1 {\displaystyle 1+G(s)} ( The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. The Nyquist method is used for studying the stability of linear systems with pure time delay. 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. s s 0 ) ( + + 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 \(\Lambda_{n s 1}=0.96438\). The argument principle from complex analysis gives a criterion to calculate the difference between the number of zeros and the number of 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. \(G(s)\) has one pole at \(s = -a\). Since we know N and P, we can determine Z, the number of zeros of The poles of \(G(s)\) correspond to what are called modes of the system. k In the case \(G(s)\) is a fractional linear transformation, so we know it maps the imaginary axis to a circle. = We will look a little more closely at such systems when we study the Laplace transform in the next topic. This reference shows that the form of stability criterion described above [Conclusion 2.] {\displaystyle G(s)} shall encircle (clockwise) the point Draw the Nyquist plot with \(k = 1\). 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 \(-18.5 / 0.96438+j 0=-19.18+j 0\). G 0.375=3/2 (the current gain (4) multiplied by the gain margin That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. = ), Start with a system whose characteristic equation is given by . F D s The poles of the closed loop system function \(G_{CL} (s)\) given in Equation 12.3.2 are the zeros of \(1 + kG(s)\). Because it only looks at the Nyquist plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane singularities must be known). 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 \(\omega\). G k j as defined above corresponds to a stable unity-feedback system when Rearranging, we have A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. We conclude this chapter on frequency-response stability criteria by observing that margins of gain and phase are used also as engineering design goals. However, the Nyquist Criteria can also give us additional information about a system. ( Additional parameters appear if you check the option to calculate the Theoretical PSF. times such that N {\displaystyle 1+GH} 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 1 If, on the other hand, we were to calculate gain margin using the other phase crossing, at about \(-0.04+j 0\), then that would lead to the exaggerated \(\mathrm{GM} \approx 25=28\) dB, which is obviously a defective metric of stability. is the number of poles of the open-loop transfer function In \(\gamma (\omega)\) the variable is a greek omega and in \(w = G \circ \gamma\) we have a double-u. is mapped to the point {\displaystyle F(s)} D 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. s The Nyquist criterion for systems with poles on the imaginary axis. The above consideration was conducted with an assumption that the open-loop transfer function G ( s ) {displaystyle G(s)} does not have any pole on the imaginary axis (i.e. poles of the form 0 + j {displaystyle 0+jomega } ). = is formed by closing a negative unity feedback loop around the open-loop transfer function For the edge case where no poles have positive real part, but some are pure imaginary we will call the system marginally stable. Any class or book on control theory will derive it for you. It is easy to check it is the circle through the origin with center \(w = 1/2\). s Section 17.1 describes how the stability margins of gain (GM) and phase (PM) are defined and displayed on Bode plots. ) Stability in the Nyquist Plot. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. ) s Based on analysis of the Nyquist Diagram: (i) Comment on the stability of the closed loop system. ) This continues until \(k\) is between 3.10 and 3.20, at which point the winding number becomes 1 and \(G_{CL}\) becomes unstable. 1 {\displaystyle 1+GH(s)} 0000001188 00000 n
) ) ( {\displaystyle -1+j0} {\displaystyle P} ( {\displaystyle 1+G(s)} In units of Hz, its value is one-half of the sampling rate. ) k s s 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. 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. ( In practice, the ideal sampler is replaced by . ) = 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. 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.. ) . For a SISO feedback system the closed-looptransfer function is given by where represents the system and is the feedback element. ) 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 \(\Lambda\)? ( ) ) There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. ( s s ( N Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. G We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. {\displaystyle G(s)} It is more challenging for higher order systems, but there are methods that dont require computing the poles. It is certainly reasonable to call a system that does this in response to a zero signal (often called no input) unstable. The Nyquist plot can provide some information about the shape of the transfer function. + To get a feel for the Nyquist plot. {\displaystyle r\to 0} Thus, it is stable when the pole is in the left half-plane, i.e. denotes the number of zeros of poles of the form To simulate that testing, we have from Equation \(\ref{eqn:17.18}\), the following equation for the frequency-response function: \[O L F R F(\omega) \equiv O L T F(j \omega)=\Lambda \frac{104-\omega^{2}+4 \times j \omega}{(1+j \omega)\left(26-\omega^{2}+2 \times j \omega\right)}\label{eqn:17.20} \]. {\displaystyle N} N By the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of ( {\displaystyle G(s)} denotes the number of poles of From the mapping we find the number N, which is the number of 17: Introduction to System Stability- Frequency-Response Criteria, Introduction to Linear Time-Invariant Dynamic Systems for Students of Engineering (Hallauer), { "17.01:_Gain_Margins,_Phase_Margins,_and_Bode_Diagrams" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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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. j The Nyquist Stability Criteria is a test for system stability, just like the Routh-Hurwitz test, or the Root-Locus Methodology. Compute answers using Wolfram's breakthrough technology & {\displaystyle A(s)+B(s)=0} D s 0000001731 00000 n
s 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 T(s)} are the poles of the closed-loop system, and noting that the poles of Give zero-pole diagrams for each of the systems, \[G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\]. 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. + On the other hand, a Bode diagram displays the phase-crossover and gain-crossover frequencies, which are not explicit on a traditional Nyquist plot. clockwise. This page titled 17.4: The Nyquist Stability Criterion is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. The factor \(k = 2\) will scale the circle in the previous example by 2. We will be concerned with the stability of the system. In 18.03 we called the system stable if every homogeneous solution decayed to 0. That is, setting l G Z u of poles of T(s)). {\displaystyle N=Z-P} The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. Let us complete this study by computing \(\operatorname{OLFRF}(\omega)\) and displaying it on Nyquist plots for the value corresponding to the transition from instability back to stability on Figure \(\PageIndex{3}\), which we denote as \(\Lambda_{n s 2} \approx 15\), and for a slightly higher value, \(\Lambda=18.5\), for which the closed-loop system is stable. Note that we count encirclements in the ( Step 2 Form the Routh array for the given characteristic polynomial. T Does the system have closed-loop poles outside the unit circle? in the right half plane, the resultant contour in the s The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s - 1)/(s + 1)}{1 + 2(s - 1)/(s + 1)} = \dfrac{s - 1}{3s - 1}.\]. Rule 2. ) Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability. {\displaystyle Z=N+P} The mathematics uses the Laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. 0 ) + A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. Practice, the ideal sampler is replaced by. a system. at such systems we. By. signal processing system and is the circle in the previous example by 2. represents the system if! 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