3173:
33:
250:
337:). This method is easily applicable even for systems with delays and other non-rational transfer functions, which may appear difficult to analyze with other methods. Stability is determined by looking at the number of encirclements of the point (−1, 0). The range of gains over which the system will be stable can be determined by looking at crossings of the real axis.
2850:
RHP zeros to "cancel out" RHP poles does not remove the instability, but rather ensures that the system will remain unstable even in the presence of feedback, since the closed-loop roots travel between open-loop poles and zeros in the presence of feedback. In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.)
4766:
4829:
defined by rational functions, such as systems with delays. It can also handle transfer functions with singularities in the right half-plane, unlike Bode plots. The
Nyquist stability criterion can also be used to find the phase and gain margins of a system, which are important for frequency domain controller design.
2849:
must be equal to the number of open-loop poles in the RHP. Any clockwise encirclements of the critical point by the open-loop frequency response (when judged from low frequency to high frequency) would indicate that the feedback control system would be destabilizing if the loop were closed. (Using
4828:
The
Nyquist stability criterion is a graphical technique that determines the stability of a dynamical system, such as a feedback control system. It is based on the argument principle and the Nyquist plot of the open-loop transfer function of the system. It can be applied to systems that are not
351:
When drawn by hand, a cartoon version of the
Nyquist plot is sometimes used, which shows the linearity of the curve, but where coordinates are distorted to show more detail in regions of interest. When plotted computationally, one needs to be careful to cover all frequencies of interest. This
4359:
4122:
4656:
253:
A Nyquist plot. Although the frequencies are not indicated on the curve, it can be inferred that the zero-frequency point is on the right, and the curve spirals toward the origin at high frequency. This is because gain at zero frequency must be purely real (on the
2812:
If the system is originally open-loop unstable, feedback is necessary to stabilize the system. Right-half-plane (RHP) poles represent that instability. For closed-loop stability of a system, the number of closed-loop roots in the right half of the
3519:
3766:
4610:
4468:
4183:
3955:
4761:{\displaystyle {\begin{aligned}Z={}&N+P\\={}&{\text{(number of times the Nyquist plot encircles }}{-1/k}{\text{ clockwise)}}\\&{}+{\text{(number of poles of }}G(s){\text{ in ORHP)}}\end{aligned}}}
4661:
722:
4933:
times clockwise along a semicircle of infinite radius. After applying this rule, the zero poles should be neglected, i.e. if there are no other unstable poles, then the open-loop transfer function
237:
Although
Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable. Techniques like
5440:
3268:
1518:
1076:
97:
501:
4175:
997:
964:
890:
857:
791:
644:
3333:
3087:
3046:
2716:
340:
The
Nyquist plot can provide some information about the shape of the transfer function. For instance, the plot provides information on the difference between the number of
3822:
1986:
3366:
2582:
2500:
2424:
1841:
1785:
1566:
1370:
1221:
1176:
3005:
2950:
2914:
2044:
2015:
1957:
1928:
1126:
555:
4911:
3947:
5134:
1899:
600:
2754:
1854:
in 1932 uses a less elegant approach. The approach explained here is similar to the approach used by Leroy MacColl (Fundamental theory of servomechanisms 1945) or by
1729:
3142:
2976:
2555:
2355:
2320:
2285:
2188:
2153:
2118:
2083:
928:
5107:
3652:
2847:
2805:
1647:
5061:
5024:
4992:
4960:
4865:
4798:
4648:
4497:
3900:
3851:
3620:
3571:
3395:
3116:
2885:
2680:
2651:
2473:
2384:
2250:
2217:
1814:
1758:
1595:
1460:
1428:
1399:
1303:
1270:
820:
754:
427:
398:
219:
4931:
4885:
4818:
3871:
3591:
3542:
3162:
2773:
2622:
2602:
2520:
2444:
1691:
1671:
1615:
1343:
1323:
1241:
2252:. By counting the resulting contour's encirclements of −1, we find the difference between the number of poles and zeros in the right-half complex plane of
4509:
3407:
3660:
5295:
4367:
5424:
130:
4354:{\displaystyle N=-{\frac {1}{2\pi i}}\oint _{u(\Gamma _{s})}{1 \over u}\,du=-{{1} \over {2\pi i}}\oint _{v(u(\Gamma _{s}))}{1 \over {v+1/k}}\,dv}
258:-axis) and is commonly non-zero, while most physical processes have some amount of low-pass filtering, so the high-frequency response is zero.
4117:{\displaystyle N=-{\frac {1}{2\pi i}}\oint _{\Gamma _{s}}{D'(s) \over D(s)}\,ds=-{\frac {1}{2\pi i}}\oint _{u(\Gamma _{s})}{1 \over u}\,du}
2920:
that the contour cannot pass through any pole of the mapping function. The most common case are systems with integrators (poles at zero).
333:
system is done by applying the
Nyquist stability criterion to the Nyquist plot of the open-loop system (i.e. the same system without its
2923:
To be able to analyze systems with poles on the imaginary axis, the
Nyquist Contour can be modified to avoid passing through the point
2120:
in the complex plane. By the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of
5335:
1847:
encirclements. That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative.
214:(LTI) systems. Nevertheless, there are generalizations of the Nyquist criterion (and plot) for non-linear systems, such as the
4619:. In fact, we find that the above integral corresponds precisely to the number of times the Nyquist plot encircles the point
3187:
Our goal is to, through this process, check for the stability of the transfer function of our unity feedback system with gain
5156:
656:
192:
5233:
561:, but this method is somewhat tedious. Conclusions can also be reached by examining the open loop transfer function (OLTF)
5113:
of the system becomes difficult and the only conclusion that can be drawn from the graph is that there exist zeros on the
306:-axis. The frequency is swept as a parameter, resulting in one point per frequency. The same plot can be described using
4994:
is stable, then the closed-loop system is unstable, if and only if, the
Nyquist plot encircle the point −1 at least once.
5257:"Inventing the 'black box': mathematics as a neglected enabling technology in the history of communications engineering"
5581:
5522:
5508:
5494:
5477:
3368:
enclosing the right half plane, with indentations as needed to avoid passing through zeros or poles of the function
3197:
1465:
1005:
5263:
191:
with right half-plane singularities. In addition, there is a natural generalization to more complex systems with
39:
439:
5181:
5561:
5326:
3273:
That is, we would like to check whether the characteristic equation of the above transfer function, given by
430:
364:, which transforms integrals and derivatives in the time domain to simple multiplication and division in the
4616:
4130:
2190:
in the right-half complex plane. If instead, the contour is mapped through the open-loop transfer function
17:
966:
is determined by the values of its poles: for stability, the real part of every pole must be negative. If
5421:
149:
5576:
969:
936:
862:
829:
763:
616:
352:
typically means that the parameter is swept logarithmically, in order to cover a wide range of values.
203:
5586:
176:
3279:
3051:
3010:
2685:
175:
of either the closed-loop or open-loop system (although the number of each type of right-half-plane
318:
of the transfer function is the corresponding angular coordinate. The
Nyquist plot is named after
3774:
1965:
3344:
2560:
2478:
2402:
1819:
1763:
1523:
1348:
1199:
1131:
5556:
2981:
2926:
2890:
2020:
1991:
1933:
1904:
1084:
513:
4890:
3908:
5217:
5116:
2858:
The above consideration was conducted with an assumption that the open-loop transfer function
1881:
1273:
564:
5196:
2721:
1696:
283:
211:
3121:
2955:
2525:
2325:
2290:
2255:
2158:
2123:
2088:
2053:
999:
is formed by closing a negative unity feedback loop around the open-loop transfer function,
898:
5540:
EIS Spectrum
Analyser - a freeware program for analysis and simulation of impedance spectra
5080:
3625:
2820:
2778:
1620:
168:
5037:
5000:
4968:
4936:
4841:
4774:
4622:
4473:
3876:
3827:
3596:
3547:
3371:
3092:
2861:
2656:
2627:
2449:
2360:
2226:
2193:
1790:
1734:
1571:
1436:
1404:
1375:
1279:
1246:
796:
730:
403:
374:
8:
5544:
5300:
5191:
1243:
plane, encompassing but not passing through any number of zeros and poles of a function
227:
153:
3172:
278:. The most common use of Nyquist plots is for assessing the stability of a system with
5404:
5378:
5347:
5343:
5110:
4916:
4870:
4803:
3856:
3576:
3527:
3398:
3147:
2917:
2758:
2607:
2587:
2505:
2429:
1676:
1656:
1650:
1600:
1328:
1308:
1226:
1187:
507:
311:
231:
223:
210:. While Nyquist is one of the most general stability tests, it is still restricted to
5534:
5518:
5504:
5490:
5473:
5408:
5396:
5351:
5287:
5229:
2853:
1859:
361:
345:
330:
323:
307:
291:
275:
188:
180:
135:
4605:{\displaystyle N=-{\frac {1}{2\pi i}}\oint _{G(\Gamma _{s}))}{\frac {1}{v+1/k}}\,dv}
3514:{\displaystyle -{\frac {1}{2\pi i}}\oint _{\Gamma _{s}}{D'(s) \over D(s)}\,ds=N=Z-P}
1858:(Network analysis and feedback amplifier design 1945), both of whom also worked for
5388:
5339:
5186:
1193:
215:
157:
118:
5074:
greater than 0) is exactly the number of unstable poles of the closed-loop system.
3902:
that appear within the contour, that is, within the open right half plane (ORHP).
3761:{\displaystyle Z=-{\frac {1}{2\pi i}}\oint _{\Gamma _{s}}{D'(s) \over D(s)}\,ds+P}
5428:
5225:
341:
271:
267:
172:
5366:
2817:-plane must be zero. Hence, the number of counter-clockwise encirclements about
5550:
5539:
5146:
4463:{\displaystyle v(u(\Gamma _{s}))={{D(\Gamma _{s})-1} \over {k}}=G(\Gamma _{s})}
610:
299:
287:
114:
5547:
for creating a Nyquist plot of a frequency response of a dynamic system model.
2604:
is the number of poles of the closed loop system in the right half plane, and
5570:
5482:
5400:
5392:
5317:
5256:
3905:
We will now rearrange the above integral via substitution. That is, setting
1851:
334:
319:
315:
145:
5176:
5171:
5166:
4500:
3338:
has zeros outside the open left-half-plane (commonly initialized as OLHP).
2220:
1855:
5161:
558:
199:
179:
must be known). As a result, it can be applied to systems defined by non-
4913:. During further analysis it should be assumed that the phasor travels
3341:
We suppose that we have a clockwise (i.e. negatively oriented) contour
647:
238:
5292:
Die elektrische Selbsterregung mit einer Theorie der aktiven Netzwerke
2322:
are the poles of the closed-loop system, and noting that the poles of
5151:
2887:
does not have any pole on the imaginary axis (i.e. poles of the form
606:
or, as here, its polar plot using the Nyquist criterion, as follows.
603:
184:
4800:
as defined above corresponds to a stable unity-feedback system when
1862:. This approach appears in most modern textbooks on control theory.
1081:
then the roots of the characteristic equation are also the zeros of
206:, as well as other fields, for designing and analyzing systems with
5383:
279:
207:
27:
Graphical method of determining the stability of a dynamical system
5030:, then for the closed-loop system to be stable, there must be one
2952:. One way to do it is to construct a semicircular arc with radius
2854:
The Nyquist criterion for systems with poles on the imaginary axis
1874:, a contour that encompasses the right-half of the complex plane:
32:
141:
5321:
249:
241:, while less general, are sometimes a more useful design tool.
1850:
Instead of Cauchy's argument principle, the original paper by
2155:
in the right-half complex plane minus the number of poles of
129:, independently discovered by the German electrical engineer
5365:
Chaffey, Thomas; Forni, Fulvio; Sepulchre, Rodolphe (2023).
5218:"Chapter 4.3. Das Stabilitätskriterium von Strecker-Nyquist"
5307:(NB. Earlier works can be found in the literature section.)
314:
of the transfer function is the radial coordinate, and the
2624:
is the number of poles of the open-loop transfer function
5077:
However, if the graph happens to pass through the point
5553:- free interactive virtual tool, control loop simulator
3164:
is the multiplicity of the pole on the imaginary axis.
717:{\displaystyle {\mathcal {T}}(s)={\frac {N(s)}{D(s)}}.}
348:
by the angle at which the curve approaches the origin.
2653:
in the right half plane, the resultant contour in the
400:; when placed in a closed loop with negative feedback
152:
in 1932, is a graphical technique for determining the
5364:
5119:
5083:
5040:
5003:
4971:
4939:
4919:
4893:
4873:
4844:
4806:
4777:
4659:
4625:
4512:
4476:
4370:
4186:
4133:
3958:
3911:
3879:
3859:
3830:
3777:
3663:
3628:
3599:
3579:
3550:
3530:
3410:
3374:
3347:
3282:
3200:
3150:
3124:
3095:
3054:
3013:
2984:
2958:
2929:
2893:
2864:
2823:
2781:
2761:
2724:
2688:
2659:
2630:
2610:
2590:
2563:
2528:
2508:
2481:
2452:
2432:
2405:
2363:
2328:
2293:
2258:
2229:
2196:
2161:
2126:
2091:
2056:
2023:
1994:
1968:
1936:
1907:
1884:
1843:
and that encirclements in the opposite direction are
1822:
1793:
1766:
1737:
1699:
1679:
1659:
1623:
1603:
1574:
1526:
1468:
1439:
1407:
1378:
1351:
1331:
1311:
1282:
1249:
1229:
1202:
1134:
1087:
1008:
972:
939:
901:
865:
832:
799:
766:
733:
659:
619:
567:
516:
442:
406:
377:
171:, it can be applied without explicitly computing the
144:
in 1930 and the Swedish-American electrical engineer
42:
5557:Mathematica function for creating the Nyquist plot
5128:
5101:
5055:
5018:
4986:
4954:
4925:
4905:
4879:
4859:
4812:
4792:
4760:
4642:
4604:
4491:
4462:
4353:
4169:
4116:
3941:
3894:
3865:
3845:
3816:
3760:
3646:
3614:
3585:
3565:
3536:
3513:
3389:
3360:
3327:
3262:
3156:
3136:
3110:
3081:
3040:
2999:
2970:
2944:
2908:
2879:
2841:
2799:
2767:
2748:
2710:
2674:
2645:
2616:
2596:
2576:
2549:
2514:
2494:
2467:
2438:
2418:
2378:
2349:
2314:
2279:
2244:
2211:
2182:
2147:
2112:
2077:
2038:
2009:
1980:
1951:
1922:
1893:
1835:
1808:
1779:
1752:
1723:
1685:
1665:
1641:
1609:
1589:
1560:
1512:
1454:
1422:
1393:
1364:
1337:
1317:
1297:
1264:
1235:
1215:
1170:
1120:
1070:
991:
958:
922:
884:
851:
814:
785:
748:
716:
638:
594:
549:
495:
421:
392:
198:The Nyquist stability criterion is widely used in
91:
4696:(number of times the Nyquist plot encircles
5568:
2050:The Nyquist contour mapped through the function
371:We consider a system whose transfer function is
5034:-clockwise encirclement of −1 for each pole of
4887:, then the Nyquist plot has a discontinuity at
3089:. Such a modification implies that the phasor
1181:
183:, such as systems with delays. In contrast to
5209:
4127:We then make a further substitution, setting
3263:{\displaystyle T(s)={\frac {kG(s)}{1+kG(s)}}}
1513:{\displaystyle \Gamma _{F(s)}=F(\Gamma _{s})}
1071:{\displaystyle G(s)H(s)={\frac {A(s)}{B(s)}}}
506:Stability can be determined by examining the
329:Assessment of the stability of a closed-loop
5470:Introduction to the Theory of Linear Systems
5280:
4650:clockwise. Thus, we may finally state that
2916:). This results from the requirement of the
234:can also be applied for non-linear systems.
5562:The Nyquist Diagram for Electrical Circuits
5310:
3622:by the same contour. Rearranging, we have
3118:travels along an arc of infinite radius by
2584:. Alternatively, and more importantly, if
92:{\displaystyle G(s)={\frac {1}{s^{2}+s+1}}}
5248:
4820:, as evaluated above, is equal to 0.
3167:
1787:. Note that we count encirclements in the
1693:are, respectively, the number of zeros of
496:{\displaystyle {\frac {G(s)}{1+G(s)H(s)}}}
5382:
5215:
4595:
4344:
4250:
4107:
4040:
3745:
3486:
195:, such as control systems for airplanes.
5336:American Telephone and Telegraph Company
5286:
5222:Lineare Regelungs- und Steuerungstheorie
4470:gives us the image of our contour under
3171:
248:
31:
5441:"12.2: Nyquist Criterion for Stability"
5415:
5316:
5254:
5063:in the right-half of the complex plane.
1816:plane in the same sense as the contour
510:of the desensitivity factor polynomial
14:
5569:
5371:IEEE Transactions on Automatic Control
4503:. We may further reduce the integral
5503:; Silesian University of Technology;
5367:"Graphical Nonlinear System Analysis"
5066:The number of surplus encirclements (
4170:{\displaystyle v(u)={\frac {u-1}{k}}}
2718:shall encircle (clockwise) the point
892:are also said to be the roots of the
646:can be expressed as the ratio of two
193:multiple inputs and multiple outputs
127:Strecker–Nyquist stability criterion
5515:Feedback Control of Dynamic Systems
4997:If the open-loop transfer function
4965:If the open-loop transfer function
4838:If the open-loop transfer function
24:
5535:Applets with modifiable parameters
5462:
5344:10.1002/j.1538-7305.1932.tb02344.x
4552:
4448:
4412:
4384:
4301:
4226:
4083:
3992:
3697:
3438:
3349:
2690:
2565:
2483:
2407:
2033:
2004:
1975:
1946:
1917:
1824:
1768:
1498:
1470:
1353:
1204:
975:
942:
868:
835:
769:
662:
622:
25:
5598:
5528:
5294:(in German). Stuttgart, Germany:
5157:Routh–Hurwitz stability criterion
3176:A unity negative feedback system
992:{\displaystyle {\mathcal {T}}(s)}
959:{\displaystyle {\mathcal {T}}(s)}
885:{\displaystyle {\mathcal {T}}(s)}
852:{\displaystyle {\mathcal {T}}(s)}
786:{\displaystyle {\mathcal {T}}(s)}
639:{\displaystyle {\mathcal {T}}(s)}
5255:Bissell, Christopher C. (2001).
4867:has a zero pole of multiplicity
1962:a semicircular arc, with radius
1325:. Precisely, each complex point
270:of a frequency response used in
5269:from the original on 2019-06-14
3593:denotes the number of poles of
3544:denotes the number of zeros of
1430:plane yielding a new contour.
244:
5489:; Cambridge University Press;
5433:
5358:
5182:Barkhausen stability criterion
5109:, then deciding upon even the
5050:
5044:
5013:
5007:
4981:
4975:
4949:
4943:
4854:
4848:
4787:
4781:
4746:
4740:
4564:
4561:
4548:
4486:
4480:
4457:
4444:
4421:
4408:
4396:
4393:
4380:
4374:
4313:
4310:
4297:
4291:
4235:
4222:
4143:
4137:
4092:
4079:
4034:
4028:
4020:
4014:
3936:
3930:
3921:
3915:
3889:
3883:
3840:
3834:
3824:has exactly the same poles as
3811:
3805:
3787:
3781:
3739:
3733:
3725:
3719:
3609:
3603:
3560:
3554:
3480:
3474:
3466:
3460:
3384:
3378:
3328:{\displaystyle D(s)=1+kG(s)=0}
3316:
3310:
3292:
3286:
3254:
3248:
3231:
3225:
3210:
3204:
3105:
3099:
3082:{\displaystyle 0+j(\omega +r)}
3076:
3064:
3041:{\displaystyle 0+j(\omega -r)}
3035:
3023:
2962:
2874:
2868:
2743:
2725:
2711:{\displaystyle \Gamma _{G(s)}}
2703:
2697:
2669:
2663:
2640:
2634:
2544:
2538:
2462:
2456:
2373:
2367:
2344:
2338:
2309:
2303:
2287:. Recalling that the zeros of
2274:
2268:
2239:
2233:
2206:
2200:
2177:
2171:
2142:
2136:
2107:
2101:
2072:
2066:
1972:
1803:
1797:
1747:
1741:
1718:
1712:
1584:
1578:
1507:
1494:
1483:
1477:
1449:
1443:
1417:
1411:
1388:
1382:
1292:
1286:
1259:
1253:
1159:
1153:
1144:
1138:
1115:
1109:
1103:
1097:
1062:
1056:
1048:
1042:
1030:
1024:
1018:
1012:
986:
980:
953:
947:
911:
905:
879:
873:
846:
840:
809:
803:
780:
774:
743:
737:
705:
699:
691:
685:
673:
667:
633:
627:
589:
583:
577:
571:
544:
538:
532:
526:
487:
481:
475:
469:
455:
449:
416:
410:
387:
381:
52:
46:
13:
1:
5327:Bell System Technical Journal
5202:
4823:
3048:and travels anticlockwise to
1865:
431:closed loop transfer function
355:
163:Because it only looks at the
4962:should be considered stable.
3817:{\displaystyle D(s)=1+kG(s)}
3573:enclosed by the contour and
3180:with scalar gain denoted by
1981:{\displaystyle r\to \infty }
7:
5140:
3361:{\displaystyle \Gamma _{s}}
2577:{\displaystyle \Gamma _{s}}
2495:{\displaystyle \Gamma _{s}}
2419:{\displaystyle \Gamma _{s}}
1836:{\displaystyle \Gamma _{s}}
1780:{\displaystyle \Gamma _{s}}
1651:Cauchy's argument principle
1561:{\displaystyle s={-1/k+j0}}
1365:{\displaystyle \Gamma _{s}}
1216:{\displaystyle \Gamma _{s}}
1182:Cauchy's argument principle
1171:{\displaystyle A(s)+B(s)=0}
150:Bell Telephone Laboratories
123:Nyquist stability criterion
10:
5603:
5224:(in German) (2 ed.).
4832:
3000:{\displaystyle 0+j\omega }
2945:{\displaystyle 0+j\omega }
2909:{\displaystyle 0+j\omega }
2522:be the number of zeros of
2446:be the number of poles of
2039:{\displaystyle 0-j\infty }
2017:and travels clock-wise to
2010:{\displaystyle 0+j\infty }
1952:{\displaystyle 0+j\infty }
1923:{\displaystyle 0-j\infty }
1185:
1121:{\displaystyle 1+G(s)H(s)}
550:{\displaystyle 1+G(s)H(s)}
204:control system engineering
4906:{\displaystyle \omega =0}
4734:(number of poles of
4617:Cauchy's integral formula
3942:{\displaystyle u(s)=D(s)}
3873:by counting the poles of
2357:are same as the poles of
1128:, or simply the roots of
360:The mathematics uses the
5582:Classical control theory
5551:PID Nyquist plot shaping
5487:Response & Stability
5468:Faulkner, E. A. (1969):
5393:10.1109/TAC.2023.3234016
5216:Reinschke, Kurt (2014).
5129:{\displaystyle j\omega }
2399:Given a Nyquist contour
1894:{\displaystyle j\omega }
1878:a path traveling up the
1520:will encircle the point
1372:is mapped to the point
1276:to another plane (named
595:{\displaystyle G(s)H(s)}
3168:Mathematical derivation
2749:{\displaystyle (-1+j0)}
1724:{\displaystyle 1+kF(s)}
1462:, which is the contour
1305:plane) by the function
894:characteristic equation
322:, a former engineer at
5472:; Chapman & Hall;
5445:Mathematics LibreTexts
5130:
5103:
5057:
5020:
4988:
4956:
4927:
4907:
4881:
4861:
4814:
4794:
4762:
4644:
4606:
4499:, which is to say our
4493:
4464:
4355:
4171:
4118:
3943:
3896:
3867:
3847:
3818:
3762:
3648:
3616:
3587:
3567:
3538:
3515:
3391:
3362:
3329:
3264:
3184:
3158:
3138:
3137:{\displaystyle -l\pi }
3112:
3083:
3042:
3001:
2972:
2971:{\displaystyle r\to 0}
2946:
2910:
2881:
2843:
2810:
2801:
2769:
2750:
2712:
2676:
2647:
2618:
2598:
2578:
2551:
2550:{\displaystyle 1+G(s)}
2516:
2496:
2469:
2440:
2420:
2380:
2351:
2350:{\displaystyle 1+G(s)}
2316:
2315:{\displaystyle 1+G(s)}
2281:
2280:{\displaystyle 1+G(s)}
2246:
2213:
2184:
2183:{\displaystyle 1+G(s)}
2149:
2148:{\displaystyle 1+G(s)}
2114:
2113:{\displaystyle 1+G(s)}
2079:
2078:{\displaystyle 1+G(s)}
2040:
2011:
1982:
1953:
1924:
1895:
1837:
1810:
1781:
1754:
1725:
1687:
1667:
1643:
1611:
1591:
1562:
1514:
1456:
1424:
1395:
1366:
1339:
1319:
1299:
1266:
1237:
1217:
1172:
1122:
1072:
993:
960:
924:
923:{\displaystyle D(s)=0}
886:
853:
816:
787:
750:
718:
640:
596:
551:
497:
423:
394:
259:
226:. Additionally, other
110:
93:
5513:Franklin, G. (2002):
5338:(AT&T): 126–147.
5322:"Regeneration Theory"
5197:Hankel singular value
5131:
5104:
5102:{\displaystyle -1+j0}
5058:
5021:
4989:
4957:
4928:
4908:
4882:
4862:
4815:
4795:
4763:
4645:
4607:
4494:
4465:
4356:
4172:
4119:
3944:
3897:
3868:
3853:. Thus, we may find
3848:
3819:
3763:
3649:
3647:{\displaystyle Z=N+P}
3617:
3588:
3568:
3539:
3516:
3392:
3363:
3330:
3265:
3175:
3159:
3139:
3113:
3084:
3043:
3002:
2973:
2947:
2911:
2882:
2844:
2842:{\displaystyle -1+j0}
2802:
2800:{\displaystyle N=Z-P}
2770:
2751:
2713:
2677:
2648:
2619:
2599:
2579:
2552:
2517:
2497:
2470:
2441:
2421:
2396:
2381:
2352:
2317:
2282:
2247:
2214:
2185:
2150:
2115:
2080:
2041:
2012:
1983:
1954:
1925:
1896:
1838:
1811:
1782:
1755:
1726:
1688:
1668:
1644:
1642:{\displaystyle N=P-Z}
1612:
1592:
1563:
1515:
1457:
1425:
1396:
1367:
1340:
1320:
1300:
1267:
1238:
1223:drawn in the complex
1218:
1173:
1123:
1073:
994:
961:
925:
887:
854:
817:
788:
751:
719:
641:
597:
552:
498:
433:(CLTF) then becomes:
424:
395:
284:Cartesian coordinates
252:
220:scaled relative graph
212:linear time-invariant
94:
36:The Nyquist plot for
35:
5501:Control fundamentals
5499:Gessing, R. (2004):
5117:
5081:
5056:{\displaystyle G(s)}
5038:
5019:{\displaystyle G(s)}
5001:
4987:{\displaystyle G(s)}
4969:
4955:{\displaystyle G(s)}
4937:
4917:
4891:
4871:
4860:{\displaystyle G(s)}
4842:
4804:
4793:{\displaystyle T(s)}
4775:
4657:
4643:{\displaystyle -1/k}
4623:
4510:
4492:{\displaystyle G(s)}
4474:
4368:
4184:
4131:
3956:
3909:
3895:{\displaystyle G(s)}
3877:
3857:
3846:{\displaystyle G(s)}
3828:
3775:
3661:
3626:
3615:{\displaystyle D(s)}
3597:
3577:
3566:{\displaystyle D(s)}
3548:
3528:
3408:
3390:{\displaystyle G(s)}
3372:
3345:
3280:
3198:
3191:, which is given by
3148:
3122:
3111:{\displaystyle G(s)}
3093:
3052:
3011:
2982:
2956:
2927:
2891:
2880:{\displaystyle G(s)}
2862:
2821:
2779:
2759:
2722:
2686:
2675:{\displaystyle G(s)}
2657:
2646:{\displaystyle G(s)}
2628:
2608:
2588:
2561:
2526:
2506:
2479:
2468:{\displaystyle G(s)}
2450:
2430:
2403:
2379:{\displaystyle G(s)}
2361:
2326:
2291:
2256:
2245:{\displaystyle G(s)}
2227:
2219:, the result is the
2212:{\displaystyle G(s)}
2194:
2159:
2124:
2089:
2054:
2021:
1992:
1966:
1934:
1905:
1882:
1820:
1809:{\displaystyle F(s)}
1791:
1764:
1753:{\displaystyle F(s)}
1735:
1697:
1677:
1657:
1621:
1601:
1590:{\displaystyle F(s)}
1572:
1524:
1466:
1455:{\displaystyle F(s)}
1437:
1433:The Nyquist plot of
1423:{\displaystyle F(s)}
1405:
1394:{\displaystyle F(s)}
1376:
1349:
1329:
1309:
1298:{\displaystyle F(s)}
1280:
1265:{\displaystyle F(s)}
1247:
1227:
1200:
1132:
1085:
1006:
970:
937:
899:
863:
830:
815:{\displaystyle D(s)}
797:
764:
749:{\displaystyle N(s)}
731:
657:
617:
565:
514:
440:
422:{\displaystyle H(s)}
404:
393:{\displaystyle G(s)}
375:
40:
5192:Control engineering
2386:, we now state the
1872:the Nyquist contour
1870:We first construct
1760:inside the contour
793:, and the roots of
5427:2008-09-30 at the
5126:
5111:marginal stability
5099:
5053:
5016:
4984:
4952:
4923:
4903:
4877:
4857:
4810:
4790:
4771:We thus find that
4758:
4756:
4640:
4602:
4489:
4460:
4351:
4167:
4114:
3939:
3892:
3863:
3843:
3814:
3771:We then note that
3758:
3654:, which is to say
3644:
3612:
3583:
3563:
3534:
3511:
3399:argument principle
3387:
3358:
3325:
3260:
3185:
3154:
3134:
3108:
3079:
3038:
2997:
2968:
2942:
2918:argument principle
2906:
2877:
2839:
2797:
2765:
2746:
2708:
2672:
2643:
2614:
2594:
2574:
2547:
2512:
2492:
2465:
2436:
2416:
2376:
2347:
2312:
2277:
2242:
2209:
2180:
2145:
2110:
2075:
2036:
2007:
1978:
1949:
1920:
1891:
1833:
1806:
1777:
1750:
1721:
1683:
1663:
1639:
1607:
1587:
1558:
1510:
1452:
1420:
1391:
1362:
1335:
1315:
1295:
1262:
1233:
1213:
1188:Argument principle
1168:
1118:
1068:
989:
956:
920:
882:
849:
812:
783:
746:
714:
636:
613:transfer function
592:
547:
493:
419:
390:
302:is plotted on the
294:is plotted on the
260:
228:stability criteria
224:nonlinear operator
189:transfer functions
181:rational functions
111:
89:
5577:Signal processing
5517:; Prentice Hall,
5377:(10): 6067–6081.
4926:{\displaystyle l}
4880:{\displaystyle l}
4813:{\displaystyle Z}
4752:
4735:
4718:
4697:
4593:
4538:
4436:
4364:We now note that
4342:
4281:
4248:
4212:
4177:. This gives us
4165:
4105:
4069:
4038:
3984:
3866:{\displaystyle P}
3743:
3689:
3586:{\displaystyle P}
3537:{\displaystyle Z}
3484:
3430:
3258:
3157:{\displaystyle l}
3007:, that starts at
2768:{\displaystyle N}
2617:{\displaystyle P}
2597:{\displaystyle Z}
2515:{\displaystyle Z}
2439:{\displaystyle P}
2390:Nyquist Criterion
2085:yields a plot of
1988:, that starts at
1860:Bell Laboratories
1686:{\displaystyle P}
1666:{\displaystyle Z}
1610:{\displaystyle N}
1338:{\displaystyle s}
1318:{\displaystyle F}
1236:{\displaystyle s}
1066:
933:The stability of
709:
557:, e.g. using the
491:
362:Laplace transform
346:transfer function
331:negative feedback
324:Bell Laboratories
308:polar coordinates
292:transfer function
276:signal processing
272:automatic control
169:open loop systems
87:
16:(Redirected from
5594:
5587:Stability theory
5456:
5455:
5453:
5452:
5437:
5431:
5419:
5413:
5412:
5386:
5362:
5356:
5355:
5320:(January 1932).
5314:
5308:
5306:
5304:
5296:S. Hirzel Verlag
5284:
5278:
5277:
5275:
5274:
5268:
5261:
5252:
5246:
5245:
5243:
5242:
5235:978-3-64240960-8
5213:
5187:Circle criterion
5135:
5133:
5132:
5127:
5108:
5106:
5105:
5100:
5062:
5060:
5059:
5054:
5025:
5023:
5022:
5017:
4993:
4991:
4990:
4985:
4961:
4959:
4958:
4953:
4932:
4930:
4929:
4924:
4912:
4910:
4909:
4904:
4886:
4884:
4883:
4878:
4866:
4864:
4863:
4858:
4819:
4817:
4816:
4811:
4799:
4797:
4796:
4791:
4767:
4765:
4764:
4759:
4757:
4753:
4750:
4736:
4733:
4728:
4723:
4719:
4717: clockwise)
4716:
4714:
4710:
4698:
4695:
4691:
4671:
4649:
4647:
4646:
4641:
4636:
4611:
4609:
4608:
4603:
4594:
4592:
4588:
4570:
4568:
4567:
4560:
4559:
4539:
4537:
4523:
4498:
4496:
4495:
4490:
4469:
4467:
4466:
4461:
4456:
4455:
4437:
4435:
4430:
4420:
4419:
4403:
4392:
4391:
4360:
4358:
4357:
4352:
4343:
4341:
4337:
4319:
4317:
4316:
4309:
4308:
4282:
4280:
4269:
4264:
4249:
4241:
4239:
4238:
4234:
4233:
4213:
4211:
4197:
4176:
4174:
4173:
4168:
4166:
4161:
4150:
4123:
4121:
4120:
4115:
4106:
4098:
4096:
4095:
4091:
4090:
4070:
4068:
4054:
4039:
4037:
4023:
4013:
4004:
4002:
4001:
4000:
3999:
3985:
3983:
3969:
3948:
3946:
3945:
3940:
3901:
3899:
3898:
3893:
3872:
3870:
3869:
3864:
3852:
3850:
3849:
3844:
3823:
3821:
3820:
3815:
3767:
3765:
3764:
3759:
3744:
3742:
3728:
3718:
3709:
3707:
3706:
3705:
3704:
3690:
3688:
3674:
3653:
3651:
3650:
3645:
3621:
3619:
3618:
3613:
3592:
3590:
3589:
3584:
3572:
3570:
3569:
3564:
3543:
3541:
3540:
3535:
3520:
3518:
3517:
3512:
3485:
3483:
3469:
3459:
3450:
3448:
3447:
3446:
3445:
3431:
3429:
3415:
3396:
3394:
3393:
3388:
3367:
3365:
3364:
3359:
3357:
3356:
3334:
3332:
3331:
3326:
3269:
3267:
3266:
3261:
3259:
3257:
3234:
3217:
3163:
3161:
3160:
3155:
3143:
3141:
3140:
3135:
3117:
3115:
3114:
3109:
3088:
3086:
3085:
3080:
3047:
3045:
3044:
3039:
3006:
3004:
3003:
2998:
2977:
2975:
2974:
2969:
2951:
2949:
2948:
2943:
2915:
2913:
2912:
2907:
2886:
2884:
2883:
2878:
2848:
2846:
2845:
2840:
2806:
2804:
2803:
2798:
2775:times such that
2774:
2772:
2771:
2766:
2755:
2753:
2752:
2747:
2717:
2715:
2714:
2709:
2707:
2706:
2681:
2679:
2678:
2673:
2652:
2650:
2649:
2644:
2623:
2621:
2620:
2615:
2603:
2601:
2600:
2595:
2583:
2581:
2580:
2575:
2573:
2572:
2556:
2554:
2553:
2548:
2521:
2519:
2518:
2513:
2501:
2499:
2498:
2493:
2491:
2490:
2474:
2472:
2471:
2466:
2445:
2443:
2442:
2437:
2425:
2423:
2422:
2417:
2415:
2414:
2385:
2383:
2382:
2377:
2356:
2354:
2353:
2348:
2321:
2319:
2318:
2313:
2286:
2284:
2283:
2278:
2251:
2249:
2248:
2243:
2218:
2216:
2215:
2210:
2189:
2187:
2186:
2181:
2154:
2152:
2151:
2146:
2119:
2117:
2116:
2111:
2084:
2082:
2081:
2076:
2045:
2043:
2042:
2037:
2016:
2014:
2013:
2008:
1987:
1985:
1984:
1979:
1958:
1956:
1955:
1950:
1929:
1927:
1926:
1921:
1900:
1898:
1897:
1892:
1842:
1840:
1839:
1834:
1832:
1831:
1815:
1813:
1812:
1807:
1786:
1784:
1783:
1778:
1776:
1775:
1759:
1757:
1756:
1751:
1730:
1728:
1727:
1722:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1648:
1646:
1645:
1640:
1616:
1614:
1613:
1608:
1596:
1594:
1593:
1588:
1567:
1565:
1564:
1559:
1557:
1544:
1519:
1517:
1516:
1511:
1506:
1505:
1487:
1486:
1461:
1459:
1458:
1453:
1429:
1427:
1426:
1421:
1400:
1398:
1397:
1392:
1371:
1369:
1368:
1363:
1361:
1360:
1344:
1342:
1341:
1336:
1324:
1322:
1321:
1316:
1304:
1302:
1301:
1296:
1271:
1269:
1268:
1263:
1242:
1240:
1239:
1234:
1222:
1220:
1219:
1214:
1212:
1211:
1194:complex analysis
1177:
1175:
1174:
1169:
1127:
1125:
1124:
1119:
1077:
1075:
1074:
1069:
1067:
1065:
1051:
1037:
998:
996:
995:
990:
979:
978:
965:
963:
962:
957:
946:
945:
929:
927:
926:
921:
891:
889:
888:
883:
872:
871:
858:
856:
855:
850:
839:
838:
821:
819:
818:
813:
792:
790:
789:
784:
773:
772:
755:
753:
752:
747:
723:
721:
720:
715:
710:
708:
694:
680:
666:
665:
645:
643:
642:
637:
626:
625:
601:
599:
598:
593:
556:
554:
553:
548:
502:
500:
499:
494:
492:
490:
458:
444:
428:
426:
425:
420:
399:
397:
396:
391:
298:-axis while the
232:Lyapunov methods
216:circle criterion
187:, it can handle
158:dynamical system
139:
119:stability theory
108:
98:
96:
95:
90:
88:
86:
73:
72:
59:
21:
5602:
5601:
5597:
5596:
5595:
5593:
5592:
5591:
5567:
5566:
5545:MATLAB function
5531:
5465:
5463:Further reading
5460:
5459:
5450:
5448:
5439:
5438:
5434:
5429:Wayback Machine
5420:
5416:
5363:
5359:
5315:
5311:
5298:
5288:Strecker, Felix
5285:
5281:
5272:
5270:
5266:
5259:
5253:
5249:
5240:
5238:
5236:
5228:. p. 184.
5226:Springer-Verlag
5214:
5210:
5205:
5143:
5118:
5115:
5114:
5082:
5079:
5078:
5039:
5036:
5035:
5002:
4999:
4998:
4970:
4967:
4966:
4938:
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2018:
1993:
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1989:
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1406:
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1347:
1346:
1345:in the contour
1330:
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1281:
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1184:
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1038:
1036:
1007:
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1003:
974:
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967:
941:
940:
938:
935:
934:
900:
897:
896:
867:
866:
864:
861:
860:
859:. The poles of
834:
833:
831:
828:
827:
798:
795:
794:
768:
767:
765:
762:
761:
756:are called the
732:
729:
728:
695:
681:
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661:
660:
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654:
621:
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618:
615:
614:
566:
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512:
511:
459:
445:
443:
441:
438:
437:
405:
402:
401:
376:
373:
372:
358:
342:zeros and poles
268:parametric plot
247:
173:poles and zeros
133:
100:
68:
64:
63:
58:
41:
38:
37:
28:
23:
22:
15:
12:
11:
5:
5600:
5590:
5589:
5584:
5579:
5565:
5564:
5559:
5554:
5548:
5542:
5537:
5530:
5529:External links
5527:
5526:
5525:
5511:
5497:
5483:Pippard, A. B.
5480:
5464:
5461:
5458:
5457:
5432:
5414:
5357:
5318:Nyquist, Harry
5309:
5279:
5247:
5234:
5207:
5206:
5204:
5201:
5200:
5199:
5194:
5189:
5184:
5179:
5174:
5169:
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5159:
5154:
5149:
5147:BIBO stability
5142:
5139:
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5137:
5125:
5122:
5098:
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5092:
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5075:
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4809:
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4786:
4783:
4780:
4769:
4768:
4751: in ORHP)
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2100:
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2074:
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2068:
2065:
2062:
2059:
2048:
2047:
2035:
2032:
2029:
2026:
2006:
2003:
2000:
1997:
1977:
1974:
1971:
1960:
1948:
1945:
1942:
1939:
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1916:
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1314:
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1261:
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1232:
1210:
1206:
1186:Main article:
1183:
1180:
1167:
1164:
1161:
1158:
1155:
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1149:
1146:
1143:
1140:
1137:
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1017:
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988:
985:
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977:
955:
952:
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907:
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842:
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811:
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771:
745:
742:
739:
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724:
713:
707:
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701:
698:
693:
690:
687:
684:
678:
675:
672:
669:
664:
635:
632:
629:
624:
611:Laplace domain
591:
588:
585:
582:
579:
576:
573:
570:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
504:
503:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
457:
454:
451:
448:
418:
415:
412:
409:
389:
386:
383:
380:
357:
354:
300:imaginary part
246:
243:
131:Felix Strecker
115:control theory
85:
82:
79:
76:
71:
67:
62:
57:
54:
51:
48:
45:
26:
9:
6:
4:
3:
2:
5599:
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5555:
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5543:
5541:
5538:
5536:
5533:
5532:
5524:
5523:0-13-032393-4
5520:
5516:
5512:
5510:
5509:83-7335-176-0
5506:
5502:
5498:
5496:
5495:0-521-31994-3
5492:
5488:
5484:
5481:
5479:
5478:0-412-09400-2
5475:
5471:
5467:
5466:
5446:
5442:
5436:
5430:
5426:
5423:
5422:Nyquist Plots
5418:
5410:
5406:
5402:
5398:
5394:
5390:
5385:
5380:
5376:
5372:
5368:
5361:
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5349:
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5341:
5337:
5333:
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5208:
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5123:
5120:
5112:
5096:
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5069:
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3692:
3685:
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3641:
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3456:
3453:
3442:
3433:
3426:
3423:
3420:
3416:
3411:
3404:
3403:
3402:
3401:states that
3400:
3381:
3375:
3353:
3339:
3322:
3319:
3313:
3307:
3304:
3301:
3298:
3295:
3289:
3283:
3276:
3275:
3274:
3251:
3245:
3242:
3239:
3236:
3228:
3222:
3219:
3213:
3207:
3201:
3194:
3193:
3192:
3190:
3183:
3179:
3174:
3165:
3151:
3131:
3128:
3125:
3102:
3096:
3073:
3070:
3067:
3061:
3058:
3055:
3032:
3029:
3026:
3020:
3017:
3014:
2994:
2991:
2988:
2985:
2965:
2959:
2939:
2936:
2933:
2930:
2921:
2919:
2903:
2900:
2897:
2894:
2871:
2865:
2851:
2836:
2833:
2830:
2827:
2824:
2816:
2809:
2808:
2794:
2791:
2788:
2785:
2782:
2762:
2740:
2737:
2734:
2731:
2728:
2700:
2694:
2666:
2660:
2637:
2631:
2611:
2591:
2569:
2557:encircled by
2541:
2535:
2532:
2529:
2509:
2487:
2475:encircled by
2459:
2453:
2433:
2411:
2395:
2394:
2392:
2391:
2370:
2364:
2341:
2335:
2332:
2329:
2306:
2300:
2297:
2294:
2271:
2265:
2262:
2259:
2236:
2230:
2222:
2203:
2197:
2174:
2168:
2165:
2162:
2139:
2133:
2130:
2127:
2104:
2098:
2095:
2092:
2069:
2063:
2060:
2057:
2030:
2027:
2024:
2001:
1998:
1995:
1969:
1961:
1943:
1940:
1937:
1914:
1911:
1908:
1888:
1885:
1877:
1876:
1875:
1873:
1863:
1861:
1857:
1853:
1852:Harry Nyquist
1848:
1846:
1828:
1800:
1794:
1772:
1744:
1738:
1731:and poles of
1715:
1709:
1706:
1703:
1700:
1680:
1660:
1652:
1636:
1633:
1630:
1627:
1624:
1617:times, where
1604:
1581:
1575:
1554:
1551:
1548:
1545:
1541:
1537:
1534:
1530:
1527:
1502:
1491:
1488:
1480:
1474:
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1440:
1431:
1414:
1408:
1385:
1379:
1357:
1332:
1312:
1289:
1283:
1275:
1256:
1250:
1230:
1208:
1196:, a contour
1195:
1189:
1179:
1165:
1162:
1156:
1150:
1147:
1141:
1135:
1112:
1106:
1100:
1094:
1091:
1088:
1059:
1053:
1045:
1039:
1033:
1027:
1021:
1015:
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1002:
1001:
1000:
983:
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917:
914:
908:
902:
895:
876:
843:
825:
806:
800:
777:
759:
740:
734:
727:The roots of
711:
702:
696:
688:
682:
676:
670:
653:
652:
651:
649:
630:
612:
607:
605:
586:
580:
574:
568:
560:
541:
535:
529:
523:
520:
517:
509:
484:
478:
472:
466:
463:
460:
452:
446:
436:
435:
434:
432:
413:
407:
384:
378:
369:
367:
363:
353:
349:
347:
343:
338:
336:
335:feedback loop
332:
327:
325:
321:
320:Harry Nyquist
317:
313:
309:
305:
301:
297:
293:
289:
285:
281:
277:
273:
269:
265:
257:
251:
242:
240:
235:
233:
229:
225:
221:
217:
213:
209:
205:
201:
196:
194:
190:
186:
182:
178:
177:singularities
174:
170:
166:
161:
159:
155:
151:
147:
146:Harry Nyquist
143:
137:
132:
128:
124:
120:
116:
107:
103:
83:
80:
77:
74:
69:
65:
60:
55:
49:
43:
34:
30:
19:
5514:
5500:
5486:
5469:
5449:. Retrieved
5447:. 2017-09-05
5444:
5435:
5417:
5374:
5370:
5360:
5331:
5325:
5312:
5291:
5282:
5271:. Retrieved
5250:
5239:. Retrieved
5221:
5211:
5177:Phase margin
5172:Hall circles
5167:Nichols plot
5071:
5067:
5031:
5027:
4827:
4770:
4615:by applying
4614:
4501:Nyquist plot
4363:
4126:
3904:
3770:
3523:
3397:. Cauchy's
3340:
3337:
3272:
3188:
3186:
3181:
3177:
2922:
2857:
2814:
2811:
2398:
2397:
2389:
2388:
2387:
2221:Nyquist Plot
2049:
1871:
1869:
1856:Hendrik Bode
1849:
1844:
1432:
1191:
1080:
932:
893:
823:
757:
726:
608:
602:, using its
505:
370:
365:
359:
350:
339:
328:
303:
295:
264:Nyquist plot
263:
261:
255:
245:Nyquist plot
236:
197:
165:Nyquist plot
164:
162:
126:
122:
112:
105:
101:
29:
18:Nyquist plot
5299: [
5162:Gain margin
1901:axis, from
1401:in the new
648:polynomials
559:Routh array
200:electronics
134: [
5571:Categories
5451:2023-12-25
5384:2107.11272
5334:(1). USA:
5273:2019-06-14
5241:2019-06-14
5203:References
4824:Importance
3949:, we have
1866:Definition
604:Bode plots
356:Background
239:Bode plots
185:Bode plots
5409:236318576
5401:0018-9286
5352:115002788
5152:Bode plot
5124:ω
5085:−
4895:ω
4701:−
4627:−
4553:Γ
4542:∮
4532:π
4520:−
4449:Γ
4425:−
4413:Γ
4385:Γ
4302:Γ
4285:∮
4275:π
4261:−
4227:Γ
4216:∮
4206:π
4194:−
4156:−
4084:Γ
4073:∮
4063:π
4051:−
3993:Γ
3988:∮
3978:π
3966:−
3698:Γ
3693:∮
3683:π
3671:−
3506:−
3439:Γ
3434:∮
3424:π
3412:−
3350:Γ
3132:π
3126:−
3068:ω
3030:−
3027:ω
2995:ω
2963:→
2940:ω
2904:ω
2825:−
2792:−
2729:−
2691:Γ
2566:Γ
2484:Γ
2408:Γ
2034:∞
2028:−
2005:∞
1976:∞
1973:→
1947:∞
1918:∞
1912:−
1889:ω
1825:Γ
1769:Γ
1634:−
1535:−
1499:Γ
1471:Γ
1354:Γ
1272:, can be
1205:Γ
288:real part
154:stability
5485:(1985):
5425:Archived
5290:(1947).
5264:Archived
5141:See also
5028:unstable
4011:′
3716:′
3457:′
3144:, where
2682:-plane,
1845:negative
822:are the
368:domain.
310:, where
280:feedback
218:and the
208:feedback
5032:counter
4833:Summary
2978:around
1653:. Here
1568:of the
344:of the
290:of the
167:of the
142:Siemens
5521:
5507:
5493:
5476:
5407:
5399:
5350:
5232:
3524:Where
2502:, and
2426:, let
1597:plane
1274:mapped
429:, the
286:, the
282:. In
121:, the
5405:S2CID
5379:arXiv
5348:S2CID
5303:]
5267:(PDF)
5260:(PDF)
5136:axis.
1192:From
824:poles
758:zeros
508:roots
316:phase
266:is a
230:like
222:of a
156:of a
138:]
99:with
5519:ISBN
5505:ISBN
5491:ISBN
5474:ISBN
5397:ISSN
5230:ISBN
1673:and
609:Any
312:gain
274:and
202:and
117:and
5389:doi
5340:doi
5026:is
2223:of
1930:to
1649:by
826:of
760:of
160:.
148:at
140:at
125:or
113:In
5573::
5443:.
5403:.
5395:.
5387:.
5375:68
5373:.
5369:.
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5332:11
5330:.
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5262:.
5220:.
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650::
326:.
262:A
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106:jω
104:=
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5411:.
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5048:s
5045:(
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4852:s
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3497:N
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3491:s
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3478:s
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3472:D
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3317:)
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3252:s
3249:(
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3208:s
3205:(
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3103:s
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2869:(
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2807:.
2795:P
2789:Z
2786:=
2783:N
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2741:0
2738:j
2735:+
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2374:)
2371:s
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2307:s
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2046:.
2031:j
2025:0
2002:j
1999:+
1996:0
1970:r
1959:.
1944:j
1941:+
1938:0
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1909:0
1886:j
1829:s
1804:)
1801:s
1798:(
1795:F
1773:s
1748:)
1745:s
1742:(
1739:F
1719:)
1716:s
1713:(
1710:F
1707:k
1704:+
1701:1
1681:P
1661:Z
1637:Z
1631:P
1628:=
1625:N
1605:N
1585:)
1582:s
1579:(
1576:F
1555:0
1552:j
1549:+
1546:k
1542:/
1538:1
1531:=
1528:s
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1503:s
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1489:=
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1257:s
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1166:0
1163:=
1160:)
1157:s
1154:(
1151:B
1148:+
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1142:s
1139:(
1136:A
1116:)
1113:s
1110:(
1107:H
1104:)
1101:s
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1060:s
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1049:)
1046:s
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1034:=
1031:)
1028:s
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1016:s
1013:(
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987:)
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915:=
912:)
909:s
906:(
903:D
880:)
877:s
874:(
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844:s
841:(
836:T
810:)
807:s
804:(
801:D
781:)
778:s
775:(
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744:)
741:s
738:(
735:N
712:.
706:)
703:s
700:(
697:D
692:)
689:s
686:(
683:N
677:=
674:)
671:s
668:(
663:T
634:)
631:s
628:(
623:T
590:)
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578:)
575:s
572:(
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542:s
539:(
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533:)
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521:+
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482:(
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382:(
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366:s
304:Y
296:X
256:X
109:.
102:s
84:1
81:+
78:s
75:+
70:2
66:s
61:1
56:=
53:)
50:s
47:(
44:G
20:)
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