{\displaystyle s={-1/k+j0}} ( 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. We consider a system whose transfer function is ). + = 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 \(\PageIndex{4}\). s Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular values of gain. However, the actual hardware of such an open-loop system could not be subjected to frequency-response experimental testing due to its unstable character, so a control-system engineer would find it necessary to analyze a mathematical model of the system. ) s ) We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. ) G 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 Double control loop for unstable systems. . But in physical systems, complex poles will tend to come in conjugate pairs.). G ) ( To simulate that testing, we have from Equation \(\ref{eqn:17.18}\), the following equation for the frequency-response function: \[O L F R F(\omega) \equiv O L T F(j \omega)=\Lambda \frac{104-\omega^{2}+4 \times j \omega}{(1+j \omega)\left(26-\omega^{2}+2 \times j \omega\right)}\label{eqn:17.20} \]. the clockwise direction. plane yielding a new contour. We will look a little more closely at such systems when we study the Laplace transform in the next topic. 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 P {\displaystyle {\mathcal {T}}(s)} ( Static and dynamic specifications. 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. poles of the form s s s , and the same system without its feedback loop). Here, \(\gamma\) is the imaginary \(s\)-axis and \(P_{G, RHP}\) is the number o poles of the original open loop system function \(G(s)\) in the right half-plane. G The Nyquist criterion allows us to answer two questions: 1. G s Answer: The closed loop system is stable for \(k\) (roughly) between 0.7 and 3.10. The mathematics uses the Laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. Draw the Nyquist plot with \(k = 1\). 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. 1This transfer function was concocted for the purpose of demonstration. Let \(\gamma_R = C_1 + C_R\). are called the zeros of So we put a circle at the origin and a cross at each pole. F {\displaystyle G(s)} A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. {\displaystyle \Gamma _{s}} {\displaystyle N} is mapped to the point The poles of the closed loop system function \(G_{CL} (s)\) given in Equation 12.3.2 are the zeros of \(1 + kG(s)\). ) T Notice that when the yellow dot is at either end of the axis its image on the Nyquist plot is close to 0. {\displaystyle \Gamma _{s}} Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. j F ( ) >> olfrf01=(104-w.^2+4*j*w)./((1+j*w). 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. are the poles of the closed-loop system, and noting that the poles of {\displaystyle P} To be able to analyze systems with poles on the imaginary axis, the Nyquist Contour can be modified to avoid passing through the point 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. {\displaystyle N(s)} . *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. G We can show this formally using Laurent series. r We may further reduce the integral, by applying Cauchy's integral formula. In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. {\displaystyle u(s)=D(s)} Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\) stable? ( for \(a > 0\). {\displaystyle GH(s)={\frac {A(s)}{B(s)}}} r s Note that the pinhole size doesn't alter the bandwidth of the detection system. plane in the same sense as the contour + ) 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. This is a case where feedback destabilized a stable system. We begin by considering the closed-loop characteristic polynomial (4.23) where L ( z) denotes the loop gain. ) -plane, For our purposes it would require and an indented contour along the imaginary axis. ( ( ( 1 0 Stability can be determined by examining the roots of the desensitivity factor polynomial Note that we count encirclements in the 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. s {\displaystyle -l\pi } ( Rule 2. is the number of poles of the closed loop system in the right half plane, and ( 0 For what values of \(a\) is the corresponding closed loop system \(G_{CL} (s)\) stable? This is just to give you a little physical orientation. j s u {\displaystyle P} has zeros outside the open left-half-plane (commonly initialized as OLHP). s ) N s A {\displaystyle {\frac {G}{1+GH}}} Complex Variables with Applications (Orloff), { "12.01:_Principle_of_the_Argument" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12.02:_Nyquist_Criterion_for_Stability" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12.03:_A_Bit_on_Negative_Feedback" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Complex_Algebra_and_the_Complex_Plane" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Analytic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Multivariable_Calculus_(Review)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Line_Integrals_and_Cauchys_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Cauchy_Integral_Formula" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Harmonic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Two_Dimensional_Hydrodynamics_and_Complex_Potentials" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Taylor_and_Laurent_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Residue_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Definite_Integrals_Using_the_Residue_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Conformal_Transformations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Argument_Principle" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Laplace_Transform" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Analytic_Continuation_and_the_Gamma_Function" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "license:ccbyncsa", "showtoc:no", "authorname:jorloff", "Nyquist criterion", "Pole-zero Diagrams", "Nyquist plot", "program:mitocw", "licenseversion:40", "source@https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FAnalysis%2FComplex_Variables_with_Applications_(Orloff)%2F12%253A_Argument_Principle%2F12.02%253A_Nyquist_Criterion_for_Stability, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Theorem \(\PageIndex{2}\) Nyquist criterion, source@https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018, status page at https://status.libretexts.org. ( , which is the contour 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. 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 \(\Lambda_{n s 1}\). , or simply the roots of Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. s 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. For the edge case where no poles have positive real part, but some are pure imaginary we will call the system marginally stable. {\displaystyle G(s)} has exactly the same poles as j + Is the closed loop system stable when \(k = 2\). 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. 1 So in the limit \(kG \circ \gamma_R\) becomes \(kG \circ \gamma\). {\displaystyle 1+GH(s)} ( L is called the open-loop transfer function. Keep in mind that the plotted quantity is A, i.e., the loop gain. The right hand graph is the Nyquist plot. {\displaystyle {\mathcal {T}}(s)} be the number of poles of . (2 h) lecture: Introduction to the controller's design specifications. + The system is stable if the modes all decay to 0, i.e. Rule 1. G 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 If the number of poles is greater than the . Calculate the Gain Margin. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Z 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. It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. The Bode plot for 0. Suppose that the open-loop transfer function of a system is1, \[G(s) \times H(s) \equiv O L T F(s)=\Lambda \frac{s^{2}+4 s+104}{(s+1)\left(s^{2}+2 s+26\right)}=\Lambda \frac{s^{2}+4 s+104}{s^{3}+3 s^{2}+28 s+26}\label{eqn:17.18} \]. s Moreover, if we apply for this system with \(\Lambda=4.75\) the MATLAB margin command to generate a Bode diagram in the same form as Figure 17.1.5, then MATLAB annotates that diagram with the values \(\mathrm{GM}=10.007\) dB and \(\mathrm{PM}=-23.721^{\circ}\) (the same as PM4.75 shown approximately on Figure \(\PageIndex{5}\)). The row s 3 elements have 2 as the common factor. H s However, the Nyquist Criteria can also give us additional information about a system. s If the answer to the first question is yes, how many closed-loop 2. 2. For a SISO feedback system the closed-looptransfer function is given by where represents the system and is the feedback element. B 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. + = We suppose that we have a clockwise (i.e. That is, the Nyquist plot is the circle through the origin with center \(w = 1\). The only pole is at \(s = -1/3\), so the closed loop system is stable. 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 new system is called a closed loop system. \(\PageIndex{4}\) 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. ) = Nyquist plot of the transfer function s/ (s-1)^3 Natural Language Math Input Extended Keyboard Examples Have a question about using Wolfram|Alpha? Physically the modes tell us the behavior of the system when the input signal is 0, but there are initial conditions. 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 \(\Lambda\)? ( ( {\displaystyle 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. Nyquist Stability Criterion A feedback system is stable if and only if \(N=-P\), i.e. Its system function is given by Black's formula, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)},\]. We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane: The Nyquist contour mapped through the function Is the closed loop system stable? = Right-half-plane (RHP) poles represent that instability. Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability.
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