In Cartesian coordinates, the real part of the transfer function is plotted on the X-axis while the imaginary part is plotted on the Y-axis. Additional parameters P On the other hand, a Bode diagram displays the phase-crossover and gain-crossover frequencies, which are not explicit on a traditional Nyquist plot. The Routh test is an efficient times such that Assessment of the stability of a closed-loop negative feedback system is done by applying the Nyquist stability criterion to the Nyquist plot of the open-loop system (i.e. . In 18.03 we called the system stable if every homogeneous solution decayed to 0. This method for design involves plotting the complex loci of P ( s) C ( s) for the range s = j , = [ , ]. The Nyquist criterion gives a graphical method for checking the stability of the closed loop system. Phase margin is defined by, \[\operatorname{PM}(\Lambda)=180^{\circ}+\left(\left.\angle O L F R F(\omega)\right|_{\Lambda} \text { at }|O L F R F(\omega)|_{\Lambda} \mid=1\right)\label{eqn:17.7} \]. Based on analysis of the Nyquist Diagram: (i) Comment on the stability of the closed loop system. Since the number of poles of \(G\) in the right half-plane is the same as this winding number, the closed loop system is stable. s 0 H Yes! {\displaystyle \Gamma _{F(s)}=F(\Gamma _{s})} In signal processing, the Nyquist frequency, named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. This results from the requirement of the argument principle that the contour cannot pass through any pole of the mapping function. / 1 If the system is originally open-loop unstable, feedback is necessary to stabilize the system. + ) s inside the contour The frequency-response curve leading into that loop crosses the \(\operatorname{Re}[O L F R F]\) axis at about \(-0.315+j 0\); if we were to use this phase crossover to calculate gain margin, then we would find \(\mathrm{GM} \approx 1 / 0.315=3.175=10.0\) dB. The new system is called a closed loop system. Static and dynamic specifications. . For these values of \(k\), \(G_{CL}\) is unstable. G ) From complex analysis, a contour \[G_{CL} (s) = \dfrac{1/(s + a)}{1 + 1/(s + a)} = \dfrac{1}{s + a + 1}.\], This has a pole at \(s = -a - 1\), so it's stable if \(a > -1\). ) That is, if the unforced system always settled down to equilibrium. Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. s If the system with system function \(G(s)\) is unstable it can sometimes be stabilized by what is called a negative feedback loop. are same as the poles of In the previous problem could you determine analytically the range of \(k\) where \(G_{CL} (s)\) is stable? 1 Another aspect of the difference between the plots on the two figures is particularly significant: whereas the plots on Figure \(\PageIndex{1}\) cross the negative \(\operatorname{Re}[O L F R F]\) axis only once as driving frequency \(\omega\) increases, those on Figure \(\PageIndex{4}\) have two phase crossovers, i.e., the phase angle is 180 for two different values of \(\omega\). {\displaystyle \Gamma _{s}} {\displaystyle F(s)} does not have any pole on the imaginary axis (i.e. {\displaystyle G(s)} The Nyquist criterion allows us to answer two questions: 1. The Nyquist Contour Assumption: Traverse the Nyquist contour in CW direction Observation #1: Encirclement of a pole forces the contour to gain 360 degrees so the Nyquist evaluation = = ( \nonumber\]. + s If, on the other hand, we were to calculate gain margin using the other phase crossing, at about \(-0.04+j 0\), then that would lead to the exaggerated \(\mathrm{GM} \approx 25=28\) dB, which is obviously a defective metric of stability. {\displaystyle {\mathcal {T}}(s)} ) Does the system have closed-loop poles outside the unit circle? Choose \(R\) large enough that the (finite number) of poles and zeros of \(G\) in the right half-plane are all inside \(\gamma_R\). 0 plane, encompassing but not passing through any number of zeros and poles of a function s This has one pole at \(s = 1/3\), so the closed loop system is unstable. Nyquist stability criterion is a general stability test that checks for the stability of linear time-invariant systems. For our purposes it would require and an indented contour along the imaginary axis. ) The MATLAB commands follow that calculate [from Equations 17.1.7 and 17.1.12] and plot these cases of open-loop frequency-response function, and the resulting Nyquist diagram (after additional editing): >> olfrf01=wb./(j*w.*(j*w+coj). G ( {\displaystyle G(s)} The reason we use the Nyquist Stability Criterion is that it gives use information about the relative stability of a system and gives us clues as to how to make a system more stable. In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. Consider a system with The system is stable if the modes all decay to 0, i.e. Z s ( G(s)= s(s+5)(s+10)500K slopes, frequencies, magnitudes, on the next pages!) \(\text{QED}\), The Nyquist criterion is a visual method which requires some way of producing the Nyquist plot. + {\displaystyle T(s)} That is, setting {\displaystyle 1+G(s)} Its system function is given by Black's formula, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)},\]. {\displaystyle \Gamma _{s}} The reason we use the Nyquist Stability Criterion is that it gives use information about the relative stability of a system and gives us clues as to how to make a system more stable. {\displaystyle F(s)} The Nyquist criterion allows us to answer two questions: 1. Nyquist and Bode plots for the above circuits are given in Figs 12.34 and 12.35, where is the time at which the exponential factor is e1 = 0.37, the time it takes to decrease to 37% of its value. Since on Figure \(\PageIndex{4}\) there are two different frequencies at which \(\left.\angle O L F R F(\omega)\right|_{\Lambda}=-180^{\circ}\), the definition of gain margin in Equations 17.1.8 and \(\ref{eqn:17.17}\) is ambiguous: at which, if either, of the phase crossovers is it appropriate to read the quantity \(1 / \mathrm{GM}\), as shown on \(\PageIndex{2}\)? The system is called unstable if any poles are in the right half-plane, i.e. According to the formula, for open loop transfer function stability: Z = N + P = 0. where N is the number of encirclements of ( 0, 0) by the Nyquist plot in clockwise direction. In the case \(G(s)\) is a fractional linear transformation, so we know it maps the imaginary axis to a circle. The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are all in the left half of the complex plane. The Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. 0000039854 00000 n This approach appears in most modern textbooks on control theory. N Let us begin this study by computing \(\operatorname{OLFRF}(\omega)\) and displaying it on Nyquist plots for a low value of gain, \(\Lambda=0.7\) (for which the closed-loop system is stable), and for the value corresponding to the transition from stability to instability on Figure \(\PageIndex{3}\), which we denote as \(\Lambda_{n s 1} \approx 1\). Note that the phase margin for \(\Lambda=0.7\), found as shown on Figure \(\PageIndex{2}\), is quite clear on Figure \(\PageIndex{4}\) and not at all ambiguous like the gain margin: \(\mathrm{PM}_{0.7} \approx+20^{\circ}\); this value also indicates a stable, but weakly so, closed-loop system. This happens when, \[0.66 < k < 0.33^2 + 1.75^2 \approx 3.17. {\displaystyle G(s)} G -plane, ) The stability of v the clockwise direction. \(\PageIndex{4}\) includes the Nyquist plots for both \(\Lambda=0.7\) and \(\Lambda =\Lambda_{n s 1}\), the latter of which by definition crosses the negative \(\operatorname{Re}[O L F R F]\) axis at the point \(-1+j 0\), not far to the left of where the \(\Lambda=0.7\) plot crosses at about \(-0.73+j 0\); therefore, it might be that the appropriate value of gain margin for \(\Lambda=0.7\) is found from \(1 / \mathrm{GM}_{0.7} \approx 0.73\), so that \(\mathrm{GM}_{0.7} \approx 1.37=2.7\) dB, a small gain margin indicating that the closed-loop system is just weakly stable. Note that a closed-loop-stable case has \(0<1 / \mathrm{GM}_{\mathrm{S}}<1\) so that \(\mathrm{GM}_{\mathrm{S}}>1\), and a closed-loop-unstable case has \(1 / \mathrm{GM}_{\mathrm{U}}>1\) so that \(0<\mathrm{GM}_{\mathrm{U}}<1\). Techniques like Bode plots, while less general, are sometimes a more useful design tool. {\displaystyle 1+GH(s)} Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. So far, we have been careful to say the system with system function \(G(s)\)'. From the mapping we find the number N, which is the number of In control theory and stability theory, the Nyquist stability criterion or StreckerNyquist stability criterion, independently discovered by the German electrical engineer Felix Strecker[de] at Siemens in 1930[1][2][3] and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932,[4] is a graphical technique for determining the stability of a dynamical system. Is the closed loop system stable when \(k = 2\). s ) {\displaystyle {\mathcal {T}}(s)={\frac {N(s)}{D(s)}}.}. ( as defined above corresponds to a stable unity-feedback system when ) If the counterclockwise detour was around a double pole on the axis (for example two ( If the number of poles is greater than the the same system without its feedback loop). {\displaystyle F(s)} Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. j {\displaystyle {\mathcal {T}}(s)} This gives us, We now note that is determined by the values of its poles: for stability, the real part of every pole must be negative. s G Nyquist plot of \(G(s) = 1/(s + 1)\), with \(k = 1\). , which is to say our Nyquist plot. + ( Nyquist stability criterion states the number of encirclements about the critical point (1+j0) must be equal to the poles of characteristic equation, which is nothing but the poles of the open loop transfer function in the right half of the s plane. ( D and poles of Thus, it is stable when the pole is in the left half-plane, i.e. So the winding number is -1, which does not equal the number of poles of \(G\) in the right half-plane. P MT-002. The Nyquist criterion allows us to assess the stability properties of a feedback system based on P ( s) C ( s) only. G All the coefficients of the characteristic polynomial, s 4 + 2 s 3 + s 2 + 2 s + 1 are positive. We will look a little more closely at such systems when we study the Laplace transform in the next topic. {\displaystyle D(s)} + ) The other phase crossover, at \(-4.9254+j 0\) (beyond the range of Figure \(\PageIndex{5}\)), might be the appropriate point for calculation of gain margin, since it at least indicates instability, \(\mathrm{GM}_{4.75}=1 / 4.9254=0.20303=-13.85\) dB. 1 travels along an arc of infinite radius by G Any class or book on control theory will derive it for you. It is informative and it will turn out to be even more general to extract the same stability margins from Nyquist plots of frequency response. F Precisely, each complex point ( s That is, if all the poles of \(G\) have negative real part. . The mathlet shows the Nyquist plot winds once around \(w = -1\) in the \(clockwise\) direction. ( Microscopy Nyquist rate and PSF calculator. ) , the result is the Nyquist Plot of ) 1 The Nyquist criterion is a frequency domain tool which is used in the study of stability. s If the answer to the first question is yes, how many closed-loop Mark the roots of b From now on we will allow ourselves to be a little more casual and say the system \(G(s)\)'. Legal. The argument principle from complex analysis gives a criterion to calculate the difference between the number of zeros and the number of poles of This continues until \(k\) is between 3.10 and 3.20, at which point the winding number becomes 1 and \(G_{CL}\) becomes unstable. Recalling that the zeros of {\displaystyle G(s)} must be equal to the number of open-loop poles in the RHP. 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The frequency is swept as a parameter, resulting in a plot per frequency. Moreover, we will add to the same graph the Nyquist plots of frequency response for a case of positive closed-loop stability with \(\Lambda=1 / 2 \Lambda_{n s}=20,000\) s-2, and for a case of closed-loop instability with \(\Lambda= 2 \Lambda_{n s}=80,000\) s-2. r However, the Nyquist Criteria can also give us additional information about a system. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Figure 19.3 : Unity Feedback Confuguration. Assessment of the stability of a closed-loop negative feedback system is done by applying the Nyquist stability criterion to the Nyquist plot of the open-loop system (i.e. drawn in the complex {\displaystyle F(s)} We know from Figure \(\PageIndex{3}\) that this case of \(\Lambda=4.75\) is closed-loop unstable. Take \(G(s)\) from the previous example. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. This page titled 12.2: Nyquist Criterion for Stability is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. u + 0000001367 00000 n L is called the open-loop transfer function. The positive \(\mathrm{PM}_{\mathrm{S}}\) for a closed-loop-stable case is the counterclockwise angle from the negative \(\operatorname{Re}[O L F R F]\) axis to the intersection of the unit circle with the \(OLFRF_S\) curve; conversely, the negative \(\mathrm{PM}_U\) for a closed-loop-unstable case is the clockwise angle from the negative \(\operatorname{Re}[O L F R F]\) axis to the intersection of the unit circle with the \(OLFRF_U\) curve. Nyquist stability criterion states the number of encirclements about the critical point (1+j0) must be equal to the poles of characteristic equation, which is nothing but the poles of the open loop G Another unusual case that would require the general Nyquist stability criterion is an open-loop system with more than one gain crossover, i.e., a system whose frequency ( For what values of \(a\) is the corresponding closed loop system \(G_{CL} (s)\) stable? G s {\displaystyle P} {\displaystyle -l\pi } s ( . %PDF-1.3 % s The Nyquist plot is the trajectory of \(K(i\omega) G(i\omega) = ke^{-ia\omega}G(i\omega)\) , where \(i\omega\) traverses the imaginary axis. 0000002305 00000 n ) ( Open the Nyquist Plot applet at. 1 plane in the same sense as the contour The Nyquist Contour Assumption: Traverse the Nyquist contour in CW direction Observation #1: Encirclement of a pole forces the contour to gain 360 degrees so the Nyquist evaluation encircles origin in CCW direction Observation #2 Encirclement of a zero forces the contour to loose 360 degrees so the Nyquist evaluation encircles origin in CW direction j j In practice, the ideal sampler is replaced by ) + ( ) Is the system with system function \(G(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}\) stable? This is possible for small systems. P is the multiplicity of the pole on the imaginary axis. in the right-half complex plane minus the number of poles of s "1+L(s)=0.". , and the roots of point in "L(s)". For example, quite often \(G(s)\) is a rational function \(Q(s)/P(s)\) (\(Q\) and \(P\) are polynomials). Nyquist plot of the transfer function s/(s-1)^3. Thus, for all large \(R\), \[\text{the system is stable } \Leftrightarrow \ Z_{1 + kG, \gamma_R} = 0 \ \Leftrightarow \ \text{ Ind} (kG \circ \gamma_R, -1) = P_{G, \gamma_R}\], Finally, we can let \(R\) go to infinity. The most common use of Nyquist plots is for assessing the stability of a system with feedback. {\displaystyle G(s)} Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. ) {\displaystyle Z} s The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s + 1)/(s - 1)}{1 + 2(s + 1)/(s - 1)} = \dfrac{s + 1}{3s + 1}.\]. + Gain margin (GM) is defined by Equation 17.1.8, from which we find, \[\frac{1}{G M(\Lambda)}=|O L F R F(\omega)|_{\mid} \mid \text {at }\left.\angle O L F R F(\omega)\right|_{\Lambda}=-180^{\circ}\label{eqn:17.17} \]. encirclements of the -1+j0 point in "L(s).". If we were to test experimentally the open-loop part of this system in order to determine the stability of the closed-loop system, what would the open-loop frequency responses be for different values of gain \(\Lambda\)? P T So, the control system satisfied the necessary condition. It is certainly reasonable to call a system that does this in response to a zero signal (often called no input) unstable. ( s j The approach explained here is similar to the approach used by Leroy MacColl (Fundamental theory of servomechanisms 1945) or by Hendrik Bode (Network analysis and feedback amplifier design 1945), both of whom also worked for Bell Laboratories. k k To be able to analyze systems with poles on the imaginary axis, the Nyquist Contour can be modified to avoid passing through the point {\displaystyle s} Let us continue this study by computing \(OLFRF(\omega)\) and displaying it as a Nyquist plot for an intermediate value of gain, \(\Lambda=4.75\), for which Figure \(\PageIndex{3}\) shows the closed-loop system is unstable. The only pole is at \(s = -1/3\), so the closed loop system is stable. 0 , as evaluated above, is equal to0. Z {\displaystyle F(s)} s {\displaystyle \Gamma _{s}} ) ( 1 It is likely that the most reliable theoretical analysis of such a model for closed-loop stability would be by calculation of closed-loop loci of roots, not by calculation of open-loop frequency response. s This should make sense, since with \(k = 0\), \[G_{CL} = \dfrac{G}{1 + kG} = G. \nonumber\]. {\displaystyle Z} + ) Suppose \(G(s) = \dfrac{s + 1}{s - 1}\). 0 + {\displaystyle {\mathcal {T}}(s)} s The assumption that \(G(s)\) decays 0 to as \(s\) goes to \(\infty\) implies that in the limit, the entire curve \(kG \circ C_R\) becomes a single point at the origin. Look at the pole diagram and use the mouse to drag the yellow point up and down the imaginary axis. Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}\) stable? G The left hand graph is the pole-zero diagram. s be the number of zeros of {\displaystyle s} + L is called the open-loop transfer function. (Using 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. A linear time invariant system has a system function which is a function of a complex variable. ) 1This transfer function was concocted for the purpose of demonstration. {\displaystyle 1+GH} ( So in the limit \(kG \circ \gamma_R\) becomes \(kG \circ \gamma\). the same system without its feedback loop). You should be able to show that the zeros of this transfer function in the complex \(s\)-plane are at (\(2 j10\)), and the poles are at (\(1 + j0\)) and (\(1 j5\)). = {\displaystyle G(s)} . that appear within the contour, that is, within the open right half plane (ORHP). ( {\displaystyle F(s)} ( are, respectively, the number of zeros of in the contour Counting the clockwise encirclements of the plot GH(s) of the origincontd As we traverse the contour once, the angle 1 of the vector v 1 from the zero inside the contour in the s-plane encounters a net change of 2radians a clockwise semicircle at L(s)= in "L(s)" (see, The clockwise semicircle at infinity in "s" corresponds to a single ( {\displaystyle H(s)} A pole with positive real part would correspond to a mode that goes to infinity as \(t\) grows. The roots of of poles of T(s)). Now, recall that the poles of \(G_{CL}\) are exactly the zeros of \(1 + k G\). . ( The theorem recognizes these. clockwise. {\displaystyle GH(s)} {\displaystyle 0+j\omega } Step 2 Form the Routh array for the given characteristic polynomial. {\displaystyle 1+G(s)} Compute answers using Wolfram's breakthrough technology & The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. 0000039933 00000 n If the number of poles is greater than the number of zeros, then the Nyquist criterion tells us how to use the Nyquist plot to graphically determine the stability of the closed loop system. ) 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. ( H {\displaystyle -1+j0} ) s The roots of b (s) are the poles of the open-loop transfer function. P Any Laplace domain transfer function The frequency is swept as a parameter, resulting in a pl {\displaystyle l} 0000000608 00000 n = ( times, where has zeros outside the open left-half-plane (commonly initialized as OLHP). We conclude this chapter on frequency-response stability criteria by observing that margins of gain and phase are used also as engineering design goals. \(G(s)\) has one pole at \(s = -a\). {\displaystyle 0+j(\omega -r)} {\displaystyle F(s)} The portion of the Nyquist plot for gain \(\Lambda=4.75\) that is closest to the negative \(\operatorname{Re}[O L F R F]\) axis is shown on Figure \(\PageIndex{5}\). -P_PcXJ']b9-@f8+5YjmK p"yHL0:8UK=MY9X"R&t5]M/o 3\\6%W+7J$)^p;% XpXC#::` :@2p1A%TQHD1Mdq!1 k G We will look a little more closely at such systems when we study the Laplace transform in the next topic. s s D Stability can be determined by examining the roots of the desensitivity factor polynomial Z We dont analyze stability by plotting the open-loop gain or {\displaystyle 1+G(s)} B The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. 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 G The system with system function \(G(s)\) is called stable if all the poles of \(G\) are in the left half-plane. The Bode plot for 0 Expert Answer. G 1 . Hence, the number of counter-clockwise encirclements about Here N = 1. have positive real part. ) N {\displaystyle N(s)} Given our definition of stability above, we could, in principle, discuss stability without the slightest idea what it means for physical systems. This is a case where feedback destabilized a stable system. 2. F {\displaystyle 1+G(s)} ) Proofs of the general Nyquist stability criterion are based on the theory of complex functions of a complex variable; many textbooks on control theory present such proofs, one of the clearest being that of Franklin, et al., 1991, pages 261-280. H|Ak0ZlzC!bBM66+d]JHbLK5L#S$_0i".Zb~#}2HyY YBrs}y:)c. ) {\displaystyle A(s)+B(s)=0} Thus, we may find The Nyquist plot can provide some information about the shape of the transfer function. Thus, this physical system (of Figures 16.3.1, 16.3.2, and 17.1.2) is considered a common system, for which gain margin and phase margin provide clear and unambiguous metrics of stability. To use this criterion, the frequency response data of a system must be presented as a polar plot in which the magnitude and the phase angle are expressed as We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. {\displaystyle v(u)={\frac {u-1}{k}}} . s denotes the number of zeros of , that starts at ( Microscopy Nyquist rate and PSF calculator. Another unusual case that would require the general Nyquist stability criterion is an open-loop system with more than one gain crossover, i.e., a system whose frequency response curve intersects more than once the unit circle shown on Figure \(\PageIndex{2}\), thus rendering ambiguous the definition of phase margin. B ( 0 {\displaystyle N=P-Z} The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. Does the system have closed-loop poles outside the unit circle? u s + ) Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. {\displaystyle P} ) Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. This is just to give you a little physical orientation. ( . The shift in origin to (1+j0) gives the characteristic equation plane. There are no poles in the right half-plane. s = ) k The same plot can be described using polar coordinates, where gain of the transfer function is the radial coordinate, and the phase of the transfer function is the corresponding angular coordinate. ; when placed in a closed loop with negative feedback We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. H F G s We may further reduce the integral, by applying Cauchy's integral formula. Contact Pro Premium Expert Support Give us your feedback s The graphical display of frequency response magnitude becoming very large as 0 is produced by the following MATLAB commands, which calculate frequency response and produce a Nyquist plot of the same numerical solution as that on Figure 17.1.3, for the neutral-stability case = n s = 40, 000 s -2: >> wb=300;coj=100;wns=sqrt (wb*coj); {\displaystyle (-1+j0)} ) ) of the This is distinctly different from the Nyquist plots of a more common open-loop system on Figure \(\PageIndex{1}\), which approach the origin from above as frequency becomes very high. s ) ( l When the highest frequency of a signal is less than the Nyquist frequency of the sampler, the resulting discrete-time sequence is said to be free of the The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. u Closed loop approximation f.d.t. s ( plane gain margin as defined on Figure \(\PageIndex{5}\) can be an ambiguous, unreliable, and even deceptive metric of closed-loop stability; phase margin as defined on Figure \(\PageIndex{5}\), on the other hand, is usually an unambiguous and reliable metric, with \(\mathrm{PM}>0\) indicating closed-loop stability, and \(\mathrm{PM}<0\) indicating closed-loop instability. This method is easily applicable even for systems with delays and other non if the poles are all in the left half-plane. This page titled 17.4: The Nyquist Stability Criterion is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. The Nyquist Stability Criteria is a test for system stability, just like the Routh-Hurwitz test, or the Root-Locus Methodology. plane yielding a new contour. ) The beauty of the Nyquist stability criterion lies in the fact that it is a rather simple graphical test. ( 0000002345 00000 n For example, Brogan, 1974, page 25, wrote Experience has shown that acceptable transient response will usually require stability margins on the order of \(\mathrm{PM}>30^{\circ}\), \(\mathrm{GM}>6\) dB. Franklin, et al., 1991, page 285, wrote Many engineers think directly in terms of \(\text { PM }\) in judging whether a control system is adequately stabilized. Such a modification implies that the phasor s s ( T When \(k\) is small the Nyquist plot has winding number 0 around -1. Rule 1. Let \(G(s) = \dfrac{1}{s + 1}\). yields a plot of Check the \(Formula\) box. r That is, the Nyquist plot is the circle through the origin with center \(w = 1\). F s trailer << /Size 104 /Info 89 0 R /Root 92 0 R /Prev 245773 /ID[<8d23ab097aef38a19f6ffdb9b7be66f3>] >> startxref 0 %%EOF 92 0 obj << /Type /Catalog /Pages 86 0 R /Metadata 90 0 R /PageLabels 84 0 R >> endobj 102 0 obj << /S 478 /L 556 /Filter /FlateDecode /Length 103 0 R >> stream These are the same systems as in the examples just above. is mapped to the point 0 We can show this formally using Laurent series. ) ) 0 Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. The algebra involved in canceling the \(s + a\) term in the denominators is exactly the cancellation that makes the poles of \(G\) removable singularities in \(G_{CL}\). P With a little imagination, we infer from the Nyquist plots of Figure \(\PageIndex{1}\) that the open-loop system represented in that figure has \(\mathrm{GM}>0\) and \(\mathrm{PM}>0\) for \(0<\Lambda<\Lambda_{\mathrm{ns}}\), and that \(\mathrm{GM}>0\) and \(\mathrm{PM}>0\) for all values of gain \(\Lambda\) greater than \(\Lambda_{\mathrm{ns}}\); accordingly, the associated closed-loop system is stable for \(0<\Lambda<\Lambda_{\mathrm{ns}}\), and unstable for all values of gain \(\Lambda\) greater than \(\Lambda_{\mathrm{ns}}\). s , can be mapped to another plane (named as the first and second order system. 0.375=3/2 (the current gain (4) multiplied by the gain margin N You have already encountered linear time invariant systems in 18.03 (or its equivalent) when you solved constant coefficient linear differential equations. ) are also said to be the roots of the characteristic equation In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. ) The poles of \(G(s)\) correspond to what are called modes of the system. Make a mapping from the "s" domain to the "L(s)" Its image under \(kG(s)\) will trace out the Nyquis plot. Natural Language; Math Input; Extended Keyboard Examples Upload Random. s That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. s N ) In \(\gamma (\omega)\) the variable is a greek omega and in \(w = G \circ \gamma\) we have a double-u. = F (At \(s_0\) it equals \(b_n/(kb_n) = 1/k\).). The answer is no, \(G_{CL}\) is not stable. Since we know N and P, we can determine Z, the number of zeros of s This reference shows that the form of stability criterion described above [Conclusion 2.] + , where , then the roots of the characteristic equation are also the zeros of Then the closed loop system with feedback factor \(k\) is stable if and only if the winding number of the Nyquist plot around \(w = -1\) equals the number of poles of \(G(s)\) in the right half-plane. Let us complete this study by computing \(\operatorname{OLFRF}(\omega)\) and displaying it on Nyquist plots for the value corresponding to the transition from instability back to stability on Figure \(\PageIndex{3}\), which we denote as \(\Lambda_{n s 2} \approx 15\), and for a slightly higher value, \(\Lambda=18.5\), for which the closed-loop system is stable. Note that we count encirclements in the Stability in the Nyquist Plot. There are two poles in the right half-plane, so the open loop system \(G(s)\) is unstable. The poles of the closed loop system function \(G_{CL} (s)\) given in Equation 12.3.2 are the zeros of \(1 + kG(s)\). The most common use of Nyquist plots is for assessing the stability of a system with feedback. Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. For this we will use one of the MIT Mathlets (slightly modified for our purposes). s {\displaystyle v(u(\Gamma _{s}))={{D(\Gamma _{s})-1} \over {k}}=G(\Gamma _{s})} ) In units of Hz, its value is one-half of the sampling rate. = Which, if either, of the values calculated from that reading, \(\mathrm{GM}=(1 / \mathrm{GM})^{-1}\) is a legitimate metric of closed-loop stability? 0 Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular values of gain. The Nyquist stability criterion has been used extensively in science and engineering to assess the stability of physical systems that can be represented by sets of linear equations. (3h) lecture: Nyquist diagram and on the effects of feedback. ) by counting the poles of negatively oriented) contour The oscillatory roots on Figure \(\PageIndex{3}\) show that the closed-loop system is stable for \(\Lambda=0\) up to \(\Lambda \approx 1\), it is unstable for \(\Lambda \approx 1\) up to \(\Lambda \approx 15\), and it becomes stable again for \(\Lambda\) greater than \(\approx 15\). Nyquist criterion and stability margins. are the poles of s s s ) denotes the number of poles of Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. and travels anticlockwise to For gain \(\Lambda = 18.5\), there are two phase crossovers: one evident on Figure \(\PageIndex{6}\) at \(-18.5 / 15.0356+j 0=-1.230+j 0\), and the other way beyond the range of Figure \(\PageIndex{6}\) at \(-18.5 / 0.96438+j 0=-19.18+j 0\). ( N {\displaystyle {\mathcal {T}}(s)} ( enclosing the right half plane, with indentations as needed to avoid passing through zeros or poles of the function G G Suppose F (s) is a single-valued mapping function given as: F (s) = 1 + G (s)H (s) s + . The Nyquist method is used for studying the stability of linear systems with pure time delay. j + ( {\displaystyle \Gamma _{G(s)}} To connect this to 18.03: if the system is modeled by a differential equation, the modes correspond to the homogeneous solutions \(y(t) = e^{st}\), where \(s\) is a root of the characteristic equation. ( are called the zeros of Describe the Nyquist plot with gain factor \(k = 2\). {\displaystyle G(s)} ( We suppose that we have a clockwise (i.e. To simulate that testing, we have from Equation \(\ref{eqn:17.18}\), the following equation for the frequency-response function: \[O L F R F(\omega) \equiv O L T F(j \omega)=\Lambda \frac{104-\omega^{2}+4 \times j \omega}{(1+j \omega)\left(26-\omega^{2}+2 \times j \omega\right)}\label{eqn:17.20} \]. has exactly the same poles as {\displaystyle \Gamma _{s}} ( ( In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). ) 0000002847 00000 n olfrf01=(104-w.^2+4*j*w)./((1+j*w). Make a system with the following zeros and poles: Is the corresponding closed loop system stable when \(k = 6\)? {\displaystyle D(s)} Transfer Function System Order -thorder system Characteristic Equation s + ) s >> olfrf01=(104-w.^2+4*j*w)./((1+j*w). s G {\displaystyle P} ) Thus, we may finally state that. ( F In using \(\text { PM }\) this way, a phase margin of 30 is often judged to be the lowest acceptable \(\text { PM }\), with values above 30 desirable.. Matrix Result This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. In this case, we have, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)} = \dfrac{\dfrac{s - 1}{(s - 0.33)^2 + 1.75^2}}{1 + \dfrac{k(s - 1)}{(s - 0.33)^2 + 1.75^2}} = \dfrac{s - 1}{(s - 0.33)^2 + 1.75^2 + k(s - 1)} \nonumber\], \[(s - 0.33)^2 + 1.75^2 + k(s - 1) = s^2 + (k - 0.66)s + 0.33^2 + 1.75^2 - k \nonumber\], For a quadratic with positive coefficients the roots both have negative real part. , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. {\displaystyle Z=N+P} Since there are poles on the imaginary axis, the system is marginally stable. One way to do it is to construct a semicircular arc with radius s right half plane. G , e.g. *(j*w+wb)); >> olfrf20k=20e3*olfrf01;olfrf40k=40e3*olfrf01;olfrf80k=80e3*olfrf01; >> plot(real(olfrf80k),imag(olfrf80k),real(olfrf40k),imag(olfrf40k),, Gain margin and phase margin are present and measurable on Nyquist plots such as those of Figure \(\PageIndex{1}\). ( ( ( + We then note that and that encirclements in the opposite direction are negative encirclements. ) D The significant roots of Equation \(\ref{eqn:17.19}\) are shown on Figure \(\PageIndex{3}\): the complete locus of oscillatory roots with positive imaginary parts is shown; only the beginning of the locus of real (exponentially stable) roots is shown, since those roots become progressively more negative as gain \(\Lambda\) increases from the initial small values. Set the feedback factor \(k = 1\). ) An approach to this end is through the use of Nyquist techniques. There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. D , which is to say. In control system theory, the RouthHurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system.A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. You can also check that it is traversed clockwise. is not sufficiently general to handle all cases that might arise. {\displaystyle 1+kF(s)} [@mc6X#:H|P`30s@, B R=Lb&3s12212WeX*a$%.0F06 endstream endobj 103 0 obj 393 endobj 93 0 obj << /Type /Page /Parent 85 0 R /Resources 94 0 R /Contents 98 0 R /Rotate 90 /MediaBox [ 0 0 612 792 ] /CropBox [ 36 36 576 756 ] >> endobj 94 0 obj << /ProcSet [ /PDF /Text ] /Font << /TT2 96 0 R >> /ExtGState << /GS1 100 0 R >> /ColorSpace << /Cs6 97 0 R >> >> endobj 95 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -568 -307 2028 1007 ] /FontName /HMIFEA+TimesNewRoman /ItalicAngle 0 /StemV 94 /XHeight 0 /FontFile2 99 0 R >> endobj 96 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 150 /Widths [ 250 0 0 500 0 0 0 0 333 333 500 564 250 333 250 278 500 500 500 500 500 500 500 500 500 500 278 0 0 564 0 0 0 722 667 667 722 611 556 722 722 333 389 0 611 889 722 722 556 0 667 556 611 722 722 944 0 0 0 0 0 0 0 500 0 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 500 500 333 389 278 500 500 722 500 500 444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 333 0 0 350 500 ] /Encoding /WinAnsiEncoding /BaseFont /HMIFEA+TimesNewRoman /FontDescriptor 95 0 R >> endobj 97 0 obj [ /ICCBased 101 0 R ] endobj 98 0 obj << /Length 428 /Filter /FlateDecode >> stream encircled by plane) by the function . s Rearranging, we have This is a diagram in the \(s\)-plane where we put a small cross at each pole and a small circle at each zero. s s j {\displaystyle -1/k} Determining Stability using the Nyquist Plot - Erik Cheever Rule 2. 0000001210 00000 n In units of ( This is a case where feedback stabilized an unstable system. s The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are {\displaystyle s={-1/k+j0}} s It is more challenging for higher order systems, but there are methods that dont require computing the poles. . k Legal. N T ) F ) {\displaystyle {\mathcal {T}}(s)} We will make a standard assumption that \(G(s)\) is meromorphic with a finite number of (finite) poles. 91 0 obj << /Linearized 1 /O 93 /H [ 701 509 ] /L 247721 /E 42765 /N 23 /T 245783 >> endobj xref 91 13 0000000016 00000 n = ( \[G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),\], \[\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}\], which is clearly analytic at \(s_0\). 2. The Nyquist stability criterion has been used extensively in science and engineering to assess the stability of physical systems that can be represented by sets of The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. 1 Lecture 1: The Nyquist Criterion S.D. This assumption holds in many interesting cases. If the number of poles is greater than the number of zeros, then the Nyquist criterion tells us how to use the Nyquist plot to graphically determine the stability of the closed loop system. F 1 ) {\displaystyle N=Z-P} D F ) In signal processing, the Nyquist frequency, named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. To use this criterion, the frequency response data of a system must be presented as a polar plot in We thus find that Another unusual case that would require the general Nyquist stability criterion is an open-loop system with more than one gain crossover, i.e., a system whose frequency response curve intersects more than once the unit circle shown on Figure 17.4.2, thus rendering ambiguous the definition of phase margin. Since they are all in the left half-plane, the system is stable. 1 Is the open loop system stable? s {\displaystyle P} s We begin by considering the closed-loop characteristic polynomial (4.23) where L ( z) denotes the loop gain. (2 h) lecture: Introduction to the controller's design specifications. Let \(G(s)\) be such a system function. Lecture 2: Stability Criteria S.D. ) H Figure 19.3 : Unity Feedback Confuguration. We will just accept this formula. the number of the counterclockwise encirclements of \(1\) point by the Nyquist plot in the \(GH\)-plane is equal to the number of the unstable poles of the open-loop transfer function. The following MATLAB commands, adapted from the code that produced Figure 16.5.1, calculate and plot the loci of roots: Lm=[0 .2 .4 .7 1 1.5 2.5 3.7 4.75 6.5 9 12.5 15 18.5 25 35 50 70 125 250]; a2=3+Lm(i);a3=4*(7+Lm(i));a4=26*(1+4*Lm(i)); plot(p,'kx'),grid,xlabel('Real part of pole (sec^-^1)'), ylabel('Imaginary part of pole (sec^-^1)'). l ) The feedback loop has stabilized the unstable open loop systems with \(-1 < a \le 0\). encircled by G v Because it only looks at the Nyquist plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane singularities must be known). ( \(G(s) = \dfrac{s - 1}{s + 1}\). {\displaystyle \Gamma _{s}} s ) The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. D The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are all in the left half of the complex plane. (There is no particular reason that \(a\) needs to be real in this example. s The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s - 1)/(s + 1)}{1 + 2(s - 1)/(s + 1)} = \dfrac{s - 1}{3s - 1}.\]. 1 G 1 = Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\) stable? {\displaystyle \Gamma _{s}} Assume \(a\) is real, for what values of \(a\) is the open loop system \(G(s) = \dfrac{1}{s + a}\) stable? {\displaystyle 1+G(s)} Calculate transfer function of two parallel transfer functions in a feedback loop. The mathematics uses the Laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. But in physical systems, complex poles will tend to come in conjugate pairs.). ) D While Nyquist is one of the most general stability tests, it is still restricted to linear, time-invariant (LTI) systems. T The Nyquist method is used for studying the stability of linear systems with G Z k The row s 3 elements have 2 as the common factor. are the poles of the closed-loop system, and noting that the poles of Answer: The closed loop system is stable for \(k\) (roughly) between 0.7 and 3.10. poles at the origin), the path in L(s) goes through an angle of 360 in We regard this closed-loop system as being uncommon or unusual because it is stable for small and large values of gain \(\Lambda\), but unstable for a range of intermediate values. represents how slow or how fast is a reaction is. = It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. ) *(26- w.^2+2*j*w)); >> plot(real(olfrf007),imag(olfrf007)),grid, >> hold,plot(cos(cirangrad),sin(cirangrad)). This case can be analyzed using our techniques. ( 20 points) b) Using the Bode plots, calculate the phase margin and gain margin for K =1. ) We draw the following conclusions from the discussions above of Figures \(\PageIndex{3}\) through \(\PageIndex{6}\), relative to an uncommon system with an open-loop transfer function such as Equation \(\ref{eqn:17.18}\): Conclusion 2. regarding phase margin is a form of the Nyquist stability criterion, a form that is pertinent to systems such as that of Equation \(\ref{eqn:17.18}\); it is not the most general form of the criterion, but it suffices for the scope of this introductory textbook. Instead of Cauchy's argument principle, the original paper by Harry Nyquist in 1932 uses a less elegant approach.

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