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