We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. In \(\gamma (\omega)\) the variable is a greek omega and in \(w = G \circ \gamma\) we have a double-u. r ) (ii) Determine the range of \ ( k \) to ensure a stable closed loop response. Section 17.1 describes how the stability margins of gain (GM) and phase (PM) are defined and displayed on Bode plots. . L is called the open-loop transfer function. *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. s G + P s ( The beauty of the Nyquist stability criterion lies in the fact that it is a rather simple graphical test. {\displaystyle \Gamma _{s}} G A {\displaystyle \Gamma _{F(s)}=F(\Gamma _{s})} G , the closed loop transfer function (CLTF) then becomes You can also check that it is traversed clockwise. {\displaystyle GH(s)={\frac {A(s)}{B(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. is the multiplicity of the pole on the imaginary axis. 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 \(\Lambda_{n s 1}=0.96438\). s The Nyquist criterion is a frequency domain tool which is used in the study of stability. Here N = 1. So in the limit \(kG \circ \gamma_R\) becomes \(kG \circ \gamma\). 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Additional parameters appear if you check the option to calculate the Theoretical PSF. ( 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 \(\Lambda\)? F 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 \(\Lambda_{n s 2} = 15.0356\). ; when placed in a closed loop with negative feedback s A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane: The Nyquist contour mapped through the function s %PDF-1.3 % That is, the Nyquist plot is the circle through the origin with center \(w = 1\). G 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 Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. 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. be the number of zeros of {\displaystyle 0+j\omega } This case can be analyzed using our techniques. 0 {\displaystyle u(s)=D(s)} The counterclockwise detours around the poles at s=j4 results in 0000001188 00000 n Take \(G(s)\) from the previous example. {\displaystyle \Gamma _{s}} ( negatively oriented) contour in the complex plane. Since we know N and P, we can determine Z, the number of zeros of \[G(s) = \dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + a_1 (s - s_0)^{n + 1} + \ ),\], \[\begin{array} {rcl} {G_{CL} (s)} & = & {\dfrac{\dfrac{1}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{1 + \dfrac{k}{(s - s_0)^n} (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \\ { } & = & {\dfrac{(b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}{(s - s_0)^n + k (b_n + b_{n - 1} (s - s_0) + \ a_0 (s - s_0)^n + \ )}} \end{array}\], which is clearly analytic at \(s_0\). + ) F s are also said to be the roots of the characteristic equation F For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero. ) l {\displaystyle 1+G(s)} {\displaystyle G(s)} = plane yielding a new contour. ) ) Now refresh the browser to restore the applet to its original state. Precisely, each complex point , which is to say our Nyquist plot. These are the same systems as in the examples just above. 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). Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. In units of Hz, its value is one-half of the sampling rate. This method is easily applicable even for systems with delays and other non and travels anticlockwise to The left hand graph is the pole-zero diagram. G In 18.03 we called the system stable if every homogeneous solution decayed to 0. , that starts at N 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. G Lets look at an example: Note that I usually dont include negative frequencies in my Nyquist plots. The Bode plot for Assume \(a\) is real, for what values of \(a\) is the open loop system \(G(s) = \dfrac{1}{s + a}\) stable? As \(k\) goes to 0, the Nyquist plot shrinks to a single point at the origin. are, respectively, the number of zeros of 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 \(0<\mathrm{GM}_{\mathrm{U}}<1\). {\displaystyle D(s)} Such a modification implies that the phasor times, where = 1 In units of The poles of \(G(s)\) correspond to what are called modes of the system. 2. by the same contour. ( s ) T = Legal. + G Suppose \(G(s) = \dfrac{s + 1}{s - 1}\). This is a case where feedback destabilized a stable system. 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. {\displaystyle G(s)} 1 The factor \(k = 2\) will scale the circle in the previous example by 2. For example, quite often \(G(s)\) is a rational function \(Q(s)/P(s)\) (\(Q\) and \(P\) are polynomials). It is informative and it will turn out to be even more general to extract the same stability margins from Nyquist plots of frequency response. {\displaystyle 1+G(s)} are same as the poles of ( Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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. ) The Nyquist criterion is an important stability test with applications to systems, circuits, and networks [1]. k However, the positive gain margin 10 dB suggests positive stability. We consider a system whose transfer function is Calculate transfer function of two parallel transfer functions in a feedback loop. ( Alternatively, and more importantly, if If we set \(k = 3\), the closed loop system is stable. T ( 1 Since \(G\) is in both the numerator and denominator of \(G_{CL}\) it should be clear that the poles cancel. ) + ( are the poles of the closed-loop system, and noting that the poles of Recalling that the zeros of s {\displaystyle N} Additional parameters There are no poles in the right half-plane. The formula is an easy way to read off the values of the poles and zeros of \(G(s)\). 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 If the system is originally open-loop unstable, feedback is necessary to stabilize the system. Since there are poles on the imaginary axis, the system is marginally stable. ( ( The feedback loop has stabilized the unstable open loop systems with \(-1 < a \le 0\). , let 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 0+j(\omega +r)} is the number of poles of the open-loop transfer function s Z The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. {\displaystyle \Gamma _{s}} and poles of {\displaystyle s={-1/k+j0}} \(\PageIndex{4}\) 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. {\displaystyle D(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 are called the zeros of l Nyquist Plot Example 1, Procedure to draw Nyquist plot in 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 \(\PageIndex{4}\). 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