{"id":2008,"date":"2019-12-07T02:31:47","date_gmt":"2019-12-07T02:31:47","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=2008"},"modified":"2021-01-14T14:35:11","modified_gmt":"2021-01-14T14:35:11","slug":"10-5evans-root-locus-construction-rule-3-asymptotic-angles-and-centroid","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/10-5evans-root-locus-construction-rule-3-asymptotic-angles-and-centroid\/","title":{"raw":"10.5\tEvans Root Locus Construction Rule # 3: Asymptotic Angles and Centroid","rendered":"10.5\tEvans Root Locus Construction Rule # 3: Asymptotic Angles and Centroid"},"content":{"raw":"This Rule deals with the asymptotic angles and centroid location. If gain K is large enough, one can see that the branches of the RL travelling towards infinity follow a straight-line path that is asymptotic to a hypothetical line, called an asymptote<\/strong>, at a certain angle, called an asymptotic angle<\/strong>. If one extended these hypothetical lines, they would all intersect at an \"anchor\" point, called a centroid<\/strong>. Evans showed that the asymptotic angles and the centroid location can be computed as shown in this Rule.\r\n\r\nWhen the test point [latex]s^*[\/latex]\u00a0 is close to the open loop singularities (poles, zeros), angles for vectors drawn from the singularity towards the point [latex]s^*[\/latex] , which are used to evaluate [latex] G(s^*) [\/latex] function, are quite different. However, as the gain K tends to approach infinity, [latex] K \\rightarrow \\infty [\/latex], which is the descriptor for asymptotic condition, point [latex] s^* [\/latex] begins to practically lie on the asymptote, and these angles all begin to look alike and approach the asymptotic angle [latex] \\theta_i [\/latex]. <\/em>\r\n\r\nRecall that the total angle of the function [latex] G(s^*) [\/latex] is equal to:\r\n\r\n\r\n\r\n
[latex] \\angle G(s^*)=\\angle_{zeros}-\\angle_{poles} [\/latex]<\/td>\r\nEquation 10-7<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe total angles, respectively, for all vectors associated with poles and all vectors associated with zeros, will be equal to:\r\n\r\n\r\n\r\n\r\n
[latex] \\angle_{poles} = n\\cdot\\theta_i [\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[latex] \\angle_{zeros}=m\\cdot\\theta_i [\/latex]<\/td>\r\nEquation 10-8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf the test point [latex] s^* [\/latex] is to belong to the root locus, the angle of [latex] G(s^*) [\/latex] has to meet the angle criterion:\r\n\r\n\r\n\r\n\r\n
[latex] \\angle G(s^*)=180^{\\circ} [\/latex]<\/td>\r\n<\/td>\r\n<\/tr>\r\n
[latex] m\\cdot\\theta_i - n\\cdot\\theta_i=180^{\\circ} [\/latex]<\/td>\r\nEquation 10-9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the formula below, the sign in the denominator is reversed, because [latex] m\\leq n [\/latex]. This will have no effect on the formula, because [latex] +180^{\\circ}=-180^{\\circ} [\/latex]. <\/em>The asymptotes need to be anchored on the plot. To do that, a so-called root locus centroid is defined, as a \"centre of gravity\" of the plot.\r\n
\r\n\r\nRule 3:<\/strong> The asymptotes are centred on the Real axis at the centroid, described by this equation:\r\n\r\n\r\n\r\n
[latex] \\sigma = \\frac{\\sum poles - \\sum zeros}{n-m} [\/latex]<\/td>\r\nEquation 10-10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe branches of Root Locus that tend to infinity converge at asymptotic angles, described by this equation:\r\n\r\n\r\n\r\n
[latex] \\theta_i = \\frac{180^{\\circ}\\pm k\\cdot360^{\\circ}}{n-m} [\/latex]<\/td>\r\n[latex] k=0,1,...,(n-m-1) [\/latex]<\/td>\r\nEquation 10-11<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nAs an example, consider RL shown in Figure 10\u20114. Centroid and asymptotes are calculated as follows:\r\n

[latex] \\sigma = \\frac{-10-5-2}{3-0}=-5.67 [\/latex], [latex] \\theta_i = \\frac{180^{\\circ}\\pm k\\cdot360^{\\circ}}{3-0}=60^{\\circ},180^{\\circ},-60^{\\circ} [\/latex]<\/p>\r\nSee how this shows on the RL plot in Figure 10\u20116.\r\n\r\n[caption id=\"attachment_2009\" align=\"aligncenter\" width=\"681\"]\"Figure Figure 10\u20116 Example of Root Locus with Centroid, Asymptotic Angles and Break-Away Point[\/caption]","rendered":"

This Rule deals with the asymptotic angles and centroid location. If gain K is large enough, one can see that the branches of the RL travelling towards infinity follow a straight-line path that is asymptotic to a hypothetical line, called an asymptote<\/strong>, at a certain angle, called an asymptotic angle<\/strong>. If one extended these hypothetical lines, they would all intersect at an “anchor” point, called a centroid<\/strong>. Evans showed that the asymptotic angles and the centroid location can be computed as shown in this Rule.<\/p>\n

When the test point [latex]s^*[\/latex]\u00a0 is close to the open loop singularities (poles, zeros), angles for vectors drawn from the singularity towards the point [latex]s^*[\/latex] , which are used to evaluate [latex]G(s^*)[\/latex] function, are quite different. However, as the gain K tends to approach infinity, [latex]K \\rightarrow \\infty[\/latex], which is the descriptor for asymptotic condition, point [latex]s^*[\/latex] begins to practically lie on the asymptote, and these angles all begin to look alike and approach the asymptotic angle [latex]\\theta_i[\/latex]. <\/em><\/p>\n

Recall that the total angle of the function [latex]G(s^*)[\/latex] is equal to:<\/p>\n\n\n\n
[latex]\\angle G(s^*)=\\angle_{zeros}-\\angle_{poles}[\/latex]<\/td>\nEquation 10-7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The total angles, respectively, for all vectors associated with poles and all vectors associated with zeros, will be equal to:<\/p>\n\n\n\n\n
[latex]\\angle_{poles} = n\\cdot\\theta_i[\/latex]<\/td>\n<\/td>\n<\/tr>\n
[latex]\\angle_{zeros}=m\\cdot\\theta_i[\/latex]<\/td>\nEquation 10-8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

If the test point [latex]s^*[\/latex] is to belong to the root locus, the angle of [latex]G(s^*)[\/latex] has to meet the angle criterion:<\/p>\n\n\n\n\n
[latex]\\angle G(s^*)=180^{\\circ}[\/latex]<\/td>\n<\/td>\n<\/tr>\n
[latex]m\\cdot\\theta_i - n\\cdot\\theta_i=180^{\\circ}[\/latex]<\/td>\nEquation 10-9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

In the formula below, the sign in the denominator is reversed, because [latex]m\\leq n[\/latex]. This will have no effect on the formula, because [latex]+180^{\\circ}=-180^{\\circ}[\/latex]. <\/em>The asymptotes need to be anchored on the plot. To do that, a so-called root locus centroid is defined, as a “centre of gravity” of the plot.<\/p>\n

\n

Rule 3:<\/strong> The asymptotes are centred on the Real axis at the centroid, described by this equation:<\/p>\n\n\n\n
[latex]\\sigma = \\frac{\\sum poles - \\sum zeros}{n-m}[\/latex]<\/td>\nEquation 10-10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The branches of Root Locus that tend to infinity converge at asymptotic angles, described by this equation:<\/p>\n\n\n\n
[latex]\\theta_i = \\frac{180^{\\circ}\\pm k\\cdot360^{\\circ}}{n-m}[\/latex]<\/td>\n[latex]k=0,1,...,(n-m-1)[\/latex]<\/td>\nEquation 10-11<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

As an example, consider RL shown in Figure 10\u20114. Centroid and asymptotes are calculated as follows:<\/p>\n

[latex]\\sigma = \\frac{-10-5-2}{3-0}=-5.67[\/latex], [latex]\\theta_i = \\frac{180^{\\circ}\\pm k\\cdot360^{\\circ}}{3-0}=60^{\\circ},180^{\\circ},-60^{\\circ}[\/latex]<\/p>\n

See how this shows on the RL plot in Figure 10\u20116.<\/p>\n

\"Figure
Figure 10\u20116 Example of Root Locus with Centroid, Asymptotic Angles and Break-Away Point<\/figcaption><\/figure>\n","protected":false},"author":156,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2008","chapter","type-chapter","status-publish","hentry"],"part":1471,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2008","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":6,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2008\/revisions"}],"predecessor-version":[{"id":2695,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2008\/revisions\/2695"}],"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\/2008\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=2008"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=2008"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=2008"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=2008"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}