So the moment your hits are high technically you are on top whenever someone is searching for you it will, you see, like for example if you search Mogra, it will top up at first because the rates are high and you know my rate to five so the more we rate you the more people rate you the more tops up. The more visible you become. You mentioned something very important when it comes to websites. So who is eligible to own a website and let's look at that first. It depends on what you want to own and as long as you are above 18 in Kenya having an idea you have to be 18 of course so as long as you are above 18 you are eligible to own a website and again those people who are selling anything can have their websites. Anyone who wants to be visible like in this world you all want to be visible so you can own a website. What about business owners who feel like my business does not require a website they become a little bit sluggish and that will be like one location is just fine by me. This is my advice to them. They will start to analogue still because they are not moving with the current flow of the world the way we are heading. If I want to do something if I am searching for Y254 how do I get to know them? I have never been here I want to know about this and this so first thing I go and type Y254 these guys are nowhere to be found now maybe these are scammers so to avoid that you need to be visible so I can tell those guys who are business owners to have websites and their systems that will be controlling all they are working from. So how do I get mentioned and linked to other websites just for easy visibility what I am doing, my product and my business just getting out there when it comes to marketing. When it comes to marketing two dimension CEOs that's where we come in as Mogra we help you to be visible so for example you are doing ABC you are like I know ladies love those nails now what we will do as we Mogra we shall develop that website for you we help you to be visible then that's it So what is the biggest mistakes when it comes to digital marketing that you see businesses that are actually coming into SMEs small businesses which are coming into the game that are making fear of starting so they feel like if I am online everyone will know what I do and again how will I secure my thing there were hackers who also steal my data or the other my competitors will do what I do so that fear lead to them not to get started but definitely you succeed because you have outcome of fear Alright so what should run why should one continue to invent when it comes to whatever they are doing even in the digital space if your business is in the e-commerce so why should I consistently be looking out to just reinvent whatever I am doing Growth Wealth So growth I mean definitely you want to move from point A to B you cannot stack from A it's like having a kid you would love to see your son and daughter grow from zero to any other level the same case as business if you don't constantly innovate or invent in whatever you are doing you will stack to point zero and that case you never grow actually so even if you are looking for any investors why would I invest in your product why are you still at point A so that's why we need to always think like an innovator for you to keep on growing Take us to one of your looking back one of your innovation that you're really proud of I have three but I would mention two of them Take us to those two of them No my first innovation was on eHealth just as I said the challenges makes you innovate I remember when I wanted to travel to Kosovo I went there without any skills on how to be there now when I get there I got sick I was like oh I wish I knew how the weather was in before now when I was coming back I thought of coming up an application that can be travel it takes you through the way that hey you are going to this place this place is called kindly carry ABC like this place is hot kindly do ABC just in the case of emergence software will track because to normal users AIAA is an efficient agent so to normal tracks according to the nearby hospitals and then it brings you all the nearby hospitals that will be and some people that will be in case of emergence I felt like that could that could solve a lot of the people for those who are travelling like me again there is another one Mogra school I love that one because it's a new learning platform that you can learn using your phone imagine we didn't know that you can learn using your phone nowadays but afraid it's possible so I came up with that application after corona hit the second month I thought it will last that rest for 18 months for us to get the cure or the vaccine that's how you thought if we are getting that for 18 months at home what will happen to our students what will happen to our kids so I thought of coming up with that application that will be helping people to learn online to their comfort zone so you want a couple of businesses that you see that will develop or come up due to covid-19 online businesses because now we are here someone else is outside there for example if you are watching us why would you be watching this show maybe you could have been in an office doing ABC that's the first disadvantage and at the same time it's the first disadvantage because of covid now we are able to learn our other abilities that we didn't know we have so the online businesses I would tell them kindly adapt to the technology because there is no way we are going back to analog that one I can assure you we still progress now we are going back so if you are owning a business or if you are a business owner kindly adapt to the technology because you have no otherwise if not you will still start to point A what is the role of technology when it comes to innovation and also the digital business rapid growth that's the first law of technology again easy of you getting your customers if I want to reach you how can I know that you even exist without technology I would go to someone and start looking can I have her number where does she work can I do this the bureaucracy is a long process to take a long process but because of technology I would go to WhatsApp then I need this and this can you deliver if I want something in a cost I travel I will spend a lot of things then I forget I don't even know whether I get you but because of technology we are able to chat with people and we are able to expand our businesses using technology what couple of lessons that you have learnt along this particular journey of just starting something of your own business and also adding value for digital businesses so what a couple of lessons that you have learnt along the way since 2018 I have learnt a lot but I love saying failures because I like making mistakes the moment you don't collect your mistake you are not learning anything as I said earlier so that's the first lesson I learnt I also learnt that cutting your ego helps a lot especially when you are in this business you tell me this thing doesn't even look good you feel embarrassed it's your product but remember you are doing things for the people not for you so the moment you think you are doing product for you you feel like you want to move ahead so the first mistake we as developers do it's when you are developing something like a software you feel proud of it but the moment you roll it out people tell you no change this you feel like if you love a customer market in your mind when creating a product before you even do anything you need to look for first customers actually that's another thing I learnt because now you cannot develop something just for fun talk to customers look for what they need then come up with something not to come up with something and then you go to look for customers no so that's the best way the best way of learning is actually through failing learning to do better because experience is a summation of all errors as long as you are able to correct your errors then you are moving you are progressing so how can people find across all the social media and see if they want to keep this conversation going and find more about innovation and how they cooperate with the digital business I'm not that active on social media okay but how can they reach out to the business to the business you can reach out as Mogra EA that is on the IG Facebook Mogra EA Twitter at Mogra EA number is 0703 916371 okay we did that one more last time 0703 916371 thank you very much Fabrice for creating time to be with us today we appreciate thanks for having me so that is Fabrice Mugabe who is the CEO of Mogra dealing with all matters pertaining to software development we were talking about innovation in digital business so make sure you stay tuned to have so much coming your way right here on why in the morning and if it's Tuesday it's entrepreneurship Tuesday at Y2F4 channel so you can find us across all our social media handles that Michelle Ashira


This reverse blow dryer has been waiting months for me to try out because I haven't been putting any heat on my hair, but I am finally ready to test it out and see how it works and see if it's worth it. This is the reverse blow dryer by Rev Air. It has this pretty hefty base to it, and then it has this long tube attachment, and then this part that you stick your hair into when you're blow-drying it, and it sucks it into the tube. So let's plug this in. Of course I have to start with the heat protectant. Always use one if you're going to put heat on your hair, if you care about the health of your hair, or if you want to maintain your curl texture. So this one is a silk blowout quick drying thermal spray by OGX. And it also says it has hydrolyzed silk and quinoa, and says it will cut blow-dry time and protect hair up to 450 degrees. So I'm definitely going to be putting less than 450 degrees on my hair. But that's really good to know that I do have protection up to that heat, and I don't actually see any heat settings on this blow dryer, but I know it's way less than 450. I'm really not playing any games when it comes to the health of my hair because I am trying to grow it out. And it's so hard, it's so long, but it can be done. So I'm just taking every precaution to make sure that my hair stays as healthy as possible so I can grow it out longer. I already detangled when I conditioned my hair, so I'm going to use my prose brush right now just to help spread out the heat protectant and make sure that it got everywhere in the hair. And also just to get it ready to have in different sections so I can stick it down the tube. So right now the only thing I have in my hair is the OGX heat protectant and water. And I even rinsed out my conditioner today. I always leave my conditioner in, but I was like, if I'm going to try out the blow dryer and straighten my hair after this, I don't want to have any product build up and I want to see how straight I can get it without having anything except for this and potentially one more thing when I straighten my hair. But that's next video. I'm going to kind of section my hair. It's not going to be anything crazy, but this just might make it a little bit easier for the workflow. So I just have this one quadrant down right now. I'm just going to turn this on and use it on level three because it's in the middle. Setting three felt really cold to me. So I don't know if it's like anything three and under is on a cold setting, but it did dry my hair somewhat. It's definitely a little damp and took a little bit longer than it would if I increased the setting. So I think I'm going to try level five. It still feels like cool air to me on setting five. So maybe it just doesn't have any heat, but this is what the piece looks like when it's blown out. I think I'm going to try level seven, which is the max just to see if it'll make it straighter or if it will go faster. It seems like the stronger the level, just the stronger the suction it is and everything is cold. So I like that I'm able to dry my hair very quickly, by the way, without any heat. And I think because it doesn't have heat, it makes it more just stretched out and it doesn't really like strain at all, which is kind of what I was expecting. And I'm going to be straining my hair after this anyway. So all I really need is for my hair to be blown dry. One thing I just thought of is what if I do this and it's heatless and then I do heatless curls after on the blown out hair. Like that would be crazy to get curls like manufactured curls on my hair without having any heat on it. Let me know if you guys want to see that. And also let me know what to do with the ends if I do try heatless curls because my ends aren't straight. So I would have to like really make sure they wrap around the curl, maybe gel. I'm not really entirely sure. So let me know and let's do the rest in my head. I just realized there are buttons on here and they control the heat. So I've had it on the O, which I'm assuming means off. And so that's why there hasn't been any heat. But if you press these two buttons, then it adds heat so you can have one amount of heat and two amounts of heat. The instructions say the zero that I've been using is no heat. The one is low temperature and the two is high temperature. So let's use the low temperature setting on the hair that I have at the front. So it looks like low temperature is pretty different than no temperature. This section is a lot straighter and flatter than these sections. These are still like kind of fluffy and then like a little bit wavy still. I think I just didn't finish drying this as well as I should have, but let's just ignore that. It's like a lot fluffier because it was done on cold. And then this one is closer to actually getting like a blowout or straightening because it has some heat. Well, let's try the high heat just so we can find out if it's any different than low heat. I'm literally vacuuming my head. High heat definitely dries it the fastest. I'm not sure if it's really better than the low heat, but it's definitely faster. But just for my hair health, I probably will stick to having it on low heat so I can get the effect. Just spend a little extra time. I need to do a better job at the roots and finish the back. So I'll do that on low heat and then I will show you the final results. I'm just going to brush out my hair a little bit. So this is the final product. This is how my hair looks after being blow dried by the Rev Air Reverse blow dryer. And as you can see, it's definitely stretched out my hair. My hair is very dry and I didn't have to put any insane amount of heat on it. And actually I did most of it without any heat and then the rest with low heat. But even then, the low heat is so low. I was even like, is this on low heat? I had to double check if it was on cold. So overall, I really enjoyed this product. I thought it was so fun to use. It's so innovative. I like that I didn't have to use heat or I had the option of using super low heat on it. It did a good job prepping my hair to be straightened and I am so excited to see what it's going to look like straight. I haven't straightened it in so many years. So I'm very much looking forward to that and I'm either going to do that tonight or tomorrow. So I will post the video as soon as it's ready so you can see how I went from this to straight. So lastly, do I recommend this product? I definitely do think it is an investment. So if you do this to your hair very often, then it might be the perfect investment for you. But if you are a naturally curly girl and you don't really blow dry your hair or when you do, you just diffuse or diffuse hover, then you don't need this. So if you're the type of person who straightens your hair regularly or if you already have straight hair and you want to dry it or if you blow dry it so you can do braids or something like that this could definitely cut down on your blow dry time and decrease heat damage. So if you're one of those people, then I would recommend it. Also, can we just talk about these raccoon tails? Like this is just a prequel to how they'll look when they're straight. But oh my goodness, this is so cute. I know this is going to eat when it's straight. So those are my thoughts on the Rev Air Reverse Blow Dryer. Let me know if you guys have any tips and what you want me to review next.


Welcome back everyone to our second lecture, we see 2-1-3 the other in Revelation chapter 6. Any questions before we move on? Yes first. Like the chapter 6 we see like it's literally the starting of the tribulation period right? Yes. And we have also discussed that rapture happens pre-tribulation and also we have said that the book of John is chronological. Revelation is chronological but if we see after 5th chapter it's starting with tribulation but there is no mention of rapture like between 5th chapter. As they say and they have to say in sequence. So rapture happens in between chapter 5 and chapter 6? You can imagine like this. Rapture has happened. Rapture has happened. So there is something happening in heaven, something happening on earth. First we will show you heaven chapter 5 and 6. Then we will come to earth chapter 6. But it's happening simultaneously but chapter I have to write 4 and 5 and then I have to write 6. Rapture has taken place before chapter 4. So only then all the saints are in heaven worshipping God. So that is 4 and 5. Yes. Chapter 4 is after rapture but chapter 4 and 5 is a picture of heaven. Chapter 6 is a picture of earth. So we will stay on earth for some time then suddenly we will also go up to heaven. Chapter 7. We will see little of heaven come back to earth. So we are trying to see what is happening there and here simultaneously. So in between we will go to heaven, we will see silence in heaven then we come back to earth. Then again we go to heaven like that we will see. We will see both sides. So basically from chapter 4 on is after the rapture. Chapter 6 is what's happening here on earth but 6 to 19. But in between 6 to 19 we are going to go up to heaven, come back here, go up. He is saying this is what's happening in heaven, this is what's happening on earth. Yes. So we will see both happening. First when we are taking this interpretation when you told this Mitya and all. So on what basis that we can interpret or translate like say if maybe after our generation maybe people will interpret or understand in a different way. Maybe because of this change in this climate and all. On what basis we can interpret and other thing is I heard people wrote some books on this. The relation by interpreting in a different ways. And they told like when we when we discussing also there is no basis for us. We just okay this is what I understood. Like on what basis we have to do and is it necessary that to do translation. Good question. So one is when something is literal when something can be literal then we will keep it as literal. And something cannot be literal then we take it as prophetic as a prophetic image representing something. But like we said there is something that is happening which is literal. But it has been communicated in the language that John has. So I give an example something that is literal which must be taken as literal. 24 elders seated around the throne that is literal. There are 24 elders as we read the book of Revelation we will see some of them are from the Old Testament and all that. So there are 24 elders literal they are seated around the throne. Something is prophetic the lamb like we saw the lamb coming. It's imaginary the lamb is image for Jesus. Lion of the tribe of Judah image for Jesus not like a literal lamb or literal lion. So there we say yeah he's seeing the lamb come to take the scroll but it's not one lamb walking there. It's Jesus coming to take the scroll that is prophetic. But then there is this in between what is the in between John is seeing things but he can only write in his language. I mean he can only describe it with what he knows. So like we talked about you know for example what we were discussing in verse 13 Revelation 6 13 and the stars of heaven fell to the earth as a fig tree drops its lake. And then we will see again in in chapter 8 we will see chapter 8 or I'm just I'll just give one more example. In chapter 9 he sees things that are like smoke of locust that is Revelation 9 verse 3. There were locusts and they were killing people on the earth and the shape of the locusts were like horses prepared for battle. The crowns or something like gold. This is Revelation 9 7. They had faces like man they had hair like woman's hair that lion wings all those things right. So what is that? We don't know but he's only describing what it is. So what exactly is that? Could it be? I don't know but we can imagine something. What could it be? Because he's only saying I saw something. It had like woman's hair teeth like lion. This was 9. Revelation 9 9 breastplates of iron sound of wings were like the sound of chariots. Many horses running into battle and they were destroying the people. So what was this actually? Now it's left to imagination because we don't know what it is. And he's not telling us oh these are drones that were made by Russians. He doesn't know. We can imagine hey this sounds like something that's flying. It looks like drone could be I don't know but or it could be something else. We don't know. And John also doesn't have language from our day. God didn't say John right drones will be flying over the people. He's just describing what he's seeing and he's writing right. So this is where we don't have a clear definite statement of what this could be. So it is open to interpretation. But there are some things very clear taken literal some things prophetic very clear. And then there are some things which are open for interpretation. But we're not saying everything in the book of revelations like that. Only a few things. Most of the things are either literal or prophetic which can be explained. And the other thing when we are seeing this Antichrist time. So if I can ask like who is this Antichrist who really is is he is a satan or is he an other angel. And the next next continuation is if he is an if he is a spirit or some angel.  And then and then he have so many names. I mean in different different context. So like a beast. And if his beast means if he is human being under we won't I mean name him like a beast. That in different different context that different different names. And is there a possibility for such like a person to come here like a human being and rule over us. So from what we can see in scripture. He is a man. Human being human person. He's a man. But he's empowered by Satan. So when you read through Daniel, Daniel talks about this man who speaks blasphemous things against God. He's a man human. But he's speaking all this against God. He has authority.   He sets himself up as God. So very clearly he's a human being. But he is empowered by the dragon. So we will see two people main actors. There is the Antichrist and there is a false prophet. Both are people. Both are empowered by the dragon. The Antichrist is also called the beast. The false prophet. So yeah, Revelation 13, let's see what's his title. He is referred to, no, so Antichrist was a first beast. And this false prophet is referred to as the second beast, another beast. But these are two people. And the dragon that is the devil is giving them power. Revelation 13 is giving them power to do the things they are doing. One way to understand it and not necessarily... There's a difference but to look at it Christ was Christ on the earth when he walked on the earth. He walked as a man. Now the difference is he was God who became man. The Antichrist is a man but empowered by the dragon. So it's not like some angel has become a man. No, he's a man empowered by the dragon. We'll carry out these things along with the false prophet. We'll carry out these things. We'll see in Revelation 13. Alright, chapter 6. Now we go to chapter 7. What happens in chapter 7? In chapter 7, very interesting. The Bible tells us, John is saying, God has marked 144,000 Jews. This is during the tribulation, 144,000 Jews. They are marked by God with the seal of God. So the seal of God is, again, we would say, we understand in the New Testament, the seal is the Holy Spirit. I also can refer to the name of Jesus, both. But one of the seal, meaning of seal of God is the Holy Spirit. 144,000 Jews are marked by God and they are servants of God. So that means during the tribulation, God is going to have 144,000 Jews. Doesn't mean they're all in Israel. They could be all over the world and they are preaching Jesus. They're serving God. 144,000 Jews. So it's on the first half of the tribulation. They start off in the first half of the tribulation. When they come to the middle of the tribulation, they are taken up into heaven. You will see, Revelation 14. But this is very interesting, 144,000 Jews. That means that the preaching of Jesus Christ is going to happen during the tribulation by many people. God has marked 144,000 Jews to do it. Now the question is why? Why God set this part up? Like why is He saying 144,000 Jews who are going to be serving Him in the tribulation? Again, the answer we can only think. Like it's not given here, but we can think that maybe it's a time when even the Jews, Jewish people are going to turn and recognizing that Jesus is indeed the Messiah. Now some Jews, not everybody, are recognized, few Jews. I say some, I don't know if few, but some Jews are recognizing. Some are not. But when all these things begin to happen, we can say that there will be lots of Jewish people who will recognize that Jesus is the Messiah. Everything that He said, everything Daniel wrote, all has come to pass. They will turn to Jesus. And Zachariah 12.10 says that there will be a great outpouring of the Holy Spirit on the house of David. That means on the Jewish people during the tribulation.  That means during the tribulation, there will be an outpouring of the Holy Spirit, especially on the Jews. And 144,000 Jewish people are going to serve God. And in the second half of the tribulation, again, now we're going to have a verse 9. So chapter 7 verse 1 to 8 on earth. Now he says, okay, let's go up to heaven. What is happening? He is seeing great multitude, again worshipping God. And then they are signing on the throne. And then verse 13, Revelation 7.13, where did they come from? So one of the elders are asking, John, John, where did they come from? John says, you know them? So verse 14 says, these are the ones who come out of the great tribulation. That means these people have been killed. So continue. He saw chapter 6, people are dying. There are again chapter 7, people are coming. But of course, when a person dies for Jesus, immediately his spirit and soul is in heaven. Right? So these people are in heaven. They've come out of the tribulation. They're worshipping God and the one who sits on the throne. And they will never have these tears anymore. So people see this. That throughout the tribulation, we'll see this again and again. People are being martyred. And they're coming to heaven. They're martyred? Coming to heaven. Martyred? Coming to heaven. So we can say that there will be lots of people who believe in Jesus, but they will lose their life. Okay? Then chapter 8. Now, the seals are over. Now we are moving into the next set of judgments. 7 trumpets. Suddenly, there is silence in heaven. And then we see chapter 8 was 1 to 6. There is lot of prayer coming up into heaven. Prayer coming up into heaven. But where is the prayer coming from? It's coming from the earth. So that's why we are saying at that time, there will be a big prayer movement on the earth. People are going to pray, calling on the name of God. Right? So it says here, there's another angel having a golden sense. It was given much incentive. He should offer prayers of all the saints. And it throws this sense. The angel took the sense and filled it with fire and threw it to the earth. That means this is lighting up this, igniting this prayer movement because there's so much of prayer coming. It's igniting it. More prayer. So you can imagine, people everywhere, they're believing in Jesus. They're being martyred. But they are praying.  So there's prayer coming up into heaven from the earth. Then the trumpet stops. First trumpet, this is verse 7, chapter 8, verse 7. All the trees and the grass was burned up. One third, one third. So these days, we hear, we can read about, maybe the last couple of years, we read about big fires happening. That huge hectares and hectares of land is just burned up. Forest fires. But here, chapter 8, verse 7, one third of the trees were burned. I mean, this is like something they haven't even seen before. But nowadays, we read about fire here, fire, then the forest fires. But this one is one third. Then the sea is struck. One third of the sea creatures die. Then the third trumpet, waters are struck. Men start, one third of the waters become bad. People die. Fourth, this is verse 12. And once again, same thing. Moon is darkened. A third of the stars become dark. Sun is darkened. So the third of the day did not shine. And likewise, the night. So things have, chaotic things are happening in the sky, chaotic things. Stars are becoming dark. Sun is becoming dark. So you can imagine what's happening. Heavens are being affected. Then chapter 9, fifth trumpet. So here, something is happening. It's saying the bottomless pit was opened. And all kinds of demonic things were released upon the earth. So we understand scorpions, locusts representing demonic spirits. They're coming on the earth. But they will not touch those who have the seal of God. So they will not be able to harm these 144,000 Jews and possibly, you know, the others who believe in Jesus. But for five months, they are tormenting people. And notice in verse 6, it says, in those days, men will seek death and will not find it.  That means it will be so bad. They want to die, but they don't know how to die. Like, you know, they can't, they want to die, but they're not able to bring themselves to that place of taking their own lives and so on.   And then he's talking about verses 7 to 12. He's describing what these demonic beings are like. And so, like we said earlier, what kind of expression they take? We don't know. But he said they came out of the bottomless pit and they're tormenting men. Exactly this. We don't know. We're giving some description here. So this is left to imagination. Somebody may say this is the, you know, COVID-19 virus or somebody may say, whatever they may say. Okay, fine. That is your imagination. But really we don't know. We just know that these creatures have been released from the bottomless pit. They're tormenting people. People want to die. They cannot die. And there they have this strange thing.  Again, we don't know. Right. So that's why I say somebody can say, oh, it looks like drones or it looks like that or whatever. Don't know. Okay. But this is happening. This is what the effect is. It is causing these creatures are causing so much of torment. That is the fifth trumpet. Sixth trumpet again is strange because there are angels that are released from the river Euphrates. And it's like an army. And these angels are tormenting people, killing people. They were released to kill. Sorry, they're killing one third of mankind. One third of human beings are killed. So previous people are being tormented. Next, people are dying. But it also says the army of this army was about, this is what 16, Revelation 916. This army was about 200 million. And he says, I saw the horses in the vision. Those who sat on them at breastplate of fiery red, highest in the blue and sulphur yellow. This is what I was talking about. You know, there's all strange colors you're seeing. Heads of the horses were like heads of lions. Out of their mouths came fire, smoke and brimstone. So he's saying fire, smoke and brimstone. So I'm just wondering, just using my imagination, what if God was showing John these nuclear bombs and things happening in the silence, causing fire, smoke and brimstone, which are killing people.  So could it be some sort of missiles coming and he's thinking like some fire is coming from the back and some brimstone and all this. So we don't know, right? We don't know. It's again, this is what John is seeing and he's writing it in his words. If God was showing him some other things that were happening today, how would he write it? So again, this is left to imagination. He's written it now. What exactly are those things? We don't know. But one third of people are dying. So what could cause this kind of death? It could be these missiles are applying, armies firing, these kinds of things against each other. He mentions an army of 200 million in verse 16. So that's why again, people say, okay, where could there be an army of 200 million people coming like this? So is it literally soldiers like this or is it something else? But generally people say, okay, this could be the outstanding army of China. They are the only ones who have such big army and all of that. Okay, maybe. We don't know. But that's what he's saying. First, killing the believers or the people who were left? So in the fifth trumpet, it's clearly said that they cannot touch those who have the seal of God in verse 4. It says they cannot touch those who have the seal of God on the foreheads. But it doesn't say that for the sixth trumpet, the seal of God, which we said represents the Holy Spirit, the name of God and the Holy Spirit, the seal of God. 144,000 Jews. So I think my thought is it represents both. It represents the 144,000 Jews whom God said specifically sealed. And I also think it represents all those who do believe in Jesus during the tribulation. Like we're seeing, people are being killed over and over again. People are being killed. So this fifth trumpet that causes the release of these locus-like, scorpion-like spirits from the bottomless spirit, which are affecting people, tormenting people for five months where they want to die but cannot die. It's saying it will not affect those who have the seal of God. So my thought is, well, it includes 144,000 Jews plus those who would believe in Jesus during that time. But it doesn't say that other judgments, you know, how it affects the believers. It's possible that believers directly or indirectly are being affected because we're all on the planet.  So in some way or the other, they may be affected. So it doesn't say that. So the sixth trumpet is interesting because, you know, it's talking about an army of 200 million horsemen seeing an army. Again, people wonder, you know, what that army is and that, you know, you might read in some books that, oh, this is China, the army from China and so on. But what is interesting, Revelation 9, 20 years, that in spite of all of this, people do not repent. There are some people who do not repent. It says in Revelation 9, 20 and 21, the rest of mankind who were not killed by these plagues did not repent of the works of their hands and they should not worship demons and idols of gold, silver, brass, stone and wood which can either seen or hear or walk, and they did not repent of their murders or their sorceries or their sexual immorality or their thefts. So it's like in the tribulation, well, all this is going on. There are some who will repent and turn to God, but there will also be some who don't repent who refuse to turn to God. So that's what he's saying here in Revelation 9, 20 and 21. Let's just do chapter 10 and then we'll pause. So in chapter 10, it's like a parenthetical chapter, meaning, okay, this is something that is happening to John while the Revelation has been given to him. So he sees one big angel with a little book and this angel comes to John says, John, time to eat. It's time. Eat this book. So John eats the book and it is sweet initially and then it becomes bitter in his stomach and then the angel says, John, this is Revelation 10 verse 11, you must prophesy again about many people, nation, tongues and kings. That means more prophecy is there, but it's bitter. I guess it's all about judgment. It's more. You have to prophesy more. So again here, it's symbolic. Like John at that moment, one angel comes and says, eat this book. He's eating. It's symbolic. And says, okay, you have to prophesy some more about nations, tongues and kings. So that's parenthetical, meaning that's not happening in the future, but it is happening to John while he's being given all these visions. Okay, so we'll pause here. You're with me so far? So now we have reached the middle of the tribulation. So first three and a half years over. John is given a book. John to speak some more, but it is going to be bitter. Second three and a half years. So chapter 11 verse one onwards. Second. So we'll pick that up next. We'll have great tribulation next week. This week only tribulation. Okay, any questions for those online? What is the symbols and the trumpet means? Seals and trumpet means. I think, again, they are representative. That means a seal is something that it's like you can imagine a letter that is stamped closed. That means still that time it's like a secret. Nobody knows it is sealed. Then it's open. This is what it is. It must happen on earth. So it's a secret that is unpacked or opened and it is shown. Trumpet is announcement. Like, okay, somebody blows the trumpet. Okay, now do this. Second announcement for something to happen. So we can see when Christ descending, there's a sound of a trumpet. Then there's a second trumpet.  So, you know, for that, for that event to happen, for Christ to descend, one trumpet is sounded. Another trumpet is sounded for the people to be caught up. So here, that is a good thing. Those are good things for us. Here, seven trumpets, but they are going to be judgment for each trumpet. Something is happening. So seals, trumpets, and then bowls, like something being poured out. So they are representing, all these are representing that a judge, something is happening, each one. People, they are using for worship trumpet. For Sunday service. Nothing done with the trumpet. Trumpet you can use. So far, people use all that. There's nothing wrong with that. But in this case, it is a moment in time saying, now this judgment must take place. So the trumpet itself is not wrong. It's only announcing something. In the Old Testament, you will see they use trumpet to call for war, to assemble the people. So they use for different things. Okay, so let's close for today. I will pick up from chapter 11 and go forward next week, just to get over here, right? So don't worry next year. In third year, we will read through Daniel verse by verse and revelation verse by verse. Now we're just giving us an idea, a sequence of events. We will read things in detail in the third year. Somebody can pray and then we will dismiss, please. Father, we thank you for this day. Lord, once again, we come to your presence and we pray as we have learned this. All things father, father, give us knowledge so we can understand deeply Father, your word, Lord Jesus, help us to understand, help us to learn more and more Lord Jesus. Thank you, Lord. In Jesus' name, I pray. Amen. Thank you. Thank you to one. We'll get back next week. Thank you. Bye now.


Hi everyone. Today I'd like to welcome you guys to Car Hacking Village of DEFCON 2020 Safe Mode. Today's presentation is going to be a throwback to the times before J1939. If you don't know what that is, that is a protocol that runs on top of CAN and is primarily used in heavy vehicle component communications. So we will be talking about how we came about the creation of a decoder slash parser for J1708 and J1587. So just a quick breakdown of how this presentation is going to pan out. First, we will let you know who we are. We're going to let you know how this project came about. And then we're going to give a brief recap of the protocols that are interesting to us. And you will then see how we came across these protocols. Afterward, we're going to talk about the decoder slash parser and we'll give you a quick demo. Last, we'll conclude this talk with the info you'll need to pick up where we are leaving off. Okay, so my name is Daniel Salom. I'm a reverse engineer at Assured Information Security. We're based out of Rome, New York. I've been with them for a little over a year and a half at this point. And I spend my days analyzing the security of things, digging into protocols and developing anything to support these efforts. You can see in the images, I'm on the right hand side. And there's an image of Thomas on the left hand side. So now you have proof that we actually work. Yeah, hi. So my name is Thomas Hayes. I'm a hardware engineer at Bendix Commercial Vehicle Systems based out of Illaria, Ohio. I spend most of my days designing hardware, so PCBs and digital and analog interfaces for brake control systems. And I've been there for almost five years now. Okay, so what are we doing here? Well, we discovered that there are more than just J1939 messages that cross the networks in heavy trucks. And within this traffic, there's some very interesting data. So we're here to show you how it came about as we needed a simple way to see it. And I can imagine you guys want to see how the decoder works as well. So is there a backstory? We know trucks that have been playing them for a while. This project started off in mid-2019 when we started to do some exploratory testing on heavy vehicle systems. We noticed multiple networks and the interconnection between all of these modules. We started to explore some tools and decoders that are already out there available for use for people in industry. And these legacy protocols of J1708 and J1587 are still being sent across the bus, even though now can J1939 is much more prevalent in newer vehicles. And it looks like it could be very interesting as it is a very important bus that happens on the vehicle with a lot of useful information that we were interested in looking at. Okay, so how do we, I'm sorry, what do we need in order to get things rolling? First is an understanding of the protocols involved. In this case, we mentioned J1708 and J1587. They kind of work together. We also need a hassle-free way of analyzing the traffic that we see. And at that point we can decide what may or may not be done with this data. What are J1708 and J1587? So they are SAE, the Society for Automotive Engineers Standards for Medium and Heavy Duty Vehicles Post-1985. J1708 describes a physical layer and data links layer and defines a bi-directional serial vehicle network. J1708 defines how the messages are structured and how they're distributed over the network. And 1587 makes up a transport application later and describes messages format and parameters. So that's almost a big dictionary for all the possible messages that can be sent over this communication protocol. So 1708 is almost always used in conjunction with the application layer protocol, J1587. It's based on an RS45 bus. We can have a max of 20 nodes on a vehicle and it runs at 9,600 BOD. A message contains a one-byte-long MID, a message identifier, followed by some data related to that MID, and a one-byte checksum. The message can be up to 21 bytes long. However, if the vehicle is stopped and no movement is detected, the messages can be longer if required. So J1708 MIDs, we go up to 0 to 127, defined by J1708, and these are defined by SAE for important vehicle systems. And then from 128 to 255 are defined by J1587, and these are used a little bit more for maintenance and information on the vehicle, more from a diagnostic side. It's very cool. The MIDs also serve as an arbitration method for the communication. So the lower your MID number is, let's say 0 to 7 for engine, that means it's a very high-important message and will be passed before perhaps the triple quarter message when MID at 56 to 61. The checksum is quite simple. It's a two-s complement of the sum of the message and the data content is defined by the applications document given by the supplier. So you can send the messages defined by SAE or possibly something defined by you as a manufacturer if you need to send something specific or proprietary across the bus. So J1587 defines the PID, the parameter identifier. So that's a little bit of a complement to the MID. So if you have an MID for the engine, a PID can be related to your engine oil or a knock sensor or something in your camshaft. So it's a little bit more information for the PID for the system to recognize if there's a fault and direct where that fault can be. So look, it's this of one MID, one or more PIDs and a checksum. The data length of 1587 messages is mostly limited to 21 bytes according to J1708. And if a message needs to be longer than 21 bytes, we can use the COTS, the TP4, so that can be used to segment and reassemble large messages if needed. So the PIDs range from 0 to 4 bytes and the unit's resolution range will be defined in the spec or by whoever is sending them out. So if you're one engine manufacturer could say the PID for oil pressure is one PSI per bit or another one could say it's five. So it just depends on the manufacturer for that PID. There's also SIDs. These are related to the MID, but these are more related to anything that can be used to identify a section that is not necessarily covered under a PID, but covered under a diagnostic MID. So the MIDs for using SIDs will begin at 128 as defined by 1587. And these are really used for something that can be troubleshooted or replaced by a field technician if needed. So what's in the data? Now we, a lot of our testing was in the communication between trucks and trailers. So the trailer has an ABS ECU and the trailer also has an ABS ECU and those two will talk to each other using this protocol. The trailer ABS can have all kinds of diagnostic message. A trough test is a test of the system. It will modulate the air solenoids a little bit and dispense a bit of air. Like when you hear our truck break, you hear a large air release sound and that's kind of just making sure that your system is working properly. You can restart the ECU, you can check the status of the ECU and most importantly demonstrate an ABS fault. So as per SAE standard, the ABS fault, if the trailer has an issue, the trailer ECU has to communicate to the truck ECU and illuminate a light on the dash, as well as a light on the trailer itself, indicating that there is an issue with the ABS and it needs to be resolved. Diving a little bit deeper. There's also a configuration options on the ECU. So wheel size, which then relate to the speed of the vehicle as there's a towing ring that counts the amount of rotations done and driver info. So where the truck is, where the trailer is going and what truck it is connected to at the lot and sensor data. So it's a speed, mileage, your odometer stored on the ABS, trailer ABS location, temperature, and many proprietary vendor messages. So who uses J-1708 and 1587? So any kind of heavy vehicles that we all know, but also school buses, all that same platform, military vehicles, and we even found that yachts will be using the same communication platform. So our use case, we were observing exchange between a tractor and a trailer. They, as we said, the brake controllers communicate over J-1708 and J-1587 and these messages actually pass over a power line, which is very cool. Trailer manufacturers did not want to add another wire or two needed for an ABS communication. So they superimpose a sinusoidal wave on the power line going from the trailer, which then gets filtered out on the trailer side and on the truck side, depending if it's using that communication. It's very cool. But Dan and I did not focus our time on this. We were working on the J-1708 and J-1587 protocol. Okay. So the question comes up now, how do we look at these messages? There's a couple of options. We can use tools that were developed for J-1939 and our backward compatible with the other two protocols we mentioned. Or we can get our hands on an RS-485 transceiver, some bubblegum, a paperclip, maybe a tinfoil hat and see what we can hack up. But what happens when our bus is a power line? In this case, you need a specialty diagnostic tool. Some of the larger companies like Nexik, Haldex, VG Tech, they provide these. But the problem with this approach is it's going to cost you some mula. And flow between programs is not really there as software is kind of tied with the hardware and it's all proprietary, which means minimal flexibility. So our solution is pretty J-1587. This is a decoder that helps with at least part of the problem. By default, this takes in hexadecimal bytes, which are common delimited, but really it can take in a stream of any bytes and a user can define how to manipulate that data to be parsed correctly by the decoder. It also supports reading from standard in, files, or sockets, UDP or TCP. And what it does is it takes this information, these messages, and it will break them down into a human tolerable format so they can be easily analyzed by an end user. So to start off, what happens is the parser will read the SAE documentation PDFs. Okay, and it will do this to set up a primary database, which is used for interpreting the data that's coming in. And this is needed because there's no simple database holding the data that we need in the convenient format. Okay, it's all in the specifications or proprietary by vendors. So there is an option for an end user to create custom messages with this tool. And this will be handy for working with proprietary data. The benefit of this using this tool is that it can be piped from or to other programs, as it's just a command line utility. So there are some requirements. What do we need to get this off the ground? Well, the first part is on you guys, you're going to need an interface to some bus to get the traffic. The next part is, it uses the program PDF to text, which is contained within the popular utus package standard on most Linux distributions. Okay, and what happens with this is we use this tool to convert the SAE portable document format specifications into formatted text files. Okay, then you need Python 2. We are hoping to port this to Python 3, but it started as a spaghetti code project that we just needed up and running. And Python 2 was default here. And that's what we stuck with until now when we realized it's actually useful. There's still some minor issues with it. The spec obviously was not written with being parsed in mind. Okay, so there's a lot of regular expressions and things that parse this information out of these files. Okay, so I think at this point, I'm going to give you guys a quick demo about how this thing works. It's a repository, which will be made public after this talk. And you can just clone it. And at that point, you will set up the configuration file to point at the text files that you've created from using PDF to text on the SAEJ 1587 and 1708 specifications. So the first thing that I'll mention is this repository within we have a tool called fuzzy messages, which I'll be using for the demo. All right, this program just creates random bytes. I believe one to 19 bytes in length. And this winds up working out as a nice test environment to create a robust program. That way it's not just failing on any sort of messages. As you might see when running us, you know, different types of hardware, you'll get funky bikes that aren't actually part of the message at times. So we've decided to make this program a little bit robust and that won't just error out on an incorrect message. It might print a warning to standard out and it'll keep chugging. Okay, so the first thing I'm going to do is I'm going to create a file with, let's say 100 messages. And I'm going to write that to temp. Okay, let me just make sure that that worked. Yep, seems like I got messages in there. All right, so now to run the tool, I just point it at that file that I created with all the messages. I'm also going to give it a flag to print the limiters between the messages. So you can see when there are errors, it gets printed to standard error. So these could be piped to DevNull if you're not interested. But you can also see here the original message, MID, in hexadecimal and these also are accompanied by decimal forms. And then the PIDs and the meanings of the PIDs along with the accompanying bytes. So that's pretty good. But if you want more information, you can give this program a little bit more verbosity like so. And you'll get details. This is the same message here that we saw previously. But now you can see it's giving details about the individual bytes related to a PID. So here we can see there's one byte and this has to do with the extended range barometric pressure. And here are the units right here. I just want to mention one thing. We are using random bytes for this data. So this MID says idle adjust system. And it's not necessarily in a real world scenario. This PID is probably not going to be matched with this MID. But we don't air out. We just want to parse the data as it comes. There's a couple of other options I wanted to mention. So some people prefer JSON. We added a flag for JSON. I will disable the output that you just saw, which can be easily grept. And I will print these messages in JSON format. And I'm also going to pipe the standard out to dev know. Okay. So you can see in here, same messages, but these are output as JSON formatted messages. This could be convenient if you use the JQ tool and you want to filter these messages the way you'd wish. But there's another option for you. And that is we have a white list here. So what we can do is specify specific PIDs, like if we are interested in certain PIDs within a message, we only want to print those messages. We use the white list option. So I will just grab one from here and let it roll. So you can see I printed out one message with that PID 48. And that is in actually decimal form for that. Okay, so there's a couple other things that this can do. If you just give it the dash H, if you use some help, one of the options is for a custom database. Now this means if you want to define your own messages and overwrite those that are provided by the spec, or let's say you're working with some proprietary messages and you have a good idea what it means, you can define that. We provide a sample file in the repo and you can see here that basically this is the format. You can just json and you just give it your definitions. You can use this file as a guide. Okay, there are still some issues. There are a few PIDs that are a little bit more tricky to work with, and those might not have their individual byte definitions over it, but it's working progress there. I wanted to mention that, okay, if you want to read from standard input, the way that you do it is you just pipe hexadecimal delimited bytes as I mentioned to this, and you will use this usual format for reading from standard input. Okay, I'll use packet delimiters here. And you can see what the meaning is, but one of my colleagues came to me and he said, well, you know what, you know, we have an issue because the output that I'm getting from my program is not set up like that. It doesn't have the comma delimited values. It's all just a string of hexadecimal bytes. So what we did was we added the option here. If you look in this file called canon functions, an end user can define a function to be called on the command line that will manipulate this format. And the only requirement is that the output would be a Python list of bytes. Okay, or a Python list of integers. So in this case, we can use the dash j option and say canon node delims, I believe I called it. And you can see that it will just parse the message in that format. Okay, so it doesn't really matter. All you need to do is know what your input looks like and figure out how to get that into a Python list of integers. Let me see if there's anything else I wanted to mention. I think that's it for now.  So getting back, what are the benefits of using this decoder? Well, there's no real software investment other than getting your hands on the SAE specs. Okay, I'm sorry, we can't just provide them for free. But SSA, sorry, SAE would not be happy with us. And they take the intellectual property very seriously. So I'm sorry about that. You can handle it. I'm sure the other benefit is that you can work with anything that outputs 1708 1587 messages. So where can you get it? If you want to get your hands on this tool. You'll go to this URL. After the talk, we will make this a public repo. We're rolling with an MIT license. So don't forget where it came from and feel free to make money on. But remember that there are a few dependencies. So you need the specs. You need PDF to text tool and Python two at the moment. We wanted to give a special thank you to NMF TA for giving us the opportunity to do this work. And our friendly motor freight carriers for allowing us to play with their trucks. Also the Hilton executive lounge for hosting a nice hack tackler atmosphere by letting this group meet and set up mobile pass benches. And we only left a few solder burns under nice wood grain tables. We just want to thank you for joining us and enjoy the rest of your DEF CON talks.


What mercury being in retrograde means to astrologists, scientists, and your roommate? The planet mercury rules communication. During the time that it's in retrograde or moving backwards, wires get crossed and miscommunications tend to happen. Well, actually that part where it's moving in reverse is an optical illusion. And even if it did move backwards, we're too far away for it to actually affect us. Those are all great theories, but I know for a fact that mercury in retrograde is the reason my rent was late. I was late to work, I got a parking ticket, and why my boyfriend won't text me back.


So, last class we have discussed the solution of finite time regulator problem. The basic steps of solution of finite time regulator problem is first you have to solve the what is called dynamic matrix Riccati equation. Once that you have to solve by what is called knowing the final time state weighting matrix, final time state weighting matrix. You have to backward integration the matrix Riccati equation which is a n into n plus 1 by 2 dynamic first order differential equation and coupled. So, you have to solve it backward integration and store the values of this information of the matrices at different instant of time from t f to t 0 you store it then you implement the control law u of t. Once you implement the control u of t then question comes then what is the our optimal cost value the objective function value what is our objective function value. That means we have a terminal cost plus integral of the quadratic terms associated with the objective function that you have to compute from 0 to t is equal to t 0 to t f. So, that we are discussing since we have obtained the optimal trajectory for control input which in turn we can get the optimal trajectory for the state trajectory and it must satisfy the solution of that our control law in turn the state trajectory must satisfy our that constraints what is that constraint x dot is equal to x plus b u that constraint you have to satisfy. That means if you take this is that side this equation must satisfy and we can show it that optimal objective function value is nothing but a half x transpose t 0 p of t 0 x of t 0 x t 0 is the initial time of the state and p t 0 when you do the backward integration of the Riccati matrix Riccati equation dynamic matrix Riccati equation from t is equal to t f to t is equal to t 0 then t 0 value what is the Riccati what is the value of p matrix that you have to use it here to get the optimal cost function value. So, this is let us see how we have achieved this one. So, that is today I will discuss how we have achieved that optimal cost function is equal to j star is equal to half x transpose t 0 into p of t 0 into x of t 0 how you have achieved. So, let us see this one and if you consider that one that our equation that basic equation what we consider x dot star of t minus a of t x star of t x star of t minus b of t u of t is equal to 0 let us call this is the equation u star is the our optimal trajectory for control input optimal trajectory that we have found out by optimizing the our performing index and correspondingly state trajectory we can get it. So, this equation that is constraint that x dot is equal to x plus b u must be satisfied by this optimal trajectory x star as well as the input optimal trajectory u star. So, this is the basic equation. So, our performing index expression is like this way this of t is equal to half x transpose t f star along the optimal trajectory that f of t which is nothing but a p of t f this is nothing but a is equal to terminal state weighting matrix this f of t which is nothing but a p of t into x t of t star this is the our terminal cost plus integral of the what is called quadratic terms associated with state and input vector is that one this 0 to t 0 to t f then this one state weighting matrix x t u of t x of t plus u star of t plus r of t u star of t plus lambda star of t into a x a function of t x star of t minus plus b star b u of t b of t u star of t this is b of t u star of t minus x dot star of t this quantity is 0 0 multiplied by this. So, objective function values will not change even if you multiply the linear multiplier factor it will not change. So, now let us see this one actually our objective function value this plus that term we have added this one because along the trajectory this values this is 0. So, it does not it will not change the objective function values of this one now let us call this is the equation number 2 as we know from the derivation of the finite time regulator problem when we derived this one we have we have seen that we know that lambda dot star of t is equal to minus q of t x star of t minus a transpose of t lambda star of t which if you take this term in the left hand side this will be a transpose of t lambda star of t is equal to minus q of t x star of t minus lambda dot star of t and which we can write it that we can write it lambda dot if you just make it this is the transpose both side left hand side right hand side you take the transpose left hand side and right hand side if you take the transpose then you will get lambda star transpose of t into a of t minus x star of t transpose q of t actually q transpose of t since q is a symmetric matrix then I can write q of t next is this is equal to lambda dot star transpose of t. Let us call that is the equation number 3 this is the equation number t this is the equation whole equation is equation number 2 and this is the equation number 3 this one. So, this is the one expression we got it further we can note it that u star note that our control input that u star of t optimal control input is minus r inverse v transpose p of t p of t actually then we have a lambda star of t when we have derived this one u star of t c at the derivation we got it minus r inverse v transpose lambda star of t. So, this we can write it is if you take this both side I multiplied by r if you both side multiplied by this is r of t u star of t is equal to minus v transpose of t lambda star of t take the transpose of both side take the transpose of both side then it will be a lambda star transpose of t v of t is equal to minus u star t transpose of t into r of t transpose since it is a symmetric matrix r transpose is equal to r. So, you can write it since it is a symmetric matrix r transpose is equal to r. So, this due to the symmetric matrix. So, this is the equation number 4 using equation number 3 equation number 3 and 4 in equation number 2 you just replace that one lambda star transpose of t into a t replace by this expression and here once again lambda star transpose v of t lambda star of t u express by this equation 4 using 3 and 4 in 2 we will get it that j star j x star of t of t is equal to half first term terminal cost will be as it is there is no change in here p of t f x of t f then half t 0 to t f then this will be a first term is x transpose x star transpose q of t then x of t transpose plus u transpose t r of t u transpose u star of t that is that first two terms as it is now I replace lambda star t transpose of t into a t by this expression.  So, u star that I am replacing by this one and what is left with this one this two terms left is lambda star transpose x dot star. So, it is left with minus then minus lambda star transpose x dot star of t the whole thing d t now see this one this you push it x in the bracket then this and this cancelled this and this cancelled agree. Then what we can write it this one this is nothing but a integration of left is integration of minus lambda dot star transpose x star of t minus lambda star transpose x dot of transpose these two things commonly we can write it into this form rewrite this equation that half x star transpose t of f p t of f x t of f then minus because both are minus term here minus term is there you see. So, I am taking minus half then this I can write it t 0 to t f d of d t lambda star transpose of t x star of t d t this. So, this two term commonly this two terms commonly I can write it into this form. So, if you take integrate this one I will get it first term will be as it is constant depending on the value of x t f and p t f this term is constant then minus then what is the limit of that one I will get it this x transpose but there is a one term this is this lambda star this x dot. So, I will get it this. So, if you do this one what will get it that one lambda star t f transpose x t f star that lower limit if you take lower limit if you take minus is there minus plus half x star transpose of t 0 x of star t 0. So, this quantity lower limit is the upper limit I have written then lower limit I have written.   So, this term is this one plus half that will be x transpose t of 0 star then p t of 0 then your what is called then x t of 0 star. So, this this cancel So, our optimum value of j star that our optimum value of j star will get it this half x transpose of this t 0 into p t 0 plus x t 0 star. Now, you see initial value of this system says this is known to us. So, this is nothing but a x star t 0 is nothing but a x of t 0 is known initial known value known and this p 0 we have completed the solution of matrix dynamic matrix Riccati equation backward integration backward in time we have integrated and we have lined up with a t is equal to 0 this value also known. So, you can we can now find out what is the optimal cost of this functional or objective function corresponding to the what is called finite time regulator problems what is the optimal cost this one this is the important relation that is your half x transpose t 0 p of t 0 x of this is the now next we consider that what is called finite time regulator problem we have considered next we will consider that is what is called infinite time linear quadratic regulator for linear time invariant systems. So, next we will consider that infinite time regulator problems infinite time linear quadratic regulator for linear time invariant systems time invariant systems. This is linear quadratic regulator problem infinite time regular linear quadratic problem also called that that horizon infinite horizon finite time regulator problem horizon infinite horizon LQR problem infinite horizon previously we have considered finite horizon linear quadratic regulator problem now infinite time LQR problem also called infinite horizon LQR problems. Then what is the problem is this different from this one previous case finite time regulator if you recollect the our final time is t is equal to t f is finite. That means at time t is equal to t f our state from initial state x t of 0 will drive will go from initial state to a final state which is close to the origin or desired position of this one by using a what is called suitable control effort we have to drive the state near to the as small as possible near to the desired values that is. So, here we consider an infinite time regulator t f tends to 0 when you are talking about t f tends to 0 final time tends to 0 by using the control law what will see that our state will go to the equilibrium position equilibrium point and equilibrium point is nothing but a linear system case is a origin. So, at t tends to infinity x t f is equal to 0. So, if you consider the terminal cost if you see the terminal cost is not coming into the picture because at t is equal to t f when t f tends to infinity this state will go to the equilibrium point which is nothing but a origin of the coordinate systems. So, that terminal cost is not coming into the picture and integral part of this the integration of the quadratic term associated with the performing index that limit will come from t 0 t is equal to t 0 to t is equal to t f where t f tends to infinity. So, that will be the our corresponding performing index. So, let us redefine that our problems of that one let us consider the plant as x dot of t is equal to a of x t plus b I we are consider in the problem we have consider linear time invariant. So, a is not a function of time now if it is a function of time still we can write it this one in general form. So, a of t b is replaced by a and b since it is linear time invariant and your initial state is given x t of 0 is given. So, and the corresponding performing index as j is equal to half 0 to infinity t 0 is equal to 0 to infinity t f is equal to infinity because t f now in infinite time is infinity then x transpose of t q x of t plus u transpose of t r of t u of t whole multiplied by d. Now, you look at the terminal cost is not coming to the picture because when our t f the final time is t f is infinity it indicates the with the control f out the our state will go to the equilibrium point and equilibrium point in the our system. So, the terminal cost x t f will be 0. So, the terminal cost term terminal cost term will not come into the picture and our t f is now is infinity. So, you have to solve this optimization problem if you recast this or problem in it will be like this way or problem is design a control law u such that this performing index is minimized subject to the condition that this dynamic equation of the system is satisfied that is our problem. So, if that what I have just mentioned it that I will just write it here for your convenience if the final time t f tends to infinity then x t f should approach to the equilibrium state either 0 or origin of the coordinate systems or in the all states will drive to the 0. So, naturally this terminal cost will not come into the problem the terminal cost in j is thus no longer necessary is no thus no longer is necessary. So, this problem is known as this problem is known as infinite time time l q r problem or infinite time, infinite time horizontal problem. So, infinite time then q r problem so our previous formulation for finite time regulator problem previous formulation for finite time regulator problem is still valid only there is an additional assumption is made that the system a and b must be controllable or at least it should be a stabilizable. So, this is the additional condition is imposed there that means in this case compared to the our finite time regulator problem a and b must be controllable then the our performance index will be a bounded performance index this one because still if it is a a and b are controllable completely controllable or stabilizable the state by using the control law that can be stabilized the state. So, our assumption from the previous one that finite time regulator that a b pair pair a b must be completely controllable or at least stabilizable. So, this condition this assumption ensures a bounded performance bounded performance because time t t is equal to 0 this performance will be bounded. That means this for whenever find out the objective function that is our optimal control if you use in this expression t is equal to time t is equal to 0 to t infinity then this expression value will be bounded if this assumption is considered that means the state trajectory of this one x of t will be stable one. So, this is our basic assumption keeping this thing in mind these two things it can be shown that performance index 2 the let us call this is the system equation this is our system equation let us call equation number 1 this is the performance index is the system number 2. So, if you see this solution of this one that means it can be shown that the performance index 2 and the assumption x t is equal to t f tends to infinity is equal to 0 the matrix the matrix limit t tends to infinity p of t f will be a p which is a constant matrix. What does it mean that if the assumption is considered a b and this t f is tends to infinity this value is reached to the equilibrium point of the system that means origin then it can be shown that our solution of p is constant when t is tends to infinity this solution of this one constant. If you recollect the finite time regulator problem that our solution we have seen it let us call t f is very large then we have seen t f is finite case we have seen it if you see these values are nature of the p is like this way this is a p of t expression value this is the t f and this is t is equal to t is equal to t 0 then this period from here to here this period is the transient response of this our solution of our matrix p dot p dot is the matrix. So, this is the transient period this from here to here is a transient period that means during this period the value of p is changing let us call up to this after that the value of p is constant even for finite time regulator similarly now if you put it in limited case t f is now increasing and approaching to the infinity. So, the solution will still will get same nature of this for short period of time it is a transient period is there p of t value will be changing and then after that it is a constant. So, if it is so then our the finite time regulator expression if you see p dot is equal to then our p dot expression is this p dot is equal to in finite time regulator problem is a transpose because our system is time invariant system means it is not a function of time this is equal to p plus p of t p of a plus p b are inverse b transpose p plus q is that one. Since p is constant you see see this values when t tends to infinity that actually this is equal to t if you write it our p of function of t most basically p of function of t if you write it and if you consider the our state weighting matrix q is constant which is which is also positive semi definite matrix and our input weighting matrix are is also what is called constant and assume it is a positive definite matrix. So, since t f is infinity then most of the time after the after the transient period is over this is constant. So, during this period from t 0 to if you see from t 0 to sufficient long time of this one p is constant. So, this term will be a 0 then I can replace that one is a transpose this p of t is constant p then p of t is constant a then p b are inverse b transpose p plus q this.  That dimension is n cross n n is the n is the order of the order of the dynamic systems systems and this is called this whole equation is called algebraic Riccati equation non-linear more more specifically going to write non-linear algebraic Riccati equation because is no dynamic equation is involved p dot is not involved for infinite time regulator problems. So, now question is if you must know how to solve that one if you can solve this one then our problem is solved. That means we know our that u is equal to minus r inverse b transpose p p is constant because most of the from t 0 to sufficiently long term this is constant only short period of time this value of p is changing with respect to time. So, this is the transient period and from here to here is a steady state value of p steady state value of p this is the. So, that u of t is equal to 0.  If it is not known we have to by some means we have to estimate or you have to get the information of the states which will call estimation or observer we one has to introduce the observer to estimate the state of this system.   So, our first step is solution of non-linear algebraic equation this is the non-linear you solve it you will get p.  Once you know k I know what is u and then you just recursively you solve this one or in real time what you will get you get the information of x multiplied by k. k you have done you have solved this equation it gets an offline and you know the information of k. So, multiplied by x. So, it is a state feedback but controller k is designed based on minimizing the what is called our objective function this which is called a infinite time regulated problems that due to some initial state disturbance this controller will drive the state to a equilibrium position or equilibrium point. So, if you see this one that what is our algorithmic steps algorithmic steps for finite time infinite time l q r problem or horizon l q r problems infinite time horizon control problems or you can say infinite time l q r problem solution for this one. So, first thing we have made the assumption we have to check it the assumption the pair a and b first assumption is pair a and b is pair a is controllable and a and c is detectable. So, what is controllable you might have read it in your first course of linear control theory that with the if the system is controllable by using and control f out we will be able to drive the state at time t is equal to 0 to a desired state at t is equal to t f at desired state we will be able to drive it if the system is controllable or at least it is a stabilizable and this a and c must be a detectable of this this one. So, this is the first assumption if this assumption is valid then and also we have made the assumption if you recollect that our state weighting matrix this is the q is the state weighting matrix state weighting matrix associate in the performing index that must be a positive semi definite and the input weighting matrix input weighting matrix must be positive definite. So, this thing our first step with this assumption if the system is not controllable then we will not proceed further that means we are the system must be controllable or at least stabilizable. Step one that compute the open loop system may be unstable, but it must be controllable or at least stabilizable. So, compute the solution of p form the algebraic edicati equation. So, a transpose p plus p a minus p b or inverse b transpose p plus q is equal to null matrix whose dimension is n cross n and if x is the number of states which number of states is n small n order of the system is n then a is the dimension n cross n b is the dimension depends on the number of inputs u. If u is m the dimension of b n cross m the dimension of b n cross m and the dimension of a is a n cross n order of the system then this immediately this dimension is n cross n the dimension of r is a m cross n it depends on the number of inputs and one has to solve this one. We will discuss later then how to solve the algebraic edicati equation which is a non-linear algebraic edicati equation what are the methods are available to solve such type of algebraic edicati equation. So, once you compute this one from this step you can compute that one the step two once I compute the solution of edicati equation solution of our equation then step two compute the controller gain compute the feedback gain matrix k is equal to r inverse b transpose p you know this is known by solution of this is nothing but a solution of algebraic edicati equation. So, this is known once you know this one you can compute step four step three you can compute the controller u of t is equal to minus k x of t assumed all the states are measurable agree all the states are measurable. Then once you know this one once you know this one then immediately you can find out that what is called the our cost objective function value that compute the optimum or compute the minimum value of the j that means j star I can compute j star is equal to half x transpose of t 0 m t 0 is equal to 0 then this then solution of the edicati equation p then x of 0. So, this you can compute. So, this is the steps of how to solve the what is called infinite time regulator problems this is the solution. So, if you see the stability analysis of the closed loop system it is exactly same that what we have consider in the finite time regulator problems. So, stability analysis of the closed loop C L closed loop system bracket for infinite infinite time L q r infinite time regulator problems that is in short it is called L q r infinite problem. So, this stability so let us see the stability of the system our system description is a is x of t a is the system is linear time invariant systems. So, b of t and our system state is given initial state is given if you see our control law u of t is equal to minus x of t where k is we have r inverse b transpose p this way you have generated p. So, what is the closed loop systems closed loop system is if you just write it use the value of k in this expression u k x in this expression then it is a closed loop system x dot closed loop system x dot is equal to a x of t plus b what is u minus k x of t. So, this is if you simplify adjust simplify this one a minus b k whole x of t. Now, let us call this is the equation number 1 and this is the equation number 2. So, this is our closed loop system in short we have denoted by a suffix c closed loop of x t you have denoted by this. Now, look at this one what is the what is the how you check it whether the system is stable or not one way of checking it since you have calculated the k put the value of k here find out the Eigen values of this one. And analytically how we will check the I told you exactly same way what we have described for finite time regulator based on the Lyapunov function technique we can check it the stability of the system. So, now consider the simplest Lyapunov function is like this way v of x t is equal to x transpose of p. And this is the quadratic form this is energy function is always what is called value is greater than equal to 0 when x is 0 will be when the states are all 0 then it will be a 0. And this value will be always greater than 0 greater than equal to 0 because p is a positive definite matrix p is a positive definite matrix. So, we have selected the energy function in like this way now v dot let us k. So, if energy if due to some initial condition the energy contain in the system is that one. Now, I will see whether this energy is decreasing with respect to time or not that how to v dot we have to find out with time as time is increasing whether the energy is decreasing or not that means v dot must be negative definite. So, what is v dot differentiate this thing with respect to time then you will get x dot t of t p x dot of t plus x transpose of t then p is constant we are not differentiating this p is constant. In earlier case p was a finite time regulator p was a function of we got a three terms, but here is two terms. So, put the value of x dot from equation two use equation two in equation three in three then we got v dot is equal to x of t is equal to a minus c this is our equation two x dot is equal to this. So, x dot transpose so it will come x transpose a minus a minus v k whole transpose. So, I am writing x transpose a minus v k whole transpose then p multiplied by x t this is the first term I am written here then next is this term I am writing here this term I am written here. So, this term is x transpose of t p then what is this one a minus v k into x of t this is x of t. So, this is this two terms I have written x transpose p and this is our x dot I have written.  So, the v dot v dot is a scalar function this will be negative definite provided this is a negative definite matrix this is a negative definite matrix and p is the solution of the Riccati equation this. So, this matrix if you can prove it that this is the negative definite matrix then our stability analysis proved that for infinite time regulator problems. Let us call this equation is 4. So, recall our recall that algebraic Riccati equations.  So, now I will just see what I am doing it here that b p b are inverse I am taking p common from the right post and inverse b transpose then whole multiplied by p.  Then you see a transpose p b are inverse b transpose this is also one then p a this is also one, but there is one additional term p minus p b are inverse b transpose. So, I will just add another term of that one p b are inverse b transpose p plus this term. So, this our equation just remain same now what we can write it this one you see a minus what you can write it for this one a minus if you just write it the transpose of that one. So, a minus b are inverse b transpose p whole thing transpose into p. So, first term you see as it is.  So, this is same as that one plus p then a minus b are inverse b transpose p and this equal to take this is that side minus q then minus will be plus because minus I am taking common. So, p b are inverse b transpose p. So, if you see this quantity is nothing but our k.  Now see the left hand side I can easily replace that let us call this is equation 5 using equation 5 in 4 in this equation using equation 5 in 4 if at this term I can replace this term is same as this one I can replace by the right hand side. So, equation 4 write it here not here equation 4 equation 4 can be rewritten using the right hand side. Equation 5 as what you can write it that equation 4 you see this equation 4. So, v dot equation 4 is v dot. So, v dot x of t of t this one is x transpose of t then I am writing minus.  So, this equation I replaced this left hand side is replaced by that one in equation 4. So, I got it that one now this change in Lyapunov function will be must be negative in order to become the system is stable. So, is a specific with a negative sign. So, this matrix must be this must be positive definite I can in the similar manner what we have shown in the finite time regulator problem same thing we can use here. So, since our assumption q e is q e is our greater than equal to 0 q e is greater than equal to positive semi definite r is positive definite matrix. Say if r is a positive definite matrix r inverse also positive definite matrix and it is specific with a some matrix and pre multiplied by some matrix post multiplied by same matrix transpose. So, this is the matrix you can say this is the matrix. So, r inverse is multiplied by pre multiplied by some matrix the combination of p and b and post multiplied by transpose of that one. So, if it is so then we can write we can also say the combined matrix is also positive definite sin r is the positive definite matrix. So, resultant of this one since is the positive definite matrix and since it is a positive semi definite matrix resultant will be positive definite matrix. So, this includes that v dot of t of t is less than 0 because this is positive negative sign is less. So, this implies that our the closed loop system is asymptotically stable. In the sense at time t is equal to 0 the states will come back to what is called equilibrium position even though there is initial condition initial disturbances given to the system at t is equal to t 0. So, the system is asymptotically stable. Next question comes in mind that how to solve that what is called a algebraic rickety equations. There are different methods are there one some methods are what is called iterative method to solve the algebraic rickety equation. Some methods are used for numerically more reliable method to solve the algebraic rickety equation. So, next class we will discuss the methods for solution of methods for solution of algebraic rickety equation. So, this is the method how this algebraic rickety equation is solved. And also we will discuss suppose a problem is given that whatever we consider the finitely regulator in state of t is equal to 0. Now, t f sorry t t is equal to t f is equal to finite time now t f is equal to tends to infinity. And our then correspondingly our rickety matrix rickety equation matrix rickety dynamic matrix rickety equation will be converted into a algebraic rickety equation. And we have to solve the algebraic rickety equation. And if you get the solution of p is positive definite we have to solve this rickety equation in such a way. So, that p must be a positive definite matrix. And then correspondingly what the solution will get it that controller gain that controller gain will stabilize the system or the states will bring down to with the controller for the states will come to equilibrium states again. So, we will stop it here now.