G ( Stability is determined by looking at the number of encirclements of the point (1, 0). The only pole is at \(s = -1/3\), so the closed loop system is stable. ) F 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 The Nyquist criterion allows us to answer two questions: 1. (There is no particular reason that \(a\) needs to be real in this example. We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane: The Nyquist contour mapped through the function The Nyquist criterion is a frequency domain tool which is used in the study of stability. But in physical systems, complex poles will tend to come in conjugate pairs.). One way to do it is to construct a semicircular arc with radius L is called the open-loop transfer function. s enclosed by the contour and You should be able to show that the zeros of this transfer function in the complex \(s\)-plane are at (\(2 j10\)), and the poles are at (\(1 + j0\)) and (\(1 j5\)). The zeros of the denominator \(1 + k G\). Now, recall that the poles of \(G_{CL}\) are exactly the zeros of \(1 + k G\). a clockwise semicircle at L(s)= in "L(s)" (see, The clockwise semicircle at infinity in "s" corresponds to a single \nonumber\]. k G The curve winds twice around -1 in the counterclockwise direction, so the winding number \(\text{Ind} (kG \circ \gamma, -1) = 2\). ) 1 While Nyquist is one of the most general stability tests, it is still restricted to linear time-invariant (LTI) systems. + 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 \(\PageIndex{2}\)? 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 s With \(k =1\), what is the winding number of the Nyquist plot around -1? It applies the principle of argument to an open-loop transfer function to derive information about the stability of the closed-loop systems transfer function. 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. 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. = ) The poles are \(\pm 2, -2 \pm i\). s = s {\displaystyle D(s)} . times, where Make a system with the following zeros and poles: Is the corresponding closed loop system stable when \(k = 6\)? 0000039933 00000 n ( A simple pole at \(s_1\) corresponds to a mode \(y_1 (t) = e^{s_1 t}\). This approach appears in most modern textbooks on control theory. (2 h) lecture: Introduction to the controller's design specifications. v Transfer Function System Order -thorder system Characteristic Equation (Closed Loop Denominator) s+ Go! (Actually, for \(a = 0\) the open loop is marginally stable, but it is fully stabilized by the closed loop.). = The counterclockwise detours around the poles at s=j4 results in If the system with system function \(G(s)\) is unstable it can sometimes be stabilized by what is called a negative feedback loop. {\displaystyle \Gamma _{s}} ( {\displaystyle {\mathcal {T}}(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. {\displaystyle G(s)} 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); 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. The following MATLAB commands, adapted from the code that produced Figure 16.5.1, calculate and plot the loci of roots: Lm=[0 .2 .4 .7 1 1.5 2.5 3.7 4.75 6.5 9 12.5 15 18.5 25 35 50 70 125 250]; a2=3+Lm(i);a3=4*(7+Lm(i));a4=26*(1+4*Lm(i)); plot(p,'kx'),grid,xlabel('Real part of pole (sec^-^1)'), ylabel('Imaginary part of pole (sec^-^1)'). ) The Nyquist method is used for studying the stability of linear systems with pure time delay. Any Laplace domain transfer function {\displaystyle D(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 } ). Legal. F {\displaystyle -1/k} By the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of ( {\displaystyle G(s)} ) . + . From the mapping we find the number N, which is the number of ) Given our definition of stability above, we could, in principle, discuss stability without the slightest idea what it means for physical systems. 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. Graphical method of determining the stability of a dynamical system, The Nyquist criterion for systems with poles on the imaginary axis, "Chapter 4.3. G There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. k *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. s , then the roots of the characteristic equation are also the zeros of 0000001503 00000 n Step 2 Form the Routh array for the given characteristic polynomial. 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 ( 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 \(-18.5 / 0.96438+j 0=-19.18+j 0\). {\displaystyle 0+j(\omega +r)} 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. = However, the positive gain margin 10 dB suggests positive stability. \[G_{CL} (s) \text{ is stable } \Leftrightarrow \text{ Ind} (kG \circ \gamma, -1) = P_{G, RHP}\]. inside the contour We will be concerned with the stability of the system. ( 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. ( ( s 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. This typically means that the parameter is swept logarithmically, in order to cover a wide range of values. F {\displaystyle {\frac {G}{1+GH}}} This assumption holds in many interesting cases. + ) ) 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. 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 \(\Lambda_{n s 1}\). ( 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. Since they are all in the left half-plane, the system is stable. {\displaystyle GH(s)} ( ) -P_PcXJ']b9-@f8+5YjmK p"yHL0:8UK=MY9X"R&t5]M/o 3\\6%W+7J$)^p;% XpXC#::` :@2p1A%TQHD1Mdq!1 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. 2. 1 s s be the number of poles of ( using the Routh array, but this method is somewhat tedious. Any clockwise encirclements of the critical point by the open-loop frequency response (when judged from low frequency to high frequency) would indicate that the feedback control system would be destabilizing if the loop were closed. ) Recalling that the zeros of . ( = If the number of poles is greater than the G For example, the unusual case of an open-loop system that has unstable poles requires the general Nyquist stability criterion. {\displaystyle 1+G(s)} ) and travels anticlockwise to So in the limit \(kG \circ \gamma_R\) becomes \(kG \circ \gamma\). + ). {\displaystyle 1+G(s)} Thus, it is stable when the pole is in the left half-plane, i.e. + 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. ) Hence, the number of counter-clockwise encirclements about {\displaystyle N} B ( In practice, the ideal sampler is replaced by We thus find that s = To use this criterion, the frequency response data of a system must be presented as a polar plot in Keep in mind that the plotted quantity is A, i.e., the loop gain. Pole-zero diagrams for the three systems. ) 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 \(\PageIndex{5}\). {\displaystyle F(s)} Suppose \(G(s) = \dfrac{s + 1}{s - 1}\). shall encircle (clockwise) the point P We will make a standard assumption that \(G(s)\) is meromorphic with a finite number of (finite) poles. drawn in the complex {\displaystyle s} have positive real part. / + plane in the same sense as the contour ) are same as the poles of ) Legal. ) For the Nyquist plot and criterion the curve \(\gamma\) will always be the imaginary \(s\)-axis. , can be mapped to another plane (named Now refresh the browser to restore the applet to its original state. So far, we have been careful to say the system with system function \(G(s)\)'. ( The frequency is swept as a parameter, resulting in a pl We will just accept this formula. Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. ( P The factor \(k = 2\) will scale the circle in the previous example by 2. G + s G F ( The Nyquist plot of Is the open loop system stable? s + The system with system function \(G(s)\) is called stable if all the poles of \(G\) are in the left half-plane. s . (0.375) yields the gain that creates marginal stability (3/2). The poles of \(G(s)\) correspond to what are called modes of the system. Hb```f``$02 +0p$ 5;p.BeqkR {\displaystyle G(s)} The Nyquist Stability Criteria is a test for system stability, just like the Routh-Hurwitz test, or the Root-Locus Methodology. It is more challenging for higher order systems, but there are methods that dont require computing the poles. ( the clockwise direction. ( 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 The argument principle from complex analysis gives a criterion to calculate the difference between the number of zeros and the number of poles of This is a case where feedback stabilized an unstable system. ( T s {\displaystyle G(s)} G H 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 However, the Nyquist Criteria can also give us additional information about a system. N Set the feedback factor \(k = 1\). This method for design involves plotting the complex loci of P ( s) C ( s) for the range s = j , = [ , ]. G plane + That is, setting It is also the foundation of robust control theory. j Since one pole is in the right half-plane, the system is unstable. Suppose that the open-loop transfer function of a system is1, \[G(s) \times H(s) \equiv O L T F(s)=\Lambda \frac{s^{2}+4 s+104}{(s+1)\left(s^{2}+2 s+26\right)}=\Lambda \frac{s^{2}+4 s+104}{s^{3}+3 s^{2}+28 s+26}\label{eqn:17.18} \]. and Since \(G_{CL}\) is a system function, we can ask if the system is stable. I'm confused due to the fact that the Nyquist stability criterion and looking at the transfer function doesn't give the same results whether a feedback system is stable or not. ( s (At \(s_0\) it equals \(b_n/(kb_n) = 1/k\).). ( ) s + The left hand graph is the pole-zero diagram. Its image under \(kG(s)\) will trace out the Nyquis plot. Take \(G(s)\) from the previous example. s ) Thus, we may find F Check the \(Formula\) box. N {\displaystyle 1+G(s)} {\displaystyle s} {\displaystyle G(s)} Lecture 1: The Nyquist Criterion S.D. ) in the new Figure 19.3 : Unity Feedback Confuguration. A ( s G r in the right-half complex plane minus the number of poles of {\displaystyle G(s)} Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. When the highest frequency of a signal is less than the Nyquist frequency of the sampler, the resulting discrete-time sequence is said to be free of the 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). + 0.375=3/2 (the current gain (4) multiplied by the gain margin For this topic we will content ourselves with a statement of the problem with only the tiniest bit of physical context. = 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 17.4.2, thus rendering ambiguous the definition of phase margin. The most common use of Nyquist plots is for assessing the stability of a system with feedback. {\displaystyle \Gamma _{s}} in the contour In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. entire right half plane. by Cauchy's argument principle. in the right-half complex plane. So, the control system satisfied the necessary condition. Such a modification implies that the phasor {\displaystyle 1+GH} Z Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. The \(\Lambda=\Lambda_{n s 1}\) plot of Figure \(\PageIndex{4}\) is expanded radially outward on Figure \(\PageIndex{5}\) by the factor of \(4.75 / 0.96438=4.9254\), so the loop for high frequencies beneath the negative \(\operatorname{Re}[O L F R F]\) axis is more prominent than on Figure \(\PageIndex{4}\). D Nyquist plot of \(G(s) = 1/(s + 1)\), with \(k = 1\). {\displaystyle {\mathcal {T}}(s)} \(G\) has one pole in the right half plane. ) , which is the contour ( (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. F Nyquist stability criterion is a general stability test that checks for the stability of linear time-invariant systems. T N 0 ) It is certainly reasonable to call a system that does this in response to a zero signal (often called no input) unstable. must be equal to the number of open-loop poles in the RHP. F Z {\displaystyle Z} The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are s The frequency is swept as a parameter, resulting in a plot per frequency. can be expressed as the ratio of two polynomials: It is easy to check it is the circle through the origin with center \(w = 1/2\). Note that we count encirclements in the Contact Pro Premium Expert Support Give us your feedback Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. has exactly the same poles as The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. {\displaystyle s={-1/k+j0}} s 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. {\displaystyle -l\pi } An approach to this end is through the use of Nyquist techniques. 91 0 obj << /Linearized 1 /O 93 /H [ 701 509 ] /L 247721 /E 42765 /N 23 /T 245783 >> endobj xref 91 13 0000000016 00000 n 1 {\displaystyle G(s)} On the other hand, a Bode diagram displays the phase-crossover and gain-crossover frequencies, which are not explicit on a traditional Nyquist plot. {\displaystyle P} Is the open loop system stable? ( {\displaystyle 1+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. ( and poles of s 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 \(\Lambda_{\mathrm{ns}}\). Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\) stable? 1 {\displaystyle 1+G(s)} (10 points) c) Sketch the Nyquist plot of the system for K =1. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. ( Look at the pole diagram and use the mouse to drag the yellow point up and down the imaginary axis. Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. Stability in the Nyquist Plot. s Nyquist criterion and stability margins. , or simply the roots of The poles of \(G\). {\displaystyle \Gamma _{s}} 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. If the answer to the first question is yes, how many closed-loop poles are outside the unit circle? ( {\displaystyle F(s)} ( {\displaystyle 1+G(s)} *(26- w.^2+2*j*w)); >> plot(real(olfrf007),imag(olfrf007)),grid, >> hold,plot(cos(cirangrad),sin(cirangrad)). {\displaystyle Z} 1 Closed loop approximation f.d.t. (3h) lecture: Nyquist diagram and on the effects of feedback. denotes the number of poles of 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 \(\omega\). Draw the Nyquist plot with \(k = 1\). 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. T j 0. So the winding number is -1, which does not equal the number of poles of \(G\) in the right half-plane. G The stability of G plane) by the function by counting the poles of L is called the open-loop transfer function. For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero. if the poles are all in the left half-plane. Z For this we will use one of the MIT Mathlets (slightly modified for our purposes). be the number of zeros of G The Nyquist criterion is a frequency domain tool which is used in the study of stability. F s F {\displaystyle D(s)=1+kG(s)} We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. The formula is an easy way to read off the values of the poles and zeros of \(G(s)\). Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. We will look a Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. >> olfrf01=(104-w.^2+4*j*w)./((1+j*w). F The correct Nyquist rate is defined in terms of the system Bandwidth (in the frequency domain) which is determined by the Point Spread Function. While sampling at the Nyquist rate is a very good idea, it is in many practical situations hard to attain. D {\displaystyle F(s)} s s ) ( s {\displaystyle {\mathcal {T}}(s)} This gives us, We now note that = around {\displaystyle F(s)} {\displaystyle N} {\displaystyle F} Open the Nyquist Plot applet at.
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