 Okay, we will start with now a little bit of a revision and some kind of a consolidation of some of the concepts and we will see how things are for the mathematical representation of control systems, okay. So we have found like now we have already done some of the parts, but now we are just consolidating these in some way. This has been done, some part of this has been done already in the past, okay, in your classes. So we just recall some of the concepts and move on from there, okay. We will get going with this kind of outline, we will see some of these properties of the system, the linearity time invariance, then we will see these three major different ways of representation and fourth one comes for the nonlinear systems. For linear systems, you have ordinary differential equation form, transfer function form and state space, okay. These are the three major forms and then how do you get solutions in each of these forms. So these are the concepts that we have already seen the nonlinear systems one is what we are dealing probably with new. Some of you might have done some nonlinear systems handling in systems in control kind of a courses. So we will not get into too much of a depth, but again the here idea is to kind of see from Macatron's application perspective how we can deal with these different concepts, okay. So we begin with the simple properties which you all already know. So the linearity of a system, the principle is important for like you know many systems. This principle of superposition is valid for the linear systems, meaning like the input if system is given input u1 and produces output y1, input u2 and output y2, then if these inputs are scaled and added or subtracted correspondingly, the outputs will get scaled and added and subtracted, okay. That is a kind of a concept that you have here. Time invariance is another very important concept. If you start at t is equal to 0 or any other time for the same initial conditions then your response is same that is what your time invariant principle says, okay. So this coefficients typically would be constant for time invariant system and in time varying systems the coefficient will be functions of time. So if you see carefully your regulation problem or tracking problem, regulation problem does not change the system property of time invariance actually. But tracking problem introduces the time trajectory that theta desired or x desired kind of a trajectory which is explicit function of time in the system. So the moment you start talking about the tracking problem in control it will have a time varying system considerations to be given, okay. So you have other examples also for a time varying system apart from this tracking problem is a space vehicle in a manner of the mass of the space vehicle is changing with respect to time, okay. Those kind of things will have to deal with little bit in a mathematically different form than usually we handle the time varying system or time invariant systems, okay. There are these kind of a different types of systems I mean they are kind of more kind of name that is given or definition that is given. So linear time invariant system typically are represented by linear ODE's so LTI systems, okay. So you have a time dependent coefficient for the linear time varying systems, okay. Then you will have this single input single output systems again they are typically terminology in the linear system domain and multiple input multiple output kind of systems MIMO systems ESO systems and then you use this LTI term for linear time invariance. And the nonlinear systems are typically represented by nonlinear ODE's or the state form of the representation that we have seen for you know getting the MATLAB numerical simulation going, okay. So this representation of the systems in different forms is what is important for development of control. So first is like you know our differential equation form which we have already seen in the feedback lecture, okay you start with the differential equation form and start applying some of the control fundamentals and analysis of control in that form only. Then we can consider a transfer function form where you have poles and zeros defined for the system and based on the poles and zeros you kind of look at the open loop kind of a stability of a system and we look by many, many different ways you look at the closed loop stability of the same system by considering say for example rule-plocus kind of analysis or board plot kind of analysis, frequency domain methods and those all kind of things which you have part of like this MAC course in theory that you have studied some of these kind of tools and techniques. So we will not get into the details of those tools and techniques you maybe if you want you can just a little bit kind of a refresh them we will not get into a lot of depth of those we see some of the applications if at all about those techniques, okay especially in the case of like you know placing the poles at appropriate location kind of a problems. Then you have a state space methods. So state space kind of a form is this x is equal to AX plus BU and Y is equal to CX plus DU. This form is used typically. So now this A, B, C, D matrices are basically as you know all constant matrices and now how do you solve system in which each of these forms is the next question and how do you obtain this form? This is again you are aware about or like you have seen already in the MATLAB kind of a simulation while doing the simulation we anyway need to get the system represented in the in this x dot is equal to fx form even if the system is linear this part here f of x of u and t will reduce to some matrix A and B and can be separated out in like you know x x related terms and u related terms, okay. So that is how like you know these different forms of the representation of the same system and can be there and you can choose to kind of if you have a linear system you can choose to work with any of the forms that you wish for development of control and we are just going to see now a little bit of a brief or revise the little bit of a you know brief of this Laplace domain representations and state space domain representations and then we will see how do you solve the systems in these different domains and that will conclude this part of the lecture, okay. So we used Laplace transform standard tables of transforms are basic fundamental properties to get the Laplace domain representation done. So you need to revise this some of these definitions of what is Laplace transform and how do you kind of take Laplace transform of a given system and things like that and some properties of the transforms and some kind of you know these are also properties like the final value theorem and convolution integral. So this convolution integral is an important concept to be looked at, okay. Although we may not use like for purpose of application or development of control, we may not use this convolution integral too much but if you want to have a solution of Laplace transform system done, you may need that convolution integral concepts for a given any given you know general input is there, you may need it is not mandatory but you may need that, okay. So this is an example of our standard differential equation of spring mass damper system and you use Laplace transform here considering so in for Laplace transform you need to kind of define what is your input and what is your output, okay. This input output definitions are your own definitions like you need to define or whatever requirements of system will dictate what is input and output. Then you take Laplace transforms and represent like you know this is output of the Laplace transform divided by input of the Laplace transform of this system will give you this kind of a transfer function and usually in transfer function we assume zero initial conditions. Suppose there are the initial conditions are non-zero then you cannot get this transfer function form, okay. This is very important because it comes from the definition of or the property of Laplace transform for a derivative. Typically the derivative will give you some kind of a derivative Laplace transform will give you some kind of a non-zero initial condition representation in the transform, okay. So to avoid that we assume initial conditions to be zero and then like the system has this kind of a nice form of like you know just Laplace transform is a division of two Laplace domain polynomials, okay. To convert this system into state space form you need in this kind of a form you need to define states first x is equal to x, x1 is equal to x and x2 is equal to x dot. So x1 dot will be equal to x2 and x2 dot will be equal to based on this equation, okay. One upon m times minus cx2 minus kx1, okay. So this is a typical process for getting the state space form, okay. You define the states because state space form has only single derivative but of a vector. So you need to kind of add you know introduce more variables in the system to to represent the system into this form, okay. So this is how like you represent system in these different forms, okay Laplace transform form and then the state space form and then you can handle the system in either way. There are control tools and techniques available in each of these domains to to think of system properties and develop control or propose control algorithms for that, okay. So this is some kind of example of non-zero initial conditions. So you can see how they they show up in the in the Laplace transform stuff. So you can get a solution with non-zero initial conditions no problem by using Laplace transform. That is no problem at all. You can use this kind of a method Laplace transform method to get a system solved. But if you see here you cannot represent it as a as a transfer function, okay. So only when the initial conditions are zero this part will go to zero and then you will have this you know this input can be brought here to kind of define a transfer function, okay. But in the state space form there is no problem of these zero initial conditions, okay. So that is how things go. Alright then next thing is finding a solution to to these different forms which we have discussed. So already the differential equation solving again I do not revise here the basic mathematics of re-solving you all know that I presume from the previous background. So if you not like we can just push up some of the fundamentals of homogeneous part of the solution and particularly integral and that those kind of things. Then for Laplace transform representation you typically like know substitute whatever is input that is given the Laplace transform the input and get the expression for Laplace transform output you know in the Laplace domain. And once that expression is available you apply the inverse Laplace transform by method of you know dividing the the polynomial into its roots, okay. So multiplication of multiple roots is the denominator and you represent these each of the roots separately and you then use the standard tables to to do the inverse transform, okay. So this method you I am presuming you all know this and you can go ahead and revise if you want to. Then the exponential for the for the state space representation of the system x dot is equal to x plus pu you typically use the matrix exponentials, okay. So we will we will go into little bit more details about this method. And then for nonlinear systems typically some systems you may get closed form solution by using this typical ordinary differential equation solving methods. But many systems may not admit that kind of a closed form solution or it may be too tough to get that. So under this scenario we can we will resort to the numerical simulations that we have seen and use those simulation techniques for understanding the dynamics and further for developing of development of control one can use this Lyapunov theory, okay. So we will go partly through this Lyapunov theory and come up with some kind of a you know universal controller for mechanical systems. So we have seen the mechanical systems in with the Grange formulation. So for those kind of a systems rigid body mechanical systems we can get a very interesting like a controller which is applicable for all systems of that sort, okay. And that is for the tracking kind of a problem. So regulation is very kind of a specific case of the tracking problem and one can kind of attempt to develop. So one can have that controller used and all the rigid body control problem would be solved. So this big domain of systems we can do the control of these nonlinear systems by using some Lyapunov theory based techniques, okay. So let us get into little bit more of this stretch based system solution. So you use what you use matrix exponential. See if you see here the homogenous part, okay. So as we know for linear systems already differential, ordinary differential equation solution have a homogenous part and a particular solution, okay. So in this case the homogenous part will be given as e to the power at times x of 0. This is like a 0 initial, this is like a initial condition. So if these initial conditions are 0 in the in the state space case then this homogenous part will be 0, okay. But in general it will be like you know represented in this kind of a fashion. Now this is like you know exponential of a matrix here. So we will see how to get to that. And then particular integral or particular solution will have this kind of a form e to the power 8 times t minus tau. So this is like a convolution integral with the with the control input, okay of the homogenous part of the solution. So this is e to the power 8 t minus tau times b u, b u is like know your control I mean the non homogenous part or control input part. So these are the this is like you know addition of these two will give you this complete solution of a state space system and you use this finally for the state space system solution, okay. So we will use this you know also for the case of digital systems when we talk of the sampling of the systems we may come back to this form of the solution, okay. So this is important for you to kind of know, okay this form of a solution would exist and now how do you find e to the power 8? So that is here. So how to determine this form e to the power 8? The first method is like you know you use Laplace transform. So if you take a Laplace transform of this system which is having like you know the homogenous part here, okay this control input is 0. So x dot is equal to a x and then then look at this. So what you need to look at is so there are some steps in mode here beyond this, okay. So we will not get into that but finally what you get see this x dot will give you s i here, okay. So see although there is no i here we have to kind of when we start subtracting we need to assume that there is a i here and also it will give you this you know by the way of derivative property you will get this kind of a initial condition 0 initial condition into the system when you take Laplace transform of x dot, okay. So by simply taking that and doing some kind of a mathematical simplification of homogenous part of this solution you can arrive at this formula, okay. So you get a Laplace transform solved and then like you need to kind of bring that Laplace on the other side and then you will get this solution. So you try it out and then like know if you get into any trouble like you know we will see again in the class if at all. So this is a typical kind of a form that you get for doing the obtaining this you know e to the power at. So you can use this so if you see that this is particular matrix which is which typically we use for Eigen solutions and that matrix inverse times that matrix you need to invert that matrix and get its Laplace inverse. Then this method 2 is based on Kali Hamilton theorem. So this Kali Hamilton theorem you need to again revise it basically allows you to express any higher power of e to the n, okay e to the power n which is higher than the dimensions, okay or n and suppose n is a dimension of a matrix n by n, okay. So a to the n and its higher powers are expressible in the form of a polynomial up to a to the power n minus 1, okay. That is what this theorem says basically, okay. So this comes from the fact that every matrix, okay, a satisfies its own characteristic equation, okay. You know characteristic equation of a matrix which we will get by you know lambda i minus a cut determinant, okay. So that is a characteristic equation in terms of lambda, okay. Now if you replace lambda by a then that equation is also satisfied that is what the Kali Hamilton theorem says and based on that you get this property which I said that, okay a to the power n or a to the power n plus 1 or a to the power n plus 2 everything can be expressed as a polynomial up to a to the power n minus 1. So if you see this exponential by the way of series expansion you have this lot of like you know this can be expressed as a powers of you know generally like you know the infinite series polynomial series with different powers for n, different powers for a and we use Kali Hamilton theorem to kind of express this as a summation of you know powers up to the coefficients will be different, coefficients will be different from what you have seen for the you know exponential series expansion. So we use the general coefficients and then coefficients are to be determined, okay. How do we determine coefficients by using like so that that equation whatever you get will also be satisfied by when you substitute lambda there, okay. So we use like you know the property of Kali Hamilton theorem only that this equation will be satisfied by if you are expressing this a equation then it will be satisfied by the lambda also like the eigenvalues also. So by using eigenvalues in the equation which expresses e to the power a t in the form of say like no series or polynomial up to a to the power n minus 1, one can now get the coefficients n coefficients which are determined by substituting n eigenvalues in these algebraic equations, okay. And once you get those coefficients substituting them will give you the value of e to the power a t. So you try it out and check it out actually for some simple system and that will kind of develop more understanding, okay. So these two ways are there for solving getting the e to the power a matrix power, okay. Then there are these concepts of stability we need to talk about or understand. So for any of these forms you have a definition for concepts of stability for system which is LTI this is basic definition of stability then one of the notions is so this is like a fundamental notion when the system is excited by a bounded input the output remains bounded, okay. That is one notion of the stability for especially for linear domain systems and other kind of a notion is in the absence of inputs output tends towards zero or equilibrium state of the system, okay irrespective of any initial conditions, okay. So this is like kind of a notion of some kind of asymptotic stability. So there are many many notions are you know definitions of stability you will find, okay. So for linear system typically like basic fundamental definition is bounded input bounded output and from there one can derive the conditions for stability, okay. So this concept or notion of stability and then conditions of stability they are two different kind of a things you do not say, okay. We get the conditions by applying these basic fundamental mathematical notions of stability, okay. So when we apply these notions for example for linear system we can get like these more tangible conditions. So the conditions are like not roots of characteristic equation or poles you know the poles of a system, right. With the Laplace transform transfer function representation you can define poles and zeros of the system and when the poles are negative real part the system is stable, okay. So if any one even only even one root of this characteristic equation has positive real part then the system is unstable, okay. And so system is marginally stable if these roots are on the imaginary axis, okay. So we will see like no little bit more about how this you know relationship between the system response and the location of poles has. So this you need to have some kind of a feel for these relationships that will be good, okay. So these are the more details about poles and zero definitions which you I presume you already know and then pole zero plots and animations you can see some some of the websites are given here to try out and I have some kind of things plotted here for you, okay. So let us we will go through that. So effect of poles is there on the response, okay. So we will see the cases different cases of these how these poles individually are like two or three poles affect the response of the system, okay. So let us see it here first you have a single real pole, okay which is on the negative part of the real axis, okay. This real axis is for the Laplace s, okay. s is imaginary number its real part is here and imaginary part is here. So when s is equal to minus a then it can be represented in on this complex explain in this kind of a form and with that typically you get the impulse response. So all these are like impulse responses which we have plotted here which is bounded or decaying exponentially, okay. So the same kind of impulse response now we are going to observe for different locations of a poles. So just to get a feel for like, okay when the location is here or location is somewhere else how the response is typically going to look like. See for example here real axis poles will not have any oscillations, oscillatory behavior. But the moment you introduce the two poles on the off the axis, okay, real axis you will get the oscillatory behavior possible, okay. Imaginary axis, imaginary part will be there for the s then you will get the oscillatory behavior. So here now we see that when the pole is shifted to the real side, positive real side then your system response exponentially increases, okay. And this is a case where like you have the two poles which are having both real parts and imaginary parts. When the imaginary part is added to the system of course you will need to consider these poles will always exist in the pair by the way. You cannot have only one kind of a pole shown up here and no pole corresponding, okay. That scenario is not there. So you will get this kind of oscillatory behavior in this case. See other thing is like you know if these poles are further shifted towards left side then these oscillations will start decreasing, okay. These oscillations will decrease little faster than these and if the pole is on the real axis then the exponential decay also will be faster. It is far away on this plane then exponential decay also will be faster. So if you see here this exponential decay is proportional to this e to the power at times like you know a. So a is larger you will get like you know the decay which is faster, okay. So this decay will be faster and faster happening if the pole is far away shifted from here to negative side of it. Now these are very interesting connotation or influence or implication in you know thinking about multiple poles. So if you have multiple poles coming up on the on this side, okay, negative real axis which will be there for many kind of real life systems. The poles which are far away would contribute lesser to the response than poles which are close by. So if the poles are placed or they are far away in the system one can understand that okay those far away poles will not have much of you know influence on a response of a system, okay. The response because of those will decay faster than response because of the poles which are close to the imaginary axis, okay. That is a kind of an important understanding we should take away from these discussions. Then pole shifted to the right half like you will see that this response will be kind of continuously growing with oscillations. So imaginary part of the solution is also there. Then poles are on the imaginary axis you will find that this is sinusoidal kind of oscillations, okay, harmonic systems basically without damping. Then double poles there on the imaginary axis, you will amplitude of the oscillations will keep on growing, okay. That is what happens. Then you have a single pole at origin you will get this kind of a straight line behavior in the response, constant behavior or impose response is a constant. So then if you have like double poles then the input again goes unbounded and then this is unstable system, okay. So now if you add zeros to this system like you know the response will have some small variations that is happening. So this is now we are considering this system with this kind of oscillatory behavior we have a zero added. So without zero and with zero added you see hardly any change in the response only this amplitude here changes and the zero added on the left half and then when you add zero on the right half then there is some kind of a you know small reversal of this one but the overall behavior of the system does not change, okay. So the zeros they do not affect the stability of system, okay. That we all know that mathematical property, okay. But this is just to kind of see okay what is the effect of zeros typically for a second order system, okay. So then other important idea that you already have probably studied and maybe we need to recall here is about the first and second order system standard kind of a response, okay. So every system can be looked at purely from the mathematical perspective without kind of giving I mean you know keeping the physical kind of interpretation apart for a while, okay. So if you are able to see the system from that perspective then like you know the basic mathematical principles that are there for these systems as a standard kind of a system responses they are applicable. For example, we have seen our motor, okay. So motor with the speed as a output is a first order system and if you know the first order system response behavior you can get to this you know directly the some of the interesting parameters for the you know especially the time okay or the characteristics time for the system immediately seen and can conclude about like okay whether that dynamics is important to be considered or not important to be considered all those kind of things we can talk about. So we are not then there we are not specifically like you know dealing with the motor system alone. Now for the same kind of a concepts will be applicable for you remember this tank filling system with the you know the flow input to the tank and then some kind of a drain output. So that kind of a tank filling system also is a first order system then heat transfer systems okay they are all first order systems. So first order systems you have some kind of a characteristics response known for example when you run a fan with the motor the fan will not go to the speed more than the maximum speed and then come back to the maximum speed okay that kind of a oscillatory behavior okay would not be observed in the first order system response okay. So this is these are some of the very important kind of a concepts from the you know understanding of the system from purely mathematical kind of a perspective and then interpretation in terms of physics okay that can be done very nicely if you at least know this behavior of the first and second order standard systems okay. So this is a first order system impulse response behavior so now this is given in terms of this characteristics time t and this t comes in transfer function in this kind of a form. So if you have some other kind of a terms on the top you just kind of divided them on the bottom side and like know produce this part sorry sorry wait wait wait not on the top side you need to have ts plus one form here first okay. So this whatever is if this term is non-zero you divide it by this term and this term okay so you get this kind of a some kind of a scalar multiple here so that will multiply this response here okay. So this otherwise this will be one over t value but if you have some scaling factor here in place of one it will be like you know that scaling factor times one over t okay that kind of a scaling will happen to the response also okay that's the only kind of a difference okay. So this is how one can get you know you are any kind of a first order system transform into this form and observe the behavior and and see what is this time t. So you can see that at some one kind of a time t the response goes down by roughly about 66% okay. So this is important kind of understanding can come up with this time constant so characteristic time constant for different systems you can see and then see okay what is the dynamics that is of importance what is dynamics what is of not so much of importance and then like you know you can take some call to simplify the system behavior. For the same first order system this is step response oh sorry 66 63.2% it is it is going so similar I don't know in a in a input response also there will be some value I think it will be similar 63.2% okay. So yeah so this is these are some of the important things to kind of note that okay your response you know already if you if you know the system like that you don't need to solve the system you know say if the if it is a first order system with this particular kind of a representation then you this is the response for that system okay. So this response characteristics will be able to use in in the in the sense of this characteristics time constants and other other other kind of a parameters which are defined for different different you know order systems. So first order system has this this t as a characteristic time constant okay if you come to the second order system okay so this is you can go through a little bit more detail for ramp response of the first order system some kind of a response will be shown here. For the second order system you have many more parameters to define and those again you will consider the standard form of this kind omega n square upon s square plus 2 zeta omega ns plus omega n square is a standard form of a second order system. So if your system is not in this form you basically like you know do some division and multiplication whatever you want to do and get it into this form then you may find some scaling factor may exist on the top that can be scaling the response instead of one this value will be that scaling factor okay and you have this different cases coming up here okay so this is typically for an under damped system you will get this kind of a response okay and then there are formulae directly for a maximum overshoot and this rise time okay it's a 0.5 or 50 percent of the maximum value that it is reaching in the rise time. So these are some of the specifications that are important from control perspective people may say okay I want to get my system settled in the in say this settling time is specified to be say five seconds or two seconds or whatever or I don't want any overshoot for my system for example robotic systems we don't allow any overshoot typically okay the system should like you can see that okay your arm should go finally and stay there it is it is very like you know annoying to see that arm is going like this and then settling to the final position and again going like this and settling to the final position okay so that's so that's to avoid that like you you you will have to have the specifications given okay in terms of the second order system response although no robotic system is is like a non-linear system you can give the specifications of this and then like no see how we can match those specifications okay and then these are like some of the formulae I will get for this so standard formulae you can check some books and our website also this form will be available okay and we can use this to design the system to satisfy some of these kind of constraints that are given okay and typically this transient response time will be specified in terms of like 5% or 2% of maximum value if you want to settle into that how much is your time okay that's how one can define okay so so these are like no in summary some kind of important concepts so what is the feedback is other so not a summary of this lecture but I'm saying generally these are the important concepts to ponder over or like no understand before we kind of start applying the these techniques and tools to actually designing control for mechatonic system okay so so feedback is important as a which we have already seen then you know what is open loop versus closed loop kind of a behavior or system and the the other thing is how do you process this desired feedback quantity we have partly seen that in terms of like no the simple PD control for ODE based system and how do you analyze it and things like that okay to come up with different kind of a control methods or algorithms and then this is important like okay what is if the goal is given how do you decide what should be control algorithm okay so you use all these techniques and technology okay which of these representations you want to use or it is applicable for a given scenario and then choose a proper control algorithm to try it out and do the analysis and get okay this is satisfying my system response you go ahead with otherwise you like you know iterate this process you choose some other kind of control algorithm and then go ahead with that okay and this is another important thing is like you know you need to look at control input what is needed to achieve your goal okay so in mechatonic system typically you will have a limit on this control input okay so you need to kind of make sure that we look at this control input and especially if the control input is too high then the system response will get modified when the saturation is put on this control input okay so like that we will look at some of these you know in little more detail for the for the especially like when the goal is given how do you choose what kind of control algorithm that we will discuss further okay in the linear domain non-linear domain and and take these discussions forward okay so this is what we will conclude for this part of the lecture and then we will continue okay thank you