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. P + 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. . Since \(G\) is in both the numerator and denominator of \(G_{CL}\) it should be clear that the poles cancel. The system is stable if the modes all decay to 0, i.e. N F s s ) ( 1 s We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. Right-half-plane (RHP) poles represent that instability. If the answer to the first question is yes, how many closed-loop = 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); s s So the winding number is -1, which does not equal the number of poles of \(G\) in the right half-plane. s Which, if either, of the values calculated from that reading, \(\mathrm{GM}=(1 / \mathrm{GM})^{-1}\) is a legitimate metric of closed-loop stability? {\displaystyle G(s)} {\displaystyle Z=N+P} While Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable. + F Thus, for all large \(R\), \[\text{the system is stable } \Leftrightarrow \ Z_{1 + kG, \gamma_R} = 0 \ \Leftrightarow \ \text{ Ind} (kG \circ \gamma_R, -1) = P_{G, \gamma_R}\], Finally, we can let \(R\) go to infinity. ( 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. Nyquist Plot Example 1, Procedure to draw Nyquist plot in 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). The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s - 1)/(s + 1)}{1 + 2(s - 1)/(s + 1)} = \dfrac{s - 1}{3s - 1}.\]. , we now state the Nyquist Criterion: Given a Nyquist contour G Since on Figure \(\PageIndex{4}\) there are two different frequencies at which \(\left.\angle O L F R F(\omega)\right|_{\Lambda}=-180^{\circ}\), the definition of gain margin in Equations 17.1.8 and \(\ref{eqn:17.17}\) is ambiguous: at which, if either, of the phase crossovers is it appropriate to read the quantity \(1 / \mathrm{GM}\), as shown on \(\PageIndex{2}\)? Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. ) s {\displaystyle Z} The following MATLAB commands, adapted from the code that produced Figure 16.5.1, calculate and plot the loci of roots: Lm=[0 .2 .4 .7 1 1.5 2.5 3.7 4.75 6.5 9 12.5 15 18.5 25 35 50 70 125 250]; a2=3+Lm(i);a3=4*(7+Lm(i));a4=26*(1+4*Lm(i)); plot(p,'kx'),grid,xlabel('Real part of pole (sec^-^1)'), ylabel('Imaginary part of pole (sec^-^1)'). will encircle the point ( 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. {\displaystyle 1+GH} in the right-half complex plane. 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 \(0<\mathrm{GM}_{\mathrm{U}}<1\). N G By the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of . 1 , which is the contour Static and dynamic specifications. 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 drawn in the complex G The beauty of the Nyquist stability criterion lies in the fact that it is a rather simple graphical test. Z Nyquist stability criterion is a general stability test that checks for the stability of linear time-invariant systems. Make a mapping from the "s" domain to the "L(s)" H|Ak0ZlzC!bBM66+d]JHbLK5L#S$_0i".Zb~#}2HyY YBrs}y:)c. . When drawn by hand, a cartoon version of the Nyquist plot is sometimes used, which shows the linearity of the curve, but where coordinates are distorted to show more detail in regions of interest. Pole-zero diagrams for the three systems. and ) s H ( Stability can be determined by examining the roots of the desensitivity factor polynomial {\displaystyle 1+G(s)} In fact, we find that the above integral corresponds precisely to the number of times the Nyquist plot encircles the point D ( Such a modification implies that the phasor {\displaystyle {\mathcal {T}}(s)} So in the limit \(kG \circ \gamma_R\) becomes \(kG \circ \gamma\). s + [@mc6X#:H|P`30s@, B R=Lb&3s12212WeX*a$%.0F06 endstream endobj 103 0 obj 393 endobj 93 0 obj << /Type /Page /Parent 85 0 R /Resources 94 0 R /Contents 98 0 R /Rotate 90 /MediaBox [ 0 0 612 792 ] /CropBox [ 36 36 576 756 ] >> endobj 94 0 obj << /ProcSet [ /PDF /Text ] /Font << /TT2 96 0 R >> /ExtGState << /GS1 100 0 R >> /ColorSpace << /Cs6 97 0 R >> >> endobj 95 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -568 -307 2028 1007 ] /FontName /HMIFEA+TimesNewRoman /ItalicAngle 0 /StemV 94 /XHeight 0 /FontFile2 99 0 R >> endobj 96 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 150 /Widths [ 250 0 0 500 0 0 0 0 333 333 500 564 250 333 250 278 500 500 500 500 500 500 500 500 500 500 278 0 0 564 0 0 0 722 667 667 722 611 556 722 722 333 389 0 611 889 722 722 556 0 667 556 611 722 722 944 0 0 0 0 0 0 0 500 0 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 500 500 333 389 278 500 500 722 500 500 444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 333 0 0 350 500 ] /Encoding /WinAnsiEncoding /BaseFont /HMIFEA+TimesNewRoman /FontDescriptor 95 0 R >> endobj 97 0 obj [ /ICCBased 101 0 R ] endobj 98 0 obj << /Length 428 /Filter /FlateDecode >> stream s To simulate that testing, we have from Equation \(\ref{eqn:17.18}\), the following equation for the frequency-response function: \[O L F R F(\omega) \equiv O L T F(j \omega)=\Lambda \frac{104-\omega^{2}+4 \times j \omega}{(1+j \omega)\left(26-\omega^{2}+2 \times j \omega\right)}\label{eqn:17.20} \]. = \(G\) has one pole in the right half plane. ) 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. For the Nyquist plot and criterion the curve \(\gamma\) will always be the imaginary \(s\)-axis. ( for \(a > 0\). Assume \(a\) is real, for what values of \(a\) is the open loop system \(G(s) = \dfrac{1}{s + a}\) stable? (Actually, for \(a = 0\) the open loop is marginally stable, but it is fully stabilized by the closed loop.). D \(\text{QED}\), The Nyquist criterion is a visual method which requires some way of producing the Nyquist plot. / of the 0 Mark the roots of b ) s + Is the open loop system stable? Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. That is, \[s = \gamma (\omega) = i \omega, \text{ where } -\infty < \omega < \infty.\], For a system \(G(s)\) and a feedback factor \(k\), the Nyquist plot is the plot of the curve, \[w = k G \circ \gamma (\omega) = kG(i \omega).\]. ) 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. (There is no particular reason that \(a\) needs to be real in this example. shall encircle (clockwise) the point The poles of Here s Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. 0 ( + u as the first and second order system. It is easy to check it is the circle through the origin with center \(w = 1/2\). ). %PDF-1.3 % ( The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are all in the left half of the complex plane. 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. ) Look at the pole diagram and use the mouse to drag the yellow point up and down the imaginary axis. s ) Set the feedback factor \(k = 1\). D {\displaystyle 1+G(s)} Take \(G(s)\) from the previous example. {\displaystyle D(s)} ) The most common use of Nyquist plots is for assessing the stability of a system with feedback. 1 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. {\displaystyle G(s)} s + ( Let us consider next an uncommon system, for which the determination of stability or instability requires a more detailed examination of the stability margins. {\displaystyle N} We will look a An approach to this end is through the use of Nyquist techniques. To use this criterion, the frequency response data of a system must be presented as a polar plot in which the magnitude and the phase angle are expressed as N 0 L is called the open-loop transfer function. Keep in mind that the plotted quantity is A, i.e., the loop gain. ) This continues until \(k\) is between 3.10 and 3.20, at which point the winding number becomes 1 and \(G_{CL}\) becomes unstable. , then the roots of the characteristic equation are also the zeros of encirclements of the -1+j0 point in "L(s).". 1 For the edge case where no poles have positive real part, but some are pure imaginary we will call the system marginally stable. ( Nyquist plot of the transfer function s/(s-1)^3. ( s This method is easily applicable even for systems with delays and other non G Calculate the Gain Margin. Lecture 2: Stability Criteria S.D. {\displaystyle P} s 0 G ( 20 points) b) Using the Bode plots, calculate the phase margin and gain margin for K =1. {\displaystyle F} We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane: The Nyquist contour mapped through the function using the Routh array, but this method is somewhat tedious. Does the system have closed-loop poles outside the unit circle? Here N = 1. In the case \(G(s)\) is a fractional linear transformation, so we know it maps the imaginary axis to a circle. 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 \(\Lambda_{n s 2} = 15.0356\). by the same contour. (iii) Given that \ ( k \) is set to 48 : a. To get a feel for the Nyquist plot. 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. F The poles are \(\pm 2, -2 \pm i\). Legal. be the number of poles of . {\displaystyle \Gamma _{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 \(\approx 15\). by counting the poles of Answer: The closed loop system is stable for \(k\) (roughly) between 0.7 and 3.10. G right half plane. Is the system with system function \(G(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}\) stable? ) s {\displaystyle 1+GH(s)} and poles of 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. u For this we will use one of the MIT Mathlets (slightly modified for our purposes). + {\displaystyle -1+j0} {\displaystyle v(u(\Gamma _{s}))={{D(\Gamma _{s})-1} \over {k}}=G(\Gamma _{s})} Check the \(Formula\) box. {\displaystyle GH(s)} Choose \(R\) large enough that the (finite number) of poles and zeros of \(G\) in the right half-plane are all inside \(\gamma_R\). plane travels along an arc of infinite radius by {\displaystyle 1+G(s)} G + The Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. {\displaystyle \Gamma _{s}} 1 This is just to give you a little physical orientation. Since \(G_{CL}\) is a system function, we can ask if the system is stable. the same system without its feedback loop). In general, the feedback factor will just scale the Nyquist plot. s Nyquist plot of the transfer function s/(s-1)^3. Looking at Equation 12.3.2, there are two possible sources of poles for \(G_{CL}\). {\displaystyle P} In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). The frequency is swept as a parameter, resulting in a pl is the multiplicity of the pole on the imaginary axis. Suppose F (s) is a single-valued mapping function given as: F (s) = 1 + G (s)H (s) 0 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. Z There is one branch of the root-locus for every root of b (s). {\displaystyle F(s)} P The algebra involved in canceling the \(s + a\) term in the denominators is exactly the cancellation that makes the poles of \(G\) removable singularities in \(G_{CL}\). s + {\displaystyle 1+G(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 ) The frequency is swept as a parameter, resulting in a plot per frequency. Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. 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 \(\Lambda_{n s 1}=0.96438\). 0 1 Draw the Nyquist plot with \(k = 1\). u So, the control system satisfied the necessary condition. That is, if all the poles of \(G\) have negative real part. 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. To use this criterion, the frequency response data of a system must be presented as a polar plot in The poles of \(G\). This case can be analyzed using our techniques. Any Laplace domain transfer function Describe the Nyquist plot with gain factor \(k = 2\). in the complex plane. 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. T {\displaystyle G(s)} s clockwise. In order to establish the reference for stability and instability of the closed-loop system corresponding to Equation \(\ref{eqn:17.18}\), we determine the loci of roots from the characteristic equation, \(1+G H=0\), or, \[s^{3}+3 s^{2}+28 s+26+\Lambda\left(s^{2}+4 s+104\right)=s^{3}+(3+\Lambda) s^{2}+4(7+\Lambda) s+26(1+4 \Lambda)=0\label{17.19} \]. ( 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 -plane, We conclude this chapter on frequency-response stability criteria by observing that margins of gain and phase are used also as engineering design goals. = G 1 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 \(\PageIndex{4}\). In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. H 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. ) The most common use of Nyquist plots is for assessing the stability of a system with feedback. G 0000039933 00000 n MT-002. s 1 For our purposes it would require and an indented contour along the imaginary axis. 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 \(\omega\). 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