The poles of \(G\). There is one branch of the root-locus for every root of b (s). Check the \(Formula\) box. ) Nyquist plot of the transfer function s/(s-1)^3. Choose \(R\) large enough that the (finite number) of poles and zeros of \(G\) in the right half-plane are all inside \(\gamma_R\). in the right-half complex plane. {\displaystyle N} N {\displaystyle H(s)} ( Note that a closed-loop-stable case has \(0<1 / \mathrm{GM}_{\mathrm{S}}<1\) so that \(\mathrm{GM}_{\mathrm{S}}>1\), and a closed-loop-unstable case has \(1 / \mathrm{GM}_{\mathrm{U}}>1\) so that \(0<\mathrm{GM}_{\mathrm{U}}<1\). The Nyquist plot of {\displaystyle N} This has one pole at \(s = 1/3\), so the closed loop system is unstable. The Routh test is an efficient Compute answers using Wolfram's breakthrough technology & 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 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. encirclements of the -1+j0 point in "L(s).". s if the poles are all in the left half-plane. Any class or book on control theory will derive it for you. {\displaystyle P} Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\) stable? This method is easily applicable even for systems with delays and other non >> olfrf01=(104-w.^2+4*j*w)./((1+j*w). We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. Look at the pole diagram and use the mouse to drag the yellow point up and down the imaginary axis. + For example, Brogan, 1974, page 25, wrote Experience has shown that acceptable transient response will usually require stability margins on the order of \(\mathrm{PM}>30^{\circ}\), \(\mathrm{GM}>6\) dB. Franklin, et al., 1991, page 285, wrote Many engineers think directly in terms of \(\text { PM }\) in judging whether a control system is adequately stabilized. 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. Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. plane) by the function The poles of \(G(s)\) correspond to what are called modes of the system. 0 s 0000001188 00000 n P If the number of poles is greater than the number of zeros, then the Nyquist criterion tells us how to use the Nyquist plot to graphically determine the stability of the closed loop system. , which is to say our Nyquist plot. The Nyquist stability criterion has been used extensively in science and engineering to assess the stability of physical systems that can be represented by sets of linear equations. Assume \(a\) is real, for what values of \(a\) is the open loop system \(G(s) = \dfrac{1}{s + a}\) stable? 1 ( ). This should make sense, since with \(k = 0\), \[G_{CL} = \dfrac{G}{1 + kG} = G. \nonumber\]. Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability. You have already encountered linear time invariant systems in 18.03 (or its equivalent) when you solved constant coefficient linear differential equations. Is the system with system function \(G(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}\) stable? ) u ) It does not represent any specific real physical system, but it has characteristics that are representative of some real systems. ( In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). Moreover, we will add to the same graph the Nyquist plots of frequency response for a case of positive closed-loop stability with \(\Lambda=1 / 2 \Lambda_{n s}=20,000\) s-2, and for a case of closed-loop instability with \(\Lambda= 2 \Lambda_{n s}=80,000\) s-2. entire right half plane. 0000002345 00000 n F But in physical systems, complex poles will tend to come in conjugate pairs.). inside the contour Suppose that the open-loop transfer function of a system is1, \[G(s) \times H(s) \equiv O L T F(s)=\Lambda \frac{s^{2}+4 s+104}{(s+1)\left(s^{2}+2 s+26\right)}=\Lambda \frac{s^{2}+4 s+104}{s^{3}+3 s^{2}+28 s+26}\label{eqn:17.18} \]. We thus find that {\displaystyle {\mathcal {T}}(s)} denotes the number of zeros of 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 \(\PageIndex{5}\)). ( {\displaystyle F(s)} We will make a standard assumption that \(G(s)\) is meromorphic with a finite number of (finite) poles. To begin this study, we will repeat the Nyquist plot of Figure 17.2.2, the closed-loop neutral-stability case, for which \(\Lambda=\Lambda_{n s}=40,000\) s-2 and \(\omega_{n s}=100 \sqrt{3}\) rad/s, but over a narrower band of excitation frequencies, \(100 \leq \omega \leq 1,000\) rad/s, or \(1 / \sqrt{3} \leq \omega / \omega_{n s} \leq 10 / \sqrt{3}\); the intent here is to restrict our attention primarily to frequency response for which the phase lag exceeds about 150, i.e., for which the frequency-response curve in the \(OLFRF\)-plane is somewhat close to the negative real axis. We draw the following conclusions from the discussions above of Figures \(\PageIndex{3}\) through \(\PageIndex{6}\), relative to an uncommon system with an open-loop transfer function such as Equation \(\ref{eqn:17.18}\): Conclusion 2. regarding phase margin is a form of the Nyquist stability criterion, a form that is pertinent to systems such as that of Equation \(\ref{eqn:17.18}\); it is not the most general form of the criterion, but it suffices for the scope of this introductory textbook. The Nyquist criterion for systems with poles on the imaginary axis. The above consideration was conducted with an assumption that the open-loop transfer function G ( s ) {displaystyle G(s)} does not have any pole on the imaginary axis (i.e. poles of the form 0 + j {displaystyle 0+jomega } ). = s has zeros outside the open left-half-plane (commonly initialized as OLHP). 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 {\mathcal {T}}(s)} ( {\displaystyle N=Z-P} T 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 right hand graph is the Nyquist plot. j Z ) , the result is the Nyquist Plot of The Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. s We will be concerned with the stability of the system. 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. s Since \(G_{CL}\) is a system function, we can ask if the system is stable. ) ) s The MATLAB commands follow that calculate [from Equations 17.1.7 and 17.1.12] and plot these cases of open-loop frequency-response function, and the resulting Nyquist diagram (after additional editing): >> olfrf01=wb./(j*w.*(j*w+coj). ) N 1 The frequency is swept as a parameter, resulting in a plot per frequency. {\displaystyle Z} Stability can be determined by examining the roots of the desensitivity factor polynomial 1 1 s The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are The Nyquist method is used for studying the stability of linear systems with {\displaystyle \Gamma _{s}} As \(k\) goes to 0, the Nyquist plot shrinks to a single point at the origin. D s Phase margin is defined by, \[\operatorname{PM}(\Lambda)=180^{\circ}+\left(\left.\angle O L F R F(\omega)\right|_{\Lambda} \text { at }|O L F R F(\omega)|_{\Lambda} \mid=1\right)\label{eqn:17.7} \]. ( In units of Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. ) = B Phase margins are indicated graphically on Figure \(\PageIndex{2}\). of poles of T(s)). G k s *(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 \(\PageIndex{1}\). {\displaystyle \Gamma _{s}} s F The roots of b (s) are the poles of the open-loop transfer function. be the number of zeros of A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. (iii) Given that \ ( k \) is set to 48 : a. For this topic we will content ourselves with a statement of the problem with only the tiniest bit of physical context. s 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 \(\PageIndex{5}\). s ( N Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. 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 \(-1+j 0\). A defective metric of stability Analysis Consortium ESAC DC stability Toolbox Tutorial January 4, 2002 Version 2.1.... ). `` Sudhoff Energy Sources nyquist stability criterion calculator Consortium ESAC DC stability Tutorial! A parametric plot nyquist stability criterion calculator a Nyquist plot is a parametric plot of a frequency response used in automatic and. 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