s 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 (\(1 j5\)). {\displaystyle 1+G(s)} j . Matrix Result This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. 0 have positive real part. ) Conclusions can also be reached by examining the open loop transfer function (OLTF) Double control loop for unstable systems. However, it is not applicable to non-linear systems as for that complex stability criterion like Lyapunov is used. F N D s + j On the other hand, a Bode diagram displays the phase-crossover and gain-crossover frequencies, which are not explicit on a traditional Nyquist plot. the clockwise direction. ( 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. {\displaystyle Z=N+P} {\displaystyle D(s)=1+kG(s)} Keep in mind that the plotted quantity is A, i.e., the loop gain. \(\text{QED}\), The Nyquist criterion is a visual method which requires some way of producing the Nyquist plot. Techniques like Bode plots, while less general, are sometimes a more useful design tool. Cauchy's argument principle states that, Where 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, \[w = k G \circ \gamma (\omega) = kG(i \omega).\]. Transfer Function System Order -thorder system Characteristic Equation G 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 In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. A s In 18.03 we called the system stable if every homogeneous solution decayed to 0. domain where the path of "s" encloses the The Nyquist criterion is a frequency domain tool which is used in the study of stability. ) Refresh the page, to put the zero and poles back to their original state. {\displaystyle G(s)} This is possible for small systems. s ) G Note on Figure \(\PageIndex{2}\) that the phase-crossover point (phase angle \(\phi=-180^{\circ}\)) and the gain-crossover point (magnitude ratio \(MR = 1\)) of an \(FRF\) are clearly evident on a Nyquist plot, perhaps even more naturally than on a Bode diagram. ) 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). P For example, Brogan, 1974, page 25, wrote Experience has shown that acceptable transient response will usually require stability margins on the order of \(\mathrm{PM}>30^{\circ}\), \(\mathrm{GM}>6\) dB. Franklin, et al., 1991, page 285, wrote Many engineers think directly in terms of \(\text { PM }\) in judging whether a control system is adequately stabilized. 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. We will look a = s Non-linear systems must use more complex stability criteria, such as Lyapunov or the circle criterion. ) P {\displaystyle G(s)} (0.375) yields the gain that creates marginal stability (3/2). ( Calculate the Gain Margin. F ( 20 points) b) Using the Bode plots, calculate the phase margin and gain margin for K =1. ( 1 s s s Let us consider next an uncommon system, for which the determination of stability or instability requires a more detailed examination of the stability margins. , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. plane) by the function Observe on Figure \(\PageIndex{4}\) the small loops beneath the negative \(\operatorname{Re}[O L F R F]\) axis as driving frequency becomes very high: the frequency responses approach zero from below the origin of the complex \(OLFRF\)-plane. as the first and second order system. In units of Hz, its value is one-half of the sampling rate. This case can be analyzed using our techniques. Rule 1. 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 Thus, we may finally state that. You have already encountered linear time invariant systems in 18.03 (or its equivalent) when you solved constant coefficient linear differential equations. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. plane yielding a new contour. T H around ( Graphical method of determining the stability of a dynamical system, The Nyquist criterion for systems with poles on the imaginary axis, "Chapter 4.3. Let \(G(s)\) be such a system function. For what values of \(a\) is the corresponding closed loop system \(G_{CL} (s)\) stable? s So the winding number is -1, which does not equal the number of poles of \(G\) in the right half-plane. ( If the system with system function \(G(s)\) is unstable it can sometimes be stabilized by what is called a negative feedback loop. 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)'). ( We will be concerned with the stability of the system. That is, if the unforced system always settled down to equilibrium. G ( ) The left hand graph is the pole-zero diagram. j The poles of \(G(s)\) correspond to what are called modes of the system. s The poles are \(-2, -2\pm i\). T s ( G {\displaystyle F(s)} {\displaystyle \Gamma _{s}} s G 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. is not sufficiently general to handle all cases that might arise. s G + {\displaystyle 0+j(\omega +r)} 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.) Legal. u ) P 0 The Nyquist criterion is a frequency domain tool which is used in the study of stability. However, the Nyquist Criteria can also give us additional information about a system. If the answer to the first question is yes, how many closed-loop Now, recall that the poles of \(G_{CL}\) are exactly the zeros of \(1 + k G\). has zeros outside the open left-half-plane (commonly initialized as OLHP). In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). T are the poles of s ( \(G(s)\) has one pole at \(s = -a\). of poles of T(s)). ) Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. This assumption holds in many interesting cases. If instead, the contour is mapped through the open-loop transfer function In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. ) We then note that 1 The negative phase margin indicates, to the contrary, instability. This has one pole at \(s = 1/3\), so the closed loop system is unstable. ( 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 \(\Lambda\)? ( Instead of Cauchy's argument principle, the original paper by Harry Nyquist in 1932 uses a less elegant approach. The frequency is swept as a parameter, resulting in a plot per frequency. {\displaystyle \Gamma _{s}} Lecture 2: Stability Criteria S.D. But in physical systems, complex poles will tend to come in conjugate pairs.). is formed by closing a negative unity feedback loop around the open-loop transfer function . s Z , which is to say. The right hand graph is the Nyquist plot. G F N The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. G ) = Let \(G(s) = \dfrac{1}{s + 1}\). using the Routh array, but this method is somewhat tedious. Z The poles of \(G\). {\displaystyle \Gamma _{G(s)}} That is, the Nyquist plot is the circle through the origin with center \(w = 1\). s From complex analysis, a contour 0000001210 00000 n G P 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. Draw the Nyquist plot with \(k = 1\). 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. The roots of 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 Physically the modes tell us the behavior of the system when the input signal is 0, but there are initial conditions. Rearranging, we have {\displaystyle GH(s)} That is, if all the poles of \(G\) have negative real part. For these values of \(k\), \(G_{CL}\) is unstable. D So we put a circle at the origin and a cross at each pole. ( T {\displaystyle Z} is the multiplicity of the pole on the imaginary axis. The stability of Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. 0000002847 00000 n Lecture 1: The Nyquist Criterion S.D. Assessment of the stability of a closed-loop negative feedback system is done by applying the Nyquist stability criterion to the Nyquist plot of the open-loop system (i.e. Nyquist Stability Criterion A feedback system is stable if and only if \(N=-P\), i.e. Figure 19.3 : Unity Feedback Confuguration. 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. {\displaystyle \Gamma _{F(s)}=F(\Gamma _{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 F be the number of zeros of . {\displaystyle v(u)={\frac {u-1}{k}}} ( are, respectively, the number of zeros of ( enclosing the right half plane, with indentations as needed to avoid passing through zeros or poles of the function 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. G 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. + 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 The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s - 1)/(s + 1)}{1 + 2(s - 1)/(s + 1)} = \dfrac{s - 1}{3s - 1}.\]. , can be mapped to another plane (named 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 l There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. s Nyquist plot of the transfer function s/ (s-1)^3 Natural Language Math Input Extended Keyboard Examples Have a question about using Wolfram|Alpha? 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. 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.. Looking at Equation 12.3.2, there are two possible sources of poles for \(G_{CL}\). {\displaystyle A(s)+B(s)=0} We know from Figure \(\PageIndex{3}\) that this case of \(\Lambda=4.75\) is closed-loop unstable. Phase and gain stability margins must use more complex stability criterion Calculator I learned about this in ELEC 341 the! Looking at Equation 12.3.2, there are two possible sources of poles for (. You have already encountered linear time invariant systems in 18.03 ( or its equivalent ) when solved! Can handle transfer functions with right half-plane singularities poles back to their original state only essence. 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