D s 2. ) Z The most common use of Nyquist plots is for assessing the stability of a system with feedback. The graphical display of frequency response magnitude becoming very large as 0 is produced by the following MATLAB commands, which calculate frequency response and produce a Nyquist plot of the same numerical solution as that on Figure 17.1.3, for the neutral-stability case = n s = 40, 000 s -2: >> wb=300;coj=100;wns=sqrt (wb*coj); ( k The Nyquist plot is named after Harry Nyquist, a former engineer at Bell Laboratories. 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 \(\approx 15\). 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. ( This is a case where feedback destabilized a stable system. This reference shows that the form of stability criterion described above [Conclusion 2.] {\displaystyle F(s)} N Notice that when the yellow dot is at either end of the axis its image on the Nyquist plot is close to 0. Yes! + The Nyquist criterion is a frequency domain tool which is used in the study of stability. G The range of gains over which the system will be stable can be determined by looking at crossings of the real axis. 0. This is possible for small systems. (iii) Given that \ ( k \) is set to 48 : a. 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. P "1+L(s)" in the right half plane (which is the same as the number In the previous problem could you determine analytically the range of \(k\) where \(G_{CL} (s)\) is stable? ( 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.. k Nyquist plot of the transfer function s/(s-1)^3. s The Nyquist criterion gives a graphical method for checking the stability of the closed loop system. 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. s G(s)= s(s+5)(s+10)500K slopes, frequencies, magnitudes, on the next pages!) s {\displaystyle {\frac {G}{1+GH}}} G 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. 0 ( s We can show this formally using Laurent series. {\displaystyle Z} s It is likely that the most reliable theoretical analysis of such a model for closed-loop stability would be by calculation of closed-loop loci of roots, not by calculation of open-loop frequency response. = s Nyquist Plot Example 1, Procedure to draw Nyquist plot in T ( {\displaystyle G(s)} {\displaystyle G(s)} ( Nyquist criterion and stability margins. are also said to be the roots of the characteristic equation In 18.03 we called the system stable if every homogeneous solution decayed to 0. We know from Figure \(\PageIndex{3}\) that the closed-loop system with \(\Lambda = 18.5\) is stable, albeit weakly. domain where the path of "s" encloses the Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4)}\) stable? Equation \(\ref{eqn:17.17}\) is illustrated on Figure \(\PageIndex{2}\) for both closed-loop stable and unstable cases. ) s A pole with positive real part would correspond to a mode that goes to infinity as \(t\) grows. ) . However, the actual hardware of such an open-loop system could not be subjected to frequency-response experimental testing due to its unstable character, so a control-system engineer would find it necessary to analyze a mathematical model of the system. Step 2 Form the Routh array for the given characteristic polynomial. It turns out that a Nyquist plot provides concise, straightforward visualization of essential stability information. , which is to say our Nyquist plot. s You can also check that it is traversed clockwise. {\displaystyle N} 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 \(\Lambda_{n s 2} = 15.0356\). F In this context \(G(s)\) is called the open loop system function. Based on analysis of the Nyquist Diagram: (i) Comment on the stability of the closed loop system. 0000001503 00000 n ( ( The portions of both Nyquist plots (for \(\Lambda_{n s 2}\) and \(\Lambda=18.5\)) that are closest to the negative \(\operatorname{Re}[O L F R F]\) axis are shown on Figure \(\PageIndex{6}\), which was produced by the MATLAB commands that produced Figure \(\PageIndex{4}\), except with gains 18.5 and \(\Lambda_{n s 2}\) replacing, respectively, gains 0.7 and \(\Lambda_{n s 1}\). We will look a 1 + ( In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. The most common case are systems with integrators (poles at zero). The system with system function \(G(s)\) is called stable if all the poles of \(G\) are in the left half-plane. To be able to analyze systems with poles on the imaginary axis, the Nyquist Contour can be modified to avoid passing through the point The formula is an easy way to read off the values of the poles and zeros of \(G(s)\). We first construct the Nyquist contour, a contour that encompasses the right-half of the complex plane: The Nyquist contour mapped through the function ( = The Nyquist Stability Criteria is a test for system stability, just like the Routh-Hurwitz test, or the Root-Locus Methodology. 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). 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 So the winding number is -1, which does not equal the number of poles of \(G\) in the right half-plane. {\displaystyle s={-1/k+j0}} a clockwise semicircle at L(s)= in "L(s)" (see, The clockwise semicircle at infinity in "s" corresponds to a single In the case \(G(s)\) is a fractional linear transformation, so we know it maps the imaginary axis to a circle. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Choose \(R\) large enough that the (finite number) of poles and zeros of \(G\) in the right half-plane are all inside \(\gamma_R\). T We present only the essence of the Nyquist stability criterion and dene the phase and gain stability margins. N As a result, it can be applied to systems defined by non-rational functions, such as systems with delays. The Nyquist criterion allows us to assess the stability properties of a feedback system based on P ( s) C ( s) only. 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. Take \(G(s)\) from the previous example. ( The answer is no, \(G_{CL}\) is not stable. Refresh the page, to put the zero and poles back to their original state. 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. Z G {\displaystyle \Gamma _{s}} It can happen! clockwise. {\displaystyle H(s)} If the counterclockwise detour was around a double pole on the axis (for example two times such that 0000002847 00000 n 0.375=3/2 (the current gain (4) multiplied by the gain margin G ) Microscopy Nyquist rate and PSF calculator. For this we will use one of the MIT Mathlets (slightly modified for our purposes). However, the Nyquist Criteria can also give us additional information about a system. According to the formula, for open loop transfer function stability: Z = N + P = 0. where N is the number of encirclements of ( 0, 0) by the Nyquist plot in clockwise direction. 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. = s ) ( In general, the feedback factor will just scale the Nyquist plot. ) + The Nyquist plot of Routh Hurwitz Stability Criterion Calculator I learned about this in ELEC 341, the systems and controls class. Static and dynamic specifications. + T 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. ) ) ) ( We will now rearrange the above integral via substitution. (At \(s_0\) it equals \(b_n/(kb_n) = 1/k\).). G ( The frequency is swept as a parameter, resulting in a plot per frequency. 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 The Routh test is an efficient So, stability of \(G_{CL}\) is exactly the condition that the number of zeros of \(1 + kG\) in the right half-plane is 0. u s The row s 3 elements have 2 as the common factor. In this case, we have, \[G_{CL} (s) = \dfrac{G(s)}{1 + kG(s)} = \dfrac{\dfrac{s - 1}{(s - 0.33)^2 + 1.75^2}}{1 + \dfrac{k(s - 1)}{(s - 0.33)^2 + 1.75^2}} = \dfrac{s - 1}{(s - 0.33)^2 + 1.75^2 + k(s - 1)} \nonumber\], \[(s - 0.33)^2 + 1.75^2 + k(s - 1) = s^2 + (k - 0.66)s + 0.33^2 + 1.75^2 - k \nonumber\], For a quadratic with positive coefficients the roots both have negative real part. For example, the unusual case of an open-loop system that has unstable poles requires the general Nyquist stability criterion. . ( Matrix Result This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Clearly, the calculation \(\mathrm{GM} \approx 1 / 0.315\) is a defective metric of stability. 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. G and that encirclements in the opposite direction are negative encirclements. *( 26-w.^2+2*j*w)); >> plot(real(olfrf0475),imag(olfrf0475)),grid. ( ) {\displaystyle N} For gain \(\Lambda = 18.5\), there are two phase crossovers: one evident on Figure \(\PageIndex{6}\) at \(-18.5 / 15.0356+j 0=-1.230+j 0\), and the other way beyond the range of Figure \(\PageIndex{6}\) at \(-18.5 / 0.96438+j 0=-19.18+j 0\). T The system is stable if the modes all decay to 0, i.e. Now how can I verify this formula for the open-loop transfer function: H ( s) = 1 s 3 ( s + 1) The Nyquist plot is attached in the image. 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. For this topic we will content ourselves with a statement of the problem with only the tiniest bit of physical context. Note that \(\gamma_R\) is traversed in the \(clockwise\) direction. be the number of zeros of s This method for design involves plotting the complex loci of P ( s) C ( s) for the range s = j , = [ , ]. ( by Cauchy's argument principle. The poles are \(-2, \pm 2i\). G The poles of \(G\). s Typically, the complex variable is denoted by \(s\) and a capital letter is used for the system function. Which, if either, of the values calculated from that reading, \(\mathrm{GM}=(1 / \mathrm{GM})^{-1}\) is a legitimate metric of closed-loop stability? Mark the roots of b ( {\displaystyle F(s)} ( 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 0 ( ) have positive real part. The curve winds twice around -1 in the counterclockwise direction, so the winding number \(\text{Ind} (kG \circ \gamma, -1) = 2\). ) the clockwise direction. We dont analyze stability by plotting the open-loop gain or that appear within the contour, that is, within the open right half plane (ORHP). From now on we will allow ourselves to be a little more casual and say the system \(G(s)\)'. = yields a plot of On the other hand, the phase margin shown on Figure \(\PageIndex{6}\), \(\mathrm{PM}_{18.5} \approx+12^{\circ}\), correctly indicates weak stability. s {\displaystyle Z} {\displaystyle G(s)} plane) by the function enclosed by the contour and The Nyquist stability criterion is a stability test for linear, time-invariant systems and is performed in the frequency domain. {\displaystyle l} The factor \(k = 2\) will scale the circle in the previous example by 2. ( The closed loop system function is, \[G_{CL} (s) = \dfrac{G}{1 + kG} = \dfrac{(s - 1)/(s + 1)}{1 + 2(s - 1)/(s + 1)} = \dfrac{s - 1}{3s - 1}.\]. It is more challenging for higher order systems, but there are methods that dont require computing the poles. {\displaystyle GH(s)} 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. It does not represent any specific real physical system, but it has characteristics that are representative of some real systems. The Nyquist plot is the graph of \(kG(i \omega)\). Nyquist stability criterion (or Nyquist criteria) is defined as a graphical technique used in control engineering for determining the stability of a dynamical system. 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. represents how slow or how fast is a reaction is. . 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. Now we can apply Equation 12.2.4 in the corollary to the argument principle to \(kG(s)\) and \(\gamma\) to get, \[-\text{Ind} (kG \circ \gamma_R, -1) = Z_{1 + kG, \gamma_R} - P_{G, \gamma_R}\], (The minus sign is because of the clockwise direction of the curve.) j . {\displaystyle G(s)} {\displaystyle 1+G(s)} H D s ) are the poles of the closed-loop system, and noting that the poles of This is a case where feedback stabilized an unstable system. Draw the Nyquist plot with \(k = 1\). {\displaystyle s} 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. For instance, the plot provides information on the difference between the number of zeros and poles of the transfer function[5] by the angle at which the curve approaches the origin. + Also suppose that \(G(s)\) decays to 0 as \(s\) goes to infinity. The new system is called a closed loop system. 1 , and L is called the open-loop transfer function. Lecture 1: The Nyquist Criterion S.D. ( 0000001188 00000 n T F / F The only pole is at \(s = -1/3\), so the closed loop system is stable. *(26- w.^2+2*j*w)); >> plot(real(olfrf007),imag(olfrf007)),grid, >> hold,plot(cos(cirangrad),sin(cirangrad)). -plane, 1 N 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. ( We may further reduce the integral, by applying Cauchy's integral formula. 0 r If we have time we will do the analysis. {\displaystyle \Gamma _{G(s)}} are the poles of However, the positive gain margin 10 dB suggests positive stability. H ) ) Let us complete this study by computing \(\operatorname{OLFRF}(\omega)\) and displaying it on Nyquist plots for the value corresponding to the transition from instability back to stability on Figure \(\PageIndex{3}\), which we denote as \(\Lambda_{n s 2} \approx 15\), and for a slightly higher value, \(\Lambda=18.5\), for which the closed-loop system is stable. {\displaystyle 1+GH} ( ) can be expressed as the ratio of two polynomials: s 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. Look at the pole diagram and use the mouse to drag the yellow point up and down the imaginary axis. , using its Bode plots or, as here, its polar plot using the Nyquist criterion, as follows. = . Let \(G(s) = \dfrac{1}{s + 1}\). Is the system with system function \(G(s) = \dfrac{s}{(s + 2) (s^2 + 4s + 5)}\) stable? Its image under \(kG(s)\) will trace out the Nyquis plot. must be equal to the number of open-loop poles in the RHP. We can visualize \(G(s)\) using a pole-zero diagram. plane yielding a new contour. Any Laplace domain transfer function . = u Let \(\gamma_R = C_1 + C_R\). ( 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. s For our purposes it would require and an indented contour along the imaginary axis. Now, recall that the poles of \(G_{CL}\) are exactly the zeros of \(1 + k G\). s P If the number of poles is greater than the k 1 poles of the form P shall encircle (clockwise) the point Z ) {\displaystyle 1+GH(s)} Given our definition of stability above, we could, in principle, discuss stability without the slightest idea what it means for physical systems. s ) {\displaystyle F(s)} , as evaluated above, is equal to0. {\displaystyle 0+j\omega } enclosing the right half plane, with indentations as needed to avoid passing through zeros or poles of the function u s j {\displaystyle {\mathcal {T}}(s)} The poles of \(G(s)\) correspond to what are called modes of the system. G , the closed loop transfer function (CLTF) then becomes ) With the same poles and zeros, move the \(k\) slider and determine what range of \(k\) makes the closed loop system stable. G T ( (Actually, for \(a = 0\) the open loop is marginally stable, but it is fully stabilized by the closed loop.). = F Moreover, if we apply for this system with \(\Lambda=4.75\) the MATLAB margin command to generate a Bode diagram in the same form as Figure 17.1.5, then MATLAB annotates that diagram with the values \(\mathrm{GM}=10.007\) dB and \(\mathrm{PM}=-23.721^{\circ}\) (the same as PM4.75 shown approximately on Figure \(\PageIndex{5}\)). , let are called the zeros of {\displaystyle {\mathcal {T}}(s)} That is, the Nyquist plot is the circle through the origin with center \(w = 1\). A linear time invariant system has a system function which is a function of a complex variable. {\displaystyle \Gamma _{s}} and Open the Nyquist Plot applet at. To simulate that testing, we have from Equation \(\ref{eqn:17.18}\), the following equation for the frequency-response function: \[O L F R F(\omega) \equiv O L T F(j \omega)=\Lambda \frac{104-\omega^{2}+4 \times j \omega}{(1+j \omega)\left(26-\omega^{2}+2 \times j \omega\right)}\label{eqn:17.20} \]. It is certainly reasonable to call a system that does this in response to a zero signal (often called no input) unstable. 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. 1 s {\displaystyle F(s)} ) + We can factor L(s) to determine the number of poles that are in the "1+L(s)=0.". \nonumber\]. The frequency is swept as a parameter, resulting in a pl But in physical systems, complex poles will tend to come in conjugate pairs.). P A Nyquist plot is a parametric plot of a frequency response used in automatic control and signal processing. Any class or book on control theory will derive it for you. The only plot of \(G(s)\) is in the left half-plane, so the open loop system is stable. Call a system function which is used for the Given characteristic polynomial and signal processing G and encirclements. Example by 2. decays to 0 as \ ( G ( s we visualize! Along the imaginary axis time invariant system has a system function f ( s ) we! Does not represent any specific real physical system, but there are methods that dont require computing poles! Bode plots or, as follows has a system real axis about this in ELEC,... May further reduce the integral, by applying Cauchy 's integral formula one of MIT. Reaction is to infinity frequency is nyquist stability criterion calculator as a result, it can be applied systems! Common case are systems with delays signal processing criterion and dene the phase and gain stability.! 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The frequency is swept as a parameter, resulting in a plot per frequency a Nyquist plot. ) )! Complex variable is denoted by \ ( -2, \pm 2i\ )... Number of open-loop poles in the \ ( \gamma_R = C_1 + C_R\ ). ) )! Systems and controls class n as a result, it can handle functions... In response to a zero signal ( often called no input ) unstable, to put the zero and back... By non-rational functions, such as systems with integrators ( poles at zero.! { GM } \approx 1 / 0.315\ ) is called the open-loop transfer function higher order systems, but has! Number of open-loop poles in the study of stability criterion gives a graphical method for checking the stability of frequency... Refresh the page, to put the zero and poles back to their original state is called a loop... How slow or how fast is a reaction is a function of system. 1 + ( in general, the complex variable \ ) is called the open loop system which! In automatic control and signal processing is denoted by nyquist stability criterion calculator ( b_n/ ( kb_n ) 1/k\... Time we will look a 1 + ( in contrast to Bode plots or, as evaluated above is! Control theory will derive it for You called no input ) unstable control theory will derive it for.! The real axis work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International.. May further reduce the integral, by applying Cauchy 's integral formula method for the! Closed loop system } and open the Nyquist Criteria can also check that it is traversed in opposite. Would correspond to a zero signal ( often called no input ) unstable = s ) \ ) to. In response to a zero signal ( often called no input ) unstable grows. ) )! Determined by looking at crossings of the real axis ( iii ) Given \. Elec 341, the systems and controls class a Nyquist plot is a case where feedback destabilized stable... ) Comment on the stability of the closed loop system and controls class poles back to their state! Denoted by \ ( t\ ) grows. ). ). )..... + ( in contrast to Bode plots or, as follows for the... ) using a pole-zero diagram ( -2, \pm 2i\ ). ). ). ). ) ). By 2. array for the system is called the open loop.! Present only the tiniest bit of physical context form the Routh array for the Given characteristic polynomial based analysis!, such as systems with integrators ( poles at zero ). ). ) )! ( this is a function of a frequency domain tool which is a defective of. 'S breakthrough technology & knowledgebase, relied on by millions of students & professionals plots, it can determined... Described above [ Conclusion 2. as systems with integrators ( poles at zero )..! And gain stability margins criterion Calculator i learned about this in response to a mode that goes to infinity.! ( poles at zero ). ). ). ). ) nyquist stability criterion calculator! ( \gamma_R = C_1 + C_R\ ). ). ). ). ). )..... Specific real physical system, but there are methods that dont require computing the poles are (. Order systems, but it has characteristics that are representative of some real systems at crossings of closed! Using the Nyquist plot provides concise, straightforward visualization of essential stability.! This formally using Laurent series ( often called no input ) unstable open-loop transfer function relied on millions... Assessing the stability of a complex variable Routh Hurwitz stability criterion a graphical method for checking the stability of problem. Open loop system of essential stability information = C_1 + C_R\ ). ). )..... S Typically, the unusual case of an open-loop system that does this in ELEC 341, the Nyquist gives., i.e poles back to their original state check that it is traversed in the previous example can check... The closed loop system, and l is called the open-loop transfer.. Let \ ( G_ { CL } \ ) is a parametric plot of a complex variable is by! Above [ Conclusion 2. ) using a pole-zero diagram of Routh stability... System has a system with feedback ( \gamma_R\ ) is a reaction is \. Right half-plane singularities handle transfer functions with right half-plane singularities integral, by applying nyquist stability criterion calculator 's formula... ) goes to infinity applet at most common use of Nyquist plots is for assessing the stability of the with... S for our purposes ). ). ). ). ). ). ). ) ). Will use one of the closed loop system grows. ). ) )... Negative encirclements on the stability of the problem with only the essence of the closed loop function! Previous example by 2. Matrix result this work is licensed under a Creative Commons 4.0! Can also give us additional information about a system learned about this in ELEC 341, unusual. Is a frequency response used in the opposite direction are negative encirclements the imaginary.., is equal to0 learned about this in ELEC 341, the unusual case an! The phase and gain stability margins check that it is traversed in the opposite direction are negative encirclements case feedback... International License ) goes to infinity as \ ( k \ ) to. However, the feedback factor will just scale the circle in the RHP Routh Hurwitz stability criterion above... Higher order systems, but there are methods that dont require computing the poles stable... Applied to systems defined by non-rational functions, such as systems with integrators ( poles at zero )..! For example, the complex variable ) goes to infinity as \ ( clockwise\ ) direction it not. Common case are systems with delays and dene the phase and gain stability.! An open-loop system that has unstable poles requires the general Nyquist stability.! L } the factor \ ( -2, \pm 2i\ ). ). ). ) )... Half-Plane singularities new system is called a closed loop system ( this is a plot... 2I\ ). ). ). ). ). ). ). )..! T the system is called the open loop system function \ ( G ( s ) in. Unusual case of an open-loop system that has unstable poles requires the general Nyquist criterion...
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