{"id":294,"date":"2019-05-18T21:11:02","date_gmt":"2019-05-19T05:11:02","guid":{"rendered":"https:\/\/linearcontrol.info\/fundamentals\/?p=294"},"modified":"2019-05-24T15:37:52","modified_gmt":"2019-05-24T23:37:52","slug":"is-a-proportional-controller-right-for-you-ask-your-control-engineer-part-iii","status":"publish","type":"post","link":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/2019\/05\/18\/is-a-proportional-controller-right-for-you-ask-your-control-engineer-part-iii\/","title":{"rendered":"Is a proportional controller right for you? Ask your control engineer! Part III"},"content":{"rendered":"\n

In Part I<\/a> and Part II<\/a> of this post we looked into the problem of finding a suitable discrete-time controller when the system being controlled is fast to reach steady-state so that it can be modeled as the single delay<\/p>\n\n\n\n

$$y(t) = G u(t – \\tau), \\quad \\tau > 0.$$<\/p>\n\n\n\n

In Part I we showed that a proportional controller is not a good choice and in Part II we found out that, surprisingly, an integral only controller is an excellent choice but requires the use of a gain $\\hat{G}$ that has to match the gain of the system $G$. In the present post we will revisit the design of a dynamic controller with the goal of showing that: <\/p>\n\n\n\n

  1. the Smith predictor from Part II also had a zero;<\/li>
  2. the integral controller is indeed an excellent choice;<\/li>
  3. the mismatch $\\hat{G} \\neq G$ is not necessarily catastrophic. <\/li><\/ol>\n\n\n\n\n\n\n\n

    Start with a generic dynamic controller of the form<\/p>\n\n\n\n

    $$u(t)=-a u(t – \\tau) + b (\\bar{y}(t) – y(t)).$$<\/p>\n\n\n\n

    The Smith predictor is a special case of the above controller in which <\/p>\n\n\n\n

    $$a = -1, \\quad b = \\hat{G}^{-1}.$$ <\/p>\n\n\n\n

    Now calculate the following quantity<\/p>\n\n\n\n

    $$y(t) + a y(t – \\tau) = G u(t – \\tau) + a G u(t – 2 \\tau) = G b \\, e(t-\\tau)$$<\/p>\n\n\n\n

    and observe that the relationship between $y$ and $e$ is described by the first-order (one delay) difference equation<\/p>\n\n\n\n

    $$y(t) + a y(t – \\tau) = G b \\, e(t-\\tau).$$ <\/p>\n\n\n\n

    This equation represents the open-loop (series) connection of the controller with the system. Because the system had one delay and the controller had another one, one would expect that the relatioship between $y$ and $e$ would have two delays. The fact that it has only one delay is the result of a pole-zero cancellation, which means that the above controller, and therefore the Smith predictor must have a zero. We will look into that in more detail in a future post. <\/p>\n\n\n\n

    In order to close the loop we need to substitute for the error to obtain<\/p>\n\n\n\n

    $$y(t) + a y(t – \\tau) = G b \\, e(t-\\tau) = G b \\, (\\bar{y}(t-\\tau) – y(t-\\tau))$$ <\/p>\n\n\n\n

    from which<\/p>\n\n\n\n

    $$y(t) + (a+Gb) \\, y(t – \\tau) = G b \\, \\bar{y}(t -\\tau)$$ <\/p>\n\n\n\n

    Proceeding as in Part I one can conclude that this system is asymptotically stable if<\/p>\n\n\n\n

    $$|a + G b| < 1.$$<\/p>\n\n\n\n

    In addition if<\/p>\n\n\n\n

    $$a=-1,$$<\/p>\n\n\n\n

    that is if the controller is an integrator, then it also displays asymptotic tracking of constant references because<\/p>\n\n\n\n

    $$G b \\, \\tilde{y} = \\tilde{y} + (-1+Gb) \\, \\tilde{y} = G b \\, \\bar{y} \\quad \\implies \\quad \\tilde{y} = \\bar{y}.$$ <\/p>\n\n\n\n

    This property is independent of the choice of $b$ or the value of $G$ as long as<\/p>\n\n\n\n

    $$|G b -1| < 1$$<\/p>\n\n\n\n

    The conclusion is that an integrator, that is $a = -1$, is an excellent choice!<\/p>\n\n\n\n

    Finally, note that the choice $b=\\hat{G}^{-1}$ when $\\hat{G}=G$ leads to<\/p>\n\n\n\n

    $$G b – 1=G \\hat{G}^{-1} -1=0$$<\/p>\n\n\n\n

    so that convergence to the constant reference is as fast as possible. If $\\hat{G}\\neq G$ nothing catastrophic ensues as long as $|G \\hat{G}^{-1} – 1|<1$. In particular, if<\/p>\n\n\n\n

    $$G \\geq \\hat{G} > (1\/2) G$$<\/p>\n\n\n\n

    then $0 \\leq G \\hat{G}^{-1} – 1 < 1$ so that the closed-loop is stable and without oscillations. If <\/p>\n\n\n\n

    $$\\hat{G} > G$$ <\/p>\n\n\n\n

    the closed loop is stable but it now oscillates because $-1 < G \\hat{G}^{-1} – 1 < 0$. Values of $\\hat{G} \\leq (1\/2) G$ lead to instability. The conclusion is that as long as $\\hat{G} > (1\/2) G$ then the closed-loop is still asymptotically stable and that if $\\hat{G} \\leq G$ then there will be no oscillations. There also seems to be no danger in overestimating $G$, as long as one can tolerate the oscillations. In fact, the onset of oscillations could even be used to estimate a suitable value of $\\hat{G}$.<\/p>\n","protected":false},"excerpt":{"rendered":"

    In Part I and Part II of this post we looked into the problem of finding a suitable discrete-time controller when the system being controlled is fast to reach steady-state so that it can be modeled as the single delay $$y(t) = G u(t – \\tau), \\quad \\tau > 0.$$ In Part I we showed … Continue reading “Is a proportional controller right for you? Ask your control engineer! Part III”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[7],"tags":[17,15,22,23],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/294"}],"collection":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/comments?post=294"}],"version-history":[{"count":39,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/294\/revisions"}],"predecessor-version":[{"id":675,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/294\/revisions\/675"}],"wp:attachment":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/media?parent=294"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/categories?post=294"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/tags?post=294"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}