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Acknowledgements

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C ONTENTS





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Chapter 1

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Chapter 1 I NTRODUCTION



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Chapter 1 I NTRODUCTION



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.  * PI.

PII. PIII.

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Chapter 2 A S TOCHASTIC A LGEBRA

2.4 Bases

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€



B  u =+B 9=+?A7  B+?F&

x

€

  b

X4+?A? b € 2   j2 b € 8  „|3†|&  bX  b   .2   #2 i' l   Ax  - l z3W +?  K  [O?@+&=+_&9 S &8 +? c<I>‚>†ˆ… „eaD}3|  x X b?I8:O?I231 +?T[$ &8 8AW  ?&rXS?T&B+7 `B+?A?I  - l &?I23&<84t& t[$ &8 8 (=+?V +?2&B$8E< B91$ >@2&7 =+B$8 S8E(2<$ &8E 2gK > 2?I81V=$8E@8=+  l W b?TB+>X B9((=$2? 8:O?I2  7 K > eTX4 2<KXS? 2A 7 7(+?T} >…}… ay† „>~Od† &}… ayJ† a  W +? ?I &8&BT[O?A B$T+? B$ ? X 7 7$[O?I2&? 27 ?I  B+? B+?e  8?I23 &B^W #+ &  , &  (', +? } >…}… ay† „>~^d† &}… ayJ† a   8 Definition.  #%$     







 

k  o ' 

e2,$9 g2.0' l '



X4+?A?i3   8.+?FjF ?AB+?I2
-

 … 

 Proof. =su6e j j5EmFu1b m=n)=Ee < =n/< =Erj b2klF<4b m=s`< Ecb . . ~  =+E==+b m5? b 1 . b m= , "iF%j r>pAhFk x 



1



x









X 7 7ˆ?ABt[O? =$8?IwW   ? $ > 

l ~  f +… | E>‚@‚ 6|  E

  



 f

# 3

 





. 3 ˆ ,  j A 1   ˆ

jA

'

' 

W

 

„   @}  |A}g „|V„  „ |3†|.>‚  ' 

   ,  #



i F%j r>pAEFk ~ S=(b i?Ahq,=+us< e bb =Ag?E

! "$#   2  ! "%# 3 "%#&              "$#(' 3 "%#&     "%#) 3 "%#&      ~~  0 0 0  ~~  ~~  ~ ~ ~          "$# &+* ' 3 "%#&                        ~   ~~          j A ˆ j A . 10  ~~    j A  j A . 1 0  ~   0 0 0   j A        





   

        

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. 3 ˆ 

< = ] < =E=A#b

jA

        

 . 10  ~~  ~  

 . 10  

u6m=(< =@u=C>@?Do? b"j ?Ecbpe A#b =(< ] < =(b/b m= j A 1 n)=(`5A=n@q#D@b m= #   `< Ecb6iF%j r>pAEsFk ~





jA

 . 6?E/b m=

4)*.,/

dFkb m=Cz=i(b F< x e Aob m=@q5?Ee E

9-| 0 ” 0 Š&” 3‘ 0 13 0 0 ”43 * ‹Š 0 “&‹ 1 # 0 “&‘ 1 1 ‹ / 3Œ7’ 0 ‹’ / š?563%“1 ‹43 / 3/” 0 ‘ / š 5”” ’” Šc / ‘ . ”43/“&‹ 0 “&‹ 1 # 0 “&‘ 16Œ (“&Š&‘ 1 ” 0 ž



l

2.5 A N A LGEBRA

G 7 +&=+_& XS? $ &?18+>X4B  :+> (?I8S g[$ &8 8.&  4rA 8S?I8=+7  8S?I &8 7 _&?AB+?A< 7 RA?IT   W.H =+[$8?9=+?AB9 =$8 _&?P+??A c<I>‚>†ˆ…„eaD}3|KxX +&=(g??A?AB$2?T] [$ &8 8?I8 Bb 8@[$ &8 8AW +? 2 &< B$ >?I8 x X 7 7^[O?g(?AB+?I x ' k 7 B.T  - 7 B.T o   €b 1W ?&r$ 7 +&=+_&XS?F?I2<+B 2A 7 7 t&B+7  B+?A?I  - l &?I23&<8PT&U+?@[$ &8 8S&  4r>XS?g2&=+7 B$ 27 =$(?  &4+?g8 Y&? 68 ?&W x

' 

h?I23&T &+



€

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x7 B.T

€



z 

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€

7 B.T



 

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3 

Example.

8S+?gjF ?AB+?I2@? b< e i=E

e E?AhF<b mF%AF< >@?jSq5?Ee E6klF<

-



                             



~

"

2.5 An Algebra



  ]> 7 q[$ &8 8 B $ B$  8TB+>X :O988 [+7 ?tu=++?ATK Y&?t+?27  $ >TXS? $ &? B „>~ ƒ9}  ‚„t& 8E(2<$ &8E 2@K > 2?I8ArO&F `|baD/c  „|bay†dct„>~ ƒ9}  ‚„  W  E=$8E ` 8F27 `rXS?@=++?A ?9= ?T K&?I23&1:+((=$23 x Bd &+  &BT+?g 7 ?I &(K?I8E< [+7 8+?I &?I23& 8:$ &2?zPX4 2<`8 > 8 $?I84 V?AX >e &K8AW  >W ^?   ( )/ ^?  ?A:+?I8?AB9P+? V>X   2 Definition. A  B a A $ B$ b  ?A:+?I8?AB9+?  2&7 =+B` B W +? >g} cjaD>‚ ‚/ #cjaw A " B$ B rO(?AB+?I A B r 8 "

A B

'

"! $# %&(' " ) ' +*-, 

a

b

 ? $ > A B    ?I2<+B 2A 7 7 t8:O?I Y B+_$r9& B 7 _&?A[+< g+?F&?I23&S:+((=$23 8+&=+7t< Y&?1EXST&:O?A< B$+8P&U+?@8 ? &?I23& 8:$ &2? B$t:+((=$2? g <t&?I23&P& $ >48:$ &2?&W  8SK [O?F &2< ?A&?I QXS?F8?' '/.QWhp >XS?A&?AIr +?@?I8=+7 <8:+?I8?AB9?I8E 7 7^+&7 X4+?AB 8 84B++?g2A &8?&W } >g} cjaD>‚V| E„ cg} cjaD>‚ ‚/ #cjai&}.…}g`„   >} c<>…+|bay†Ja&8aD}3|T„>… Proposition.  „|<|A/ c† e„ ay† >}„>~ ƒ9}  ‚„   C ‘ 0 ’#Œ7”‘ 0 Š&”(“ 0 ‘ “2‘ 0 ‘ ’ 1 ‹ŒA: / ”“&”1 ‹’ “&Šc 0 “2“&‹$ * 3 . . š / ‹‘ 14ž 

l

Chapter 2 A S TOCHASTIC A LGEBRA Proof.

 m=(< = ? < =7klF%r)< ? ‡)e F%>pEu6me im}>/rEcbsq,=7E ? b e Ec`5=nJ~

I  P

TI.

S=(b A



.

/

N Q(P

NU



.  1



P,N N P

?A5n

 B C   A B A C ~ 1  ~  S =(b  < = ] < =E=A#b b m= ,  B C  a A



iF%j r>pAEsFk B C ~

A

B

 

TII.

< = ] < =E=A#b b m=

& ' ) ' ) '    & '  ) '   & '  ) '   2  & '  ) '   &('  ) '     & '  ) '    & '  ) '   2   2 

 2   2   2 

C



 c 

Fk A ?A5n b

< Fu

 a  b 

c

a

b

a

c

a

b

a

c

a

b

A B

a

c

A C



P,N N P  A  B   C  A C  B C ~   . 1 1  ~ S=(b   ]  ] S=(b A B  ?A5n C  a b < = < =E=A#b4b m=, 7< Fu6EsFk A B ?A5n c < = < =E=A#b4b m= . iF%j r>pACFk C ~ T7N H,P

/

N Q(P

NU





 



A



B  C

 2  2  2  2 

S=(b

&(' &(' ) '     &('  ) '  2 & '  ) '      &('  ) '   & '  ) '      &('  ) '    2  &('  ) '     a b  c   a

c

b

c

a

c

b

c

a

c

A C



b





Fk B ~

c

B C

 VP,N )V%N MRP,N SO  . 0 A   ,( 0 B   .( 10  A B ~ .  1 ?A5n 1  ~S=(b  < = ] < =E=A#b2b m=,  < Fu A B a

K5MRSVR

TIII.







Fk A ?A5n b < = ] < =E=A#b2b m=

.



iF%j r>pA

&' )' &(' ) '   - &('  ) '   . 2   .( 10 2  &('  ) '  

  2   .  a  -  b   -  2   . a  b  

. 0 A   ,(0 B 



a



b

a

b

 .( 10  A B 

TIV.

QQ S

M5N RP,N N P  A  B   C  A   B  C ~ .    S=(b A    1 _ B    1  _S?A5n C       ~ S=(b a   ?A5n b < = ] < =E=A#b b m= < Fu6EFk A ?A5n B  ~ S=(b b ?A5n c < = ] < =E=A#bsb m= . "iF%j r>pAEsFk B ? A5n C ~











2 

 a  b   

A B

c

u6m=(< =  A  B    < = ] < =E=A#b Esb m=-, /< Fu„Fk A  B ~  =+m5?z= b m5? b



 



&(' ) '  C  0  4 &('  ) '   2  

 A B   C 





  

  A B    c  &  a   b ˆ  0 0 0  a   b    1   j A  ˆ 0 0 0j A    ! ! #   ,#  , !     a   ˆ j A 0 b & " 0  ˆ b  

 )





, 

lIm

2.5 A N A LGEBRA

!

!    0  b  F iF%> ] j =(b =6b m= ] < F#FkSu= / > rEcb2EmFu1b m5? b ˆ jA  B  C   < = ] < =E=A#b Eb m= . "iF%j r>pAhFk B  C ~  + = m5?z=



,!  

ˆ

!

jA



 0

!

 ,#  & b " 0 



ˆ



!

& 

ˆ

# 

0



u6m=(< =



#



     - ' "  -  " - 3  '   ' "   2  " '    "   3 " '     "     2   &('  ) '  

 2

1

b



c



  B C  

u6m=(< =r ] %F A

 A B   C 

2

&' &'

    a   -  ' "  a  0 B C 4  

 2  

 0

)'

b

) '-

& ' 0 

 

'

b

) '  

A  B C 

 m=(< =(klF< =%_,?Ag?EEF#ie ? b e z=d?j f%=q)<?~ 



  B C  

" -  

  !  # ! #  3  ' 

  b   2  ˆ  ˆ





ˆ b

#



?F$ >4+?@:+&:O?AEKh2&V=(< >  E ( ?I8B+4+&7 B 8F|baD/c  „|bay†dcT„>~ ƒ9}  ‚„>W +?A? 8 t‡I}‚1&+?@&?I23&:+((=$23IWhC" A   +?AB Lemma. 

A

A



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' 

$ > 8Ar>+?4RA?A 8f+?p(…$† >‚ f f „eay‚3† &W +?  ?AB$8 &B$86  V=$8Eh[O? 2<+98?ABVF[O? 2&B$8 8E?AB9 X +?@&?I23&:+((=$23IW +?A? 8 Bbd† &}…ay†Ja  &+?@&?I23&:+((=$23IWhC" A    +?AB Lemma. 

A

A 

'

A

'

A

$ > 8Ar++? (?AB9 E 8+?`} >…}…ay† „>~Q†d&}…ay†Ja  W +?V ?AB$8 &B$8   V=$8E1[O?V2<+98?ABt[O? 2&B$8 8E?AB94X   +?@&?I23&:+((=$23IW   ( )     +?A? 8 Bq†ˆ… >}‚<|A} &88(2  >?I`X d+?T&?I23& $ # Definition. :+((=$23IW6C" A   +?AB A

A

A

A

'



' 

X4+?A? A 8V+? >}gcjaD>‚ ‚/#cjag†ˆ… >}‚<|A}T A W +?d ?AB$8 &B$8T  V=$8E[O?d2<+98?ABcb[O? 2&B$8 8E?AB94X `+?g&?I23&4:+((=$23IW ?&r(+? B &?A<8?@8 V[O&7 8+?g8 ?g &8$ >4& K > e 7 _&?A[+< +W@C"@X 7 7h[O?t27 ?I 1& 2&B9?e FX4 2< B & ?A<8?g`=$8?&WH :O?I2 O2A 7 7 &rw+? B &?A<8?TS B  8E(2<$ &8E 2K > eX 7 7 7 X (8h=$8?1 g8E(2<$ &8E 2 B &A? <8?&WfG &V=+7 F&.+? B &?A<8?4X 7 7$[O?1_ &?AB B` 7 >?A 8?I23 &B^W

LM

Chapter 2 A S TOCHASTIC A LGEBRA

2.6 An Outer Product

C" X 7 7^[O?@?e ?A?A7 t=$8?=+7wK(?$B+?g Bc#8aD}‚ ‚ /#cjayW )  ( )/ ^? ' k T  o  .2 ' k V  o 4W +?]#8aD}‚ ‚/#cja. Definition.  $ x y x  M B$ y r$(?AB+?I x   y r 8   '      ( x y X4+?A?PXS?SB+?S$ > x   y    W +?S&=(?A6:+((=$23h2A BV[O?_&?AB+?A< 7 RA?IFF8E(2<$ &8E 2.K > 2?I8 xB+ E=$8E 2&7 =+B$8
 

B

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' 



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* €

 

* €

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X4+?A?@XS?gB+?@$ > A   B   W + ?g&=(?A1:+((=$23 2A B`[O?g=++?A1_&?AB+?A< 7 RA?It B$27 =$(? ‚}<„>~  +(7 ?@K > ewW ^?,' k    o   ^W A





B



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* " €

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* " €

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+ ?@&=(?A4:+((=$23 X 7 7^[O?I2&?F?e ?A?A7  =$8?=+7^X4+?ABd(?$B B+_K:+ E?I23 & B$8S&B9K K8=+[$8:$ &2?&WhG 8?A ?I8 (?AB9  ?I84?AB$8=+?I8AW ^? v ' k  o2 x ' k T  o   b B$ u ' k h  o2 y ' k V  o2 z ' k   o  4W ^? 2  1W  2   W x  z QI. x y z  y z x  x   y   z   ! "! "%#    z Proof.  ) ,+   * $ "

% ( & ' "$- -   #" 2

x

 

QII. x

QIII.

x

y

z

z



Proof. x

   

z

y 

Proof.

x x   zz

y

+  2  ! "%#   "$   "$- -    +  2   "$   "!- -     2   y z    y z 0 2    y z 0 x

W

x .Qz A x x   yz3W

x

  y  . 0 x

z

  y z 0 x   z y 0 x   x   z 

  (0 y  

z

  ( 0 y z  0. 0 x    y .( -0 z  0 x   x   y     .( -0 z    . ( -0  x   y    z

 

y



Ll

2.7 S UBSPACES

x xz

x u yz

 

QIV. v



v

Proof.

x v   uz x v   yz x x   uz x x   yz3W

  x   u  y

u

 y z 0  v  x   u z 0 v   u z 0 x   y z 0 v   y z 0 x    v   u   v   y   x   u   x   y 

 z



 z

2.7 Subspaces

b?@$ &? ?I8E< [+7 8+?I $ >4+?g8?4h8E(2<$ &8E 2 K > 2?I8 >r 

84 T&?I23& 8:$ &2?F6 ?AB$8 &B - l W b? 8 X ?I 7 ?Af$ >.XS?4K g?e(:+?I88f?A&?Ag&?I23& B  ` &8h =+B 9=+?47 B+?I .2&V[ B$ > &BT - l [$ &8 86&?I23&<8AW b?  B+ &+(?I886+? 88=+? wX4$ >S$ :+:O?AB$8.X4+?ABK&B+7 &T[$ &8 8.&?I23&<8. ? > 7 [+7 ?&r&X4+?A? M & bW +?@?I8=+7  84 |  | „Ec<}  4r>h ?AB$8 &B &W

^?  ($9 ^' l   &[O?F T8?SQ7 B+?I 7  B$(?A:O?AB$(?AB94&?I23&<8AWf 1? $B+?1+?F8? 2   j2  X4+?A?  b€ b* b  &8  * M'  '     2  <$9 :' l   g&,#    x x b 

Proposition. Proof. u6m=(< = SI.

SII.

.

}g|A}ja

€

„|+&}.…}g „   >}V†|V„K|  | „Ec<}TE  ' 

 m=(< =@? < =/b m)< ==p? ‡)e F%>pE+u6me im{>/rEcb7q,="E ? b e Ec`5=nJ~S=(b x  (  +* ,   ~ S=(b ~  *

$ OJM5V

   I  I)MP

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QN SO 

V SQ 

I

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V SQ 

I

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x

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x SIII.





4r



y

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,    ˆ   .  1 0 b  

 m=(< =(klF< =%_,?/ErqE ] ?i=%~

b?g2A B` 78K8:O?I Yt6 K8=+[$8:$ &2?F &8S+?|



„>…K6[$ &8 8&?I23&<88=$2< $ >

M'

8:  b € 2   j2 b*  W

365 0 :#)1 ” 0 Š&”3‘ 0 13 0 0 ”43 * ‹Š 0 “&‹ 1 # 0 “&‘ 1 1 ‹ / 3Œ7’ 0 ‹’ / š 563%“1 ‹43 / 3/” 0 ‘ / š 5” ” ’” Šc / ‘ . ”43d“&‹ 0 “&‹ 1 # 0 “&‘ 1Œ (“&Š&‘ 1 ” 0 ž

L&L

Chapter 2 A S TOCHASTIC A LGEBRA

2.8 Projections

+ ?A?T ?V8?A&?A< 7 :O&< B9 :+7 2A > &B$81.XS&Y B+_tX u B B+B+?AF:+((=$23F8:$ &2?&W +? $ <8E 8 $ > XS? K  ?I  >?A7  82=$88S+?g2&B$2?A:(4h ‚ }gcjay† >…$W Definition.

+? ‚ }gcjay† >…tf&?I23&

(   

:+ 

2



'

x

x 

]&B9B+&B

"RA?AT&?I23& 

p 



b?@K  78(?$B+?F+? >‚ba  ƒ9>…„>~:c< f! ~ } f }…aQ6 T:+ E?I23 &B` &8S&7 7 >X 8AW * ,  8+&, ( & / ^? 2 4W ^? 'i:+ Definition.  x p N y 

x 9

:+ E?I23 &B  x &B9 p W +? >‚ba  ƒ9>…„>~:c< f! ~ } f }…a^6&?I23& y r$(?AB+?I y  p 8 '



x

8

x p 2 p p p



p

y







[O?T+?

y

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x x  yz ' x  2  ' M y y  >+?AV$ Bq:+ E?I23 ]&?I23&V&B9b B++?AV&?I23&Ir6XS?2A B  78u:+ E?I23 &B9b u8=+[$8:$ &2?&W  8TX 7 7 ?9= ?@ ‚ }gcjay† >… f „eay‚3† &W (       ^? ' 8:  € 2    2  X4+?A?  $9 ' l   &`[O? Definition. b b* b   `8=+[$8:$ &2?  W ^? B ' k b €  b* o M * $ &? &8@2&7 =+B$8 +?t[$ &8 8 &?I23&<8F W +? ‚ g} cjay† >… f e„ ay‚3† &r P    r$ &88(2  >?ItX  8 y

y



P

X4+?A? 

2



B B 

Proposition.

:+ x .†|

k  b 2 b "o   ' 

* *



y

B

 

2



B B 



B

W

} ‚ }gcjay† >… E > }gcjaD>‚



'

'

x 





>… |  | „Ec<}

:+ x ' P x  „eay‚3† „|<|A/c† „eaDg}  h†Ja  

  }‚}

P

†| a  } ‚ }gcjay† >…

Proof.

Ee Af">@? b< e i=En)=(`5A=nh?q,F z=%~S=(b y q,=+b m= ] < F ! c=i(b e F%AgFk x F%A#b F

f

y

u m=(< =db m= 6 b m5? b



,   

E„| &}.…}gc„   >} &}…eaD}g

< F   ]  

x

,  0  b

 



~  m=A

ˆ

? < =db m=diF#F<n)e A5? b =E Fk y ~  =dE=j =i(bd?AFb m=(< ? < qe b<? <DLz=i(b F<_ v ! v

,    ˆ  0 b





_Erim

L

2.8 P ROJECTIONS





u6m=(< = b m=  ,    n)e irj ? <6b F v Erimhb m5? b

? < = b m=+iF#F<n)e A5? b =E6Fk v ~  m=7F<b mF%f%F%A5?jSiF%> ] j =>p=A#bFk y >/rEcbq,= ] =(< ] =A)v

 v x & y

!

,

   ! ˆ  0 b x & y      ˆ  b x & y 

 ,  

e Ai= v us?E6E=j =i(b =ng? < qe b<? < e j Dh>/rEcb6m5?z=

 b x & y   b x 



 

 b x 

u6m=(< =

 

b m5? b

 b y  !



 b , 

   ˆ  b b 

 b x 

&Ah>@? b< e ‡pklF< > b me E6q,=iF%>p=E

 

!

 ˆ  0 b 

      b x        b b    b x  !   !  )  b x  )   b b     B B      * ~ !  m=Ahu=7m5?z=   0 b  !     b x  0 b ˆ 

b b

,



ˆ



ˆ!

,

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  0 b x  0 b 

   0 b 



b



x

,  !   0 b    b   ˆ  

,    ˆ

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x

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2.9 A N I SOMORPHISM

2.9 An Isomorphism

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Chapter 2 A S TOCHASTIC A LGEBRA



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2.10 D ETERMINANT, R ANK , I NVERSE



2.10 Determinant, Rank, Inverse

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Chapter 2 A S TOCHASTIC A LGEBRA

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2.11 T HE E IGEN P ROBLEM

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Chapter 2 A S TOCHASTIC A LGEBRA

2.12 A Calculus

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2.13 Probability Distributions

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2.14 J OINT P ROBABILITY D ISTRIBUTIONS

2.14 Joint Probability Distributions

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Chapter 2 A S TOCHASTIC A LGEBRA

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2.15 T HE B IASED I NVERSE

2.15 The Biased Inverse

C"fXS?@2&B$8 (?A A    [O?g T2&B$  &B$ 7w:+&[$ [ 7 Et 8E [+=( &B +?AB`XS? K  8:O?I YK y ' Ax X4+?A? ' k .fx  z"of B$ ' k .fx   "z of B$ ' k .fx    z"o"W y x A    $ #     ^? A ' k .fx    z"o    W +?  † „|A}g†ˆ… >}‚<|A}r^(?AB+?I A r Definition. 8  ' k .f x    z"o A X4+?A? A    W +?V[  &8?I B &?A<8?@P t8E(2<$ &8E 2@K > e 81&?A :O&< B91X4+?AB]XS&Y B+_K[$ &2?A7 &r A 2A B+B+V[O?2&:+=(?Ic ?I237 b& A &8gXS? (]B+ Y B+>X +?g2&?I23[  &8AW G18S+?@B$ ?I84 7 7 =$(?&r(+?A? 8 2&B+B+?I23 &B[O?EXS?A?AB+?@[  &8?I B$ =+B [  &8?I B &?A<8? &:O?A< >&<8AW } (…  † „|Ag} d†ˆ… >}‚<|A}K†|K‚}~ e„ aDg} aD L  a  }  † „|Ag} d†ˆ… >}‚<|A}t†ˆ9 … a  } >~ˆ~  h†ˆ… ƒ P„  >‚ Proposition.   „>… f e„ ay‚3† A ' k   o     xyL(W l z A  A X   }‚} X     c<„>…  } a  # &ƒ  aEg„|g„  † „|   d† c   „|  }<}… >g} cjaD>‚3† „>~ˆ~  E„ E&g} (aDn  a  p} (…  † „|Ag}  †ˆ… >}‚<|A}  Proof. :)< F%>

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Chapter 2 A S TOCHASTIC A LGEBRA

2.16 A Nonlinear Operation \] > ebV=+7  :+7 2A > &B €  :+7 (8g `7 _&?K&7 ? Bs 8g< ?AXS&YWGa8E(2<$ &8E 2K > esK u[O?K+&=+_&9V

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2.17 T HE U NBIASED I NVERSE

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2.17 The Unbiased Inverse

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Chapter 2 A S TOCHASTIC A LGEBRA

2.18 Information Theory

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Chapter 2 A S TOCHASTIC A LGEBRA

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Chapter 3 S TOCHASTIC C ONTROL S YSTEMS



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Chapter 3 S TOCHASTIC C ONTROL S YSTEMS

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X4+?A? h*+.-/ $&n' l   ) X4 2<:+((=$2? 7 (2A 7w2&B9&78Ar w* r(_ &?ABt+?F:$ &8E7 (2A 7&[$8?A>   &B$86[O+?@_&7 &[$ 7w8(8E?A 8E< >?&r y* r+ B$ +?@_&7 &[$ 7w8(8E?A 2&B9&7yr v* W 

 h€ k 1 o' w* h*  y* v*

2   j 2  2 h k 1 o2   j2 y* k M>o2 k 1 - l o2   j 2

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k M>o



x  W L &z

  2?@$ > ?I &2<]2&B9&7 7 ?A 8 =+B$23 &B]6+?V+?A 2&B9&7 7 ?A<8AW  8 81?I B9  B$ 2A >?g&B+?gE :O?g 2&=+:+7 B+_T$ > (2A2=+<8[O?EXS?A?AB`2&B9&7 7 ?A<8SX4 2<X 7 7^[O?@?e(:+7 B+?I8+&7 &W 

3.3.4 Separation

+ ?A? 81=+B$+ ?AB9< 76?I8=+7 FX4 2< 8 ˆ?ABu?e(:+7  ?I Bb2?AB9< 7 RA?I]2&B9&7yrB$ ?A7 ]8?A:$ < > &BdS8E< >? ?I8E K >&.(?I8 _&BK B$K2&B9&7(7 X ( ?I8 _&B^WfC"S 7 7 >X 86&B+?g(?I8 _&BK F=+7 ?4X4 2<?I8E K >?I86+? _&7 &[$ 7+8(8E?A 2>C ”–|‹Š 3 “ 36: / ”42–2‘ / / 5”(3 0 ”436“&‹3” 0 1 Š&‘ 5”| Š&‹436: ‹)*Š 5‘ “&ŠcŠ š 0 ‘ . ” ‹)*) 0 0 ‹ 1 ‘ (“&”43 Œ (“&Š&‘ 1 ” 0 ž 5 “J‘ 0 3” Š&‘ • ”43 * Š&‹Œ 3‘ ’7%   › 0 ”*= % 



› 0 ”:7% 



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›5ž žž

 

3.3 M ARKOV D ECISION P ROBLEMS

E8 < >?&r x k 1 o"r B$(?A:O?AB$(?AB97 `6+?T2&B9&7^7 X@r$X4 2<d:+((=$2?I8 2&B9&784[$ &8?I`&B` 818E< >?g?I8E K >?V8=$2< $ > kx 1 o ' h y k 1 o 2-   j2 y k M>o2 x  WL n z ku 1 l o 2    2 u k M>o k 1 o ' h    x k 1 o  x  WL  z u 

X4+?A?

h

 



8+?g8E< >?F?I8E K >& B$

8+?g2&B9&77 X@W



h  #

… c<}…ay‚„>~ † ‡I}g c<>…ay‚>~ a  }t&}3|3† ƒ…cEV„|baD„eaD}T}3|bay† f „eaD>‚ E  A| }‚ >}‚  #8aP†ˆ…&} }…&}…ay~  EZa  }t&}3|3† ƒ…cEg„ c<>…ay‚>~w~ „  

Theorem 1.

Proof.



 me E6us?E6EmFu6A}q#D \  cE b< F%>

 !"  klF<

 

f

„   t} c<„>‚3‚3† }g 

B   E~

 86?A7 ?I8f&Bg+?S &236$ >6+?4:$ &8E6_&7 &[$ 7(2&B9&78Ar u k 1 - l o 2   j2 u k M>o"r ?SY B+>X4BV:O?A?I237 gX4 2< 86=$8=$ 7 7  T_& ( &88=+:( &B & 2?AB9< 7 RA?I2&B9&7yW p >XS?A&?AIr^&T(?I2?AB9< 7 RA?Ic2&B9&7. 8 8gB+TB+?I2?I88  7 b8$W   7 ? ]2&B9&7 7 ?AgK b$ &?t:O?A?I23 ?I2A 7 7$ <8P>X4Bt:$ &8EP7 (2A 72&B9&78Ar w* k 1 - l o 2   j2 w* k M>o"r SK B+S$ &? &2A2?I88hg+?1:$ &8ES7 (2A 72&B9&78  +?+?An  ) - l 2&B9&7 7 ?A<8AW  8 8TX4 b+? c<>… ay‚>~ €  |A}‚ >e„ ay† >… f „eay‚3†dc<}3|V ? Bq+?` 1?I2A5 \]

5 ((?A7yW]G c<>… ay‚>~}3b| ay† f e„ aD>‚FK s 78d[O? B+?I2?I88 d]?I8E K >? X4$ >V_&7 &[$ 7P2&B9&78@XS?A?K=$8?I Bc+? :$ &8EIW N =++?A&?&rP+?u2&B9&74?I8E K >& 2A B [O?u b=+B$23 &B F+?]2&B9&7 7 X 8 +?]+?A _&?AB9<8 x&B+?b2A B _&=+?I88K+?s2&B9&78t@+?b+?A` _&?AB9<8 [O??A V&B+?u$ &8 &2A2?I88Kq+? d2&B9&717 X 8?T?I8E K >&@ B$b 2&B9&7f?I8E K >&@ ?T? 9= ?I Bu+?98EF_&?AB+?A< 7 2A &8?&W N =++?A&?&r+?t(?I8 _&BbP+?t2&B9&76?I8E K >& 8@ =+B$23 &BsP+?7 (2A 7.2&B9&767 X 8 4 7 7 ) 2&B9&7 7 ?A<88V$ > 

  h   € 2    2 h     " " k 1 o ' h * x* k 1 o2 x W L 9z u* "  k 1 o2    2 v* k M>o v* k 1 o2    2 y* k M>o2  k 1o ' y*  x WL m z x* h * k 1 - lo " u* k 1o '    k1 o x W M9z w* h * x*  " X4+?A? h * ! 8g+?2&B9&7.?I8E K >&Ir h *   8g+?8E< >? ?I8E K >&V B$wr h   *  8g+?  ! " " " 2&B9&7Q7 X5$& ' l   q ) W  =$8 Bu(?I2?AB9< 7 RA?Id2&B9&7yr$+?T(?I8 _&Bdh+?T8E< >?V?I8E K >&<8 812&=+:+7 ?I`







 









+?(?I8 _&BdS2&B9&7 7 ?A<8+&=+_&`+?2&B9&7Q?I8E K >&<8AW  8 81[O?I2A =$8?V?I &2<u2&B9&7 7 ?A1( ?I81B+ $ &? &2A2?I881+?K:$ &8Eg2&B9&78 P+?K+?A0) - l 2&B9&7 7 ?A<8F B$]$ >g [O??Ag?I8E K >?KP 8@=+B+Y B+>X4B B(&K > &BKK T[O?F &2< ?A&?ITX  Y B+>X47 ?I(_&?4w+?1+?A2&B9&7+7 X 8AW  8S2&=+:+7 B+_gK Y&?I8h+?1_&?AB+?A< 7 (?I2?AB9< 7 RA?I@2A &8?PV=$2<g&?P t2=+7 f$ BT8 :+7 @8&7  B+#_ )a:+&[+7 ?AK8Q(+?S2?AB9< 7 RA?Ig t2=+7 E&W +?A? ?&r(+>XS?A&?AIr TB =+V[O?A 68 :+7 O2A > &B$8X4 2<d2A B[O?@K &(?F?I2>&?AP+?g8?A:$ < > &B :+&:O?AE&W +? $<8EP8 :+7 O2A > &BKX4 2<K?A_9 B$8.8?A:$ < > &B 8hV &88=+?$ >P+?12&B9&7+?I8E K >&<8.(@B+P$ &? &2A2?I886F+? 2&B9&7(7 X 8AW CDBK(?I2?AB9< 7 RA?I2&B9&7yr&X4+?ABt2&B9&7(?I8E K >&<8.(@B+P$ &? &2A2?I886g2&B9&7

n

Chapter 3 S TOCHASTIC C ONTROL S YSTEMS

7 X 8Ar(XS?F$ &? k 1 o2 x  W +l z u* h * k 1 o 2    2 k M>o " v* v* k 1 o 2    2 k M>o2 k 1 o ' h * y * - y * x  W  L&z x* k 1 lo " u* k 1o '    k1 o x  W   z w* h * x *  " X4+?A?`XS?d8?A?`$ > 8?A:$ < > &B 1+?](?I8 &_ B F2&B9&747 X 8ArS8E< >?`?I8E K  >&<8ArS B$ 2 &B9&7?I8E K >&<8 ?I8=+7 <8AW G B++?A8 :+7 O2A > &BqX4 2<q?A_9 B$8g8?A:$ < > &B 8gb &88=+?K$ >T+?2&B9&7 7 ?A<8g ? [+7 ?Ku:O?A  ?I237 &[$8?A&? ?I &2<K+?A<8.:$ &8E2&B9&78Ar&:O988 [+7 T+&=+_&t2&V=+B 2A > &B^W CDBt(?I2?AB9< 7 RA?It2&B9&7yr&X4+?AB 2&B9&7 7 ?A<8S$ &? :O?A?I231 &2A2?I88P+?@:$ &8E 2&B9&78Sf+?@+?A 2&B9&7 7 ?A<8PXS?F$ &? '  ' '  ' x  W   z D€ D D 1 ?AB$(?A B+_V+?g2&B9&7w?I8E K >&=+B+B+?I2?I88 &W b?F+?AB`$ &? k 1 o ' h * y* k 1 o 2 -    2 y* k M>o 2 x  W  &z x* k 1 l o " u* k 1 o ' h   *  x * k 1 o  x  W &n z w* " X4+?A?FXS?@8?A?F$ > 8?A:$ < > &B f+?g(?I8 _&Bh2&B9&77 X 84 B$8E< >?F?I8E K >&<8?I8=+7 <8AW G B++?A.8 :+7 O2A > &BX4 2<K2A BT[O?4K &(? 86$ >.&B+7 t‚}}.2&B9&7 86@[O?4=$8?IwW 4?I &23 &?42&B9&7 ??A<8Q@ F2&B9&7 7 ?Af&B+7 g=$8 B+_ <8698Eh?I2?AB9h&[$8?A> > &B BV+? 8E< >??I8E K >?&W CDB(?I2?AB9< 7 RA?Ig?I &23 &? 2&B9&7yr9+?g2&B9&7w?I8E K >& 84=+B+B+?I2?I88 t B$ XS?@$ &? k 1 o ' h *  y * k 1 o  x  W  z x* " k 1 o ' h   *  x * k 1 o  x  W  9z w* " X4+?A?FXS?@8?A?F$ > 8?A:$ < > &B f+?g(?I8 _&Bh2&B9&77 X 84 B$8E< >?F?I8E K >&<8?I8=+7 <8AW k1 o

'

x*







3.4 State Estimation

b?

$<8EV2&B$8 (?AF+?t2A &8?42?AB9< 7 RA?Iu8E< >?K?I8E K > &B^W +?K_&9 7h48E< >??I8E K > &B 8Fd2&?K=+: X d B?I8E K >?@f+?F_&7 &[$ 7^8E< >?Ff+?g8(8E?A`r x k 1 oQ[$ &8?I &B:$ &8E4&[$8?A> > &B$8 B$`2&B9&78 kx 1 o' h y k 1 o2 -    2 y k M>o2 x  W &m z ku 1 l o 2   j2 u k M>o CDB+?g8E(2<$ &8E 2F 7 _&?A[+< +r K8 B+_&7 ? D8E?A:bx& 8E< > 2Iz8E< >?@?I8E K >&42A B[O?g2&B$8E=$23?I &8 k 1 Ho ' C y k 1 Ho ' C Cx k 1 o x  W 9M9z x $7 ?AK+&=+_& X4 2< +?`=+?d8E< >? 8K&[$8?A&?IwW G18XS?d2A B 8?A?&r X4+?A? C C K q[O?`+&=+_&9K@ &8  8@?I8E K >& 8F[$ &8?Iu&B xyL(W L&z‚ f ~   † „|Ag} ]†ˆ… >}‚<|A} & H ?I23 &BcL(W l (rQ5h _&?  (W  8g8 B+_&7 ? 





3.5 C ONTROL L AWS

8E?A:`8E< >?@?I8E K >?FK K[O?g2&V[ B+?I X  d B&78E< >?F?I8E K >?FK  &? > T?I2=+<8 &?F8E< >?F?I8E K >& k G1 8E&`r lIm&n Ao k 1 l o' A k 1 o x k 1 o C y k 1 l o x  W l z x o ' A x u k 1 oz3W   2?`+?]&7 ?dF+?]:$ &8Et_&7 &[$ 712&B9&7 u k 1 o B (??A B B+_ A k 1 o"W p ?A?d+? X4+?A? A k 1 n ?I8E K >?@ >S+?F&7K ?F$ &8S[O?A?AB`[+&=+_&9P&X < B  ?F=$8 B+_V+? < B$8  &B K > e B$t[O?I2&?I8S+? [  &8&4+?g?I8E K >?g >4+?gB+?AX  ?&W b?V &88=+? x k M>o ' TW  8 84X4+?A?F+?@K B] Z?A?AB$2?@[O?EXS?A?AB 2?AB9< 7 RA?Id8E< >?V?I8E K > &Bu B$u(?I2?AB9< 7 RA?Id8E< >?V?I8E K > &B](2A2=+<8AW1CDB]+?T(?I2?AB9< 7 RA?Id&?A<8 &B`XS? $ &? A k 1 Ho ' A x u k 1 oz6X4+?A?+? :$ &8EP_&7 &[$ 7+2&B9&7 8hB+>X ?I8E K >?I &8.?I &2<K2&B9&7 7 ?A.( ?I8hB+PB+?I2?I88  7  Y B+>X X4 2<`2&B9&78SXS?A?F?Z?I23?I[ K+?@+?A! ) - l 2&B9&7 7 ?A<8AW 3.5 Control Laws

H &? 82=$88 &BsV=$8ET[O? _ &?ABsuX4$ >2&B$8E =(?I8V :+:+&:+  >?K[O?A$  &=+V+?`8(8E?A`W  88?AB9  7 7  +?A?b ?dEXS :O988 [ 7  ?I8AW N <8EIr XS?uX 8  =$8?b2&B9&78 _&? +?s8(8E?A B9 q:$  2=+7 t_&9 7 8E< >?&W  8 8S+?g:+&[+7 ?Aa!c<>…ay‚>~ˆ~ „  †ˆ~ †Ja  W4H ?I2&B$wr(+?A? 8 8&?g2=+V=+7 > &?F?AX <K=+B$23 &Bd>&?AS+? 8E< >?I8 >2&B9&78X4 2<dV=$8E [O?gK >e  RA?IwW  8 84+? 9=+?I8E &Bd1 ay† f „>~ c<>…ay‚>~ W N &1B+>X XS?V &88=+? +?t7 >?AV8 =$ > &Bs[+=(T8&?t 82=$88 &BbX 7 7P[O?K_ &?ABb]2&B9&7 7 [ 7 E]7 >?AIW  =$8g &88=+?+?A? 8g 8E< > &B$ K?AX <K=+B$23 &B^r x x 2 uz3r+>&?AS+?g8E< >?I8 >2&B9&78 k  







x  W  L&z



N =++?A&?&r( &88=+? XS?@ ?1 B+_T B  RA? +? B $ B ?F+& RA&B2=+V=+7 > &?@ 82&=+B9?I 8=+`r    r(  8=+B$23 &B  '

'

 

x x k M>o2 u k M>oz  x x k l o2 u k l oz   x k 1 o2 k 1 oz x u

 



x  W  z

X4+?A?   k M2 l z 8+?F=(=+?g 82&=+B9 2&B$8E< B9IW S?A7 7 K B k lIm   oh8+>XS?Iwrw=$8 B+_`( B$  2T:+&_&<  B+_$rO$ >F& +?K2?AB9< 7 RA?Iu2A &8?TX  C ' 1 x B \]

5Szg$ >K+?`&:( K 7 2&B9&77 X 8K8E< > &B$  xˆ ? B >   B93zV B$q$ >+?A? 8K 7 X (8V&B+?`[O?I8E x&T ?AXU? 9=$ 7 7 b_& ($zF &23 &B$8@][O?t< Y&?ABb& ?I &2<q8E< >?&W +? _&?AB+?A< 7.2?AB9< 7 RA?Ic2A &8?]x C ' 2 1 . 5 \]

5Sz 8fV=$2<T&?S2&:+7 2A >?IT B$ B &&7 &?I8^< B$8E& B+_ +?:+&[+7 ?A B9F 2&B9 B =+&=$868E< >?\]

5 X4 2< 8g=+7 7 s&[$8?A> [+7 ? k G1 8E&`r lIm&n Ao"W +?2&B9 B =+&=$8g8E< >? \]

5i:+&[+7 ?A X &8g8&7 &?Is[ sp >X < k lIm&n M>o"W 4?I2?AB9.?I8=+7 <8 k \] &+ B w?1 7yW r lIm&m&m o"r9+>XS?A&?AIr>$ &?48+>X4BT$ >.+? B $B ? "+& RA&BK5 \]

5c2A &8? c[O?]_&?AB+?A< 7 7 q=+B$(?I2 + [+7 ?qx :O988 [+7 ?dc:+>&?d c2&B9&747 X ?e 8E<8 B +?]_&?AB+?A< 7 5 \]

5U2A &8?z3W  ? +?A? ? +?A2=+V=+7 > &? ?AX <b=+B$23 &B$8 > 7 [+7 ? B$27 =$ B+_s &?A< _&?t2=+V=+7 > &? ?AX <c B$ $B  ? "+& RA&B2=+V=+7 > &?F?AX <wW  8?I8=+7 <8 B` Z?A?AB94E :O?I868&7 =( &B$8PT+?@2?AB9< 7 RA?I :+&[+7 ?A`W b?X 7 76?e+  B+?&B+7 `+? B $B ? "+& RA&Bb2A &8? &8  8F?I &8 ?A1K B$ _&?T 8E< > &B$ d2&B9&7f7 X@W +? (?I2?AB9< 7 RA?I ?I8=+7 <8ST&7 7 >X 8+&=+7 ?e ?AB$ +?I8?F+?A 2A &8?I8AW CDBu+?t 1?I2A5 \]

5 < ?AXS&YrO+?A?t ?TB+g2=+?AB97 u B d?I8=+7 <8F:+> B+_ +?K?e 8E?AB$2?(?I2?AB  < 7 RA?Is2&B9&767 X 8F&g Bc [ < d:+&[+7 ?A`W G18g?AB9 &B+?I Bb+? B9((=$23 &Bud 8g2<$ :(?AIr^+?



Chapter 3 S TOCHASTIC C ONTROL S YSTEMS

:+:+9 &2<+?A? X 7 7O[O?1g:+?I8?AB98 =$ > &B$8.X4+?A?1(?I2?AB9< 7 RA?IK2&B9&7 7 ?A<8.2A B 2A V&=( (?AB9 2A 7$_&7 &[$ 7 2&B9&78St K2?AB9< 7 RA?I`2&B9&7 7 ?AIW  =$8Ar+XS?@K  [O?V [+7 ?Ft2A : < 7 RA?@&B`?e 8E?AB$2?@:+  8&4+?g2?AB  < 7 RA?IK2A &8?&W6G18PXS? X 7 78?A?&r& 8.X 7 7 B &&7 &? 2&V=+B 2A > &BK[O?EXS?A?ABK+? (?I2?AB9< 7 RA?I2&B9&7 7 ?A<8AW +? B+?e  8?I23 &B`X 7 7^?I > +?V8:O?I2 O2g2A &8?@‚}<„Ecjay† >}12&B9&7yr+X4+?A?@&B+7 t+?g98E1?I2?AB9 &[$8?A> > &B$84 ? =$8?I B +?g2&B9&7w:+(2?I88AW 3.6 Reactive Control

& ? 2&B9&7 B`8&?F(?A:(^W.S?A7 7 K B 8 k lIm   o &:( K 7w8E< > &B$ K2&B9&77 X 2A B[O?@?A:+?I8?AB9?I &8 1

b?@2&B$8 (?AS+?@2A &8?1f?I &23

&+?g2A &8?F

C

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u

k 1 oH'

Fx

x  W   z

k1 o

X4+?A? F    8f++? c<>… ay‚>~+~ „  f e„ ay‚3† &W b?$ &?42<+98?ABgF?A:+?I8?AB96 86K > eT &86_&?AB+?A< 7 7 g8E(2<$ &8E 2 &8@+?A?2A Bc[O? &?K$ Bq&B+?t?9=$ 7 7 b_& (c &23 &Bcd&7 7 >X & d:$  2=+7 V8E< >?&W  8gX 7 7S$ &? <  O2A > &B$8f&P(?I2?AB9< 7 RA?I2&B9&7 k S&=( 7 ?AIr lIm&m&n o"W6p >XS?A&?AIr BK?A&?AV2A &8?4w&?$ B&B+? ?9=$ 7 7  _& (d &23 &B]XS?V2&=+7]2<+ 98?@&B+?Vh+?I8?gK&7 7 >X &B+7 d B$`+?V?I8=+7  B+_ 2&B9&7Q7 X@r F     XS&=+7 8E 7 7^[O?@&:( K 7 k 1 o ' Fx k 1 o x  W &z u  81X 7 76:+> (?V t=$8?=+7h8 :+7 O2A > &B]7 >?A [+=( @8+&=+7d[O?TB+?I`$ > FK  B &&7 &?g&?VXS&Y B +?t(?I8 _&Bs F WtS?&?K> B+_&Bb](?I2?AB9< 7 RA?Ib2&B9&7hXS?K Y&?T+?t&[$8?A> > &B]$ >V 7 +&=+_&  X &8 &88=+?I`$ > C ' 1 `2&?T=+:uX u 8F8&7 =( &B^r+?A?T2&=+7]?e 8E C  E  ' 2 1 $ >@8E 7 7h 7 7 >X [O? :+7 ?A?AB9?I]:O?A?I237 &W@CDB] 8F2A &8?&r$+?K2&B9&7Q7 X V=$8EF[O?T[$ &8?Id&Bb 8E< >?V?I8E K >?T< >+?A F $ B+?@:O?A?I2318E< >?@8T$ > k 1 Ho ' Fx k 1 o x  W n z u vP&V[ B B+_dx  W  n zPX cx  W 9M9zPXS?@  &?F > u

k 1 o'



FC Cx

x  W  z

k1 o

vP&:$  B+_`x  W  n zPX cx  W   zPXS?@  &?F >+?F&7 7 >X B+_K8E< >?A?AB9IW Theorem 2.

} a&  +

~}

†|V} „Ecjay~  † f! ~ } f }…aD„  ~ } "†ˆ…

a

}  

 F2 C   c<„|A}K†  

F  FC C

„Ecja&$„>~f

 †|ta  }(c<>…ay‚>~6~ „  f „eay‚3†  &}3|3† ƒ…}g@>‚ta   |A}‚ >„eya † >… ‚ }gcjay† >… f „eay‚3† %

Proof.

rqEcb e b r)b e Af F

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F  



FC C e A#b F

  ~ $   < =Erj b E6e A   ~ $!$ (~

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C

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5q ?I237 +z 842&:+=(< > &B$ 7 7 tV=$2<`&? B &&7 &?IwW C"$XS? &88=+?P+?42&B$  &BV +?A&?A  868 > 8 $?I xˆ&6+?2?AB9< 7 RA?IV2A &8?z3r+?ABV+?A? 86 1K :+: B+_ [O?EXS?A?ABK+?F  _&< K8ArOx  W l W  z. B$]x  W  &z3r( B$KXS?1K K8?A? +?F2&:+7 ??1&7 ?1Q2&V=+B 2A > &B B (?I2?AB  < 7 RA?I?I &23 &?@2&B9&7yW +?@:+(2?I88.(?I2?AB9< 7 R B+_K2&B9&7^2A B`[O?g2&=+B9?A?I[ t+?V2&V=+B 2A > &B x B$K =$842 &< B$ > &BOzhQ7 (2A 7w2&B9&78P[O?EXS?A?AB2&B9&7 7 ?A<8AW +?F:+(2?I88PQ:$   7 7 &[$8?A B+_T &23 &B$8 K t[O?g2&=+B9?A?I [ K+?g2&V=+B 2A > &Bsx 2 &< B$ > &BOzP6&[$8?A> > &B$8S[O?EXS?A?ABd2&B9&7 7 ?A<8AW +?A? 84 B 2?g(=$ 7w[O?EXS?A?ABd2&B9&7w B$ &[$8?A> > &B^r([O`[O? B+_K8 :+7 $?I +&=+_&`2&V=+B 2A > &B^W 3.6.3 Decoupling the System

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Chapter 3 S TOCHASTIC C ONTROL S YSTEMS

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Chapter 3 S TOCHASTIC C ONTROL S YSTEMS

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5P8 B$`+? 82?? y ?V&?A<8 &Bdh+?8(8E?A ?I >?I B k s B+_K B$

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Chapter 4

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Chapter 4 C OLLECTIVE R OBOTICS

N _&=+? $ W l 4k x7 ?ˆ3z H e&[ 7 ?F&[O<8K Y B+_=+:+? 4CH  +B ?EXS&YWx _&93z N ?I >=+?I86 K8 B+_&7 ?F&[OIW B 2&7 7 ?I23 &? &[O 2A8Arf+?_&9 7 8gs(?I8 _&Bc+? B9?A< &23 &B$8g[O?EXS?A?AB &[O<8V8=$2<q$ >T< &8Y(8g?9=  B+_ c<I }‚„eay† >… K  [O?b8=$2A2?I88E=+7 7  2&:+7 ??IwW C"  _&9 [O?]&?u &2A2=+< >?q??AKc 8 8 =$ > &B &8 c<I }‚e„ ay† >} ‚  eay†dc3|F &8 +?A?K ?TK B u2&B$2? > [+7 ?8 =$ > &B$8 BsX4 2<b_&&=+:$8 &[O<8F ?TB+@ B+_ s2 &:O?A< >?bx?&W _$W rS2&:O?  &BOz3W  82<$ :(?A &88=+?I8V$ >T+?`&[O<8T ?  B+_]c2 &:O?A< >?&rhB+ 2&:O??&W +?TE :O?I8FS:O988 [+7 ? B9?A< &23 &Bu[O?EXS?A?ABb&[O<8F ?TK B &W +?A? 8 Bu &23g Bb?AB9 ?K8:O?I23=+  B9?A< &23 &Bd:O988 [ 7  ?I8AW N & ?e+ :+7 ?&rw > &B+?V?AB$dh+?8:O?I23=+{&[O<8 < B$(&7 [+=+: B+_ B9 ?I &2< +?ASK [O? 2&B$8 (?A?I B9?A< &23 &BKX4 7 ?F gXS?A7 7O?I8E< [+7 8+?It2&V=+B 2A > &Bt:+(2&7$X4 2< 8P?A:+7 >&?I ? 9=+?AB97 ] B$d7 ?I &?I8 7 7 ? & &  8 B9?A:+?< > &B x?&W _$W r^  ?I23F< &  7 B+Y`[O?EXS?A?ABu&[O<8 +?+?A?AB$wWqCDB9?A< &23 &B$8V8+&=+7q[O?d2A >?A_&& RA?IsB+[ s+? < B$8 88 &Bq?I =+ x &8 8T8&? ?I8 (&B+?zS[+=( < >+?A[ t+? B(&K > &B$ 7w2&B9?AB9< B$8 ?IwW H (2  7 B$8?I23<8 ?18E Y B+_g?e+ :+7 ?I8P^[ &7 &_ 2A 7$&_9 B 8K8hX4 2<t$ &?1&?ATXS?A7 7O?I8E< [+7 8+?It2&V=  B 2A > &B]:+(2&78 $ >@ ?g< B$8 ?I`+&=+_&d+??AB  &B+?AB9IWFG 8 B+_&7 ?T8:O?I2 ?I8 S B9 K `$ &?T &8 K B d &8FLM  Z?A?AB9@2<+?A 2A 768 _&B$ 78 2A 7 7 ?I  }‚ f >…}3|1X4 2<u?AB$ [+7 ?T2&:+7 ?e B9?A D8:O?I2 ?I8 2&V=+B  2A > &B^W65.+?A&&B+?I8. ? 8?I2??IT& _&7 B$+8h&BK&B+?1 B9S B$K ? B=+B (??I23?IK[ V+?1 _&+7 8?AB$8  &? &B+_&=+?I8S B$wr &? 2&&B+7 &r B9?AB+B$ ?1Q+?ASB+?I [  B9<8AW +?1?A |  }‚>‚yƒ9„>…$†| f $ &8S[O?A?AB =$8?I (?I82 [O?S2&7 &B ?I8QO B9<8Sx yW ?&W r>=+:g  7 7 &B$8f B$  (=$ 78 &B >Z&<(?I[  :++?A&&B+?g2&V=+B 2A > &B^W +? WX _&_&7 ?@+ B$2/? XT.8&?@[O?A?I8 81 B++?A4?e+ :+7 ?gh XS?A7 7 ?I8E< [+7 8+?I`:+(2&7yW +? B9 2A >?@ &Bd6 81+ B$2?V2A B]2&V=+B 2A >?@[Od+?V 8E< B$2?V B$d B+_&7 ? K T (8&=+<2? &U+?@ &?&W C" 8 XS?A7 7 Y B+>X4B B B$ >=+?&r $ >`_&&=+:$8tg&[ 7 ?s _&?AB9<8ux&_9 B 8K8 &B :+(2&78 B&<(?AP B9?A< &23 € W b?g8=+_&_&?I8ES$ >+?FB$ >=+?F^+?F< B$8 88 &Bt?I =+ &h2&V=+B 2A > &B 8 ?A7 ?A> B96X4+?ABT(?I8 _&B B+_F2 &:O?A< > &?P&[O<86 B$@$ >6+?98Eh2&B &?AB ?AB9f?I =+ 8+&=+7[O?1=$8?IwW @ 8P?AB$wr9< & @((?AK8P$ &?4[O?A?ABt=$8?I B+? (?I8 _&BKQ g8K 7 7$_&&=+:^2 &:O?A< > B+_ &[ 7 ?@&[O<8@x 8?A? N _&=+? $W l z3WhCDB 8 8(8E?A 8+&4?I88 _&? :$ &2?A7  l p R&W +?I8?F?I88 _&?I8 7 7 >X +?@&[O<8PK[Od8$ ?@8?AB$8& B(&K > &B` B$2 &< B$ >? +?  &23 &B`2<+ 2?I8 B`&<(?AS[O?g8=$2A2?I88E=+7^&B`2 &:O?A< > &?1< &8Y(8AW ˆ2‹“&”›%‘ “‘ 0 ’ ‹“–2‘ “ ‘ ’+“ ” 0 1 ‹4:”2‹)*“ ‘ 0 “ ” 0 ‘ 0 “&‹ 3”5#(“&”4– ”“ ” Š(5‘ ‹ / ‹‘ 1 / ‹Š  ’‘ 0 Œ 0 Š&” 1 ‹‹4:” Šc(“&‘ ’ ‹Š 1 ‹ŒA:”“&‘ ’’3 “ 3 0 –|” 0 ‘ ŒA: / š 0 “c(“&”s“ #(“2“ ”š"Š&”‘ ’ “&” Šc)1“&‘ ’ž

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Chapter 4 C OLLECTIVE R OBOTICS

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N _&=+? $W 8k^prBACHs!G8\HIUBA?:k6H eg ?"7 :$8?Ig< ?I86  &[O<8f& B+_F :O?AB9< _&&BVX &?A 2?I8 & B`+?VK Y&?Id2 <27 ?&W1p ?A?@+?T&[O<8 V=$8E 2&B9 B =$ 7 7 `8$ ?g:O98  &B B(&K > &B B]&<(?A 8:$ &2?V+?AK8?A7 &?I81&=(F?A7 > &?A7 `&B]+?2 <27 ?&W  ? +?A +?8:O?I2 O2T7 (2A > &B]P?I &2<]&[O B+&@+? & ?AB9< > &Bc+? ?I8=+7  B+_]:O?AB9< _&&B 8V:+?I8:O?I2 $?IwW +?t&[O<8,W(?I2 (?/Xd+>X  &U+?g8$ :O?&W

m&m

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Chapter 4 C OLLECTIVE R OBOTICS

N &_ =+?$W n kZp QB^[ IJ?AK x 2Izbk@H e] ?"7 :$8?I]< ?I8  t&[O<8 (2X4BOz3W.p ?A? +?@&[O<8SV=$8E &?F&BX4+K &B+_98E+?A 8+&=+7 [O?F+? 7 ?I &(?AFP+? (2X $ >g?A7 ?I23?Iu7 ?I &(?Ag &=+B$d+?t2 <27 ?&W  8F< &8Y 8@&?A ( B$  2 B$  t2=+7 IW   2?+? (2&?A<8 _&< &2?=+7 7 &W

4.6 S UMMARY

lMl

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&  q2&V=+B 2A > &Bix& B  +?A XS?A7 71?I8E< [+7 8+?I :+(2&7ˆzK 7 7 >X 8&?Aq _&97  2&=+:+7 ?I 2&7 7 ?I23 &? &[O 2A8K< &8Y(8K [O?b &2A2&:+7 8+?I X  q?I(=$2?I [+=+<(?AB &B < &  &B$ 7 8?AB$8&<8t B$ 2&:+=(< > &B$ 7  &2 7  ?I8AW  8X &8K(?A&B$8E< >?Iq+&=+_& >  &=$8T< &8Y(8 B &&7  B+_ $&?`&[ 7 ?`&[O<8K2&V=+B 2A > B+_ X `< & ((?AK8AW +?.< &8Y(8 B [+7 ? $W & 27 &88?I8wCC3rCCC3r> B$@C  XS&=+7@B+f[O?S:O988 [+7 ? x&QXS&=+7@[O?S _&?I >f(?I 79&?  t2=+7 3zSX +&=( XS?A7 7w?I8E< [+7 8+?I B9?A&[O42&V=+B 2A > &B :+(2&78AW6G18  82=$88?I [O>&?&r9?I8&S 

l M9L

Chapter 4 C OLLECTIVE R OBOTICS

8&? & f2?AB9< 7 RA?IK2&B9&7OXS&=+7K?9= ?1 V2&V=+B 2A > &BtB+?EXS&YV B+:+=(48?AB$8& B(&K > &BK B$b&=(:+=(V &23 &B$8 & +?t2?AB9< 7 RA?Ib2&B9&7 7 ?AIWp ?A?KXS?$B$b$ >g&B$2?K `2&V=+B 2A > &BbB+?EXS&Y 8 Bb:+7 &2?&rw+?A? 8FB+`?I &8&Bd`=$8?K2?AB9< 7 RA?Iu2&B9&7yW  8F&?A $?I8x Bb 8@2&B9?e 3z +?K27 {$ > (?I2?AB9< 7 RA?Id2&B9&7 7 ?A<84X ]+?gEXSKE :O?I81 B9?A&[O 2&V=+B 2A > &B](?I82 [O?I] [O>&?V2A B]:O?A& ?e+ &237 V+? 8 ?4< &8Y(8. &8P g2?AB9< 7 RA?IK2&B9&7 7 ?AIWQCDB &23Ir BtK B 2A &8?I8 x?&W _$W r 27 &88.Cf< &8Y(8 (?I 7 &?P? t2 ?AB96 =$8?(?I2?AB9< 7 RA?IV2&B9&7 &8Q7 ?I886+ >< 1B+?A?I+8Q [O?[+9 &+2A &8Ef>&?A^+?42&V=+B 2A > &B B+?EXS&YW N &.+? 4CH  8(8E?A`r( g:+=+?A7 K2?AB9< 7 RA?IK2&B9&7 7 ?APXS&=+7KB+S[O?   [+7 ? (=+? <8 B+? t2 ?AB9S=$8? f+?@7  ?I[$ B$(X  d2&V=+B 2A > &B$8AW CDB9?A&[OT2&V=+B 2A > &B 2A Bq[O?&=+_&+7 s:+7 &2?I B9]EXSb2A >?A_&& ?I 8 k|A}…+|A>‚  |  „>‚3†ˆ… ƒ` B$ E„ cjay† >… c<I>‚ >†ˆ…e„ ay† >…$W + ? $<8E` 7 7 >X 8+?b8?AB$8 B+_ 2A :$ [ 7  ?I8 g?I &2< &[Otq[O?s8:$ >  7 7  ?e ?AB$(?I X4 7 ? +?`8?I2&B$ 7 7 >X 8g&[O<8gb_&7 &[$ 7 7 s2 &< B$ >? &23 &B$8g+&=+_& 8&?t& 1& B+_$WcScE :O?I8T B9?A&[O62&V=+B 2A > &BV ?2&:$ > [+7 ?PX  (?I2?AB9< 7 RA?I@:+ 7 98&:+ @ B$V2A BV?I &8 7 F[O? :+7 ?A?AB9?I BT[O?A$  &=+ "[$ &8?I@2&B9&7 7 ?A<8^=$8 B+_F_&?AB+?A 2P2&V=+B 2A > &BV((=+7 ?I8AW +? t2=+7 EgO2&7 7 ?I23 &?S&[O 2A8 < &8Y(842A B[O? E=$(_&?I[ K+?FE :O?g B$ &=+B94 B9?A&[O 2&V=+B 2A > &B ? 9= ?It& 8=$2A2?I88AW C" 812&B &?AB ?AB94 8:O?I Y 6+?V27 &88?I84.2&7 7 ?I23 &?@&[O 2A8< &8Y(81(? $B+?Id [O>&?@[+=(1XS?gV=$8E 8E?I88 $ > dX 7 7FB+d 7 X (8t[O?b:O988 [+7 ?u 27 ?I 7  :+7 &2?b q< &8Y B9 &B+?bg+?I8?s2A >?A_&& ?I8AW +?u< &8Y(8 (?I82 [O?Ib+?A?KXS?A?K(?A7 [O?A< >?A7 b(?I8 _&B+?I]dK Y&?+?A ?I &8d]2A >?A_&& RA?&W +? 27 &88?I8@ ?[$ &8?Iu&B +? E :O?12&V=+B 2A > &Bq? 9= ?IcX4 2<^rP &8V 82=$88?Iq?I 7 ?AIrh2A Bq[O?I2&?` t2=+7 T E=$(_&?` >T+? :+7 ?A?AB9< > &B]7 ?A&?A7yW N & ?e+ :+7 ?&r >F t _&]7 ?A&?A7Q+?2&V=+B 2A >?I B(&K > &Bd[O?EXS?A?ABu&[O<8 X &8 Bt @?A:+?I8?AB9< > &B?I &8 7 V?I2&_&B RA?IT[ T @ =+K B]x?&W _$W rUx TA2V+z.2 &< B$ >?I8.+?1 _ < 7$< & @:$ &2K+?]_&?AB+?A< 7 B+ &B F_9 =+_ B+_b+?u t2=+7 E  2&7 7 ?I23 &?F&[O 2A8P< &8Y(8[ K+?@B$ >=+?@Q+? B9?A< &23 &B$8? 9= ?It[O?EXS?A?AB`&[O<8PK$ &? _&?I >4?A IW 1B+?t4+? 98ET?A7  [+7 ?t:$ <8g4+? 4CH  8(8E?A 8@+?`2&V=+B 2A > &Bs< ?AXS&YW +? _ < 7 ?I88 _&? :$ &2 &Bqs+ >?&W  & d((?A :+:O?I <8F][O? d&?A]&[+=$8EV?+(s :+7 ?A?AB9 B+_ B9?A&[OV2&V=+B 2A > &B^W`GST < B$8 88 &Bd< >?gS :+:+e K >?A7  l p R&r?A&?AB S K:$ &2&?AIr FX 7 7f98E 7 Y&?A7  ?I2? &? +?@B+?e 1&B+?&W +?@&[O<84 ?@B+1 [+7 ?FK>&?   ?AB+&=+_& B]8=$2<d 8+&4 ?@:O?A (&  8S$ &? V=$2<?Z?I23IW +n ? a P>€ ay† f }3 | c<„>~ } &88=+:( &B` :+:O?I <8PT[O?@> 7 t&S 848(8E?A 7 7 >X B+_V&  _&+7 t?A7  [+7 ?@8?AB$8&t8$  B+_ B$` &23 &B`2 &< B$ > &B &Bd 7 7< &8Y(8AW 1B+? 2&B$2?AB B 2&7 7 ?I23 &?4&[O 2A8 8h+? 8E&:+: B+_ 88=+?&WfC"^2&B9&7 8P(?I2?AB9< 7 RA?Iwr&+>X (F+?1&[O<8 Y B+>X X4+?ABt+?1< &8Y 8 $B 8+?I   +?A?F ?1 g?AX X (8hT7  &YK >S 8AW N <8EIr K B+[O? B+?I2?I88 T& +?Aat &23=$ 7 7  8E&:?e(?I2=( B+_T+?g 7 _&& +   +?F< &8YtXS&=+72A K&B B$(? $B ?A7 &WSH ?I2&B$(7 &r$8E&:+: B+_ 2A BK[O?1[+= 7  B9g+? 7 _&& + &8 BK+?4&K > &B< &8Y(?I82 [O?IK [O>&? 9&B$2?4+? (?I8 ?IK8$ :O? 8.K &(? +?K&[O<8@8E&:b> B+_?A&?ABu+&=+_&b+?Au2&B9 B =+?Td?e(?I2=(?T+?t 7 _&& +`W +? :O&< B9@:O B9@ K Y&? 8$ >12&V=+B 2A > &Bd 7 7 >X 84 B 2?AB9< 7 RA?It< &8YtK[O?V2A  ?I &=( B] K(?I2?AB9< 7 RA?It &8 &B^WPC" 8E&:+: B+_ 8 <8?A7 .2&B$8 (?A?I t[O?T T< &8Y +?ABd 812A B] 78K[O?V &2< ?A&?I[ ` K2&7 7 ?I23 &Bdh&[O<84X4 2< 2&V=+B 2A >?&W +?4(?I8 _&Bw2&7 7 ?I23 &?[O?A$  &=+<8 BK F_&&=+:V&[O<8 8h?I &8 7 @ &2 7 < >?IT[ @+? &[+=$8EIr[O& "=+:^r (?I2?AB9< 7 RA?Ib :+:+9 &2<b &(&(2A >?Iu+?A?&WKC" 8@+&:O?Iu 8V8E=$(u B9?A&[Og2&V=+B 2A > &BbX 7 7.:+=$8 +?1[O&=+B$+  ?I8PQ2&7 7 ?I23 &? &[O 2A8.?I8?I <2<X4 7 ? >P+?F8 ?4 ? +?A7 : B+_gV:+> (? B$8 _&9<8 B9g+? [ &7 &_ 2A 7w2&V=+B 2A > &B:+(2&78S[ tX4 2<`XS?@ ? B$8: ?IwW

6  3 $7        '3    |#'(*,'&$(3  0 -'(  &  '($ $J-     &-86 ! )-. 2 ! 0 -'(  & 0 *,'$d' ) s +03   3      4 0/'-  6'($d 7  , &3   + 6  3 $#   3  '$ %'(06 (.-'  .'($s 6  0/'3   @ 06%# '(     )  5'& 3 p$(     J   0  40 -'(  &0 ) 23 '&     8 ]  ! 9 m)< e Ecb F m=(< ?Afb F%A  P,N #N M5N RSV N #ISWX Y Y



Chapter 5

S ELF -O RGANIZATION  8K2<$ :(?A(?I82 [O?I8 s2 &< B$ > &Bq?I2<$ B 8 &K b8(8E?A F2&V=+B 2A > B+_s _&?AB9<8VX  8:$ <8? 2&B+B+?I23 &B$8 k    B$  7 ?A=(?A $r$LM&M l o"W  8 8T d&?Ab_&?AB+?A< 7h &7SX4 2<qK b[O? =$8?IsuK Y&? (?I2?AB9< 7 RA?I(?I2 8 &B$8AW +?@?I2<$ B 8 8[$ &8?I &Bd 8E(2<$ &8E 2 &?A<8 &B h2?A7 7 =+7  =(&K >< `xyH$vSG z3WOG :$ < ??A4 B$ 7 &_&&=$8PK WE?A:O?A< >=+?/XT2A B`[O?F=+B+?I K> K+?@[O?A$  &=+4Q+?V8(8E?A`WhC" 8&=+B$ $ >+?F[O?I8E?A:O?A< >=+? &42 &< B$ > &Bt(2A2=+<8SB+?I 4 V:+$ &8? < B$8  &B [O?EXS?A?AB &<(?A4 B$ 2<$ 98AW6S (?I8 _&B^rw2 &< B$ > &B]( ?I81B+ ?A7 `&Bu B :$  2=+7  8E=$23=+?Vh+?2&B+B+?I23 &B$81[O?EXS?A?ABu _&?AB9<8Ar$ =$8  K t[O?g :+:+7 2A [+7 ? K 7 _&?F < h8:$ <8?A7 t2&V=+B 2A > B+_&[ 7 ?F&[O<8&48ˆEX ?F _&?AB9<8AW G18TXS?$ &?8?A?AB B :+?A &=$8T2<$ :(?A<8Arh+? ?A &g} c<}… ay‚„>~ † ‡Ig} u|  b| aD} f ?AB$2&:$ &88?I8V7 _&? [O( ?I8T XS&Y & ?AB+_ B+?A?A B+_$r2&:+=(?Ah82 ?AB$2?&r B$VK >+?AK > 2A8AW  e+ :+7 ?I8 B$27 =$(?SB+?EXS&Y(8QO&[ 7 ?S&[O<8 k \] ><  2r lIm&m LAo"r&8ˆEX ? _&?AB9<8 k S&B$ [O?I =?1 7yW r lIm&m o"r& B$2?A7 7 =+7 h =(&K >< k b&7 < `r lIm o"W6G 2&  &B`+?I & B] 7 7f(?I2?AB9< 7 RA?Id8(8E?AK8 84+? 88=+?g# c<I>‚ >†ˆ…e„ ay† >…$W p >X 81 K7 _&?@B =+V[O?A1| „>‚<|A}~  2&V=+B 2A > B+_t _&?AB9<8 [+7 ?@t:+((=$2?V t2&+?A?AB9 _&7 &[$ 7^[O?A$  &=+ =$8 B+_ 8 :+7 ?g=+7 ?I8 cG B$8XS?A B+_ 8 9=+?I8E &B X 7 7wB+4&B+7 K:O?A +?g2&B$8E=$23 &Bt B9?A?I8E B+_ B$ =$8?=+7^  O2  7w8(8E?AK8S[+=(4K t 7 7 >X =$8K=+B$(?A<8E< B$ &?@ [O&=(4+?@B$ >=+< 7wXS&7wW.G B9<8 B$t+?g+?A 8(2  7 B$8?I23<8 ?F:O?A?I23 ?e+ :+7 ?I8  7 (2A 7 B9?A< &23 &B :+((=$2 B+_c s2&+?A?AB9K_&7 &[$ 7[O?A$  &=+IW C" 8K:O988 [+7 ?&K 7 7 &B$8@ B9<8Tc &23 &8V s|  }‚>‚yƒ9„>…$†| f +&=+_&c7 (2A 7P:++?A&&B+?t2&V=+B 2A > &B^W b?8?A?AY]u?A:+((=$2? 8T [ 7 Eu&Bq =+B$+ ?AB9< 7^7 ?A&?A7 B`&<(?ASK2 &< B$ >?@  O2  7w8(8E?AK8AW C"T2A Bc[O? _&=+?Iu$ > c<}~ˆ~ (~ „>‚`#„ 8aD f e„ aD„dx"vSG zV ?+?8 :+7 ?I8EV?e+ :+7 ?t dK >+?AK > 2A 7S(?I2?AB  < 7 RA?Ib8(8E?A`W 1 _ B$ 7 7 ]8E=$ ?Iu[ u&&B  ?A=+K B+B k lIm&n&n o"r+??A vSG 8F=$8?I]](?I82 [O?K8(8E?AK8  8:$ <8?A7 c2&=+:+7 ?Iq Z?A?AB$2?? 9=$ > &B$8AWq 1?I8: ? +? K8 :+7 ??I2<$ B 2A8ArP8&? ?e ?A?A7  B9?A?I8E B+_ lM



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Chapter 5 S ELF -O RGANIZATION

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&8g? 7yW k lIm&m oS8E=$(u+?8 ? 2 &< B$ > &Bb< &8Ys &8gX 7 7S[O?t?e+  B+?I +?A? Bb+?t2A &8?4(??A B 8E 2vSGgWQp >XS?A&?AIrw+? V :+:+9 &2< 8Fd=$8?K `_&?AB+? 2K 7 _&& +{d?A&&7 &? (??A B 8E 2F=+7 ?I8 8=$2A2?I88E=+7^ >4+?F< &8Y X4+?A?I &8S+?A?@$ B$ D2((?I8E(2<$ &8E 2 =+7 ?I8 ?@(?I82 [O?IwW +?A &=+B$ $ >t+?u[O?I8E`8&7 =( &B$8tXS?A?] [+7 ?d 8?AB$ 7 &B+_q< B+_&? „>b‚ ayd† c~ }3|]x 8  7 tq+98? B +? „ f } E 6† }z k G B$(?@? 7yW r lIm&m  o B &<(?AVs &2< ?A&? 2 &< B$ > &B^W +?I8?:$  27 ?I8V?A7 c&Bc+?`=+B$(?A7  B+_ 8E=$23=+?P+?t2&B+B+?I23 &B$8 [O?EXS?A?ABc2?A7 78ArQ8:O?I2 O2A 7 7 d$ >g?I &2<b2?A7 7 8@2&B+B+?I23?I] <8@B+? _& [O&=+<8 B B (?AB9 2A 7SK B+B+?AIW +?2=+?AB9gXS&Yb &88=+?I8@$ >TB+u8=$2<c=+B$(?A7  B+_u8E=$23=+?tK b[O? ?e(:+7  ?I B$ $ >t+?u8 ?d?I2<$ B 8 8+&=+7 XS&Yc&  Z?A?AB92&B+B+?I23 &?] <2< ?I23=+?I8AW +?u298EK&K 8 B$2?I &8?Ib&?A<8 > 7 E 8@$ >V+?t?I8=+7  B+_]=+7 ?I8g ?K7 ?I88g? t2 ?AB9x Bc?AK8@ ?tu2 &< B$ >?z1$ B +?  :$  27 ? "[$ &8?I 2&=+B9?A:$ <8AW B$ Y>  

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80

70

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80

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Chapter 5 S ELF -O RGANIZATION

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Chapter 5 S ELF -O RGANIZATION

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Chapter 6

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Chapter 6 S YNTHESIS 

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Chapter 6 S YNTHESIS

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Chapter 6 S YNTHESIS 

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Chapter 6 S YNTHESIS

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6.1 C ONCLUSIONS

6.1 Conclusions   ! 1   -%, !$"1 &   /2"1U-D  & s3'.$k1s!$-4N2  - D  . U1   ,G  &    I.     )*  ! 6& U1-2"1 " -2"7-D &  & 



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IV

m1/2 ,  7 ! 1 -2 , -  /,F51 2 )*    /2 2 #  1/2 ,  7 !In7. / & Y -',_" 2'Qe1- ('2 1  - 2 x   3  2'Q !     S. 2"!  !  &   6 &!$ j*'', * -W )-D71- ('2 1  - 2   -R3& , ) /2"3-, #R!  2"!  1- -,F! 2   1 -"2  

V

VI

m1/2 ,  7 ![1-2 , -a1 2    ("1F& )-,   1 /2M" )-D71- ('2 1  -2[D 1     & 2C , 1 # 1/2  ,  7  !51-2 , -  & )-'2X-Dj1- ('2 1  ! 2$DE-,  -2     ! /* /2"!/2 l7-  &P /2"3-, #H3& , 2'Q "2 ! 1 -2P1- -,F! 2  -2 , UY .  . 1F& 2   2 , -. - F  &   2   2 % # 2 ! -" +1/2 ,  7  !_1-2 , -



2+7-,  -D, 

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VII

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VIII

IX

X



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4

1-2 , -  !Z. # 1 /2 ,  Q /2"1#C.'  &   2 xE'2'-."  /, Y ! aD     1   

XI

6&/,  , UDE'2"!  /2  3*  1Fs-D7  D( -, Q 2 ! Q2_-D ,  ^ 1 i! 1/2 ,  7  !_3#$   

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Chapter 6 S YNTHESIS

6.2 Contributions h Y 2'Q%3 , 7 !  & &  2"!R  !  &S1-2"1 " -2"/G & ! /2  ^ ! 



2[1-2 , .' -2"N-D & 07-,  , +2'-4

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6& 

S TOCHASTIC A LGEBRA



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I   -4XN -$1F&  1M!$# 2  1     -2" -. N,   /2 2     2'-4N2  Q/.', 1NDE,   7-,  

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III

IV V

VI

,  - Y%3#$   /, M1  2 2   -$1F&  1  Q/.', N& 1F&    , O%DE-,(   -2    L3&'-4N2  & !1/2 ,  7  -2)-D ,  -YU1-2 , -'*', -.    *  1 +7-U1-2" , 2Fa-2  & 3#$   HG    Y  /2  -[,  )-Y 2'Q[ /2"3-,F 2"! 1   -,F $ /2 ,  7 !Z1-2 , -     Y  /2  ! 1 /2  ,  7 !_1-2  , - /  &R1- ('2 1  -2  '-',L1    -D]1 /2  ,  7 ! ,  -Ys,  1 Y1-2 , -  /,F /, !^"2!(.  !U-2 # *  2"! )-'2 -DM1 - ('2 1  -2g2  ! !  -C.  O 1  # ) *    /2  .  H. #g!1/2 ,  7 !g1-2 , -  /,F_xA6&/-A ,   K  

2  v  -$1F&   1_1 41   "  S"  !   -  2  , 7 _. -  & ,  -YV1F& 2" 2"! ,  -YV! 1  -2 *', -$1     2  & +2  Q& . -', &'- -$!5-D 2     .',   *', -. .  #5!   , .'  -2 

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1-2'2 1  -" 2    ! +. + /2R  .  #CxE" 2' Q  & (  -$1F&   1  Q /2_*', - .   7-D 2     n.  ,  t!   , .'  -2 2" !  & 1 ,   -2a!   , "1  -2)-D 2$DE-,   -2RxE-,F! /,G /2  , -* #  2   -$1F&   1 ,  -Y%3#$    

C OLLECTIVE R OBOTICS I

II

2 7-,  -D )-.   , -. - F

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6.3 O PEN Q UESTIONS S ELF -O RGANIZATION I II III

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6.3 Open Questions 6& X  1 -2H!  1"  * -W  .   7DE-,DE ', 7-,  N&    &+2O ! 4   &1)-, M*'&  -W3-*'& 1   3  B 2   !  -M1-2  2    - ! /2  DE# 2"!('2"! /,F  2"! &.  1  1F& 2  r-D]  D( -, Q 2 7  -2 " -$1F&  1   Q /.', x 2"!S1 41   "  7-,     &)!  1 ,  N    2"!S!  1,      oaO  /2" -2" -s. -  &1-2 2 '-"j    2" !     #). s* -W  .   $-2" ! /, 2'Q(   (D  -, Q 2 7  -2SDE, -   &M1-2 , - * /,F3* 1 Y     s2  ',     #P!  1" M. -  & 2  /, 2 rD  !$. 1  2"! O  /, 2 j*' ,  3' ,  6& sDE,   7-,    -4Xs. -  &Z, /Y/,F .    2"! , , /Y /,F .   , k 1  -, /G  & +DE-, Q   2'Q)-D 2  1-2"!  -2" 2"!H1 ,   -2_-Dj2     , "1  ',  ,  -Y  3#$   /G'1   2R  -$1F&   1  Q /.', G *', -Y !  3-  !5DE,   7-,  DE-,DE' ,  & /,!  1"  -2_-D  &   U1-2"1 /* F  hX4-  1 2  )*', -Y )-2  & 5  -$1F&   1)! 1 /2  ,  7 !C! 1  -2   2'QR*', -$1    B 51 2C^",F (  "!$# ! \ / , /2 5m   *"   -H     & )* -W  .  SY ,  -2 2CQ, -'*[. /& Y -', 6&       

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K

Chapter 6 S YNTHESIS

1- -,F! 2  -2 Q&. s )*  ^ ! :2  &/Y/2-D   DE'2"1 -2iGW2 d, -. - F5    -   1   # )-YN'*   &   !'! /,  -, /*  1  & ,XD  2'Q  !/,F  6& X    2  ', I, -   -   MDE-, ,  ^ 1 3-$1   2"!  k-,* -W  .  #_DE-,XDE ',  7-,  2C1-  1 YU, -. -  1/ " "!  2 &  ,   # 43-%*', -Y ! 2" Q&  2  -  & +*' , /Y  /2"1M-D &  # * M-Dj-, Q 2 7  -2 2H2  ',   "$1  2'QM'*iG a!  1"  !  2 " 1 -2 K"G   k-,  3 2 , QN!1/2 ,  7 !S3#$   $ 2X! Q2   3#$    (-2 _ 1   2"!  & /2 *' *   #  &   2'-  &/, X,  &/, %Q/2/,  1  2'Q[,   -2"3& *"N& 1F&  #5. ('2"! /,F  - -$! 2"!H"   !  -%*' , ! 1   & S 1  !_'*R. /& Y -',C6&U, -. -  1/X3#$  uDE, -  $& * /,XK  U!  Q2  !  -   4-  1 / ,  2 )-'2 U-DX 1  2'QVx  Q G . U!  Q2 !PDE-,  5, -. - Fs.'.  2'-       1  X*   1 / ,   2  /Y    & M1 - ('2 1  -2. 2"!  !  & L^"2   2"!  ,  3 -   N& 7-2  1F&%, -. -  .  2'Q .    -  , 2"    - /Y /, #[-  & /,+, -. -  6& "  -DN3* ,F  1-2'2 1  -2"M. + /2 Q /2 F  .       /Y    & 1 - ('2 1  -2S. 2"!  !  &*', - .   HG.'  5  L   / 2   &%  -$1F&   1N1     ,   -   G  &    L1- -,F! 2  - 2 )- , ! 1    L , Q /,L3#$      

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6.4 F INAL T HOUGHTS 



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Stochastic ecentralizeD Systems by Timothy David ...

For example, an i nverted pendulum wants to fall one way or the other and we would l i ke to prevent th i s from occurr i ng ; the system i s naturally unstable and we are try i ng to stab i l i ze i t. In the. Markov i an control framework d i scussed here, the natural tendency of the system i s to head towards zero. ( i t i s stable).

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