Well, the session is going to be moderated by Mr. Rahul Mishra, head of marketing and comms, Shimaru, please can I have a huge round of applause and him on stage please. And now I am going to be calling up our panel members. We have our first panel member, Ashish Pratap Singh, head of digital marketing, Exigo, Varun Kumar, CMO and head of growth, Jayraj Jadav, a vice president, head of marketing and digital business, Tata AIG. I'll move on to the next name. We have Bhargavi S, head e-commerce ecosystem, ICICI bank, please a huge round of applause. Next, Radha Prasad, vice president, performance marketing, went his avenues. All right, I'll throw over to you. Hello. It's working. So Jayraj is on his way. When he comes, he comes and joins us right in the middle of a very interesting discussion we're going to have over the next 40 minutes today. Firstly, I think thank you to the audience for being patient post lunch and being part of this enriching discussion we're going to have today. Thank you to my fellow panel members. I think we've got a great mix in the panel here today. Some experts in the digital marketing space, people who have been running performance marketing to the T, as we say companies which are sort of built on performance marketing. Plus we've got representation from the agency side as well from Ventus as well, so they've been helping brands achieve their performance marketing metrics as well. ICICI bank, then we've got representation from Zoom car, again a very strong digital focused car rental company doing a lot of work on digital. And of course, Xigo, which represents a huge chunk of the travel advertising pie. They have a fiercely competitive space and it'll be good to get a sense from all my fellow panel members today and how to look at this particular challenge. So the topic today is very interesting. I just made some quick notes last night and I'm going to, I think just to give a sense to the audience here and to all of us here, I think just to put a context of what we mean by performance marketing because it's a very, very broad term. Performance marketing has been existing before digital marketing even came to place. There's a KPI attached to every marketing dollar we spent as marketeers. Essentially that's what performance marketing is, but for our conversation here today, just want to give a quick context to what we're talking about before I open up the mic to my fellow panel members. So in the traditional form of advertising, advertisers would pay a huge amount of dollars and sort of put it out, splash it out all there and that wouldn't be linked to any performance. That'll be linked to a certain brand love, brand growth, a funnel being created which is going to eventually lead to revenue for the company and that could actually mean that you're spending a lot of money in the short term or a medium term without having any conversions as well. With performance marketing, advertisers can actually only pay for a conversion. Conversion for an advertiser could mean actual sale for a lot of e-commerce companies or online based companies, sale happening online itself. It could also... Great, so we've got Jayraj as well. Welcome. Great, welcome Jayraj. So I've just said in the context of what the discussion is going to be today on performance marketing, so right time. So performance marketing advertisers only pay for successive conversion which could be a CPS, a cost per lead, a cost per acquisition, whatever the metrics it is. So you actually only pay for that part of the advertising you're spending on and thanks to data driven approaches by most firms right now, this is getting more and more used across companies a lot in India, a lot of a huge bank of customers available online, how do we convert them, how do we make them buy our products is what focus of both brands is and it's become 100% measurable right now. The entire industry has actually been able to bring in a lot of technology in place that are various networks that are various ad fraud companies and auditors who help brands get their right metrics in place. Also what happens is a lot of marketing right now doesn't have to wait for the big budget approval coming from the management to go live with the product. You've got a product right now, it's a digital product, I can actually run a performance marketing campaign immediately because I know if I invest X, I'm going to get two X of that. So I don't need to wait for my board to approve a huge marketing spend for me to go do it. So we can actually see a lot of movement happening very quickly thanks to performance marketing. It's a good space, it's got a lot of merits. I think the idea today is to discuss and learn from the experts as well as to what their thoughts are. So I'm going to, the first question I'm going to actually open up to the panel is going to be, are brands becoming more focused on performance marketing with the hope that anyway bring them brand love? So essentially what I want to understand is that the brands you are representing, are you focused a lot more on performance marketing? Are you putting a lot of money behind performance because you're reaching out to a huge set of audience with the hope that the audiences you're reaching out to also love your brand, not just look at a transactional basis brand conversion. So I'll actually open up the discussion. Let's start with Varun. We'd like to hear your thoughts on that. Sure. Thanks. Thanks, Rahul. Hi, everyone. That's a great question. So let me take a step back, right? Imagine like quickly for next, like probably 30 seconds, imagine like let's go in past like 10 years from now in the history. I mean, Facebook was still struggling for traffic. Amazon was still selling books. And by the way, iPhone was just two year old. So point is that cut to, you know, 2019, I mean, marketing has changed so much in the last five years than in past 50. And this is because the root cause of all of this is that earlier, even 10 years ago, the businesses were actually brand centric and everything was outbound.  So in that process, some things were measurable, some things were not measurable. But now there is a definitely a paradigm shift because you know, businesses are consumer centric.  So you go to all these, you know, popular shopping e-commerce sites, and there are multiple micro-million moments of truth. And when there is continuous interaction of a consumer with the platform, so he's researching, he or she is researching online, then, you know, buying offline, which I famously call as robo effect or reverse robo, that researching offline and then buying online. So what happens is when there are multiple micro-million moments of truth, you have to catch the customer wherever he is. And hence, you have to deploy a multi-channel marketing strategy. And since the beauty is that each of the channel is so highly measurable that there is no choice as a marketer that you have to, you know, really see as to where this customer is in, you know, the entire consumer decision journey, basically the path to purchase journey. And hence, everything becomes measurable. Hey, by the way, when was the first time the customer came to my side? When was the second, did he make a purchase? Did he made, you know, do, you know, add to a shopping cart? Like, and then did he dropped off? Why did he dropped out?  So the holy grail, the genesis here is obviously to get him convert and then him repeat, then repeat. And basically, like you can actually build a model.  So just to like, yeah, probably that sets a good tone for the, that's very interesting actually insights. You've got a, you've got your own sort of matrix you've developed in house, which you use to measure success of your campaigns and your retargeting and the programmatic. Excellent. I think the next, I think I'd like to open the question to Ashish and we'd like to hear your thoughts at Xigo. I will, you know, we were just chatting up before the, before we met here. I think Xigo is very focused on performance. You know, I just want to get a sense from you. How is your journey in performance been? Are you seeing your brand being more accepted because of your performance campaigns? And just to get a sense from you, what that means to you? So as Varun highlighted that advertising change in last 10 years from, you know, a non attributable spends to now attributable spends. So in performance, the main keyword that is, that's attribution, right? So if we attribute every money, every penny spent on marketing to my business metrics, that can be a simple app install or a simple website visit to lead or to a conversion or to a repeat, whichever is the matrix for that business. So that's actually, you know, the crux of the performance marketing, right? And people also working, you know, people also working towards, you know, measuring radio advertisements or let's say TV advertisements and link it to the, link it back to every penny spent on the slots and all. So that's the thing. So as an Xigo, we have, you know, built a brand Xigo on performance marketing, purely on performance marketing. And what we feel is if you, if you do the product right, and if you do the marketing right, and if you keep retargeting that customer, keep pushing your brand values, why you should, why the user should use your product versus other products, then definitely you can, you know, kind of build a brand on performance marketing and that, that, you know, push or why are retargeting mediums that's available and the kind of data that we are gathering and the kind of power in retargeting we are adding with all these data points. It can be a success matrix for a business and it can be, it can drive actually organic growth also if you do it right. So essentially driving business and the brand together is what you're able to achieve through performance, right? That's great. That's nice to hear. I think Bhargavi, I would like to just get some initial thoughts from you as to what your overall vision from the company you represent is on performance and how do you look at it as your business which you manage right now? Okay. So upfront, I can't comment on the marketing of ICSFM because I'm not from the marketing team. However, we work extensively, I work extensively with clients who are, who are built on the basis of performance marketing, right? The cost of customer acquisition significantly makes or breaks their panel and what we have learned is that there is an extremely overarching, I think across the board, it's universal. I don't think there is any company today that is talking cash burn to acquire a customer. Every single penny that a large entity, a large client, spends to a younger, newer startup, they are very careful about where they're spending their money and even before typically as bankers, we are conservative, known to be, but the clients today start off conversations saying this was how we were spending earlier, this was a cost of, I'm not getting into specifics on clients or how much. So if my cost of acquisition was 1,000, today my cost of acquisition of that same customer, and they're talking cost of acquisition of the customer, the same customer is something like there's a 60, so 1,000 to now 600. So there's a good 40% drop across clientele, I'm saying. So I think what each of these organizations are doing are pointing in exactly the same direction. And I, since I'm now in banking and I'm looking at banking, how my banking counterparts are doing, so Citibank has publicly stated that it ran a campaign, and if you saw a few days ago, Facebook had taken out this really big attribution last page of the economic times, telling us what we should be doing. But I'm not going to talk about India, I'm going to talk about Thailand, where they spoke about, they're saying that they got 75% increase in conversions with a 74%, let's take 75% reduction in cost. So you're telling me earlier you spent 100 rupees to acquire a customer, today you're spending, if you do the math, 14 to 15 rupees to acquire the same customer. I mean, that's efficiency, that's super crazy, and I love it. Interesting. Wow, that's some good learnings here. I think let's get a sense from the agency perspective here right now. So at Ventus, you obviously help a lot of brands do that. So do you see a lot of pressure from the brands you work with to get their performance metrics better and better on a weekly daily basis, and how do you sort of enable that for some of the brands you work with? Yeah, true. So when we're dealing with any brand, particular brands, so every brand has their own perspective of conversion goals. Like a couple of brands like in the VFS segment, they are looking out of the end level convergence. In let's say ODD segment, they look out for the views, they look out for customers who are actually viewing the ads, because that's where they're earning money. When you're talking about the gaming client, they will be talking about how many levels you are completing. So every brand is having their own perspective or own goals set initially. What we do is we actually set up a basis on our targeting. Like we actually strategize everything on a phase by phase. So whenever there is a performance campaign launched in the market, when we strategize it, the first phase should happen like an awareness, if you are not a big brand, obviously the user should know what is all you are talking about. If you are offering something, then obviously the user should know what exactly the offer is all about. So we create a strategy for every phase. Of course there are lots of challenges that we are facing from a brand, saying that your deadline or your attribution window is seven days, 30 days, it all happens. But over a period of time, in between that frame, how exactly you are reaching out to the same users and monetizing them and also helping them in re-engaging with your brand. That's how we are actually achieving the conversion for all prospective. So high pressure job. That's great. Okay. I think, Cerej, again one of the biggest digital spenders in the market. You guys have been running a lot of branding and performance campaigns, I would say, on digital. Am I right to say that? You run a mix of that. From your perspective, how do you see this performance marketing space evolving and where is the right mix for you? Where do you see, sort of, here I need to build a brand for 10 years and here I want revenue today. How do you manage that sort of space? So I think, fundamentally, the way we probably look at it is we don't have, especially, we don't look at something called brand and a performance as a separate thing. It's probably about building the audience, engaging the audience and probably converting them at some point of time. So it's a funnel management for your business. Whether you look at bottom of the funnel or the top of the funnel, that's how we look at it. Also, I think, the whole performance management, performance media has been kind of evolved from perspective where it's been becoming complicated every year by many players coming into kind of competition, moving into it, spending more money onto the same plus kind of martech, at tech evolution that has happened. It's making every year it becomes complicated, right? But it's like probably any of the games that you play, every level it becomes complicated. But that's where also by reaching to that, you're also becoming expert to get yourself kind of master the game. So it's a part of our DNA as IIG and I think we look at it purely as performance, everything. And it's building funnel for us, building audience for us. Okay, great. That's interesting. I'm going to sort of change the sort of space we're discussing right now. I think this is going to be a little more from your each of your business perspective. You can sort of give me a quick one line response or a longer response if you wish. You know, with ad blockers becoming very rampant globally. That's actually coming to India in a big way as well right now. Are you seeing any threat to some of the performance campaigns you run at this point of time? And my second part of the question is, because of ad blockers getting more and more rampant, influencers are becoming athletes now. So there are influencers on social media who are able to give you your performance metrics in a much better way. So there is now traditionally we only had affiliates or a programmatic running performance campaigns. Now we have a set of influencers who can actually help us achieve the same results. And for the benefit of all of us and from any learnings you've had in the past, could you want to share what your thoughts are on this space? I think I'll give you a number on the ad blocker thing. So typically it's still a single digit number. You can even Google it. But as far as my best knowledge is concerned, it's something close to three percent. So ad blockers still in India is very, very nascent stages. Of course globally it is on a rising spree. So I mean it is also a chicken and egg situation because obviously Google being the big daddy over there and if you like they're almost on DBM, they were their own YouTube right. So point is like they would definitely come around with counter solutions where you know if you if you skip even if I will give you an example of YouTube if you skip an ad if you don't want to view it you can actually in your settings. I'm not sure if many of you know this in your settings you can actually turn off videos and one of the things which you know YouTube want to monetize say monetize is on the YouTube premium which they have recently launched. So you completely get all the things ad freight. So that's one thing. I think the second like the sub answer to your question is that you know a lot of co-curated content is happening which is a very you know like refined form of advertorials right. So it is not the on the face like just kharilo kharilo besh lo besh lo right. Not that kind of advertising anymore. The third I think the question with like interesting question which you asked was around influences. So baspot on. So like at Zoom car like if you can like all of them I'm sure you have your Instagram handles you just go to Instagram and just search for hashtag Zoom car. I mean we have more than 100k followers and the kind of content that we get from our you know consumers is humongous. So all of this is user generated content UGC. I mean I know brands literally pay to influencers to actually post UGC and we are actually you know like Zoom car being a selfie brand being you know in that space travel leisure space. I mean we get this a lot of love from our customers. I think the answer and the kind of trend that we have seen is that we have seen a rise of micro influencers. So it's okay. So because there are two things to have you have you ever given the same performance metric you would give to an affiliate to a micro influencer for conversion and what is your response to that. Absolutely as it was the same way. Absolutely because like like you respect Mr. Amitabh Bachchan but if Amitabh Bachchan will tweet something it looks like an advertisement. But did you pay Amitabh Bachchan. No no no not at Zoom car. But if genuinely a micro influencer who's actually used a service because if you're searching for a flight ticket on XeGo or if you're actually searching for Zoom car you might actually you know go on a trip advisor and might trust someone who has given who has a four or five star rating and you actually trust those people then probably a celeb who is saying something on Instagram right. So typically what we also like we like our cost per acquisition is like something like 200 rupees cost per booking. So we get like almost one and a half lakh bookings per month on the selvi space and like 10 percent of it is actually driven by influencers and these are like most of the organic channels. That's great that's sort of almost from zero is not 10 percent in the last one or two years I'm imagining. Anybody Jairaj have you had any opportunity or Ashish. So on the ad blockers front for those right so we have not you know in the phase of worrying for the ad blockers but on the mark influencers front right. So influencers at XeGo we you know take really seriously the influencer and especially the micro influencers right. So we designed a nice program the social mojo which actually you know takes care of looks at your social cloud and gives you certain points and then you can book you know your travel with those points. And we actually attribute it same as our performance channels and it's you know if you talk about performance and cost per conversion it's at par with our best performing performance channels. So you know and with all these new social apps like Tiktok and share chat and all these micro influencers getting more and more par and more and more you know reach to extend to a brand to drive performance or to you know to drive a brand campaign through that. Excellent. Okay. Okay. So like Varun and Ashish told obviously the ad blocker is completely it's not a trade right now in India. So like Varun actually told the right figure it's actually 3% it's 2.79% which is triggered in last quarter of 2018. If you're looking at the figure that 2.3% is only for mobile desktop is quite high desktop you will be seeing almost 40% ad blocking happens. But why exactly these ad blockers have been placed in picture. So the reason being most of our million are actually looking at the creative placement. So how engaging your creative are basis on that decides. So we'll be going on recent studies by Google's only. So they have told like 60% of the users who are into 16 to 34 age group they actually prefer to see a content more on the traditional media than on digital media because of the content that you're playing on the digital space and they're ready to go for the ad filtration instead of ad blocker. So ad filtration is a concept wherein you can actually personalize your own ad interest areas and your concept realizing that I am to showcase this kind of ads to my users. So personally you can actually block what kind of ad you want to see and what kind of ad you don't want to see. So that kind of feature is already available. And coming back to the influencer is always there in the market. So previously it was called as let's say incentivizing reference check referral cross-example. Now you're calling it so because of social media part we're calling it an influencer marketing. Obviously influencer is all together different games so whatever the convergence they're doing they're doing into that own frame of like audience or even frame of followers. But in the affiliate market when you're putting a benchmark to achieve something obviously we are looking at a gross level like how mass how targeting can be missed that kind of feature we are already looking at. Interesting. Okay. Okay. So in your roles as marketeers and heading businesses you know have you ever in the last maybe recent times I would say have you been have you ever been under the threat of a network? An ad network because you're dependent so much on performance right now for your metrics. You could run your own performance channel but you have to work with various networks and they help you enable businesses. Like in our case from Shimaru's point of view we're on an OTT platform as well and there are times we've had conversations with networks where we've been asked to shell out 100% of the acquisition money of the sale transaction is what the network is is looking at which probably doesn't make business sense why would I get into performance then but yeah it gives me numbers not as much to vent this but I'm generally saying that there are there are industry norms right now and all of you in your respective sectors have facing the same.  I think that's you're actually taking out all the secrets right so point is that like without naming the networks see one has to understand the audience pool is common whether it is being hatched by network A network B network C or network D. I think you're actually hitting in the right direction you need to spend some significant amount of time into affiliate marketing to understand what is quality traffic what's not quality traffic what is bot traffic because like typically how network work is like there will be a big network under big network there'll be sub publishers and then sub publishers and then sub publishers and then actually the one who's running your campaign so all the network will come to your doorstep saying oh you know by the way tell us what you want and we'll give you but give us exclusive you know contract and you know we'll get you this inventory I think with networks the holy grail is to maintain discipline if you master the art of discipline I think you've cracked affiliate networks so instead of going impressions burn and cost per click I that was so 2016 you have to be really an apartment language you have to be really anal about the performance metric because you have to you know like the way you know you work for with other channels you have to work on say cost per first booking or cost per like second transaction or repeat transaction I know there will be a lot of hiccups from the network saying oh you know when Google with Google and Facebook you still work on cost per click and then you find an arbitrage they will give you all sort of arguments but yeah it's chicken and egg that you have to master an art of discipline while dealing with that that's a very honest responsive testing any jara you want to fill in something sorry probably you know we've never used network okay at the same time so I would not probably want so how do you how do you run your performance I think as I said you know for us it's a probably audience building and we look at using you know certain set of media or you know things like probably content marketing to build top of the funnel and then you know kind of get the audience engaged that's over the period of time then do the remarketing at the bottom of the funnel we use adobe so I kind of talk about the product but so we use you know sort of automation there at the bidding part of it and that's how probably we've been spending huge amount but we've never used network from that perspective oh great that's interesting actually great I think we are almost going to run out of time I would just like to you know get some thoughts from you overall on the performance marketing piece before we end this discussion so from from a so I'm a marketer so how I look at performance piece is a little differently actually so I feel performance marketing is great for immediate marketing efficiency you want to drive numbers go ahead do that but it actually ignores how brand actually grow and make money over a period of time a long-term interest brands do not go by targeting high-probability consumers they actually go by targeting low-probability consumers and converting them to consumers keeping that in mind and if you if you may agree or disagree with me what are your thoughts on performance marketing especially these companies which have been built on performance marketing how do you see yourself in the future taking forward so I agree on that I mean the brands in a long-term build on the from top of the funnel right when user starts looking for that actually starts planning for that product so let's say if you talk about a house maybe when you start a job then I can you know keep aspiring you are when you if you talk about travel maybe three months before the travel season we can aspire you so that's what I said there are two things one is we go all out in brand marketing throughout the year one is that way second is you know target your customers when they are actually in the planning stage right so specifically if you talk about travel target users you know when two three months before the season starts do a good nice branding campaign based on you know content nice content and put money on the branding and keep retargeting those users sorry targeting is the key right attribute those users and keep retargeting them with nice offers and maybe good aspirational content and all so that that's my point of you and that's how we you know see it we'll pass the mic to you Arun what are your thoughts I think it's a staircase function right so the beauty of performance marketing especially performance marketing is digitally driven even if you're a startup I mean you with a very limited budget you can start you know piloting not doing a lot of baby test and you know get initial success once you hit an initial success there is no two way about about it that you once you reach an inflection point or probably a critical mass what after that because you will hit a ceiling for sure so in order to then expand the ecosystem for your brand you have to invest in brand activity and especially like mass media activities and atl could be a good thing if it fits into your annual operating plan so it is basically a staircase function that you and of course you can measure atl as well so it is like performance in digital then you want to grow exponentially you invest in atl and you mass media then again performance then digital so it's basically a staircase function until unless the last thing what we say that there is there is something which you know we famously as performance market is called as ROI ROAS or cost to income ratio until unless if that number is under a good range you will keep the board of directors happy it's good to hear Bhagavi what are your thoughts and so since I am the odd one out and I it's business that I look at not just marketing sweet me marketing is the incoming funnel I think what you've said everything right and to build on that so you have your digital marketing you hit a certain what you call threshold then you try and do mass market overall campaigns and then you take it forward combined with this I think goes efficiencies which anybody who runs a business will talk about wherein I need a sales conversion efficiency my CRM has to talk to the lead gen tool right which today many of the old-fashioned tools are two separate things and you're tracking different things and I'm selling on different things and the final revenue or profit of an organization is made on what is tractable in the CRM so that is one so for example in BFSI right we found that somebody says I want to go alone or I want an educational I want a personal loan right I want to go on somewhere and book a travel holiday and he's interested and it's coming to me via a partner right or if somebody saying I'm traveling so I want travel insurance if you don't call the chap within 10 minutes like I was telling you earlier that lead is gone so the intent was there the it was genuine customer high quality partner but you've managed to ruin it so I think combined with each of these the true success of marketing is in getting other people to realize the value that they are giving and having the support of this functions which is what I think within the organizations especially that's something that you'll have to be I'm sure you are working along with right with your sales teams especially no direct sales teams yeah so your direct sales teams you're basically direct to customers so you would have to do this to get any benefit from your top line yeah sorry yeah she right mentioned I think as much of you know time and energy that you you kind of invest on to acquisition I think it is also important to kind of streamline customer journeys to get the right ROI metrics correct I think that's the way to look at it also just to probably add I know running out of time but on the same line you know for us you know what we've realized is while you know as much as investment that you're able to do at the bottom of the funnel is fine but you know at some point of time you'll reach you know kind of threshold there the way we have over the period kind of you know build the whole thing is typically for a travel insurance if you look at hardly one percent of total population who travel by travel insurance so there is always a scope to kind of increase the bucket out there right and how we've kind of done our journey in last couple of years is we've started our engagement with the audience around three four year three four months before he actually buy insurance right so you know travel insurance journey it's typically the guy decided to you know kind of travel destination around three four months back probably in January and end up traveling in April May June earlier we used to do only performance marketing only in these three months now what we have done is we've expanded our bucket into content marketing display social engagement and started mapping user journey even before he decided to buy very interesting uh jeraj what are your sort of thoughts on on the brand piece and so so when we call about the performance marketing we normally when we do our networks or agencies uh are in goal majorly coming across like new user acquisitions but when we do the similar kind of strategy with facebook and google we actually created strategies for a couple of months so obviously every brand having their own uh like convergent flow it's a complex journey of the convergent flow it depends on how uh like how your flow is lengthy or something like if your convergent like he said for travel and insurance the convergent flow ends at three or four months back so probably in a shorter flow you might have been depending on the uh like our networks or agencies but in the longest journey you are actually depending on the facebook and google so when we are uh working with any ad networks or agency we actually forget to talk about the lifetime value this particular ad network is bringing on we actually not looking at the post attribution matrix that post attribution matrix that is followed by these ad networks that is one reason we are not able to exactly attribute whether this particular ad network or agency has contributed to my conversion not so that is something wherein most of these brands have to walk on or along with the agency or networks you have to walk on the targeting retargeting at the at the same time on the data driven strategies then only you will be able to achieve the convergence problem i think yeah i think some really very interesting conversation and very i would say very candid uh responses by all the fellow panels here today panel members today uh i think what we're gonna all take out from this is that performance marketing is the sort of uh way to go for uh businesses to invest in at this point of time uh thanks to the growing digital marketing ecosystem but at the same time we need to find a balance between uh the long term value of a customer which we can also look at attributes uh beyond uh performance to build that uh i think on that note i'd like to thank my fellow panel members for joining us today uh do we have time for any questions from the audience uh excuse me do we have any time for any questions from the audience offline okay fine so i think we will uh you know end it now thank you so much for your time it's great being here thank you


So how do we relate passivity to Lyapunov stability when we want to study the stability in this sense? So meaning that we have now a model, I have been telling you that passivity is all about inputs and outputs, but now let's say that we also consider the state of the system, okay? And we want to relate passivity with Lyapunov stability. Well, we have that if the system is passive with a certain, with a storage function or energy function V that is positive definite, then the origin for this system without inputs, because now we are talking about Lyapunov stability, so we disregard the input, then we have stability, okay? So think of again the pendulum without friction, it will just be oscillating, that's a stable behavior, it just keeps oscillating, right? If there is friction, then we will have output-strict passivity. I didn't speak about L2 and all the outputs, so forget about this. If I have output-strict passivity, that means I have friction in my system, and then the system will be asymptotically stable if in addition to being output-strictly passive, it is also serious state observable, okay? So serious state observability, I don't know if you have seen this, but it means, so serious state observability means that y equal to zero implies that x is equal to zero. So if you have worked with Lyapunov, essentially what we have in this situation of output-strict passivity is that if V is my energy equation, we will have V dot because it is output-strictly passive, we will have minus y square minus plus y or u or actually, usually it's written the other way around, plus uy, and then we forget for Lyapunov, for the purpose of Lyapunov stability, we just want to zero the input, so forget about this guy, so you have only this. Let's say that, yeah, due to this property, y is going to zero, then you just set it equal to zero and check if the state is equal to zero. We'll do that for some examples, so in case you, this doesn't ring a bell. But all this to say that, yeah, output-strictly passive systems, output-strict passivity essentially leads to asymptotic stability. Yeah, if state is strictly passive, so we have this term here, it's appearing with the whole state, then what? Then we also have asymptotic stability right away because once again, you could use this as a Lyapunov function, forget about this guy, and you will have a Lyapunov function that is negative definite. So the moral of all this is that you, if you think in terms of passivity, you have to figure out the energy function of your system and use that as a Lyapunov function essentially and then try to come up with some quadratic negative terms in your derivative, but the good thing is now you have an interpretation for this term. Now internally, Lyapunov stability, well, we saw that the interconnection of passive systems is passive, well, here because if these systems are passive, when we interconnect them, this will be passive, and due to what I just said over here, how passivity relates to Lyapunov stability, we will be able to also say things about this passive interconnection. This theorem here is taken by, it's taken from, well, it's a known resolver, it's taken from a book by Van der Schaft. It's Springer lecture notes. I can send you the complete reference. So it just essentially says that if you have these two systems that are passive, the closed loop system will passive, will be passive so you will have stability. Now it is left, of course, to determine stability of what, of which equilibrium. Well, normally when you are trying to control a system to a set point, for instance, you would like that this equilibrium be a set point that you chose, right? So the only thing that is left to do is to choose, to manipulate the energy of these systems in a way that the energy functions have minima at the equilibrium that you want to stabilize, right? So you want the energy of this guy to have a minimum but takes one and the energy of the second guy, you want it to have a minimum at another equilibrium point, right? If in addition you have output of strict passivity and the equilibrium is unique, then you will have asymptotic stability, okay? If provided that you have this property of detectability, yeah, so detectability is the following. Detectability means that y equal to zero implies that x goes to zero. So this is a weaker property than observability. We just want to be sure that basically if you know how to bring the output to zero, you want to be sure that bringing the output to zero also implies that the whole state will go to zero. If zero is the equilibrium you want to stabilize, otherwise just replace the equilibrium to the origin, okay? So yeah, I'm almost done with this set of slides. Essentially now we are, well, I presented you this a little bit fast but I will show you how this works in the other slides. But what we want to do now is passivity-based control. So in passivity-based control essentially what you have is a plant that let's suppose that it is passive, okay? Or maybe it isn't, but yeah, let's say it is passive, it has some passivity property. So you want to design a controller that you will put here in the feedback loop. And as I said, when you interconnect systems that are passive in feedback, the resulting system will be passive. If the plant is not passive initially, well, you apply a controller pro if possible to render it passive, right? So you will want this. At the end of the day anyway, what you want is to enforce the passivity or to create passivity in your system. So passivity-based control is about designing a controller in a way that the closed loop system be passive, okay? That's what you want to do. And if you want in addition really asymptotic stability of your closed loop system, you will want to be this passivity to be strict. So yeah, if you have seen these inequalities for non-linear systems, what we are going to try to do is to design you so that in closed loop you still have a passive system. So to continue speaking about passivity, what we want is to have you, you should have one component that makes the system passive. And another component that we were going to call you new just to, you know, like I did for the circuit, you just need a new input from which to you define your passive map, right? Because otherwise if you don't have this, you close the loop with some function of the state, there is no new input then you cannot speak about passivity if there is no new input, right? So you want to design you so that the closed loop system with this part here, so that will be this, should be less or equal than the new input that you're injecting, multiplied by the output, okay? In that case, you will have a passive map from the new input, okay? If you tilde in addition injects dumping through this output, then you will have this inequality here that will give you a strict passivity, okay? This on the other side is the, what is that, the energy, well, not the energy, the power in the system. If you integrate all that, you will get the energy balance equation, right? The supplied energy, the dissipated energy and the difference between the initial and final and not final, but energy at any time, too. And yeah, let me finish this set of slides with this very important theorem that says the following. This is taken from a paper by Beren Cicidori Williams, three very big names in Cicidori. And it says the following. If you have this system and it is passive and you have some energy function with respect to which it is passive and also it is zero state detectable. So I already said zero state detectability means that when y is equal to zero, that implies that x goes to zero. If you have these conditions, all you have to do is to re-inject the output into it to add the dumping and then your system will be asymptotically stable in those. So the control should look just by this, like this simple expression, just a function that is strictly contained in the first and third quadrant, right? So there. So of course, just a constant times the output will do, but you can also use a saturation. You can use many non-linear functions, right? The saturation will look like this. Bound it under and so on. So you can use many things. So let's see how passivity-based control works now. So now that we know the definitions of passivity and we know the Kalman-Yakovich Popov statement for linear systems and its equivalent version for non-linear systems, again, taken from Halil. Now we'll be working with these notations. So Lfv is the directional derivative of v in the direction of f, right? So that's partial of v with respect to x times f. That's this. The output is defined like this. So that's reminiscent of that for linear systems. This one is reminiscent of this one here. So here we have q that is positive definite and here we have a class k function of the norm of alpha. So we are asking a strict positive realness in the linear systems. It comes to asking this negative definiteness of v dot in the direction of f, okay? So basically we are asking the system to be state is strictly passive. You see, we have v dot less or equal than minus alpha plus uy. So the energy balance inequality gives me this. Supplied energy is larger than the energy dissipation through the whole state plus a constant which is actually negative minus v zero. So that's the energy at the initial time. And now, so that's what we want to work with for to use passivity to design to design controllers. This is just a reminder of passivity for basically what we saw in the previous slides with different notation to confuse you a little bit. Well, I already mentioned that this integral is known as inner product which can also be written in compact form like this. And we speak of passivity if this integral is bounded from below by a constant, a real constant. So that's basically this number here, okay? Output strict passivity, we have this guy and the state strict passivity, we have this guy. So let's see passivity based control in its general form. So this method goes back to the work of Peter Kokotovich and Hector Sussman in 89. And I don't know the story very well, but Ortega, so Kokotovich was kind of the big boss in Illinois in 89. And Ortega and Spong, Spong I think had just been recruited assistant professor and Romeo Ortega was a positive there. So yeah, Kokotovich came up with this method of passivity based control and Romeo and Spong more or less in the same time. They wrote a paper that has been cited a lot on adaptive control of robot manipulators and they coined the term passivity based control. In this paper, it's called feedback passivation or something like that. Anyway, the method is here just for the little story. And essentially it consists in the following. So they were studying systems that have this form. So these systems basically come from, you have a non-linear system, we are in the 80s. In the 80s, the people were mostly working with feedback linearization. I guess you have studied that feedback linearization. So you want to find an output and the feedback such that a coordinate transformation that brings your system into a linear form. So there are some structural conditions to be able to do that. And sometimes for a certain output, you cannot feedback linearize the system completely. So write it in linear form, but only partially. And you end up with a system that has a linear part and a non-linear part. So these guys came to this system from working with feedback linearization. So here what we have is after applying the preliminary feedback and transformation and so on, let's suppose you end up with this system. So you already applied a preliminary feedback, but you say, okay, I'm going to add another feedback to complete the stabilization task because I didn't manage to linearize this part. This is what people know in feedback linearization as zero dynamics with respect to the output Xi. So meaning that if you partially feedback linearize your system and it became like this, it is fairly easy just to design use so that if A and B is stabilizable, you just bring Xi to zero and that's great. But once you bring Xi to zero, what will happen with the zero dynamics that is the dynamics when Xi here is equal to zero. So maybe they are unstable and there is nothing you can do about it through the controller priori because well, it's just sitting there and there is no control input there. So what can you do? So these guys came up with this nice method that applies of course to a certain class of systems, not just any, a certain class of systems for which F satisfies certain properties. So we need to lay out some assumptions and definitions and stuff from this paper to understand what they did. First, they say, well, our result will hold for systems for which when we take Xi equal to zero, we know that the origin for the resulting system is gas and it's a non-minimal phase and the nonlinearity can be written in this form. Sorry, I have a doubt about this. So anyway, the nonlinearity has to be written in this form. So basically you want to, from this nonlinearity you want to extract one part that they call F zero plus the sum. So this is a bit particular of a bunch of nonlinearity that are multiplying certain outputs and they call an output feedback positive real if it has a property that there exists a feedback like this. So minus k Xi to place the poles of this guy here and an extra input such that when you apply this feedback to the linear system here, you get a strictly positive real system, meaning that the matrix A minus Bk will be Horowitz and the closed loop system with this feedback will be strictly positive real. Okay, so it will satisfy the Kalman-Jakovic problema with these for the closed loop system, right? So with the feedback matrix here. So you take A minus Bk, that should be Horowitz and plus the system should be of relative degree, relative degree one. Now, what does it mean a feedback positive real output? Well, there are a lot of definitions and it's a bit hard to read the paper, but it's a very important paper. So they call this feedback positive real output, essentially if it's an output such that the non-linearity, well, such that you can write this non-linearity like that. So they say in general terms, if f belongs to the span of y. So basically y is a vector of outputs, right? There is yi here, so there is y. So it's a vector of output. So you can consider that as if you are divide, you are splitting the non-linearity into a linear combination of a bunch of other non-linearity in which the coefficients that are multiplying your other functions are also functions of the state. So it's essentially this guy here. So again, you want the output to be composed of y1 up to ym and each of these is multiplying a non-linearity. So you have f0 plus f1, y1 plus f2, y2 plus f3, y3, et cetera. So the non-linearity has to have this special structure. And then they say, well, I'm going to call f, a stable FPR stories span the composition, essentially if you have that, the first non-linearity in f is globally a asymptotically state. So yeah, the origin for this system, okay? So x, your system is x dot equals f0 of x and psi plus y1, f1 plus up to ym, fm. Now, essentially what they are doing is assuming that this guy is already gas, so there is not, they don't have to worry about that. So basically we want to worry about the effect of these non-linearity here. And what the passivity-based control created by Kokotović is doing is defining U in a way that it takes care of the effect of the non-linearity. So the non-linearity here, defining U in a way that it takes care of the effect of these non-linearity in the system. Okay, so that's what we are aiming at. But for that they need all these assumptions. So as I was saying, this has to be gas, and not only gas, but also globally asymptotically stable uniform link psi, what does that mean? It means that we should have a Lyapunov function that is positive definite and radially bounded It's a Lyapunov function that depends also on psi, but the bounds do not depend on psi. And V should be negative definite along F0, also independently of psi. So it's a very strong condition, but yeah, you can have it for many systems. And then what? They say if we have all these conditions, essentially if F I can decompose it like this, then the system is globally smoothly stabilizable. So smoothly emphasized because again in the 80s, there were already some results using non-smooth controls and constructing Lyapunov functions that are not smooth, et cetera. So the big deal was to do all this smooth. So yeah, if your system, if your F can decompose as I said, and you have these properties, these guys say the following. Well, all you have to do is go and take that V that you already know that works for the first piece of the system that is F0. Take that F, take its derivative. The F0 term will give you something negative.  Because F is F0 plus all that. So you will have this. Well, that's where this equation comes into play. The next thing you want to do is design an additional input in that system that will take care of those non-linearities. And all you have to do is to add to your Lyapunov function that you designed that you have initially for F0, right? Sorry, I wrote it as a function of X, but it's also a function of psi. So you use that function, right? No, sorry. Yeah, I think, yeah, yeah, yeah. Yeah, but the psi appears in the non-linearity. I think it should be there in V. Anyway, it's a function V that you use for this part of the system, okay?  What you want to do next is to add a quadratically Lyapunov function in function of the state of psi, which is the system this below. And when you take the derivative of that, what will you obtain? Because you designed K for this to be Horowitz. Well, since this is Horowitz, it generates a function P, right? Metrics P that you put here, and you take the derivative of this new function. So the function you initially had plus this new quadratic function here, okay? You will obtain that the derivative is, well, is less or equal than this guy that comes from V. Now, this one, you know that it is bounded from above by a class K function. Then you have all these guys that you don't know a priori what to do with them, and then you have, of course, this guy that comes from Horowitz of these metrics. But then, because you added a new input that you are still going to design now, you will get this term here. Xy transpose times P times B times the new input, okay? And what will happen next is that you see PB by construction because your system here is strictly positive real. So it has this structure property that PB is equal to C. C being exactly the matrix that defines this output here, what you will have is that actually this term here is nothing else than the sum of all the outputs, Yi times the inputs Vi, okay? So it's essentially why transpose the new input. So you can see easily that all what you have left to do is say I'm going to take V to be equal to that with a minus sign, so because V appears there and V and Y appears here, Y appears here.  And you will be left only with this. As I said, this by assumption generates an alpha term and this one comes already from the orbits of that. So yeah, and all the rest you just eliminated it. And you end up with this very nice inequality. So yeah, in case you don't see the passivity here what you have is before eliminating all the terms you have this guy here, so essentially you have W dot less or equal to minus alpha 3 X, right? Then this guy minus one half psi transpose Q psi, then plus this guy plus Y transpose V, sorry it doesn't fit, plus all the bunch of things that you want to eliminate. So if you integrate this and just for the sake of argument, forget about this guy, you see that you have this passivity property between V and Y, right? So if this thing that is bothering you was not there you have a passive system. Now essentially you have a passive system with something else that is coming into your system but it's coming into your system through the right channel because it's multiplying exactly the passive output. So the perturbation that is coming into your system and so multiplying this output that is from which you have passivity so with the extra input you can actually go and kill the terms that are the perturbation that is coming into your system.