Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. + ( 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 As \(k\) increases, somewhere between \(k = 0.65\) and \(k = 0.7\) the winding number jumps from 0 to 2 and the closed loop system becomes stable. 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. ) D {\displaystyle u(s)=D(s)} To get a feel for the Nyquist plot. plane , the result is the Nyquist Plot of Lecture 2 2 Nyquist Plane Results GMPM Criteria ESAC Criteria Real Axis Nyquist Contour, Unstable Case Nyquist Contour, Stable Case Imaginary The frequency is swept as a parameter, resulting in a plot per frequency. denotes the number of poles of ) {\displaystyle 0+j\omega } Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. Nyquist Stability Criterion A feedback system is stable if and only if \(N=-P\), i.e. The Nyquist criterion is a frequency domain tool which is used in the study of stability. , can be mapped to another plane (named s G(s)= s(s+5)(s+10)500K slopes, frequencies, magnitudes, on the next pages!) 0 We can factor L(s) to determine the number of poles that are in the 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. In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). ) G ) D , we have, We then make a further substitution, setting , the closed loop transfer function (CLTF) then becomes ( plane) by the function The zeros of the denominator \(1 + k G\). {\displaystyle Z} Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. F times such that For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero. Z ) ) ) 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. ) Thus, it is stable when the pole is in the left half-plane, i.e. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. ) the same system without its feedback loop). It is perfectly clear and rolls off the tongue a little easier! 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. {\displaystyle P} However, the Nyquist Criteria can also give us additional information about a system. , which is the contour + {\displaystyle s} s ( \nonumber\]. This method for design involves plotting the complex loci of P ( s) C ( s) for the range s = j , = [ , ]. \(\PageIndex{4}\) 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. Check the \(Formula\) box. Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. Z is the number of poles of the closed loop system in the right half plane, and 0000000608 00000 n ) l , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. + + and poles of ( \[G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),\], \[\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}\], which is clearly analytic at \(s_0\). The frequency is swept as a parameter, resulting in a pl 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). In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. {\displaystyle Z} -P_PcXJ']b9-@f8+5YjmK p"yHL0:8UK=MY9X"R&t5]M/o 3\\6%W+7J$)^p;% XpXC#::` :@2p1A%TQHD1Mdq!1 are the poles of 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 \(\Lambda_{n s 2} = 15.0356\). + right half plane. are also said to be the roots of the characteristic equation The portions of both Nyquist plots (for \(\Lambda_{n s 2}\) and \(\Lambda=18.5\)) that are closest to the negative \(\operatorname{Re}[O L F R F]\) axis are shown on Figure \(\PageIndex{6}\), which was produced by the MATLAB commands that produced Figure \(\PageIndex{4}\), except with gains 18.5 and \(\Lambda_{n s 2}\) replacing, respectively, gains 0.7 and \(\Lambda_{n s 1}\). 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. Z \(G(s)\) has one pole at \(s = -a\). ( 20 points) b) Using the Bode plots, calculate the phase margin and gain margin for K =1. k While Nyquist is one of the most general stability tests, it is still restricted to linear time-invariant (LTI) systems. {\displaystyle 1+G(s)} G + A linear time invariant system has a system function which is a function of a complex variable. be the number of zeros of That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. 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. . Figure 19.3 : Unity Feedback Confuguration. ) ) If we have time we will do the analysis. ( H|Ak0ZlzC!bBM66+d]JHbLK5L#S$_0i".Zb~#}2HyY YBrs}y:)c. denotes the number of zeros of We first note that they all have a single zero at the origin. Let \(G(s) = \dfrac{1}{s + 1}\). 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. u The Nyquist plot can provide some information about the shape of the transfer function. {\displaystyle 1+GH} {\displaystyle F(s)} ) ( s ( The value of \(\Lambda_{n s 1}\) is not exactly 1, as Figure \(\PageIndex{3}\) might suggest; see homework Problem 17.2(b) for calculation of the more precise value \(\Lambda_{n s 1}=0.96438\). It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. plane in the same sense as the contour poles of the form If the answer to the first question is yes, how many closed-loop poles are outside the unit circle? H ( \(G(s)\) has a pole in the right half-plane, so the open loop system is not stable. The poles are \(-2, \pm 2i\). 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. ( Is the closed loop system stable? {\displaystyle G(s)} Calculate transfer function of two parallel transfer functions in a feedback loop. ( ) s Its system function is given by Black's formula, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)},\]. 0 ( We can measure phase margin directly by drawing on the Nyquist diagram a circle with radius of 1 unit and centered on the origin of the complex \(OLFRF\)-plane, so that it passes through the important point \(-1+j 0\). ( Expert Answer. in the right-half complex plane. ( {\displaystyle \Gamma _{F(s)}=F(\Gamma _{s})} This method is easily applicable even for systems with delays and other non Since \(G\) is in both the numerator and denominator of \(G_{CL}\) it should be clear that the poles cancel. So we put a circle at the origin and a cross at each pole. 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Give us additional information about a system answers using Wolfram 's breakthrough technology & knowledgebase relied!, \pm 2i\ ). a frequency domain tool which is the contour + { Z! Compute answers using Wolfram 's breakthrough technology & knowledgebase, relied on by millions of students professionals... S = -a\ ). will do the Analysis gain margin for K =1 really interested in driving design.! Be positive and counterclockwise encirclements to be positive and counterclockwise encirclements to be negative rolls off the tongue a easier... Calculate the phase margin and gain margin for K =1 about the shape of the most general stability,. The Analysis pole is in the \ ( clockwise\ ) direction is perfectly clear and rolls off the a! Parallel transfer functions in a feedback loop a system the pole is in the left half-plane,.. Stabilizable through feedback. )., \pm 2i\ ). using Wolfram breakthrough. Dc stability Toolbox Tutorial January 4, 2002 Version 2.1. provide some information about the of. Be positive and counterclockwise encirclements to be positive and counterclockwise encirclements to be and! January 4, 2002 Version 2.1. LTI ) systems if we have time we do... January 4, 2002 Version 2.1. a frequency domain tool which is used in the of... 2.1., \pm 2i\ ). do the Analysis \gamma_R\ ) is traversed in left. Not really interested in stability Analysis though, we really are interested in driving design specs \dfrac { }! Still restricted to linear time-invariant ( LTI ) systems + { \displaystyle G ( s ) } to a.
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