16 CHAPTER 2.NORMS FOR SIGNALS AND SYSTEMS Table 2.1:Output norms and pow for two inputs Now suppose that u is not a fixed signal but that it can be any signal of 2-norm<1.It turns out that the least upper bound on the 2-norm of the output,that is, sup{lyll2:lu2≤1， which we can call the 2-norm/2-norm system gain,equals the oo-norm of G;this provides entry (1,1)in Table 2.2.The other entries are the other system gains.The oo in the various entries is true as long as G≠0，that is,as long as there is some w for which G(jw)≠0. ul2 lulloo pow(u) l2 IlGIl oo 0∞ 00 lylloo IG12 IG 0 pow (y) 0 ≤IG创 IG创 Table 2.2:System Gains A typical application of these tables is as follows.Suppose that our control analysis or design problem involves,among ot her things,a requirement of disturbance attenuation:The controlled system has a dist urbance input,say u,whose effect on the plant output,say y,should be small.Let G denote the impulse response from u to y.The controlled system will be required to be stable,so the transfer function G will be stable.Typically,it will be strictly proper,too (or at least proper). The tables tell us how much u affectsy according to various measures.For example,if u is known to be a sinusoid of fixed frequency (maybe u comes from a power source at 60 Hz),then the second column of Table 2.1 gives the relative size of y according to the three measures.More commonly, the dist urbance signal will not be known a priori,so Table 2.2 will be more relevant. Notice that the oo-norm of the transfer function appears in several entries in the tables.This norm is therefore an important measure for system performance. Example A system with transfer function 1/(10s+1)has a disturbance input d(t)known to have the energy bound dl2<0.4.Suppose that we want to find the best estimate of the oo-norm of the output y(t).Table 2.2 says that the 2-norm/oo-norm gain equals the 2-norm of the transfer function,which equals 1/v20.Thus lglx≤ 0.4 The next two sections concern the proofs of the tables and are therefore optional. 2.4 Power Analysis (Optional) For a power signal u define the autocorrelation function R()=7J 1 u(t)u(t+T)dt
CHAPTER NORMS FOR SIGNALS AND SYSTEMS Table Output norms and pow for two inputs Now suppose that u is not a xed signal but that it can be any signal of norm It turns out that the least upper bound on the norm of the output that is supfkyk kuk g which we can call the norm norm system gain equals the norm of G this provides entry in Table The other entries are the other system gains The in the various entries is true as long as G that is as long as there is some for which G j kuk kuk powu kyk kG k kyk kG k kGk powy kG k kG k Table System Gains A typical application of these tables is as follows Suppose that our control analysis or design problem involves among other things a requirement of disturbance attenuation The controlled system has a disturbance input say u whose e ect on the plant output say y should be small Let G denote the impulse response from u to y The controlled system will be required to be stable so the transfer function G will be stable Typically it will be strictly proper too or at least proper The tables tell us how much u a ects y according to various measures For example if u is known to be a sinusoid of xed frequency maybe u comes from a power source at Hz then the second column of Table gives the relative size of y according to the three measures More commonly the disturbance signal will not be known a priori so Table will be more relevant Notice that the norm of the transfer function appears in several entries in the tables This norm is therefore an important measure for system performance Example A system with transfer function s has a disturbance input dt known to have the energy bound kdk Suppose that we want to nd the best estimate of the norm of the output yt Table says that the norm norm gain equals the norm of the transfer function which equals p Thus kyk p The next two sections concern the proofs of the tables and are therefore optional Power Analysis Optional For a power signal u dene the autocorrelation function Ru lim T T Z TT utut dt
2.4.POWER ANALYSIS (OPTIONAL) 17 that is,Ru(r)is the average value of the product u(t)u(t+).Observe that Ru(0)=pow(u)2≥0. We must restrict our definition of a power signal to those signals for which the above limit exists for all values of r,not just r=0.For such signals we have the additional property that |Ru(r)川≤Ru(O). Proof The Cauchy-Schwarz inequality implies that l,aoss(aera)(,a） Set v(t)=u(t+7)and multiply by 1/(2T)to get 房e+s(a(a+r” 1/2 Now let T-oo to get the desired result. Let S denote the Fourier transform of Ru.Thus Su(jw)= Ru(r)e jdT, -C0 Ru(T)= 1 2xJ-0 Su(jw)e dw, pow(u)2 = Ru(0)= 2玩/ Su(jw)dw. From the last equation we interpret Su(jw)/2 as power density.The function Su is called the power spectral density of the signal u. Now consider two power signals,u and v.Their cross-correlation function is R()=27- 1 u(t)u(t+T）dt and Suv,the Fourier transform,is called their cross-power spectral density function. We now derive some useful facts concerning a linear system with transfer function G,assumed stable and proper,and its input u and output y. 1.Ruy =G*Ru Proof Since y()= G(a)u(t-a)da (2.1) we have u()ve+r)-f G(a)u(t)u(t+r-a)da
POWER ANALYSIS OPTIONAL that is Ru is the average value of the product utut Observe that Ru powu We must restrict our denition of a power signal to those signals for which the above limit exists for all values of not just For such signals we have the additional property that jRu j Ru Proof The CauchySchwarz inequality implies that Z TT utvtdt Z TT ut dt Z TT vt dt Set vt ut and multiply by T to get T Z TT utut dt T Z TT ut dt T Z TT ut dt Now let T to get the desired result Let Su denote the Fourier transform of Ru Thus Suj Z Ru ej d Ru Z Sujej d powu Ru Z Sujd From the last equation we interpret Suj as power density The function Su is called the power spectral density of the signal u Now consider two power signals u and v Their crosscorrelation function is Ruv lim T T Z TT utvt dt and Suv the Fourier transform is called their crosspower spectral density function We now derive some useful facts concerning a linear system with transfer function G assumed stable and proper and its input u and output y Ruy G Ru Proof Since yt Z G ut d we have utyt Z G utut d