Chapter 2 Norms for Signals and Systems One way to describe the performance of a control system is in terms of the size of certain signals of interest.For example,the performance of a tracking system could be measured by the size of the error signal.This chapter looks at several ways of defining a signal's size (i.e.,at several norms for signals).Which norm is appropriate depends on the situation at hand.Also introduced are norms for a system's transfer function.Then two very useful tables are developed summarizing input-output norm relationships. 2.1 Norms for Signals We consider signals mapping (-00,oo)to R They are assumed to be piecewise continuous.Of course,a signal may be zero for t<(i.e.,it may start at time t=0). We are going to introduce several different norms for such signals.First,recall that a norm must have the following four properties: ()lu‖≥0 (i)lul‖=0台u(t)=0, t (i)‖aull=la‖lu, a∈R (iv)lu+v‖≤‖u+lwl The last property is the familiar triangle inequality. 1-Norm The 1-norm of a signal u(t)is the integral of its absolute value: llul:=lu(e)ldt. -no 2-Norm The 2-norm of u(t)is 1/2 lul2= For example,suppose that u is the current through a 1 n resistor.Then the instantaneous power equals u(t)2 and the total energy equals the integral of this,namely,u.We shall generalize this 11
Chapter Norms for Signals and Systems One way to describe the performance of a control system is in terms of the size of certain signals of interest For example the performance of a tracking system could be measured by the size of the error signal This chapter looks at several ways of dening a signals size ie at several norms for signals Which norm is appropriate depends on the situation at hand Also introduced are norms for a systems transfer function Then two very useful tables are developed summarizing inputoutput norm relationships Norms for Signals We consider signals mapping to R They are assumed to be piecewise continuous Of course a signal may be zero for t ie it may start at time t We are going to introduce several di erent norms for such signals First recall that a norm must have the following four properties i kuk ii kuk ut t iii kauk jajkuk a R iv ku vkkuk kvk The last property is the familiar triangle inequality Norm The norm of a signal ut is the integral of its absolute value kuk Z jutjdt Norm The norm of ut is kuk Z ut dt For example suppose that u is the current through a resistor Then the instantaneous power equals ut and the total energy equals the integral of this namely kuk We shall generalize this
12 CHAPTER 2.NORMS FOR SIGNALS AND SYSTEMS interpretation:The instantaneous power of a signal u(t)is defined to be u(t)2 and its energy is defined to be the square of its 2-norm. oo-Norm The oo-norm of a signal is the least upper bound of its absolute value: luoo:=sup u(t儿. For example,the oo-norm of (1-e-)1(t) equals 1.Here 1(t)denotes the unit step function. Power Signals The average power of u is the average over time of its inst ant aneous power: 27 u(t)2dt. The signal u will be called a power signal if this limit exists,and then the squareroot of the average power will be denoted pow(u): 1/2 pow(u):= 2元 u(t)2dt Note that a nomero signal can have zero average power,so pow is not a norm.It does,however, have properties (i),(iii),and (iv). Now we ask the question:Does finiteness of one norm imply finiteness of any ot hers?There are some easy answers 1.If u2<oo,then u is a power signal with pow(u)=0. Proof Assuming that u has finite 2-norm,we get 1 ut)2dt 2元143. But the right-hand side tends to zero as T→oo.■ 2.If u is a power signal and ullo<oo,then pow(u)<luloo. Proof We have 分TueP≤u立t=alk Let T tend to oo.■ 3.If lu<and lul<oo,then u2<(lu)1/2,and hence u<. Proof uoPt=b-≤uloleh·
CHAPTER NORMS FOR SIGNALS AND SYSTEMS interpretation The instantaneous power of a signal ut is dened to be ut and its energy is dened to be the square of its norm Norm The norm of a signal is the least upper bound of its absolute value kuk sup t jutj For example the norm of ett equals Here t denotes the unit step function Power Signals The average power of u is the average over time of its instantaneous power lim T T Z TT ut dt The signal u will be called a power signal if this limit exists and then the squareroot of the average power will be denoted powu powu lim T T Z TT ut dt Note that a nonzero signal can have zero average power so pow is not a norm It does however have properties i iii and iv Now we ask the question Does niteness of one norm imply niteness of any others There are some easy answers If kuk then u is a power signal with powu Proof Assuming that u has nite norm we get T Z TT ut dt T kuk But the righthand side tends to zero as T If u is a power signal and kuk then powu kuk Proof We have T Z TT ut dt kuk T Z TT dt kuk Let T tend to If kuk and kuk then kuk kukkuk and hence kuk Proof Z ut dt Z jutjjutjdt kukkuk
2.2.NORMS FOR SYSTEMS 13 pow Figure 2.1:Set inclusions. A Venn diagram summarizing the set inclusions is shown in Figure 2.1.Note that the set labeled "pow"contains all power signals for which pow is finite;the set labeled "1"contains all signals of finite 1-norm;and so on.It is instructive to get examples of functions in all the components of this diagram (Exercise 2).For example,consider 0. ift≤0 ui(t 1/WE,if0<t≤1 0. ft>1. This has finite 1-norm: Its 2-norm is infinite because the integral of 1/t is divergent over the interval [0,1].For the same reason,ui is not a power signal.Finally,u is not bounded,so is infinite.Therefore,u lives in the bottom component in the diagram. 2.2 Norms for Systems We consider systems that are linear,time-invariant,causal,and (usually)finite-dimensional.In the time domain an input-output model for such a sy stem has the form of a convolution equation, y=G*山， that is, u)-clr r)ulr)dr. Causality means that G(t)=0 for t<0.Let G(s)denote the transfer function,the Laplace transform of G.Then G is rational(by finite-dimensionality)with real coefficients.We say that G is stable if it is analytic in the closed right half-plane (Re s>0),proper if G(j2 is finite (degree of denominator degree of numerator),strictly proper if G(j2 )=0(degree of denominator degree of numerator),and biproper if G and G are both proper(degree of denominator=degree of numerator)
NORMS FOR SYSTEMS pow Figure Set inclusions A Venn diagram summarizing the set inclusions is shown in Figure Note that the set labeled pow contains all power signals for which pow is nite the set labeled contains all signals of nite norm and so on It is instructive to get examples of functions in all the components of this diagram Exercise For example consider ut if t p t if t if t This has nite norm kuk Z p t dt Its norm is innite because the integral of t is divergent over the interval For the same reason u is not a power signal Finally u is not bounded so kuk is innite Therefore u lives in the bottom component in the diagram Norms for Systems We consider systems that are linear timeinvariant causal and usually nitedimensional In the time domain an inputoutput model for such a system has the form of a convolution equation y G u that is yt Z Gt u d Causality means that Gt for t Let G s denote the transfer function the Laplace transform of G Then G is rational by nitedimensionality with real coecients We say that G is stable if it is analytic in the closed right halfplane Re s proper if G j is nite degree of denominator degree of numerator strictly proper if G j degree of denominator degree of numerator and biproper if G and G are both proper degree of denominator degree of numerator
14 CHAPTER 2.NORMS FOR SIGNALS AND SYSTEMS We introduce two norms for the transfer function G. 2-Norm oo-Norm IlGo:=sup IG(jw) Note that if G is stable,then by Parseval's theorem 1a=(2 UP）-(cPe)”. The oo-norm of G equals the distance in the complex plane from the origin to the farthest point on the Nyquist plot of G.It also appears as the peak value on the Bode magnitude plot of G.An important property of the oo-norm is that it is submult iplicative: IGlo≤G创oalo. It is easy to tell when these two norms are finite. Lemma 1 The 2-norm of G is finite iff G is strictly proper and has no poles on the imaginary aris;the oo-norm is finite iff G is proper and has no poles on the imaginary aris. Proof Assume that G is strictly proper,with no poles on the imaginary axis.Then the Bode magnit ude plot rolls off at high frequency.It is not hard to see that the plot of c/(rs+1)dominates that of G for sufficiently large positive c and sufficiently small positive 7,that is, |c/(rjw+1)川≥lG(w)儿， Yw. But c/(rs+1)has finite 2-norm;its 2-norm equals c/v2r (how to do this computat ion is shown below).Hence G has finite 2-norm. The rest of the proof follows similar lines. How to Compute the 2-Norm Suppose that G is strictly proper and has no poles on the imaginary axis(so its 2-norm is finite). We have G(ju)2dw 1 G(-s)G(s)ds fa(-u 1 The last integral is a contour integral up the imaginary axis,then around an infinite semicircle in the left half-plane;the contribution to the integral from this semicircle equals zero because G is
CHAPTER NORMS FOR SIGNALS AND SYSTEMS We introduce two norms for the transfer function G Norm kG k Z jG jjd Norm kG k sup jG jj Note that if G is stable then by Parsevals theorem kG k Z jG jjd Z jGtjdt The norm of G equals the distance in the complex plane from the origin to the farthest point on the Nyquist plot of G It also appears as the peak value on the Bode magnitude plot of G An important property of the norm is that it is submultiplicative kGH k kG kkH k It is easy to tell when these two norms are nite Lemma The norm of G is nite i G is strictly proper and has no poles on the imaginary axis the norm is nite i G is proper and has no poles on the imaginary axis Proof Assume that G is strictly proper with no poles on the imaginary axis Then the Bode magnitude plot rolls o at high frequency It is not hard to see that the plot of c s dominates that of G for suciently large positive c and suciently small positive that is jcj jjG jj But c s has nite norm its norm equals cp how to do this computation is shown below Hence G has nite norm The rest of the proof follows similar lines How to Compute the Norm Suppose that G is strictly proper and has no poles on the imaginary axis so its norm is nite We have kG k Z jG jjd j Z j j G sG sds j I G sG sds The last integral is a contour integral up the imaginary axis then around an innite semicircle in the left halfplane the contribution to the integral from this semicircle equals zero because G is
2.3.INPUT-OUTPUT RELATIONSHIPS 15 strictly proper.By the residue theorem,G equals the sum of the residues of G(-s)G(s)at its poles in the left half-plane. Example 1 Take G(s)=1/(Ts+1),T >0.The left half-plane pole of G(-s)G(s)is at s =-1/T. The residue at this pole equals 1 111 lim 8-1/ T -T8+1T8+1=2 Hence‖G2=1/v2r. How to Compute the oo-Norm This requires a search.Set up a fine grid of frequency points, tw1;...,WN}. Then an estimate for Gloo is max.|G(jwk)儿. 1<k<N Alternatively,one could find where G(jw)is maximum by solving the equation dGy(jo）=0. dw This derivative can be computed in closed form because G is rational It then remains to compute the roots of a polynomial. Example 2 Consider -密 with a,b>0.Look at the Bode magnitude plot:For a>b it is increasing (high-pass);else,it is decreasing (low-pass).Thus Ja/b,a≥b 1. a<b. 2.3 Input-Output Relationships The question of interest in this section is:If we know how big the input is how big is the output going to be?Consider a linear system with input u,output y,and transfer function G,assumed stable and strictly proper.The results are summarized in two tables below.Suppose that u is the unit impulse,6.Then the 2-norm of y equals the 2-norm of G,which by Parseval's theorem equals the 2-norm of G;this gives entry (1,1)in Table 2.1.The rest of the first column is for the oo-norm and pow,and the second column is for a sinusoidal input.The oo in the (1,2)entry is true as long asG(jw)≠0. u(t)=6(t)u(t)=sin(wt) yl2 IG12 00 lll.o IGlloo IG()1 pow(y） 0 方86oy
INPUTOUTPUT RELATIONSHIPS strictly proper By the residue theorem kG k equals the sum of the residues of G sG s at its poles in the left halfplane Example Take G s s The left halfplane pole of G sG s is at s The residue at this pole equals lim s s s s Hence kG k p How to Compute the Norm This requires a search Set up a ne grid of frequency points fN g Then an estimate for kG k is max kN jG jkj Alternatively one could nd where jG jj is maximum by solving the equation djG j d j This derivative can be computed in closed form because G is rational It then remains to compute the roots of a polynomial Example Consider G s as bs with a b Look at the Bode magnitude plot For a b it is increasing highpass else it is decreasing lowpass Thus kG k ab a b a b InputOutput Relationships The question of interest in this section is If we know how big the input is how big is the output going to be Consider a linear system with input u output y and transfer function G assumed stable and strictly proper The results are summarized in two tables below Suppose that u is the unit impulse Then the norm of y equals the norm of G which by Parsevals theorem equals the norm of G this gives entry in Table The rest of the rst column is for the norm and pow and the second column is for a sinusoidal input The in the entry is true as long as G j ut t ut sint kyk kG k kyk kGk jG jj powy p jG jj