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. 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. + 0000001731 00000 n ) are the poles of ( G Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. $G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),$, $\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}$, which is clearly analytic at $$s_0$$. s {\displaystyle GH(s)} Z {\displaystyle N=P-Z} Lets look at an example: Note that I usually dont include negative frequencies in my Nyquist plots. In the case $$G(s)$$ is a fractional linear transformation, so we know it maps the imaginary axis to a circle. j 0000039854 00000 n = the clockwise direction. Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. clockwise. ( 1 In the previous problem could you determine analytically the range of $$k$$ where $$G_{CL} (s)$$ is stable? has exactly the same poles as 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. F {\displaystyle 1+kF(s)} Note that a closed-loop-stable case has $$0<1 / \mathrm{GM}_{\mathrm{S}}<1$$ so that $$\mathrm{GM}_{\mathrm{S}}>1$$, and a closed-loop-unstable case has $$1 / \mathrm{GM}_{\mathrm{U}}>1$$ so that $$0<\mathrm{GM}_{\mathrm{U}}<1$$. But in physical systems, complex poles will tend to come in conjugate pairs.). Since we know N and P, we can determine Z, the number of zeros of Note that the pinhole size doesn't alter the bandwidth of the detection system. ; when placed in a closed loop with negative feedback of poles of T(s)). Clearly, the calculation $$\mathrm{GM} \approx 1 / 0.315$$ is a defective metric of stability. s s Natural Language; Math Input; Extended Keyboard Examples Upload Random. times, where 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 $$-1+j 0$$. s + ) {\displaystyle Z=N+P} ( 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. 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). l MT-002. {\displaystyle P} We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane: The Nyquist contour mapped through the function G ( Describe the Nyquist plot with gain factor $$k = 2$$. ) 1 2. j P T The factor $$k = 2$$ will scale the circle in the previous example by 2. 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. It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. {\displaystyle F} G r For the edge case where no poles have positive real part, but some are pure imaginary we will call the system marginally stable. s 0 H ) 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. G {\displaystyle {\frac {G}{1+GH}}} Phase margins are indicated graphically on Figure $$\PageIndex{2}$$. 1 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. When drawn by hand, a cartoon version of the Nyquist plot is sometimes used, which shows the linearity of the curve, but where coordinates are distorted to show more detail in regions of interest. The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. The Nyquist criterion is an important stability test with applications to systems, circuits, and networks . ) Lecture 1: The Nyquist Criterion S.D. {\displaystyle A(s)+B(s)=0} = {\displaystyle G(s)} T There is one branch of the root-locus for every root of b (s). ) is formed by closing a negative unity feedback loop around the open-loop transfer function Any Laplace domain transfer function Gain $$\Lambda$$ has physical units of s-1, but we will not bother to show units in the following discussion. 0 Thus, we may find 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. $$G(s)$$ has a pole in the right half-plane, so the open loop system is not stable. ) / {\displaystyle s={-1/k+j0}} F A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. For a SISO feedback system the closed-looptransfer function is given by where represents the system and is the feedback element. s The frequency is swept as a parameter, resulting in a pl 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. H The system is called unstable if any poles are in the right half-plane, i.e. {\displaystyle F(s)} {\displaystyle T(s)} {\displaystyle u(s)=D(s)} v + ) Rule 2. ) s Lecture 2 2 Nyquist Plane Results GMPM Criteria ESAC Criteria Real Axis Nyquist Contour, Unstable Case Nyquist Contour, Stable Case Imaginary We will just accept this formula. Since $$G$$ is in both the numerator and denominator of $$G_{CL}$$ it should be clear that the poles cancel. Suppose F (s) is a single-valued mapping function given as: F (s) = 1 + G (s)H (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. 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. ( P We know from Figure $$\PageIndex{3}$$ that the closed-loop system with $$\Lambda = 18.5$$ is stable, albeit weakly. ) Which, if either, of the values calculated from that reading, $$\mathrm{GM}=(1 / \mathrm{GM})^{-1}$$ is a legitimate metric of closed-loop stability? 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 present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. {\displaystyle P} encirclements of the -1+j0 point in "L(s).". as defined above corresponds to a stable unity-feedback system when {\displaystyle 1+GH} ) s ) -plane, The poles of $$G$$. From the mapping we find the number N, which is the number of G + {\displaystyle F(s)} encircled by N The poles of $$G(s)$$ correspond to what are called modes of the system. A Z times such that Note that the phase margin for $$\Lambda=0.7$$, found as shown on Figure $$\PageIndex{2}$$, is quite clear on Figure $$\PageIndex{4}$$ and not at all ambiguous like the gain margin: $$\mathrm{PM}_{0.7} \approx+20^{\circ}$$; this value also indicates a stable, but weakly so, closed-loop system. With $$k =1$$, what is the winding number of the Nyquist plot around -1? So the winding number is -1, which does not equal the number of poles of $$G$$ in the right half-plane. Is the closed loop system stable when $$k = 2$$. 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. D We can show this formally using Laurent series. {\displaystyle {\mathcal {T}}(s)} = You have already encountered linear time invariant systems in 18.03 (or its equivalent) when you solved constant coefficient linear differential equations. (3h) lecture: Nyquist diagram and on the effects of feedback. ) ( s + Nyquist Plot Example 1, Procedure to draw Nyquist plot in It is certainly reasonable to call a system that does this in response to a zero signal (often called no input) unstable. 0 A simple pole at $$s_1$$ corresponds to a mode $$y_1 (t) = e^{s_1 t}$$. It can happen! drawn in the complex {\displaystyle D(s)=1+kG(s)} The Nyquist plot of 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 1+G(s)} There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. Z 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. 0000000608 00000 n 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. {\displaystyle \Gamma _{s}} is the number of poles of the open-loop transfer function 0000039933 00000 n Calculate transfer function of two parallel transfer functions in a feedback loop. {\displaystyle (-1+j0)} The right hand graph is the Nyquist plot. H So, the control system satisfied the necessary condition. s We will now rearrange the above integral via substitution. \nonumber\]. This is possible for small systems. s G(s)= s(s+5)(s+10)500K slopes, frequencies, magnitudes, on the next pages!) ) ) 1 . ) in the contour As $$k$$ increases, somewhere between $$k = 0.65$$ and $$k = 0.7$$ the winding number jumps from 0 to 2 and the closed loop system becomes stable. 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 Nyquist method is used for studying the stability of linear systems with pure time delay. The mathlet shows the Nyquist plot winds once around $$w = -1$$ in the $$clockwise$$ direction. The stability of s The Nyquist criterion allows us to assess the stability properties of a feedback system based on P ( s) C ( s) only. Set the feedback factor $$k = 1$$. We consider a system whose transfer function is *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. Z T enclosed by the contour and {\displaystyle \Gamma _{s}} ( Transfer Function System Order -thorder system Characteristic Equation 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. the same system without its feedback loop). The negative phase margin indicates, to the contrary, instability. However, the positive gain margin 10 dB suggests positive stability. {\displaystyle F(s)} has zeros outside the open left-half-plane (commonly initialized as OLHP). Assume $$a$$ is real, for what values of $$a$$ is the open loop system $$G(s) = \dfrac{1}{s + a}$$ stable? To use this criterion, the frequency response data of a system must be presented as a polar plot in H|Ak0ZlzC!bBM66+d]JHbLK5L#S\$_0i".Zb~#}2HyY YBrs}y:)c. {\displaystyle {\mathcal {T}}(s)} The beauty of the Nyquist stability criterion lies in the fact that it is a rather simple graphical test. s In fact, we find that the above integral corresponds precisely to the number of times the Nyquist plot encircles the point The Nyquist plot can provide some information about the shape of the transfer function. 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 $$\omega$$. and poles of Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular values of gain. 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 $$\Lambda_{n s 2} = 15.0356$$. 1 {\displaystyle G(s)} The closed loop system function is, $G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s + 1)/(s - 1)}{1 + 2(s + 1)/(s - 1)} = \dfrac{s + 1}{3s + 1}.$. The feedback loop has stabilized the unstable open loop systems with $$-1 < a \le 0$$. (At $$s_0$$ it equals $$b_n/(kb_n) = 1/k$$.). Static and dynamic specifications. Suppose that the open-loop transfer function of a system is1, $G(s) \times H(s) \equiv O L T F(s)=\Lambda \frac{s^{2}+4 s+104}{(s+1)\left(s^{2}+2 s+26\right)}=\Lambda \frac{s^{2}+4 s+104}{s^{3}+3 s^{2}+28 s+26}\label{eqn:17.18}$. 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 G . ( u 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. 1 D 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 1+G(s)} Nyquist criterion and stability margins. The theorem recognizes these. 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. ) The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. *(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 $$\PageIndex{1}$$. s {\displaystyle 1+G(s)} ( Z Now, recall that the poles of $$G_{CL}$$ are exactly the zeros of $$1 + k G$$. entire right half plane. s To be able to analyze systems with poles on the imaginary axis, the Nyquist Contour can be modified to avoid passing through the point G 0 and travels anticlockwise to Z The frequency is swept as a parameter, resulting in a plot per frequency. {\displaystyle Z} ( T The argument principle from complex analysis gives a criterion to calculate the difference between the number of zeros and the number of poles of k 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 F(s)} Draw the Nyquist plot with $$k = 1$$. ) The most common use of Nyquist plots is for assessing the stability of a system with feedback. + Section 17.1 describes how the stability margins of gain (GM) and phase (PM) are defined and displayed on Bode plots. 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. In practice, the ideal sampler is replaced by Let $$G(s)$$ be such a system function. s F in the right half plane, the resultant contour in the P We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. The counterclockwise detours around the poles at s=j4 results in N Since $$G_{CL}$$ is a system function, we can ask if the system is stable. 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 $$\PageIndex{2}$$, thus rendering ambiguous the definition of phase margin. 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 $$\approx 15$$. This page titled 12.2: Nyquist Criterion for Stability is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 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