1 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. Give zero-pole diagrams for each of the systems, \[G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\]. ) s 0.375=3/2 (the current gain (4) multiplied by the gain margin = This is possible for small systems. + In \(\gamma (\omega)\) the variable is a greek omega and in \(w = G \circ \gamma\) we have a double-u. Thus, we may finally state that. {\displaystyle 1+GH(s)} s We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. {\displaystyle F(s)} the number of the counterclockwise encirclements of \(1\) point by the Nyquist plot in the \(GH\)-plane is equal to the number of the unstable poles of the open-loop transfer function. is formed by closing a negative unity feedback loop around the open-loop transfer function ( ) While Nyquist is one of the most general stability tests, it is still restricted to linear time-invariant (LTI) systems. Legal. The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s + 1)/(s - 1)}{1 + 2(s + 1)/(s - 1)} = \dfrac{s + 1}{3s + 1}.\]. , then the roots of the characteristic equation are also the zeros of Transfer Function System Order -thorder system Characteristic Equation = All the coefficients of the characteristic polynomial, s 4 + 2 s 3 + s 2 + 2 s + 1 are positive. We can factor L(s) to determine the number of poles that are in the The only pole is at \(s = -1/3\), so the closed loop system is stable. For the Nyquist plot and criterion the curve \(\gamma\) will always be the imaginary \(s\)-axis. s Section 17.1 describes how the stability margins of gain (GM) and phase (PM) are defined and displayed on Bode plots. ( s inside the contour 0 ) ) Stability in the Nyquist Plot. Natural Language; Math Input; Extended Keyboard Examples Upload Random. That is, if all the poles of \(G\) have negative real part. 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. Lets look at an example: Note that I usually dont include negative frequencies in my Nyquist plots. ( This assumption holds in many interesting cases. Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. For what values of \(a\) is the corresponding closed loop system \(G_{CL} (s)\) stable? The other phase crossover, at \(-4.9254+j 0\) (beyond the range of Figure \(\PageIndex{5}\)), might be the appropriate point for calculation of gain margin, since it at least indicates instability, \(\mathrm{GM}_{4.75}=1 / 4.9254=0.20303=-13.85\) dB. Such a modification implies that the phasor T ( ) + This method for design involves plotting the complex loci of P ( s) C ( s) for the range s = j , = [ , ]. 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. In 18.03 we called the system stable if every homogeneous solution decayed to 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 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. s s 1 H The answer is no, \(G_{CL}\) is not stable. For our purposes it would require and an indented contour along the imaginary axis. ) . %PDF-1.3
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G Z We may further reduce the integral, by applying Cauchy's integral formula. 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\newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 17.3: The Practical Effects of an Open-Loop Transfer-Function Pole at s = 0 + j0, Virginia Polytechnic Institute and State University, Virginia Tech Libraries' Open Education Initiative, source@https://vtechworks.lib.vt.edu/handle/10919/78864, status page at https://status.libretexts.org. . + {\displaystyle D(s)} 1 The row s 3 elements have 2 as the common factor. {\displaystyle N} ) s Terminology. negatively oriented) contour In this case, we have, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)} = \dfrac{\dfrac{s - 1}{(s - 0.33)^2 + 1.75^2}}{1 + \dfrac{k(s - 1)}{(s - 0.33)^2 + 1.75^2}} = \dfrac{s - 1}{(s - 0.33)^2 + 1.75^2 + k(s - 1)} \nonumber\], \[(s - 0.33)^2 + 1.75^2 + k(s - 1) = s^2 + (k - 0.66)s + 0.33^2 + 1.75^2 - k \nonumber\], For a quadratic with positive coefficients the roots both have negative real part. 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 \Gamma _{s}} F be the number of poles of F s There are no poles in the right half-plane. ) {\displaystyle D(s)=1+kG(s)} + {\displaystyle l} Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability. 1 These are the same systems as in the examples just above. ( \(G(s) = \dfrac{s - 1}{s + 1}\). >> olfrf01=(104-w.^2+4*j*w)./((1+j*w). Physically the modes tell us the behavior of the system when the input signal is 0, but there are initial conditions. T We begin by considering the closed-loop characteristic polynomial (4.23) where L ( z) denotes the loop gain. Given our definition of stability above, we could, in principle, discuss stability without the slightest idea what it means for physical systems. s + We thus find that ) T We will be concerned with the stability of the system. 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); Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. N . -P_PcXJ']b9-@f8+5YjmK p"yHL0:8UK=MY9X"R&t5]M/o 3\\6%W+7J$)^p;%
XpXC#::` :@2p1A%TQHD1Mdq!1 ) As \(k\) goes to 0, the Nyquist plot shrinks to a single point at the origin. 0 The frequency-response curve leading into that loop crosses the \(\operatorname{Re}[O L F R F]\) axis at about \(-0.315+j 0\); if we were to use this phase crossover to calculate gain margin, then we would find \(\mathrm{GM} \approx 1 / 0.315=3.175=10.0\) dB. 0 {\displaystyle {\frac {G}{1+GH}}} ( Nyquist plot of the transfer function s/(s-1)^3. in the right half plane, the resultant contour in the Thus, it is stable when the pole is in the left half-plane, i.e. \[G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),\], \[\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}\], which is clearly analytic at \(s_0\). ( s G , we now state the Nyquist Criterion: Given a Nyquist contour G ) 1 gain margin as defined on Figure \(\PageIndex{5}\) can be an ambiguous, unreliable, and even deceptive metric of closed-loop stability; phase margin as defined on Figure \(\PageIndex{5}\), on the other hand, is usually an unambiguous and reliable metric, with \(\mathrm{PM}>0\) indicating closed-loop stability, and \(\mathrm{PM}<0\) indicating closed-loop instability. . There are two poles in the right half-plane, so the open loop system \(G(s)\) is unstable. 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 plane in the same sense as the contour , can be mapped to another plane (named ( 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. Microscopy Nyquist rate and PSF calculator. has zeros outside the open left-half-plane (commonly initialized as OLHP). From now on we will allow ourselves to be a little more casual and say the system \(G(s)\)'. Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. G Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\) stable? The Nyquist criterion gives a graphical method for checking the stability of the closed loop system. Nyquist stability criterion is a general stability test that checks for the stability of linear time-invariant systems. Nyquist plot of the transfer function s/ (s-1)^3 Natural Language Math Input Extended Keyboard Examples Have a question about using Wolfram|Alpha? {\displaystyle -1/k} s (0.375) yields the gain that creates marginal stability (3/2). 1 The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are 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. We can visualize \(G(s)\) using a pole-zero diagram. Nyquist criterion and stability margins. The poles of \(G(s)\) correspond to what are called modes of the system. s G 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 You can also check that it is traversed clockwise. , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. 0 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. 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. s G times such that ) s s 1 Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. = ( ) ) 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. + s {\displaystyle F(s)} T {\displaystyle T(s)} k 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. s P B The most common use of Nyquist plots is for assessing the stability of a system with feedback. The Nyquist criterion allows us to answer two questions: 1. 0 F P 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. ( 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. It is certainly reasonable to call a system that does this in response to a zero signal (often called no input) unstable. yields a plot of Calculate the Gain Margin. ( 0000000701 00000 n
( s Z According to the formula, for open loop transfer function stability: Z = N + P = 0. where N is the number of encirclements of ( 0, 0) by the Nyquist plot in clockwise direction. 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. Figure 19.3 : Unity Feedback Confuguration. ) s D s When \(k\) is small the Nyquist plot has winding number 0 around -1. ) v Note that the pinhole size doesn't alter the bandwidth of the detection system. s ) This case can be analyzed using our techniques. ) 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 \(\approx 15\). , and s Gain margin (GM) is defined by Equation 17.1.8, from which we find, \[\frac{1}{G M(\Lambda)}=|O L F R F(\omega)|_{\mid} \mid \text {at }\left.\angle O L F R F(\omega)\right|_{\Lambda}=-180^{\circ}\label{eqn:17.17} \]. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. are also said to be the roots of the characteristic equation 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. The Nyquist method is used for studying the stability of linear systems with + That is, the Nyquist plot is the image of the imaginary axis under the map \(w = kG(s)\). {\displaystyle 1+GH} G(s)= s(s+5)(s+10)500K slopes, frequencies, magnitudes, on the next pages!) It can happen! 0000002305 00000 n
In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. 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, \[w = k G \circ \gamma (\omega) = kG(i \omega).\]. 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. A P 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 Routh test is an efficient 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. 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 If instead, the contour is mapped through the open-loop transfer function j Legal. 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 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. ) Calculate transfer function of two parallel transfer functions in a feedback loop. ) 0 for \(a > 0\). For gain \(\Lambda = 18.5\), there are two phase crossovers: one evident on Figure \(\PageIndex{6}\) at \(-18.5 / 15.0356+j 0=-1.230+j 0\), and the other way beyond the range of Figure \(\PageIndex{6}\) at \(-18.5 / 0.96438+j 0=-19.18+j 0\). To get a feel for the Nyquist plot. 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. 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