In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). G 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 poles are \(-2, -2\pm i\). 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 olfrf01=(104-w.^2+4*j*w)./((1+j*w). , where Is the closed loop system stable when \(k = 2\). j 0000002345 00000 n \(G(s) = \dfrac{s - 1}{s + 1}\). Gain margin (GM) is defined by Equation 17.1.8, from which we find, \[\frac{1}{G M(\Lambda)}=|O L F R F(\omega)|_{\mid} \mid \text {at }\left.\angle O L F R F(\omega)\right|_{\Lambda}=-180^{\circ}\label{eqn:17.17} \]. F ) ) 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 Nyquist method is used for studying the stability of linear systems with pure time delay. Legal. To simulate that testing, we have from Equation \(\ref{eqn:17.18}\), the following equation for the frequency-response function: \[O L F R F(\omega) \equiv O L T F(j \omega)=\Lambda \frac{104-\omega^{2}+4 \times j \omega}{(1+j \omega)\left(26-\omega^{2}+2 \times j \omega\right)}\label{eqn:17.20} \]. + We will now rearrange the above integral via substitution. j Is the closed loop system stable? 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. ( G = *(26- w.^2+2*j*w)); >> plot(real(olfrf007),imag(olfrf007)),grid, >> hold,plot(cos(cirangrad),sin(cirangrad)). k Is the closed loop system stable when \(k = 2\). s F Lecture 1: The Nyquist Criterion S.D. is formed by closing a negative unity feedback loop around the open-loop transfer function Additional parameters 0. (Actually, for \(a = 0\) the open loop is marginally stable, but it is fully stabilized by the closed loop.). right half plane. {\displaystyle Z} Suppose \(G(s) = \dfrac{s + 1}{s - 1}\). 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 \(\Lambda_{n s 1}\). in the contour Natural Language; Math Input; Extended Keyboard Examples Upload Random. In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. ) . ) There is one branch of the root-locus for every root of b (s). does not have any pole on the imaginary axis (i.e. 0000039854 00000 n + Given our definition of stability above, we could, in principle, discuss stability without the slightest idea what it means for physical systems. {\displaystyle G(s)} = 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. j 0.375=3/2 (the current gain (4) multiplied by the gain margin 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. are the poles of s 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 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.. 1 For instance, the plot provides information on the difference between the number of zeros and poles of the transfer function[5] by the angle at which the curve approaches the origin. The Nyquist bandwidth is defined to be the frequency spectrum from dc to fs/2.The frequency spectrum is divided into an infinite number of Nyquist zones, each having a width equal to 0.5fs as shown. Nyquist plot of \(G(s) = 1/(s + 1)\), with \(k = 1\). s Hence, the number of counter-clockwise encirclements about If the system with system function \(G(s)\) is unstable it can sometimes be stabilized by what is called a negative feedback loop. (There is no particular reason that \(a\) needs to be real in this example. , which is to say our Nyquist plot. , that starts at Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. {\displaystyle {\mathcal {T}}(s)} ( 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 \(-1+j 0\). All the coefficients of the characteristic polynomial, s 4 + 2 s 3 + s 2 + 2 s + 1 are positive. This gives us, We now note that Section 17.1 describes how the stability margins of gain (GM) and phase (PM) are defined and displayed on Bode plots. {\displaystyle s} >> olfrf01=(104-w.^2+4*j*w)./((1+j*w). (iii) Given that \ ( k \) is set to 48 : a. Our goal is to, through this process, check for the stability of the transfer function of our unity feedback system with gain k, which is given by, That is, we would like to check whether the characteristic equation of the above transfer function, given by. Z G For this we will use one of the MIT Mathlets (slightly modified for our purposes). 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. As \(k\) goes to 0, the Nyquist plot shrinks to a single point at the origin. Pole-zero diagrams for the three systems. This is a case where feedback destabilized a stable system. = {\displaystyle G(s)} {\displaystyle G(s)} The poles of {\displaystyle G(s)} {\displaystyle F(s)} It applies the principle of argument to an open-loop transfer function to derive information about the stability of the closed-loop systems transfer function. There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. Let \(G(s)\) be such a system function. {\displaystyle Z} 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 + + ) 1 Instead of Cauchy's argument principle, the original paper by Harry Nyquist in 1932 uses a less elegant approach. {\displaystyle D(s)=0} (2 h) lecture: Introduction to the controller's design specifications. the same system without its feedback loop). The factor \(k = 2\) will scale the circle in the previous example by 2. 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. 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. + s ( 20 points) b) Using the Bode plots, calculate the phase margin and gain margin for K =1. That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. ( T A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. 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. With the same poles and zeros, move the \(k\) slider and determine what range of \(k\) makes the closed loop system stable. G s Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. The above consideration was conducted with an assumption that the open-loop transfer function The only plot of \(G(s)\) is in the left half-plane, so the open loop system is stable. of poles of T(s)). 1 by counting the poles of u 1 G 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. {\displaystyle N=P-Z} are also said to be the roots of the characteristic equation Give zero-pole diagrams for each of the systems, \[G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\]. . 1 s {\displaystyle 1+kF(s)} G Legal. Z Choose \(R\) large enough that the (finite number) of poles and zeros of \(G\) in the right half-plane are all inside \(\gamma_R\). , which is to say. This results from the requirement of the argument principle that the contour cannot pass through any pole of the mapping function. To use this criterion, the frequency response data of a system must be presented as a polar plot in In \(\gamma (\omega)\) the variable is a greek omega and in \(w = G \circ \gamma\) we have a double-u. This method is easily applicable even for systems with delays and other non {\displaystyle F(s)} H , the closed loop transfer function (CLTF) then becomes On the other hand, a Bode diagram displays the phase-crossover and gain-crossover frequencies, which are not explicit on a traditional Nyquist plot. s ( ) 1 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. So, stability of \(G_{CL}\) is exactly the condition that the number of zeros of \(1 + kG\) in the right half-plane is 0. That is, if the unforced system always settled down to equilibrium. . 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 P} {\displaystyle (-1+j0)} L is called the open-loop transfer function. The only pole is at \(s = -1/3\), so the closed loop system is stable. *(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 \(\PageIndex{1}\). 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. The roots of b (s) are the poles of the open-loop transfer function. + For a SISO feedback system the closed-looptransfer function is given by where represents the system and is the feedback element. We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. ) F Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. We consider a system whose transfer function is s represents how slow or how fast is a reaction is. Note on Figure \(\PageIndex{2}\) that the phase-crossover point (phase angle \(\phi=-180^{\circ}\)) and the gain-crossover point (magnitude ratio \(MR = 1\)) of an \(FRF\) are clearly evident on a Nyquist plot, perhaps even more naturally than on a Bode diagram. By the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of s Determining Stability using the Nyquist Plot - Erik Cheever G G Compute answers using Wolfram's breakthrough technology & have positive real part. + G s One way to do it is to construct a semicircular arc with radius "1+L(s)=0.". T We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. That is, the Nyquist plot is the circle through the origin with center \(w = 1\). F (At \(s_0\) it equals \(b_n/(kb_n) = 1/k\).). This has one pole at \(s = 1/3\), so the closed loop system is unstable. s This reference shows that the form of stability criterion described above [Conclusion 2.] s by Cauchy's argument principle. ( s The most common use of Nyquist plots is for assessing the stability of a system with feedback. 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