{"id":434,"date":"2019-05-22T19:36:52","date_gmt":"2019-05-22T19:36:52","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=434"},"modified":"2021-01-12T18:42:52","modified_gmt":"2021-01-12T18:42:52","slug":"2-4determining-stable-range-for-proportional-controller-operations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/2-4determining-stable-range-for-proportional-controller-operations\/","title":{"raw":"2.4\tDetermining Stable Range for Proportional Controller Operations","rendered":"2.4\tDetermining Stable Range for Proportional Controller Operations"},"content":{"raw":"Consider a closed loop system under Proportional Control in Figure 2\u20112:\r\n\r\n[caption id=\"attachment_436\" align=\"aligncenter\" width=\"750\"]\"Figure Figure 2-2 Non-unit Feedback Closed Loop System under Proportional Control[\/caption]\r\n\r\nThe closed loop transfer function and the system Characteristic Equation are:\r\n\r\n\r\n\r\n
[latex]G_{cl}(s) = \\frac{K_{prop}G(s)}{1 +K_{prop}G(s)\\cdot H(s)}[\/latex]\r\n\r\n[latex]1+ K_{prop}G(s)\\cdot H(s) = 0[\/latex]<\/td>\r\n\r\n

Equation 2\u20115<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe variable parameter - the Proportional Controller Gain - is now a part of the system Characteristic Equation and will feature in the Routh Array. This allows us to define the required condition for a stable, safe, range of Controller operations.\r\n

\u00a0<\/strong>Example<\/strong><\/h3>\r\nConsider the following control system:\r\n

\"\"<\/p>\r\n\r\n

    \r\n \t
  • Is this system open-loop stable?<\/li>\r\n \t
  • Determine a range of gains K required for a stable operation of this closed loop system.<\/li>\r\n<\/ul>\r\nSolution:<\/strong>\r\n

    Part 1: Open Loop<\/strong><\/h4>\r\nOpen loop characteristic equation is:\r\n\r\n[latex]Q(s) = s^{3}+ 11s^{2}+ 38s+ 4 = 0[\/latex]\r\n

    \"\"<\/p>\r\nThe system is open-loop stable, based on the Routh-Hurwitz Stability Criterion. We can also find numerically (MATLAB), what the poles of the open system are: -5.45 +j2.68, -5.45 - j2.68, -0.11.\r\n\r\nThe system Open Loop Transfer Function components are:\r\n

    [latex]G(s)=\\frac{4K_{prop}}{s^{3} +11s^{2}+ 38s+ 4},\u00a0 H(s) = 1[\/latex]<\/p>\r\n\r\n

    Part 2: Closed Loop<\/strong><\/h3>\r\nThe system Closed Loop Transfer Function is:\r\n

    [latex]G_{cl} = \\frac{G(s)}{1 +G(s)H(s)}=\\frac{\\frac{4K_{prop}}{s^{3}+ 11s^{2}+ 38s +4}}{1 +\\frac{4k_{prop}}{s^{3} +11s^{2}+ 38s+ 4}} = \\frac{4K_{prop}}{s^{3}+ 11s^{2} +38s +(4K_{prop} +4)}[\/latex]<\/p>\r\nApply the Routh-Hurwitz criterion to the closed loop characteristic equation:\r\n

    [latex]s^{3} +11s^{2}+ 38s +(4K_{prop}+ 4) = 0[\/latex]<\/p>\r\nThe Necessary Condition is:\r\n

    [latex]4 +4K_{prop} > 0,\u00a0 \u00a0K_{prop} >-1[\/latex]<\/p>\r\nThe Sufficient Conditions from Routh Array:\r\n

    \"\"<\/p>\r\nThe resulting conditions are:\r\n

    [latex]\\frac{11\\cdot 38 - (4+ 4K_{prop})}{11} > 0,\u00a0 4K_{prop} < 414,\u00a0 K_{prop} < 103.5 [\/latex]<\/p>\r\nThe total condition is: [latex]-1 < K_{prop} < 103.5[\/latex]\r\n

    \"\"<\/p>\r\nNote that the practical range of stable controller operations, as opposed to the previous purely mathematical condition, is [latex]0<K_{prop}<103.5[\/latex] remember that we do not want to run systems with a negative gain!\r\n\r\nExtreme values of the determined range are called Critical Gain <\/em>values. Again, of practical interest is only the upper, positive Critical gain value. What happens when the Controller gain reaches that Critical gain value?\r\n\r\nWhen [latex]K_{prop} = 103.5[\/latex] :\r\n

    \"<\/p>\r\n[latex]Q_{aux}(s) = 11s^{2} +418[\/latex]\r\n\r\n[latex]Q_{aux}(s) = 0[\/latex]\r\n\r\n[latex]s^{2} +38 = 0[\/latex]\r\n\r\n[latex]s_{1} = j\\sqrt{38} = j6.16[\/latex]\r\n\r\n[latex]s_{2} = -j\\sqrt{38} = -j6.16[\/latex]\r\n\r\nAs the pole-zero map illustrates, the location of marginally stable poles on the Imaginary axis corresponds to the frequency of sustained oscillations, \\[latex]\\omega_{crit} = 6.16 rad\/sec[\/latex]. This corresponds to the period of oscillations equal to 1.02 seconds. Step response of the system when [latex]K_{prop}=103.5[\/latex] is also shown.\r\n

    \"\"<\/p>\r\n

    \"\"<\/p>\r\n ","rendered":"

    Consider a closed loop system under Proportional Control in Figure 2\u20112:<\/p>\n

    \"Figure
    Figure 2-2 Non-unit Feedback Closed Loop System under Proportional Control<\/figcaption><\/figure>\n

    The closed loop transfer function and the system Characteristic Equation are:<\/p>\n\n\n\n
    [latex]G_{cl}(s) = \\frac{K_{prop}G(s)}{1 +K_{prop}G(s)\\cdot H(s)}[\/latex]<\/p>\n

    [latex]1+ K_{prop}G(s)\\cdot H(s) = 0[\/latex]<\/td>\n

    		\n

    Equation 2\u20115<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    The variable parameter – the Proportional Controller Gain – is now a part of the system Characteristic Equation and will feature in the Routh Array. This allows us to define the required condition for a stable, safe, range of Controller operations.<\/p>\n

    \u00a0<\/strong>Example<\/strong><\/h3>\n

    Consider the following control system:<\/p>\n

    \"\"<\/p>\n

      \n
    • Is this system open-loop stable?<\/li>\n
    • Determine a range of gains K required for a stable operation of this closed loop system.<\/li>\n<\/ul>\n

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

      Part 1: Open Loop<\/strong><\/h4>\n

      Open loop characteristic equation is:<\/p>\n

      [latex]Q(s) = s^{3}+ 11s^{2}+ 38s+ 4 = 0[\/latex]<\/p>\n

      \"\"<\/p>\n

      The system is open-loop stable, based on the Routh-Hurwitz Stability Criterion. We can also find numerically (MATLAB), what the poles of the open system are: -5.45 +j2.68, -5.45 – j2.68, -0.11.<\/p>\n

      The system Open Loop Transfer Function components are:<\/p>\n

      [latex]G(s)=\\frac{4K_{prop}}{s^{3} +11s^{2}+ 38s+ 4},\u00a0 H(s) = 1[\/latex]<\/p>\n

      Part 2: Closed Loop<\/strong><\/h3>\n

      The system Closed Loop Transfer Function is:<\/p>\n

      [latex]G_{cl} = \\frac{G(s)}{1 +G(s)H(s)}=\\frac{\\frac{4K_{prop}}{s^{3}+ 11s^{2}+ 38s +4}}{1 +\\frac{4k_{prop}}{s^{3} +11s^{2}+ 38s+ 4}} = \\frac{4K_{prop}}{s^{3}+ 11s^{2} +38s +(4K_{prop} +4)}[\/latex]<\/p>\n

      Apply the Routh-Hurwitz criterion to the closed loop characteristic equation:<\/p>\n

      [latex]s^{3} +11s^{2}+ 38s +(4K_{prop}+ 4) = 0[\/latex]<\/p>\n

      The Necessary Condition is:<\/p>\n

      [latex]4 +4K_{prop} > 0,\u00a0 \u00a0K_{prop} >-1[\/latex]<\/p>\n

      The Sufficient Conditions from Routh Array:<\/p>\n

      \"\"<\/p>\n

      The resulting conditions are:<\/p>\n

      [latex]\\frac{11\\cdot 38 - (4+ 4K_{prop})}{11} > 0,\u00a0 4K_{prop} < 414,\u00a0 K_{prop} < 103.5[\/latex]<\/p>\n

      The total condition is: [latex]-1 < K_{prop} < 103.5[\/latex]\n\n\n

      \"\"<\/p>\n

      Note that the practical range of stable controller operations, as opposed to the previous purely mathematical condition, is [latex]0Critical Gain <\/em>values. Again, of practical interest is only the upper, positive Critical gain value. What happens when the Controller gain reaches that Critical gain value?<\/p>\n

      When [latex]K_{prop} = 103.5[\/latex] :<\/p>\n

      \"\"<\/p>\n

      [latex]Q_{aux}(s) = 11s^{2} +418[\/latex]<\/p>\n

      [latex]Q_{aux}(s) = 0[\/latex]<\/p>\n

      [latex]s^{2} +38 = 0[\/latex]<\/p>\n

      [latex]s_{1} = j\\sqrt{38} = j6.16[\/latex]<\/p>\n

      [latex]s_{2} = -j\\sqrt{38} = -j6.16[\/latex]<\/p>\n

      As the pole-zero map illustrates, the location of marginally stable poles on the Imaginary axis corresponds to the frequency of sustained oscillations, \\[latex]\\omega_{crit} = 6.16 rad\/sec[\/latex]. This corresponds to the period of oscillations equal to 1.02 seconds. Step response of the system when [latex]K_{prop}=103.5[\/latex] is also shown.<\/p>\n

      \"\"<\/p>\n

      \"\"<\/p>\n

       <\/p>\n","protected":false},"author":118,"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-434","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/434","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\/118"}],"version-history":[{"count":60,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/434\/revisions"}],"predecessor-version":[{"id":2631,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/434\/revisions\/2631"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/434\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=434"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=434"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=434"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=434"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}