Wiley signals and systems e book TLFe BO 153

138 Solving Initid Condition Problems with the I ~ , p l ; ~Tkansforrn e ~ N system with oiie input and oiic o u t p ~ t tthat i.; givcri by the state-space drscril)t ion ( 3 ) ( ct kiaw any iiitcrnal st ructulc, aiid so also any eoefticicnts for mslriws To me the Latpliice timsform o11 t l i i s vystem of N first-ordcr differciitial eyiiwt ioiis we only nccd t hc differentiation thcwcrii tor tlic first clc.rix-:ttivr Used on the state vector z ( l ) it, i s VVc start with tlic iirtlrr C ( k ( t ) }= S Z ( ) - z ( ) ('7.59) 'I'he Laplace iratisform here rcf~rsto the individual component\ of the st att w c ?ors (Rj - ~ ( )= A Y(s) = @Z(S)4DX(s) (7.61) (7.62) e ok tlic matrix (51-A) must s t a v on tlrt left 1)rc.ausctlie iriatrix producl 13y i r i w r w Liiplace transfoiriii-Atioii (7.63),tlie t i m e bchaviour of ihe state wsiwe R ~ Clookiiig foi the output aides can b p drterrriiired il it is of intcmst H s i g i d , y ( t ) , however, c ~ i so ~ dwt-' siibstitutJr (7.tj3) into (7.6%).The result is r -/