 So, last class we are discussing about the singular value decomposition. Suppose, A is a matrix whose dimension is m cross n which is in which is rectangular matrix and this matrix A can be real or complex. Then we can decompose the matrix A into 3 product of matrices U sigma V transpose. For real matrix A, U is a orthogonal matrix. If A is a complex matrix then U is a unitary matrix and sigma is a diagonal matrix and each element of the diagonal elements are real and positive. And they rearrange in descending order the diagonal elements and V also orthogonal in case of A is real matrix and it will be unitary matrix if it is a complex matrix A. Then we have come to conclusion with this derivation of these things then the U columns of U are nothing but a eigen vector of the matrix A A transpose in case of A is real. A into A H means Hermitian matrix means A H means A transpose A conjugate transpose or A transpose conjugate if it is a complex matrix. That means the columns of U are the eigen vectors of the matrix A A transpose or A H A similarly columns of V are the eigen vectors of A transpose A or is it a for real matrix A transpose A or if it is a complex matrix H A H into A and A H is what A conjugate transpose and sigma about the sigma that the sigma matrix is a diagonal elements. Each element of the diagonal elements is non zero or non zero means it may be a zero some elements or some elements will be all elements will be positive. And these are the square root of the eigen values of A transpose A or A H A that we have shown it how it is coming. Then the structure of sigma if you see the structure of sigma as I told you sigma 1 this is not one sigma 1 sigma 2 dot dot sigma k the next is not a sigma 0 0 the sigma k plus 1 dot dot sigma m. If I consider that the rectangular matrix m dimension is less than n this the rectangular matrix whose number of rows is less than a number of columns then it would be there. And this is your n columns and this is your m rows and this is rearrange in such a arrange in such a way sigma which is a positive quantity is greater than equal to sigma 2 and sigma 2 is greater than equal to sigma 3 if up to sigma m plus m sigma of m this m is the number of rows in the that matrix or the dimension of matrix m into n if m is less than n. So, now we will discuss about the system what is called H infinity norm. As you we have discussed earlier also that let us call what do you mean by system norm system that we will describe by this. Now, singular values of the system what is this the system g is system a dynamic system which is described in a transformation matrix g of s. So, the singular values of a transformation matrix g of s m cross or you write it p cross m number of outputs is p and number of is this is you can number of outputs outputs and this is the m is the number of inputs. And naturally the dimension of the transformation matrix p cross m. So, the dimension of this is the transformation of means are defined as the square root of the Eigen values of the product of its frequency response g of j omega. We know the physical interpretation of physical what is called frequency response of multi input multi output system we have discussed last class by its conjugate. What is it mean singular values of g j omega is equal to root over lambda i g of j omega g of minus j omega transpose. If you recollect this one we have also for a real matrix we have also sigma i how to compute the singular values of this lambda i then a transpose or lambda i a transpose a. It depends on whether a transpose a or a transpose a depends on the minimum of m comma p m comma m comma n in our case will be is equal to minimum of you see the minimum of m p m is the number of inputs p is the number of inputs this or one can write it g i this is i lambda i g minus j omega transpose g of j omega. It depends on the which one is minimum out of this. So, let us call here this dimension is this is the lambda of that one that bracket is from here to here that bracket is from here to here let us call g is dimension is what p by m g transpose dimension will be m by p. If p is p is minimum of between m and m p is minimum then you will use this expression. If m is minimum see this expression this one g transpose means m comma p and p comma m. So, if m is minimum of between these two things then we will use this is you cannot p you write it some other let us call r minimum of this. So, we can find out the what is called single values of this one how you g f of s is the complex rational transformation matrix that is the conjugate of rational conjugate of g this is the this is also complex rational transformation matrix product of this one will be real. So, you can find out the Eigen values of this one and Eigen values will be real quantities of this one. So, this so I can write the singular values are positive or null real numbers. So, this symbol we are using sigma bar maximum singular values this is of what the matrix g of j omega when you are at particular frequency omega you have sigma bar maximum singular which is nothing but a by our definition g of j omega. This is the maximum singular value if you see that last class we have just read this is the maximum singular value next maximum singular value this one next maximum singular and so descending order of this one, but all r diagonal elements are positive quantity. So, this is the singular values of g j omega and which is maximum is denoted by a sigma bar is greater than equal to sigma 2 j j omega find out the singular values of g j omega the transformation matrix the second maximum singular value will be less than the first singular values that this that less than equal to sigma 3 g of j omega and so on and that is less than equal to sigma m g of j omega which is denoted by which is denoted by this is denoted by sigma underscore g of j omega. So, maximum singular value is denoted by sigma bar and minimum singular value is denoted by sigma underscore g j omega. So, now we can so this is the singular value of a transformation matrix now let us call system as we are talking about the system g of s agree that what does it mean that this singular values of g j omega has a physical interpretation then what is that physical interpretation. So, this is the system a transformation matrix which dimension let us call p cross m. So, if you see g dimension is p cross m which we can decompose u p cross small p cross p sigma p cross m and v h because this is for a particular value of g j omega particular frequency I will decompose into this one. So, these are the complex quantity of this one. So, this is the matrix and this dimension is m cross m agree. So, let us call we omit that g j omega here g j omega here is g j omega this is real and this is g j omega. So, omit g j omega of this one. So, we will get it g post multiplied by that v. So, it will be a v then is equal to u sigma v h v and this is the identity matrix because v is a unitary matrix. So, this dimension you see p cross m and this dimension m cross m agree. So, now this is the dimension of this. So, what is this one I can write it u whose dimension is p cross p and sigma whose dimension is m cross p cross m. So, what we can write it this in details form we can write it this in next g is a system matrix at particular frequency we know the g j omega agree this v is how many what is the dimension of v m cross m. So, we have a let us call m vectors are there v 1 v 2 dot dot v m and dimension of v 1 is m cross 1 this is also m cross 1 in such a way we have a m vectors. So, this is equal to we write it u 1 u 2 dot dot u p and this dimension is p cross 1 p cross 1 and so on this into the sigma structure as you have seen it sigma 1 is a non zero quantities may be positive quantity is a positive quantity sigma 2 dot dot sigma p and others are 0 0 0 0 0 0 0 0 this is 0s and this is all are 0s. So, what is the dimension of that one dimension of that one is your p cross m that means this is the number of columns are m and number of rows are p. Now, you multiply what will get it in right hand side sigma 1 sigma 1 is scalar quantity u 1 next column the second column multiplied by this it will be a sigma 2 u 2 and in this way if you go it I will get it sigma p u p and other columns are 0 0 0. So, this into this is 0 then this into this is 0 and so on last is that one. So, what is this one if you multiply by this u 1 by sigma 1. So, this will get it a vector m cross this is p cross 1 vector similarly p cross 1 vector p cross 1 vector this is also p cross 1 vector and in this way if you multiply by this and this what is the resultant vector this is p cross p p cross m ultimately it will come because you have a m vectors are there like this way. So, if I equate left hand side and right hand side I can write in short v i g v i is equal to u i sigma i and i varies from 1 to dot dot p. So, this we can write it then what does it mean this one this is our system and v is the vector input vector and this input vector has a direction because elements of this dimension of this one is m cross 1 all are you can think of it is a sinusoidal quantity of this one. So, this is system is an input vector I am getting u i into sigma. So, this you can write it input in the direction v i and this dimension is just now I mention is m cross 1 and this one then this is our you see system transduction multiplied by input vector is we are getting the output. So, this is u you can say u is the output in the direction u i whose dimension is p cross 1 and further since u and v are what is called unitary matrix then we can write u v u i a clarion norm or second norm of u i is equal to 1 and second norm of v is equal to 1 since they are unitary matrix and v i u 1 u 2 all these things are the column of unitary matrix and each is what is called a clarion norm or distance is a clarion norm or distance as norm 2 is 1 because this u 1 u 2 u 3 are the columns of that our unitary matrix u v 1 v 2 are the columns of our unitary matrix v. So, there two norms are equal to 1. So, that is also you can write it. So, now see this one note this is important statement note the ith singular value sigma i directly gives the gain of the system matrix g in the direction v i in other words what you can write it. So, now look at this one g v i is equal to sigma i u i. So, this is the input and output when the system is excited g of s excited with the input vector v i input vector v i whose dimension is m cross 1 then we are getting the output is p cross 1 this output. So, now let us see this one what we can write it. So, we can write it this g is a complex matrix of this one. So, you write it g v i into g v i whole h transpose because it is a complex one. So, you have to go conjugate and then transpose is what is this one I am writing v i v i h then g h agree. So, this is the identity matrix this is a scalar is one this is one agree this is one. So, what we can write it g into g h or you can write it same thing we can write g v i h pre multiplied by this g v i. So, what you what we can write it g h v i g h no this or what this is that one we are getting then what we can write it that g h g h v i into v i h let us. So, this what we can write it then we will write it this one what you can write it for this one that this equal to this one and left hand side this is the left hand side sorry this is the left hand side and right hand side what we can write it right hand side you can write it you see g v i is nothing but a sigma i into u i and g v i transpose is nothing but a u i h sigma i because it is a scalar quantity. So, it is nothing but a this is a one this is nothing but a sigma i square. So, right hand side is equal to left hand side agree. So, what we can write it for this one that this quantity that we can write it sigma i is equal to square root of lambda i g h g into g h of that one. So, how we are writing this one you see this is equal to one and this is equal to this and if you just put it this what is that one we got it u transpose u sigma v h this is the g and we can g h is what v sigma transpose and g u transpose. Now, this and this is identity matrix when you have multiplied by this and this this and this it will be sigma square sigma i square diagonal form and it will be a again it will be a identity matrix u u transpose u h sorry u h is a identity matrix. So, if you take the Eigen value both sides then it will be sigma i is nothing but a lambda i g h. Once again I am telling you this one we have left hand side we have written we got it this one what is singular value decomposition of g u sigma v h what is the singular value decomposition conjugate transpose. So, this v sigma transpose u. So, this equal this quantity is a identity matrix. So, this quantity is identity matrix then it is u and this is the diagonal form. So, it will be a sigma this is a sigma transpose into u and if you expand this one you will get what is called diagonal form of that one agree. So, now you see the Eigen values of this is equal to Eigen values of that one is something like a this is a matrix I am doing the transformation of this one is the Eigen values of the and from there I can write it it is nothing but a sigma i is nothing but a lambda i Eigen values of g g h similarly I can write it this is nothing but a same as same we can write it sigma i g h g. How I can write it this one because this must be a square matrix of this one full rank of this one. So, now this if you write it this one it this dimension is g dimension is p cross m and this g h dimension is m cross p. So, dimension p by p and in this case dimension is m by p and this is p by m agree. So, g that should be a full rank of that one agree. So, this both the cases the singular values will be same. So, this one can see this one if you see this one this is nothing but a this quantity sigma i g now we can write it is nothing but a lambda i g h of g is same as sigma lambda i g of g h now this thing equivalently I can write it if you see this one this we can write v i sigma 2. What is v i sigma 2 you see v you take the norm of this norm of this and if you take the norm of this then this quantity is 1 and this quantity norm is sigma agree. So, sigma is what you see from here I can write it take the norm g of v i norm 2 is equal to sigma i sigma i u i norm 2. So, this value is 1 again. So, sigma i is nothing but a g v i norm g v i that vector norm 2. So, sigma i is that one we got it agree. So, this divided by and sigma i v sigma i is what v into sigma is output. So, this output divided by the input norm v i is the input and that value is equal to 1. So, this quantity I divided by 1 it does not matter. So, it is nothing but a norm of v i. So, norm of what is called norm 2 of output vector norm 2 of input vector is nothing but a gain. So, what is gain sigma i is nothing but a that g i g into v i norm 2 divided by v i norm 2 this is nothing but a the output norm and this is nothing but a input vector norm. This is nothing but a input vector input vector norm 2 and this is nothing but a output vector this is nothing but a output vector norm 2. How I got it just now I told you that you see take the norm both side of this one this and this is a sigma if you take the norm of this one it will be sigma i norm of 2. So, this is sigma i is nothing but a sigma i is nothing but a norm of g into v i vector norm 2 and this is I got it this is nothing but a 1 I can write v i. So, this is the output vector norm divided by input vector norm. So, you see clearly it depends on the ratio of output vector and input vector this. So, the gain of the system how many sigma i is there is sigma 1 sigma 2 dot dot sigma n at the output. So, the gain depends on the direction of input vector that v i. So, maximum gain what you will get it what is the maximum gain this sigma bar is equal to sigma 1 g is nothing but a v corresponding sigma 1 corresponding vector is v i norm of this divided by v 1 norm of this and this is equal to 1 and this you will get. So, maximum singular value of the transformation matrix will give you the maximum gain of the systems. So, this is our conclusion of that one. So, now I will go for what is called for multi input for multi input multi output system g of s. So, just now we have seen gain of the system depends on not only on the frequency but also for multi input case also depends on the direction of the input vector agree. So, we can write it gain of multi input multi output case multi output multi output transformation system depends not only on the frequency also, but also on the input direction vector input direction vector v i whose dimension is m cross 1. So, now if you have a given for multi output multi outputs it what is the h infinity norm what is the h infinity norm of write it is a j omega what is the h infinity norm of a transformation matrix this is nothing but a maximum singular value of h infinity norm maximum singular value of g j omega matrix. So, in other words I can write a supremum of omega varies from this maximum singular value of g j omega what does it mean h infinity norm of that one you have a particular frequency g I can find out the singular values of g out of this what is the maximum value I recorded the another frequency I will find out g j omega which is a complex matrix rational functions I will find out is singular value maximum singular value recorded agree similarly another value of frequency w is equal to w 3 omega is equal to omega 3 I will find out the singular value of transformation matrix and from there maximum singular value of a picked up in this way I will sweep the frequency from 0 to infinity and out of this which one is maximum that will give you the h infinity norm of the system means in that frequency if you place the input vector v 1 in that direction or the system is excited with a v 1 in that direction at that particular frequency you will get maximum gain of the multi input multi output system from the multi output system. So, it is a supremum so and you see the supremum is nothing but a this is this meaning is nothing but a highest value of the systems gain let us call I am just what does it mean this expression I will find out that omega is equal to let us call omega 0 then I will find g j omega 0 for this one I will find out the maximum singular value of g j omega maximum singular value of this one and this one let us call next is this is omega 0 to g then omega is equal to omega 1 I will find out g j omega 1 and maximum singular value of g of j omega 1 and so on you go on changing omega is equal to let us call omega n then I will find out g of j omega and find out the maximum singular value of g of j omega and omega n and out of this all singular value which one is maximum that will give you the h infinity norm of transfer function matrix this dimension is p cross m that means at that frequency what are the frequency out of all this thing let us call frequency is omega is equal to omega x that frequency I am getting the maximum singular value of the transfer function matrix it means in that frequency agree if you find out the s b d of that transfer function agree and from there if you find out the maximum singular value of this one agree and that maximum singular value will give you the maximum gain of the system when the system g j omega is excited with that particular frequency corresponding the input vector v i agree. So, this is for multi input multi output case as you know from single input single output case that our for single input single output case the our g of this h infinity norm of single input single output case one by one p is equal to one m is equal to one is nothing but a supremum of omega varies from 0 to infinity real quantity and this is the absolute value of g j omega that means it indicates nothing but a maximum gain of the system when you are sweeping the frequency from 0 to infinity maximum gain of this one and for multi input multi output case is there for particular frequency if you do the singular value decomposition of that transfer function matrix you will get the sigma 1 sigma 2 dot sigma m sigma p agree then you find out that what is the maximum singular value of this one stored it another frequency find out the maximum singular value of the stored it and in this way you sweep the frequency from 0 to infinity and out of this which one is maximum that is the you would call h infinity norm of the transfer function physically it interprets that at that frequency you will get maximum gain out of the system when the system is excited with a corresponding v i v 1 if you excite that one at corresponding the frequency what frequency will get in maximum gain of the system corresponding v i if you excite the system you will get maximum gain from the system. So, this is the most important situation for the when you will die when you will design the h infinity controller for this one. So, this is nothing but a maximum gain of the systems agree. So, if you see this one what I did it for this one like that how I find out the singular values is relation with the conjugate g multiply transfer function multiplied by g conjugate and transpose or g h conjugate of g transpose agree. So, that is what we have seen it next is that what we will now talk about the our control problem the control problem statement in h infinity framework. So, you recollect that we any system we can put into this framework we have a exogenous input w is the exogenous input j is the regulated output and y is the measurement output which will be used for designing a controller or generating the control law u. So, it is the output feedback we are considering and this is the our u of t w of t z of t and this and it is our p block which is a 4 block representation 4 block representation our 4 block what sense this is the 2 inputs 2 output any system we can transfer into this structure. We are assuming for the time being that our what is called perturbation model perturbation that is shown by dotted delta is equal to is there agree. So, this is 0. So, and this is you know this w is exogenous input the exogenous input indicates the sensor noise that disturbances external disturbances and the common input and regulated output is states error state and the control state error signal states and control signals and measurement output is used for control purpose means to generate the u. So, our this is our controller. So, now we will state the problems what is our optimal h infinity control problems optimal h infinity control problem. The problem statement is find stabilizing controller k of s that makes the closed loop system internally stable internally stable means all internal signals should be stable in presence of noise disturbances all these things stable and minimize the transfer function t z of w of s infinity by the stabilizing controller k s stabilizing. So, what is mean according to this 4 block represent we are going to design a controller k s such that the closed loop system will be closed loop system represented if it is a state based model it is represented we will convert into a that is called 4 block representation this is a standard for problem. If the we have to design a controller in such a way the closed loop system is internally stable not only that we want to minimize the what is called exogenous effect of exogenous input to the regulated output this is the regulated output nothing, but a states error signals and the controlled signals this is the regulated. The effect of w on z that we have to minimize so, if you can minimize the transfer what is called transfer function that one then internet statement and minimize the h infinity norm of the transfer functions. What is the transfer function between the w to z between w to z what is the transfer function you find out and that h infinity norm of this one you minimize this one by selecting the stabilizing controller k s this is the optimal h infinity control bar, but this problem is very solution of this problem is very tedious. So, people are looking for a more practical solution of h infinity control problem that is called standard h infinity control problems. So, more practical standard problem standard h infinity control problem which is sub optimal sub optimal h infinity control problem this is also called sub optimal h infinity control what is this again find a stabilizing controller k of s such that k of s that makes the closed loop system internally stable. Only difference here in this case from optimal control problem optimal control problem is the this statement stable such that the h infinity norm T w of this h infinity of transfer function between the w to z this will be less than some gamma and gamma is gamma is some gamma is greater than 0 gamma is positive quantity positive quantity and this is called sub optimal sub optimal this statement sub optimal h infinity controller because this I am not going to optimize this one I am just continue this h infinity norm of this one will be less than some gamma positive quantity. So, that you can write less than gamma now next is what is this is called the standard h infinity controller that means we are looking for a controller k such that the closed loop system will be a internally stable not only that that by selecting the controller by selecting the controller we are finding out the h infinity norm is less than some gamma k gamma is a positive quantity of that one some positive quantity because why you have considered this if you consider if you consider this is you can say this quantity sub optimal controller is more practical since other real constraint real constraint taken into the real constraint problem considerations. So, because in the when you are designing controller there may be some constraint impose in the problem formulation if you do this one then our we cannot do that state way the as I told you the optimal control problem solution h infinity controller is the very tedious. So, we are we are introducing some constraint in the problem and now then we are not solving the optimization optimal control problem, but we are solving the sub optimal control problem by selecting k s in such that that h infinity norm between the w to z will be less than some gamma pre specified positive quantity. You can write it pre specified this is a pre specified positive quantity. So, now next we will go for the solution of h infinity control problem solution of h infinity control problem and let us call the system is described into state phase formulation. We are solving this one with the state phase formulation solution of h infinity controller solution of h infinity control problem is done through state phase formulation. So, you know already if you recall our earlier lecture plant description. So, x dot is equal to whose dimension is n cross 1 is equal to a x of t plus b 1 w of t plus b 2 u of t. And this dimension I consider if you recollect this one this is your m 1 in cross 1 this w sorry w exogenous input dimension m 1 cross 1 and the input dimension is m 2 cross 1 and x dimension in n cross 1. So, this is the state equation x dot then your what is called your measurement equation regulated the output y of t is equal to c 1 x of t plus d 1 1 w of t and d 1 2 u of t. Once you know the dimension of w u you can know the dimension of an x you know dimension of b v 2 a everything similarly c 1 d 1 d 1 2 you also know. So, this is our regulated output next is your what is called our measured output c 2 x of t plus d 2 1 w of t plus d 2 2 u of t. Once you know the dimension w u and x I know the dimension c 2 d 1 d 2 this and if you recollect this one our earlier discussion that how to describe that in compact notation from state phase to what is called trans function model agree. So, your state phase to trans function model if you description is there suppose I have a say let us call x dot is equal to x plus b u and y is equal to c x plus d u this is a state phase representation this is a state phase representation. Now, what is the trans function representation of this one g of s is equal to c s i minus a whole inverse b plus d if d is 0 is there so trans function is strictly power trans function when d is 0 the trans function g of s is 0 strictly power trans functions. Now, this notation one can just if this is called packed notation one model is given you can find out the other things state phase model. So, this packed notation is like this way a b c d. That means if you are the state phase model a b c d you know you can find out g s how g s is c into s i minus a whole inverse b plus d agree. So, this is the packed notation similarly, for our this case what is our packed notation to find out the trans function of that one corresponding to this packed notation corresponding to our system description our if you recollect that our four block represent the p s is equal to this a packed notation is form a then b b is what our exogenous input as well as our control input. So, b 1 b 2 then our regulated output c 1 then our d 1 1 d 1 2 d is our d 1 1 d 1 2 d 2 1 and our measurement output c 2 is equal to d 2 1 d 2 2 if you see the partition wise that one this is the a matrix this whole thing is our b matrix this whole this is c and this is our d this is the packed notation and what is our p s once I know this p s is what a c c is c 1 c 2 then s i minus a s i minus a whole inverse and what is b b is our b 1 b 2 this is our b 2 this is our b if you see this is if you compare to other case then this is our you can think of it this is b this is our c and this is our d then this plus our d d 1 1 d 1 2 d 2 1 d 2 2. So, this is our that our trans function matrix of this whose dimension you can find out this what is the dimension of that that one. So, where again we partitioned because it is a two inputs two outputs two inputs two outputs is there we can partition like this way p s p 1 2 s p 2 1 s p 2 2 s. So, we will discuss tomorrow in details all these things.