(Actually, for \(a = 0\) the open loop is marginally stable, but it is fully stabilized by the closed loop.). This typically means that the parameter is swept logarithmically, in order to cover a wide range of values. 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). ) s ) G Does the system have closed-loop poles outside the unit circle? encirclements of the -1+j0 point in "L(s).". denotes the number of poles of be the number of zeros of ) For what values of \(a\) is the corresponding closed loop system \(G_{CL} (s)\) stable? However, the gain margin calculated from either of the two phase crossovers suggests instability, showing that both are deceptively defective metrics of stability. Calculate transfer function of two parallel transfer functions in a feedback loop. {\displaystyle G(s)} 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 \(\omega\). ( The Nyquist plot is the trajectory of \(K(i\omega) G(i\omega) = ke^{-ia\omega}G(i\omega)\) , where \(i\omega\) traverses the imaginary axis. ) {\displaystyle G(s)} 1 , or simply the roots of {\displaystyle \Gamma _{F(s)}=F(\Gamma _{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. + + ) On the other hand, the phase margin shown on Figure \(\PageIndex{6}\), \(\mathrm{PM}_{18.5} \approx+12^{\circ}\), correctly indicates weak stability. ( 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. + If the system is originally open-loop unstable, feedback is necessary to stabilize the system. Z , where This assumption holds in many interesting cases. The only plot of \(G(s)\) is in the left half-plane, so the open loop system is stable. enclosed by the contour and 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). s 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. {\displaystyle P} For a SISO feedback system the closed-looptransfer function is given by where represents the system and is the feedback element. ) The range of gains over which the system will be stable can be determined by looking at crossings of the real axis. If the system with system function \(G(s)\) is unstable it can sometimes be stabilized by what is called a negative feedback loop. A linear time invariant system has a system function which is a function of a complex variable. k + ( D Moreover, if we apply for this system with \(\Lambda=4.75\) the MATLAB margin command to generate a Bode diagram in the same form as Figure 17.1.5, then MATLAB annotates that diagram with the values \(\mathrm{GM}=10.007\) dB and \(\mathrm{PM}=-23.721^{\circ}\) (the same as PM4.75 shown approximately on Figure \(\PageIndex{5}\)). poles at the origin), the path in L(s) goes through an angle of 360 in Recalling that the zeros of G \(G_{CL}\) is stable exactly when all its poles are in the left half-plane. (0.375) yields the gain that creates marginal stability (3/2). ) \(G\) has one pole in the right half plane. s 1 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. plane in the same sense as the contour ) Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. s 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 \(\PageIndex{4}\). s This results from the requirement of the argument principle that the contour cannot pass through any pole of the mapping function. 0000002847 00000 n ( {\displaystyle 1+G(s)} We suppose that we have a clockwise (i.e. k Cauchy's argument principle states that, Where 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. Describe the Nyquist plot with gain factor \(k = 2\). We dont analyze stability by plotting the open-loop gain or Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. {\displaystyle GH(s)} + P Compute answers using Wolfram's breakthrough technology & Nyquist criterion and stability margins. \(G(s)\) has one pole at \(s = -a\). In practice, the ideal sampler is replaced by 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. P G in the right half plane, the resultant contour in the This case can be analyzed using our techniques. L is called the open-loop transfer function. s 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. ) Hence, the number of counter-clockwise encirclements about ( While Nyquist is one of the most general stability tests, it is still restricted to linear time-invariant (LTI) systems. ) F ) ( 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. The Nyquist plot is the graph of \(kG(i \omega)\). = Nyquist plot of the transfer function s/(s-1)^3. That is, \[s = \gamma (\omega) = i \omega, \text{ where } -\infty < \omega < \infty.\], For a system \(G(s)\) and a feedback factor \(k\), the Nyquist plot is the plot of the curve, \[w = k G \circ \gamma (\omega) = kG(i \omega).\]. . Since \(G_{CL}\) is a system function, we can ask if the system is stable. 0000002305 00000 n ) represents how slow or how fast is a reaction is. 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 Z s Nyquist stability criterion (or Nyquist criteria) is defined as a graphical technique used in control engineering for determining the stability of a dynamical system. ( Check the \(Formula\) box. + Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. 1 ) Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. 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. ) s Here N = 1. 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. can be expressed as the ratio of two polynomials: {\displaystyle F(s)} {\displaystyle F(s)} ( + 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. 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. . ( Then the closed loop system with feedback factor \(k\) is stable if and only if the winding number of the Nyquist plot around \(w = -1\) equals the number of poles of \(G(s)\) in the right half-plane. Now refresh the browser to restore the applet to its original state. {\displaystyle G(s)} inside the contour j plane) by the function In the previous problem could you determine analytically the range of \(k\) where \(G_{CL} (s)\) is stable? {\displaystyle G(s)} 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 \(-18.5 / 0.96438+j 0=-19.18+j 0\). {\displaystyle l} ) ) be the number of poles of Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. {\displaystyle -1/k} (10 points) c) Sketch the Nyquist plot of the system for K =1. 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. ( The above consideration was conducted with an assumption that the open-loop transfer function It is perfectly clear and rolls off the tongue a little easier! and travels anticlockwise to Techniques like Bode plots, while less general, are sometimes a more useful design tool. s ) {\displaystyle G(s)} {\displaystyle G(s)} s {\displaystyle Z} "1+L(s)" in the right half plane (which is the same as the number Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability. ) Section 17.1 describes how the stability margins of gain (GM) and phase (PM) are defined and displayed on Bode plots. That is, the Nyquist plot is the image of the imaginary axis under the map \(w = kG(s)\). s N The zeros of the denominator \(1 + k G\). s We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. There are two poles in the right half-plane, so the open loop system \(G(s)\) is unstable. ( G 0 A ( if the poles are all in the left half-plane. ( {\displaystyle {\mathcal {T}}(s)} 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 \(\PageIndex{2}\)? 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 F(s)} ) 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 \(\Lambda_{n s 2} = 15.0356\). 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. The Nyquist criterion allows us to answer two questions: 1. If the counterclockwise detour was around a double pole on the axis (for example two ( is determined by the values of its poles: for stability, the real part of every pole must be negative. 1 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 1+kF(s)} ) Figure 19.3 : Unity Feedback Confuguration. Step 1 Verify the necessary condition for the Routh-Hurwitz stability. plane yielding a new contour. s So, the control system satisfied the necessary condition. times, where The poles are \(\pm 2, -2 \pm i\). Rule 2. The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. {\displaystyle \Gamma _{s}} Here s , then the roots of the characteristic equation are also the zeros of The right hand graph is the Nyquist plot. ) ( {\displaystyle D(s)=1+kG(s)} Double control loop for unstable systems. 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. However, it is not applicable to non-linear systems as for that complex stability criterion like Lyapunov is used. ) N G , which is to say our Nyquist plot. = = The other phase crossover, at \(-4.9254+j 0\) (beyond the range of Figure \(\PageIndex{5}\)), might be the appropriate point for calculation of gain margin, since it at least indicates instability, \(\mathrm{GM}_{4.75}=1 / 4.9254=0.20303=-13.85\) dB. 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 } ). Lecture 1: The Nyquist Criterion S.D. In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. Nyquist plot of the transfer function s/ (s-1)^3 Natural Language Math Input Extended Keyboard Examples Have a question about using Wolfram|Alpha? . 0000001367 00000 n ) = 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. The mathlet shows the Nyquist plot winds once around \(w = -1\) in the \(clockwise\) direction. While Nyquist is one of the most general stability tests, it is still restricted to linear, time-invariant (LTI) systems. Note that we count encirclements in the Figure 19.3 : Unity Feedback Confuguration. u For the Nyquist plot and criterion the curve \(\gamma\) will always be the imaginary \(s\)-axis. Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. 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. G {\displaystyle {\frac {G}{1+GH}}} 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); So far, we have been careful to say the system with system function \(G(s)\)'. where \(k\) is called the feedback factor. Thus, it is stable when the pole is in the left half-plane, i.e. Transfer Function System Order -thorder system Characteristic Equation We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Thus, we may finally state that. T Legal. s 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 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. , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. is mapped to the point {\displaystyle P} This is a diagram in the \(s\)-plane where we put a small cross at each pole and a small circle at each zero. Instead of Cauchy's argument principle, the original paper by Harry Nyquist in 1932 uses a less elegant approach. , and {\displaystyle G(s)} ) G ) When plotted computationally, one needs to be careful to cover all frequencies of interest. {\displaystyle D(s)} {\displaystyle D(s)} With the same poles and zeros, move the \(k\) slider and determine what range of \(k\) makes the closed loop system stable. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. {\displaystyle Z} It is easy to check it is the circle through the origin with center \(w = 1/2\). {\displaystyle G(s)} A pole with positive real part would correspond to a mode that goes to infinity as \(t\) grows. 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. s = In units of v s k 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. , which is to say. Thus, we may find The answer is no, \(G_{CL}\) is not stable. 1This transfer function was concocted for the purpose of demonstration. The factor \(k = 2\) will scale the circle in the previous example by 2. If the answer to the first question is yes, how many closed-loop as defined above corresponds to a stable unity-feedback system when {\displaystyle D(s)=0} {\displaystyle N} j P Now how can I verify this formula for the open-loop transfer function: H ( s) = 1 s 3 ( s + 1) The Nyquist plot is attached in the image. ) 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. This approach appears in most modern textbooks on control theory. The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. Which, if either, of the values calculated from that reading, \(\mathrm{GM}=(1 / \mathrm{GM})^{-1}\) is a legitimate metric of closed-loop stability? + ( . u In control theory and stability theory, the Nyquist stability criterion or StreckerNyquist stability criterion, independently discovered by the German electrical engineer Felix Strecker[de] at Siemens in 1930[1][2][3] and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932,[4] is a graphical technique for determining the stability of a dynamical system. ( The poles are \(-2, -2\pm i\). T ) has exactly the same poles as Stability in the Nyquist Plot. This reference shows that the form of stability criterion described above [Conclusion 2.] ) \(\PageIndex{4}\) includes the Nyquist plots for both \(\Lambda=0.7\) and \(\Lambda =\Lambda_{n s 1}\), the latter of which by definition crosses the negative \(\operatorname{Re}[O L F R F]\) axis at the point \(-1+j 0\), not far to the left of where the \(\Lambda=0.7\) plot crosses at about \(-0.73+j 0\); therefore, it might be that the appropriate value of gain margin for \(\Lambda=0.7\) is found from \(1 / \mathrm{GM}_{0.7} \approx 0.73\), so that \(\mathrm{GM}_{0.7} \approx 1.37=2.7\) dB, a small gain margin indicating that the closed-loop system is just weakly stable. This is just to give you a little physical orientation. {\displaystyle \Gamma _{s}} using the Routh array, but this method is somewhat tedious. j G The Nyquist method is used for studying the stability of linear systems with pure time delay. We thus find that around (There is no particular reason that \(a\) needs to be real in this example. Let us begin this study by computing \(\operatorname{OLFRF}(\omega)\) and displaying it on Nyquist plots for a low value of gain, \(\Lambda=0.7\) (for which the closed-loop system is stable), and for the value corresponding to the transition from stability to instability on Figure \(\PageIndex{3}\), which we denote as \(\Lambda_{n s 1} \approx 1\). ( s {\displaystyle 1+GH} {\displaystyle F(s)} r domain where the path of "s" encloses the + Determining Stability using the Nyquist Plot - Erik Cheever We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. {\displaystyle N(s)} is formed by closing a negative unity feedback loop around the open-loop transfer function This gives us, We now note that The Nyquist criterion is a frequency domain tool which is used in the study of stability. ; when placed in a closed loop with negative feedback = Mark the roots of b G D / s 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. F Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. However, the positive gain margin 10 dB suggests positive stability. Nyquist Stability Criterion A feedback system is stable if and only if \(N=-P\), i.e. Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. ). We will look a little more closely at such systems when we study the Laplace transform in the next topic. 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. -plane, ( 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 \(\Lambda_{n s 1}=0.96438\). We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. In this context \(G(s)\) is called the open loop system function. ) 1 Its system function is given by Black's formula, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)},\]. H ( ) {\displaystyle \Gamma _{G(s)}} 1 G , as evaluated above, is equal to0. s Give zero-pole diagrams for each of the systems, \[G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\]. If the number of poles is greater than the r The left hand graph is the pole-zero diagram. T + H Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. Natural Language; Math Input; Extended Keyboard Examples Upload Random. 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. plane, encompassing but not passing through any number of zeros and poles of a function ( "1+L(s)=0.". ( 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
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