State space systems analysis
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1 State pace ytem aalyi Repreetatio of a ytem i tate-pace (tate-pace model of a ytem To itroduce the tate pace formalim let u tart with a eample i which the ytem i dicuio i a imple electrical circuit with a curret ource Let deote the voltage over the capacitor, the curret through the iductor ad y i the voltage over the reitor The Kirchoff' curret ad voltage law for thi circuit are writte dow a C = u L = R We alo have y = R The three equatio ca be writte a = + u C C R = + u L C y = + R + u which yield the tate equatio directly a d C = C u A Bu dt = = + R + L L y = [ R] = C Thi i called the tate pace repreetatio of thi electrical circuit Notice that the tate variable ( -voltage over the capacitor ad -curret through the iductor are coected with the two elemet i the circuit which ca tore eergy Thu the tate pace repreetatio of a ytem it i a atural form of repreetig the iformatio o the eergy of a ytem I the followig we will alo ee that the tate-pace decriptio of a ytem provide more iformatio o the ytem dyamic tha a imple iput-output repreetatio of the ytem (ie a trafer fuctio repreetatio
2 May phyical ytem ca be modeled i term of the liear time-ivariat (LTI tatepace equatio = A + Bu y = C + Du with R the iteral tate, u m R the cotrol iput, ad y p R the meaured output igal The ytem or plat matri i A R ad it decribe the ytem m iteral dyamic, B R p i the cotrol iput matri, C R i the output or p m meauremet matri, ad D R i the direct feed matri A iitial coditio vector ( ad a cotrol iput u(t mut be pecified to olve the differetial equatio for the traectorie of the ytem tate R ad the ytem output yt ( Sometime the tate-pace ytem i imply deoted by (A,B,C,D p R Frequecy domai olutio To olve thi equatio i the frequecy domai, tae the Laplace traform to obtai X ( ( = AX ( + BU ( Y ( = CX ( + DU ( Now rearrage the tate equatio to obtai ( I AX ( = ( + BU( X( = ( ( + ( BU( Oe alo ha Y ( = C( ( + [ C( B + D] U ( Thee are the two mai relatio which provide the olutio of the tate equatio Both have two part: a The firt term deped oly o the iitial coditio ( ad are the oly term preet if the iput u(t i zero Therefore, they are ow a the zero-iput (ZI repoe b The ecod term deped oly o the iput u(t ad are the oly term preet whe the iitial tate i equal to zero Therefore, they are ow a the zero-tate (ZS repoe Defiitio The trafer fuctio of the ytem i give by Y ( = H ( U ( (we alo Y( write H( = whe the iitial coditio are equal to zero The trafer fuctio i U( a compact repreetatio of the ytem effect over the iput igal (ie it how how the ytem iput i modified to become the ytem output Therefore, the trafer fuctio i give by H ( = C( B + D The deomiator of thi trafer fuctio i the characteritic polyomial ad it i the determiat of I A, deoted
3 ( = I A The root of the characteritic equatio ( = I A = are the ytem pole Let p, =, deote the pole of the ytem, the the atural mode of the ytem are pt e, =, We will ee later that thee epoetial compoet appear i the repoe of the ytem to every iput igal ut (, a they are aociated with the fied iteral dyamic of the ytem The quatity Φ( = ( i ow a the reolvet matri I term of the reolvet matri oe may write X( =Φ ( ( +ΦBU( Y( = CΦ ( ( + [ CΦ ( B+ DU ] ( = CΦ ( ( + H( U( H( = CΦ ( B+ D See that the output i equal to the trafer fuctio throughput, H( U(, plu a part that deped o the iitial coditio Let U( be the Laplace traform of a impule iput I thi cae U( = The the Laplace traform of the impule repoe of the ytem i Y( = H( Thu the trafer fuctio decribig the ytem dyamic i i fact the Laplace traform of the impule repoe of the ytem I time domai thi mea that the iput ut ( i covolved with the impule repoe of the ytem ht ( to fid the output of the ytem yt ((ie the ytem repoe to that give iput ut ( Time Domai Solutio To obtai the time domai olutio of the ytem of equatio oe ue the ivere Laplace traform o the olutio obtaied i frequecy domai, owig that e = L [ Φ( ], or L[ e ] = Φ( The matri epoetial φ ( t e i ow a the tate traitio matri ad deoted with Uig the ivere Laplace traform o the tate-pace olutio for X(, Y( foud above we obtai the time-domai olutio t A( t τ = e ( + e Bu( τ dτ y( t = Ce ( + Ce t A( t τ Bu( τ dτ + Du( t 3
4 Recall that the product of two Laplace traform repreet covolutio i the time domai Note that, a i the frequecy-domai olutio, the olutio have two part, the ZI part ad the ZS part, t = Uig the hiftig property of the uit impule (Kroecer delta u( t =, oe, t may write the output a t A( t τ y = Ce ( + [ Ce B + Du ( t τ ] u( τ dτ Recall that the iput i covolved with the impule repoe to fid the output Thi idetifie the impule repoe a h ( t Ce = B + Du ( t Note that the impule repoe i give a the ivere Laplace traform of H( To compute the tep repoe rt (, oe may imply calculate r( t = L [ H ( / ] Problem a Coiderig the electrical ytem that we dicued i the begiig, calculate the characteritic polyomial, the reolvet matri ad the trafer fuctio (do ot ue value for the elemet of the ytem but the literal otatio R, L, C b Compare the reultig characteritic polyomial with the tadard form ( = + α + ω = + ζω + ω ad calculate the decay term, the time cotat, the atural frequecy, dampig ratio, ad ocillatio frequecy (loo at Lecture c Selectig value of L= h, R= 3 Ω, C= 5 f calculate the pole ad the atural mode of the ytem Calculate the reolvet matri, the trafer fuctio ad the tate traitio matri Remember from Liear Sytem cla that the tate traitio matri ca be calculated by performig ivere Laplace traform o each elemet of Φ( The reult will how that the matri epoetial i alway epreed i term of a liear matri combiatio of the atural mode d Determie the impule repoe of the ytem by taig ivere Laplace traform of H( e Fid the output y(t give iitial coditio of (= v, (= A, ad a iput of u(t= e -3t u - (t Sytem repoe to a epoetial iput I Lecture we have dicued the repoe of firt ad ecod order ytem to tep ad impule iput There we have ued the iformatio provided by the tep ad impule repoe to fid the parameter of the ytem which produced that give repoe Let u loo ow at the repoe of a ytem to a, more geeral, epoetial iput 4
5 Let the trafer fuctio of a igle-iput igle-output (SISO ytem be where N ( = b ( z ; ( = a ( p i i= = N ( H( = ( Let the iput of the ytem be the epoetial ut ( = αe u ( t with the Laplace α traform U( = (Note that if λ =, α = the ut ( i othig but a uit tep λ Taig ivere Laplace of Y( = H( U( oe obtai pt λt where = yt ( = [ ce + de ] u ( t b ( p z i i= α p p a ( p pl l= c = lim ( p H( U( =, =, λ i= = ( λ z d = lim( λ H( U( = α = H( λα λ b a i ( λ p Deote with yt( t = [ ce ] u ( t the traiet repoe, which deped oly o the = pt pole ad zero of the ytem ad the pole of the iput igal Thi repoe correpod to the chage i the iteral equilibrium of the ytem due to the iput igal λt Deote with yp ( t = de u ( t the teady-tate repoe, which deped o the ytem iput ad the pole ad zero of the ytem λt y ( t = H( λα e u ( t = H( λ ut ( P The traiet part of the repoe y ( t i udeirable but uavoidable ad determied T by the ytem dyamic, while the teady tate repoe yp ( t i deired to follow the iput ut ( Notice that yt ( t a t ad yt ( yp ( t if ad oly if all the pole of the trafer fuctio have egative real part Thi coditio i equivalet to the tability property of a ytem Thu if it i deired that the output of a ytem follow the iput the the ytem mut be table Later we will dicu about thi i more detail alo i the cotet of table cotrol ytem deig Note that if λ = zi the H ( λ = ad thu yp ( t = (ie the epoetial iput igal i ot viible i the ytem output We ay that the zero of a ytem are blocig the λt 5
6 tramiio of certai epoetial iput We will dicu about thi agai i the cotet of zero of a ytem Relative degree ad zero of tate pace ytem (part Let u ow loo i more detail at the trafer fuctio of a ytem give i tate pace form The trafer fuctio of a tate-pace ytem (A,B,C,D i give by H ( = CΦ( B + D = C( B + D C[ ad( ] B C[ ad( ] B + D I A = + D = I A I A N(, = ( where ad( deote the adoit of a matri Oe ee that the pole, which are the root of the deomiator of H(, are give oly i term of A Note that all the iformatio o the feedbac loop i cotaied i A The zero geerally deped o all four matrice A,B,C,D The ytem pole are the root of the ytem characteritic polyomial Δ( The trafer fuctio pole are the root of the trafer fuctio deomiator AFTER pole/zero cacellatio (ie after all the commo term of N ( ad ( have bee implified Notice that ome of the ytem pole might ot appear a pole of the ytem trafer fuctio Thu the teady tate decriptio of a ytem provide more iformatio o the ytem dyamic tha the trafer fuctio of the ytem (Looig ahead, we ote that the et of ytem pole (ie the olutio of ( = i equal to the et of trafer fuctio pole if the ytem i completely cotrollable ad obervable which mea that the ytem ha o decouplig zero A ote about thi will be made i Lecture 3 The relative degree of H( i the degree of the deomiator miu the degree of the umerator If A i a matri, the the degree of I A i, while the degree of ad( i at mot - if D= Therefore, if D= the the relative degree of H( mut be greater tha or equal to If D i ot zero, the the trafer fuctio ha relative degree of zero Thi mea there i a direct-feed term (ie there i itataeou trafer of iformatio from the ytem iput to the ytem output The umber of fiite zero i equal to the degree of the umerator There are pole ad zero If the umber of fiite zero i ot equal to the ay miig zero are at ifiity The umber of ifiite zero i equal to the relative degree 6
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