We consider a system whose transfer function is ) G {\displaystyle G(s)} 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. Techniques like Bode plots, while less general, are sometimes a more useful design tool. ) The Nyquist method is used for studying the stability of linear systems with pure time delay. The mathematics uses the Laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. {\displaystyle N} 0000001367 00000 n {\displaystyle 1+G(s)} ) j and that encirclements in the opposite direction are negative encirclements. ( . . ( ) 0000039933 00000 n P G ( Is the closed loop system stable when \(k = 2\). s One way to do it is to construct a semicircular arc with radius j can be expressed as the ratio of two polynomials: 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. The Nyquist criterion is a frequency domain tool which is used in the study of stability. s For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero. F a clockwise semicircle at L(s)= in "L(s)" (see, The clockwise semicircle at infinity in "s" corresponds to a single For our purposes it would require and an indented contour along the imaginary axis. Nyquist plot of \(G(s) = 1/(s + 1)\), with \(k = 1\). + The value of \(\Lambda_{n s 2}\) is not exactly 15, as Figure \(\PageIndex{3}\) might suggest; see homework Problem 17.2(b) for calculation of the more precise value \(\Lambda_{n s 2} = 15.0356\). ) The Nyquist plot of We will look a little more closely at such systems when we study the Laplace transform in the next topic. ( the same system without its feedback loop). H s The Routh test is an efficient 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. s ) This is a case where feedback destabilized a stable system. D Right-half-plane (RHP) poles represent that instability. , where ( must be equal to the number of open-loop poles in the RHP. s {\displaystyle F(s)} {\displaystyle {\mathcal {T}}(s)} s 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. Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}\) stable? 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. {\displaystyle s={-1/k+j0}} We can factor L(s) to determine the number of poles that are in the 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. Given our definition of stability above, we could, in principle, discuss stability without the slightest idea what it means for physical systems. In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. In the case \(G(s)\) is a fractional linear transformation, so we know it maps the imaginary axis to a circle. Does the system have closed-loop poles outside the unit circle? F ( . -P_PcXJ']b9-@f8+5YjmK p"yHL0:8UK=MY9X"R&t5]M/o 3\\6%W+7J$)^p;% XpXC#::` :@2p1A%TQHD1Mdq!1 {\displaystyle \Gamma _{F(s)}=F(\Gamma _{s})} We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. = s The above consideration was conducted with an assumption that the open-loop transfer function 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 Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular values of gain. ) The Nyquist plot can provide some information about the shape of the transfer function. {\displaystyle {\mathcal {T}}(s)={\frac {N(s)}{D(s)}}.}. Stability in the Nyquist Plot. G Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. {\displaystyle {\frac {G}{1+GH}}} If we have time we will do the analysis. The shift in origin to (1+j0) gives the characteristic equation plane. 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. {\displaystyle Z} F D , or simply the roots of The most common case are systems with integrators (poles at zero). We conclude this chapter on frequency-response stability criteria by observing that margins of gain and phase are used also as engineering design goals. 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. Since there are poles on the imaginary axis, the system is marginally stable. (Actually, for \(a = 0\) the open loop is marginally stable, but it is fully stabilized by the closed loop.). F is mapped to the point s \(G(s) = \dfrac{s - 1}{s + 1}\). 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 \(\PageIndex{5}\). Since one pole is in the right half-plane, the system is unstable. , let G H Graphical method of determining the stability of a dynamical system, The Nyquist criterion for systems with poles on the imaginary axis, "Chapter 4.3. ( k 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. When plotted computationally, one needs to be careful to cover all frequencies of interest. {\displaystyle 0+j(\omega -r)} To get a feel for the Nyquist plot. be the number of zeros of For this we will use one of the MIT Mathlets (slightly modified for our purposes). N G That is, the Nyquist plot is the image of the imaginary axis under the map \(w = kG(s)\). , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. = ( v enclosing the right half plane, with indentations as needed to avoid passing through zeros or poles of the function is determined by the values of its poles: for stability, the real part of every pole must be negative. The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are That is, \[s = \gamma (\omega) = i \omega, \text{ where } -\infty < \omega < \infty.\], For a system \(G(s)\) and a feedback factor \(k\), the Nyquist plot is the plot of the curve, \[w = k G \circ \gamma (\omega) = kG(i \omega).\]. {\displaystyle -1/k} Lets look at an example: Note that I usually dont include negative frequencies in my Nyquist plots. But in physical systems, complex poles will tend to come in conjugate pairs.). Z r Compute answers using Wolfram's breakthrough technology & Here N = 1. {\displaystyle u(s)=D(s)} ( So we put a circle at the origin and a cross at each pole. The only plot of \(G(s)\) is in the left half-plane, so the open loop system is stable. s Answer: The closed loop system is stable for \(k\) (roughly) between 0.7 and 3.10. s T 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. 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 F(s)} {\displaystyle {\mathcal {T}}(s)} + ) We first note that they all have a single zero at the origin. 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. Static and dynamic specifications. This is just to give you a little physical orientation. H However, the gain margin calculated from either of the two phase crossovers suggests instability, showing that both are deceptively defective metrics of stability. The Nyquist method is used for studying the stability of linear systems with G(s)= s(s+5)(s+10)500K slopes, frequencies, magnitudes, on the next pages!) ( Legal. "1+L(s)" in the right half plane (which is the same as the number Z G ) Thus, we may find j ( G 1 inside the contour {\displaystyle G(s)} {\displaystyle G(s)} Since they are all in the left half-plane, the system is stable. 0 Determining Stability using the Nyquist Plot - Erik Cheever D ( D Thus, we may finally state that. 0 G 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. Nyquist Stability Criterion A feedback system is stable if and only if \(N=-P\), i.e. Looking at Equation 12.3.2, there are two possible sources of poles for \(G_{CL}\). Suppose that the open-loop transfer function of a system is1, \[G(s) \times H(s) \equiv O L T F(s)=\Lambda \frac{s^{2}+4 s+104}{(s+1)\left(s^{2}+2 s+26\right)}=\Lambda \frac{s^{2}+4 s+104}{s^{3}+3 s^{2}+28 s+26}\label{eqn:17.18} \]. The system is stable if the modes all decay to 0, i.e. + s {\displaystyle Z=N+P} 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. ) that appear within the contour, that is, within the open right half plane (ORHP). s L is called the open-loop transfer function. We will be concerned with the stability of the system. / , and . for \(a > 0\). All the coefficients of the characteristic polynomial, s 4 + 2 s 3 + s 2 + 2 s + 1 are positive. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. 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. ) as the first and second order system. ). Alternatively, and more importantly, if ( In general, the feedback factor will just scale the Nyquist plot. + Nyquist Plot Example 1, Procedure to draw Nyquist plot in This should make sense, since with \(k = 0\), \[G_{CL} = \dfrac{G}{1 + kG} = G. \nonumber\]. 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. In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). u T Note that we count encirclements in the , which is the contour ) ) While Nyquist is one of the most general stability tests, it is still restricted to linear time-invariant (LTI) systems. 0000000608 00000 n ) s G = ( j If the number of poles is greater than the To use this criterion, the frequency response data of a system must be presented as a polar plot in 0 {\displaystyle G(s)} However, the Nyquist Criteria can also give us additional information about a system. (At \(s_0\) it equals \(b_n/(kb_n) = 1/k\).). It can happen! ) {\displaystyle P} 0000002345 00000 n = is the number of poles of the closed loop system in the right half plane, and {\displaystyle G(s)} s Recalling that the zeros of Natural Language; Math Input; Extended Keyboard Examples Upload Random. Such a modification implies that the phasor ( s ) 1 Lecture 1: The Nyquist Criterion S.D. F Z The Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. 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. Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. 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); as defined above corresponds to a stable unity-feedback system when Let \(\gamma_R = C_1 + C_R\). ) 1 = 0000039854 00000 n ) , we have, We then make a further substitution, setting {\displaystyle N=Z-P} . If we set \(k = 3\), the closed loop system is stable. Now how can I verify this formula for the open-loop transfer function: H ( s) = 1 s 3 ( s + 1) The Nyquist plot is attached in the image. + s 1 For this topic we will content ourselves with a statement of the problem with only the tiniest bit of physical context. ) the clockwise direction. {\displaystyle Z} P This case can be analyzed using our techniques. T ) 1This transfer function was concocted for the purpose of demonstration. ( G {\displaystyle Z} G ( Since \(G_{CL}\) is a system function, we can ask if the system is stable. The Bode plot for {\displaystyle D(s)} Z ) B and , as evaluated above, is equal to0. + l {\displaystyle N=P-Z} ( The roots of b (s) are the poles of the open-loop transfer function. 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