Chap_1_Static Optimization_1.1_2014.ppt

8/9/2019 Chap_1_Static Optimization_1.1_2014.ppt

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OPTIMAL CONTROL

Chapter_1 Static Optimization

1.1 Optimization Without Constraints, 11.2 Optimization With Equality Constraints, 5

1.3 Numerical Solution Methos, 21

!asics " 2,3#$%lots


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The goals of this course are to epose the stu!ents to the mathematical tools ofparametric an! !"namic optimization an! their uses in !esigning optimall"#eha$ing !"namic s"stems% The nonlinear s"stems course co$ers the follo&ingtopics'1. Static Optimization

2.& Optimal Control o' #iscrete$(ime Systems

2.1 )inear *uaratic +eulator

2.2 Steay$State Close$loop Control o' Su-$Optimal ee-ac/

2.3 (he (rac/in %ro-lem

2.0 +eulator ith unction o' inal State ie

3.& Optimal Control o' Continuous$(ime Systems

3.1 )inear *uaratic +eulator

3.2 Steay$State Close$loop Control o' Su-$Optimal ee-ac/

3.3 (he (rac/in %ro-lem3.0 +eulator ith unction o' inal State ie

3.5 inal$(ime$ree %ro-lem

3. Constraine 4nput %ro-lem

0.& #ynamic %rorammin0.1 #iscrete$(ime


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4ntrouctionThe state fee!#ac( an! o#ser$er !esign approach is a fun!amental tool in thecontrol of state e)uation s"stems% *o&e$er+ it is not al&a"s the most usefulmetho!% Three o#$ious !ifficulties are'

, The translation from !esign specifications -maimum !esire!o$er an! un!ershoot+ settling time+ etc%. to !esire! poles is notal&a"s !irect+ particularl" for comple s"stems/ &hat is the #estpole configuration for the gi$en specifications0, In MIMO s"stems the state fee!#ac( gains that achie$e a gi$enpole configuration is not uni)ue% hat is the #est 2 for a gi$enpole configuration0, The eigen$alues of the o#ser$er shoul! #e chosen faster than

those of the close!3loop s"stem% Is there an" other criteriona$aila#le to help !eci!e one configuration o$er another0

The metho!s that &e &ill no& intro!uce gi$e ans&ers to these

)uestions% e &ill see ho& the state fee!#ac( an! o#ser$er gainscan #e 'oun in an optimal ay.


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(he !asic Optimal Control %ro-lem

hat !oes optimal mean0 Optimal means !oing a 4o# in the #estpossi#le &a"% *o&e$er+ #efore starting a search for an optimalsolution+

, theo-must #e !efine!, a mathematical scale must #e esta#lishe! to )uantif" &hat

&e mean #" -est, the possi-lealternati$es must #e spelle! out%

5nless these )ualifiers are clear an! consistent+ a claim that as"stem is optimal is reall" meaningless%

A cru!e+ inaccurate s"stem might #e consi!ere! optimal #ecauseit is inepensi$e+ is eas" to fa#ricate+ an! gi$es a!e)uateperformance%

Con$ersel"+ a $er" precise an! elegant s"stem coul! #e re4ecte!as non3optimal #ecause it is too epensi$e or it is too hea$" or

&oul! ta(e too long to !e$elop%


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6urther+ the control as &ell as the state $aria#les are generall"su#4ect to constraints+ &hich ma(e man" pro#lems in optimal

control non3classical+ since pro#lems &ith path constraints canhar!l" #e han!le! in the classical calculus of $ariations%

That is+ the pro#lem of optimal control can then #e state! as'7etermine the control signals that &ill cause a s"stem to satisf"the ph"sical constraints an!+ at the same time+ minimize -ormaimize. some performance criterion%%

7espite its successes+ ho&e$er+ optimal control theor" is #" nomeans complete+ especiall" &hen it comes to the )uestion of&hether an optimal control eists for a gi$en pro#lems%


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S(6(4C 7E+S8S #9N6M4C O%(4M4:6(4ON;

Optimization < is the process o' minimizin ormaimizing the costs8#enefits of some action%

Static Optimization < re'ers to the process o'

minimizing or maimizing the costs8#enefits of some o#4ecti$efunction for one instant in time onl"%

#ynamic Optimization < re'ers to the process o'

minimizing or maimizing the costs8#enefits of some o#4ecti$e

function o$er a perio! of time% Sometimes calle! optimal control%

In #oth cases e)ualit" an! ine)ualit" constraints can #e enforce!%


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1% Static Optimization is concerne! &ith controlling a plantun!er stea!" state con!itions+ i%e%+ the s"stem $aria#les are not

changing &ith respect to time% The plant is then !escri#e! #"alge#raic e)uations% Techni)ues use! are or!inar" calculus+Lagrange multipliers+ linear an! nonlinear programming%

9% 7"namic Optimization concerns &ith the optimal control ofplants un!er !"namic con!itions+ i%e%+ the s"stem $aria#les arechanging &ith respect to time an! thus the time is in$ol$e! ins"stem !escription% Then the plant is !escri#e! #" !ifferential -or!ifference. e)uations% Techni)ues use! are search techni)ues+!"namic programming an! $ariational calculus%


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Static Optimization #ynamic Optimization

Optimal state+ :+ an! control

+u:+ are fie!+ i%e%+ the" !o notchange o$er time -for oneinstant in time onl".

; Optimal state an! control $ar"

o$er time -o$er a perio! oftime.% Some times calle!optimal control%

,


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1.1 Optimization Without Constraint

, Pro#lem' ; A scalarperformance index L-u. is gi$en that is afunction of a control or decision vector u; e &ant to'

Min )>u?

, Solution' L-u du . = L-u. dL (increment)

3Ta"lor epansion to 9nd or!er in !u

&ith gra!ient $ector Luan! cur$ature matri Luu

3Stationar" point u:0

dL = B to1st or!er term for ar#itrar" du


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The eponential function -in #lue.+ an! the sum of thefirst n1 terms of its Ta"lor series at B -in re!.%

(aylor seriesis a representation of afunction as an infinite

sum of terms calculate!from the $alues of its!eri$ati$es at a singlepoint%


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(aylor series


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Plot of f-x. = sin-9x. from 8D to E8D/ note fFs secon! !eri$ati$eis f-x. = ;Dsin-9x.% Tangent is #lue &here cur$e is conca$e up-a#o$e its o&n tangent.+ green &here conca$e !o&n -#elo& itstangent.+ an! re!at inflection points' B+ 89 an!


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http://mathforum.org/mathimages/index.php/Image:Taylor_Main.gif


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G

!t!
http://mathforum.org/mathimages/index.php/Image:TwoApproximations.gifhttp://mathforum.org/mathimages/index.php/Image:TwoApproximations.gifhttp://mathforum.org/mathimages/index.php/Image:TwoApproximations.gifhttp://mathforum.org/mathimages/index.php/Image:Taylor_Main.gifhttp://en.wikipedia.org/wiki/File:Animated_illustration_of_inflection_point.gif


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, Hraphical Interpretation of solution
http://en.wikipedia.org/wiki/File:Animated_illustration_of_inflection_point.gifhttp://en.wikipedia.org/wiki/File:Animated_illustration_of_inflection_point.gif


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A !iti f : t # i i f J- :. B ll !


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, A necessar" con!ition for : to #e a maimum is f J-:. = B+ calle!such an : a critical point%

, A sufficient con!itionfor : to #e a maimum is that it is a critical

point an! f JJ-:. K B %

That is+ if : satisfies #oth e)uations+ then it must #e a maimum%The con!ition is also calle! a secon!3or!er con!ition%

The height of the function at a isgreater than -or e)ual to. the heightan"&here else in that inter$al%

Or+ more #riefl"' f-a. f-. for all inthe inter$al In other &or!s+ there isno height greater than f-a.%

Note' f-a. shoul! #e insi!e theinter$al+ not at one en! or the other%

C t


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Comments', f ma" ha$e man" critical points% The sufficient con!itions helpsus pic( the maimum out of the set of all critical points%, oth con!itions onl" esta#lish that : is a local ma% That is+

there is no point close to : that has a higher f3$alue% ut therema" #e other points not close to : that !o% In that case theremust #e multiple points fulfilling #oth con!itions an! &e simpl"calculate &hich of these has the highest f3$alue%

, If f is strictl" conca$e+ there is onl" one critical point an! thismust #e the glo#al ma% The reason is that f JJ-. K B e$er"&here-that is &hat it means to #e strictl" conca$e.% Thus+ there can onl"#e one point &here f J-. = B an! that satisfies the sufficientcon!ition automaticall"%

@ l Th f ti http 88en i(ipe!ia org8 i(i8Ma ima an! minima


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@amples' The function

, 9has a uni)ue glo#al minimum at = B%, has no glo#al minima or maima% Although the first!eri$ati$e -9. is B at = B+ this is an inflection point%,

has a uni)ue glo#al maimum at = e figure%, 3has a uni)ue glo#al maimum o$er the positi$e realnum#ers at = 18e%, 8 has first !eri$ati$e 9 1 an! secon! !eri$ati$e 9%

Setting the first !eri$ati$e to B an! sol$ing for gi$esstationar" points at 1 an! 1% 6rom the sign of the secon!!eri$ati$e &e can see that 1 is a local maimum an! 1 is

a local minimum% Note that this function has no glo#almaimum or minimum%

http'88en%&i(ipe!ia%org8&i(i8Maima_an!_minima

C t l Th 6 ti


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Cont% eamples' The 6unction, QQ has a glo#al minimum at =B that cannot #e foun! #"ta(ing !eri$ati$es+ #ecause the !eri$ati$e !oes not eist at =B%, cos-.has infinitel" man" glo#al maima at B+ 9+ D+ +

an! infinitel" man" glo#al minima at + + %, 9 cos-. has infinitel" man" local maima an! minima+ #ut

no glo#al maimum or minimum%, cos-.8 &ith B%11%1 has a glo#al maimum at

=B%1 -a #oun!ar".+ a glo#al minimum near =B%+ a localmaimum near = B%U+ an! a local minimum near =1%B%

, 991 !efine! o$er the close! inter$al -segment.>D+9? has a local maimum at = 1VEW+ a local

minimum at = 1VEW+ a glo#al maimum at = 9 an! a glo#al

minimum at = D%

C t


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The #ottom part of the !iagram sho&s some contour lines &ith astraight line running through the location of the maimum $alue%The cur$e at the top represents the $alues along that straight line

hen the lines areclose together themagnitu!e of thegra!ient is large' the$ariation is steep%

Contour

Contour


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Minimize an A#solute Criterion
http://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svg


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Minimize an A#solute Criterion

, Achie$e a specific o#4ecti$e; Minimum time; Minimum fuel

; Minimum financial cost, to achie$e a goal

6or a continuous3time linear s"stem+ !efine! on + !escri#e! #"

&ith a )ua!ratic cost function !efine! as
http://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svghttp://en.wikipedia.org/wiki/File:Courbe_niveau.svg


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@ample

Slope #ecomes gra!ient $ector'
http://en.wikipedia.org/wiki/File:Singularptfn.JPGhttp://en.wikipedia.org/wiki/Partial_derivativehttp://en.wikipedia.org/wiki/Differential_calculus


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3Slope #ecomes gra!ient $ector'

3 9n! or!er !eri$ati$e #ecomes *essian -or cur$ature matri.'


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, 6urther @tension' $ector function of $ector

3 SlopeY of '-u. &ith respect to u is a


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3Meaning of matri LuuZB or KB0

& positi$e !efinite -&$%. if x#&x K B x [ B

& negati$e !efinite -&'%. if

& positi$e semi!efinite -& B. ifx#&x B x [ B

& in!efinite ifx#&x Z B for somex [ B an!x#&x K B for somex [ B

& negati$e semi!efinite -& B. ifx#&x B x [ B

-Appen!i A%.


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3Testing Positi$e8Negati$e 7efinitenessMetho! 31% " @igen$alues ]iof &

& positi$e !efinite -&$%. if all Z B+ i =1+L+n ]

& negati$e !efinite -&'%. if - B. if all B+ i 1+ + n & positi$esemi!efinite & ]i = L

B an! some B+ i 1+ +n i L = K Z ] ] i & in!efinite if some& negati$e semi!efinitex#&x Z B x [ B


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Metho! 39% " 7eterminant

: Testing positi$e8negati$e3!efinitenessO#tain lea!ing minors of

&'

, Testing semi3positi$e8negati$e3!efiniteness ' O#tain Principal


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,Testing semi3positi$e8negati$e3!efiniteness ' O#tain Principalminor of &'

* & is in!efinite if none of the a#o$e cases are satisfie!

Some sef l Matri Calc l s form la -Appen!i A D.


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, Some useful Matri Calculus formula -Appen!i A%D.

, If & is s"mmetric


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, If & is s"mmetric

; Some useful *essians


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Some useful *essians

, If & is s"mmetric

I : 8 i i t f L( ) 0


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3Is u: ma8min point of L(u) 0

dL at stationar" point u* ecomes


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; Optimize! $alue of performance in!e'


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: Stationar" point

Plug in the a#o$e e)uation

Example 1.1-2: Optimization by Scalar Manipulations


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e ha$e !iscusse! optimization in terms of $ectors an! the gra!ient% As an alternati$eapproach+ &e coul! !eal entirel" in terms of scalar )uantities%

6in! L-1+31. 0


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efore start

section 1%9

*o& to !ra&mesh+ contoursplots

G an! ^ arra"s for 37 plots

G an! ^ arra"s for 37 plots


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f -+". = 9 "9% Then an eas" calculation sho&s that f -9+ 1. = E%Therefore+ -9+ 1+ E. is an or!ere! triple that lies on the graph of f %This point is sho&n in 6igure%

6igure Point -9+ 1+ E. is on the graph of f %To plot the graph of f -. = 9in the plane+ &e #egin #" ma(ing ata#le of points that satisf" the e)uation+ as sho&n in Ta#le

Ta#le A ta#le of points satisf"ing f (x) " x+

transforms the !omain specifie! #" $ectors an! " into arra"s Gan! ^ that can #e use! for the e$aluation of functions of t&o$aria#les an! 37 surface plots

6or creating a ta#le of points that satisf" the e)uation f -+". = 9 "9% Matla#


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Outputis easil" un!erstoo! if one superimposes the matri ^ onto the matri G to o#taina gri! of or!ere! pairs%

g p " ) - ". "accomplishes this &ith the meshgri! comman!%ZZ >G+^?=meshgri!->1+9++D+E?.

Therefore+ Ta#le contains a set of points in the plane that &e &ill su#stitute into thefunction f (x,y) " xy+ Matla#Fs arra" smart operators ma(e this an eas" proposition%

The ro&s of the output arra" G are copies of the $ector an! the columns of the outputarra" ^ are copies of the $ector "%

ZZ >G+^?=meshgri!->1+9++D+E?.


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ZZ =G%`9 % 9

It is no& an eas" tas( to plot the surface to &hich these points #elong%

ZZ mesh-G+^+.

6igure ' Plotting the surfacef (x,y) "x y+

ZZ =3'


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= 3 39 31 B 1 9

ZZ "=3'" = 3 39 31 B 1 9

ZZ >G+^?=meshgri!-+".

ZZ =G%`9 % 9

ZZ mesh-G+^+.

1

1

1

1

The surface - " x y on t.e domain

The plot of z = x2+ y2.


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ZZ =3'%9'/ZZ "=3'%9'/ZZ >G+^?=meshgri!-+"./ZZ =G%b9^%b9/

ZZ mesh-G+ +.ZZ la#el-3aisF.ZZ "la#el-"3aisF.ZZ zla#el-z3aisF.ZZ title-The plot of z = b9 "b9%F.

-4

-2

02

4

-4

-2

0

2

40

5

10

15

20

x-axisy-axis

z-axis

close all1 1 1


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"=31'%1'1/z=31'%1'1/>""+zz?=meshgri!-"+z./=zz%b/mesh-+""+zz./la#el-d3aisd."la#el-d"3aisd.zla#el-dz3aisd.title-dThe plot of G = b9$ie&->1B+DB?.


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efore start section 1%9

Sol$e section'1%1 @amplesan!

@n! Chapter Pro#lems ofsection 1%1

@ample -Not in #oo(.


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s"ms "

f=b"b:b93:"b93

f=!iff-f+./

f"=!iff-f+"./

s=sol$e-f+f"./

>s%+s%"?

p - .

Ma Min0

Classification of critical points


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Classification of critical points

alue of f-+". at critical pointsf-B+B.=3 f-39+B.=-3.B19B3=3D

f-39+9.= 3193193=3 f-B+9.= BB3193=19

Luu=BI7

LuuKB7

LuuZB37

Mesh Surface


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>+"?=meshgri!-3'%9'./z=%b"%b:%b93:"%b93/mesh-+"+z.la#el-3ais."la#el-"3ais.

7ifficult to$ie& MeshSurface

Contour map

>+"?=meshgri!-3'%9'./z=%b"%b:%b93:"%b93/contour-+"+z.la#el( x-axis )ylabel( y-axis )

-4

-2

0

2

4

-4

-2

0

2

4-100

-50

0

50

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

>+"?=meshgri!-3'%9'./ 3


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> "? g - .z=%b"%b:%b93:"%b93/contour-+"+z+9B.la#el- 3ais .

"la#el- "3ais .

>+"?=meshgri!-3'%9'./z=%b"%b:%b93:"%b93/>c+h?=contour-+"+z+31D'3D./

cla#el-c+h.la#el- 3ais ."la#el- "3ais .

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

x-axis

y-ax

is

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

Slop 00

Approach from an"!irection "ou are


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x-axis

y-ax

is

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

Approach from an"!irection "ou are &al(ing

uphill crossing contour at3+3U+3E an! finall" local

maimum at -39+B.

!irection "ou are&al(ing !o&nhill

crossing contour at3+3+311 an! finall" local

minimum at -B+9.

If "ou mo$e from point -B+B.or -39+9. in an" one !irection

the height of the contour

increases an! !ecreases if"ou mo$e in opposite!irection

Xuestion'9 Assignment


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6in! the minimum $alue of L-1+9.=19319991/ let = L-+".=93""9

s"ms "f=%b93:""%b9:/

f=!iff-f+./

f"=!iff-f+"./

s=sol$e-f+f"./

>s%+s%"?

>+"?=meshgri!-3'%9'./

f=%b93:""%b9:/mesh-+"+f.la#el-3ais."la#el-"3ais.

>+"?=meshgri!-3'%9'./

f=%b93:""%b9:/contour-+"+f.la#el( x-axis )ylabel( y-axis )

>+"?=meshgri!-3'%9'./f=%b93:""%b9:/>c+h?=contour-+"+f+31D'3D./cla#el-c+h.la#el- 3ais .

"la#el- "3ais .

>+"?=meshgri!-3'%9'./f=%b93:""%b9://contour-+"+f+EB.la#el- 3ais ."la#el- "3ais .


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>+"?=meshgri!-3'%9'./z=%b"%b:%b93:"%b93/u = z-+"./)ui$er-+"+u. / gri! on/ ais e)ual

Chec( for )ui$er-+"+u.

ANIMATION _ +"?=meshgri!-3'%9'./f=%b93:""%b9://contour-+"+f+EB.

la#el- 3ais ."la#el- "3ais .

Pro#lem 1%93D -oo(.


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s"ms " l

f=b9:3U3"/

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Chap_1_Static Optimization_1.1_2014.ppt