Frequency Response Methods

Transcription

1 Frequecy Respose Methods The frequecy respose Nyquist diagram polar plots Bode diagram magitude ad phase Frequecy domai specificatios

2 Frequecy Respose Methods I precedig chapters the respose ad performace of a system have bee described i terms of the complex frequecy variable s ad the locatio of the poles ad zeros o the s-plae. A very practical ad importat alterative approach to the aalysis ad desig of a system is the frequecy respose method. The frequecy respose of a system is defied as the steady-state respose of the system to a siusoidal iput sigal. The siusoid is a uique iput sigal, ad the resultig output sigal for a liear system as well as sigals throughout the system, is siusoidal i the steady state the out of the system; it is differs from the iput waveform oly i amplitude ad phase agle.

3 Frequecy Respose Cosider the followig u t R cos t α R cos t x ss t Bcos t φ Bcos t φ B B e B φ θ G G e G θ α R R e R α G is kow as the gai ad θ φ α is kow as the phase of the system.

4 Frequecy Respose For example for a system with dyamics described by the differetial equatio d y d y dy a a a a y... 0 dt dt dt u The trasfer fuctio from u to y is : G s as a s... a o This meas a frequecy depedet system gai or trasfer fuctio: G... a a a o

5 Frequecy Respose The frequecy respose ca be obtaied from the trasfer fuctio by substitutig s. Brief derivatio: u t Re, y Be Write u ad y as ad substitute ito DE: t d y d a a dt dt a Be y... a a dy dt a Be... a Be t t t t t 0 [ a a... a a ] Be Re 0 y a 0 Be t Re t This meas that the frequecy depedet system gai or trasfer fuctio is : G Be... t t Re a a a o Hece the s substitutio rule is ustified.

6 Frequecy Respose Alteratively, cosider the system where pi are assumed to be distict poles. The i partial fractio form we have Takig the iverse Laplace trasform yields l where α ad β are costats which are problem depedet.

7 Frequecy Respose If the system is stable, the all pi are have positive ozero real parts, poles are pi, ad l sice each expoetial term decays to zero as t. l Thus the steady-state output sigal depeds oly o the magitude ad phase of T at a specific frequecy. Notice that the steady state respose as described the above is true oly for stable systems, Ts.

8 Laplace vs. Fourier Trasform Laplace trasform: where Fourier trasform: Settig s i Fs yields the Fourier trasform of f t But Fourier Trasform is ofte used for sigals that exist for t<0

9 Jea Baptiste Joseph Fourier Bor i 768 i Auxerre, Frace Died i 830 i Paris Was early guillotied i 794 Was taught by Laplace, Lagrage ad Moge Created Cairo Istitute Developed Fourier series while prefect i Greoble

10 Advatage of Frequecy respose method The frequecy respose method is the ready availability of siusoid test sigals for various rages of frequecies ad amplitudes. Thus the experimetal determiatio of the frequecy respose of a system is easily accomplished ad is the most reliable ad ucomplicated method for the experimetal aalysis of a system. Furthermore the desig of a system i the frequecy domai provides the desiger with cotrol of the badwidth of a system ad some measure of the respose of the system to udesired oise ad disturbace.

11 Trasfer Fuctio i Frequecy Respose Method The frequecy respose method is that the trasfer fuctio describig the siusoid steady-state behavior of a system ca be obtaied by replacig s with i the system trasfer fuctio Ts. The trasfer fuctio represetig the siusoidal steady-state behavior is the a fuctio of the complex variable ad is itself a complex fuctio T. Direct correlatios betwee the frequecy respose ad the correspodig trasiet respose characteristics i the time domai are somewhat teuous very weak.

12 Frequecy Respose Methods The siusoid is a uique iput sigal, ad the resultig output sigal for a liear system as well as sigals throughout the system, is siusoidal i the steady state the out of the system; it is differs from the iput waveform oly i amplitude ad phase agle. The importat issue i frequecy respose methods is how to descript the amplitude ad phase agle of the system. We will study differet methods to represet amplitude ad phase.

13 Frequecy Respose Plots G Re Im φ ta Im Re Review Appedix G i textbook

14 Frequecy Respose Plots st Order system φ ta Im Re ta / ta

15 Polar Plot or Nyquist Diagram G s s K s τ rd system φ ta Im Re ta K K τ ta τ

16 Harold Nyquist Bor i 889 i Swede Died i 976, USA Yale PhD, 97 Career at Bell Labs 38 patets Nyquistdiagram, criterio, samplig theorem Laid the foudatio for iformatio theory, data trasmissio ad egative feedback theory

17 Frequecy respose diagrams The Nyquist diagram of G s 3 s s s G Nyquist Diagrams Gs/s 3 s s Imagiary Axis is plotted for [ 0,,0] Real Axis

18 Bode Diagram Plots of 0log0 G ad of Φ as a fuctio of log Gai i decibels db RC filter:

19 Hedrik Wade Bode , USA PhD from Columbia i 935 Etire career at Bell Labs Iveted magitude ad phase frequecy plots i 938 May other cotributios i electrical egieerig ad cotrol

20 Bode Diagram -3dB Break or corer frequecy

21 Bode Diagram decade Advatage of logarithmic plot is that multiplicative factors are coverted ito additive oes We ca the decompose a high order trasfer fuctio ito a product of simple stadard compoets to sketch the broad features of the Bode diagram

22 Form of the Trasfer Fuctio We treat real ad complex poles ad zeros separately, ad use oe of two stadard forms. We the plot magitude i db ad phase o a log scale of frequecy By usig db for magitude, we ca use additio to combie the effects of each pole or zero sice the product becomes a sum of log terms. The phase effect is already a liear combiatio The log frequecy scale allows piecewise liear approximatios with reasoable accuracy Bode plots work best for real poles ad zeros. The complex case is less accurate ad requires careful treatmet.

23 How to draw a Bode plot for a give trasfer fuctio? There are oly 8 types of factors i ay trasfer fuctio - which are these? Ls w y w k k k k y m i i Ls y y m i i Ls m Ls m m m m e s s p s s z s k e p s s z s k e p s p s p s z s z s z s k e a a s s a s a b b s s b s b G s ζ L w y w k k k k p y m i zi L y pi y m i zi L p p p zm z z L m m m m e T T K e T T K e T T T T T T K e a a a a b b b b ζ G If we kew the look of the Bode plots for each of the 8 types, we could add up the Bode plots from them.

24 Bode plot aalysis techiques Factorizatio G m K i y w w y T p k T zi ζ k k k e L Gai i db : Lm G 0log G 0log K 0log T 0log T... 0log T 0ylog z 0log T 0log T... p p z ζ 0log T 0log... ζ w 0log w p y w w zm

25 Bode plot aalysis techiques w w w w y p p p zm z z T T T T T T K ζ ζ π G Phase: The laborious procedure of plottig the amplitude ad the phase by meas of substitutig several values of ca be avoided whe drawig Bode diagrams, because we ca use several short cuts. These short cuts are based o simplifyig approximatios, which allow us to represet the exact, smooth plots with straight-lie approximatios. The differece betwee actual curves ad these asymptotic approximatios is small, ad ca be added as a correctio.

26 The 8 types of factors i a trasfer fuctio Costat K Trasport delay L e

27 The 8 types of factors i a trasfer fuctio Itegrators y Differetiators r

28 The 8 types of factors i a trasfer fuctio First order Lead terms real zeros T zi r First order lag terms real poles T p r

29 The 8 types of factors i a trasfer fuctio Quadratic lag terms complex poles ζ k k k r Quadratic lead terms complex zeros ζ k k k r

30 Bode Diagram Plots of 0log0 G ad of Φ as a fuctio of log Gai i decibels db RC filter:

31 Bode plot aalysis techiques Factorizatio G m K i y w w y T p k T zi ζ k k k e L Gai i db : Lm G 0log G 0log K 0log T 0log T... 0log T 0ylog z 0log T 0log T... p p z ζ 0log T 0log... ζ w 0log w p y w w zm

32 Bode plot aalysis techiques w w w w y p p p zm z z T T T T T T K ζ ζ π G Phase: The laborious procedure of plottig the amplitude ad the phase by meas of substitutig several values of ca be avoided whe drawig Bode diagrams, because we ca use several short cuts. These short cuts are based o simplifyig approximatios, which allow us to represet the exact, smooth plots with straight-lie approximatios. The differece betwee actual curves ad these asymptotic approximatios is small, ad ca be added as a correctio.

33 Detailed examiatio of the 8 factors System type correspods to itegrators for 0 type there is ot itegrator factor Diagram of a costat Lm K 0log G K m K i y w w y Tp k db π T zi ζ k k k e L K<0 K>0

34 Detailed examiatio of the 8 factors Diagram of itegrators Lm y 0log y 0log 0y log 0y log y 90y y

35 Detailed examiatio of the 8 factors Bode diagram of a differetiator Lm y y 0log 0y log 0y log y 90y y

36 Detailed examiatio of the 8 factors Bode diagram of a first order lag term Lm 0log 0 log 0 log T T T T ta T T 0log T T <<. Lm 0log 0dB T T >>. Lm T 0log T 0logT

37 Detailed examiatio of the 8 factors Bode diagram of a first order lag term frequecy error Corer frequecy badwidth -3dB At half the corer frequecy -db At a quarter of the corer frequecy -0.6 db

38 Detailed examiatio of the 8 factors First order lead term T <<. Lm T 0 log T 0 log 0log T ta T T T T 0log 0dB Lm T >>. T 0log T 0logT Lm

39 Detailed examiatio of the 8 factors ζ < ζ 0log 0log Lm ζ ζ ζ / / ta ζ ζ Quadratic secod order Lag

40 0dB 0log Lm ζ For small ζ 40log 0log Lm For large Detailed examiatio of the 8 factors Quadratic secod order Lag

41 Detailed examiatio of the 8 factors Quadratic secod order Lag For ζ < there is a resoat peak at with peak size M m ζ ζ m ζ

42 Detailed examiatio of the 8 factors Trasport Lag Lme L 0, e L L

43 No-Miimum Phase System A trasfer fuctio is called miimum phase if all its zeros lie i the left-had plae. It is called o-miimum phase if it has zeros i the right-had plae.

44 Root locus examples GH s s s 3s Apply steps -4 ad step 5 for breakaway poit Imag Axis Real Axis

45 Root locus examples GH s s s 3s Apply steps -4, Step 4 for crossig ad Step 5 for breakaway poit Imag Axis Real Axis

46 No-Miimum Phase System Note that G is idetical, but the phase is differet

47 Drawig the Bode Diagram 0log54-0dB -40dB? 40dB/De?

48 Drawig the Bode Diagram 0log54-0dB -40dB? 40dB/De

49 Drawig the Bode Diagram < 0log G 0log 5 0log > 0log 0.5 >0 0log 0. >50 0log 0.6 / 50 / 50

50 Drawig the Bode Diagram

51 Drawig the Bode Diagram

52 Usig Matlab

53 Example G s 30 s 5

54 Example G

55 Example G

56 Performace Specificatios i the Frequecy Domai The basic disadvatage of the frequecy respose method for aalysis ad desig is the idirect lik betwee the frequecy ad the time domai. The for give a set of time-domai trasiet performace specificatios, how do we specify the frequecy respose? Direct correlatios betwee the frequecy respose ad the correspodig trasiet respose characteristics are somewhat teuous very weak. However, we eed to develop a method to evaluate the performace i the frequecy respose method. Like i the time-domai approach, we oly cosider the performace of a simple secod order system to a step iput.

57 Step respose for secod order systems-i the time domai ζ yt e t si βt cos ζ, β ζ β T s PO 4 for δ % ζ T p 00 e π ζ ζπ / ζ Settlig time Percetage overshoot Peak time T r.6 ζ 0. 6 Rise time 0% -90%

58 Performace Specificatios i the Frequecy Domai Cosider a secod order system The closed-loop trasfer fuctio i the frequecy domai: T s. s ζs The Bode diagram of the frequecy respose of this feedback system is show i Fig

59 0dB 0log Lm ζ For small ζ 40log 0log Lm For large Detailed examiatio of the 8 factors Quadratic secod order Lag

60 Performace Specificatios i the Frequecy Domai T s s ζ s. r value of the frequecy respose, M p, is attaied. B At the resoat frequecy,, a maximum The badwidth,, is a measure of a system s ability to faithfully reproduce a iput sigal. The badwidth is the frequecy, B, at which the frequecy respose has declied 3 db from its lowfrequecy value.

61 Performace Specificatios i the Frequecy Domai The resoat frequecy r ad 3dB badwidth ca be related to the speed of the trasiet respose. Thus as the badwidth B icrease, the rise time of the step respose of the system will decrease. Furthermore the overshoot to a step iput ca be related to, M p through the dampig ratio ζ. by M p t e ζπ ζ. The resoat peak M p idicates the relative stability of a system

62 Performace Specificatios i the Frequecy Domai The badwidth of a system B as idicated o the frequecy respose ca be approximately related to the atural frequecy of the system. Figure 8.6 shows the ormalized badwidth, B /, versus ζ for the secod-order system

63 Performace Specificatios i the Frequecy Domai Thus desirable frequecy-domai specificatios are as follows:. Relatively small resoace magitude: M p <. Relatively large badwidths so that the system time costat is sufficietly small τ /ζ.5, for example.

64 Performace Specificatios i the Frequecy Domai The usefuless of these frequecy respose specificatios ad their relatio to the actual trasiet performace deped upo the approximatio of the system by a secod-order pair of complex poles, that is the domiat roots. If the frequecy respose is domiated by a pair of complex poles, the relatioships betwee the frequecy respose ad the time respose discussed i this sectio will be valid. Fortuately a large proportio of cotrol system satisfied this domiat secod-order approximatio i practice.

65 Steady-state error costats The steady-state error specificatio ca also be related to the frequecy respose of a closed-loop system. As we kew, the steady-state error for a specific test iput sigal ca be related to the gai ad umber of itegratios poles at the origi of the ope-loop trasfer fuctio, i.e., the type of the system. I frequecy respose method, the type of the system determies the slop of the logarithmic gai curve at low frequecy, sice steady-state error is defied at s 0, i.e., 0. Thus, iformatio cocerig the existece ad magitude of the steady-state error of a cotrol system to a give iput ca be determied from the observatio of the low-frequecy regio of the logarithmic gai curve.

66 Determie of static positio error costats. For type 0 system N0, we have lim lim 0 0 G s G K s P Cosider the trasfer fuctio as follows. Q k k N M i i K G τ τ For type 0 N0 system, at the low frequecy, we have K G or K G K P lim 0 Q k k M i i Q k k M i i K K G 0 τ τ τ τ

67 Determie of static positio error costats. K P lim G 0 Hece, we ca determie the steady-state positio error by measure the value from its logarithmic gai curve let 0logKc, K K p c/ 0 0logK / logk

68 Determie of static velocity error costat For type system N, we have lim lim 0 0 G s sg K s v Cosider the trasfer fuctio as follows. Q k k N M i i K G τ τ Accordig to the defiitio, we have K G K v lim 0 at the low frequecy. τ τ K K G Q k k M i i

69 Determie of static velocity error costat 0 log K v 0 log K v. Also, we ca fid out Kv usig the fact that the itersectio of the iitial 0dB/decade segmet or its extesio with the 0dBlie has a frequecy umerically equal to Kv K v or K v At the itersectio of the iitial 0dB/decade segmet or its extesio with the 0-dB lie, the horizotal coordiate, i.e., the frequecy is umerically equal to the. K v

70 Determie of static acceleratio error costat K G K a lim 0 For type system N, we have Cosider the trasfer fuctio as follows lim lim 0 0 G s G s K s a. Q k k N M i i K G τ τ at the low frequecy. τ τ K K G Q k k M i i

71 Determie of static acceleratio error costat Ka 0log 0log K a a The frequecy at the itersectio of the iitial -40db/decade segmet or its extesio with the 0-dB lie gives the square root of K a umerically. K 0log a 0log 0 which yields a K or K. a a a

72 Desig Example: Egravig Machie The goal is to select a appropriate gai K, utilizig frequecy respose method, so that the time respose to step commads is acceptable

73 Desig Example: Egravig Machie To represet the frequecy respose of the system, we will first obtai the ope-loop ad closed-loop Bode diagram. G s s s

74 Desig Example: Egravig Machie The we use the closed-loop Bode diagram to predict the time respose of the system ad check the predicted result with the actual result T s. 3 s 3s s T s. 3 0 logm p 0log T 5 db at r or M p r

75 Desig Example: Egravig Machie If we assume that the system has domiat secod-order roots, we ca approximate the system with a secod-order frequecy respose of the form show i Fig. 0 log M p 5 or M p r r 0.8 ζ 0.9

76 Desig Example: Egravig Machie M p.78 r 0.8 ζ 0.9 r / Sice we are ow approximatig Ts as a secod-order system, we have T s s ζ s s s

77 Desig Example: Egravig Machie T s s ζ s s s The overshoot to a step iput as 37% for ζ 0. 9 The settlig time to withi % of the fial value is estimated as T 4 ζ s 5.7 sec ods.

78 Desig Example: Egravig Machie The actual overshoot for a step iput is 34%, ad the actual settlig time is 7 secods. We see that the secod-order approximatio is reasoable i this case ad ca be used to determie suitable parameters o a system. If we require a system with lower overshoot, we would reduce K to ad repeat the procedure.

79 Disk Drive Read System The disk drive uses a flexure suspesio to hold the reader head mout, as show i Fig we will iclude the effect of the sprigy flexure withi the model of the motor-load system. We model the flexure with the mouted head as a mass M, a sprig k, ad a slidig frictio b.

80 Disk Drive Read System The trasfer fuctio of a sprig-mass-damper was developed i Chapter, where Y s U s G 3 s s ζ s ζs / s / ζ A typical flexure ad head has ad atural resoace at f 3000Hz or 3.

81 Disk Drive Read System The sketch is a plot of the magitude characteristics for the ope-loop Bode diagram,or 0log K G G G3, K400 Note the resoace at We wish to avoid excitig This resoace.

82 Disk Drive Read System Plots of the magitude of the ope-loop Bode diagram ad the closed-loop Bode diagram are show i followig Mp where As log as of the system. ζ 0.8 T K 400, 4. B 000rad/ sec s 5ms ζ the resoace is outside the badwidth

83 Problems: Experimetal determiatio of trasfer fuctio of a system based o its frequecy respose Determie the trasfer fuctio of the system that has the followig frequecy respose: 5 s

84 Problems: Experimetal determiatio of trasfer fuctio of a system based o its frequecy respose Determie the trasfer fuctio of the system that has the followig frequecy respose: 0.s 0.0s s 0. s 0 0.0s s

85 Summary I this chapter we have cosidered the represetatio of a feedback cotrol system by its frequecy respose characteristics. The frequecy respose of a system was defied as the steady-state respose of the system to a siusoidal iput sigal. Several alterative forms of frequecy plots were cosidered, icludig the polar plot of the frequecy respose of a system G ad logarithmic plots, ofte called Bode plots, ad the value of the logarithmic measure was illustrated. The ease of obtaiig a Bode plot for the various factors of G was oted, ad a example was cosidered i detail. The asymptotic approximatio for sketchig the Bode diagram simplifies the computatio cosiderably

86 Summary of fiftee typical Bode plots.. 3.

87 Summary of fiftee typical Bode plots 5.

88 Summary of fiftee typical Bode plots 7. 9.

89 Summary of fiftee typical Bode plots 0...

90 Summary of fiftee typical Bode plots 3. 4.