{"id":1512,"date":"2019-12-02T22:33:21","date_gmt":"2019-12-02T22:33:21","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=1512"},"modified":"2021-01-14T16:12:59","modified_gmt":"2021-01-14T16:12:59","slug":"13-6lead-lag-controller","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/13-6lead-lag-controller\/","title":{"raw":"13.6\tLead-Lag Controller","rendered":"13.6\tLead-Lag Controller"},"content":{"raw":"Lead-Lag Control combines the benefits of both the Lead and the Lag Controllers. The transfer function of the Lead-Lag Controller is as follows:\r\n\r\n\r\n\r\n
[latex] G_c(s) = K_c \\cdot \\frac{s\\tau_1 +1}{s\\alpha_1\\tau_1+1}\\cdot\\frac{s\\alpha_2\\tau_s +1}{s\\tau_2+1} [\/latex]<\/td>\r\nEquation 13-27<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nKc corresponds to the DC gain of the controller and both [latex] \\alpha_1 < 1 [\/latex], [latex] \\alpha_2 <1 [\/latex]. There are two zeros, at [latex] s_1 = -\\frac{1}{\\tau_1} [\/latex], and [latex] s_2 = -\\frac{1}{\\alpha_2\\tau_2} [\/latex], and two poles, at [latex] s_3 = -\\frac{1}{\\alpha_1\\tau_1} [\/latex] and [latex] s_4 = -\\frac{1}{\\tau_2} [\/latex]. Note that what makes this compensator \"tick,\" is its sequence: POLE-ZERO-ZERO-POLE, as shown in Figure 13\u201129. In the frequency domain, the four corner frequencies are:\r\n

[latex] \\omega_1 = \\frac{1}{\\tau_1} [\/latex], [latex] \\omega_2 = \\frac{1}{\\alpha_2\\tau_2} [\/latex], [latex] \\omega_3 = \\frac{1}{\\alpha_1\\tau_1} [\/latex], [latex] \\omega_4 = \\frac{1}{\\tau_2} [\/latex]<\/p>\r\nA frequency response plot of the lead-lag compensator is shown in Figure 13\u201130. Again, note the sequence: POLE-ZERO-ZERO-POLE, as shown in Figure 13\u201129. This structure is sometimes also referred to as the Lag-Lead Controller - the Lag block comes first on the frequency plot, followed by the Lead block as Figure 13\u201130 shows. We will however use the name Lead-Lag Controller, based on the sequence in which its components are used in the design - the Lead component is used first, then the Lag component.\r\n\r\n[caption id=\"attachment_1682\" align=\"aligncenter\" width=\"540\"]\"Figure Figure 13\u201129: Pole Zero Map for Lead-Lag Controller[\/caption]\r\n\r\n[caption id=\"attachment_1683\" align=\"aligncenter\" width=\"510\"]\"Figure Figure 13\u201130: Frequency Response Plots for Lead-Lag Controller[\/caption]\r\n\r\n13.6.1 Simplified Lead-Lag Controller Design<\/span>\r\n\r\nFigure 13\u201131 shows the values significant for the design procedure which is as follows: choose the compensator gain [latex] K_c [\/latex], based on the steady state error requirements for the closed loop operation. Re-plot the open loop frequency response, including the required \"gain lift\":\r\n\r\n\r\n\r\n
[latex] G_{open}(j\\omega)=K_cG(j\\omega)H(j\\omega) [\/latex]<\/td>\r\nEquation 13-28<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAssume the necessary phase margin [latex] \\Phi_m [\/latex], based on the required Percent Overshoot. Determine the crossover frequency [latex] \\omega_{cp} [\/latex], from the settling time requirement.\r\n\r\nDetermine the necessary phase lead lift [latex] \\theta [\/latex] at this frequency (add an extra 5 degrees, since the Lag Controller block will be used):\r\n\r\n\r\n\r\n
[latex] \\theta= \\phi_{max} = -180^{\\circ} + \\Phi_m + 5^{\\circ}-\\angle GH(\\omega_{cp}) [\/latex]<\/td>\r\nEquation 13-29<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_1684\" align=\"aligncenter\" width=\"540\"]\"Figure Figure 13\u201131: How to Use Lead-Lag Compensator in Simplified Design[\/caption]\r\n\r\nCalculate the Lead parameter [latex] \\alpha_1 [\/latex]:\r\n\r\n\r\n\r\n
[latex] \\alpha_1 = \\frac{1-\\sin{\\phi_{max}}}{1+\\sin{\\phi_{max}}} [\/latex]<\/td>\r\nEquation 13-30<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCalculate the Lead time constant [latex] \\tau_1 [\/latex] from:\r\n\r\n\r\n\r\n
[latex] \\omega_{cp}=\\omega_0=\\frac{1}{\\sqrt{\\alpha_1}\\tau_1} [\/latex]<\/td>\r\nEquation 13-31<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCalculate (or measure from the plot) the total open loop gain at the crossover frequency:\r\n\r\n\r\n\r\n
[latex] M_{open}(j\\omega_{cp})=\\left | G(j\\omega_{cp})H(j\\omega_{cp}) \\right | \\cdot K_c\\cdot\\frac{1}{\\sqrt{\\alpha_1}} [\/latex]<\/td>\r\nEquation 13-32<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCalculate the Lag parameter [latex] \\alpha_2 [\/latex] from a necessary gain reduction at this frequency:\r\n\r\n\r\n\r\n
[latex] \\alpha_2 = \\frac{1}{M_{open}} [\/latex]<\/td>\r\nEquation 13-33<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCalculate the Lag time constant [latex] \\tau_2 [\/latex] from:\r\n\r\n\r\n\r\n
[latex] \\omega_{cp}=\\frac{10}{\\alpha_2\\tau_2} [\/latex]<\/td>\r\nEquation 13-34<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nComment:<\/strong>\r\n\r\nThis design theoretically meets all three typical performance requirements - accuracy, speed, and lack of oscillations. Whether it will work well, depends on how closely the compensated closed loop transfer function resembles our standard second order under-damped model, on which the design was based. Always run simulations of the closed loop system response under this compensation scheme - the design may require iterations to improve its performance.","rendered":"

Lead-Lag Control combines the benefits of both the Lead and the Lag Controllers. The transfer function of the Lead-Lag Controller is as follows:<\/p>\n\n\n\n
[latex]G_c(s) = K_c \\cdot \\frac{s\\tau_1 +1}{s\\alpha_1\\tau_1+1}\\cdot\\frac{s\\alpha_2\\tau_s +1}{s\\tau_2+1}[\/latex]<\/td>\nEquation 13-27<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Kc corresponds to the DC gain of the controller and both [latex]\\alpha_1 < 1[\/latex], [latex]\\alpha_2 <1[\/latex]. There are two zeros, at [latex]s_1 = -\\frac{1}{\\tau_1}[\/latex], and [latex]s_2 = -\\frac{1}{\\alpha_2\\tau_2}[\/latex], and two poles, at [latex]s_3 = -\\frac{1}{\\alpha_1\\tau_1}[\/latex] and [latex]s_4 = -\\frac{1}{\\tau_2}[\/latex]. Note that what makes this compensator “tick,” is its sequence: POLE-ZERO-ZERO-POLE, as shown in Figure 13\u201129. In the frequency domain, the four corner frequencies are:\n\n\n

[latex]\\omega_1 = \\frac{1}{\\tau_1}[\/latex], [latex]\\omega_2 = \\frac{1}{\\alpha_2\\tau_2}[\/latex], [latex]\\omega_3 = \\frac{1}{\\alpha_1\\tau_1}[\/latex], [latex]\\omega_4 = \\frac{1}{\\tau_2}[\/latex]<\/p>\n

A frequency response plot of the lead-lag compensator is shown in Figure 13\u201130. Again, note the sequence: POLE-ZERO-ZERO-POLE, as shown in Figure 13\u201129. This structure is sometimes also referred to as the Lag-Lead Controller – the Lag block comes first on the frequency plot, followed by the Lead block as Figure 13\u201130 shows. We will however use the name Lead-Lag Controller, based on the sequence in which its components are used in the design – the Lead component is used first, then the Lag component.<\/p>\n

\"Figure
Figure 13\u201129: Pole Zero Map for Lead-Lag Controller<\/figcaption><\/figure>\n
\"Figure
Figure 13\u201130: Frequency Response Plots for Lead-Lag Controller<\/figcaption><\/figure>\n

13.6.1 Simplified Lead-Lag Controller Design<\/span><\/p>\n

Figure 13\u201131 shows the values significant for the design procedure which is as follows: choose the compensator gain [latex]K_c[\/latex], based on the steady state error requirements for the closed loop operation. Re-plot the open loop frequency response, including the required “gain lift”:<\/p>\n\n\n\n
[latex]G_{open}(j\\omega)=K_cG(j\\omega)H(j\\omega)[\/latex]<\/td>\nEquation 13-28<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Assume the necessary phase margin [latex]\\Phi_m[\/latex], based on the required Percent Overshoot. Determine the crossover frequency [latex]\\omega_{cp}[\/latex], from the settling time requirement.<\/p>\n

Determine the necessary phase lead lift [latex]\\theta[\/latex] at this frequency (add an extra 5 degrees, since the Lag Controller block will be used):<\/p>\n\n\n\n
[latex]\\theta= \\phi_{max} = -180^{\\circ} + \\Phi_m + 5^{\\circ}-\\angle GH(\\omega_{cp})[\/latex]<\/td>\nEquation 13-29<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\"Figure
Figure 13\u201131: How to Use Lead-Lag Compensator in Simplified Design<\/figcaption><\/figure>\n

Calculate the Lead parameter [latex]\\alpha_1[\/latex]:<\/p>\n\n\n\n
[latex]\\alpha_1 = \\frac{1-\\sin{\\phi_{max}}}{1+\\sin{\\phi_{max}}}[\/latex]<\/td>\nEquation 13-30<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Calculate the Lead time constant [latex]\\tau_1[\/latex] from:<\/p>\n\n\n\n
[latex]\\omega_{cp}=\\omega_0=\\frac{1}{\\sqrt{\\alpha_1}\\tau_1}[\/latex]<\/td>\nEquation 13-31<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Calculate (or measure from the plot) the total open loop gain at the crossover frequency:<\/p>\n\n\n\n
[latex]M_{open}(j\\omega_{cp})=\\left | G(j\\omega_{cp})H(j\\omega_{cp}) \\right | \\cdot K_c\\cdot\\frac{1}{\\sqrt{\\alpha_1}}[\/latex]<\/td>\nEquation 13-32<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Calculate the Lag parameter [latex]\\alpha_2[\/latex] from a necessary gain reduction at this frequency:<\/p>\n\n\n\n
[latex]\\alpha_2 = \\frac{1}{M_{open}}[\/latex]<\/td>\nEquation 13-33<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Calculate the Lag time constant [latex]\\tau_2[\/latex] from:<\/p>\n\n\n\n
[latex]\\omega_{cp}=\\frac{10}{\\alpha_2\\tau_2}[\/latex]<\/td>\nEquation 13-34<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Comment:<\/strong><\/p>\n

This design theoretically meets all three typical performance requirements – accuracy, speed, and lack of oscillations. Whether it will work well, depends on how closely the compensated closed loop transfer function resembles our standard second order under-damped model, on which the design was based. Always run simulations of the closed loop system response under this compensation scheme – the design may require iterations to improve its performance.<\/p>\n","protected":false},"author":156,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1512","chapter","type-chapter","status-publish","hentry"],"part":1043,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1512","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":10,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1512\/revisions"}],"predecessor-version":[{"id":2726,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1512\/revisions\/2726"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/parts\/1043"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1512\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=1512"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=1512"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=1512"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=1512"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}