From around the globe, it's theCUBE with digital coverage of AWS re-invent 2020. Sponsored by Intel and AWS. Hi, we are theCUBE live covering AWS re-invent 2020. I'm Lisa Martin and I've got one of our CUBE alumni back with me, Mike Miller is here, general manager of AWS AI devices at AWS. Mike, welcome back to theCUBE. Hi, Lisa, thank you so much for having me. It's really great to join you all again at this virtual re-invent. Yes, I think last year you were on set. We have always had two sets at re-invent and you had the deep racer car. And so we're obviously socially distanced here, but talk to me about deep racer, what's going on some of the things that have gone on in the last year that you're excited about? Yeah, I'd love to tell you a little bit about what's been happening. We've had a tremendous year. Obviously COVID has restricted our ability to have our in-person races. So we've really gone gangbusters with our virtual league. So we have monthly races for competitors that culminate in a championship at re-invent. So this year we've got over a hundred competitors who have qualified and who are racing virtually with us this year at re-invent. They're participating in a series of knockout rounds that are being broadcast live on Twitch over the next week. That'll whittle the group down to a group of 32, which will have a series of single elimination brackets leading to eight finalists who will race Grand Prix style, five laps, eight cars on the track at the same time and will crown the champion at the closing keynote on December 15th this year. Exciting, so you're bringing a re-enforcement learning together with sports that so many of us have been missing during the pandemic. Talk to me a little bit about some of the things that you've improved with DeepRacer and some of the things that are coming next year. Yeah, absolutely. So first of all, DeepRacer not only has been interesting for individuals to participate in the league, but we continue to see great traction and adoption amongst big customers. And they're using DeepRacer for hands-on learning for machine learning. And many of them are turning to DeepRacer to train their workforce in machine learning. So over 150 customers from the likes of Capital One, Moody's, Accenture, DBS Bank, JP Morgan Chase, BMW, and Toyota have held DeepRacer events for their workforces. And in fact, three of those customers, Accenture, DBS Bank, and JP Morgan Chase have each trained over a thousand employees in their organizations because they are just super excited and they find that DeepRacer is a way to drive that excitement and engagement across their customers. We even have Capital One expanded this to their families. So Capital One ran a DeepRacer Kids Cup, a family-friendly virtual competition this past year where over 250 children and 200 families got to get hands-on with machine learning. So I envisioned this being a big facilitator during the pandemic when there's been this massive shift to remote work has you seen an uptick in it for companies that talking to me about training need to be able to hire many, many more people remotely but also train them as DeepRacer facilitator of that? Yeah, absolutely. DeepRacer has a core component of the experience which is all virtualized. So we have a console and integration with other AWS services so that racers can participate using a 3D racing simulator. They can actually see their car driving around a track in a 3D world simulation. We're also selling the physical devices. So if participants want to get one of those devices and translate what they've done in the virtual world to the real world, they can start doing that. And in fact, just this past year, we made our DeepRacer car available for purchase internationally through the amazon.com website to help facilitate that. So how many DeepRacers are out there? I'm just curious. Oh, thousands. And what we've seen is some companies will purchase them in bulk and use them for their internal leagues just like JP Morgan Chase and DBS Bank. These folks have their own tracks and racers that they'll use to facilitate both the in-person as well as the virtual racing. I'm curious, with this shift to remote that we mentioned a minute ago, how are you seeing DeepRacer as a facilitator of engagement? You mentioned engagement and that's one of the biggest challenges that so many teams and DevOps processes have without being co-located with each other. Does DeepRacer help with that? I mean, from an engagement perspective? I think so. What we've seen is that DeepRacer is just fun to get your hands on and we really lower the learning curve for machine learning and in particular, this branch called Reinforcement Learning, which is where you train this agent through trial and error to learn how to do a new complex task. And what we've seen is that customers who have introduced DeepRacer as an event for their employees have seen a very wide variety of employee skill sets kind of get engaged. So you've got not just the hardcore deep data scientists or the ML engineers, you've got web front-end programmers. You even have some non-technical folks who want to get their hands dirty and learn about machine learning. And DeepRacer really is a nice gradual introduction to doing that. You can get engaged with it with very little kind of coding knowledge at all. So talk to me about some of the new services and let's look at some specific use case, customer use cases with each service. Yeah, absolutely. So just to set the context, Amazon's got hundreds, AWS has hundreds of thousands of customers doing machine learning on AWS. Customers of all sizes are embedding machine learning into their core business processes. And one of the things that we always do at Amazon is we're listening to customers, 90 to 95% of our roadmaps are driven by customer feedback. And so as we've been talking to these industrial and manufacturing customers, they've been telling us, hey, we've got data, we've got these processes that are happening in our industrial sites. And we just need some help connecting the dots. Like how do we really most effectively use machine learning to improve our processes in these industrial and manufacturing sites? And so we've come up with these five services that are focused on industrial manufacturing customers. Two of the services are focused around predictive maintenance and the other three services are focused on computer vision. And so let's start with the predictive maintenance side. So we announced Amazon Monotron and Amazon Lookout for Equipment. So these services both enable predictive maintenance powered by machine learning in a way that doesn't require the customer to have any machine learning expertise. So Monotron is an end-to-end machine learning system with sensors, a gateway, and an ML service that can detect anomalies and predict when industrial equipment will require maintenance. And I've actually got a couple examples here of the sensors and the gateway. So this is Amazon Monotron. These little sensors, this little guy is a vibration and temperature sensor that's battery operated and wireless connects to the gateway, which then transfers the data up to the ML service in the cloud. And what happens is the sensors can be connected to any rotating machinery like a pump or a fan or a compressor and they will send data up to the machine learning cloud service, which will detect anomalies or sort of irregular kind of sensor readings and then alerts via a mobile app, just a tech or a maintenance technician at an industrial site to go have a look at their equipment and do some preventative maintenance. So it's super streamlined end-to-end and easy for a company that has no machine learning expertise to take advantage of. Really helping them get on board quite quickly. Yeah, absolutely. It's simple to set up. There's really very little configuration. It's just a matter of placing the sensors, pairing them up with the mobile app and you're off and running. Excellent, I like easy. So some of the other use cases? Yeah, absolutely. So we've seen, so Amazon fulfillment centers actually have enormous amounts of equipment. You can imagine the size of an Amazon fulfillment center, 28 football fields long, miles of conveyor belts and Amazon fulfillment centers have started to use Amazon monotron to monitor some of their conveyor belts. And we've got a fulfillment center in Germany that has started using these with a thousand sensors and they've already been able to do predictive maintenance and prevent downtime, which is super costly for businesses. We've also got customers like Fender, who makes guitars and amplifiers and musical equipment here in the U.S. They're adopting Amazon monotron for their industrial machinery to help prevent downtime, which again, can cost them a great deal as they kind of hand manufacture these high-end guitars. Then there's Amazon Lookout for Equipment, which is one step further from Amazon monotron. Amazon Lookout for Equipment provides a way for customers to send their own sensor data to AWS in order to build and train a model that returns predictions for detecting abnormal equipment behavior. So here we have a customer, for example, like GPS in South Korea, or I'm sorry, GS EPS in South Korea, they're an industrial conglomerate and they've been collecting their own data. So they have their own sensors from industrial equipment for a decade and they've been using just kind of basic rules-based systems to try to gain insight into that data. Well, now they're using Amazon Lookout for Equipment to take all of their existing sensor data have Amazon for Equipment automatically generate machine learning models and then process the sensor data to know when their abnormalities or when some predictive maintenance needs to occur. So you've got the capabilities of working with customers and industry that don't have any ML training to those that do have been using sensors. So really everybody has an opportunity here to leverage this new Amazon technology, not only for predictive, but one of the things I'm hearing is contactless, being able to understand what's going on without having to have someone physically there unless there is an issue in contactless is not one of the words of 2020, but I think it probably should be. Yeah, absolutely. And in fact, that was some of the genesis of some of the next industrial services that we announced that are based on computer vision. What we saw and what we heard when talking to these customers is they have what we call human inspection processes or manual inspection processes that are required today for everything from monitoring, like workplace safety to quality of goods coming off of a machinery line or monitoring their yard and sort of their truck entry and exit and they're looking for computer vision to automate a lot of these tasks. And so we just announced a couple of new services that use computer vision to do that, to automate these once previously manual inspection tasks. So let's start with AWS Panorama, uses computer vision to improve those operations and workplace safety. And AWS Panorama comes in two flavors. There's an appliance, which is a box like this. It basically can go get installed on your network and it will automatically discover and start processing the video feeds from existing cameras. So there's no additional capital expense to take AWS Panorama and have it apply computer vision to the cameras that you've already got deployed. So customers are seeing that computer vision is valuable but the reason they want to do this at the edge and put this computer vision on site is because sometimes they need to make very low latency decisions where if you have like a fast moving industrial process you can use computer vision but I don't really want to incur the cost of sending data to the cloud and back. I need to make a split second decision. So we need machine learning that happens on premise. Sometimes they don't want to stream high bandwidth video or they just don't have the bandwidth to get this video back to the cloud. And sometimes there are data governance or privacy restrictions that restrict a company's ability to send images or video from their site offsite to the cloud. And so this is why Panorama takes this machine learning and makes it happen right here on the edge for customers. So we've got customers like Cargill who uses or who's going to use Panorama to improve their yard management. They want to use computer vision to detect the size of trucks that drive into their granaries and then automatically assign them to an appropriately sized loading dock. You've got a customer like Siemens Mobility who works with municipalities on traffic and other transport solutions. They're going to use AWS Panorama to take advantage of those existing kind of traffic cameras and build machine learning models that can improve congestion, allocate curbside space, optimize parking. We've also got retail customers. For instance, Parkland is a Canadian fuel station and retailer like a little quick stop and they want to use Panorama to do things like count the people coming in and out of their stores and do heat maps like where are people visiting my store so I can optimize retail promotions and product placement. That's fantastic. The number of use cases is just, I imagine if we had more time, like you could keep going and going, but thank you so much for not only sharing what's going on with DeepRacer and the innovations, but also for show and tell, even though we weren't in person at Reinvent this year. Great to have you back on theCUBE, Mike. We appreciate your time. Yep, thanks Lisa for having me. I appreciate it. For Mike Miller, I'm Lisa Martin. You're watching theCUBE's live coverage of AWS Reinvent 2020.


So, last class first we have discussed the time optimal control means initial state, if the initial state x of 0 is here, we have to drive the state by proper control action so that it reached to the origin x t f, x t is equal to t f it reached to the origin, but that time requires must be optimal. So, we have discussed the time optimal control by using bang-bang control.  The time optimal control we have discussed then we have seen the how to really implement in hardware or in a software how to implement the bang-bang control to drive the initial state to final state in a minimum time. Next we have considered if you recollect that next we have considered that suppose we have a constraint on the control input and also in the state variables, but we are not dealing with the time optimal control, but minimizing the performance index which involved in minimizing the control effort or control energy that we want to minimize. So, our statement of the problem is that consider first let us call consider the system A x dot is equal to A x plus B u. Our problem is our problem is to minimize the performance index which is equal to the what is called your control effort minimization subject to the constraint that u of t magnitude belongs to u of A that for all t 0 to t f let us call this as t f and the constraint on the state variable is this one this is the constraint in the state variable we have made at l constraints are there and this constraint are differentiable with respect to t that is the time. So, our problem statement in this case is given the system which is described in state phase model our problem is to minimize the control effort or the control energy we have to minimize subject to the condition condition on input and also constraint on the state. Then this formulation is one of the simplest technique is that you introduce a new variable that if the number of the states in the system is n that we have discussed is here if it is a n then we introduce one additional variable let us call n plus 1 th state variable we introduce that this one that n plus 1 dot is equal to this is the constraint on the state the first constraint square multiplied by u s and u s is nothing but a strictly unit step function similarly, second constraint square multiplied by another unit strictly unit step function this step function when this psi of 2 value is greater than 0 then this value is 1 when it is less than equal to 0 that function value is 0. Similarly the l th constraint we have written that constraint square into strictly unit step function this. So, this is the new additional state is introduced state introduced this. So, we have made the all inequality constraint now is a equality constraint of that one what we have already discussed about the u s that is this u s is the when it is less than equal to 0 this value is 0 and when it is greater than 0 that it is a strictly unit and step function means 1. Now, look at this expression of 3 and 4 suppose this may be a situation that psi value some cases may be 0 less than equal to 0 or some cases 0 some cases may be greater than 0. So, whatever the this situation is there this expression is x dot of t that expression is less than less than 0 or less than equal to 0 this for all t this one. The another situation may arise this situation will arise when all the constraints are not satisfied then this situation is 0 when all the constraint state constraints are satisfied under this condition this will be equal to 0 x n plus 1 dot t is 0 for all times when all the constraints are satisfied. So, yesterday only we have come up to this point. So, we will now continue rest of the part. So, now if you now specify the boundary condition for if now we specify if we specify now specify the boundary condition for equation 3 the equation 3 is that one equation 3 is this equation the new additional state vector this is equation. Now, you introduce some specific boundary condition for equation 3 that x n plus 1 of t 0 initial time t 0 is equal to 0 and x n plus 1 t f is equal to 0 because this our things all the inequality constraint must be satisfied. So, at the terminal condition that must be satisfied that means this we made an assumption that boundary condition of x 1 t 0 0 and this is 0. If this is 0 and this is 0 then immediately I can write if x n plus 1 of t 0 is equal to 0 and x n plus 1 t f is equal to 0 then x n plus 1 t f is equal to 0. We can write x n plus 1 t is equal to 0 for all t 0 for all t belongs to t 0 to t f. How it is because if you see the dynamic equation x n plus 1 the dynamic equation this is equal to 0 because it must satisfy this is equal to 0 when all the constraints are satisfied. If you integrate this one you will get it this equal to what is called 0 provided x t and 0 is 0 and final condition also 0. So, for all the time this will be 0. So, this is x n plus 1 t is the solution of equation 3 if x n plus 1 dot 0 is equal to 0. So, x n plus 1 dot of t is equal to 0 for all t belongs to t 0 to t f this is the condition for this one. So, now we know that at x n t f also 0 x t 0 is 0 with this assumption we can make it x n plus 1 t is 0 for all time t if x dot of t is 0 when it will be 0 if all the constraints are satisfied from 0 to t f under this question. So, solution of the problem along with the solution of that problem along with the that the state equation constraint and additional boundary condition this is the boundary condition and with the additional boundary condition we will be able to solve the problem. Which problem that problem is given the state equation and we have to minimize a performance index means minimizing the control effort or control energy subject to the what is called constraint on the input as well as constraint on the state that problem we will be able to solve with if there is a constraint on the state we have to introduce a new additional state variables x n plus 1 and with the knowledge of the constraint equation we can write x n plus 1 dot is equal to that expression x n plus dot is equal to that expression and this expression x n dot plus 1 this expression is equal to 0 when all the constraints are satisfied over the interval t 0 to t f this condition. And with this assumption we made it we specify the boundary x t n 0 is 0 and x t f is 0 with this condition with this condition boundary condition of this one this x t n is 0 provided x n plus 1 t is 0 for all time this all time this is equal to when it will be 0 when the inequality constraint inequality constraint is satisfied. So, let us solve a problem of such type of problems that mean when input there is a constraint on the input as well as there are constraint on the state variables. So, example so minimize j is equal to 0 to t f half t square of t u t square or d t. So, you see here clearly this is not the time optimal control that means we do not want to drive the state from initial state to final state in a minimum time. Our objective is to minimize the control effort u square t control effort or control energy over the interval 0 to t f and at t f the state will the original state will drive to the final state at origin and what should be the sequence of control law for that situation. And subject to what x 1 dot is equal to x 2 this is the dynamic equation of the system and x 2 dot is equal to minus 2 x 1 of t plus u of t and our initial condition is given x of 0 which is nothing but a x 1 of 0 plus x 2 of 0 which is given as a 0 and 1. So, what is our problem transform transfer the initial state to the final state x 1 dot is equal to x of t f x 2 of t f is equal to 0 in minimum control effort with the following constraints. So, this is our problem what is the constraints are given this constraint is first constraint is u of t mod must be equal to less than 1 that mod of u is equal to minus 1. So, mellow will be less than 1 for all t 0 to belongs to all t belongs to for all t belongs to 0 to t f and second condition is state variable constraint state variable constraint what is this constraint x 1 is greater than equal to 0 and minus 1 x 2 is greater than minus 1 less than is equal to 2. So, this type of problems we have done in static optimization problem where we are minimizing the performance index subject to the constraints agree and there that equality constraint inequality constraint equality and inequality constraint was there. So, we have solved that type of problems earlier, but now this problem is dynamic optimization problem in sorry not dynamic dynamic that is the dynamic optimization problem our problem is to minimize the that performance index is nothing but a control effort minimize control effort. Subject to the constraint this is the dynamic equation and what we have to do that transform the initial state x what is this x of 0 initial state that is x of 0 to the final state at origin in minimum control effort with the following state constraints that following constraints input constraint and the state constraint. Next question is let us call this is the equation number one this is the equation number two and this constraints are equation three and this is equation four agree. So, our problem how to solve this problem. So, we have told you that all inequality constraint we have to write in this structure phi psi of this less than equal to 0 psi i i is equal to for our case i is equal to 1 2 3 now how psi 1 I am reading. So, x 1 is solution. So, x 1 now solution. So, psi 1 first I am writing psi 1 is what psi 1 x of t of t is nothing but a our things is what we can write this equal to that minus x 1 of t that is less than equal to 0. So, this problem we have to solve by augmenting one state x and plus one state we have to augment before that I am writing that inequality constraint in the usual form I am writing psi 1 is x 1 another condition is that one if you take the x t in this side that minus 1 minus this is x 2 minus 1 minus x 2 less than equal to 0. So, psi 2 of x of t of t is equal to minus x 2 of t minus 1 bracket close is equal to less than equal to 0. This is the second condition inequality constraint in the state third you can write it x 2 2 you bring it that side is less than equal to 0. So, psi 3 of x t of t is equal to x 2 minus 2 is that equal to less than equal to this and that is equation number 5 and each constraint is differentiable with respect to x of t. So, if you write it better small x because I have written the small x possibly yes I have written small x is equal to x 1 x 2. So, write it small x so that all this thing is differentiable with respect to x of t and time this is the thing. Now, we will form the new additional state the new additional state the new additional state. So, we have a secondary system x 1 and x 2. So, third state is x 3 dot is equal to we will write it minus that psi 1 of x of t comma t this whole square into u of s psi 1 x of t of t then minus psi 2 x of t and t whole square u of s psi 2 x of x 1 this is x. So, write it x of t or t this whole square of t this is function of x means x 1 x 2 this we have seen the inequality constraint is a function of x 1 x 2, but in your case some cases is a function of x 1 only some cases function of x 2. And some cases may be function of x 1 and x 2 also that if it is state constraint either it will be a state constraint is there either that the way we are writing here is a function of x 1 or function of x 2 if you this way. So, you can write more specifically it is a function of x 1 that second one is a function of x 2 and third is psi 3 x of 3 of t whole square u of s psi 3 x of t into comma t. So, you know already what is the value of this value this what is psi 1 of x just now we have calculate from this psi 1 of x is minus x 1 t minus x 1 t. So, psi 1 is minus x 1 t. So, minus x 1 t square that means x 1 square of t into u s psi 1 x t of t.  So, it will be ultimately square if you take x 2 of t plus 1 whole square u s psi 2 dot function of x of comma t. Then third one is this is x 2 minus 2 value x 2 minus 2. So, x 2 minus 2 whole square u s psi 3 of 3. Let us call this is the equation number 6. So, we have written the equality constant values here and square this all this case and this turns out to be like this way. Now, so equation you see the equation 6 is 0 when all the constant is 0 when all the constant in 5 that means all the constant in 5 this are true. If all the constant psi 1 psi 2 is true that means it is less than equal to 0 that means this value will be 0 this value will be 0 this value will be 0. So, equation 6 is 0 when all the constant state constant state variable constants are satisfied. When the state variable constants are satisfied less than 0 that this quantity is less than 0 that means that step function strictly step function is valid when this x psi of t value is greater than 0 or greater than 0 then it is valid. And equation 6 and equation 6 is less than equal to 0 when the constants are not satisfied when for all t when for all t the for all t when constants 5 are not constant 5 are not satisfied. So, this is the we made a conclusion from this one that that equation say x 3 dot is 0 when all the constants will satisfy x 3 dot is not equal to is less than equal to 0 is not equal to 0 means less than equal to 0 when this equal to that is constant are not satisfied. Now, we specify our boundary conditions specify the boundary condition what is the boundary conditions x 3 of 0 our t 0 is 0 is 0 and x 3 t f which you do not know is equal to 0. Then immediately you can write it then x 3 t is equal to 0 since for all t belongs to that t 0 to t f t 0 to t f since x 3 is equal to x 3 dot of t is equal to 0 why x 3 is equal to 0 because during this interval if all constants are satisfied then x 3 is 0. Now, you consider the Hamilton equation that our Hamilton Hamiltonian Hamiltonian equation including the new stage new state variable including the new state x 3 of t. So, now what is the Hamiltonian function Hamiltonian equation or Hamiltonian function for Hamiltonian function not equation function what is this one h x 3 of t this now dimension is x 3 that means x 3 is also involved x 3 of t then u of t then lambda 1 is equal to lambda 1 because 3 states are there 3 multipliers are there I am writing this one 3 multipliers and then t what is this one see the performing index our that integral the part inside the integral is that one. So, you have to write it this one half u square plus lambda 1 lambda 2 lambda 3 which is a function of time lambda transpose means lambda 1 to this multiplied by the state equations x dot is equal to x plus b u that I am writing state equation that first state equation x 1 dot is equal to x 2 this I am writing now you just say that I am writing this equation this equation x 1 dot this I can write it x 1 dot x 2 dot x 3 dot in terms of a x plus b u form. So, I am writing this multiplied by x 2 of t the next equation x 2 dot is equal to minus 2 x 1 of t plus u of t plus third equation is if you see the third equation this is that one I am writing. So, that is it will lend the expression, but I am writing let us call I am writing with a star and next page I am writing that what is that star that means 3 1 position that is the 3 1 position the space will I will not get it. So, I will write it that 3 1 position then what is this position is that one minus see what I am writing it that one minus x 1 square minus x 1 square u of s. So, I am writing minus x 1 the psi 1 square dot square then what is this one u s is psi 1 x is what minus x 1 of t. So, this minus psi 2 dot square u s and that value is you can write it that psi 2 or you can write it psi 2 psi 2 value is what minus x minus minus x 2 minus 1 that is the psi 2 value then this then third is minus psi 3 square dot u s what is this psi 3 value psi 3 value is x 2 minus 2. So, this is the this is the value of star that means 3 1 position that is the value of star that means 3 1 position that means 3 1 that that is the value of star. So, now you see from this expression what is the Hamiltonian function that half u square t lambda 1 t lambda 1 t multiplied by x 2 lambda 2 to e multiplied by minus x 1 t plus u t lambda 3 t multiplied by star that one. So, I am writing this Hamiltonian function h which is equal to that x of 3 which one x of t u of t lambda 1 t lambda 2 t lambda 3 t and t which is equal to I am writing half u square minus lambda 1. Now, see this one what is the value of lambda 2 into lambda 1 into x 2 lambda 1 into x 2 plus lambda 1 into x 2 plus lambda 2 into what is the value of that lambda 2 this one lambda 2 sigma psi 2 value is what psi 2 value is minus 1 lambda 2 psi 2 value is minus x 2 psi 2 value you just see it here the very psi 2 value is you see the psi 2 value. So, minus x 2 minus 1 so if you take minus common x 2 plus 1 minus specific with minus that square.   So, see this one this lambda 1 into x 2 lambda 1 into x 2 I have written this one then this into this lambda 2 into minus 2 x 1 plus u 2 lambda 2 minus 2 x 1 plus u 2 this then lambda 3 next is lambda 3 lambda 3 is lambda 3 into that 3 1 position is that that one. So, this is the minus lambda 3 t and this will be what psi 1 square means x 1 square psi 1 is minus x 1. So, that square is minus x 1 square u s minus x 1 of t then minus if you take the minus common from this this will be plus psi 2 is your what minus 1 minus x 2 that means if you take minus common and square that will be plus.  So, this is this is the Hamiltonian function we have written now let us call that is the equation number 8 what is the equation number is this is 7 6 this is the equation number 6 we have given no this is the page number and last equation number is 5 and this is 6 and this is the equation number 5 and that is we have given the equation number 6 let us call this is we give the equation number 7 this all boundary condition equation number 7 and this is the equation number 8. Now, we can write our necessary condition the function to be optimized that means control afford to be minimized. So, what is the our problem is minimize with respect to minimize this with respect to all admissible control vector or in our case is a scalar in your case scalar because single input we have considered that u of t belongs to u of a for all t 0 to t f all for all for all t belongs to 0 to t f again that is for constant. So, minimize 0 with respect to all admissible control vector this to obtain u star of t in terms of x of t lambda of t and t. So, let us see what is our necessary condition del h del u of t is equal to 0 and which implies if you see the del h del u if you differentiate this Hamiltonian function with respect to u that means u of t is equal to this term there is no term u here is u. So, lambda of t there is u term is first term and third term no other terms u is involved. So, this will be a u of t is equal to u of t plus lambda t u of t plus lambda t lambda 2 t is equal to 0 and therefore, u of t is equal to minus limit 2 of this. So, let us call this is equation number 9 since our according to the our problem since u of t mod is less than equal to 1 according to our problem is less than equal to our 1. This expression that u of t value will get less than 1 when mod of lambda 2 t is less than 1 when mod of 1 for lambda 2 t is less than 1 when mod of 1 for lambda 2 t is less than 1 when lambda 2 of t mod will be less than equal to 1. So, then what is our control sequence is there suppose if it is a less than 1. However, when u of t when lambda 2 of t mod is greater than 1 the following control minimizes H Hamilton function. What is this? Now look at this is the condition necessary condition is there since we know the mod of u of t is less than equal to 1 this will be less than equal to 1 when lambda 2 mod is less than equal to 1, but there what is the guarantee that lambda 2 always will be less than equal to 1. It may be a situation that lambda 2 of t mod may be greater than 1 in that situation what should be the control law we should adopt. So, to minimize the what is called the Hamiltonian function. So, our control strategy should be like this way this equal to minus 1 for lambda 2 of t when is greater than equal to 1. If it is greater than equal to 1 now you see here if it is greater than equal to 1 now you see I am replacing u is minus 1. So, it is within the limit of that constant 1 control input now this is plus 1 for lambda 2 is less than minus 1, but it is lambda 2 of 2 for if for or if minus 1 greater than equal to lambda 2 will greater than minus 1 is less than equal to 1. So, this is nothing but a set function this is we call. So, we will apply the control sequence like this way if lambda 2 is mod of lambda 2 is less than equal to 1 we will apply whatever the lambda 2 is there we will apply same thing there.  So, this is the control so set function this is called set function is defined like this way it is minus 1 when x is greater than 1 x of t it and when x of mod of x t is less than equal to 1 when it is a plus minus this is a plus 1 when it is sorry it is a plus 1 when you greater than this and when it is a this minus 1 when x of t is less than minus 1. So, when it is x of t x of lambda t is greater than 1 this is a plus 1 when this is less than this is minus 1 and this is when it is a this and it will be a simply lambda x of t not this x of t this is the set function we can write it. So, now see this situation when x is greater than 1 when it is greater than 1 we are putting is plus 1 that means u is minus 1 when it is less than x 2 is less than 1 then we are putting with minus minus means plus. So, in this way we are making the control sequence of that one now how to solve this problems our problem is now to solve once again I am telling you this that this is the necessary condition of this one now you see that this we got it lambda. So, u of t will be minus lambda of t so in order to make this thing what is called that mod of u less than 1 the mod of lambda 2 must be less than equal to 1 then we will get it that 1 mod of this less than equal to this, but when suppose lambda of t is positive lambda of t is positive then u is is negative u is coming negative. Suppose lambda of t less than 1, but negative then u of t is coming positive. So, in this way you have to see now when we will do this one now let us see this one lambda 2 is greater than 1 if lambda 2 is greater than 1 then I have to make it lambda 2 if it is greater than 1 then I have to make it with a minus sign sorry I made it this mistake here minus sign because u t is equal to minus of lambda t when lambda 2 is less than 1 less than 1 means that minus 2 let us call again. So, I have to make it is a positive sign u of t is positive 1 I have to switch to positive 1 only. So, because if it is a less than minus 1 that minus minus plus so I have to switch to minus 1 because I have to I need not to cross the limit of bound of 1.  Anyway our problem is when x is in within the range when lambda 2 is within the range minus 1 to plus 1 which switch to lambda 2 and this. So, now next what we have to do it then we have to solve these problems.  Now, this we got it now you find out the state equation the another what is the rest of the necessary condition del h del lambda of t is equal to del lambda of t is equal to x dot of t if you do this one then you will get x 1 dot is equal to x 2 of t x 2 dot is equal to minus x 1 of t plus u star of t plus x 3 dot of t is equal to minus x 1 square of t plus u s minus x 1 of t minus x 2 of t plus 1 whole square u of t minus x 2 of t minus 1 then minus x 2 of t minus 2 minus x 2 of t minus whole square u of s x 2 of 2 this one that is what we will get it this is the state equation and then you find out costate vector equation costate vector equation how will you take costate vector equation then del h del lambda of t is equal to x dot of t minus x 2 of t minus that is your del x of t is equal to minus lambda dot of t. So, just differentiate what is called Hamilton function with respect to x of t and x of t dimension 3 cross 1 then you will write lambda 1 dot of t expression that will be finally, it will come like this way lambda 2 plus twice lambda 3 of t and lambda x 1 of t is equal to minus plus u s minus x of t then lambda 2 dot of t is equal to you will get it minus lambda 1 of t minus lambda 3 of t into 2 x 2 of t plus 1 into that bracket is this one into u s x 2 of t minus 2 this is the x 2 of t minus 1 this is the this is the minus 1 and here also minus 1 then next is your lambda 3 t minus into 2 x 2 of t minus 2 of t minus 2 u s x 2 of t minus 2 this. So, this is the expression similar and then lambda 3 of 3 is equal to 0. So, this is the costate equation we got costate equation state vector equation and we need boundary equation to solve this problem. What are the boundary equations are there boundary conditions are x 1 of 0 is 0 given x 2 of 0 is one initial condition is given then we want to drive the state x 1 at t f is equal to 0 x 2 at t f is equal to 0 then we are not x 3 state at 0 is equal to 0 x 3 t f is equal to 0 this is the boundary condition in addition to this there are boundary condition on where we will get another we will get the that is called for free end point system terminal condition. So, boundary condition for free end point systems. So, that I will write in next class that what is this one and once you know this costate equation you know the what is called the state equation and boundary condition you can solve this problem and depending upon the lambda 2 value if it is lambda 2 mod of lambda 2 is less than equal to 1 then put the u value is minus lambda 2 if it is greater than 1 the lambda 2 lambda 2 is greater than 1 switch to minus 1 and if it is lambda 2 is less than minus 1 switch to plus 1 and in this way you will drive the state into a origin in a minimum control effort if you have a both constant in the control input as well as in the state. So, we will complete this one next class thank you.


From around the globe, it's theCUBE with digital coverage of IBM Think 2021, brought to you by IBM. Hello everyone and welcome back to IBM Think 2021, the virtual edition of theCUBE's continuous coverage and we're excited to talk about people. How do you align people and technology? And of course there's a lot of process in between. Those are the hard things. Technology is sometimes easy. Amy Wright is here. She's managing partner of talent transformation at IBM. Amy, great to see you. Thanks, great to be here Dave. Yeah, you know, we love to talk tech and sometimes we kind of sweep the really hard stuff under the rug and we talk about transformation. I mean, it's ongoing.  We had to transform overnight. You know, the vast majority of leaders, I think that figures like close to 95, 96% say that they've accelerated their digital transformation by half a decade. And of course that was a lot of, it was like I say, it's a forced march, so it wasn't really planned full but now they've got time to plan about a digital first approach and how to deal with remote workers. I wonder if you could talk about the role that people play in that digital transformation. Right, thanks Dave. And happy to, you know, a lot of people think of digital transformation about being technology oriented. It's a total shift in tech and it is, but it really can't be successful with just tech. So you're right, what the pandemic has done for digital transformation is it really, it pushed us to these technology extremes more than anyone could have anticipated, particularly with our ways of working being remote. It also pushed us to extremes and highlighted the role that humanity played place. We'll continue to play. So we've been pushed to reimagine jobs, pushed to reimagine workplaces, pushed to reimagine how technology can deliver this connected enterprise, you know, through virtuality. And virtual working wasn't really something that was accepted before, but now we've been, you know, forced to accept it, which is really great for the digital transformation because it accelerated that. So the connected enterprise though isn't really just working virtually, it's these new levels of productivity and decision-making that are enabled by intelligent workflows and cloud and data. And so technology is absolutely critically important, but automation doesn't have empathy. So it takes people to turn these insights that are brought to us through technology and automation. It takes people to turn them into action. And it's that human technology partnership that's required for the digital transformation to get to that desired impact. So when you think about, when you think about people and the role they play and, you know, the pivotal role they play, it's really multi-part, it's kind of three parts. One is people are the ones that build the tech and so they influence whether or not the automation's gonna work, whether it meets the needs of the enterprise, if it takes advantage of the latest thinking and if it's, you know, it is irresistible, if you will. The second is the people use the technology to gain this meaningful insight and turn it into action. And then the third is the people are the ones that embed this tech change into culture so that it's actually sustainable. So to be able to drive this sustainable digital transformation, the people, it requires the people to make it happen. So if you look at healthcare, Dave, think about the dramatic shift in healthcare in the past year, where doctors have shifted to telemedicine, nurses have shifted to using iPads as caregivers with their patients. That not only required a shift in the tech, but an adoption of caregivers of a new way of working that, again, couldn't have been successful unless they adopted and embedded, and you know, embedded a different way of working and a different culture in everything that they do. You know, what you said is really important. Especially, we talk a lot about what machines can do that people can't, what people can do that machines can't and you just nailed it with empathy. And when you think about too, the remote work, I think prior to the pandemic, it was probably around 15, 16% of workers were remote. And when we do surveys with a partner, ETR out of New York, and they project based on these surveys that that's going to double somewhere between 33, 35%. But people don't really know. When you talk to people, they kind of like working at home. Other people say, I can't wait to get back to the office. So people are obviously a critical part of the digital transformation. But how do you think about creating those meaningful experiences at work, whether that's remote, part-time remote, you know, full-time back at work? So this is a really great question because I think our point of view on this has changed. So first of all, most enterprises we talk to, we'll move back to some hybrid kind of environment. We're never going to be everybody back in the office. Again, that's not who we will be moving forward. But the expectations of employees have changed. We all know that, you know, think about your consumer lives and how we experience that personalization when we go to buy something online. That's now bled over into the workplace. So the employees expect that exact same personalized experience at work, but it's now so much more than that. Now it's not only personalization, which, you know, obviously tech enables quite dramatically, but the experience is broader to look at a holistic relationship between the employer and the employee. That's a little bit less, it's less transactional. Like I do my job and my company pays me for doing this set of activities, but it's more supportive and integrated with their personal selves. So, you know, we did a recent study in which we looked at consumers and employees and their highest priority areas for the expectations that they now have for their employers is career and skill advancement opportunities with speed. Second is work-life balances that might take the form of what hours they work, their ability to, you know, manage with what they're doing in their home with their families and children, you know, their ability to be camera ready or not at all times of day and night and actually where they work from. So people are now working not only at home, but they're moving to different cities and want that flexibility. And then third, a high area of priority now is ethics and values. So not only diversity, equity, inclusion, obviously critically important, but ways of working and meaningful and purposeful work. So when you look at all of those together, the employee experience has grown to be not only that of personalization like we have in our consumer world that is critically important, but now it's all of these other things as well. As a matter of fact, they become so important, Dave, that in our recent research, it shows that one in four employees will change employers in 2021, one in four. One in four will also change professions in 2021. And while about 75% of employers, companies, believe that they are doing a good job of meeting the needs, these expanded needs of their employees, less than half of employees feel the same way. So there's a lot of work to be done. So you asked the question, why is this people experience so important? It's important because it's required for the digital transformation and it's so much broader than what we used to think that it's now a competitive differentiator for employers as they try to not only achieve their digital transformation, but as their organization is disrupt over and over again, it's a requirement in order to meet their enterprises objectives. So those are great stats, Amy, you just put out there. I mean, the career advancement, I feel like it's always been there, but it's now much more front and center, employees are more vocal about it. The work-like life balance, same thing. I mean, you're seeing some organizations, 100-hour weeks, we're revolting. And then the ethics and values piece to me is one of the most interesting. I often joke, Milton Friedman is rolling over in his grave because he was the economist. I said, oh, it's just about shareholder value, that's it. And that's not anymore. In fact, there's clearly a relationship between shareholder value and ESG and ethics and young people are very, very concerned about it. So here's the question. Who's accountable for making sure that you have a positive employee experiences occur? Yeah, really, really, really good question. And the thing is, this is what makes it so hard, there's not one group or one person. It's actually all of us. And I know that answer sounds like a little bit like a cop-up, but this is why it makes it so hard. Every leader is responsible for the employee experience. Every manager is responsible for the employee experience. Every employee is actually now responsible for the experience of their teammates and actually speaking up if the experience isn't using inclusiveness as an example. If it's not inclusive, every employee has the responsibility to speak up. So some companies actually have employee experience leaders. Some companies have digital transformation leaders that embodies that, that includes that employee experience. But most actually start this journey through the partnership between IT and HR. So IT is responsible for this technology architecture, the cloud strategy, the data strategy, architectural framework, all those pieces that put together the foundation and the building blocks and the security that help to modernize this employee experience. And by the way, they're doing this at the same time where they're modernizing their entire way in which the IT function operates. So you've got IT that's kind of setting the stage and the foundation for what's possible.  So things like design thinking that puts the employee at the center of the way we architect and create these experiences using rapid iterative design principles with, again, with the employee at the center. Making that the cultural norm across the enterprise is a really big deal. So HR is usually in the lead on making that happen. But again, this is a cultural shift, not just I have a project, I kind of have a project plan and here's what I'm going to execute on. Leadership roles. So HR is the steward of leadership. And those characteristics of leaders now are changing very dramatically to be more, even in a big enterprise, large global enterprise, entrepreneurial, transparent, co-creation really at the core of everything. So being transparent with your teams and be able to co-create for the future. So data and AI, we can use data and AI now to actually predict the impact that the workforce and the cultural will have on business results. Predict attrition. Predict what different workforce design scenarios will look like to the supply chain. Predict the speed of hiring and how that will impact a literally bottom line business result. So you said it right when you talked about shareholder value. The people is at the center of shareholder value now. So our functions need to be modernized, but it's really this partnership between HR and IT that's going to be able to make it happen in a big way. It's interesting, I'm just thinking that AI as well can be a canary in a coal mine when there's potential problems. And I love this transparency, that's critical co-creation. So okay, so tech is a key part of that, especially in terms of when you go from analog to digital, taking friction out of the system, shows the employees that we're investing in your experience, but it's more than that. You're saying it's cultural and makes it so kind of fuzzy. Cultural, it takes a village, so that's part of what makes it so hard. How do you think about the journey? Where do you start and how do you keep iterating? You're never done in this world, are you? Yeah, that's a question everybody's asking now is where do I start? So as you said, this is very hard and it's hard. One of the reasons it's hard is it's because it revolves culture. It's not only about technology, they are hard, technologies are harder in their own right. It's not just about data, that's harder in its own right. But once you involve technology, it makes it even harder. And of course the people aspect unless done very proactively and meaningfully, it can be kind of a wild card, right? On who's going to adopt what? So where do you start? So the way we like to think about giving advice to enterprises regarding where we've seen this work well is to pick a business problem. So what's a business problem that if you solved, you can actually make an impact not only for your people but for your people, but for the enterprise. So if you could pick a business problem and actually fix it using data, using cloud, using people experiences, using a cultural shift, then you'll get that buy-in. You'll get the buy-in that yes, we can do this. This is very doable. We can repeat it, it's repeatable over and over again and it has an impact on our culture. That's a great place to start. Okay, so then you say if that's a place to start, how do we actually, there's gotta be foundational things that have to be in place to make that work. So one of them is a consistency in data and the use of AI and the ability to make insights meaningful, that come through data and AI. And the other part that's really important once you pick your business problem is the shift in the way of working. The shift so that it can impact cultural change, a shift so that there's co-creation with your people and there's transparency. So each one of these business problems and the way companies pick to fix them, they won't all work. And the way you get that trust and transparency with your people is as scary as it is to share with them what you're attempting to do and share with them how you're doing along the journey. And if it fails, okay, it fails. Pick yourself back up and start again. That trust and transparency with your people, that's the way we all make this cultural impact. So kind of the net of this is to be, to make sustainable change, we can all make short-term change, we can do projects, but to make sustainable change, the humanity aspect has to come to life in these digital transformations. And that only comes to life with this cultural shift. Amy, right, you've thought about this a lot. Deep expertise in the area. I really appreciate you sharing it with our audience. And thanks for coming on theCUBE. Dave, my pleasure. All right, keep it right there. But this is Dave Vellante. You're watching IBM Think 2021, the virtual edition from theCUBE.


This is just a really quick Misty number 39 update video I guess to tell you about what I'm doing here and also about what's coming up. On Facebook I got involved with this, it's called I believe the postcard exchange experiment or PEE. This woman named Diane who is running the program sent me this postcard and it's part of the Mystery Singles Theater 3000 which I'd, yeah I didn't know that was part of this. I joined the Misty Singles group on Facebook for a brief while and I didn't see any value in it for me so I left but this is an example of the kind of postcard they're sending out. Anyway the backside of the postcard has the MST3K silhouettes on it. I thought I would use that as a cue to create my own postcards rather than buy a postcard or get a blank one and draw something on it. I thought I would make postcards that have my existing drawings on them. On my Tumblr blog called One Sketch Today I used to draw Mystery Science Theater sketches all the time so I opened those up and I made a set of four postcards that have the first four Misty sketches that I ever did. The first ones were, this was actually the first one here. Yeah it's really sketchy but that's what I'm going to send out. I got my exchange ease address by email so I'm going to send this out to him. And on the back, and this is something that I expected to happen but I didn't know it was going to be quite so bad was that the printing doesn't go full bleed. It has like a quarter inch edge all the way around it. So at first I was just going to cut these up and send these out the way they are but then I realized there was a quick way to fix this by putting the bottom edges together and the rest of the edges of the postcard didn't need to go flush at all so I can just cut these up along the center and then I've got my postcards. So yeah there we go. This one here, the Crotea Robot Sketch is going to be my first launch into the MST3KP experiment. Today is Thursday the 27th and tomorrow Friday evening here in Portland the live MST3K show is going to be showing tomorrow evening. I don't have tickets for that one because I wanted to see the secret movie. Tomorrow here in Portland they're showing EGA but then Saturday evening in Seattle they're showing EGA followed by the secret movie and I don't know what it's going to be so I wanted to see that so I have tickets to in Seattle. Of course I'm going to make some video about it and do a review and everything but something else really interesting came up so it's not just going to be me involved in the review video. You'll see that soon enough because it's only two days from now and it's going to be on the 50th Street channel not part of my MST39 thing here on ThinkBall. It's going to be the launch of a bigger deal on the 50th Street channel so look for that. Okay this is MST number 39 saying so long.