) N the same system without its feedback loop). {\displaystyle (-1+j0)} = . Is the system with system function \(G(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}\) stable? 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. 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. s , where Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. F (There is no particular reason that \(a\) needs to be real in this example. 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. From the mapping we find the number N, which is the number of {\displaystyle -1/k} We suppose that we have a clockwise (i.e. ( The only pole is at \(s = -1/3\), so the closed loop system is stable. G 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 is the number of poles of the open-loop transfer function 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. N = {\displaystyle A(s)+B(s)=0} s is the multiplicity of the pole on the imaginary axis. j P {\displaystyle G(s)} {\displaystyle T(s)} H Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. For our purposes it would require and an indented contour along the imaginary axis. point in "L(s)". 1 Let \(G(s) = \dfrac{1}{s + 1}\). In using \(\text { PM }\) this way, a phase margin of 30 is often judged to be the lowest acceptable \(\text { PM }\), with values above 30 desirable.. Notice that when the yellow dot is at either end of the axis its image on the Nyquist plot is close to 0. L is called the open-loop transfer function. 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. That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. This gives us, We now note that We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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). encirclements of the -1+j0 point in "L(s).". + Phase margin is defined by, \[\operatorname{PM}(\Lambda)=180^{\circ}+\left(\left.\angle O L F R F(\omega)\right|_{\Lambda} \text { at }|O L F R F(\omega)|_{\Lambda} \mid=1\right)\label{eqn:17.7} \]. {\displaystyle \Gamma _{s}} {\displaystyle -l\pi } . We thus find that + 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 G \(G_{CL}\) is stable exactly when all its poles are in the left half-plane. j k We then note that Suppose F (s) is a single-valued mapping function given as: F (s) = 1 + G (s)H (s) Since \(G\) is in both the numerator and denominator of \(G_{CL}\) it should be clear that the poles cancel. {\displaystyle {\mathcal {T}}(s)} This is a case where feedback stabilized an unstable system. The frequency is swept as a parameter, resulting in a pl Check the \(Formula\) box. 0000002305 00000 n 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 is stable if the modes all decay to 0, i.e. %PDF-1.3 % The formula is an easy way to read off the values of the poles and zeros of \(G(s)\). 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. \(G(s) = \dfrac{s - 1}{s + 1}\). {\displaystyle P} Rearranging, we have enclosed by the contour and This should make sense, since with \(k = 0\), \[G_{CL} = \dfrac{G}{1 + kG} = G. \nonumber\]. . {\displaystyle G(s)} This case can be analyzed using our techniques. If \(G\) has a pole of order \(n\) at \(s_0\) then. ( s ) {\displaystyle N=P-Z} F G T Z ( j by Cauchy's argument principle. {\displaystyle G(s)} + The Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. But in physical systems, complex poles will tend to come in conjugate pairs.). For the edge case where no poles have positive real part, but some are pure imaginary we will call the system marginally stable. Make a mapping from the "s" domain to the "L(s)" ) ). , we now state the Nyquist Criterion: Given a Nyquist contour ) \[G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),\], \[\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}\], which is clearly analytic at \(s_0\). , e.g. and that encirclements in the opposite direction are negative encirclements. This is a case where feedback destabilized a stable system. Rule 2. 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 stability of and G 1 Note that the phase margin for \(\Lambda=0.7\), found as shown on Figure \(\PageIndex{2}\), is quite clear on Figure \(\PageIndex{4}\) and not at all ambiguous like the gain margin: \(\mathrm{PM}_{0.7} \approx+20^{\circ}\); this value also indicates a stable, but weakly so, closed-loop system. 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. and poles of is determined by the values of its poles: for stability, the real part of every pole must be negative. Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. . of poles of T(s)). We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. 0 {\displaystyle 1+G(s)} Its system function is given by Black's formula, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)},\]. F s 1 ) s 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. ) j 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. Mark the roots of b 1 The argument principle from complex analysis gives a criterion to calculate the difference between the number of zeros and the number of poles of ( Cauchy's argument principle states that, Where In general, the feedback factor will just scale the Nyquist plot. s have positive real part. \(G\) has one pole in the right half plane. Does the system have closed-loop poles outside the unit circle? ( 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, \[s^{3}+3 s^{2}+28 s+26+\Lambda\left(s^{2}+4 s+104\right)=s^{3}+(3+\Lambda) s^{2}+4(7+\Lambda) s+26(1+4 \Lambda)=0\label{17.19} \]. We can factor L(s) to determine the number of poles that are in the 0 G ) + 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 \(\Lambda_{n s 1}\). 0000001210 00000 n {\displaystyle Z} {\displaystyle Z} ( Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. F F 1 ( To use this criterion, the frequency response data of a system must be presented as a polar plot in Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. be the number of poles of gives us the image of our contour under 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. ( This typically means that the parameter is swept logarithmically, in order to cover a wide range of values. The right hand graph is the Nyquist plot. {\displaystyle {\mathcal {T}}(s)} ( When plotted computationally, one needs to be careful to cover all frequencies of interest. 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