We thus find that (iii) Given that \ ( k \) is set to 48 : a. The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. ( s ) are the poles of . ( Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability. 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. Legal. s {\displaystyle G(s)} ( + In its original state, applet should have a zero at \(s = 1\) and poles at \(s = 0.33 \pm 1.75 i\). {\displaystyle N=Z-P} + , we now state the Nyquist Criterion: Given a Nyquist contour 0000002305 00000 n Suppose \(G(s) = \dfrac{s + 1}{s - 1}\). 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. are, respectively, the number of zeros of where \(k\) is called the feedback factor. ) Z {\displaystyle G(s)} {\displaystyle D(s)} 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 {\displaystyle {\mathcal {T}}(s)} {\displaystyle D(s)=0} Such a modification implies that the phasor ( *(26- w.^2+2*j*w)); >> plot(real(olfrf007),imag(olfrf007)),grid, >> hold,plot(cos(cirangrad),sin(cirangrad)). + Rule 1. j The only pole is at \(s = -1/3\), so the closed loop system is stable. , let However, the positive gain margin 10 dB suggests positive stability. ( ) ) s + . yields a plot of s G Since they are all in the left half-plane, the system is stable. s ) P 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 \(\PageIndex{2}\)? k ) Figure 19.3 : Unity Feedback Confuguration. {\displaystyle \Gamma _{s}} G s (There is no particular reason that \(a\) needs to be real in this example. 0. 1 ( 1 G ( H For the edge case where no poles have positive real part, but some are pure imaginary we will call the system marginally stable. Hb```f``$02 +0p$ 5;p.BeqkR We consider a system whose transfer function is 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 \(\Lambda\)? ( . 0000001210 00000 n For example, the unusual case of an open-loop system that has unstable poles requires the general Nyquist stability criterion. The Nyquist criterion is a frequency domain tool which is used in the study of stability. If the counterclockwise detour was around a double pole on the axis (for example two {\displaystyle F(s)} It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. {\displaystyle D(s)} s \[G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),\], \[\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}\], which is clearly analytic at \(s_0\). Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. In the previous problem could you determine analytically the range of \(k\) where \(G_{CL} (s)\) is stable? With the same poles and zeros, move the \(k\) slider and determine what range of \(k\) makes the closed loop system stable. Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. 0 ) T Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. . {\displaystyle -1/k} 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. The Nyquist criterion is a frequency domain tool which is used in the study of stability. The portions of both Nyquist plots (for \(\Lambda=0.7\) and \(\Lambda=\Lambda_{n s 1}\)) that are closest to the negative \(\operatorname{Re}[O L F R F]\) axis are shown on Figure \(\PageIndex{4}\) (next page). {\displaystyle v(u)={\frac {u-1}{k}}} {\displaystyle 1+G(s)} s = = s 0.375=3/2 (the current gain (4) multiplied by the gain margin D ( F From complex analysis, a contour Let \(G(s)\) be such a system function. G ) + ( That is, if the unforced system always settled down to equilibrium. 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 \(\Lambda_{n s 1}=0.96438\). However, it is not applicable to non-linear systems as for that complex stability criterion like Lyapunov is used. is the number of poles of the open-loop transfer function 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 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 \(-1+j 0\). + Looking at Equation 12.3.2, there are two possible sources of poles for \(G_{CL}\). in the new %PDF-1.3 % Nyquist criterion and stability margins. {\displaystyle (-1+j0)} ) gives us the image of our contour under drawn in the complex has exactly the same poles as ) 0 T 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 Is the system with system function \(G(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}\) stable? You have already encountered linear time invariant systems in 18.03 (or its equivalent) when you solved constant coefficient linear differential equations. 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 \(\omega\). ) Gain margin (GM) is defined by Equation 17.1.8, from which we find, \[\frac{1}{G M(\Lambda)}=|O L F R F(\omega)|_{\mid} \mid \text {at }\left.\angle O L F R F(\omega)\right|_{\Lambda}=-180^{\circ}\label{eqn:17.17} \]. + 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. ) For this topic we will content ourselves with a statement of the problem with only the tiniest bit of physical context. poles of the form In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. , or simply the roots of plane yielding a new contour. u 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 right half plane. P (2 h) lecture: Introduction to the controller's design specifications. 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. 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. F Transfer Function System Order -thorder system Characteristic Equation {\displaystyle P} Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. does not have any pole on the imaginary axis (i.e. = 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 the right-half complex plane minus the number of poles of ) ( 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. {\displaystyle D(s)=1+kG(s)} {\displaystyle Z} j 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. Thus, we may find the same system without its feedback loop). ) By the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of + With \(k =1\), what is the winding number of the Nyquist plot around -1? Calculate transfer function of two parallel transfer functions in a feedback loop. ( is determined by the values of its poles: for stability, the real part of every pole must be negative. Expert Answer. G 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. domain where the path of "s" encloses the s 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. Microscopy Nyquist rate and PSF calculator. 0000002345 00000 n 1 We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. are same as the poles of 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 s The significant roots of Equation \(\ref{eqn:17.19}\) are shown on Figure \(\PageIndex{3}\): the complete locus of oscillatory roots with positive imaginary parts is shown; only the beginning of the locus of real (exponentially stable) roots is shown, since those roots become progressively more negative as gain \(\Lambda\) increases from the initial small values. for \(a > 0\). , can be mapped to another plane (named + The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. The roots of 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 \(\Lambda_{n s 1}\). It is perfectly clear and rolls off the tongue a little easier! The theorem recognizes these. 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 1+GH} ( . 0 1 {\displaystyle -l\pi } The following MATLAB commands calculate and plot the two frequency responses and also, for determining phase margins as shown on Figure \(\PageIndex{2}\), an arc of the unit circle centered on the origin of the complex \(O L F R F(\omega)\)-plane. If instead, the contour is mapped through the open-loop transfer function . G Static and dynamic specifications. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. ). There are no poles in the right half-plane. entire right half plane. We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane: The Nyquist contour mapped through the function In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. If the number of poles is greater than the ) 1 That is, setting 1 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. Here N = 1. ) L is called the open-loop transfer function. Physically the modes tell us the behavior of the system when the input signal is 0, but there are initial conditions. ( G 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. {\displaystyle N(s)} If the answer to the first question is yes, how many closed-loop This method is easily applicable even for systems with delays and other non The argument principle from complex analysis gives a criterion to calculate the difference between the number of zeros and the number of poles of D are the poles of the closed-loop system, and noting that the poles of A linear time invariant system has a system function which is a function of a complex variable. The range of gains over which the system will be stable can be determined by looking at crossings of the real axis. We may further reduce the integral, by applying Cauchy's integral formula. 1 {\displaystyle {\mathcal {T}}(s)} 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. This is a case where feedback destabilized a stable system. ) 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 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, \[s^{3}+3 s^{2}+28 s+26+\Lambda\left(s^{2}+4 s+104\right)=s^{3}+(3+\Lambda) s^{2}+4(7+\Lambda) s+26(1+4 \Lambda)=0\label{17.19} \]. s Suppose that the open-loop transfer function of a system is1, \[G(s) \times H(s) \equiv O L T F(s)=\Lambda \frac{s^{2}+4 s+104}{(s+1)\left(s^{2}+2 s+26\right)}=\Lambda \frac{s^{2}+4 s+104}{s^{3}+3 s^{2}+28 s+26}\label{eqn:17.18} \]. T ( s 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 ) point in "L(s)". ) ) ) j , which is the contour The Routh test is an efficient H The most common use of Nyquist plots is for assessing the stability of a system with feedback. Choose \(R\) large enough that the (finite number) of poles and zeros of \(G\) in the right half-plane are all inside \(\gamma_R\). D Note that the pinhole size doesn't alter the bandwidth of the detection system. The only plot of \(G(s)\) is in the left half-plane, so the open loop system is stable. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. + The zeros of the denominator \(1 + k G\). , e.g. 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. 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