[Music] what do we mean by these models and how do we get them and what do we use them for right so that's something which we need to understand so once again I am revisiting our idea of what we mean by a system right so for us a system is a mapping between U of T and Y of T now suppose if let me call this as problem one okay the first problem I want to solve is that given U of T and Y of T we are asked to find the mapping s okay that relates the input U of T and the output Y of T okay this problem is what is called as the mauve problem of synthesis that is suppose let's say you don't take C for let me take an example of a room that is being ad conditioned right so let us say if I take the example of a room that is being a heart condition then you suppose if I want to regulate the temperature of the room right so the temperature of air in the room is the output of the system and let's say the input to the system is the amount of cold air okay that I blow into the room okay so let's say you know like let's look at this let's say we have a room air conditioner right so let us say the temperature of air in the room is my output and let's say the amount of cold air provided right to the room through the AC vents is the input of the system so what would this synthesis problem entitled you know like the problem of synthesis essentially means that I provide varying quantities of the input to the system and then measure the output and try to get a mapping between the two okay so that's what I do so that I get a mathematical model for the system so in this case you know like if I provide various problems of cold air to my gonna say air conditioner and I measure the corresponding temperature and I get a relationship between the two so then I will know how temperature of air in this particular room would vary you know like I say provide varying amounts of cold air right so that's something which I won't know so once I do this first problem of synthesis I can go and do problem number two which is going to be the following given the mapping s once we find the mapping s and given a particular input U of T find y of T suppose I have got a mathematical model for temperature of an in this room right equipped with an air conditioner so tomorrow let's say Anana KY I want to essentially change that conditioner right so and let's say I have five choices so I know like what what is the capacity of or to say blowing cold air into the room of all the five a C's right without even purchasing them I can use the model which I have done to figure out how temperature would vary and settle down to a desired value by doing this analysis right once I have the mathematical model for this particular process of air conditioning so the advantage is that like I can predict the temperature variation without even essentially the requirement of buying the air conditioner installing it and testing it right so that I can make a well-informed choice right as far as which AC to buy so this problem is what is called as the analysis problem or the prediction problem okay so that's that's essentially the analysis or the prediction problem okay so and let us say we have looked at these two problems you know like another problem which we can do once we complete the synthesis problem is that given the mapping s and a desired output Y of T find U of T okay that is the third problem that we can solve so this problem is what is called as the control problem okay so what do I mean by this in onyx suppose let's say you know I come into a room with an air conditioner I essentially want to would say regulate the temperature of air in this room right so I said the temperature of air as 25 degree Celsius the question is that like what is the amount of cold air which I should blow I know which would get my temperature to 25 degree Celsius right that is the problem of control so I know what is a mapping of the dynamic system and I want to design output what is the input which I have to provide okay so in a certain sense it's the inverse problem of what we do in pran over 1 right so here you know Anik are inverse of what we do in problem number 2 right so so this mapping helps us to find a way of T given on U of T in the problem of control what we do is that like given away of T you know like I want to use this mapping to find what you of T will get me that output Y of T right so that is the problem of control okay so what are a few examples of control you know like that we can we can essentially look at Anan ik we have already looked at in on ik room temperature control all right so that's that's one example that we have discussed right let's say you know like we have also looked at an example where we want to control the motor speed of a motor right okay and our human body is a marvelous controller say for example you know like our human body maintains our body temperature in a very very narrow band right so even like if it does if the temperature goes 200 degree Fahrenheit right so we are in trouble correct so so essentially our body maintains our internal body temperature you know like in a very narrow band irrespective what that environment temperature is right so it doesn't matter whether we are in winter or in summer you know the human internal body temperature is maintained at almost a constant value and that's a great control system you know like and for example the blood pressure you know like that is maintained by a body you know it should be in a good range blood sugar levels right so that's another modulus control mechanism and in fact like given our heartbeat right so even our heart which is a pump essentially that pumps you know like blood throughout the body you know like it needs to work repeatedly again and again right so for you know like as far as we live right so that's extremely important for us right so our human body is filled with more or less controllers you know which regulate various variables you know like our own desired values very accurately and for a long time you know that's something which is marvelous right so so essentially we are going to look at various case studies as we go along and then like we will see how to formulate practical problems as a control problem and solve them okay so before we were to say get into the mathematics I just want to give a more another physical perspective as far as what we were to say how we classify how we look at control and so on right so essentially in a very broad sense a people classify control and what is called as open-loop control and closed-loop control so what are these terms so let us say we consider the example of a ceiling fan right so if I have a ceiling fan what's going to happen I have a fan I have a regulator I have a switch right I could come switch on the fan and I can vary the speed setting maybe indiscreet what is the quantities by using the fan speed regulator and that's about it right so I just get some output which is a fan rpm okay so that's an open-loop controller in the sense that you Nanak it cannot account for any disturbances right so which come during the operation of the fan say for example if there's a voltage fluctuation right then you know like the output of the fan or the speed of rotation of the fan is going to fall down okay or go up depending on whether the voltage is falling down what going up alright so then you know the question is are like I lose certain performance right but the question we need to ask ourselves is that like is it important for a ceiling fan right so typically ceiling fans are open-loop control systems where the speed settings are calibrated in the factory and the regulator essentially are just the I would say fan rpm in discrete I'm going to say in a discrete range of values right set of values and essentially in the presence of disturbances obviously the system response gets affected right so that's a what to say characteristic of open-loop control okay so open-loop control means in onic what are the various I want to say characteristics there is no feedback we will shortly define what is called as feedback okay so as a result there is no feedback in open-loop control so essentially it cannot tolerate disturbances okay it's not robust to disturbances and so on okay but on the flip side the cost and complexity are lower on the other hand suppose you know like you fee want a a fan in awning whose rpm needs to be maintained at what is a very baseline value let's say 100 rpm right and I cannot afford a huge variation in the rpm right for some industrial application right what would I do what I would do is that like I would measure the actual rpm using a speed sensor and then like I would take the actual rpm and each and every instant of time compare it with what is a desired value and then take the difference quantify the difference between what I desire and what is actually or being obtained take that error and pass it through what is called as a controller and the controller will then adjust the electrical input to the fact all right then then that happens you know we have what is called as closed loop control okay and closed loop control systems have feedback okay feedback is the process of measuring variables that need to be regulated or control so that corrective action can be taken okay that is the process of feedback so consequently closed-loop control I want to say is more tolerant or robust to uncertainties that come in disturbances and so on okay and typically what we called us during the design process unmodeled dynamics and so on okay so but anyway let me qualify them to group them under uncertainties okay uncertainties disturbances and all but the flipside of closed-loop control is in I were to say the cost and complexity are high okay so in this particular course we are going to deal with closed-loop control okay so that's what we are going to deal with in this particular course okay so let me just draw a construct an example to just how to say explain what we did suppose let's say you know like we have let's say a DC motor right so the DC motor is our a system okay or plant okay and - this DC motor let us say I provide a voltage V of T as my input and Omega T is my output so this is my input and this is my output suppose let us say I want a desired output okay let's say a desired rpm of rotation okay from the motor so this is what we call as a reference input okay and what we do then is a following so suppose if I measure the actual output then what I do is that I go and compare then I take the difference between what I desire and what I measure which is what is called as an error and then let's say we pass it through an element called as a controller and the controller then calculates what should be the input that should be provided to the DC motor okay when the controller calculates the input that should be provided to the system the objective control is added to the term input okay so the input to the system becomes what is called as a control input okay so this is feedback okay so this is a typical layout of a closed loop control system with what is called as negative feedback okay okay so this is essentially closed loop control system with negative feedback why it's negative feedback because of the summing Junction you can see that we are we are subtracting you know like the feedback signal is subtracted from the desired reference input right so that's why it's called as negative feedback okay so we are taking the difference between the reference input on the actual output okay and calculating the error okay this is what is called as a closed loop control system with negative feedback okay so we can see that this is what we are going to essentially do in this particular course okay so some more terms that I want to or to say introduce you is that like this process of essentially taking this output measurement and feeding it back giving it back is what is called as feedback this is what is called as a feedback path typically you know like we can have a mapping in the feedback path see for example say for example you know like we can have some sensor dynamics coming into play here right so let's say I use a sensor for measuring speed you know like that may have its own dynamic characteristics I mean use I may need to use a filter to filter out noise then the mapping comes in the feedback path okay so if a mapping comes in the feedback path you know like we call it as a when the mapping is not what we call it as non-unity feedback okay so when this mapping in the feedback path is one we call it as unity feedback okay like we will see how this affects our analysis later on right and many times you know like when the controller provides you know like calculates and provides a control signal to the system it is typically realized by what is called as an actuator see for example you know like the let's say you know I have to essentially move I said design a motion control system that essentially displaces a workpiece along one axis right let's say simple translation of a workpiece in a in a machine machining system right so let us say you know like I want to regulate the position of the workpiece right I give a what to say a voltage signal to a let's say an electric motor drive system which essentially provides translational motion to the workpiece right suppose if I control a calculate that at this point of time provide five volts of input right to the electric motor okay or if it calculates and tells me that look you know like provide you know like ten Newton's of force to the workpiece in order to essentially move by some distance X right so the controller may say provide ten Newtons on this instant of time but then there may be a small or to say response time before the 10 Newtons is actually realized in practice through the electric motor and drive system right so that if that is important then you know we need to figure out a figure in what is called as actuator dynamics in the design process okay and sometimes you know like we can have disturbances coming into the system right let's say you know like I have to say sudden load that may come on a DC motor right for example you know that I mean to model as a disturbance and then let's see how we can overcome such disturbances and so on right so one can see that you know we can add varying levels of complexity to this feedback path okay as a feedback system okay we are going to study the basic feedback loop you know like that's what we are going to do in this particular course okay and as we go along maybe when you go to some case studies we'll add some more blocks okay and then see how the design varies okay to summarize you know like what what we we we are going to do in this course is the following okay so as a summary of all this discussion that we did so our the title of our course is control systems right so what we are going to do what we are going to learn is the following right low slope feedback control of sizzle LTI causal dynamic systems okay so that's what we are going to do in this particular course okay so that's the class of systems that we are going to do so alt a causal and dynamic systems and we are going to do what is called as closed-loop feedback control of this class of systems okay and it's so turns out that this class of systems right so LTI causal dynamic systems the the mathematical models that are typically used to characterize this class of systems okay usually take the form of a linear linear ordinary differential equations okay what what are abbreviated as Oh des with constant coefficients okay so typically the mathematical models that are used to characterize this class of systems that we are going to study in this particular course take the form of linear ordinary differential equations with constant coefficients okay so that's the class of equations that we would be focusing on and another important aspect is that the kind of mathematical models we are going to essentially look at are what are called as spatially homogeneous that means that B we do not consider or to say variations of variables with space right so we only consider temporal variations of variables see for example let us say I want analyze the variation of temperature of air in this room right so obviously the temperature of air in this room can be different you know like depending on the point where I am measuring and also the time at which I measure right so but then what we do is that like we lump all the points in this room that is the temperature of all the point in this room into a single entity which essentially can be represented as a function of time okay so in a certain sense you know like what we are doing is that we are lumping the effects okay as for a spatial variation is concerned and we are going to assume that the entire room can be characterized by a single temperature which is only a function of time okay so spatially homogeneous means you know like we are essentially following sort of a lump parameter approach right towards our modeling right so spatially homogeneous we are going to have what are called continuous time continuous time dynamic deterministic mathematical moments okay that's the class of models we are going to use right so essentially in a certain sense you know like continuous time means you know like we treat time as a continuous variable variable so from automatic perspective it just means that we are going to get all the ease right ordinary differential equations dynamic means we are going to have derivatives in the equations you know like that's what essentially it means it basically it represents so what is a dynamic model it's something which explicitly considers or future states of the system right so and it essentially incorporates what is going to happen to the system in future right through by incorporating derivatives or the variables so that's a dynamic model deterministic means we essentially and neglect any stochastic effects okay we do not consider variables as random variables we consider all variables as deterministic variables and we are going to have deterministic models so okay so this is a class of models that we are going to use so in summary you know like what we are going to do in this course is closed loop feedback control of C so LT a causal dynamic systems using spatially homogeneous continuous time dynamic deterministic mathematical models okay so that's what we are going to essentially learn how to do in this course okay so what we would subsequently do is essentially have a brief recap of what is the mathematical background that is required to do this analysis okay so that's something which we are going to learn okay which we are going to recap you know it's assumed that some mathematical courses have already been completed before one comes to this particular course particularly causes on complex variables ordinary differential equations and plus transform so we would quickly recap some of these or to say tools and then we will move forward okay so that's going to be the plan of action fine okay so thank you [Music] you