( ( Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. Refresh the page, to put the zero and poles back to their original state. 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. We will look a little more closely at such systems when we study the Laplace transform in the next topic. {\displaystyle GH(s)} ) Das Stabilittskriterium von Strecker-Nyquist", "Inventing the 'black box': mathematics as a neglected enabling technology in the history of communications engineering", EIS Spectrum Analyser - a freeware program for analysis and simulation of impedance spectra, Mathematica function for creating the Nyquist plot, https://en.wikipedia.org/w/index.php?title=Nyquist_stability_criterion&oldid=1121126255, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License 3.0, However, if the graph happens to pass through the point, This page was last edited on 10 November 2022, at 17:05. This results from the requirement of the argument principle that the contour cannot pass through any pole of the mapping function. , as evaluated above, is equal to0. If we set \(k = 3\), the closed loop system is stable. If the number of poles is greater than the number of zeros, then the Nyquist criterion tells us how to use the Nyquist plot to graphically determine the stability of the closed loop system. ( ( Yes! are also said to be the roots of the characteristic equation 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 If the counterclockwise detour was around a double pole on the axis (for example two by Cauchy's argument principle. The poles are \(-2, -2\pm i\). {\displaystyle F(s)} ) In 18.03 we called the system stable if every homogeneous solution decayed to 0. T We dont analyze stability by plotting the open-loop gain or G Stability in the Nyquist Plot. *(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 \(\PageIndex{1}\). {\displaystyle \Gamma _{s}} The most common case are systems with integrators (poles at zero). {\displaystyle {\mathcal {T}}(s)} ( {\displaystyle N=P-Z} G From the mapping we find the number N, which is the number of 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. Recalling that the zeros of s In the case \(G(s)\) is a fractional linear transformation, so we know it maps the imaginary axis to a circle. 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 \(\Lambda\)? Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. A Choose \(R\) large enough that the (finite number) of poles and zeros of \(G\) in the right half-plane are all inside \(\gamma_R\). ( That is, the Nyquist plot is the circle through the origin with center \(w = 1\). . 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 However, the actual hardware of such an open-loop system could not be subjected to frequency-response experimental testing due to its unstable character, so a control-system engineer would find it necessary to analyze a mathematical model of the system. ) s F 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 \(\Lambda_{\mathrm{ns}}\). \(G(s) = \dfrac{s - 1}{s + 1}\). P {\displaystyle r\to 0} 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. {\displaystyle D(s)=1+kG(s)} 1 {\displaystyle s={-1/k+j0}} + v 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 \(\omega\). s For our purposes it would require and an indented contour along the imaginary axis. Static and dynamic specifications. 1 Our goal is to, through this process, check for the stability of the transfer function of our unity feedback system with gain k, which is given by, That is, we would like to check whether the characteristic equation of the above transfer function, given by. However, it is not applicable to non-linear systems as for that complex stability criterion like Lyapunov is used. = enclosing the right half plane, with indentations as needed to avoid passing through zeros or poles of the function ( Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}\) stable? ) 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 \(-18.5 / 0.96438+j 0=-19.18+j 0\). s ) ) ) F have positive real part. {\displaystyle u(s)=D(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. In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. Nyquist criterion and stability margins. A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. s 0 . Any Laplace domain transfer function 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. Draw the Nyquist plot with \(k = 1\). the same system without its feedback loop). s To get a feel for the Nyquist plot. Now, recall that the poles of \(G_{CL}\) are exactly the zeros of \(1 + k G\). Answer: The closed loop system is stable for \(k\) (roughly) between 0.7 and 3.10. The following MATLAB commands calculate [from Equations 17.1.12 and \(\ref{eqn:17.20}\)] and plot the frequency response and an arc of the unit circle centered at the origin of the complex \(OLFRF(\omega)\)-plane. {\displaystyle F(s)} The poles are \(\pm 2, -2 \pm i\). where \(k\) is called the feedback factor. {\displaystyle 1+GH} u For a SISO feedback system the closed-looptransfer function is given by where represents the system and is the feedback element. So far, we have been careful to say the system with system function \(G(s)\)'. , e.g. We can show this formally using Laurent series. Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. Accessibility StatementFor more information contact us [email protected] check out our status page at https://status.libretexts.org. 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). Calculate transfer function of two parallel transfer functions in a feedback loop. However, the gain margin calculated from either of the two phase crossovers suggests instability, showing that both are deceptively defective metrics of stability. Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. ( Accessibility StatementFor more information contact us [email protected] check out our status page at https://status.libretexts.org. {\displaystyle P} D s s If the system is originally open-loop unstable, feedback is necessary to stabilize the system. G 2. The stability of B Z Lecture 1: The Nyquist Criterion S.D. 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 \(\approx 15\). ( j {\displaystyle P} In \(\gamma (\omega)\) the variable is a greek omega and in \(w = G \circ \gamma\) we have a double-u. {\displaystyle Z} We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. The formula is an easy way to read off the values of the poles and zeros of \(G(s)\). s s ) The Nyquist plot is the graph of \(kG(i \omega)\). \(G_{CL}\) is stable exactly when all its poles are in the left half-plane. ( T 1 Assume \(a\) is real, for what values of \(a\) is the open loop system \(G(s) = \dfrac{1}{s + a}\) stable? 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 -1/k} s 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. The Nyquist Contour Assumption: Traverse the Nyquist contour in CW direction Observation #1: Encirclement of a pole forces the contour to gain 360 degrees so the Nyquist evaluation encircles origin in CCW direction Observation #2 Encirclement of a zero forces the contour to loose 360 degrees so the Nyquist evaluation encircles origin in CW direction j j + s ) The mathematics uses the Laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. Here N = 1. T D The Nyquist Stability Criteria is a test for system stability, just like the Routh-Hurwitz test, or the Root-Locus Methodology. {\displaystyle F(s)} Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. ) The beauty of the Nyquist stability criterion lies in the fact that it is a rather simple graphical test. This continues until \(k\) is between 3.10 and 3.20, at which point the winding number becomes 1 and \(G_{CL}\) becomes unstable. 0.375=3/2 (the current gain (4) multiplied by the gain margin 1 ) using the Routh array, but this method is somewhat tedious. (At \(s_0\) it equals \(b_n/(kb_n) = 1/k\).). s . ( u It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. s (Using RHP zeros to "cancel out" RHP poles does not remove the instability, but rather ensures that the system will remain unstable even in the presence of feedback, since the closed-loop roots travel between open-loop poles and zeros in the presence of feedback. ( It is easy to check it is the circle through the origin with center \(w = 1/2\). {\displaystyle \Gamma _{s}} Compute answers using Wolfram's breakthrough technology & The poles of the closed loop system function \(G_{CL} (s)\) given in Equation 12.3.2 are the zeros of \(1 + kG(s)\). G 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. {\displaystyle 0+j(\omega -r)} Phase margins are indicated graphically on Figure \(\PageIndex{2}\). The algebra involved in canceling the \(s + a\) term in the denominators is exactly the cancellation that makes the poles of \(G\) removable singularities in \(G_{CL}\). s Given our definition of stability above, we could, in principle, discuss stability without the slightest idea what it means for physical systems. The mathematical foundations of the criterion can be found in many advanced mathematics or linear control theory texts such as Wylie and Barrett (1982), D'Azzo and does not have any pole on the imaginary axis (i.e. The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s + 1)/(s - 1)}{1 + 2(s + 1)/(s - 1)} = \dfrac{s + 1}{3s + 1}.\]. G ) 0000002305 00000 n
( Let \(G(s) = \dfrac{1}{s + 1}\). {\displaystyle G(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. T Nyquist stability criterion states the number of encirclements about the critical point (1+j0) must be equal to the poles of characteristic equation, which is nothing but the poles of the open loop transfer function in the right half of the s plane. 1 {\displaystyle 1+GH(s)} If the number of poles is greater than the number of zeros, then the Nyquist criterion tells us how to use the Nyquist plot to graphically determine the stability of the closed loop system. It does not represent any specific real physical system, but it has characteristics that are representative of some real systems. s s s Graphical method of determining the stability of a dynamical system, The Nyquist criterion for systems with poles on the imaginary axis, "Chapter 4.3. We will make a standard assumption that \(G(s)\) is meromorphic with a finite number of (finite) poles. It is certainly reasonable to call a system that does this in response to a zero signal (often called no input) unstable. 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. enclosed by the contour and {\displaystyle F(s)} ) Lecture 2 2 Nyquist Plane Results GMPM Criteria ESAC Criteria Real Axis Nyquist Contour, Unstable Case Nyquist Contour, Stable Case Imaginary This has one pole at \(s = 1/3\), so the closed loop system is unstable. is mapped to the point 0000039933 00000 n
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XpXC#::` :@2p1A%TQHD1Mdq!1 A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. ) Figure 19.3 : Unity Feedback Confuguration. s D We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. H ) The Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. s G We conclude this chapter on frequency-response stability criteria by observing that margins of gain and phase are used also as engineering design goals. s {\displaystyle 0+j(\omega +r)} k , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\) stable? 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. N Thus, for all large \(R\), \[\text{the system is stable } \Leftrightarrow \ Z_{1 + kG, \gamma_R} = 0 \ \Leftrightarow \ \text{ Ind} (kG \circ \gamma_R, -1) = P_{G, \gamma_R}\], Finally, we can let \(R\) go to infinity. Legal. 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 {\displaystyle \Gamma _{s}} {\displaystyle N(s)} {\displaystyle GH(s)={\frac {A(s)}{B(s)}}} are called the zeros of Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. 1 We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 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. Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. 0 Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. ) 0000001367 00000 n
While Nyquist is one of the most general stability tests, it is still restricted to linear time-invariant (LTI) systems. The argument principle from complex analysis gives a criterion to calculate the difference between the number of zeros and the number of poles of + The value of \(\Lambda_{n s 2}\) is not exactly 15, as Figure \(\PageIndex{3}\) might suggest; see homework Problem 17.2(b) for calculation of the more precise value \(\Lambda_{n s 2} = 15.0356\). and poles of = k s drawn in the complex r ) So, stability of \(G_{CL}\) is exactly the condition that the number of zeros of \(1 + kG\) in the right half-plane is 0. The Nyquist criterion is a frequency domain tool which is used in the study of stability. 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. ( You can also check that it is traversed clockwise. N s For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero. On the other hand, a Bode diagram displays the phase-crossover and gain-crossover frequencies, which are not explicit on a traditional Nyquist plot. Z Hb```f``$02 +0p$ 5;p.BeqkR Hence, the number of counter-clockwise encirclements about ( P Rearranging, we have So the winding number is -1, which does not equal the number of poles of \(G\) in the right half-plane. 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. ( G {\displaystyle Z=N+P} The new system is called a closed loop system. The system with system function \(G(s)\) is called stable if all the poles of \(G\) are in the left half-plane. represents how slow or how fast is a reaction is. s {\displaystyle Z} 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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