{"id":1990,"date":"2019-12-07T02:01:08","date_gmt":"2019-12-07T02:01:08","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&#038;p=1990"},"modified":"2021-01-14T14:25:48","modified_gmt":"2021-01-14T14:25:48","slug":"10-2evans-root-locus-construction-rules-introduction","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/10-2evans-root-locus-construction-rules-introduction\/","title":{"raw":"10.2\tEvans' Root Locus Construction Rules - Introduction","rendered":"10.2\tEvans&#8217; Root Locus Construction Rules &#8211; Introduction"},"content":{"raw":"To use analytic techniques to solve Equation 10\u20112 for [latex] 0&lt;K&lt;\\infty [\/latex], is time consuming, since we would have to tediously solve and plot the resulting magnitudes and phase angles of G(s) that satisfy the magnitude and phase criteria. Also, note that no formulae exist for roots of polynomials of order higher than third, and the roots for polynomials of higher orders have to be found using iterative numerical methods, such as Newton-Raphson.\r\n\r\nHowever, as it turns out, such analytical solutions are not necessary because Root Loci follow a set of simple construction rules that were formulated in 1948 by Walter R. Evans, who was working in the field of guidance and control of aircraft. These rules stem from Equation 10\u20112 that must be satisfied for every point on the Root Locus, and constitute an orderly process for sketching an approximation of the root loci for [latex] 0&lt;K&lt;\\infty [\/latex].\r\n\r\nEvans' Root Locus construction method utilizes the graphical evaluation of a function in the s-plane:\r\n<ul>\r\n \t<li>For any point in the s-plane, the open loop function [latex] G(s^*) [\/latex] can be evaluated;<\/li>\r\n \t<li>The angle criterion for\u00a0[latex] G(s^*) [\/latex] can be checked;<\/li>\r\n \t<li>If the angle criterion is met, the point [latex] s^* [\/latex] belongs to a system Root Locus.<\/li>\r\n<\/ul>\r\nConstruction rules developed by Evans dealt with:\r\n\r\n<strong>Rule 1: <\/strong>Beginning and end of Root Locus plot, symmetry;\r\n\r\n<strong>Rule 2: <\/strong>Points on the Real axis;\r\n\r\n<strong>Rule 3: <\/strong>Asymptotic angles and centroid;\r\n\r\n<strong>Rule 4: <\/strong>Break-away (break-in) points;\r\n\r\n<strong>Rule 5: <\/strong>Crossovers with Imaginary axis;\r\n\r\n<strong>Rule 6: <\/strong>Angles of departure (arrival) from\/to complex poles (zeros).\r\n\r\nEvans also established how to determine the Proportional Gain used in the closed loop system operation that corresponds to any particular point <span style=\"font-size: 1em\">[latex] s^* [\/latex]<\/span>\u00a0on the Root Locus by:\r\n<ul>\r\n \t<li>Preparing an accurate Root Locus plot<\/li>\r\n \t<li>Using the magnitude criterion to evaluate the gain at the point <span style=\"font-size: 1em\">[latex] s^* [\/latex]<\/span><\/li>\r\n<\/ul>\r\n<p style=\"text-align: center\">[latex] 1+K^*G(s^*)=0\\rightarrow K^*=\\frac{1}{|G(s^*)|} [\/latex]<\/p>\r\nNOTE: In Matlab, to plot Root Locus plots and to evaluate the gains on the plots we will use the <strong>rlocus.m<\/strong> and <strong>rlocfind.m<\/strong> subroutines. Figure 10\u20113 shows where the starting points of RL are, the crossover with the Imaginary axis, asymptotes, centroid and break-away point on RL.\r\n\r\n[caption id=\"attachment_1991\" align=\"aligncenter\" width=\"540\"]<img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_3.png\" alt=\"Figure 10\u20113 Components of a Root Locus Plot\" width=\"540\" height=\"400\" class=\"wp-image-1991 size-full\" \/> Figure 10\u20113 Components of a Root Locus Plot[\/caption]\r\n\r\nAs a MATLAB example, consider a unit feedback closed loop control system under Proportional Gain, where the process transfer function G(s) is as shown below. The Root Locus plot is obtained by MATLAB and shown in Figure 10\u20114.\r\n<p style=\"text-align: center\">[latex] G(s)=\\frac{10}{s^3+17s^2+80s+100}=\\frac{10}{(s+10)(s+5)(s+2)} [\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_1992\" align=\"aligncenter\" width=\"859\"]<img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_4.png\" alt=\"Figure 10\u20114 MATLAB Example of a Root Locus Plot\" width=\"859\" height=\"593\" class=\"wp-image-1992 size-full\" \/> Figure 10\u20114 MATLAB Example of a Root Locus Plot[\/caption]","rendered":"<p>To use analytic techniques to solve Equation 10\u20112 for [latex]0<K<\\infty[\/latex], is time consuming, since we would have to tediously solve and plot the resulting magnitudes and phase angles of G(s) that satisfy the magnitude and phase criteria. Also, note that no formulae exist for roots of polynomials of order higher than third, and the roots for polynomials of higher orders have to be found using iterative numerical methods, such as Newton-Raphson.\n\nHowever, as it turns out, such analytical solutions are not necessary because Root Loci follow a set of simple construction rules that were formulated in 1948 by Walter R. Evans, who was working in the field of guidance and control of aircraft. These rules stem from Equation 10\u20112 that must be satisfied for every point on the Root Locus, and constitute an orderly process for sketching an approximation of the root loci for [latex]0<K<\\infty[\/latex].\n\nEvans&#8217; Root Locus construction method utilizes the graphical evaluation of a function in the s-plane:\n\n\n<ul>\n<li>For any point in the s-plane, the open loop function [latex]G(s^*)[\/latex] can be evaluated;<\/li>\n<li>The angle criterion for\u00a0[latex]G(s^*)[\/latex] can be checked;<\/li>\n<li>If the angle criterion is met, the point [latex]s^*[\/latex] belongs to a system Root Locus.<\/li>\n<\/ul>\n<p>Construction rules developed by Evans dealt with:<\/p>\n<p><strong>Rule 1: <\/strong>Beginning and end of Root Locus plot, symmetry;<\/p>\n<p><strong>Rule 2: <\/strong>Points on the Real axis;<\/p>\n<p><strong>Rule 3: <\/strong>Asymptotic angles and centroid;<\/p>\n<p><strong>Rule 4: <\/strong>Break-away (break-in) points;<\/p>\n<p><strong>Rule 5: <\/strong>Crossovers with Imaginary axis;<\/p>\n<p><strong>Rule 6: <\/strong>Angles of departure (arrival) from\/to complex poles (zeros).<\/p>\n<p>Evans also established how to determine the Proportional Gain used in the closed loop system operation that corresponds to any particular point <span style=\"font-size: 1em\">[latex]s^*[\/latex]<\/span>\u00a0on the Root Locus by:<\/p>\n<ul>\n<li>Preparing an accurate Root Locus plot<\/li>\n<li>Using the magnitude criterion to evaluate the gain at the point <span style=\"font-size: 1em\">[latex]s^*[\/latex]<\/span><\/li>\n<\/ul>\n<p style=\"text-align: center\">[latex]1+K^*G(s^*)=0\\rightarrow K^*=\\frac{1}{|G(s^*)|}[\/latex]<\/p>\n<p>NOTE: In Matlab, to plot Root Locus plots and to evaluate the gains on the plots we will use the <strong>rlocus.m<\/strong> and <strong>rlocfind.m<\/strong> subroutines. Figure 10\u20113 shows where the starting points of RL are, the crossover with the Imaginary axis, asymptotes, centroid and break-away point on RL.<\/p>\n<figure id=\"attachment_1991\" aria-describedby=\"caption-attachment-1991\" style=\"width: 540px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_3.png\" alt=\"Figure 10\u20113 Components of a Root Locus Plot\" width=\"540\" height=\"400\" class=\"wp-image-1991 size-full\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_3.png 540w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_3-300x222.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_3-65x48.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_3-225x167.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_3-350x259.png 350w\" sizes=\"auto, (max-width: 540px) 100vw, 540px\" \/><figcaption id=\"caption-attachment-1991\" class=\"wp-caption-text\">Figure 10\u20113 Components of a Root Locus Plot<\/figcaption><\/figure>\n<p>As a MATLAB example, consider a unit feedback closed loop control system under Proportional Gain, where the process transfer function G(s) is as shown below. The Root Locus plot is obtained by MATLAB and shown in Figure 10\u20114.<\/p>\n<p style=\"text-align: center\">[latex]G(s)=\\frac{10}{s^3+17s^2+80s+100}=\\frac{10}{(s+10)(s+5)(s+2)}[\/latex]<\/p>\n<figure id=\"attachment_1992\" aria-describedby=\"caption-attachment-1992\" style=\"width: 859px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_4.png\" alt=\"Figure 10\u20114 MATLAB Example of a Root Locus Plot\" width=\"859\" height=\"593\" class=\"wp-image-1992 size-full\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_4.png 859w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_4-300x207.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_4-768x530.png 768w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_4-65x45.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_4-225x155.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_4-350x242.png 350w\" sizes=\"auto, (max-width: 859px) 100vw, 859px\" \/><figcaption id=\"caption-attachment-1992\" class=\"wp-caption-text\">Figure 10\u20114 MATLAB Example of a Root Locus Plot<\/figcaption><\/figure>\n","protected":false},"author":156,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1990","chapter","type-chapter","status-publish","hentry"],"part":1471,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1990","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/users\/156"}],"version-history":[{"count":4,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1990\/revisions"}],"predecessor-version":[{"id":2688,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1990\/revisions\/2688"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/parts\/1471"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1990\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=1990"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=1990"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=1990"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=1990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}