Copyright FL Lewis 7 All rights reserved Updated: Moday, November 1, 7 EE 4314 - Cotrol Systems Bode Plot Performace Specificatios The Bode Plot was developed by Hedrik Wade Bode i 1938 while he worked at Bell Labs Here we shall show how performace specificatios i terms of Bode plots i the frequecy domai are related to time domai performace Badwidth ad Rise Time The Bode plot of the trasfer fuctio α 1 H() s = = s+ α s+ 1 is show The break frequecy occurs at 1 rad/sec, the magitude of the pole -3db -5-1 System: sys Frequecy (rad/sec): 998 Magitude (db): -3-15 - -5-3 -35-4 -3 Magitude (db) B -6-9 1-1 1 1 1 1 1 3 Frequecy (rad/sec) The 3dB cutoff frequecy, or badwidth, B is the frequecy at which the frequecy magitude respose has decreased by 3dB from its low frequecy value I this example B = α = 1 rad / s αt t/ τ The impulse respose of this system is ht () = e = e, where the time costat is τ = 1/ α = 1/ B 1
The step respose rise time is give by tr = τ The settlig time is ts = 5τ The time costat is iversely related to the badwidth Therefore, as badwidth icreases, the system respose becomes faster COMPLEX POLE PAIR A trasfer fuctio with a complex pair of poles ad o fiite zeros ca be writte as H ( = = s + αs + s s Δ( + ζ + The umerator is chose to scale the trasfer fuctio so that the DC gai (eg set s=) is equal to oe The deomiator is the Characteristic polyomial which ca be writte i several atural or caoical forms, icludig Δ ( = s + α s + = s + ζ s + Oe may also write Δ( = s + α s + = ( s + α) + β where β + α = These variables mea somethig i terms of time domai performace as we have see They also mea somethig i the frequecy domai, particularly the dampig ratio ζ ad the atural frequecy 5 The Bode plot for H() s = is show s + s+ 5 4 M P 3-3db Magitude (db) 1-1 - System: sys Frequecy (rad/sec): 778 Magitude (db): -3-3 -4-5 -6 r B -45-9 -135-18 1 1 1 1 Frequecy (rad/sec)
Recall that for complex poles, the step respose is faster tha τ due to the oscillatory compoets However, the settlig time is ts = 5τ ad is closely related to the badwidth; it decreases as badwidth icreases The resoat frequecy is give for ζ 77 by The maximum value of the Bode plot at resoace is give by 1 M p = ζ 1 ζ r = 1 ζ These fuctios are show i the figure From either of these, oe may compute the dampig ratio ad hece the percet overshoot i the time domai Figure 811 from Dorf ad Bishop editio 1 The quality factor 1 Q = = ζ α measures the sharpess of the resoat peak i the Bode plot Note that this is effectively determied solely by the dampig ratio The poles are complex if Q> 1/ I terms of the quality factor oe may write the characteristic polyomial i the odimesioal form 3
s Δ( = 1 s + + 1 Q Figure from Dorf ad Bishop, Moder Cotrol Systems, editio 8 Bode Desig i terms of the Ope-Loop Gai Cosider the trackig cotroller give i the figure The plat is H( ad the compesator K(; the feedback gai is k The fuctio of the tracker is to make the output y(t) follow the commad or referece iput r(t) by makig the trackig error e(t)=r(t)-y(t) small The disturbace is d(t) r(t) e(t) d(t) H( y(t) The closed-loop trasfer fuctio is 4
Y ( H ( T ( = = R( 1+ H ( The deomiator is Δ ( s ) = 1+ H ( 1+ kg( where G(=K(H( is the ope-loop gai Note that we use the same symbol for the deomiator of T( as for the state-variable characteristic polyomial Δ ( = si A However, 1+kG( is actually a polyomial fractio, whose umerator is the system characteristic polyomial May desig techiques rely o tryig to determie closed-loop properties from opeloop properties I root locus desig, oe uses the ope-loop gai G( to estimate the locatios of the closed-loop poles, which are the roots of the umerator of Δ ( s ) = 1+ kg( The key poit of RL desig is that it is easier to plot the locatios of the closed-loop poles versus the feedback gai parameter k tha it is to fid the actual closed-loop poles themselves This was extremely importat i days before digital computers whe fidig roots of high-order polyomials was difficult, ad it also gives great isight ito the properties of the closed-loop system Similarly, Bode desig uses the Bode plots of the ope-loop trasfer fuctios H( ad K(H( to select the compesator K( to give desirable closed-loop properties icludig stability, good POV, ad fast trasiet respose Steady-State Error Recall that the system is type N if there are N itegrators (ie N poles at s=) i the feedforward path K(H(=G( For zero steady-state error i respose to a uit step commad, or a uit step disturbace, oe requires the system to be of type 1 The Bode plot of the itegrator compesator K(= 1/s is give i the figure It has a costat slope of =1, or - db/decade, ad a agle of -9 o Therefore, a system of type oe has a slope of =-1 at low frequecies To get zero steady-state error i respose to a uit step, oe must add a itegrator to obtai such a slope, uless the Bode plot of H( already has this slope at low frequecies 5 Magitude (db) -5-1 -15 - -89-89 -894-896 -898-9 -9-94 -96-98 -91 1 1 1 Frequecy (rad/sec) 5
Crossover Frequecy The crossover frequecy c is where the loop gai G(= K(H( has a gai of uity, ie kg( j c ) = 1 The closed-loop trasfer fuctio is Y ( H ( T ( = = R( 1+ H ( Whe << c oe has kg( j ) >> 1 so that T( j) 1 ad the closed-loop gai is uity Whe >> c oe has kg( j ) << 1 so that T( j) kg( j) ad T( j) falls off like kg( j ) Therefore, the (closed-loop) badwidth B is about equal to the (ope loop gai) crossover frequecy To get faster resposes, oe eeds to icrease the crossover frequecy c Gai Margi, Phase Margi The gai margi ad phase margi deped o both of the ope-loop gai Bode plots, magitude ad phase By extractig iformatio from both plots, GM ad PM provide closed-loop stability ad performace criteria The closed-loop deomiator is Δ ( s ) = 1+ H ( 1+ kg( The closed-loop poles are give by Δ ( = 1 + H( 1 + kg( = or kg( s ) = 1 Therefore, stability may be studied i terms of whe kg( j ) has a magitude of oe ad a phase of 18 o The gai margi is the gai icrease required to make kg( j ) = 1 whe its phase is -18 o The phase margi is the phase shift required to make phase( kg( j )) = 18 o whe kg( j ) = 1 If the magitude of kg( j ) is α whe its phase is -18 deg, the the gai margi is 1/α The logarithm of 1/α is egative the logarithm of α Therefore, i terms of db, if α is d db, the 1/α is simply d db Note that kg( j ) = 1 meas kg( j ) = db 6
The Bode plot of kg() s = ss ( + )( s+ 3) is show Lookig at the crossover frequecy c, where the magitude is equal to oe (ie db), the phase margi = -155 (-18)= 745 deg Lookig at the frequecy where the phase is -18 deg, the gai margi is -35 db 5 Gm = 35 db (at 45 rad/sec), Pm = 745 deg (at 37 rad/sec) 35 db Magitude (db) -5-1 c 1/GM -155 o -15-9 -135-18 -5 PM -7 1-1 -1 1 1 1 1 The dampig ratio icreases with phase margi accordig to the figure Therefore, to icrease dampig ratio, we eed to icrease phase margi Relatio betwee PM ad dampig ratio for a secod order system Figure 91 from Dorf ad Bishop ed 1 7