{"id":655,"date":"2019-05-24T15:29:45","date_gmt":"2019-05-24T23:29:45","guid":{"rendered":"https:\/\/linearcontrol.info\/fundamentals\/?p=655"},"modified":"2019-06-03T07:24:08","modified_gmt":"2019-06-03T15:24:08","slug":"balance-scales-and-integral-action-part-ii","status":"publish","type":"post","link":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/2019\/05\/24\/balance-scales-and-integral-action-part-ii\/","title":{"rendered":"Balance scales and integral action. Part II"},"content":{"rendered":"\n

In a previous post<\/a> we discussed how the principles of operation of a balance scale<\/a> could be understood as a feedback loop with integral action. We left off after analyzing the impact of measurement disturbances in the loop and mentioned that input disturbances could be used to model “operator errors.”<\/p>\n\n\n\n\n\n\n\n

\"\"
Fig. 1.13: Feedback loop with input disturbance<\/figcaption><\/figure><\/div>\n\n\n\n

The motivation for considering “operator errors” is that it is very likely that a human operating the scale will not be able to produce weights, that is the signal $u$, that are exactly proportional to the error signal $e$. In that case an input disturbance $w$ entering the loop as in Fig. 1.13 can be used to model the “operator error” in producing the “control signal” $u$.<\/p>\n\n\n\n

With the addition of the input disturbance, and following the same reasoning as in Part I of this post<\/a>, it is possible to describe the loop in Fig. 1.13 by the recursive equations<\/p>\n\n\n\n

$$y(k+1)=y(k) + u(k) + w(k), \\quad u(k) = K e(k), \\quad e(k) = \\bar{y} – y(k).$$<\/p>\n\n\n\n

As before, it is possible to obtain a recursion to describe the error signal $e$ as in<\/p>\n\n\n\n

$$
\\begin{aligned}
e(k+1) &=\\bar{y} – y(k+1) \\\\
&= \\bar{y} – (y(k) +u(k) + w(k)) \\\\
&= \\bar{y} – (\\bar{y}-e(k)) -K e(k)- w(k) \\\\
&= (1 – K) e(k) – w(k)
\\end{aligned}
$$<\/p>\n\n\n\n

Note how assuming that the reference signal $\\bar{y}(k)=\\bar{y}$ is constant has the effect of cancelling $\\bar{y}$ from the recursion. This is integral action once again at work!<\/p>\n\n\n\n

A reasonable assumption on the disturbance $w(k)$ might be that it is small<\/em> when compared with $u(k)$. For example, that $w(k)=\\delta(k) \\, u(k)$ and that $\\delta(k)$ is bounded<\/em>, say $|\\delta(k)| \\leq \\gamma$. Note that we are not saying how small $\\gamma$ should be! That’s a question that we will seek to answer based on what we know about the system. Indeed, because<\/p>\n\n\n\n

$$
\\begin{aligned}
|e(k+1)| &= |(1 – K) e(k) – w(k)| \\\\
&= |(1 – K) e(k) + \\delta(k) u(k)| \\\\
&= |1 – K – \\delta(k) K| |e(k)|,
\\end{aligned}
$$<\/p>\n\n\n\n

the error $e(k)$ converges to zero even in the presence of operator errors as long as<\/p>\n\n\n\n

$$|1-K+\\delta(k) K|\\leq|1-K|+|K||\\delta(k)|\\leq |1-K|+|K|\\gamma<1.$$<\/p>\n\n\n\n

In other words, if<\/p>\n\n\n\n

$$\\gamma<\\frac{1-|1-K|}{|K|}.$$<\/p>\n\n\n\n

One conclusion is that, since $K > 0$ and $|1-K|<1$, it will always be possible to tolerate some operator error. Better yet, if the operator is cautious<\/em>, that is if $0<K<1$ then <\/p>\n\n\n\n

$$\\gamma<\\frac{1-|1-K|}{|K|} = \\frac{1 – 1 + K}{K} = 1$$<\/p>\n\n\n\n

no matter the value of $K$. That is, the balance scale can tolerate up to 100% of operator error, and it will still converge!<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

The type of analysis performed above is covered in much of Chapter 8. Indeed, the above condition in terms of the $\\gamma$ can be obtained via the application of a discrete-time counterpart for the small gain<\/em> argument provided in Section 8.3, in which one can think of the above model for the operator error as been a particular case of the block-diagram in Fig. 8.3 in which the block $\\Delta$ is the $\\delta(k)$ above.<\/p>\n","protected":false},"excerpt":{"rendered":"

In a previous post we discussed how the principles of operation of a balance scale could be understood as a feedback loop with integral action. We left off after analyzing the impact of measurement disturbances in the loop and mentioned that input disturbances could be used to model “operator errors.”<\/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":[53,8,34,6,35,36],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/655"}],"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=655"}],"version-history":[{"count":13,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/655\/revisions"}],"predecessor-version":[{"id":670,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/655\/revisions\/670"}],"wp:attachment":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/media?parent=655"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/categories?post=655"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/tags?post=655"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}