State-space behaviours 2 using eigenvalues
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1 1 Stat-spac bhaviours 2 using ignvalus J A Rossitr Slids by Anthony Rossitr
2 Introduction Th first vido dmonstratd that on can solv 2 x x( ( x(0) Th stat transition matrix Φ( can b computd using Laplac mthods, although this is tdious ( L 1 [( si A) 1 ] This vido looks at an altrnativ drivation using similarity transforms and ignvalu/vctor dcompositions. Slids by Anthony Rossitr
3 Intrim summary W xpct th mods of bhaviour of a stat spac modl to b dtrmind by th ignvalus of th A matrix. This vido sris will not gt sid trackd by spcial cass with mbddd pol/zro cancllations, rpatd pols and non-simpl Jordan forms and th lik. ( Slids by Anthony Rossitr L 1 [( si A) 1 ] Pols com from dtrminant of (si-a) which ar clarly th sam as th ignvalus of A. 3
4 1 st ordr xampl and xtnsion Considr th cas whr thr is only on stat. In this cas th stat spac modl rducs to a standard 1 st ordr diffrntial quation whos bhaviours ar wll undrstood. 4 x ax at x( x(0) Can w driv an quivalnt solution for matrics? x x( x(0) Slids by Anthony Rossitr
5 Rmark From th prvious vido w know that 5 x x ( x(0) ( L 1 ( si A) 1 A simplistic statmnt could b to dfin th following as th maning of matrix xponntial. x x( 0) ( x(0) ( Slids by Anthony Rossitr
6 Altrnativ insight/ky rsult For now ignor th systm input and considr th systm dynamics (transition matrix). 6 x x x(0) This dfinition of Φ( accords wll with th ruls for diffrntiation of xponntials of scalars. d dt A Typical txt books us Maclaurin xpansions to prov this cor rsult. Slids by Anthony Rossitr
7 Dfinition of 7 Whr it xists (distinct ignvalus), it may b asir to us an ignvalu/vctor dcomposition. d dt A x x Wz A W z x z AWz; z z z t z( 0) x W t x(0) Slids by Anthony Rossitr W t
8 Dfinition of Tak th rsult from th prvious pag and not that th middl matrix is diagonal. W t ; t diag[ 1t,, nt ] 8 Thrfor: d dt d dt W t 1 W 0 t 1 n 0 t n W t A t W W A d dt Slids by Anthony Rossitr
9 Stat transition matrix It is wll accptd that: 9 x x x(0) Th stat transition matrix Φ( can b dfind as follows using an ignvalu/vctor dcomposition. ( W t L 1 ( si A) 1 This is usful as it mphasiss th rol of th ignvalus in th dynamics of th solution and also xploits scalar computations whr that is hlpful. Slids by Anthony Rossitr
10 Summary Th bhaviours of a stat-spac systm ar govrnd by th ignvalus of th A matrix. 10 x x( x(0) This rsult follows dirctly from a Laplac transform analysis and also from a similarity transform using th ignvctors. For distinct ignvalus, th stat transition matrix is givn as: W t L 1 1 [( si A) ] Slids by Anthony Rossitr
11 Anthony Rossitr Dpartmnt of Automatic Control and Systms Enginring Univrsity of Shffild Univrsity of Shffild This work is licnsd undr th Crativ Commons tribution 2.0 UK: England & Wals Licnc. To viw a copy of this licnc, visit or snd a lttr to: Crativ Commons, 171 Scond Strt, Suit 300, San Francisco, California 94105, USA. It should b notd that som of th matrials containd within this rsourc ar subjct to third party rights and any copyright notics must rmain with ths matrials in th vnt of rus or rpurposing. If thr ar third party imags within th rsourc plas do not rmov or altr any of th copyright notics or wbsit dtails shown blow th imag. (Plas list dtails of th third party rights containd within this work. If you includ your institutions logo on th covr plas includ rfrnc to th fact that it is a trad mark and all copyright in that imag is rsrvd.)
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