s Z Make a system with the following zeros and poles: Is the corresponding closed loop system stable when \(k = 6\)? s has zeros outside the open left-half-plane (commonly initialized as OLHP). N The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. D 1 Open the Nyquist Plot applet at. {\displaystyle N=Z-P} {\displaystyle 0+j(\omega +r)} ) A G A We conclude this chapter on frequency-response stability criteria by observing that margins of gain and phase are used also as engineering design goals. ) \[G_{CL} (s) = \dfrac{1/(s + a)}{1 + 1/(s + a)} = \dfrac{1}{s + a + 1}.\], This has a pole at \(s = -a - 1\), so it's stable if \(a > -1\). s Determining Stability using the Nyquist Plot - Erik Cheever 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. This results from the requirement of the argument principle that the contour cannot pass through any pole of the mapping function. Since \(G\) is in both the numerator and denominator of \(G_{CL}\) it should be clear that the poles cancel. This is a case where feedback destabilized a stable system. Stability in the Nyquist Plot. is mapped to the point v ( ( 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 If instead, the contour is mapped through the open-loop transfer function \[G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),\], \[\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}\], which is clearly analytic at \(s_0\). can be expressed as the ratio of two polynomials: + r as the first and second order system. Let \(\gamma_R = C_1 + C_R\). j 0000002847 00000 n This is a diagram in the \(s\)-plane where we put a small cross at each pole and a small circle at each zero. are same as the poles of F The Nyquist method is used for studying the stability of linear systems with pure time delay. 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 \(\Lambda\)? 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. {\displaystyle N} With a little imagination, we infer from the Nyquist plots of Figure \(\PageIndex{1}\) that the open-loop system represented in that figure has \(\mathrm{GM}>0\) and \(\mathrm{PM}>0\) for \(0<\Lambda<\Lambda_{\mathrm{ns}}\), and that \(\mathrm{GM}>0\) and \(\mathrm{PM}>0\) for all values of gain \(\Lambda\) greater than \(\Lambda_{\mathrm{ns}}\); accordingly, the associated closed-loop system is stable for \(0<\Lambda<\Lambda_{\mathrm{ns}}\), and unstable for all values of gain \(\Lambda\) greater than \(\Lambda_{\mathrm{ns}}\). Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. ) 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. For instance, the plot provides information on the difference between the number of zeros and poles of the transfer function[5] by the angle at which the curve approaches the origin. ( The system with system function \(G(s)\) is called stable if all the poles of \(G\) are in the left half-plane. That is, if all the poles of \(G\) have negative real part. ) "1+L(s)" in the right half plane (which is the same as the number \nonumber\]. The system is stable if the modes all decay to 0, i.e. We may further reduce the integral, by applying Cauchy's integral formula. Which, if either, of the values calculated from that reading, \(\mathrm{GM}=(1 / \mathrm{GM})^{-1}\) is a legitimate metric of closed-loop stability? as defined above corresponds to a stable unity-feedback system when *(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 \(\PageIndex{1}\). + s ( ( s s We will be concerned with the stability of the system. We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. ( The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s + 1)/(s - 1)}{1 + 2(s + 1)/(s - 1)} = \dfrac{s + 1}{3s + 1}.\]. Draw the Nyquist plot with \(k = 1\). {\displaystyle T(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 \(0<\mathrm{GM}_{\mathrm{U}}<1\). Choose \(R\) large enough that the (finite number) of poles and zeros of \(G\) in the right half-plane are all inside \(\gamma_R\). ), Start with a system whose characteristic equation is given by {\displaystyle v(u(\Gamma _{s}))={{D(\Gamma _{s})-1} \over {k}}=G(\Gamma _{s})} ( s 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, \[w = k G \circ \gamma (\omega) = kG(i \omega).\]. Equation \(\ref{eqn:17.17}\) is illustrated on Figure \(\PageIndex{2}\) for both closed-loop stable and unstable cases. Lecture 1: The Nyquist Criterion S.D. This page titled 12.2: Nyquist Criterion for Stability is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 0 The shift in origin to (1+j0) gives the characteristic equation plane. {\displaystyle F(s)} The Nyquist criterion is a frequency domain tool which is used in the study of stability. Since we know N and P, we can determine Z, the number of zeros of s As per the diagram, Nyquist plot encircle the point 1+j0 (also called critical point) once in a counter clock wise direction. Therefore N= 1, In OLTF, one pole (at +2) is at RHS, hence P =1. You can see N= P, hence system is stable. s *(26- w.^2+2*j*w)); >> plot(real(olfrf007),imag(olfrf007)),grid, >> hold,plot(cos(cirangrad),sin(cirangrad)). Thus, we may find ) 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); {\displaystyle F(s)} 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, \[s^{3}+3 s^{2}+28 s+26+\Lambda\left(s^{2}+4 s+104\right)=s^{3}+(3+\Lambda) s^{2}+4(7+\Lambda) s+26(1+4 \Lambda)=0\label{17.19} \]. G The frequency is swept as a parameter, resulting in a plot per frequency. P in the complex plane. So we put a circle at the origin and a cross at each pole. {\displaystyle \Gamma _{G(s)}} s ) s (ii) Determine the range of \ ( k \) to ensure a stable closed loop response. H 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. G(s)= s(s+5)(s+10)500K slopes, frequencies, magnitudes, on the next pages!) ) s Let \(G(s) = \dfrac{1}{s + 1}\). 1 Nyquist criterion and stability margins. T . ) and Lecture 2: Stability Criteria S.D. s H|Ak0ZlzC!bBM66+d]JHbLK5L#S$_0i".Zb~#}2HyY YBrs}y:)c. {\displaystyle Z} Notice that when the yellow dot is at either end of the axis its image on the Nyquist plot is close to 0. is the multiplicity of the pole on the imaginary axis. You have already encountered linear time invariant systems in 18.03 (or its equivalent) when you solved constant coefficient linear differential equations. . Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. We then note that The curve winds twice around -1 in the counterclockwise direction, so the winding number \(\text{Ind} (kG \circ \gamma, -1) = 2\). 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 \(\PageIndex{5}\). The mathematics uses the Laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. k 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. Here 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. 1 The negative phase margin indicates, to the contrary, instability. The stability of = j That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. Microscopy Nyquist rate and PSF calculator. ) If the answer to the first question is yes, how many closed-loop is the number of poles of the open-loop transfer function Nyquist plot of the transfer function s/(s-1)^3. (At \(s_0\) it equals \(b_n/(kb_n) = 1/k\).). For the Nyquist plot and criterion the curve \(\gamma\) will always be the imaginary \(s\)-axis. 0 Let us consider next an uncommon system, for which the determination of stability or instability requires a more detailed examination of the stability margins. Static and dynamic specifications. Let us begin this study by computing \(\operatorname{OLFRF}(\omega)\) and displaying it on Nyquist plots for a low value of gain, \(\Lambda=0.7\) (for which the closed-loop system is stable), and for the value corresponding to the transition from stability to instability on Figure \(\PageIndex{3}\), which we denote as \(\Lambda_{n s 1} \approx 1\). Phase margin is defined by, \[\operatorname{PM}(\Lambda)=180^{\circ}+\left(\left.\angle O L F R F(\omega)\right|_{\Lambda} \text { at }|O L F R F(\omega)|_{\Lambda} \mid=1\right)\label{eqn:17.7} \]. The Routh test is an efficient ( We can show this formally using Laurent series. Figure 19.3 : Unity Feedback Confuguration. are called the zeros of Stability is determined by looking at the number of encirclements of the point (1, 0). {\displaystyle 1+kF(s)} Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. H Z s s The only plot of \(G(s)\) is in the left half-plane, so the open loop system is stable. ) 0000001188 00000 n \(G(s) = \dfrac{s - 1}{s + 1}\). 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 \(\Lambda_{n s 1}=0.96438\). Precisely, each complex point In 18.03 we called the system stable if every homogeneous solution decayed to 0. does not have any pole on the imaginary axis (i.e. the clockwise direction. If {\displaystyle {\mathcal {T}}(s)} T . *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. Refresh the page, to put the zero and poles back to their original state. s This method for design involves plotting the complex loci of P ( s) C ( s) for the range s = j , = [ , ]. In practice, the ideal sampler is replaced by {\displaystyle -1+j0} j ( Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s - 1)/(s + 1)}{1 + 2(s - 1)/(s + 1)} = \dfrac{s - 1}{3s - 1}.\]. = H j L is called the open-loop transfer function. ) ) Step 2 Form the Routh array for the given characteristic polynomial. inside the contour 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. For example, the unusual case of an open-loop system that has unstable poles requires the general Nyquist stability criterion. drawn in the complex (3h) lecture: Nyquist diagram and on the effects of feedback. So in the limit \(kG \circ \gamma_R\) becomes \(kG \circ \gamma\). 1 Is the open loop system stable? ) ) 0 Techniques like Bode plots, while less general, are sometimes a more useful design tool. This case can be analyzed using our techniques. 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 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. Microscopy Nyquist rate and PSF calculator. 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. On the other hand, the phase margin shown on Figure \(\PageIndex{6}\), \(\mathrm{PM}_{18.5} \approx+12^{\circ}\), correctly indicates weak stability. We will make a standard assumption that \(G(s)\) is meromorphic with a finite number of (finite) poles. , or simply the roots of {\displaystyle 0+j(\omega -r)} {\displaystyle 1+G(s)} Also suppose that \(G(s)\) decays to 0 as \(s\) goes to infinity. The zeros of the denominator \(1 + k G\). j Is the system with system function \(G(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}\) stable? Pole-zero diagrams for the three systems. are, respectively, the number of zeros of ) For these values of \(k\), \(G_{CL}\) is unstable. ( We will look a little more closely at such systems when we study the Laplace transform in the next topic. + right half plane. . 0 , e.g. Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability. ( is not sufficiently general to handle all cases that might arise. The poles are \(-2, \pm 2i\). 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. In this context \(G(s)\) is called the open loop system function. ( . 0000039933 00000 n T ( {\displaystyle s} A linear time invariant system has a system function which is a function of a complex variable. 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. {\displaystyle G(s)} Additional parameters 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 \(\Lambda_{n s 1}\). ) N ( / encircled by {\displaystyle -l\pi } {\displaystyle H(s)} P Thus, it is stable when the pole is in the left half-plane, i.e. For what values of \(a\) is the corresponding closed loop system \(G_{CL} (s)\) stable? 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 P} Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}\) stable? 0 H Expert Answer. 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