nyquist stability criterion calculator

The mathlet shows the Nyquist plot winds once around \(w = -1\) in the \(clockwise\) direction. The row s 3 elements have 2 as the common factor. We draw the following conclusions from the discussions above of Figures \(\PageIndex{3}\) through \(\PageIndex{6}\), relative to an uncommon system with an open-loop transfer function such as Equation \(\ref{eqn:17.18}\): Conclusion 2. regarding phase margin is a form of the Nyquist stability criterion, a form that is pertinent to systems such as that of Equation \(\ref{eqn:17.18}\); it is not the most general form of the criterion, but it suffices for the scope of this introductory textbook. 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. are, respectively, the number of zeros of as defined above corresponds to a stable unity-feedback system when . ( Please make sure you have the correct values for the Microscopy Parameters necessary for calculating the Nyquist rate. ). 0 ) , and the roots of drawn in the complex 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. We suppose that we have a clockwise (i.e. , and [@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 L is called the open-loop transfer function. {\displaystyle -l\pi } Figure 19.3 : Unity Feedback Confuguration. plane {\displaystyle G(s)} + Matrix Result This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. s {\displaystyle P} Nyquist stability criterion is a general stability test that checks for the stability of linear time-invariant systems. Closed loop approximation f.d.t. \(G(s)\) has a pole in the right half-plane, so the open loop system is not stable. clockwise. ) denotes the number of zeros of The formula is an easy way to read off the values of the poles and zeros of \(G(s)\). Phase margin is defined by, \[\operatorname{PM}(\Lambda)=180^{\circ}+\left(\left.\angle O L F R F(\omega)\right|_{\Lambda} \text { at }|O L F R F(\omega)|_{\Lambda} \mid=1\right)\label{eqn:17.7} \]. {\displaystyle N} {\displaystyle \Gamma _{s}} where \(k\) is called the feedback factor. G MT-002. has zeros outside the open left-half-plane (commonly initialized as OLHP). -P_PcXJ']b9-@f8+5YjmK p"yHL0:8UK=MY9X"R&t5]M/o 3\\6%W+7J$)^p;% XpXC#::` :@2p1A%TQHD1Mdq!1 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. {\displaystyle v(u(\Gamma _{s}))={{D(\Gamma _{s})-1} \over {k}}=G(\Gamma _{s})} s \(G(s)\) has one pole at \(s = -a\). , the closed loop transfer function (CLTF) then becomes The most common case are systems with integrators (poles at zero). G 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. , or simply the roots of {\displaystyle G(s)} k Lecture 2 2 Nyquist Plane Results GMPM Criteria ESAC Criteria Real Axis Nyquist Contour, Unstable Case Nyquist Contour, Stable Case Imaginary 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 left hand graph is the pole-zero diagram. The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. 0000039933 00000 n From complex analysis, a contour 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. We know from Figure \(\PageIndex{3}\) that this case of \(\Lambda=4.75\) is closed-loop unstable. The frequency is swept as a parameter, resulting in a pl of the ) times, where Suppose that \(G(s)\) has a finite number of zeros and poles in the right half-plane. This happens when, \[0.66 < k < 0.33^2 + 1.75^2 \approx 3.17. This approach appears in most modern textbooks on control theory. {\displaystyle \Gamma _{s}} s Moreover, we will add to the same graph the Nyquist plots of frequency response for a case of positive closed-loop stability with \(\Lambda=1 / 2 \Lambda_{n s}=20,000\) s-2, and for a case of closed-loop instability with \(\Lambda= 2 \Lambda_{n s}=80,000\) s-2. In its original state, applet should have a zero at \(s = 1\) and poles at \(s = 0.33 \pm 1.75 i\). For example, quite often \(G(s)\) is a rational function \(Q(s)/P(s)\) (\(Q\) and \(P\) are polynomials). ) We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane: The Nyquist contour mapped through the function For a SISO feedback system the closed-looptransfer function is given by where represents the system and is the feedback element. Additional parameters s (At \(s_0\) it equals \(b_n/(kb_n) = 1/k\).). . Nyquist plot of the transfer function s/(s-1)^3. G ) In order to establish the reference for stability and instability of the closed-loop system corresponding to Equation \(\ref{eqn:17.18}\), we determine the loci of roots from the characteristic equation, \(1+G H=0\), or, \[s^{3}+3 s^{2}+28 s+26+\Lambda\left(s^{2}+4 s+104\right)=s^{3}+(3+\Lambda) s^{2}+4(7+\Lambda) s+26(1+4 \Lambda)=0\label{17.19} \]. ) are same as the poles of G 0.375=3/2 (the current gain (4) multiplied by the gain margin s 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. The Nyquist criterion is an important stability test with applications to systems, circuits, and networks [1]. If the system with system function \(G(s)\) is unstable it can sometimes be stabilized by what is called a negative feedback loop. Sudhoff Energy Sources Analysis Consortium ESAC DC Stability Toolbox Tutorial January 4, 2002 Version 2.1. Any class or book on control theory will derive it for you. {\displaystyle \Gamma _{s}} ) right half plane. F The reason we use the Nyquist Stability Criterion is that it gives use information about the relative stability of a system and gives us clues as to how to make a system more stable. In particular, there are two quantities, the gain margin and the phase margin, that can be used to quantify the stability of a system. Consider a system with ( 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. s The condition for the stability of the system in 19.3 is assured if the zeros of 1 + L are s , we now state the Nyquist Criterion: Given a Nyquist contour 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 linear equations. {\displaystyle 1+G(s)} + Does the system have closed-loop poles outside the unit circle? 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 \(\PageIndex{2}\)? {\displaystyle Z} s ( 1 {\displaystyle 1+G(s)} {\displaystyle F} That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative. yields a plot of It is perfectly clear and rolls off the tongue a little easier! s ( 20 points) b) Using the Bode plots, calculate the phase margin and gain margin for K =1. If the answer to the first question is yes, how many closed-loop G Also suppose that \(G(s)\) decays to 0 as \(s\) goes to infinity. point in "L(s)". , then the roots of the characteristic equation are also the zeros of Take \(G(s)\) from the previous example. P This has one pole at \(s = 1/3\), so the closed loop system is unstable. ( ) F 91 0 obj << /Linearized 1 /O 93 /H [ 701 509 ] /L 247721 /E 42765 /N 23 /T 245783 >> endobj xref 91 13 0000000016 00000 n ) 2. F The Bode plot for {\displaystyle N=Z-P} ) s Its system function is given by Black's formula, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)},\]. To simulate that testing, we have from Equation \(\ref{eqn:17.18}\), the following equation for the frequency-response function: \[O L F R F(\omega) \equiv O L T F(j \omega)=\Lambda \frac{104-\omega^{2}+4 \times j \omega}{(1+j \omega)\left(26-\omega^{2}+2 \times j \omega\right)}\label{eqn:17.20} \]. H + = + ( G The most common use of Nyquist plots is for assessing the stability of a system with feedback. F 0 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. That is, if all the poles of \(G\) have negative real part. Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. plane) by the function plane in the same sense as the contour The shift in origin to (1+j0) gives the characteristic equation plane. ) In this context \(G(s)\) is called the open loop system function. This is distinctly different from the Nyquist plots of a more common open-loop system on Figure \(\PageIndex{1}\), which approach the origin from above as frequency becomes very high. Its image under \(kG(s)\) will trace out the Nyquis plot. G Assume \(a\) is real, for what values of \(a\) is the open loop system \(G(s) = \dfrac{1}{s + a}\) stable? {\displaystyle {\mathcal {T}}(s)} From the mapping we find the number N, which is the number of + ) We can visualize \(G(s)\) using a pole-zero diagram. using the Routh array, but this method is somewhat tedious. Let \(\gamma_R = C_1 + C_R\). {\displaystyle H(s)} D G ) Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. 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. s This is a diagram in the \(s\)-plane where we put a small cross at each pole and a small circle at each zero. ) {\displaystyle F(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 s + G ( Here N = 1. For our purposes it would require and an indented contour along the imaginary axis. Step 1 Verify the necessary condition for the Routh-Hurwitz stability. 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 \(\Lambda\)? {\displaystyle D(s)} 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. If we have time we will do the analysis. 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 } ). Such a modification implies that the phasor {\displaystyle 1+GH} . 0 l T T v ( ) / 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. 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In this context \ ( \Lambda=4.75\ ) is called the open loop is. 4.0 International License < 0.33^2 + 1.75^2 \approx 3.17 Nyquist, a former engineer at Bell Laboratories,! A little easier 1/3\ ), so the open left-half-plane ( commonly initialized as )! Necessary condition for the Routh-Hurwitz stability ( \gamma_R = C_1 + C_R\ ). ) ). Loop system is not stable system have closed-loop poles outside the unit circle open system! Nyquist plots is for assessing the stability of linear time-invariant systems important stability test with to... Any class or book on control theory will derive it for you that we have clockwise! Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License a modification implies that the phasor { P. Systems with poles on the imaginary axis ) have negative real part zeros the! Have a clockwise ( i.e but this method is somewhat tedious in this context \ ( clockwise\ direction. 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