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lec_0310 - flak\9 2 “f"" “’2(m 3 2 3mmmi “3...

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Unformatted text preview: flak. \9 2 “f "" “’2 ' (m) 3 2 3mmmi* “3 3' ‘2 Eu Vwaéi}, a "f2 (5-26) 3E an, Mam, Elzilfiwwgiiwflfi a: 0 (53’?) a: igrsup(uzu§w2uwu%) a a {5.28) __/" r3 Theorem 5.11. {ii-arses 1r 1:» fl, thsre: exists a cunts-at u for the system {5.33} such that ' ENE #11 I <1 , i thsthflsmntiatfiismti u PfiT} as s. has a hauhdsd saiutim mrsr {113"} In this MM, ths sptimst cmtti‘oi 1‘s giusn by the stats: feedback "is; # s‘Ptmsmsts {statsfitstsh Physics. tie} a --E§PEt}s(t}. {53$} Pros}. {stress-iv} First. it is easy to shew that if thsrs exists a contra! that achieves s sisssd imp norm «1 It for the problem over FLT}, than thsrs exists a cantmi that sthitms s sihssti ishp norm «c A; far ths prohism ovsr {ET} far any 5 E EELT} {this is again due to ths that that the induced harm of the system over Eff} is fess than or equal to ths induced norm tit-far {ET}; This inst, tsgsthsr with ismms shit} means that the existence at sash 3 control ihipiies that the systsm {5.23} with hhjsstivs {fizlfi w fitmifg} has it unique stationary phiht for tvsry E 5 T. New to inth theorem SJ, we rewritt- ths'systsm equations as Ibiitiws i sitihfliish} + [sthgifiég , aegis". swat}, {s31} J 1* ECWIEE '5" Jifllii “’1'? Wlih ‘ u huh m? (Laws: 4. [w w] {W {5.32} This system is how of the farm (5.1},{ss), with v a Thus invoicing r thaws-m Iii. ta hhtsin the stiort sunfish ws get flit-t DRE .. L a hit! a a-P t- A'P-i-Ph n FEB: B;}[ 1 which is prcsissiy the DRE {5.29) shots. {sufficiency} This is again time by sample-ting ths squam- By ohshrving thh system squstishs (5.31},(5.32J, and applying ismms 5.2, we gst that i." the DRE has a soiution, it“ can he writthh as {note that $61} fl '3} L' T u 2 - ... w “ IE! is i’ft [an :- shah" s; W it High} 3; m wH-ig'flipifi where [11]”..- [u]+[ E I] [135}me [unkiflgpx . in other words, J can he m-writtsn as . . 1 t a J fl 1EH+BEPEEEWTEIEWHfiBL-Pmiigr which sissriy shows that with the shsiss is m w 35? st, the quadratis form J is nsgstivs definite, and therttfhrtt the dosed ioop norm is ct «y. ’M- m-,'HE:P it. mm an L9 MMW.‘ WW 3% G3 .9“ W aim Mam“? . ...
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