View
3
Download
0
Category
Preview:
Citation preview
StabilitklyapunovFunctionsRecapI Ox tie O C 112 unknown
go.ee output regulation e A o as eUct bounded htt 30
rig lo u Iti x
Candidate Lyapunov fan x 8 Lax't 18 ofE E
co I 1 x 0
Equilibria e 8 sito
x o I C IR
all points on the line CO G 8 EIR are
equilibriao f 0 for any 8 E IRnot enough info to conclude Xcel 30
Consider an autonomous systemi f x x o x C R
f Rn IRN continuous
f107 0 equilibrium at 0
Candidate Lyapunov fan V 112 co V is C
Thm_ weak Lyapunov stability criterion
Suppose F a continuous fan W IR co s t
x E W x fo for all XERDThen for every solution xttremainsbounded
for all C 20 W x t o as e7
Remarks1 boundedness of Xlt is important2 if V is radially unbounded VIX 20 as ki sadthen all solutions are bold automatically Khalil3 V is not required to be p d 4 7 can still be
0 even if o
4 Krasovskii LaSalle thm every bold trajectoryapproaches the largest positively invariant setinside the set x C Rn V x o
A set D C IR is positively invariant iflo C D 3 It C D htt 30
If D E x lx7 o is the largest p.in setthen
xlt bold him dist Xlt D oe a
Proffwill need a lemma along the wayV xlt V Nol Joticxishds V E W
E V x o JotwkishdsWCxistlds E x lol VCxitt 30
E r x lol to
Vtzo O E W xishds Vlxco Leo W 30
fgfwlxithdt fimffwcxislldsaffisff.itWe would like to conclude that W x El 0
Cannot do this in general
Lemmy Barbalat's lemma Let f o a co 3be a uniformly continuous f ca s t
ffeldt figs fixishds
exists and is finite Then ftt o as e a
Remart1 Continuity us uniform continuityf is continuous if tt V E O F Se 81k o
sit It t1 s If Ct't fit l8 will depend on Beth E and t
f is uniformly continuous if V E o 78 8 e o
s t It t1 S If ti fit I E
S depends only on E
2 f t can take negative values the result stillholds Khalil's book has the proof
3 uniform continuity is necessaryXflt
FH Eso it
fits finds It t t t t
f t 17 0 b c f'ft is unbounded
Proofyl of BarbalatBy contradictionAssume f It 13 o astra
Then FE o and a sequence th sit the o
and th 32 as a s E f Ith Z E ok
Xflts
E i
ii E
By uniform continuity FS o 5 t
It tht LS If ftp7 f t71CEl2 V kThen for all k 30 and all E Ctf Getsf t f Ith t f ft flea If HEI f All
EhE 7 42for It ticks42 Vk 70
tuts
f fftldtz.IE sitidt cannotexist
the
E s
Corollary Let Xlt rilt be bold Wcontinuousand
f Wlxcelldtexists and is finite Then WINED o as
a
Back to proof of main thou
S Whatnot L
where Xlt is a bald trajectory of X fixSince f is continuous and c t is boldflectl is bold x t is bold
So WCxit o as e a by Barbalat B
Backtoex mple izlo
x 8 text tee 5x 8 x
wcx.io x
x4t dt exists and is finite
o E got x4s7ds E Vlxco htt o
Wl x is continuous
Xlt t Xlt t are bold
1221 At EIGHT 05 EV xco7 Glo
xlt Icel Gtfo
t Ott bold
filth _Ico Ect D cell LI th ft e4t a
By weak Lyapunov criterion WlxltD x4t 30
ect as E soo
On the other hand can't say anything aboutconvergence of Itt
fwhy LaSalle Celt IHI 0,8 ER
Preview x Ext is actually useful toshow Xlt 30 as C a
connection to observability ECE Sls review
observability Gramians etc
Recommended