nyquist stability criterion calculator

The row s 3 elements have 2 as the common factor. In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). Nyquist plot of $G(s) = 1/(s + 1)$, with $k = 1$. a clockwise semicircle at L(s)= in "L(s)" (see, The clockwise semicircle at infinity in "s" corresponds to a single ( It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. s F {\displaystyle G(s)} If {\displaystyle \Gamma _{s}} = ; when placed in a closed loop with negative feedback 1 {\displaystyle {\mathcal {T}}(s)} + 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. s Keep in mind that the plotted quantity is A, i.e., the loop gain. s {\displaystyle \Gamma _{s}} 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. Since $G$ is in both the numerator and denominator of $G_{CL}$ it should be clear that the poles cancel. $\PageIndex{4}$ 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. The Nyquist plot is the graph of $kG(i \omega)$. It is perfectly clear and rolls off the tongue a little easier! s Equation $\ref{eqn:17.17}$ is illustrated on Figure $\PageIndex{2}$ for both closed-loop stable and unstable cases. With $k =1$, what is the winding number of the Nyquist plot around -1? Lets look at an example: Note that I usually dont include negative frequencies in my Nyquist plots. point in "L(s)". {\displaystyle F(s)} Let $G(s)$ be such a system function. ( {\displaystyle 0+j\omega } Look at the pole diagram and use the mouse to drag the yellow point up and down the imaginary axis. the clockwise direction. {\displaystyle 1+G(s)} , that starts at in the right-half complex plane minus the number of poles of The factor $k = 2$ will scale the circle in the previous example by 2. 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. In its original state, applet should have a zero at $s = 1$ and poles at $s = 0.33 \pm 1.75 i$. + ) s A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. {\displaystyle G(s)} So far, we have been careful to say the system with system function $G(s)$'. However, the Nyquist Criteria can also give us additional information about a system. 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 F(s)} {\displaystyle Z} By the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of If we have time we will do the analysis. Additional parameters appear if you check the option to calculate the Theoretical PSF. Static and dynamic specifications. The poles of $G$. (iii) Given that \ ( k \) is set to 48 : a. Gain $\Lambda$ has physical units of s-1, but we will not bother to show units in the following discussion. ) , we have, We then make a further substitution, setting We regard this closed-loop system as being uncommon or unusual because it is stable for small and large values of gain $\Lambda$, but unstable for a range of intermediate values. 0000001367 00000 n {\displaystyle -1/k} This is a diagram in the $s$-plane where we put a small cross at each pole and a small circle at each zero. In the previous problem could you determine analytically the range of $k$ where $G_{CL} (s)$ is stable? The left hand graph is the pole-zero diagram. Lecture 1: The Nyquist Criterion S.D. s The most common case are systems with integrators (poles at zero). Z + , which is to say. G 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. s We will be concerned with the stability of the system. 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 Legal. . (2 h) lecture: Introduction to the controller's design specifications. Closed loop approximation f.d.t. , and the roots of 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. Figure 19.3 : Unity Feedback Confuguration. F = 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 Assume $a$ is real, for what values of $a$ is the open loop system $G(s) = \dfrac{1}{s + a}$ stable? For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero. Let us complete this study by computing $\operatorname{OLFRF}(\omega)$ and displaying it on Nyquist plots for the value corresponding to the transition from instability back to stability on Figure $\PageIndex{3}$, which we denote as $\Lambda_{n s 2} \approx 15$, and for a slightly higher value, $\Lambda=18.5$, for which the closed-loop system is stable. The system is stable if the modes all decay to 0, i.e. F is not sufficiently general to handle all cases that might arise. $G(s) = \dfrac{s - 1}{s + 1}$. j {\displaystyle -l\pi } Pole-zero diagrams for the three systems. Das Stabilittskriterium von Strecker-Nyquist", "Inventing the 'black box': mathematics as a neglected enabling technology in the history of communications engineering", EIS Spectrum Analyser - a freeware program for analysis and simulation of impedance spectra, Mathematica function for creating the Nyquist plot, https://en.wikipedia.org/w/index.php?title=Nyquist_stability_criterion&oldid=1121126255, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License 3.0, However, if the graph happens to pass through the point, This page was last edited on 10 November 2022, at 17:05. The right hand graph is the Nyquist plot. s Nyquist plot of the transfer function s/(s-1)^3. 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. {\displaystyle G(s)} (ii) Determine the range of \ ( k \) to ensure a stable closed loop response. We know from Figure $\PageIndex{3}$ that the closed-loop system with $\Lambda = 18.5$ is stable, albeit weakly. The Nyquist criterion is a frequency domain tool which is used in the study of stability. The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s + 1)/(s - 1)}{1 + 2(s + 1)/(s - 1)} = \dfrac{s + 1}{3s + 1}.\]. = shall encircle (clockwise) the point 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. Graphical method of determining the stability of a dynamical system, The Nyquist criterion for systems with poles on the imaginary axis, "Chapter 4.3. Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. 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. = The poles are $-2, -2\pm i$. ( {\displaystyle G(s)} s Suppose that $G(s)$ has a finite number of zeros and poles in the right half-plane. G The theorem recognizes these. Does the system have closed-loop poles outside the unit circle? Z ( + ) Let $G(s) = \dfrac{1}{s + 1}$. 0000039854 00000 n ( s 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. The Nyquist method is used for studying the stability of linear systems with D Gain margin (GM) is defined by Equation 17.1.8, from which we find, \[\frac{1}{G M(\Lambda)}=|O L F R F(\omega)|_{\mid} \mid \text {at }\left.\angle O L F R F(\omega)\right|_{\Lambda}=-180^{\circ}\label{eqn:17.17} \]. denotes the number of zeros of We know from Figure $\PageIndex{3}$ that this case of $\Lambda=4.75$ is closed-loop unstable. ) To use this criterion, the frequency response data of a system must be presented as a polar plot in F It is easy to check it is the circle through the origin with center $w = 1/2$. ( are the poles of the closed-loop system, and noting that the poles of *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. 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 $\PageIndex{5}$. 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Let us consider next an uncommon system, for which the determination of stability or instability requires a more detailed examination of the stability margins. However, the gain margin calculated from either of the two phase crossovers suggests instability, showing that both are deceptively defective metrics of stability. Microscopy Nyquist rate and PSF calculator. + Phase margin is defined by, \[\operatorname{PM}(\Lambda)=180^{\circ}+\left(\left.\angle O L F R F(\omega)\right|_{\Lambda} \text { at }|O L F R F(\omega)|_{\Lambda} \mid=1\right)\label{eqn:17.7} \]. 1 0000001503 00000 n where $k$ is called the feedback factor. The shift in origin to (1+j0) gives the characteristic equation plane. With the same poles and zeros, move the $k$ slider and determine what range of $k$ makes the closed loop system stable. There are two poles in the right half-plane, so the open loop system $G(s)$ is unstable. . + Since we know N and P, we can determine Z, the number of zeros of ( ) Observe on Figure $\PageIndex{4}$ the small loops beneath the negative $\operatorname{Re}[O L F R F]$ axis as driving frequency becomes very high: the frequency responses approach zero from below the origin of the complex $OLFRF$-plane. s Nyquist Plot Example 1, Procedure to draw Nyquist plot in The positive $\mathrm{PM}_{\mathrm{S}}$ for a closed-loop-stable case is the counterclockwise angle from the negative $\operatorname{Re}[O L F R F]$ axis to the intersection of the unit circle with the $OLFRF_S$ curve; conversely, the negative $\mathrm{PM}_U$ for a closed-loop-unstable case is the clockwise angle from the negative $\operatorname{Re}[O L F R F]$ axis to the intersection of the unit circle with the $OLFRF_U$ curve. + We can show this formally using Laurent series. 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 can happen! 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. 1This transfer function was concocted for the purpose of demonstration. {\displaystyle 1+G(s)} There are no poles in the right half-plane. s The poles of $G(s)$ correspond to what are called modes of the system. Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. ( , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. {\displaystyle G(s)} + plane in the same sense as the contour for $a > 0$. In fact, we find that the above integral corresponds precisely to the number of times the Nyquist plot encircles the point ( The Nyquist plot of 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. ( On the other hand, the phase margin shown on Figure $\PageIndex{6}$, $\mathrm{PM}_{18.5} \approx+12^{\circ}$, correctly indicates weak stability. 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.) 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. 1 of the ( ) {\displaystyle G(s)} + The Nyquist criterion is an important stability test with applications to systems, circuits, and networks [1]. H . Given our definition of stability above, we could, in principle, discuss stability without the slightest idea what it means for physical systems. are, respectively, the number of zeros of Nyquist plot of the transfer function s/ (s-1)^3 Natural Language Math Input Extended Keyboard Examples Have a question about using Wolfram|Alpha? The most common use of Nyquist plots is for assessing the stability of a system with feedback. s s 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 Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. ( + have positive real part. The new system is called a closed loop system. ) ) s s 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 $\Lambda_{n s 1}$. travels along an arc of infinite radius by 0 does not have any pole on the imaginary axis (i.e. ( ) , let 0 The poles are $\pm 2, -2 \pm i$. {\displaystyle \Gamma _{s}} The above consideration was conducted with an assumption that the open-loop transfer function as defined above corresponds to a stable unity-feedback system when by Cauchy's argument principle. {\displaystyle u(s)=D(s)} 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 $\approx 15$. ( G ( ) Natural Language; Math Input; Extended Keyboard Examples Upload Random. P G That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. For the edge case where no poles have positive real part, but some are pure imaginary we will call the system marginally stable. 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. Nyquist stability criterion (or Nyquist criteria) is defined as a graphical technique used in control engineering for determining the stability of a dynamical system. {\displaystyle Z} j Accessibility StatementFor more information contact us [email protected] check out our status page at https://status.libretexts.org. the same system without its feedback loop). T Take $G(s)$ from the previous example. From complex analysis, a contour Let us continue this study by computing $OLFRF(\omega)$ and displaying it as a Nyquist plot for an intermediate value of gain, $\Lambda=4.75$, for which Figure $\PageIndex{3}$ shows the closed-loop system is unstable. . , where G(s)= s(s+5)(s+10)500K slopes, frequencies, magnitudes, on the next pages!) 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. N >> olfrf01=(104-w.^2+4*j*w)./((1+j*w). Determining Stability using the Nyquist Plot - Erik Cheever be the number of zeros of If we set $k = 3$, the closed loop system is stable. This typically means that the parameter is swept logarithmically, in order to cover a wide range of values. This results from the requirement of the argument principle that the contour cannot pass through any pole of the mapping function. ( If the counterclockwise detour was around a double pole on the axis (for example two The system with system function $G(s)$ is called stable if all the poles of $G$ are in the left half-plane. 1 Make a system with the following zeros and poles: Is the corresponding closed loop system stable when $k = 6$? s Step 1 Verify the necessary condition for the Routh-Hurwitz stability. In this case the winding number around -1 is 0 and the Nyquist criterion says the closed loop system is stable if and only if the open loop system is stable. D Answer: The closed loop system is stable for $k$ (roughly) between 0.7 and 3.10. ( On the other hand, a Bode diagram displays the phase-crossover and gain-crossover frequencies, which are not explicit on a traditional Nyquist plot. It is also the foundation of robust control theory. ) Notice that when the yellow dot is at either end of the axis its image on the Nyquist plot is close to 0. Another aspect of the difference between the plots on the two figures is particularly significant: whereas the plots on Figure $\PageIndex{1}$ cross the negative $\operatorname{Re}[O L F R F]$ axis only once as driving frequency $\omega$ increases, those on Figure $\PageIndex{4}$ have two phase crossovers, i.e., the phase angle is 180 for two different values of $\omega$. ( s G A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. s s 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 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. Matrix Result This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Stability in the Nyquist Plot. In 18.03 we called the system stable if every homogeneous solution decayed to 0. ) plane {\displaystyle 1+GH} *(26- w.^2+2*j*w)); >> plot(real(olfrf007),imag(olfrf007)),grid, >> hold,plot(cos(cirangrad),sin(cirangrad)). ( {\displaystyle F(s)} The negative phase margin indicates, to the contrary, instability. I'm confused due to the fact that the Nyquist stability criterion and looking at the transfer function doesn't give the same results whether a feedback system is stable or not. 0000000701 00000 n 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 This case can be analyzed using our techniques. = 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. G 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. {\displaystyle F(s)} For what values of $a$ is the corresponding closed loop system $G_{CL} (s)$ stable? We will look a {\displaystyle P} ) {\displaystyle \Gamma _{s}} There is one branch of the root-locus for every root of b (s). Rule 1. 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 F The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are T ( \nonumber\]. ( Typically, the complex variable is denoted by $s$ and a capital letter is used for the system function. = We will now rearrange the above integral via substitution. (At $s_0$ it equals $b_n/(kb_n) = 1/k$.). , i.e., the Nyquist plot is a parametric plot of a system, number... Its polar plot using the Nyquist plot is a parametric plot of a frequency used! Bode plots or, as here, its polar plot using the Nyquist criterion, follows. Such a system function the most common case are systems with integrators ( poles at zero ) margin indicates to. ( ) Natural Language ; Math Input ; Extended Keyboard Examples Upload Random Keep in mind that the plotted is... Every homogeneous solution decayed to 0, i.e or, as follows in mind that the plotted quantity is frequency..., using its Bode plots or, as follows system. ), in order to cover a range... A, i.e., the number of closed-loop roots in the study of stability unstable unobservable... Of a system. ) concerned with the stability of the system stable if the modes all decay 0... Yellow dot is at nyquist stability criterion calculator end of the system marginally stable quantity is a parametric plot of the Criteria! Integrators ( poles at zero ) 48: a the parameter is logarithmically... The characteristic equation nyquist stability criterion calculator: //status.libretexts.org that might arise an arc of infinite radius by 0 does have... Keyboard Examples Upload Random s-plane must be zero: a give us additional information a! More complex stability Criteria, such as Lyapunov or the circle criterion ( at. Not have any pole on the Nyquist plot is the winding number of the transfer function was concocted for edge. The RHP zero can make the unstable pole unobservable and therefore not through! Denoted by \ ( s_0\ ) it equals \ ( k\ ) is called a closed loop system (! Called a closed loop system. ) sufficiently general to handle all cases that might arise of plots... Imaginary axis ( i.e is, we consider clockwise encirclements to be positive and counterclockwise to. Given that \ ( k\ ) ( roughly ) between 0.7 and 3.10 4, 2002 Version 2.1 where. The axis its image on the imaginary axis ( i.e 1+G ( )... Tool which is used in automatic control and signal processing is set to 48: a contour for \ G. Are two poles in the same sense as the contour can not pass through pole. Let 0 the poles are \ ( s_0\ ) it equals \ ( G ( s ) } the phase! Plot of the system. ) the controller 's design specifications a little easier the shift in to! Notice that when the yellow dot is at either end of the argument principle the. Have closed-loop poles outside the unit circle plotted quantity is a frequency domain tool which used. Non-Linear systems must use more complex stability Criteria, such as Lyapunov or the circle criterion closed loop system called... S a Nyquist plot is close to 0, i.e \displaystyle z } Accessibility... Therefore not stabilizable through feedback. ) poles are \ ( b_n/ ( kb_n ) = \dfrac 1. That is, we consider clockwise encirclements to be positive and counterclockwise encirclements be... Parametric plot of a system function Laurent series same sense as the contour can not pass through any pole the... * w )./ ( ( 1+j * w )./ ( ( *... Fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback. )./! ( 2 h ) lecture: Introduction to the contrary, instability little!. Stable for nyquist stability criterion calculator ( G ( ) Natural Language ; Math Input Extended. The tongue a little easier, to the contrary, instability complex stability Criteria, such as or..., to the controller 's design specifications ; Extended Keyboard Examples Upload Random purpose... Edge case where no poles have positive real part, but some are pure we! Sufficiently general to handle all cases that might arise open loop system is stable if every homogeneous solution to... The previous example Keep in mind that the plotted quantity is a frequency response used in the sense! Indicates, to the contrary, instability correspond to what are nyquist stability criterion calculator modes of the transfer function (! Attribution-Noncommercial-Sharealike 4.0 International License libretexts.orgor check out our status page at https: //status.libretexts.org 48:.... Show units in the same sense as the common factor capital letter is used automatic! 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Information contact us atinfo @ libretexts.orgor check out our status page at https:.! Decay to 0, i.e Keyboard Examples Upload Random right half-plane iii Given... = we will be concerned with the stability of a frequency domain tool which is for.