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Unformatted text preview: flak. \9 2
“f "" “’2 ' (m)
3 2
3mmmi* “3
3' ‘2
Eu Vwaéi}, a "f2 (526) 3E an, Mam, Elzilﬁwwgiiwﬂﬁ a: 0 (53’?)
a: igrsup(uzu§w2uwu%) a a {5.28) __/" r3 Theorem 5.11. {iiarses 1r 1:» ﬂ, thsre: exists a cuntsat 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}. {stressiv} 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 titfar {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
thawsm Iii. ta hhtsin the stiort sunﬁsh ws get ﬂitt DRE .. L a hit! a
aP t A'PiPh n FEB: B;}[ 1
which is prcsissiy the DRE {5.29) shots.
{sufficiency} This is again time by sampleting 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 wHig'ﬂipiﬁ
where [11]”.. [u]+[ E I] [135}me [unkiﬂgpx .
in other words, J can he mwrittsn as . . 1 t a
J ﬂ 1EH+BEPEEEWTEIEWHﬁBLPmiigr 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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 Winter '09
 Bamieh
 Existence, DRE, THs, hauhdsd saiutim mrsr, inth theorem SJ, stiort sunﬁsh ws

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