{"id":2107,"date":"2019-12-16T02:29:27","date_gmt":"2019-12-16T02:29:27","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=2107"},"modified":"2021-01-14T16:23:50","modified_gmt":"2021-01-14T16:23:50","slug":"14-4-1-gain-margin-from-polar-plot","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/14-4-1-gain-margin-from-polar-plot\/","title":{"raw":"14.4 Gain and Phase Margins vs. Polar Plots","rendered":"14.4 Gain and Phase Margins vs. Polar Plots"},"content":{"raw":"14.4.1 Example Gain Margin from Polar Plot<\/b>\r\n\r\nLet the crossover frequency be defined as [latex] \\omega_{cg}[\/latex], the frequency at which the phase plot crosses over the [latex]-180^ \\circ[\/latex] line. On the polar plot, this corresponds to the plot crossing the negative part of Real axis. Remember the definition of Gain Margin [latex] G_{m} [\/latex] :\r\n\r\n\r\n\r\n
\r\n

[latex]G_m = \\frac{K_{crit}}{K}[\/latex]<\/p>\r\n

[latex]G_m>1[\/latex] system stable<\/p>\r\n

[latex]G_m<1[\/latex] system unstable<\/p>\r\n<\/td>\r\n

Equation 14-6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
\r\n

[latex]G_m = \\frac{1}{ \\mid A \\mid}[\/latex]<\/p>\r\n<\/td>\r\n

Equation 14-7<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOn the polar plot, Gain Margin [latex]G_m[\/latex] can be found as an inverse of the coordinate A of the polar plot crossover with the Real axis, as shown next. If the crossover is to the right of (-1, j0) point,[latex] \\mid A \\mid < 1[\/latex] ,[latex] G_{m}>1[\/latex] , and the system is stable. If the crossover is to the left of (-1, j0) point,[latex] \\mid A \\mid>1[\/latex] ,[latex]G_m[\/latex] [latex]<1[\/latex] , and the system is unstable.\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n14.4.2 Example Phase Margin from Polar Plot<\/b>\r\n\r\nThe crossover frequency defined as [latex] \\omega_{cp}[\/latex], is the frequency at which the polar plot crosses over the unit circle ([latex]0[\/latex] dB = 1 Volt\/Volt). Phase Margin [latex]\\Phi_m[\/latex] is defined as [latex] \\Phi_m=180 ^ \\circ + \\angle GH( \\omega_{cp} ). [\/latex] Therefore, on the polar plot Phase Margin [latex] \\Phi_m[\/latex] can be found as the angle between the Real axis and the crossover of the polar plot with the unit circle, as shown in Error! Reference source not found.<\/strong> If this angle is above the Real axis, the system is unstable, if this angle is below the Real axis, the system is stable.\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n14.4.3 Solved Example<\/b>\r\n\r\nConsider a unit feedback closed loop system where the open loop transfer function [latex]G(s)[\/latex] is known to be unstable and its transfer function [latex]G(s)[\/latex] is known as [latex]G(s) = \\frac{s+2}{s(s-2)} .[\/latex] Such system can be stabilized by using an appropriately large value of the controller gain. We need to establish the critical gain [latex]K_{crit}[\/latex] .\r\n\r\nSolution Part 1: <\/strong>Let's tackle this problem in s-domain. The system closed loop characteristic equation is:\r\n

[latex]1+KG(s)=0[\/latex]<\/p>\r\n

[latex] 1+K \\frac{s+2}{s(s-2)} =0[\/latex]<\/p>\r\n

[latex]s^2-2s+Ks+2K=0[\/latex]<\/p>\r\n

[latex]s^2+(K-2)s+2K=0[\/latex]<\/p>\r\nFor the 2nd<\/sup> order system the necessary and sufficient condition of stability is that all coefficients of the characteristic polynomial are positive:\r\n

[latex]K-2>0[\/latex]<\/p>\r\n

[latex]K>0[\/latex]<\/p>\r\nThe critical value of the gain is [latex]K_{crit}=2[\/latex] and the range of gains for stable system performance is:\r\n

[latex]2<K< \\infty[\/latex]<\/p>\r\nThe frequency of oscillations [latex] \\omega_{osc}[\/latex] at the critical gain is equal to [latex]2[\/latex] rad\/s:\r\n

[latex]s^2+(K-2)s+2K=0[\/latex]<\/p>\r\n

[latex]K_{crit}=2[\/latex]<\/p>\r\n

[latex]s^2+4=0[\/latex]<\/p>\r\n

[latex]s= \\pm j2[\/latex]<\/p>\r\n

[latex] \\omega_{osc} =2[\/latex]<\/p>\r\nThe upper limit of the gain range will be determined by practical issues such as saturation.\r\n\r\nSolution Part 2: <\/strong>Now let's try to apply the Gain Margin and Phase Margin definitions to this system. The open loop frequency response has to be simulated as the system is open-loop unstable and no measurements on the open loop are possible. From the plot shown in Error! Reference source not found.<\/strong> we read:\r\n

[latex]G_m=+6dB=2[\/latex]<\/p>\r\n

[latex] \\omega_{cg}=2[\/latex]<\/p>\r\n \r\n

\"\"<\/p>\r\n \r\n\r\nThe positive Gain Margin [latex]G_m= 6 dB = 2[\/latex] Volt\/Volt is measured at the crossover frequency [latex] \\omega_{cp} [\/latex] = 2 rad\/s. This would have to be interpreted as the system being stable for gains lower <\/strong>than 2, which as we know from the s-domain analysis is not correct. On the other hand, the Phase Margin [latex] \\Phi_m[\/latex] is negative, indicating the system is unstable for gains [latex]< 2[\/latex]. This is an example of when the Gain and Phase Margin definitions cannot be applied consistently to determine the system stability limits. A new, more general frequency domain based stability criterion will now be defined.","rendered":"

14.4.1 Example Gain Margin from Polar Plot<\/b><\/p>\n

Let the crossover frequency be defined as [latex]\\omega_{cg}[\/latex], the frequency at which the phase plot crosses over the [latex]-180^ \\circ[\/latex] line. On the polar plot, this corresponds to the plot crossing the negative part of Real axis. Remember the definition of Gain Margin [latex]G_{m}[\/latex] :<\/p>\n\n\n\n
\n

[latex]G_m = \\frac{K_{crit}}{K}[\/latex]<\/p>\n

[latex]G_m>1[\/latex] system stable<\/p>\n

[latex]G_m<1[\/latex] system unstable<\/p>\n<\/td>\n

Equation 14-6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
\n

[latex]G_m = \\frac{1}{ \\mid A \\mid}[\/latex]<\/p>\n<\/td>\n

Equation 14-7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

On the polar plot, Gain Margin [latex]G_m[\/latex] can be found as an inverse of the coordinate A of the polar plot crossover with the Real axis, as shown next. If the crossover is to the right of (-1, j0) point,[latex]\\mid A \\mid < 1[\/latex] ,[latex]G_{m}>1[\/latex] , and the system is stable. If the crossover is to the left of (-1, j0) point,[latex]\\mid A \\mid>1[\/latex] ,[latex]G_m[\/latex] [latex]<1[\/latex] , and the system is unstable.\n\n \n\n\"\"<\/p>\n

 <\/p>\n

14.4.2 Example Phase Margin from Polar Plot<\/b><\/p>\n

The crossover frequency defined as [latex]\\omega_{cp}[\/latex], is the frequency at which the polar plot crosses over the unit circle ([latex]0[\/latex] dB = 1 Volt\/Volt). Phase Margin [latex]\\Phi_m[\/latex] is defined as [latex]\\Phi_m=180 ^ \\circ + \\angle GH( \\omega_{cp} ).[\/latex] Therefore, on the polar plot Phase Margin [latex]\\Phi_m[\/latex] can be found as the angle between the Real axis and the crossover of the polar plot with the unit circle, as shown in Error! Reference source not found.<\/strong> If this angle is above the Real axis, the system is unstable, if this angle is below the Real axis, the system is stable.<\/p>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

14.4.3 Solved Example<\/b><\/p>\n

Consider a unit feedback closed loop system where the open loop transfer function [latex]G(s)[\/latex] is known to be unstable and its transfer function [latex]G(s)[\/latex] is known as [latex]G(s) = \\frac{s+2}{s(s-2)} .[\/latex] Such system can be stabilized by using an appropriately large value of the controller gain. We need to establish the critical gain [latex]K_{crit}[\/latex] .<\/p>\n

Solution Part 1: <\/strong>Let’s tackle this problem in s-domain. The system closed loop characteristic equation is:<\/p>\n

[latex]1+KG(s)=0[\/latex]<\/p>\n

[latex]1+K \\frac{s+2}{s(s-2)} =0[\/latex]<\/p>\n

[latex]s^2-2s+Ks+2K=0[\/latex]<\/p>\n

[latex]s^2+(K-2)s+2K=0[\/latex]<\/p>\n

For the 2nd<\/sup> order system the necessary and sufficient condition of stability is that all coefficients of the characteristic polynomial are positive:<\/p>\n

[latex]K-2>0[\/latex]<\/p>\n

[latex]K>0[\/latex]<\/p>\n

The critical value of the gain is [latex]K_{crit}=2[\/latex] and the range of gains for stable system performance is:<\/p>\n

[latex]2\n

The frequency of oscillations [latex]\\omega_{osc}[\/latex] at the critical gain is equal to [latex]2[\/latex] rad\/s:<\/p>\n

[latex]s^2+(K-2)s+2K=0[\/latex]<\/p>\n

[latex]K_{crit}=2[\/latex]<\/p>\n

[latex]s^2+4=0[\/latex]<\/p>\n

[latex]s= \\pm j2[\/latex]<\/p>\n

[latex]\\omega_{osc} =2[\/latex]<\/p>\n

The upper limit of the gain range will be determined by practical issues such as saturation.<\/p>\n

Solution Part 2: <\/strong>Now let’s try to apply the Gain Margin and Phase Margin definitions to this system. The open loop frequency response has to be simulated as the system is open-loop unstable and no measurements on the open loop are possible. From the plot shown in Error! Reference source not found.<\/strong> we read:<\/p>\n

[latex]G_m=+6dB=2[\/latex]<\/p>\n

[latex]\\omega_{cg}=2[\/latex]<\/p>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

The positive Gain Margin [latex]G_m= 6 dB = 2[\/latex] Volt\/Volt is measured at the crossover frequency [latex]\\omega_{cp}[\/latex] = 2 rad\/s. This would have to be interpreted as the system being stable for gains lower <\/strong>than 2, which as we know from the s-domain analysis is not correct. On the other hand, the Phase Margin [latex]\\Phi_m[\/latex] is negative, indicating the system is unstable for gains [latex]< 2[\/latex]. This is an example of when the Gain and Phase Margin definitions cannot be applied consistently to determine the system stability limits. A new, more general frequency domain based stability criterion will now be defined.\n<\/p>\n","protected":false},"author":162,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2107","chapter","type-chapter","status-publish","hentry"],"part":2068,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2107","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\/162"}],"version-history":[{"count":18,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2107\/revisions"}],"predecessor-version":[{"id":2735,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2107\/revisions\/2735"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/parts\/2068"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2107\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=2107"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=2107"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=2107"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=2107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}