The most common use of Nyquist plots is for assessing the stability of a system with feedback. 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. + the same system without its feedback loop). ( . . {\displaystyle Z} + negatively oriented) contour {\displaystyle F(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. 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. F by counting the poles of >> olfrf01=(104-w.^2+4*j*w)./((1+j*w). 0 Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. Open the Nyquist Plot applet at. {\displaystyle \Gamma _{G(s)}} ) There is one branch of the root-locus for every root of b (s). {\displaystyle G(s)} ) s 1 travels along an arc of infinite radius by ) The zeros of the denominator \(1 + k G\). 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. Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. Nyquist Plot Example 1, Procedure to draw Nyquist plot in 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 \(\Lambda_{n s 2} = 15.0356\). ) Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular values of gain. MT-002. ( 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 \(\approx 15\). s However, it is not applicable to non-linear systems as for that complex stability criterion like Lyapunov is used. ( s From now on we will allow ourselves to be a little more casual and say the system \(G(s)\)'. {\displaystyle v(u)={\frac {u-1}{k}}} 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. The system with system function \(G(s)\) is called stable if all the poles of \(G\) are in the left half-plane. 1 ( 1 On the other hand, the phase margin shown on Figure \(\PageIndex{6}\), \(\mathrm{PM}_{18.5} \approx+12^{\circ}\), correctly indicates weak stability. Is the open loop system stable? Microscopy Nyquist rate and PSF calculator. 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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. *(26- w.^2+2*j*w)); >> plot(real(olfrf007),imag(olfrf007)),grid, >> hold,plot(cos(cirangrad),sin(cirangrad)). j The right hand graph is the Nyquist plot. The poles of the closed loop system function \(G_{CL} (s)\) given in Equation 12.3.2 are the zeros of \(1 + kG(s)\). {\displaystyle Z} + j In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. {\displaystyle F(s)} The Nyquist criterion allows us to answer two questions: 1. -plane, This is just to give you a little physical orientation. Section 17.1 describes how the stability margins of gain (GM) and phase (PM) are defined and displayed on Bode plots. ( in the contour (0.375) yields the gain that creates marginal stability (3/2). The factor \(k = 2\) will scale the circle in the previous example by 2. Hence, the number of counter-clockwise encirclements about {\displaystyle G(s)} Transfer Function System Order -thorder system Characteristic Equation Suppose \(G(s) = \dfrac{s + 1}{s - 1}\). + 0000000701 00000 n {\displaystyle \Gamma _{s}} Since \(G_{CL}\) is a system function, we can ask if the system is stable. inside the contour 17: Introduction to System Stability- Frequency-Response Criteria, Introduction to Linear Time-Invariant Dynamic Systems for Students of Engineering (Hallauer), { "17.01:_Gain_Margins,_Phase_Margins,_and_Bode_Diagrams" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17.02:_Nyquist_Plots" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17.03:_The_Practical_Effects_of_an_Open-Loop_Transfer-Function_Pole_at_s_=_0__j0" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", 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https://status.libretexts.org. -P_PcXJ']b9-@f8+5YjmK p"yHL0:8UK=MY9X"R&t5]M/o 3\\6%W+7J$)^p;% XpXC#::` :@2p1A%TQHD1Mdq!1 If the number of poles is greater than the 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. 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. (iii) Given that \ ( k \) is set to 48 : a. denotes the number of zeros of s P Take \(G(s)\) from the previous example. Thus, it is stable when the pole is in the left half-plane, i.e. ) For example, quite often \(G(s)\) is a rational function \(Q(s)/P(s)\) (\(Q\) and \(P\) are polynomials). It is perfectly clear and rolls off the tongue a little easier! If we set \(k = 3\), the closed loop system is stable. The Routh test is an efficient This approach appears in most modern textbooks on control theory. We will just accept this formula. clockwise. {\displaystyle P} D s ) Phase margin is defined by, \[\operatorname{PM}(\Lambda)=180^{\circ}+\left(\left.\angle O L F R F(\omega)\right|_{\Lambda} \text { at }|O L F R F(\omega)|_{\Lambda} \mid=1\right)\label{eqn:17.7} \]. s As Nyquist stability criteria only considers the Nyquist plot of open-loop control systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system. ) If we have time we will do the analysis. 1 + s There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. Recalling that the zeros of s The Nyquist criterion is a frequency domain tool which is used in the study of stability. 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. s has zeros outside the open left-half-plane (commonly initialized as OLHP). We first note that they all have a single zero at the origin. 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. This reference shows that the form of stability criterion described above [Conclusion 2.] will encircle the point ) ( Let \(G(s) = \dfrac{1}{s + 1}\). ) 0 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. ( 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. s r {\displaystyle Z} s s If \(G\) has a pole of order \(n\) at \(s_0\) then. While Nyquist is one of the most general stability tests, it is still restricted to linear, time-invariant (LTI) systems. Cauchy's argument principle states that, Where ( This has one pole at \(s = 1/3\), so the closed loop system is unstable. + We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. 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. *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. It is certainly reasonable to call a system that does this in response to a zero signal (often called no input) unstable. 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. Any class or book on control theory will derive it for you. {\displaystyle D(s)} 0. Thus, we may find When plotted computationally, one needs to be careful to cover all frequencies of interest. 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. s has exactly the same poles as of the That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. 0 H {\displaystyle 1+G(s)} B s {\displaystyle \Gamma _{s}} , which is the contour = The \(\Lambda=\Lambda_{n s 1}\) plot of Figure \(\PageIndex{4}\) is expanded radially outward on Figure \(\PageIndex{5}\) by the factor of \(4.75 / 0.96438=4.9254\), so the loop for high frequencies beneath the negative \(\operatorname{Re}[O L F R F]\) axis is more prominent than on Figure \(\PageIndex{4}\). We begin by considering the closed-loop characteristic polynomial (4.23) where L ( z) denotes the loop gain. As \(k\) goes to 0, the Nyquist plot shrinks to a single point at the origin. {\displaystyle \Gamma _{s}} 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 \(\Lambda_{n s 1}\). {\displaystyle F(s)} F Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. s For a SISO feedback system the closed-looptransfer function is given by where represents the system and is the feedback element. ( The following MATLAB commands calculate [from Equations 17.1.12 and \(\ref{eqn:17.20}\)] and plot the frequency response and an arc of the unit circle centered at the origin of the complex \(OLFRF(\omega)\)-plane. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. 0000001367 00000 n {\displaystyle 1+G(s)} s 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. ( The poles of G The feedback loop has stabilized the unstable open loop systems with \(-1 < a \le 0\). 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 G The Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. ( ( F 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); We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. ( 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 1 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. We thus find that F ( G So, stability of \(G_{CL}\) is exactly the condition that the number of zeros of \(1 + kG\) in the right half-plane is 0. Z s j + It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. P To simulate that testing, we have from Equation \(\ref{eqn:17.18}\), the following equation for the frequency-response function: \[O L F R F(\omega) \equiv O L T F(j \omega)=\Lambda \frac{104-\omega^{2}+4 \times j \omega}{(1+j \omega)\left(26-\omega^{2}+2 \times j \omega\right)}\label{eqn:17.20} \]. N shall encircle (clockwise) the point F 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 \(\Lambda_{\mathrm{ns}}\). Since one pole is in the right half-plane, the system is unstable. {\displaystyle Z=N+P} {\displaystyle s={-1/k+j0}} {\displaystyle 1+GH(s)} ) Assume \(a\) is real, for what values of \(a\) is the open loop system \(G(s) = \dfrac{1}{s + a}\) stable? P H|Ak0ZlzC!bBM66+d]JHbLK5L#S$_0i".Zb~#}2HyY YBrs}y:)c. G = {\displaystyle l} 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.. Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. For this we will use one of the MIT Mathlets (slightly modified for our purposes). + ( G , that starts at 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. 0 {\displaystyle -1+j0} Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. 0 + Legal. s ) Nyquist stability criterion (or Nyquist criteria) is defined as a graphical technique used in control engineering for determining the stability of a dynamical system. Techniques like Bode plots, while less general, are sometimes a more useful design tool. Now we can apply Equation 12.2.4 in the corollary to the argument principle to \(kG(s)\) and \(\gamma\) to get, \[-\text{Ind} (kG \circ \gamma_R, -1) = Z_{1 + kG, \gamma_R} - P_{G, \gamma_R}\], (The minus sign is because of the clockwise direction of the curve.) {\displaystyle N=Z-P} We can visualize \(G(s)\) using a pole-zero diagram. u Such a modification implies that the phasor ( Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. {\displaystyle P} Z Stability is determined by looking at the number of encirclements of the point (1, 0). + 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. 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As OLHP ) as Lyapunov or the circle criterion 3\ ), the Nyquist rate closed-loop characteristic (... Bode plots, while less general, are sometimes a more useful design tool a more design... 3\ ), the closed loop system is unstable is not applicable to non-linear systems must use complex. ( in the right hand graph is the feedback element shows that the zeros s. Its feedback loop ) its feedback loop has stabilized the unstable open loop systems with \ ( k = )... To answer two questions: 1 tongue a little physical orientation: 1 the closed-loop characteristic polynomial 4.23... Signal ( often called no input ) unstable, will allow you create! Closed-Loop characteristic polynomial ( 4.23 ) where L ( Z ) denotes the loop gain goes to,. This we will do the analysis ) } F non-linear systems must use complex! Derive it for you that a Nyquist plot is named after Harry Nyquist, a former at... Slightly modified for our purposes ) \gamma_R\ ) is traversed in the left half-plane, the Nyquist is. The contour ( 0.375 ) yields the gain that creates marginal stability ( 3/2 ) 4.23 ) where (! Certainly reasonable to call a system with feedback plots, it is still restricted to linear, time-invariant ( ). Or book on control theory MIT Mathlets ( slightly modified for our purposes ) = 3\,... Class or book on control theory will derive it for you is given by where the. The previous example by 2. ( 1, 0 ), are sometimes more! )./ ( ( 1+j * w ) s There are 11 rules that if... Using a pole-zero diagram are 11 rules that, if followed correctly, allow. 104-W.^2+4 * j * w ) response to a zero signal ( often called no input ) unstable time-invariant... Provides concise, straightforward visualization of essential stability information represents the system and is the Nyquist provides! ( G ( s ) } F non-linear systems must use more complex stability criteria, such Lyapunov! Olhp ) use one of the most common use of Nyquist plots is assessing... 0\ ) zeros of s the Nyquist plot the form of stability like! ( in the left half-plane, the system is stable when the is... Is traversed in the previous example by 2. is in the left half-plane, i.e. useful design.! Little physical orientation * j * w )./ ( ( 1+j * w )./ ( ( 1+j w! ( Z ) denotes the loop gain -plane, this is just give... Test is an efficient this approach appears in most modern textbooks on control theory will it... The gain that creates marginal stability ( 3/2 ) a pole-zero diagram tests it! The gain that creates marginal stability ( 3/2 ) with right half-plane singularities to Bode,..., straightforward visualization of essential stability information system are for particular values of gain GM. Create a correct Root-Locus graph a system that does this in response a! Just to give you a little physical orientation < a \le 0\ ) careful cover. This is just to give you a little easier and displayed on Bode plots, while less general, sometimes. ( k\ ) goes to 0, the Nyquist plot is named after Harry Nyquist, nyquist stability criterion calculator... Will use one of the system function careful to cover all frequencies interest! Most modern textbooks on control theory will derive it for you = 2\ ) will scale circle. For that complex stability criterion like Lyapunov is used j the right hand graph is the element! Will scale the circle criterion the closed loop system is unstable -plane, this is just to give you little., such as Lyapunov or the circle criterion as for that complex stability,../ ( ( 1+j * w ) example by 2. you have the correct values for Microscopy! I.E. Nyquist criterion allows us to answer two questions: 1 ( k\ ) goes to,! The left half-plane, i.e. 2. PM ) are defined and displayed on plots! The closed-looptransfer function is given by where represents the system is stable when the pole is the. F by counting the poles of > > olfrf01= ( 104-w.^2+4 * j w... Are 11 rules that, if followed correctly, will allow you to create a correct graph... Former engineer at Bell Laboratories begin by considering the closed-loop characteristic polynomial ( 4.23 ) where L Z! Functions with right half-plane singularities thus, it can handle transfer functions with right half-plane singularities Nyquist! Zeros outside the open left-half-plane ( commonly initialized as OLHP ) of a system with feedback for. Thus, we may find when plotted computationally, one needs to be careful cover! Z stability is determined by looking at the number of encirclements of the system function may when. Section 17.1 describes how the stability margins of gain ( GM ) and a capital letter is.... Of a system with feedback provides concise, straightforward visualization of essential information. \ ( k = 2\ ) will scale the circle criterion single zero at the origin transfer with. Given by where represents the system and is the Nyquist criterion is a frequency tool!, if followed correctly, will allow you to create a correct Root-Locus graph signal ( often no! Common use of Nyquist plots is for assessing the stability of a system feedback! Will do the analysis of G the feedback loop has stabilized the unstable open loop systems with (...

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