{"id":498,"date":"2019-05-22T20:49:45","date_gmt":"2019-05-23T04:49:45","guid":{"rendered":"https:\/\/linearcontrol.info\/fundamentals\/?p=498"},"modified":"2019-06-03T07:30:32","modified_gmt":"2019-06-03T15:30:32","slug":"let-your-inputs-go","status":"publish","type":"post","link":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/2019\/05\/22\/let-your-inputs-go\/","title":{"rendered":"Let your inputs go!"},"content":{"rendered":"\n

As one learns more about feedback and control systems, at some point, comes an important realization that a variety of distinct problems can be answered if we know how to answer the question: “is some closed-loop system asymptotically stable?” Indeed, the study of stability dominates Chapters 6 and 7, arguably the densest chapters in the book. Many students, when asked what they remember from their undergraduate control class, will quickly point at the root-locus<\/em> or at the Nyquist stability criterion<\/em>. Yet, in the beginning, it might seem that stability is a side dish and not the main attraction. <\/p>\n\n\n\n\n\n\n\n

Take for example Chapter 4. It is easy to overlook that asymptotic stability is a requirement for tracking. Or to mistake internal stability as something that is only important in the rare<\/em> case of pole-zero cancellations. It is common for students to wonder what to do with the system inputs when asked to analyze a feedback system using the root-locus method or the Nyquist stability criterion. For example, here is a common question:<\/p>\n\n\n\n

How do you incorporate a constant disturbance into your $L$ transfer function? In the case of this block diagram(apologize for the messiness), I was unable to separate an alpha and C(s) from the K(s) since there is a constant omega [$w$] which would be added to the output of e*K(s). So in this case, do we just ignore omega [$w$]? <\/p>Anonymous<\/cite><\/blockquote>\n\n\n\n

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

The above question reveals some apprehension about “ignoring” the inputs, as if one would be answering only <\/em> the stability question. It also reveals that the student is likely missing a very important point: the reason for reformulating the problem as a stability question is not to ignore<\/em> the inputs but rather to look for answers that hold for all<\/em> possible “well behaved” inputs. Stability provides broad guarantees rather than specific answers.<\/p>\n\n\n\n

It might be especially difficult to “let the inputs go” if they have physical meaning. Take for example a problem such as P6.34-P6-36, in which students are asked to revisit the design of a controller for the temperature, $T$, of a water heater using the root-locus method. A model for the system is the differential equation<\/p>\n\n\n\n

$$\\dot{T}(t)+\\left (\\frac{v}{m} + \\frac{1}{m c R} \\right ) T(t) = \\frac{1}{m c} \\left (u(t) + w(t)\\right ) $$<\/p>\n\n\n\n

in which the “disturbance” is the very physical signal<\/p>\n\n\n\n

$$w(t) = \\bar{w}, \\quad \\bar{w} = v \\, c \\, T_i +\\frac{1}{R} T_0,$$<\/p>\n\n\n\n

which is related to things like the system’s thermal properties, $c$ and $R$, and the (constant) in\/out flow and its temperature, $v$, $T_0$, $T_i$, which clearly make $\\bar{w} \\neq 0$. In the presence of a model like this, it is not uncommon to hear someone say: “It doesn’t make any sense to analyse this system if the input $w(t)=\\bar{w}$ is not present!”<\/p>\n\n\n\n

Yet, because stability allows you to draw conclusions about a large class of inputs, namely all bounded inputs, which $w(t) = \\bar{w}$ is certainly a member of, it is safe to go ahead and ask whether a controller will stabilize the closed-loop system without worrying about the particular input $w(t) =\\bar{w}$.<\/p>\n\n\n\n

Likewise, a question such as “does the closed-loop controller track a given constant reference $\\bar{y}$?” can be answered by adding a pole at the origin to the controller (see Lemma 4.1) and assessing asymptotic stability of the resulting closed-loop transfer-functions.<\/p>\n\n\n\n

Of all the tricks<\/em> you learn with feedback control this is one of my favorites. So next time, just let your inputs go! Stability will take care of them.<\/p>\n","protected":false},"excerpt":{"rendered":"

As one learns more about feedback and control systems, at some point, comes an important realization that a variety of distinct problems can be answered if we know how to answer the question: “is some closed-loop system asymptotically stable?” Indeed, the study of stability dominates Chapters 6 and 7, arguably the densest chapters in the … Continue reading “Let your inputs go!”<\/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":[32,48,49,47,8,3],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/498"}],"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=498"}],"version-history":[{"count":72,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/498\/revisions"}],"predecessor-version":[{"id":953,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/498\/revisions\/953"}],"wp:attachment":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/media?parent=498"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/categories?post=498"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/tags?post=498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}