Nyquist stabilit criterion aleri Ougrino vski August   Abstract This note giv es brief in tro duction to the closed lo op system stabilit analysis based on the frequency resp onse of the system op en
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Nyquist stabilit criterion aleri Ougrino vski August Abstract This note giv es brief in tro duction to the closed lo op system stabilit analysis based on the frequency resp onse of the system op en

Nyquist olar plots olar plot of the frequency resp onse pro duces curv with frequency as parameter relating the oin ts on the olar plot to the appropriate oin ts on the Bo de plots see Fig 1 The Nyquist plot con tains the same information as the Bo

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Nyquist stabilit criterion aleri Ougrino vski August Abstract This note giv es brief in tro duction to the closed lo op system stabilit analysis based on the frequency resp onse of the system op en




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Nyquist stabilit criterion aleri Ougrino vski August 9, 2006 Abstract This note giv es brief in tro duction to the closed lo op system stabilit analysis based on the frequency resp onse of the system op en lo op gain. Nyquist olar plots olar plot of the frequency resp onse pro duces curv with frequency as parameter relating the oin ts on the olar plot to the appropriate oin ts on the Bo de plots, see Fig. 1. The Nyquist plot con tains the same information as the Bo de plot. Th us, one can use ap- pro ximate Bo de plots to reconstruct the shap of the Nyquist plot. or instance,

the Bo de and Nyquist plots of 5( 1) 10) are sho wn elo in Figure 1. Arro ws on the Nyquist plot indicate the direction in whic the plot hanges as the frequency increases from 1 to (or equiv alen tly sw eeps from the to along the imaginary axis). Note the symmetric branc hes of the Nyquist plot corresp onding to ositiv and negativ frequencies. Unlik Bo de plots, oth ositiv and negativ frequencies are used for plotting Nyquist curv es; the reasons for this will ecome clear later. The Matlab command to pro duce accurate Nyquist plots is as follo ws: >> nyquist(G,w); The second argumen is

optional; it allo ws ou to sp ecify an arra of frequencies of in terest to capture some details of the plot. or example, to pro duce the Nyquist plot in Fig 1, ou need to yp >> G=tf(5*[1 1],[1 10]); >> nyquist(G); The command nyquist can also used for computing the frequency resp onse of system lo op gain for ositiv frequencies only No plot is pro duced, hence the command plot ust used: >> G=tf(5*[1 1],[1 10]); >> [reG,imG]=nyquist(G,{0.01 ,100 0}) >> plot(squeeze(reG),squeeze (imG ))
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Bode Diagram Frequency (rad/sec) Phase (deg) Magnitude (dB) 10 10 10 −90 −45 45

System: G Frequency (rad/sec): 2.83 Phase (deg): −1.66e−14 −30 −20 −10 10 System: G Frequency (rad/sec): 2.83 Magnitude (dB): 7.96 Nyquist Diagram Real Axis Imaginary Axis 1 0.5 0.5 1.5 2.5 1.5 1 0.5 0.5 1.5 System: G Real: 5e06 Imag: 0.005 Freq (rad/sec): 1e+03 System: G Real: 0.5 Imag: 0 Freq (rad/sec): 0 System: G Real: 2.5 Imag: 0.00186 Freq (rad/sec): 2.83 System: G Real: 5e06 Imag: 0.005 Freq (rad/sec): 1e+03 Figure 1: Bo de and Nyquist plots of the lo op gain transfer function 5( +1) +2 +10
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The graphical metho can used to obtain an appro

ximate Nyquist plot. Consider an example. Let 100( 10) 1) 100) This transfer function has the single nonmin um um phase zero at 10, the double ole at (real), and the pair of complex conjugate oles at 05 10 First let us sk etc the magnitude Bo de plot. PSfrag replacemen ts dB 60 40 20 -20 -40 -60 -80 0.1 10 100 Note that zero at 10 and the pair of complex conjugate oles at 05 10 yield the same corner frequency 10 rad/sec. Hence the net hange of slop at this corner frequency is +20 40 20 dB/decade. Also, note that the oles 05 10 are ery underdamp ed, hence it is appropriate to dra eak at the

corresp onding corner frequency no pro ceed to phase calculations. rite the transfer function in \standard form": 100( 10) 1) 100) Note the gain factor of 100. The sign means that the +180 or 180 term should tak en in to accoun t, let’s use 180 The phase of can expressed as 180 the angles are sho wn in the Figure elo w. Note the factor of in fron of this factor is to reect the con tribution of the double ole at 1.
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PSfrag replacemen ts Figure 2: rad/sec 180 180 -90 90 0.1 180 45 -90 90 -90 180 90 -90 90 -180 9.5 135 90 -90 90 -225 9.95 135 90 -45 90 -270 10 135 90 90

-315 10.05 135 90 45 90 -360 10.5 135 90 90 90 -405 100 90 90 90 90 -450 90 90 90 90 -450 This agrees with the true Bo de plot sho wn in Figure 5. Note the rapid hange of 180 around the resonan frequency (natural undamp ed frequency) 10rad/sec. rom the ab analysis, can sk etc the Nyquist plot; see Figure 3. The true plot is sho wn in Figure (p ositiv frequencies only). It as obtained using nyquist command: >> num=100*[-1 100]; >> den=conv([1 0.2 0.01],[1 0.1 100]); >> [re,im]=nyquist(num,den,{ 0.01 ,10 0}); >> plot(squeeze(re),squeeze( im)) >> grid >> On Figure (left plot) the general shap of

the plot can seen but the lo op near the origin is in visible. In Figure on the righ t, the axis limits ha een set so that the eha vior of the olar plot near the frequency 10 can observ ed. One metho of \zo oming in" the picture is to apply the commands >> a=[-10 20 -10 20]; >> axis(a); In the rst command, \zo om windo w" is dened as [Xmin Xmax Ymin Ymax] The second command scales the axes to sho the part of the gure within the desired windo w.
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Figure 3: Nyquist sk etc for the example 200 200 400 600 800 1000 700 600 500 400 300 200 100 100 10 5 10 15

20 10 5 10 15 20 Figure 4: rue Nyquist plot: complete and detailed
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10 2 10 1 10 10 10 80 60 40 20 20 40 60 10 2 10 1 10 10 10 500 400 300 200 100 , rad/sec Figure 5: Bo de plots The Nyquist stabilit criterion The Nyquist criterion is essen tially graphical metho and allo ws to establish the stabilit of the close lo op system from the Nyquist plot of the op en lo op gain transfer function ). PSfrag replacemen ts Figure 6: Consider for simplicit the system sho wn in Figure 6. The closed lo op system has the transfer function cl Hence the oles of the closed lo op system are the

ro ots of ): or, after dividing (1) (2) Hence, stabilit of the closed lo op system can determined from the analysis of the righ half-plane zeros of the function ).
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Consider con tour in the righ half-plane of the -plane; see gure 7. or the set of complex alues of the function forms another con tour in the complex plane. a) Re G(s) b) Im G(s) −1 PSfrag replacemen ts Figure 7: The plane and plane. The Principle of the Argumen states: If con tour in the complex plane encircles zeros and oles of and the tra ersal is in the clo kwise direction along the con tour, then

the corresp onding con tour encircles the oin in the complex plane exactly times in the clo kwise direction. The Principle of the Argumen allo ws us to establish the um er of oles of the closed lo op system simply lo oking at the Nyquist plot of ). Select con tour to large enough to include the en tire righ half of the complex plane: PSfrag replacemen ts This con tour is often called D-con tour ecause of its shap e. Note that the D-con tour ust oid imaginary axis oles of ), as sho wn in the gure. or stabilit should not ha ro ots in the righ half plane, i.e, need 0. Also, since has the

same oles as ), then equals the um er of unstable oles of ). This leads to the follo wing principle, called the Nyquist stabilit criterion: The Nyquist stabilit criterion The closed lo op system is stable if and only if the net um er of anticlo ckwise encirclemen ts of
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the oin the Nyquist plot of the system lo op gain transfer function is equal to the um er of oles of in the righ half-plane: The equiv alen form ulation of the Nyquist stabilit criterion The closed lo op system is stable if and only if the net um er of clo ckwise encirclemen ts of the oin the Nyquist plot of the

system lo op gain transfer function plus the um er of oles of in the righ half-plane is zero: Examples 3.1 Example Consider the feedbac system in Figure 6, in whic +5 The Nyquist D-con tour for this system is sho wn in Figure 8. -5 PSfrag replacemen ts Figure 8: sk etc the Nyquist plot of the lo op gain transfer function, use the graphical metho d: rad/sec Gain Phase deg. 45 90 N/A N/A The Nyquist diagram of is sho wn in Figure 9. The Nyquist diagarm nev er encircles the critical oin for an 0. I.e., the um er of an ti-clo kwise encirclemen ts (none) the um er of unstable oles of (none). Th us,

the closed lo op system is stable for an 0. No let use the same Nyquist D-con tour. Use the graphical metho to sk etc the Nyquist plot of the lo op gain transfer function: rad/sec Gain Phase deg. 180 135 90 N/A N/A
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1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 Nyquist Diagram Real Axis Imaginary Axis K=5 K/5 1+j0 = =0) PSfrag replacemen ts Figure 9: The Nyquist plot of for and is sho wn in Figure 10. The lo op gain transfer function has one unstable ole, 1. When 4, the Nyquist plot do es not encircle the critical oin 0, 0; i.e, the um er of coun ter-clo

kwise encirclemen ts the um er of unstable oles of ): Th us, the closed lo op system is unstable for 4. When 6, the Nyquist diagarm mak es one circle around the critical oin in the an ti-clo kwise direction, 1, i.e., the um er of an ti-clo kwise encirclemen ts the um er unstable oles of ), Th us, the closed lo op system is stable for 6. Indeed, cl Hence if the closed lo op system has the unstable ole at 1. If the closed lo op system has the stable ole at 1. 1.4 1.2 1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 Nyquist Diagram Real Axis Imaginary Axis 1+j0 K/(5) =0) = =+ K=6 K=4 PSfrag

replacemen ts Figure 10:
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3.2 Example Consider the feedbac system in Figure 6, with +1)( +2) The Nyquist D-con tour for this system is sho wn in Figure 11a. Note that the ole at is oided. Since this ole is exluded from the righ half plane the hosen Nyquist con tour, for stabilit need the Nyquist plot not to encircle 0. sk etc the Nyquist plot mapping 1 2 pole at 0 PSfrag replacemen ts 1 +0 1 3/4 1/6 PSfrag replacemen ts 1 +0 (a) (b) Figure 11: the con tour on to the plane, consider four regions denoted 1,2, and as lab eled in the gure. Image of region

1. In this region, where 0. rad/sec Gain Phase deg. +0 +0 90 10 165 (4 10 195 270 Since the hanges from 90 to 270 then the plot ma or ma not encircle the oin nd out for sure, need to calculate the lo cation of the in tersection of the plot and the real axis. The in tersection is haracterized the condition 1)( 2) Calculate +1)( +2) 1)( 2) 1)( 2) 1)( 4) 2) 1)( 4) 10
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1)( 2) 1)( 4) at 2. Hence there are only suc frequencies rad/sec (not in region 1), and rad/sec. frequency rad/sec, 1)( 2) 1)( 4) Th us, the plot crosses the real axis on the righ of 0. Note that at 2, 180

i.e, could determine the crosso er oin from the Bo de plot of lo oking at the gain at the phase crosso er frequency Note also that as 0, then Image of region 2. This region maps in to the origin on the plane. Image of region 3, The plot is mirror image of the plot corresp onding to region 1. Image of region 4, 0. Note that go es from 90 to 90 an ti-clo kwise. Also, 1)( 3) Th us, as 0, and hanges from 90 to 90 clo kwise. Hence the tra ersal along the arc maps to curv of Nyquist plot whic has the ‘radius =r and swings from to in the clo kwise direction. It is orth while to hec ourselv es

computing at 0: 1)( 2) The appro ximate Nyquist diagram is sho wn in Fig. 11b. The Nyquist plot do es not circle around 0, i.e, and is equal to the um er of oles of inside the D-con tour, hence the closed lo op system is stable. 3.3 Example Consider the feedbac system in Figure 6, with +1) 1)( +2) Note that the lo op gain transfer function has one ole in the righ half of the complex plane at 1. Th us, for the closed-lo op system to stable, the Nyquist diagram of ust mak exactly one turn around in the an ti-clo kwise direction. The Nyquist D-con tour for this problem is the standard Nyquist

D-con tour; see Figure 8. 0, 0) 2, 180 0, (1 1) 2). 0, arg 90 The Nyquist diagram of the frequency resp onse function is sho wn in Figure 12 (plot (a) is the appro ximate, plot (b) is the exact plot for 1). rom the plot (a), conclude that the closed lo op system is stable if and only if ecause in this case the Nyquist plot encircles the oin The plot in gure (b) illustrates the Nyquist plot of corresp onding to the unstable system. In the plot (b), the Nyquist plot do es not encircle the oin Since has one unstable ole, then the function has one zero in the righ half of the complex

plane. 11
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-1 PSfrag replacemen ts 1 Real Axis Imaginary Axis Nyquist Diagrams 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 PSfrag replacemen ts 1 (a) (b) Figure 12: urther reading [1] Dorf Bishop. Mo dern Contr ol Systems Chapter 9. Sections 9.1-9.3, 9.10, 9.12 are ust. Section 9.12 includes table of most common transfer functions and their Nyquist and Bo de plots; ou ma nd it useful. Sections 9.5, 9.8, 9.11 are optional. 12