So here is a recurrent question. You have some model that has a constant showing up on the differential equation and you’re not sure how to handle it. For example, your model looks like:<\/p>\n\n\n\n
$$\\dot{y}(t)=ay(t)+bu(t)+c$$<\/p>\n\n\n\n
where $c$ is a constant. There are multiple ways you can handle this situation.<\/p>\n\n\n\n\n\n\n\n
One way is to treat it as a disturbance signal<\/strong>. What do I mean by that is to compare the above equation with the one<\/p>\n\n\n\n $$\\dot{y}(t)=ay(t)+b(u(t)+w(t))$$<\/p>\n\n\n\n in which $w(t)$ is a disturbance signal. Of course when $w(t)=c\/b$, $t \\geq 0$, then you recover your original equation. The advantage is that you can use the standard block-diagrams in the book, e.g. Fig 4.2, and their associated transfer-functions. You can also use linearity to calculate the combined response to $u$ and $w$, or $\\bar{y}$ in case of a closed-loop. That’s one way among many others. Here’s another one.<\/p>\n\n\n\n How about a change-of-variables? If you let $z(t)=y(t)+c\/a$ then <\/p>\n\n\n\n $$\\dot{z}(t)=az(t)+bu(t)$$<\/p>\n\n\n\n and voila, the constant is gone! One limitation in this case is that it only works if $c$ is a constant, whereas the previous idea can be extended to handle any disturbance signal.<\/p>\n\n\n\n Variations of the above idea can be used to work with initial conditions in pretty much the same way.<\/p>\n","protected":false},"excerpt":{"rendered":" So here is a recurrent question. You have some model that has a constant showing up on the differential equation and you’re not sure how to handle it. For example, your model looks like: $$\\dot{y}(t)=ay(t)+bu(t)+c$$ where $c$ is a constant. There are multiple ways you can handle this situation.<\/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":[51,48,8],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/94"}],"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=94"}],"version-history":[{"count":5,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/94\/revisions"}],"predecessor-version":[{"id":491,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/94\/revisions\/491"}],"wp:attachment":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/media?parent=94"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/categories?post=94"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/tags?post=94"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}