s {\displaystyle G(s)} {\displaystyle Z} Z s 0 That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. plane in the same sense as the contour 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. ( . Z A linear time invariant system has a system function which is a function of a complex variable. {\displaystyle v(u(\Gamma _{s}))={{D(\Gamma _{s})-1} \over {k}}=G(\Gamma _{s})} We dont analyze stability by plotting the open-loop gain or G Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. From now on we will allow ourselves to be a little more casual and say the system \(G(s)\)'. In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. j The poles of the closed loop system function \(G_{CL} (s)\) given in Equation 12.3.2 are the zeros of \(1 + kG(s)\). T B In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.). Closed loop approximation f.d.t. T G G {\displaystyle G(s)} , the result is the Nyquist Plot of ) 1 Cauchy's argument principle states that, Where Determining Stability using the Nyquist Plot - Erik Cheever 0 ( Additional parameters appear if you check the option to calculate the Theoretical PSF. As a result, it can be applied to systems defined by non-rational functions, such as systems with delays. s L is called the open-loop transfer function. s l F 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. 0000000608 00000 n There are no poles in the right half-plane. u D We can show this formally using Laurent series. Keep in mind that the plotted quantity is A, i.e., the loop gain. ( We can measure phase margin directly by drawing on the Nyquist diagram a circle with radius of 1 unit and centered on the origin of the complex \(OLFRF\)-plane, so that it passes through the important point \(-1+j 0\). The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. have positive real part. {\displaystyle H(s)} Legal. 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. ) If the counterclockwise detour was around a double pole on the axis (for example two -plane, Pole-zero diagrams for the three systems. = = ( T s *(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 \(\PageIndex{1}\). + Nyquist plot of the transfer function s/(s-1)^3. {\displaystyle Z} {\displaystyle D(s)=0} in the right half plane, the resultant contour in the , as evaluated above, is equal to0. 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. 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. ( The Nyquist criterion allows us to answer two questions: 1. ( 0000039933 00000 n ( The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s - 1)/(s + 1)}{1 + 2(s - 1)/(s + 1)} = \dfrac{s - 1}{3s - 1}.\]. (3h) lecture: Nyquist diagram and on the effects of feedback. + in the new Since they are all in the left half-plane, the system is stable. G + F ( Gain margin (GM) is defined by Equation 17.1.8, from which we find, \[\frac{1}{G M(\Lambda)}=|O L F R F(\omega)|_{\mid} \mid \text {at }\left.\angle O L F R F(\omega)\right|_{\Lambda}=-180^{\circ}\label{eqn:17.17} \]. the same system without its feedback loop). H that appear within the contour, that is, within the open right half plane (ORHP). There are two poles in the right half-plane, so the open loop system \(G(s)\) is unstable. 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 only plot of \(G(s)\) is in the left half-plane, so the open loop system is stable. {\displaystyle {\mathcal {T}}(s)={\frac {N(s)}{D(s)}}.}. 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). 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. ) 1 = To use this criterion, the frequency response data of a system must be presented as a polar plot in Now refresh the browser to restore the applet to its original state. Matrix Result This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. The stability of . {\displaystyle 1+GH(s)} plane) by the function 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. The formula is an easy way to read off the values of the poles and zeros of \(G(s)\). {\displaystyle \Gamma _{s}} s {\displaystyle N} Stability can be determined by examining the roots of the desensitivity factor polynomial *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. 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. {\displaystyle G(s)} We will just accept this formula. 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Thus, we may find To be able to analyze systems with poles on the imaginary axis, the Nyquist Contour can be modified to avoid passing through the point ), Start with a system whose characteristic equation is given by enclosed by the contour and 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. 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. The theorem recognizes these. 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. That is, the Nyquist plot is the circle through the origin with center \(w = 1\). We consider a system whose transfer function is {\displaystyle {\mathcal {T}}(s)} In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes. If the number of poles is greater than the F s Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. s as the first and second order system. The Bode plot for 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 } ). Proofs of the general Nyquist stability criterion are based on the theory of complex functions of a complex variable; many textbooks on control theory present such proofs, one of the clearest being that of Franklin, et al., 1991, pages 261-280. From complex analysis, a contour ) {\displaystyle -1+j0} Conclusions can also be reached by examining the open loop transfer function (OLTF) , which is to say our Nyquist plot. s j Natural Language; Math Input; Extended Keyboard Examples Upload Random. 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). + ( Looking at Equation 12.3.2, there are two possible sources of poles for \(G_{CL}\). ) From the mapping we find the number N, which is the number of F s s F The factor \(k = 2\) will scale the circle in the previous example by 2. ) Here N = 1. 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)'). 1 We will look a {\displaystyle N=Z-P} ) 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 \(\PageIndex{4}\). {\displaystyle Z=N+P} {\displaystyle \Gamma _{s}} For what values of \(a\) is the corresponding closed loop system \(G_{CL} (s)\) stable? In units of Hz, its value is one-half of the sampling rate. , let The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. j travels along an arc of infinite radius by ) 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. "1+L(s)=0.". ( 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. ) H When \(k\) is small the Nyquist plot has winding number 0 around -1. ) {\displaystyle 0+j(\omega -r)} ) \nonumber\]. Z {\displaystyle G(s)} ( , we have, We then make a further substitution, setting . Instead of Cauchy's argument principle, the original paper by Harry Nyquist in 1932 uses a less elegant approach. 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. {\displaystyle {\frac {G}{1+GH}}} = Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability. {\displaystyle u(s)=D(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. In control system theory, the RouthHurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system.A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. {\displaystyle D(s)} . s \(G(s)\) has a pole in the right half-plane, so the open loop system is not stable. 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