+ {\displaystyle N=P-Z} The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. While Nyquist is one of the most general stability tests, it is still restricted to linear, time-invariant (LTI) systems. ( (There is no particular reason that \(a\) needs to be real in this example. ) j ( (2 h) lecture: Introduction to the controller's design specifications. 1 , we now state the Nyquist Criterion: Given a Nyquist contour Suppose that the open-loop transfer function of a system is1, \[G(s) \times H(s) \equiv O L T F(s)=\Lambda \frac{s^{2}+4 s+104}{(s+1)\left(s^{2}+2 s+26\right)}=\Lambda \frac{s^{2}+4 s+104}{s^{3}+3 s^{2}+28 s+26}\label{eqn:17.18} \]. {\displaystyle T(s)} 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 \(0<\mathrm{GM}_{\mathrm{U}}<1\). Now refresh the browser to restore the applet to its original state. 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. Is the closed loop system stable when \(k = 2\). 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 s ( , the result is the Nyquist Plot of {\displaystyle A(s)+B(s)=0} For our purposes it would require and an indented contour along the imaginary axis. in the contour shall encircle (clockwise) the point The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. Since we know N and P, we can determine Z, the number of zeros of 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. G ( The Nyquist criterion is an important stability test with applications to systems, circuits, and networks [1]. {\displaystyle D(s)=1+kG(s)} ) -P_PcXJ']b9-@f8+5YjmK p"yHL0:8UK=MY9X"R&t5]M/o 3\\6%W+7J$)^p;% XpXC#::` :@2p1A%TQHD1Mdq!1 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. j Nyquist plot of the transfer function s/(s-1)^3. the number of the counterclockwise encirclements of \(1\) point by the Nyquist plot in the \(GH\)-plane is equal to the number of the unstable poles of the open-loop transfer function. ( G k For this we will use one of the MIT Mathlets (slightly modified for our purposes). Z for \(a > 0\). Additional parameters The Nyquist method is used for studying the stability of linear systems with pure time delay. This case can be analyzed using our techniques. 1 ( and travels anticlockwise to {\displaystyle s} [@mc6X#:H|P`30s@, B R=Lb&3s12212WeX*a$%.0F06 endstream endobj 103 0 obj 393 endobj 93 0 obj << /Type /Page /Parent 85 0 R /Resources 94 0 R /Contents 98 0 R /Rotate 90 /MediaBox [ 0 0 612 792 ] /CropBox [ 36 36 576 756 ] >> endobj 94 0 obj << /ProcSet [ /PDF /Text ] /Font << /TT2 96 0 R >> /ExtGState << /GS1 100 0 R >> /ColorSpace << /Cs6 97 0 R >> >> endobj 95 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -568 -307 2028 1007 ] /FontName /HMIFEA+TimesNewRoman /ItalicAngle 0 /StemV 94 /XHeight 0 /FontFile2 99 0 R >> endobj 96 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 150 /Widths [ 250 0 0 500 0 0 0 0 333 333 500 564 250 333 250 278 500 500 500 500 500 500 500 500 500 500 278 0 0 564 0 0 0 722 667 667 722 611 556 722 722 333 389 0 611 889 722 722 556 0 667 556 611 722 722 944 0 0 0 0 0 0 0 500 0 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 500 500 333 389 278 500 500 722 500 500 444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 333 0 0 350 500 ] /Encoding /WinAnsiEncoding /BaseFont /HMIFEA+TimesNewRoman /FontDescriptor 95 0 R >> endobj 97 0 obj [ /ICCBased 101 0 R ] endobj 98 0 obj << /Length 428 /Filter /FlateDecode >> stream The most common use of Nyquist plots is for assessing the stability of a system with feedback. ( ( is not sufficiently general to handle all cases that might arise. We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. 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. N Figure 19.3 : Unity Feedback Confuguration. ( Pole-zero diagrams for the three systems. ( + From now on we will allow ourselves to be a little more casual and say the system \(G(s)\)'. s 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. *(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 \(\PageIndex{1}\). If we set \(k = 3\), the closed loop system is stable. 0.375=3/2 (the current gain (4) multiplied by the gain margin , where . {\displaystyle D(s)=0} s The negative phase margin indicates, to the contrary, instability. = F T encircled by = F Note that we count encirclements in the l poles at the origin), the path in L(s) goes through an angle of 360 in The algebra involved in canceling the \(s + a\) term in the denominators is exactly the cancellation that makes the poles of \(G\) removable singularities in \(G_{CL}\). If the system is originally open-loop unstable, feedback is necessary to stabilize the system. Let \(G(s)\) be such a system function. 1 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. 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 \(\Lambda_{\mathrm{ns}}\). G right half plane. 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. s Answer: The closed loop system is stable for \(k\) (roughly) between 0.7 and 3.10. ) We will just accept this formula. \[G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),\], \[\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}\], which is clearly analytic at \(s_0\). v The new system is called a closed loop system. 0 ( G We begin by considering the closed-loop characteristic polynomial (4.23) where L ( z) denotes the loop gain. When plotted computationally, one needs to be careful to cover all frequencies of interest. D ( ( ) the clockwise direction. {\displaystyle 0+j\omega } \[G_{CL} (s) = \dfrac{1/(s + a)}{1 + 1/(s + a)} = \dfrac{1}{s + a + 1}.\], This has a pole at \(s = -a - 1\), so it's stable if \(a > -1\). ( + We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. Figure 19.3 : Unity Feedback Confuguration. Physically the modes tell us the behavior of the system when the input signal is 0, but there are initial conditions. ) That is, the Nyquist plot is the circle through the origin with center \(w = 1\). 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. , and The system is called unstable if any poles are in the right half-plane, i.e. negatively oriented) contour ) 0000001210 00000 n The theorem recognizes these. However, the gain margin calculated from either of the two phase crossovers suggests instability, showing that both are deceptively defective metrics of stability. Nyquist and Bode plots for the above circuits are given in Figs 12.34 and 12.35, where is the time at which the exponential factor is e1 = 0.37, the time it takes to decrease to 37% of its value. G ( ( (Actually, for \(a = 0\) the open loop is marginally stable, but it is fully stabilized by the closed loop.). 2. ) 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. s , that starts at 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. Let us consider next an uncommon system, for which the determination of stability or instability requires a more detailed examination of the stability margins. {\displaystyle D(s)} Set the feedback factor \(k = 1\). 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.. if the poles are all in the left half-plane. ) Determining Stability using the Nyquist Plot - Erik Cheever 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. Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular values of gain. does not have any pole on the imaginary axis (i.e. {\displaystyle Z} in the right half plane, the resultant contour in the {\displaystyle \Gamma _{s}} %PDF-1.3 % In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. {\displaystyle 1+GH(s)} s 1 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. P Transfer Function System Order -thorder system Characteristic Equation (Closed Loop Denominator) s+ Go! ) {\displaystyle G(s)} {\displaystyle G(s)} D ) ) The right hand graph is the Nyquist plot. Nyquist Stability Criterion A feedback system is stable if and only if \(N=-P\), i.e. j / 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 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. u {\displaystyle Z} 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 s = ( 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. {\displaystyle \Gamma _{s}} s 0 v 0 ) s plane / 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 Bode plot for s s The Nyquist criterion is widely used in electronics and control system engineering, as well as other fields, for designing and analyzing systems with feedback. G 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. From the mapping we find the number N, which is the number of 1This transfer function was concocted for the purpose of demonstration. {\displaystyle N(s)} N ( ) + T Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. Stability in the Nyquist Plot. {\displaystyle Z} 0 Give zero-pole diagrams for each of the systems, \[G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}, \ \ \ G_1(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\]. in the right-half complex plane. s If the number of poles is greater than the F D Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. 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. 1 ( G 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 \(\Lambda_{n s 1}=0.96438\). ) Alternatively, and more importantly, if T , let 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 Thus, it is stable when the pole is in the left half-plane, i.e. We first note that they all have a single zero at the origin. {\displaystyle 0+j\omega } is formed by closing a negative unity feedback loop around the open-loop transfer function The formula is an easy way to read off the values of the poles and zeros of \(G(s)\). Since \(G\) is in both the numerator and denominator of \(G_{CL}\) it should be clear that the poles cancel. . = {\displaystyle F(s)} If \(G\) has a pole of order \(n\) at \(s_0\) then. The roots of b (s) are the poles of the open-loop transfer function. Terminology. 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. The mathematics uses the Laplace transform, which transforms integrals and derivatives in the time domain to simple multiplication and division in the s domain. s s 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. The significant roots of Equation \(\ref{eqn:17.19}\) are shown on Figure \(\PageIndex{3}\): the complete locus of oscillatory roots with positive imaginary parts is shown; only the beginning of the locus of real (exponentially stable) roots is shown, since those roots become progressively more negative as gain \(\Lambda\) increases from the initial small values. 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 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 \(\PageIndex{5}\). If the counterclockwise detour was around a double pole on the axis (for example two 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 Expert Answer. = The poles of \(G\). the same system without its feedback loop). The feedback loop has stabilized the unstable open loop systems with \(-1 < a \le 0\). 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