直流电机调速系统的设计方法探讨_英文_

*

中山大学研究生学刊 ( 自然科学、医学版 )

第 28 卷第 2 期 J OURNAL OF T H E GRADUAT ES VOL 28 2

2 007 S UN YAT SEN UN I V ER S I TY ( NATU R A L SC I E NC E S 、M E D I C I N E ) 2 007

直流电机调速系统的设计方法探讨

姜 宇 ( 华南理工大学自动化科学与工程学院, 广东 广州 510640)

摘 要: 本文首先介绍了双闭环直流电机调速系统的数学模型, 并为之设置

了具体参数。随后使用经典的控制器设计方法分别对电流环和转速环由内而外

的进行了设计。然后建立了使用现代控制理论的状态空间方法描述的数学模

型。为同时达到渐进跟踪和抑制扰动的目的, 通过引入了内模理论, 将原问题

转化为增广系统的镇定性问题, 从而只需为增广系统配置合适的状态反馈增

益, 这一点可通过特征根配置或者 LQR 方法来实现。接着设计了 H 无穷状态

反馈控制器, 这种方法会产生静差, 但可以不使用积分器。所有的设计方法均

在 M A TL A B /S i m u li n k 环境下进行了数值仿真并列出各自的特点。最后提出了

更进一步的设计问题, 包括观测器, 滤波器的相关问题以及混合使用各种方法

讨论。 关键词: 状态空间; 极点配置; LQR ; H 无穷

1 In t roducti on

1 1 DC M o tor

T h e dc m otor is a po w er actu ator de v ice that d elivers energy to a load . It converts d irect cu rren t e lectrical en er gy in to r o tati o n alm ech an i c al energy . A m a j o r fraction of th e t o rqu e g en er ated i n th e r o t o r ( ar m ature) of th e m otor is a v a il a b le to d rive an e x ter n a l l o ad . B ecau se of f ea tu res su ch as h igh t o rqu e , speed c o n trollab ility over a w ide range , p o rtab ility , w ell b ehaved speed t o rqu e ch aracteris tics , and ad ap tab ility to v ari o u s types of c o n tr o l m eth o d s , dc m o t o rs are st ill w ide l y u sed i n nu m er ous c o n trol app licati o n s , i n clud i n g r o b o tic m an i p u lators , tap e tran s p o rt m echan is m s , d i sk d r ives , m ach i n e too,l and servo valve act u a tors . [ 1, pp 54 - 55 ]

T h e re are t w o t y pes o f d c m o t o rs : the f ie l d cu r ren t c o n t r o lled m o t o r an d th e ar m atu r e c o n tr o lled d c m o tor . I n th i s rep o r,t w e on ly d i s cu s s th e ar m at u r e con tr o lled dc m otor , w h i ch u s es t h e ar m ature cu r ren t as th e c o n t r o l variab le .

* 收稿日期: 2007 - 04 - 03

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F i g 1 DC M o t o r

1 2 D o ub l e Loop C o n t r o l S y ste m

Th is t y p e of c o n trol syste m c o ns ists of th ree e x terna l parts bes i d es the m o tor itsel:f they are t h e un ifor m c o n tr o lled rect ifier ( U C R ) , the ad jus tab le sp eed re g u lat o r ( A S R ) , the ad j u stab le cu rren t re g u lator ( ACR ), sep arate l y . U s ing class ical m ethod , th e A S R an d ACR are usua ll y P I c o n tr o llers , wh ich can ach ieve e x cellen t perfor m an ce . H o w ever , i n th is rep o r,t I have m ad e s o m e atte m p t s at des i g n i ng c o n t rollers v ia state sp a ce m eth o d s .

2 M o de li ng

2 1 B lock D iagra m o f A r m ature C o n t r o lled D cM otor

F i g 2 D o u b l e L oop C o n t r o l S y s t e m of D c N o t o r

直流电机调速 系统的设计方法探讨

2 2 P ara m eters S e tt i n g

T ab le1 P ara m eters S e t t i n g

Sy m b o l D escrip t i on V alue

U n R e f eren c e i n p u t vo lta g e 1V

K s

Am p lify ra t i o o f U CR 40

T l

A r m a t u r e t i m e c o n s tan t 0 03 s

T s

UCR t i m e c o n s tan t 0 00 16 7 s

T m

R otor In e rt i a 0 18N m s 2 / rad

R

A r m atu r e res i s t an c e 0 5

C e

R at i o of e . m. f and sp e ed 0 1 32N m /A

V o lta g e - S p e ed rat i o 0 0 1

V o lta g e cu r ren t ra t i o

0 0 5

n O u t pu t sp e ed U n / = 100 0 r /m i n

= 0V t 3s U V o ltage d i s t u r ban c e

I dL

E q u i v a len t load cu r ren t

= 1V t> 1s

= 1A 0< t 4s = 3A t > 4 s

I d

A r m a t u r e cu r ren t I d / = 20A

L i m it o f th e A SR o u t p u t

[ - 1, 1]

2 3 P erfor m ance A na l y s is

In order to p rotect t h e ar m ature cu rren ,t A S R m u st sat u ra te w hen c o m es to a large vo ltage i n pu ,t hen ce , it i s non tri v ial to d iscu ss the s tart i n g response sep arated fro m the d is t u rb ance re ject i o n w h en the m o tor is w o rk ing at a g iven speed . Fu rther m o re , th ere are t w o sorts o f d is t u rb an c e : th e equ i v a len t l o ad cu r ren t and the vo ltage un d u l at i on.

T h erefore , w e m ain ly f o cu s on th e f o ll o w ing resp o n ses to e v a l u ate th e p er f o r m an ce of th e c o n t r o ller w e des i g ned .

> T h e start i n g resp o n s e

> T h e cu r ren t d istu r ban c e re jection

> T h e vo ltage d i s t u r b a nce rejecti o n

3 C lass i ca lM et h o d s

3 1 P ri n c ip l es

In c lass ica l m ethods , th e curren t l o op an d sp eed f eedb ack loop are des i g n ed sep arate l y . Because A SR m ay satu r ate w h e n s t art i n g , A CR m us t h a ve p r oper d y n a m ic p e r f o r m an c e to li m it

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t h e ar m ature cu rren t and short ris i n g ti m e . H o w ever , fr o m th e c o n f i g u ration of the syste m, w e note th at th e cu rren t feedback loop is c o u p led w ith b ack e . m. .f , hen ce w e have to do s o m e pred i g es t i o n s as sho w n i n th e follo w i n g f i g u r e :

F i g 3 S i m p l i f i cat i on o f th e In n e r L oo p By n o ticing , it can b e found that w h en the curren t has a l m ost been ad j u sted , the b ack e . m. .f m ay ch a nge sli g h t ly . T herefore , w e can assu m e it to b e a step d i stu r ban c e , w h i ch de c o up l es the o ri g i n s y s t e m . F o ll o w ing

class ical m ethods , w e can d e s ign th e inn e r l o op th e n th e o u t er o n e .

3 2 P I C o n t rollers

In th e app r oach o f des i g n i n g P I c o n t r o llers , there is a co mm on w ay i n w h i ch w e m o d i fy th e o p e n loop tran s fer f un c ti o n to a so ca lled T ype 1 T y p i cal For m .

F i g 4 T y p e 1 T y p i ca l F o r m T h e n its closed l o op d y n a m i c resp o n s e can be calcu lated b y setti n g p a ra m eters K an d T.

A n d th e P I c o n t r o ller can b e chosen th r ough t h e g i v en T ab l e in [ re f eren c e 2, pp . 143 ]

W e set the P I para m eters as sho w n i n th e f i g u r e d r a wn by M A TL A B /S i m u l i n k. N o te th e

A S R i s c o n f i n e d w ith a saturati o n b l ock .

直流电机调速 系统的设计方法探讨

F ig 5 P I C o n t r o l l e rs D ra w n by S i m u link

F ig 6 S i m u lat i on b e t w een 0 s ~ 0 5 s F i g 7 S i m u l at i on b e t w e en 0 s ~ 6 s

F ro m th e s i m u lation resu l,t w e can f i n d that the P I c o n trollers can ach ieve ad equ ate s tart i n g perfor m an ce as w ell as th e d istu rban ce re ject i o n . S ince they are easy to des i g n i n p ract i c al use , it can e x p la i n w h y th is type of con tr o llers i s m ost w i d ely u sed to c o n trol th e speed of d c m o t o rs in t h e rea l w orl d .

3 3 C o m pen s ators

F o llo w i n g th e s i m ilar p r o cedu re of t h e des i g n of P I c o n tr o llers , w e can c o n struct th e co m pen sators fro m th e i nn er l o op to th e o u ter o n e . In order to ach ieve as y m p t o t ic track ing an d d is tu r ban c e re ject i o n , w e add each loop an in t e g rator b e fore d e sign i n g t h e c o m pen s ator .

T h e M A TL A B t oo l b ox c o n ta i n s a GU I d es ign t oo l wh ich is c o n v en ien t i n se lecting e x p ected c o m p e nsators by e v aluat i n g t h e resp o n s e both i n the f requ e n c y an d ti m e d o m a i n .

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F i g 8 D es i gn C o m p e ns a tors w it h t h e S i s o t oo l o f M ATLAB

F ig 9 C o m p e n s a t o rs

F ig 10 S i m u l at i on o f the S y s t e m C o n t r o lled by C o m pen s at o rs

直流电机调速 系统的设计方法探讨

T T

T h e resu lt is n o t satis f actory , s i n c e i n c lass i ca l con tr o l th e ory , there is n o su c h a w ay in w h i ch w e can choose the b e st c o m p e nsator , rath e r than d e s i gn by error and trial m eth o d .

4 S t ate S pace M et h o d s

4 1 M o d e li n g in S tate S p a ce

F ig 11 S e lect i on of the S t ate V a riab l es

T hu s , th e state sp a ce m o de l can be d e ri v ed as th e f o llo w i n g equat i o ns :

X = AX + b 1 U + b 2 I dL + b 3 U C

( 1) y = c X

W here

X = ( x 1, x 2, x 3 ) ,

- 598 8

0 0 A = 66 67

- 33 33 - 66 67 , 0 2 7778 0

T

b 1 = ( 0 66 67 0)

b 2 = ( 0 0 2 77 78) T

b 3 = ( 23952 0 0)

c = ( 0 0 0 07576 )T ;

It can be v eri f ied th a t (A, b 3 ) is c o n t r o llab l e and ( A, c ) is o b s er v ab l e .

4 2 In t erna lM o de l P rinc i p le

In order to ach ieve as y m p t o t ic track i n g and d istu r b a n c e rejection s i m u l tan e o u s l y , m ere l y des i g n i n g state f eedb a ck g a i n o f th e th r ee s t ate variab les m en ti o n ab o ve w ill not b e enough. T h e refore , w e u s e in t erna l m o de l p r i n c ip l e th e ory to c o n s tr u c t an aug m en t ed syste m, and stab i li z e it b y sett i n g f eedback g ains , an d th e n th e track i n g and reject i n g go als w ill be n a turall y o b

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tained.

S i n c e th e re f eren c e inpu t and d istu r b a n c e s i g n a ls are all assu m ed t o b e step o nes . W e de rive both sides o f equ a ti o n ( 1) respect to ti m e , w h i ch y ields :

X = AX + b 3 U C

W e in t roduce a n e w state variab le :

e = r - y

together w it h ( x 1, x 2, x 3 ) as a set o f ne w v ariab l es to c o n s tr u c t a au g m en t ed syste m as sho w n

i n th e follo w i n g equ a ti o n s : e 0 - C =

e 0 + U c ( 2) X 0 - C 0 A X b 3

0 It tu r n s o u t that th e pa ir , 0 A b 3

is c o n t r o llab l e . [ 3, pp . 244 - 245 ] W h i ch m ean s w e can stab ilize ( 2 ) by selecting f eedb a ck g a i n s su c h t h a t U C = (K a K ) e

X

( 3) I n t egrating b o th s i d e s of ( 3) y ields :

U C = - KX - K a ed t

Th is ind i cates the f o ll o w ing con f i g u r ation :

F i g 1 2 S t a te F eedb a ck In c lud i ng In t erna l M o d e l

H en ce , the as y m p t o tica ll y track ing and d istu rbance re ject i o n pr o b le m is tran s ferred to a stab ili z i n g one . T h e o n ly th in g w e have t o w o n der is ho w to set th e f eedb ack g a i n s i n order to a ch i e ve expected p e r f o r m an c e . A nd th i s w ill b e ill u s trated i n the n e xt p a r.t 4 3 P o les P lace m en t V ia S tate Feedb a ck I f w e p lace a ll t h e p o les ( e i g en v al u e s) at o n e p o i n t or g r oup t h e m i n a very s m a ll re g i o n, t h e n u s u a ll y th e resp o n s e w ill b e sl o w and th e actu a ti n g s i g na l w ill be lar g e . T h e re f o re it is bet

直流电机调速 系统的设计方法探讨

ter t o p lace all ei g en v a l u e s e v en l y ar o und a circle w ith rad i u s r i n s ide t h e sect o r as sh o w n i n re f eren c e [ 3, pp . 239 ].

W e ch o ose the e x p e cted p o les as { - 100 10 0i , - 100 2 5i }, and th e n u s e M A TL A B to calcu late the f eedb a ck g a i n :

k = - 632 33 29 - 0 0097 0 0 365 1 365 5

U s i n g th e calcu lated f eedb a ck la w , th e resp o n s e is s i m u l ated i n th e follo w i n g f i g u r e :

F ig 13 S i m u l at i on R esu l t o f P o l es P lace m en t D es i g n A lthough th is t y p e of des i g n can re ject d i stu r bance , be th e y cau s ed b y voltage or cu r ren ,t t h e re e x is ts a pr o b l e m w h i ch can n o t b e ignored th a t th e ov ersh o o t o f the ar m atu r e cu r ren t is so lar ge th a t it m ay b r eak t h e m ach i n e .

O n e w ay to s o l v e t h e p r ob le m is add i n g a satu r ation b l ock to li m it th e cu r ren t n o t m ore th a n 20A. F ig 14 S t ate F eedb a ck w ith A S a tu r at i on B l o ck

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-

F ig 15 S i m u l at i on R esu l t a f te r A dd i ng S a tu r at i o n

H o w ever , w e can see fro m th e resp o n se cu r ves th a,t a lthough the cu rren t is p rotected , it seri o us ly affects the o u tpu t sp eed , du e to the li m it of inpu t p o w er . H ence the i m p r ov e m ethod is not sat i s f act o r y .

4 4 LQR M ethod

T h e idea of LQR is to f i n d a f eedb a ck c o n t r o l la w t h a t m i n i m ize th e perfor m an c e i n d e x

J = 0 (X QX + u R u )d t

T

t

It tu r n s o u t thatK = R 1BP

W here is th e s o l u t ion o f th e so called A l g eb r aic R iccat i Equ a tion [ 5 ] :

T A P + PA - PBR - 1 - 1 B P + Q = 0

F o rtun a te l y , t h is can b e o b t ai n e d si m p l y b y u s in g M A TL A B f un c ti o n s .

K = - 3 1623 0 0 599 1 18 37 0 4181

T h e b l ock d i a gra m of th e LQR m et h o d d i ffers o n l y in the para m eters of the f eedb a ck gains .

F i g 1 6 S i m u l a t i on R esu lt of T h e LQR M et h o d

直流电机调速 系统的设计方法探讨

d L

F r o m th e s i m u l ati o n, it can b e f o und tha t th e L Q R m eth o d can o b t ain a m uch better p e r f o r m an c e c o m pared w ith m erely p lac i n g p o les .

4 5 H i n fi n it y M ethod

C o n s i d e r th e f o llo w i n g syste m:

X = AX + b 1 I dL + b 2 U c

y = c X

C 1X

z = D U C

T h e idea o f H in f in i t y m ethod is to find a c o n t r o l la w U c = - kx th a t m i n i m izes th e i n f in i ty n o r m of the trans f er f un c tion fro m the i n pu t d istu r b a n c e to th e genera lized o u t pu ,t or : T - 1 S i m ilarl y , w e setK = - (D D ) T b 2

p

m i n : T I U

Z W here is th e s o l u t ion o f th e so called A l g eb r aic R iccat i Equ a tion [ 5 ]: T 1 T T - 1 T T A P + PA + 2p b 1 b 1 p - p b 2 (D D ) b 2P + C 1 C 1 = 0

A n d is a s m all given p o s i t i ve nu m b e r .

W e then u s e t h e f o ll o w in g M A TLAB c o mm an d s to solve :

k = 0 0 185 0 00 01 0 000 2

N ow w e d e s i g n t h e b l ock d iagra m i n s i m u l ink w ith th e calcu l ated H in f i n ity f eedb a ck g ai n s :

F i g 1 7 T h e C o n f i g u r at i on of H i n f i n it y M ethod D ra w n b y S i m u l i nk

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F ig 18 S i m u l at i on R esu l t s of the H i n f i n it y M ethod

T hu s t h is t y p e of des i g n can p rotect ar m at u re cu rren ,t and m in i m ize the o u tpu t error caused b y d istu rbance . A lt h ough th e m otion error can n o t be e li m i n ated , it requ ires th e least c o n t r o l equ i p m en .t F or exa mp le , no i n t egrator is u s ed , c o m pare w ith o th e r c o n t r o l m et h o d s .

5 C o m p a r i son

In above d iscuss ion , w e have tr i e d five d i ff eren t m ethods to des i g n th e d c m otor c o n t roller . H en c e , it is t h e ti m e w e e v al u a te th e m.

C o m parab ly speak i ng , th e d e sign of P I c o n t r o llers as A CR an d A S R h a s b e en de v el o p e d m ore re liab le and p r actica;l hen c e th i s m ethod is th e m o st w i d e ly u s ed. Bu t w h e n th e re f erence i n p u t or th e d i stu r ban c e si g n a ls are m ore c o m p l ex , f o r e x a m p l e , s i nus o i d a l si g n a ls , th e n it m ay be m u c h m ore inconven i e n t to des i g n . Lead / lag n e t w ork can ach i eve b e tter perfor m ance th e o ret ica lly , b u t it is n o n tr i v ial to des i g n th e c o m p e nsator . S t ate sp a ce m eth o d offers w ays to ca l cu l ate th e f eedb a ck gain ; b y add i ng in t er n a l m o d e ,l th e track i n g and re j e cting d i stu r bance pr o b l e m has b e en tran s ferred i n t o a state stab ili z ing o n e . It is better to p lace the p o les e v en l y a r o u n d a circ le i n th e g iven region i n c o m p lex p l a i n , th a n group the m together . H o w ever , w h e n satu r ated , th e static error is so large that it m ay seri o u s ly affect th e o u t pu .t LQR is a better ap proach , if o n l y w e can offer the p r o p e r w e i g h i ng m atrices , i t w ill y ield e x ce llen t perfor m an c e . T h e H in f i n ity is a m ethod to d e sign c o n t r o llers w h i ch m in i m ize th e tran s fer f un c t i o n fr o m d is tu r ban c e i n p u t to th e speed o u t pu t in t h e H il b e rt space . It m ay cause stat ic error , bu t th is m eth o d requ ires no i n t egrators .

I have also c o n c luded th e above d i scu s s i o n i n th e f o llo w i n g tab l e :

直流电机调速 系统的设计方法探讨

T a b l e 2. Com p a r i s on o f D i ff e r e n t C on tro l M e thod s

6 F u rther D iscu s s i on

6 1 Es t i m ator

In rea lity , w e m ay n o t b e ab le to u t ilize ever y s tate variab le , du e to the d i ff ic u lt ies to

m easu re the m. Bu t w e can design an es ti m ator to es ti m ate t h e m, p r ov i d ed th at th e p air is ob ser vab le . ( So m eti m es w e on l y requ ire th e pa ir to be pred ictab le). H ere w e d es ign a f u ll state esti m a tor for t h e LQR c o n t roller sh o wn i n the f o ll o w i n g f i gure .

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F i g 19 F eedb a ck U s i ng A n E s t i m ator

F i g 20 C o m par ison o f th e R ea l an d E s t i m ated S t a te

F o r m th e c o m p a ris o n b e t w een the rea l speed an d th e es t i m ated v al u e s , w e can f i nd the est i m ated v ari a b l es converge to t h e tru e states i m m ed i a tely , re g ard l ess the i n itial data .

6 2 C o m p o u n d i ng D es i g n

I d id not try to c o m pound d i ff eren t c o n tr o l m eth o d s in one des i g n , b ecau se th ere are too m u ch sorts o f co mb i n ation. H ow ever , p r o b ab l y e x ce llen t effects w ill b e p roduced w h en u s i n g d i ff eren t m et h o d s together . A nd I t h i n k that is a good w ay to i n v est i g ate if w e w an t to d iscu ss t h e des i g n p r o b l e m f u r ther .

6 3 F ilters i n the Feedb a ck L oo p s

P ractica ll y , the tran s fer f un cti o n s on th e f eedb ack loops are not s i m p l y c o n stan ts su ch as t h e , w e set i n the above d iscuss i o n s , bu t o n e o rd e r filters i n th e for m o :f

直流电机调速 系统的设计方法探讨

T on s + 1

H en ce it is ab le to filter alter n ating s i g n als cau sed by th e ar m atu re r o tat i o n . For c o n v en ien ,t w e set a feed forw ar d tran sfer f u nct i o n to a vo i d the d elaying ph eno m ena b et w een the re f eren ce i n pu t and the f eedb ack s i g na ls . T here f o re , the o n ly d i ff eren ce bet w een the s y s te m w it h filters in t h e f eedb ack l o o p s and the syste m w e d iscu ss above is a n e g lectab le p ole o f th e closed loop tran sfer f un cti o n , caus i n g on l y s li g h tly d ifferences . B u t w e can a lso ad d the f ilters w h en s i m u la t i n g , in order to ach i eve m ore accu r a te resu l ts .

B i b li og r a ph y

[ 1 ] R i chard C . D or.f , R o be rt H. B ishop . , M odern C o n t r o l S y stem s ( N i n t h E d iti o n ) [ J ] Pren t i ce

H a l,l 2 00 1.

[ 2 ] Y an g G eng , L u o Y i n g l.i , E lectr i ca l M ach i n e ry and M o t i on C o n t r o l S y s t e m s [ J ] B e ij i ng : T s i n g hu a

U n i v ers i ty P res s , 2 00 6. 3.

[ 3 ] C h e n , C . T. , L i n e ar Sy s t em T h e o ry an d D es i gn [ J ] N ew Y o rk : O x f o rd U n i v e rs i ty P res s , 19 99.

[ 4 ] G en e F . F rank lin . , J . D a v i d P o w e l.l , A bb a s E m a m i N ae i n .i , F eedb a ck C o n t r o l o f D y n a m ic Sy s

te m s [ J] USA: A d d i son W es l ey P ub l i sh i n g C o m p a ny , 198 5.

[ 5 ] F ran k L . Lew i s . , O p t i m a l C o n t r o l [ J] USA: John W ile y & So n s , 198 6.

So m e A tte m pts a t the D esign o f DC M o t or Speed C on t r o l Sy s t e m

J iang Y u

(C o ll e g e of A u t o m a t ion S cience and E ng i n e ering, S o u t h C h i na U n i versity of T ec hn o l o g y ,

G u a ng z h o u 51 0640, Ch in a )

A b s t r a c t : I n th i s rep o rt , a m ath e m at i ca l m o d e l of ar m atu r e c o n t r o l l ed d c m otor i s i n tr o du c ed , w it h sp e ci f i ed p a ra m eters . T hen class i ca l m eth o d s are us e d to d e s i gn c o n t r o llers f o r each f eedb a ck loop , from th e inn e r o n e t o the o u t er one , a f t er so m e s i m p li f i ca t i on . T h e n the s t ate space m o d e l i s d e v e l o p e d . In t ern a l m o d e l p r i n c ip l e is in t r o d u ced h e re i n order to ach ieve as y m p t o t i ca l track i ng and d i stu r b a n c e re j ect i on .

B y ad d i n g w h i ch , the p r o b le m can b e t r an s f e rred to a s t ab i l i z i ng one . T h e n t h e on l y p r o ced u r e i s to s e t the p r oper f eedb a ck g a i n for t h e augm en ted s y s t e m. Th is can b e ob tained b y p o le p l ace m en t o r ca lcu l a t i ng LQR c o n t rol la w. T h e f eedb a ck g a i n o f H in f i n i ty m ethod is also ob tained , b u t it m ay p r o du c e s t at i c err o rs . A ll the d e s i gn m eth o d s are s i m u l a ted w it h M ATLAB / S i m u l i nk . A fe w d i s c u s s i o n s are m ad e f o r th e s i m u l at i on resu lts . T hen co m p a ris o n s o f t h e f i v e d ifferen t m eth o d s a re l i s t . F i na l l y , s o m e f u rt h er issu es su ch as es t i m a tor , f ilters and c o m p o u nd i n g d i f f e ren t m eth o d s a re pu t for w ard .

K ey w o r d s : s t ate sp a ce ; p o les p l ace m en t ; LQR; H i n f i n ity

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