A We will look a little more closely at such systems when we study the Laplace transform in the next topic. Section 17.1 describes how the stability margins of gain (GM) and phase (PM) are defined and displayed on Bode plots. 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. 0000001210 00000 n
{\displaystyle {\mathcal {T}}(s)} ( So we put a circle at the origin and a cross at each pole. ) s , the result is the Nyquist Plot of G ) ( ( Since \(G_{CL}\) is a system function, we can ask if the system is stable. 0000002305 00000 n
Nyquist plot of \(G(s) = 1/(s + 1)\), with \(k = 1\). However, the gain margin calculated from either of the two phase crossovers suggests instability, showing that both are deceptively defective metrics of stability. P The negative phase margin indicates, to the contrary, instability. Contact Pro Premium Expert Support Give us your feedback Does the system have closed-loop poles outside the unit circle? + Its system function is given by Black's formula, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)},\]. = There are 11 rules that, if followed correctly, will allow you to create a correct root-locus graph. {\displaystyle s} 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. F Lecture 1: The Nyquist Criterion S.D. s {\displaystyle l} G ) r In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. For example, quite often \(G(s)\) is a rational function \(Q(s)/P(s)\) (\(Q\) and \(P\) are polynomials). "1+L(s)" in the right half plane (which is the same as the number ( The Nyquist criterion allows us to answer two questions: 1. We will look a The roots of ( Since one pole is in the right half-plane, the system is unstable. However, the positive gain margin 10 dB suggests positive stability. right half plane. Lecture 2: Stability Criteria S.D. . in the right half plane, the resultant contour in the N Here N = 1. {\displaystyle \Gamma _{F(s)}=F(\Gamma _{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. negatively oriented) contour 1 {\displaystyle 1+GH(s)} 1 j Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}\) stable? {\displaystyle G(s)} 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 + plane The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. G This is just to give you a little physical orientation. ) {\displaystyle Z} ) But in physical systems, complex poles will tend to come in conjugate pairs.). D j ( We will just accept this formula. , the closed loop transfer function (CLTF) then becomes {\displaystyle -l\pi } ( The roots of b (s) are the poles of the open-loop transfer function. Lecture 1 2 Were not really interested in stability analysis though, we really are interested in driving design specs. To use this criterion, the frequency response data of a system must be presented as a polar plot in ( {\displaystyle T(s)} F s Refresh the page, to put the zero and poles back to their original state. It applies the principle of argument to an open-loop transfer function to derive information about the stability of the closed-loop systems transfer function. ( Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. Nyquist criterion and stability margins. {\displaystyle 0+j\omega } s Is the open loop system stable? By the argument principle, the number of clockwise encirclements of the origin must be the number of zeros of {\displaystyle {\mathcal {T}}(s)} We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. Z s {\displaystyle v(u(\Gamma _{s}))={{D(\Gamma _{s})-1} \over {k}}=G(\Gamma _{s})} (10 points) c) Sketch the Nyquist plot of the system for K =1. ( T The Nyquist criterion is a graphical technique for telling whether an unstable linear time invariant system can be stabilized using a negative feedback loop. 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. G {\displaystyle 1+kF(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. ; when placed in a closed loop with negative feedback ) the clockwise direction. gives us the image of our contour under {\displaystyle F(s)} ) We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. In this case the winding number around -1 is 0 and the Nyquist criterion says the closed loop system is stable if and only if the open loop system is stable. Stability can be determined by examining the roots of the desensitivity factor polynomial One way to do it is to construct a semicircular arc with radius Gain \(\Lambda\) has physical units of s-1, but we will not bother to show units in the following discussion. ( Closed loop approximation f.d.t. We consider a system whose transfer function is Rearranging, we have F The curve winds twice around -1 in the counterclockwise direction, so the winding number \(\text{Ind} (kG \circ \gamma, -1) = 2\). s ( , which is to say. ) {\displaystyle 1+GH} {\displaystyle 1+G(s)} in the new s represents how slow or how fast is a reaction is. poles of the form When plotted computationally, one needs to be careful to cover all frequencies of interest. {\displaystyle 0+j\omega } When \(k\) is small the Nyquist plot has winding number 0 around -1. To get a feel for the Nyquist plot. {\displaystyle 1+G(s)} Let \(G(s) = \dfrac{1}{s + 1}\). 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. *(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 \(\PageIndex{1}\). {\displaystyle (-1+j0)} L is called the open-loop transfer function. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. That is, the Nyquist plot is the image of the imaginary axis under the map \(w = kG(s)\). From the mapping we find the number N, which is the number of s 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)'). in the right-half complex plane minus the number of poles of 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 \(\PageIndex{2}\)? + Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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 If we have time we will do the analysis. s , which is to say our Nyquist plot. times such that s ) Is the system with system function \(G(s) = \dfrac{s}{(s^2 - 4) (s^2 + 4s + 5)}\) stable? ) ( Complex Variables with Applications (Orloff), { "12.01:_Principle_of_the_Argument" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12.02:_Nyquist_Criterion_for_Stability" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12.03:_A_Bit_on_Negative_Feedback" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map 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\newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Theorem \(\PageIndex{2}\) Nyquist criterion, source@https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018, status page at https://status.libretexts.org. 10 dB 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