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S  6$  `P    P*    6  `    R*  H  0޽h ? ̙3380___PPT10.6  0 0 (     6  @ ~FSTABILIZING a NONLINEAR SYSTEM with LIMITED INFORMATION FEEDBACKGG  08 S KDaniel Liberzon  x33L     # @ D    <A ??/ o  Z   s *0  Z   s * (   <   vCoordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign U.S.A.wnw  0$ Jj "CDC  03H  0޽h ? ̙33r8 0 ((<48  '(  4 4 6X-  p@ > MOTIVATION  4 6D pU,$D 0  Limited communication capacity many systems/tasks share network cable or wireless medium microsystems with many sensors/actuators on one chip@!q!p] ) 4 B` t,$D 0 q5 Need to minimize information transmission (security) 65R 4 B ^ ,$D 0 P Event-driven actuators PWM amplifier manual car transmission stepping motor@98l @W  :4W @ ,$D 0  4 B  h  7EncoderrB  4 BDjJ 6 6l 4 <jJ @v  #4 dA  txp_fig0W P  SOURCEz\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \begin{document} $i\in\{1,\dots,N\} $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH129.8756PICTUREFILESIZE5706l  v @ ;4v @,$D 0 "4 Bd    7Decoderl !4 <jJ  v @il p  <4p ,$D 0rB  4 BDjJ 6@6s@ p  94p  $4 dAtxp_fig r SOURCEh\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \begin{document} $z\in\R^k $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 62.8756PICTUREFILESIZE3194rB %4 BDjJ6@6 4 B(  0K  = QUANTIZER r ,4 BjJ  rB &4 BDjJ @66 (4 dA txp_fig   SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \begin{document} $q_i(z)\in\mathcal{Q} $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 87.8756PICTUREFILESIZE4046 *4 B5  p P  Nfinite subset of   +4 dA  txp_figd  | SOURCE`\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \begin{document} $\R^k $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 27.8756PICTUREFILESIZE1390`B /4 0D 0  H 4 0޽h ? ̙33y___PPT10Y+2D'  = @B DP' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<4%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:4%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;4%(++0+40 ++0+40 ++0+40 +D 0 ><6<+r4  ;(    6O  p U!ACTIVE PROBING for INFORMATION""l @ h(,$D 0l - <jJP` Il . <jJ l / <jJ 9 rB 0 BDjJ  9lB 1B <DjJrB 3 BDjJ 5 HT jJPh J APLANT   6 HY jJ 0* E QUANTIZER   7 H^ jJ  > CONTROLLER  lB 8B <DjJb  rB 9B BDjJ  lB ; <DjJ  r < dA txp_fig@v SOURCEZ\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \begin{document} $u $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue&ORIGWIDTH124PICTUREFILESIZE390x = dAtxp_figG D v SOURCEZ\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \begin{document} $x $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 10.754PICTUREFILESIZE390 > dAtxp_fig s| SOURCE`\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \begin{document} $q(x) $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 38.8756PICTUREFILESIZE2054ZB W s *DjJl   x  o  x ,$D 0`B \ 0D V( Vx  ] Bz   8  ;dynamic`B ` 0D @ @ p a B  8 X  ;dynamicl 8`8  n8`8 ,$D 0 c 6   `8  K(changes at sampling times) d BP   8X  >(time-varying)Ll f  q f,$D 0 @ B     7EncoderrB A BDjJ &]  ] l B <jJ  & C dA  txp_figz 8 SOURCEz\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \begin{document} $i\in\{1,\dots,N\} $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH129.8756PICTUREFILESIZE5706 E B  ) t  7Decoderl F <jJ  rB H BDjJ ] ] rB K BDjJ F] ] r M BjJ  VrB N BDjJ ] F]  T dA txp_figf    SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $x $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 10.754PICTUREFILESIZE390 U dA txp_fig   SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $q(x) $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 38.8756PICTUREFILESIZE2054l  y  r y ,$D 0 Z B y  > very small `B m 0D (  @H  0޽h ? ̙33___PPT10+`|D'  = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*r%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*o%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*n%(+!' 0 !Ta(  T T 6  p CLINEAR SYSTEMSx !T <D 4 ^(Baillieul, Brockett-L, Hespanha et. al., Nair-Evans, Petersen-Savkin, Tatikonda, and others)_x_t    H T 0޽h ? ̙33 @ 0 z5r5 ^0 4(    6  p CLINEAR SYSTEMS|l ` ;` 0,$D 0@ P   4`fB  6D3f1P4 4`B ! 0D3f1d`B " 0D3f1  d`B ) 0D3f1d / ^Artxp_figp  SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $t_0 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue&ORIGWIDTH184PICTUREFILESIZE902 1 ^Aitxp_fig " SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $t_0+\tau $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 58.756PICTUREFILESIZE2590 3 ^Altxp_fig  $ SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $t_0+2\tau $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 70.756PICTUREFILESIZE3222ZB 5B s *D ZB 6 s *D  ZB 7 s *D  8 <x _  @sampling timesl @`h X`@h,$D 0   Bd  ` MZoom out to get initial boundf 9 6jJ @hl <  Y< ,$D 0  dA_ txp_fig8< 0( SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\hat x(t_0):=0 $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 98.8756PICTUREFILESIZE4378`2  0  8  [  B  Z G Example: " f3@  QTB  c $D88TB   c $DZB   s *DZB   s *Dhh Z ^Avtxp_figJPH SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\dot x=Ax+ Bu,\ n=2,\ N=9=3^n $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 334.758PICTUREFILESIZE 13966) A q  \# Aq ,$D 0 ] <@+ A  KBetween sampling times, let ^ ^Aetxp_figH  q <4 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\dot{\hat x}=A\hat x+Bu $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH1276PICTUREFILESIZE5978H  0޽h ? ̙33: 2 ___PPT10 +0SD '  = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(+6h! 0 OO,c$(  1O(  $ I$ 0 F,$D 0 $ 6:  p CLINEAR SYSTEMSl A   E$P Ap ,$D 0 "$ B> A   8Consider  #$ dAftxp_fig  " SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $e:=\hat x-x $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue&ORIGWIDTH966PICTUREFILESIZE3286l H p ?  G$ H ?P ,$D 0 $$ dAgtxp_figH q *" SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\dot{x}=A x+Bu $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH1276PICTUREFILESIZE5162f %$ 6p   &$ dAhtxp_figy0 ? >6 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\Rightarrow \dot e=Ae $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 92.8756PICTUREFILESIZE3494 *$ BZ h ,$D  0 W% is divided by 3 at the sampling time&& $ B^  Z G Example: " f3L  $# TB $ c $D88TB $ c $DZB  $ s *DZB  $ s *Dhh) A q   $# Aq ,$D 0  $ <pe A  KBetween sampling times, let  $ ^Aetxp_figH  q <4 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\dot{\hat x}=A\hat x+Bu $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH1276PICTUREFILESIZE5978^ $ 6jJ@hX2  $ 0p2 B$ H6?O X2 C$ 0@` p D$ dAstxp_figP; SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\hat x $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 10.754PICTUREFILESIZE558l 8 Px \$ 8 H,$D 0TB R$ c $D X xTB S$ c $D P pTB U$ c $D8 TB V$ c $D8  W$ s *' H,$D 0Dl ( 8  Z$( ,$D 0`2 J$ 0 h `2 X$ 0( 8 l  ` ^$8 X,$D  0l2 [$ < ` ]$ ^Astxp_figP SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\hat x $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue,ORIGWIDTH 10.754PICTUREFILESIZE558 a$ ^Avtxp_figJPH SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\dot x=Ax+ Bu,\ n=2,\ N=9=3^n $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue.ORIGWIDTH 334.758PICTUREFILESIZE 13966l F  c$ F,$D 0q `0 P  '$# h ,$D 0 ($ <x `0 P  uC grows at most by the factor in one periodDD )$ ^Aktxp_fig` >  RJ SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $\Lambda:=e^{\|A\|_\infty \tau} $ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue0ORIGWIDTH116.8756PICTUREFILESIZE5374 ,$ B  F   8The norm  b$ ^Atxp_fig " P ^V SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand {\R}{\mathcal R} \newcommand{\pp}{p\in\P} \newcommand{\dvdx}{\frac{\partial V}{\partial x}} \newcommand{\eps}{\varepsilon} \newcommand{\K}{\mathcal K} \newcommand{\Ki}{\mathcal K_\infty} \newcommand{\KL}{\mathcal K\mathcal L} \renewcommand{\P}{\mathcal P} \renewcommand{\d}{\Delta} \begin{document} $$\|e\|_\infty=\max_{1\le i\le n}|e_i|: $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300"TIMEOUT156BITMAPFORMATbmpmono8 DEBUGINTERACTIVETrue(ORIGWIDTH1748PICTUREFILESIZE 12666H $ 0޽h ?/ B$[$ ̙33___PPT10+ Di'  = @B D$' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I$