1 Input-Output Stability

Transcription

1 Inut-Outut Stability Inut-outut stability analysis allows us to analyz th stability of a givn syst without knowing th intrnal stat x of th syst. Bfor going forward, w hav to introduc so inut-outut athatical odls.. Stability W considr a syst whos inut-outut rlation is rrsntd by whr H is so aing or orator that scifis y in trs of u. Th inut u blongs to a sac of signals that a th ti intrval [ 0, ) into th Euclidian sac v ;.g. u :0, [ ) R. Exals ar th sac of icwis continuous, boundd functions; that is, su u() t <, and th stat sac of icwis continuous, suar-intgrabl t 0 T functions; that is, u () () 0 y = Hu t u t dt <. To asur th siz of a signal, w introduc th nor function u, which satisfis th following thr rortis: ) Th nor of a signal is zro if and only if th signal is idntical zro and is strictly ositiv othrwis ) Scaling a signal rsults in a corrsonding scaling of th nor; that is, au = a u for any ositiv constant a and vry signal u 3) Th nor satisfis th triangl inuality u+ u u + u for any signals u and u. For th sac of icwis continuous, boundd functions, th nor is dfind as u t 0 = su u t < and th sac is dnotd by. For th sac of icwis continuous, suar-intgrabl functions, th nor is dfind by () () T u u t u t dt 0 = < and th sac is dnotd by. Mor gnrally, th sac for < is dfind as

2 th st of all icwis continuous functions u :0, [ ) R such that Th subscrit in rfrs to th -nor usd to dfin th sac, whil th surscrit is th dinsion of th signal u. If w think of u as a stabl inut, th ustion to ask is whnvr th outut y will b stabl in th sns that y, allowing for diffrnt nubrs of oututs to inuts. A syst having th rorty that any stabl inut will gnrat a stabl outut is dfind as a stabl syst. Howvr, w cannot dfin H as a aing fro to, bcaus w hav to dal with systs which ar unstabl, in that an inut u ay gnrat an outut y that dos not blong to. Thrfor, H is usually dfind as a aing fro an xtndd sac to an xtndd sac, whr is dfind by and u is a truncation of u dfind by u () t ( () ) 0 u = u t dt < {, [ 0, )} = u u u( t), 0 t = 0, t > Th xtndd sac is a linar sac that contains th unxtndd sac as a subst. It allows us to dal with unboundd vr-growing signals. Exal.: Th signal u() t = t dos not blongs to th sac, but its truncation u () t t, 0 t = 0, t > blongs to for vry finit. Hnc, u( t) = t blongs to th xtndd sac. Dfinition.: A syst is causal if th valu of th outut is only dtrind by ast inuts, that is

3 Hu = Hu Or uivalntly, that for all tis, for any inut signals u, v, if Error! Objcts cannot b cratd fro diting fild cods., thn Error! Objcts cannot b cratd fro diting fild cods. Rark.: All systs of th for x& = f ( x, u), y h( x, u) = ar causal Dfinition.: α :0, [ a) [ 0, ), continuous is a class K function if onoton incrasing. :0, [ a) [ 0, ) α ( r) as r. Dfinition.3: A aing H : α 0 = 0 and is α is a class K if α is class K, a = and α, dfind on [ 0, ), and a nonngativ constant β such that is stabl if thr xists a class K function for all u 0,. It is finit-gain stabl if thr xists nonngativ constants γ and β such that for all and [ ) and [ 0, ) u. ( Hu) ( u ) α + β ( Hu) u γ + β Not that β is calld bias tr and th sallst ossibl γ is calld th gain of H. Exal.: A orylss, ossibly ti-varying, function h :0, [ ) R R can b viwd as an orator H that assigns to vry inut signal u() t th outut signal () h t, u() t y t t =. W us this sil orator to illustrat th dfinition of stability. for so nonngativ constants a, b and c. Using th fact ' h u cu cu hu = a+ btanh cu= a+ b cu cu 4bc = bc cu cu ( + ), u R +

4 w hav hu a+ bcu, u R Hnc, H is finit-gain stabl with γ = bc and β = a. Furthror, if a = 0, thn for ach [, ), Thus, for ach [, ], th orator H is finit-gain stabl with zro bias and γ = bc ( ()) hut dt bc ut dt 0 0. t h b a ti-varying function that satisfis htu (, ) au, t 0, u R for so ositiv constant a. For ach [, ], th orator H is finit-gain stabl with zro bias and γ = a. Finally lt Sinc = u hu H and stabl with zro bias and α r = r. It is not finit-gain stabl bcaus th function hu = u cannot b boundd by a straight lin of th for hu all u R. su hut ( ()) su ut () t 0 t 0 γ u+ β, for. Th Sall-Gain Thor Th foralis of inut-outut stability is articular usful in studying stability of intrconnctd systs, sinc th gain of a syst allows us to track how th nor of a signal incrass or dcrass as it asss through th syst. This is articularly so for th fdback connction in th nxt figur.

5 Hr, w hav two systs : H and : H. Suos both systs ar finit-gain stabl; that is y γ + β, y γ + β, [ ) [ ), 0,, 0, Thor.: Undr th rcding assutions, th fdback connction is finit-gain stabl if γγ <. Proof Assuing xistnc of th solution w can writ =, = u ( H ) u H Thn, Sinc γγ <, for all [ 0, ). Siilarly, u + H u + γ + β u + γ u + γ + β + β = γγ u + u + γ u + β + γ β u + γ u + β + γ β γ γ

6 for all [ 0, ) u + γ u + β + γ β γ γ. Th roof is coltd by noting that +, which follows fro th triangl inuality. Rark.: This is a sufficint but not ncssary condition. Rark.3: Rstricting ourslvs to linar systs, thr is a convrs rsult. For any linar syst H with gain γ, thr xists a dstabilizing linar syst H with gain γ =. γ Th fdback connction in th figur rovids a convnint stu for studying robustnss issus in dynaical systs. Quit oftn, dynaical systs subjct to odl uncrtaintis can b rrsntd in th for of a fdback connction with H, say, as a stabl noinal syst and H as a stabl rturbation. Thn, th ruirnt γ γ < is satisfid whnvr γ is sall nough, rsctivly, by controlling th syst H to hav th sallst gain, it bcos robust to th largst urturbations. Thus, th sall-gain thor rovids a conctual frawork for undrstanding any of th robustnss rsults that aris in th study of dynaical systs, scially whn fdback is usd.

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