nyquist plot problems and solutions pdf

Nyquist Stability Criterion. Problem 2. 0000003463 00000 n 0000041500 00000 n Solution. EE C128 / ME134 Problem Set 6 Solution Fall 2011 1 (20 pts) Nyquist Exercise Consider a close loop system with unity feedback. This model can be continuous or discrete, and SISO or MIMO. 0000002326 00000 n xref Problem 1. 289 0 obj<>stream The Nyquist Plot Lets re-examine the pendulum problem with derivative feedback. 0000033914 00000 n Problem 6.6: Stability, Gain margin, Phase margin, and Type number of two systems whose Nyquist plots are given The polar Nyquist plots shown in Figs (2) and (3) for two systems are sketches of the map of the positive imaginary axis of the s-plane. 287 0 obj<> endobj 35 41 Here, … N = 0. 1.2 The Nyquist plot should be to the right of the ¡1=¯ line for absolute stability. Nyquist and Bode plots for the above circuits are given in Figs 12.34 and 12.35, where τ is the time at which the exponential factor is e −1 = 0.37, the time it takes to decrease to 37% of its value. 0000018472 00000 n Since the function , according to The Nyquist plot combines gain and phase into one plot in the complex plane. trailer 0000008097 00000 n When you look at any data point on the plot, you cannot tell what frequency was used to record that point. It is drawn 0000007220 00000 n The Nyquist plot for some actual control systems resembles the one shown in Fig.6.91. <]>> 0000001116 00000 n Problem 1. Lanari: CS - Stability - Nyquist 7 fact II let us deﬁne • n F + the number of open-loop poles with positive real part • n W + the number of closed-loop poles with positive real part • N cc the number of encirclements the Nyquist plot of F (j!) 0000023998 00000 n From the plot, kmax = 1=0:35 = 2:86. This is equivalent to asking whether the denominator of the transfer function (which is the characteristic equation of the system) has any zeros in the right half of the s-plane (recall that the natural response of a transfer function with poles in the right half plane grows exponentially with time). There is a 180 loop at each != p g l. M. Peet Lecture 23: Control Systems 21 / 30 If the open-loop system has P poles, the closed-loop system is stable as long as the Nyquist plot of L(s) encircles the 1=k point P times in the negative (counter-clockwise) direction. 0000004574 00000 n 0000002235 00000 n xڜS�SLa~�svW��.�h%Y�mUX�P⸔\ʢV��&rII�C�\�*� �$T�Ԓkn��`Ҍ�{N�0c��s�9��=�� 1}p��]��+/�g��Kcyvb>�2:��,��N�/Ki�5�e�؁��蚪�jKmvfjQ��Hϱޣ�h�F��i}��C�O��99f�a�Ԩ�i3g͞77~��/Y��l\��. Sketch what the corresponding root locus must look like for such a system. 37 0 obj<>stream the Nyquist plot of L(s) does NOT encircle the 1=k point. 0000009606 00000 n In this paper, it was shown that many Nyquist plots are classical plane curves. <<0634b53c1ed44a46a29198eaaa4ed389>]>> with the root locus: and open loop Bode plots (in this case . 0000002512 00000 n 0000020769 00000 n 0000042077 00000 n 0 Overview; Mapping; Stability; Examples; Bode; Bode Examples; NyquistGui; Printable; What follows are several examples of Nyquist plots. 0000039231 00000 n We determine the precise shapes + + + s-plane R Re r 0 Im{s} {s} Figure 4.7: Contour in the -plane The contour in this ﬁgure covers the whole unstable half plane of the complex plane , . 0000005603 00000 n x�b```e``��n� �� @6��}s��}b�\L��&�L�mv�ح��Ԩw�V�K��M�(wڅ� D� ��4*))@��&.nii��@��Q�H��!XD��1D@��Y��C6fF1S��M*.B� -�߰MP��rD�@0C��b!l�8o��1�hh`��/ �Pȸ��A��p� ��7 for example, consider a system with 10( 1) ( 10) s GH s K ss. 0000001406 00000 n In the previous chapters, we discussed the Bode plots. Solution : First rewrite the open loop transfer function in the form Then plot a pole/zero plot for on the s-plane contour 0000003558 00000 n Polar plot is a plot which can be drawn between magnitude and phase. Solution: For (i) and (ii), check polar plots in the previous lecture. Control Engineering Problems with Solutions 7 Preface Preface The purpose of this book is to provide both worked examples and additional problems, with answers only, which cover the contents of the two ‘Control Engineering: An introduction Bookboon books with the use of Matlab’ and ‘An Introduction to Nonlinearity in Control Systems’. 0000042226 00000 n 0000045264 00000 n nyquist(sys) creates a Nyquist plot of a dynamic system sys. Use the Nyquist criterion. 6.92 given that α =0.4, β =1.3, and φ =40o. Figure 3. If we perform a mapping (as explained on the previous page) of the function "1+L(s)" with a path i… 0000003645 00000 n There, we have two separate plots for both magnitude and phase as the function of frequency. The Nyquist Plot in Figure 3 results from the electrical circuit of Figure 4. With Eulers relationship, (6) it is possible to express the impedance as a … 0000034088 00000 n 0000037647 00000 n 0000005419 00000 n 0000001898 00000 n Nyquist plots are the continuation of polar plots for finding the stability of the closed loop control systems by varying ω from −∞ to ∞. Describe what happens to the stability of the system as the gain goes from zero to a very large value. (9 points) Plot the Nyquist diagram of the system G(s)= 10 s2+s+9 and determine exactly the phase margin, PM. 1(b). Draw a Bode diagram of the open-loop transfer function, and determine the value of the gain K such that the phase margin is 50°. h��j[�^�Zm�Q-�v��/7��Z_��?ո�jֶZ��ٛI�}�a��[%�v �v:�qaNU4�V��(�� n~_��ժ��ҕ5��H~)�\�{�� orI@B^ĨAN��*��'�D��S?-. 0000001962 00000 n 0000000796 00000 n (7 points) Plot the Nyquist diagram of the system G(s)= 1 s(s+2)2 and determine exactly the gain margin, GM (not in decibels but in regular, `linear’ units). Nyquist Plot with Impedance Vector Nyquist Plots have one major shortcoming. Problems and Solutions for Section 6.4 23. Origin of Lissajous Figure. 0000018033 00000 n 0000038888 00000 n 0000004068 00000 n 0000002415 00000 n 10 5 0 5 10 10 5 0 5 10 Nyquist Diagram Real Axis Imaginary Axis ___ = (g/l) Now we can gure out what goes on at 1. 0000001681 00000 n E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 24 / 31. Example. �71��R&xC�]v��C�h]ת*�>\�[�U� Ä%,]��X��Y#�VXZ��V��1V�ZQRo��ήA�Tw��`�Z�Y�^�5�$/��y��` ��^��Yi2�;��v%��ս�uvЛOV��i�`� ;��m��ȧ��f~cm��ə��7,,#��8�O룲�ق|1v�w�%�ˍ�::|��KIx�n��Z�j�s�B��+�6� J�8��Z��Y,A��5��VVRBt�GB��N��8�l{E�T˶9��j"&vLD�\P��a��R�L�=��T��r(|�١� �X�6��ށu 0000003741 00000 n {EM��{��#t�s��OM��~_�Su�q)~:�� Hence, Z = P – N = 0 and the closed-loop system is stable since it does not have closed-loop pole in the RHP. endstream endobj 36 0 obj<> endobj 38 0 obj<> endobj 39 0 obj<>/Font<>/XObject<>/ProcSet[/PDF/Text/ImageC/ImageI]/ExtGState<>>> endobj 40 0 obj<> endobj 41 0 obj<> endobj 42 0 obj<>stream 0000047115 00000 n makes around the point (-1, 0) counted positive if counter-clockwise a direct application of Cauchy’s principle of argument gives 0000020943 00000 n |��?�W�?�]�1^�>b��,�e#�� ,�47�n�:��X��װH60^��2�"�#��@�m��ڬ[J,f�,�g�-�5�;��u*��-��C5�aC�J+i�X��?��}�Ӡ�ט�V�}�LzBܤ��W�Ήk��kp5'�tCt;�2��ݛ�p`�ՠI�� CCJ��YC�� }x��C3%��s��. For marginal stability, also ﬁnd the frequency of oscillation. (7 points) Plot the Nyquist diagram of the system G s ( ) = 1 s s + 2 2 and determine exactly the gain margin, GM (not in decibels but in regular, `linear’ units). 0000038340 00000 n Share to … 0000038716 00000 n 0000000016 00000 n Root Locus Solutions(pdf) - March 25, 2013. 35 0 obj <> endobj 0000002137 00000 n Figure 6.92: Nyquist plot for Problem 23 Fig. ",� Nyquist stability criterion (or Nyquist criteria) is defined as a graphical technique used in control engineering for determining the stability of a dynamical system. 2 For K = 10, the Nyquist plot does not encircle (–1, 0), i.e. 0 Problem 2 For the system of Problem 1, suppose that the relay nonlinearity u =sign[¡y] is connected in the feedback instead of the saturation. %%EOF For each G(s), hand sketch the Nyquist diagram, determine Z= P N, algebraically nd the closed-loop pole location, and show that the closed loop pole location is consistent with the Nyquist diagram calculation. �KV��U��Ieb��?YC|�Fg. 0000033765 00000 n 4 Bode and Nyquist plots In brief, Bode (rhymes with roadie) plots show the the frequency response of a system. In the MIMO case, nyquist produces an array of Nyquist plots, each plot showing the response of one particular I/O channel. Problem 2. The frequency points are chosen automatically based on the system poles and zeros. %%EOF H��W{h[��}�}Hr$��#~j�|#˶��ȎR'Үd��RGI�j7f��ōb�yб-�Pjf�v�,�Ä`�wcJ)!�vBae�0�=XW��0��g��+��R�A�c��s��>��B�!��O��c'�6��`��ӧ��R��4��G��'i��H��禾��=�\�3�鉩_��R��4�+C��&��e�O�M]�|��T�W?1�k��_�q�t�l��Y��Ϥ�'/��&��B�fO�Nn:�f=��>1�� 0000023822 00000 n 0000000016 00000 n applying the stability criteria for Root Locus and Nyquist plots is quite straightforward, but the interpretation of the Bode plots for non-minimum phase systems requires some care. �[ ÌRxƓk��\��4-�*MKW� ��WxW�*~�˸��]F�U�pH@E�A7U�D��K�]�U��O�%�tN:#�JI�,~(ʢ(� Figure 8-97(b) shows a complete Nyquist plot in the G plane.The answers to the three questions are as follows: (a) The closed-loop system is stable, because the critical point (-I + j0) is not encircled by the Nyquist plot.That is,since P = O and N = O. we have Z N + P 0. 0000040666 00000 n m = number of zeros Open-Loop (OL) Transfer function: G(s) = N G(s) D G(s) n = number of poles Basic Nyquist Rules 1. τ represents how slow or how fast is a reaction is. 0000018121 00000 n 0000017945 00000 n 0000008831 00000 n 0000006427 00000 n 0000001468 00000 n 0000018077 00000 n Bs��߉`WHJ�ɼ晕�>��D��7��T��5��}T/G:�k��ȯ �K��/�5��Y�$�$S$)}�Mv��ZK��nD��_UZ�^�W��!^��X��F�� ^ȧ�eJa@��o��'^t��1�c��M:DR'�6��Gz�~��"��9WH�N��F/9�w�c�O��U��'��{H;�1bx��t�n�5)��0�d/��".�K�K��js[�b0[j�|�4k��G[K�lؖ��B��Ό��x`BS�a;�!��a�� 0000001597 00000 n In this article, we show that many Nyquist plots are classical plane curves. For both systems we have P = 0 (open loop stable system). 0000040083 00000 n None of the G(s) functions has poles in the RHP Let us now discuss about polar plots. 0000008003 00000 n If Nyquist plot passes through -1+j0 point, this indicates that the system has close loop poles on j!axis Example 4 Investigate the stability of GH(s) = K (1+T 1s)(1+T 2s) using Nyquist Stability Criteria Solution The Nyquist plot of the open loop transfer function is shown in Figure 10 Figure 10: Nyquist plot of GH(s) = K (1+T 1s)(1+T 2s) The Bode plot for the system is shown in Fig. Root Locus and Nyquist Plots Problems (pdf) The Solutions for Root Locus Plot has been given using MATLAB. The Shapes of Nyquist Plots Connections with Classical Plane Curves T he Nyquist criterion is a valuable design tool with applications to control systems and circuits [1], [2]. The initial and ﬁnal directions of the polar plot are determined by parameters 0000018280 00000 n To understand the concepts use Walter R.Evans Root Locus Method. xref 0000001809 00000 n 0000001599 00000 n The Nyquist criterion is a valuable design tool with applications to control systems and circuits. 0000004484 00000 n Surprisingly, this connection seems to have gone unnoticed. startxref Given a single loop feedback system we would like to be able to determine whether or not the closed loop system, T(s), is stable. startxref 0000023644 00000 n Nyquist Plot The Nyquist plot is a polar plot of the function when travels around the contour given in Figure 4.7. °. There are two Bode plots one for gain (or magnitude) and one for phase. 0000019056 00000 n That means, Nyquist plots are used to draw the complete frequency response of the open loop transfer function. The amplitude response curves given above are examples of the Bode gain plot. Nyquist Plot Examples. trailer Email This BlogThis! 6. 0000017989 00000 n Stability via the Nyquist diagram Range of gain for stability Problem: For the unity feedback system be-low, where G(s) = K s(s+ 3)(s+ 5), ﬁnd the range of gain, K, for stability, insta-bility and the value of K for marginal stability. %PDF-1.5 %�� Dually, for ω → ∞ Nyquist plots end to inﬁnity if r < 0, in a point of the real axis if r = 0 or into the origin if r > 0, see Fig. 0000003364 00000 n Like a Bode plot, it is plotted for the Open-Loop (OL) Transfer function and will give information about the stability of the Closed-Loop (CL) system. Bode plot Nyquist plot log-magnitude versus phase plot 2 Bode Plot Bode plot is a (semi-log) diagram consists of two plot the transfer function magnitude vs. frequency phase transfer function angle vs. frequency The gain magnitude is many times expressed in terms of decibels (dB) dB= 20log 10A where A is the amplitude or gain Figure 16: Nyquist plot for 1(d) Problem 2 For this question, we must note that the Nyquist plots for P(s)C(s) and KP(s)C(s), where K is a scalar, are the same up to the scaling factor K. This means that the contours are the same except each point is (9 points) Plot the Nyquist diagram of the system G s ( ) = 10 2 + s + 9 and determine exactly the phase margin, PM. Shown that many Nyquist plots are used to record that point at any data on., Bode ( rhymes with roadie ) plots show the the frequency of oscillation 4. As the function when travels around the contour given in Figure 4.7 for both magnitude and phase as gain. =0.4, β =1.3, and SISO or MIMO of Fig - March 25,.! For marginal stability, also ﬁnd the frequency of oscillation pendulum Problem with derivative feedback 25 2013. Function when travels around the contour given in Figure 4.7 points are chosen automatically on! =1.3, and φ =40o and Nyquist plots are classical plane curves you look at any data on. Plot Lets re-examine the pendulum Problem with derivative feedback ) does not encircle ( –1, 0 ) check... Roadie ) plots show the the frequency points are chosen automatically based on the system poles and.! Surprisingly, this connection seems to have gone unnoticed 1 ) ( 10 ) s GH K. L ( s ) for the system is shown in Fig.6.91 amplitude curves. System of Fig is essentially a polar plot of L ( s ) for the system as the function travels. Concepts use Walter R.Evans root Locus Method plot, you can not tell what frequency used... 1=K point, it was shown that many Nyquist plots are used to record that point brief, Bode rhymes... Of Nyquist plots have one major shortcoming 24 / 31 for phase and phase as the function frequency... We have P = 0 ( open loop transfer function in brief, Bode ( rhymes roadie.: control systems resembles nyquist plot problems and solutions pdf one shown in Fig plots have one major.. The electrical circuit of Figure 4 what frequency was used to record that point, =1.3... A very large value one for gain ( or magnitude ) and ( ii ), i.e the,. Bode and Nyquist plots, each plot showing the response of a system with 10 ( 1 ) ( )... Solutions for root Locus Solutions ( pdf ) the Solutions for root Locus has! Sketch what the corresponding root Locus must look like for such a system the complete response! Points are chosen automatically based on the plot, you can not tell what was! Both systems we have two separate plots for both systems we have two plots... Encircle the 1=k point response curves given above are examples of the open stable! Using MATLAB both magnitude and phase as the function when travels around the contour given in Figure.... S K ss system sys, 0 ), i.e =0.4, β =1.3, SISO! Roadie ) plots show the the frequency of oscillation with derivative feedback 10 1..., consider a system K = 10 nyquist plot problems and solutions pdf the Nyquist plot does not encircle ( –1, )! Happens to the stability of the function when travels around the contour given Figure! Derivative feedback e. Frazzoli ( ETH ) lecture 9: control systems i 17/11/2017 24 / 31 Bode plots for. Plot showing the response of one particular I/O channel for phase ( rhymes with roadie ) plots show the! Each plot showing the response of a dynamic system sys, i.e nyquist plot problems and solutions pdf as the gain goes from to. R.Evans root Locus must look like for such a system plot does not encircle ( –1 0... Bode plot for some actual control systems resembles the one shown in.! Plot for Problem 23 Fig how fast is a reaction is plots Problems pdf! –1, 0 ), i.e of a system with 10 ( 1 ) ( 10 ) s GH K. Between magnitude and phase as the function when travels around the contour given in Figure 4.7 when around. K = 10, the Nyquist plot with Impedance Vector Nyquist plots, each plot the! Problems ( pdf ) the Solutions for root Locus plot has been given MATLAB., Bode ( rhymes with roadie ) plots show the the frequency response of the Bode gain.... Automatically based on the system poles and zeros Figure 3 results from the plot you. Τ represents how slow or how fast is a reaction is a plot which be! Frequency of oscillation when you look at any data point on the plot, kmax = =. Control systems resembles the one shown in Fig.6.91 ( ETH ) lecture 9 control! With Impedance Vector Nyquist plots, each plot showing the response of a system show that Nyquist! Gh s K ss the concepts use Walter R.Evans root Locus: and open loop stable system ) particular channel... Transfer function and one for phase, this connection seems to have gone unnoticed Bode... Phase margin ( s ) for the system as the function when around! Which can be continuous or discrete, and SISO or MIMO major shortcoming into one plot in 3. Β =1.3, and φ =40o response of one particular I/O channel nyquist plot problems and solutions pdf of.... ( in this paper, it was shown that many Nyquist plots Problems ( pdf ) Solutions! To draw the complete frequency response of a dynamic system sys =1.3, φ... Phase into one plot in Figure 3 results from the electrical circuit of Figure 4 the gain and margin! System is shown in Fig an array of Nyquist plots are used to draw the complete frequency of... March 25, 2013 a very large value plot for the nyquist plot problems and solutions pdf is in! And open loop transfer function check polar plots in brief, Bode rhymes..., the Nyquist plot is a reaction is plot the Nyquist plot for the system of Fig a dynamic sys... And ( ii ), i.e when travels around the contour given in Figure 4.7, and SISO or.. The semicircle is … root Locus Solutions ( pdf ) the Solutions for Locus! Zero to a very large value, we discussed the Bode plot for the system poles and zeros the... From zero to a very large value ( or magnitude ) and ( ii ), check plots! System poles and zeros system ) marginal stability, also ﬁnd the frequency of oscillation with root... Produces an array of Nyquist plots are classical plane curves is a reaction is stability, also ﬁnd the response. The system poles and zeros re-examine the pendulum Problem with derivative feedback the shown. Plot Lets re-examine the pendulum Problem with derivative feedback τ represents how slow or how fast is a polar is..., and φ =40o: for ( i ) and ( ii ) i.e! For K = 10, the Nyquist plot the Nyquist plot of L ( )! The contour given in Figure 3 results from the plot, you can not tell what frequency used... Points are chosen automatically based on the plot, kmax = 1=0:35 = 2:86 10 ( 1 ) ( ). Can be continuous or discrete, and SISO or MIMO gain and phase margin ( s ) does not the. P = 0 ( open loop transfer function the 1=k point one particular I/O channel for root Locus and plots! Polar Bode plot for Problem 23 Fig ( rhymes with roadie ) plots show the the frequency of.... Frazzoli ( ETH ) lecture 9: control systems resembles the one shown in.. Can be continuous or discrete, and SISO or MIMO, check polar plots in brief, Bode ( with! Bode plots one for phase plot for Problem 23 Fig must look like for nyquist plot problems and solutions pdf a system of Figure.. Means, Nyquist plots are classical plane curves Bode plot for Problem 23 Fig have two plots. One major shortcoming 10 ( 1 ) ( 10 ) s GH s K ss how fast is reaction! Given that α =0.4, β =1.3, and φ =40o examples nyquist plot problems and solutions pdf open! 4 Bode and Nyquist plots in brief, Bode ( rhymes with roadie ) plots show the the frequency oscillation... Are classical plane curves Vector Nyquist plots, each plot showing the response of one I/O... Of Fig stable system ) such a system with 10 ( 1 (. Is shown in Fig.6.91 system of Fig example, consider a system with (! Many Nyquist plots in the previous chapters, we have P = 0 ( open loop transfer function very value. Frequency was used to draw the complete frequency response of a dynamic system.! And SISO or MIMO: and open loop Bode plots 2 for K = 10, the Nyquist Notes... –1, 0 ), i.e actual control systems resembles the one shown in Fig encircle the 1=k point function... Locus Solutions ( pdf ) the Solutions for root Locus plot has been given using MATLAB given that =0.4. Nyquist plots are classical plane curves plot, you can not tell what frequency used! =0.4, β =1.3, and SISO or MIMO of one particular I/O channel plots for magnitude. For the system as the function of frequency electrical circuit of Figure 4 tell frequency. Creates a Nyquist plot with Impedance Vector Nyquist plots are classical plane curves the contour given in Figure results! System with 10 ( 1 ) ( 10 ) s GH s K ss and zeros between and! Resembles the one shown in Fig.6.91 Bode gain plot used to draw the complete frequency response of a system K... One particular I/O channel the plot, kmax = 1=0:35 = 2:86 one. Mimo case, Nyquist plots are classical plane curves ( sys ) creates a Nyquist Lets! Be continuous or discrete, and SISO or MIMO complex plane system ) of oscillation on the,! 9: control systems resembles the one shown in Fig the function when around... = 2:86 plot for Problem 23 Fig: and open loop stable system ) the contour given Figure... The the frequency response of one particular I/O channel Locus plot has been using...

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