{"id":590,"date":"2019-05-25T06:10:49","date_gmt":"2019-05-25T14:10:49","guid":{"rendered":"https:\/\/linearcontrol.info\/fundamentals\/?p=590"},"modified":"2019-06-03T07:17:49","modified_gmt":"2019-06-03T15:17:49","slug":"asymptotic-tracking-a-constant-reference-without-a-pole-at-zero","status":"publish","type":"post","link":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/2019\/05\/25\/asymptotic-tracking-a-constant-reference-without-a-pole-at-zero\/","title":{"rendered":"Asymptotic tracking a constant reference without a pole at zero?"},"content":{"rendered":"\n

On a previous post<\/a> we have discussed how by invoking stability one can solve problems without having to worry about particular inputs. The contrast between a solution that holds for a particular input versus a solution that holds for a class of inputs is highlighted in the present post by revisiting the problem of asymptotic tracking for a simple example.<\/p>\n\n\n\n\n\n\n\n

\"\"
Fig. 4.2: Standard feedback diagram<\/figcaption><\/figure><\/div>\n\n\n\n

Say one would like to design a controller $K$ in the standard feedback diagram of Fig. 4.2 to asymptotically track a particular<\/em> constant input $\\bar{y}(t) = \\bar{y}$, $t \\geq 0$. By particular I mean that there is only one possible value for $\\bar{y}$ that one is interested in tracking. For example, let $G$ be the linear model represented by the ordinary differential equation<\/p>\n\n\n\n

$$\\dot{y}(t) = a y(t) + u(t)$$<\/p>\n\n\n\n

and consider the problem of asymptotically tracking the particular given constant $\\bar{y}$, that is the problem of designing a controller $K$ such that<\/p>\n\n\n\n

$$\\lim_{t \\rightarrow \\infty} y(t) = \\bar{y}.$$<\/p>\n\n\n\n

Let $a > 0$ to make the open-loop unstable and make things more interesting.<\/p>\n\n\n\n

If one follows the discussion in Chapter 4, a solution to this problem must be in the form of a dynamic controller that has a pole at zero. In the present case, because $a > 0$ is such that the open-loop tranfer-function<\/p>\n\n\n\n

$$G=\\frac{1}{s-a}$$<\/p>\n\n\n\n

is unstable, one will also also need a zero. That is, the simplest possible controller is the PI controller<\/p>\n\n\n\n

$$K(s)=\\frac{K_p s + K_i}{s}.$$<\/p>\n\n\n\n

According to Lemma 4.1, all that needs to happen is for $K_p$ and $K_i$ to be picked so that the transfer-function<\/p>\n\n\n\n

$$S=\\frac{1}{1 + K G} = \\frac{1}{1 + \\frac{K_p s + K_i}{s} \\frac{1}{s -a}}= \\frac{s (s – a)}{s^2 +(K_p – a) s + K_i}$$ <\/p>\n\n\n\n

is asymptotically stable. From Section 6.1, any choice so that<\/p>\n\n\n\n

$$K_p > a, \\quad K_i > 0$$<\/p>\n\n\n\n

is stabilizing. Note how $a > 0$ means that it is essential for the controller to have a zero, that is that $K_p > 0$. The above solution not only guarantees asymptotic tracking for the particular $\\bar{y}$ in question but for all possible non-zero $\\bar{y}$!<\/p>\n\n\n\n

Now go back to the original question. If one particular input is all that one cares for, how about the following solution to this problem? Define<\/p>\n\n\n\n

$$u(t) = \\tilde{u}(t) – \\bar{u}$$<\/p>\n\n\n\n

and use the fact that <\/p>\n\n\n\n

$$e(t) = \\bar{y} – y(t) \\quad \\implies \\quad y(t) = \\bar{y} – e(t)$$ <\/p>\n\n\n\n

to write<\/p>\n\n\n\n

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

Therefore the feedback controller<\/p>\n\n\n\n

$$u(t) = \\tilde{u}(t) – \\bar{u}, \\quad \\tilde{u}(t) = K e(t), \\quad \\bar{u} = a \\bar{y}$$<\/p>\n\n\n\n

with $K > a$ is such that <\/p>\n\n\n\n

$$\\dot{e}(t) = a e(t) – \\tilde{u}(t), \\quad \\tilde{u}(t)=K e(t),$$<\/p>\n\n\n\n

and<\/p>\n\n\n\n

$$\\dot{e}(t) = (a – K) e(t)$$<\/p>\n\n\n\n

is asymptotically stable. In other words,<\/p>\n\n\n\n

$$\\lim_{t\\rightarrow \\infty}e(t) = 0 \\quad \\implies \\quad \\lim_{t \\rightarrow \\infty} y(t) = \\bar{y}.$$<\/p>\n\n\n\n

This means asymptotic tracking even though neither the system nor the controller have a pole at zero!<\/p>\n\n\n\n

So where’s the catch? The latter tracking property holds only if the controller is built with $\\bar{u}$ that matches $a \\bar{y}$ exactly. A small change in the value of the reference input or in the system parameter $a$ and gone is asymptotic tracking. In that way, this solution is akin to an open-loop solution, even though in this present case feedback was necessary to stabilize the closed-loop system. See Section 8.6 for a related discussion in the context of feedforward control.<\/p>\n\n\n\n

By contrast, the former solution guarantees tracking for any possible $\\bar{y} \\neq 0$, thanks to the virtues of invoking asymptotic stability to handle a class of inputs<\/a> rather than worrying about a particular input. <\/p>\n","protected":false},"excerpt":{"rendered":"

On a previous post we have discussed how by invoking stability one can solve problems without having to worry about particular inputs. The contrast between a solution that holds for a particular input versus a solution that holds for a class of inputs is highlighted in the present post by revisiting the problem of asymptotic … Continue reading “Asymptotic tracking a constant reference without a pole at zero?”<\/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":[32,48,3],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/590"}],"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=590"}],"version-history":[{"count":22,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/590\/revisions"}],"predecessor-version":[{"id":696,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/590\/revisions\/696"}],"wp:attachment":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/media?parent=590"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/categories?post=590"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/tags?post=590"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}