 So, last class we have seen that if you have a system physical system is there and if there is any perturbation is there and that perturbation we have considered in additive perturbation form. Then we can convert that system into general p delta structure, since in our case is the additive perturbation. So, our structure is p delta a it is not m it is a p delta structure. So, then we have seen that that p is a p m is the 4 by 4 block. So, it is also core 4 block representation of a dynamic systems where the perturbation is taken out from the system the outer loop. So, if you see the basic structure of there are two loops are there the two outer loops one is one is the upper outer loop and another is lower bottom or lower upper lower loop. So, if you are interested to merge the perturbation block into a p block then we have to eliminate w from this expression. So, ultimately you are eliminating w we got this expression then you got this expression that the deletion w z then u this that p delta structure in p block the delta the perturbation block is entered here. So, how will you get it to remember this relationship between output and the input this that since we are the outer loop is merged to the p block then p block last row that means p 2 2 it will start with p 2 2. So, it is a first index p 2 2 then next will be p 2 1 then delta a perturbation bracket i what is called i is the identity matrix of proper dimension then p 1 1 then delta a whole inverse. So, this first index already gain. So, the second index will come last. So, it will be a p 1 2 multiplied by this. So, it is called the linear fractional transformation LFT since the upper block is merged to a p block. So, it is called upper linear fractional transformation this one similarly if you like to that lower loop you want to merge to p block. So, naturally that change p block will be a function of the controller k. So, that will be called again it is a linear fractional transformation but this upper that lower block is converted into a merged into a that block p block. So, it is called lower LFT. So, now we will consider that lower LFT how to get it the lower LFT. So, in order to get the lower LFT that means in turn we will get the transformation between between z and w between w to z w is the exogenous input and z is the regulated output exogenous input and this is the z is the regulated output. So, let us see from the basic equation we call that z is equal to p 1 1 w p 1 2 u and y is equal to y is equal to p 2 2 w plus p sorry p 2 1 w p 2 2 u. Then from this equation let us call this is equation number 1 this is equation number 2 and in addition to this we know another equation that u is equal to k into y that is that one if you see this basic diagram that one this one k into s k I am omitting the s for convenience this k into y. So, now after manipulation we will find out the relationship between w and z input is w and z thus. So, it is the transformation between the signal w and z. So, you can write now if you use this one I can write this one y is equal to p 2 1 w plus p 2 2 k y. So, if you bring it that side it will be i minus p 2 2 k of y I am bringing to that side then p 2 1 w. So, therefore, y is equal to that i minus p 2 2 k whole inverse p 2 1 w. So, let us call this is equation number 4. Now, from equation 3 let us call this is equation number 3 that I missed. So, from equation 3 you can write it now from 3 we know u is equal to k into y we already know the expression of y you see y expression of y is this one you will put it there. So, it will be k into i minus p 2 2 k whole inverse p 2 1 w. So, replacing u in from equation 1 from 1 then only we will get the relationship between the input signal w and the z the regulated signal z and the input signal is w which is exogenous input that relation we can find out from there we can find out the transformation. And that transformation is very useful when we will design the h infinity controller for a dynamic system in transformation domain. So, from 1 we can write z is equal to p 1 1 you see then w then p 1 2 p 1 2 u put the value of u p 1 1 w p 1 2 k then i minus I am putting the value of u i minus p 2 2 k whole inverse p 2 1 w. So, if you take the w come on exogenous input. So, p 1 1 plus p 1 2 k this then i minus p 2 2 k whole inverse p 2 1 into w. So, this is this whole thing is nothing but a function of the our p block and k is w. So, this we denoted by the transformation this one you see z and w z is the input z is the output. So, it is the transformation between the z w and z. So, it is denoted by transformation between input and output z this into w. So, what is our transformation between this and this t z w between the input and output is equal to p 1 1 plus p 1 2 k i minus 2 2 k 2 2 k whole inverse p 2 1. So, this is the our transformation and this transformation is very useful when you will design the h infinity controller. Then you have as how you remember this expression you see as I told you earlier also look the basic block 4 block representation of system. So, this loop the outer loop that 4 block the lower loop we want to merge it in p block. So, you want to merge in p block that means you have to start with the upper row of that p block p 1 1. So, I have started with p 1 1 then first index I have written then p 1 2 another index is left with p 1 2 then controller then i minus. So, p 1 1. So, it will be p 2 2 k whole inverse then it will come the what is called second index last. So, first index it is come first and second index will come last. So, it will be a p 2 1. So, this is the transformation of this one. So, we know how first thing we know how to represent in general if you have a dynamic system is there. There are basic two inputs two output one input is exogenous input which consist of that noises then disturbances then common input and the another input is control signal input which will regulate the system response. So, output we have a z that is called regulated output that output is affected by W. That regulated output may be a state error then you may be a control signal control signal all these things then another is your why is the output that is the measured output that output will be used for controller design to regulate the system response. So, this is the basic block diagram structure. So, any system we can represent into four block representation in the sense that any system we can represent by W, U, Z and Y and if there is a perturbation is there perturbation block will come here agree. And if your controller is there controller is connected to be here the a of s and this is the perturbation. So, this is the basic structure of that before we go little bit details in the h infinity controller design we must know what is the frequency response of multi input multi output systems. So, next is our discussion is multi input multi output system response frequency response frequency response of a dynamical system G s bracket multi input multi output system. We have discussed little bit single input single output we know in under graduate class also what is single input single output. So, briefly I will discuss first frequency response of single input single output. Suppose a dynamic system is there which is described by a trans function model G s. So, I want to get the frequency response of this one. So, and you assume the system is linear time invariant invariant system. The frequency response is this one that your input is a sinusoidal signal magnitude of the input is kept constant. Then frequency we are changing at each frequency we will see what is the output magnitude and frequency will be same, but there will be change in phase angle each input or each frequency will observe that one. So, let us call our input is we have since you have a frequency response u 0 sin omega t of alpha this is the input. So, naturally our output response also will be sinusoidal, but magnitude is different from u 0 and phase angle will different from b. So, let us call this will be u 0 magnitude of G j omega of this and sin omega t same frequency minded and beta. So, where G of j omega magnitude is nothing but a gain of the systems gain of the system at frequency omega. So, naturally from this expression you see how I am telling it is gain I will explain later here now you see it is dependent on frequency. So, what do you mean by the gain at a particular frequency what is gain input magnitude is u 0 output magnitude is this whole thing is y 0. So, what is gain of this one y 0 by u 0. So, this y 0 by u 0 if you do this one it is nothing but a G of j omega it represent the gain of the systems. So, gain of the system it does not depend on the input because u 0 output is u 0 G j omega and input is u 0 u 0 u 0 whatever the input magnitude is change corresponding output magnitude is will be change this is the property of linear systems. And phase angle phase there frequency will be remain same in frequency only phase angle is change. So, that you can say output amplitude of the signal output amplitude of the signal divided by input amplitude of the signal. So, it is basically it is nothing but a G of j omega. So, and what is the phase angle associated with this one we can easily see phase angle associated with G of j omega is angle associated with G of j omega is nothing but a beta minus alpha is the phase shift. Now, how you see we know the phasor quantity what is this if you represent this phasor quantity I can say that one is something like this way this is u 0 u of t is equal to u 0 sin omega t plus alpha. If you represent in the phasor quantity means a vector which is rotating with a constant angular velocity omega this is also another vector which is rotating with a constant angular velocity omega both are same. So, I can say this I can represented by this one u 0 e to the power of j alpha. And this is the vector and how you represent this vector magnitude u 0 phase angle is it is something like this magnitude is u 0 phase angle is alpha and that vector is rotating with angular velocity omega you can say this is the real and this is the imaginary axis of this one. How you represent that one y 0 that y of t is equal to y of 0 you know y of 0 is what this part is y of 0 sin omega t plus beta. So, how you represent in phasor quantity form this is will be y of 0 e to the power of j beta and that vector is rotating with angular velocity omega. And that phase angle is you can think of it as may be like this way this this is your u 0 this is y 0 and phase angle is beta and that is also rotating with angular velocity of this. So, if you see this one ratio of this two phasor quantities of this one u 0 y 0 and phase angle will be beta minus alpha is nothing but this one if you see y 0 y 0 e to the power of j beta this is the output signal phasor quantity and this is u 0 e to the power of j alpha which is equal to y 0 e 0 and this is nothing but e to the power of j beta minus alpha. So, its magnitude is if this is a gain is nothing but this quantity is nothing but if you see the g of j omega absolute value this gain and what is the phase angle associated with beta minus this. So, this is the what it means in a linear system for single input signal the input signal is changed by a factor is a input signal u output signal is what the input signal u magnitude is changed by a factor this factor and its phase angle is changed by a angle. So, I can write it when this signal you can say when signal u t is passed through a system again that is input magnitude signal is changed by a factor u 0 by a factor of that one and its phase angle is changed or and it is see and its phase is shifted from input by phi which is equal to beta minus alpha. Now, let us scale this system this concept can be extended for a multi input multi output systems. So, let us scale for multi input multi output systems. So, what is this we have a systems g s and g s is a multi input multi output system. So, let us call the number of outputs is p and number of inputs is m. So, this is a transfer function matrix not a transfer function only transfer function matrix and p stands for number of outputs and m stands for the number of inputs. So, you can think of it if more than one inputs are there if I represent by double line to differentiate from this one that is u of t and its magnitude what is the how many input vectors are there input vector dimension m cross 1 or in Laplace domain it is a u s which dimension is m cross 1 and output is y of t whose dimension pre cross 1 or in transfer function domain y of s is a p cross 1. So, if you write the transfer function matrix relationship input output relation we can write it y of s is equal to g of s into u of s. You cannot write for multi input multi output say transfer function y of s divided by u of s because why in your case is a vector p cross 1 this is m cross 1 and in turn this will be a p cross m. So, you cannot divide that one you have to write into this section some transfer function matrix multiplied by input is equal to output vector some transfer function matrix multiplied by input vector is equal to output vector. So, this is the expression for this one now let us see this. So, if you write how many outputs are there p outputs how many inputs are there m input if I write more in general vector and matrix notation form we have a how many outputs p outputs are there y p of s is equal to and how many the g of s what is the dimension p rows m columns m columns. So, g 1 1 s g 1 2 s dot dot g 1 m of s g 2 1 s second row g 2 2 s and g 2 m s and in this way the last output expression this is first output second output this last output p means last output p th output is equal to p into p 1 p 1 s g p 2 s dot dot dot dot g p m s that multiplied by input vector how many input vector dimension is how much u 1 of s m u 2 of s dot dot u m of s. So, this is the basic equation of this one. So, that vector is the input vector and that is the your output vector and this is our transfer function matrix whose dimension is p cross m. So, this is you can call input channel vector input channel and this is your output channel. So, input channel i th component that is first component second component and i th component if you write it this thing. So, now look at this one what is y 1 s y 1 s is nothing but a g 1 1 s u 1 s plus g 1 2 s u 2 s plus g 1 3 s u 3 s and so on plus g 1 f u s. So, if I write it that i th that output channel i th component of i th channel i th component output of the i th channel. So, if you write it the i th component of output channel how I can write it w y i of w is equal to g i 1 because it is a i th component i th component let us start with g i 1 g i 2 g i m. So, g i 1 of w g i 1 of w u 1 of w then g 2 of w g i 2 of w u 2 of w plus dot dot g g i m of j w. So, I missed it j j j and this y i of j s is replaced by that this one. So, you write it j. So, this is the i th component and i varies from 1 to dot dot p because there are output channel there are p outputs are there. So, we can write it this summation of i is equal to or j is equal to 1 to m g i j j omega u j j omega this one where u j j omega is nothing but a you see what is this one I can represent this is a nothing but a what is called phasor quantity form this is nothing but a u j 0 e to the power of j alpha j first component let us call first component u j u 1 u 1 of t what we can write it u 1 of 0 sin omega t plus alpha 1 and in phasor quantity what we will write it I will write it u 1 0 e to the power of j alpha and that vector is rotating with angular velocity omega. So, this is because this is the function of omega at the imaginary component of this vector imaginary component of this vector represents that one if you see this is the that vector u 1 0 e to the power of j alpha and it is rotating with angular velocity omega and this is real and this is imaginary power and imaginary component the vertical component of this one is nothing but a this. So, this is a function of omega is change omega is going is the constant speed. So, I can write it g j omega is that one and for and similarly y i j omega I can write it y i 0 e to the power of j beta beta i and this is j varies from 1 2 dot dot m and i varies 1 2 dot dot p. So, how many vectors are there phasor vectors that m vectors are there how many output vector how many vectors are there there is a output vector output channel vector is there how many components are there p components the input vector how many components are there for input vector m components. So, input vector dimension m cross 1 and output vector dimension p cross 1. So, we can write it this is a if you write it this one this is a function of this is not j this is a function of this. So, I can write it that one that u of w is nothing but a u of w is nothing but a u 1 again u 1 u 1 of w u 1 of w u 2 of w dot dot u m of w. And which we can write it what we can write it this one you put the value of u 1 of w that is your u 1 of 0 e to the power of j alpha 1 u 2 of 0 e to the power j alpha 2 and so on u m of 0 e to the power j alpha m. So, this is the this similarly I can write u whose dimension is m cross 1 this dimension is p cross 1 which is nothing but a this is the phasor quantity of this one I can write representing that one. And this is I write in y 1 0 e to the power j beta 1 y 2 0 e to the power j beta 2 and so on y p of 0 e to the power j beta p. So, this is our input vectors input. So, this is you can write it this one it is a input vector of sinusoidal sinusoidal signals this is the your output vector vector of sinusoidal signal okay. So, we have represented now this is the complex number. So, this is a complex number representing this one you see this is the complex number u j is a complex number. Similarly, y x is also complex number representing at each frequency omega represented at each frequency why I made mediated frequency if frequency is change because we are frequency response when frequency we are changing keeping magnitude is constant. So, the vector will be that u i y 0 of this one this with dependent on the frequency of that one vector. So, this is a this is changing with a that is that is rotating with a angular velocity frequency. Suppose, supply frequency is omega 1 it will rotating with a angular velocity omega 1 if supply is change from omega 1 to it will rotating with a angular velocity omega 2. So, this is that is why it is written function of omega. So, now, we can write it this expression what is the magnitude of the input vector what is the magnitude of input, but a complex quantity magnitude is what if the complex number is there multiplied by it conjugate then take the square root of this one or you know the magnitude of this one is nothing, but a u 1 square of this one. So, our vector is that one this is the input vector I want to find out the magnitude of the input vector. So, magnitude of input vector signal u of w again u of w what is this one magnitude is nothing, but a vector is nothing, but a Euclidean norm is nothing, but a root over u 1 0 square u 2 0 square plus dot u m square of 0. Suppose, you consider this one one element what is this one magnitude of this one one element u 1 0 square, but it is a vector of m elements. So, individual square sum of individual square plus square root of that one agree. So, this is the or you can say u w into u star transpose that will give you the magnitude square that I need magnitude. So, it is nothing, but a Euclidean norm of this signal this one. Similarly, magnitude of output vector signal y of w is equal to y of w Euclidean norm distance 2 norms or Euclidean norm Euclidean norm is equal to square root of y 1 0 square y 2 0 square plus dot dot y p 0 square agree. So, what is the gain you have applied input vector to a system and m input we have applied system and we got it output is y p vectors. So, what is the gain of the systems the output we know input ratio of this one is the gain. So, the gain of the system g s which is p cross m systems transformation gain of the system in transformation matrix this for a particular frequency particular input vector u of w is given by norm of the input vector magnitude that vector magnitude is a gain or you can say physically it is not a energy content in the output signal divided by energy content input signal is a gain. So, this is you have just now we have seen this is nothing, but a g of j omega into u of j omega of this divided by this is our output agree. And this is equal to u of omega of this and just now this is nothing, but our if you see this is nothing, but a y w of y w of t I have written this one u of w as it is this one. So, this is nothing, but a square root of 0 1 0 square square root of second output square magnitude square then dot dot plus y p 0 square whole divided by square root of u 1 0 square plus u 2 0 square plus dot dot u m 0 square this. So, this is nothing, but again you can think of it suppose you have a two signals are there sinusoidal signals. And these two signals if you add together then what is the energy content in this signal is nothing, but a square root of r m s that r m s square of individual signal r m s square sum of r m s square it is something like this only because you though it is a maximum value I can divide by what is called root 2 square both side then it will be r m s only. So, the gain of the system is nothing, but a for multi input output is nothing, but a input output vectors magnitude of output vector and magnitude of the input vectors look at this expression this is a dependent on frequency the gain is dependent on frequency same as the single input single output case in addition to that we have a another freedom that the gain also depends on the direction of the input vector that we will see later. So, we can say remarks again the again the gain depends on the frequency and independent of magnitude independent of the input magnitude input vector that means u u of w of this further it also for multi input multi output for multi input multi output case the gain depends on multi and multi gain depends on the direction of input vector direction of input vectors input vector u of t m cross 1. So, this is another freedom in adjusting the rather in order to get change the gain of the system. So, this is an additional freedom additional degree of freedom to have different gain of the system to control rather to control the gain of the system to control the gain of the systems. So, multi input multi output we have a another freedom it depends on the direction of the vector the gain of the system also depends on the what is called the direction of the vector and also it depends on the frequency, but in the single input there is only one input is there. So, it depends on the gain depends on the frequency for multi input it depends on the frequency we have a another freedom to control the gain of the system that is the direction of the input vector. So, let us see now how to get the so as I all these things background we have to give it before we discuss the what is called design of h infinity controller. So, next is singular value decomposition in short it is written as b d. So, singular value decomposition is there I am just instead of writing this one if you have a matrix is a non square matrix is there this matrix may be real or complex this matrix we can decompose into three matrices m cross m one matrix then sigma whose dimension is m cross n what is m first that first matrix will be m cross n. Then what is the second matrix it will come the dimension same as that one then third matrix is your b transpose n cross n the dimension of that one. So, for real matrix a which is rectangular this is this matrix is your orthogonal matrix orthogonal matrix you know the properties of orthogonal matrix and this is the rectangular matrix agree and this is also orthogonal matrix. But if it is a complex matrix is there then this matrix if it is a complex. So, I am writing in the black complex or sorry blue then this matrix will be a unitary matrix the complex matrix instead of orthogonal matrix to differentiate that one it is a unitary matrix and that is also unitary matrix. And in that case we have to write this is I am writing in bracket U sigma V H and V H is the unitary matrix this may be indicates this indicates V is a complex things conjugate then transpose V H indicates. So, note that V H indicates V H indicates that V star transpose and so any real matrix or complex matrix I can decompose this into this form three matrices. In real case it is orthogonal matrix V and U are orthogonal matrix and this is a rectangular matrix agree and each elements are real the elements of this one elements are real. And this rectangular only diagonal elements are the present only diagonal elements are present and only diagonal diagonal elements are present whether it is a complex or real is diagonal elements present and that are real and that are real both cases. So, there are few properties of the statement from this one we can make few statements the following statements the following statements are true it is obvious you look at this one look at this I since it is a orthogonal matrix or Hermitian matrix we can write it let us call real matrix A is a real matrix what we can write it for this one you see the columns of U 1 the columns of U m cross m are the Eigen vectors of the matrices of the matrix A into A and for if it is A is a complex matrix then it will be A into A H if it is a complex what is A H A H is the complex A H in A A conjugate transpose this indicates A H indicates the bracket is considering when A E is a complex this indicates is nothing but A A conjugate transpose is equal to A H agree the columns of U are the Eigen vectors of the matrices this is the and this is obviously 2 how you see this one U I am multiplied by suppose real case I am multiplied by A into A transpose A is U sigma V transpose this is A A transpose is what reverse order V sigma transpose then U U transpose now you see U sigma V transpose V sigma transpose and U transpose agree this is your identity matrix this is orthogonal property then what you can write remaining U sigma sigma transpose agree then U U transpose. So, now what is this dimension is m cross n and this dimension is n cross n. So, this will be m so what you can write it this you multiplied by this things by U both pre multiplied or post multiplied by U then it is A A transpose U is equal to U sigma sigma transpose. Now, see if A is a matrix this whole thing is a matrix multiplied by the vector U H is a matrix each column then it will be a is called what you will get it this one U into that I told you each element is only the diagonal elements of the it will get sigma 1 square sigma 2 square in this way that way. So, you know the definition of Eigen value Eigen vector if A is a Eigen value A is a matrix X is a V is a vector then this is the real. So, using the definition of Eigen value Eigen vector we can prove that statement of the first one. Similarly, we can say the columns of V n into n this is V n into n are the Eigen values of Eigen vectors sorry Eigen vectors of the matrix A transpose A which is A H for complex A H A. Similarly, you can prove this one using that one and third is the diagonal entries of sigma the diagonal entries of sigma m cross n the singular values of A m cross n it is denoted by sigma A are the non negative square roots of the Eigen values of Eigen values lambda i of A transpose A or A H for complex A H of A which one is nothing but a sigma i is nothing but a square root of lambda i A transpose A A transpose or it will be lambda i A transpose A agree and i varies from 1 2 dot dot p and p is the minimum of m cross n. You see this one if A what is the dimension of A into A first find out what is the dimension of A it is a dimension of A is m cross n and n transpose is n cross m. So, if m is less than n then you will do this one so Eigen values of square matrices suppose n is less than n is less than m then you do it this one A transpose A then it will get the Eigen values of sigma 1 sigma 2 all this thing that is why I have written p is minimum of that one agree that in this way that singular values you can find basically if you see that structure of sigma if you see that structure of sigma is like this way sigma 1 sigma 2 dot dot sigma k then 0 0 then this is equal to sigma m suppose it is m by m and this is will be 0 0 dot dot 0 0 0 0 0 m 0 0 0 0 0 all are 0. So, this is m cross n when I am considering m is greater than m is less than n, but sigma 1 sigma 2 sigma 3 sigma k after that it may be all are up to m may be distinct non 0 some of the may be 0. So, in general I am writing is k so in another is they are rearranged like this way sigma 1 sigma 1 greater than equal to sigma 2 greater than equal to sigma 3 greater than equal to dot dot dot sigma m you can easily draw you can easily draw when m when n is less than m this structure. So, this is the your that and all sigma 1 sigma 2 all are positive non 0 or non negative quantity non negative quantity up to this m and using this structure we can find out the what is called the h infinity norm of a multi input multi output system norm. We know the h infinity norm of a system for single input single output case it is nothing but a supremum of omega where whether is this is a real and what is the supremum value of this g of j omega for this is for single input single output case. That means, you do the frequency response of this one at what frequency you will get the maximum value of the gain that is the h infinity norm of the system for single input single output based on this concept we will see what is the what is the h infinity norm of the multi input multi output same supremum means least upper bound. You can say the supremum is nothing but a least upper bound of this one the frequency response of this one means gain you gain at it frequency you see the gain next frequency you see the gain and over the sweeping the frequency from 0 to infinity you see at what frequency you are getting the maximum gain that is the h infinity norm of the system for single input single output system. So, we will stop it here and next is multi input multi output system what is the what is the h infinity norm that we will see later and based on this one h infinity controller also will give you the basic idea behind this what h infinity norm of a systems I will stop it here.