Metodología híbrida para resolver el control predictivo de sistemas (lineales) afines a tramos

 , A  #  w  B  W  O  W  + W #    +  + , 5  w V , + , w    A +  w  + , # W  A W V  # , 5 W 5 A ,   5 E w W  ,  w , 5   6 W  , 5  A +    5 P l j x h o H 1 S h  q d > H g x d u g r I 1 F d p d f k r > V d q g u d S l  q r q p s h q d C l q d x w 1 x q v m 1 h g x 1 d u > h g x d u g r C h v l 1 x v 1 h v > v s l q r q C f d u w x m d 1 x v 1 h v G h s d u w d p h q w r g h V l v w h p d v | D x w r p d w l f d + H V L , X q l y h u v l g d g g h V h y l o o d + i t  4 i ? H v w h d u w  l f x o r s u h v h q w d x q s u r f h g l p l h q w r k  l e u l g r s d u d u h v r o y h u h o F r q w u r o S u h g l f w l y r + P S F , g h v l v w h p d v + o l q h d o h v , d  q h v d w u d p r v + S Z D , 1 H o s u r f h g l p l h q w r x w l o l } d o r v f r q f h s w r v g h f r q m x q 0 w r v d o f d q } d e o h v / f r q m x q w r v f r q w u r o d e o h v | j u d  f r v g h w u d q v l f l r q g h h v w d g r v + V W J , s d u d u h g x f l u h o q  x p h u r g h s u r e o h p d v g h s u r j u d p d f l r q f x d g u d w l f d + T S , q h f h v d u l r v s d u d r e w h q h u x q r s w l p r j o r e d o 1 H o d o j r u l w p r s u r s x h v w r u h g x f h f r q v l g h u d e o h p h q w h h o q  x p h u r g h h { s o r u d f l r q h v t x h v h q h f h v l w d q g x 0 u d q w h o d e  x v t x h g d g h x q p  l q l p r j o r e d o | g h h v w d i r u p d h o w l h p s r u h t x h u l g r s r u h o P S F v h u h g x f h d x q d s h t x h  q d i u d f f l r q g h o w l h p s r t x h v h u h t x l h u h s d u d h o s u r e o h p d r u l j l q d o 1 S d o d e u d v f o d y h = F r q w u r o S u h g l f w l y r / v l v w h p d v k  l e u l g r v / v l v w h p d v + o l q h d o h v , d  q h v d w u d p r v 1  W  A +  # N   W   H q o r v  x o w l p r v d  q r v v h k d q u h s r u w d g r g l i h u h q w h v p  h w r g r v g h d q d o l v l v | g l v h  q r g h f r q w u r o d g r u h v s d u d v l v w h p d v k  l e u l g r v + K V , ^ 4 3 ` / ^ 5 ` 1 H q w u h h v w r v / o d f o d v h g h f r q w u r o d g r u h v r s w l p r v h v x q d g h o d v p d v h v w x g l d g d v 1 O d p d | r u f d q w l g d g g h o r v w u d e d m r v t x h v h k d q s x e o l f d g r w u d w d q v r e u h h o f r q w u r o g h v l v w h p d v k  l e u l g r v + K V , f r q w l q x r v | h o f d o f x o r g h v r o x f l r q h v r s w l p d v r v x e r s w l p d v ^ 8 ` / ^ ; ` 1 D x q t x h d o j x q d v w  h f q l f d v s d u d g h w h u p l q d u o h | h v g h f r q w u r o s r u u h d o l p h q w d f l r q s d u h f h q v h u p x | s u r p l v r u l d v / p x f k d v g h h o o d v v x i u h q s u r e o h p d v g h g l p h q f l r q d 0 o l g d g 1 H o f r q w u r o s u h g l f w l y r + P S F , k d v l g r d f h s 0 w d g r f r p r h v w d q g d u s d u d u h v r o y h u s u r e o h p d v g h f r q w u r o f r p s o h m r v f r q u h v w u l f f l r q h v s d u d o d l q g x v 0 w u l d g h s u r f h v r v 1 S h u r s r u r w u d s d u w h / d  x q k r | h { l v w h q d o j x q d v o l p l w d f l r q h v s d u d t x h o r v s u r f h v r v f r q w u r o d g r v s r u P S F s x h g d q v h u x w l o l } d g r v 1 H v w r v h g h e h d t x h o d r s w l p l } d f l r q h q o  l q h d u h t x l h u h x q d j u d q f d q w l g d g g h f d o f x o r 1 S r u h v w d u d } r q v h k d q g h g x f l g r y d u l d v v r o x f l r q h v h { s o  l f l w d v s d u d h o s u r e 0 o h p d g h f r q w u r o P S F g h v l v w h p d v o l q h d o h v f r q u h 0 v w u l f f l r q h v / o d v f x d o h v s x g l h u d q l q f u h p h q w d u h o d u h d g h x v r s d u d h v w h w l s r g h f r q w u r o d g r u + ^ 6 ` / ^ 9 ` | ^ < ` , 1 G h v d i r u w x q d g d p h q w h / h v w d v d s u r { l p d f l r q h v w d p 0 e l  h q v r q f r p s o h m d v s d u d o r v K V 1 O d d s o l f d f l r q g h o P S F d K V u h t x l h u h u h v r o y h u x q s u r e o h p d g h r s w l 0 p l } d f l r q f r q y d u l d e o h v g h g h f l v l r q u h d o h v | h q w h u d v p h } f o d g d v 1 S d u d u h v r o y h u h v w h s u r e o h p d v h s x h g h q x w l o l } d u d o j r u l w p r o o d p d g r v P L T S ^ 5 ` 1 Y d u l r v d o j r 0 u l w p r v k d q v l g r s u r s x h v w r v | d s o l f d g r v f r q  h { l w r s d u d o r v s u r e o h p d v P L T S g h d s o l f d f l r q g h p h 0 g l d q d | j u d q h v f d o d ^ 7 ` 1 V l q h p e d u j r o r v P L T S v v r q j u d q g h v f r q v x p l g r u h v g h w l h p s r | s r u w d q 0 w r s u r k l e l w l y r v h q w l h p s r u h d o s d u d o d p d | r u  l d g h o r v f d v r v 1 F r q h o r e m h w r g h u h g x f l u h o w l h p s r g h f d o f x o r ^ 4 ` s u h v h q w d x q d h v w u d w h j l d e d v d g d h q o d s r 0 g d g h o d v u d p d v g h x q d u e r o t x h u h v x h o y h h o P L T S l q y r o x f u d q g r h y r o x f l r q h v g h w l h p s r g h x q p r g h o r o l q h d o f r q g l q d p l f d v p h } f o d g d v + P O G , 1 H v w h w u d e d m r s u h v h q w d x q s u r f h g l p l h q w r k  l e u l g r s d u d u h r o y h u h o P S F g h v l v w h p d v S Z D | u h 0 g x f l u h o w l h p s r g h f d o f x o r q h f h v d u l r s d u d u h r o y h u h o s u r e o h p d 1 O d d s u r { l p d f l r q s u h v h q w d g d d t x  l q r s h u w h q h f h d o d f o d v h g h p  h w r g r v g h s r g d v g h u d p d v g h x q d u e r o d p s o l d p h q w h f r q r f l g r f r p r E u d q f k d q g E d r x q g + E ) E , / h q v x o x j d u v h x w l 0 o l } d x q p  h w r g r g h h q x p h u d f l r q | x q d o j r u l w p r h v w d q g d u g h s u r j u d p d f l r q f x d g u d w l f d + T S , s d u d u h v r o y h u h o s u r e o h p d g h r s w l p l } d f l r q f r q u h d o h v | h q w h u r v p h } f o d g r v 1 H o s u r f h g l p l h q w r x w l o l } d o r v f r q f h s w r v g h f r q m x q w r d o f d q } d e o h | f r q m x q 0 w r f r q w u r o d e o h ^ : ` f r p e l q d g r f r q h o f r q f h s w r g h j u d  f r g h w u d q v l f l r q g h h v w d g r + V W J , / s d u d u h 0 g x f l u h o q  x p h u r g h s u r e o h p d v T S v t x h v h q h f h v l 0 w d q s d u d u h v r o y h u h o d o j r u l w p r g h r s w l p l } d f l r q 1 H o d o j r u l w p r s u r s x h v w r u h g x f h f r q v l g h u d e o h p h q w h h o q  x p h u r g h h { s o r u d f l r q h v t x h v h q h f h v l w d q g x u d q w h o d e  x v t x h g d g h x q p  l q l p r j o r e d o | d v  l h o w l h p s r g h f d o f x o r u h t x h u l g r s r u h o P S F s x h g h u h g x f l u v h d x q d s h t x h  q d i u d f f l r q g h o w l h p s r u h t x h u l g r v l h o d o 0 j r u l w p r g h r s w l p l } d f l r q v h d s o l f d v h g l u h f w d p h q w h v r e u h h o s u r e o h p d r u l j l q d o 1 H o w u d e d m r h v w d r u j d q l } d g r f r p r v l j x h = h q o d V h f 0 f l r q 5 v h g h v f u l e h h o v l v w h p d S Z D | v h g h v d u u r o o d o d h v w u d w h j l d g h P S F x w l o l } d g d 1 X q h m h p s o r g h v l p x o d f l r q h v p r v w u d g r h q o d V h f f l r q 6 | s r v w h u l 0 r u p h q w h h q o d V h f f l r q 7 v h g d q o d v f r q f o x v l r q h v 1 2 6  +  N w   W   # , w  +   w ,   Y d u l d v i r u p d v g h p r g h o d g r g h K V h q w l h p s r g l v 0 f u h w r h v w d q | d h v w d q g d u l } d g d v / h q w u h h o o d v v h h q 0 f x h q w u d h o S Z D 1 X q v l v w h p d S Z D v h g h  q h f r p r { n . 4 @ D l { n . E l x n . i l | n @ F l { n . j l i r u =  { n x n  5 [ l + 4 , g r q g h i [ l j v l @ 4 h v x q d s d u w l f l r q s r o l  h g u l f d g h o h v 0 s d f l r g h h v w d g r | o d v h q w u d g d v 1 F d g d [ l h v w d g d g d s r u [ l /   { n x n  m T l  { n x n   t l  g r q g h { n / x n > | n g h q r w d q o r v y h f w r u h v g h h v w d 0 g r v / g h o d v h q w u d g d v | g h o d v d o l g d / u h v s h f w l y d 0 p h q w h 1 F d g d v x e v l v w h p d V l g h  q l g r s r u o d : 0 x s o d + D l > E l > F l > i l > j l > T l > t l , / l 5 i 4 > 5 > = = = > v j h v g h q r 0 p d l q d g r f r p s r q h q w h g h o v l v w h p d S Z D 1 H q + 4 , / D l 5 U q  q > E l 5 U q  p > | + D l > E l , h v x q s d u f r q 0 w u r o d e o h 1 F l 5 U u  q / T l 5 U s l  + q . p , / i l > j l | t l v r q y h f w r u h v f r q v w d q w h v d g h f x d g r v 1 Q r w h t x h q h v h o q  x p h u r g h h v w d g r v / p h v h o q  x p h u r g h h q 0 w u d g d v / u h v h o q  x p h u r g h v d o l g d v | s l h v h o q  x p h u r g h k l s h u s o d q r v t x h g h  q h h o s r o l h g u r l 1 H q h v w h w u d e d m r v h v x s r q h t x h h q h o l q v w d q w h g h w l h p s r n v h g l v s r q h g h w r g d v o d v p h g l f l r q h v g h o r v h v w d g r v 1 O d p d | r u  l d g h o d v i r u p x o d f l r q h v g h o P S F u h t x l u h q t x h h o s u r e o h p d X @ d u j + p l q X M , + 5 , w r = M @ Q [ l @ 4 t l l + | n . l m n  z n , 5 . Q  4 [ l @ 4 u l l x 5 n + 6 , v 1 w 1 = x p l q  x n . l  x p d { n @ 4 > = = = > P + 7 , v h u h v x h o y d h q f d g d w l h p s r n / g r q g h | n . l m n g h q r w d h o y h f w r u g h v d o l g d v s u h g l f k r h q h o w l h p s r n . l / r e w h q l g r o x h j r g h d s o l f d u o d v h f x h q f l d g h h q w u d g d X /  x n > 1 1 1 / x n . Q  4  d o p r g h o r + 4 , f r p h q } d q g r g h v g h h o h v w d g r { n v x m h w r d u h v w u l f f l r q h v 1 Q r w h v h t x h z n h v o d v d o l g d g h u h i h u h q f l d 1 H q w r q f h v / v h o h d s o l f d d o s u r f h v r o d s u l p h u d h q w u d g d g h f r q w u r o 1 H q o d s u r { l p d p x h v w u d / o d v p h g l f l r q h v v r q x w l o l } d g d v s d u d d f w x d o l } d u h o s u r e o h p d g h r s w l p l } d f l r q | o d r s w l p l } d f l r q v h u h s l w h 1 G h h v w d i r u p d v h r e w l h q h x q d d s u r { l p d f l r q d o d } r f h u u d g r 1 F r q v l g h u h p r v h o s u r e o h p d g h s u h g l f f l r q d v r f l d g r d o P S F h q h o f d v r g h x q v l v w h p d S Z D 1 H o v x e v l v 0 w h p d t x h g h v f u l e h h o s u r f h v r h v f r q r f l g r v l { n h v f r q r f l g r / s h u r o r v v l j x l h q w h v v x e v l v w h p d v g h s h q 0 g h q g h o d v h f x h q f l d g h f r q w u r o d s o l f d g d 1 S x h g h f r q v l g h u d u v h t x h h q w u h x q l q v w d q w h g h p x h v w u h r | h o s u r { l p r v h s u r g x f h x q f d p e l r + w u d q v l f l r q , g h o p r g h o r 1 H q j h q h u d o v h d f w l y d x q d v h f x h q f l d g h v x e v l v w h p d v L @ i L n / L n . 4 / = = = / L n . Q j 1 V r o r v h f r q r f h h o y d o r u l q l f l d o g h h v w d v h f x h q f l d L n @ L n + { n , 1 V l q r h { l v w h q u h v w u l f f l r q h v / h o q  x p h u r g h v h f x h q f l d v s r v l e o h v s d u d x q k r u l } r q w h g h s u h g l f 0 f l r q Q h v v Q  4 1 S d u d u h v r o y h u h o s u r e o h p d P S F + 6 , v h d  q d g h o d v h f x h q f l d g h r s w l p l } d f l r q f r p r y d u l d e o h g h g h f l v l r q 1 H o s u r e o h p d g h r s w l p l } d f l r q u h v x o w d q w h s x h g h h { s u h v d u v h h q w r q f h v f r p r X @ d u j + p l q X > L M , + 8 , g r q g h o d v u h v w u l f f l r q h v u h o d f l r q d q o d g h s h q g h q f l d g h o d v s r v l e o h v v h f x h q f l d v X h L g h e h q v h u d  q d g l g d v l 1 h 1 = T L n . m { n . 4  t L n . m > m @ i 4 > = = > Q j + 9 , G h e l g r d o d q d w x u d o h } d h q w h u d g h o d v h f x h q f l d L / h o s u r e o h p d g h h q f r q w u d u h o r s w l p r s x h g h u h v r o y h u v h h q f r q w u d q g r h o r s w l p r g h o d v v r o x f l r q h v s d u d w r g d s r v l e o h v h f x h q f l d g h L / l 1 h 1 X @ d u j  p l q L  p l q X  M T L X x  t L X    + : , g r q g h T L X x  t L X l q g l f d o d v u h v w u l f f l r q h v g h e l g r d o d g h s h q g h q f l d h q w u h L | X 1 5 1 4 U h v r o x f l r q g h o S u r e o h p F r p s x w d f l r q d o O d h f x d f l r q + 6 , s x h g h h v f u l e l u v h f r p r M @ +  |   z , W  T +  |   z , .  x W  U  x + ; , g r q g h  U @ g l d j ^ u l l ` / +  U @  U W " 3 , /  T @ g l d j ^ t l l ` / +  T @  T W " 3 , 1 Q r w h t x h  T + p d 0 w u l } t x h s h v d h o h u u r u , h v g l i h u h q w h d T l + p d w u l } t x h g h  q h h o s r o l h g u r , /  z @  z W n . 4    z W n . Q  W /  x @  x W n    x W n . Q  4  W | h o y h f w r u g h v d o l g d s u h g l 0 f k r  | @  | W n . 4    | W n . Q  W s x h g h v h u h v f u l w r f r p r  | @ I | { n . K |  x . i r | + < , I | @ F | I { > K | @ F | K { > i r | @ F | i r { g r q g h = F | @ g l d j + F L n . 4 > F L n . 5 >    > F L n . Q , K { @  k 4 k 5    k Q  I { @ 5 9 9 9 7 D L n D L n . 4 D L n 1 1 1 D L n . Q  4 D L n . 4    D L n 6 : : : 8 k 4 @ 5 9 9 9 7 E L n D L n . 4 E L n 1 1 1 D L n . Q  4 D L n . Q  5    D L n . 4 E L n 6 : : : 8 k 5 @ 5 9 9 9 9 9 7 3 E L n . 4 D L n . 5 E L n . 4 1 1 1 D L n . Q  4 D L n . Q  5    D L n . 4 E L n . 4 6 : : : : : 8 k Q @ k 3 3 3 1 1 1 + E L n . Q  4 , W l W i r { @  i 4 i 5    i Q  i 4 @ 5 9 9 9 7 L D L n . 4 1 1 1 D L n . Q  4 D L n . Q  5    D L n . 4 6 : : : 8 i 5 @ 5 9 9 9 9 9 7 3 L D L n . 5 1 1 1 D L n . Q  4 D L n . Q  5    D L n . 5 6 : : : : : 8 i Q @  3 3 3    L  W Q r w h t x h v h f x p s o h q o d v v l j x l h q w h v l j x d o g d g h v  { @ I { { n . K {  x . i r { + 4 3 ,  | @ F |  {  { @  { W n . 4 { W n . 5    { W n . Q  W U h h p s o d } d q g r + < , h q + ; , h o  l q g l f h h v M + L > X , @  x K T S  x . i W T S  x . j T S + 4 4 , g r q g h K T S @  K W |  T K | .  T  i W T S @  5 { n I W |  T K | . 5 i W r  T K | . 5  z W  T K |  j T S @ { W n I W |  T I | { n . 5 i W r  T I | { n . 5 i W r  T  z . i W r  T i r  5  z W  T I | { n .  z W  T  z O d u h v w u l f f l r q v r e u h h o f r q w u r o + 7 , s x h g h h v f u l e l u v h f r p r T x  x  t x / T x @ ^  L Q  Q L Q  Q ` W > t x @ k  + x p l q , W + x p d { , W l W | o d v u h v w u l f f l r q h v g h e l g r d o d g h s h q g h q f l d g h L | X + 9 , s x h g h q h v f u l e l u v h f r p r T L {  { L  t L { + 4 5 , g r q g h  { L @  { W n . 4 / 1 1 1 / { W n . Q  4  W / T L { @ g l d j + T L n . 4 / = = = > T L n . Q  4 , /  { L @ k  t L n . 4  W / 1 1 1 /  t L n . Q  4  W l W /  { L s x h g h h v f u l e l u v h f r p r  { L @ F {  { + 4 6 , g r q g h F { @  L q  Q  q  Q 3  1 U h h p s o d } d q g r + 4 6 , | + 4 3 , h q + 4 5 , / o d v u h v w u l f f l r q h v g h e l g r d o d g h s h q g h q f l d h q w u h X h L u h v x o w d q h q T L X  x  t L X / T L X @ T L { F { K { / t L X @ t L {  T L { F { I { { n  T L { F { i r { 1 V l o d v u h v w u l f f l r q h v h q o d v d f f l r q h v g h f r q w u r o v h f r q v l g h u d q w d p e l  h q h q w r q f h v T T S  x  t T S + 4 7 , , N  >, N @ 4 5 V   5 5 V   v 5 V   >, N , N @ >, N , N , N @ I l j x u d 4 = J u d  f r g h w u d q v l f l r q h v 1 T T S @  + T x , W + T L X , W  W | t T S @  + t x , W + t L X , W  W 1 S r u w d q w r / x q d y h } t x h v h  m d o d v h f x h q f l d L / h o s u r e o h p d s x h g h u h v r o y h u v h p l q l p l } d q g r + 4 4 , v x m h 0 w r d o d v u h v w u l f f l r q h v + 4 7 , 1 5 1 5 F r q m x q w r D o f d q } d e o h / F r q m x q w r F r q w u r o d e o h | D o j r u l w p r g h W u d q v l f l r q g h H v w d g r v O d l g h d i x q g d p h q w d o g h o d o j r u l w p r s u r s x h v w r h v g h w h u p l q d u h o f r q m x q w r g h s r v l e o h v u h j l r q h v t x h s x h g h q d o f d q } d u v h g h v g h o d u h j l r q d f w x d o h q x q r v s r f r v w l h p s r v g h p x h v w u h r v l j x l h q w h v 1 S d u d h v w h s u r s r v l w r v h x w l o l } d h o f r q f h s w r g h f r q m x q w r d o f d q 0 } d e o h ^ : ` 1 H o f r q m x q w r g h u h j l r q h v t x h s x h g h q d o 0 f d q } d u v h g h v g h x q d g h w h u p l q d g d u h j l r q s x h g h s u r 0 j u d p d u v h f r p r x q J u d  f r g h W u d q v l f l r q g h H v w d 0 g r v + V W J , w d o | f r p r s x h g h y h u v h h q o d  j x u d 4 1 H q w r q f h v s x h g h r u j d q l } d u v h x q d e  x v t x h g d g l u l j l g d s r u h v w h j u d  f r 1 H v g h f l u / w r g d v h f x h q f l d t x h q r s x h g d r e w h q h u v h v l j x l h q g r h o j u d  f r g h w u d q v l f l r q / q r v h f r q v l g h u d 1 G h  q l f l r q 4 = ^ : ` F r q m x q w r D o f d q } d e o h + U V , U u +  , = G d g r x q v l v w h p d g l q d p l f r g l v f u h w r { n . 4 @ i + { n > x n > z n , g r q g h z n 5 Z  U g h v x q d s h u w x u 0 e d f l r q g h v f r q r f l g d + Z v r q f r q m x q w r v f h u u d g r v h q U g , | x n 5 X  U p + X h v x q f r q m x q w r f r p s d f w r h q U p , 1 H o f r q m x q w r U u +  , h v x q f r q m x q w r g h h v 0 w d g r v h q U q h q h o f x d o h o v l v w h p d v h h q f r q w u d u d h q h o s u r { l p r s d v r g h w l h p s r / s d u d d o j  x q { n   / h q w u d g d g h f r q w u r o | s h u w x u e d f l r q d g p l v l e o h v / l 1 h 1 U u +  , /  { n . 4 5 U q m < { n 5  > x n 5 X > z n 5 Z = { n . 4 @ i + { n > x n > z n ,  + 4 8 , H o U V s d u d x q v l v w h p d o l q h d o g l v f u h w r { n . 4 @ D { n . E x n g h x q f r q m x q w r g h s r o l w r s r  g h o h v 0 s d f l r g h h v w d g r / v l v h f r q v l g h u d t x h o d d f f l r q g h f r q w u r o x n 5 X  U p + X h v x q s r o l w r s r + f r p 0 s d f w r , h q U p , s x h g h f d o f x o d u v h f r p r x q d v x p d g h P l q n r z v n l g h o d s u r | h f f l r q r u w r j r q d o g h o f r q 0 m x q w r  v r e u h D | h o X v r e u h E 1 Y h d ^ : ` s d u d p d v g h w d o o h v 1 E d v d g r v h q o d l g h d g h o r v f r q m x q w r v d o f d q } d e o h s x h g h g h  q l u v h h o Y h f l q r d o f d q } d e o h d x q s d v r + 4 0 V U Q , 1 H o v x e v l v w h p d V m h v 4 0 V U Q g h o v x e 0 v l v w h p d V l / g h q r w d g r s r u V l $ 4 V m / v l h o f r q m x q 0 w r [ m s x h g h v h u d o f d q } d g r g h v g h h o f r q m x q w r [ l h q x q s d v r 1 H v s r v l e o h g h w h u p l q d u v l V l $ 4 V m g h 0 e l g r d V l $ 4 V m + , U + [ l , _ [ m 9 @ > 1 G h o d p l v 0 p d p d q h u d h v w h v h s x h g h g h w h u p l q d u d g r v s d 0 v r v + 5 0 V U Q , v l v h f r q v l g h u d q o r v f r q m x q w r v d o f d q 0 } d e o h v d g r v s d v r v U 5 + [ l , @ U + U + [ l , , f r p r V l $ 5 V m + , U 5 + [ l , _ [ m 9 @ > 1 \ d v  l v x f h v l y d 0 p h q w h s d u d h o q 0 V U Q 1 G h  q l f l r q 5 = F r q m x q w r D o f d q } d e o h d q 0 s d v r v + q 0 V U Q , V l $ q V m 1 V l $ q V m + , U q + [ l , _ [ m 9 @ > H o  l q g l f h g h w r g r v o r v v x e v l v w h p d v q 0 V U Q g h o v x e 0 v l v w h p d l i r u p d q h o f r q m x q w r  l q g l f h g h q 0 V U Q 1 G h  q l w l r q 6 = F r q m x q w r  l q g l f h g h q 0 V U Q Q l q Q l q / q n 5 ^ 4 > = = = > v ` @ V l $ q V n r O d w u d q v l f l r q g h h v w d g r v g h L m d L m . 4 s h u p d q h f h d f r w d g d s r u o r v 4 0 V U Q v g h o f r u u h v s r q g l h q w h v x e 0 v l v w h p d L m 1 H v w r s h u p l w h s r g d u h o j u d  f r g h w u d q 0 v l f l r q + V W J , = o d v w u d q v l f l r q h v g h v g h L m d L m . 4 q r v r q f r q v l g h u d g d v v l L m . 4 @ 5 Q L m 4 1 V l h v w h f r q f h s 0 w r v h h { w l h q g h / h o d u e r o g h e  x v t x h g d s x h g h v h u s r g d g r f r q v l g h u d q g r d g h p d v t x h L m . n g h e h u  l d s h u w h q h f h u d Q L m n 1 S r u w d q w r / v l v r o r v h f r q v l g 0 h u d q o r v 4 0 V U Q s d u d x q k r u l } r q w h g h s u h g l f f l r q Q / h o q  x p h u r g h s r v l e o h v v h f x h q f l d v d s u r e d u s r u h o d o j r u l w p r T S h v v + L n , v + L n . 4 , = = = v + L n . Q  4 , + h q o x 0 j d u g h v Q / v + L n ,  v , / g r q g h v L n h v h o q  x p h u r g h 4 0 V U Q v f r u u h v s r q g l h q w h v d o v x e v l v w h p d L n 1 G h e h q r w d u v h t x h o d g h w h u p l q d f l r q g h o 4 d q 0 V U Q v s d u d f d g d v x e v l v w h p d s x h g h k d f h u v h i x h u d g h o  l q h d 1 O d o l v w d g h y h f l q r v s d u d f d g d x q r g h o r v v x e v l v 0 w h p d v v h l q f o x | h h q o d g h v f u l s f l r q g h o p r g h o r 1 S r u w d q w r / f d g d v x e v l v w h p d h v w d g h  q l g r s r u o d + : . Q , 0 x s o d + D l > E l > F l > i l > j l > T l > t l > Q l 4 > = = = > Q l Q , / l 5 i 4 > 5 > = = = > v j g r q g h / Q l h v o d o l v w d t x h f r q w l h q h o r v y h f l q r v d o f d q } d e o h v g h o v x e v l v w h p d l 1 H o V W J s x h g h s r g d u v h p d v d x q v l v h f r q v l g h u d q o r v f r q 0 m x q w r f r q w u r o d e o h v d x q s d v r + 4 0 V F V , 1 G h  q l f l r q 7 = ^ : ` F r q m x q w r F r q w u r o d e o h d x q s d v r + 4 0 V F V , T f +  , 1 H o f r q m x q w r T f +  , h v h o f r q m x q w r g h h v w d g r v h q U q s d u d h o f x d o h { l v w h x q d h q w u d g d g h f r q w u r o d g p l v l e o h t x h j d u d q w l } d u d t x h h o v l v w h p d v h d o o h y d g r d  h q x q s d v r / s d u d w r g d s h u w x u e d f l r q s h u p l v l e o h / l 1 h 1 T f +  , /  { n 5 U q m < x n 5 X = i + { n > x n > z n , 5  > ; z n 5 Z  + 4 9 , H q ^ : ` v h p x h v w u d x q p  h w r g r s d u d f d o f x o d u h o 4 0 V F V 1 H v l p s r u w d q w h k d f h u  h q i d v l v h q t x h h o 4 0 V F V | h o U V r s h u d q h q g l i h u h q w h v g l u h f f l r q h v 1 H o 4 0 V F V             -2              [ 1 5  ;   LQW5  ;  ;   4   4   4   [  I l j x u d 5 = F r q m x q w r v f r q w u r o d e o h v d x q r / g r v | w u h v s d v r v g h o v h f w r u < v r e u h h o v h f w r u 4 5 h v h o f r q m x q w r g h h v w d g r v g h v g h h o f x d o h o v l v w h p d s x h g h v h u f r q g x f l g r d x q f r q m x q w r g d g r 1 H o U V h v h o f r q m x q w r g h h v w d g r v d o f x d o h o v l v w h p d s x h g h v h u f r q g x f l g r g h v g h x q f r q m x q w r g d g r 1 Q r h { l w h x q d u h o d f l r q h { s o  l f l w d h q w u h h v w r v g r v f r q m x q w r v 1 H o 4 0 V F V g h o v x e v l v w h p d m v r e u h h o v x e v l v w h p d l T m @ l 4 / h v h o v x e f r q m x q w r g h [ l g h v g h g r q g h h o v x e 0 v l v w h p d m + [ m , s x h g h v h u d o f d q } d g r h q x q s d v r 1 H v w r h v / T m @ l 4 /  { n 5 [ l m < x n 5 X = i + { n > x n > z n , 5 [ m > ; z n 5 Z  H v w h f r q m x q w r s x h g h f d o f x o d u v h f r p r T m @ l 4 @ T 4 + U 4 + [ l , _ [ m , _ [ l H v w h f r q f h s w r s x h g h v h u h { w h q g l g r g h v g h 4 k d v w d q s d v r v f r p r T m @ l q 1 Q r w h t x h o d s u r { l p d i x q f l r q g h h v w d g r f r u u h v s r q g l h q w h d o v l v w h p d l h v w d l p s o l f d g d h q h o f d o f x o r g h o U V d v  l f r p r h q h o h o f d o f x o r g h o 4 0 F V 1 O d  j x u d 5 p x h v w u d h o f r q m x q w r g h 4 / 5 | 6 s d v r v s d u d h o v h f w r u < v r e u h h o v h f w r u 4 5 1 S d u d u h g x f l u h o q  x p h u r g h h y d o x d f l r q h v g h o s u r e o h 0 p d T S / v h f r q v l g h u d h o h v w d g r g h { n @ 5 T L n . 4 @ L n 4 1 V l h v w d s u r s r v l f l r q v h v d w l v i d f h / h q w r q f h v h o h v w d g r { n . 4 q r s h u w h q h f h d o d w u d q v l f l r q L n . 4 > s r u w d q w r / o d w u d q v l f l r q g h v g h L n k d v w d L n . 4 q r h v w d s h u p l w l 0 g d 1 S d u d d s o l f d u h v w r v f r q f h s w r v / g h e h d  q d g l u v h o d g h  q l f l r q g h o f r q m x q w r T m @ l 4 / 1 1 1 / T m @ l Q d o d g h  q l 0 f l r q g h f d g d v x e v l v w h p d 1  , a ,   w  W w N 5 A +  A W V  O d h v w u d w h j l d g h h { s o r u d f l r q g h o d u e r o s u r s x h v w d h q h v w h w u d e d m r k d v l g r d s o l f d g d d o f r q w u r o s u h g l f 0 w l y r g h o v l v w h p d g h x q s  h q g x o r v l p s o h 1 V h k d x w l 0 o l } d g r f r p r p r g h o r d o d h f x d f l r q o l q h d o l } d g d g h o d g l q d p l f d g l v f u h w d g h o v l v w h p d g h o s  h q g x o r v l p s o h 1                   [   N [   N [   N [   N [   N I l j x u d 6 = U h s u h v h q w d f l r q g h o p r g h o r S Z D g h o s  h q g x o r 1 F r q v l g h u h h o v l j x l h q w h v l v w h p d q r o l q h d o p o   . n o b  . p j v l q +  , @  o g r q g h  h v h o d q j x o r g h o s  h q g x o r / o h v o d o r q j l w x g g h o s  h q g x o r / p h v o d p d v d g h o s  h q g x o r / j h v o d i x h u d j u d y l w d f l r q d o / n h v x q f r h  f l h q w h g h i u l f f l r q |  h v h o w r u t x h d s o l f d g r 1 H q w r q f h v / h o p r g h o r h q h o h v s d f l r g h h v w d g r g l v f u h w l } d g r h q x q w l h p s r g h p x h v w u h r W 3 h v { n . 4 4 @ { n 4 . W 3 { n 5 { n 4 @  4  W 3 n p  { n 5  W 3 j o v l q  { n 4  . W 3 o 5 p x n g r q g h x n @  + n , / { n 4 @  + n , | { 5 n @ b  + n , 1 X w l o l } d q g r p @ 4 / o @ 4 / n @ 3 = 8 / j @ < = ; / W 3 @ 3 = 3 5 f r p r s d u d p h w u r v s d u d h o p r g h o r 1 D s d u w l u g h x q p r g h o r g l v f u h w l } d g r v h k d r e w h q l g r x q v l v w h p d S Z D f r p r + 4 , x w l o l } d q g r x q d o l q h d 0 o l } d f l r q s r u v h f w r u h v v r e u h x q d j u l o o d x q l i r u p h g h o h v s d f l r g h h v w d g r 1 V h g h  q h x q d s d u w l f l r q [ l /  { n m T l { n  t l  o x h j r g h r e w h q h u 8 9 v h f w r u h v v r e u h h o h v s d f l r g h h v w d g r 1 V h f r q v l g h u d q l q y d 0 u l d q w h v d o d v p d w u l f h v F l | E l / l 1 h 1 = E l @ E @  3 W 3 o 5 p  W | F l @ F @ L + p > p , 1 O d  j x u d 6 p x h v 0 w u d o d i x q f l r q g h o l q h d o l } d f l r q g h h v w d g r v 1 X q d y h } p d v v h r e w l h q h h o p r g h o r o l q h d o l } d g r / o r v y h f l q r v f r u u h v s r q g l h q w h v d f d g d v h f w r u v h s x h g h q h q f r q 0 w u d u x w l o l } d q g r h o d o j r u l w p r g h f r q m x q w r v d o f d q 0 } d e o h v 1 H q h v w h h m h p s o r / o d v f r w d v g h o w r u t x h v r q 4 3 = : ;    4 3 = : ; + 4 3 = : ; @ 4 = 4 p j o , / g r q g h h o w r u t x h p d { l p r h q h o h v w d g r h v w d f l r q d u l r s d u d d o 0 f d q } d u f x d o t x l h u s r v l f l r q h v  @ p j o 1 W r g d v o d v s u x h e d v g h v l p x o d f l r q i x h u r q k h f k d v h q P d w o d e 1 O d W d e o d 4 p x h v w u d h o g h v h p s h  q r g h o f r p s x w d 0 g r u x w l o l } d g r 1 O d W d e o d 5 p x h v w u d o d v v l j o d v g h o p  h w r g r g h s r g d x w l o l } d g r | x q d e u y h d f o d u d f l r q g h o p l v p r 1 V h k d f r q v l g h u d g r x q k r u l } r q w h g h s u h g l f f l r q Q @ 7 1 O r v s h v r v g h o h u u r u | g h o d d f 0 f l r q g h f r q w u r o v r q  T { 4 @ 4 3 3 3 /  T { 5 @ 4 3 3 / U @ 4 / u h v s h f w l y d p h q w h 1 O d  j x u d 7 p x h v w u d o r v u h v x o w d g r v s d u d x q d u h 0 i h u h q f l d z @ ^ 3 3 ` | s d u d g l i h u h q w h v f r q g l f l r q h v l q l f l d o h v 1 O d v o  l q h d v v r o l g d v | g h s x q w r v p x h v w u d q W d e o d 4 = H v w d q g d u g h f r p s d u d f l r q g h P d w o d e + i x q f l r q % e h q f k % , = g h v h p s h  q r g h o d S F e d m r Z l q 0 g r z < ; / P d w o d e 8 1 6 R G H O X V s d u f h 6 0 G 5 0 G 3 1 6 6 3 1 6 ; 3 1 6 < 4 1 5 4 4 1 5 4 W d e o d 5 = V l j o d v g h o p  h w r g r 1 V U Q @ y h f l q r v d o f d q 0 } d e o h v d q s d v r v > V F R V @ f r q m x q w r v f r q w u r o d e o h v d q s d v r v g h x q v h f w r u v r e u h r w u r 5  } * @  U * @ h @ U  L ?    L ? t  _ i h @ t L * @ 4 i ? | i 4  5 +    2   L ? t  _ i h @ 4 c 5 c 6  5 +        L ? t  _ i h @ 4  5 +  ) 4  5   5      2  L ? t  _ i h @ 4  5 +  ) 4 c 5  5   5      2   L ? t  _ i h @ 4  5 +  ) 4 c 5 c 6  5   5   2      L ? t  _ i h @ 4 c 5 c 6  5 +  ) 4  5   5   2     2  L ? t  _ i h @ 4 c 5 c 6  5 +  ) 4 c 5  5   5   2     2   L ? t  _ i h @ 4 c 5 c 6  5 +  ) 4 c 5 c 6  5   5 h o f r p s r u w d p l h q w r g h o v l v w h p d s d u d  @ 3 + o d } r d e l h u w r , | h o v l v w h p d f r q w u r o d g r / u h v s h f w l y d p h q w h 1 R e v h u y h t x h h o h v w d g r h q o d } r d e l h u w r g l y h u j h s d u d o d v f r q g l f l r q h v l q l f l d o h v { 3 @ ^ 6 6 ` W 1 O d  j x u d 8 p x h v w u d x q d f x u y d g h h v w d g r s d u d p h w u l } d g d h q h o w l h p s r | h o q  x p h u r g h h y d 0 o x d f l r q h v T S s d u d h o p  h w r g r g h u h g x f f l r q + s r g d , Q 4 5 6 1 H q h v w h f d v r h o q  x p h u r g h h y d o x d f l r q h v g h T S h v x q d i x q f l r q h { f o x v l y d g h o v x e v l w h p d d o f x d o s h u w h q h f h 1 O d  j x u d 9 f r p s d u d o d v u h v s x h v w d v g h o p  h w r g r + Q 4 , f r q w u d + Q 4 5 6 , 1 Q r w h t x h h q j h q h u 0 d o h o q  x p h u r g h s u r e o h p d v T S g h f u h f h f x d q g r v h x w l o l } d h o p  h w r g r Q 4 5 6 1 O d  j x u d ; p x h v w u d f r p r v l v h x w l o l } d h o p  h w r g r Q 4 0 F 4 5 6 / h o q  x p h u r g h h y d o x d f l r q h v g h T S g h 0 f u h f h f x d q g r h o h v w d g r l q l f l d o h v w d o h m r v g h o d u h j l r q g h o d i u r q w h u d / h v w r h v / f x d q g r { 3 q r s h u w h q h f h d q l q j  x q f r q m x q w r f r q w u r o d e o h | v x v y h f l q r v 1 H q h v w h v h f w r u h o q  x p h u r g h h y d o x d f l r q h v T S g h f u h f h f x d q 0 g r h o p  h w r g r Q 4 0 F 4 v h f r p s d u d f r q Q 4 0 F 4 5 6 + I l j 1 : , 1 O d  j x u d < f r p s d u d h o q  x p h u r g h h y d o x d f l r 0 q h v T S + p  l q l p r / p d { l p r | s u r p h g l r , s d u d g l i h u 0 h q w h v p  h w r g r v g h s r g d 1 Q r w h t x h f x d q g r f x d q g r v h f r q v l g h u d x q d o w r q  x p h u r g h f r q m x q w r v f r q w u r 0 o d e o h v d x q t x h g h f u h f h q o d f d q w l g d g g h h y d o x d f l r q h v p  l q l p d | s u r p h g l r / o d p d { l p d f d q w l g d g g h h y d 0 o x d f l r q h v s h u p d q h f h f r q v w d q w h 1 e     w N 5 W   , 5 H q h v w h w u d e d m r v h k d s u r s x h v w r x q d q x h y d d s u r { l 0 p d f l r q s d u d u h v r o y h u h o f r q w u r o s u h g l f w l y r g h v l v 0 w h p d v + o l q h d o h v , d  q h v d w u d p r v 1 H o d o j r u l w q r v h e d v d h q o d h { s o r u d f l r q g h j u d  f r v g h w u d q v l f l r q g h h v w d g r v | o r v f r q f h s w r v g h f r q m x q w r v d o f d q } d e o h v -4 -3 -2 -1 01234 -10 -8 -6 -4 -2 0 2 4 6 8 10 x 0=[-3,3] x 0=[2,1] x 0=[3,3] x 2 x 1 I l j x u d 7 = S o d q r i d v h f r q z @ ^ 3 3 ` | { 3 @ ^ 5 4 ` W / { 3 @ ^ 6 6 ` W / { 3 @ ^  6 6 ` W 0           W>VHJ@           143(YDOXDWLRQV [  [  W  W  W  t=0.6 W  t=1 W  W   W  W  W  W  W  0HWKRG1 27              W  W  W  W  W  W  W  W  W  W  t=2 t=2.2 W  0HWKRG1 I l j x u d 8 = S o d q r g h i d v h | q  x p h u r g h h y d o x d f l r q h v T S s d u d p h w u l } d g r h q h o w l h p s r + Q 4 5 6 , 1 W>VHJ@                   143(YDOXDWLRQV W>VHJ@       &RPSXWLQJ7LPH>VHJ@ 0HWKRG 1 1 I l j x u d 9 = W l h p s r g h f d o f x o r | q  x p h u r g h h y d o x d 0 f l r q h v g h T S ^ + 0 , Q 4 5 6 + = , Q 4 `       W>VHJ@ 0      W>VHJ@        &RPSXWLQJ7LPH>VHJ@       143(YDOXDWLRQV 0HWKRG 1& 1& I l j x u d : = W l h p s r g h f d o f x o r | q  x p h u r g h h y d o x d 0 f l r q h v g h T S ^ + 0 , Q 4 0 F 4 + 0 0 , Q 4 0 F 4 5 6 `            W>VHJ@            143(YDOXDWLRQV [  [                W  W  W  W  W  W  W  W  W  W  W  W  W     W  W  W  W  W  W  W  W  t=1.6 W  W  W  W  0HWKRG 1& I l j x u d ; = S o d q r g h i d v h | q  x p h u r g h h y d o x d f l r q h v T S s d u d p h w u l } d g r h q h o w l h p s r + Q 4 0 F 4 5 6 , 1        [143(YDOXDWLRQV0LQ 143(YDOXDWLRQV0D[ R143(YDOXDWLRQV0HDQV 1XPEHURI43(YDOXDWLRQV 1& 1& 1& 1& 1& 1& 1 1 0HWKRG I l j x u d < = W d e o d g h f r p s d u d f l r q g h o q  x p h u r g h h y d o x d f l r q h v T S 1 | f r q w u r o d e o h v 1 S d u d o d h v w u d w h j l d g h h { s o r u d f l r q g h o d u e r o v h k d q h o h j l g r o r v f r q m x q w r v d o f d q } d e o h v s d u d g l v p l q x l u h o q  x p h u r g h s r v l e o h v u h d o l } d f l r q h v g h o d v y d u l d e o h v h q w h u d v | g h h v d i r u p d u h g x f l u f r q 0 v l g h u d e o h p h q w h h o q  x p h u r g h s u r e o h p d v T S t x h v h q h f h v l w d u h v r o y h u 1 D j u d g h f l p l h q w r v O r v d x w r u h v d j u d g h f h q d o r v P l q l v w h u l r v g h H g x 0 f d f l r q | F x o w x u d | G h s r u w h | g h F l h q f l d | W h f 0 q r o r j  l d + H v s d  q d , | d o F r q v h m r Q d f l r q d o L q y h v w l j d 0 f l r q h v F l h q w  l  f d v | W h f q r o r j l f d v + D u j h q w l q d , s r u  q d q f l d u h v w h w u d e d m r e d m r o r v s u r | h f w r v V E 5 3 3 3 0 3 5 ; 8 d q g G S L 5 3 3 5 0 3 7 6 : 8 0 F 3 6 0 3 4 + i u i h i ? U  @ t ^ 4 ` D 1 E h p s r u d g / G 1 P l j q r q h / d q g P 1 P r u d u l 1 D q h  f l h q w e u d q f k d q g e r x q g d o j r u l w k p i r u v w d w h h v w l p d w l r q d q g f r q w u r o r i k | e u l g v | v 0 w h p v 1 L q S u r f 1 H x u r s h d q F r q w u 1 F r q i / N d u o 0 v u x k h / J h u p d q | / 4 < < < 1 ^ 5 ` D 1 E h p s r u d g d q g P 1 P r u d u l 1 F r q w u r o r i v | v 0 w h p v l q w h j u d w l q j o r j l f / g | q d p l f v / d q g f r q 0 v w u d l q w v 1 D x w r p d w l f d / 6 8 + 6 , = 7 3 : ~ 7 5 : / 4 < < < 1 ^ 6 ` D 1 E h p s r u d g / P 1 P r u d u l / Y 1 G x d / d q g H 1 S l v 0 w l n r s r x o r v 1 W k h h { s o l f l w o l q h d u t x d g u d w l f u h j 0 x o d w r u i r u f r q v w u d l q h g v | v w h p v 1 L q S u r f 1 D p h u 1 F r q w u 1 F r q i / s d j h v ; : 5 ~ ; : 9 / F k l f d j r / L O / 5 3 3 3 1 ^ 7 ` U 1 I o h w f k h u d q g V 1 O h |  h u 1 Q x p h u l f d o h { s h 0 u l h q f h z l w k o r z h u e r x q g v i r u p l t s e u d q f k 0 d q g 0 e r x q g 1 W h f k q l f d o u h s r u w / X q l y h u v l w | r i G x q g h h / G h s w 1 r i P d w k h p d w l f v / V f r w o d q g / X 1 N 1 / 4 < < 8 1 ^ 8 ` V 1 K h g o x q g d q g D 1 U d q w } h u 1 R s w l p d o f r q w u r o r i k | e u l g v | v w h p v 1 L q S u r f 1 6 ; w k L H H H F r q i 1 r q G h f l v l r q d q g F r q w u r o / s d j h v 6 < : 5 ~ 6 < : 9 / S k r h q l { / D ] / X V D / 4 < < < 1 ^ 9 ` W 1 M r k d q v v h q / L 1 S h w h u v h q / d q g R 1 V o x s s k d x j 1 H { s o l f l w v x e r s w l p d o o l q h d u t x d g u d w l f u h j x o d 0 w l r q z l w k l q s x w d q g v w d w h f r q v w u d l q w v 1 L q S u r f 1 r i 6 < w k 1 L H H H F r q i 1 r q G h f l v l r q d q g F r q w u r o / V l g q h | / 5 3 3 3 1 ^ : ` H 1 N h u u l j d q 1 U r e x v w F r q v w u d l q w V d w l v i d f w l r q = L q y d u l d q w v h w v d q g s u h g l f w l y h f r q w u r o 1 S k G w k h v l v / X q l y h u v l w | r i F d p e u l g j h / 5 3 3 3 1 ^ ; ` S 1 U l h g l q j h u / I 1 N u d w } / F 1 L x q j / d q g F 1 ] d q q h 1 O l q h d u t x d g u d w l f r s w l p l } d w l r q i r u k | e u l g v | v w h p v 1 L q S u r f 1 6 ; w k L H H H F r q i 1 r q G h f l v l r q d q g F r q w u r o / S k r h q l { / D ] / X V D / 4 < < < 1 ^ < ` P 1 V h u r q / M 1 G r q / d q g J 1 J r r g z l q 1 J o r e d o d q d o l w l f d o p r g h o s u h g l f w l y h f r q w u r o z l w k l q s x w f r q v w u d l q w v 1 L q S u r f 1 r i 6 < w k 1 L H H H F r q i 1 r q G h f l v l r q d q g F r q w u r o / V l g q h | / 5 3 3 3 1 ^ 4 3 ` F 1 W r p o l q / M 1 O | j h u r v / d q g V 1 V d v w u | 1 D j d p h w k h r u h w l f d s s u r d f k w r f r q w u r o o h u g h v l j q i r u k | e u l g v | v w h p v 1 L q S u r f 1 r i w k h L H H H / y r o 0 x p h ; ; / M x o | 5 3 3 3 1