 So, last class we have discussed about the signal size. Suppose we have a signal t in this direction and x of t is in this direction. Suppose signal is like this way, this is x of t. This signal tends to 0 as t tends to infinity. So, in that situation size of the signal is measured by the energy content in that signal. So, we have seen how to find out the energy content in the signal. Suppose the signal x does not tends to 0 just like this way something like this x of t which does not x of t which does not tends to 0 as t tends to infinity. Then we cannot give the size of the signal by a energy signal. So, we have to define by a what is called power signal. And power signal we have defined how to find out the signal size in power signal form that is limit t tends to infinity 1 by 2 t minus t 2 plus 3 x of t mod square d t. That mod is taken for if the signal is a complex then mod otherwise x t whole square is a positive quantity always in this way. This is nothing but a average power this expression that we have seen it. Then directly we went to find out the system signal norm in case of system what is the what do you mean by system in case of system what do you mean by signal norm. So, suppose we have a system which is described in the trans function domain where u is the input in time domain u of t and s domain u of s y is the output in time domain and this that system that signal may be a scalar may be vector. And we have defined and this signal is continuous and piecewise continuous that is assumption is made and this is the general definition of the p norm l norm is nothing but a absolute value of this signal from minus infinity to plus infinity that quantity when p is equal to 1. When p is equal to 2 we call is a 2 norm of a signal in case of scalar signal it is like this way note we represent the double line both sides x 2 norm is minus infinity to plus infinity in this expression you put p is equal to 2. So, this is square is nothing but a square of the signals magnitude and this 1 by half the whole that is missed here 1 by half. So, this is nothing but a square root of square root of the sum of this squares of the signals and l 2 l infinity norm is simply called infinity norm of a signal is nothing but a supremum value of this signal over the time t tends to infinity supremum that means list upper bound. Then what do you mean by list upper bound we have seen it that if you have a signal like this way the list upper bound of this thing is 1. Again that this value is will be that function value of signal function will be always less than 1 and which upper bound is 1. So, it is the list upper bound is of the signal is 1 that is what and is another way and this is a supremum and infimum value of a signal is nothing but a greatest lower bound. That means lower bound what is the lower bound greatest lower bound that is called the infimum of a signal. So, similarly this can be extended if the signal is a vector where the x of t has a n components are there they are also we can represent into similar manner that is what we have then we have discussed norm of a system just now we have considering in case of system what is the norm of a signal we have discussed. Now, we will discuss norm of a system and if you see once again if you have a system g of x when it is accepted with a input which is a sinusoidal a sin omega t the output also we expect a sinusoidal with different magnitude and different phase angle, but the frequency will be same because it is a linear see linear time invariant systems. Suppose now see input signal is a sin omega t and output signal will get it a mod of g j omega sin omega t plus 5 and what is the gain input output by input if you see gain is g j omega. So, the system norm in a psi was a system in a trans function is usually measured by either h infinity norm or h 2 norm that let us see what does it mean Percival's theorem. That means suppose we have a signal is there if you want to find out the what is the energy contained in that signal again this you can find out by using Percival's theorem. Now, let us call we have a two signal x and y it that voltage and current if you consider it is nothing, but a power integration of power minus infinity what does it mean. Then ultimately we have shown it this one if x and y are y is equal to x then we have seen the integral of minus infinity to plus infinity is the energy contained in the signal is equal to nothing, but a 1 by 2 pi minus infinity to plus infinity mod of x omega whole square d of this again. So, this indicates the this x of this is the Fourier transform of the signal. So, what this indicates this indicates that this indicates that we can write that x of t x of t small x of t norm this square is nothing, but a 1 by 2 minus infinity to plus infinity Fourier transform of that signal square d omega that is what we got it plus plus if you see last class we have minus infinity is nothing, but a 1 by t minus infinity to omega d omega. What is this indicates shows this indicates Percival's theorem tells that energy contained in that signal in time domain it just implies the energy contained in the signal in time domain it can be related with a energy contained in that signal in frequency domain descriptions. This is the frequency domain you find out the Fourier transform of the signal mod of this whole square mod of this whole square means you is nothing, but a this same thing whole square over the interval of frequency minus infinity what is the energy contained this signal is same as in frequency domain and the time domain. So, this Percival's theorem the Percival's theorem relates the time domain of the energy time domain of the energy in a signal to the frequency domain description to the frequency domain descriptions. The Fourier transform of the signal mod of this whole square and integrate minus infinity to plus infinity with agree. Now, we will go to the system norm next is what topics is the system norm. So, let us call we have a system g of s and output if the system is excited with the input u of t and output is this is the output and the s to norm of the system g of s we want to find out what is the s to norm of g of s. We assume this trans function is strictly proper trans function proper trans function if you do not assume this one then we will not be able to get the h and p norm of the system g of s is not a finite it will be infinite that is what this assumption is made it and most of the particular system is strictly proper trans function and low pass filter type. So, and also assume the system is asymptotically stable asymptotically stable this system can be described also into a state space model state space description of the system what is the x dot of is equal to a x of t plus b u of t and our output y of t is equal to c x of t. This is the state space description of the system and we know the correspondingly that g of s if you want to find out the trans function model g of s from the knowledge of the matrices a b c we can find out c s i minus a inverse b and since there is a part d part is there d we assume 0 because of the function the trans function is strictly proper trans function. So, the d term will not be there since it is a strictly proper trans function. So, now what do you mean by that system norm of this one that I told you that g of s physical interpretation of g of s is nothing but a the gain of the system at different frequency if omega is equal to some omega 1 then magnitude of g j omega is nothing will give you the gain of the system at that particular frequency. So, the h 2 norm of the system this is defined as and we will see the physical meaning of that one now 1 by twice pi integration minus infinity to plus infinity t r trace this means trace of a matrix g j omega h or star g j omega d omega this indicates note g j omega h indicates sometimes it is denoted by g j omega h indicates sometimes it is denoted by g j omega star is nothing but a g of minus j omega whole transpose in complex because g of j omega is a complex system and complex matrix or scalar quantity we assume from now onwards let us call in general g of j omega is a matrix means we if you have a multi input multi output case the dimension m cross n m cross p agree this is the p cross m you just write it p cross m indicates number of outputs is p number of inputs is m agree this and in case of the multi input multi output case h infinity norm of this one is 1 by 2 pi minus infinity to minus infinity trace of this matrix and if you recollect that our definition of what is called Frobenius this is called Frobenius norm norm is nothing but a we mention it here nothing but a this is summation of and this dimension of matrix if you consider m cross n this is nothing but a i is equal to 1 to m j is equal to 1 to n this is 1 to m 1 to n then a i j whole square then whole of this is nothing but a is same as if somebody wants to write it this whole expression is same as trace of a matrix a transpose a is same as transpose trace of a into a transpose this is a Frobenius norm here also same thing if you have a g is a transformation of multi input multi output then you find out the Frobenius norm of that one agree and then do the integration minus infinity to plus infinity this and 1 by 2 pi this indicates is nothing but a energy contained in the signals because c is the that is g of s is the matrix. So, this can be written as 1 by 2 pi 1 by 2 pi integration minus infinity to plus infinity trace in terms of you can write it trace g of minus j whole transpose in place of this one I am writing this then g of j omega and that is differentiated and that is if somebody wants to write it in terms of s you can also write into this one 1 by 2 pi minus j infinity to plus j infinity this is infinity g of s minus g of s transpose g of s d s and this quantity if you see this quantity is nothing but a Frobenius norm trace of this one is nothing but a g of j omega whole Frobenius norm of the matrix agree. So, which is nothing but a this is nothing but a in other words is nothing but a i is equal to i j or you can write i is equal to 1 to m j is equal to 1 to p agree and g j g i j of omega square this you can write it. So, i is equal to 1 to p this is 1 to p this is j is equal to 1 to m number of inputs is we have considered m number of outputs is p in this see here in this expression. So, one can write into this form also that this expression this is the trace that is missed here trace t or trace of that one. So, this thing the trace of this I can easily trace of this one I can easily nothing but a Frobenius norm of g or summation of i is equal to 1 to p or j is equal to 1 to p that each element of the transfer function at a frequency omega is equal to omega 1 that is also and one can write into this is also in time domain g of s norm is equal to h to h to norm g of t is nothing but a this is equal to we can write it integration of integration of 0 to infinity trace g of tau transpose g of tau whole thing then d tau this is the and this physically this mean a measure of energy a measure of the square root square root this is the square root of the square root of the integral of the integral square value of the output signal when the system is excited with a impulse input when the input is an impulse. So, I know if the g of s is a transfer function if we excited this system with a impulse input the output response of this one is nothing but a Laplace inverse of g of s is that one. So, this physically this indicate that square root of square root of the integral square this is minus infinity to plus infinity you can write it minus infinity minus infinity to plus infinity and if you consider this is a causal system then it will be a coming from 0 to infinity the square root of the integral square of the output signal when the input is impulse. So, now question is how to compute that things or how to compute this in terms of g s how to compute this one next question is how to compute this integration of that one. So, that is the next question or you can write it g of s norm 2 norm is equal to root over 1 by twice pi i and integration the close integral of this one trace g of minus s transpose g of s whole d s now question is how to compute this expression to how to compute this expression to find that h 2 norm of a of the transfer functions of the systems. So, this is and this can note this can be done by using the Cauchy integral principle. So, the contour integral that run up the imaginary axis and then around the infinite semicircle in the left half plane left half s plane and the contribution of the integral from the for the left half semicircle is 0. The contribution integral contribution the contribution to the integral from this semicircle equals to 0 if g is strictly proper transfer functions. So, now we have to integrate because whole semicircle part that left half of the spline is 0. So, only the things we have to integrate from minus j on the along the imaginary axis from j from minus j infinity to plus j infinity. So, the our problem now boils down to this competition of this one equals by using the residue term equals the sum of the residues of g minus s transpose g s at its stable or stable poles or left half poles of s. So, this let us call we taken one example we taken one example then see how to find out the we taken one example and how to find out g of that norm of norm find out the norm of h 2 norm of g of s is 1 by suppose the system is 1 plus tau s this is. So, h 2 norm this indicates that you have to find out 1 by twice pi j minus j to infinity to plus infinity g of minus s transpose g of s d s. So, I told you the whole complete left half of the spline semicircle again that part is 0 only along the imaginary axis and along the imaginary axis nothing but a sum of the residues compute for this matrix at stable poles at stable poles. So, let us call find out g minus s transpose of g s is nothing but a s minus s tau plus 1 this is g s and transpose has no meaning because it is a single input single output case then s tau plus 1 then what is the residue of this one at time at stable poles s is equal to minus s is equal to minus 1 by tau. So, find out the residue s is equal to minus or you can find out a plus s tau plus 1 plus b minus s tau plus 1. So, you have to find out the residue at stable poles a the value of a will be equal to 1 by 2 tau by using the residue. So, what is the value of that one this composition of that one is that whole thing compositions is find out the residue of that one along this one. This is nothing but a residue of this is nothing but a residue of g transpose minus s g of s at stable poles. So, this implies that our g of s the norm of this 2 is equal to square root of 1 by 2 tau is nothing but a 1 by square root of 2 tau, but if it is a multi input multi output case then it becomes very tedious and complex. So, that can be handled by using in state pace model for multi input multi output case for multi input multi output case composition of g j g of s this is not straight forward is not straight forward. So, how to compute this one. So, let us call g of s which is a multi input multi output which a m is p is the number of outputs p is the p is the number of whatever m is the number of inputs. And that state phase representation state phase representation converting to state phase representation that x dot is equal to a x plus b u and y is equal to c x of t. And since I have considered number of outputs is p and number of inputs is m then let us call this is the state is n. So, first you find out the either controllability Gramian matrix. So, how to find out that for multi input multi output case h infinity sorry h 2 norm of the systems h 2 norm of the system it does not it means the energy contained in the system for different signals of this one. I told you g of s is a when the input is the what is called the impulse input the output that what you are getting corresponding to input is the output signal. And what is the output signal contents in the system that can be find out with the help of h infinity h 2 norm of a g s for a impulse input. So, now two methods two way we can two ways to compute g of s when g s is the multi input multi output case. So, let us call we first find out the controllability Gramian the controllability can be u e is equal to 0 to infinity. And we are assuming the system is a causal that e of t this b b transpose a of t whole transpose d of t. So, this is the controllability Gramian and this matrix must be a this whole expression must be a value will must be a your positive nonzero values. If it is nonzero value for all t then system is controllable if you recollect that one when we are considering the linear time invariant systems then with the knowledge of system matrices b a we found out a b then a square b then then a n minus 1 b the rank of this matrix must be equal to n where n is the dimension of the matrix a n cross n. So, this in other words you can check the Gramian matrix whose value is greater than 0 for all t then system is controllable in the sense that n is there exist some controller over the interval t 0 to t f by which we can transfer that state from initial condition to the final state with the help of controller. And there exist some control if this condition is satisfied. So, what will do it. So, you solve this Lyapunov equation. So, this is the Lyapunov equation you solve it or you can consider that this is a what is called algebraic eddy type equation type also. So, solve this one once you solve this one then find out that norm of that is nothing but a h 2 norm of the system is nothing but a trace of c u c transpose trace of a matrix trace of a matrix is nothing but a sum of all diagonal elements. So, that will give you the what is called h 2 norm of the systems the physical interpretation h 2 norm of system means the system is there you are excited the system with a impulse input then output response is nothing but a Laplace transform or Laplace inverse of inverse Laplace transform of g of s and that is the in time domain what is the energy content in the output signal that it can be found out with this one. This is method one for whether it is a single input or single input single output multi input multi output does not matter you have to solve this equation. Method two this is called controllability Grameon matrix. Now, we will consider the observability Grameon matrix that s is equal to 0 to infinity that you can get it from this one a is replaced by a transpose b is replaced by c transpose. If you a replaced by a transpose it will be a t whole transpose then b is replaced by c transpose then c transpose c into a a t whole a is transpose by a is replaced by a transpose a transpose whole transpose is a a and d t. And this Grameon observability Grameon matrix this is called observability Grameon matrix if the observability Grameon matrix value is greater than 0 for all t then system is observable that means we can estimate the state information at time t is equal to 0 by processing the input output data of the system. Then we can estimate the initial value of the states if the system is observable observability Grameon matrix is greater value is greater than 0 because whole quantity is a scalar quantity for greater than 0 for all t. So, with this one I can also compute the what is called solve the again the Riccati equation that Riccati equation is a transpose s that s a in this equation in this equation replace a by a transpose u in state of Grameon matrix u observability Grameon matrix. So, this plus c transpose c is equal to 0. So, this is the Lyapunov type equation one can solve by standard technique by solving this one or you can use for solving this one you can use algebraic Riccati equation solution if you know we have discussed when you are considering the LQR problem or LQG problem in our optimization technique that is dynamic optimization technique you can use that method to solve this one. So, once you get it this one that h 2 norm of the system this is nothing, but a trace of a t r you write it b transpose s b. So, this way you can compute it. So, next is your this is the h 2 norm and next is our h infinity norm of a system g of s. Let us call we assume because h infinity norm concept is coming from the singular value for multi input multi input that concept is different. So, let us consider we have a single input single output case u of t y of t is the output. So, it is a single input single output systems it is defined as g of s that whole infinity is equal to supremum value of this function omega not omega u of t not equal to 0 y of t. This is the time domain I am expressing u of t when input is impulse input the output response of the impulse output response y of t due to the impulse input this or any input you can write it the what is the energy content into the system output signal energy content also divided by energy content in the input signal that ratio that what value of u when u is not equal to 0 that this quantity will be maximum that that supremum value of this ratio will give you the h infinity norm. But, if you talk about the system in terms of frequency domain this is the input and this is the output and if you consider for stable system the h infinity norm of the system g s is nothing but a g of s supremum w g of j omega. What does it mean that g of j omega you find out the frequency response of the system when the omega is varying from 0 to infinity and find out at what frequency it gives the maximum value of this amplitude or gain and that is nothing but a h infinity norm of a system when this is single input single output case. So, you restrict our discussion with a single input single output case because multi input multi output case the concept of singular values decomposition singular value decomposition technique will be adopted here singular values concept of singular values will be adopted here this quantity the whole quantity is the maximum maximum gain of the system or in fact it is a resonance at what frequency resonance is occur the maximum peak is occurred this is and. So, this is about the system mark because system norms signal norm will be used when we are discussing the robust design of controllers h infinity controllers or h 2 controllers when we will discuss this and that concept will be used. Now, so far we have seen that if you have a system is there if you excite the system with input and you will get the output the system stability is studied from the knowledge of that what is called based on input output stability criteria we are checking the stability of the systems that is in turn it will call input output stability. So, in true sense the system stability is studied we have to study all the internal signal of the system must be stable then it will be a call the system is stable in true sense, but input output stability does not give the clear picture of the system stability. So, let us see with an example that we have a systems let us call we have a systems g of s is equal to s minus 1 s sorry s plus 1 s plus 1. So, this system there is a pole zero cancellation is there, but stable pole zero cancellation you see the stable pole zero cancellation is there this implies mathematically, mathematically in distinguishable it cannot distinguish from the constant transaction g of s is equal to 1 same thing I can do, but here is stable pole zero cancellation is there. So, long stable pole zero cancellation is there there is no problem except the system properties will change either it may not be controllable completely or it may not be observable completely, but in similar wise like similarly if g s is equal to s minus 1 s plus sorry s minus 1 s minus 1 still this is mathematically indistinguishable from the constant transaction this mathematically it is same as the constant transaction because if you cancel this one this will be a 1 only, but let us call for the sake of our interest we assume that this pole at 1 and pole at pole at 1 and 0 at 1 they are not exactly overlapping that slightly change in state of pole at 1 I am taking pole at 0.9999 and now immediately you will say the system is unstable because they are not canceling each other the system is unstable because you can logically you can think of it the system parameters may deteriorate with day day years. So, this parameters after that it may not be 1 exactly it may be 0.99. So, let us call this type of pole 0 cancellation unstable 0 and unstable pole cancellation is not allowed. So, unstable pole 0 cancellation in practical plant stroke controller is not acceptable. Let us see why it is not this example when it is a s is equal to instead of 1 it will be 0.99991 you can say system is unstable for any kind of signals even the signal magnitude is very small. Let us call we have a system this systems this is the output y s is the output and this signal is feedback here negative feedback this is R of s reference signal and this signal is E of s and this signal we are considering E of s. Now, see this one that E of s if you can find out the trans function between the this signal and this signal this is the output signal this is the input signal trans function this will you will get s plus 1 plus s plus 2 R of s indicates stable signal E of s this signal is stable, but if you find out the trans function between R s between y s and R s that means U s signal you want to check it then this is nothing but a this you know expression this expression you multiplied by 1 by s. So, this is considered C of s this is G of s if I consider. So, it is a C of s nothing but a 1 plus G of s into C of s into R of s and that value will come if you see 1 minus s this is I have written C of s 1 by s and that value is always already there this is s plus 2 into s plus 1. So, this you can say E of s is nothing but a E of s is nothing but a 1 by G of s into C of s into R of s and put the value of C of s yes it will get it and see this signal this R of s this signal this is this indicates for any kind of input or noise or signal R s reference input indicates the signal U of s is unstable. So, this signal the internal signal internal signal is unstable for this type system. So, the concept of internal stability is and other things is coming to the picture when you will study the stability of the systems. So, internal stability of systems. So, as I mentioned the input output stability input output stability does not granted does not guarantee that the other signal other internal signal internal signal are bounded. In this example you see this internal stability you have to check all input signal all summer output signal you have to check it all internal signal this is the signal this is also signal all this thing if you see the output this this cancel and you will find it is stable, but before that this will be a unstable signal. So, in general now if you have a block diagram like this way C s is our controller then we have a block of that one this is U of s this is E of s and this is our plant G of s this is our output I will explain that one this is the reference input and this is the controller and this is the plant and this is your D e o input disturbance this is called input disturbance input disturbance this is called output disturbance output disturbance output disturbance output disturbance which is denoted by d y and this is your d n measurement noise this d n is the measurement noise. So, now we say we have a system reference input signal then controller then plant the controller output is corrupt with the input disturbance and output of the system actual output of the system is corrupted with a disturbance output disturbance and the measured signal is corrupted with a noise generally the reference input disturbances are a low frequency signals and measurement and noise are generally high frequency signals. So, you have to study the stability of the system it does not mean the stability of this system this this this signals we do not have any control, but inherently inside the system may act. So, if you just study the stability from input and output that y of t that does not mean the system is stable or not that input output stability does not grant t that other stability signal inside the system is also stable there if you study the stability between the this in signal and this signal even it is stable it does not mean that all other internal signal also will be bounded or stable. So, generally we take the output from each summer output is the output of the signals is summer output. So, this is u p of s this is this summer output is this and summer output is y this is also getting y this is the output. So, you have a how many outputs are there there are four summers are there 1 2 then 3 then 4 let us call this signal is y n. So, you have to study the stability of this one. So, let us call I am considering the system is single input single output system. Same thing you can extend for multi output multi output system provided if you take care of the matrix operations vector operation during the process. So, let us call we consider input signal or r this is the input signal where in time to r then d u d suffix u d y and your d n this is the input signal input signal input signal. Then output signal is your e u p y d y and y n these signals are not in your hand again the output signal is if you see here the output signal is summer output this one this one y this one and this one that output signal. So, we have to study the stability of the system internal stability of the system that means we have to find out the input output relationship. So, you see the our output is what our output is e of s u p of s y of s and y n of s this is our output and this is and our input is here r of s d u of a capital d u of capital d u of s d y of s and d n of s. Now, this is our output Laplace transform Laplace domain this output this input is this is the input and this is the output this input are mapped with a some matrix in a output signal the output signal the input signal. So, this input are mapped through the system in a another set of signal which will call output signal we will find out what is the matrices are there naturally you can see there are 4 inputs 4 outputs the dimension of the transform matrix will be 16 4 by 4 4 by 4 that means 16 elements are there you have to find out that 16 elements here. So, we will discuss tomorrow the rest of the part.