nyquist plot problems and solutions pdf

287 25 (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). xref Nyquist stability criterion (or Nyquist criteria) is defined as a graphical technique used in control engineering for determining the stability of a dynamical system. 0000004068 00000 n Solution. 0000037647 00000 n 0000001809 00000 n 0000041500 00000 n �KV��U��Ieb��?YC|�Fg. In this article, we show that many Nyquist plots are classical plane curves. 0000037996 00000 n Share to … Polar plot is a plot which can be drawn between magnitude and phase. 0000004241 00000 n There are two Bode plots one for gain (or magnitude) and one for phase. + + + 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 , . 0000002137 00000 n 1.2 The Nyquist plot should be to the right of the ¡1=¯ line for absolute stability. 0000001116 00000 n 0000018077 00000 n With Eulers relationship, (6) it is possible to express the impedance as a … Root Locus Solutions(pdf) - March 25, 2013. trailer Surprisingly, this connection seems to have gone unnoticed. 0000033765 00000 n 0000001597 00000 n In the MIMO case, nyquist produces an array of Nyquist plots, each plot showing the response of one particular I/O channel. 0000023822 00000 n 0000017989 00000 n Problems and Solutions for Section 6.4 23. 0000018033 00000 n 0000020943 00000 n 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. 35 0 obj <> endobj Overview; Mapping; Stability; Examples; Bode; Bode Examples; NyquistGui; Printable; What follows are several examples of Nyquist plots. startxref {EM��{��#t�s��OM��~_�Su�q)~:�� 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. with the root locus: and open loop Bode plots (in this case . Hence, Z = P – N = 0 and the closed-loop system is stable since it does not have closed-loop pole in the RHP. 0000001468 00000 n <<0634b53c1ed44a46a29198eaaa4ed389>]>> 0000039231 00000 n 289 0 obj<>stream 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�� 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. 1(b). The Nyquist plot for some actual control systems resembles the one shown in Fig.6.91. 0000002415 00000 n Problem 1. Nyquist Plot The Nyquist plot is a polar plot of the function when travels around the contour given in Figure 4.7. for example, consider a system with 10( 1) ( 10) s GH s K ss. 0000045264 00000 n 0000044271 00000 n %%EOF τ represents how slow or how fast is a reaction is. 287 0 obj<> endobj 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). the Nyquist plot of L(s) does NOT encircle the 1=k point. Origin of Lissajous Figure. Email This BlogThis! 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. Problem 1. Nyquist Stability Criterion. 0000003741 00000 n ECE 3510 Nyquist Plot Notes A Nyquist plot is essentially a polar Bode plot. 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. |��?�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��. (9 points) Plot the Nyquist diagram of the system G s ( ) = 10 2 + s + 9 and determine exactly the phase margin, PM. The Bode plot for the system is shown in Fig. 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. 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 0000002326 00000 n 0000023644 00000 n 0000018121 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. Let us now discuss about polar plots. The amplitude response curves given above are examples of the Bode gain plot. Solution : First rewrite the open loop transfer function in the form Then plot a pole/zero plot for on the s-plane contour 0000042226 00000 n The frequency points are chosen automatically based on the system poles and zeros. 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. Example. 0000001599 00000 n 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. 0000002512 00000 n Use the Nyquist criterion. 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. Describe what happens to the stability of the system as the gain goes from zero to a very large value. 0000038340 00000 n Consider the system shown in Figure Q3. %PDF-1.5 %�� 0000005511 00000 n Introduction to the Nyquist criterion The Nyquist criterion relates the stability of ... which is exactly the polar plot of G(s)H(s). °. E. Frazzoli (ETH) Lecture 9: Control Systems I 17/11/2017 24 / 31. trailer 0000020769 00000 n 0000001681 00000 n 0000008003 00000 n The semicircle is … 0000005603 00000 n 0000040083 00000 n 6. 0000000796 00000 n (9 points) Plot the Nyquist diagram of the system G(s)= 10 s2+s+9 and determine exactly the phase margin, PM. 0000000016 00000 n 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. (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). 6.91 given that α=0.4,β=1.3,and φ=40o.Describe what happens to the stability of the system as the gain goes from zero to a very large value. We determine the precise shapes Here, … 0000003558 00000 n This model can be continuous or discrete, and SISO or MIMO. 0000002235 00000 n 0000003364 00000 n Figure 3. 0000004484 00000 n h��j[�^�Zm�Q-�v��/7��Z_��?ո�jֶZ��ٛI�}�a��[%�v �v:�qaNU4�V��(�� n~_��ժ��ҕ5��H~)�\�{�� orI@B^ĨAN��*��'�D��S?-. 0000017945 00000 n 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°. Nyquist Plot with Impedance Vector Nyquist Plots have one major shortcoming. 0000001898 00000 n Root Locus and Nyquist Plots Problems (pdf) The Solutions for Root Locus Plot has been given using MATLAB. 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\��. 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 The Nyquist plot combines gain and phase into one plot in the complex plane. 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’. There, we have two separate plots for both magnitude and phase as the function of frequency. �[ ÌRxƓk��\��4-�*MKW� ��WxW�*~�˸��]F�U�pH@E�A7U�D��K�]�U��O�%�tN:#�JI�,~(ʢ(� When you look at any data point on the plot, you cannot tell what frequency was used to record that point. %PDF-1.5 %�� In this paper, it was shown that many Nyquist plots are classical plane curves. 37 0 obj<>stream The Nyquist criterion is a valuable design tool with applications to control systems and circuits. For marginal stability, also ﬁnd the frequency of oscillation. To understand the concepts use Walter R.Evans Root Locus Method. %%EOF The Nyquist Plot in Figure 3 results from the electrical circuit of Figure 4. 0000001962 00000 n There is a 180 loop at each != p g l. M. Peet Lecture 23: Control Systems 21 / 30 0000004574 00000 n 0 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. 0000033914 00000 n 35 41 For both systems we have P = 0 (open loop stable system). The initial and ﬁnal directions of the polar plot are determined by parameters What are the gain and phase margin(s) for the system of Fig. Figure 6.92: Nyquist plot for Problem 23 Fig. 0000038716 00000 n 0000023308 00000 n 0000042077 00000 n 0000009606 00000 n 0000019056 00000 n 0000040666 00000 n 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 N = 0. 0000023998 00000 n 0000005419 00000 n 0000001406 00000 n <]>> 0 The Nyquist Plot Lets re-examine the pendulum problem with derivative feedback. Problem 2. 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]. Bs��߉`WHJ�ɼ晕�>��D��7��T��5��}T/G:�k��ȯ �K��/�5��Y�$�$S$)}�Mv��ZK��nD��_UZ�^�W��!^��X��F�� ^ȧ�eJa@��o��'^t��1�c��M:DR'�6��Gz�~��"��9WH�N��F/9�w�c�O��U��'��{H;�1bx��t�n�5)��0�d/��".�K�K��js[�b0[j�|�4k��G[K�lؖ��B��Ό��x`BS�a;�!��a�� m = number of zeros Open-Loop (OL) Transfer function: G(s) = N G(s) D G(s) n = number of poles Basic Nyquist Rules 1. 0000034088 00000 n Nyquist Plot Examples. �71��R&xC�]v��C�h]ת*�>\�[�U� Ä%,]��X��Y#�VXZ��V��1V�ZQRo��ήA�Tw��`�Z�Y�^�5�$/��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 Solution: For (i) and (ii), check polar plots in the previous lecture. 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) 0000003645 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 ∞. 0000038888 00000 n 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!) EE C128 / ME134 Problem Set 6 Solution Fall 2011 1 (20 pts) Nyquist Exercise Consider a close loop system with unity feedback. 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 In the previous chapters, we discussed the Bode plots. Since the function , according to Problem 2. 0000018472 00000 n From the plot, kmax = 1=0:35 = 2:86. 0000047115 00000 n 0000008097 00000 n It is drawn 4 Bode and Nyquist plots In brief, Bode (rhymes with roadie) plots show the the frequency response of a system. 0000008831 00000 n That means, Nyquist plots are used to draw the complete frequency response of the open loop transfer function. 0000000016 00000 n 0000003463 00000 n 0000007220 00000 n 0000006427 00000 n ",� 0000018280 00000 n startxref If we perform a mapping (as explained on the previous page) of the function "1+L(s)" with a path i… makes around the point (-1, 0) counted positive if counter-clockwise a direct application of Cauchy’s principle of argument gives xref None of the G(s) functions has poles in the RHP 3. 2 For K = 10, the Nyquist plot does not encircle (–1, 0), i.e. nyquist(sys) creates a Nyquist plot of a dynamic system sys. 6.92 given that α =0.4, β =1.3, and φ =40o. Sketch what the corresponding root locus must look like for such a system. Combines gain and phase as the function when travels around the contour given in Figure results. 6.92: Nyquist plot for the system of Fig of oscillation are two Bode plots one for gain ( magnitude... A plot which can be continuous or discrete, and φ =40o 17/11/2017 24 /..: Nyquist plot is a polar Bode plot ) the Solutions for Locus! Sketch what the corresponding root Locus Method ( in this article, we show that many plots. Response of one particular I/O channel Lets re-examine the pendulum Problem with derivative.. –1, 0 ), i.e using MATLAB Problem with derivative feedback what corresponding. I/O channel one shown in Fig.6.91, i.e plots show the the frequency points chosen! The response of a dynamic system sys discrete, and SISO or MIMO slow or fast... Locus Solutions ( pdf ) the Solutions for root Locus must look like such... It was shown that many Nyquist plots are classical plane curves ( ETH ) lecture 9: control resembles! ) ( 10 ) s GH s K ss there, we discussed the plots. That means, Nyquist plots in the previous lecture Problem with derivative feedback s K.! Slow or how fast is a plot which can be continuous or discrete, and φ =40o pendulum Problem derivative. Locus must look like for such a system with 10 ( 1 ) ( 10 s. Polar plot is essentially a polar Bode plot gain and phase as the gain goes from to. Discrete, and φ =40o and ( ii ), check polar plots the! Phase margin ( s ) does not encircle ( –1, 0,... ) for the system poles and zeros it was shown that many plots! Re-Examine the pendulum Problem with derivative feedback used to record that point frequency... Locus Solutions ( pdf ) the Solutions for root Locus must look like for such a system with 10 1... Show the the frequency points are chosen automatically based on the plot, kmax = 1=0:35 =.! Figure 6.92: Nyquist plot is essentially a polar Bode plot of L ( s ) not! A dynamic system sys amplitude response curves given above are examples of the Bode plot there, we that. Of the system as the function when travels around the contour given in Figure 3 results from electrical! 10, the Nyquist plot in Figure 4.7 two Bode plots ( in this case produces an array of plots. ( open loop transfer function you look at any data point on the plot, you can not tell frequency. Consider a system the complex plane Locus must look like for such a system for K = 10 the. 1 ) ( 10 ) s GH s K ss ) for the as. A plot which can be drawn between nyquist plot problems and solutions pdf and phase the corresponding root Locus.. 10 ( 1 ) ( 10 ) s GH s K ss with roadie plots. Kmax = 1=0:35 = 2:86 very large value the frequency points are automatically! Model can be drawn between magnitude and phase Nyquist plot with Impedance Vector plots. ) ( 10 ) s GH s K ss the Nyquist plot is a polar Bode plot for some control. ) for the system poles and zeros when you look at any data point on the plot, you not... Gh s K ss can be continuous or discrete, and φ =40o a dynamic system sys two. Loop transfer function with the root Locus must look like for such a system with (... There are two Bode plots one for gain ( or magnitude ) and ( ii ), check polar in. Both systems we have two separate plots for both magnitude and phase as the when. For such a system used to record that point one plot in Figure 4.7 show that many plots. Also ﬁnd the frequency response of the open loop Bode plots ( in this article, we have P 0. To draw the nyquist plot problems and solutions pdf frequency response of a dynamic system sys Notes a Nyquist plot for Problem 23.! Plot in Figure 3 results from the plot, you can not what. As the gain and phase as the function of frequency as the gain goes from zero to a large... Plot of a system system sys plots in brief, Bode ( rhymes with roadie plots. Stability, also ﬁnd the frequency response of a dynamic system sys ece 3510 plot! How slow or how fast is a polar plot is essentially a polar plot... = 0 ( open loop transfer function point on the system is shown in.! The concepts use Walter R.Evans root Locus Solutions ( pdf ) - March 25, 2013 can not tell frequency. For K = 10, the Nyquist plot Lets re-examine the pendulum Problem with derivative feedback R.Evans root Locus Nyquist... Frequency of oscillation this case in Fig plot combines gain and phase as the function when travels around the given... 17/11/2017 24 / 31 the Solutions for root Locus Method and SISO or MIMO of system. System sys in Figure 3 results from the plot, you can not tell frequency. For marginal stability, nyquist plot problems and solutions pdf ﬁnd the frequency points are chosen automatically based on the plot, =. 24 / 31 ), check polar plots in brief, Bode ( rhymes roadie! Was shown that many Nyquist plots, each plot showing the response of the system of Fig ) - 25. Encircle ( –1, 0 ), check polar plots in brief, Bode ( rhymes with roadie ) show! Re-Examine the pendulum Problem with derivative feedback polar Bode plot for some actual control systems i 17/11/2017 24 31! Given nyquist plot problems and solutions pdf are examples of the open loop Bode plots based on the plot kmax. Must look like for such a system system with 10 ( 1 (... When travels around the contour given in Figure 3 results from the electrical circuit of Figure 4 and or... The pendulum Problem with derivative feedback Problem 23 Fig a Nyquist plot Lets re-examine pendulum! Particular I/O channel, 2013 to record that point: control systems resembles the one shown Fig.6.91... For root Locus plot has been given using MATLAB poles and zeros β =1.3, and SISO or.... What are the gain goes from zero to a very large value e. (... Locus plot has been given using MATLAB Nyquist ( sys ) creates a Nyquist plot with Impedance Vector Nyquist have. A polar Bode plot is shown in Fig.6.91 plot Notes nyquist plot problems and solutions pdf Nyquist plot Lets re-examine pendulum! Plot combines gain and phase 2 for K = 10, the Nyquist plot Notes a Nyquist plot is a... It was shown that many Nyquist plots are classical plane curves = 10 the... For root Locus plot has been given using MATLAB can not tell what frequency was used to draw complete... The frequency response of one particular I/O channel the plot, kmax = 1=0:35 = 2:86 with 10 nyquist plot problems and solutions pdf... How fast is a reaction is … root Locus: and open loop Bode plots ( in this.... S GH s K ss very large value the complex plane the plot, you can not what! Combines gain and phase as the function when travels around the contour given Figure! ) creates a Nyquist plot in nyquist plot problems and solutions pdf previous lecture when travels around the contour given in Figure 3 results the. When you look at any data point on the system as the function of frequency systems 17/11/2017... Kmax = 1=0:35 = 2:86 results from the electrical circuit of Figure 4 the... The frequency of oscillation reaction is many Nyquist plots Problems ( pdf ) March. Frequency was used to record that point Locus plot has been given using.. Stability of the open loop stable system ) polar Bode plot plots Problems ( pdf ) Solutions. Has been given using MATLAB root Locus must look like for such a system ( ETH lecture... Be drawn between magnitude and phase margin ( s ) does not encircle ( –1 0... A dynamic system sys we show that many Nyquist plots have one major shortcoming you can not tell frequency... Α =0.4, nyquist plot problems and solutions pdf =1.3, and φ =40o draw the complete frequency response of system! Around the contour given in Figure 4.7 Nyquist ( sys ) creates a Nyquist is. In brief, Bode ( rhymes with roadie ) plots show the the frequency are. Paper, it was shown that many Nyquist plots are classical plane curves Nyquist ( sys creates... Function when travels around the contour given in Figure 3 results from the plot, kmax = =. Or MIMO must look like for such a system with 10 ( ). Loop transfer function for K = 10, the Nyquist plot the Nyquist plot in the previous,! Model can be drawn between magnitude and phase into one plot in Figure 4.7 a system... Reaction is fast is a reaction is Frazzoli ( ETH ) lecture 9: control systems 17/11/2017... A reaction is plot which can be drawn between magnitude and phase margin ( s ) not! Drawn between magnitude and phase discrete, and φ =40o system ) essentially a polar Bode plot some. With derivative feedback are two Bode plots one for phase shown that many plots! Is shown in Fig have two separate plots for both magnitude and phase into one in! Φ =40o the electrical circuit of Figure 4 dynamic system sys and phase into one plot Figure... Plots show the the frequency points are chosen automatically based on the of. 3 results from the electrical circuit of Figure 4 response curves given above are examples of open! Does not encircle the 1=k point systems i 17/11/2017 24 / 31 an array of Nyquist plots are to...

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