{"id":994,"date":"2019-06-09T12:52:13","date_gmt":"2019-06-09T20:52:13","guid":{"rendered":"https:\/\/linearcontrol.info\/fundamentals\/?p=994"},"modified":"2019-06-10T08:31:11","modified_gmt":"2019-06-10T16:31:11","slug":"step-by-step-nyquist-plot-example-part-iii","status":"publish","type":"post","link":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/2019\/06\/09\/step-by-step-nyquist-plot-example-part-iii\/","title":{"rendered":"Step-by-step Nyquist plot example. Part III"},"content":{"rendered":"\n

One more on Nyquist plots this time for a non-minimum-phase transfer-function with a pole at the origin.<\/p>\n\n\n\n\n\n\n\n

In a previous post<\/a> we sketched the magnitude and phase of the frequency response of the third-order non-minimum phase transfer-function with a pole at the origin:<\/p>\n\n\n\n

$$G(s) = \\frac{s+2}{s(s^2-9s -10)}= \\frac{s+2}{s(s+1)(s-10)} .$$<\/p>\n\n\n\n

The diagrams obtained are in the following figures:<\/p>\n\n\n\n

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

We start the Nyquist plot at $90^\\circ$ and infinite radius with a decreasing phase:<\/p>\n\n\n\n

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

From there the phase crosses the imaginary axis ($90^\\circ$) at about $0.05$ ($\\approx -25$dB) with a growing phase:<\/p>\n\n\n\n

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

to finally reach $180^\\circ$ with a zero radius:<\/p>\n\n\n\n

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

We connect all these points:<\/p>\n\n\n\n

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

and draw the mirror image<\/p>\n\n\n\n

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

complete with markings denoting the limits at $0^+$ and $0^-$. As discussed in detail in Section 7.6, because there is only one pole at the origin these two large magnitudes should be connected by a clockwise $180^\\circ$ arc starting at $0^-$ and ending at $0^+$ as in:<\/p>\n\n\n\n

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

to complete the Nyquist plot. Compare this hand sketch with the one produced by the Matlab command nyquist<\/code>:<\/p>\n\n\n\n

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

It amazes me that a hand sketch can be much more informative than a plot produced by a professional software! But this seems to be the case in almost all interesting Nyquist plots, which are hard to scale well.<\/p>\n\n\n\n

For closed-loop analysis with $L(s)=G(s)$, note that $P_\\Gamma = 1$ and that for any $\\alpha> 0$ there will always be one clockwise encirclement, therefore<\/p>\n\n\n\n

$$Z_\\Gamma=P_\\Gamma\\,-\\, \\frac{1}{2 \\pi} \\Delta_\\Gamma^{-1\/\\alpha} L(s) = 1 – (-1) = 2$$ <\/p>\n\n\n\n

poles on the right-hand side of the complex plane. As before, the 2 poles on the right-hand side can be visualized in the root-locus:<\/p>\n\n\n\n

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

If one wants to stabilize the above system in closed-loop then one needs more than a gain.<\/p>\n","protected":false},"excerpt":{"rendered":"

One more on Nyquist plots this time for a non-minimum-phase transfer-function with a pole at the origin.<\/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":[55],"tags":[47,54,45,33,3],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/994"}],"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=994"}],"version-history":[{"count":9,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/994\/revisions"}],"predecessor-version":[{"id":1020,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/994\/revisions\/1020"}],"wp:attachment":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/media?parent=994"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/categories?post=994"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/tags?post=994"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}