 So, we were talking about information structures and information structures which were of two of two kinds classical and the ones that were not classical as non-classical the any information structure we have studied so far which is all our stochastic control problems they involved they involved classical information structure. And the question I had in the previous at the end of the previous lecture was does it make sense to ask practically to study any problem that involves in non-classical information structure. So, I will show you now a typical control system diagram and let us think let us what we will do is let us think carefully about it and let us see if there is any reason for us to think that a non-classical information structure might emerge. So, a typical control system diagram has something like this that is a system which is what we need to control the system is controlled through actions that are sent to it by an actuator. So, the actuator may be a physical actuator and that actuator is actually present on the system itself. So, it is an act the actuator is in fact is some kind of a physical this thing say it may be a plunger or an arm or something like that in that is actually or a motor or a gear or something which is physically attached to the system. So, we can think of this as something that is connected to the system itself directly physically. The actuation comes to is decided but what signal how much actuation is to be provided is decided by some kind of a computer computational device say for example, that is your controller a microcontroller a chip or something like that and this is the entity that does the computation. So, let us call this as a decision block this is what decides what needs to be what the actuator needs to do. That is what sends its decisions to the actuator. Now, this decision block may or may not be physically associated present with the actuator oftentimes controllers require a different physical environment than what is what the physical then what the system actually is operating in. So, the actuator may be present on the system itself but the decision block here the control the computational block that I have drawn here decision and computation block here may not be physically attached to may not be physically present on this on the actuator itself. So, usually this is something this is located separately from the actuator. We can ask also how is it that this decisions how are these decisions in fact arrived well these decisions are arrived based on readings and based on observations that we get right. So, these are these are readings and observations readings or observations now who gives us these readings and observations these observations these come from a sensor these this sensor is what senses the this sensor is what senses the system and provides us with observations. And again usually for physical reasons this sensor may be attached to the system may not be attached to the system. But nonetheless one thing is for sure that it is not present at the decision block it is it is it is the sensor the sensing block is is closer to the system and and from the all the observations that we get from the sensor are then sent to the decision block here. So, what does this mean then we can think that what we are actually doing are two different decisions in a in a broad sense. We are here on first at the first step we are doing some something called something that we call sensing or observation. There is in fact regardless of what you know usually when model sensors as a black box you know you have a system state and we get a certain reading we just model model that as a transformation. But remember inside a sensor is itself a some kind of an algorithm which or some kind of a decision algorithm which is which is coming up with the reading that is to be sent eventually to the decision maker. So, there is therefore we can think of this here as stage one of our decision problem we are this is where we take the first decision which is the which is to get the and this decision is to come up with the readings that are to be sent to the decision maker. So, as I said usually when we think of in terms of hardware we do not have the option to design the way this readings are to be sent we do not have the option to do that. So, we take these as fixed but technically I am now think trying to think of this problem in a fresh light in you know without the baggage of current practices. So, if one thinks of the problem in this way then you would realize that this also is a decision you know which is the decide which is taking the taking the observations that come out of the system taking or sensing the system and deciding what is it that is to be sent to the decision block. So, this here is a decision. The other thing that is happening is because these are specially separated the reading block and the and a decision block there is implicit in this a communication channel right this is a there is a implicit in this implicitly present here a communication channel communication medium let me let me call it that. And if these are wireless media they usually involve noise and corruption or delays or something of that sort and whatever comes to the to the decision maker will be corrupted by all of this. So, we have stage one here which is which is reading and observations we have then stage two which is which is taking where in one takes decisions. Now, since we have since we have a communication medium and the communication medium could be adding noise we can say well let us find a way of beating the noise in the system right defeating the noise that is present by using using some kind of noise cancellation techniques. So, which means that there is implicit in if you if you want to do that right if one wants to take those techniques if you want to do any kind of noise cancellation here then this diagram again has to change then this here of course there is a communication medium I will just draw this again this is there is still a communication medium here, but whatever is but we the techniques for canceling noise have to basically do some transformations on the signals that are coming into the medium and the signals that are going out of the medium. These are usually called encoding and decoding but let us let us not worry about those but there is there would usually be a transmitter here which is deciding what to transmit knowing that there is going to be noise in the medium this is your transmitter and then there is a receiver which receives what comes out of the communication medium and then sends that decides and then sends something onwards to the decision block. So, now what have we got now as a result of this this is not stage 2 anymore I wrote this as stage 2 earlier when there was only a communication medium and the medium was was adding noise but now really there is what has changed is that this is not stage 2 anymore because really unknowingly we have again added a two more decision stages we have added a transmitter here and added a receiver here. Now these again less like we said we take sensors as black boxes and take them for granted oftentimes one also takes the transmission and the communications infrastructure for granted. But remember we are doing a fresh end to end design by looking at everything holistically then in that case the design of the transmitter and the design of the receiver should also be considered as part of the design of this entire entire system here. So, therefore implicitly what has what has happened as a result is that this therefore has become stage 2 this has become stage 3 and this has become stage 4. So, what has happened because of this by introducing a transmitter and receiver is that we have now basically have now more decision entities in the problem. So, what is what this diagram represents is the is one where we are designing where we are doing joint design of communication sensing and control the usual control systems paradigm assumes that we are we do a only a control system design keeping holding for keeping constant or taking for taking as fixed the sensing technology and the communication methodologies involved. But once we think holistically about the problem we realize that all of these elements have to be designed jointly. Now, so you might say okay well this is too complicated I would not want to I would not want these transmitter and receiver in the picture I would want to get rid of them and I would go back to the earlier problem then well if you are going back to the earlier problem in that case you only have the communication medium in that case all you have is this block here you do not have the transmitter and receiver block. Now, let us look at either of these stages either of these cases in either of these cases whether you have one whether you have the transmitter and receiver as part of the design or you ignore them and you have only a communication medium in either of the cases what is there on the left of the communication medium is not perfectly known to what is there on the right. This is the fact of the communication medium even whatever is comes in to the communication medium gets corrupted by noise and then what comes out is not what was sent into the communication medium. The communication the transmitter and receiver what they do is they what they tend to do is they try to ensure that what they try to ensure the transmitter and receiver try to ensure that what the sensor wanted to send is what is received by the decision by the decision block. So, they try to ensure that this end to end transmission happens in a noise free manner but this here is never is never noiseless never noiseless. So, if we because that because the communication in the reason it is never noiseless is because there is communication there is a communication medium in between. So, whether you have the decision whether you have the transmitters and receiver in the picture or you do not have them in the picture and just have a communication medium regardless of that there will always be two entities in the in the system. Two entities in the system such that there is two entities where there is going to be some entity let us say at time k or time t suppose who has information i t which is the information that that entity has and that there is another entity at time s where s is greater than t. So, an entity that acts later and it will be the case that this the information available at with the entity at time t is not what is available with the entity at time s. So, whatever it is that so once there is a there is noise in this medium you have this issue that that whatever is the information available on the left of that medium is not available on the right at the somewhere there is a compromise there is some loss. Right now this may have the entities could be the entities here could be the sensor and the decision maker and the the the loss may be happening because of the communication medium the entity could the entities could be the transmitter and the receiver and once again the loss is happening because of the communication medium. So, regardless of what we how we say think of this there is there is going to be some two entities where this this inequality breaks down. And now you recall what was our definition of the classical information structure our classical information structure was that this in equality this this inclusion must hold for all times. So, there is the the i k minus 1 is all is a subset of i k for all k. So, this so once there is noise somewhere here this inclusion does not hold. So, the the classical information structure breaks down. So, the so classical inform so this results in non classical information structure. So, you can see the roots of non classical information structure in us in a typical control problem control systems or the first is that is that our attitude of that we are not going to look at we said we are not going to look at any of these problems in the way they are usually posed which is where they take sensing and and and communication for granted. We are going to think of sensing and communication as part of the overall design of the system. So, the control when we design the control system we are really designing not just the actuation and we are not just designing the control signals and the actuation we are designing also the sensing and the communication right. So, morally that is what one needs to do when one has to design a control system right. So, that is part of the design. So, the optimal design involves designing all of these things together. So, that is one thing the second is the there is there is a spatial separation there is a spatial separation between sensing and sensing and computing and as a result of that communication is involved once communication is involved whether you like it or not somewhere there will be two entities at least that do not where the entity acts that acts before it does not the information available to an entity that acts before is not available to one that acts later right and this information loss continues even further as well. So, this so once there is a time once there is spatial separation spatial separation plus joint design of sensing communication and control this in this is giving us a non-classical information structure. Now, here so this now has leaves us with at an interesting stage. So, we have we have found that once there is spatial separation and we are talking of joint design of sensing communication and control then there is always a non-classical information structure at play. But one might ask why should one look at joint design why not look at the separate design right why not take the best sensor you have in the market put the best sensor in why not take the best receivers and transmitters that we have put the best them the best ones in. So, in other words you you you put in the sensor that has that is optimal for sensing take the transmitter and receiver which are which are optimal for communication plug those in and then take then assume that the sensing is not a decision communication is is not a decision it is in fact sensed observations are available perfectly and then assuming that let us one can design then a control. So, this paradigm is what is called a separate design this way of thinking is called a separate design. So, separate design is basically design sensing design sensor for optimal sensing plus transmitter and receiver for optimal for optimal communication and then and design design controller assuming assuming the above are deployed. So, this is what is separate design now unfortunately the separate turns out although so this in fact this is this is what is called separate design right and this is how industrial control systems are actually designed one takes the best you know components which are designed for component wise excellence. So, the sensor is you take the sensor which is designed to be the best sensor it can be you take a communication device which is defined designed to do the best quality communication best reliable most reliable communication put those together and then you you you put together a controller you then build on that a control logic or a control architecture unfortunately all of this design is is just design it is not necessarily known to be optimal. So, the if you want had to find the optimal design then one had to basically are look at the joint design problem the one that I have the one that I that is written out here where all four stages are to be optimized together right and then come up with the with the design for that particular for that particular for that particular problem for the problem at hand whereas on the other hand here right what we are doing is we are doing separate design with where each component is being designed separately and then putting it together there is no guarantee that the separate design is in fact in fact going to be optimal for is going to be the optimal design. So, as a result of this it is not clear at it is not obvious at all that while separate design may work in the sense that it achieves the goals that we want we we we have set out from from the control from the problem but it may not be mathematically optimal as far as the if one looks at the system holistically. So, what this means is that any kind of any kind of fresh slate clean slate control system design necessarily has to think of all these components together right. But there is still something amiss here which is that we have not really proved that the separate design is in fact not optimal or or is away from optimal. This here is where theory stood for some time until a brilliant paper was written by a by a scientist called Wittsenhausen and he showed that in fact there is a gap that separate design is not necessarily the best the best joint the best joint design that joint design offers orders of magnitude improvement over separate design. What this meant therefore is that our hope that one could just achieve the optimal optimal performance for a control system like this by using optimal components is not necessarily true right. Now one way in which this this separation of sensing and control actually come manifests in in problems that we have already seen is the is the LQG problem itself. The LQG problem remember we had that the the structure of the optimal controller was that it involved a superposition of an estimator which we we said we can do via Kalman filtering and then a controller which was just the same controller that we had for a deterministic problem. So this superposition helped us to say well we could do an optimal estimation on the one hand and optimal control on the other hand put the two together and that thing performs optimally. The what Wittsenhausen showed what was that this is not true once the information structure is not classical. So if the information structure is not classical there is no separation principle that can be applied which means the estimation and the control has to be jointly designed and in fact the joint design far is far far better than any separate design that one could do. So this this here is an epoch making result by Wittsenhausen and it will take us a good part of the coming lectures to discuss all the implications of it. So that is what is coming up next as in the next part of the course.