![\"\"](\"https:\/\/en.wikipedia.org\/wiki\/File:Stability_Diagram.png\")
<\/a>In particular, the usual stability condition for continuous-time linear systems (all eigenvalues in the open left half-plane) can be reduced to a simple criteria expressed in terms of \\( \\mbox{Tr} A, \\mbox{det} A\\): $$\\max\\mbox{Re}(\\Lambda(A))<0 \\Leftrightarrow \\max(\\mbox{Tr} A, -\\mbox{det} A)<0.$$<\/p>\n\n\n\n
This condition is particularly useful for control theorists, as it greatly simplifies the characterization of the stabilizing state-feedback gains \\(K\\) ensuring that a closed-loop system (with \\(A_{cl}=A+BK\\)) is stable, allowing for quick, back-of-the-envelope calculations.<\/p>\n\n\n\n
Although unsurprising, it is rarely brought up that a similar stability condition holds for second-order discrete-time linear systems: $$x_{k+1}=Ax_k, \\quad A\\in \\mathbb{R}^{2\\times 2}.$$<\/p>\n\n\n\n
Indeed, the usual stability condition (the spectral radius of \\(A\\) is smaller than \\(1\\), thus all eigenvalues are in the unit disc) can also be reduced to a simple criteria expressed in terms of \\( \\mbox{Tr} A, \\mbox{det} A\\): $$\\rho(A)<1 \\Leftrightarrow \\max( |\\mbox{Tr} A| – \\mbox{det} A, \\mbox{det} A)<1.$$<\/p>\n\n\n\n Interestingly, this criterion defines a triangular region on the \\((\\mbox{Tr} A, \\mbox{det} A)\\)-plane, sometimes referred to as the ‘stability triangle.’ This triangle is represented hereabove, together with a few sample trajectories produced by systems corresponding to points within it. Analogous to the continuous-time Poincar\u00e9 diagram, oscillatory behaviors associated with complex eigenvalues occur in the green region, where \\((\\mbox{Tr} A)^2 < 4\\mbox{det} A \\). In the red region, only scaling, symmetry flips, and accumulation along eigenvector directions occur.<\/p>\n","protected":false},"excerpt":{"rendered":" Let us take a second-order continuous-time linear system: $$\\dot{x}(t)=Ax(t), \\quad A\\in \\mathbb{R}^{2\\times 2}.$$ For such linear systems, the qualitative behavior of \\(x(t)\\) is determined by the eigenvalues of \\(A\\). What makes the \\(2×2\\) case interesting is that these eigenvalues can be written explicitly in terms of the trace and determinant of \\(A\\): $$\\Lambda(A)= \\left\\{\\frac{\\mbox{Tr} A}{2} […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,5,6],"tags":[],"class_list":["post-1","post","type-post","status-publish","format-standard","hentry","category-control-theory","category-linear-systems","category-trivia"],"_links":{"self":[{"href":"https:\/\/gbainier.com\/index.php?rest_route=\/wp\/v2\/posts\/1","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gbainier.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gbainier.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gbainier.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gbainier.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1"}],"version-history":[{"count":46,"href":"https:\/\/gbainier.com\/index.php?rest_route=\/wp\/v2\/posts\/1\/revisions"}],"predecessor-version":[{"id":97,"href":"https:\/\/gbainier.com\/index.php?rest_route=\/wp\/v2\/posts\/1\/revisions\/97"}],"wp:attachment":[{"href":"https:\/\/gbainier.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gbainier.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gbainier.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}<\/figure>\n\n\n\n